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Advances in

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS

V O L U M E 53

Editors

PAUL R. BERMAN

University of MichiganAnn Arbor, Michigan

CHUN C. LIN

University of WisconsinMadison, Wisconsin

ENNIO ARIMONDO

University of PisaPisa, Italy

Editorial Board

C. JOACHAIN

Université Libre de BruxellesBrussels, Belgium

M. GAVRILA

F.O.M. Insituut voor Atoom- en MolecuulfysicaAmsterdam, The Netherlands

M. INOKUTI

Argonne National LaboratoryArgonne, Illinois

Founding Editor

SIR DAVID BATES

Supplements

1. Atoms in Intense Laser Fields, Mihai Gavrila, Ed.2. Cavity Quantum Electrodynamics, Paul R. Berman3. Cross Section Data, Mitio Inokuti, Ed.

ADVANCES IN

ATOMIC,MOLECULAR,AND OPTICALPHYSICSEdited by

G. RempeMAX-PLANCK INSTITUTE

FOR QUANTUM OPTICS

GARCHING, GERMANY

and

M.O. ScullyTEXAS A&M UNIVERSITY

AND

PRINCETON UNIVERSITY

Volume 53

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Dedicated to HERBERT WALTHER

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Contents

CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiPREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

Non-Classical Light from Artificial Atoms

Thomas Aichele, Matthias Scholz, Sven Ramelow and Oliver Benson

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Single Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. Single-Photon Generation . . . . . . . . . . . . . . . . . . . . . . . . . 74. A Single Photon as Particle and Wave . . . . . . . . . . . . . . . . . . 135. A Multi-Color Photon Source . . . . . . . . . . . . . . . . . . . . . . 166. Multiplexed Quantum Cryptography on the Single-Photon Level . . . 227. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Quantum Chaos, Transport, and Control—in Quantum Optics

Javier Madroñero, Alexey Ponomarev, André R.R. Carvalho, Sandro Wimberger,Carlos Viviescas, Andrey Kolovsky, Klaus Hornberger, Peter Schlagheck,Andreas Krug and Andreas Buchleitner

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342. Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343. Dynamics and Transport . . . . . . . . . . . . . . . . . . . . . . . . . 414. Control through Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . 595. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Manipulating Single Atoms

Dieter Meschede and Arno Rauschenbeutel

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762. Single Atoms in a MOT . . . . . . . . . . . . . . . . . . . . . . . . . . 773. Preparing Single Atoms in a Dipole Trap . . . . . . . . . . . . . . . . 824. Quantum State Preparation and Detection . . . . . . . . . . . . . . . . 845. Superposition States of Single Atoms . . . . . . . . . . . . . . . . . . 86

vii

viii Contents

6. Loading Multiple Atoms into the Dipole Trap . . . . . . . . . . . . . . 897. Realization of a Quantum Register . . . . . . . . . . . . . . . . . . . . 918. Controlling the Atoms’ Absolute and Relative Positions . . . . . . . . 949. Towards Entanglement of Neutral Atoms . . . . . . . . . . . . . . . . 99

10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10111. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10212. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Spatial Imaging with Wavefront Coding and Optical CoherenceTomography

Thomas Hellmuth

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062. Enhanced Depth of Focus with Wavefront Coding . . . . . . . . . . . 1073. Spatial Imaging with Optical Coherence Tomography . . . . . . . . . 1204. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

The Quantum Properties of Multimode Optical Amplifiers Revisited

G. Leuchs, U.L. Andersen and C. Fabre

1. General Linear Input–Output Transformation for a Linear Optical De-vice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

2. The Phase-Insensitive Amplifier . . . . . . . . . . . . . . . . . . . . . 1413. The Multimode Phase Insensitive Amplifier . . . . . . . . . . . . . . . 1434. The Nature of the Ancilla Modes . . . . . . . . . . . . . . . . . . . . . 1445. An Optical Amplifier Working at the Quantum Limit . . . . . . . . . . 1476. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Quantum Optics of Ultra-Cold Molecules

D. Meiser, T. Miyakawa, H. Uys and P. Meystre

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522. Molecular Micromaser . . . . . . . . . . . . . . . . . . . . . . . . . . 1533. Passage Time Statistics of Molecule Formation . . . . . . . . . . . . . 1634. Counting Statistics of Molecular Fields . . . . . . . . . . . . . . . . . 1685. Molecules as Probes of Spatial Correlations . . . . . . . . . . . . . . . 1736. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1817. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1828. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Contents ix

Atom Manipulation in Optical Lattices

Georg Raithel and Natalya Morrow

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1872. Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . 1903. Review of One-Dimensional Lattice Configurations for Rubidium . . 1964. Periodic Well-to-Well Tunneling in Gray Lattices . . . . . . . . . . . 2085. Influence of Magnetic Fields on Tunneling . . . . . . . . . . . . . . . 2136. Sloshing-Type Wave-Packet Motion . . . . . . . . . . . . . . . . . . . 2197. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2228. Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2239. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Femtosecond Laser Interaction with Solid Surfaces: Explosive Ablation andSelf-Assembly of Ordered Nanostructures

Juergen Reif and Florenta Costache

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2282. Energy Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2293. Secondary Processes: Dissipation and Desorption/Ablation . . . . . . 2334. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2465. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2496. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Characterization of Single Photons Using Two-Photon Interference

T. Legero, T. Wilk, A. Kuhn and G. Rempe

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2542. Single-Photon Light Fields . . . . . . . . . . . . . . . . . . . . . . . . 2563. Two-Photon Interference . . . . . . . . . . . . . . . . . . . . . . . . . 2604. Jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2705. Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 2776. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2867. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2878. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

Fluctuations in Ideal and Interacting Bose–Einstein Condensates: Fromthe Laser Phase Transition Analogy to Squeezed States and BogoliubovQuasiparticles

Vitaly V. Kocharovsky, Vladimir V. Kocharovsky, Martin Holthaus,C.H. Raymond Ooi, Anatoly Svidzinsky, Wolfgang Ketterle and Marlan O. Scully

x Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2932. History of the Bose–Einstein Distribution . . . . . . . . . . . . . . . . 2983. Grand Canonical versus Canonical Statistics of BEC Fluctuations . . 3154. Dynamical Master Equation Approach and Laser Phase-Transition

Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3285. Quasiparticle Approach and Maxwell’s Demon Ensemble . . . . . . . 3576. Why Condensate Fluctuations in the Interacting Bose Gas are Anom-

alously Large, Non-Gaussian, and Governed by Universal Infrared Sin-gularities? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3908. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3949. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395A. Bose’s and Einstein’s Way of Counting Microstates . . . . . . . . . . 395B. Analytical Expression for the Mean Number of Condensed Atoms . . 397C. Formulas for the Central Moments of Condensate Fluctuations . . . . 399D. Analytical Expression for the Variance of Condensate Fluctuations . . 401E. Single Mode Coupled to a Reservoir of Oscillators . . . . . . . . . . . 402F. The Saddle-Point Method for Condensed Bose Gases . . . . . . . . . 404

10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

LIDAR-Monitoring of the Air with Femtosecond Plasma Channels

Ludger Wöste, Steffen Frey and Jean-Pierre Wolf

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4132. Conventional LIDAR Measurements . . . . . . . . . . . . . . . . . . . 4153. The Femtosecond-LIDAR Experiment . . . . . . . . . . . . . . . . . . 4194. Nonlinear Propagation of Ultra-Intense Laser Pulses . . . . . . . . . . 4215. White Light Femtosecond LIDAR Measurements . . . . . . . . . . . 4276. Nonlinear Interactions with Aerosols . . . . . . . . . . . . . . . . . . 4337. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4378. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4389. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443CONTENTS OF VOLUMES IN THIS SERIAL . . . . . . . . . . . . . . . 453

CONTRIBUTORS

Numbers in parentheses indicate the pages on which the author’s contributions begin.

THOMAS AICHELE (1), Nano Optics, Physics Department, Humboldt-Universität zuBerlin, 10117 Berlin, Germany

MATTHIAS SCHOLZ (1), Nano Optics, Physics Department, Humboldt-Universität zuBerlin, 10117 Berlin, Germany

SVEN RAMELOW (1), Nano Optics, Physics Department, Humboldt-Universität zu Berlin,10117 Berlin, Germany

OLIVER BENSON (1), Nano Optics, Physics Department, Humboldt-Universität zu Berlin,10117 Berlin, Germany

JAVIER MADROÑERO (33), Physik Department, Technische Universität München, James-Franck-Straße, D-85747 Garching, Germany; Max-Planck-Institut für Physik kom-plexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany

ALEXEY PONOMAREV (33), Max-Planck-Institut für Physik komplexer Systeme, Nöth-nitzer Str. 38, D-01187 Dresden, Germany

ANDRÉ R.R. CARVALHO (33), Max-Planck-Institut für Physik komplexer Systeme, Nöth-nitzer Str. 38, D-01187 Dresden, Germany

SANDRO WIMBERGER (33), Dipartimento di Fisica Enrico Fermi and CNR-INFM, Uni-versità di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy

CARLOS VIVIESCAS (33), Max-Planck-Institut für Physik komplexer Systeme, NöthnitzerStr. 38, D-01187 Dresden, Germany

ANDREY KOLOVSKY (33), Max-Planck-Institut für Physik komplexer Systeme, Nöth-nitzer Str. 38, D-01187 Dresden, Germany

KLAUS HORNBERGER (33), Arnold-Sommerfeld-Zentrum für Theoretische Physik,Ludwig-Maximilians-Universität München, Theresienstr. 37, D-80333 München, Ger-many

PETER SCHLAGHECK (33), Theoretische Physik, Universität Regensburg, D-93040 Re-gensburg, Germany

xi

xii Contributors

ANDREAS KRUG (33), Max-Planck-Institut für Physik komplexer Systeme, NöthnitzerStr. 38, D-01187 Dresden, Germany

ANDREAS BUCHLEITNER (33), Max-Planck-Institut für Physik komplexer Systeme,Nöthnitzer Str. 38, D-01187 Dresden, Germany

DIETER MESCHEDE (75), Institut für Angewandte Physik, Universität Bonn, Wegelerstr.8, D-53115 Bonn, Germany

ARNO RAUSCHENBEUTEL (75), Institut für Angewandte Physik, Universität Bonn,Wegelerstr. 8, D-53115 Bonn, Germany

THOMAS HELLMUTH (105), Department of Optoelectronics, Aalen University of AppliedSciences, Germany

G. LEUCHS (139), Max Planck Research Group of Optics, Information and Photonics,University of Erlangen-Nürnberg, Erlangen, Germany

U.L. ANDERSEN (139), Max Planck Research Group of Optics, Information and Photon-ics, University of Erlangen-Nürnberg, Erlangen, Germany

C. FABRE (139), Laboratoire Kastler-Brossel, Université Pierre et Marie Curie et EcoleNormale Supérieure, Place Jussieu, cc74, 75252 Paris cedex 05, France

D. MEISER (151), Department of Physics, The University of Arizona, 1118 E. 4th Street,Tucson, AZ 85705, USA

T. MIYAKAWA (151), Department of Physics, The University of Arizona, 1118 E. 4thStreet, Tucson, AZ 85705, USA

H. UYS (151), Department of Physics, The University of Arizona, 1118 E. 4th Street,Tucson, AZ 85705, USA

P. MEYSTRE (151), Department of Physics, The University of Arizona, 1118 E. 4th Street,Tucson, AZ 85705, USA

GEORG RAITHEL (187), FOCUS Center, Department of Physics University of Michigan,Ann Arbor, MI 48109, USA

NATALYA MORROW (187), FOCUS Center, Department of Physics University of Michi-gan, Ann Arbor, MI 48109, USA

JUERGEN REIF (227), Brandenburgische Technische Universität Cottbus, Konrad-Wachs-mann-Allee 1, 03046 Cottbus, Germany BTU/IHP JointLab, Erich-Weinert-Strasse 1,03046 Cottbus, Germany

FLORENTA COSTACHE (227), Brandenburgische Technische Universität Cottbus, Konrad-Wachsmann-Allee 1, 03046 Cottbus, Germany BTU/IHP JointLab, Erich-Weinert-Strasse 1, 03046 Cottbus, Germany

T. LEGERO (253), Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748Garching, Germany

Contributors xiii

T. WILK (253), Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748Garching, Germany

A. KUHN (253), Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748Garching, Germany

G. REMPE (253), Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748Garching, Germany

VITALY V. KOCHAROVSKY (291), Institute for Quantum Studies and Department ofPhysics, Texas A&M University, TX 77843-4242, USA; Institute of Applied Physics,Russian Academy of Science, 600950 Nizhny Novgorod, Russia

VLADIMIR V. KOCHAROVSKY (291), Institute of Applied Physics, Russian Academy ofScience, 600950 Nizhny Novgorod, Russia

MARTIN HOLTHAUS (291), Institut für Physik, Carl von Ossietzky Universitat, D-2611Oldenburg, Germany

C.H. RAYMOND OOI (291), Institute for Quantum Studies and Department of Physics,Texas A&M University, TX 77843-4242, USA

ANATOLY SVIDZINSKY (291), Institute for Quantum Studies and Department of Physics,Texas A&M University, TX 77843-4242, USA

WOLFGANG KETTERLE (291), MIT-Harvard Center for Ultracold Atoms, and Departmentof Physics, MIT, Cambridge, MA 02139, USA

MARLAN O. SCULLY (291), Institute for Quantum Studies and Department of Physics,Texas A&M University, TX 77843-4242, USA; Princeton Institute for Material Sci-ence and Technology, Princeton University, NJ 08544-1009, USA

LUDGER WÖSTE (413), Physics Department, Freie Universität Berlin, Arnimallee 14,14195 Berlin, Germany

STEFFEN FREY (413), MIT, Department of Earth, Atmospheric, and Planetary Sciences,77 Massachusetts Avenue, Cambridge, MA 02139, USA

JEAN-PIERRE WOLF (413), GAP-Biophotonics, University of Geneva, 20, rue de l’Ecolede Médecine, 1211 Geneva 4, Switzerland


HERBERT WALTHER


PREFACE

Prof. Herbert Walther is a quantum optics star of galactic magnitude! Experimen-tal physicists admire his ability to conduct experiments previously consideredimpossible. Theoretical physicists eagerly look forward to the stunning resultsthat come out of his laboratory. His discoveries have brought increasingly newlife to both the theoretical and experimental quantum optical physicists. The sci-entific methods developed in his laboratory have become a mainstay to quantumoptics laboratories all over the world.

Three qualities of Herbert Walther stand out most clearly: His enormous en-ergy, his unique dedication to science and his special eye for scientific quality.He obviously must subscribe to the German motto: “Die Probleme existieren, umüberwunden zu werden” (problems exist to be overcome). This statement holdstrue not only for scientific matters but also for science policy. Three examplesillustrating his qualities offer themselves:

In the early 1980s the Max-Planck-Society inherited the Ringberg castle lo-cated in the picturesque Bavarian mountains next to Lake Tegernsee. The Max-Planck Institute for Quantum Optics was one of the first institutes that startedusing this facility as a retreat to review its progress in the various groups and ini-tiate novel research directions. It was during one of these early meetings whenHerbert Walther’s group was discussing the new possibilities in cavity quantumelectrodynamics offered by the unique combination of Rydberg atoms and high-Qmicrowave resonators. Herbert Walther proposed to build a new type of maserdriven by a single atom. However, fresh ideas are rarely received with enthu-siasm, especially by those who have to transfer the Gedanken experiments intoreal experiments. It was argued that too many novel techniques, such as, atomicbeams and cryogenic equipment, which had only worked separately before, nowhad to be combined into one single experiment: it was considered impossible tomake all these experimental tricks work at the same time. Herbert Walther triedto convince the nay sayers about the feasibility of the experiment—without suc-cess. Finally he decided to follow a different route and attract students to do thework. Indeed, several students, starting with Dieter Meschede, Gerhard Rempe,Ferdinand Schmidt-Kaler, Georg Raithel, Oliver Benson, and Ben Varcoe, now allfaculty members at different scientific institutions, together with other students,postdocs and visitors, planned and implemented today’s famous research line ofthe micromaser.

Second, Herbert Walther is a great institution builder. He was a main drivingforce responsible for building up the Max-Planck Institute for Quantum Optics to

xvii

xviii PREFACE

one of the top institutions in the field worldwide and in fact a Mecca for manyinternational scientists visiting it religiously. Today it is hard to believe that in thelate 1970s the Institute was an institute on probation: The Max-Planck Societyhad installed a research group in the newly emerged field of laser physics, theso called “Projektgruppe für Laserforschung” (project group for laser research).Herbert Walther was hired as one of the directors. In no time he was able to attractmany bright students to his group and bring the high society of laser physics tothe project group. Clearly enough, the Max-Planck Society then was given littlechoice but to found a full fledged Max-Planck Institute. Many years later, HerbertWalther had the unique opportunity to repeat this story of success on a much largerscale: As Vice President of the Max-Planck Society, he was a leading authoritywhile setting up the new institutes in East Germany after reunification.

A third example illustrating Herbert Walther’s lack of fear was the hiring ofProf. Theodor Hänsch who had previously turned down several offers from Ger-man universities. It seemed to be a hopeless task to lure Hänsch away from Stan-ford. Nevertheless, Herbert Walther was not afraid to compete. He first arrangeda Humboldt prize for Hänsch to get him used again to German life. He then man-aged to arrange an offer which could not be refused—a chair at the Universityof Munich together with a directorship at the MPQ. In this way, Herbert Waltherachieved the impossible.

It is difficult to describe the impact of his works in just a few words. Those ofus who have been associated with him consider ourselves very fortunate, havingbenefited from the relationship in many different ways. Needless to say, HerbertWalther has trained a large number of students and other researchers, many ofwhom have become authorities in the field. His students and colleagues have wonNobel prizes; few are able to boast this line. But his humbleness and generosityhave no bounds and the optical community knows and appreciates all that he hasdone for it.

We hope that some of the many shining aspects of his scientific life are reflectedin the present volume. All articles have been written by Herbert Walther’s formerstudents and collaborators, now grown up and dedicated to their own research.But clearly enough, the nucleus of their work lies in Herbert Walther’s laboratory.To our minds, when history is written, then one would find that many of the dis-coveries made in Herbert Walther’s laboratory will stand out as some of the mostfundamental discoveries in the discipline of quantum optics.

So with the preceding in mind, appreciation and admiration in our hearts, and aspecial applause to his endearing spouse Margot, we dedicate this volume to youHerbert. Vielen Dank!

Girish Agarwal Wolfgang SchleichGerhard Rempe Marlan Scully

ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 53

NON-CLASSICAL LIGHT FROMARTIFICIAL ATOMS*

THOMAS AICHELE†, MATTHIAS SCHOLZ, SVEN RAMELOWand OLIVER BENSON

Nano Optics, Physics Department, Humboldt-Universität zu Berlin, 10117 Berlin, Germany

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Single Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. Single-Photon Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1. Correlation Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2. Micro-Photoluminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3. InP Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4. A Single Photon as Particle and Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135. A Multi-Color Photon Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166. Multiplexed Quantum Cryptography on the Single-Photon Level . . . . . . . . . . . . . . 22

6.1. A Single-Photon Add/Drop Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2. Application to Quantum Key Distribution . . . . . . . . . . . . . . . . . . . . . . . . 26

7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1. Introduction

A photon is the fundamental excitation of the quantized electro-magnetic field.Its introduction helped to get a deeper, yet more intuitive understanding of thephenomenon light. The year 2005 celebrates the 100th anniversary of Einstein’singenious explanation of the photoelectric effect using the concept of the photon.Until today, the photon is a workhorse to test the foundation of quantum physicsagainst recurring efforts of a purely classical interpretation of nature [1,2]. More

* We would like to dedicate this article to Prof. Herbert Walther on behalf of his 70th birthday. Hepioneered quantum optics with single quantum systems and drew our attention to the beauty of thesingle photon. The experiment we report in Section 4 of our article was motivated by his wonderfulexperiments with single ions.

† Present address: CEA/Université J. Fourier, Laboratoire Spectrométrie, Grenoble, France.

© 2006 Elsevier Inc. All rights reservedISSN 1049-250X

DOI 10.1016/S1049-250X(06)53001-0

1

2 T. Aichele et al. [1

FIG. 1. Photon number distributions of (a) thermal light, (b) a coherent state, and (c) a sin-gle-photon source (with 25% efficiency).

recently, single photons entered the stage to play an important role in the field ofquantum information processing. Bennett and Brassard [3] suggested, that datacan be transmitted without the possibility of eavesdropping, if information is en-coded in the quantum state of single particles (for a review see [4] and referencestherein). For transmission over large distances, the photon is currently the onlyreasonable carrier of quantum information. Knill et al. [5] proposed an implemen-tation of all-optical quantum gates for quantum computation using solely linearoptics and single-photons which is based on non-deterministically prepared entan-gled states and quantum teleportation [6]. Single photons have also been discussedas transmitters of quantum information [7] between different knots of stationary,matter-based qubits, such as ions [8–11], atoms [12,13], quantum dots [14,15],and Josephson qubits [16–19].

In spite of their fundamental character, single photons cannot be generated eas-ily by a classical light source. As photons obey Bose–Einstein statistics, classicalsources tend to emit photons in bunches. Figure 1 shows probability distributionsfor various classical and non-classical states of light. Thermal light fields (a),such as light from a bulb, have a smeared distribution with significant prob-abilities for larger photon numbers. Even laser light (b), which possesses thenarrowest classically obtainable photon number distribution, shows Poissonianstatistics pn = exp(−μ)μn/n! with average photon number μ. However, for ap-plications in quantum information processing, single-photon operation of the lightsource is crucial: For photonic quantum gates [5], but also quantum repeaters [20]and quantum teleportation [21], multi-photon states may lead to wrong detectionevents that cause wrong interpretations of the outcome of a quantum operation. Inquantum cryptography, an eavesdropper may split off additional photons to gainpartial information.

In contrast, an ideal single-photon source has a probability of one to measureexactly one photon at a time (pn = δ1,n). Such sub-Poissonian distributions—with a width narrower than a Poissonian of the same average photon number—are known to be non-classical and have to be described by means of quantummechanics. Additionally to the single-particle character, high purity of the spatial

1] NON-CLASSICAL LIGHT FROM ARTIFICIAL ATOMS 3

and temporal mode is often required in many applications, for example, Hong–Ou–Mandel-type [22] two-photon interference plays an important role in linearoptical quantum gates [5]. For useful operation, highly efficient photon generationand the ability to trigger the emission time are also desirable. Real single-photonsources show various loss mechanisms, like emission into uncontrolled opticalmodes or absorption, so that a more realistic photon number distribution has acertain zero-photon probability, as the one in Fig. 1(c) [23].

There are many ways to realize single-photon sources. The easiest is to ap-proximate single-photon states by highly attenuated laser pulses: Due to theirPoissonian photon number distribution, the multi-photon probability scales linearwith the mean photon number, p�2 ≈ μp1/2, which approximates a single-photon state for μ � 1. However, the single-photon probability scales in the sameway, p1 ≈ μ for μ � 1, which makes this method highly inefficient. Anotherwidely used method to generate single-photon states is spontaneous paramet-ric down-conversion in non-linear crystals. Presently, these sources are the mostpractical and brightest sources for non-classical light, such as entangled photonpairs [24]. However, due to the stochastic nature of this process, only a limitedoverall efficiency is offered while an increase of pump power to improve the pho-ton rate leads to an increased probability to generate two-photon pairs. These factslimit the potential of this method for future commercial quantum applications thatneed high single-photon rates.

Another method is to use spontaneous emission from a single quantum emit-ter. Suppressing non-radiative decay mechanisms, these emitters represent, inprinciple, single-photon sources with 100% efficiency. To make use of this highefficiency, a strong control of the spatial emission mode is required which setsa technical but no fundamental limit to today’s maximally achievable photonefficiency. The variety of possible quantum emitters offered by nature allows amultitude of realizations.

Discrete electronic transitions in atoms were the first to be investigated in 1977by Kimble et al. [25]. Recent experiments used single atoms [26,27] and singleions [28] coupled to microcavities to exploit effects of cavity quantum electrody-namics. In this way, not only the emission time, but also the spatial and temporalmode of single photons can be controlled. The emission of an isolated single atomis free from additional broadening due to coupling to the environment. Addition-ally, identical atoms emit identical photons which is a requirement for possibleapplications in quantum information. Radiative cascades in atoms have also beenused for entangled-photon generation [1]. Moreover, the generation of stationary,single- or few-photon Fock-states was demonstrated using Rydberg atoms and su-perconducting ultra high-Q cavities [29,30]. One drawback of atomic systems is,however, the complexity of today’s atom traps or atomic beam experiments.

Transitions in single molecules and single nanocrystals also produce singlephotons [31–33]. Nanocrystals are semiconductor crystals in the size of a few


nanometers which are chemically produced as colloids [34]. Similar to quantumdots (see below), nanocrystals show discrete energy levels, in contrast to bulkcrystals, leading to single-photon transitions. Molecules and nanocrystals havesimilar properties with respect to single-photon emission. Both systems can beoperated even at room temperature which makes them cheap and easy to handle.Their drawback is their susceptibility for photo-bleaching and blinking [34]. Thelatter describes the effect of interrupted emission even on large timescales due tothe presence of long-lived dark states. However, this problem may be reduced byimproved synthesis.

In experiments where the single-photon character is the only important prop-erty, nitrogen–vacancy defect centers in diamonds are advantageous. These struc-tures show room temperature single-photon emission without optical instabilities,like blinking and bleaching [35,36], but have a broad optical spectrum at roomtemperature together with comparably long lifetimes (12 ns).

This article focuses on single-photon generation using self-assembled singlequantum dots. Quantum dots are few-nanometer sized semiconductor structuresshowing discrete electronic energy levels, in contrast to energy bands in bulksemiconductors.1 Many properties of quantum dots (emission spectrum, elec-tronic structure, etc.) resemble features known from atoms. For this reason quan-tum dots are also referred to as artificial atoms. To suppress electron–phononinteraction and thermal ionization, quantum dots mostly need to be operated atcryogenic temperatures, but experiments at increasingly higher temperature havealso been reported [37,38]. The emission from a quantum dot combines nearlylifetime-limited narrow spectral lines and short transition lifetimes. In contrastto its nanocrystal counterpart, quantum dots are optically very stable. They alsooffer the possibility of electric excitation [39] and the implementation in inte-grated photonic structures [40]. Due to the variety of possible materials, quantumdots have shown single-photon emission throughout the ultraviolet, visible, andinfrared spectrum. Moreover, it was proposed and demonstrated to use quantumdot multi-photon cascades for the generation of entangled photon pairs [41,42].

2. Single Quantum Dots

For the calculation of electronic states in quantum dots (or artificial atoms), sev-eral schemes have been used at different levels of sophistication [43]. Figure 2illustrates the simplest approach which assumes a spherical potential trap for elec-trons and holes. When the quantum dot is occupied by several quasi-free charge

1 Although colloidal nanocrystals are also quantum dots by this definition, to avoid confusion, herethe terminology of quantum dots is used for quantum dot structures grown on semiconductor sub-strates.


FIG. 2. Excitations in a quantum dot: (a) Exciton formed by an electron–hole pair, (b) biexcitoncontaining two electron–hole pairs, generally with a different energy than the exciton. (c) Schematicterm scheme for the exciton and biexciton decay cascade. The two dark excitons are indicated by graylines. Numbers indicate the electron, hole, and total spin.

FIG. 3. (a) Micro-photoluminescence image of InP quantum dots in GaInP. (b) Spectrum of asingle InP/GaInP quantum dot with the spectral lines of exciton and biexciton decay.

carriers (electrons or holes), Coulomb interaction has to be taken into account, aswell. While equally charged carriers repel each other, the energy of the system islowered for an electron–hole pair, and an exciton is formed (Fig. 2(a)). The recom-bination of the exciton leads to the emission of a single photon. Correspondingly,two electron–hole pairs form a biexciton, but generally with a different energydue to Coulomb interaction (Fig. 2(b)). When decaying, first one electron–holepair recombines, leading to the emission of a first photon. The remaining excitonin the quantum dot emits a second photon with a different wavelength (Fig. 2(c)).The quantum dot fine structure (see, for example, [44]) reveals a single biexci-ton ground state and four exciton ground states. Two of them are dark states andparticipate in neither the biexciton nor the exciton decay. Figure 3 shows a pho-toluminescence image of an ensemble of InP quantum dots. The image was takenthrough a bandpass filter to suppress excitation stray light from the optical excita-tion. The spectrum of a single InP quantum dot with two dominant spectral lines,originating from the exciton and biexciton decay, is displayed in Fig. 3.


Quantum dot samples can be fabricated by a variety of methods startingfrom higher dimensional semiconductor heterostructures, like etching pillars inquantum well systems or forming intersections of quantum wells or quantumwires [43]. The growth of nanostructures on patterned substrates, such as groovesor pyramids, led to successful quantum dot formation [45] and single-photonemission [46], as well. These fabrication methods allow a high degree of posi-tion control which is advantageous for coupling the quantum dot to microcavitiesand photonic devices.

So-called natural quantum dots [47] are formed by thickness fluctuationsmainly of quantum wells, but also in nanotube systems. In this environment, ex-citons are trapped in broader regions of the quantum well, where the confinementenergy is lowered, so that a potential minimum is formed. Such excitons exhibitlarge oscillator strengths leading to short radiative lifetimes [48] as the lateral sizeof natural quantum dots is usually much larger than the exciton Bohr radius.

The experiments described in this article are performed on self-assembledquantum dots. These quantum dots are fabricated by epitaxial growth of onecrystal type on top of another. If the lattice constants differ noticeably, dislo-cations due to strain are created, and material islands are formed to minimizethe strain. A thin layer, which is known as the wetting layer, will remain, cov-ering the substrate completely. This growth mode is called Stranski–Krastanovgrowth. The wetting layer forms a quantum well which usually shows photolumi-nescence at energies above the quantum dot emission. There are different epitaxialtechniques like Molecular Beam Epitaxy (MBE) or Metal–Organic Vapor PhaseEpitaxy (MOVPE).

The InAs/GaAs material system is by far the most studied of all quantum dotsystems. Work on single-photon emission reported to date has predominatelybeen done on InAs dots emitting in the 900–950 nm region [39,49], but also at1250 nm [50] and 1300 nm [51,52]. Photon correlation measurements at thesewavelengths require good infrared single-photon detectors. Single-photon gener-ation on demand at 1300 nm will be very useful for quantum cryptography viaoptical fibers. Nitride quantum dots (GaN in AlN) emit single photons in the ul-traviolet region [53].

II–VI-type quantum dots have the advantage of short lifetimes (∼100 ps com-pared to ∼1 ns for the previously described III–V systems) which reduces theprobability of decoherence during the emission process and enables the generationof single photons on demand with a small time uncertainty [37,54]. This suggestsa much higher maximum single-photon emission rate than for III–V dots. Thissystem also shows a larger energy splitting between the exciton and the biexcitonthan the InAs/GaAs material system, which is useful to achieve a better filteringof the exciton emission, enabling operation at higher temperatures. The refractiveindex of ZnSe is lower than of GaAs which reduces photon losses due to totalinternal reflection at the sample surface.


In the experiments reported here, InP dots in a GaInP matrix are used to gener-ate single photons in the 640–690 nm range as well as photon pairs and triplets.In principle, this material system can be used to generate single photons between620 nm and 750 nm which fits to the maximum efficiency of silicon avalanchephoto diodes (over 70% at around 700 nm).

3. Single-Photon Generation

3.1. CORRELATION MEASUREMENTS

The measurement of intensity correlations is a standard method for testing single-photon emission: The intensity correlation of a light field is detected at two pointsin time, resulting in the second-order coherence function g(2)(t1, t2). In the caseof stationary fields, it has the form

(1)g(2)(τ = t1 − t2) = 〈 : I (0)I (τ ) : 〉〈I (0)〉2

,

where : : denotes normal ordering of the operators. This function is proportionalto the joint probability of detecting one photon at time t = 0 and another at t = τ .This function has several characteristic properties: As each random process is as-sumed to become uncorrelated after a sufficiently long timescale, the normalizedcorrelation function tends to a value of unity for large times. It can further beshown [55] that for all classical fields g(2)(0) � 1 and g(2)(0) � g(2)(τ ) hold. Forclassical light fields, this prohibits values smaller than unity.

The case g(2)(0) > 1 is characteristic for thermal light sources. In this case, thephotons are bunched, which means that there is an increased probability to detecta second photon soon after a first one (Fig. 4(a)). For coherent light fields, such ascontinuous laser light, g(2)(τ ) = 1 for all τ which indicates a Poissonian photonnumber distribution and photons arriving randomly (Fig. 4(b)).

If, however, the probability to detect a second photon soon after a first detectionevent is reduced compared to an independent process, g(2)(0) < 1 (Fig. 4(c)).This effect is called anti-bunching. As mentioned before, this case is reservedto non-classical states with sub-Poissonian photon statistics. For photon numberstates |n〉, with exactly n photons, g(2)(0) = 1 − 1/n and in the special case of asingle-photon state (n = 1), g(2)(0) = 0. For statistical mixtures of one- and two-photon states (or more), intermediate values can also be obtained. In the case of apulsed source, the second-order coherence function possesses a peaked structure.Here, a missing peak at τ = 0 indicates the generation of one and only one photonper pulse (Fig. 4(d)).

A straightforward method to measure the second-order coherence functionwould be to simply note the times of detector clicks and to compute the correlation


FIG. 4. Top: illustrative distribution of the photon arrival time, bottom: second-order coherencefunction g(2)(τ ) of (a) a thermal light source (for example, a light bulb), (b) coherent light (laserlight), (c) a continuously driven single-photon source, and (d) a pulsed single-photon source.

FIG. 5. (a) Scheme of the Hanbury Brown–Twiss setup. (b) Correlation measurement of a spectralline of a single quantum dot over a timescale that is large compared to the average time betweendetection events (7.7 µs for the black and 20 µs for the gray curve). The logarithmic scale emphasizesthe exponential behavior. The dip at delay time τ = 0 (see zoom in the inset) indicates single-photonemission.

function according to Eq. (1). However, this approach prevents the measure-ment of timescales shorter than the detector’s dead time (≈50 ns for avalanchephoto detector modules [56]). To overcome this problem, a Hanbury Brown–Twiss arrangement is chosen [57] as depicted in Fig. 5(a), consisting of two photodetectors monitoring the two outputs of a 50:50 beam splitter. With this setup,the second detector can be armed right after the detection event of the first. Forsmall count rates, one can neglect the case, where the first detector is alreadyarmed while the second one is still dead. Losses, like photons leaving the wrongbeam splitter output or undetected photons, simply lead to a global decrease ofthe measured, non-normalized correlation function.


Technically, it is very difficult to acquire absolute detection times with a resolu-tion in the nanosecond regime. Additionally, the sheer amount of detection eventsneeded to reach a reasonable statistics (107 events are typical for a count rate of105 s−1) makes the computation of the correlation function very time-consuming.Instead, only the time differences between detection events are usually registeredand binned together in a histogram. An electronic delay shifts the time origin andenables the observation of asymmetric cross-correlation functions (see Section 5).Using a time-to-amplitude converter, time differences can be measured very pre-cisely. This method has the additional advantage that the measurement can betracked online. However, the function measured in this way is the waiting timedistribution [58] d(t) rather than the second-order correlation function g(2)(t).The function d(t) is defined as:

d(t) = (Prob. density to measure a stop event at t

after a start event at time 0)

× (Prob. that no stop det. occurred before)

(2)= (T g(2)(t)+ rD)(

1 −t∫

0

d(t ′)dt ′),

where the transmission T was introduced to account for possible photon lossesand rD describes the detector dark count rate. For large t it follows:

d(t) = (T g(2)(t)+ rD)

exp

(−

t∫0

(T g(2)(t ′)+ rD

)dt ′)

(3)≈ const × e−(rc+rD)t .

The second line indicates that for long time differences, the measured histogramdecays exponentially on a timescale given by the detector count rate. Figure 5(b)shows such a large-time measurement of a single-photon source (see also theinset). Only if the average arrival time of the photons t = r−1

c (rc: photon countrate) is much larger than the observed time t between start and stop event, theprobability, that no stop detection has occurred before, is approximately 1 andg(2)(t) ≈ d(t).

3.2. MICRO-PHOTOLUMINESCENCE

In order to perform experiments with single quantum dots, several require-ments concerning the setup have to be fulfilled: As self-assembled quantum dotsamples—even on a so-called low-density sample—have quantum dot densities


FIG. 6. Basic scheme of a micro-PL setup (FMs: mirrors on flip mounts, DM: dichroic mirror,PH: pinhole, BP: narrow bandpass filter, APDs: avalanche photo detectors).

of 108 . . . 1011 cm−2, a sufficient spatial resolution is required to select a sin-gle quantum dot or at least only as few as possible. At the same time, a highcollection efficiency is preferred to gain a maximum amount of photons. Thesetwo requirements can be combined by choosing a micro-photoluminescence (PL)setup which is a well-established setup in single molecule spectroscopy.

Figure 6 shows the experimental setup. The sample is mounted inside acontinuous-flow liquid Helium cryostat which can be cooled down to 4 K. Op-tical access for the excitation of the sample and collection of the emitted light isprovided through a thin glass window. The sample is excited by either a pulsed(Ti:Sapphire, pulse width 400 fs, repetition rate 76 MHz, frequency-doubled to400 nm) or a continuous wave (Nd:YVO4, 532 nm) laser. Thus, the excitation isoff-resonant and creates charge carriers in the continuum which are subsequentlycaptured by the quantum dot. The laser light is sent into the microscope objec-tive via a dichroic mirror. The microscope system with a numerical aperture ofNA = 0.75 has a lateral resolution of 0.5 µm which allows the resolution of in-dividual quantum dots on the sample. The collected PL light is filtered spatiallyby imaging onto a pinhole in order to block stray and PL light from neighbor-ing sites. Spectral filtering can be performed with a narrow bandpass interferencefilter. The light transmitted through these filters is directed onto a CCD camerafor imaging or to a grating spectrograph for spectral analysis (Fig. 3 shows acorresponding image and spectrum). Finally, a Hanbury Brown–Twiss correla-tion setup is used to measure the second-order coherence function. It consists oftwo avalanche photo diodes (APDs) and correlation electronics that collects thetime differences between start and stop detector in a histogram. The overall timeresolution of the correlation setup is 800 ps.


3.3. INP QUANTUM DOTS

InP quantum dots grown in a GaInP matrix are a particularly interesting sys-tem for generating single photons for free-space experiments as their emissionwavelength around 690 nm allows the highest possible detection efficiency ofcommercial Si APDs. However, they show disadvantages in fiber-coupled ap-plications because their losses in glass fibers are much higher than at infraredwavelengths. On the other hand, infrared photo detection suffers from low effi-ciency and bad signal-to-noise ratio. Thus, single photons from InP quantum dotsare particularly important for free-beam experiments.

The sample used in this section was grown by Metal–Organic Vapor PhaseEpitaxy (MOVPE).2 Figure 7(a) shows the structure of the sample. On a GaAswafer, a 300 nm thick GaInP layer was deposited, followed by 1.9 mono-layers ofInP which form the quantum dots and another 100 nm layer of GaInP. The densityof dots emitting around 690 nm was estimated to be about 108 cm−2 by imagingthrough a narrow bandpass filter. In order to increase light extraction efficiency,a 200 nm thick Al layer was deposited on top of the sample to form a mirror. Thesample was then glued upside down with epoxy onto a Si substrate, and the GaAssubstrate was removed using a selective wet etch. For the purpose of increasing thelight extraction efficiency, the use of a metallic mirror is preferable to a distributedBragg reflector (DBR), as metal mirrors reflect strongly at all angles, resulting ina larger integrated reflectivity if a point-like emitter is assumed.

Figure 7(b) shows a spectrum of a single InP quantum dot at 10 K. The lowergraph is an unfiltered spectrum. The excitation power density was adjusted to haveonly one dominant emission line. By measuring the intensity of this spectral line,

FIG. 7. (a) Structure of the InP/GaInP sample. (b) PL spectra taken on a single InP quantum dotat cw excitation and a temperature of 10 K. The bottom spectrum was taken without filtering, the topspectrum through a narrow bandpass filter. An offset was added for separating the graphs. The blackline in the inset is a spectrum over a larger wavelength range, the gray line shows the efficiency of thesingle-photon detectors.

2 This sample was provided by the group of Prof. W. Seifert from Lund University (Sweden).


FIG. 8. Measurement of the g(2)-function at continuous excitation. The gray curve is the expectedcorrelation function for an ideal single-photon source, but limited time resolution. The right graph isa magnification of the dip in the left plot.

a linear dependency on the excitation power was observed, indicating an excitontransition. Additional emission lines appear with increasing laser power. A spec-trum taken over a wider wavelength range is displayed in the inset of Fig. 7(b) andshows that all the emission within a very broad wavelength range originates fromthe dot under study. The inset also shows the detection efficiency of the APDswith its maximum right at the emission wavelength of the quantum dot. Whenplacing a narrow 1 nm bandpass filter into the beam path, only light from a singletransition of this quantum dot is transmitted (upper curve of Fig. 7(b)).

Figure 8 shows a correlation function (see Eq. (1)) measured on the excitonspectral line of this dot performed at continuous excitation [59]. The total countrate was 1.1 × 105 counts per second. The dashed gray line in this figure is thecalculated correlation function obtained by taking into account the limited timeresolution of the Hanbury Brown–Twiss setup. This function is modelled as aconvolution of the expected shape of the ideal correlation function g(2)(τ ) = 1 −exp(−γ τ) and a Gaussian distribution with a width of the system’s time resolutionof 800 ps. 1/γ is the timescale of the anti-bunching dip and depends on boththe transition lifetime and the excitation timescale. This timescale is used as a fitparameter here. A zoom into the region around the origin of the left graph is givenin the right graph of Fig. 8. The excellent agreement between calculations andmeasurement indicates that the quantum dot device generates single photons andthat the minimum dip value of 5% (relative to the value at large time differences)is mostly due to the limited time resolution. The characteristic timescale 1/γ ofthe anti-bunching dip is fitted to 2.3 ns. The measurement in Fig. 8 is the quantumdot counterpart of first measurements performed with single trapped ions [60].

In Fig. 9, correlation measurements at pulsed excitation are displayed [59]. Thetotal count rate was 4.4×104 counts per second for the measurement in Fig. 9(a).It is observed that the peak at zero delay time is vanishing almost completely. Thisamounts to single-photon generation on demand: Upon each laser pulse creatingan exciton, one and only one photon is produced.


FIG. 9. Second-order coherence function measured on a single InP quantum dot at pulsed excita-tion: (a) at 8 K and (b) between 20 and 50 K.

This measurement is a prerequisite for succeeding single-photon experimentsbecause it proofs single-photon emission and excludes the possibility to observelight from several quantum dots, leading to ensemble averaging of the results.Additionally, for quantum cryptography experiments, only a normalized area ofthe peak zero delay time clearly below 0.5 ensures secure transmission of theencryption key.

The graphs in Fig. 9(b) show measurements at higher temperatures. When in-creasing the temperature, the emission intensity of the quantum dot decreaseswhich can be attributed to thermal carrier escape. Moreover, broadening of thespectral lines due to phonon interactions [61–63] leads to an increased incoherentbackground when other spectral lines start to overlap with the filter transmissionwindow. Both of these effects deteriorate the quality of single-photon generation.However, up to 27 K, the peak zero delay time at is still almost completely sup-pressed. With increasing temperature, this peak slowly starts to grow, but it has arelative area still below 0.5 even at 50 K, indicating that a single quantum dot’stransition still dominates the emission.

4. A Single Photon as Particle and Wave

The wave–particle duality lies at the heart of quantum mechanics. With respect tolight, the wave-like behavior is perceived as being classical and the particle aspectas being non-classical while for massive microscopic objects, like neutrons andatoms, the opposite holds. The occurrence of an interference pattern is a manifes-tation of the wave-nature of matter.

Already in 1909, soon after the introduction of the concept of light ‘quanta’,Taylor observed experimentally that there is no deviation from the classically pre-dicted interference pattern if a double-slit interference experiment is performedwith very weak light, even if the intensity is so small that on average only asingle photon is present inside the apparatus [64]. Later, this observation was


FIG. 10. Experimental setup for the simultaneous Michelson and Hanbury Brown–Twiss experi-ment.

accounted for by quantum mechanics and was confirmed by more precise exper-iments [65,66]. There exists an exact correspondence between the interference ofthe quantum probability amplitudes for each single photon to travel along eitherpath of an interferometer, on the one hand, and the interference of the classicalfield strengths on the different paths, on the other hand. Therefore, the outcomeof any first-order interference experiment can be obtained by describing light asa classical electromagnetic wave, independent of the statistical distribution of theincident photons. In more recent experiments, Grangier et al. [67] performed aseries of experiments with single photons from atoms. In a first step, they showedthe single-photon character of the atomic emission by observing the correspond-ing anti-bunched behavior of the intensity correlation function. In a second step,they inserted the photons into a Mach–Zehnder interferometer and observed aninterference pattern with varying path difference, a feature that displays the wavenature of light. Braig et al. [68] implemented a similar experiment using a di-amond defect center as the emitter and observed single-photon statistics afterdetecting interference in a Michelson interferometer.

In this section, an experiment is described that combines these two experimen-tal techniques in a single step for simultaneous observation of interference andanti-bunching of the quantum dot fluorescence [69]. Therefore, a Michelson in-terferometer and the Hanbury Brown–Twiss setup were set in series as displayedin Fig. 10.

The clicks of the detectors (APDs) can be evaluated in two ways: (I) Whenlooking for coincidences between clicks from start and stop APD, anti-bunchingis observed, revealing the particle nature of light. The only effect from the changebetween constructive and destructive interference on this measurement is an over-


FIG. 11. Measured correlation function and interference pattern, respectively, for a single quantumdot at pulsed excitation ((a) and (b)) and at cw excitation ((c) and (d)).

all change of the coincidence rate, independent of the delay time between start andstop events. Since the latter is short compared to the timescale of the arm lengthvariation in the Michelson interferometer, the non-normalized second-order co-herence function changes just by a constant factor. (II) On the other hand, onecan count the clicks of either APD while the arm length of the Michelson inter-ferometer is changed. A modulation of the single detector count rate representsinterference which directly demonstrates a wave-feature of light. Since the detec-tor produces a classical electrical pulse that can, after detection of a photon, beeasily split into two parts, it is also possible to perform these two measurementssimultaneously.

Figure 11 displays the results of such a combined measurement when excitinga quantum dot with a pulsed ((a) and (b)) and a cw laser ((c) and (d)), respectively.In Figs. 11(a) and (c), the autocorrelation functions of the two measurementsare plotted, expressed by the number of coincidences. The non-classical anti-bunching effect is clearly visible since the number of coincidences exhibits apronounced minimum at zero delay time. In contrast, Figs. 11(b) and (c) depict thesingle-detector count rate at an integration time of 10 ms, dependent on the pathdifference on the interferometer. It shows the expected first-order interference pat-tern that reveals the wave-like nature of the emitted single-photon radiation.

The described combination of the two experimental techniques, Michelson in-terferometer and Hanbury Brown–Twiss correlation setup, forms an extension to


the experiments of Grangier et al. [67], as one and the same photon contributesto both the measured interference pattern and the anti-bunched correlation func-tion. In this sense, the described experiment is similar to the classic experimentof Taylor [64] and to the experiments described in Refs. [65,66], but gives an un-equivocal evidence of the particle nature of light: Instead of using weak lightfields with classical photon number statistics (with a super-Poissonian photonnumber distribution), the anti-bunching effect shows that the quantum dot pho-toluminescence represents number states that can only be described within theframe of quantum mechanics. In a similar work, Höffges et al. [70] simultaneouslyperformed heterodyne and photon correlation measurements in the resonance flu-orescence of a single ion.

5. A Multi-Color Photon Source

The potential of single semiconductor quantum dots as emitters in photonic de-vices is not only the generation of single photons on demand. Quantum dots arepromising candidates for the generation of entangled photon pairs. It was pro-posed [41] and demonstrated [42] to make use of polarization correlations in thebiexciton–exciton cascade. But as will be demonstrated in the next section, evenwithout entanglement formation, multi-photon cascades find applications in quan-tum communication experiments.

Here, intensity cross-correlations between several different quantum dot transi-tions on the InP quantum dot sample are measured. Similar exciton–biexcitoncross-correlation measurements have also been reported on InAs quantumdots [23,71,72] and II–VI quantum dots [73,74]. Such experiments answer sev-eral purposes: First, they are an important tool for identifying the nature of theinvestigated spectral lines, such as resulting from an exciton, biexciton, triexciton,or emerging from the same or different quantum dots. Second, they give informa-tion about the different decay and excitation timescales in multi-photon cascades.Finally, polarization resolved cross-correlations form a first step towards the ob-servation of entangled photon pairs.

In order to detect correlations between different transitions, a variant of thesecond-order coherence function is considered. The cross-correlation function isdefined in a similar way as in Eq. (1), but with the intensity operators assigned todifferent field modes α and β:

(4)g(2)αβ (τ ) =

〈: Iα(t)Iβ(t + τ) :〉〈Iα(t)〉〈Iβ(t)〉 .

In the experiments described here, the two modes represent spectral lines oftwo quantum dot transitions. In order to distinguish this cross-correlation from


FIG. 12. Modified Hanbury Brown–Twiss setup for cross-correlation experiments. The differentcolors of the two beams represent two spectral lines selected by the bandpass filters.

the second-order coherence function, the latter is also referred to as the auto-correlation function.

The experiment is performed by filtering of the two spectral lines for each photodetector in the Hanbury Brown–Twiss system individually. Here, this was realizedby placing narrow bandpass filters directly in front of each APD, as sketched inFig. 12. In this way, the resulting correlation function will show an asymmetrywith respect to the time origin, as start and stop detection events now arise fromdifferent processes and a change in sign of the time axis accords with an effectiveexchange of start and stop detector.

To get a first idea about the origin of the distinct spectral lines, their differentscaling with the excitation intensity was investigated. Figure 13 shows PL spectraof a quantum dot, taken at various excitation intensities at 8 K. In the follow-ing, the reference excitation power density P0 was kept constant at 1 nW/µm2.The spectral behavior in Fig. 13 is typical in terms of line spacing and power de-pendence, for an InP dot emitting in this energy range. At low excitation powerdensity, a single sharp emission line at 686.3 nm (1.8155 eV) is present in thespectrum (X1). As the excitation power is increased, a second line (X2) appearsabout 0.6 nm (1.5 meV) beside the exciton emission. When further increasing theexcitation power, additional lines appear. The integrated photoluminescence in-tensity of X1 increases linearly with the excitation intensity whereas X2 shows aquadratic dependence. This behavior is a good indication of excitonic and biexci-tonic emission, respectively. The lines appearing at high excitation power density,such as X3, are attributed to a multi-exciton of higher complexity. Especially,X3 is assumed to originate from a triexciton, as will be proven later.

For such a complex excitation as the triexciton, it is necessary to invoke addi-tional states to the single-particle ground states of the quantum dot. Figure 14(a)shows two of the possible triexciton decays together with a simplified decay chainof a triexciton in (b). The triexciton X3 recombines to an excited biexcitonX∗

2 that


FIG. 13. Power dependent spectroscopy of a single InP quantum dot showing the lines used inthe correlation measurements. The excitation intensity is given as a multiple of P0 = 1 nW/µm2.Lines X1, X2, and X3 are assigned to different excitations, as described in the text.

rapidly relaxes to the biexciton ground state X2 which in turn recombines via theexciton X1 to the empty ground state G of the quantum dot [44].

After the different spectral lines have been characterized and pre-identified,additional information can be gained by performing cross-correlation measure-ments between these emission lines. Figure 15(a) shows the cross-correlations ofthe exciton and biexciton line of that dot at different cw excitation power den-sities [75]. A strong asymmetric behavior is observed: At positive times, whenthe detection of a biexciton photon starts the correlation measurement and thedetection of an exciton photon stops it, photon bunching occurs, as here the de-tection of the starting biexciton photon projects the quantum dot into the excitonstate which has now an increased probability of recombining shortly after. On theother hand, if the correlation measurement is started by the exciton photon, whichprepares the dot in the ground state, and stopped by the biexciton photon (neg-ative times in Fig. 15(a)), a certain time is needed until the dot is re-excited. Inthis measurement, effectively the recycling time of the quantum dot is observedwhich explains the strong anti-bunching for negative times. The population of thebiexciton state is dependent on the laser power, and the excitation time decreaseswhen the laser power increases. Similar measurements have also been performed


FIG. 14. Illustration of the multi-exciton cascade in quantum dots. (a) Occupation of electron andhole states in the decay of the triexciton state X3 to the biexciton ground state X2 and to an excitedstate X∗

2 . (b) Decay cascade model used in the discussions and the rate equation approach. Dashedarrows indicate excitation, solid arrows radiative decay, and the open arrow a non-radiative relaxation.The dashed state C symbolizes an effective cut-off state as explained in the text.


FIG. 15. Measured cross-correlation functions (a) between the exciton and biexciton line and(b) between the biexciton and triexciton line of the quantum dot that was also used for Fig. 13 atdifferent excitation intensities (P0 = 1 nW/µm2). (c) Exciton–biexciton cross-correlations of a sec-ond quantum dot.

on another quantum dot (depicted in Fig. 15(c)). While its timescales are similar,it shows a more pronounced bunching peak.

In the same way, the cross-correlation of the biexciton emission with the triex-citon emission was measured. This is shown in Fig. 15(b). Its behavior is similarto the exciton–biexciton case, but with different timescales apparent.

The presence of the combined bunching/anti-bunching shape is a unique hintfor observing a decay cascade of two adjacent states. In contrast, the cross-correlation function of spectral lines of two independent transitions (for example,from two quantum dots) would show no (anti-)correlations, at all. It can be con-cluded that there is a three-photon cascaded emission from the triexciton viabiexciton and exciton to the quantum dot ground state. Together with the infor-


mation of the different scaling of the spectral lines with excitation power, thisjustifies the previous assignments to these lines.

In order to support the interpretation of the obtained correlation data, the photoncascade was analyzed using a common rate model [23,71]. The rate equationscorrespond to the scheme shown in Fig. 14(b) where only two transition typesaccount for the dynamics of the excitonic states: spontaneous radiative decay andre-excitation at a rate proportional to the excitation power. As the excitation isperformed above the quantum dot continuum, the relaxation of the charge carriersinto the multi-exciton states, as well as the relaxation of the excited biexcitonafter the triexciton decay, should also be taken into account. But as this processhappens on a much faster timescale (several 10 ps [76]) than the state lifetimes(≈ns), it is neglected in this consideration. The according rate equation ansatzthen reads:

(5)d

dtn(t) =

⎛⎜⎜⎜⎝−γE γ1 0 0 0γE −γE − γ1 γ2 0 00 γE −γE − γ2 γ3 00 0 γE −γE − γ3 γC0 0 0 γE −γC

⎞⎟⎟⎟⎠ n(t)

with n(t) = (nG(t), n1(t), n2(t), n3(t), nC(t)) and γi = τ−1i . Here n1, n2, and

n3 represent the populations of the exciton, biexciton, and triexciton, respectively,with corresponding decay times τ1, τ2, and τ3. nG is the population of the emptyground state, and τ−1

E is the excitation rate. In order to truncate the ladder of statesconnected by rates in this model, an effective cut-off state with population nC andlifetime τC was introduced. This accounts for population and depopulation of allhigher excited states via excitation and radiative decay, respectively.

This rate equation can be solved analytically [77]. The general solution is asum of decaying exponentials with different time constants. The initial conditionsare defined by the transition that forms the start event in the Hanbury Brown–Twiss measurement which prepares the quantum dot in the next lower state α, sothat nα(0) = 1 and nγ =α(0) = 0. On the other hand, the detection of a photonfrom the stop transition dictates the shape of the cross-correlation function, asg(2)αβ (t) ∝ nβ(t). Therefore it is clear that the cross-correlation function on the

positive and negative side is described by two completely different functions witha possible discontinuity at τ = 0. In the experiment (Fig. 15), this discontinuityis washed out, due to the finite time resolution of the detectors. Because of thissmoothing of the experimental data, the minima in the graphs of Fig. 15 are shiftedtowards the anti-bunching side, as well.

The model was used to describe the auto- and cross-correlation data in Figs. 14and 15. The results are shown as gray lines in these graphs. The lifetimes of ex-citon, biexciton, and triexciton were taken from independent measurements. Theexcitation rate was chosen to optimally fit the correlation functions in these fig-ures, but was kept linear to the experimental excitation power P throughout the


graphs. In this way, apart from the one-time initialization of the experimentallyinaccessible values τ3, τC , and τE , the normalization was the only real fit pa-rameter in all graphs. No vertical offset was used to compensate the lift of theanti-bunching dips. Apparently, the model describes the experimental data verywell. Minor deviations can be explained by the long-term variation of the excita-tion power due to a spatial drift of the sample while taking data or by the presenceof additional states neglected in this model.

6. Multiplexed Quantum Cryptography on the Single-PhotonLevel

6.1. A SINGLE-PHOTON ADD/DROP FILTER

Among the requirements for single-photon sources, high efficiencies and highemission rates are a major priority in order to raise the statistical significance ofexperimental outcomes or to enhance the bandwidth for quantum communica-tion protocols. The overall efficiency can be improved by using passive opticalelements such as integrated mirrors (compare the InP sample Fig. 7(a)), solid im-mersion lenses to enhance the optical collection efficiency [78], or by resonanttechniques that embed the quantum emitters in microcavities [79,80]. The lattermethod exploits the Purcell effect [81] in order to enhance the emission rate ina certain well-defined resonant cavity mode. The Purcell effect can also substan-tially modify the overall spontaneous emission rate. For a single-photon source,which relies on the decay of an excited state, the (modified) spontaneous lifetimedetermines the maximum photon generation rate.

In classical communications, multiplexing is a well-established technique toincrease the transmission bandwidth. It is the transmission and retrieval of morethan one signal through the same communication link (sketched in Fig. 16). Thisis usually accomplished by marking each signal with a physical label, such as thewavelength. At the receiver, the signals are identified by using filters tuned to thecarrier frequencies [82]. Losses when merging and separating the signals can becompensated by amplification of the classical signal. For single-photon channels,

FIG. 16. Transmission of N optical signals distinguished by their wavelengths through the samefiber using multiplexing.


the no-cloning theorem [83] prevents the amplification of qubit information sothat losses have to be kept minimal and effective separation of photons with dif-ferent wavelengths is required. Moreover, a multi-color single-photon source isneeded to provide distinguishable photons.

In this section, an interferometric technique is described to perform multiplex-ing on a single-photon level (see also Ref. [84]). The biexciton–exciton cascadein quantum dots provides an excellent source for the required photon pairs withwell-separated energies and strong correlation in the emission time. As a proof-of-principle, a quantum key distribution experiment using the BB84 protocol [3]was performed.

In order to use several independent qubits in a single communication channelsimultaneously, they have to be distinguishable by at least one physical property,and a method is needed to merge and divide them at the sender and receiver side,respectively. For photons, a reasonable choice would be to distinguish them bytheir wavelength and to use their polarization to encode quantum information.A common way to separate light with different wavelengths uses diffractive orrefractive optics. However, these techniques are unfavorable, especially in inho-mogeneously broadened systems, like a sample of self-organized quantum dots.Here, the wavelength of the two photons as well as their wavelength differencemay vary from dot to dot. When using diffractive and refractive optics, a completerealignment of the beam paths for each individual quantum dot under consider-ation would be required. Moreover, diffractive optics suffer from losses due todiffraction into different orders.

A superior method is to use interferometric techniques, like the one sketchedin Fig. 17(a). Two photons with different wavelengths λ and λ + �λ enter aMichelson interferometer with variable arm lengths. Retro-reflector prisms areused to obtain a lateral shift between input and output beam. Due to the differ-ence in wavelength, the two photons undergo different interference conditions.As long as the path difference s between the two interferometer arms is signif-icantly smaller than the coherence length scoh, the probability to find a photon(with wavelength λ) at one of the two interferometer output ports is:

(6)p1,2(s, λ) = 1

2± 1

2cos(2πs/λ).

The signs + and − correspond to the interferometer output ports labeled as 1and 2 in Fig. 17, respectively.

In order to illustrate how this can be used to separate the two photons, the in-terference pattern p1(s) for two different wavelengths is plotted in Fig. 17(b). Fors ≈ 0, each wavelength shows the same interference pattern. But for increas-ing path difference, they run out of phase and at a certain position (indicatedby the dotted line in Fig. 17(b)) the two interference patterns are in oppo-site phase, i.e. each photon interferes constructively at a different output port.


FIG. 17. Sketch of the experimental setup: (a) Two photons with different energies enter a Michel-son interferometer that consists of a 50:50 beam splitter and two retro-reflectors. (b) Scheme of theintensity interference pattern at one interferometer output for two distinct wavelengths versus the pathdifference between the two interferometer arms. (c) Combining the separated photons: The two in-terferometer output ports are coupled to an optical fiber each, one of them delayed by half of theexcitation laser repetition time, and recombined at a beam splitter, again.

The smallest path difference for which such a wavelength separation occurs iss0 = λ(λ + �λ)/(2�λ). As long as s0 � scoh, such a situation can always beachieved. Note that this condition simply reflects the spectral distinguishabilityof the two photons which is a general limit for wavelength separation. When s0is in the order of or bigger than the coherence length, the interference visibilitydecreases and none or only poor photon separation is performed. In this case, thesetup would basically act as a 50:50 beam splitter.

It can be seen that such an interferometric technique is highly insensitive tochanges in wavelength difference. For example, when changing the quantumemitter or in case of spectral drifts, these effects can be compensated for by cor-recting the interferometer arm lengths which causes no change in the exiting, finalbeam direction. Moreover, as long as s0 � scoh, the main losses that occur in sucha system are caused by partial back-reflections at the interfaces of optical compo-nents which can be strongly suppressed by appropriate anti-reflection coatings.

Figure 18 shows a set of spectra taken from one interferometer output port.For Fig. 18(a), white light illumination was used, and the path difference wasset to s = 40 µm. According to Eq. (6), a sine-like modulation with the period�λ = 5.5 nm is observed in the spectrum. Switching to spectra with discrete lines,the full power of this method becomes evident. In the top graph of Fig. 18(b),


FIG. 18. (a) Spectrum of a white light source observed through the Michelson interferometer.(b) Spectrum of a few InP quantum dots through the interferometer. In the top graph, one arm wasblocked resulting in the original spectrum. In all five graphs, the intensity axes are equally scaled, forcomparison. The numbers indicate three arbitrarily chosen spectral lines whose brightness depends onthe path difference in the interferometer.

an unfiltered few-quantum dot spectrum is displayed with several well-separatedspectral lines. It was obtained by blocking one interferometer arm. But when un-blocking and exposing the lines to interference, it was possible to align the pathdifference for selectively switching on and off individual lines. This is the case inthe lower four graphs, indicated by three arbitrarily selected spectral lines. In theother interferometer output, the opposite picture would be visible. In this way, aninventive spectral switch can also be achieved.

In a further step, the setup was expanded by the part sketched in Fig. 17(c). Byadjusting the arm length difference in the unblocked Michelson interferometer,a situation was achieved where the exciton and biexciton lines show constructiveinterference in either output port and destructive in the other. After filtering, thesetwo output beams were coupled into two multi-mode fibers. The different fiberlengths provided a relative delay time of 6.6 ns (half the repetition time of thepulsed Ti:Sa laser). Behind the fibers, the two beams were merged and detectedby the Hanbury Brown–Twiss detectors. In this way, the train of exciton photonswas shifted in between the biexciton photon train. This enables the simultaneousobservation of the two photon sources and leads to a stream of single photonswith a doubled repetition rate. Ideally, a second, inverted Michelson arrangementwill be used to merge the two beams leaving the fibers. However, for simplicity,a 50:50 beam splitter was used instead.

Figure 19(b) shows correlation measurements of the light merged behind thefibers. For comparison, an exciton correlation function is displayed in Fig. 19(a).Both figures exhibit the characteristics of a pulsed single-photon source. Whilethe impinging photons in Fig. 19(a) have a time separation of 13.2 ns determinedby the excitation laser repetition rate of 76 MHz, the photon stream in Fig. 19(b)possesses only half the repetition time. Still, clear anti-bunching is visible.


FIG. 19. Intensity correlation of (a) the exciton spectral line and (b) the multiplexed signal.

The last graph demonstrates how the maximum emission rate of a single-photonsource based on spontaneous emission is limited. As the photo detectors cannotdistinguish between the energies of the two photons, a similar correlation mea-surement would have been obtained if photons from two excitonic transitions hadbeen recorded, but at a doubled excitation rate. In both situations, the time pe-riod between excitation events approaches the spontaneous lifetime which is inthe order of 1 ns for the quantum dot transitions as reflected by the peak widths.Thus, the peaks start to overlap, and the photons cannot be assigned to individualexcitation pulses any more which is vital for their use in quantum communica-tions. The result in Fig. 19(b) is already on the onset of this process. However, inthe presented kind of experiment, adjacent photons remain distinguishable withrespect to their wavelength, and an assignment of each photon to a certain pulsecan be preserved.

6.2. APPLICATION TO QUANTUM KEY DISTRIBUTION

An important application of single-photon multiplexing is quantum key distribu-tion. In this techniques, single photons (as required by the BB84 protocol [3]) orentangled photon pairs (as used by the Ekert protocol [85]) were used to secretlydistribute cryptography keys among distant parties. Eavesdropping is preventedas the no-cloning theorem [83] forbids to copy the quantum states of the dis-tributed photons. A review and detailed discussion is given by Gisin et al. [4].Long distance experiments have been successfully realized with weak coherentlaser pulses [4,86] and down-converted entangled photon pairs [87]. Realizationsof the BB84 protocol with single-photon states were performed using diamonddefect centers [88] and single quantum dots [89].


FIG. 20. Possible implementations of the multiplexer using the BB84 protocol. In both schemes,the polarization is modulated between rectilinear and circular polarization using the polarizer P withan electro-optic modulator (EOM). Bob’s detection side is realized by another EOM, a polarizingbeam splitter (PBS), and detectors (D). ω1 and ω2 indicate the two energies of the photons, and M isthe Michelson arrangement. In (a), the two photons are recombined without delay time, in (b), a delaytime is introduced, and the two beams are then recombined with a beam splitter BS.

For most protocols, information is stored in the photon’s polarization whereasthe exact wavelength is unimportant. Thus, multiplexing—as described previ-ously—can provide an increased communication bandwidth without loss of se-curity. Figure 20(a) shows a possible implementation of interferometric multi-plexing in the BB84 protocol. On Alice’s side, a cascaded photon source, such asa single quantum dot, provides two closely emitted single photons with differentenergies upon each excitation pulse. In a first Michelson interferometer, the twophotons are separated. In the same way as in the conventional protocol, polariz-ers define a fixed polarization3 and electro-optic modulators (EOMs) randomlymodulate between (H, V,L,R) polarization for each photon. This destroys anypolarization correlation between the two photons, thus providing independentqubits. An inverted Michelson interferometer recombines the two photons in asingle channel for transmission. On the other side, Bob uses the same arrange-ment to separate the photons. A second set of EOMs is used to randomly changethe bases. Polarizing beam splitters in combination with two APDs detect the po-larization state of each photon. In this way, the transmission rate will be doubledcompared to a protocol using only one photon per pulse.

To demonstrate this application, a simplified proof-of-principle experiment assketched in Fig. 20(b) was set up: The excitonic and biexcitonic photons from the

3 For already polarized photons, the polarizers would be ideally aligned for optimum transmission.


quantum dot single-photon source were separated by a Michelson interferometer,fiber-coupled, and delayed (see also Fig. 17). The photon pulse rate was doubled,but the average photon number per pulse was halved. Deviating from the proposedscheme, Bob’s detection consisted of a second EOM, an analyzing polarizer, andan APD, with the EOM randomly switched between the two bases. Since Bobmeasures the same state as Alice only in one fourth of the cases, a reduction ofthe effective count rate of 50% follows, compared to a scheme with two detectors.In this configuration, two-photon events can create a possible insecurity, but in oursetup, the collection efficiency is estimated to be p ≈ 10−3 [77]. The probabilityto collect two adjacent photons is p11 = p2 ≈ 10−6 and thus much smaller thanthe probability to collect one photon and loose the next, p10 = p(1−p) ≈ 10−3.At higher collection efficiency, the setup in Fig. 20(a) is favored.

The transmission distance was 1 m. For exciting the quantum dots, a pulseddiode laser (λ = 635 nm, pulse width 125 ps) with a repetition rate of 10 kHz wasused which was adapted to the modulation rate of the EOM drivers. These driversconsist of a digital-to-analog converter steered by a computer card and a sub-sequent high-voltage amplifier to supply the EOMs with half- and quarter-wavevoltages. A rectangular voltage signal acts as a trigger for the laser pulses, theEOM switching, and the detection gate for acquiring Bob’s detection events. Thetrigger and the detection gate were shifted towards the end of the EOM switchingperiod in order not to affect photon polarization by initial voltage spikes of theEOM driver. The presence of these spikes dictated the maximum modulation rate.The choice of the random bases and data acquisition were controlled by a Lab-view program. An improved software-based random number generator providedthe randomness of the bases.

In the images of Fig. 21, the results of a quantum key distribution are visual-ized. In a first step, Alice and Bob exchanged quantum information resulting ina common sequence of random bits. A series of random number tests checkedand confirmed the randomness of the key. This key then encrypted Fig. 21(a) byapplying an exclusive-OR (XOR) operation between every bit of image and key.The result is shown in Fig. 21(b) into which the randomness of the key was trans-

FIG. 21. Visualization of the quantum key distribution. After exchanging the key, Alice encryptsimage (a), a photography of Berlin’s skyline taken out of our lab window, and sends the encryptedimage (b) to Bob. After decryption with his key, he obtains image (c).


ferred. Then, Fig. 21(b) was classically submitted to Bob who decrypted it byapplying another XOR operation with his received key, yielding Fig. 21(c).

Altogether, the experiment was run with the following parameters: After theelectronic gating of Bob’s detector signals, the rate of usefully exchanged pho-tons is found to be 30 s−1 whereas the dark count rate is reduced to 0.75 s−1. Theprobability to transmit photons through the two EOMs with crossed polarizationswas measured to be 6.8%. After comparing Alice’s and Bob’s keys, an error rateof 5.5% was found. The presence of transmission errors leads to the necessity oferror correction. This requires the exchange of redundant data which opens aneavesdropping loophole for gaining partial information of the message. With theexperimental parameters, the number of secure bits per pulse is 5×10−4 (follow-ing Lütkenhaus [90]) which is a typical value for current single-photon quantumcryptography experiments (≈1 × 10−3 secure bits per pulse, see Refs. [88,89]).

The Michelson add/drop filter might also find applications in linear opticalquantum computation (LOQC). Since gates in LOQC have only limited successprobabilities, parallel processing may increase the efficiency of gates or at leastimprove the statistical significance of a computational result. The method, whichwas demonstrated here, also allows the spatial separation of two polarizationentangled photons (produced, for example, according to the proposal describedin [41]) without destroying their entanglement. Thus, they can be subsequentlyused in a multitude of experiments and applications.

7. Summary

In this article, we have described single-photon generation with single InP quan-tum dots which emit in the visible spectrum around 690 nm. At this wavelength,highest detection efficiencies with Si-based photo detectors are currently avail-able which makes InP quantum dots preferable for free-space quantum opticalapplications. The single-photon character of this source enables the performanceof fundamental quantum optics experiments, where the wave- and particle-aspectof light can be observed simultaneously. We demonstrated single-photon statisticsand cross-correlations of various transitions from multi-excitonic states includingbiexciton and triexciton decays. Multi-photon generation from single quantumdots may find applications in quantum cryptography devices since a higher rateof photons also enhances the maximum transmission rate of the quantum infor-mation. Therefore, a method to perform multiplexing was presented, similar tothe classical technique, but on the single-photon level. A typical application—the BB84 quantum key distribution protocol—was performed to demonstrate thismethod.

Quantum dot single-photon sources have reached a state where they can be im-plemented as ready-to-use non-classical light sources in a number of experiments.


The on-demand character of the emission together with the potential for entan-gled pair generation will be extremely useful in all-solid-state implementations ofquantum information devices. Quantum cryptography and quantum computing,but also interfacing of (small-scale) quantum information systems will be futuretasks of ‘quantum photonics’.

8. Acknowledgements

We acknowledge W. Seifert for providing the quantum dot sample. We thankV. Zwiller, G. Reinaudi, and J. Persson for valuable assistance. This work wassupported by Deutsche Forschungsgemeinschaft (SFB 296) and European Union(EFRE).

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QUANTUM CHAOS, TRANSPORT,AND CONTROL—IN QUANTUM OPTICS*

JAVIER MADROÑERO1,2, ALEXEY PONOMAREV2,ANDRÉ R.R. CARVALHO2, SANDRO WIMBERGER3, CARLOS VIVIESCAS2,ANDREY KOLOVSKY2, KLAUS HORNBERGER4, PETER SCHLAGHECK5,ANDREAS KRUG2,† and ANDREAS BUCHLEITNER2

1Physik Department, Technische Universität München, James-Franck-Straße, D-85747 Garching,

Germany2Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany3Dipartimento di Fisica Enrico Fermi and CNR-INFM, Università di Pisa, Largo Pontecorvo 3,

I-56127 Pisa, Italy4Arnold-Sommerfeld-Zentrum für Theoretische Physik, Ludwig-Maximilians-Universität München,

Theresienstr. 37, D-80333 München, Germany5Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342. Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.1. Parametric Level Dynamics and Universal Statistics . . . . . . . . . . . . . . . . . . . 352.2. Spectral Signatures of Mixed, Regular-Chaotic Phase Space Structure . . . . . . . . . 40

3. Dynamics and Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1. Atomic Conductance Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2. Web-Assisted Transport in the Kicked Harmonic Oscillator . . . . . . . . . . . . . . . 463.3. Ericson Fluctuations in Atomic Photo Cross Sections . . . . . . . . . . . . . . . . . . 493.4. Photonic Transport in Chaotic Cavities and Disordered Media . . . . . . . . . . . . . 513.5. Directed Atomic Transport Due to Interaction-Induced Quantum Chaos . . . . . . . . 55

4. Control through Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1. Nondispersive Wave Packets in One Particle Dynamics . . . . . . . . . . . . . . . . . 614.2. Nondispersive Wave Packets in the Three Body Coulomb Problem . . . . . . . . . . . 634.3. Quantum Resonances in the Dynamics of Kicked Cold Atoms . . . . . . . . . . . . . 66

5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

* We dedicate this paper to Herbert Walther, at the occasion of his 70th anniversary, in reverenceto his contributions to the foundations of quantum optics, as well as to identifying the “quantumsignatures of chaos” in the lab. Happy birthday!

† Present address: Siemens Medical Solutions, Erlangen, Germany.


DOI 10.1016/S1049-250X(06)53002-2

33

34 J. Madroñero et al. [2

AbstractChaos implies unpredictability, fluctuations, and the need for statistical modelling.Quantum optics has developed into one of the most advanced subdisciplines ofmodern physics in terms of the control of matter on a microscopic scale, and, inparticular, of isolated, single quantum objects. Prima facie, both fields therefore ap-pear rather distant in philosophy and outset. However, as we shall discuss in thepresent review, chaos, and, more specifically, quantum chaos opens up novel per-spectives for our understanding of the dynamics of increasingly complex quantumsystems, and of ultimate quantum control by tailoring complexity.

1. Introduction

Quantum optics has nowadays largely accomplished its strictly reductionist pro-gram of preparing, isolating and manipulating single quantum objects—atoms,ions, molecules, or photons—such as to access the very fundaments of quan-tum theory, from quantum jumps [2,3] over the measurement process [4] anddecoherence [5], to quantum nonlocality and entanglement [6], in the laboratory.The field turns “complex” now, by building up—or “engineering”—complexityfrom the bottom, with nonlinear Hamiltonian dynamics [7], particle–particle in-teractions [8,9], disorder [10] or noise [11,12] as essential ingredients. Somewhatunexpectedly, quantum optics therefore makes contact with quantum chaos—thetheory of finite size, strongly coupled quantum systems.

While for a long time under the suspicion of rather mathematical interest, com-ing up with “large fluctuations and hazardous speculations”, quantum chaos [1]now finds an ever expanding realm of experimental applications [7,13–30]. In ad-dition, it provides novel tools for the understanding and the robust control [14,28,29,31,32] of the dynamics of increasingly “complex” quantum systems. In thepresent review, we recollect some of the generic features encountered within such“chaotic” quantum systems, and spell out their potential for the control of quan-tum dynamics in light-matter interaction.

2. Spectral Properties

There are different ways to approach quantum chaos. Possibly the most sugges-tive one proceeds along the semiclassical line, juxtaposing classical phase spacestructures or dynamics on the one side, and the quantum spectral density or wavefunction evolution in phase space, on the other [33–35]. The specific motivationof this program lies in the intricate nature of the semiclassical limit (“h → 0”,

2] QUANTUM CHAOS, TRANSPORT, AND CONTROL 35

meaning the vanishing of Planck’s quantum when compared to typical classicalactions on macroscopic scales), and, hence, of the emergence of classical fromquantum dynamics at sufficiently large actions. This is an extremely attractiveapproach, with a beautiful mathematical and theoretical machinery, leading toimportant practical consequences, such as the rather recent semiclassical elucida-tion of the helium spectrum [36–41]. However, it is—by construction—bound toquantum systems with a well-defined classical counterpart, since it derives quan-tum features from the backbone of the underlying classical dynamics.

While we shall adopt the semiclassical perspective for the motivation or inter-pretation of some of the results to be discussed in this paper, we will often dealwith systems which lack a well-defined classical analog. Therefore, most of ourobservations will be derived directly from the quantum spectrum of the specificsystems under study.

2.1. PARAMETRIC LEVEL DYNAMICS AND UNIVERSAL STATISTICS

On the spectral level, quantum chaos is tantamount to the destruction of goodquantum numbers [42,43]. Since the latter express symmetries, or dynamical in-variants, of the specific system under study, quantum chaos occurs when thesesymmetries are destroyed, e.g., by the nonperturbative coupling of initially sepa-rable degrees of freedom. If a well-defined classical Hamiltonian dynamics un-derlies the quantum dynamics, good quantum numbers are inherited from theclassical constants of the motion, and their destruction is paralleled by the in-vasion of classical phase space by chaotic motion.

Good quantum numbers can be considered, in a bounded system with a discretespectrum, as the labels attributed to individual eigenvalues of the Hamiltonian.Symbolically, we may write for a system with three degrees of freedom:

(1)H(λ)|n � m〉(λ) = E(λ)n � m|n � m〉(λ).

These labels are good labels in the sense that, if H(λ) depends parametrically on areal scalar λ, the eigenvectors |n � m〉(λ) do not (ex)change their specific characterover a finite interval of λ.

The corresponding good quantum numbers loose their significance for the iden-tification of individual eigenstates as soon as different eigenstates of H(λ) arestrongly mixed by a perturbation which couples at least two of the degrees offreedom represented by the quantum numbers n, �, and m, on arbitrarily smallintervals of λ—they are “destroyed” by the perturbation-induced coupling.

In the jargon of quantum chaos, the parametric evolution of the eigenval-ues E(λ) of some Hamiltonian H(λ) parametrized by the real scalar λ is called“regular level dynamics” if completely classifiable by good quantum numbers.“Chaotic” or “irregular level dynamics” (also “level spaghetti”) is encountered


when all good quantum numbers are destroyed. Such irregular level dynamicsalone is one possible indicator of quantum chaos, without any recourse to someanalogous classical dynamics.1

A nice illustration of the transition from regular to irregular level dynamics isprovided by the Floquet–Bloch spectrum generated by the Bose–Hubbard Hamil-tonian under static tilt,

(2)HB = −J

2

(L∑l=1

a†l+1al + h.c.

)+ F

L∑l=1

dlnl + W

2

L∑l=1

nl(nl − 1).

The Hamiltonian is formulated in terms of the creation and annihilation opera-tors a†

l and al of a bosonic atom at the lattice site l, with the associated num-ber operators nl . It describes the dynamics of N ultracold bosonic atoms in aone-dimensional optical lattice of length L and lattice constant d . The implicitsingle band approximation assumes that no excitations to the first conductionband of the lattice can be mediated by the tilt, Fd � �Egap, nor by ther-mal activation, kT � �Egap, with �Egap the band gap. J and W quantify thestrength of the nearest neighbor tunneling coupling J , and of the on-site inter-action strength W between the atoms, respectively, which compete with a staticforcing of strength F . A suitable gauge transform reestablishes the translationalinvariance in space apparently broken by the static field term in (2), and addi-tionally introduces an explicit, periodic time dependence with the Bloch periodTB = 1/F [44]. The time evolution operator for one Bloch cycle in this timedependent coordinate frame is the Floquet–Bloch operator associated with HB.

Figure 1 displays the level dynamics of the one cycle propagator, parametrizedby F , for different values of the ratio of tunneling coupling to interaction strength.Clearly, when J and W become comparable, the eigenstates of the Floquet–Blochoperator interact strongly for any value of F , while in the limit W � J (andequally so for J � W ) individual eigenstates are clearly identifiable over largeintervals of F . In this specific model—which is actually realized in laboratoryexperiments which load Bose Einstein condensates (BEC’s) into periodic op-tical lattices [8,45]—the transition from regular dynamics to quantum chaos isapparent and unambiguous. Yet, this interacting multiparticle system has no well-defined classical counterpart! Further down in this review (see Section 3.5), wewill analyze the dynamical (and experimentally highly relevant) consequences ofthis transition. At present, it is enough to state that the qualitative transition ob-served in Fig. 1 is actually qualitatively underpinned by the cumulative spacing

1 The term “dynamics” is motivated by considering the parameter λ as some generalized time, with

the eigenvalues E(λ) some generalized particle position evolving under variations of λ.


FIG. 1. Spectrum of the Floquet–Bloch operator generated by HB as defined in (2), as a functionof 1/F , for N = 4 particles distributed over a lattice with L = 7 wells (periodic boundary condi-tions). Only states with quasimomentum κ = 0 are shown, in order to separate different symmetryclasses [44]. The particle–particle interaction strength and the tunneling coupling are set equal toW = 0.032, and J = 0.00076 (top) and J = 0.038 (bottom), respectively. As we tune the tunnelingcoupling to a value comparable to the interaction strength, the “individuality” of the energy levelsdrowns in an irregular pattern: isolated avoided crossings between different energy levels which canbe labeled by the interaction energy between the different particles of a given multiparticle eigenstatein the lattice [44] (for weak tunneling coupling, the distribution of the particles over the lattice char-acterizes a given eigenstate very well, except for resonant tunneling enhancements at isolated valuesof F ) are replaced by strongly interacting levels, for arbitrary values of F .


FIG. 2. Cumulative level spacing distribution of the Floquet–Bloch operator generated by HB(Eq. (2)), for N = 7 bosonic atoms distributed over a lattice of length L = 9 (periodic boundaryconditions), static tilt F = 0.01, tunneling strength J = 0.038, interaction strength W = 0.032(full line). The statistics is obtained from the unfolded spectrum [43] with the symmetry class definedby quasimomentum κ = 0 [44]. The dashed and dash-dotted line indicate the RMT prediction forPoissonian and Wigner–Dyson statistics, respectively.

distribution,

(3)I (s) =s∫

0

P(s′)ds′,

with P(s) the probability distribution of the (normalized and unfolded, see,e.g., [43]) spacings s between adjacent eigenphases of the Floquet–Bloch op-erator [44]. Inspection of Fig. 2 clearly shows that I (s) (and equally so P(s),but the comparison of I (s) with the random matrix prediction is known to bemore reliable, in particular in the vicinity of s = 0) exhibits Poissonian sta-tistics, P(s) = exp(−s), in the regular limit, and Wigner–Dyson statistics,P(s) = πs exp(−π

4 s2)/2, in the chaotic limit (more precisely, the level spacings

faithfully reproduce the COE statistics of random matrices of the circular (C) or-thogonal (O) ensemble (E) [46]). Hence, by simply tuning the ratio of J and W , inthe perfectly deterministic Hamiltonian (2), we induce a spectral structure whichenforces a statistical description if we seek for a robust, quantitative descriptionof the system dynamics.

Another example of chaotic level dynamics is shown in Fig. 3, where we dis-play the parametric evolution of the eigenphases of the Floquet operator of thekicked harmonic oscillator. The Floquet operator—or one cycle propagator—U = exp(−i ∫ τ0 H(t ′) dt ′/h), with τ the kicking period, is generated by the


FIG. 3. Spectrum of the Floquet operator generated by Hkho in (4), as a function of theLamb–Dicke parameter η, for a fixed phase space structure indicated by the single trajectory runs over40,000 kicks in the insets (u and v are suitably defined, canonical phase space variables, see [32]).Only eigenphases with an overlap larger than 10−3 with the initial state |ψ0〉 are represented. Filledcircles represent |ψ0〉 = |0〉, while dots refer to a displaced vacuum centered at (1.3, 3.0) (top) and(1.2, 2.0) (bottom).

Hamiltonian

(4)Hkho = hνa†a +Kmν

k2

{cos[η(a + a†)]} ∞∑

n=0

δ(t − nτ).


This is a paradigmatic example of a quantum chaotic system which, on the classi-cal level, does not obey the Kolmogorov–Arnold–Moser (KAM) theorem (whichguarantees stability with respect to small perturbations) [47], due to the degener-acy of the unperturbed spectrum of the harmonic oscillator. In (4), a and a† rep-resent the annihilation and creation operators of the harmonic oscillator modes ofthe translational degree of freedom (for a particle of mass m), and K measuresthe strength of the kicking mediated by the periodically flashed standing wavepotential with wave vector k. η = k

√h/2mν is the experimentally easily tun-

able Lamb–Dicke parameter, which essentially measures the ratio of the widthof the harmonic oscillator ground state in units of the wave length of the kickingpotential.Hkho can be realized in semiconductor heterostructures [48] as well as with

cold, harmonically trapped ions, and allows for unlimited, superdiffusive energygrowth (i.e., for trapped ions, unlimited heating) under rather precisely definedconditions, as we will see further down in this review. This specific dynamicalbehavior has once again its root in the largely irregular level dynamics shown inFig. 3, which is here illustrated for two different ratios q = 2π/τν = 5 (top)and q = 6 (bottom) of kicking period τ and oscillator period 1/ν, under variationof η. These two choices correspond to a crystalline and quasicrystal [49] sym-metry of the classical phase space structure, as indicated by the classical sampletrajectories shown in the corresponding insets. The crystal case still bears someremnants of regularity, with regularly aligned avoided crossings coexisting withapparently randomly distributed anticrossings of variable size. The quasicrystalcase, in contrast, exhibits an extremely complicated level structure, with no ap-parent regularity left. The details and structure of the level dynamics remain tobe understood, but part of its peculiarities can already be exploited for novel per-spectives of quantum control, as we shall see further down in Section 3.2.

2.2. SPECTRAL SIGNATURES OF MIXED, REGULAR-CHAOTIC PHASE SPACE

STRUCTURE

In quantum systems with a well-defined classical analog which exhibits mixedregular chaotic phase space structure [21,31,36,50–62], the parametric evolu-tion of the eigenenergies does not exhibit an unambiguously “chaotic” structure.Eigenenergies associated with eigenstates that are localized in phase space do-mains of regular motion are only weakly affected by the adjacent chaotic phasespace component and evolve, in general, smoothly under variations of some con-trol parameter λ. Since regular domains of phase space are associated with localdynamical invariants, these states can actually be labeled with good quantumnumbers, and undergo, in general, only locally avoided crossings with states livingon the chaotic phase space component. Consequently, such states “go straight” in


FIG. 4. Parametric evolution of the spectrum of a microwave driven hydrogen atom, in suitablyrescaled energy units, under variation of the driving field amplitude F0 (measured in units of theCoulomb field experienced by the Rydberg electron propagating along an unperturbed Kepler orbitwith principal quantum number n0) [31]. Two energy levels, which anticross at F0 � 0.036, clearly“go straight” in this plot, and only weakly interact with the “level spaghetti” background: They repre-sent eigenstates of the atom in the field which are localized on elliptic regions in the classically mixedregular-chaotic phase space, and are therefore shielded against strong interaction with states living inthe chaotic phase space component.

the energy level dynamics, with almost constant slope, as displayed in Fig. 4 forthe (quasi)energy level associated with a wave packet eigenstate of a microwave-driven Rydberg state of atomic hydrogen (see also Section 4.1 below). In arather abstract sense, such states can therefore sometimes be attributed solitoniccharacter [63]—they anticross with “chaotic” eigenstates without changing theircharacteristic features like localization properties, dipole moments, or the like.Conversely, the soliton-like motion under variations of λ can serve as an identifierfor eigenstates which are shielded from the irregular part of the spectrum, even inthe absence of an unambiguous classical dynamics—examples are found, e.g., inmicrowave driven Rydberg states of alkali atoms [64], with their nonhydrogenicmultielectron core which induces quantum mechanical diffraction effects on topof the semiclassical Rydberg dynamics [57,65].

3. Dynamics and Transport

The specific spectral structure of a given quantum system fully determines theassociated time evolution. If we initially prepare our system in the state |ψ0〉, the


action of the time evolution operator is given by

(5)U(t)|ψ0〉 =∑n

exp(−iEnt/h)|En〉〈En|ψ0〉,

where we assume, for simplicity, a discrete spectrum {En} of H . Alternatively,the energies En may be thought of as complex eigenvalues En − iΓn/2 of someeffective Hamiltonian, with the decay rates Γn representing, for instance, thenonvanishing coupling to a continuous part of the spectrum [66–69]. In most ex-periments, some sort of (auto)correlation signal like

C(t) = 〈ψ0|U(t)|ψ0〉 =∑n

∣∣〈ψ0|φn〉∣∣2 exp(−iEnt/h)

(6)→∑n

∣∣〈ψ0|φn〉∣∣2 exp(−iEnt/h) exp

(−Γn

2t

)is measured [70,71], which, besides the purely spectral ingredients En and Γnalso includes a local “probe” |〈ψ0|φn〉|2 of the spectrum, in the vicinity of thestate |ψ0〉 with which the time evolved wave function is to be correlated. Alsoionization or survival probabilities which are often encountered in atomic ion-ization experiments or in model systems which probe quantum mechanical phasespace transport are closely related to such correlation functions, possibly amendedby an additional summation over a (discrete or continuous) set of “test functions”|ψ0〉 [53,72–75].

3.1. ATOMIC CONDUCTANCE FLUCTUATIONS

It is immediately clear from the form of (6) that the dynamics of a chaotic quan-tum system in the sense of chaotic level dynamics as illustrated in Section 2 willexhibit a sensitive parameter dependence, reflecting the parametric evolution ofthe spectrum. A nice example is provided by the ionization yield of one electronRydberg states under microwave driving—which probes the asymptotic electrontransport induced by the external perturbation. In such type of experiments [21,50,76–88], one electron Rydberg states (with excitations to principal quantumnumbers around n0 � 70) are exposed to a microwave field of frequency ω andamplitude F , for an adjustable interaction time t . The experimentally easily ac-cessible ionization yield Pion is formally given [73] by

(7)Pion = 1 −∑j

∣∣〈ψ0|φj 〉∣∣2 exp(−Γj t).

The sum extends over the complete spectrum of the atom dressed by the field,though weighted by the overlap of the (field free) initial state with the atomic


FIG. 5. Typical distribution of the ionization rates Γj and local weights Wj = |〈ψ0|φj 〉|2 enteringthe expression (7) for the ionization yield Pion of an atomic Rydberg state under electromagneticdriving. In the upper plot, 500 spectra of a one-dimensional model atom initially prepared in theRydberg state |n0 = 100〉 are accumulated, for driving field frequenciesω/2π = 13.16 . . . 16.45 GHz,at fixed photonic localization length � = 1 (see Eq. (9)). In the lower plot, one single spectrum ofthe three-dimensional hydrogen atom initially prepared in the state |n0 = 70 �0 = 0 m0 = 0〉, atω/2π = 35.6 GHz and � = 1 is shown. There is no apparent correlation between ionization rates andlocal weights—which also manifests in the parameter dependence of Pion itself, see Fig. 6.

dressed states for the specific choice of ω and F . Typically, several hundreds tothousands dressed states contribute to the representation of |ψ0〉 [89,90].

Under changes of ω or F , not only the decay rates Γj of the individual dressedstates will fluctuate, but, equally important, the local weights |〈ψ0|φj 〉|2—as acorollary of the destruction of good quantum numbers in the realm of quan-tum chaos: The characteristic properties of the system eigenstates vary rapidlywith the control parameter (here ω or F ), and so does the decomposition of the(parameter-independent) initial state |ψ0〉. In general, the fluctuations of decayrates and overlaps are uncorrelated, as illustrated in Fig. 5, for typical driving fre-quencies and amplitudes, and for a one-dimensional model of the driven atom,as well as for the real, three-dimensional system. While one might believe thatthese fluctuations average out under the summation in (7), this is actually notthe case—Fig. 6 shows the ionization yield of atomic hydrogen, initially pre-pared in the unperturbed n0 = 100 Rydberg state, under microwave drivingwith variable frequency. Indeed, Pion fluctuates rapidly with the scaled frequencyω0 = ω × n3

0 [93] in this plot, at fixed n0. This is the dynamical manifestationof the sensitive ω0-dependence of the quantities which determine Pion, accordingto (7). While this sensitive dependence shows that the mere ionization yield for


FIG. 6. Ionization yield Pion, Eq. (7), of a one-dimensional Rydberg atom launched in the Rydbergstate |n0 = 100〉, as a function of the scaled driving field frequency ω0 = ω×n3

0, at fixed localizationlength � = 1 (see Eq. (9)). The strong fluctuations of the signal under variations of ω0 are characteristicof a strongly localized (in the sense of Anderson [91]) transport process (here on the energy scale, andinduced by the external driving) in disordered media [92].

given ω and F does not provide a robust characterization of the electronic trans-port process induced by the external drive, a statistical analysis allows for someinsight: The atomic conductance [94]

(8)gatom = 1

�

∑j

∣∣〈ψ0|φj 〉∣∣2Γj ,

formally equivalent to the time derivative of the ionization yield at t = 0 (with �the average spacing between adjacent energy levels), exhibits a log-normal dis-tribution, i.e., ln gatom is normally distributed, when sampled for a fixed photoniclocalization length [95]

(9)� = �E

ω= 6.66F 2

0 n0

ω7/30

(1 − n2

0

n2c

)−1

.

The latter is a measure of the typical decay length of the electronic populationdistribution over the near resonantly coupled Rydberg states away from the atomicinitial state |ψ0〉, and determines the asymptotic continuum transport on average,according to [93]:

(10)〈ln gatom〉 ∼ 1/�.

In particular, this proportionality relation together with the lognormal distribu-tion for fixed localization length, which are established in Figs. 7 and 8 for aone-dimensional hydrogen atom (which is a reliable model for the description ofreal 3D hydrogen under external microwave driving, when initially prepared in


FIG. 7. Average value of the natural logarithm of the atomic conductance g vs. the inverse pho-tonic localization length 1/�, for a one-dimensional Rydberg atom initially prepared in the state |n0〉with principal quantum number n0 = 40, 60, 70, 90, 100 (from top to bottom). Clearly, the direct pro-portionality (10) predicted by the Anderson picture is very well satisfied for sufficiently large valuesof n0 [93].

FIG. 8. Distribution (histograms) of the atomic conductance g of a one-dimensional Rydbergatom [93,94,97], sampled over 500 different spectra with photonic localization length � = 0.2, inthe frequency range ω0 = 2.0 . . . 2.5, for initial principal quantum number n0 = 40 (left) andn0 = 100 (right). The log-normal fit is excellent for n0 = 100, in perfect quantitative agreementwith the Anderson picture. Finite size effects lead to discrepancies between the numerical distributionof ln g and the lognormal fit at lower excitations around n0 = 40.

an extremal parabolic state [96,97]), provide strong quantitative support for theanalogy between electronic transport along the energy axis in periodically drivenatomic Rydberg states and electronic transport across one-dimensional disordered


wires [92,94,98–100]: Destructive quantum interference of the many transitionamplitudes connecting the initial atomic state to the atomic continuum, in theatomic problem, and the left and the right edge of the disordered wire, in themesoscopic problem, leads to an exponential suppression of the quantum trans-port, as opposed to diffusive transport in a classical description. This phenomenonis known as Anderson localization [91,101–104] (also strong localization), andwas baptized dynamical localization [17,19,105–115] in the realm of quantumchaos, where dynamical chaos substitutes for disorder.

3.2. WEB-ASSISTED TRANSPORT IN THE KICKED HARMONIC OSCILLATOR

An alternative scenario for the detection of chaos-induced fluctuations on the levelof quantum transport properties is provided by cold, harmonically trapped ionsunder periodic kicking. We already have seen in Section 2.1 that the energy leveldynamics of the kicked harmonic oscillator which is realized in such a settingexhibits many avoided crossings of variable size. Indeed, if we launch a wavepacket in the harmonic oscillator ground state and monitor its mean energy astime evolves, the energy growth rate is found to depend sensitively on the pre-cise value of the Lamb–Dicke parameter η, which is easily tuned in state of theart ion trap experiments. Figure 9 shows such behavior, for three different valuesof η, at fixed classical phase space structure (η ∼ √

h determines the effectivesize of h with respect to the typical classical action of the harmonic oscillator;also see Fig. 3). Correspondingly, the mean energy extracted by the atoms fromthe kicking field, after a fixed interaction time, exhibits strong, apparently randomfluctuations with the Lamb–Dicke parameter, as illustrated in Fig. 10. Once again,this can be directly associated with the avoided crossings in the energy level dia-gram in Fig. 3, and is strongly reminiscent of the atomic conductance fluctuationsencountered in Fig. 6. Note, however, that the classical phase space structure ofthe kicked harmonic oscillator is different from the phase space structure of theharmonically driven Rydberg atom, since we are here dealing with a non-KAMsystem. The signature of this non-KAM structure in the spectral statistics is hith-erto unexplored, and represents a formidable challenge, both for random matrixtheory, as well as for computational physics.

We can nonetheless precisely identify the universal cause of the locally en-hanced energy absorption of the trapped ions from the kicking field, by inspectionof the eigenstates which undergo the specific avoided crossing, at a given valueof η: Fig. 11 shows the Husimi phase space projections [69] of those eigenfunc-tions which account for the dominant part in the decomposition of the ionic initialstate |ψ0〉 = |0〉 in the vicinity of η = 0.464 (the associated level anticrossingis shown by the inset in Fig. 9), i.e., at a value where strongly enhanced heatingof the ions is observed. While for Lamb–Dicke parameters slightly below and


FIG. 9. Mean energy of the kicked harmonic oscillator, Eq. (4), for crystal symmetry, q = 6,kicking strength K = 2.0, and initial state |ψ0〉 = |0〉. Tiny changes of the Lamb–Dicke parameterfrom η = 0.459 (a) over η = 0.464 (b) to η = 0.469 (c) lead to a locally dramatic enhancement of theenergy absorption by the trapped particle from the kicking field, with respect to the classical heatingprocess. This local enhancement can actually be traced back to an avoided crossing of the continuationof the eigenphase associated with |ψ0〉 in the level dynamics (inset) with a “web-state” (see Fig. 11)reaching far out to high energies in the harmonic oscillator phase space. The above values of η areindicated by the corresponding labels, in the inset. Filled black circles indicate an overlap of morethan 1% of the associated eigenstate with the initial state |ψ0〉.

FIG. 10. Mean energy (left vertical axis) after 600 (full line) kicks vs. the Lamb–Dicke parame-ter η. The classical phase space structure is fixed by K = 2.0 and q = 6. Locally strongly enhancedenergy absorption can always be traced back to avoided crossings of the initial state with web states,as apparent from the underlaid energy level dynamics (right vertical axis).


FIG. 11. Husimi representations of the eigenstates associated with the labels a (left column) andc (right column) in the inset of Fig. 9, in the rescaled phase space coordinates v/2η = −60 . . .+60and u/2η = −60 . . .+60 of the insets of Fig. 3. The top left and bottom right plot represent webstates associated with the top left and bottom right branch of the avoided crossing shown in the insetof Fig. 9. At η = 0.464, i.e., at the center of that avoided crossing, they strongly mix with the contin-uation (bottom left and top right branch of the avoided crossing, and bottom left and top right Husimirepresentation in the present figure) of |ψ0〉, thus giving rise to efficient transport from the trap centerto high energy states of the harmonic oscillator, along the stochastic web of the underlying classicalphase space flow. Since the avoided crossing of the web state with the localized state occurs at fixedphase space structure, this is a pure quantum tunneling effect, without classical analog.

slightly above this critical value the eigenstate which is strongly localized in thevicinity of the origin of phase space has the largest weight in the initial state de-composition, an eigenstate localized on the stochastic web has equal weight rightat η = 0.464. The existence of such web states is a peculiarity of non-KAM sys-tems and is at the very origin of the observed enhanced energy growth, simplysince the stochastic web reaches out to infinity, and therefore provides an effi-cient transport channel to high energy states of the oscillator. Since the avoidedcrossing which mediates the coupling of the initial state to the web state occursunder changes of the effective value of h (via η), at fixed phase space structure,we have here—much as in the above case of strong localization in the ionizationprocess of periodically driven atoms—a pure quantum effect without classicalanalog, leading now to a dramatic enhancement of the asymptotic transport, as


compared to the classical dynamics. A closely related phenomenon has been ob-served in the conductance across semiconductor superlattices, in the presence ofa tunable magnetic field [48]. Since there the magnetic field allowed to switchbetween localized and delocalized (i.e., web-) states, web-states mediate, in somesense, metal-insulator like transitions.

3.3. ERICSON FLUCTUATIONS IN ATOMIC PHOTO CROSS SECTIONS

In the preceding two subsections, we encountered examples of a sensitive de-pendence of asymptotic transport on some control parameter, typical of quantumchaotic systems, in explicitly time dependent transport processes. As a third exam-ple, we now consider the continuum decay of Rydberg electrons induced by staticexternal fields, which can be probed through the photoabsorption cross sectionfor a probe laser beam from the atomic ground state into the Rydberg spectrum.Indeed, an atomic one electron Rydberg system exposed to perpendicularly ori-ented, static electric and magnetic fields, allows us to realize such a situation: TheHamiltonian reads

(11)HExB = p2

2+ Vatom(r)+ B

2Lz + B2

8

(x2 + y2)+ Fx,

in atomic units, with F and B the strength of the electric and magnetic field, re-spectively, and Lz the angular momentum projection on the magnetic field axis.If Vatom(r) is given by the hydrogenic Coulomb potential, the diamagnetic termin (11) is known to induce chaotic motion in the bound space dynamics of theRydberg electron. For B = 0 the electric field, while leaving the dynamics com-pletely integrable, induces a Stark saddle and, hence, strong coupling of the boundeigenenergies with the continuum part of the spectrum. If both external fields arepresent, all symmetries of the unperturbed Coulomb problem are destroyed, andone faces a truly three-dimensional problem which exhibits dynamical chaos. Inthe case of alkali atoms, the additional presence of a multielectron core is not ex-pected to suppress the signature of the classically chaotic Coulomb dynamics, onthe spectral level [57,116,117].

Due to the suppression of the ionization threshold by the electric field, the highlying Rydberg states can acquire relatively large autoionization rates Γj , withan average value Γ which can become larger than the mean level spacing � ofthe (quasi)discrete energy levels Ej , i.e., Γ > �. In this regime of overlappingresonances, Ericson fluctuations [118–122] are expected in the photoabsorptioncross section

(12)σ(E) = 4π(E − E0)

chIm∑j

|〈g|T |Ej 〉|2Ej − iΓj /2 − E


FIG. 12. Distribution of the resonance widths Γj which contribute to the photo cross section σ(E),Eq. (12), in an energy interval which covers the experimentally [27,123] scanned region. The dashedline indicates the average (local) spacing � of the resonance states on the energy axis. Approx. 65%of them exhibit overlapping widths, Γj > �.

from the atomic ground state |g〉 into the Rydberg regime at energy E: Bound-continuum transition amplitudes which mediate the decay of individual reso-nances couple to overlapping intervals of continuum states, and thus may inter-fere. Consequently, one expects interference structures in the cross section whichcan no more be attributed to individual resonance eigenstates with a specificwidth Γj , but are rather due to the interference of several decay channels, andexhibit typical widths smaller than Γ . If a classical analog dynamics is available,these structures are predicted to be correlated on an energy scale which is deter-mined by the dominant Lyapunov exponent of the classically chaotic dynamics,i.e., by the shortest decorrelation time scale of the classical dynamics [120].

Indeed, the transition into the Ericson regime has recently been observed inthe photoionization cross section of rubidium Rydberg states in the presence ofcrossed fields [27,123]. A detailed theoretical analysis of the experimental situ-ation shows that the laboratory results indeed entered the regime of overlappingresonances, and approx. 65% of all resonance eigenstates contributing to the pho-toabsorption signal have widths which are larger than the mean level spacing �.Figure 12 shows the numerically calculated distribution of resonance widths overthe energy range probed by the experiment, under precisely equivalent conditionsas in the experiment (fixed by the strength of the magnetic and electric fields).Besides the strongly fluctuating background signal, the cross section σ(E) dis-played in Fig. 13 also shows some narrow resonances on top, which stem fromisolated resonances with Γj < �. However, many of the structures with a widthsmaller than Γ can no more be associated with single isolated resonances, andthus indicate the interference of different decay amplitudes.

Thus, we observe the coexistence of individually resolved resonances withEricson fluctuations. This it is not too surprising, since the original Ericson sce-


FIG. 13. Numerically obtained photo cross section (12) of rubidium Rydberg states in crossedelectric and magnetic fields [124], deduced from a parameter free diagonalization of the Hamil-tonian (11), using exactly the experimental parameters [27], B = 2.0045 T, F = 22.4 kV/m.

nario was inspired by highly excited compound nuclei with a large number ofessentially equally weighted decay channels, while we are here dealing with alow-dimensional atomic decay problem, where different decay channels (e.g.,through different angular momentum channels) have certainly different weightsand equally different effective bound-continuum coupling constants.

Once again, due to the underlying chaotic level structure—here additionallycomplicated by resonance overlap—the experimentally accessible cross sectionshows erratic fluctuations, essentially uncorrelated on energy scales which arelarger than the inverse of the characteristic life time of the ion–electron com-pound (which, in a classical picture, is determined by the largest Lyapunov expo-nent).

3.4. PHOTONIC TRANSPORT IN CHAOTIC CAVITIES AND

DISORDERED MEDIA

In the previous section, we showed how the fine interplay between overlappingand isolated resonances determines the nature of the fluctuations in the transportproperties of chaotic systems. In this section, we shall consider a novel kind ofsystems for which this interplay has also a determinative role: random lasers.

In contrast to standard lasers, random lasers do not possess mirrors. They area class of nonlinear amplifiers realized in disordered dielectrics with a fluctuatingdielectric constant that varies randomly in space. Light amplification is providedby an active optical medium, while the multiple chaotic scattering of photons inthe random medium constitutes the feedback mechanism. Due to multiple scat-tering, the time spent by the light inside the active medium is enhanced. This, inturn, increases the probability of stimulated emission, making the field amplifica-tion efficient. Laser oscillations emerge when the radiation losses are overcomeby the light amplification.


In recent years, several experiments on random lasers (see Ref. [125] for areview) as well as on lasers in chaotic resonators [126,127] have attracted consid-erable interest in the characterization of the properties of light emitted by thesedevices. Most striking are the generic signatures of the underlying disorder ofthe random media in the emission spectra: In samples with a low density of scat-terers [128], light is only weakly confined and we expect the resonant modes tooverlap. Once the pump energy exceeds the laser threshold, the onset of lasing issignaled by a collapse of the thermal emission spectrum into a single broad peakwith a width of a few nanometers at the center of the amplification bandwidth.For samples with a high density of scatterers [128], on the other hand, some well-resolved resonant modes exist. As soon as the laser enters the operation regimeabove threshold, several very sharp peaks appear, the frequencies (within the am-plification bandwidth) and strengths of which fluctuate strongly from sample tosample.

The above-mentioned features of the emission spectra cannot be explained bystandard laser theory [129–131]. The reasons are twofold: First, in random lasersthe spatial structure of the resonant modes as well as their frequencies dependon the statistical properties of the disordered medium. Random lasers, therefore,must be analyzed in an statistical fashion. Second, due to the absence of mir-rors, light in random lasers is only weakly confined, giving rise to spectrallyoverlapping resonances. Recently, based on a field quantization method for opensystems with large outcoupling losses [132–134], a quantum theory of randomlasing incorporating both effects, random scattering of light and mode overlap,was proposed [135].

For a random laser with an active medium composed of two-levels atoms, thequantum Langevin equations of motion for the field variables are

(13)aλ(t) = −i∑λ′

Hλλ′aλ′(t)+∑p

g∗λpσ−p(t)+ Fλ(t).

Here, aλ is the annihilation operator of the field mode λ, and σ−p is the dipoleoperator of the pth atom. The coupling amplitudes gλp between field and atomsare proportional to the atomic dipole d and to the field amplitude u(r) at the po-sition of the atom, gλp ∝ duλ(rp). Equation (13) should be complemented withthe equations of motion for the atomic operators, which we have omitted as theyremain the same as those found in standard laser theory [129]. There are drasticdifferences between Eq. (13) and the independent-oscillator equations of standardlaser theory. They arise from the fact that in order to account for the strong cou-pling of the field with the outside, all internal modes must now leak into the sameexternal channels, i.e., they are coupled to the same bath. Hence, the internal dy-namics of the field is determined by the non-Hermitian operator H, accounting forthe system’s losses due to the coupling with the exterior, and coupling the different


modes aλ. Additionally, and consistently with the fluctuation–dissipation theo-rem, the noise operators Fλ of the different modes are correlated, 〈F †

λFλ′ 〉 = δλλ′(the expectation value is defined with respect to the state of the bath).

The emission properties of random lasers are determined by the complex eigen-values ωk − iΓk/2, and the nonorthogonal eigenfunctions R(r) of H. Due to thestrong correlation among modes, the relation between the mean frequency sepa-ration � of the real frequencies ωk , and the average decay rate Γ of the modesis of crucial relevance for the emission spectra. In the regime of overlapping res-onances, Γ > �, typically many broad modes will contribute to the emittedradiation. The resulting spectrum is then a smooth function of the frequency. Onthe contrary, in the regime of isolated resonances, Γ < �, the spectrum consistsof a set of sharp peaks located at the resonant frequencies of the system. Morestriking, however, is the effect of the mode correlations on the coherence timeof the random laser emission. For single mode lasing the coherence time δτ isinversely proportional to the laser line width δω. The latter was first calculatedfor standard lasers by Schawlow and Townes [136], by taking into account thespontaneous emission noise, and was found to decrease for increasing output in-tensities, δωST ∼ 1/I . In random lasers, however, the noise correlation betweendifferent modes leads to an enhancement of the line width. One then has [132]

(14)δω = KδωST,

where K � 1 is the so called Petermann factor [137–139]. K can be related tothe self-overlap of the nonorthogonal laser mode R(r), and is a measure of thecorrelations in the system. Hence, the coherence time of a random laser is smallerthan the coherence time of a standard laser with the same output intensity.

The signatures of the underlying disorder in random lasers are also present inthe photon statistics of the emitted light. Though for light propagating in a disor-dered material the photon statistics below threshold is well understood [140,141],only recently the nonlinear optical regime above threshold has been investi-gated [142–145]. As an example, we evaluate the mean photocount of the emittedfield from a chaotic laser resonator in the regime of single-mode lasing [142]. Weconsider the coupling of the cavity to the outside to be weak, so that all resonancesin the cavity are well defined. In this perturbative limit, the non-Hermitian oper-ator H in Eq. (13) becomes diagonal, and the laser mode a decouples from allother modes. Moreover, since the cavity opening is small, we can replace R(r) bythe orthogonal close cavity modes u(r). In chaotic resonators the amplitude u(r)at a point r behaves like a Gaussian random variable, and is uncorrelated with theamplitude at any other point, provided it lies further apart than an optical wavelength λ [146,147]. As we shall show, these spatial fluctuations induce strongmode-to-mode fluctuations in the laser emission.

In its steady-state, the laser is characterized by three parameters comprisingthe effects of the active medium on the field: The linear gain A, the nonlinear


saturation B, and the total loss rate C. The photon number distribution giving theprobability to find the laser field at a time t with n photons is [130]

(15)Pn = N (Ans/C)n+ns

(n+ ns)! ,

where the symbol N stands for a normalization constant, and the nonlinear satu-ration B enters through the so called saturation photon number ns = A/B. Whenthe number of atoms in the active medium is large, A and B are shown to acquiresharp values.C = Γ +κ , on the other hand, is the sum of the photon escape rate Γdue to the cavity opening, and the absorption rate κ accounting for all other lossmechanisms of the radiation inside the resonator. While here κ may be consideredfixed, the photon escape rate depends on the resonator mode u. Thus, inasmuch asthe resonator mode represents wave chaos, Γ , and therefore C, become randomnumbers. The distribution P(Γ ) over an ensemble of modes in time-reversal in-variant cavities is a well-know result from random-matrix theory [148,149], and isgiven by the χ2

ν distribution. Here, ν is an integer, counting the number of escapechannels at the opening of the resonator. For the case ν = 1, the correspondingdistribution is known as the Porter–Thomas distribution.

For a single-mode laser, the mean output intensity is given by I = Γ 〈n〉, where〈n〉 is the mean photon number inside the cavity. Over an ensemble of chaoticcavity modes the mean output intensity fluctuates from one mode to the other. Itsdistribution is given by

(16)P(I) =∫

dΓ P (Γ )δ(I − Γ 〈n〉).

Note that the right-hand side involves a twofold average, the quantum opticalaverage with the distribution Pn (represented by the brackets 〈. . .〉) and the en-semble average over the cavity modes with distribution P(Γ ). We evaluate nu-merically P(I). The results for an ensemble of chaotic cavities with one escapechannel are plotted in Fig. 14, for two different sets of parameters. In both casesA > C, i.e., they correspond to lasers above threshold in the ensemble aver-age. We note that all distributions are strongly non-Gaussian. They are all peakedas I−1/2 at small intensities, and present a second peak for maximal intensity.Furthermore, for one of the parameter sets (dashed lines) the distribution P(I)

displays a shoulder for submaximal I . This last feature is seen to be a signature ofspontaneous emission [142]. Thus, for lasers in resonators with irregular shape thechaotic nature of the cavity modes gives rise to fluctuations of the photocount ontop of the quantum optical fluctuations known from laser theory. Chaos-inducedfluctuations are found when a single-mode photodetection is performed over anensemble of modes.

In recent years, in the light of nonlinear optical effects, the investigation on mul-tiple scattering of photons has received new impetus. A fresh and fertile field for


FIG. 14. Distribution P(I/Imax) as a function of the dimensionless mean intensity I/Imax, forone escape channel and two sets of parameters. Rates are given in units of A ≡ 1, the nonlinearity isB = 0.005. The solid line corresponds to κ = 0.7, Γ = 0.02; the dashed line to κ = 0.7, Γ = 0.2.

interesting physics is found in this region where nonlinear optics and wave chaosintersect. Random lasers are just one example of the kind of problems encoun-tered there. Other relevant examples constitute studies of coherent backscatteringof light by a cloud of cold atoms. In these system, for sufficiently high intensitiesof the incident light, nonlinearities becomes relevant and a new class of coherenteffects are seen to arise [150,151]. In the near future, new questions concerningthe consequences of nonlinear effects for the strong localization of light are likelyto move into focus.

3.5. DIRECTED ATOMIC TRANSPORT DUE TO INTERACTION-INDUCED

QUANTUM CHAOS

All the above examples of transport in quantum chaotic systems stem from therealm of one (active) particle dynamics—where we also include the phenomenaobserved with alkaline atoms, since the multielectron atomic core only inducesadditional quantum diffraction effects, which can be accounted for on the oneparticle level. In our last example, we consider now an interacting many-particleproblem, which is motivated by recent progress in the manipulation of ultracoldatoms loaded into optical lattices, and which establishes, in some sense, the exper-imentally “controlled” version of multiparticle quantum chaos originally thoughtof by Bohr [152] and Wigner [153] when they modelled compound nuclear reac-tions.

One of the prominent models to describe the dynamics of matter waves in opti-cal potentials is defined by the Bose–Hubbard Hamiltonian (2) which we alreadyencountered above. Indeed, it can be shown that (2) exhibits Wigner–Dyson sta-tistics in a broad interval of tunneling coupling J and interaction strength W , for


filling factors n = N/L, N the particle number and L the lattice length, even inthe absence of any static forcing, i.e., for F = 0 [154]. Surprisingly, this was real-ized only recently, despite the fact that (2) is a standard “working horse” for quitea big community—which, however, is mostly interested in ground state propertiesrather than dynamics. Only recent experiments in quantum optics laboratories [8,155–160] have triggered enhanced interest in dynamics, and hence in the excita-tion spectrum of the many-body Hamiltonian.

On the dynamical level, the chaotic character of the Bose–Hubbard spec-trum induces the rapid decay of single particle Bloch oscillations across a one-dimensional lattice, for not too large static forcing (such that the static term in (2)does not dominate the symmetry of the problem) [161,162]. The single particledynamics can be defined equally well by the reduced single particle wave func-tion of the bosonic ensemble, or by a second, spin-polarized fermionic componentloaded into the lattice [163]. We shall here adopt the latter scenario, where non-interacting fermionic atoms interact with a bosonic “bath”. The correspondingtwo-component Hamiltonian writes

(17)HFB = HF +HB +Hint,

and decomposes into the (single particle) fermionic part

(18)HF = −JF

2

(L∑l=1

|l + 1〉〈l| + h.c.

)+ Fd

L∑l=1

|l〉l〈l|,

the (many particle) bosonic part

(19)HB = −JB

2

(L∑l=1

a†l+1al + h.c.

)+ WB

2

L∑l=1

nl(nl − 1),

and a term which mediates the collisional interaction between fermions andbosons,

(20)Hint = WFB

L∑l=1

nl |l〉〈l|.

Here we built in the assumption that only the fermions experience the externalstatic force—this can be arranged by preparing the fermionic and bosonic compo-nent in appropriate internal electronic states, which couple differently to externalfields.

Since in (17) there is a clear separation between “system” (the fermions) and“bath” (the bosons), we can derive a master equation for the time evolution inthe fermionic degree of freedom, in Markovian approximation [163]. A crucialingredient for this derivation is the chaotic level dynamics of the bath degree offreedom, what ensures a broad distribution of frequencies of the bath modes, such


as to act as a Markovian environment, with a rapid decay of the bath correlations,on the relevant time scales of the system dynamics [164]. One ends up with

(21)∂ρ

(F)l,m

∂t= − i

h

[HF(t), ρ

(F )]l,m

− γ (1 − δl,m)ρ(F)l,m ,

where ρ is the fermionic one particle density matrix, and the relaxation rate γ iscompletely determined by the parameters of our original Hamiltonian (17):

(22)γ = τ n2W 2FB

h2� 3n2W 2

FB

hJB.

In other words, we can “engineer” incoherent Markovian dynamics in a perfectlyHamiltonian system, (17), by exploiting the chaotic dynamics of one system com-ponent. The resulting decay of the fermionic Bloch oscillations is illustrated inFig. 15, where perfect agreement of the actual decay rate (resulting from an ex-act numerical propagation of the dynamics generated by (17)) with the analyticalexpression (22) is observed.

The collisional interaction of the fermions with the bosonic bath provides arelaxation mechanism which, in the theory of electronic conductance across a pe-riodic potential, is the necessary ingredient for observing a net current across thelattice [165]. Yet, in Fig. 15 we do not observe any net drift of the electrons. This

FIG. 15. Bloch oscillations of the fermionic mean velocity in the optical lattice, under a statictilt Fd = 0.57 × JF, with JF = JB, and WFB = 0.101 × JF, 0.143 × JF, 0.202 × JF (from top tobottom). The bosonic bath, which is the source of the collisionally induced damping of the oscillations,is composed of N = 7 particles, distributed over a lattice of length L = 9. v0 = JFd/h. The typicaltime scale of the interaction induced decay fits the time scale predicted by Eq. (22) (dash-dotted lines)very well [163].


is due to the fact that we are here dealing with a perfectly closed system, withoutattaching any leads—in particular, we are dealing with a finite size bath, which,consequently, has a finite heat capacity. Therefore, the initial state of the bathplays a crucial role for the effective fermionic transport across the optical lattice:If prepared in the thermalized state (as in Fig. 15), with equal population of allenergy levels of the bath, no net energy flux can occur from the fermionic into thebosonic degree of freedom, and, hence, no net drift velocity of the fermions canemerge. In contrast, if we prepare the bath in a low temperature state, with only theground state and few excited states initially populated, the bath can absorb energyfrom the fermions, via collisions, and the fermionic component acquires a non-vanishing drift—which lasts until the bath is fully thermalized. This is illustratedin Fig. 16, together with the corresponding energy increase of the bath. Figure 17shows the resulting current (fermionic drift velocity v) voltage (static tilt F expe-rienced by the fermionic component) characteristics under variations of F , whichdisplays a marked transition from Ohmic behavior (small F ) to negative differ-ential conductance (large F )! Note that such behavior was earlier predicted forsemiconductor superlattices [166], on the basis of a semiclassical theory with aphenomenologically determined relaxation rate γ , whereas the present scenarioallows for the experimental tuning of the relaxation rate, on the basis of our mi-croscopic theory (with crucial input from the theory of quantum chaos).

FIG. 16. Mean velocity v(t) of the fermionic component (top, solid line) for a low temperature(kBT � 2.86×JB) bath, under static tilt Fd = 0.143×JF, with WFB = 0.143×JF, WB/JB = 3/7,JB = JF, N = 7, L = 9. The solid line in the bottom plot shows the associated time evolution ofthe mean energy EB of the bath. Dashed lines in both plots indicate the result for a thermalized bath(kBT � 150 × JB), when no net energy exchange between the fermions and the bosons is possible.Clearly, only for the low temperature bath do we observe a nonvanishing drift velocity (i.e., a directedcurrent) of the fermions across the lattice.


FIG. 17. Current–voltage (expressed as drift velocity v vs. tilt Fd) characteristics for the directedfermionic current across the optical lattice (stars) [163], for the same parameters as in Fig. 16. Thecontinuous line shows the prediction of a phenomenological model of charge transport in semicon-ductor superlattices [166], with the relaxation rate γ extracted from Eq. (22). A clear transition fromOhmic to negative differential conductance at large tilt potentials is observed.

4. Control through Chaos

We have seen in the preceding sections that quantum chaos is tantamount to strongcoupling of the various degrees of freedom of a given quantum system, of thedestruction of good quantum numbers, and that all this usually leads to large fluc-tuations of various observables under slight changes of some control parameter,or to decoherence-like reduced dynamics. Though, does quantum chaos provideus with any means not only to describe, but also to control complex quantumsystems in a robust way?

Indeed, there is a positive response to this question, at least for periodicallydriven quantum systems with a mixed regular-chaotic structure of the underlyingclassical dynamics. The phase space of such systems decomposes into domainsof regular and of chaotic motion, see Fig. 18, which are associated with ellip-tic (i.e., stable) and hyperbolic (i.e., unstable) periodic orbits. Elliptic periodicorbits are surrounded by elliptic islands in phase space, which define regionsof regular, i.e., integrable classical motion. A classical particle launched withinsuch an island cannot leave it (or, in higher dimensions, only on rather long timescales [167,168]), and the only way for a quantum particle to leave the island is bytunneling. It is rather obvious on semiclassical grounds [169], and has also beenrealized by approximating the quantum dynamics in elliptic islands by a quantumpendulum [170], that such regular regions in classical phase space lend supportfor quantum eigenstates localized on top of them, provided the island’s volume


FIG. 18. Example for the surface of section of the classically mixed regular-chaotic phase space ofa periodically driven system in a one-dimensional configuration space—here derived from the equa-tions of motion of a one-dimensional hydrogen atom under periodic electromagnetic driving (in dipolecoupling) [31]. The phase space—spanned by the classical action I (measured in units of some ref-erence action n0) and the conjugate angle θ—decomposes into essentially three main components:a near-integrable (weakly perturbed) part (for actions below approx. 0.9), a prominent resonance is-land structure centered around (θ = π, I/n0 = 1.0), and a chaotic region—the complement ofnear-integrable and island domain.

is large enough to accommodate the typical phase space volume hf (with f thenumber of degrees of freedom) of a quantum state. Later on it was realized that, inperiodically driven systems, these quantum eigenstates faithfully follow the timeevolution of the elliptic trajectory they are anchored to [20,171–173], and thattheir localization properties are preserved by the elliptic island—i.e., by the un-derlying nonlinearity of the classical dynamics—thus protecting them against theusual dispersion of quantum wave packets in unharmonic systems. Hence, ellipticislands in classical phase space give rise to the emergence of nondispersive wavepackets on the quantum level [31]. The only mechanism which limits their lifetime (as long as incoherent processes can be excluded [31,174,175]) is tunnelingfrom the island into the surrounding chaotic sea, which, however, is strongly sup-pressed in the semiclassical limit of large classical actions as compared to h [176].

Since elliptic structures in mixed regular chaotic classical dynamics are ubiq-uitous, so are nondispersive wave packets in the microscopic world. And theclassical nonlinear dynamics bears yet another blessing: The KAM theorem guar-antees that elliptic islands in classical phase space are extremely robust againstperturbations—i.e., for sufficiently small perturbations, an elliptic island is possi-bly slightly distorted in phase space, though preserves its topology. While KAMmight appear of essentially mathematical interest on a first glance, this statementhas indeed very far-reaching consequences on the experimental level: Note that it


is very hard to prevent conventional Rydberg wave packets, built, e.g., on a Starkmanifold (by exciting a coherent superposition of the Stark levels, with a laserpulse, from the atomic ground state) from dispersion [71]—since any small (un-controlled) perturbation shifts the Stark levels and thus induces an unharmonicityin the spectrum, leading to dispersion of the wave packet. In contrast, a nondis-persive wave packet anchored to an elliptic island in classical phase space isessentially inert against any perturbation which is not strong enough to destroythe island, as a consequence of KAM. In other words, the KAM theorem as one ofthe fundamental theorems of classical nonlinear dynamics shields nondispersivewave packets against technical noise (alike stray fields, etc.). It is this robustnesswhich allows the experimentalist to realize and manipulate nondispersive wavepackets in the laboratory [28], over time scales which exceed “traditional” wavepacket dynamics by orders of magnitude!

4.1. NONDISPERSIVE WAVE PACKETS IN ONE PARTICLE DYNAMICS

The simplest realization of nondispersive wave packets is provided by an unhar-monic, bounded, one-dimensional system under periodic driving, described bythe Hamiltonian

(23)Hwp = H0(z)+ λV (z) cos(ωt).

Transformation to the action-angle variables (I, θ) of H0 allows one to rewritethis as

(24)Hwp = H0(I )+ λ

m=+∞∑m=−∞

Vm(I) cos(mθ − ωt),

where we assumed, for simplicity, that the Fourier amplitudes Vm(I) are real [31].Reminding ourselves of θ = Ωt , with Ω the classical roundtrip frequency alongthe unperturbed trajectory with action I , we immediately realize that choices ofthe driving frequency ω such that sθ − ωt � 0, for some term m = s in theabove sum in (24), will lead to a separation of time scales in the time evolutiongenerated by Hwp. While all terms in (24) except the one with m = s will os-cillate rapidly, a resonance will occur between the external drive at frequency ω

and the unperturbed motion along the trajectory with sΩ(I) = ω. In other words,proper choice of the driving frequency allows one to selectively address a spe-cific trajectory of the unperturbed dynamics, via this resonance condition. Fors = 1, a suitable coordinate transformation, followed by a secular approxima-tion (which averages over the rapidly oscillating terms in (24), at resonance), anda final quadratic expansion around the action of the resonantly driven classicalorbit yields a pendulum Hamiltonian, which establishes the backbone of the typ-ical phase space structure of an elliptic island at weak perturbation amplitudes,


FIG. 19. Typical phase space structure in the vicinity of a resonantly driven trajectory of abounded, one-dimensional system, in action-angle coordinates I and θ . I is measured in units ofsome reference action n0. The external driving frequency is chosen such as to match the unperturbedroundtrip frequency of the trajectory with action I/n0 = 1.0. The consequent separation of timescales in (24) induces an onion-like, elliptic island structure centered around (θ = π, I/n0 = 1.0),already at weak perturbation strengths λ. With increasing λ chaos invades phase space, at the expenseof the elliptic island and of near integrable regions at low actions. However, comparison with Fig. 18also shows that the center of the elliptic island survives (actually to rather large values of λ [31,63]),what is a consequence of the KAM theorem, and identifies elliptic islands as very robust topologicalstructures in classical phase space.

FIG. 20. Electronic density of a nondispersive electronic wave packet in a periodically driven,one-dimensional Rydberg atom. The wave packet starts (at phase ωt = 0 of the driving field) at theouter turning point of the classical eccentricity one orbit, is reflected from the Coulomb singularityat ωt = π , and precisely refocuses at the outer turning point, without dispersion, after one completefield cycle.

displayed in Fig. 19. The KAM theorem essentially guarantees that the core ofthis structure survives even a considerable increase of λ, whilst all the remainingphase space volume may undergo a dramatic metamorphosis, as evident from acomparison of Figs. 18 and 19.

Figure 20 shows the configuration space representation of a nondispersive wavepacket launched along the Rydberg orbit with principal quantum number n0 = 60,for the one-dimensional Coulomb problem [20]. This model describes the dynam-


ics of quasi one-dimensional (i.e., extremal parabolic) Rydberg states of atomichydrogen in a near resonant field reasonably well [96,97]. Such a nondispersiveelectronic wave packet propagating without dispersion along a highly excited Ry-dberg orbit has recently been excited and probed in laboratory experiments withlithium atoms [28,29]. In particular, these experiments succeeded to demonstratethe extremely long life time of these objects, by probing the electron’s position onits Rydberg orbit after 15,000 cycles of the driving microwave field. This is equiv-alent to 15,000 Kepler orbits of the unperturbed Coulomb dynamics, and thus byapproximately three orders of magnitude longer than the life time of any Rydbergwave packet so far generated in the laboratory. Furthermore, the experimentallymeasured life time only gives a lower bound for the wave packet’s endurance,since longer probing times were not possible due to the geometry of the exper-imental setup. Theory predicts life times of approx. 106 Kepler orbits, at theseexcitations [31,176].

4.2. NONDISPERSIVE WAVE PACKETS IN THE THREE BODY

COULOMB PROBLEM

The above scenario of nondispersive one particle wave packets can be general-ized for the three body Coulomb problem, naturally realized in the helium atom.A very nontrivial complication arises here from the fact that the electron–electroninteraction term in the helium Hamiltonian

(25)HHe = p12

2+ p2

2

2− 2

r1− 2

r2+ 1

|r1 − r2| ,

generates classically chaotic dynamics even in the absence of any external per-turbation [39]. This is nowadays identified as the cause of the failure of the earlysemiclassical quantum theory to come up with a quantitative description of the he-lium spectrum [41]. Furthermore, doubly excited states of helium have a finite au-toionization probability, again due to the electron–electron interaction [177,178].Hence, the helium atom itself has to be treated as an open system, and bearssome similarity with the crossed fields problem which we discussed in Section 3.3above. Indeed, Ericson fluctuations are also expected in the photoabsorption crosssection of helium [179], for sufficiently high excitations, though the required en-ergy range has not yet been reached in the lab [23].

Thus, since the classical phase space structure of the helium atom is globallychaotic, our above motivation of the typical elliptic island structure on whichto build nondispersive wave packets is not straightforward, since there are noglobal action-angle variables for irregular classical dynamics. However, we canfocus on specific regular domains in the classical phase space of the helium


FIG. 21. Characteristic frozen planet trajectory of the unperturbed three body Coulomb problem.The inner electron precesses on highly eccentric ellipses, with a rapid Kepler oscillation between theinner and the outer turning point. Upon average over the inner electron’s rapid motion, Coulomb at-traction due to the screened Coulomb potential of the nucleus and electron–electron repulsion conspiresuch as to create an adiabatic, shallow binding potential for the outer electron [181]. Consequently,the outer electron is locked upon the precessing motion of the inner electron, leading to a strongcorrelation of both electrons’ positions.

FIG. 22. Phase space structure for the outer electron of the (collinear) frozen planet configura-tion [182], in the absence (a) and in the presence (b) of an external, near resonant driving field. If theexternal field frequency is chosen to match a resonance condition with the unperturbed outer electron’smotion, secondary resonance islands emerge as in (b).

atom, which are elliptic islands themselves.2 These lend support for stable eigen-states of the unperturbed helium atom—the most prominent thereof being thefrozen planet configuration [36,180]. Figures 21 and 22 show a typical clas-sical, highly correlated two-electron trajectory, and the phase space structureof the frozen planet configuration, respectively. Given the regular phase spacestructure with well-defined, stable periodic orbits as shown in Fig. 22, we areback to our original setting: If we apply an external field with a frequency near

2 Indeed, by mapping an f degrees of freedom system on a periodically driven f − 1 degrees offreedom system, where the periodic time dependence of the drive is provided by the periodic timedependence of the remaining degree of freedom, these islands can be made formally equivalent tothose considered above [31,47].


FIG. 23. Top: Husimi representation of a nondispersive two-electron wave packet propagatingalong the collinear frozen planet orbit of the planar helium atom [184], in the phase space coordinatesof the outer electron along the quantization axis defined by the linear field polarization vector, for dif-ferent phases ωt = 0 (left), π/2 (middle), and π (right). Very clearly, the electronic density faithfullytraces the resonantly driven frozen planet trajectory, as obvious from a comparison with the classicalphase space structure shown below (on identical scales).

resonant with one of the stable periodic orbits of the classical phase space ofthe unperturbed system, we induce elliptic islands which propagate along theunperturbed trajectory, phase-locked on the period of the drive. Consequently,for sufficiently high excitations, we find nondispersive two-electron wave pack-ets [182,183] propagating along the frozen planet trajectory, as illustrated inFig. 23 for an excitation to the fifth autoionization channel (in other words,the inner electron is launched along an extremal parabolic orbit with principalquantum number N = 6). Note that a quantum treatment of the planar threebody Coulomb problem (an accurate treatment of the fully three-dimensionalproblem is hitherto out of reach, due to the size of Hilbert space when manyangular momenta are coupled by the driving field) predicts life times of ap-prox. 1000 driving field periods (or, due to the resonance condition on driveand unperturbed two-electron orbit, 1000 frozen planet periods) for these wavepacket eigenstates [184,185]. This prediction can be expected to be reliable, onthe basis of a comparison of typical He autoionization rates in 1D, 2D, and3D configuration space [186]. The predicted two-electron wave packet’s lifetimes are considerably less than the life times predicted for the one electronproblem considered in the previous section, though still much longer than lifetimes of conventional Rydberg wave packets, and thus eligible for applicationsin coherent control. Recently, the excitation of another type of nondispersivetwo-electron wave packets has been suggested, with both electrons far from thenucleus [187].


4.3. QUANTUM RESONANCES IN THE DYNAMICS OF KICKED COLD ATOMS

Nondispersive wave packets as those discussed above are ubiquitous, and can berealized in any driven quantum system with an unharmonic spectrum (the un-harmonicity guarantees the selectivity of the addressing of a specific classicaltrajectory by the near resonant drive) and mixed regular-chaotic phase space [31].

Importantly though, their creation is not necessarily restricted to the realm ofsemiclassical physics, where h becomes small in comparison to the classical ac-tions of the dynamics. This has been realized recently, in the treatment of quantumresonances [188] and quantum accelerator modes [189] in the translational degreeof freedom of periodically kicked cold atoms loaded into one-dimensional opticallattices which are flashed periodically. Such quantum resonances occur due to theclose similarity of the kicked atom Hamiltonian

(26)HKA = p2

2−K cos(kx)

+∞∑m=−∞

δ(t −mτ)

with the kicked rotor, apart from the different boundary conditions (an infinite pe-riodic lattice in the atomic problem, a circle in the case of the kicked rotor [188]).They are excited by kicking periods τ = 2π�, � integer, since then the kicks aresynchronized with the exact revivals of the free evolution of the rotor dynamics(we omit here the discussion of the specific value of the atomic quasimomentum,which implies further restrictions, though is not indispensable for our present ar-gument), leading to ballistic energy growth, for the appropriately prepared initialquasimomentum state of the atoms [188].

If one considers the quantum dynamics close to the resonance condition, i.e., atτ = 2π�+ε, with a small detuning ε, it turns out [188,189] that the time evolutiongenerated by the Hamiltonian (26) can be obtained from the formal quantizationof some well-defined classical dynamics described by a map, with the detuningε taking the role of h ≡ τ (which itself remains constant and can be arbitrarilylarge!).

The quantum accelerator modes are created when an additional static potential(such as provided by gravity) is added to the Hamiltonian of Eq. (26). For appro-priate parameters, this Stark field allows the experimentalist to design classicalnonlinear-resonance islands (classical in the above sense of ε taking the role of h)embedded in a surrounding chaotic sea. These islands support ballistic transport,which—in contrast to the ballistic motion at quantum resonance—is directed dueto the destruction of the translational invariance by the Stark field (see the accel-erated tail of the atoms’ momentum distribution in Fig. 24).

In this generalized classical picture, both quantum resonances and quantum ac-celerator modes are nothing but quantum eigenstates anchored to elliptic islandsin the phase space of that classical map, i.e., a variant of our above nondispersive


FIG. 24. (Courtesy of Gil Summy.) Time dependence (measured by the number of pulses or kicks)of the atomic momentum distribution under periodic kicks along the gravitational field [15], in areference frame freely falling with the atoms. Besides the bulk of the atomic ensemble, which doesnot acquire momentum, there is an atomic component which exhibits ballistic acceleration. This is theexperimental signature of a quantum accelerator mode.

wave packets. This mode-locking of the external drive to the intrinsic charac-teristic frequency of the system allows the experimentalist to efficiently transferlarge momenta to the atoms. Once again, these modes are robust against perturba-tions [190], are clearly identifiable in laboratory experiments [14–16], see Fig. 24,and offer a variety of experimental applications, such as for high precision mea-surements of the gravitational constant [14].

5. Conclusion

As quantum optics addresses the dynamics of more and more complex quantumsystems, methods imported from quantum chaos provide useful tools for identi-fying statistically robust quantities for their description, and also to control theirtime evolution. In this review, we have seen examples for characteristic universalfeatures of chaotic quantum systems on the spectral as well as on the dynamicallevel, in such different settings like ultracold atoms in periodic optical potentials,excitation and ionization processes of one and two-electron atoms subject to sta-tic or oscillating external fields, random laser theory, and cold atoms kicked bystanding light fields. The chosen examples are far from exploring all the diver-sity of current experimental and theoretical activities at the interface of quantum


optics and chaos—we did not discuss here the recently predicted and observeduniversal ionization threshold of one electron Rydberg states under microwavedriving [84,117], weak and strong localization phenomena in the scattering ofphotons off clouds of cold (or ultracold) trapped atoms (with close connections torandom lasing) [191], nor the complementary scenario of matter wave transportin disordered optical or magnetic potentials [10,115], or the role of incoherentprocesses which might compete with coherent quantum transport in complex dy-namics [12,86,174,175]. Nonetheless, we hope that the examples treated alreadygive a flavor of the potential applications of quantum chaos, from the microscopicmodelling of an atomic current across a periodic potential, by using a chaoticbosonic system as a bath which provides the necessary relaxation processes, tonondispersive, one and two-electron wave packets which, due to their extraordi-narily long life times and robustness against technical noise (inherited from theKAM theorem), might find applications in robust quantum control schemes or asquantum memory, in the context of quantum information processing. In partic-ular, the analogies between quantum chaos and quantum transport in disorderedsystems are currently coming into focus, and hold a panoply of intriguing chal-lenging questions, to be tackled in the near future.

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MANIPULATING SINGLE ATOMS

DIETER MESCHEDE and ARNO RAUSCHENBEUTEL

Institut für Angewandte Physik, Universität Bonn, Wegelerstr. 8, D-53115 Bonn, Germany

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762. Single Atoms in a MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.1. Magneto-Optical Trap for Single Atoms . . . . . . . . . . . . . . . . . . . . . . . . 772.2. Dynamics of Single Atoms in a MOT . . . . . . . . . . . . . . . . . . . . . . . . . . 792.3. Beyond Poissonian Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3. Preparing Single Atoms in a Dipole Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 824. Quantum State Preparation and Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 845. Superposition States of Single Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866. Loading Multiple Atoms into the Dipole Trap . . . . . . . . . . . . . . . . . . . . . . . . 897. Realization of a Quantum Register . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918. Controlling the Atoms’ Absolute and Relative Positions . . . . . . . . . . . . . . . . . . . 94

8.1. An Optical Conveyor Belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.2. Measuring and Controlling the Atoms’ Positions . . . . . . . . . . . . . . . . . . . . 958.3. Two-Dimensional Position Manipulation . . . . . . . . . . . . . . . . . . . . . . . . 98

9. Towards Entanglement of Neutral Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . 999.1. An Optical High-Finesse Resonator for Storing Photons . . . . . . . . . . . . . . . . 999.2. A Four-Photon Entanglement Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 1009.3. Cold Collisions in Spin-Dependent Potentials . . . . . . . . . . . . . . . . . . . . . 100

10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10111. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10212. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

AbstractNeutral atoms are interesting candidates for experimentally investigating the tran-sition from well-understood quantum objects to many particle and macroscopicphysics. Furthermore, the ability to control neutral atoms at the single atom levelopens new routes to applications such as quantum information processing andmetrology. We summarize experimental methods and findings in the preparation,detection, and manipulation of trapped individual neutral atoms. The high efficiencyand the observed long coherence times of the presented methods are favorable forfuture applications in quantum information processing.


DOI 10.1016/S1049-250X(06)53003-4

75

76 D. Meschede and A. Rauschenbeutel [1

1. Introduction

Neutral atoms have played an outstanding role in our understanding of the micro-scopic world through quantum physics. Countless details of quantum mechanicshave been discovered and experimentally investigated with dilute gases of atoms.With the advent of tunable, narrowband lasers around 1970, it became possibleto use laser light as an agent to control not only the internal quantum state ofatoms but also the motional degrees of freedom. The first observation of individ-ual atomic particles was successful in 1978 by P. Toschek and collaborators [1].The experimenters realized essential premises to observe individual Barium ions:A strong electromagnetic radio frequency trap (Paul trap) to store ions in a smallvolume and for extended periods of time, and an efficient optical detection byresonance fluorescence from a narrowband tunable laser.

As a result of this breakthrough, trapped ions became prime objects for study-ing and illustrating light–matter interactions at the ultimate microscopic level, i.e.,single particles interacting with well-controlled light fields. Interesting advancesin the 1980s include the observation of quantum jumps [2–4], anti-bunching inresonance fluorescence [5], ion crystals [6,7], and more.

A similar degree of control was achieved for neutral atoms beginning in 1994[8–10]. The origin for this delay with respect to ions is straightforwardly associ-ated with the much weaker trapping forces available for a neutral atomic particlein comparison with a charged particle. Neutral atoms can be localized in space byexerting radiation pressure (magneto-optical trap, MOT), in the effective poten-tial of an optical dipole trap (DT), or by magnetic traps (MT) if the atom carriesa permanent magnetic moment. A simple calculation shows that for typical laserbeam intensities trapping depths do not exceed 1 K for the MOT, 10 mK for DTs,and 1 K for typical MT designs [11].

Experimental accomplishments in handling microscopic particles since 1980have led to the demonstration of many quantum processes at an elementary level.Perhaps even more importantly they have initiated new lines of research wherethe control of atomic systems—and in particular atom–atom interactions—haveopened the route to study novel many particle systems. The celebrated realizationof Bose–Einstein condensation with neutral atoms in 1995 [12,13] has catapultedexperiments with neutral atoms into a central and unique role: they allow thestudy of many particle systems with tailored interactions in a highly controlledenvironment. It has already been shown with ultracold samples of atoms contain-ing 10,000s of atoms, that novel quantum states, for instance, induced by quantumphase transitions, can be realized and investigated [14]. A combination of thesemethods with an experimental access to the atomic constituents at the single par-ticle level promises deep insight into the physics of many particle systems andtheir application, e.g., in quantum simulation and quantum information process-ing [15].

2] MANIPULATING SINGLE ATOMS 77

It is the aim of this article to describe the state of art in the manipulation ofsingle neutral atoms. It is focused on well-known optical traps for neutral atoms,usually employed for trapping much larger samples of atoms. In an alternativeapproach, single neutral atoms can be prepared through the interaction with asingle mode of a low loss optical resonator which is of relevance for the field ofcavity-QED. For more information about this field we refer to [16].

2. Single Atoms in a MOT

2.1. MAGNETO-OPTICAL TRAP FOR SINGLE ATOMS

The magneto-optical trap, proposed by J. Dalibard and realized by D. Pritchardand coworkers in 1987 [17], has revolutionized experimental work in atomic andoptical physics, because it allows to directly prepare and confine cold, i.e., lowvelocity atoms from a background gas at room temperature. The MOT relies onspatially modulated, velocity dependent radiation pressure forces exerted by reddetuned laser beams in combination with a magnetic quadrupole field. It remainsto this day the work horse of physics with cold atoms and serves in nearly allexperiments to initially prepare an ensemble of atoms at very low velocities.

The MOT capture rate is determined by the gradient of the magnetic quadru-pole field, the diameter and the detuning of the trapping laser beams, as well as thepartial pressure of the atomic species to be trapped [18]. The loss rate, on the otherhand, is determined by collisions with the residual gas and exothermic intra-trapcollisions. In a conventional MOT with a quadrupole field gradient of 10 G/cm,cm-wide beams, and a red detuning of the trapping laser beams of about −2γ ,where γ is the natural linewidth of the atomic resonance line, typically 109 atomsare captured with characteristic temperatures below 1/2 the Doppler tempera-ture. For Caesium atoms, which are used in the experiments described here, theDoppler temperature is TDopp = hγ /2kB = 125 µK.

Single atom preparation and observation in a MOT is achieved by taking severalMOT parameters to the limits [8–10]: Since atom capture is mostly determinedby the time available for radiation pressure deceleration, the trapping rate isdramatically reduced by small laser beam diameters (≈1 mm) and strong fieldgradients (up to several 100 G/cm) [19], and of course, very low partial pres-sure (<10−14 mbar) of the trapped atomic species. Very low residual gas pressure(≈10−11 mbar) also makes storage times of order 1 min and more possible. In ourexperiment, the magnetic field gradient can be ramped up and down within typi-cally 20–30 ms time scale which allows to actively control trap loading processes(see Section 2.3).

Resonance fluorescence is collected from a 2.1% solid angle by a self-mademicroscope objective with a diffraction limit below 2 µm [20], and recorded with


FIG. 1. Schematic of experimental setup of the magneto-optical trap. A diffraction limited mi-croscope objective (working distance 36 mm, NA = 0.29) collects fluorescence from a 2.1% solidangle and directs half of the signal towards an intensified CCD camera (ICCD, approx. 10% quantumefficiency at 852 nm, one detected photon generates about 350 counts on the CCD chip). The otherhalf of the fluorescence signal is transmitted by the beamsplitter and focused onto an avalanche pho-todiode (APD, 50% quantum efficiency). Alternatively, the ICCD can be replaced by a second APD inorder to measure photon correlations (see below). The ICCD image shows the fluorescence of a singleCaesium atom trapped in the MOT. One pixel corresponds to approximately 1 µm, exposure time is1 s. Interference and spatial filters (IF, SF) are used to suppress background.

either an intensified CCD camera or with avalanche photodiodes. Spectral as wellas spatial filtering helps to suppress stray light and reduces background to typi-cally below 20,000 counts/s while the fluorescence of a single atom contributestypically R = 60,000 counts/s to the fluorescence signal. The “portrait” of a sin-gle Caesium atom illuminated with trapping laser beams at the 852 nm D2 line isshown in Fig. 1 for a 1 s exposure time.

The rate of photons recorded by the APDs reflects the time evolution of thenumber of trapped atoms in Fig. 2: Prominent upward steps indicate loading,downward steps disappearance of an individual atom from the trap. Neglect-ing background, the number of counts is proportional to the atom number Nthrough CT = N · f · T , where f is the fluorescence rate detected from indi-vidual atom and T is the integration time of the counter. The width �CT of theindividual steps in Fig. 2 is dominated, to better than 99%, by shot noise, i.e.,


FIG. 2. Left: Time chart clip of resonance fluorescence from neutral atoms trapped in a MOT.Well-resolved equidistant fluorescence levels (step size f · T , see text) correspond to integer numbersof atoms. Right: Distribution of count rates shows shot noise limited detection, here for an average ofabout 2 atoms.

�CT � √CT = √

NfT . In order to distinguish N from N + 1 atoms with betterthan 99% confidence, the step size f T must be larger than the peak widths bya factor of ≈5, i.e., f T/5 �

√NfT . Thus the minimal time to detect N atoms

with negligible background is T � 25N/f , which for f = 6 · 104 results inT � N · 400 µs, many orders of magnitude shorter than the storage and hence theprocessing time, see the next section.

For purely random loading and loss processes, the distribution of the occur-rences for atom numbers N should exhibit a Poissonian distribution. In reality,deviations are observed as a result of atom–atom interactions as discussed belowin more detail.

An interesting application of the single atom MOT has been developed by Z. Luand coworkers [21]: The ATTA method (Atom Trap Trace Analysis) makes use ofextreme selectivity of the magneto-optical trap with respect to atom species andspatial detection. The sensitivity of the method for the detection of rare species isessentially limited by the number of atoms that can be sent through the trappingvolume only.

2.2. DYNAMICS OF SINGLE ATOMS IN A MOT

In the MOT, trapped atoms continuously scatter near-resonant light. During theseexcitation and de-excitation processes, the atoms are optically pumped from onestate to another in their multilevel structure. Furthermore, due to the randomtransfer of momentum in each scattering event, they undergo diffusive motionin the trap volume. Finally, the interaction between atoms in the presence of near-resonant light can induce inelastic collisions causing departure from the trap.

Substantial information about all relevant dynamical processes can be retrievedfrom photon correlations in the resonance fluorescence which are imposed by the


atomic dynamics. We analyze photon correlations either by the classic configura-tion introduced by Hanbury Brown and Twiss [22], in order to overcome detectordead times at the shortest nanosecond time scale, or by directly recording photonarrival times with a computer and post-processing.

From this data, second order auto- or cross-correlation functions are derived.In the photon language, g(2)(τ ) describes the conditional probability to observe asecond photon with a delay τ once a first photon was observed:

g(2)AB(τ ) =

〈nA(t + τ)nB(t)〉〈nA(t)〉〈nB(t)〉 ,

where 〈. . .〉 denotes time averaging, and A and B symbolize the two quantitiescorrelated with each other.

The dynamics of a single (or a few) Caesium atoms trapped in the MOT can bederived from these measurements at all relevant time scales [23]:

(a) Rabi-Oscillations. Excitation and de-excitation of electronic atomic transi-tions occurs at the nanosecond time scale. The corresponding measurement of theauto-correlation function is shown in Fig. 3(a) and shows (after substraction ofthe background) the famous phenomenon of anti-bunching, i.e., the second ordercorrelation function shows non-classical behavior at τ = 0, g(2)(0) = 0 [5,24].Damping of the Rabi oscillations occurs at the 30 ns free space lifetime of theexcited Caesium 6P level. The data also show that with increasing number ofatoms the rate of stochastic coincidences rapidly increases: Anti-bunching can beobserved at the level of a single or very few atoms only.

(b) Optical Pumping. It is known that optical pumping of multi-level atomsplays a central role for the realization of sub-Doppler temperatures in MOTs andoptical molasses [25,26]. The single atom MOT has allowed to directly observeoptical pumping by measuring, e.g., the cross-correlation g

(2)lr (τ ) for left- and

right-hand circularly polarized fluorescent light, see Fig. 3(b): Observation of alefthanded photon projects the atom into a strongly oriented quantum state fromwhich the observation of right-handed photons is significantly reduced. Atomicmotion through the spatially varying polarization of the near-resonant trappinglight field induces optical pumping and causes this orientation to relax. From thedata one can estimate that it takes several microseconds for an atom to travel adistance of λ/2, i.e., the length over which typical polarization variations occur.

(c) Diffusive dynamics. If one half of the image of the trapping volume isblocked, the intensity measured at the detector indicates the presence of the atomin the open or in the obstructed half of the trapping volume: If an atom is detectedin the visible part of the MOT, it will stay there and continue to radiate into thedetector until it vanishes into the oblique part by diffusion. Fig. 3(c) shows thiseffect in the intensity autocorrelation measurement of a single atom moving aboutin a MOT. A diffusion model agrees well with the observations, showing that theso-called position relaxation time is of the order of 1 ms, as directly seen from the


FIG. 3. Time domain measurements of atomic dynamics in a MOT by photon correlations (a)–(c)and direct observation (d). See text for details.

experimental data. The average kinetic energy and hence the diffusion constant ofthe atom is controlled by the detuning of the trapping laser beams.

(d) Cold collisions. The time chart of Fig. 3(d) shows the slow load and lossdynamics at the seconds to minutes time scale similar to the one which has al-ready been presented in Fig. 2. One of the most interesting properties is theobservation of two-atom losses (arrows), which occur much more frequently thanwhat can be expected if one assumes Poissonian-distributed, i.e., independent,one-atom losses [27]. The analysis of the occurrence of such two-atom losses re-veals that their rate is proportional to N(N − 1), where N is the total number ofatoms trapped in the MOT. Its origin thus clearly stems from a two-body process.A detailed examination shows that inelastic collisions which are induced by thetrapping laser light, so-called radiative escape processes [28], are the dominantmechanism for these two-atom losses. This experiment shows that atom–atominteractions can be observed at the level of only two atoms.

2.3. BEYOND POISSONIAN LOADING

Stochastic loading of the MOT is acceptable for applications with very small num-bers of atoms. For instance, if MOT parameters are such that on average a single


atom populates the trap, Poissonian statistics predicts about 37% probability ofsingle atom events. For many experiments, implementation of control loops doesnot offer a significant advantage in this case.

Some of the most interesting future routes of research with neutral atoms sys-tems, however, will be directed towards small (“mesoscopic”) systems of neutralatoms with controlled interactions. In experiments it will thus be essential to loadan exactly known number of, e.g., 5–20 atoms in a much shorter time than offeredby stochastic fluctuations of the atom number. In the MOT the random loadingprocess can be manipulated by controlling the magnetic field gradient, the trap-ping laser beam properties, or the flux of atoms entering the trap volume. Severalstrategies for controlling the exact number of trapped atoms have already beeninvestigated or are currently studied:

In the experiment by Schlosser et al. [29] an optical trap providing very strongconfinement was superposed with the MOT (see also Section 3). Light assistedatom–atom interaction prevents presence of more than one atom in the trap whichthus fluctuates between 0 and 1 atom occupation numbers only. Suppression oftwo-atom occupation of a purely magnetic trap was also observed by Willems etal. [30].

An active feedback scheme for a single Cr atom MOT has been introducedby McClelland and coworkers [31]: If the trap is empty, rapid loading (≈5 ms)is achieved by directing the flux from a source of Cr atoms through light forcesinto the MOT volume. Using the MOT fluorescence as the indicator loading isterminated when a single atom is detected in the trap, and it is dumped if thetrap contains more than one atom. An average single atom occupation probabilityexceeding 98% has been demonstrated in this experiment. The authors estimatethat such a device may deliver individual atoms up to a rate of about 10 kHz.

In our laboratory, we have begun to explore a loading scheme, where we rapidlyload a preset mean number of atoms into our MOT by temporarily lowering itsmagnetic field gradient. After this forced loading, the magnetic field gradient isramped up again and the actual number of trapped atoms is determined by an-alyzing the level of fluorescence with a software discriminator [32]. As a resultof this analysis, the atoms are either loaded into an optical dipole trap for furtherexperiments, see Section 6, or, in case the MOT does not store the desired atomnumber, the atoms are discarded and the forced loading of the MOT is repeated.

3. Preparing Single Atoms in a Dipole Trap

While the MOT is an excellent device for the preparation of an exactly knownnumber of neutral atoms, it relies on spontaneous scattering of near-resonant laserlight which is highly dissipative and makes precise quantum state control of thetrapped atoms impossible. We have found in our experiments that preparation of


FIG. 4. Scheme of the experimental set-up. See text for details.

a sample of an exactly known number (1–30) of atoms in a MOT and subsequenttransfer to an optical dipole trap (DT) makes a very efficient instrument for exper-iments investigating quantum control of small ensembles of neutral atoms. A verytightly confining dipole trap for similar objectives was demonstrated by Schlosseret al. [29].

In our experiment (Fig. 4), the DT is generated by a focused and far off resonantNd:YAG or Yb:YAG laser beam at λ = 1.06 µm and 1.03 µm, respectively. Thelaser beam is split into two arms and can be used in a single beam configuration(traveling wave), or in a configuration of two counterpropagating beams (stand-ing wave). We routinely reach transfer efficiencies from the MOT into the DT andvice versa in excess of 99% [33]. The dipole trap provides an approximately con-servative, harmonic potential with bound oscillator quantum states for the neutralatoms. Focusing of the trapping laser beam power of several Watts to a 10–30 µmwaist provides strong confinement of the atom in the transverse direction, and ap-plication of a standing wave with 0.5 µm modulation period exerts even strongerforces in the longitudinal direction. The dipole trap provides a typical potentialdepth of order UTrap/k ≈ 1 mK. After transfer from the MOT, we measure tem-peratures of 50–70 µK, significantly below the 125 µK Doppler temperature forCaesium atoms [34]. Sub-Doppler cooling is enhanced during transfer from theMOT into the dipole trap since the atomic transition frequencies are light shiftedtowards higher frequencies and hence the cooling lasers are effectively further reddetuned.


FIG. 5. Left: ICCD-image of atomic fluorescence in the optical dipole trap under continuous il-lumination with molasses beams, exposure time 0.5 s. In the horizontal direction, the width of thefluorescent spot is determined by the resolution of our imaging system. In the vertical direction thespot shows the extension of atomic trajectories corresponding to a temperature of about 50–70 µK inthe trap of depth 1 mK. Right: Characteristic parameters of the dipole trap. Shaded areas schematicallyindicate MOT and molasses laser beams.

We have also realized a method to continuously illuminate an atom in the dipoletrap with an optical molasses and to observe its presence through fluorescencedetection. The laser cooling provided by the molasses in this case balances theheating forces. In Fig. 5 we show an ICCD image of a trapped atom as well ascharacteristic parameters of the dipole trap.

4. Quantum State Preparation and Detection

Neutral atoms are considered to be one of several interesting routes towards theimplementation of quantum information processing. Fundamental informationprocessing operations such as the famous quantum CNOT gate must be real-ized through physical interaction of the qubits [35]. For neutral atoms, severalconcepts, including photon exchange mediated by cavity-QED [36–38], or coldcollisions [39,40] have been proposed. Each of these concepts relies on tight con-trol of the quantum evolution of atomic qubits which already poses importantexperimental challenges.

In our experiments, hyperfine ground states of the Caesium atom are employedas qubits, the elementary units of quantum information storage. It is well knownfrom the Caesium atomic clock that the microwave transition operated at νhfs =9.2 GHz between the long lived |F = 4〉 and |F = 3〉 hyperfine states provides


efficient means of internal quantum state manipulation. It is thus expected thatspecific hyperfine states of the Caesium atom are excellent candidates to serve asqubit states with, e.g., |0〉 = |F = 4〉 and |1〉 = |F = 3〉. The first step in theseapplications is to prepare and detect (“write” and “read”) arbitrary quantum stateinto Caesium prepared in the DT.

During the transfer from the MOT into the dipole trap, an atom is normallyprepared in the |F = 4〉 state. This is achieved by switching off the MOT coolinglaser, near resonant with the |F = 4〉 → |F ′ = 5〉 transition, a few millisecondsbefore switching off the MOT repumping laser, resonant with the |F = 3〉 →|F ′ = 4〉 transition. After this transfer, we can populate the |F = 4,mF = 0〉magnetic substate using resonant optical pumping on the |F = 4〉 → |F ′ = 4〉and |F = 3〉 → |F ′ = 4〉 transition of the λ = 852 nm D2-line multiplet for about5 ms with linear π-polarized light. In the mF = 0 states, the influence of ambientmagnetic field fluctuations is strongly suppressed, a favorable condition for theobservation of long dephasing times described in Section 5. On the other hand,using circular σ−-polarized light, atoms can be pumped to the |F = 4,mF = −4〉state. This state allows fine tuning of its energy level by external magnetic fieldswhich is essential for position selective addressing and the implementation of aneutral atom quantum register (see Section 7). Finally, an initial pure |F = 3〉quantum state can be prepared by switching off the MOT repumping laser about10 ms before switching off the MOT cooling laser. In this way, the |F = 4〉 stateis depleted while transferring the atom from the MOT into the DT. In our trap,residual light scattering of the DT lasers causes relaxation of the hyperfine statepopulations of the |F = 3〉 and |F = 4〉 Caesium ground states at a time scale ofseveral seconds or more, depending on the trapping laser intensity.

For unambiguous detection of the hyperfine state of the trapped atoms, we cur-rently use a destructive “push-out” method [41], which discriminates the F = 3and F = 4 levels with excellent contrast of better than 1:200 (Fig. 6). Discrimina-tion is realized by ejecting atoms from the trap if and only if they are in the F = 4state and by monitoring the presence or absence of the atom after this procedure.For this purpose, a saturating laser beam resonant with the F = 4 → F ′ = 5cycling transition is applied transversely to the dipole trap axis. When the trapdepth is lowered to approximately 0.12 mK, atoms in F = 4 are pushed out inless than 1 ms by scattering on average 35 photons. Atoms in the |F = 3〉 stateare not affected by the push-out laser. In the last step, the remaining atoms areeither detected at a given dipole trap site by imaging with the ICCD camera, orby observing their fluorescence after recapture in the MOT. A fluorescing site in-dicates projection to the F = 3 quantum state, an empty site that was occupiedbefore is equivalent to projection to the F = 4 quantum state.


FIG. 6. Detecting the quantum state of a single neutral atom. Upper trace: An atom is prepared inthe MOT and transferred to the dipole trap in state |F = 4〉. A resonant push-out laser removes theatom from the trap. When the MOT lasers are switched on again, stray light is observed only. Lowertrace: In the dipole trap, the atom is transferred to the dipole trap in state |F = 3〉. The push-out laseris invisible for an atom in |F = 3〉. After switching on the MOT lasers the 1 atom fluorescence levelis recovered. See text for details on atom state preparation.

5. Superposition States of Single Atoms

The two hyperfine states form a pseudo spin-1/2 system, which can be manip-ulated by spin rotations, induced by shining in microwave radiation resonantwith the atomic clock transition. For instance, spin-flips are caused by so-calledπ-pulses (|0〉 π→ |1〉, |1〉 π→ −|0〉), where for a given magnetic field ampli-tude B⊥ and transition moment μ the microwave pulse duration τ is defined byΩτ = (μB⊥/h)τ = π . We have found that in our geometrically complex ap-paratus, the power of our 33 dBm microwave source is most efficiently directedat the experimental region with a simple open ended waveguide. We find a min-imal pulse length of 16 µs for a π-pulse. Arbitrary quantum state superpositionscos(Ωτ/2)|0〉+ eiφ sin(Ωτ/2)|1〉 can be generated by varying the pulse area Ωτ

and phase φ, and a π/2-pulse generates superpositions with even contributions ofthe two quantum eigenstates.

Future applications of the trapped atom quantum states as qubits depend cru-cially on the question whether coupling to the environment (“decoherence”) orto technical imperfections and noise (“dephasing”) can be suppressed to such adegree that coherent quantum evolution is preserved at all relevant time scales.


Promisingly long coherence time in dipole traps have been first observed byDavidson et al. [42].

In the Bloch vector model, the longitudinal and transversal relaxation time con-stants T1 and T2, are introduced phenomenologically. T1 describes the relaxationof the population difference of the two quantum states to their thermal equilib-rium, T2 the relaxation of the phase coherence between the two spin states. Whilespontaneous decay is completely negligible, the hyperfine state of the Caesiumatom can be changed by spontaneous Raman scattering. In our current setup, wemeasure typically T1 � 3 s [33]. With the exception of the trap life time of order1 min this time is longer than all other relaxation times. It can be further increasedby reducing the trapping laser power.

Several mechanisms contribute to transversal relaxation described by the timeconstant T2. Here, we distinguish reversible contributions with time constant T ∗

2arising from inhomogeneities of the measured ensemble, and irreversible con-tributions (T ′

2), which affect the ensemble homogeneously. The total transversalrelaxation time constant is thus composed of two different time constants withT −1

2 = T ∗−12 +T ′ −1

2 . Using Ramsey’s method of separated oscillatory fields [43]we have experimentally determined the atomic coherence properties with regardto dephasing in the dipole trap [44]. A detailed analysis can be found in [41].

Figure 7 shows an example of Ramsey spectroscopy, i.e., the evolution of themF = 0 hyperfine state under the action of two π/2 microwave pulses as a func-tion of the delay time between the pulses. If the microwave is resonant with thehyperfine transition, one expects perfect transfer from one to the other hyperfinestate. The “Ramsey-fringes” observed here result from a small, intentional detun-

FIG. 7. Population oscillation showing hyperfine coherences of optically trapped Caesium atoms:Dephasing Ramsey fringes and spin echo signal. The |F = 3,mF = 0〉 state is coupled to the|F = 4, mF = 0〉 state by 9.2 GHz microwaves. The solid line corresponds to a theoretical predictionbased on the thermal energy distribution of the atoms in the dipole trap only. For details see [41,44].


ing from perfect resonance. The initially observed coherent oscillation collapsesafter a dephasing time T ∗

2 ≈ T2, where longer dephasing times are observed formore shallow dipole potentials. This dephasing is caused by the thermal distribu-tion of atomic motional states in the dipole trap which causes an inhomogeneousdistribution of light shifts: “Cold” atoms with low kinetic energy near the po-tential minimum, or intensity maximum of the dipole trap experience on averagestronger light shifts than “hot” atoms with larger kinetic energy.

The phase evolution of the internal atomic quantum state depends on the exter-nal, motional degrees of freedom since binding forces are caused by the light shiftof the internal energy levels. Since the two hyperfine states F = 3 and F = 4 ex-perience a small but significant relative light shift of order νhfs/νD2 = η � 10−4,the phase evolution of any superposition state is affected by this difference andcauses dephasing depending on the trajectory of the atom in the trap. In a semi-classical model, we have assumed that the free precession phase accumulated byan atomic superposition state between the two π/2-pulses depends on the averagedifferential light shift only and calculated the thermal ensemble average yieldingthe solid line in Fig. 7. A quantum mechanical density matrix calculation of thesame observable reproduces this result within a few percent. The deviation canbe attributed to the occurrence of small oscillator quantum numbers nosc � 7 inthe stiff direction of the trap. We find that the envelope of the collapse of the ini-tial oscillation corresponds to the Fourier transform of the thermal oscillator statedistribution [41].

It is known that a “spin-echo” can be induced by application of a rephasingpulse [45]. Application of a π-pulse at time Tπ induces an echo of the Ramseysignal with a maximum amplitude at time 2Tπ . The revival of the oscillation isalso shown in Fig. 7. We have measured a 1/e decay time T ′

2 � 0.15 s for therevival amplitude. We have experimentally analyzed in detail the origin of thisirreversible decay. We have found that currently the dominating sources of deco-herence are the lack of beam pointing stability as well as intensity fluctuations ofthe trapping laser beams, while other effects such as magnetic field fluctuationsand heating are negligible [41]. All relevant relaxation and dephasing times arerecapitulated in Table I. Since no fundamental source of decoherence has beenfound which could not be reduced by technical measures, it should be possibleto further increase the time span of coherent quantum evolution of the trappedatoms.

Alternatively, we have also employed resonant two-photon Raman transitionsin order to introduce pseudo-spin rotations. In Fig. 8 we show a measurementof population oscillations (Rabi oscillations) between the F = 4 and F = 3Caesium hyperfine ground states [46]. Efficient two-photon Rabi rotations are al-ready achieved with relatively low power levels below 1 mW in each laser beam,e.g., in Fig. 8 the two-photon Rabi frequency exceeds 10 kHz. It is routine todayto use focused Raman laser beams in order to address an individual particle out


Table IMeasured hyperfine relaxation times of atoms in our dipole trap

Trelax Umax/k

(mK)mF Value Limiting mechanism

T1 1 0, −4 8.6 s spontaneous Raman scattering

T ∗2 0.1 0 3 ms thermal motion, scalar light shift

0.04 0 19 ms thermal motion, scalar light shift0.1 −4 270 µs thermal motion, vector light shift

T ′2 0.1 0 34 ms beam pointing instability

0.04 0 150 ms beam pointing instability0.1 −4 2 ms without gradient: thermal motion,

vector light shift0.1 −4 600 µs with gradient: thermal motion,

inhomogeneous magnetic field

FIG. 8. Population (Rabi) oscillation showing hyperfine coherences of optically trapped Caesiumatoms induced by resonant two-photon Raman transitions [46]. On the left side, details of the Caesiumquantum states involved and the power levels of the Raman laser beams are given.

of a string of trapped ions [47] and to induce quantum coherences. This method,which has significantly contributed to the first successful operations of fundamen-tal quantum gates with in these systems [48,49], is straightforwardly transferredto systems of neutral atoms. However, in Section 7 we will show that, with neutralatoms, a gradient method providing spatial resolution via spectral resolution canbe applied which eliminates the need for focused laser beams.

6. Loading Multiple Atoms into the Dipole Trap

When atoms are transferred from the MOT into the dipole trap, they are distrib-uted randomly across a 10 µm stretch of the standing wave, corresponding to


FIG. 9. (a) After the transfer from the MOT, the atoms are trapped in the potential wells of thestanding wave dipole trap at random positions. The spatial period of the schematic potential wellsis stretched for illustration purposes. (b) Fluorescence image of five optically resolved atoms in thestanding wave dipole trap (trap axis is horizontal) after the 1D expansion detailed in the text. Integra-tion time is 0.5 s.

about 20 antinodes or potential wells. With 5 atoms, the average separation isonly 2 µm, too small to be optically resolved by our imaging system.

In order to improve the addressability, we have adopted a modified transferprocedure: After the transfer from the MOT into the standing wave dipole trap,formed by the two counterpropagating laser beams, we switch off one of the twobeams within 1 ms. The potential of the resulting running wave dipole trap, cre-ated by one focussed laser beam, has Lorentzian shape with a FWHM of about1 mm in the longitudinal direction. We let the atoms expand longitudinally for1 ms such that they occupy a length of ≈100 µm. Then, we switch the secondtrapping laser beam on again within 1 ms, so that the atoms are “arrested” by thestanding wave micropotentials at the position they have reached during the expan-sion. Exposure to the optical molasses warrants low temperatures of the trappedatoms. The 5 fluorescent spots in Fig. 9 correspond to a single atom each, spreadout across 50 µm in this case with easily resolvable spatial separations.

As has been pointed out in Section 2.3, we have recently started to operatea feedback scheme for loading a preset number of atoms into our DT. For this,the MOT is rapidly loaded with a selectable mean number of atoms, which areonly transferred into the DT if the desired number of atoms is detected in theMOT. This is particularly useful if one seeks to carry out experiments with alarger number (>3) of atoms. In this case, loading the DT with a Poissonian dis-tributed number of atoms and postselection of the events with the desired atomnumber dramatically increases data acquisition time. First results obtained withthis scheme are presented in Fig. 10: Part (a) shows the accumulated uncondi-tional MOT fluorescence histogram for a large number of MOT loading cycleswith a mean atom number of about 3. Part (b), on the other hand, corresponds tothose events, where three atoms have been detected in the MOT, loaded into theDT, and retransferred into the MOT. The resulting conditional histogram clearlyshows that we manage to controllably load three atoms into the DT with a goodefficiency. In the course of these experiments, we have also found that single atom


FIG. 10. Selectively loading 3 atoms. (a) Binned fluorescence signal detected by the APD after alarge number of MOT loading processes. Part (b) contains all events, where three atoms were detectedby the feedback loop. These atoms were then transferred into the DT and back into the MOT, see textfor details.

occupation of the 1D lattice sites is generally preferred over multiple occupationfavoring a regular, non-Poissonian distribution of the atoms. Details will be pub-lished in [32].

7. Realization of a Quantum Register

A quantum register consists of a well-known number of qubits that can be individ-ually addressed and coherently manipulated. Our quantum register is composedof a string of neutral atoms, provided by the procedures described in the previoussections, which can be selectively prepared in arbitrary quantum states.

In ion traps selective addressing is achieved by means of focused Raman laserbeams [47]. As discussed in Section 5, we have shown that Raman pulses can


FIG. 11. Sequence of operations to generate and detect a |01010〉 quantum register state in a stringof five atoms. The whole sequence lasts 1.5 s.

be used to create coherent superpositions of hyperfine states of the atoms trappedin our experiment [46]. However, in the experiments presented here, we use analternative technique where we apply microwave radiation which is made resonantwith an atom at a selected site only by means of magnetic field gradients. Inthis method, spatial selectivity is indeed realized in the same way as in magneticresonance imaging (MRI) [50].

We can currently operate our register in the following way [51], see Fig. 11:We load between 2 and 10 atoms into our dipole trap. We then take a camerapicture and determine the positions of all atoms with sub-micrometer precision.In the next step all atoms are optically pumped into the same |F = 4,mF = −4〉quantum state as described in Section 4 to initialize the register.

Individual addressing is now realized by tuning the microwave frequency tothe exact transition frequency corresponding to the known individual atomicsites where the relationship is controlled by an external B-field gradient ofB ′ � 0.15 µT/µm along the DT axis. The atomic resonance frequency is shiftedby the linear Zeeman effect according to ν = νhfs + 24.5 kHz/µT, and we finda spatial frequency shift of dν/dz = 3.7 kHz/µm. We also apply a homoge-neous magnetic field of about 0.4 mT in order to provide guiding for the angularmomenta and to reduce the influence of transversal magnetic field gradients. InFig. 11 we show the result of two selective inversion operations (π-pulses) carriedout with a string of five atoms stored in our dipole trap array.

We have furthermore measured the resolution of the magnetic field gradientmethod. Figure 12 shows the result for the longest pulses applied (83 µs FWHM).The solid line is obtained from a numerical solution of the Bloch equations andreproduces the measurement very well. The spatial resolution is limited by the


FIG. 12. Measured spatial resolution of the addressing scheme. The data were obtained by delib-erately addressing positions offset from the actual atom site. For each point approximately 40 singleatom events were analyzed. The Gaussian microwave π -pulse has a FWHM length of 83 µs.

Fourier width of the microwave pulse. Our method clearly demonstrates that wecan address atoms for separations exceeding 2.5 µm (i.e., atoms are separated byabout 5 empty sites). The resolution of the magnetic method in our current set-upis thus comparable to addressing by optical focusing. Neighboring atoms experi-ence of course a phase shift due to non-resonant interaction with the microwaveradiation. However, this phase shift is known and can be taken into account infurther operations.

We have furthermore explored the coherence properties of atoms, now in themagnetically most sensitive mF = −4 states instead of the mF = 0 states. Theresults are displayed in Table I of Section 5. It is not surprising that dephasingtimes are much shorter in this case and are indeed dominated by fluctuations andinhomogeneities of the magnetic field. However, they are already now much largerthan simple operation times for, e.g., π-pulses and technical improvements willfurther enhance the time available for coherent evolution.

The method described requires very precise timing of the microwave pulses inorder to guarantee a precise control of the evolution from one quantum state to an-other. As an alternative, we have also applied quantum state control by means ofrapid adiabatic passage [52]: In this case, the frequency of an intense microwavepulse is swept through resonance thereby transforming an initial into a final eigen-state of the system, in our case realized for the |F = 4〉 and |F ′ = 3〉 hyperfineground states. In a gradient magnetic field we have analyzed the transfer proba-bility as a function of the resonance position of the sweep center frequency withrespect to the trapped atom for a fixed sweep width. The result in Fig. 13 showsthe expected flat top profile indicating the reduced sensitivity to the precise settingof the center frequency and the sweep width [53]. The width of the edges whichdrop to zero within 3 µm is a measure of the spatial resolution of this method andcomparable to the resonant addressing scheme described above.


FIG. 13. Position-dependent adiabatic population transfer of individual atoms in an inhomoge-neous magnetic field. The graph shows the population transfer as a function of the position offset �xalong the trap axis. Each data point is obtained from about 40 single atom measurements. The solidline is a theoretical fit [53].

Summarizing, in this section we have demonstrated procedures to experimen-tally realize both write and read operations at the level of a single neutral atom.We have demonstrated individual addressing of the atoms within a string of storedatoms with excellent resolution, and we are able to prepare arbitrary quantum su-perpositions on an individual atomic, or qubit site. In conclusion we have demon-strated the operation of a neutral atom quantum register, including the applicationof spin rotations, i.e., Hadamard gates in the language of quantum informationprocessing.

8. Controlling the Atoms’ Absolute and Relative Positions

Considering the ratio between the experimentally measured 2.5 µm addressingresolution presented above and the 1 mm Rayleigh zone of our standing waveDT, our neutral atom quantum register could in principle operate on more than100 individually addressable qubits. Methods for the regularization of the distrib-ution of atoms by controlling their absolute positions in the trap must be realized,however, in order to manage larger quantum registers. Tight position control isfurthermore essential to realize the necessary controlled atom–atom interaction.In optical cavity QED, for example, this interaction is mediated by the field ofan ultrahigh finesse Fabry–Perot resonator [36,54]. The field mode sustained bysuch a resonator has a typical transverse dimension of 10 µm so that the atom pairwill have to be placed into this mode with a submicrometer precision while the


distance between the atoms has to be controlled at the same level. We have demon-strated such a submicrometer position control for individual neutral atoms [55].

8.1. AN OPTICAL CONVEYOR BELT

The position of the trapped atoms along the DT axis can be conveniently manip-ulated by introducing a relative detuning between the two counter-propagatingdipole trap laser beams. A detuning by �ν causes the standing wave pattern tomove in the laboratory frame with a speed �νλDT/2, where λDT is the wave-length of the DT laser. As a result, the trapping potential moves along the DTaxis and thereby transports the atoms [56–58]. In the experiment, the relative de-tuning between the DT beams can be easily set with radiofrequency precision byacousto-optic modulators (AOMs, Fig. 4). They are placed in each beam and aredriven by a phase-synchronous digital dual-frequency synthesizer. A phase slip ofone cycle between the two trapping laser beams corresponds to a transportationdistance of λDT/2.

We can realize typical accelerations of a = 10,000 m/s2 and hence acceleratethe atoms to velocities of up to 5 m/s (limited by the 10 MHz bandwidth of theAOMs) in half a millisecond. Thus, for typical parameters, a 1 mm transport takesabout 1 ms. At the same time, the displacement of the atoms is controlled witha precision better than the dipole trap laser wavelength since this scheme allowsus to control the relative phase of the two trapping laser beams to a fraction ofa radian.

Using continuous illumination, we have imaged the controlled motion of oneand the same or several atoms (Fig. 14) transported by the conveyor belt [58]with observation times exceeding one minute. Recently, it was shown that op-tical dipole traps similarly to our arrangement can be used to transport neutralatoms into high finesse resonators for cavity-QED experiments with very goodprecision [59,60].

8.2. MEASURING AND CONTROLLING THE ATOMS’ POSITIONS

If one wants to take ultimate advantage of the optical conveyor belt transportabove in order to place atoms at a predetermined position, the atoms’ initial posi-tion along the dipole trap axis has to be known with the highest possible precision,ideally better than the distance between two adjacent potential wells. This can beachieved by recording and analyzing an ICCD fluorescence image of the trappedatoms. We have shown that by fitting the corresponding fluorescence peaks with aGaussian, the atoms’ position can be determined with a ±150 nm precision froman ICCD image with 1 s exposure time [55].


FIG. 14. Transport of 3 atoms by an optical conveyor belt: Snapshots of the movie publishedin [58]. In the first image, 3 atoms are stored in the MOT from where they are loaded into the conveyorbelt formed by two counterpropagating laser beams. The frequency difference of the laser beams iscontrolled with two AOMs driven by a phase-coherent RF-source. At 40 s and 65 s the direction oftransport is reversed. The atoms are lost from the conveyor belt by random collisions with thermalresidual gas.

Furthermore, we have demonstrated that by means of our optical conveyor belttechnique, we can place an atom at a predetermined position along the dipole trapaxis with a ±300 nm accuracy. Such a position control sequence is exemplified in


FIG. 15. Active position control. (a) After transferring a single atom from the MOT into the dipoletrap its initial position is determined from an ICCD image and its distance with respect to the targetposition is calculated. (b) The atom is then transported to the target position and its final position isagain measured from an ICCD image.

Fig. 15. After loading one atom from the MOT into the dipole trap, its position hasa ±5 µm uncertainty, corresponding to the diameter of the MOT. We determinethe atom’s initial position from a first ICCD fluorescence image and calculate itsdistance L from the desired target position. The atom is then transported to thistarget position and the success of the operation is verified by means of a secondICCD image.

In order to measure the distance between two simultaneously trapped atoms,we determine their individual positions as above. From one such measurementwith 1 s integration time, their distance can thus be inferred with a precision of√

2 × 150 nm. This precision can even be further increased by taking more thanone image of the atom pair and by averaging over the measurements obtainedfrom these images. Now, since the atoms are trapped inside a periodic potential,their distance d should be an integer multiple of the standing wave period: d =nλDT/2; see Fig. 9(a). This periodicity is clearly visible in Fig. 16, where thecumulative distribution of atomic separations is given when averaging over more


FIG. 16. Cumulative distribution of separations between simultaneously trapped atoms inside thestanding wave potential. The discreteness of the atomic separations due to the standing wave potentialis clearly visible.

than 10 distance measurements for each atom pair. The resolution of this distancemeasurement scheme is ±36 nm, much smaller than the standing wave period.We directly infer this value from the width of the vertical steps in Fig. 16. Thisresult shows that we can determine the exact number of potential wells separatingthe simultaneously trapped atoms [55].

8.3. TWO-DIMENSIONAL POSITION MANIPULATION

A single standing wave optical dipole trap allows to shift the position of a stringof trapped atoms as a whole in one dimension along the dipole trap axis using theoptical conveyor belt technique presented above. If one seeks to prepare stringswith a well-defined spacing or to rearrange the order of a string of trapped atoms,however, a two-dimensional manipulation of the atomic positions is required. Forthis reason, we have set up a second standing wave dipole trap, perpendicular tothe first one, which acts as optical tweezers and which allows us to extract atomsout of a string and to reinsert them at another predefined position.

Figure 17 shows a first preliminary result towards this atom sorting and distancecontrol scheme [61]. We start with a string of three randomly spaced atoms whichhas been loaded from the MOT into the horizontal (conveyor belt) dipole trap.In Fig. 17(a), the string has already been shifted such that the rightmost atom isplaced at the position of the vertical (optical tweezers) dipole trap. This atom isthen extracted with the vertical dipole trap and, after shifting the remaining twoatoms along the horizontal dipole trap, we place it 15 µm to the left of the initiallyleftmost atom of the string; see Figs. 17(b)–(d). Repeating this procedure a secondtime, we prepare a string of three equidistantly spaced atoms, where the order of


FIG. 17. Rearranging a string of three atoms using two perpendicular standing wave dipole traps.See text for details.

the string has been modified according to (1, 2, 3) → (3, 1, 2) → (2, 3, 1); seeFigs. 17(e)–(h).

9. Towards Entanglement of Neutral Atoms

There is a plentitude of proposals of how to implement a two-qubit quantumgate with neutral atoms which suggest the coherent photon exchange of twoatoms inside a high-finesse optical resonator [36,54,59,62]. The experimentalchallenges for their realization are quite demanding. Although there has been anumber of successes in optical cavity-QED research recently, including the trans-port of atoms into a cavity [59,60], trapping of single atoms inside a cavity [63],single photon generation [64,65], feedback control of the atomic motion in a cav-ity [66,67], and cooling of atoms inside a cavity [68–70], the realization of atwo-qubit quantum gate with ground state atoms remains to be shown.

9.1. AN OPTICAL HIGH-FINESSE RESONATOR FOR STORING PHOTONS

Our goal is the deterministic placement of two atoms inside an optical high-finesseresonator. For this purpose, we have already set up and stabilized a suitable res-


onator [71]. We plan to transport atoms from the MOT, which is a few millimetersaway from the cavity, into the cavity mode using our optical conveyor belt. Em-ploying the imaging techniques and the image analysis presented above, we wererecently able to control the position of the trapped atoms along the trap axis witha precision of ±300 nm [55]. This should allow us to reliably place the atomsinto the center of the cavity mode, which has a diameter of 10 µm. Since themicrowave-induced one-qubit operations on the quantum register demonstratedin Section 7 do not require optical access to the trapped atoms, they can even takeplace inside the cavity.

9.2. A FOUR-PHOTON ENTANGLEMENT SCHEME

One of the most promising schemes to create entanglement between two atoms inoptical cavity QED was proposed by L. You et al. [54] and is the basis for the re-alization of a quantum phase gate [72]. It relies on the coherent energy exchangebetween two atoms stimulated by a four-photon Raman process involving the cav-ity mode and an auxiliary laser field. We have determined optimized theoreticalparameters and calculated the expected fidelity according to this proposal for ourparticular experimental conditions. With a maximum fidelity of F = 85%, whichcan be expected from this calculation. The demonstration of entanglement andthe implementation of a quantum gate thus seems feasible with our experimentalapparatus.

9.3. COLD COLLISIONS IN SPIN-DEPENDENT POTENTIALS

We plan to investigate small strings of collisionally interacting neutral atomsfor applications in quantum information processing. The atoms are stored, oneby one, in a standing wave dipole trap and the interaction between the atoms,necessary for the implementation of quantum gates, will be realized through con-trolled cold collisions [39,40] which have been demonstrated with large sampleof ultracold atoms already but without addressability of the individual atomicqubit [15]. For this purpose, we will employ the technique of spin dependenttransport [39,40] at the level of individual atoms. This technique will allows us to“manually” split the wave functions of the trapped atoms in a deterministic andfully controlled single atom Stern–Gerlach experiment, where the dipole trap pro-vides the effective magnetic field. By recombining the atomic wave function, wewill then realize a single atom interferometer and directly measure the coherenceproperties of the splitting process. A sequence of splitting operations, carried outon a single atom, will result in a quantum analogue of the Galton board, where theatom carries out a quantum walk. Such quantum walks have recently been pro-posed as an alternative approach to quantum computing [73]. Our ultimate goal


is the implementation of fundamental quantum gates using controlled cold col-lisions within a register of 2–10 trapped neutral atoms. A parallel application ofsuch quantum gates should then open the route towards the preparation of smallcluster states [74] consisting of up to 10 individually addressable qubits.

10. Conclusions

In this overview, we have presented experimental techniques and results concern-ing the preparation and manipulation of single or a few optically trapped neutralCaesium atoms. We have shown that a specially designed magneto-optical trap(MOT) can store a countable number of atoms. Information about the dynamicsof these atoms inside the MOT can be gained at all relevant timescales by analyz-ing photon correlation in their resonance fluorescence. Furthermore, using activefeedback schemes, the Poissonian fluctuations of the number of atoms in the MOTcan be overcome, making such a MOT a highly deterministic source of an exactlyknown number of cold atoms.

For coherent manipulation, we transfer the atoms with a high efficiency fromthe dissipative MOT into the conservative potential of a standing wave dipoletrap (DT). The quantum state of atoms stored in this DT can be reliably preparedand detected at the level of single atoms. We have examined the coherence prop-erties of the atoms in the DT and identified the dephasing mechanisms in thissystem. The experimentally measured long coherence times show that the atomichyperfine ground states are well suited for encoding and processing coherent in-formation.

A string of such trapped Caesium atoms has thus been used to realize a quan-tum register, where individual atoms were addressed with microwave pulses incombination with a magnetic field gradient. Using this method, we have demon-strated all basic register operations: initialization, selective addressing, coherentmanipulation, and state-selective detection of the individual atomic states.

We have furthermore demonstrated a high level of control of the atoms’ exter-nal degrees of freedom. Our DT can be operated as an “optical conveyor belt”that allows to move the atoms with submicrometer precision along the DT. In ad-dition, we have measured the absolute and relative positions of the atoms alongthe dipole trap with a submicrometer accuracy. This high resolution allows us tomeasure the exact number of potential wells separating simultaneously trappedatoms in our 532 nm-period standing wave potential and to transport an atom to apredetermined position with a suboptical wavelength precision.

Finally, using a second dipole trap operated as optical tweezers, we have ob-tained first results towards an active control of the atoms’ relative positions withinthe string. This will allow us to prepare strings with a preset interatomic spacingand to rearrange the order of atoms within the string at will.


The presented techniques are compatible with the requirements of cavity QEDand controlled cold collision experiments. In our laboratory, we now actively worktowards the implementation of such experiments in order to realize quantum logicoperations with neutral ground state atoms.

11. Acknowledgements

We wish to thank the Deutsche Forschungsgemeinschaft, the Studienstiftung desDeutschen Volkes, the Deutsche Telekom Stiftung, INTAS, and the EuropeanCommission for continued support. Furthermore, we are indebted to numerousenthusiastic coworkers and students at the Diplom- and doctoral level who haveparticipated in this research: W. Alt, K. Dästner, I. Dotsenko, L. Förster, D. Frese,V. Gomer, D. Haubrich, M. Khudaverdyan, S. Knappe, S. Kuhr, Y. Mirosh-nychenko, S. Reick, U. Reiter, W. Rosenfeld, H. Schadwinkel, D. Schrader,F. Strauch, B. Ueberholz, and R. Wynands.

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SPATIAL IMAGING WITH WAVEFRONTCODING AND OPTICAL COHERENCETOMOGRAPHY∗

THOMAS HELLMUTH

Department of Optoelectronics, Aalen University of Applied Sciences, Germany

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062. Enhanced Depth of Focus with Wavefront Coding . . . . . . . . . . . . . . . . . . . . . . . 107

2.1. Abbe’s Theory of the Microscope and Wavefront Coding . . . . . . . . . . . . . . . . 1112.2. Partial Coherent Illumination and Wavefront Coding . . . . . . . . . . . . . . . . . . 1152.3. Wavefront Coding with Variable Phase Plates . . . . . . . . . . . . . . . . . . . . . . 117

3. Spatial Imaging with Optical Coherence Tomography . . . . . . . . . . . . . . . . . . . . 1203.1. Time Domain Optical Coherence Tomography (TDOCT) . . . . . . . . . . . . . . . . 1203.2. Linear Optical Coherence Tomography (LOCT) . . . . . . . . . . . . . . . . . . . . . 1303.3. Spectral Domain Optical Coherence Tomography (SDOCT) . . . . . . . . . . . . . . 133

4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

AbstractWith wavefront coding the wavefront in the pupil plane of an optical imaging systemis modified by introducing a phasemask. The resulting image intensity distributionis processed with an inverse digital filter providing an image of the object with en-hanced depth of focus. Optical coherence tomography provides three-dimensionalinformation about the object. The depth resolution is only determined by the co-herence length of the light source. New applications and methods based on thesetechniques are presented.

∗ I am very pleased to dedicate this review article to Prof. Herbert Walther. I have learnt from himhow to get an understanding of complicated phenomena both in quantum and in classical optics interms of simple pictures and key experiments. Happy birthday.


DOI 10.1016/S1049-250X(06)53004-6

105

106 T. Hellmuth [1

1. Introduction

It is easy to measure the height of an object with a microscope by focusing firstonto the top plane and then onto the bottom plane of the object and measuring thedisplacement of the probe stage. However, it is more difficult to find the positionof best focus the smaller the numerical aperture of the objective is because ofits large depth of focus. Furthermore, due to the low numerical aperture lateralresolution is also low. Thus, high depth of focus seems to be associated with lowlateral resolution and low depth resolution.

In fact, according to linear optical system theory first established by Ernst Abbein 1881 with his theory of the microscope depth resolution is directly related tolateral resolution of the optical system [4]. For a diffraction limited optical systemboth depth of focus DOF = λ/NA2 and lateral resolution limit dmin = λ/2NA aredetermined by the numerical aperture NA of the objective and the wavelength λ.

Many optical systems are intrinsically characterized by a large depth of focus.For example, in ophthalmoscopy the optically usable pupil diameter of the eye isonly two millimeters. With dilation of the pupil higher pupil diameters are possi-ble. But due to the optical aberrations of the eye the dilated pupil diameter cannotbe fully utilized for imaging the fundus. With the normal eye length of 24 millime-ters and the mean refractive index of the aqueous and vitreous humor of n = 1.34the effective numerical aperture of fundus imaging systems is practically limitedto NA = 0.06 with λ = 0.55 µm giving a depth of focus of 0.15 mm and a lateralresolution limit of 5 µm. Although lateral resolution fulfills most diagnostic needsdepth of focus of 0.15 mm forbids resolution of the microscopically thin layerstructures of the retina with classical optical sectioning techniques like confocalimaging.

Similar restrictions of the numerical aperture of imaging systems limit alsodepth resolution in industrial metrology. The working distance has to be large toavoid collisions and the aperture angle of the optical system has to be small to fitinto narrow apertures. On the other hand surface structures of workpieces haveto be measured within the manufacturing process with resolutions in the order ofmicrons.

In other applications like image processing large numerical apertures areneeded to get enough light to the CCD-target. The prize to pay is small depthof focus. As a consequence the position of the object has to be controlled bycomplex autofocus systems.

These practical examples show that the complementarity of depth of focus ver-sus lateral resolution on one side and the complementarity of depth of focus versusdepth resolution on the other severely limit the performance of optical systems inmany situations.

2] SPATIAL IMAGING 107

Within the last 15 years two new approaches have been brought up which havein common to provide high depth of focus without restricting either lateral reso-lution or depth resolution.

An optical imaging technique with both enhanced depth of focus and highlateral resolution was first invented by Dowski et al. in 1995 using wavefront cod-ing [1–3]. With this approach the optical transfer function of an imaging systemis modified at the pupil plane. An inverse digital filter is applied to the image torestore an image with high depth of focus and almost no loss of lateral resolution.

Optical coherence tomography invented by Fujimoto et al. in 1991 is an imag-ing technique providing both high depth of focus and high depth resolution [8].This method makes use of short coherence interferometry where depth resolutionis no longer limited by the numerical aperture of the optical system but by thecoherence length of the light source.

Both wavefront coding and optical coherence tomography essentially dependon the coherence properties of the illumination. Wavefront coding only workswith illumination characterized by a low degree of spatial coherence whereas op-tical coherence tomography is based on the low temporal coherence of the lightsource.

In the following sections various new applications and extensions of the twomethods are discussed which have been developed and investigated at Aalen Uni-versity of Applied Sciences.

2. Enhanced Depth of Focus with Wavefront Coding

Wavefront coding is a technique which provides high depth of focus without lossof lateral resolution. A cubic phaseplate (Fig. 1) is located at the exit pupil of theoptical system.

The phaseplate is a transparent plate with one flat surface on one side and acubic surface on the other. The surface sag of the cubic surface can be described bythe sag function h(x, y) = α(x3+y3). The parameter α determines the “strength”of the phaseplate. Because the phaseplate modifies the wavefront Φ(x, y) in theexit pupil the optical transfer function OTF of the optical system is modified. TheOTF is the autocorrelation function of the pupil function p(x, y) [6]:

(1)p(x, y) = t (x, y)e−iΦ(x,y),(2)OTF = p(x, y)⊗ p(x, y)

with “⊗” symbolizing the correlation operation. It is t (x, y) the stop functionwhich is 1 within the stop aperture and 0 outside. The modulation transfer functionMTF = |OTF| describes the image contrast as a function of the spatial frequencyof a periodic object.

108 T. Hellmuth [2

FIG. 1. Measured surface of cubic phaseplate.

The cubic phaseplate reduces the image contrast but does not reduce the band-width of the optical system. In addition, with the cubic phaseplate in place theOTF and the MTF do not change significantly when the object is defocused overmany depths of focus of the optical system. Because the OTF is invariant over alarge depth of field it is possible to apply an inverse filter OTF−1 to the Fouriertransform of the acquired image with the phaseplate in place to get an unblurredimage with a large depth of focus.

Figure 2 shows the MTF of a focused diffraction limited optical system withand without phaseplate. Below the respective MTFs are shown for the defocusedobject. Whereas the bandwidth of the MTF dramatically shrinks when the opticalsystem without phaseplate is defocused both the shape and the bandwidth of theMTF of the optical system with cubic phaseplate remain constant. In addition theMTF of the defocused optical system without phaseplate is zero for certain spatialfrequencies, in other words, these frequencies are not transmitted to the image atall.

Figure 3 shows the point spread function (PSF) of the optical system with thecubic phaseplate in place. The PSF is related to the OTF by the Fourier transformOTF = FT{PSF}. The PSF can be measured by using a transilluminated pinholeas an object and registering the image with a CCD camera.

Figure 4 (left) shows the blurred image of a defocused barcode pattern. The cu-bic phaseplate is not inserted. Figure 4 (right) shows the inversely filtered imagewhen the cubic phaseplate is inserted. This image is Fourier transformed to getthe spatial frequency spectrum. The complex spectrum function is inversely fil-tered by the inverse optical transfer function OTF−1 = 1/FT{PSF} of the opticalsystem with the cubic phaseplate in place. Finally the inversely filtered spectrumis multiplied by the focused optical transfer function of the optical system with-


FIG. 2. Ideal MTF of a focused diffraction limited system (top left). MTF of the same opticalsystem with phaseplate in place (top right). MTF of defocused optical system without phaseplate(bottom left). MTF of defocused optical system with phaseplate (bottom right).

out phaseplate OTFideal. The restored image is received by the inverse Fouriertransform

(3)imagerestored = FT−1{FT{image} · OTF−1 · OTFideal}.

Because the OTF of the optical system with cubic phaseplate does not signifi-cantly change over a large depth of field the inverse filter is able to restore theunblurred image even if the object is defocused.

A typical application shows Fig. 5. A barcode reading system needs a highaperture to receive enough light from the object. Barcode readers are mainly used

110 T. Hellmuth [2

FIG. 3. Point spread function of the optical system with cubic phaseplate. It is registered with aCCD target. The object is a transilluminated pinhole with a diameter considerably smaller than theresolution of the objective.

FIG. 4. Image of defocused barcode. Objective without cubic phaseplate (left). Inversely filteredimage of defocused barcode. Objective with cubic phaseplate (right).

in logistic applications, for example, to identify pieces of luggage on conveyorbelts in airports. Due to the high aperture which is necessary to accept enoughlight within a short image acquisition time the depth of focus of an ordinary cam-era becomes too small compared to the variable object distances one has to copewith. Fast autofocus systems have to track the barcode. An alternative is depth offocus enhancement with cameras equipped with cubic phaseplates.

The theory of wavefront coding can be explained in terms of ambiguity func-tions [2]. Although this approach is very useful for the optimization and the designof cubic phaseplates it is of a more formal nature. Therefore, in the following sec-tion the theory shall be explained in terms of Abbe’s theory of the microscope.


FIG. 5. Barcode reading system for luggage identification.

2.1. ABBE’S THEORY OF THE MICROSCOPE AND WAVEFRONT CODING

Abbe’s theory of the microscope is originally based on coherent illumination(Fig. 6) [11]. The light from a point light source is collimated by the condenserlens illuminating a diffraction grating. The diffraction orders are plane wavespropagating into different directions which are focused into the back focal planeof the microscope objective where the aperture stop of the objective is located.The tube lens collimates again the various diffraction orders to bring them to in-terference at the image plane where a CCD may be located.

The cubic phaseplate is inserted at the exit pupil of the objective which is in thiscase the aperture stop plane. However, the cubic phaseplate would not have muchinfluence on the imaging properties of the system in this coherent illuminationcase. It would only cause a shift of the interference pattern in the image planebecause the phaseplate thickness is different at the locations where the diffractionorders are focused.

In order to understand the behavior of an optical system with a cubic phaseplateit is necessary to take the partial coherence properties of the illumination intoaccount.

Instead of a point light source the light source is now extended. The variousdiffraction orders of the grating object generate images of the light source in theback focal plane of the objective (Fig. 7). The light field illuminating the object isno longer a coherent wave but has to be described by a spatial coherence functionG(x, y) in the object plane [36]. According to the van Zittert–Zernike theoremthis coherence function can be calculated as the Fourier transform of the intensitydistribution of the light source located in the front focal plane of the condenser

112 T. Hellmuth [2

FIG. 6. Microscope setup with coherent illumination. The object is a diffraction grating locatedin the front focal plane of the objective. The diffraction orders are focused into the back focal plane(aperture plane) of the objective. The tube lens collimates the waves emanating from the foci. Theplane waves interfere at the image plane generating the image of the grating object.

FIG. 7. Partial coherent illumination. The extended light source is imaged into the back focal planeof the objective. Each diffraction order generates an image of the light source in the objective apertureplane.

lens. The coherence function G(x, y) is modulated by the amplitude object trans-mission function Fobj(x, y). The objective lens generates the Fourier transform ofthe coherence function multiplied by the transmission function of the objectivein the back focal plane of the objective lens. The final image is generated in theimage plane with the tubelens by a further Fourier transform.

If the shape of the incoherent light source can be described by f (r) = 1 withina circle of radius 1 and f (r) = 0 outside of the circle the coherence functionof the illuminating light field in the object plane can be described by the shiftinvariant function G(x, y) = J1(r)/r (J1(r) is the respective Bessel function offirst order and r is the radial variable). This is the same function which describesthe amplitude of a focused laser beam with a diameter equal to the source diameterof the incoherent light source (Fig. 8). The diffraction grating splits the focusedlaser beam into various diffraction orders which are collimated by the objective


FIG. 8. Laserscanning microscope. The object is shifted laterally (object scan). The diffractionorders interfere within the aperture. A detector (not shown) integrates the light within the aperture.The detector signal is recorded as a function of the scan position.

lens into parallel light bundles laterally shifted relative to each other proportionalto the spatial frequency of the grating object.

The first orders interfere with the zeroth order in the overlapping areas indicatedas interference domains in Fig. 8. When the object is laterally shifted (objectscan) the phase of the first diffraction orders are shifted relative to the phase ofthe zeroth order according to the shift theorem of Fourier transform theory. Thedetector (not shown in Fig. 8) integrates the energy across the whole aperture.The detector signal is recorded as a function of the scan position. The modulationcontrast of the detector signal decreases with increasing grating constant becausethe area of the interference zone decreases. The MTF describes the contrast as afunction of the spatial frequency ν. The MTF is proportional to the area of theinterference zone. It is given by the formula [6]

MTF(ν) = 2

π

[arccos

(ν

νmax

)− ν

νmax

√1 − (ν/νmax)2

],

where νmax = 2NA/λ is the resolution limit which is reached when the zeroth andfirst order do not overlap any more, in other words, the OTF is the autocorrelationfunction of the pupil function (see Eq. (2)).

If the object is defocused the first diffraction orders constitute plane waveswhich are tilted relative to the plane wave of the zeroth order. Within the in-terference zones interference fringes appear. Their fringe density increases withincreasing defocus. When the object is shifted in the lateral direction (scan) the

114 T. Hellmuth [2

FIG. 9. Wavefront W(x) across the pupil with defocus without phasemask (top, left), focused withphasemask (center, left) and with defocus and phasemask (bottom, left). Modulation frequency f (x)

of the interference pattern within the interference zone corresponding to the diagrams on the left. Itis f (x) the second derivative of W(x). On the right: Corresponding interference patterns within theinterference zone.

phase of the interference pattern changes but not the fringe density. The modula-tion contrast of the resulting signal decreases with defocus because the detectorintegrates across the aperture. Figure 9 (first row, second diagram) shows thefrequency of the interference fringes across the interference zone. The spatialmodulation frequency of the interferogram f (x) is constant but increases withincreasing defocus.

Instead of the plane waves of the diffraction orders of a grating object the imagegeneration process can be also discussed in terms of the spherical wave emanatingfrom a point object. In the focused case the spherical wave is transformed into aplane wave by the objective lens and becomes a spherical wave in the defocusedcase. In the approximation of paraxial optics the spherical wave can be approxi-mated by a paraboloid which is a parabola W(x) ∼ x2 in one dimension (Fig. 9,first row, left diagram). The spatial modulation frequency of the interferogramf (x) is related to the wavefront W(x) by

f (x) ∼ d2W(x)/dx2.

If a cubic phaseplate is located at the aperture plane the wavefront W(x) is acubic function (Fig. 9, second row, left diagram). The second derivative is a linearfunction (Fig. 9, second row, central diagram). That means that the modulationfrequency of the interference fringes within the interference zone between the ze-


roth and first order is described by a linear chirp function. The zero crossing off (x) in Fig. 9 indicates the area where the interference pattern is not modulated.The integrating detector averages the intensity across the interference zone. How-ever, the averaged energy across the aperture is stronger modulated compared tothe case without cubic phaseplate because of the broad zone where the fringe den-sity is low. The broad dark zone in Fig. 9 (second row, right diagram) contributesmuch to the modulation contrast. It oscillates between bright and dark when theobject is scanned. The dense interference fringes do not contribute to image con-trast because they are averaged out. The dark zone is laterally shifted when theobject is defocused but does not change in size. Thus, the MTF is insensitive todefocus. Of course, the contrast of the detector signal modulation finally vanisheswhen the defocus is so high that the zero crossing leaves the interference zone.

2.2. PARTIAL COHERENT ILLUMINATION AND WAVEFRONT CODING

The qualitative description of wavefront coding in terms of Abbe’s theory asshown in the last section can be described quantitatively with Hopkins’ theoryof partial coherent imaging [5].

The Fourier transform (object spectrum) of the object transmission functionFobj(x, y) is

(4)Fobj(f, g) =+∞∫∫−∞

Fobj(x, y)ei2π(f x+gy) dx dy

with f and g as the spatial object frequency in the x- and y-direction, respectively.The image intensity distribution spectrum with partial coherent illumination is

Jimage(f, g) =+∞∫∫−∞

T(f ′ + f, g′ + g, f ′, g′

)(5)× Fobj

(f ′ + f, g′ + g

)F ∗

obj

(f ′, g′)df ′ dg′

with the bilinear transfer function

T(f0, g0; f ′

0, g′0

) = +∞∫∫−∞

Jcond(f , g)K(f + f0, g + g0

)(6)× K∗(f + f ′

0, g + g′0)df dg.

It is Jcond(f, g) the circular pupil function of the condenser. It is K(f, g) thecomplex pupil function of the objective. It is

(7)K(f, g) = t (f, g) exp[iΦ(f, g)

]

116 T. Hellmuth [2

with the wavefront function Φ and the transmission function t of the objectiveaperture which is zero outside the objective aperture and 1 inside. For one-dimensional objects (e.g. edge) we get g = g′ = 0. Thus, the bilinear transferfunction in Eq. (6) becomes

T(f ′ + f, 0, f ′, 0

) = T(f, f ′)

(8)=+∞∫∫−∞

Jcond(f , g)K(f + f ′ + f, g

)K∗(f + f ′, g

)df dg.

Setting Fobj(f′ + f, 0) = Fobj(f

′ + f ) and F ∗obj(f

′, 0) = F ∗obj(f

′) we finally getfor the image spectrum in Eq. (5),

(9)Jimage(f ) =+∞∫

−∞T(f, f ′)Fobj

(f ′ + f

)F ∗

obj

(f ′) df ′.

The intensity distribution of the image is then

(10)J(x′) = +∞∫

−∞Jimage(f )e

−i2πf x′ df.

Figure 10 shows the inversely filtered intensity distribution of the image of abar pattern illuminated with partial coherent light in comparison with the simula-tion [16]. The inverse filter function is derived from the pointspread function of apinhole object. The degree of coherence of the illumination does not influence thePSF. However, the imaging of the bar pattern is significantly influenced. There-fore, the inverse filter based on the PSF cannot compensate the image artefacts

FIG. 10. The circular condenser aperture is smaller than the quadratic objective aperture (partialcoherent illumination). The inversely filtered image (with phaseplate) shows typical fringes both inexperiment and simulation which are due to the high degree of spatial coherence of illumination.


introduced by the coherence properties of the illumination. As a consequence theinversely filtered image exhibits artefacts both in the experiment and the simula-tion (edge fringes) as shown in Fig. 10.

2.3. WAVEFRONT CODING WITH VARIABLE PHASE PLATES

So far the effect of cubic phasemasks with fixed strength parameter α has beendiscussed. A large value of the parameter α is associated with a large depth offocus but also with several drawbacks.

With a strong phaseplate the OTF decreases. Thus the inverse filter becomesstrong at higher frequencies where the signal is noisy. As a consequence the in-verse filtered image shows a high noise level. Therefore, the strength of the cubicphaseplate should be chosen only as high as it is necessary. A phaseplate withvariable parameter α which can be adapted to the required depth of focus is analternative [15]. Figure 11 shows the setup of a variable phaseplate system. It con-sists of two phaseplates. Phaseplate 1 has a convex surface which can be describedby the surface function

(11)f (x, y) = κ(x4 + y4)

(the bottom side is flat). Phaseplate 2 has a concave surface which can be de-scribed by the surface function

(12)g(x, y) = −κ(x4 + y4).Both phaseplates can be shifted relative to each other by the displacement para-meter � in the diagonal direction (45◦ to the x-axis). As a result one obtains an

FIG. 11. Variable phaseplate system with surface functions f (x, y) = ±κ(x4 + y4). The dis-placement � generates an effective optical performance of the system equivalent to the performanceof a cubic phasemask.

118 T. Hellmuth [2

effective wavefront Φ(x, y),

Φ(x, y) ∼ f (x −�, y −�)+ g(x +�, y +�)

(13)= 8κ�(x3 + y3)+ 8κ�3(x + y)

corresponding to the wavefront produced by a phaseplate with an effective cubicparameter α = 8κ� which can be adjusted by appropriately setting the dis-placement parameter �. The “linear” term in Eq. (13) (x + y) leads only to adisplacement of the image which can be easily compensated by digital processing.

An alternative setup of a variable phaseplate system shows Fig. 12. It consistsof 4 phaseplates with “cylinder-like” surfaces.

Phaseplate 1 and 2 can be shifted in the x-direction (perpendicular to the opticalaxis) in opposite directions. Thus, phaseplate 1 is shifted by −� and phaseplate 2by +�. Phaseplate 1 has a convex surface on one side and a flat surface on theother. The convex surface can be described by

(14)f1(x, y) = κ · x4.

Phaseplate 2 has a concave surface which can be described by

(15)f2(x, y) = −κ · x4.

Phaseplate 3 and 4 can be shifted along the y-axis. Phaseplate 3 has a convexsurface on one side and a flat surface on the other. The convex surface can bedescribed by

(16)f3(x, y) = κ · y4.

Phaseplate 4 has a concave surface which can be described by

(17)f4(x, y) = −κ · y4.

FIG. 12. Variable phaseplate system with cylinder like surface function f (x, y) = ±κx4 and±κy4, respectively.


When the phaseplates are shifted by� one obtains an effective wavefrontΦ(x, y),

Φ(x, y) ∼ f1(x −�, y)+ f2(x +�, y)+ f3(x, y −�)+ f4(x, y +�)

(18)∼ 8κ�(x3 + y3)+ 8�3κ(x + y).

With α = 8κ� the strength of the cubic phaseplate can be adjusted via the shiftparameter �. Again, the linear term x + y only causes a slight image shift.

This system consisting of 4 elements is more complicated than the two part so-lution described above. However, it is easier to manufacture cylinder like surfacesthan free-form surfaces required in the two part solution.

The phase plates discussed above are not rotational symmetric. Classical grind-ing and polishing processes which are used for spherical glass surfaces cannot beused. At the Center of Optical Technology at Aalen University of Applied Sci-ence optical surfaces of arbitrary shape can be polished with a polishing robot(Fig. 13) [17]. Figure 14 shows the interferometer measurements of the convex

FIG. 13. Polishing robot.

120 T. Hellmuth [3

FIG. 14. Interferometer measurement of the concave surface κ(x4 + y4) and its convex counter-part.

and concave surface of the (x4 + y4)-phaseplate in glass. Similar results are ob-tained by making phaseplates in PMMA with a diamond turning machine.

3. Spatial Imaging with Optical Coherence Tomography

Optical coherence tomography (OCT) is a noninvasive imaging technique pro-viding subsurface imaging of biological tissue with micrometer-scale resolution.OCT was first used for imaging of the retina [8,9,21] and is now applied to avariety of medical fields to gain morphological [23,24,26,38,41] and functionaldata [27]. All OCT sensors either work in the time or Fourier domain. In the timedomain the depth gating of the sample is achieved by using a low coherence lightsource, a Michelson interferometer setup and a reference optical delay line. AnOCT image (B-scan) is built up of several scans of the optical delay line in the ref-erence arm (A-scans) [32]. In the Fourier domain depth information of the sampleis obtained by investigating the spectrum of the interferometer output [33] or byusing a tunable laser and a single photodiode as sensor [34,35].

We have investigated a third approach [10,12,13]. It is an OCT-sensor withoutusing a variable reference optical delay line, a spectrometer or a tunable laser. Themain item of the interferometer is a two-pinhole device built of two monomodefibers aligned in parallel. Light emerging from these two fibers interferes on alinear CCD-array similar to Young’s two-pinhole experiment. For this reason thesetup is called linear OCT sensor (LOCT). Depth gating is achieved by detectingthe interference signal on the CCD-array. Different positions of the interferencesignal on the CCD-array correspond to different depths inside the sample. There-fore a complete A-scan can be derived from a single readout of the CCD-array.

3.1. TIME DOMAIN OPTICAL COHERENCE TOMOGRAPHY (TDOCT)

Figure 15 shows the classical setup of a time domain optical coherence tomo-graph. The light source is a superluminescent diode with a short coherence length


FIG. 15. Time domain OCT.

FIG. 16. Linear OCT interference signal.

in the order of 10 µm. Because the light from a SLD is diffraction limited likelaser light it can be focused into a monomode fiberoptic interferometer with highefficiency. The interferometer consists of a 3 dB coupler splitting the incominglight from the SLD into a reference path and a probe path. The light of the refer-ence arm is reflected back by a retroreflecting prism into the fiber. The prism ismounted on an electromechanical scanner (galvoscanner) moving the prism backand forth. The light of the probe arm hits the sample which may be a multilayerstructure as, for example, the different tissue layers of the retina of a patient’seye. The light reflected from an individual layer interferes with the light fromthe reference arm only if the arm length of the reference arm corresponds to thedistance between the probe arm fiber exit and the respective layer of the sample.Because of the moving retroreflecting prism the interference signal occurs as aburst (Fig. 16). The signal can be also recorded with a demodulating logarithmicamplifier providing the envelope function of the interference signal. Another op-tion is to calculate the envelope function of the interferogram with the Hilbert

122 T. Hellmuth [3

transform [25]. The analog OCT-signal in Fig. 16 can be regarded as the real partof an analytical function h+(t) = hRe(t) + i · hIm(t). The imaginary part hIm(t)

is related to the real part by the Hilbert transform

(19)hIm(t) = 1

π

+∞∫−∞

hRe(t′)

t ′ − tdt ′.

The envelope function of the interference signal is

f (t) =√h2

Re + h2Im

as shown in Fig. 17. The OCT signal can be interpreted as the optical echo fromthe object. It is analogous to the A-scan signal in ultrasound imaging. However,the resolution of OCT is better than ultrasound by at least a factor 10. If the lightbeam is focused onto the object and scanned laterally the subsequent A-scans canbe arranged to a B-scan map which displays a tomographic view of the object.Figure 18 shows a cross-sectional view of the fundus of an eye [31]. The signalintensity is shown in false color contrast.

FIG. 17. Envelope of OCT signal calculated from signal shown in Fig. 16 with Hilbert transform.

FIG. 18. OCT tomogram from retina.


We have studied various applications and new methods based on OCT at AalenUniversity of Applied Sciences which shall be discussed in the following sections.

3.1.1. Determination of Blood Oxygenization with OCT

The main application of OCT is the tomographic imaging of the retinal layers ofthe human eye for diagnostic purposes. But this provides only morphologic infor-mation. Many diseases of the retina occur before any morphologic changes areobservable. As an example glaucoma is usually detected by perimetry where thevisual function of the retina is registered. Another age related disease of the macu-lar region of the fundus is macular degeneration. Treatment with lasers is difficultand can only retard the progression of the disease. Both glaucoma and maculardegeneration are supposed to be related to metabolic and blood supply disordersof the fundus which can only be understood if functional imaging techniques areavailable which can identify changes of concentration of metabolic substanceslike oxygen, glucose or cholesterol concentrations. Whereas it is difficult to iden-tify glucose and cholesterol spectra without fluorescent markers oxygenizationof the red blood cells can be detected by measuring the blood spectrum. Fig-ure 19 shows schematically the absorption coefficients of oxygenized hemoglobin(HbO2) and deoxygenized hemoglobin (Hb) (see also [18]). The crossover of thetwo spectra is the isobestic point. Its wavelength is around 800 nm.

Figure 20 shows the OCT signal for two wavelengths (680 nm and 815 nm)from oxygenized and deoxygenized blood pumped through a transparent tube [40].The light from two OCT interferometers equipped with a 680 nm SLD and a815 nm SLD, respectively, is combined with a dichroic mirror. The two wave-length beams are collinear to collect the OCT signal simultaneously from the

FIG. 19. Absorption coefficient of oxygenized (HbO2) and deoxygenized (Hb) blood.

124 T. Hellmuth [3

FIG. 20. In vitro measurement of OCT signal with 680 nm and 815 nm for oxygenized and de-oxygenized blood.

FIG. 21. In vivo OCT tomograms of the retina.

same object point. It can be seen from Fig. 20 that the OCT signals from the de-oxygenized blood probe and the oxygenized blood probe exhibit the same signalstrength at 815 nm because this wavelength is near the isobestic point of the bloodspectrum. At 680 nm the OCT signal from the oxygenized blood probe is higherthan the signal from the deoxygenized blood in agreement with Fig. 19. It can alsobe seen that the penetration of the SLD light into the blood sample is higher forthe longer wavelength 815 nm. This is because the backscattering cross-sectionof the blood cells is lower at 815 nm than at 680 nm.

Figure 21 shows an in vivo OCT measurement of the retina of a human eye at815 nm and 680 nm [40]. The OCT beam is scanned across a vein and an artery


of the retinal bloodvessel system. It can be seen that the OCT signals at 815 nmfrom the artery and the vein do not differ significantly in comparison to the OCTsignals at 680 nm which show a much smaller signal from the vein. This is inagreement with the spectral properties of HbO2 and Hb shown in Fig. 19.

3.1.2. Bloodflow Measurement and OCT

The blood supply of the retina consists of two separate vascular systems. Thechoroid provides the blood supply for the outer one half of the sensory retina. Thechoroid consists of a dense network of capillaries. It is supplied from the posteriorciliary artery. The inner portion of the retina is supplied by the branches of thecentral retinal artery which enters the eye at the nerve head (blind spot). Thesetwo blood supply systems are independent. Both circulations must be intact tomaintain retinal function.

Laser Doppler velocimetry is a technique which allows to measure the bloodflow at the fundus of the eye in vivo [19,20,27]. A laser goes through one halfof the eye pupil (Fig. 22). The angle of incidence of the laser at the fundus isproportional to the lateral offset of the laser at the pupil. Due to the finite angleof incidence there is a finite component of the k-vector of the laser parallel to thefundus. The laser light is scattered at the moving blood cells. Thus, the frequencyof the reflected k-vector component parallel to the fundus is shifted due to theDoppler effect. The frequency shift is �ν = ν0 · v/c, where ν0 is the nominalfrequency of the laser, v is the velocity of the blood cells and c is the speed oflight. But the same k-vector component is also scattered at the retinal tissue whichis at rest. The light backscattered from the retina passes the pupil and is finallydetected with a photodiode. The light scattered at the moving blood cells and at

FIG. 22. Setup for laser Doppler velocimetry of the retinal blood flow.

126 T. Hellmuth [3

FIG. 23. OCT-signal from the retina (A-scan).

the stationary tissue interferes at the detector. Because of the Doppler shift of thelight scattered at the moving blood cells the interference signal is modulated withthe Doppler shift frequency �ν which is in the range of some kHz depending onthe angle of incidence at the fundus and on the bloodflow velocity. The strengthof the Doppler signal is proportional to the density of the blood cells at the laserfocus on the retina. Therefore the product of the strength and the frequency shiftof the Doppler signal is proportional to the blood flow [19].

However, this technique measures the integral bloodflow of both the choroidalcirculation (supplied by the posterior ciliary artery) and the inner retinal circu-lation (supplied by the central retinal artery). OCT is a technique which candifferentiate between the two circulation systems because the various parts of theOCT signal from the retina can be attributed to the various retinal layers.

The OCT beam is coupled into the pupil in the same decentered way as the laserbeam with laser Doppler velocimetry. The OCT signal is shown in Fig. 23. Thefirst peak originates from the retinal sheet nurtured by the ciliary artery the secondfrom the choroid. The two signal peaks are Fourier transformed separately to pro-vide the Fourier spectra of the two signals shown in Fig. 24 [40]. The choroidalsignal frequency is shifted by 8 kHz relative to the signal frequency from a sta-tionary mirror as a reference object. The retinal signal frequency is shifted by5 kHz. Taking the angle of incidence of the OCT beam at the fundus into accounta bloodflow velocity of 4 cm/s can be estimated from the Doppler shift of thechoroidal signal and 2.5 cm/s for the retinal bloodflow.


FIG. 24. OCT Doppler velocimetry of the retinal blood flow (in vivo). Both the spectra of thechoroidal signal and of the retinal signal are shifted relative to the stationary mirror signal. Thechoroidal spectrum is shifted more than the retinal spectrum indicating a higher bloodflow velocity inthe choroidal tissue. It can also be seen that both the retinal and the choroidal spectrum is broader thanthe mirror spectrum which is caused by the velocity distribution of the blood cells.

3.1.3. Eye Length Measurement with OCT

Short coherence length interferometry can be used to measure the length of thehuman eye which is an important parameter for cataract surgery [22]. There, it isnecessary to select the correct intraocular lens before removing the eye lens. Theeye length has to be measured through the turbid eye lens in a contactless mode.The method described in [22] brings the reflex from the cornea and the retina tointerference by sending the light from the eye into a Michelson interferometerwith different arm lengths. The length difference is chosen so that it correspondsto the optical length of the eye bringing the corneal reflex and the retinal reflexto interference. The length difference of the interferometer is mechanically varieduntil the interference between the retinal reflex and the corneal reflex occurs.

We have investigated an alternative approach making use of a time domainOCT setup as shown in Fig. 25 [14]. Of course the range for the movement of theretroreflector in the TDOCT interferometer described above would be too longto cover the whole eye length in a short time. Short acquisition time is necessaryto avoid artefacts due to the saccadic eye movements. Another problem is thedispersion of the aqueous and vitreous humor of the eye causing a spreading ofthe interferogram and thus a reduction of the signal to noise ratio.

In the setup of Fig. 25 the retroreflector is scanned periodically in the range of2 mm. In addition a PMMA rod with an optical length corresponding to the opticallength of the standard human eye is periodically flipped in and out (as shown inFig. 25) of the reference path of the OCT interferometer. With the rod flipped

128 T. Hellmuth [3

FIG. 25. Setup of OCT interferometer for eye length measurement.

FIG. 26. OCT signal for eye length measurement. The corneal reflex consists of the front reflex(left diagram, left peak) and the reflex at the rear side of the cornea (left diagram, right peak). Thefundus reflex consists of the signal scattered by the retinal nerve fiber layer (right diagram, left peak)and by the choroid (right diagram, right peak).

out the signal from the cornea (corneal reflex) is recorded (Fig. 26, left diagram).With the rod flipped in the fundus reflex is recorded (Fig. 26, right diagram).

The corneal reflex consists of two peaks. The first peak corresponds to the re-flection at the front surface of the cornea the other to the reflection at the interfacesurface between cornea and the anterior chamber of the eye. The retinal reflexalso consists of two peaks. One is the reflex from the nerve fiber layer in front of


the receptor layer the other is generated by the choroid tissue behind the receptorlayer providing the blood supply for the retina. For the eye length measurementthe distance between the front corneal reflex and the receptor layer is relevant asshown in Fig. 26. Due to the similar dispersion properties of the PMMA rod thedispersion of the aqueous and vitreous humor is compensated. The optical eyelength can be determined with an accuracy of 20 microns which corresponds toan error in refractive power of the optical system of the eye of less than 0.05 D.The residual uncertainty of the eyelength measurement is due to the variability ofthe dispersion of the aqueous and vitreous humor because the measurement wave-length is 830 nm but the wavelength of the visible spectrum is around 550 nm.The clinical practice, however, shows that these uncertainties are negligible.

3.1.4. OCT in Optical Manufacturing

Within the last 15 years computer numerical controlled (CNC) machines havecompletely changed optical manufacturing. In particular the manufacturing of as-pheres has become an interesting alternative to spherical lenses both under costand functional aspects. Aspheric surfaces provide additional degrees of freedomin lens design where several spherical surfaces would be necessary. Therefore,aspheres permit lighter and smaller objectives for projection systems, sensors orphotographic systems. Aspheres are manufactured in three steps. First, the grind-ing machine has to shape the lens with an accuracy of 1 µm. In a second stepthe lens is polished. The polishing process smoothes the surface. In a third stepthe polished surface is measured interferometrically with an accuracy in the or-der of 10 nm. Deviations of the surface from the nominal design data are locallycorrected in the polishing process. For that purpose a polishing robot is used toguide the polishing tool directly to the zone of the aspheric surface which is to becorrected.

The grinding of the lens is the most crucial step in the whole process because nomeasurement feedback permits inline corrections. Because the surface of the lensis not reflecting standard interferometric techniques cannot be applied. Geomet-rical optical techniques as used in autofocus sensors are not applicable becausethey need large numerical apertures and small working distances.

Depth resolution of an OCT sensor depends only on the coherence length of thelight source. Therefore, an objective with low numerical aperture and long work-ing distance can be used. Figure 27 shows the experimental setup [30]. It consistsof a fiber optic TDOCT-setup as described above. The objective in the probe armis fixed to the tool mount (not shown) which is moved across the surface of the ob-ject which is to be measured. When the tool is moved the fiber is bent causing anindex change in the fiber. The OCT signal is shifted due to the index change in theorder of several tens of microns corrupting the measurement results. Therefore,a reference mirror is installed which is also fixed to the tool mount of the grind-ing machine. This reference signal is shifted in the same way as the probe signal

130 T. Hellmuth [3

FIG. 27. Setup of OCT interferometer for aspheric profile measurement in grinding machine.

induced by the bending fiber. The movement of the probe signal relative to this ref-erence signal finally provides the information about the surface topography whenthe objective is moved across the asphere by the CNC grinding machine. Figure 28shows the profile of a grinded plane surface derived from the peak positions of theOCT signals. The result is compared with the measurement of the surface with aultraprecision tactile mechanical measurement machine (Zeiss UPMC).

The signal noise of the OCT-measurement is primarily due to the surface rough-ness. The lateral resolution of the OCT-measurement is in the order of 100 micron(SLD-focus) whereas the lateral resolution of the tactile measurement machine islimited by the probe ball with a diameter of about 2 mm which is scanned acrossthe surface. The working distance between the objective and the surface is 10 mm.Because of the scattering properties of the rough surface the OCT signal can beregistered also from tilted surfaces with a tilt angle up to 20 degrees.

3.2. LINEAR OPTICAL COHERENCE TOMOGRAPHY (LOCT)

Time domain OCT uses a galvoscanner as a moving device for the retroreflec-tor. Mechanical devices are limited in their frequency bandwidth thus limitingthe image acquisition speed. Another important drawback is the limited lifetime


FIG. 28. Profile measurement of a grinded plane surface derived from the peak positions of theOCT signals. The result is compared with the reference measurement of the surface with a ultrapreci-sion tactile mechanical measurement machine (Zeiss UPMC).

FIG. 29. Linear optical coherence tomograph.

of electromechanical parts. An alternative approach comprises an interferometricsetup shown in Fig. 29 [12,13].

132 T. Hellmuth [3

FIG. 30. The position of the interferogram on the CCD-line depends on the pathlength difference.

Again a superluminescent diode is used as low coherence source. The lightpasses a first coupler with a splitting ratio of 90:10. In order to improve the sen-sitivity of the setup, 90% of the light is directed to the sample arm, while 10% ofthe light is routed to the reference arm. Additional 50:50 couplers are placed inthe sample and reference arm, respectively. In each interferometer arm, the lightpasses the two 50:50 couplers. The reference beam is reflected from a referencemirror whereas the sample beam is reflected from the sample that is to be imaged.The position of the reference mirror is fixed, no optical delay line scanning isneeded. The backreflected light from the reference mirror and the sample againpasses the 50:50 couplers and is then routed to a fiber two-pinhole device. Thedistance between the two fibers can be varied between 250 µm and 8 mm. Outsidethe two fibers, the light propagates in a solid angle determined by the numericalaperture of the monomode fibers (NA = 0.11). A linear CCD-array is locatedin a distance of 25 cm from the two fibers and is illuminated with the light fromthe two fibers. An efficient illumination of the CCD-array is achieved by usinga cylindrical lens optics. Therefore the circular light cone emerging from eachfiber is transformed into two overlapping lines at the position of the CCD-array,collinear with the active area. The signal processing is done using a bandpassfilter, a logarithmic amplifier, a demodulator, an AD-converter and a computer.

Figure 30 shows how the position of the interference structure on the CCD-linedepends on the path length difference of the probe arm and the reference armof the interferometer. If these path lengths are equal the interference pattern is


located at the center of the CCD line. Then the path lengths between the two fiberoutputs are equal.

If the probe arm length differs from the reference arm length by � the interfer-ence pattern appears on the CCD line where the distance between the maximum ofthe interference pattern and fiber output A differs from the corresponding distanceto fiber output B by �.

It can be shown that the theoretical limit of the signal to noise ratio is equivalentto that of TDOCT [10]. Another special aspect of LOCT is that the fringe oscil-lation frequency can be set independently from the image acquisition time andthe wavelength by choosing an appropriate separation of the two fiber outputs Aand B.

3.3. SPECTRAL DOMAIN OPTICAL COHERENCE TOMOGRAPHY (SDOCT)

Besides TDOCT and LOCT a third method has become an interesting alternativewhich is known as spectral domain optical coherence tomography (SDOCT) [22,33,37]. The principle is shown in Fig. 31. The light from the superluminescentdiode is split into two beams by a beamsplitter (or a 3dB-coupler). The refer-ence beam is reflected from the reference mirror. The probe beam is reflectedfrom the sample. The two reflected beams are superimposed and hit a diffractiongrating. The dispersed beam is finally focused onto a CCD-line. The registeredspectrum represents basically the spectrum of the light source. However, the spec-trum is modulated due to the interference of the probe beam and the referencebeam (Fig. 32) [39]. The modulation frequency is proportional to the armlengthdifference of the reference arm and the probearm. Computing the Fourier trans-

FIG. 31. Spectral domain optical coherence tomograph.

134 T. Hellmuth [3

FIG. 32. Modulated spectrum of SLD.

FIG. 33. Fourier transform of signal in Fig. 32. The position of the peak depends on the armlengthdifference of the interferometer.

form finally provides the signal (Fig. 33). The signal is equivalent to the signal ofTDOCT or LOCT. The position of the peak depends on the armlength differenceand thus on the position of the sample object. However, as it will be shown inthe next section the SDOCT signal is superior to the TDOCT and LOCT signalconcerning the signal to noise ratio.

3.3.1. Comparison of Noise in Spectral Domain OCT and Time Domain OCT

In [28,29] TDOCT and SDOCT are compared in respect of their noise perfor-mance. It is shown that if photon shot noise is the relevant noise source SDOCTis superior to TDOCT and LOCT. The following simulation shall illustrate thisremarkable result. The results can also be directly applied to the noise analysis ofline spectrometers and Fourier transform spectrometers because of their analogyto SDOCT and TDOCT. Only if amplifier noise is the dominant noise source line


FIG. 34. Comparison of Poissonian noise in spectral domain OCT and time domain OCT.

spectrometers and Fourier transform spectrometers are equivalent regarding noiseperformance.

In the following simple model it is assumed that the SDOCT signal is generatedon a CCD line with 512 pixels with 50 photons per pixel on the average. Thewhole photon budget is therefore 25,600 photons. The modulated spectrometersignal is given by

fm = A · sin2(2πνsm)

with m = −N/2 . . . + N/2. The simulated signal is shown in Fig. 34 (top left)with νs = 0.4, the number of pixels N = 512 and A = 100. The values fmexhibit Poissonian noise generated by a standard Poissonian noise generation al-gorithm [7]. The Fourier transform of fm is given by the N values

fn =+N/2∑

m=−N/2

e−i2πnmfm

with n = −N/2 . . .+N/2. The SDOCT-Signal is then given by the N values

sn =√∣∣fn∣∣2 =

√√√√√∣∣∣∣∣+N/2∑−N/2

e−i2πnmfm

∣∣∣∣∣2

136 T. Hellmuth [5

as shown in Fig. 34 (bottom, left). For comparison the corresponding TDOCT-signal is given by the N values

tm = A1

2

[1 + e−(m/b)2 sin(2πνtm)

]shown in Fig. 34 (top, right) with νt = 0.2 and the coherence length of the lightsource b = 20. Also Poissonian noise is introduced. The mean photon budget isas in the SDOCT case 25,600. The envelope function can be calculated with theHilbert transform (Fig. 34 bottom, right). Comparing the SDOCT and TDOCTsignal it is obvious that the signal to noise ratio is better for SDOCT than forTDOCT although the same number of signal photons has been taken for the sim-ulation.

4. Conclusion

Both wavefront coding and optical coherence tomography are commercially usedtechniques. Although in principle many fields of applications are open to thesemethods wavefront coding is mainly used for specific sensor applications likebarcode readers. OCT is used mainly in ophthalmology.

The limiting factors for wavefront coding are mainly artefacts introduced bythe inverse filtering. Thus, this technique is not yet found in the field of high qual-ity imaging techniques like microscopy and digital photography although firstcommercial trials have been undertaken in microscopy. Because of the low man-ufacturing costs of the phaseplates and the progress in the field of fast signal andimage processing many new applications in medium quality photography (e.g.mobile phone cameras or surveillance cameras) and in sensor technology can beexpected.

The limiting factor in OCT is mainly the coherence length and the brightnessof the light source. It has been proven in various applications that high resolu-tion OCT tomograms can be obtained by using femtosecond lasers, rapidly swepttunable laser sources [42] or supercontinuum laser sources generated with fem-tosecond laser pulses in photonic crystals [43]. As soon as these light sources areavailable at low cost in high quantities OCT will have a great future not only inophthalmology as today but also in such important fields of medical applicationslike endoscopy.

5. Acknowledgements

The projects have been supported by the “Landesstiftung Baden-Württemberg”and “Bundesministerium für Bildung und Forschung”. We gratefully appreciate


valuable discussions and cooperation with Dr. C. Hauger and Dr. H. Gross (bothCarl Zeiss AG, Oberkochen).

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THE QUANTUM PROPERTIES OFMULTIMODE OPTICAL AMPLIFIERSREVISITED

G. LEUCHS1,* , U.L. ANDERSEN1 and C. FABRE2

1Max Planck Research Group of Optics, Information and Photonics,University of Erlangen-Nürnberg, Erlangen, Germany2Laboratoire Kastler-Brossel, Université Pierre et Marie Curie et Ecole Normale Supérieure,

Place Jussieu, cc74, 75252 Paris cedex 05, France

1. General Linear Input–Output Transformation for a Linear Optical Device . . . . . . . . . 1402. The Phase-Insensitive Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1413. The Multimode Phase Insensitive Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . 1434. The Nature of the Ancilla Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445. An Optical Amplifier Working at the Quantum Limit . . . . . . . . . . . . . . . . . . . . . 1476. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

AbstractThere are a number of physically different realizations of an optical amplifier andyet they all share the same fundamental quantum limit as far as their noise character-istics are concerned. We review the underlying mathematical formalism without therestriction of a minimum number of modes being involved, its physical implicationsand relate it to the phenomenological models for the various amplifiers.

The study of optical amplifiers and their properties started with the invention ofthe maser and the laser, based on stimulated emission [1–5]. These first opticalamplifiers belong to the class of phase insensitive amplifiers. Modern semicon-ductor optical amplifiers, and especially the Erbium doped fiber amplifiers whichare widely used in telecommunications [6], belong to the same category [7,8].There is a second class of optical amplifiers which are based on non-linear opticalprocesses such as stimulated Brillouin, Rayleigh or Raman processes [9] or three-

* E-mail: [email protected].


DOI 10.1016/S1049-250X(06)53005-8

139

mailto:[email protected]

140 G. Leuchs et al. [1

and four-wave mixing [10,11], which are also, in most cases, phase-insensitive.However, there are some configurations where the amplification depends on thephase of the input signal. The degenerate parametric amplifier, or the parametricamplifier with the input signal is injected on both the signal and idler modes, areexamples of such phase-sensitive amplifiers [12].

In the simplest case, well below saturation, all these amplifiers are linear de-vices, i.e. the field operators describing the output beams are linear combinationsof the field operators at the input side. Nevertheless, these seemingly simple math-ematical relations allow for a fairly complex scenario as described in the paperson the quantum behavior of linear amplifiers by Haus and Mullen [13] and byCaves [14]. All the properties including the quantum aspects can be traced backto these linear field operator transformations. We will discuss these relations in ageneral framework, with the possibility of involving many modes in the device.Next we will treat the amplifier more phenomenologically and identify the exper-imental nature of the modes the existence of which is required by the unitarity ofthe field operators. To some extent the different types of amplifiers can be relatedto the different types of attenuators.

In a recent demonstration of quantum cloning with continuous variables, theessential ingredient was an optical amplifier working at the quantum limit [15].This amplifier does not use any non-linear optical process nor stimulated emissionbut just linear optical elements, detectors, modulators and electronic feed-forwardcircuits. The amplifier set-up with its modular structure makes it easy to identifythe various origins of the noise figure of the optical amplifier and to compare itwith the general performance limitation.

1. General Linear Input–Output Transformation for a LinearOptical Device

Let us consider a linear optical device, which can be an amplifier, an attenuatoror a quantum gate. From a basic point of view it is well known that there canbe no such device which transforms an input field described by the field operatora to an output field described by a field operator a′ = βa with |β| = 1. Thiswould violate the requirement that all free space field operators fulfill the bosoniccommutation relation

(1)[a, a†] = [a′, a′†] = 1.

The conservation of such a commutation relation leaves two possibilities:

• The device is single mode, but of the form

(2)a → a′ = β1a + β2a†

2] MULTIMODE OPTICAL AMPLIFIERS 141

with

(3)|β1|2 − |β2|2 = 1.

• The device is multimode and couples N > 1 input modes to N output modes.We will first consider here the simplest and basic case of a two mode device,and call b the annihilation operator of this second, “ancilla”, mode. The mostgeneral transformation fulfilling the requirement (1) is the two-mode Bogoli-ubov transformation [16]

(4)a, b → a′ = β1a + β2a† + γ1b + γ2b

†

with

(5)|β1|2 − |β2|2 + |γ1|2 − |γ2|2 = 1.

The terms proportional to β2 and γ2 correspond to “spontaneous emission”processes, where photons are produced at the output even in absence of any pho-ton in the corresponding input mode.

In the first, single mode case, the device is necessarily phase-sensitive: itmultiplies the input mean field by a factor which depends on the quadratureXφ = eiφa + e−iφa† of the input signal. When the input is a coherent or vac-uum state, the output is a squeezed state [17]. Although being formally a linearprocess, the phase sensitive amplifier associated with the squeezing operation isan intrinsically non-linear process and requires an optically non-linear interac-tion [17].

In the two-mode case, relation (4) leads to phase insensitivity of a new typewhich is present even if β2 = 0. In this latter case,

(6)|γ1|2 − |γ2|2 = 1 − |β1|2.The device is an amplifier when |β1| > 1, and an attenuator when |β1| < 1.Relation (6) implies that at least γ2 must be different from zero in the amplifiercase, and that at least γ1 must be different from zero in the attenuator case. But inthe general case both coefficients are different from zero and mode a may, e.g.,be coupled to a squeezed vacuum. Attenuation and amplification are merely twolimiting cases.

2. The Phase-Insensitive Amplifier

The general phase insensitive amplifier of energy gain G > 1 contains an un-known mixture of γ1 and γ2. It is characterized by the transformation

(7)a, b → a′ = √G a + g1b + g2e

iαb†,


where all quantities G, g1, g2 and α are real. The phase factors of β1 = √G

and γ1 have been included in the annihilation operators a and b. The term ‘phaseinsensitive’ relates to the fact that with β2 = 0 the amplification of the meanvalues of the quadrature components is phase insensitive which still leaves roomfor phase sensitive noise. Relation (4) implies that g1 and g2 have to fulfill thecondition

(8)g22 − g2

1 = G− 1.

The quantum statistical properties of this amplifier can be described by themoments of the field quadratures. An important scenario is the case where theinput mode a is in a coherent state |α〉 and mode b is in the vacuum state |0〉 whichwe will write as |α, 0〉. The input signal is a modulation at a given frequency ofthe amplitude or the phase of the input wave, or more generally of any quadratureXφ = eiφa + e−iφa† of the input signal. The second moment of this quadratureat the output of the device is

(9)⟨δX′ 2

φ

⟩ = G+ g21 + g2

2 + 2g1g2 cos(α + 2φ).

Let us recall that the variance for any quadrature of a coherent state is 1 with thepresent notations. Using relation (8), this gives the following value for the noisefigure F , which is in the present case the output noise 〈δX′ 2

φ 〉 divided by the gain

(10)F = 2 − 1

G+ 2g1

g1 +√g2

1 +G− 1 cos(α − 2φ)

G.

In the usual parametric amplifier case, the ancilla mode is the idler mode andg1 = 0. The noise figure reduces to [13,14]

(11)F = 2 − 1

G,

and in the limit of large gain we recover the familiar result F = 2, well known asthe 3 dB quantum limit of the phase insensitive optical amplifier.

One notices that when g1 = 0 the noise of the amplifier, and therefore its noisefigure, is phase sensitive, whereas the gain for the mean value is phase insensitive.Its minimum value is obtained for φ = π−α

2 ,

Fmin = 2 − 1

G+ 2g1

g1 −√g2

1 +G− 1

G

(12)= 2 − 1

G− 2

g1

g1 +√g2

1 +G− 1

G− 1

G.

An interesting limiting case is when g1 � √G, for which, according to (12),

Fmin is getting close to 1. One has in this case a two-mode noiseless amplifier.


A possible implementation is to use a regular non-degenerate parametric amplifierand to insert a perfect squeezer at the input of the idler mode. This configurationhas been experimentally studied in [18].

So far we discussed the case where either the signal mode or the ancilla modeare supposed to be unique in the amplification or attenuation process. We will con-sider in the following sections the possibility of linear coupling between multiplemodes.

3. The Multimode Phase Insensitive Amplifier

Another important class of optical amplifiers is the multi-mode amplifier, that islikely to amplify simultaneously several orthogonal modes, for example, imageamplifiers [19–21]. For the simplicity of the discussion, we will take here thesimple example of a two-mode phase insensitive amplifier having the same gainfor any combination of the two modes a1 and a2. As in the single mode case, re-lation (1) requires the existence of at least one ancilla mode b. Let us first assumethat there is only one such mode. We can then write

a1, a2, b → a′1 = βa1 + γ1b + γ2b†,

(13)a′2 = βa2 + γ ′1b + γ ′

2b†

and using the commutators of the various field operators

|γ2|2 − |γ1|2 = ∣∣γ ′2

∣∣2 − ∣∣γ ′1

∣∣2 = |β|2 − 1,

(14)γ2γ′ ∗2 − γ1γ

′ ∗1 = 0.

A straightforward derivation shows that the relations (14) cannot be simultane-ously fulfilled, so that a second ancilla mode is needed. The demonstration caneasily be extended to the N -mode amplifier. The conclusion of this simple butgeneral reasoning is that one needs at least as many ancilla modes as there areinput signal modes in a multimode amplifier.

In the two-mode amplifier, calling b1 and b2 these two required modes, and inthe simple case where only creation operators for the ancilla modes are involvedin the input-output relation, one has

a1, a2, b → a′1 = βa1 + γ11b†1 + γ12b

†2,

(15)a′2 = βa2 + γ21b†1 + γ22b

†2,

and

|γ11|2 + |γ12|2 = |γ21|2 + |γ22|2 = |β|2 − 1,

(16)γ11γ∗21 + γ12γ

∗22 = 0,


which correspond to a unitary transformation in the two-mode ancillary space:

γ11 =√|β|2 − 1 cos θeiψ ,

γ12 =√|β|2 − 1 sin θe−iψ ,

γ21 = −√|β|2 − 1 sin θeiψ ,

(17)γ22 =√|β|2 − 1 cos θe−iψ ,

where θ and ψ are arbitrary angles. If one performs the inverse of this transfor-mation on the two-mode signal space, the gain will not be changed for the twonew modes, as the gain matrix is proportional to the identity, and the ancilla termswill be diagonalized, a single ancilla mode being associated to each amplified sig-nal mode: the only possible configuration for a two-mode amplifier is thereforetwo independent single mode amplifiers with identical gains. As a result the noisefigure will be the same as in the single mode case, and independent of the com-binations of input modes used as a signal. These conclusions are no longer validin the more complicated case where the gain is different for the two amplifiedmodes.

4. The Nature of the Ancilla Modes

If one now turns to physical implementations of the optical amplifier one mightask the question: what is the additional mode b which is so essential in the math-ematical description of the amplifier?

In the case of the parametric amplifier the answer is straightforward. Modeb is the idler mode which has to be in the vacuum state for the standard phaseinsensitive amplifier (Fig. 1).

Next one asks the same question for the prototype of all amplifiers, the mediumwith population inversion which gives rise to amplification by stimulated emis-sion. Here the identification of mode b is not straightforward. There are many

FIG. 1. Sketch of a parametric amplifier. The pump is taken to be a classical field and the para-metric amplifier couples mode a and b.


FIG. 2. Panel (a) shows the signal field mode amplified by the medium and two of the N − 1vacuum modes scattering into the signal mode. Panel (b) represents a formal sketch of the situation.

In the text two (N ×N ) matrices are used, β+mn couples a†n to a′m and βmm couples an to a′m.

more than just two modes which one has to consider. A related multi-mode for-malism is used to describe multiple scattering in inhomogeneous media [22]. Allbut the signal mode are taken to be in the vacuum state. Each of these vacuummodes may couple into the signal mode (or any other vacuum mode) by scatter-ing off one of the inverted molecules in the medium via spontaneous emission(Fig. 2a). The linear coupling of the N input modes to the N output modes isdescribed by an N ×N matrix

(18)a′m = βmmam +∞∑n=m

βmnan + β+mna†n

with

(19)|βmm|2 +∑n=m

|βmn|2 −∑n=m

|β+mn|2 = 1.

If m is the signal mode then |βmm|2 = G and∑

n=m |β+mn|2 −∑n=m |βmn|2 =G− 1. Under these conditions the noise figure for the stimulated emission ampli-


fier is

(20)F = 1 + 1

G

∑n=m

(βmnβ+mn + β∗

mnβ∗+mn + |βmn|2 + |β+mn|2

).

In the case of the ideal amplifier (i.e. βmn = 0 if m = n), the formula reduces to

(21)F = 1 + 1

G

∑n=m

|β+mn|2 = 2 − 1

G,

and with the sum rule (19) we retrieve the familiar results for the single modecase (11). This shows that a multimode manifold of vacuum modes effectively hasthe same impact as the one vacuum mode in the single mode case. The quantumlimit of the ideal amplifier results from the admixture of the creation operator ofone “super” vacuum mode b, which is actually the linear combination

(22)b =∑

n=m βmnan√∑n=m |βmn|2

.

In the case of the stimulated emission amplifier, the an modes being the planewave modes in which spontaneous photons are likely to be emitted. For a single

FIG. 3. Sketch of a 2 × 2 beam splitter (a) and a lossy element (b) such as a neutral density filterin which case the loss channels n are absorbed inside the medium (not shown).


atom or molecule of the amplifying medium this super mode b is nothing but thedipole wave which is emitted by the atom or which couples best to the atom [23,24]. The discussion also shows that confining the amplifying medium, e.g., to thecore of a photonic crystal fiber [25] will reduce the number of modes but will notaffect the quantum noise limit of the amplifier.

It is worth noting that the amplifier and the phenomenological descriptionabove can be mapped to the familiar case of an optical attenuator. It is immediatelyclear that the single mode case, i.e. the counterpart to the parametric amplifier, isthe ubiquitous beam splitter. The stimulated emission amplifier however resem-bles a neutral density attenuator where the light is also coupled to many modes(see Fig. 3). Again with a similar line of arguments these many modes can betreated effectively as one mode for the purpose of noise consideration.

5. An Optical Amplifier Working at the Quantum Limit

In a recent experiment it was shown that an optical amplifier working at the quan-tum limit can be demonstrated using just linear optical elements, detectors, anamplifying electronic circuit and optical modulators for amplitude and phase [15].The scheme, which is an extension of the intensity modulation amplifier [26], issketched in Fig. 4.

The signal input field is split at the first beam splitter and mixed with the aux-iliary mode v1. The split off part is measured according to the scheme of Arthurand Kelly [27] where x and p denote the amplitude and phase quadratures of thefield measured for example with a local oscillator (not shown). The detected sig-nals are amplified and fed forward to an amplitude and a phase modulator. If theamplification factor λ is chosen properly in dependence on the splitting ratio of

FIG. 4. Sketch of optical amplification with beam splitters, detectors and modulators. The detectorsignals have to be amplified and fed forward to the detector (see [15]).


the first beam splitter, then the input v1 does not influence the noise characteristicsof the amplifier. Note that this is a universal scheme. For a theoretical quantumtreatment of the electro-optic feed forward see [28–30]. In the spirit of the abovediscussion this amplifier can be described with the single additional mode whichis readily identified with the vacuum input field v2.

6. Conclusion

We have discussed the fundamental noise limit of optical amplification in whichmany modes are involved, either for the input signal or for the ancilla modes.We have considered various physical implementations: the parametric ampli-fier, the stimulated emission amplifier and the quantum electro-optic feedforwardamplifier. Other amplifiers such as a Raman amplifier [31,32] can be discussedfollowing the same arguments. Quantum noise considerations are relevant, e.g.,to optical communication [33,34]. The main point here was to physically identifythe ancillary field modes required for the mathematical description.

7. References

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ture in the microwave spectrum of NH3, Phys. Rev. 95 (1954) 282.[5] A.L. Schawlow, C.H. Townes, Infrared and optical masers, Phys. Rev. 112 (1958) 1940.[6] E. Desurvire, “Erbium Doped Fiber Amplifiers”, Wiley Interscience, New York, 1993.[7] M. Shtaif, G. Eisenstein, Noise properties of nonlinear semiconductor optical amplifiers, Opt.

Lett. 21 (1996) 1851.[8] E.S. Björlin, Noise figure of vertical-cavity semiconductor optical amplifiers, IEEE J. Quantum

Electron. 38 (2002) 61.[9] R.W. Boyd, “Nonlinear Optics”, Academic Press, 2003.

[10] Y.R. Shen, “The Principle of Nonlinear Optics”, Wiley, 1984.[11] I. Protsenko, L. Lugiato, C. Fabre, Spectral analysis of the degenerate optical parametric oscilla-

tor as a noiseless amplifier, Phys. Rev. A 50 (1994) 1627.[12] P.A. Franken, A.E. Hill, C.W. Peters, G. Weinreich, Generation of optical harmonics, Phys. Rev.

Lett. 7 (1961) 118.[13] H.A. Haus, J.A. Mullen, Quantum noise in linear amplifiers, Phys. Rev. 128 (1962) 2407–2413.[14] C.M. Caves, Quantum limits on noise in linear amplifiers, Phys. Rev. D 26 (1982) 1817–1839.[15] U.L. Andersen, V. Josse, G. Leuchs, Unconditional quantum cloning of coherent states with

linear optics, Phys. Rev. Lett. 94 (2005) 240503.[16] N.N. Bogoliubov, J. Phys. (USSR) 11 (1947) 23.[17] W. Schleich, “Quantum Optics in Phase Space”, Wiley-VCH, Weinheim, 2001.[18] Z.Y. Ou, S.F. Pereira, H.J. Kimble, Quantum noise reduction in optical amplification, Phys. Rev.

Lett. 70 (1993) 3239.


[19] S.-K. Choi, M. Vasilyev, P. Kumar, Noiseless optical amplification of images, Phys. Rev. Lett. 83(1999) 1938.

[20] A. Mosset, F. Devaux, E. Lantz, Spatially noiseless optical amplification of images, Phys. Rev.Lett. 94 (2005) 223603.

[21] S. Gigan, L. Lopez, V. Delaubert, N. Treps, C. Fabre, A. Maitre, cw phase sensitive parametricimage amplification, J. Modern Opt., Special Issue on Quantum Imaging (2005), in press.

[22] C.W.J. Beenakker, Thermal radiation and amplified spontaneous emission from a randommedium, Phys. Rev. Lett. 81 (1998) 1829.

[23] S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, Focusing light to a tighter spot, Opt.Commun. 179 (2000) 1–7.

[24] C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, “Photons and Atoms”, Wiley, 1997.[25] J.C. Knight, T.A. Birks, P.J.S. Russell, et al., All-silica single-mode optical fiber with photonic

crystal cladding, Opt. Lett. 21 (1996) 1547.[26] A.V. Masalov, A.A. Putilin, Quantum noise of a modulation optical amplifier, Opt. Spectrosc. 82

(1997) 823.[27] E. Arthur, J.L. Kelly, Bell Syst. Tech. J. 44 (1965) 725.[28] H.M. Wiseman, G.J. Milburn, All-optical versus electro-optical quantum-limited feedback, Phys.

Rev. A 49 (1994) 4110.[29] B. Julsgaard, J. Sherson, J.I. Cirac, J. Fiurasek, E.S. Polzik, Experimental demonstration of quan-

tum memory for light, Nature 432 (2004) 482, see supplementary note.[30] U.L. Andersen, V. Josse, N. Lütkenhaus, G. Leuchs, Experimental quantum cloning with continu-

ous variables, in: N. Cerf, G. Leuchs, E.S. Polzik (Eds.), “Quantum Information with ContinuousVariables of Atoms and Light”, Imperial College Press, London, 2006.

[31] E.B. Tucker, Amplification of 9.3-kMc/sec ultrasonic pulses by MASER action in Ruby, Phys.Rev. Lett. 6 (1961) 547.

[32] E. Garmire, F. Pandarese, C.H. Townes, Coherently driven molecular vibrations and light modu-lation, Phys. Rev. Lett. 11 (1963) 160.

[33] W.S. Wong, H.A. Haus, L.A. Jiang, P.B. Hansen, M. Margalit, Photon statistics of amplifiedspontaneous emission noise in a 10-Gbit/s optically preamplified direct-detection receiver, Opt.Lett. 23 (1998) 1832–1834.

[34] M. Meißner, K. Sponsel, K. Cvecek, A. Benz, S. Weisser, B. Schmauß, G. Leuchs, 3.9 dB OSNRgain by a NOLM based 2-R regenerator, IEEE Phot. Techn. Lett. 16 (2004) 2105.



QUANTUM OPTICS OF ULTRA-COLDMOLECULES

D. MEISER, T. MIYAKAWA, H. UYS and P. MEYSTRE

Department of Physics, The University of Arizona, 1118 E. 4th Street, Tucson, AZ 85705, USA

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522. Molecular Micromaser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

2.1. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1542.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

3. Passage Time Statistics of Molecule Formation . . . . . . . . . . . . . . . . . . . . . . . . 1634. Counting Statistics of Molecular Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

4.1. BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1684.2. Normal Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1704.3. Fermi Gas with Superfluid Component . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5. Molecules as Probes of Spatial Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 1735.1. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1755.2. BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1775.3. Normal Fermi Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1795.4. BCS State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180


AbstractQuantum optics has been a major driving force behind the rapid experimental de-velopments that have led from the first laser cooling schemes to the Bose–Einsteincondensation (BEC) of dilute atomic and molecular gases. Not only has it providedexperimentalists with the necessary tools to create ultra-cold atomic systems, but ithas also provided theorists with a formalism and framework to describe them: manyeffects now being studied in quantum-degenerate atomic and molecular systems finda very natural explanation in a quantum optics picture. This article briefly reviewsthree such examples that find their direct inspiration in the trailblazing work carriedout over the years by Herbert Walther, one of the true giants of that field. Specifi-cally, we use an analogy with the micromaser to analyze ultra-cold molecules in adouble-well potential; study the formation and dissociation dynamics of moleculesusing the passage time statistics familiar from superradiance and superfluorescencestudies; and show how molecules can be used to probe higher-order correlations inultra-cold atomic gases, in particular bunching and antibunching.


DOI 10.1016/S1049-250X(06)53006-X

151

152 D. Meiser et al. [1

1. Introduction

Quantum optics plays a central role in the physics of quantum-degenerate atomsand molecules. Laser light and its coherent and incoherent interactions withatoms are ubiquitous in these experiments, and the tools that have culminated inthe achievement of Bose–Einstein condensation (BEC) (Anderson et al., 1995;Bradley et al., 1995; Davis et al., 1995) were first studied and understood inquantum optics. Indeed, the deep connection between quantum optics and coldatom physics was realized well before the first experimental realizations of BEC,both at the experimental and theoretical levels. On the theory side, there are(at least) two important reasons why quantum optics methods are well suitedfor the study of cold atoms systems. First, bosonic fields have direct analogsin electromagnetic fields, which have been extensively studied in quantum op-tics. Second, for fermions the Pauli Exclusion Principle restricts the occupa-tion of a given mode to zero or one, and these two states—mode occupied orempty—can often be mapped onto a two-level system, as we shall see. As aresult, many situations familiar from quantum optics are also found in cold-atom systems, including matter–wave interference (Andrews et al., 1997), atomlasers and matter–wave amplifiers (Inouye et al., 1999b; Ketterle and Miesner,1997; Kozuma et al., 1999; Law and Bigelow, 1998) matter–wave beam splitters(Burgbacher and Audretsch, 1999) four-wave mixing (Christ et al., 2003; Lenz etal., 1993; Meiser et al., 2005a; Miyakawa et al., 2003; Moore et al., 1999; Rojoet al., 1999; Search et al., 2002b), and Dicke superradiance (Inouye et al., 1999a;Moore and Meystre, 1999), to name a few.

At the same time the physics of ultra-cold atoms is much richer than itsquantum-optical counterpart since atoms can be either fermions (DeMarco andJin, 1998, 1999; Hadzibabic et al., 2003) or bosons and have a rich internalstructure. In addition, the interaction between atoms can be tuned relativelyeasily on fast time scales using for instance Feshbach resonances (Duine andStoof, 2004; Dürr et al., 2004; Inouye et al., 1998, 2004; Stan et al., 2004;Timmermans et al., 1999) or two-photon Raman transitions (Theis et al., 2004;Wynar et al., 2000). Indeed, some of the most exciting recent developments inthe physics of ultra-cold atoms are related to the coherent coupling of atomsto ultra-cold molecules by means of Feshbach resonances (Dürr et al., 2004;Regal et al., 2003), and photo-association (Kerman et al., 2004; Wynar et al.,2000). Both bosons and fermions have been successfully converted into mole-cules. In both cases BEC of molecules has been observed (Donley et al., 2002;Greiner et al., 2003; Jochim et al., 2003; Zwierlein et al., 2003), and the long-standing question of the BEC-BCS crossover is being investigated experimentallyand theoretically in those systems (Bartenstein et al., 2004; Holland et al., 2001;Ohashi and Griffin, 2002; Regal et al., 2004; Timmermans et al., 2001; Zwierlein

2] QUANTUM OPTICS OF ULTRA-COLD MOLECULES 153

et al., 2004). Other developments with close connections with quantum optics in-clude the trapping of atoms in optical lattices (Greiner et al., 2002a, 2002b; Jakschet al., 1998), which play a role closely related to a high-Q resonator in cavity QED(Search et al., 2004; Walther, 1992) and leads in addition to fascinating connec-tions with condensed matter physics and quantum information science.

With so many close connections between the physics of quantum-degenerateatomic and molecular systems and quantum optics, it is natural and wise togo back to the masters of that field to find inspiration and guidance, and thisis why Herbert Walther’s intellectual imprint remains so important. This briefreview illustrates this point with three examples. Section 2 shows that the con-version of pairs of fermions into molecules in a double-well potential can bedescribed by a generalized Jaynes–Cummings model. Using this equivalence,we show that the dynamics of the molecular field at each site can be mappedto that of a micromaser, one of Herbert Walther’s most remarkable contribu-tions (Meschede et al., 1985). Section 3 further expands on the mapping ofultra-cold fermion pairs onto two-level atoms to study the role of fluctuationsin the association and dissociation rates of ultra-cold molecules. We show thatthis system is closely related to Dicke superradiance, and with this analogy asa guide, we discuss how the passage time fluctuations depend sensitively on theinitial state of the system. In a third example, inspired by Herbert Walther’s workon photon statistics and antibunching (Brattke et al., 2001; Krause et al., 1989;Rempe et al., 1990; Rempe and Walther, 1990) Section 4 analyzes how the statis-tics of their constituent atoms affects the counting statistics of molecules formedby photo-association. We compare the three cases where the molecules are formedfrom a BEC, an ultra-cold Fermi gas and a Fermi system with a superfluid compo-nent. The concept of quantum coherence developed by R.J. Glauber and exploitedin many situations by H. Walther and his coworkers, in particular in their stud-ies of resonance fluorescence, are now applied to characterizing the statisticalproperties of the coupled atom–molecule system. Finally, Section 5 further elabo-rates on these ideas to probe spatial correlations and coherent properties of atomicsamples, and we find that the momentum distribution of the molecules containsdetailed information about the second-order correlations of the initial atomic gas.

2. Molecular Micromaser

Ultra-cold atoms and molecules trapped in optical lattices provide an excitingnew tool to study a variety of physics problems. In particular, they provide re-markable connections with the condensed matter of strongly correlated systemsand with quantum information science, a very well controlled environment tostudy processes such as photo-association (Ryu et al., 2005), and, from a point-of-view more directly related to quantum optics, can be thought of as matter–


wave analog of photons trapped in high-Q cavities. In particular, the high degreeof real-time control of the system parameters offers the opportunity to directlyexperimentally study some of the long-standing questions of condensed matterphysics, such as the ground state structure of certain models and many-bodydynamic properties (Jaksch and Zoller, 2005). The coherent formation of mole-cules in an optical lattice via either Feshbach resonances and two-photon Ra-man photo-association has been studied both theoretically (Jaksch et al., 2002;Damski et al., 2003; Esslinger and Molmer, 2003; Molmer, 2003; Moore andSadeghpour, 2003) and experimentally (Köhl et al., 2005; Rom et al., 2004;Ryu et al., 2005; Stöferle et al., 2005). In particular, the experiment of Ref. (Ryuet al., 2005) observed reversible and coherent Rabi oscillations in a gas of coupledatoms and molecules.

The idea of the molecular micromaser (Search et al., 2003) relies on the obser-vation that, as a consequence of Fermi statistics, the photo-association of fermi-onic atoms into bosonic molecules can be mapped onto a generalized Jaynes–Cummings model. This analogy allows one to immediately translate many ofthe results that have been obtained for the Jaynes–Cummings model to atom–molecule systems. In addition, the molecular system possesses several propertiesthat have no counterpart in the quantum optics analog, giving rise to interest-ing generalizations of the original micromaser problem (Filipowicz et al., 1986;Guzman et al., 1989; Meschede et al., 1985; Rempe et al., 1990). One of thesenew features is the inter-site tunneling of atoms and molecules between adjacentlattice sites, leading to a system that can be thought of as an array of molecularmicromasers (Search et al., 2004).

To see how this works, rather than treating a full lattice potential we con-sider the dynamics of the molecular field in the simpler model of a coupledatom–molecule system in a double-well potential. We first show that inter-welltunneling enhances number fluctuations and eliminates trapping states in a man-ner similar to thermal fluctuations. We also examine the buildup of the relativephase between the two molecular states localized at the two wells due to the com-bined effect of inter-well tunneling and two-body collisions. We identify threeregimes, characterized by different orders of magnitude of the ratio of the two-body collision strength to the inter-well tunneling coupling. The crossover of thenon-equilibrium steady state from a phase-coherent regime to a phase-incoherentregime is closely related to the phase locking of condensates in Josephson-typeconfigurations (Leggett, 2001), while we consider an open quantum system withincoherent pump and molecular loss which results in a dissipative steady state.

2.1. MODEL

We consider a mixture of two hyperfine spin states |σ =↑,↓〉 of fermionic atomsof massmf trapped in a double-well potential at temperature T = 0, which can be


coherently combined into bosonic molecules of mass mb via two-photon Ramanphoto-association. If the band-gap of the lattice potential is much larger than anyother energy scale in the system, the fermions and molecules occupy only thelowest energy level of each well and the number of fermions of a given spin stateis at most one in each well.

In the tight binding approximation, the effective Hamiltonian describing thecoupled atom–molecule system is

(1)H =∑i=l,r

(H0i + HI i

)+ HT ,

where

H0i = h(ωb + δ)ni + hωf (n↑i + n↓i )+ 1

2hUbni(ni − 1)

(2)+ hUxni(n↑i + n↓i )+ hUf n↑i n↓i ,

(3)HI i = hχ(t)b†i c↑i c↓i + H.c.,

(4)HT = −hJbb†l br − hJf

(c

†↑l c↑r + c

†↓l c↓r)+ H.c.

Here cσ i and bi , i = l, r , are the annihilation operators of fermionic atoms andbosonic molecules in the left (l) and right (r) wells, respectively. The correspond-ing number operators ni = b

†i bi and nσ i = c

†σ i cσ i have eigenvalues ni and nσi ,

respectively, and hωb and hωf are the energies of the molecules and atoms in theisolated wells.

The terms proportional to Ub, Ux , and Uf in H0i describe on-site two-body in-teractions between molecules, between atoms and molecules, and between atoms,respectively. The interaction Hamiltonian HI i describes the conversion of atomsinto molecules via two-photon Raman photo-association. The photo-associationcoupling constant χ(t) is proportional to the far off-resonant two-photon Rabi fre-quency associated with two nearly co-propagating lasers (Heinzen et al., 2000),and δ is the two-photon detuning between the lasers and the energy difference ofthe atom pairs and the molecules. The tunneling between two wells is describedby the parameters Jb and Jf in the tunneling Hamiltonian HT .

The molecular field is “pumped” by a train of short photo-association pulses ofduration τ , separated by long intervals T � τ during which the molecules aresubject only to two-body collisions and quantum tunneling between the potentialwells, as well as to losses due mainly to three-body collisions and collisional re-laxation to low-lying vibrational states. In the absence of inter-well tunneling, thisseparation of time scales leads to a situation very similar to that encountered inthe description of traditional micromasers, with the transit of individual two-levelatoms through the micromaser cavity replaced by the train of photo-associationpulses.


The dynamics of the molecular field in the double-well system is governed bythe following four mechanisms:

(i) Coherent pumping by injection of pairs of fermionic atoms inside thedouble-well. This process is the analog of the injection of two-level atoms intoa micromaser cavity. The injection of pairs of fermionic atoms into the double-well potential can be accomplished, e.g., by Raman transfer of atoms from anuntrapped internal state (Jaksch et al., 1998; Mandel et al., 2004). This results inthe pumping of fermions into the double well at a rate Γ (Search et al., 2002a).We assume that for times T � Γ −1, a pair of fermions has been transferred to thetwo wells with unit probability, that is, the state of the trapped fermions in well iis

(5)|ei〉 = c†↓i c

†↑i |0〉.

(ii) Molecular damping, which is the analog of cavity damping. During the timeintervals T when the photo-association lasers are off, the molecular field decaysat rate γ (Search et al., 2003). The decay of the molecules is due to Rayleighscattering from the intermediate molecular excited state, three-body inelastic col-lisions between a molecule and two fermions, and collisional relaxation from avibrationally excited molecular state to deeply bound states. These loss mecha-nisms can be modeled by a master equation (see, e.g., Meystre and Sargent III,1999; Miyakawa et al., 2004; Scully and Zubairy, 1997).

(iii) The application of a train of photo-association pulses. This mechanism isformally analogous to the Jaynes–Cummings interaction between the single-modefield and a sequence of two-level atoms traveling through the microwave cavityin the conventional micromaser. As already mentioned, we assume that these aresquare pulses of duration τ and period T + τ , with τ much shorter than all othertime scales in this model, τ � J−1

b,f , γ−1. This assumption is essential if we are

to neglect damping and tunneling while the photo-association fields are on. Thechange in the molecular field resulting from atom–molecule conversion is givenby

(6)Fi(τ )ρb ≡ Tra[Ui(τ )ρab(t)U

†i (τ )],

where ρab is the total density operator of the atom–molecule system and Tra[ ]denotes the trace over the atomic variables, Ui(τ ) = exp (−ihiτ/h) being theevolution operator for a single-well Hamiltonian,

hi = H0i + HI i .

The key observation that allows us here and below to build a bridge from the coldatoms and molecular system to quantum optics systems is that by means of themapping (Anderson, 1958)

σ−i = c↑ic↓i ,


σ+i = c†↓ic

†↑i ,

(7)σzi = c†↑ic↑i + c

†↓ic↓i − 1,

the atomic degrees of freedom take the form of a fictitious two-level system. Theoperators σ+i , σ−i , and σzi can be interpreted as the raising and lowering oper-ators of the fictitious two-level atom and the population difference, respectively.The single-well Hamiltonian takes the form

hi = h (ωb + Ux) ni + h(ωf + Uxni)σzi

(8)+ h(χ(t)b

†i σ−i + χ∗(t)bi σ+i

)+ h

2Ubni(ni − 1),

where we have dropped constant terms and we have redefined ωb and ωf accord-ing to ωb + δ → ωb and ωf + Uf /2 → ωf .

The Hamiltonian hi is Jaynes–Cummings-like, and for χ = const, the re-sulting dynamics can be determined within the two-state manifolds of each well{|ei, ni〉, |gi, ni+1〉} by a simple extension of the familiar solution to the Jaynes–Cummings model. Within each manifold the system undergoes Rabi oscillations.

Since tunneling is neglected during the photo-association steps, the two wellsare independent of each other and identical to each other. The resultant moleculargain is then modeled by independent coarse-grained equations of motion for thereduced density matrices of each molecular mode.

(iv) The unitary time evolution of the molecular field under the influence oftwo-body collisions and quantum tunneling, a process absent in conventional mi-cromasers. During the intervals T it is governed by

(9)∂ρb

∂t= − i

h

[Hb, ρb

],

where the Hamiltonian

(10)Hb = −hJb(b

†l br + b†

r bl)+ h

Ub

4(nl − nr )

2

contains tunneling and collisions. In Eq. (10), we have neglected terms that arefunctions only of N = nl + nr , a step justified as long as the initial density matrixis diagonal in the total number of molecules in the two wells.

Combining the coherent and incoherent processes (i) to (iv), we obtain the fullevolution of the molecular field

(11)∂ρb

∂t=∑l,r

Li ρb + 1

T

∑l,r

[Fi(τ )− Ii

]ρb − i

h[Hb, ρb],

where ρb is the reduced density matrix of the molecules. The initial condition forthe molecules is taken to be the vacuum state. Because the molecular pumpingand decay is the same in both wells, the density matrix ρ remains diagonal in the


total number of molecules in the two wells for all times. This is a generalizationof the micromaser result that the photon density matrix will remain diagonal if itis initially diagonal in a number state basis (Filipowicz et al., 1986).

2.2. RESULTS

The master equation describing the molecular micromaser dynamics contains sixindependent parameters: the number of photo-association cycles per lifetime ofthe molecule,Nex = 1/γ T ; the “pump parameter”Θ = √

Nex |χ |τ ; the two-bodycollision strength and tunneling coupling strength per decay rate, ub = Ub/γ andtJ = Jb/γ ; and finally, the detuning parameter η = (2ωf − ωb)/2|χ | and thenonlinear detuning parameter β ≡ (2Ux − Ub)/2|χ |.

In our model, the atomic and molecular level separations in the wells are re-quired to be much larger than the relevant interaction energies,

(12)hωb � Ub〈ni〉(〈ni〉 − 1

), |χ |√〈ni〉,

〈ni〉 being the average number of molecules in well i. A comparison with actualexperimental parameters (Greiner et al., 2002a; Jaksch et al., 1998; Miyakawaet al., 2004) shows that these conditions are satisfied as long as the number ofmolecules does not exceed 10. In addition, the neglect of inter-well tunneling anddamping effects during the photo-association pulses requires that

(13)τ � J−1b , γ−1.

This condition is satisfied in typical experiments.In the remainder of this section, we discuss the dynamics of the molecular field

obtained by direct numerical integration of the master equation with a Runge–Kutta algorithm until a dissipative steady state is reached. For simplicity, weconfine our discussion to exact resonance only, η = β = 0, and a fixed valueof Nex = 10.

2.2.1. Single-Well Molecular Statistics

We first discuss the statistics of a single-well molecular mode, which is given bytracing over the full density matrix with respect to degrees of freedom of the otherlocalized mode as

(14)P(nl{r}) = Trr{l}[ρ(nl, nr ;ml,mr)

] =∑nr{l}

ρ(nl, nr ; nl, nr).

We note that off-diagonal elements of the density matrix for a single well are zero.Since the initial state of the molecules in each well is the same, i.e. the vacuumstate, and Hb is invariant with respect to the interchange l ↔ r , the moleculestatistics for left and right wells are identical, P(nl) = P(nr) ≡ P(n).


FIG. 1. 〈ni 〉 versus Θ/π for ub = 0 and Nex = 10, and for (a) tJ = 0 and (b) tJ = 5.

Figure 1 shows the steady-state average number 〈ni〉 plotted as a function ofthe pump parameter Θ for ub = 0, Nex = 10. In the absence of inter-welltunneling, corresponding to Fig. 1(a), the result reproduces that of conventionalmicromasers, with a “lasing” threshold behavior at around Θ ≈ 1 and an abruptjump to a higher mean occupation at about the first transition point, Θ � 2π .The former effect is not affected by the tunneling coupling as shown in Fig. 1(b).However, the latter abrupt jump disappears in the presence of inter-well tunneling.This is because the coupling to the other well leads to fluctuations in the numberof molecules in each well and has an effect similar to thermal fluctuations in thetraditional micromaser theory. The enhancement of fluctuations can also be seenin Fig. 2 where the Mandel Q-parameter

Q = 〈n2i 〉 − 〈ni〉2

〈ni〉 − 1

is plotted as a function of Θ .It is known that in the usual micromaser the sharp resonance-like dips in 〈ni〉

and Q are attributable to trapping states, which are characterized by a sharp pho-ton number. For the specific value of Θ = √

5π , as shown in Fig. 3(a), thenumber probability does not reach beyond number state |ni = 1〉 in the case oftJ = 0. As shown in Fig. 3(b), the tunneling coupling makes possible transitionsinto higher number states and eliminates the trapping state in a manner similar tothermal fluctuations in the conventional micro maser.


FIG. 2. Q parameter versus Θ/π for ub = 0 and Nex = 10, and for (a, solid line) tJ = 0 and(b, dashed line) tJ = 5.

FIG. 3. Molecular number statistics P(ni ) for Θ = √5π , ub = 0 and Nex, and for (a) tJ = 0

and (b) tJ = 5.


2.2.2. Phase Coherence between Two Micromasers with Tunneling Coupling

So far we have discussed the single-well molecule statistics and how it is af-fected by inter-well tunneling. Now we turn to a more detailed discussion ofthe phase coherence between the two localized modes. It is very useful to di-vide the parameter space of the ratio of the two-body collision strength to theinter-well tunneling coupling into three regimes (Leggett, 2001): “Rabi-regime”(ub/tJ � 〈N〉−1); “Josephson-regime” (〈N〉−1 � ub/tJ � 〈N〉); and “Fock-regime” (〈N〉 � ub/tJ ); where 〈N〉 denotes the average total molecule number.

The analysis of the relative coherence of the molecular fields in the two wellsis most conveniently discussed in terms of the angular momentum representation

J+ = Jx + iJy = b†l br ,

J− = Jx − iJy = b†r bl ,

Jz = 1

2

(b

†l bl − b†

r br),

(15)J 2 = N

2

(N

2+ 1

).

The symmetry of the density matrix with respect to the two wells furthermoreimplies that 〈Jz〉 = 〈Jy〉 = 0. The first-order coherence between the molecularfields in the left and right potential wells is then given by 〈Jx〉. Figure 4 shows thenormalized steady-state first-order coherence 〈Jx〉/〈nj 〉 as a function of ub/tJ forΘ = π and tJ = 2.5. 〈Jx〉 is suppressed in both the Rabi and Fock regimes andhas an extremum at |ub|/tJ ∼ 0.6. In the Fock regime, |ub|/tJ � 〈N〉 ∼ 10, the

FIG. 4. 〈Jx 〉/〈nj 〉 versus ub/tJ for Θ = π and tJ = 2.5.


non-linearity in Hb dominates and reduces the coherence between the localizedstates of each well. We note that the average occupation numbers for each wellare relatively unaffected by ub/tJ , with 〈nj 〉 = 〈N〉/2 = 4.78–4.87 for |ub|/tJ =102–10−2.5.

The reason why the first-order coherence is suppressed in the weak couplinglimit, ub = 0, can be understood as follows. The expectation value 〈Jx〉 cor-responds to the difference in occupation numbers between the in-phase, bs =(bl + br )/

√2, and out-of-phase, ba = (bl − br )/

√2, states of the localized states

of each well, Jx = b†s bs − b

†aba . Since the bandwidth of the photo-association

pulse is larger than their energy splitting, 1/τ � Jb, those states are equally pop-ulated, resulting in 〈Jx〉 = 0 for ub = 0. Thus, the origin of the mutual coherencebetween two molecular modes is due solely to two-body collisions. Furthermore,we remark that a semiclassical treatment results in 〈Jx〉 = 0 for all times and allvalues of ub/tJ (Miyakawa et al., 2004). Hence, we conclude that the build-up of〈Jx〉 is a purely quantum-mechanical effect due to quantum fluctuations.

The phase distribution of the two wells can be studied using the Pegg–Barnett phase states (Barnett and Pegg, 1990; Javanainen and Ivanov, 1999;Luis and Sanchez-Soto, 1993; Pegg and Barnett, 1988). Since the density matrixis diagonal in the total number of molecules it is sufficient to consider the relativephase. Figure 5 shows the time evolution of the relative phase distribution in threedifferent regimes: (a) Rabi, ub/tJ = 0.0032, (b) Josephson, ub/tJ = 0.5623,and (c) Fock ub/tJ = 56.23, for Θ = π , tJ = 2.5. Since the vacuum stateis taken as the initial state, the relative phase at t = 0 is randomly distributed,P(φn) = const.

In the Rabi regime, corresponding to Fig. 5(a), bimodal phase distribution withpeaks around both 0 and ±π builds up in the characteristic time γ−1 needed toreach a steady state (Filipowicz et al., 1986). In the Josephson regime, the relativephase locks around 0 (±π), for repulsive (attractive) two-body interactions, seeFig. 5(b). In contrast to these two regimes, in the Fock regime the relative phasedistribution becomes almost random for all times, and the localized modes in thetwo wells evolve independently of each other.

FIG. 5. Time evolution of P(φn) for Θ = π , tJ = 2.5 and for (a) ub/tJ = 0.0032,(b) ub/tJ = 0.5623, (c) ub/tJ = 56.23.


The three regimes of phase distributions correspond to different orders of mag-nitude of the ratio ub/tJ . The crossover of the non-equilibrium steady state froma phase-coherent regime to the random-phase situation is reminiscent of thesuperfluid-Mott insulator phase transition for the ground state of an optical lat-tice (Fisher et al., 1989; Jaksch et al., 1998). Since we consider just two sites,however, there is no sharp transition between these regimes.

3. Passage Time Statistics of Molecule Formation

We now turn to a second-example that illustrates the understanding of the dynam-ics of quantum-degenerate atomic and molecular systems that can be gained fromquantum optics analogies. Here, we consider the first stages of coherent molecularformation via photo-association. Since in such experiments the molecular field istypically in a vacuum initially, it is to be intuitively expected that the initial stagesof molecule formation will be strongly governed by quantum noise, hence sub-ject to large fluctuations. One important way to characterize these fluctuations isin terms of the so-called passage time, which is the time it takes to produce, ordissociate, a predetermined number of molecules. Quantum noise results in fluc-tuations in that time, whose probability distribution can therefore be used to probethe fluctuations in the formation dynamics.

Because of the analogy between pairs of fermionic atoms and two-level systemsthat we already exploited in the discussion of the molecular micromaser, one canexpect that the problem at hand is somewhat analogous to spontaneous radiationfrom a sample of two-level atoms, the well-know problem of superradiance. Inthis section we show that this is indeed the case, and use this analogy to study thepassage time statistics of molecular formation from fermionic atoms.

We consider again a quantum-degenerate gas of fermionic atoms of mass mf

and spin σ =↑,↓, coupled coherently to bosonic molecules of mass mb = 2mf

and zero momentum via photo-association. Neglecting collisions between fermi-ons and assuming that for short enough times the molecules occupy a single-modeof the bosonic field, this system can be described by the boson–fermion modelHamiltonian

H =∑k

1

2hωk(c

†k↑ck↑ + c

†−k↓c−k↓

)(16)+ hωbb

†b + hχ∑k

(b†ck↑c−k↓ + bc

†−k↑c

†k↓),

where b†, b are molecular bosonic creation and annihilation operators andc

†kσ , ckσ are fermionic creation and annihilation operators describing atoms of

momentum hk and spin σ . The first and second terms in Eq. (16) describe the


kinetic energy hωk/2 = h2k2/(2mf ) of the atoms and the detuning energy of themolecules respectively, and the third term describes the photo-association of pairsof atoms of opposite momentum into molecules.

Introducing the pseudo-spin operators (Anderson, 1958) analogous to Eq. (7),

σ zk = 1

2

(c

†k↑ck↑ + c

†−k↓c−k↓ − 1

),

(17)σ+k = (σ−

k )† = c

†−k↓c

†k↑,

the Hamiltonian (16) becomes, within an unimportant constant (Barankov andLevitov, 2004; Meiser and Meystre, 2005),

(18)H =∑k

hωkσzk + hωbb

†b + hχ∑k

(b†σ−

k + bσ+k

).

This Hamiltonian is known in quantum optics as the inhomogeneously broad-ened (or non-degenerate) Tavis–Cummings model (Tavis and Cummings, 1968).It describes the coupling of an ensemble of two-level atoms to a single-modeelectromagnetic field. Hence the mapping (17) establishes the formal analogy be-tween the problem at hand and Dicke superradiance, with the caveat that we aredealing with a single bosonic mode (Andreev et al., 2004; Barankov and Levitov,2004; Javanainen et al., 2004; Meiser and Meystre, 2005; Miyakawa and Meystre,2005; Pazy et al., 2005). Instead of real two-level atoms, pairs of fermionic atomsare now described as effective two-level systems whose ground state correspondsto the absence of a pair, |gk〉 = |0k↑, 0−k↓〉 and the excited state to a pair of atomsof opposite momenta, |ek〉 = |1k↑, 1−k↓〉, in close analogy to the treatment of theatoms in the previous section.

The initial condition consists of the molecular field in the vacuum state and afilled Fermi sea of atoms

(19)|F〉 =∏

k�|kF|σ+k |0〉,

where kF is the Fermi momentum. As such, the problem at hand is in direct anal-ogy to the traditional superradiance problem where one starts from an ensembleof excited two-state atoms, as expected from our previous comments. Later on wewill also consider an initial state containing only molecules and no atoms. Thisis an important extension of the traditional Dicke superradiance system, wherethe two-level atoms are coupled to all modes of the photon vacuum a situation,thereby precluding the possibility of an initial state containing a single, macro-scopically occupied photon mode unless the system us prepared in a high-Qcavity.

We assume from now on that the inhomogeneous broadening due to the spreadin atomic kinetic energies can be ignored. This so-called degenerate approxi-mation is justified provided that the kinetic energies are small compared to the


atom–molecule coupling energy, β = εF/(hχ) � 1, where εF is the Fermi en-ergy. It is the analog of the homogeneous broadening limit of quantum optics,and of the Raman–Nath approximation in atomic diffraction. A comparison withtypical experimental parameters (Heinzen et al., 2000) shows that the degenerateapproximation is justified if the number of atoms does not exceed ∼102–103 (Uyset al., 2005).

Limiting thus our considerations to small atomic samples, we approximate allωk’s by ωF and introduce the collective pseudo-spin operators

(20)Sz =∑k

σ zk , S± =∑k

σ±k ,

obtaining the standard Tavis–Cummings Hamiltonian (Miyakawa and Meystre,2005; Tavis and Cummings, 1968)

(21)H = hωFSz + hωbb†b + hχ

(bS+ + b†S−).

This Hamiltonian conserves the total spin operator S2. The total number of atoms

is twice the total spin and hence is also a conserved quantity. Sz measures thedifference in the numbers of atom pairs and molecules.

Equation (21) can be diagonalized numerically with reasonable computationtimes even for relatively large numbers of atoms. One can, however, gain signif-icant intuitive insight in the underlying dynamics by finding operator equationsof motion and then treating the short-time molecular population semiclassically,〈nb〉 → nb. To this end we introduce the “joint coherence” operators

Tx = (bS+ + b†S−)/2,

(22)Ty = (bS+ − b†S−)/2i,

and find the Heisenberg equations of motion

(23)˙nb = −2χTy,

(24)˙T x = δTy,

(25)˙T y = −δTx − χ

(2Sznb + S+S−),

where δ = ωb − ωF , so that 2χTx + δnb is a constant of motion.In the following, we confine our discussion to the case of δ = 0 for simplicity.

We thus neglect the contribution of Tx in Eq. (25). In order to better understandthe short time dynamics we reexpress S+S− as

(26)S+S− = −n2b + (2S − 1)nb +N.

This shows that the operator S+S− is non-vanishing when the molecular field is ina vacuum and hence can be interpreted as a noise operator. Indeed Eqs. (23)–(25)


FIG. 6. Short-time dynamics of 〈nb〉. From left to right, the curves give the linearized solution(28) and the full quantum results for N = 500, N = 250, and N = 100, respectively. Figure takenfrom Ref. (Uys et al., 2005).

show that the buildup of the molecular field is triggered only by noise if nb = 0initially. By keeping only the lowest-order terms in nb we can eliminate Ty toobtain the differential equation

(27)¨nb ≈ 2Nχ2(2nb + 1)

which, for our initial state, may be solved to yield

(28)⟨nb(t)⟩ ≈ sinh2(χ√N t

).

Figure 6 compares the average molecule number 〈nb〉 obtained this way, with thefull quantum solution obtained by direct diagonalization of the Hamiltonian (21)for various values of N . The semiclassical approach agrees within 5% of the fullquantum solution until about 20% of the population of atom pairs has been con-verted into molecules in all cases.

Next we turn to the passage time statistics. In Fig. 7 we show (solid line)the distribution of times required to produce a normalized molecule numbernrefb /N = 0.05 from a sample initially containing N = 500 pairs of atomic fermi-

ons, as found by direct diagonalization of the Hamiltonian (21). This distributiondiffers sharply from its counterpart for the reverse process of photodissociationfrom a molecular condensate into fermionic atom pairs, which is plotted as thedashed line in Fig. 7. In contrast to photo-association, this latter process sufferssignificantly reduced fluctuations.

To understand the physical mechanism leading to this reduction in fluctuationswe again turn to our short time semi-classical model. Within this approximation,the Heisenberg equations of motion (23)–(24) can be recast in the form of a New-tonian equation (Miyakawa and Meystre, 2005)

(29)d2nb

dt2= −dU(nb)

dnb,


FIG. 7. Passage time distribution for converting 5% of the initial population consisting of onlyatoms (molecules) into molecules (atoms) for N = 500. For initially all atoms: solid line, for initiallyall molecules: dashed line.

FIG. 8. Effective potential for a system with N � 1. The circle (square) corresponds to an initialstate with all fermionic atoms (molecules). The part of the potential for nb < 0 is unphysical. Figuretaken from Ref. (Uys et al., 2005).

where the cubic effective potential U(nb) is plotted in Fig. 8. (Note we have nowkept all orders in nb.) In case the system is initially composed solely of fermi-onic atoms, nb(0) = 0, the initial state is dynamically unstable, with fluctuationshaving a large impact on the build-up of nb. In contrast, when it consists initiallysolely of molecules, nb = N , the initial state is far from the point of unstableequilibrium, and nb simply “rolls down” the potential in a manner largely insen-sitive to quantum fluctuations. This is a consequence of the fact that the bosonicinitial state provides a mean field that is more amenable to a classical description.Hence, while the early stages of molecular dimer formation from fermionic atomsare characterized by large fluctuations in formation times that reflect the quantumfluctuations in the initial atomic state, the reverse process of dissociation of a con-densate of molecular dimers is largely deterministic. The diminished fluctuations


in this reversed process is peculiar to the atom–molecule system and not normallyconsidered in the quantum optics analog of Dicke superradiance.

4. Counting Statistics of Molecular Fields

An important quantum mechanical characteristic of a quantum field is its counting(or number) statistics. In this section we show how the similarity of the coherentmolecule formation with quantum optical sum-frequency generation can be usedto determine the counting statistics of the molecular field. In particular we showhow the counting statistics depends on the statistics of the atoms from which themolecules are formed. Besides being interesting in its own right, such an analysisis crucial for an understanding of several recent experiments that used a “projec-tion” onto molecules to detect BCS superfluidity in fermionic systems (Regal etal., 2004; Zwierlein et al., 2004). Our work shows that the statistical propertiesof the resulting molecular field indeed reflect properties of the initial atomic stateand are a sensitive probe for superfluidity.

As before, we restrict our discussion to a simple model in which all the mole-cules are generated in a single mode. We use time dependent perturbation theoryto calculate the number of molecules formed after some time t , n(t), as well asthe equal-time second-order correlation function g(2)(t, t). We also integrate theSchrödinger equation numerically for small numbers of atoms, which allows usto calculate the complete counting statistics Pn.

4.1. BEC

Consider first a cloud of weakly interacting bosons well below the condensationtemperature Tc. It is a good approximation to assume that all atoms are in thecondensate, described by the condensate wave function ψ0(x). The coupled sys-tem of atoms and molecules is described by the effective two-mode Hamiltonian(Anglin and Vardi, 2001; Javanainen and Mackie, 1999)

(30)HBEC = hδb†b + hχ(b†c2 + bc†2),

where b, b† and c, c† are the bosonic annihilation and creation operators for themolecules and for the atoms in the condensate, respectively, δ is the detuningbetween the molecular and atomic level, and hχ is the effective coupling constant.

Typical experiments start out with all atoms in the condensate and no mole-cules, corresponding to the initial state,

(31)|ψ(t = 0)〉 = c†Na

√Na! |0〉,


FIG. 9. Number statistics of molecules formed from a BEC with Nmax = 30 and δ = 0.

where Na = 2Nmax is the number of atoms, Nmax is the maximum possible num-ber of molecules and |0〉 is the vacuum of both molecules and atoms. We cannumerically solve the Schrödinger equation for this problem in a number basisand from that solution we can determine the molecule statistics Pn(t). The resultsof such a simulation are illustrated in Fig. 9, which shows Pn(t) for 30 initialatom pairs and δ = 0. Starting in the state with zero molecules, a wave-packet-like structure forms and propagates in the direction of increasing n. Near Nmaxthe molecules begin to dissociate back into atom pairs.

We can gain some analytical insight into the short-time dynamics of moleculeformation by using first-order perturbation theory (Kozierowski and Tanas, 1977;Mandel, 1982). We find for the mean molecule number

(32)n(t) = (χt)22Nmax(2Nmax − 1)+O((χt)2)

and for the second factorial moment

(33)g(2)(t1, t2) = 1 − 2

Nmax+O(N−2

max

).

For Nmax large enough we have g(2)(t1, t2) → 1, the value characteristic of aGlauber coherent field. From g(2) and n(t) we also find the relative width of the


molecule number distribution as

(34)

√〈(n− n)2〉n

=√g(2) + n−1 − 1.

It approaches n−1/2 in the limit of large Nmax, typical of a Poisson distribution.This confirms that for short enough times, the molecular field is coherent in thesense of quantum optics.

4.2. NORMAL FERMI GAS

We now turn to the case of photo-association from two different species of non-interacting ultra-cold fermions. The two species are again denoted by spin up anddown. At T = 0, the atoms fill a Fermi sea up to an energy μ. Weak repulsiveinteractions give rise only to minor quantitative modifications that we ignore. Werefer to this system of non-interacting Fermions as a normal Fermi gas (NFG)(Landau et al., 1980).

As before we assume that atom pairs are coupled only to a single mode ofthe molecular field, which we assume to have zero momentum for simplicity.Then, using the mapping to pseudo spins Eq. (17) we find that the system isagain described by the inhomogeneously broadened Tavis–Cummings Hamil-tonian Eq. (18). However, in contrast to the previous case, we do not assume thatthe fermionic energies are approximately degenerate, in order to be able comparethe results to the BCS case, where the kinetic energies are essential.

Figure 10 shows the molecule statistics obtained this way. The result is clearlyboth qualitatively and quantitatively very different from the case of moleculeformation from an atomic BEC. From the Tavis–Cummings model analogy weexpect that for short times the statistics of the molecular field should be chaotic,or “thermal”, much like those of a single-mode chaotic light field. This is be-cause each individual atom pair “emits” a molecule independently and withoutany phase relation with other pairs. That this is the case is illustrated in the insetof Fig. 10, which fits the molecule statistics at selected short times with chaoticdistributions of the form

(35)Pn,thermal = e−n/〈n〉∑n e

−n/〈n〉 .

The increasing ‘pseudo-temperature’ 〈n〉 corresponds to the growing averagenumber of molecules as a function of time.

As before we determine the short-time properties of the molecular field in first-order perturbation theory. We find for the mean number of molecules

(36)n(t) = (χt)22Na.


FIG. 10. Number statistics of molecules formed from a normal Fermi gas. This simulation is forNa = 20 atoms, the detuning is δ = 0, the Fermi energy is μ = 0.1hχ and the momentum of theith pair is |ki | = (i − 1)2kF /(Na/2 − 1). The inset shows fits of the number statistics to thermaldistributions for various times as marked by the thick lines in the main figure.

It is proportional to Na , in contrast to the BEC result, where n was proportionalto N2

a , see Eq. (32). This is another manifestation of the independence of all theatom pairs from each other: While in the BEC case the molecule production is acollective effect with contributions from all possible atom pairs adding construc-tively, there is no such collective enhancement in the case of Fermions. Each atomcan pair up with only one other atom to form a molecule. For the second factorialmoment we find

(37)g(2)(t1, t2) = 2

(1 − 1

2Na

)which is close to two, typical of a chaotic or thermal field.

4.3. FERMI GAS WITH SUPERFLUID COMPONENT

Unlike repulsive interactions, attractive interactions between fermions have a pro-found impact on molecule formation. It is known that such interactions give rise


FIG. 11. Number statistics of molecules formed from a Fermi gas with pairing correlations. Forthis simulation the detuning is δ = 0, the Fermi energy is μ = 0.1g and the background scatteringstrength is V = 0.03χ resulting in Na ≈ 9.4 atoms and a gap of � ≈ 0.15χ . The momenta of theatom pairs are distributed as before in the normal Fermi gas case.

to a Cooper instability that leads to pairing and drastically changes the qualitativeproperties of the atomic system. The BCS reduced Hamiltonian is essentially theinhomogeneously broadened Tavis–Cummings Hamiltonian (18) with an addi-tional term accounting for the attractive interactions between atoms (Kittel, 1987),

(38)H =∑k

hωkσzk + hωbb

†b + hχ∑k

(b†σ−

k + bσ+k

)− V∑k,k′

σ+k σ

−k′ .

The approximate mean-field ground state |BCS〉 is found by minimizing 〈HBCS −μN〉 in the standard way. The dynamics is then obtained by numerically inte-grating the Schrödinger equation with |BCS〉 as the initial atomic state and themolecular field in the vacuum state.

Figure 11 shows the resulting molecule statistics for V = 0.03hχ , which cor-responds to a gap of � = 0.15hχ for the system at hand. Clearly, the moleculeproduction is much more efficient than it was in the case of a normal Fermi gas.The molecules are produced at a higher rate and the maximum number of mole-cules is larger. The evolution of the number statistics is reminiscent of the BEC


case. This also shows that the qualitative differences seen between the normalFermi gas and a BEC in the previous section cannot be attributed to inhomoge-neous broadening and the resulting dephasing alone but are instead a result of thedifferent coherence properties of the atoms.

The short-time dynamics is again obtained in first-order perturbation theory,which gives now

(39)n(t) ≈ (χt)2[(

�

V

)2

+Na

].

In addition to the term proportional to Na representing the incoherent contributionfrom the individual atom pairs that was already present in the normal Fermi gas,there is now an additional contribution proportional to (�/V )2. Since (�/V ) canbe interpreted as the number of Cooper pairs in the quantum-degenerate Fermigas, this term can be understood as resulting from the coherent conversion ofCooper pairs into molecules in a collective fashion similar to the BEC case. Thecoherent contribution results naturally from the non-linear coupling of the atomicfield to the molecular field. This non-linear coupling links higher-order correla-tions of the molecular field to lower-order correlations of the atomic field. Forthe parameters of Fig. 11 �/V ≈ 6.5 so that the coherent contribution from theCooper pairs clearly dominates over the incoherent contribution from the unpairedfermions. Note that no signature of that term can be found in the momentum dis-tribution of the atoms themselves. Their momentum distribution is very similar tothat of a normal Fermi gas. The short-time value of g(2)(t1, t2), shown in Fig. 12,decreases from the value of Eq. (37) for a normal Fermi gas at � = 0 down to oneas � increases, underlining the transition from incoherent to coherent moleculeproduction.

5. Molecules as Probes of Spatial Correlations

The single-mode description of the molecular field of the previous section resultsin the loss of all information about the spatial structure of the atomic state. Inthis final section we adopt a complementary view and study the coupled atom–molecule system including all modes of the molecular and atomic field so as toresolve their spatial structure. This problem is too complex to admit an exactsolution, hence we rely entirely on perturbation theory.

One of the motivations for such studies are the on-going experimental effortsto study the so-called BEC-BCS crossover. A difficulty of these studies has beenthat they necessitate the measurement of higher-order correlations of the atomicsystem. While the momentum distribution of a gas of bosons provides a clearsignature of the presence of a Bose–Einstein condensate, the Cooper pairing be-tween fermionic atoms in a BCS state hardly changes the momentum distribution


FIG. 12. g(2)(0+, 0+) as a function of the gap parameter �. (Figure taken from Ref. (Meiser andMeystre, 2005).)

or spatial profile as compared to a normal Fermi gas. This poses a significantexperimental challenge, since the primary techniques for probing the state of anultra-cold gas are either optical absorption or phase contrast imaging, which di-rectly measure the spatial density or momentum distribution following ballisticexpansion of the gas. In the strongly interacting regime very close to the Feshbachresonance, evidence for fermionic superfluidity was obtained by projecting theatom pairs onto a molecular state by a rapid sweep through the resonance (Regalet al., 2004; Zwierlein et al., 2004). More direct evidence of the gap in the excita-tion spectra due to pairing was obtained by rf spectroscopy (Chin et al., 2004) andby measurements of the collective excitation frequencies (Bartenstein et al., 2004;Kinast et al., 2004). Finally, the superfluidity of ultra-cold fermions in the stronglyinteracting regime has recently been impressively demonstrated via the generationof atomic vortices (Zwierlein et al., 2005).

Still, the detection of fermionic superfluidity in the weakly interacting BCSregime remains a challenge. The direct detection of Cooper pairing requires themeasurement of second-order or higher atomic correlation functions. Several re-searchers have proposed and implemented schemes that allow one to measure


higher-order correlations (Altman et al., 2004; Bach and Rzazewski, 2004; Burtet al., 1997; Cacciapuoti et al., 2003; Hellweg et al., 2003; Regal et al., 2004) butthose methods are still very difficult to realize experimentally. While the measure-ment of higher-order correlations is challenging already for bosons, the theory ofthese correlations has been established a long time ago by Glauber for photons(Glauber, 1963a, 1963b; Naraschewski and Glauber, 1999). For fermions how-ever, despite some efforts (Cahill and Glauber, 1999) a satisfactory coherencetheory is still missing.

From the previous section we know that one can circumvent these difficul-ties by making use of the non-linear coupling of atoms to a molecular field.The non-linearity of the coupling links first-order correlations of the moleculesto second-order correlations of the atoms. Furthermore the molecules are alwaysbosonic so that the well-known coherence theory for bosonic fields can be usedto characterize them. Considering a simplified model with only one molecularmode, it was found that the molecules created that way can indeed be used as adiagnostic tool for second-order correlations of the original atomic field.

We consider the limiting case of strong atom–molecule coupling as comparedto the relevant atomic energies. The molecule formation from a Bose–Einsteincondensate (BEC) serves as a reference system. There we can rather easily studythe contributions to the molecular signal from the condensed fraction as well asfrom thermal and quantum fluctuations above the condensate. The cases of a nor-mal Fermi gas and a BCS superfluid Fermi system are then compared with it.We show that the molecule formation from a normal Fermi gas and from the un-paired fraction of atoms in a BCS state has very similar properties to those of themolecule formation from the non-condensed atoms in the BEC case. The state ofthe molecular field formed from the pairing field in the BCS state on the otherhand is similar to that resulting from the condensed fraction in the BEC case. Thequalitative information gained by the analogies with the BEC case help us gain aphysical understanding of the molecule formation in the BCS case where directcalculations are difficult and not nearly as transparent.

5.1. MODEL

We consider again the three cases where the atoms are bosonic and initially form aBEC, or consist of two species of ultra-cold fermions (labeled again by σ =↑,↓),with or without superfluid component. In the following we describe explicitly thesituation for fermions, the bosonic case being obtained from it by omitting thespin indices and by replacing the Fermi field operators by bosonic field operators.

Since we are primarily interested in how much can be learned about the second-order correlations of the initial atomic cloud from the final molecular state, wekeep the physics of the atoms themselves as well as the coupling to the mole-cular field as simple as possible. The coupled fermion–molecule system can


be described by the Hamiltonian (Chiofalo et al., 2002; Holland et al., 2001;Timmermans et al., 1999)

H =∑k,σ

ωk

2c

†kσ ckσ +

∑k

ωka†kak + V −1/2

∑k1,k2,σ

Utr(k2 − k1)c†k2σ

ck1σ

+ U0

2V

∑q,k1,k2

c†k1+q↑c

†k2−q↓ck2↓ck1↑

(40)+ hg

(∑q,k

a†qcq/2+k↓cq/2−k↑ + H.c.

).

The kinetic energies ωk are defined as before, V is the quantization volume,Utr(k) = V −1/2

∫Vd3xe−ikxUtr(x) is the Fourier transform of the trapping po-

tential Utr(r) and U0 = 4πh2a/mf is the background scattering strength witha the background scattering length. The coupling constant g between atoms andmolecules is, up to dimensions, equal to χ of the previous sections.

We assume that the trapping potential and background scattering are relevantonly for the preparation of the initial state before the coupling to the moleculesis switched on at t = 0 and can be neglected in the calculation of the subsequentdynamics. This is justified if hg

√N � U0n, hωi , where n is the atomic density,

N the number of atoms, and ωi are the oscillator frequencies of the atoms in thepotential Utr(r) that is assumed to be harmonic. Experimentally, the interactionbetween the atoms can effectively be switched off by ramping the magnetic fieldto a position where the scattering length is zero, so that this assumption is fulfilled.

Regarding the strength of the coupling constant g, two cases are possible:hg

√N can be much larger or much smaller than the characteristic kinetic energies

involved. For fermions the terms broad and narrow resonance have been coinedfor the two cases, respectively, and we will use these for bosons as well. Bothsituations can be realized experimentally, and they give rise to different effects.For strong coupling the conversion process needs not satisfy energy conservationbecause of the energy time uncertainty relation. For weak coupling energy con-servation is enforced. This energy selectivity can be useful in certain situationsbecause it allows one to resolve additional structures in the atomic state. Theanalysis of this case is fairly technical, however. Therefore we only consider thecase of strong coupling and refer the interested reader to (Meiser et al., 2005b) fordetails of the calculations and the case of weak coupling.

First-order time-dependent perturbation theory requires that the state of theatoms does not change significantly and consequently, only a small fraction ofthe atoms are converted into molecules. It is reasonable to assume that this is truefor short interaction times or weak enough coupling. Apart from making the sys-tem tractable by analytic methods there is also a deeper reason why the coupling


should be weak: Since we ultimately wish to get information about the atomicstate, it should not be modified too much by the measurement itself, i.e. the cou-pling to the molecular field. Our treatment therefore follows the same spirit asGlauber’s original theory of photon detection, where it is assumed that the light–matter coupling is weak enough that the detector photocurrent can be calculatedusing Fermi’s Golden rule.

5.2. BEC

We consider first the case where the initial atomic state is a BEC in a sphericallysymmetric harmonic trap. We assume that the temperature is well below the BECtransition temperature and that the interactions between the atoms are not toostrong. Then the atomic system is described by the field operator

(41)ψ(x) = ψ0(x)c + δψ(x),

where ψ0(x) is the condensate wave function and c is the annihilation operator foran atom in the condensate. In accordance with the assumption of low temperaturesand weak interactions we do not distinguish between the total number of atomsand the number of atoms in the condensate. The fluctuations δψ(x) are small andthose with wavelengths much less than RTF will be treated in the local densityapproximation while those with wavelengths comparable to RTF can be neglected(Bergeman et al., 2000; Hutchinson and Zaremba, 1997; Reidl et al., 1999).

We are interested in the momentum distribution of the molecules

(42)n(p, t) = ⟨b†p(t)bp(t)

⟩which for short times, t , can be calculated using perturbation theory. In the broadresonance limit we ignore the kinetic energies and find

nBEC(p, t) = (gt)2N(N − 1)V∣∣ψ2

0 (p)∣∣2

(43)+ (gt)24N∫

d3x

V

⟨δc†

p(x)δcp(x)⟩,

where the expectation value in the last term includes a thermal average. From thisexpression we see that our approach is justified if (

√N gt)2 � 1 because for

such times the initial atomic state can be assumed to remain undepleted. The firstterm in Eq. (43) is the contribution from condensed atoms and the second termcomes from uncondensed atoms above the condensate. The contribution from thecondensate can be evaluated in closed form in the Thomas–Fermi approximationfor a spherical trap. The contribution from the thermal atoms can be calculatedusing the local density approximation. The details of this calculation can be foundin Ref. (Meiser et al., 2005b).


FIG. 13. Momentum distribution of molecules formed from a BEC (dashed line) with a = 0.1aoscand T = 0.1Tc and a BCS type state with kFa = 0.5 and aosc = 5k−1

F (0) (solid line), both for

N = 105 atoms. The BCS curve has been scaled up by a factor of 20 for easier comparison. Theinset shows the noise contribution for BEC (dashed) and BCS (solid) case. The latter is simply themomentum distribution of molecules formed from a normal Fermi gas. The local density approxima-tion treatment of the noise contribution in the BEC case is not valid for momenta smaller than 2π/ξ(indicated by the dotted line in the inset). Note that the coherent contribution is larger than the noisecontribution by five orders of magnitude in the BEC case and three orders of magnitude in the BCScase.

The momentum distribution (43) is illustrated in Fig. 13. The contribution fromthe condensate is a collective effect, as indicated by its quadratic scaling with theatom number. It clearly dominates over the incoherent contribution from the fluc-tuations, which is proportional to the number of atoms and only visible in theinset. The momentum width of the contribution from the condensate is roughlyh2π/RTF which is much narrower than the contribution from the fluctuations,whose momentum distribution has a typical width of h/ξ , where ξ = (8πan)−1/2

is the healing length. This is a case where coherence properties of the atoms canbe read off the momentum distribution of the molecules: The narrow momen-tum distribution of the molecules is only possible if the atoms were coherent overdistances ∼RTF. At this point this is a fairly trivial observation and the same infor-mation could have been gained by looking directly at the momentum distributionof the atoms, which is after all how Bose–Einstein condensation was detected al-ready in the very first experiments (Anderson et al., 1995; Bradley et al., 1995;


Davis et al., 1995). Still we mention it because it will be very interesting (indeedinteresting enough to motivate this whole work!) to contrast this situation to theBCS case below.

Using the same approximation scheme we can calculate the second-order cor-relation. If we neglect fluctuations we find

(44)g(2)BEC(p1, t1; p2, t2) = 1 − 6

N+O(N−2).

For N → ∞ this is very close to 1, which is characteristic of a coherent state.This result implies that the number fluctuations of the molecules are very nearlyPoissonian. The fluctuations lead to a larger value of g(2), making the molecularfield partially coherent, but their effect is only of order O(N−1).

5.3. NORMAL FERMI GAS

We treat the gas in the local density approximation where the atoms locally fill aFermi sea

(45)|NFG〉 =∏

|k|<kF(x)

c†k|0〉

with local Fermi momentum hkF(x) and |0〉 being the atomic vacuum. It is re-lated to the local density of the atoms in the usual way (Butts and Rokshar, 1997;Landau et al., 1980).

The momentum distribution and second-order correlation function are readilyfound in perturbation theory. The momentum distribution is shown in the insetin Fig. 13. The total number of molecules scales only linear with the numberof atoms, meaning that, in contrast to the BEC case, the molecule formation isnon-collective. Each atom pair is converted into a molecule independently of allthe others and there is no collective enhancement. Furthermore the momentumdistribution of the atoms is much wider than in the BEC case. It’s width is ofthe order of hn1/3

0 indicating that the atoms are correlated only over distancescomparable to the inter atomic distance.

Similarly, we find for the local value of g(2) at position x,

(46)g(2)loc(p, x, t) ≡ g

(2)loc(p, t; p, t, x) = 2

(1 − 1

Neff(p, x)

),

whereNeff is the number of atoms that are allowed to form a molecule on the basisof momentum conservation. For large Neff g

(2) approaches 2 which is character-istic of a thermal field. Indeed, using the analogy with an ensemble of two-levelatoms coupled to every mode of the molecular field provided by the Tavis–Cummings model, it is easy to see that the entire counting statistics is thermal.


5.4. BCS STATE

Let us finally consider a system of fermions with attractive interactions, U0 < 0,at temperatures well below the BCS critical temperature. It is well known thatfor these temperatures the attractive interactions give rise to correlations betweenpairs of atoms in time reversed states known as Cooper pairs. We assume thatthe spherically symmetric trapping potential is sufficiently slowly varying that thegas can be treated in the local density approximation. More quantitatively, thelocal density approximation is valid if the size of the Cooper pairs, given by thecorrelation length

λ(r) = vF(r)/π�(r),

is much smaller than the oscillator length for the trap. Here, vF(r) is the velocityof atoms at the Fermi surface and �(r) is the pairing field at distance r from theorigin, which we take at the center of the trap. Loosely speaking, in the local den-sity approximation the ground state of the atoms is determined by repeating thevariational BCS-calculation of the previous section in a small volume at every po-sition x. A thorough discussion of this calculation can be found in Ref. (Houbierset al., 1997).

We find the momentum distribution of the molecules from the BCS-type stateby repeating the calculation done in the case of a normal Fermi gas. For the BCSwave function, the relevant atomic expectation values factorize into normal andanomalous correlations. The normal terms are proportional to densities and arealready present in the case of a normal Fermi gas while the anomalous contri-butions are proportional to the gap parameter. The momentum distribution of themolecules becomes

(47)nBCS(p, t) ≈ (gt)2∣∣∣∣∑k

〈cp/2+k,↓cp/2−k,↑〉∣∣∣∣2 + nNFG(p, t).

The first term is easily shown to be proportional to the square of the Fouriertransform of the gap parameter. Since the gap parameter is slowly varying overthe size of the atomic cloud, this contribution has a width of the order of h/RTF, incomplete analogy with the BEC case above. The total number of atoms in the firstcontribution is proportional to the square of the number of Cooper pairs, whichis a macroscopic fraction of the total atom number well below the BCS transitiontemperature. That means that this contribution is a collective effect. The secondterm is the wide and incoherent non-collective contribution already present in thecase of a normal Fermi gas. It is very similar to the thermal noise in the BEC caseas far as its coherence properties are concerned.

For weak interactions such that the coherent contribution is small compared tothe incoherent contribution, the second-order correlations are close to those of anormal Fermi gas given by Eq. (46), g(2)(p, x, t) ≈ 2. However, in the strongly


interacting regime, kF|a| ∼ 1, and large N , the coherent contribution from thepaired atoms dominates over the incoherent contribution from unpaired atoms. Inthis limit one finds that the second-order correlation is close to that of the BEC,g(2)(p, x, t) ≈ 1. The physical reason for this is that at the level of even-ordercorrelations the pairing field behaves just like the mean field of the condensate.This is clear from the factorization property of the atomic correlation functions interms of the normal component of the density and the anomalous density contri-bution due to the mean field. In this case, the leading-order terms in N are givenby the anomalous averages.

To summarize, molecules produced from an atomic BEC show a rather narrowmomentum distribution that is comparable to the zero-point momentum width ofthe BEC from which they are formed. The molecule production is a collectiveeffect with contributions from all atom pairs adding up constructively, as indi-cated by the quadratic scaling of the number of molecules with the number ofatoms. Each mode of the resulting molecular field is to a very good approxi-mation coherent (up to terms of order O(1/N)). The effects of noise, both dueto finite temperatures and to vacuum fluctuations, are of relative order O(1/N).They slightly increase the g(2) and cause the molecular field in each momentumstate to be only partially coherent.

In contrast, the momentum distribution of molecules formed from a normalFermi gas is much broader with a typical width given by the Fermi momentum ofthe initial atomic cloud, i.e. the atoms are only correlated over an interatomic dis-tance. The molecule production is not collective as the number of molecules onlyscales like the number of atoms rather than the square. In this case, the second-order correlations of the molecules exhibit super-Poissonian fluctuations, and themolecules are well characterized by a thermal field.

The case where molecules are produced from paired atoms in a BCS-like stateshares many properties with the BEC case: The molecule formation rate is collec-tive, their momentum distribution is very narrow, corresponding to a coherencelength of order RTF, and the molecular field is essentially coherent. The non-collective contribution from unpaired atoms has a momentum distribution verysimilar to that of the thermal fluctuations in the BEC case.

6. Conclusion

In this paper we have used three examples to illustrate the profound impact ofquantum optics paradigms, tools and techniques, on the study of low-density,quantum-degenerate atomic and molecular systems. There is little doubt that theremarkably fast progress witnessed by that field results in no little part from theexperimental and theoretical methods developed in quantum optics over the lastdecades. It is therefore fitting, on the occasion of Herbert Walther’s seventieth


birthday, to reflect on the profound impact of the field that he has helped invent,and where he has been and remains so influential, on some of the most excitingdevelopments in AMO science.

7. Acknowledgements

This work was supported in part by the US Office of Naval Research, by the Na-tional Science Foundation, by the US Army Research Office, and by the NationalAeronautics and Space Administration.

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ATOM MANIPULATION IN OPTICALLATTICES*

GEORG RAITHEL and NATALYA MORROW

FOCUS Center, Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1872. Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

2.1. Light Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1902.2. Atom-Field Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1912.3. Quantum Monte-Carlo Wave-Function Simulations . . . . . . . . . . . . . . . . . . . 194

3. Review of One-Dimensional Lattice Configurations for Rubidium . . . . . . . . . . . . . . 1963.1. Red-Detuned Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1973.2. “Gray” Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2003.3. Magnetic-Field-Induced Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2043.4. Related Laser-Cooling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

4. Periodic Well-to-Well Tunneling in Gray Lattices . . . . . . . . . . . . . . . . . . . . . . . 2084.1. Experimental and Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 2084.2. Analysis Based on Band-Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

5. Influence of Magnetic Fields on Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 2135.1. Motivation and Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . 2135.2. Interpretation of the Results Based on Two Models . . . . . . . . . . . . . . . . . . . 217

6. Sloshing-Type Wave-Packet Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2196.1. Wave-Packets Localized in Single Lattice Wells . . . . . . . . . . . . . . . . . . . . . 2196.2. Experimental Study of Sloshing-Type Motion in a Magnetized Gray Lattice . . . . . 220

7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2228. Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2239. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

1. Introduction

Optical lattices are periodic light-shift potentials for cold atoms created by the in-terference of multiple laser beams. Atoms can be laser-cooled and localized in the

* This chapter has been prepared in dedication to Professor Herbert Walther and his 70th birthday.His vision and excellence as an experimental physicist has lead to many discoveries in atomic physicsand quantum optics, and has inspired our research in many ways.


DOI 10.1016/S1049-250X(06)53007-1

187

188 G. Raithel and N. Morrow [1

sub-micrometer-sized potential wells of optical lattices, leading to exciting possi-bilities in fundamental studies of quantum mechanics and applications of quantumtheory. The field has been reviewed by Jessen and Deutsch (1996). The local-ization of atoms in lattice wells has initially been shown through spectroscopicstudies by Verkerk et al. (1992) and Jessen et al. (1992). Due to the flexibilities inthe choice of atomic transitions, the number of lattice laser beams, their angles,intensities and detunings, a wide variety of lattice geometries and potentials canbe realized, as shown in a systematic manner by Petsas et al. (1994). Applicationsof optical lattices are many-fold. In many laboratories, optical lattices are em-ployed to laser-cool atoms and to localize and store these atoms in microscopicwells. Optical lattices have been utilized in a wide range of experiments on tran-sient phenomena, including experiments on Landau–Zener tunneling (Niu et al.,1996), Bloch oscillations (Dahan et al., 1996), Wannier–Stark states (Wilkinsonet al., 1996), wave-packet revivals (Raithel et al., 1998), and tunneling in near-resonant lattices (Dutta et al., 1999) and far-off-resonant lattices (Haycock et al.,2000). A recent account of applications of optical lattices in atom lithographyis provided by Bradley et al. (1999). Optical lattices have further been proposedas platforms for quantum information processing by Brennen et al. (1999) andJaksch et al. (1999). The dynamics of quantum gases in optical lattices has be-come a field of high interest. This area and some of its implications on quantuminformation processing have recently been reviewed by Bloch (2004).

In this chapter, we focus on applications of optical lattices on the field of wave-packet preparation and manipulation. In all experimental schemes presented, thelattices are used two-fold, namely as an initialization tool to prepare suitableinitial states of the quantum system, and as a platform on which the actual wave-packet experiments are performed. Therefore, the utilized lattices provide fast androbust laser-cooling of cold atoms into the lowest few quantum states of the lat-tice, and the decoherence rate of the trapped atoms caused by the fluorescence ofthe atoms in the lattice light is sufficiently low that coherent wave-packet motioncan be observed over a number of periods of the motion.

In the presented work, we deal with two types of wave-packet motion. In onetype, referred to as sloshing-type motion, wave-packets oscillate back and forthin individual lattice wells. This type of oscillation has been observed by Kozumaet al. (1996) and Raithel et al. (1998) using a photon-exchange method. Thesloshing-type motion occurs within the spatial range of a single lattice well, withvanishing coupling between neighboring wells, on a length scale of a few tenths ofthe laser wavelength used to form the lattice. The motion can be excited by a sud-den small displacement of the laser-cooled and localized atoms from the minimaof the lattice wells or, equivalently, a sudden displacement of the lattice under-neath the atoms. In a simplified harmonic model of the lattice wells and under theassumption that the atoms are initially in the lattice ground state, this procedurecorresponds to the generation of coherent wave-packet states by a shift operation.

1] ATOM MANIPULATION IN OPTICAL LATTICES 189

The sloshing-type wave-packet oscillation that ensues after the shift operationcan be measured in a non-destructive manner, revealing the lattice oscillation fre-quency, the anharmonicity of the lattice potential and decoherence rates (Raithelet al., 1998). Also, measurements on the wave-packet motion can be employedto apply real-time feedback onto the wave-packet motion, allowing one to studyfeedback-controlled cold-atom systems (Morrow et al., 2002). The investigationsdealing with sloshing-type wave-packet motion require lattices that support atleast three to four bound states of the center-of-mass motion in lattice wells. Fur-ther, in order to be able to observe the wave-packet motion, the anharmonicity ofthe wells should be sufficiently low to avoid wave-packet dispersion during thefirst couple of oscillation periods. Also, the fluorescence-induced decoherencerate should be much less than the oscillation frequency. These requirements aresatisfied by a quite large class of lattices.

The second type of wave-packet motion we discuss in this chapter is periodicwell-to-well tunneling. Tunneling measurements have, for instance, been used byDutta et al. (1999) to investigate gauge potentials that were predicted by Dumand Olshanii (1996). In this chapter, we present measurements on the influenceof magnetic-field-induced level shifts on the tunneling behavior. For several rea-sons, the observation of tunneling is considerably more demanding than the studyof wave-packets evolving in single wells. The tunneling frequency of heavy atomssuch as rubidium tends to be very low, as can be seen both by simple estimatesof tunneling frequencies using approximate Gamow factors as well as by accurateband-structure calculations. Tunneling frequencies of at least 103 s−1 are requiredso that the tunneling can be observed on an experimentally feasible time scale.Further, the lattice must provide efficient laser-cooling, because cooling is re-quired to initialize the atoms in the lattice before the tunneling is measured. Also,during the tunneling process the fluorescence-induced decoherence rate must belower than the tunneling rate. Among the lattice types we consider in this chapter,only gray optical lattices, discussed in Section 3.2, satisfy all three conditions.

In Section 2 we describe the theoretical methods that we use to evaluate latticetypes with regard to their suitability for wave-packet experiments. These methodsare also used to model experimental data in detail and to obtain physical insight.In Section 3 we provide an overview over the lattice types that are, in principle, atour disposal. There, we compare the laser-cooling performance of the lattice typesand discuss their suitability for wave-packet and tunneling experiments. This sec-tion of the chapter can serve as a guide of how to select an optical-lattice type for awave-packet or tunneling experiment. In Section 4 we then describe experimentson well-to-well tunneling of atoms in a one-dimensional gray optical lattice. InSection 5 we investigate the modifications in tunneling behavior that result fromthe addition of weak magnetic fields to the lattice. It is discussed how magneticfields in the range of a few tens of milli-Gauss can be used to tune the systemthrough several tunneling resonances. We also find that somewhat stronger mag-


netic fields in the range of 100 mG suppress tunneling and induce more tightlybound levels in a subset of lattice wells. Consequently, as shown in Section 6,magnetic fields can be used to enable sloshing-type wave-packet motion in grayoptical lattices. The presented results are summarized and future prospects arediscussed in the conclusion (Section 7).

2. Theoretical Considerations

2.1. LIGHT FIELDS

In this chapter, we are concerned with one-dimensional lattice structures formedby pairs of counter-propagating laser beams with wave-vectors kL = ±(2π/λ)ez,where we use ex , ey and ez for the Cartesian unit vectors. The electric field canbe written in the form

E(Z, t) = exp(−iωt)[e+(A++ exp(ikLZ)+ A+− exp(−ikZL)

)(1)+ e−

(A−+ exp(ikLZ)+ A−− exp(−ikLZ)

)]+ c.c.

with spherical unit vectors e± = ∓(ex ± iey)/√

2, atomic center-of-mass coordi-nate Z, and c.c. referring to the complex conjugate. The field amplitudes A carrytwo superscript indices, namely a first one identifying the circular-polarizationcomponent, and a second one for propagation direction.

Throughout most of this chapter, we consider counter-propagating fields withorthogonal polarizations, E(Z, t) = E0[ex cos(kLZ−ωt)+ ey sin(−kLZ−ωt)],where E0 denotes the electric-field amplitude of a single beam. For this field, it isseen that

(2)A++ = −E0/(2√

2), A+− = A−+ = A−− = E0/

(2√

2).

This field generates the most widely known type of sub-Doppler laser cooling,“Sisyphus cooling”, which was experimentally observed by Lett et al. (1988)and by Shevy et al. (1989), and explained by Dalibard and Cohen-Tannoudji(1989) and further analyzed in detail by Finkelstein et al. (1992) and Guo andBerman (1993). Another case of interest is that of two counter-propagatingcircularly polarized beams with helicities that are the same in a fixed (beam-independent) frame. Such beams form a standing wave of circular polarization(e.g., A++ = A+− = E0/2 and A−+ = A−− = 0). In combination witha transverse static magnetic field, this field can generate magnetic-field-inducedlaser-cooling (Sheehy et al., 1990) and localization of atoms in a lattice structure.Counter-propagating circularly polarized beams with helicities that are oppositein a fixed (beam-independent) frame form a light-field with spatially rotatinglinear polarization (e.g., A++ = A−− = E0/2 and A−+ = A+− = 0). This


field does not produce modulated lattice potentials or intensities, but is character-ized by a linear-polarization vector whose tip outlines the shape of a corkscrew.The corkscrew configuration generates sub-Doppler laser cooling (Dalibard andCohen-Tannoudji, 1989) and, under certain conditions, velocity-selective coher-ent population trapping (Aspect et al., 1988).

2.2. ATOM-FIELD INTERACTION

In the following, we review the methods that we use throughout this chapter inorder to model optical lattices. In the electric-dipole approximation, the atom-field interaction is −ED, where the electric-dipole operator D = −|e|(−r+e− −r−e++rzez). The operators r+ = −(x+iy)/

√2 and r− = (x−iy)/

√2 and rz act

on the internal (electronic) degree of freedom. Using the orthogonality relationse+e+ = e−e− = 0, e+e− = e−e+ = −1, and noting that e∗+ = −e− ande∗− = −e+, the atom-field interaction at a center-of-mass location Z is

−ED = |e| exp(−iωt)[r+{A++ exp(ikLZ)+ A+− exp(−ikLZ)

}(3)+ r−

{A−+ exp(ikLZ)+ A−− exp(−ikLZ)

}]+ h.c.,

where h.c. is the Hermitian conjugate. Using the excited-state and ground-statewave-functions |ψe〉 and |ψg〉, the Schrödinger equation reads

ih∂t |ψe〉 = −ED|ψg〉 +(hω0 − i

hΓ

2

)|ψe〉,

(4)ih∂t |ψg〉 = −ED|ψe〉,where hω0 is the energy of the excited state, and the term −i hΓ2 accounts forthe decay of the excited state (Γ is the excited-state decay rate). Note that thisterm represents a weak anti-Hermitian contribution in the effective Hamiltonian,causing the norm of the wave-function to decay. The norm decay reflects thespontaneous emission associated with the probability of finding the atom in theexcited state. |ψe〉 and |ψg〉 are spinor wave-functions that have 2F ′ + 1 and2F + 1 magnetic sublevels, where F ′ and F denote the respective excited-stateand ground-state angular momenta.

To transform into a rotating frame, we use new wave-functions |Ψe〉 =exp(iωt)|ψe〉 and |Ψg〉 = |ψg〉. After making the rotating-wave approximation, inwhich terms ∝ exp(±i2ωt) are neglected, the transformed Schrödinger equationreads

ih∂t |Ψe〉 = |e|[r+{A++ exp(ikLZ)+ A+− exp(−ikLZ)}

+ r−{A−+ exp(ikLZ)+ A−− exp(−ikLZ)

}]|Ψg〉


−(hδ + i

hΓ

2

)|Ψe〉,

ih∂t |Ψg〉 = |e|[−r−{A++∗ exp(−ikLZ)+ A+−∗ exp(ikLZ)}

(5)− r+{A−+∗ exp(−ikZ)+ A−−∗ exp(ikZ)

}]|Ψe〉,where the laser-atom detuning δ = ω − ω0. Also, note that r− is the Hermitianconjugate of −r+. Usually, the excited-state part of the wave-function reaches aquasi-steady-state as the atom moves (slowly) through the lattice. In this case,the excited state can be adiabatically eliminated by setting ∂t |Ψe〉 = 0. We canthen express |Ψe〉 in terms of |Ψg〉 and insert the result in the lower part of theSchrödinger equation (5). We obtain

(6)ih∂t |Ψg〉 = V (Z)|Ψg〉with an effective Hamiltonian V (Z). For the case of counter-propagating beamswith orthogonal polarizations, which we are mostly interested in, it is

V (Z) = −e2E20

2(hδ + ihΓ /2)

{r+r− cos2(kLZ)+ r−r+ sin2(kLZ)

(7)+ i(r2− − r2+

)sin(kLZ) cos(kLZ)

}.

Considering the center-of-mass position Z a fixed classical parameter, V (Z)can be evaluated in the internal ground-state Hilbert space {|g, F,m〉 | m =−F,−F + 1, . . . , F }. Thereby, the internal-state operators r+ and r− occur inproducts that couple a ground-state vector into an excited-state one and back intothe ground state. All non-zero matrix elements of the operator r+ are of the type〈e, F ′,m′ = m + 1|r+|g, F,m〉 or 〈g, F,m|r+|e, F ′,m′ = m − 1〉, with theexited-state Hilbert space being {|e, F ′,m′〉 | m′ = −F ′,−F ′ + 1, . . . , F ′}.

The matrix elements of r+ and r− are products of a radial matrix element anda Clebsch–Gordan coefficient. The Clebsch–Gordan coefficients can be calcu-lated for all hyperfine transitions of interest and normalized such that the “cyclingtransition” |g, F,m = F 〉 ↔ |e, F ′ = F + 1,m′ = F + 1〉 has a Clebsch–Gordan coefficient of one. They can then be arranged in matrices that representoperators c+ and c− which essentially are the same as the r+ and r− except thatthe radial matrix element has been factored out. We also reverse the sign of c+(meaning that c+ and c− are the Hermitian conjugate of each other). This proce-dure allows one to write V (Z) in the convenient form

V (Z) = hΩ21

2(δ + iΓ/2)

{c+c− cos2(kLZ)+ c−c+ sin2(kLZ)

(8)+ i(c2+ − c2−

)sin(kLZ) cos(kLZ)

},


where the Rabi frequency Ω1 = Γ

√I1

2Isat. There, I1 is the single-beam intensity

and Isat the saturation intensity of the transition |g, F,m = F 〉 ↔ |e, F ′ =F + 1,m′ = F + 1〉. For 87Rb, which we use in the experiments describedin Sections 4–6, Isat = 1.6 mW/cm2 and F = 2. As before, the internal-state operators c+ and c− occur in products that couple a ground-state vectorinto an excited-state one and back into the ground state. Further, c+ (c−) al-ways increases (decreases) the magnetic quantum number. Therefore, the terms inEq. (8) involving c+c− and c−c+ yield contributions that are diagonal in the basis{|g, F,m〉 | m = −F, . . . , F } and can be regarded as the primary light-shift ef-fect of the lattice on the atomic m-levels. The terms involving c2+ and c2− producecontributions on the second off-diagonal that can be interpreted as stimulated Ra-man transitions driven by the lattice beams that cause mixing between states them-values of which differ by 2. Finally, recalling that c+ and c− are Hermitianconjugates of one another, it is also seen that the effective Hamiltonian V (Z) isHermitian for the case Γ = 0 (as required).

Since in alkali atoms typically more than one excited-state hyperfine level F ′are important, we usually sum the Hamiltonian V (Z) over the relevant valuesof F ′. Thereby, for different F ′ different values of δ apply, given by the laserfrequency and the hyperfine splittings of the utilized transition. Also, the Clebsch–Gordan coefficients entering into the c+ and c− are different for different F ′.The effect of a small magnetic field B can also be included by adding a termPggFμB F · B to Eq. (8), where gF is the ground-state g-factor, μB the Bohrmagneton, F the angular-momentum vector operator, and Pg a projector on theground-state manifold.

The eigenvalues and eigenvectors of V (Z) yield the adiabatic lattice poten-tials Vα(Z), plotted frequently in this chapter, and adiabatic internal states Ψα(Z)

of the lattice (the label α = 1, 2, . . . , 2F + 1). In a classical description ofthe center-of-mass motion, the atoms move on these lattice potentials underthe influence of an AC electric-dipole force given by the gradient d

dZVα(Z).Thereby, the internal state of the atom adiabatically follows the internal-statevector Ψα(Z). This notion can be used to qualitatively explain atom motionin optical lattices, but it does not apply in spatial regions where the adiabaticpotentials Vα(Z) are not well separated and exhibit narrow anti-crossings. Atnarrow anti-crossings between Vα(Z), the lattice-light-induced Raman couplingbetween different m-levels is quite inefficient, and atoms traveling through theanti-crossing region tend to move on the so-called diabatic potentials, Vm(Z) =〈g, F,m|V (Z)|g, F,m〉 (i.e. the diagonal components of the Hamiltonian V (Z)

in Eq. (8) in m-state representation). In this situation, atoms undergo Landau–Zener transitions between different adiabatic potentials.

It is fairly straightforward to obtain the band structure and the Bloch states ofthe lattices. We quantize the center-of-mass motion by considering the variable Z


in Eq. (8) an operator acting on center-of-mass momentum states. Noting thatEq. (8) only couples momentum states that differ by multiples of 2kL, a productHilbert space that is well suited to represent the atom-field interaction in Eq. (8)with center-of-mass quantization is {|g, F,m〉 | m = −F,−F + 1, . . . , F } ⊗{|(2n + q)kL〉 | n = 0,±1,±2 . . .}. There, the momentum states |(2n + q)kL〉include a fixed quasimomentum q restricted to a range −1 � q � 1. Theatom-field interaction potential Eq. (8), with Z taken as an operator, is the po-tential part Hpot of the full Hamiltonian. The kinetic-energy term is Hkin =∑

m,n[h2(2n + q)2k2L/(2M)]|m, (2n + q)kL〉〈m, (2n + q)kL|, where M is the

atom mass and an abbreviated notation for the product states of internal and ex-ternal degrees of freedom is used. We represent H = Hpot + Hkin in the aboveproduct Hilbert space, diagonalize the resultant matrix, and sort the real parts ofthe eigenvalues. Note that q is held fixed in any given diagonalization. The bandstructure of lattice is then obtained by plotting the lowest eigenvalue vs q, thesecond-lowest eigenvalue vs q, and so on. Following an analogous procedure,the decay rates of the Bloch states, given by twice the imaginary parts of theenergy eigenvalues, can be plotted. Numerous examples of band structures andcorresponding decay-rate plots are shown in Section 3 of this chapter. If desired,the band-structure calculation also yields the periodic Bloch functions of the lat-tice, 〈Z|Ψ (q,k)〉 = exp(iqZ)

∑n,m c

(q,k)n,m exp(i2nkLZ)|m〉, where the c

(q,k)n,m are

the Fourier coefficients of the Bloch function and k is a band label. Note that thesefunctions are spinor functions, as we sum over the magnetic quantum number m.

2.3. QUANTUM MONTE-CARLO WAVE-FUNCTION SIMULATIONS

The atom dynamics in optical lattices can be simulated using the quantum Monte-Carlo wave-function method (QMCWF), which was introduced by Dalibard et al.(1992) and used by Marte et al. (1993) to model laser cooling. The simulationsemploy a fully quantum-mechanical description of the internal and center-of-massdegrees of freedom of the atoms. We employ QMCWF to gain insight into thelaser-cooling dynamics, wave-packet evolution and coherence decay times. Thesimulations also allow us to determine the spatial and momentum distributions ofthe atoms, including the degree to which the atoms become localized in the wellsof the optical-lattice potentials. We have found that the QMCWF simulations pro-vide perfect modeling for our lattice experiments.

In the QMCWF method, the evolution of the density matrix describing theatoms in the lattice is obtained by forming averages over N quantum trajectories|Ψi(t)〉, each of which is a realization of a single-atom wave-function evolution:

(9)ρ(t) = 1

N

N∑i=1

|Ψi(t)〉〈Ψi(t)|〈Ψi(t)||Ψi(t)〉 .


Averages are taken over ensembles of typically N = 104 to N = 105 quantumtrajectories. We usually do not store the full density matrix but only the expec-tation values of observables of interest vs time, such as kinetic energy, degree oflocalization in the wells, average momentum and average magnetization, positionand momentum distributions, etc.

Each quantum trajectory |Ψi(t)〉 consists of periods of deterministic Hamil-tonian wave-function evolution, connected by discrete quantum jumps. TheHamiltonian evolution is governed by an effective Hamiltonian H = Hkin + Hpotwith kinetic and potential operators as explained in Section 2.2. The effectiveHamiltonian describes the coherent interaction between atoms and light fields,as well as wave-function damping caused by photon scattering. The damping isimplemented by the imaginary part in the energy denominator of Eq. (8), whichleads to a gradual decay of the wave-function norm. The quantum trajectoriesare represented in the basis {|m, (2n + q)kL〉 | m = −F,−F + 1, . . . , F andn = 0,±1,±2, . . . ± nmax}, where nmax determines the cutoff value of themomentum states included in the simulation. The range of momentum statesthat becomes populated depends on the lattice parameters. Therefore, we choosefrom cutoff numbers nmax = 16, 32 or 64, dependent on the physical situa-tion. The continuous quasimomentum q, which satisfies −1 < q < 1, does notchange during the Hamiltonian portions of the wave-function evolution, as canbe seen by inspection of Eq. (8). The Hamiltonian evolution is carried out nu-merically in discrete time steps. We use a split-operator method (Kosloff, 1988;Leforestier et al., 1991), in which the kinetic-energy operator, which is diagonalin the momentum basis, is applied in the momentum basis, while the atom-fieldinteraction, which is diagonal in position, is applied in a position basis. Thus,at each time step of the integration the quantum trajectory is transformed backand forth between position and momentum representations using fast Fouriertransformations. Consequently, both the position and the momentum probabil-ity distributions of the quantum trajectory |Ψ 〉 can be obtained without numericaloverhead at any time of the wave-function evolution.

The periods of Hamiltonian evolution are interrupted by discrete, instantaneousquantum jumps, which simulate the effect of the spontaneous scattering of latticephotons, and polarization- and direction-sensitive photon detection. A quantumjump is invoked when the norm of a quantum trajectory, which continuously de-cays during the Hamiltonian portions of the evolution, drops below a randomnumber that is picked at the beginning of each time segment of Hamiltonianevolution. The time instants and effects of the quantum jumps are governed byquantum-mechanical probability laws. In each quantum jump, random numbersare drawn to select the type of transition and the direction of the spontaneouslyemitted photon. In each quantum jump, the quasimomentum value can change byany value −1 < �q < 1. The applied value of�q follows from a random numberand the radiation pattern that corresponds to the type of scattered photon. In each


jump, the wave-function is modified in a well-defined manner, determined by thewave-function prior to the jump, the rules of quantum measurement, and severalrandom numbers. After quantum jumps, the quantum trajectory |Ψ 〉 is normalizedand entered into the next segment of Hamiltonian evolution.

We have found that a considerable degree of detail in the QMCWF is re-quired in order to reproduce experimental data. Our QMCWF take all ground-and excited-state hyperfine levels of the system into account. This implies thatleak transitions and the effect of a re-pumping laser, which is usually requiredin the experiments, are fully taken into account in the QMCWF. Magnetic fields,gravity, and additional laser beams that are used to optically pump the atoms incertain lattices (see Section 4) can also be included in the simulations.

3. Review of One-Dimensional Lattice Configurationsfor Rubidium

In this chapter, we are concerned with one-dimensional lattice structures of ru-bidium (transition wavelengths λ = 795 nm for the D1-line 5S1/2 ↔ 5P1/2 andλ = 780 nm for the D2-line 5S1/2 ↔ 5P3/2). We concentrate on 87Rb, which hasground-state hyperfine components F = 1 and F = 2. The lattices are mostly butnot always operated on the F = 2 level. In the presented calculations, all coupledhyperfine levels of the excited states are taken into account. The results translateto many atomic species with similar hyperfine structure and transition linewidths.

Since the lattices are to be used to both cool the atoms and to perform wave-packet and tunneling experiments on them, two figures of merits exist:

• Cooling time scale. Since we use one-dimensional lattices, spatial diffusiontransverse to the lattice-beam directions causes the atoms to escape the atom-field interaction region, which is defined by the diameter of the laser beams.Typically, the atoms remain in the atom-field interaction region for 1–2 mil-liseconds. Therefore, the lattice type selected for each experiment needs to cooland localize the atoms in the lattice wells within about 1 ms. Since generallythe time scale of laser cooling is given by the photon scattering rate, we seekconfigurations that yield high initial photon scattering rates.

• Low steady-state temperature and long coherence time. These conditions,which are a pre-requisite for wave-packet and tunneling experiments, are pri-marily achieved by choosing configurations in which the photon scattering rateconverges towards small values once most of the atom cooling has occurred.One may also exploit coherence preservation due to the Lamb–Dicke effect(Dicke, 1953).

In the following, we use potential and band-structure calculations as well as quan-tum Monte-Carlo simulations in order to evaluate various one-dimensional lattice


configurations with regard to their atom-cooling speed, steady-state temperature,degree of atom localization, and steady-state photon scattering rate. Also, thewell-to-well tunneling rates of atoms cooled deeply into the lattices are estimatedbased on Gamow factors and derived from the band structure of the lattices.

3.1. RED-DETUNED LATTICES

The most common type of sub-Doppler Sisyphus cooling occurs in counter-propagating fields of orthogonal polarization, referred to as “lin-perp-lin” con-figuration (Dalibard and Cohen-Tannoudji, 1989). We first consider a lattice thatis red-detuned with respect to a closed transition of the type F ↔ F ′ + 1, suchas the 87Rb 5S1/2 F = 2 ↔ 5P3/2 F = 3 transition. This field is equivalentto two circularly polarized standing waves of opposite helicity and a λ/4 spatialdisplacement.

In Fig. 1 we show simulation results for a red-detuned lattice on the 87Rb5S1/2 F = 2 ↔ 5P3/2 F ′ = 3 transition obtained from 104 quantum trajecto-ries. The single-beam intensity is I1 = 10 mW/cm2 and the laser detuning −6Γrelative to the utilized atomic transition (upper-state decay rate Γ = 2π×6 MHz).Figure 1(a) shows the adiabatic potentials Vα(Z) of the lattice in units of the recoilenergy, ERec = h2k2

L/(2M) (for the 87Rb D1-transition, ERec = h × 3.77 kHz).The lowest adiabatic potential is of particular interest, because atoms broughtinto the lattice become rapidly optically pumped onto that potential. The minimaof the lowest adiabatic potential correspond to locations of maximal intensity ofthe circularly polarized standing-wave components of the lattice field. Since eachstanding wave has a λ/2 period, and since the two standing waves are shiftedrelative to each other by λ/4, the field maxima have λ/4 separation and alternat-ing σ+ and σ−-polarizations. At the σ+-maxima, the internal atomic state |Ψα=1〉associated with the lowest adiabatic potential Vα=1 is practically identical with|F = 2,m = 2〉, and at the σ−-maxima it is |F = 2,m = −2〉. Since the σ±-transitions 5S1/2|F = 2,m = ±2〉 ↔ 5P3/2|F = 3,m = ±3〉 have the largestClebsch–Gordan coefficient (namely 1), the value of Vα=1 at the field maxima ismaximal and negative.

The band structure, displayed in Fig. 1(b), shows that under the conditions ofFig. 1 the lowest adiabatic potential supports about five tightly bound oscilla-tory states. For energies above the maxima of the lowest adiabatic potential, theband structure is quite complicated, because bands associated with multiple po-tentials begin to overlap and mix. Considering the lowest five tightly bound bandsin Fig. 1(b), it is noted that the separation between adjacent bands decreases withincreasing energy. This trend reflects the anharmonicity of the lowest adiabatic po-tential. The potential anharmonicity causes wave-packet dispersion, as has beenobserved in breathing-mode (Raithel et al., 1997) and sloshing-type wave-packets(Raithel et al., 1998).


FIG. 1. Simulation of laser cooling in a red-detuned optical lattice of rubidium (detailed parame-ters provided in text). (a) Adiabatic potentials Vα in recoil energies ERec = h2k2

L/(2M) vs position.

(b) Band structure Ek vs quasimomentum q. (c) Kinetic energy of cooled atoms vs time t . (d) De-gree of atom localization in the wells, as defined in text, vs time t . (e) Single-atom fluorescence ratevs time t . The exponential fits (dashed lines) in panels (c)–(e) yield the rates at which the respectivequantities approach a steady-state.

Under the absence of decoherence, atoms prepared in the tightly bound bandswould tunnel between neighboring wells at rates given by the width of the bands,which increases with increasing excitation energy in the lattice wells. For the low-est band in Fig. 1(b), the width and thus the well-to-well tunneling rate νT amountto only 3 s−1. (This bandwidth cannot be resolved on the scale of Fig. 1(b), butis evident from the numerical data used for the figure.) This tunneling rate is inqualitative agreement with a basic estimate

νT ≈ Ωosc

2πexp(−2G),

(10)G =√

2M

h2

b∫a

√Vα=1(Z)− E1 dZ,


where Ωosc2π is the center-of-mass oscillation frequency of the atoms in the wells

(≈140 kHz in Fig. 1), G is the Gamow factor, E1 is the average energy of the low-est lattice band (≈−390 ERec in Fig. 1), Vα=1(Z) is the lowest adiabatic potential,and the locations a and b denote the left and right boundaries of the tunneling bar-rier (i.e. a pair of locations where Vα=1(Z) = E1). While the integral in Eq. (10)could be easily calculated numerically, a brief survey of Fig. 1(a) allows us toquickly estimate the integral by ∼100 ERec × 0.1λ, leading to νT ∼ 1 s−1. Thisqualitative value agrees well with the exact value obtained from the band structurecalculation.

The laser-cooling performance and other properties of the lattice are studied us-ing QMCWF simulations. Atoms are entered into the simulation with a Gaussianvelocity distribution that corresponds to an average kinetic energy of 100 ERec.Due to Sisyphus cooling, the atoms become cooled into the lowest few oscillatorylevels of the wells. As seen in Fig. 1(c), most of the cooling occurs within about100 µs, and a steady-state energy of about 30 ERec is reached. Comparing thisvalue with the band structure in Fig. 1(b), and noting that in view of the virialtheorem the total energy relative to the potential minimum is about twice the ki-netic energy, it is seen that in steady-state the laser-cooled atoms mostly residein the lowest two or three tightly-bound vibrational states of the lowest adiabaticpotential.

The degree of localization of the atoms in the lattice reaches �Z =√〈Ψ |(Z − Z0)2|Ψ 〉 ≈ λ/18, where Z0 is the location of the nearest potentialminimum (see Fig. 1(d)). The laser-cooling and localization dynamics in red-detuned optical lattices has been studied in detail by Raithel et al. (1997). There,it has been found that the laser-cooling and localization rates are about the same,as is evident in Figs. 1(c) and (d), and are proportional to the fluorescence rate ofthe atoms in the lattice field. For the parameters of Fig. 1, the cooling rate equalsabout 1/30 of the single-atom fluorescence rate. This ratio is fairly typical forlaser-cooling in one-dimensional red-detuned lattices.

While red-detuned lattices exhibit good initial laser-cooling performance anda high degree of atom localization in the lattice wells, the steady-state kineticenergy does not reach very low values because the atoms settle at locations ofmaximal light scattering rate, as can be seen in Fig. 1(e). Comparing Figs. 1(c)and (e) it is actually noticed that the degree to which the photon scattering hasapproached its steady-state value reflects on the progress in cooling. The photonscattering rate reaches a steady-state value of about 1.4 × 106 s−1. This value isclose to the value one can calculate for atoms at the maxima of the two circularstanding waves the lattice is composed of, γ = Γ

22I1/Isat

1+4(δ/Γ )2= 1.6×106 s−1. This

fluorescence rate exceeds the tunneling rate on the lowest band, which is 3 s−1,by about six orders of magnitude. The disparity between fluorescence-inducedcoherence decay rate and tunneling rate renders red-detuned lattices unsuitable


for experiments on well-to-well tunneling (and similar experiments that wouldrequire long wave-packet coherence times).

We add that red-detuned optical lattices are still suitable to study wave-packets that are confined to single lattice wells, which have length scales smallerthan λ. Single-well wave-packets include sloshing-type wave-packets, observedby Kozuma et al. (1996) and Raithel et al. (1998), and breathing-mode wave-packets (Raithel et al., 1997). In such cases, the decay rate of coherences betweenthe lowest few oscillatory states can be considerably less than the photon scatter-ing rate due to the Lamb–Dicke effect (Dicke, 1953). A reasonable estimate is thatthe coherence decay rate is reduced relative to the photon scattering rate by a fac-tor of order of the Lamb–Dicke factor, η2 = ERec

hΩosc= k2

L�Z2, where Ωosc/2π is

the sloshing frequency of the atoms in the wells. This reduction factor is of order0.1 for the parameters in Fig. 1.

3.2. “GRAY” LATTICES

Efficient laser cooling occurs on the blue-detuned side of F ↔ F ′ = F res-onances, as predicted by Guo and Berman (1993) and Grynberg and Courtois(1994), and observed by Hemmerich et al. (1995). To evaluate the suitabilityof blue-detuned lattices for tunneling and other wave-packet experiments, wefirst consider the case of a lin-perp-lin lattice with single-beam intensity I1 =10 mW/cm2 that is blue-detuned by 6Γ with respect to the F = 2 ↔ F ′ = 2component of the 87Rb D2-line. The potential diagram of this lattice, shown inFig. 2(a), exhibits a lowest adiabatic potential that is quite shallow and fairly wellseparated from the higher-lying potentials. The lowest adiabatic potential wouldbe identical zero if the only relevant transition were the F = 2 ↔ F ′ = 2 tran-sition, because an F ↔ F ′ = F transition always has one dark state (i.e. a statewith zero light shift and fluorescence rate) regardless of the light polarization. Thenegative light shift and the small modulation of the lowest adiabatic potential seenin Fig. 2(a) result from residual interactions on the F = 2 ↔ F ′ = 3 transition,which is blue-shifted by 45Γ relative to the F = 2 ↔ F ′ = 2 transition, andthe F = 2 ↔ F ′ = 1 transition, which is red-shifted by 26Γ relative to theF = 2 ↔ F ′ = 2 transition.

As in the case discussed in Section 3.1, in the present case the atoms are alsorapidly optically pumped onto the lowest adiabatic potential and subsequentlycooled via Sisyphus cooling. The cooling and localization performance are dis-played in Figs. 2(d) and (e), respectively. The achieved steady-state temperature isconsiderably lower than that of the example considered in Section 3.1, while theachieved degree of localization is somewhat less (λ/14 here vs λ/18 above). Also,both the cooling and localization rates are about half of those in Section 3.1. Thediminished degree of steady-state localization obviously is a result of the larger


FIG. 2. Simulation of laser cooling in a blue-detuned optical lattice of 87Rb on the D2-line (de-tailed parameters provided in text). (a) Adiabatic potentials Vα vs position. (b) Band structure Ek vsquasimomentum q. (c) Fluorescence-induced decay rates γk of the lattice bands. (d) Kinetic energy ofcooled atoms vs time t . (e) Degree of atom localization in the wells vs time t . (f) Single-atom fluores-cence rate vs time t . The exponential fits (dashed lines) in panels (d)–(f) yield the rates at which therespective quantities approach a steady-state.

size of the localized quantum states supported by the lowest potential in Fig. 2(a),which is shallower than the one in Fig. 1(a). The reduced temperature and coolingrates are both a consequence of the low photon scattering rate (compare Fig. 1(e)with Fig. 2(f)). In the present case, the fluorescence rate strongly decreases vstime, because in the process of laser cooling atoms accumulate in the lowest fewbands supported by the lowest adiabatic potential. These bands have very lowfluorescence-induced decay rates because they inherit the “almost” dark charac-ter of the internal adiabatic states |Ψα=1(Z)〉 associated with the lowest adiabaticpotential. The very low steady-state fluorescence rate, evident from Fig. 2(f), hasgiven rise to the term “gray optical lattice” used for lattices that are blue-detunedwith respect to F ↔ F ′ = F transitions. The low temperatures and fluorescencerates afforded by the gray lattice are achieved on the expense of reduced coolingand localization rates. The gray lattice in Fig. 2 cools about half as fast as thered-detuned lattice in Fig. 1.


The band structure, displayed in Fig. 2(b), reveals two tightly bound bands.The lack of a larger number of tightly bound states renders this type of latticeunsuitable for single-well wave-packet experiments. The lowest band has a widthof about 103 s−1, meaning that the well-to-well tunneling rate of atoms preparedin that band is about 103 s−1. This value is in qualitative agreement with estimatesthat can be made based on Eq. (10). The decay rates of the Bloch states, shownin Fig. 2(c), are of order forty times as large as the tunneling rate associated withthe lowest band. While this factor is much smaller than the corresponding factorfor the red-detuned lattice studied in Section 3.1, which was of order 106, it is notlow enough for experiments on well-to-well tunneling.

An improved version of a gray lattice can be realized by elimination of the mainsource of fluorescence of atoms localized on the lowest adiabatic potential. Theseatoms are in internal states |m = +F 〉 near the maxima of the σ+-polarizedstanding-wave component of the lattice field, and |m = −F 〉 near the maximaof the σ−-component. Therefore, the residual fluorescence of the Bloch states inthe lowest few bands of Fig. 2(c) mostly occurs via off-resonant excitation intosub-states of the 5P3/2 F ′ = 3 hyperfine level. This level can be eliminated byusing the D1-line—which only has F ′ = 1 and F ′ = 2 hyperfine components—instead of the D2-line. Further, the F ′ = 1 and F ′ = 2 components of the D1-lineare separated by 138Γ , which is much larger than the corresponding separationin the D2-line. Thus, in D1-lattices there is less perturbation due to the F ′ = 1component. These advantages lead to lower residual fluorescence rates and lowertemperatures.

In Fig. 3 we show simulation results for a lin-perp-lin lattice of 87Rb withsingle-beam intensity I1 = 10 mW/cm2 that is blue-detuned by 6Γ with re-spect to the F = 2 ↔ F ′ = 2 component of the D1-line. To allow for a directcomparison of Fig. 3 with Fig. 2, in the respective simulations we have used thesame wavelength (λ = 780 nm). The lowest adiabatic potential of the D1 opticallattice (see Fig. 3(a)) is considerably shallower than that of the D2-lattice. Also,in the D1-lattice all potentials are positive. These differences reflect the absenceof the F ′ = 3 hyperfine component in the D1-lattice. The small residual modu-lation of the lowest adiabatic potential in the D1 lattice is caused by off-resonantinteraction with the F ′ = 1 hyperfine component. The lower temperatures andfluorescence rates afforded by the D1 gray lattice, evident from Figs. 3(d) and (f),are achieved on the expense of reduced cooling and localization rates. The D1gray lattice cools about half as fast as the D2 gray lattice. Nevertheless, the cool-ing rate of the D1 gray lattice is still high enough to allow for comfortable coolingunder typical experimental conditions.

The band structure of the D1 gray lattice (Fig. 3(b)) exhibits only one tightlybound band. Due to the shallow potential barrier on the lowest adiabatic poten-tial, the width of the lowest band and the well-to-well tunneling rate are fairlyhigh (7× 103 s−1). Further, the fluorescence rate of the Bloch states in the lowest


FIG. 3. Simulation of laser cooling in a blue-detuned optical lattice of 87Rb on the D1-line.(a) Adiabatic potentials Vα vs position. (b) Band structure Ek vs quasimomentum q. (c) Fluores-cence-induced decay rates γk of the lattice bands. (d) Kinetic energy of cooled atoms vs time t .(e) Degree of atom localization in the wells vs time t . (f) Single-atom fluorescence rate vs time t .The exponential fits (dashed lines) in panels (d)–(f) yield the rates at which the respective quantitiesapproach a steady-state.

band, shown in Fig. 3(c), is only of order 103 s−1, entailing two important conclu-sions. First, most of the residual steady-state scattering seen in Fig. 3(f) is causedby a small percentage of atoms that are not laser-cooled into the lowest band ofthe lattice. Second, for the atoms that are cooled into the lowest band the tunnel-ing rate is of order seven times higher than the fluorescence rate. Thus, the D1gray optical lattice is suited to perform both efficient laser cooling and to observecoherent well-to-well tunneling over multiple periods. No other type of latticeswe have studied offers this combination of possibilities.

Gray lattices can also be realized on the lower ground-state hyperfine compo-nent of alkali atoms (F = 1 for 87Rb). We have observed cooling (experimen-tally and in simulations) on lin-perp-lin lattices of 87Rb that are blue-detunedby amounts of order 5Γ with respect to the F = 1 ↔ F ′ = 1 compo-nent of the D1-line. Since those lattices require a re-pumper laser tuned to theF = 2 ↔ F ′ = 1 or 2 transition, some care needs to be taken to avoid the trap-


ping of atoms in F = 2 states that are dark with regard to the re-pumping laser.While this issue is solvable, the observed cooling performance of gray lattices onthe lower ground-state hyperfine component is not as good as on the upper one.

3.3. MAGNETIC-FIELD-INDUCED LATTICES

There exist some less known types of laser cooling in lattices that are quite robust.Some of these are reviewed in this subsection.

We consider a standing wave of well-defined helicity driving an atom blue-detuned from an F ↔ F ′ = F transition. The lattice potentials produced by thisfield are curves ∝ cos2(2kLZ), with proportionality constants given by squaresof Clebsch–Gordan coefficients. For an isolated F ↔ F ′ = F transition, i.e. ifthere are no other coupled hyperfine levels, one of the potentials that correspondto the outmost m-sublevels is identical zero, reflecting the presence of an exactdark state. If there is a perturbing excited-state hyperfine level F ′ = F + 1, thelowest potential exhibits some added spatially modulated light shift. The internal-state wave-functions associated with the lattice potentials are equivalent to the|m〉-states, since the circularly polarized standing wave does not couple differ-ent |m〉-states. If a weak transverse magnetic field B pointing in x-direction isadded, the degeneracy of the lattice potentials near the nodes of the light fieldbecomes lifted, as seen in Fig. 4(a), and the internal states associated with thepotentials become coherent mixtures of different |m〉-states. Obviously, at the ex-act nodes of the field the adiabatic states are given by the eigenstates |mx〉 of thex-component of angular momentum, Fx , rotated by π/2 about the y-axes intothe z-direction. Laser-cooling results from a Sisyphus-type mechanism that in-volves optical pumping from the higher onto the lowest adiabatic potential nearthe maxima of the adiabatic potentials, where the field intensity is maximum,and non-adiabatic transitions of atoms from the lowest potential back onto higheradiabatic potentials in the node region of the laser field. Since the non-adiabaticmixing near the field nodes is instrumental in closing the Sisyphus-type coolingcycle, the cooling only works with a transverse magnetic field present. Therefore,this type of cooling is known as magnetic-field-induced laser cooling (MILC, see(Sheehy et al., 1990)).

The efficiency of this cooling can be optimized by tuning the strength of thetransverse magnetic field, and it is fairly easy to achieve steady-state temper-atures and laser-cooling rates that closely rival those achieved with the previ-ously discussed cooling methods (see Figs. 4(d) and (f)). For the case of theD2-MILC-lattice studied in Fig. 4, the off-resonant interaction with the excited-state hyperfine level F ′ = 3, which is 45Γ above the F ′ = 2 level, causes thelowest adiabatic potential to exhibit a moderately deep potential well. This wellleads to considerable accumulation of atoms in the region near the field maxima,


FIG. 4. Simulation of magnetic-field-induced laser cooling in a blue-detuned optical lattice of87Rb on the D2-line. The single-beam lattice intensity is I1 = 10 mW/cm2 and the laser detuning4Γ relative to the F = 2 ↔ F ′ = 2 transition. A transverse magnetic field of 0.1 Gauss is applied.(a) Adiabatic potentials Vα vs position. (b) Band structure Ek vs quasimomentum q. (c) Fluores-cence-induced decay rates γk of the lowest ten lattice bands. (d) Kinetic energy of cooled atoms vstime t . (e) Steady-state spatial distribution of the atoms in the five m-levels. (f) Single-atom fluores-cence rate vs time t . The exponential fits (dashed lines) in panels (d) and (f) yield the rates at whichthe respective quantities approach a steady-state.

as can be seen in Fig. 4(e). MILC-type lattices are somewhat reminiscent of graylattices, because the fluorescence rate drops as the cooling progresses (compareFigs. 4(d) and (f)). The residual photon scattering rate is limited by the magnetic-field-induced mixing into non-dark m-states and off-resonant excitation into theF ′ = 3 hyperfine level. For the lowest few bands, the fluorescence rate has valuesaround 50 × 103 s−1 (see Fig. 4(c)), which exceeds the width of the lowest bandby many orders of magnitude. In fact, since in the MILC-type lattice the well-to-well separation is λ/2, as opposed to λ/4 in Figs. 1–3, the tunneling rates inMILC-type lattices are extremely small (only about 0.5 s−1 in the case of Fig. 4).Therefore, MILC-type lattices are not suited for tunneling experiments. As anaside, it is noted that the issue of how to measure coherent tunneling in MILC-typelattices, even if it were present, would impose further problems, because atoms inneighboring wells cannot be distinguished by their magnetic moments.


The number of tightly bound levels in Fig. 4(b) is only about two, renderingMILC-type lattices rather unsuitable for wave-packet experiments in single wells.In calculations and experiments not presented, we have also investigated cases ofMILC-type lattices on the D1-transition of rubidium. We have found no signifi-cant difference between MILC-type lattices on the D1- and the D2-transitions.

3.4. RELATED LASER-COOLING METHODS

3.4.1. “Corkscrew” Cooling

The last configuration we consider consists of two counter-propagating laserbeams of opposite helicity (in a fixed frame), which produce a light-field ofposition-independent intensity and spatially rotating and temporally fixed linearpolarization. Due to this polarization geometry, laser-cooling in this configuration(Dalibard and Cohen-Tannoudji, 1989) has been dubbed “corkscrew cooling”.Since in the corkscrew field geometry there are no spatially dependent light-shiftpotentials, this geometry is not suited for wave-packet and tunneling experiments.

As shown in Fig. 5(a) for the case I1 = 10 mW/cm2 and laser detuning δ =−6Γ relative to the F = 2 ↔ F ′ = 3-component of the 87Rb D2-line, corkscrewlaser-cooling performs not quite as well as cooling in comparable red-detunedlin-perp-lin lattices (see Fig. 1). For the case in Fig. 5(a), the fluorescence rate isabout 400 × 103 s−1 and does not significantly depend on time.

3.4.2. Velocity-Selective Coherent Population Trapping

We have found that optical lattices that exhibit dark or nearly dark states generallyexhibit some degree of velocity-selective coherent population trapping (VSCPT),which was first observed by Aspect et al. (1988) in the cooling of metastablehelium. VSCPT refers to optical pumping into coherent superpositions of entan-gled states of the internal and external degrees of freedom of the trapped atomsthat exhibit particularly low fluorescence rates. Atoms tend to accumulate in suchstates. In our one-dimensional lattice geometry, coherently coupled states havecenter-of-mass momenta that differ by integer multiples of 2kL.

Even the lin-perp-lin D1-gray lattice studied in detail in Section 3.2 and Fig. 3,the cooling is a combination of Sisyphus cooling and VSCPT, as evidenced bythe presence of separated peaks of the momentum distribution at integer multiplesof 2kL (see Fig. 5(b)). These separated peaks indicate that atoms tend accumulatein certain Bloch states that are particularly long-lived. The accumulation of atomsin selected Bloch states implies some amount of well-to-well coherence, as hasbeen observed elsewhere (Teo et al., 2002). Signs of VSCPT are further observedon the blue side of F ↔ F ′ = F transitions in corkscrew field configurationsproduced by counter-propagating circularly polarized beams with opposite helic-ity (in a fixed frame). An example is shown in Fig. 5(c), where I1 = 4 mW/cm2


FIG. 5. (a) “Corkscrew cooling” for a field that is red-detuned relative to the F = 2 ↔ F ′ = 3component of the 87Rb D2-line. The plot shows average energy vs time (solid) and an exponentialfit (dashed). (b)–(f) Indications of velocity-selective coherent population trapping in the momentumdistributions for various field configurations explained in the text.

and the field is detuned by 3Γ relative to the F = 2 ↔ F ′ = 2 component ofthe 87Rb D1-line. VSCPT in fairly clean form occurs if the field has a corkscrewpolarization configuration, is weak, and is on-resonant with an F = 1 ↔ F ′ = 1or F ′ = 0 transition so that other types of laser cooling are absent. The casein Fig. 5(d), which is for I1 = 0.2 mW/cm2 and zero detuning relative to theF = 1 ↔ F ′ = 1 component of the 87Rb D1-line, exhibits very clear VSCPTmomentum peaks at ±hkL, as is typical for VSCPT (Aspect et al., 1988). Mea-surements on VSCPT of rubidium on the F = 1 ↔ F ′ = 1 component of theD1-line have been performed by Esslinger et al. (1996). On the same transitionof the D2-line, the VSCPT is less pronounced due to off-resonant fluorescenceon the F = 2 ↔ F ′ = 3 transition (see Fig. 5(e); lattice intensity and detuningsame as in Fig. 5(d)). The additional fluorescence causes increased coherence lossand therefore washes out the VSCPT momentum peaks. Finally, three-component


VSCPT occurs on F = 2 ↔ F ′ = 2-transitions, as evident in Fig. 5(f) (latticeintensity and detuning same as in Fig. 5(d)). There, the three peaks at momen-tum values 0 and ±2kL reflect the accumulation of atoms in a gray, coherentsuperposition of the three states |m = −2, k = −2kL〉, |m = 0, k = 0〉 and|m = 2, k = 2kL〉. Various related configurations that lead to VSCPT have beendiscussed in detail by Aspect et al. (1989) and Papoff et al. (1992).

4. Periodic Well-to-Well Tunneling in Gray Lattices

In the following, we discuss experiments on periodic well-to-well tunneling of87Rb atoms in optical lattices that exhibit efficient sub-Doppler laser cooling.From Section 3 it is concluded that within this class of lattices only gray opticallattices on the D1-line are a reasonable choice to conduct tunneling experiments,because these are the only lattice type that combines a relatively large tunnelingfrequency (of order 104 s−1) with a decoherence rate low enough for the tunnelingto become observable.

4.1. EXPERIMENTAL AND SIMULATION RESULTS

In each cycle of our experiment (Dutta et al., 1999), 87Rb atoms are collected for14 ms in a standard vapor-cell magneto-optic trap (MOT). After switching off theMOT magnetic field, the atomic cloud is further cooled for about 1 ms in a three-dimensional corkscrew optical molasses. The atoms are then loaded into the one-dimensional gray optical lattice formed by two counter-propagating laser beamswith orthogonal linear polarizations (lin-perp-lin lattice). As in Fig. 3, the lattice isblue-detuned by δ = 6Γ from the D1 F = 2 to F ′ = 2 hyperfine component (λ =795 nm). A re-pumping laser tuned to the transition F = 1 → F ′ = 2 is requiredto re-pump atoms scattered into the F = 1 ground-state hyperfine level back intothe F = 2-level the lattice is operating on. It takes ∼1 ms to cool most atoms intothe lowest potential Vα=1(Z). We have seen in our QMCWF simulations that atsingle-beam intensities around I1 = 5 mW/cm2 � 60% of the atoms are preparedin the lowest band of the lattice. Atoms in wells with predominantly σ+- (σ−-)polarized light are predominantly in the |m = 2〉 (|m = −2〉) state. In order toensure that the σ+- and σ−-wells of the lattice are equally deep, great care hasbeen taken to reduce environmental magnetic fields to values of order 1 mG orless.

To initiate observable tunneling between the σ+- and σ−-wells of the lattice,the atom distribution needs to be initialized such that atoms are present in onlyone type of wells. The initialization is accomplished as follows. The re-pumpinglaser is turned off, and a σ+-polarized laser, which is co-linear with the lattice


beams and resonant with the D2 F = 2 → F ′ = 2 transition, is turned onfor 15 µs (intensity ∼0.1 mW/cm2). This laser pulse removes most atoms fromthe σ−-lattice wells by optical pumping into the F = 1 ground state, whereasthe atoms in the σ+-wells mostly survive. The intensity and duration of this ini-tialization laser is adjusted such that about half the atoms are removed from thelattice. The removed atoms remain inactive for the remainder of the tunnelingexperiment. Coherent well-to-well tunneling of the atoms left over in the latticecommences at the end of the initialization pulse.

A tunneling event from a σ+-well into a σ−-well is associated with an ex-change of 4h angular momentum between the atom and the lattice field, amount-ing to an exchange of two photon pairs between the σ+- and σ−-components ofthe lattice beams. All ∼106 atoms in the lattice tunnel in phase, because theirwave-functions were prepared identically by the initialization laser. While thephoton exchange rate caused by a single tunneling atom would not be detectable,the exchange rate caused by the whole atomic sample is substantial. We can mea-sure the tunneling current by separating the lattice beams after their interactionwith the atoms into their σ -polarized components, and measuring the differenceof the powers of the σ+- and σ−-components. The power exchange between thecomponents can be described as

(11)�Pσ = Nhc

λ

d

dt〈m〉,

where N is the number of atoms, and ddt 〈m〉 is the rate of change of the aver-

age magnetic quantum number. Based on the measured tunneling period and theknown atom number, the maximum tunneling-induced power exchange can be es-timated to be of order 10−3 to 10−4 of the incident power. The tunneling-inducedpower exchanges observed in the experiment are of that order of magnitude.

It is noted that the periodic tunneling current is measured in real-time and thatthe measurement is non-destructive. A single experimental cycle yields the time-dependence of the tunneling current over the whole time interval of interest, whichis of order one millisecond. Since it is possible to observe the tunneling signal onan oscilloscope without data averaging, experimental parameters such as back-ground magnetic fields and lattice-beam alignments can be optimized quite easily.Averages over of order 1000 realizations, which take less than a minute to accu-mulate, yield noise-free measurements of the tunneling current.

In Fig. 6, left panel, a set of experimental data taken at different lattice inten-sities is shown. The curves exhibit oscillations with a period of 150 to 200 µs;these oscillations reflect the periodic tunneling current of atoms trapped in thelowest band of the optical lattice. As estimated in Section 3.2, the coherence ofthe tunneling lasts quite long: the tunneling signal is noticeable over at least fivetunneling periods. At intensities above ≈5 mW/cm2, a higher-frequency periodiccontribution to the tunneling current is observed. To show this more clearly, the


FIG. 6. Experimental (left) and simulated (right) periodic well-to-well tunneling currents in a graylin-perp-lin optical lattice. The curves are shifted vertically by amounts proportional to the respectivesingle-beam intensities I1. In the experiment, I1 is varied up to 15.5 mW/cm2, and in the QMCWFsimulations up to 15 mW/cm2. (Reprinted with permission from (Dutta et al., 1999).)

FIG. 7. Theoretical (a) and experimental (b) result for the tunneling current at I1 = 10 mW/cm2.The derivative of the magnetic-dipole autocorrelation function (c), d/dτ 〈〈ψ |Fz(t + τ)Fz(t)|ψ〉〉, ex-hibits a time-dependence similar to that of curves (a) and (b). (Reprinted with permission from (Duttaet al., 1999).)

data taken at 10 mW/cm2 are displayed in more detail in Fig. 7, curve (b). Thehigher-frequency oscillations, the maxima of which are highlighted in Fig. 7 byvertical lines, are due to tunneling on the first excited lattice band. This band has


a larger width than the lowest band, resulting in faster tunneling oscillations. Wesimulated the experiment using QMCWF simulations, shown in the right panel ofFig. 6 and in Fig. 7, curve (a). The agreement between theory and experiment inboth the long-period and short-period oscillations is excellent.

The tunneling should also be visible in the autocorrelation function of thez-component of the magnetic-dipole, Fz. At integer multiples of the tunnelingperiod, atoms should mostly reside in the type of wells they resided in initially,and the magnetic-dipole correlation should be positive. Conversely, at half-integermultiples of the tunneling time the atoms should mostly reside in the oppositetype of wells, leading to negative values of the magnetic-dipole correlation func-tion. Very importantly, this correlation should even exist if the atoms were notinitialized in one type of wells. The QMCWF method allows the computation oftwo-time correlation functions, as explained by Marte et al. (1993). Our result forthe parameters of Fig. 7, shown in curve (c) of the figure, confirms our expec-tation: the time derivative d

dτ 〈ψ |Fz(t + τ)Fz(t)|ψ〉〉 is very similar to the actualtunneling current. Note, however, an important difference: the tunneling current,shown in curves (a) and (b) of Fig. 7, reveals tunneling oscillations only after suit-able initialization of the atoms in one type of wells, while curve (c) is obtainedby simulating the evolution of quantum trajectories under steady-state conditions,i.e. without any initialization procedure.

4.2. ANALYSIS BASED ON BAND-STRUCTURE

In the following, we analyze the described experiment using the band structure ofthe system. In the initialization process, most atoms are removed from one typeof wells. Without loss of generality we may assume that the atoms remaining inthe lattice are located in the σ+-wells (see Fig. 8). Also, most of the observedsignal is due to atoms in the lowest pair of bands. Inspecting the eigenfunctionsassociated with the band structure plotted in Fig. 8, it is found that the Bloch statesfor q = 0 in the lowest two bands approximately are

|1〉 ≈ (−1 − sin(2kLZ))|m = −2〉 + (1 − sin(2kLZ)

)|m = 2〉,(12)|2〉 ≈ (1 + sin(2kLZ)

)|m = −2〉 + (1 − sin(2kLZ))|m = 2〉

with respective energies of E1 = 3.76 ERec and E2 = 5.54 ERec. The initial-ization process amounts to the generation of a symmetric superposition of Blochstates, 1√

2(|1〉 + |2〉) ∝ (1 − sin(2kLZ))|m = 2〉, corresponding to the localiza-

tion of most atoms in the σ+-wells (dashed curve in Fig. 8(b)). The time-evolvedstate, 1√

2(exp(− iE1t

h)|1〉 + exp(− iE2t

h)|2〉), is identical with the initial state—up

to an irrelevant global phase—for times that are integer multiples of hE2−E1

. Thisvalue is the well-to-well tunneling period. At half-integer multiples of the tun-neling period, the wave-function is ∝(|1〉 − |2〉) = (1 + sin(2kLZ))|m = −2〉,


FIG. 8. (a) Lowest four lattice bands for I1 = 10 mW/cm2 and detuning 6Γ . The initializationprocedure applied in the experiment amounts to the preparation of symmetric superpositions of Blochstates from the lowest band pair, as indicated by the black circles for the case q = 0. (b) Lowestadiabatic potential Vα=1(Z) (solid line, left axis) and approximate spatial probability distributions ofthe atoms in state |m = 2〉 immediately after the initialization (dashed line, right axis) and in state|m = −2〉 half a tunneling period later (dotted line).

corresponding to a localization of most atoms in the σ−-wells (dotted curve inFig. 8(b)).

The described situation obviously parallels that of periodic tunneling in adouble-well potential, but there are some differences. The above description interms of Bloch functions properly accounts for the periodicity of the superpo-sition state: in the Bloch-state description, a delocalized atom does not tunnelbetween individual wells, but tunnels back and forth between all σ+-wells of thelattice and all σ−-wells of the lattice. Consequently, there is no directionality inthe tunneling process (as opposed to the case of a double-well potential). Further,since the quasimomentum q is quite randomly distributed, the tunneling period

hE2−E1

is not fixed but follows a quite random probability distribution. Inspectingthe band structure in Fig. 8(a) it becomes obvious that the tunneling period vs qexhibits a broad minimum for q = 0. Therefore, the tunneling signals produced bya fairly large group of atoms with q ∼ 0 will add up, while atoms with quasimo-menta very different from zero will produce tunneling signals that tend to canceleach other. The experimentally observed tunneling signal should therefore equal

hE2−E1

evaluated at q = 0. Detailed numerical modeling confirms this assessmentfor all lattice intensities we have studied (Dutta et al., 1999). For the specific caseof Fig. 8, the tunneling period is expected to be TT = h

ERec(5.54−3.76) = 150 µs.This value is experimentally observed (see Fig. 7, where the lattice parameters arethe same as in Fig. 8).


In the following, we comment on the dependence of the tunneling period onthe lattice intensity. As observed in Fig. 6, the tunneling period exhibits a broadminimum around I1 ∼ 5 mW/cm2. While this behavior accords with predictionsbased on the band structure and QMCWF simulations, it evades an immediateexplanation based on the adiabatic potentials of the system. The lowest adia-batic light-shift potential Vα=1(Z) continuously decreases with decreasing inten-sity. As a result, one would expect the tunneling time to continuously decreasewith decreasing intensity. Such a trend is observed experimentally in the rangeI1 > 5 mW/cm2. However, in the range I1 < 5 mW/cm2 the experimental obser-vation contradicts the expected trend. The discrepancy can be partially attributedto the presence of a gauge potential that needs to be added to the lowest adiabaticpotential in order to improve the description of the tunneling behavior. The gaugepotential, predicted to occur in light-shift potentials by Dum and Olshanii (1996)and observed by Dutta et al. (1999), is

(13)Gα=1(Z) = −(h2/2M)⟨Ψα=1(Z)

∣∣ d2

dZ2

∣∣Ψα=1(Z)⟩,

where |Ψα=1(Z)〉 is the position-dependent internal state associated with thelowest adiabatic potential. The potential Gα=1(Z) is always positive and is, inthe present case, intensity-independent and peaks at the maxima of the low-est adiabatic potential Vα=1(Z). Adopting the notion that the sum potentialVα=1(Z)+Gα=1(Z) determines the tunneling time, the gauge potential increasesthe tunneling time and improves the agreement between the observed tunnel-ing time and the tunneling time one may estimate based on a simple potentialpicture. However, the addition of an intensity-independent potential Gα=1(Z) toVα=1(Z) cannot explain why at the lowest intensities in Fig. 6 the tunneling timeincreases with decreasing intensity. It turns out that in this low-intensity domainthe notion that the wave-function adiabatically evolves (and tunnels) on a sin-gle potential breaks down. The underlying reason for this breakdown is that theBorn–Oppenheimer approximation separating the dynamics of internal and ex-ternal degrees of freedom fails. A more detailed discussion of this effect can befound in (Dutta and Raithel, 2000).

5. Influence of Magnetic Fields on Tunneling

5.1. MOTIVATION AND EXPERIMENTAL OBSERVATIONS

In Section 4 we have studied well-to-well tunneling for the case of zero appliedmagnetic field. In that case, the σ+- and σ−-wells of the lattice are identical, andthe tightly bound states in them are degenerate with each other. This symmetrycan be lifted by the application of a magnetic field parallel to the lattice-beam


FIG. 9. (a) Lowest adiabatic potential (left) and band structure (right) of a gray lin-perp-lin op-tical lattice with single-beam intensity I1 = 11 mW/cm2 and detuning 6.3Γ with respect to the5S1/2, F = 2 → 5P1/2, F

′ = 2 component of the 87Rb D1-line for B‖ = 0 mG. (b) Same as (a),except that a longitudinal magnetic field B‖ = 12 mG is applied. The arrows in the band structuresindicate the coherences that dominate the experimentally observed well-to-well tunneling signals,shown below in Fig. 10.

directions, B‖. In this section, we study the effect of such longitudinal magneticfields on the tunneling dynamics.

In Fig. 9 we show the influence of a weak longitudinal magnetic field on thelowest adiabatic potential and the band structure of a typical gray lattice. Thesingle-beam lattice intensity is I1 = 11 mW/cm2 and the detuning 6.3Γ withrespect to the 5S1/2, F = 2 → 5P1/2, F

′ = 2 component of the 87Rb D1-line.Panel (a) shows a situation similar to that in Fig. 8. In panel (b), a magnetic fieldB‖ = 12 mG parallel to the quantization axis of the lattice is added. This fieldcan be accounted for in Eq. (8) and in QMCWF simulations by adding a position-independent term

∑m gFμBB‖m|m〉〈m| to the atom-field interaction, where the

g-factor gF = 1/2 for 5S1/2, F = 2 of 87Rb. According to the predominantmagnetic states in the σ+- and σ−-wells of the lattice, indicated in the left panelsof Fig. 9, the longitudinal magnetic field lowers the σ−-wells (left well) by anenergy of h × 1.4 kHz = 0.386 ERec per mG and raises the σ+-wells (rightwell) by the same amount. Due to these shifts, clearly seen in the left panel inFig. 9(b), the localized states in the two types of wells are tuned out of resonancewith each other. Using selected magnetic-field values, higher-lying states in the


down-shifting type of wells can be brought into resonance with lower-lying statesin the up-shifting wells. In Fig. 9(b), for instance, the lowest (and only) state inthe up-shifted well is resonant with the first excited state in the down-shifted well.These resonances manifest themselves in the tunneling behavior of the system.

In order to measure the dependence of the tunneling frequency on B‖, weapply a variable, well-defined (to within 1 mG) magnetic field parallel to the lat-tice laser beams using a set of Helmholtz coils. Atoms are laser-cooled into theslightly magnetized lattice using the same procedure as described in Section 4.In order to observe the well-to-well tunneling, the atoms are initialized into oneset of wells via application of a 10 µs long σ -polarized pulse resonant with the5S1/2, F = 2 → 5P3/2, F

′ = 2 transition. This pulse selectively removes atomsfrom one type of lattice wells into the 5S1/2, F = 1 level and thereby initial-izes the remaining atoms in the other type of wells. At the end of the initializationpulse the remaining atoms begin to tunnel in-phase between the wells. The tunnel-ing current is measured non-destructively and in real-time by detecting the powerexchange between the σ+- and σ−-polarized components of the lattice beamsafter their interaction with the atom cloud.

Figure 10(a) shows the tunneling current measured for the indicated values ofthe applied magnetic field B‖. The signals are approximately symmetric about thevalue B‖ = 0, and the tunneling frequency generally increases with |B‖|. The ex-perimental data are compared with corresponding results of QMCWF simulations,shown in Fig. 10(b). We observe satisfactory agreement between experiment andsimulations. The simulations show more high-frequency modulations than the ex-perimental data; this difference may be attributed to the limited bandwidth of thephotodiode detector used in the experiment. Also, the simulated data are moreasymmetric about B‖ = 0 than the experiment. In this regard, it is noted that acertain degree of asymmetry is to be expected, because the polarization of theinitialization pulse is kept fixed. Consequently, for one field polarity the deeperwells are depleted of atoms during the initialization, while for the other polaritythe less deep wells are depleted. This asymmetry in the initialization sequencecauses some differences between tunneling signals observed for magnetic fieldsB‖ of equal magnitude but opposite polarity.

The tunneling frequencies νT of the experimental data in Fig. 10 have beendetermined graphically and are represented as a function of B‖ in Fig. 11. Asexpected from the picture presented in Fig. 9, the tunneling frequency is symmet-ric in B‖ (even though the underlying measured tunneling-current curves are notentirely symmetric in B‖). The tunneling frequency generally increases with themagnitude of B‖. However, at values B‖ = ±12 mG secondary minima are ob-served. This behavior can be qualitatively explained by considering a double-wellpotential with two bound states in each of the wells. In the following, we firstdiscuss such a simplified double-well model that largely reproduces Fig. 11. Wethen turn to a more rigorous analysis based on the band structure of the lattice.


FIG. 10. (a) Tunneling currents measured for the indicated values of the longitudinal magneticfield B‖ and lattice parameters as in Fig. 9. (b) Corresponding results obtained from QMCWF simu-lations.

FIG. 11. Tunneling frequencies νT vs applied longitudinal magnetic field B‖ obtained from theexperimental data in Fig. 10(a).


5.2. INTERPRETATION OF THE RESULTS BASED ON TWO MODELS

In a simplified double-well picture, the situation of Fig. 9 can be described bylocalized ground and excited states |0L〉 and |1L〉 in the left well of a double-wellpotential, and two analogous states |0R〉 and |1R〉 in the right well. The tunneling-induced coupling between the ground states, |0L〉 and |0R〉, is denoted c00, andthat between the excited states c11. The off-resonant couplings between |0L〉and |1R〉 and |1L〉 and |0R〉 are both c01. The band structure in Fig. 9 suggeststhe use of the following values for the coupling constants: c00 = h × 3 kHz,c11 = h× 12 kHz, and c01 = h× 7 kHz. The magnetic-field-free energies of thelocalized ground (excited) states are estimated as 0.5h×fosc and 1.5h×fosc withan oscillation frequency fosc = 35 kHz. In analogy with the situation in Fig. 9,the effect of a magnetic field B‖ is that the states in the left well are down-shiftedby d × B‖ with d = 1.4 kHz per mG, while the states in the right well are up-shifted by that same amount. The corresponding Hamiltonian, represented in thebasis {|0L〉, |0R〉, |1L〉, |1R〉},

(14)H =

⎛⎜⎜⎝12fosc − d × B‖ c00 0 c01

c0012fosc + d × B‖ c01 0

0 c0132fosc − d × B‖ c11

c01 0 c1132fosc + d × B‖

⎞⎟⎟⎠has eigenvalues vs B‖ as shown in Fig. 12(a). The lowest tunneling frequenciesin this system are given by the energy differences between the lowest two of

FIG. 12. (a) Energy levels of the double-well model system discussed in the text vs B‖. Thetwo arrows identify the energy differences that correspond to the lowest tunneling frequencies of thesystem. (b) Lowest tunneling frequencies as indicated in (a) vs B‖.


FIG. 13. Frequencies of the coherences between the lowest band pair at q = 0 (squares) andbetween the first- and second-excited band at q = ±1 (triangles) vs magnetic fieldB‖. The frequenciesare determined as shown by the arrows in Fig. 9.

eigenvalues and the next-higher pair of eigenvalues, as indicated by the two arrowsin Fig. 12(a). Plotting these two tunneling frequencies vs B‖ and assuming thatthe lower one will be dominant in an experimental observation of the tunneling,we obtain a plot that closely resembles the actual experimental observation in thelattice (compare Fig. 11 with Fig. 12(b)).

For a more rigorous description using the band structure of the system, we firstrecall that the experimentally observed tunneling signal is due to coherences be-tween neighboring bands, which are generated by application of the initializationpulse. Since the atoms in the lattice follow a fairly random distribution in qua-simomentum q, the coherences that produce observable effects will come fromregions in the band structure where the energy difference between neighboringbands is stationary in q. The coherences at q-values as identified by the arrowsin Fig. 9 are likely to produce a signal, because their frequencies exhibit a broadmaximum as a function of quasimomentum q. Plotting the frequency differencebetween the lowest band pair at q = 0 and between the first- and second-excitedband at q = ±1 vs B‖, we obtain the curves shown in Fig. 13. The lower envelopeof these curves, identified by the filled symbols, agrees well with the experimen-tally obtained result shown in Fig. 11.

The findings obtained in this section lead to the following summarizing as-sessment. For B‖ = 0, the atoms tunnel resonantly on the lowest band betweenthe σ+- and σ−-wells of the lattice. In the range 0 < |B‖| < 6 mG thetunneling frequency increases, because the tunneling becomes increasingly off-resonant with increasing |B‖|. The observed tunneling frequency has a maximumat |B‖| ≈ 6 mG, because at this field value the tunneling between the ground


bands associated with the σ+- and σ−-wells is off-resonant by about the sameamount as tunneling from the ground band of the up-shifted wells into the firstexcited band of the down-shifted wells. Around |B‖| ≈ 12 mG the tunnelingfrequency has as shallow secondary minimum because at that field value the tun-neling from the ground band of the up-shifted wells into the first excited bandof the down-shifted wells is resonant. The observed dependence of the tunnelingfrequency on B‖ therefore reflects the passage of the system through a couple oftunneling resonance. At each resonance, a minimum of the tunneling frequencyoccurs.

6. Sloshing-Type Wave-Packet Motion

6.1. WAVE-PACKETS LOCALIZED IN SINGLE LATTICE WELLS

So far, we have studied well-to-well tunneling in gray lattices initiated by an ini-tialization laser pulse that removes atoms from one type of wells. In the following,we will be interested in wave-packets evolving in single wells of the lattice, withnegligible tunneling-induced well-to-well coupling. An optical lattice must haveat least a couple of tightly bound bands so that meaningful wave-packets in singlewells can be formed. Lattices of the red-detuned type discussed in Section 3.1usually are deep enough to support a number of tightly bound states that is suffi-cient for the excitation of single-well wave-packets. In contrast, gray lattices in avanishing magnetic field support only one tightly bound band and will thereforenot allow one to form superposition states in single wells (see Sections 3.2 and 4).This situation changes when a longitudinal magnetic field of order B‖ ∼ 100 mGis added, because one type of lattice wells is deepened by the magnetic field,while the other type of wells essentially disappears. For sufficiently large B‖, thedeepened wells support a sufficient number of tightly bound bands to form wave-packets in single wells.

In this section, we will consider atomic center-of-mass wave-packets that areexcited by a sudden displacement of the lattice, which causes a subsequentsloshing-type wave-packet motion that takes place in single lattice wells (Raithelet al., 1998). In particular, we will be interested in the role of the longitudinalmagnetic field B‖ in enabling the formation of these sloshing-type wave-packets.

It is noted that, while we refer to the sloshing-type atomic states as wave-packets, these are not pure quantum states but are more properly described by atime-dependent density operator. The atoms are initially prepared by laser coolingin an incoherent, quasithermal density operator ρ0 with most population residingin the lowest oscillatory states of the lattice wells. The wave-packet initializa-tion, which is implemented via a lattice shift, amounts to a certain excitation ofhigher-lying oscillatory states. The lattice-shift-induced excitation is an entirely


coherent process. In a harmonic approximation of the lattice wells, the excitationis described by the application of the usual shift operator Dα = exp(αa† − α∗a)on ρ0, yielding the density operator after the shift,

(15)ρ = Dαρ0D†α.

There, a and a† are the lowering and raising operators, the complex number αis the shift parameter, and |α|2 is an approximate measure for how high up theharmonic energy ladder the state is shifted. After the shift is applied to the lat-tice, fluorescence of the atoms and the associated laser cooling cause damping ofthe sinusoidal oscillation, Tr[ρ(t)Z]. In the experiment, damping can be directlyobserved as a decay of a sinusoidal signal. The experimentally measured signaldecay is not only due to coherence decay but, in large parts, also due to the anhar-monicity of the lattice wells (i.e. the harmonic-oscillator formalism can merelyserve to provide a qualitative discussion). In this context, we also re-iterate thatQMCWF simulations one may perform in order to model optical-lattice experi-ments are entirely accurate in that they yield an approximation to the evolution ofa density operator (as opposed to the evolution of a single wave-function). Keep-ing these clarifications in mind, we will continue to use the term “wave-packets”for the states discussed this section.

6.2. EXPERIMENTAL STUDY OF SLOSHING-TYPE MOTION

IN A MAGNETIZED GRAY LATTICE

In order to experimentally initiate sloshing-type motion, we apply a voltagechange to a phase modulator positioned in one of the lattice beams after a steady-state of laser-cooling is achieved. The resultant sudden shift in lattice position,amounting to 0.1λ, initializes the center-of-mass wave-packet motion on thelowest adiabatic potential. To measure the subsequently occurring sloshing-typewave-packet motion, the lattice beams are directed onto photodiodes after theyhave interacted with the atomic cloud. The power difference signal �P(t) be-tween the photodiodes is then measured. The average electric-dipole force actingon the atoms is related to the measured power exchange �P(t) via

(16)�P(t) = Nc〈F 〉,where 〈F 〉 is the electric-dipole force averaged over the ensemble of N atoms(Raithel et al., 1998).

In Fig. 14(a), sloshing-type wave-packet oscillations measured for the indi-cated values of B‖ are represented. The utilized lattice is a gray optical latticewith single-beam intensity I1 = 11 mW/cm2 and a detuning of +6.3Γ relativeto the F = 2 ↔ F ′ = 2 component of the 87Rb D1-line (same as in Section 5).As evident in Fig. 14(a), there are two regimes of the sloshing-type motion. In


FIG. 14. (a) Wave-packet oscillations for the indicated values of B‖. (b) Wave-packet oscillationsfor low B‖-values varied in small steps of 1.2 mG. (c) Frequency νslosh of the wave-packet oscilla-tions obtained from the experimental data (squares) and from the geometry of the adiabatic potentials(triangles). The experimental uncertainty �νslosh = 15%.

a low-magnetic-field regime, |B‖| < 10 mG, there is no clear signature of asloshing-mode wave-packet oscillation. There are, however, some reproduciblelow-amplitude higher-frequency structures that were also observed in QMCWFsimulations (not shown here). The absence of sloshing oscillations in the low-magnetic-field regime, |B‖| < 10 mG, can be attributed to two factors. First, thelattice wells are not deep enough to support more than one tightly bound state.Therefore, it is not possible to form wave-packet states that oscillate in individualwells. Second, in shallow lattices such as gray lattices the sloshing-type motioncompetes with rapid tunneling between neighboring wells. Tunneling leads to arapid spread of wave-packets over multiple lattice wells. The dipole force 〈F 〉 av-eraged over spread-out wave-packets will always be near zero, as the averagingwill extend over regions of both polarities of the force. Therefore, in the low-magnetic-field regime any net sloshing-type wave-packet signal that might still


appear will be very weak and it will reflect a complicated superposition of effectscaused by both tunneling and sloshing-type dynamics. This is demonstrated in thedetailed plot in Fig. 14(b).

With increasing magnetic field, the σ+- and σ−-wells of the lattice become in-creasingly asymmetric, leading to a suppression of well-to-well tunneling on thelowest tightly bound bands and to the appearance of more and more tightly boundbands in the deepening wells. Therefore, for large enough |B‖| we expect to ob-serve the signatures of well-defined sloshing-mode oscillations. Fig. 14(a) showsthat the high-magnetic field regime in which the gray lattice supports sloshing-mode oscillations approximately is |B‖| > 20 mG. As expected in this regime,the frequency of the sloshing-mode oscillations is dependent on the shape of thelowest adiabatic potential near its minima. In Fig. 14(c), the solid squares repre-sent the frequencies νslosh of the sloshing oscillations as a function of the magneticfield obtained from the experimental data. The observed trend reflects the fact thatboth the depth and the curvature of the wells increase with increasing magneticfield. Approximating the lattice wells by harmonic potentials that match the curva-tures of the lowest adiabatic potential at the minima, we find estimated oscillationfrequencies shown by the triangles in Fig. 14(c). The estimated frequencies are inquite good agreement with the experimental values, but show a systematic trendof being ∼20% larger. The deviation may be attributed to the anharmonicity of thelattice wells. The anharmonicity of the wells also is the main cause of the signaldecay that is observed to take about five wave-packet oscillations (see Fig. 14(a)).

7. Conclusion

In this chapter we have compared different types of one-dimensional opticallattices with regard to their laser-cooling performance and their suitability forexperiments on well-to-well tunneling and sloshing-type wave-packet motion.The theoretical models used have been explained in some detail. While numer-ical results are provided for rubidium, the results and conclusions are expectedto be representative for optical lattices of many atomic species (alkaline atoms,metastable noble gases, etc.). Only one type of lattice has been identified thatprovides reasonably fast and efficient laser cooling, high tunneling rates, andsteady-state coherence decay rates that are significantly lower than the tunnelingrate.

In the second half of the chapter, we have presented typical results on wave-packet motion in a gray optical lattice. The main findings can be summarized asfollows. In a regime of very low magnetic fields parallel to the lattice beams, thepredominant dynamics of atoms is due to well-to-well tunneling. We have ex-plained the magnetic-field dependence of the tunneling using a simplified double-well potential model as well as the exact band structure of the system. We found


that the tunneling current vs applied magnetic field exhibits signatures of a cou-ple of tunneling resonances. In a domain of higher magnetic fields, the tunnelingrate between the lowest localized center-of-mass states of the lattice generallydecreases, and the number of localized states in the lattice wells increases. Con-sequently, in the domain of higher magnetic field the gray lattices are found tosupport sloshing-type wave-packet oscillations.

In future research, we intend to study non-linearities in the discussed types ofwave-packet motion. We have observed a significant dependence of sloshing-typewave-packet oscillations on the average atom density in the lattice. This depen-dence is due to the back-action of the wave-packet oscillation on the refractiveindex which the oscillating atomic ensemble presents to the lattice beams. The re-sultant position- and time-dependence of the lattice phase and its coupling to theatomic motion amounts to a non-linear atom-field coupling and to the presenceof long-range atom-atom interactions in the lattice. Interesting avenues for furtherresearch include the study of wave-packet motion of Bose–Einstein condensates(BECs) in lattices. In this case, additional non-linearities will arise from position-and time-dependent mean-field potentials. The natural continuation of the dis-cussed work on tunneling will be to investigate spinor-BECs in spin-dependentoptical lattices.

8. Acknowledgement

This work was supported by the National Science Foundation (PHY-0245532).

9. References

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FEMTOSECOND LASER INTERACTIONWITH SOLID SURFACES: EXPLOSIVEABLATION AND SELF-ASSEMBLY OFORDERED NANOSTRUCTURES*

JUERGEN REIF† AND FLORENTA COSTACHEBrandenburgische Technische Universität Cottbus, Konrad-Wachsmann-Allee 1,03046 Cottbus, Germany

BTU/IHP JointLab, Erich-Weinert-Strasse 1, 03046 Cottbus, Germany

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2282. Energy Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

2.1. Absorption by Ionization of Valence Band Electrons: Multiphoton and Tunneling Ion-ization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

2.2. Impact Heating/Free Carrier Absorption . . . . . . . . . . . . . . . . . . . . . . . . . 2333. Secondary Processes: Dissipation and Desorption/Ablation . . . . . . . . . . . . . . . . . 233

3.1. Desorption/Ablation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2343.2. Femtosecond Laser Ablation from Silicon . . . . . . . . . . . . . . . . . . . . . . . . 2393.3. Recoil Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2403.4. Surface Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2465. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2496. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

AbstractThe fundamentals of interaction between intensive laser pulses and solid surfacesare reviewed. In order to distinguish the relevant phenomena from secondary ef-fects, e.g., laser heating of the plasma plume formed upon ablation, emphasis isplaced on the action of ultrashort pulses. The present picture of energy absorptionand dissipation dynamics is discussed, and transient and permanent modification ofthe surface, in particular its morphology, are considered.

* It is a great pleasure and honor to dedicate this contribution to Prof. Herbert Walther on the oc-casion of his 70th birthday. Not only did his innumerable contributions to the basic understanding ofand deep insight into quantum and optical physics open the way for tackling the work presented here,but also he was and still is an outstandingly inspiring teacher with an ever continuing impact. Happybirthday!

† E-mail: [email protected].


DOI 10.1016/S1049-250X(06)53008-3

227


228 J. Reif and F. Costache [1

1. Introduction

Among the many specific types of laser interaction with matter, studied at present,those exploiting the very high optical fields attainable play a peculiar role [1].At intensities in the range of 1012 . . . 1018 W/cm2, easily accessible with presentfemtosecond laser systems, the corresponding electric field of ≈109 . . . 1012 V/mis no longer small compared to the intra-atomic Coulomb field (≈1011 V/m forthe hydrogen 1s-electron). Consequently, the interaction cannot be considered asa weak perturbation of the irradiated matter anymore, with a linear response. In-stead, a transient state is created, where the electrons “feel” a combination ofthe nuclear Coulomb field with the electric field of the incident radiation. Inatomic physics this led to the theoretical “dressed atom” approach [2], initiallydeveloped for a strong monochromatic cw-field driving a two level system [3].A typical observation under these conditions is that the cross-section for multi-photon processes approaches or even exceeds that of linear interaction [4], suchas for high-harmonics generation [5–7], above threshold ionization [8–11], or theemergence of relativistic effects [12].

In condensed matter, in particular in solids with quasi-localized electrons, e.g.,dielectrics or semiconductors, this transient high-field state can have even moredramatic effects than in free atoms. Since the crystalline structure is a conse-quence of equilibrium binding conditions for the atoms’ outer electrons, anychange of electronic configuration will strongly influence the crystalline stabil-ity [13,14]. Thus, an excitation faster than any electron-phonon collision time,i.e. on a sub-picosecond time scale, tends to almost immediately “soften” thematerial [15–17], long before any thermodynamic melting sets in [18,19] via anequilibration between electron and lattice temperature [20]. A consequence of thisbreakdown of crystal stability is the desorption or ablation of particles from thematerial surface [21].

For technical applications, this type of interaction plays a very important role asthe basis for most materials processing techniques, e.g., drilling, cutting, shaping,and for medical use, e.g., (eye) surgery, dermatology, etc. However, the nature anddynamics of energy coupling and dissipation as well as subsequent processes ona microscopic scale are still subject of ongoing research.

In most practical cases, for long-pulse excitation (> several ps), the basicprocesses are masked by secondary effects. Typically, already during the pulseduration material removal takes place. Thus, strong interaction of the laser withthe ablation (plasma) plume is expected, consuming a substantial part of incidentenergy for heating the plasma via inverse bremsstrahlung [22,23]. Then, plasmaerosion of the surface is no longer negligible. Also, thermodynamic processes,like mere target heating, must be taken into account. These secondary effectsmake a study of fundamental mechanisms and dynamics rather complex and com-plicated. Fortunately, today’s ultrafast laser sources open the way of separating

2] FEMTOSECOND LASER ABLATION 229

first and second order effects, thus getting closer to the “atomic” laboratory, sosuccessfully studied for free particles. In fact, transient instabilities, typical forhigh field interaction, can show up easily under ultra-short pulse irradiation [15,24–26] where, for certain conditions, Coulomb explosion of an electrostaticallyunstable surface is the basic ablation channel.

In the following, we review the present picture of laser ablation dynamicsfrom transparent dielectric crystals and semiconductors (e.g., silicon). We willstart with current models and their experimental equivalent for the energy cou-pling, continue with sketching follow-up processes of energy dissipation andparticle emission, and, finally, discuss the consequences occurring after the ab-lation/desorption process. There, i.e. well after the laser pulse termination, wewill show, relaxation dynamics are far from (thermodynamic) equilibrium for afree evolving system.

2. Energy Coupling

We will start with some considerations about the basic mechanisms of energycoupling between laser and target, in particular for dielectric materials. First, how-ever, a very peculiar feature of experiments on laser ablation/desorption1 from asolid crystal must be considered. Unlike in experiments with free atoms, where themicroscopic “laboratory” is always well-characterized and, in most cases, doesnot change its intrinsic properties during repetitive interaction, the crystalline sur-face changes with each particle removal. Each loss of a particle results in thegeneration of a microscopic defect at the surface, associated with a change in theenergy band structure, e.g., the introduction of defect states within the bandgap.Further, increasing surface erosion may result in a surface roughening sufficientto give rise to a local enhancement of the optical electric field. These effects cansubstantially change the coupling efficiency for sub-bandgap photons from pulseto pulse. Only after a certain number of desorption events, the surface “decompo-sition” reaches a kind of steady state, and additional desorption does not changethe average number of defects in the irradiated area any more. The irradiationphase before reaching this steady state is usually termed “incubation” [27,28] andis shown, typically, in Fig. 1 [29].

The incubation does, however, not only serve to generate a “stable” defect den-sity at the surface. It also helps to overcome another problem usually encountered:a typical surface of a solid target does, usually, not consist of the actual target ma-terial. Instead, it is often covered with thin films of contaminants, e.g., water,

1 Here, we apply the following convention: “desorption” denotes the taking away of individual par-ticles from the crystal surface whereas “ablation” refers to more massive material removal.


FIG. 1. Effect of incubation for the ablation from BaF2 (emission of Ba+ ions) at different laserintensities [29].

oxides, CO2, etc., both chemisorbed and physisorbed. During incubation, thesefilms are removed and, if the experiments are conducted under sufficiently goodvacuum conditions, do not re-grow during the course of subsequent investigations.

In the following, we will only consider nonmetallic targets with an incubatedsurface, i.e. with a stable average defect density and practically free from conta-minations.

2.1. ABSORPTION BY IONIZATION OF VALENCE BAND ELECTRONS:MULTIPHOTON AND TUNNELING IONIZATION

The fundamental understanding of the energy coupling between valence bandelectrons in a solid and a strong electromagnetic wave at sub-bandgap frequency


FIG. 2. Ionization yield for high laser intensity according to the Keldysh model.

has been developed already in 1965 by Keldysh [30,31]. He discusses two prin-cipal mechanisms, multiphoton ionization and tunneling ionization, the latterprevailing at very high light intensity as shown in Fig. 2.

2.1.1. Multiphoton Ionization

The possibility to bridge energy gaps larger than the photon energy by the simul-taneous interaction/absorption of several photons with a sufficient sum energyhas been termed “multiphoton” interaction and has been studied extensively dur-ing the last four decades, mainly in atomic and molecular systems but as well insolids. The interaction becomes possible if the photon density is sufficiently highfor a reasonable probability for several photons being at the same spatio-temporalinteraction site simultaneously.

In a semiclassical description this situation is equivalent to a reasonably highelectric field of the incident wave, which still can be introduced into the target’sHamiltonian via a perturbational ansatz. There, in principle, the light electricfield induces a periodic deformation of the—initially symmetric—Coulomb fieldbinding the electrons (cf. Fig. 3).

This can be considered as the generation of an oscillating dipole. In the Hamil-tonian, the increased anharmonicity leads to a coupling of two Eigenstates of theunperturbed Hamiltonian. The induced dipoles are equivalent to a polarization ofthe medium, P, which, for conventional light intensities, is just proportional to thelight electric field, E, where α is the polarizability of the medium:

(1)P = αE.


FIG. 3. Influence of a strong light field on the binding potential of an electron (perturbationmodel): (a) unperturbed near-harmonic potential, two “Eigenstates” are indicated by highlighted disks;(b) perturbed potential. The spherical symmetry is perturbed in one direction, leading to the induceddipole μ. The two Eigenstates are coupled (one in x- one in y-direction).

For higher intensity, the anharmonicity becomes increasingly larger. In Perturba-tion Theory, this is accounted for by developing the polarization in a power seriesof the electric field:

(2)P = (α + α(2)E + α(3)EE + α(4)EEE + · · ·)Ewhich involves contributions of higher harmonics of the incident field, as caneasily be seen when taking the electric field as E = E0 exp[i(ωt − kr)]. In theHamiltonian, this is equivalent to the coupling of more and more Eigenstates inthe resulting wave function.

For the case of absorption, it must be considered that the electromagnetic fieldenergy is given by the square of the field, i.e. the nonlinear contributions are pro-portional to the square of the respective term in the polarization (2), with thecoupling given by the imaginary part of the polarizability. Correspondingly, theprobability for an n-photon transition is found to be proportional to the nth powerof the incident intensity:

(3)P(n) ∝(P(n)E)2 ∝ (Im(α(n))En−1E

)2 = (Im(α(n)))2E2n ∝ In.

As a consequence, in typical experiments multiphoton transitions can be identifiedby plotting the absorbed energy (e.g., the ionization rate) as a function of incidentintensity in a log-log plot: straight lines are obtained with the slope indicating thenumber of photons involved [32] (cf. Fig. 2).

In a real system, the situation may be complicated by the detailed energy struc-ture of the material, which is contained in the explicit shape of α. According toFermi’s Golden Rule [33], resonance denominators can enhance the polarizability[34] and even reduce the nonlinearity, if an intermediate (m-photon) resonance isdirectly met.


FIG. 4. Tunneling ionization: (a) unperturbed near-harmonic potential with corresponding Eigen-states (b). Influence of a strong electric field (indicated by the dash-dotted line): The potential wellis—on one side—decreased so much that several of the unperturbed energy levels are coupled to thevacuum and thus directly ionized.

2.1.2. Tunneling Ionization

When the field of the incident light increases further, it might approach theCoulomb field binding the electrons (for the hydrogen atom this is at the orderof 1011 V/m, corresponding to an intensity of 1016 W/cm2). Already at about10% of that intensity, the perturbational treatment appears no longer justified. Inthis the regime, the potential may be considered to be so strongly changed, that di-rect (above-barrier) or tunneling ionization appears to be the dominant ionizationprocess (Fig. 4) [30,35–37].

2.2. IMPACT HEATING/FREE CARRIER ABSORPTION

Different from free particles, the ionization does not simply result in an escapeof the excited electrons. Only those close to the surface, i.e. within the averageinelastic mean free path [38] according to the “universal curve” (Fig. 5), can reallyleave the sample (if they are not held back by space charge effects, see below). Allother electrons will be free carriers in the conduction band where they can absorbadditional energy (“free carrier absorption”) [39]. In fact, the energy gained canbe larger than the bandgap, and the electrons can generate further conduction bandelectrons by impact ionization [30,35,36,40–43]. This process is characterized bythe absorption of very substantial amounts of energy in an avalanche process.

3. Secondary Processes: Dissipation and Desorption/Ablation

In the following, we will only consider only processes leading to the removal ofparticles from the surface of the irradiated target, i.e. ablation or, at a low parti-


FIG. 5. The “Universal Curve” of electrons’ inelastic mean free path (IMFP) in a solid in depen-dence on their kinetic energy.

cle emission rate, desorption. Whatever happens within the bulk of the materialis beyond the scope of this contribution. For these considerations, we refer toexperiments with laser pulses of 100 fs duration at a wavelength of 800 nm.

At moderate incident intensity, i.e., below ≈1013 W/cm2, the desorbed par-ticles are mostly positive ions, even of highly electronegative species typicallyforming anions. This ion emission is strongly coupled to the ejection of electronsfrom the surface [44]. In this regime (desorption regime), well below the so-called“ablation threshold” [45], a very large number of incident pulses is required forobservable surface damage to occur. Only at higher fluence, a more massive mate-rial removal sets in [46,47] (ablation regime), characterized by a considerable andeven prevailing contribution of neutrals, and even negative ions can be detected[48,49].

In the following section, some detailed results from these two regimes will beresumed.

3.1. DESORPTION/ABLATION DYNAMICS

First, we consider the desorption regime, i.e. at moderate particle emission wellbelow the classical damage threshold. It is characterized by a strongly nonlinearcoupling of the incident laser energy to the irradiated material, as it is displayedin Fig. 6.

The relevant absorption process can be identified as multiphoton surface ion-ization. This is obvious from the close connection to the observed emission ofelectrons, shown in Fig. 6(b). There, above ≈0.6×1012 W/cm2, the dominant ion-ization process corresponds to a band-to-band transition. Below, the lower slopeindicates ionization of an occupied surface defect state [50–52].

In Fig. 7, the dramatic enhancement of the laser-surface coupling via defectstates within the bandgap is shown, exemplarily, for the ionization of an Al2O3


FIG. 6. Yield of electrons and Ba+ ions from BaF2 irradiated with intense fs laser pulses [51].

FIG. 7. Electron emission from Al2O3 for excitation with 1.5 eV photons. (a) Yield in depen-dence on the incident intensity. Obviously, the nonlinearity of 4 is lower than expected from the 9 eVbandgap, corresponding to a 6-photon transition [51]. (b) Energy structure of Al2O3, indicating defectstates within the bandgap [52].

surface. As indicated in the schematic in Fig. 7(b), the observed 4-photon nonlin-earity corresponds to a transition in the F-center defect, i.e. an oxygen vacancy.The defect can serve as a relay for the ionization and thus reduces the order of


FIG. 8. Positive ion yield from BaF2: cluster emission [25]. The signal for masses above 250 amu(right of the dashed line) are magnified by a factor of 17.

nonlinearity by two. This interpretation is corroborated by the fact, that blue flu-orescence at ≈3 eV was detected in the experiment, corresponding to an internalrelaxation of the F-center excitation via the 3P–1S transition [52].

This significant role of defects, such as missing anions, for the enhancement ofthe coupling efficiency confirms the observed incubation effect (Fig. 1). Indeed,an increasing density of induced defects leads to an increased coupling efficiency,correspondingly.

An analysis of the desorbed positive ions by Time-of-Flight (ToF) spectroscopyreveals that not only monoatomic ions are emitted but also larger clusters. Thisobservation is, as the previous ones, independent on the specific materials understudy, as shown in Fig. 8 for a dielectric (BaF2) and in Section 3.2 for a semicon-ductor (Si).

Closer inspection shows that all these clusters have the same kinetic energy(Fig. 9(c)). This means that their velocities are different, thus excluding gas phaseinteraction as the origin of cluster formation. Consequently, they are emitted intactfrom the surface, indicating massive surface breakdown. In fact, the ions’ velocity,as derived from retarding field measurements (Fig. 9(a), (b)), does not correspondto a thermal Maxwellian distribution. More likely, we find a narrow distributionsuperimposed on a large drift velocity, similar to a seeded molecular beam and


FIG. 9. Kinetic energy of ions emitted from Al2O3. (a) Retarding voltage transmission;(b) corresponding velocity distribution (solid line), compared to a Maxwell–Boltzmann distribution(dash-dotted) and a shifted Maxwellian (dotted line, cf. Eq. (4)); (c) kinetic energy of different des-orbed clusters [28].

described by a modified Maxwellian:

(4)f (t) = A

t4·{N · exp

[−m · (D

t− u)2

2kTu

]}with the drift time t and the “Maxwellian” velocity u.

The resulting kinetic energies are rather high, at the order of 100 eV, with anarrow distribution of only ≈1 eV, indicating a fast, monochromatic ion beam!

The strong coupling between ion and electron emission, the fact that the des-orbed particles are almost exclusively positive ions, and the emitted ion dynamics


FIG. 10. General behavior of desorbed ion yield as a function of incident intensity, for two dif-ferent targets (normalized to the transition between multiphoton-ionization/Coulomb-explosion tohyperthermal particle emission [47]. (Note that the total particle emission, including neutrals increasessubstantially!)

suggest the following desorption scenario [24–26]: the main action of the laser israpid (surface) ionization, with the excited electrons in the surface region rapidlyleaving the sample. This results in a fast positive surface charging, inducing anelectrostatic instability. Much faster than any charge equilibration can take place(in a dielectric or semiconductor), the instable surface decomposes via Coulombexplosion (this explains the identical kinetic energies for all singly charged surfacefragments).

At increasing incident fluence, the increase of ion yield with intensity appearsto saturate (Fig. 10). At the same time, the total ablation rate increases dramati-cally, indicating that other than ionic species start to make up most of the ablatedmaterial, pointing more towards a different ablation mechanism than to a changein energy coupling, e.g., avalanche processes. This general behavior does not de-pend on the material investigated, as shown in Fig. 10 where normalized data fromtwo different materials are superimposed.

At the same threshold, also the distribution of the ions’ kinetic energy changes(cf. also [46,47]). This can be seen in the drift-mode2 ToF spectra of positive

2 In drift-mode, the ToF spectrometer is operated without an extraction field between sample andthe spectrometer which, usually, is applied to compensate an initial kinetic energy distribution by alarger drift velocity. Note, that only a moderate mass resolution is obtained, which makes it difficultto distinguish between the different species.


FIG. 11. Drift-mode ToF spectra from BaF2, at intensity (a) close to, (b) well above saturationthreshold (cf. Fig. 10). The solid lines in (a), (b) are a fits assuming identical kinetic energies (fastpeaks), respectively, temperatures (slow peaks) for both species, Ba+ and F+ [47].

ions from BaF2 in Fig. 11: Above the threshold a second peak of slower ionsappears which, with increasing intensity becomes more and more important. Inthis regime, the excitation density in the irradiated volume becomes so high, thata sufficient density of hot electrons is created in the conduction band, which canbe further heated by free carrier absorption [39] and then transfer their energy byelectron–phonon collisions to the crystal lattice [20,39,53]. The associated rapidheating results in new ablation mechanisms to occur, such as phase explosion [54,55]. Consequently, the ablation plume does not only consist of positive ions butalso, and particularly, of neutrals and even negative ions (Fig. 12), which may bethe result of electron capture within the plume [48].

3.2. FEMTOSECOND LASER ABLATION FROM SILICON

Below the single-shot ablation threshold of silicon the high electronic excitationleads to a nonthermal, ultrafast phase change within less than 1 ps. Here, thepercentage of fast ions ejected from the silicon surface increases [56]. Indeed, theresulting mass spectrum reveals positive atomic ions and clusters (Fig. 13, left).The ion kinetic energies distribution shows fast (several tens of eV) and slow(down to few eV) contributions suggesting a superposition between a nonthermalmechanism (such as Coulomb explosion [57,58]) and a thermal-mechanism suchas phase explosion [56].


FIG. 12. Negative ions observed during ablation from CaF2 (panel (a): mass spectrum). As can beseen panel (b), the negative ions’ distribution is much broader and slower than for the positive ions,indicating a different ablation mechanism. In fact, the negatives’ arrival time cannot only be explainedby a lower drift velocity but also a later generation time, for instance, in the ablation plume [49].

FIG. 13. Positive ion mass spectrum from a silicon surface irradiated by ∼100 fs laser pulses(left); nonmass resolved spectrum (drift-mode): peaks attributed to Si+ fast ions and Si+ slow ions.

3.3. RECOIL PRESSURE

The emission of many particles at substantial kinetic energies is associated, inturn, with a considerable recoil pressure onto the sample. This results in a non-negligible pressure load on the interaction region. For silicon, it is known thatlocalized high pressure in the GPa range results in phase transformations in thecrystal lattice [59,60]. In simple words, some of the atoms are pushed out of theirusual position and squeezed into the surrounding part of the lattice, thus chang-ing coordination and distances. The resulting new phases, e.g., hexagonal, bcc,


FIG. 14. Raman spectra from the ablated area on p-doped Si(100), taken at different areas of aspot of a few µm depth [57]: From bottom to top, the traces are taken 1 at a virgin area (reference),2 in the center of a flat crater, 3 in the ripples area, and 4 at the steep wall of a deep (several 10 µm)crater.

or rhombohedral silicon, can be detected by micro-Raman spectroscopy of thecorresponding, new phonon frequencies [61–63] as shown, exemplary, in Fig. 14.Note that, due to the penetration depth of the 532-nm Raman laser of ≈1 µm insilicon, all spectra are dominated by the TO-phonon peak of crystalline silicon at520.7 cm−1.

Interestingly, the different spectra show distinctly different behavior in the re-gion close to the TO-phonon peak (cf. Fig. 15). In Fig. 16, a more detailed analysisof this situation is presented, fitting the experimental curves to a sum of knowncontributions from different silicon structures [63], namely a broad peak dueto amorphous silicon [a-Si (TO)] at 475 cm−1, a peak attributed to zincblende(Wurtzite) structure (Si-IV) at 516 cm−1, and a contribution from polycrystal-lites, resulting in a broadening and red-shift of the c-silicon TO peak. Obviously,in the ripples zone, a significant amount of the Si-IV, polymorph is generated,whereas at the crater wall the presence of micro- and nanocrystallites is indicated.

In fact, molecular-dynamics calculations for a Coulomb explosion of sili-con upon highly-charged ion impact [64] have demonstrated, that the massivepositive-ion ejection at high kinetic energies results, indeed, in considerable re-coil pressures of up to 103 GPa, initially, and falling down to about 10 GPa after360 fs.


FIG. 15. Micro Raman spectra of reference (1 in Fig. 14), ripples area (3 in Fig. 14), and craterwall (4 in Fig. 14) in the vicinity of the TO-peak of crystalline silicon (Si-I) at 520.7 cm−1. Note thedifferent asymmetries, at the low frequency side, between spectra 3 and 4.

3.4. SURFACE MORPHOLOGY

Both ablation mechanisms, Coulomb and phase explosion, are associated with astrong transient perturbation of the target in the interaction volume. The inter-action volume, determined by the average phonon mean path, is by far not inthermal equilibrium with the surrounding matrix, with a strong gradient betweenboth regimes. Consequently, the subsequent relaxation is very unlikely to occurvia thermodynamic processes like crystallization or glass formation. Instead, non-linear dynamics models offer possible relaxation pathways.

As can be seen in Fig. 17, regular, aligned periodic structures have developedat the bottom of the ablation crater after several thousand shots, with no obvi-ous relation to the underlying crystal structure [65]. Instead, the laser polarizationappears to play an important role for the orientation. Similar features, termed“ripples” have been known in laser ablation for more than three decades [66,67],classically attributed to an inhomogeneous energy input due to an interference ofthe incident light with a surface scattered wave from the same pulse [68]. Closerinspection of the ripples structures as in Fig. 17 reveals, however, that they arenot compatible with such model: the periodicity can be substantially smaller (atthe order of 100. . . 300 nm) than the wavelength of 800 nm; the regularity is mul-tiply interrupted and interconnected, no dependence on angle of incidence andwavelength can be established.

More likely, the local intensity or irradiation density has a marked influenceon the structure width. This is shown, impressively, in Fig. 18 [69]: even at oneablated spot, the periodicity changes between the center (high intensity, wide


FIG. 16. Fit of Micro Raman spectra (solid gray lines) of (a) ripples area (3 in Fig. 14) and(b) crater wall (4 in Fig. 14) to a combination (-+- lines) of c-Si (TO; 520.7 cm−1/HWHM 2 cm−1),respectively, c-Si+crystallites (520.0 cm−1/HWHM 5.15 cm−1, a-Si (475 cm−1/HWHM 70 cm−1),and Si-IV (Wurtzite; 516 cm−1/HWHM 4 cm−1), the only fit parameters being the relative abun-dance. The different contributions, divided by a factor of 10 for visibility, are indicated in the lowerpart of (a), (b).

spacing) and the edge region (low intensity, narrow spacing). Interestingly, thetransition between both features is abrupt and does not follow the intensity distri-bution.

For really high irradiation density, the feature shape changes dramatically, ex-hibiting wide, flat crests and very narrow, very deep valleys instead of the almostsinusoidal variation at lower intensity. Also, the alignment changes from long,parallel lines to a more meandrous appearance (Fig. 19). Though in the right twopanels, at first sight, the surface appears almost like refrozen from a liquid melt,the deep valleys in between the broad flat crests indicate, that this seems to bevery unlikely. The very narrow trenches between the large flat areas are about


FIG. 17. Typical “ripples” patterns at the bottom of the ablation crater after several thousandpulses (normal incidence) at low ablation rate: regular ordered structures of sub-micron feature size.The double arrows indicate the laser polarization [65].

FIG. 18. Change in ripple spacing across one ablation spot in CaF2 (The double arrow denotesthe direction of laser polarization). Indicated below is a schematic of a corresponding beam profile.Note the abrupt transition between the narrow (≈200 nm) and the coarse (≈450 nm) spacing despitethe smooth intensity profile [69].

1 µm deep, thus exhibiting an aspect ratio of about 10 or more, which is not ex-pected from a refreezing liquid.

On the other hand, at low irradiation dose, i.e. comparably low irradiance or,at very high fluence, very few pulses, only arrays of aligned nanoparticles areobserved, very similar to what is observed for the debris outside the actual ablationcrater (Fig. 20). This suggests a formation of the ripples via an agglomeration ofthese nanoparticles in a scenario similar to a percolation process.


FIG. 19. Change in ripple spacing, shape and orientation on Si(001) with irradiation density (leftpanel: 60,000 pulses, 0.4×1012 W/cm2; middle and right panels: 20,000 pulses, 1.6×1012 W/cm2).The laser polarization is vertical [57].

FIG. 20. Agglomeration of nanoparticles in the ablation area (CaF2: left and middle panels; AFMpictures) and in the debris outside the crater (Si: right panel; SEM picture). (Left panel: 3 pulses,middle panel 5 pulses at 8 × 1013 W/cm2). The dotted line in the middle panel indicates the traceanalyzed by atomic force microscopy, shown in the lower panel and yielding an average particle sizeof ≈200 nm [65].

A very interesting feature, important for an interpretation of the origin of theobserved structures, is shown in Fig. 21: the ripples are not simple, parallel linesbut exhibit very many bifurcations, as are typical for self-organization phenom-ena.


FIG. 21. Bifurcations at the bottom of an ablation crater on CaF2. The double arrow indicates thedirection of the laser polarization [69].

4. Discussion

Bringing all observed features together, the following picture of the laser-materialinteraction evolves: the main action of the incident laser energy is a massiveionization at the irradiated surface. The corresponding positive surface chargingresults in a Coulomb explosion of the positive ions, at a femtosecond time scale,much faster than any intrinsic charge transfer times. This situation is very similarto that induced at the surface by the impact of highly charged positive ions. Theo-retical model calculations [64] demonstrate that the ion ejection starts during thefirst 40 femtoseconds after ionization and continues for several 100 femtoseconds.The target surface is left behind in a state of extreme thermal nonequilibrium andinstability. Also at higher etch rate, when sufficient electrons are created via freecarrier absorption and their energy is transferred to the lattice [39], i.e. substan-tial amounts of also neutral particles are taken away (cf. Fig. 10), such instabilityshould be expected [13–17].

Similar results are found in completely different experiments, namely in typi-cal ion etching configurations [70], where it was shown that this instability tendsto relax to self-assembled structures with a typical feature size in a few-100-nmrange, very similar to those shown above (Figs. 17–20). A particular clue to as-sume a self-organization at the origin of the observed morphology is given inFig. 21, where many bifurcations are shown, typical for such nonlinear-dynamics.Further, the experiments at lower irradiation dose (Fig. 20) indicate a possible wayfor the development of the long, parallel structures: it appears that, first, nanopar-ticles form with a typical size below 200 nm. These particles do not only occurinside the illuminated spot but are also observed in the debris precipitated around,which may have two reasons: either, such particles are already contained in the


FIG. 22. Model of a corrugated thin liquid film, homogeneously charged. The arrows at two high-lighted ions represent their ejection probability.

ejected material, or they coagulate from the pre-formed clusters. As shown inFig. 20, the nanoparticles tend to arrange in long, parallel arrays.

Also, the dependence of the ripples width on the incident intensity and dosesuggests the role of self-organization processes. Assuming the excitation densityto correspond to a “perturbation depth”, this thickness of the instability mightcontrol the order parameters for the determination of the feature size, similar as,e.g., the structure width in a Benard–Marangoni instability [71]: the thicker theinstability layer, the larger is the structure. However, the stepwise variation inFig. 18 cannot be fully explained this way.

In first numerical simulations [72], an attempt is made to simulate the unstablesurface by a thin liquid-like layer (Fig. 22). Then, the first laser pulses are neces-sary to produce a randomly corrugated surface (cf. the incubation). Subsequently,assuming, e.g., surface ionization and Coulomb explosion as the possible ablationmechanism, each laser pulse results in a homogeneous charging of the surface.For positive ions sitting in a valley of the corrugation, the number of next neigh-bor positive charges (holes) is much larger than for an ion sitting on the hill ofthe corrugation. Thus, the desorption probability is much higher in the valleys,resulting in an increased erosion of the valleys and a growing surface roughen-ing. On the other hand, at the hill of the film the surface is particularly stretched.The resulting surface tension acts to minimize the surface by refilling the valleys.Thus, we have a competition between surface roughening (erosion of the valleys)and surface smoothening (diffusion from the hills), well reflecting the postulatedunstable surface.

This situation is similar to that in ion beam erosion, postulating a surface insta-bility after massive erosion of ions [73,74] relaxing by the formation of regularpatterns like those observed in this contribution. It can be studied using well-known formalisms of nonlinear hydrodynamics of thin films [75–77]. An equationof the Cahn–Hilliard or Kuramoto–Sivashinsky type [78,79] can describe thelinear growth of periodic structures (stripes, squares) which turns into a typicalcoarsening upon increased dose of interaction. This equation is of the KPZ type


FIG. 23. Surface structure as a result of a numerical solution of Eq. (6).

(Kadar et al. [80]):

(5)∂

∂th = −V [h]

√1 + (∇h)2 −D�2h,

where the variation of the corrugation, V (h) explicitly contains erosion andsmoothening parameters in integral form. Considering that the interaction onlyis rather local, (5) can be reduced to a partial differential equation

∂

∂th = −V0 + γ

∂h

∂x+ vx

∂2h

∂x2+ vy

∂2h

∂y2−D�2h

(6)+ λx

2

(∂h

∂x

)2

+ λy

2

(∂h

∂y

)2

+ higher orders.

Such equations are well known from nonequilibrium physics and may be studiedby analytical (stability and bifurcation analysis, spectral analysis) and numeri-cal methods, showing as transient solutions similar structures as those observedexperimentally (Fig. 23).

An open question concerns the orientation of the ripples structures. It appearsthat, at least for moderate intensities, the laser polarization plays an important rolewhereas the underlying crystal structure seems to have no influence. Experimentswith circularly polarized light [81], however, show a similar structure of orderedripples (Fig. 24), without the possibility of the laser electric field as a controlparameter. Further, the meandering structures at high irradiation dose cannot yetbe understood. Up to now, no reliable model has been found to account for thestructures’ orientation.

Ongoing work concentrates on a more detailed analysis of the instabilities in-volved and the mechanisms responsible for the orientation of the self-organizednanostructures. Further, investigations are aimed at the possibility to control thestructures for possible applications.


FIG. 24. Ripples structures on CaF2 after ablation with circularly polarized light [81].

5. Acknowledgements

We gratefully acknowledge fruitful collaboration and discussions with and valu-able support by Tz. Arguirov, J. Bertram, M. Bestehorn, S. Eckert, M.E. Gar-cia, I. Georgescu, M. Henyk, W. Kautek, M. Ratzke, R.P. Schmid, W. Seifert,O. Varlamova, D. Wolfframm, and L. Zhu. The BTU/IHP JointLab is supportedby an HWP grant, a joint initiative of the German Federal Government andthe Land of Brandenburg. We also gratefully acknowledge support from theLand-Brandenburg-Schwerpunktprogramm “Qualitätsforschung” and the Euro-pean Funds for Regional Development (EFRE).

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CHARACTERIZATION OF SINGLEPHOTONS USING TWO-PHOTONINTERFERENCE∗

T. LEGERO†, T. WILK, A. KUHN‡ and G. REMPE§

Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2542. Single-Photon Light Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

2.1. Frequency Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2572.2. Spatiotemporal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2582.3. Single-Photon Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

3. Two-Photon Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2603.1. Quantum Description of the Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . 2613.2. Principle of the Two-Photon Interference . . . . . . . . . . . . . . . . . . . . . . . . . 2623.3. Temporal Aspects of the Two-Photon Interference . . . . . . . . . . . . . . . . . . . . 2633.4. Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2643.5. Two-Photon Interference without Time Resolution . . . . . . . . . . . . . . . . . . . 2663.6. Time-Resolved Two-Photon Interference . . . . . . . . . . . . . . . . . . . . . . . . . 267

4. Jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2704.1. Frequency Jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2714.2. Emission-Time Jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2734.3. Autocorrelation Function of the Photon’s Shape . . . . . . . . . . . . . . . . . . . . . 276

5. Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2775.1. Single-Photon Source and Experimental Setup . . . . . . . . . . . . . . . . . . . . . . 2775.2. Average Detection Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2795.3. Time-Resolved Two-Photon Interference . . . . . . . . . . . . . . . . . . . . . . . . . 2805.4. Interpretation of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285


∗ It is a pleasure for us to dedicate this paper to Prof. Herbert Walther, a pioneer in quantum opticsfrom the very beginning. The investigation of the amazing properties of single photons both in themicrowave and the optical domain has always been a central theme in his research. We wish him allthe best in the years to come!

† Now at Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany.‡ Now at Clarendon Laboratory, Oxford University, Parks Road, Oxford OX1 3PU, United King-

dom.§ Corresponding author. E-mail: [email protected].


DOI 10.1016/S1049-250X(06)53009-5

253


254 T. Legero et al. [1

1. Introduction

Four decades after the pioneering work on optical coherence and photon statisticsby Glauber (1965), the controlled generation of single photons with well-definedcoherence properties is now of fundamental interest for many applications inquantum information science. First, single photons are an important ingredientfor quantum cryptography and secure quantum key distribution (Gisin et al.,2002). Second, the realization of quantum computing with linear optics (LOQC),which was first proposed by Knill et al. (2001), relies on the availability ofdeterministic single-photon sources. And third, various schemes have been pro-posed to entangle and teleport the spin of distant atoms, acting as emitters ofsingle photons, by means of correlation measurements performed on the single-photon light fields (Cabrillo et al., 1999; Bose et al., 1999; Browne et al., 2003;Duan and Kimble, 2003). Therefore, in recent years, a lot of effort has been madeto realize single-photon sources. As a result, the controlled generation of singlephotons has been demonstrated in various systems, as summarized in a reviewarticle of Oxborrow and Sinclair (2005).

Using the process of spontaneous emission from a single quantum system isthe simplest way to realize a single-photon source. In this case, the quantum sys-tem is excited by a short laser pulse and the subsequent spontaneous decay of thesystem leads to the emission of only one single photon. This has been success-fully demonstrated many times, e.g., using single molecules (Brunel et al., 1999;Lounis and Moerner, 2000; Moerner, 2004), single atoms (Darquié et al., 2005),single ions (Blinov et al., 2004), single color centers (Kurtsiefer et al., 2000;Brouri et al., 2000; Gaebel et al., 2004) or single semiconductor quantum dots(Santori et al., 2001; Yuan et al., 2002; Pelton et al., 2002; Aichele et al.,2004). If the quantum system radiates into a free-space environment, the direc-tion of the emitted photon is unknown. This limits the efficiency of the source.To overcome this problem, the enhanced spontaneous emission into a cavityhas been used. The system is coupled to a high-finesse cavity and the pho-ton is preferably emitted into the cavity mode, which defines the direction ofthe photons. Although a cavity is used, most properties of the photons, likethe frequency, the duration and the bandwidth, are given by the specific quan-tum system. Only if the generation of single photons is driven by an adiabaticpassage, these spectral parameters can be controlled. This technique uses theatom-cavity coupling and a laser pulse to perform a vacuum stimulated Raman-transition (STIRAP), which leads to the generation of one single photon. Up tonow, this has been demonstrated with single Rubidium atoms (Kuhn et al., 2002;Hennrich et al., 2004), single Caesium atoms (McKeever et al., 2004) and singleCalcium ions (Keller et al., 2004) placed in high-finesse optical cavities.

The characterization of a single-photon source usually starts with the investi-gation of the photons statistics, which is done by a g(2) correlation measurement

1] SINGLE PHOTONS USING TWO-PHOTON INTERFERENCE 255

using a Hanbury Brown and Twiss (1957) setup. The observation of antibunchingshows that the source emits only single photons. However, the requirements ona single-photon source for LOQC and for the entanglement of two distant atomsgo far beyond the simple fact of antibunching. The realization of these proposalsrelies on the indistinguishability of the photons, so that even photons from dif-ferent sources need to be identical with respect to their frequency, duration andshape. Therefore it is desirable to investigate the spectral and temporal proper-ties of single photons emitted from a given source. We emphasize that propertieslike bandwidth or duration always deal with an ensemble of photons and cannotbe determined from a measurement on just a single photon. Therefore any mea-surement of these properties requires a large ensemble of successively emittedphotons. Several methods have been employed to characterize these.

The first measurement of the duration of single photons has been performedby Hong et al. (1987). In this experiment, the fourth-order interference of twophotons from a parametric down-conversion source was investigated by superim-posing the signal and the idler photon on a 50/50 beam splitter. The coincidencerate of photodetections at the two output ports of the beam splitter was measuredin dependence of a relative arrival-time delay between the two photons. Indistin-guishable photons always leave the beam splitter together, so that no coincidencecounts can be observed. If the photons are slightly different, e.g., because theyimpinge on the beam splitter at slightly different times, the coincidence rate in-creases. Therefore, as a function of the photon delay, the coincidence rate showsa minimum if the photons impinge simultaneously on the beam splitter, and forotherwise identical photons the width of this dip is the photon duration. The min-imum in the coincidence rate goes down to zero if the photons are identical. Anydifference between the two interfering photons reduces the depth of this dip. Thefirst demonstration of such a two-photon interference of two independently emit-ted photons from a quantum-dot device has been shown by Santori et al. (2002).

In addition to the two-photon coincidence experiments, a correlation measure-ment between the trigger event and the detection time of the generated photon canbe used to determine the temporal envelope of the photon ensemble (Kuhn et al.,2002; Keller et al., 2004; McKeever et al., 2004). This latter method is insensitiveto the spectral properties of the photons. In general, it does not reveal the shapeof the single-photon wavepackets, unless all photons are identical. In case of vari-ations in the photon duration or a jitter in the emission time, only the temporalenvelope of the photon ensemble is observed. No conclusions can be drawn onthe envelope of the individual photons.

The standard way to determine the coherence time of a given light source is themeasurement of the second-order interference using a Mach–Zehnder or Michel-son interferometer. This measurement can also be done with single photons, sothat each single photon follows both paths of the interferometer and interfereswith itself. The detection probability of the photons at both outputs of the inter-


ferometer shows a fringe pattern if the length of one arm is varied. The visibilityof this pattern depends on the length difference of both arms and determines thecoherence length (or the coherence time) of the photons. This method has beenused by Santori et al. (2002) and Jelezko et al. (2003) to measure the coherencetime of their single-photon sources. However, this method is hardly feasible forphotons of long duration, because the length of one arm of the interferometermust be varied over large distances. Furthermore, the measurement depends onthe mechanical stability of the whole setup, i.e. the interferometer must be stablewithin a few per cent of the wavelength of the photons, which might not be easy.

Only recently, adiabatic passage techniques have allowed the generation ofphotons which are very long compared to the detector time resolution. There-fore the detection time of a photon can be measured within the duration of thesingle-photon wavepacket. As a consequence, the two-photon interference can beinvestigated in a time-resolved manner, i.e. the coincidence rate can be measuredas a function of the time between photodetections (Legero et al., 2003, 2004). Thetheoretical analysis shows that this method not only gives information about theduration of single photons, but also about their coherence time. Here we discusshow to use this method for a spectral or temporal characterization of a single-photon source.

The article is organized as follows: After a brief summary of the nature ofsingle-photon light fields (Section 2), we discuss the interference of two photonson a beam splitter and introduce the time-resolved two-photon interference (Sec-tion 3). Thereafter, we show how a frequency and an emission-time jitter affectsthe results of a time-resolved two-photon interference experiment (Section 4). Onthis basis, the experimental characterization of a single-photon source, based onan adiabatic passage technique, is discussed (Section 5).

2. Single-Photon Light Fields

The quantum theoretical description of light within an optical cavity is well under-stood (Meystre and Sargent III, 1998). The electromagnetic field between the twomirrors is subject to boundary conditions which lead to a discrete mode structureof the field. Each mode can be labeled by a number l and is characterized by itsindividual frequency, ω. These eigenfrequencies are separated by �ω = 2πc/L,whereL is the round-trip length of the cavity. The quantization results in a discreteset of energies, En = hω(n + 1/2), and the appropriate eigenstates are definedby means of creation, a†

l , and annihilation, al , operators. The energy eigenstates

(1)|n〉 = (a†l )n

√n! |0〉


are states with a fixed photon number, n. In this context, photons are the quantaof energy in the modes of the cavity.

In the limit of L → ∞ and �ω → 0, the mode spectrum becomes continu-ous. In this case it is convenient to introduce continuous-mode operators a†(ω)

and a(ω) according to

(2)a†l → (�ω)1/2a†(ω).

These operators create and annihilate photons as quanta of monochromatic wavesin free space. These waves of infinite spatial extension do not have any begin-ning or any end. However, photons generated in a laboratory are characterized bya certain frequency bandwidth or a finite spatial extension. Therefore it is desir-able to define operators which create or annihilate photons in modes of a givenbandwidth, or, in other words, of a well-defined spatiotemporal spread.

2.1. FREQUENCY MODES

In contrast to modes describing monochromatic waves, it is possible to definefield modes of a given frequency distribution. These modes represent wavepack-ets travelling with the speed of light c through the vacuum, and the bandwidth κ

of such a mode determines the duration δt of the wavepacket. A frequency dis-tribution is described by a normalized complex function χ(ω) which is called themode function of the field. The operators a†(ω) and a(ω) can be used to definea new set of operators for the creation and annihilation of photons in these newmodes (Blow et al., 1990). The creation operator, e.g., is given by

(3)b†χ =∫

dω χ(ω)a†(ω).

Note that the mode function χ(ω) can be written as the product of a real am-plitude, ε(ω), and a complex phase, exp (−iΦ(ω)). The phase term includes theemission time τ0 and the propagation of the wavepacket. In the following, we re-strict our discussion to Gaussian wavepackets centered at the frequency ω0. Theirmode functions read

(4)χ(ω) = 4

√2

πκ2exp

(− (ω − ω0)

2

κ2

)exp(−iω(τ0 + z/c)

).

For an ideal single-photon source which always produces identical photons, thelight field is always described by the same quantum mechanical state vector. Inother words, the state vector is given by the creation operator b†(χ) acting on thevacuum state |0〉 for every single photon:

(5)|1χ 〉 = b†χ |0〉.


We emphasize that such an ideal source is hardly feasible. Usually the generationprocess cannot be controlled perfectly and therefore the mode function is subjectto small variations. To take this into account the light field must be described bya quantum mechanical density operator

(6)� =∫

dϑ f (ϑ)|1χ(ϑ)〉〈1χ(ϑ)|.Here we assume that the source produces single photons with a Gaussian fre-quency distribution and the parameters of this distribution are subject to smallvariations, according to a distribution function f (ϑ). The parameter ϑ stands forthe center frequency, ω0, the bandwidth, κ , or the emission time, τ0, of the photon,or a combination of these.

2.2. SPATIOTEMPORAL MODES

Due to the Fourier theorem, each mode with a certain frequency distribution χ(ω)can be assigned to a temporal wavepacket which is travelling through space.A mode with the Gaussian frequency distribution given by Eq. (4) therefore be-longs to a spatiotemporal mode ξ(t−z/c) of Gaussian shape. With the substitutionq := t − z/c this mode is given by the function

ξ(q) = 4

√2

πδt2exp

(− q2

δt2

)exp(iω0(τ0 − q)

)(7)≡ ε(q) exp

(iω0(τ0 − q)

).

The duration δt of this Gaussian wavepacket is given by the reciprocal bandwidthof the frequency distribution, δt = 2/κ . Blow et al. (1990) have shown that cre-ation and annihilation operators can also be assigned to spatiotemporal modes. Inorder to do that, one has to define the Fourier-transformed operators

(8)a†(q) = (2π)−1/2∫

dω a†(ω)e−iωq,

(9)a(q) = (2π)−1/2∫

dω a(ω)eiωq .

By means of these operators we define a flux operator a†(q)a(q). Its expectationvalue has the unit of photons per unit time. We need this operator in the nextsubsection to describe the detection of single photons.

Equations (8) and (9) are only valid if the bandwidth of the modes is muchsmaller than the frequency of the light, κ � ω0. This also limits the localizationof a single photon in such a spatiotemporal mode. In case of optical frequencies,this condition is usually fulfilled.


In analogy to Eq. (3) the Fourier-transformed operators can be used to definecreation and annihilation operators for photons of spatiotemporal modes ξ(q):

(10)c†ξ =∫

dq ξ(q)a†(q).

To take fluctuations into account, one can again write the density operator of thelight field as in Eq. (6), but using spatiotemporal modes. In this case, ϑ stands forany combination of ω0, δt , and τ0.

2.3. SINGLE-PHOTON DETECTION

Choosing spatiotemporal modes for describing the state of a single-photon lightfield simplifies the formal description of the detection of a photon. We assume adetector with quantum efficiency η placed at the position z = 0. The responseof the detector within a time interval [t0 − dt0/2, t0 + dt0/2] is given by theexpectation value of the flux operator:

(11)P (1)(t0) = η

t0+dt0/2∫t0−dt0/2

dt tr[�a†(t)a(t)

].

In case of single-photon wavepackets, the function P (1)(t0) gives the probabilityto detect this photon within the considered time interval. In practice, the lowerlimit of the duration dt0 is given by the detector time resolution T , i.e. dt0 � T .

If the photon duration is much longer than the detector time resolution, δt � T

and dt0 = T , Eq. (11) can be simplified to

(12)P (1)(t0) = ηT tr[�a†(t0)a(t0)

].

The measurement of the detection probability requires a large ensemble of singlephotons. In the following, we therefore assume a periodic stream of single photonsemitted one-after-the-other, so that the photons always hit the detector one by one.If all photons of this stream are identical, the light field can simply be describedby a state vector |1ξ 〉 and the density operator is given by � = |1ξ 〉〈1ξ |, with|1ξ 〉 = c

†ξ |0〉. In this case, the average detection probability of the ensemble of

photons is given by the square of the absolute value of the mode function, ξ(q),and is therefore identical to the shape of each individual photonic wavepacket,

(13)P (1)(t0) = ηT∣∣ξ(t0)∣∣2 = ηT ε2(t0).

As already discussed, the photons may differ from one another, and the densityoperator is given according to Eq. (6). The average detection probability is thengiven by


(14)P (1)(t0) = ηT

∫dϑ f (ϑ)ε2(t0, ϑ).

To obtain this equation, we assume that trace and integration can be exchanged.Obviously the average detection probability for the photon ensemble differs fromthat for individual photons. The average detection probability is, in general, af-fected by the variation, f (ϑ), of the parameters of the mode function, ξ(t).Therefore it shows only a temporal envelope of the photon ensemble. However,the effect of each parameter onto P (1)(t0) can be very different. A variation ofthe frequency, e.g., does not affect the real amplitude of the mode function, sothat the average detection probability, P (1)(t0), is simply given by Eq. (13). Thisis not the case for variations of the other parameters, as will be shown in Sec-tion 4.

3. Two-Photon Interference

We now consider two independent streams of Gaussian-shaped single photonsthat impinge on a 50/50 beam splitter such that always two photons are super-imposed. As we show in Section 5, these two streams can originate from onesingle-photon source by directing each photon randomly into two different pathsof suitable length, so that successively generated photons hit the beam splitterat the same time. Here we ask for the probability to detect the photons of eachpair in different output ports of the beam splitter. In case of identical photons,the joint detection probability is zero. With polarization-entangled photon pairsemitted from a down-conversion source, this effect has first been used by Alleyand Shih (1986) to test the violation of Bell’s inequality by joint photodetections,and one year later, Hong et al. (1987) have used it to measure the delay betweentwo photons with sub-picosecond precision. Recently, two-photon interferencephenomena have successfully been employed to test the indistinguishability ofindependently generated single photons (Santori et al., 2002). To illustrate thisinterference effect, we first assume that each photon of a given stream can be de-scribed by the same quantum mechanical state vector, |1ξ 〉, but allow the statevectors of the two considered streams to differ from one another. In Section 4we generalize this discussion to streams of photons which show a variation in theparameters of the mode functions, e.g., a variation in the photon frequency. Fi-nally we show that the interference of photon pairs reveals information about thevariations of the mode functions.

In Sections 3.1 and 3.2 we start with a brief discussion of the beam splitterand the principle of the two-photon interference. Afterwards we analyze the jointdetection probability for photons in the limits of a photon that is either very shortor very long compared to the detector time resolution.


3.1. QUANTUM DESCRIPTION OF THE BEAM SPLITTER

The beam splitter is an optical four-port device with two inputs and two outputs.The principle of the beam splitter is shown in Fig. 1. As discussed by Leonhardt(1997), each port has its own creation and annihilation operators, and the outputoperators can be expressed by the input operators using a unitary transformationmatrix B. This relation is valid for creation and annihilation operators a(ω) ofmonochromatic waves as well as for operators of spatiotemporal modes, bχ or cξ .It reads:

(15)

(a3a4

)= B(a1a2

)and(a

†3, a

†4

) = (a†1, a

†2

)B∗.

In the following discussion, we assume an ideal lossless and polarization inde-pendent beam splitter with transmission coefficient

√σ . The matrix of this beam

splitter is given by

(16)B =( √

σ√

1 − σ

−√1 − σ

√σ

).

The opposite signs of the off-diagonal terms reflect the phase jump of π for thereflection at one side of the beam splitter.

The transmission of photons from the input side to the output side of the beamsplitter can be understood as a quantum mechanical evolution of the system. This

FIG. 1. The ideal lossless beam splitter is fully characterized by its transmission coefficient σ .The reflection coefficient is then given by

√1 − σ . Light can enter the beam splitter through two

different input ports 1 and 2. According to the transmission and the reflection coefficient, it is dividedinto the output ports 3 and 4. For one of the reflections, the light is subject to a phase jump of πwhich is indicated by the minus sign. In the Heisenberg picture (a), one accounts for this process bytransforming the creation and annihilation operators of the two input modes (1 and 2) into suitableoperators of the output modes (3 and 4), whereas in the Schrödinger picture (b), the action of thebeamsplitter is expressed by the unitary operator B†, which acts on the wavevector and couples thetwo through-going modes (1 and 2).


evolution can be described in two equivalent pictures corresponding to the Heisen-berg and the Schrödinger picture in quantum mechanics (Campos et al., 1989;Leonhardt, 2003). In the Heisenberg picture, the evolution is described by thecreation and annihilation operators. The output operators are considered as theevolved input operators whereas the state vector of the field remains unchanged(see Fig. 1(a)). Using the unitary operators B and B†, this evolution can also beexpressed by(

a3a4

)= B(a1a2

)=: B(a1a2

)B† and

(17)B∗(a1a2

)=: B†

(a1a2

)B.

Alternatively, in the Schrödinger picture, the evolution can be calculated using thestate vector of the light field. In this case, the state vector of the input side |Ψin〉evolves to a state vector at the output side, |Ψout〉 = B†|Ψin〉, while the modesthemselves do not change, that is modes 1 and 2 are defined as the through-goingmodes (see Fig. 1(b)). In the next subsection the Schrödinger picture is used toillustrate the principle of the two-photon interference.

3.2. PRINCIPLE OF THE TWO-PHOTON INTERFERENCE

We consider two identical photons that impinge on a 50/50 beam splitter. Theinput state of the light field is given by |Ψin〉 = |1〉1H |1〉2H , where the indiceslabel the two input ports and the polarization of the photons. Here, we assumetwo photons of horizontal polarization. In the Schrödinger picture we describethe evolution of the state using the unitary operator B† as follows:

(18)B†|1〉1H |1〉2H = B†a†1H a

†2H |0〉.

Since BB† is equal to the identity operator 1 and B†|0〉 = |0〉 we can write

(19)B†a†1H a

†2H |0〉 = B†a

†1H BB

†a†2H B|0〉,

which according to Eq. (17) gives

B†a†1H BB

†a†2H B|0〉 = 1/

√2(a

†1H − a

†2H

)1/

√2(a

†1H + a

†2H

)|0〉= 1/2(a

†21H − a

†22H + a

†1H a

†2H − a

†2H a

†1H

)|0〉.Each term in this sum of creation operators corresponds to one of four possiblephoton distributions in the beam splitter output ports, shown in Fig. 2. In the firsttwo cases, (a) and (b), both photons are found in either one or the other output,whereas in the cases (c) and (d), the photons go to different ports. The last twocases are indistinguishable, but the two expressions leading to cases (c) and (d)


FIG. 2. Two impinging photons lead to four possible photon distributions at the beam-splitteroutput. In the first two cases (a) and (b) the photons would be found together. In the remaining twocases (c) and (d) the photons would leave the beam splitter through different ports. Since the quantumstates of the cases (c) and (d) show different signs, they interfere destructively.

have opposite sign. Therefore the two possibilities interfere destructively. As aconsequence, the two photons always leave the beam splitter as a pair and theoutput state is given by the superposition

(20)B†|1〉1H |1〉2H = 1√2

(|2〉1H |0〉2H − |0〉1H |2〉2H).

This quantum interference occurs only if the photons are identical. If the photonswere distinguishable, no interference takes place. For example, two photons oforthogonal polarization, |Ψin〉 = |1〉1H |1〉2V , give rise to four different outputstates which are distinguishable by the photon polarization. In this case the overalloutput state can be written as a product state, e.g.,

B†|1〉1H |1〉2V = 1√2

(|1〉1H |0〉2 − |0〉1|1〉2H)

⊗ 1√2

(|1〉1V |0〉2 + |0〉1|1〉2V),

which describes the state of two independently distributed photons.Note that all temporal aspects of the light field are neglected in the above

discussion. In the next sections, the two-photon interference is discussed underconsideration of the photon duration and the time resolution of the detectors.

3.3. TEMPORAL ASPECTS OF THE TWO-PHOTON INTERFERENCE

We now take into account that the photodetections in the output ports of the beamsplitter might occur at different times, t1 and t2. We use the Heisenberg picture tocalculate the probability of a joint photodetection from the second-order correla-tion function,

(21)G(2)(t1, t2) =∑s,s′

tr[�1,2A3s,4s′(t1, t2)

],


where �1,2 describes the two-photon input state and the operator A3s,4s′(t1, t2) isgiven by

(22)A3s,4s′(t1, t2) := a†3s(t1)a

†4s′(t2)a4s′(t2)a3s(t1) and s, s′ ∈ {H,V }.

The probability for a photodetection at the first detector within the time interval[t0 − dt0/2, t0 + dt0/2] and at the second detector within a time interval shiftedby τ , [t0 + τ − dτ/2, t0 + τ + dτ/2], is then given in analogy to Eq. (11),

(23)P (2)(t0, τ ) = η3η4

t0+dt0/2∫t0−dt0/2

dt1

t0+τ+dτ/2∫t0+τ−dτ/2

dt2 G(2)(t1, t2).

Here we assume that the detectors have different efficiencies, η3 and η4. In anal-ogy to Section 2.3 the smallest duration of the detection intervals is given by thedetector time resolution, T , so that dt0 � T and dτ � T . In the following, wecalculate the joint detection probability in the limit of very short and very longphotons.

If the photons are very short compared to the time resolution of the detectors,δt � T , the limits of the integration in Eq. (23) can be extended to infinity, sothat

(24)P (2) = η3η4

∫∫dt1 dt2 G

(2)(t1, t2)

gives the probability of a coincidence of photodetections.For very long photons with δt � T , the integration in Eq. (23) leads to

(25)P (2)(t0, τ ) = η3η4G(2)(t0, t0 + τ) dt0 dτ.

Therefore the probability of a joint photodetection can be studied as a functionof the two detection times, t0 and t0 + τ . In practice, only the time difference, τ ,between two photodetections is relevant. Therefore we integrate P (2)(t0, τ ) overthe time t0 of the first photodetection. This gives

(26)P (2)(τ ) = η3η4T

∫dt0 G

(2)(t0, t0 + τ),

where dτ is substituted by the detector time resolution T . The second-order cor-relation function, G(2), plays a central role in the calculation of the joint detectionprobability. It is now analyzed taking the polarization and the spatiotemporalmodes of the photons into account.

3.4. CORRELATION FUNCTION

We calculate the correlation function G(2) for two photons characterized by twomode functions, ξ1 and ξ2. Without loss of generality, we assume that both pho-tons are linearly polarized with an angle ϕ between the two polarization direc-tions. The state of the photons is then given by |1ξ1〉1H and cosϕ|1ξ2〉2H |0〉2V +


sinϕ|0〉2H |1ξ2〉2V , respectively. The density operator, �1,2 = |Ψin〉〈Ψin|, is givenby the input state

(27)|Ψin〉 = cosϕ|1ξ1〉1H |1ξ2〉2H + sinϕ|1ξ1〉1H |1ξ2〉2V ,

which is a superposition of the cases in which the impinging photons are paralleland perpendicular polarized to each other. The correlation function can then bewritten as a sum of two expressions G(2)

HH and G(2)HV ,

(28)G(2) = cos2 ϕG(2)HH + sin2 ϕG

(2)HV ,

where the first function G(2)HH accounts for the input state in which both pho-

tons have the same polarization and the second function G(2)HV accounts for the

perpendicular polarized case. Taking the mode functions into account, these twoexpressions read

(29)G(2)HH (t1, t2) =

|ξ1(t1)ξ2(t2)− ξ2(t1)ξ1(t2)|24

,

(30)G(2)HV (t1, t2) =

|ξ1(t1)ξ2(t2)|2 + |ξ1(t2)ξ2(t1)|24

.

We emphasize that the correlation function for parallel polarized photons is al-ways zero for t1 = t2, even if the mode functions ξ1(t) and ξ2(t) are not identical.As a consequence, the probability of a joint photodetection, Eq. (26), is alwayszero for τ = t2 − t1 = 0, i.e. no simultaneous photodetections are expected evenif the photons are distinguishable with respect to their mode functions.

As already mentioned in Section 2.1, the mode function can be written as theproduct of a real amplitude and a complex phase, ξj (t) = εj (t) exp (−iΦj (t))

with j ∈ {1, 2}. Since the correlation function G(2)HV for perpendicular polarized

photons is independent of the phase, it can be written as

(31)G(2)HV (t1, t2) =

(ε1(t1)ε2(t2))2 + (ε1(t2)ε2(t1))

2

4.

This is not the case for the correlation function of parallel polarized photons whichcarries a phase-dependent interference term. It can be expressed as G(2)

HH (t1, t2) =G(2)HV (t1, t2)− F(t1, t2), with

F(t1, t2) := ε1(t1)ε2(t2)ε1(t2)ε2(t1)

2× cos(Φ1(t1)−Φ1(t2)+Φ2(t2)−Φ2(t1)

).

However, this phase-dependency is only relevant, if Φ1(t) and Φ2(t) display adifferent time evolution. Otherwise the sum over the phases is always zero. Sucha difference in the time evolution is given if, e.g., the frequencies of the pho-tons are different. In that case, the interference term oscillates with the frequency


difference, which gives rise to an oscillation in the joint photodetection probabil-ity, P (2)(τ ). This will be discussed further in Section 3.6.

Taking the interference term into account, the overall correlation function,Eq. (28), can be summarized to

(32)G(2)(t1, t2) = G(2)HV (t1, t2)− cos2 ϕF(t1, t2),

where the effect of the interference term depends on the angle ϕ between the twophoton polarizations.

In the next two subsections, the joint detection probability, Eq. (23), is analyzedfor very long and very short photons.

3.5. TWO-PHOTON INTERFERENCE WITHOUT TIME RESOLUTION

First we assume Gaussian-shaped photons which are very short compared to thetime resolution of the photodetectors, δt � T . In this case one can only decidewhether there is a coincidence of detections within the time interval T or not,and the coincidence probability is given by Eq. (24). With a possible frequencydifference � := ω02 − ω01 and an arrival-time delay δτ := τ02 − τ01 of thephotons, the coincidence probability is given by

(33)P (2) = 1

2

(1 − cos2 ϕ exp

(− δt2

4/�2

)exp

(−δτ 2

δt2

)),

where we assume that the photons hit perfect photodetectors with η3 = η4 = 1.We analyze the coincidence probability as a function of the photon delay δτ fordifferent photon polarizations and frequency differences. This is shown in Fig. 3.

As already discussed in Section 3.2, perpendicular polarized photon pairs, ϕ =π/2, show no interference at all. Therefore, the probability for detecting photonsat different output ports of the beam splitter is always 1/2, independent of thephoton delay, δτ .

If the photons have identical polarizations, ϕ = 0, and identical frequencies,� = 0, the coincidence probability shows a Gaussian-shaped dip centered atδτ = 0. The minimum of P (2) is zero, indicating that the photons never leave thebeam splitter through different output ports. If the photon pairs show any differ-ence in their polarization, shape or frequency, there is no perfect interference andthe minimum of the dip is no longer zero. Therefore the two-photon interferencecan be used to test the indistinguishability of photons.

The first measurement of the coincidence rate as a function of the relativephoton delay was performed by Hong et al. (1987) using photon pairs from a para-metric downconversion source. They controlled the relative delay of the photonsby shifting the position of the 50/50 beam splitter. The frequencies and band-widths of the photons were adjusted by using two identical optical filters, so that


FIG. 3. Coincidence probability as a function of the arrival time delay, δτ , of two linear polarizedphotons. In case of perpendicular polarized photons (ϕ = π/2) there is no interference at all and thecoincidence probability shows the constant value 1/2. If the photons are parallel polarized (ϕ = 0)and have identical frequency (� = 0), there is a Gaussian-shaped dip which goes down to zero forsimultaneous impinging photons, δτ = 0. Any difference in polarization or frequency leads to areduced depth of this dip.

the coincidence rate dropped nearly to zero for simultaneously impinging pho-tons. As one can see from Eq. (33), the width of the dip is identical to the photonduration, δt . Therefore this experiment was used to measure the duration andbandwidth of the photons.

So far, most two-photon interference experiments were performed with veryshort photons. Therefore the joint detection probability was only considered as afunction of the photon delay, δτ . However, if the photon duration is much largerthan the detector time resolution, the time τ between the photodetections in thetwo output ports can be measured and the joint detection probability can addition-ally be analyzed in dependence of the detection-time difference.

3.6. TIME-RESOLVED TWO-PHOTON INTERFERENCE

We now assume, that the photon duration is much larger than the detection timeresolution, δt � T . Since the time, τ , between the photodetections can be mea-sured within the photon duration, the joint detection probability can be analyzed


as a function of this detection-time difference. Using Eq. (26) and assumingGaussian-shaped photons of identical duration, δt , the joint detection probabil-ity is given by

P (2)(τ, δτ ) = T√π δt

[1 − cos2 ϕ cos(�τ)

2+ sinh2

(τδτ

δt2

)](34)× exp

(−δτ 2 + τ 2

δt2

).

In Fig. 4 the joint detection probability is shown as a function of the photonarrival-time delay, δτ , and the detection time difference, τ .

The sign of τ indicates which detector clicks first. Similarly, the sign of δτdetermines which photon arrives first at the beam splitter. Note that the joint de-tection probability can only be different from zero if |τ | ≈ |δτ |. This leads to thecross-like structure in Fig. 4(a–c). Since the photons only interfere if the relativedelay is smaller than the photon duration, we focus our attention to the center ofFig. 4(a–c).

Again, we start our analysis with perpendicular polarized photon pairs. Obvi-ously, no interference takes place, and as one can see in Fig. 4(a), even simultane-ously impinging photons (with δτ = 0) can be detected in different output portsof the beam splitter. The joint detection probability shows therefore a Gaussian-shaped peak. According to Eq. (34), the width of this peak is identical to thephoton duration, δt . Since the photons are distinguishable by their polarization, anadditional frequency difference, �, does not affect this result. Assuming photonpairs with identical mode functions, the joint detection probability of perpendic-ular polarized photons can be used to determine the photon duration.

Figure 4(b) shows the joint detection probability for parallel polarized photonsof identical frequency, � = 0. For simultaneously impinging photons the jointdetection probability is always zero, which indicates that the photons coalesceand leave the beam splitter always together.

If the parallel polarized photons show a frequency difference, the joint detectionprobability oscillates as a function of the detection time difference, τ . This isshown in Fig. 4(c). As one can see from Eq. (34), the frequency difference, �,determines the periodicity of this oscillation. We emphasize that the oscillationalways leads to a minimum at τ = 0, independent of �, so that even photons ofdifferent frequencies are never detected simultaneously in different output ports.Furthermore, the joint detection probability at the maxima is always larger thanthe joint detection probability for perpendicular polarized photons.

Without time resolution, the detection-time difference cannot be measured andthe joint detection probability, P (2)(τ, δτ ), has to be integrated over τ . This linksthe results of a time-resolved two-photon interference to the discussion of Sec-tion 3.5. In case of perpendicular polarized photons, the τ -integrated function


FIG. 4. Joint detection probability, P (2), as a function of the relative delay between the pho-tons, δτ , and the time difference between photodetections, τ , for perpendicular polarized photons (a)and parallel polarized photons (b). In (c) the parallel polarized photon pairs have a frequency differ-ence �, which leads to an oscillation in the joint detection probability. All times and frequencies arenormalized by the photon duration, δt .


P (2)(δτ ) shows the constant value 1/2. If the photons are identical, the integra-tion leads to a Gaussian-shaped dip, which was already discussed in Section 3.5.However, the oscillation in the joint detection probability for photon pairs with afrequency difference is no longer visible. The integration leads, in accordance toEq. (33), only to a reduced depth of the dip in P (2)(δτ ).

4. Jitter

Up to now, we assumed that all photons of a given stream can be described bythe same state vector |1ξ 〉. However, this requires a perfect single-photon source,which is able to generate a stream of photons without any variation in the parame-ters of the Gaussian mode functions. Here, we consider a more realistic scenario,in which a stream of single photons shows a jitter in the parameters, ϑ . Thequantum mechanical state of the photons is then given by the density operatorof Eq. (6).

Such a jitter in the mode functions has important consequences on the resultsof measurements which can be performed on the single-photon stream. On theone hand, as already discussed in Section 2.3, it affects the average detectionprobability of the photons in a way that its measurement does in general not revealinformation about the duration or shape of each single photon. On the other hand,variations in the mode functions of photon pairs have an influence on the jointdetection probability in two-photon interference experiments. This is discussed insome detail in the following two subsections.

To analyze the effect of jitters on the two-photon interference, we considertwo streams of Gaussian-shaped photons with a variation in the parameters oftheir mode functions. In analogy to Eq. (6) the density operator for photon pairsimpinging on the beam splitter is given by

(35)�1,2 =∫∫

dϑ1 dϑ2 f1(ϑ1)f2(ϑ2)|1ξ1〉|1ξ2〉〈1ξ1 |〈1ξ2 |,so that, using Eq. (21), the correlation function reads

G(2)(t0, t0 + τ) =∫∫

dϑ1 dϑ2 f1(ϑ1)f2(ϑ2) tr(�(ξ1, ξ2)A(t0, t0 + τ)

).

(36)

Here the expression �(ξ1, ξ2) substitutes |1ξ1〉|1ξ2〉〈1ξ1 |〈1ξ2 |. In Eq. (35) we as-sumed that all photons have identical polarization and that they are completelyindependent from each other. Therefore the density operator has only diagonalelements. The parameters of the mode functions ξ1 and ξ2 of the two streams aresummarized by ϑ1 and ϑ2, respectively. In general, all parameters of the modefunctions could be subject to a variation, and all the variations could in principle


depend on each other. However, in the following two subsections, we focus ourattention only on two examples of jitters and analyze the detection probability ofphotons, P (1), for a single photon stream as well as the joint detection probabil-ity, P (2), of two streams superimposed on a beam splitter.

First, in Section 4.1, we consider streams of photons which are characterizedby a variation of the center frequency, ω0j , so that each photon pair exhibits avariation of the frequency difference,� = ω02−ω01. All remaining parameters ofthe mode functions, e.g., the duration of the photons, are assumed to be identical.In Section 4.2, we consider photons, which show only a variation in their emissiontime, so that photon pairs are characterized by a variation in their arrival-timedelay δτ = τ02 − τ01.

4.1. FREQUENCY JITTER

We start our discussion of a frequency jitter by analyzing its effect on the aver-age detection probability, P (1)(t0), for a perfect photodetector with the detectionefficiency η = 1. If the frequency variation in the stream of single photons isdescribed by a normalized distribution function, f (ω), the average detection prob-ability is given, according to Eq. (14), by the integral

(37)P (1)(t0) = T

∫dω f (ω)

∣∣ξ(t0, ω)∣∣2.Since only the phase of the Gaussian mode functions depends on the frequency,the absolute value, |ξ(t0, ω)|2 = ε2(t0), is independent of ω. Thus, the averagedetection probability is not affected by any frequency jitter and is entirely deter-mined by the spatiotemporal mode function of each single photon.

However, a frequency jitter affects the joint detection probability of photonpairs superimposed on a beam splitter. To illustrate this, we assume two indepen-dent streams of photons, each fluctuating around a common center frequency ω0according to a normalized Gaussian frequency distributions, f1(ω01) and f2(ω02).Hence, the frequency difference of the photon pairs, � = ω02 − ω01, shows alsoa normalized Gaussian variation,

(38)f (�) = 1√πδω

exp(−�2/δω2),

with width δω depending on the widths of the frequency distributions of both

streams, δω =√δω2

01 + δω202. The density operator of the photon pairs can then

be expressed in terms of the distribution function of the frequency difference,

(39)�1,2 =∫

d�f (�)�(ξ1, ξ2).


FIG. 5. Joint detection probability as a function of the detection-time difference, τ , for simulta-neously impinging photons, δτ = 0, of identical polarization. The photons are subject to a frequencyjitter of width δω.

As the operations of trace and integration are exchangeable, the correlation func-tion, according to Eq. (36), can be written as

(40)G(2)(t0, τ ) =∫

d�f (�) tr(�(ξ1, ξ2)A(t0, t0 + τ)

).

For photons which are very long compared to the detector time-resolution, thejoint detection probability is given by Eq. (26). In case of simultaneously imping-ing photons, δτ = 0, this leads to

(41)P (2)(τ ) = T

2√πδt

[1 − cos2 ϕ exp

(− τ 2

4/δω2

)]exp

(− τ 2

δt2

).

For photons of parallel polarization, the result is shown in Fig. 5. In the limitof δω → ∞ the joint detection probability shows a Gaussian-shaped peak ofwidth T1 = δt , which is the photon duration. As one can see from Eq. (41), thejoint detection probability is always zero for τ = 0 as long as the width of thefrequency distribution, δω, is finite. In fact, as one can deduce from Eq. (41), thisleads to a dip in the joint detection probability around τ = 0 that is T2 = 2/δωwide. Note that T2 represents a coherence time which must not be mixed up withthe duration of each single photon, δt .


To determine the amount of a frequency jitter from a time-resolved two-photoninterference experiment, one has to perform two measurements. First, the jointdetection probability of perpendicular polarized photons, ϕ = π/2, reveals thephoton-duration, δt . Afterwards, the joint detection probability of parallel polar-ized photons, ϕ = 0, is used to measure T2 and derive the width of the frequencyjitter, δω. This is shown in detail in Section 5.

If the photons are very short compared to the detector time-resolution, thecoincidence probability must be calculated using Eq. (24). The coincidence prob-ability is then a function of the relative photon delay, δτ , and is given by

(42)P (2)(δτ ) = 1

2

(1 − 2 cos2 ϕ√

4 + δt2δω2exp

(−δτ 2

δt2

)).

In analogy to Section 3.5, a frequency jitter, δω, now leads to a decreased depthof the Gaussian-shaped dip, while the width of this dip is not affected and alwaysidentical to the photon duration.

In principle, it is possible to derive the frequency jitter also from a two-photoninterference experiment without time-resolution, but there are some major disad-vantages. First, the depth of the dip depends not only on a frequency jitter, butalso on the mode matching of the transversal modes of both beams. A nonperfectmode matching leads to a factor comparable to cos2 ϕ in Eq. (42). Therefore, incontrast to the time-resolved measurement, one cannot distinguish between a non-perfect mode matching or a frequency jitter. Second, in case of two independentstreams of photons from two different single-photon sources, it is impossible todecide whether a constant frequency difference or a frequency jitter is the reasonfor a decreased dip depth. And third, if the frequency jitter is large, the depth ofthe dip is very small, whereas in a time-resolved measurement, the dip-depth re-mains unchanged. As it is much more reliable to determine a small width ratherthan a small depth, the time-resolved method is much more powerful.

4.2. EMISSION-TIME JITTER

Now we assume a stream of single photons which shows a jitter in the emissiontime of each photon. This variation of the emission time is assumed to be given bya normalized Gaussian distribution function, f (τ0). The average detection prob-ability of the photons for an ideal photodetector with η = 1 is again given byEq. (14),

(43)P (1)(t0) = T

∫dτ0 f (τ0)

∣∣ξ(τ0 − t0)∣∣2.

This is a convolution of the detection probability, |ξ(t0)|2, of each single pho-ton and the emission-time distribution, f (τ0), of the photon stream. Therefore


the average detection probability is always broader than the detection probabilitywhich would arise solely from the duration of each single photon. This shows thata variation in the parameters of the spatiotemporal mode functions can alter thedetection probability of the photons. Therefore, in general, the average detectionprobability is not identical to the detection probability of individual photons.

To investigate the influence of an emission-time jitter on the joint detectionprobability in a two-photon interference experiment, we now assume two streamsof photons with a Gaussian emission-time distribution of identical width, �τ . Inthis case, the photon pairs are characterized by a jitter in the arrival-time delay ofthe photons, which is again given by a Gaussian distribution,

(44)f (δτ) = 1√π�τ

exp(−δτ 2/�τ 2).

The correlation function G(2)(t0, t0 + τ) can be written in analogy to Eq. (40),using only the variation of the relative photon delay,

(45)G(2)(t0, τ ) =∫

d(δτ) f (δτ) tr(�(ξ1, ξ2)A(t0, t0 + τ)

),

and the joint detection probability has to be calculated according to Eq. (26). Incase of simultaneously impinging photons, this leads to

P (2)(τ ) = T

2√π√δt2 +�τ 2

[1 − cos2 ϕ exp

(− τ 2

δt2 + δt4/�τ 2

)](46)× exp

(− τ 2

δt2 +�τ 2

).

In contrast to the previous case, now the width of the Gaussian-shaped peak in thejoint detection probability of perpendicular polarized photons is no more iden-tical to the photon duration. The variation in the emission time affects also theamplitude of the spatiotemporal mode functions and affects the joint detectionprobability even without interference. This can be seen in Fig. 6, which showsthe joint detection probability for photon pairs characterized by a distribution ofthe photon arrival-time delay. The width, T1, of the Gaussian-shaped peak is nolonger identical to the photon duration, δt . As one can derive from Eq. (46), it isnow broadened by the width of the variation in the photon delay, �τ :

(47)T1 =√δt2 +�τ 2.

Furthermore, the width of the dip in case of parallel polarized photons is notindependent of the width T1, but is given by

(48)T2 =√δt2 + δt4/�τ 2 = δt

�τT1.


FIG. 6. Joint detection probability as a function of the detection time difference, τ , for photon pairswith a variation in their relative photon delay, δτ . The photons are assumed to be (a) perpendicular and(b) parallel polarized to each other. Curve (c) is a Gaussian mode function of width δt . The width, T1,of the Gaussian peak in (a) is broadened by the variation of the relative photon delay, �τ , which ishere assumed to be 2δt .

A time-resolved two-photon interference experiment can again be used to deter-mine the variation in the emission time of the photon streams. However, sincethe shapes of the joint detection probabilities for a frequency and an emission-time jitter are identical, it is in general not possible to distinguish between thetwo. Nonetheless, one can determine the maximum values of both jitters, as wellas all pairs of frequency and an emission-time jitters matching the data. This isdiscussed in the next section.

Note that in case of very short photons, the coincidence probability has againto be calculated using Eq. (24). The width of the Gaussian-shaped dip in the co-incidence probability

(49)P (2)(δτ ) = 1

2

(1 − cos2 ϕ√

1 +�τ 2/δt2exp

(− δτ 2

δt2 +�τ 2

))is broadened by the emission-time jitter, �τ , and the depth of the dip is also de-creased by�τ . Again it is possible to determine the emission-time jitter from sucha two-photon interference experiment without time resolution. The disadvantagesof such a procedure have already been discussed in Section 4.1.


4.3. AUTOCORRELATION FUNCTION OF THE PHOTON’S SHAPE

We now consider only one source of single photons that we want to characterizeusing a two-photon interference experiment. We assume that the stream of pho-tons generated by this source is split up in such a way that each single photonis randomly directed along two different paths. These two paths are of differentlength and the repetition rate of the source is chosen in such a way that only suc-cessively generated photons impinge on the beam splitter at the same time. Thedetails of such an experiment are discussed in Section 5.

In general, we must distinguish between the photon ensemble of the wholestream and the subensemble of photon pairs superimposed on the beam splitter.The latter consists only of successively generated photons and the characteriza-tion of a single-photon source by a two-photon interference experiment takes intoaccount only this subensemble. Since the jitter in the subensemble of succes-sive photon pairs does not have to be identical to the jitter in the whole photonstream, we need a method to decide whether the results of a two-photon interfer-ence experiment can be generalized to all photons generated by the single-photonsource. In case of very long photons, this can be done by comparing the jointdetection probability, P (2)(τ ), for perpendicular polarized photons and the auto-correlation function, A(2)(τ ), of the average detection probability of the wholephoton stream.

We start our discussion with a stream of identical single photons, so that theaverage detection probability is simply given by the square of the amplitude ofthe spatiotemporal mode function, P (1)(t0) = T ε2(t0). On the one hand, the au-tocorrelation function of P (1)(t0) reads

(50)A(2)(τ ) =∫

dt0 P(1)(t0)P

(1)(t0 + τ) = T 2∫

dt0(ε(t0)ε(t0 + τ)

)2.

On the other hand, the joint detection probability for perpendicular polarized pho-ton pairs of this stream is given by Eq. (31), which leads to

(51)P (2)(τ ) ∝ T 2∫

dt0(ε(t0)ε(t0 + τ)

)2.

Therefore the joint detection probability for perpendicular polarized photons andthe autocorrelation function have the same shape.

However, if the photons show a variation in their spatiotemporal modes, thetwo functions are no longer equal. In the following, we assume that the vari-ations in the whole photon stream are described by a normalized distributionfunction f (ϑ), whereas the variations in the subensemble of successively emittedphotons is given by f (ϑ). In general, both functions do not have to be identical,i.e. the jitter in the subensemble can be smaller than the jitter in the whole photonstream. The autocorrelation function of the average detection probability P (1)(t0),


see Eq. (14), is given by

A(2)(τ ) = T 2∫

dt0

∫∫dϑ1 dϑ2 f (ϑ1)f (ϑ2)

(ε(t0, ϑ1)ε(t0 + τ, ϑ2)

)2.

(52)

The joint detection probability,

P (2)(τ ) ∝ T 2∫

dt0

∫∫dϑ1 dϑ2 f (ϑ1)f (ϑ2)

(ε(t0, ϑ1)ε(t0 + τ, ϑ2)

)2(53)

is therefore only equal to A(2)(τ ), if (ε(t0, ϑ1)ε(t0+τ, ϑ2))2 is either independent

of ϑ1 and ϑ2, or if the distribution function f (ϑ) of the whole photon stream isidentical to the distribution function f (ϑ) of successively emitted photons.

The comparison of the joint detection probability of perpendicular polarizedphoton pairs and the autocorrelation function of the average detection probabilitytherefore answers the question whether the results of a two-photon interferenceexperiment can be generalized to the whole photon stream.

5. Experiment and Results

In the previous three sections we discussed the theoretical background for char-acterizing single photons using two-photon interference. Now, we show how touse this method to experimentally characterize single photons that are emittedfrom only one source. This single-photon source has been realized using vacuum-stimulated Raman transitions in a single Rb atom located inside a high-finesseoptical cavity. In Section 5.1 we briefly review the principle of this source anddiscuss the experimental setup, which was used to investigate the two-photon in-terference. For further details concerning the single-photon generation, we referto Kuhn et al. (2002) and references therein. The measurement of the average de-tection probability of a stream of photons emitted from this source is discussedin Section 5.2. As already discussed, we use the autocorrelation function of theaverage detection probability to determine whether the results of a two-photoninterference experiment can be generalized to the whole photon stream. Since theduration of the photons is much longer than the time resolution of the detectors,the interference of successively emitted photons is measured in a time-resolvedmanner. The results and the interpretation of these measurements are discussed indetail in Sections 5.3 and 5.4.

5.1. SINGLE-PHOTON SOURCE AND EXPERIMENTAL SETUP

A sketch of the single-photon source and the experimental setup that we use tosuperimpose successively generated photons on a 50/50 beam splitter is shown


FIG. 7. Single-photon source and experimental setup used to investigate the two-photon inter-ference of successively emitted photons. The photons are generated in an atom-cavity system by anadiabatically driven stimulated Raman transition. A polarizing beam splitter directs the photons ran-domly into two optical fibers. The delay from photon to photon matches the travel-time difference inthe two fibers, so that successively emitted photons can impinge simultaneously on the 50/50 beamsplitter. Using a half-wave plate, the polarization of the photons can be chosen parallel or perpendic-ular to each other. The photons are detected using avalanche photodiodes with a detection efficiencyof 50% and a dark-count rate of 150 Hz.

in Fig. 7. The single-photon generation starts with 85Rb atoms released from amagneto-optical trap. The atoms enter the cavity mostly one-at-a-time (the proba-bility of having more than one atom is negligible). Each atom is initially preparedin |e〉 ≡ |5S1/2, F = 3〉, while the cavity is resonant with the transition between|g〉 ≡ |5S1/2, F = 2〉 and |x〉 ≡ |5P3/2, F = 3〉. On its way through the cavity,the atom experiences a sequence of laser pulses that alternate between trigger-ing single-photon emissions and repumping the atom to state |e〉: The 2 µs-longtrigger pulses are resonant with the |e〉 ↔ |x〉 transition and drive an adiabaticpassage (STIRAP) to |g〉 by linearly increasing the Rabi frequency. This transi-tion goes hand-in-hand with a photon emission. In the ideal case, the durationand pulse shape of each photon depend in a characteristic manner on the temporalshape and intensity of the triggering laser pulses (Keller et al., 2004). As we willdiscuss in Section 5.3, the photon-frequency can be chosen by an appropriate fre-


quency of the trigger laser (Legero et al., 2004). Between two photon emissions,another laser pumps the atom from |g〉 to |x〉, from where it decays back to |e〉.While a single atom interacts with the cavity, the source generates a stream of sin-gle photons one-after-the-other. The efficiency of the photon generation is 25%.

As described in detail in Legero (2005), the source has been optimized withrespect to jitters by compensating the Earth’s magnetic field inside the cavity andby adding to the recycling scheme an additional π-polarized laser driving thetransition |5S1/2, F = 3〉 ↔ |5P3/2, F = 2〉 to produce a high degree of spin-polarization in 5S1/2, F = 3, mF = ±3. This results in an increased coupling ofthe atom to the cavity. We have characterized the emitted photons by two-photoninterference measurements before and after this optimization.

To superimpose two successively emitted photons on the 50/50 beam splitter,they are directed along two optical paths of different length. These paths are real-ized using two polarization maintaining optical fibers with a length of 10 m and1086 m, respectively. Since the photon polarization is a priori undefined, a polar-izing beam splitter is used to direct the photons randomly into the long or shortfiber. The time between two trigger pulses is adjusted to match the travel-timedifference of the photons in the two fibers, which is �t = 5.28 µs. With a prob-ability of 25%, two successively emitted photons therefore impinge on the beamsplitter simultaneously. In addition, we use a half-wave plate to adjust the mutualpolarization of the two paths.

5.2. AVERAGE DETECTION PROBABILITY

First, we investigate the average detection probability of the photon stream. Forthis measurement, the long fiber is closed and the detection times of about 103

photons are recorded with respect to their trigger pulses. From these photons,we calculate the probability density for a photodetection, shown in Fig. 8. Themeasurement has been done (a) before and (b) after optimizing the single-photonsource. The autocorrelation functions of both curves, shown in the inset of Fig. 8,were calculated using Eq. (50). The width of these curves is (a) 1.07 µs and(b) 0.81 µs. In Section 5.4 we compare both results with the joint detection prob-ability of perpendicular polarized photon pairs.

Note that one obtains no information on the shape or duration of individualphotons from a detection probability that is averaged over many photodetections.As already stated in Section 4.2, such a measurement does not exclude that thephotons are very short so that the average probability distribution reflects only anemission-time jitter. Only from a time-resolved two-photon interference experi-ment, one obtains information on the duration of the photons. This is discussed inthe next subsection.


FIG. 8. Probability density for photodetections averaged over 103 photons (a) before and (b) afteroptimizing the single-photon source. The data are corrected for the detector dark-count rate. The insetshows the corresponding autocorrelation functions, A(2)(τ ).

5.3. TIME-RESOLVED TWO-PHOTON INTERFERENCE

The detection times of about 105 photons are registered by the two detectors inthe output ports of the beam splitter, while the photons in each pair impinge si-multaneously, i.e. with δτ = 0. The number of joint photodetections, N(2), is thendetermined from the recorded detection times as a function of the detection-timedifference, τ (using 48 ns to 120 ns long time bins). To do that, the photon dura-tion must exceed the time resolution of the detectors. Otherwise, joint detectionprobabilities could only be examined as a function of the arrival-time delay, δτ ,like in most other experiments.

We have performed these two-photon interference experiments before and af-ter optimizing the single-photon source. The results are shown in Figs. 9 and 10,respectively. Each experiment is first performed with photons of (a) perpendicularand then with photons of (b) parallel polarization, until about 105 photons are de-tected. Note that the number of joint photodetections is corrected for the constantbackground contribution stemming from detector dark counts.

In case of perpendicular polarization, no interference takes place. In accordancewith Section 3.6, the number of joint photodetections shows a Gaussian peak cen-tered at τ = 0. This signal is used as a reference, since any interference leads to


FIG. 9. Number of joint photodetections in 120 ns-long time bins as a function of the detec-tion-time difference, τ , before optimizing the single-photon source. The results are shown for photonsof (a) perpendicular polarization and (b) parallel polarization. In both cases, the data is accumulatedfor a total number of 73,000 photodetections. The solid lines are numerical fits of the theoreticalexpectations to the data. T1 is the width of the Gaussian peak in (a), and the width of the dip forparallel polarized photons (b) is given by T2. The dotted curve shows the T3-wide autocorrelationfunction, A(2)(τ ), of the average detection probability.


FIG. 10. Number of joint photodetections in 48 ns long time bins after optimizing the sin-gle-photon source. The data is shown in (a) for perpendicular and in (b) for parallel polarized photons.In both cases, the data is accumulated for a total number of 139,000 photodetections. The dotted curveshows the T3-wide autocorrelation function of the average detection probability. Compared to the re-sults of Fig. 9, the width of the Gaussian peak in (a) is decreased to T1 = 0.64 µs and it is clearlysmaller than T3 = 0.81 µs. With parallel polarizations, a dip of increased width, T2 = 0.44 µs, isobserved. All results are discussed in Section 5.4.


a significant deviation. If we now switch to parallel polarization, identical pho-tons are expected to leave the beam splitter as a pair, so that their joint detectionprobability should be zero for all values of τ . In the experiment, however, thesignal does not vanish completely. Instead, we observe a pronounced minimumaround τ = 0, which complies well with the behavior one expects for varyingspatio-temporal modes, as shown in Figs. 5 and 6. In analogy to Eqs. (41) and(46), respectively, the number of joint photodetections is then given by

(54)N(2)(τ ) = N(2)0 exp

(− τ 2

T 21

)[1 − cos2 ϕ exp

(− τ 2

T 22

)].

N(2)0 is the peak value at τ = 0 that we measure for perpendicular polarized pho-

tons, ϕ = π/2. The time T1 is the width of this Gaussian peak, whereas T2 givesthe width of the dip for photons of parallel polarization, ϕ = 0. In Section 5.4,both numbers are used to deduce the frequency and the emission-time jitter. Thetwo widths, T1 and T2, are obtained from a fit of Eq. (54) to the measured data.This is done in two steps. First, we obtain T1 and N(2)

0 from a fit to the data takenwith perpendicular polarized photons. We then keep these two values and obtainthe dip width T2 from a subsequent fit to the data with parallel polarized pho-tons. In this second step, we use a polarization term of cos2 ϕ = 0.92 to take intoaccount that we have a small geometric mode mismatch. This is well justifiedsince mode mismatch and non-parallel polarizations affect the signal in the samemanner. The value of ϕ has been obtained from an independent second-order in-terference measurement.

As one can see by comparing Figs. 9 and 10, the compensation of the Earth’smagnetic field and the improved recycling scheme lead to a decreased width T1of the Gaussian peak in (a) and a broader dip T2 in (b). As we discuss in thefollowing, these results show that this optimization has successfully reduced thejitter in the mode function of the photons.

Moreover, as shown in Fig. 11(a) and (b), we resolve a pronounced oscillationin the number of joint photodetections when a frequency difference, �, is delib-erately introduced between the interfering photons (Legero et al., 2004). This isachieved by driving the atom-cavity system with a sequence of trigger pulses thatalternate between two frequencies. The frequency difference between consecu-tive pulses is either (a) 2π × 2.8 MHz or (b) 2π × 3.8 MHz. In accordance withSection 3.6, the oscillation in the joint detection probability always starts with aminimum at τ = 0, and the maxima exceed the reference signal that we measurewith perpendicular polarized photons.

The latter experiment has been performed with the optimized single-photonsource. The number of detected photons (corrected for the number of dark counts)equals the photon number in Fig. 10. Therefore N(2)

0 , T1 and T2 are well knownfrom this previous measurement. As shown by Legero (2005), the only remaining


FIG. 11. Two-photon interference of parallel polarized photons with a frequency difference, �,of (a) 2π × 2.8 MHz and (b) 2π × 3.8 MHz. The number of joint photodetections is accumulatedover (a) 210,000 and (b) 319,000 detection events. It oscillates as a function of the detection-timedifference. The solid curves represent numerical fits to the data with a frequency-difference of(a) � = 2π × 2.86 MHz and (b) � = 2π × 3.66 MHz. The dotted curve shows the referencesignal measured with perpendicular polarized photons.


parameter one can obtain from a fit of the joint detection probability,

(55)N(2)(τ ) = N(2)0 exp

(− τ 2

T 21

)[1 − cos2 ϕ cos(τ�) exp

(− τ 2

T 22

)],

to the data is the frequency difference �. Its fit value agrees very well with the fre-quency differences that we imposed on consecutive pulses. We therefore concludethat the adiabatic Raman transition we use to generate the photons allows us to ad-just the single-photon frequency. Moreover, the oscillations in the joint detectionprobability impressively demonstrate that time-resolved two-photon interferenceexperiments are able to reveal small phase variations between the photons, like,e.g., the frequency difference we have deliberately imposed here.

5.4. INTERPRETATION OF THE RESULTS

We start our analysis by comparing the autocorrelation function of the averagedetection probability with the result of the two-photon coincidence measure-ment with perpendicular polarized photons, shown in Fig. 9(a) and Fig. 10(a).The shape of the autocorrelation function, A(2)(τ ), is commensurable with thedata obtained in the two-photon correlation experiment before the optimizationof the source, but it differs significantly afterwards. As discussed in Section 4.3,the different widths of both curves, T1 = 0.64 µs and T3 = 0.81 µs, indicatethat the photon stream is subject to variations in the spatiotemporal mode func-tions that are much less pronounced in the subensemble of consecutive photons.These variations cannot be attributed to a jitter in the photon frequency, since theautocorrelation function and the joint detection probability, given by Eqs. (52)and (53), depend only on the frequency-independent amplitude of the mode func-tions. Therefore the emission time and/or the duration of the photons must besubject to a jitter. As a consequence, the average detection probability shown inFig. 8 cannot represent the shape of the underlying single-photon wavepackets.In particular the width of the measured photon detection probability is broadeneddue to the emission-time jitter.

Moreover, the discrepancy between A(2)(τ ) (Fig. 8) and the Gaussian peak(Fig. 9(a) and Fig. 10(a)) shows that the variations in the whole photon streamare larger than the variations in the subensemble of consecutive photons. There-fore the following analysis of the times T1 and T2 of the two-photon interferencecannot be generalized to the whole photon stream.

To figure out the photon characteristics that can explain the measured quantum-beat signal, we restrict the analysis of T1 and T2 to a frequency and an emission-time jitter. If we assume that successively emitted photons show only a varia-tion of their frequencies, then T1 is identical to the photon duration, whereasT2 = 2/δω is solely due to the frequency variation. In this case, T2 is identical


to the coherence time, which one could also measure using second-order interfer-ence (Santori et al., 2002; Jelezko et al., 2003). As shown in Figs. 9 and 10, opti-mizing the single-photon source reduces the bandwidth of the frequency variationfrom δω/2π = 1.03 MHz to δω/2π = 720 kHz. The remaining inhomogeneousbroadening of the photon frequency can be attributed to several technical reasons.First, static and fluctuating magnetic fields affect the energies of the Zeeman sub-levels and spread the photon frequencies over a range of 160 kHz. Second thetrigger laser has a linewidth of 50 kHz, which is also mapped to the photons. Andthird, diabatically generated photons lead to an additional broadening.

Another explanation for the measured quantum-beat signal assumes photonsof fixed frequency and shape, but with an emission-time jitter. In this case, wehave T1 = √

δt2 +�τ 2 and T2 = T1δt/�τ . From these two equations, one cancalculate δt , which is the lower limit of the photon duration, and �τ , which is themaximum emission-time jitter. In our experiment, the optimization of the single-photon source led to an increase of δt from 0.29 µs to 0.36 µs, and at the sametime to a reduction of the maximum emission-time jitter from �τ = 0.82 µs to�τ = 0.53 µs.

However, in general, both the frequency and the emission time are subject to ajitter. If these fluctuations are uncorrelated, a whole range of (δω,�τ)-pairs canexplain the peak and dip widths, T1 and T2. For our two sets of data (before (a)and after (b) the optimization of the source), this is illustrated in Fig. 12. All pairsof frequency and emission-time jitters that are in agreement with the measuredvalues of T1 and T2 lie on one of the two solid lines. From this figure, it is evidentthat the values for δω and �τ deduced above represent the upper limits for therespective fluctuations. Moreover, it is also nicely visible that our optimizationof the source significantly improved the frequency stability and emission-timeaccuracy of our single-photon source.

We emphasize again that this information about the photons can be obtainedfrom time-resolved two-photon interference experiments, but not from a mea-surement of the average detection probability.

6. Conclusion

We have shown that time-resolved two-photon interference experiments are anexcellent tool to characterize single photons. In these experiments, two photonsare superimposed on a beam splitter and the joint detection probability in the twooutput ports of the beam splitter is measured as a function of the detection-timedifference of the photons. This is only possible if the photons are long compared tothe detector time resolution. For identical photons, the joint detection probabilityis expected to be zero. Variations of the spatiotemporal modes of the photons leadto joint photodetections except for zero detection-time difference. Therefore the


FIG. 12. Frequency jitter, δω, and emission-time jitter, �τ , before (a) and after (b) optimizingthe single-photon source. The two curves represent all pairs of jitters that match the two widths, T1and T2, found in the two-photon interference experiments.

joint detection probability shows a pronounced dip. From the width of this dip,one can estimate the maximum emission-time jitter and the minimum coherencetime of the photons. In addition, a lower limit of the single-photon duration can beobtained. This is not possible by just measuring the average detection probabilitywith respect to the trigger producing the photons. Moreover, we have shown thata frequency difference between photons leads to a distinct oscillation in the jointdetection probability. This does not only demonstrate that we are able to adjustthe frequencies of the photons emitted from a single-photon source, but also thatone is sensitive to very small frequency differences in time-resolved two-photoninterference measurements.

7. Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft (SPP 1078and SFB 631) and the European Union (IST (QGATES) and IHP (CONQUEST)programs).


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FLUCTUATIONS IN IDEAL ANDINTERACTING BOSE–EINSTEINCONDENSATES: FROM THE LASERPHASE TRANSITION ANALOGYTO SQUEEZED STATES ANDBOGOLIUBOV QUASIPARTICLES*

VITALY V. KOCHAROVSKY1,2, VLADIMIR V. KOCHAROVSKY2,MARTIN HOLTHAUS3, C.H. RAYMOND OOI1, ANATOLY SVIDZINSKY1,WOLFGANG KETTERLE4 and MARLAN O. SCULLY1,5

1Institute for Quantum Studies and Department of Physics, Texas A&M University,

TX 77843-4242, USA2Institute of Applied Physics, Russian Academy of Science, 600950 Nizhny Novgorod, Russia3Institut für Physik, Carl von Ossietzky Universitat, D-2611 Oldenburg, Germany4MIT-Harvard Center for Ultracold Atoms, and Department of Physics, MIT, Cambridge,

MA 02139, USA5Princeton Institute for Materials Science and Technology, Princeton University,

NJ 08544-1009, USA

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2932. History of the Bose–Einstein Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 298

2.1. What Bose Did . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2992.2. What Einstein Did . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3032.3. Was Bose–Einstein Statistics Arrived at by Serendipity? . . . . . . . . . . . . . . . . 3072.4. Comparison between Bose’s and Einstein’s Counting of the Number of Microstates W 314

3. Grand Canonical versus Canonical Statistics of BEC Fluctuations . . . . . . . . . . . . . 3153.1. Relations between Statistics of BEC Fluctuations in the Grand Canonical, Canonical,

and Microcanonical Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3163.2. Exact Recursion Relation for the Statistics of the Number of Condensed Atoms in an

Ideal Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

* It is a pleasure to dedicate this review to Prof. Herbert Walther, our guide in so many fieldsof physics. His contributions to atomic, molecular and optical physics are enlightened by the deepinsights he has given us into the foundations of quantum mechanics, statistical physics, nonlineardynamics and much more.


DOI 10.1016/S1049-250X(06)53010-1

291

292 V.V. Kocharovsky et al.

3.3. Grand Canonical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3214. Dynamical Master Equation Approach and Laser Phase-Transition Analogy . . . . . . . 328

4.1. Quantum Theory of the Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3294.2. Laser Phase-Transition Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3344.3. Derivation of the Condensate Master Equation . . . . . . . . . . . . . . . . . . . . . 3354.4. Low Temperature Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3424.5. Quasithermal Approximation for Noncondensate Occupations . . . . . . . . . . . . 3444.6. Solution of the Condensate Master Equation . . . . . . . . . . . . . . . . . . . . . . 3454.7. Results for BEC Statistics in Different Traps . . . . . . . . . . . . . . . . . . . . . . 3504.8. Condensate Statistics in the Thermodynamic Limit . . . . . . . . . . . . . . . . . . . 3544.9. Mesoscopic and Dynamical Effects in BEC . . . . . . . . . . . . . . . . . . . . . . . 355

5. Quasiparticle Approach and Maxwell’s Demon Ensemble . . . . . . . . . . . . . . . . . . 3575.1. Canonical-Ensemble Quasiparticles in the Reduced Hilbert Space . . . . . . . . . . 3595.2. Cumulants of BEC Fluctuations in an Ideal Bose Gas . . . . . . . . . . . . . . . . . 3615.3. Ideal Gas BEC Statistics in Arbitrary Power-Law Traps . . . . . . . . . . . . . . . . 3645.4. Equivalent Formulation in Terms of the Poles of the Generalized Zeta Function . . . 370

6. Why Condensate Fluctuations in the Interacting Bose Gas are Anomalously Large, Non-Gaussian, and Governed by Universal Infrared Singularities? . . . . . . . . . . . . . . . . 3726.1. Canonical-Ensemble Quasiparticles in the Atom-Number-Conserving Bogoliubov

Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3726.2. Characteristic Function and all Cumulants of BEC Fluctuations . . . . . . . . . . . . 3746.3. Surprises: BEC Fluctuations are Anomalously Large and Non-Gaussian Even in the

Thermodynamic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3756.4. Crossover between Ideal and Interaction-Dominated BEC: Quasiparticles Squeezing

and Pair Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3796.5. Universal Anomalies and Infrared Singularities of the Order Parameter Fluctuations

in the Systems with a Broken Continuous Symmetry . . . . . . . . . . . . . . . . . . 3837. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3908. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3949. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395A. Bose’s and Einstein’s Way of Counting Microstates . . . . . . . . . . . . . . . . . . . . . 395B. Analytical Expression for the Mean Number of Condensed Atoms . . . . . . . . . . . . . 397C. Formulas for the Central Moments of Condensate Fluctuations . . . . . . . . . . . . . . . 399D. Analytical Expression for the Variance of Condensate Fluctuations . . . . . . . . . . . . 401E. Single Mode Coupled to a Reservoir of Oscillators . . . . . . . . . . . . . . . . . . . . . 402F. The Saddle-Point Method for Condensed Bose Gases . . . . . . . . . . . . . . . . . . . . 404

10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

AbstractWe review the phenomenon of equilibrium fluctuations in the number of condensedatoms n0 in a trap containing N atoms total. We start with a history of the Bose–Einstein distribution, a similar grand canonical problem with an indefinite totalnumber of particles, the Einstein–Uhlenbeck debate concerning the rounding of themean number of condensed atoms n0 near a critical temperature Tc, and a discus-sion of the relations between statistics of BEC fluctuations in the grand canonical,canonical, and microcanonical ensembles.

First, we study BEC fluctuations in the ideal Bose gas in a trap and explain whythe grand canonical description goes very wrong for all moments 〈(n0 − n0)

m〉,except of the mean value. We discuss different approaches capable of providing

1] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 293

approximate analytical results and physical insight into this very complicated prob-lem. In particular, we describe at length the master equation and canonical-ensemblequasiparticle approaches which give the most accurate and physically transparentpicture of the BEC fluctuations. The master equation approach, that perfectly de-scribes even the mesoscopic effects due to the finite number N of the atoms in thetrap, is quite similar to the quantum theory of the laser. That is, we calculate a steady-state probability distribution of the number of condensed atoms pn0(t = ∞) froma dynamical master equation and thus get the moments of fluctuations. We presentanalytical formulas for the moments of the ground-state occupation fluctuations inthe ideal Bose gas in the harmonic trap and arbitrary power-law traps.

In the last part of the review, we include particle interaction via a generalized Bo-goliubov formalism and describe condensate fluctuations in the interacting Bose gas.In particular, we show that the canonical-ensemble quasiparticle approach worksvery well for the interacting gases and find analytical formulas for the characteristicfunction and all cumulants, i.e., all moments, of the condensate fluctuations. The sur-prising conclusion is that in most cases the ground-state occupation fluctuations areanomalously large and are not Gaussian even in the thermodynamic limit. We alsoresolve the Giorgini, Pitaevskii and Stringari (GPS) vs. Idziaszek et al. debate on thevariance of the condensate fluctuations in the interacting gas in the thermodynamiclimit in favor of GPS. Furthermore, we clarify a crossover between the ideal-gas andweakly-interacting-gas statistics which is governed by a pair-correlation, squeezingmechanism and show how, with an increase of the interaction strength, the fluctu-ations can now be understood as being essentially 1/2 that of an ideal Bose gas.We also explain the crucial fact that the condensate fluctuations are governed by asingular contribution of the lowest energy quasiparticles. This is a sort of infraredanomaly which is universal for constrained systems below the critical temperatureof a second-order phase transition.

1. Introduction

Professor Herbert Walther has taught us that good physics unifies and unites seem-ingly different fields. Nowhere is this more apparent than in the current studies ofBose–Einstein condensation (BEC) and coherent atom optics which draw fromand contribute to the general subject of coherence effects in many-body physicsand quantum optics. It is in this spirit that the present paper presents the re-cent application of techniques, ideas, and theorems which have been developedin understanding lasers and squeezed states to the condensation of N bosons.Highlights of these studies, and related points of BEC history, are described inthe following paragraphs.

(1) Bose [1,2] got the ball rolling by deriving the Planck distribution withoutusing classical electrodynamics, as Planck [3] and Einstein [4] had done. Instead,

294 V.V. Kocharovsky et al. [1

he took the extreme photon-as-a-particle point of view, and by regarding theseparticles as indistinguishable obtained, among other things, Planck’s result,

(1)nk = 1

eβεk − 1,

where nk is the mean number of photons with energy εk and wavevector k, β =(kBT )

−1, T is the blackbody temperature, and kB is Boltzmann’s constant.However, his paper was rejected by the Philosophical Magazine and so he sent

it to Einstein, who recognized its value. Einstein translated it into German and gotit published in the Zeitschrift für Physik [1]. He then applied Bose’s method toatoms and predicted that the atoms would “condense” into the lowest energy levelwhen the temperature was low enough [5–7].

Time has not dealt as kindly with Bose as did Einstein. As is often the case inthe opening of a new field, things were presented and understood imperfectly atfirst. Indeed Bose did his “counting” of photon states in cells of phase space in anunorthodox fashion. So much so that the famous Max Delbrück wrote an inter-esting article [8] in which he concluded that Bose made a mistake, and only gotthe Planck distribution by serendipity. We here discuss this opinion, and retracethe steps that led Bose to his result. Sure, he enjoyed a measure of luck, but hismathematics and his derivation were correct.

(2) Einstein’s treatment of BEC of atoms in a large box showed a cusp in thenumber of atoms in the ground state, n0, as a function of temperature,

(2)n0 = N

(1 −(T

Tc

)3/2 )for T � Tc, whereN is the total number of atoms, and Tc is the (critical) transitiontemperature.

Uhlenbeck [9] criticized this aspect of Einstein’s work, claiming that the cuspat T = Tc is unphysical. Einstein agreed with the Uhlenbeck criticism but arguedthat in the limit of large numbers of atoms (the thermodynamic limit) everythingwould be okay. Later, Uhlenbeck and his student Kahn showed [10] that Einsteinwas right and put the matter to rest (for a while).

Fast forward to the present era of mesoscopic BEC physics with only thou-sands (or even hundreds) of atoms in a condensate. What do we now do with thisUhlenbeck dilemma? As one of us (W.K.) showed some time ago [11], all that isneeded is a better treatment of the problem. Einstein took the chemical potentialto be zero, which is correct for the ideal Bose gas in the thermodynamic limit.However, when the chemical potential is treated more carefully, the cusp goesaway, as we discuss in detail, see, e.g., Figs. 1a and 3.

(3) So far everything we have been talking about concerns the average numberof particles in the condensate. Now we turn to the central focus of this review:fluctuations in the condensate particle number. As the reader will recall, Einstein


used the fluctuation properties of waves and particles to great advantage. In par-ticular he noted that in Planck’s problem, there were particle-like fluctuations inphoton number in addition to the wave-like contribution, i.e.,

(3)�n2k = n2

k(wave) + nk(particle),

and in this way he argued for a particle picture of light.In his studies on Bose–Einstein condensation, he reversed the logic arguing that

the fluctuations in the ideal quantum gas also show both wave-like and particle-like attributes, just as in the case of photons. It is interesting that Einstein was ledto the wave nature of matter by studying fluctuations. We note that he knew ofand credited de Broglie at this point well before wave mechanics was developed.

Another important contribution to the problem of BEC fluctuations came fromFritz London’s observation [12] that the specific heat is proportional to the vari-ance of a Bose–Einstein condensate and showed a cusp, which he calculated asbeing around 3.1 K. It is noteworthy that the so-called lambda point in liquidHelium, marking the transition from normal to superfluid, takes place at around2.19 K.

However, Ziff, Uhlenbeck and Kac [13] note several decades later that there isa problem with the usual treatment of fluctuations. They say:

[When] the grand canonical properties for the ideal Bose gas are derived, it turns outthat some of them differ from the corresponding canonical properties—even in the bulklimit! . . . The grand canonical ensemble . . . loses its validity for the ideal Bose gas inthe condensed region.

One of us (M.H.) has noted elsewhere [14] that:

This grand canonical fluctuation catastrophe has been discussed by generations ofphysicists. . .

Let us sharpen the preceding remarks. Large fluctuations are a feature of thethermal behavior of systems of bosons. If n is the mean number of noninteractingparticles occupying a particular one particle state, then the mean square occupa-tion fluctuation in the grand canonical picture is n(n+ 1). If, however, the systemhas a fixed total number of particles N confined in space by a trapping poten-tial, then at low enough temperature T when a significant fraction of N are inthe ground state, such large fluctuations are impossible. No matter how large N ,the grand canonical description cannot be even approximately true. This seemsto be one of the most important examples that different statistical ensemblesgive agreement or disagreement in different regimes of temperatures. To avoidthe catastrophe, the acclaimed statistical physicist D. ter Haar [15] proposed thatthe fluctuations in the condensate particle number in the low temperature regime(adapted to a harmonic trap) might go as

(4)�n0 ≡√⟨(n0 − n0)2

⟩ = N − n0 = N

(T

Tc

)3

.


This had the correct zero limit as T → 0, but is not right for higher temperatureswhere the leading term actually goes as [N( T

Tc)3]1/2. The point is that fluctuations

are subtle; even the ideal Bose gas is full of interesting physics in this regard.In this paper, we resolve the grand canonical fluctuation catastrophe in several

ways. In particular, recent application of techniques developed in the quantumtheory of the laser [16,17] and in quantum optics [18,19] allow us to formulatea consistent and physically appealing analytical picture of the condensate fluctu-ations in the ideal and interacting Bose gases. Our present understanding of thestatistics of the BEC fluctuations goes far beyond the results that were formulatedbefore the 90s BEC boom, as summarized by Ziff, Uhlenbeck and Kac in theirclassical review [13]. Theoretical predictions for the BEC fluctuations, which areanomalously large and non-Gaussian even in the thermodynamical limit, are de-rived and explained on the basis of the simple analytical expressions [20,21]. Theresults are in excellent agreement with the exact numerical simulations. The exis-tence of the infrared singularities in the moments of fluctuations and the universalfact that these singularities are responsible for the anomalously large fluctuationsin BEC, are among the recent conceptual discoveries. The quantum theory oflaser threshold behavior constitutes another important advance in the physics ofbosonic systems.

(4) The laser made its appearance in the early 60s and provided us with a newsource of light with a new kind of photon statistics. Before the laser, the statisticsof radiation were either those of black-body photons associated with Planck’sradiation, which for a single mode of frequency ν takes the form

(5)pn = e−nβhν(1 − e−βhν

),

or when one considers radiation from a coherent oscillating current such as a radiotransmitter or a microwave klystron the photon distribution becomes Poissonian,

(6)pn = nn

n! e−n,

where n is the average photon number.However, laser photon statistics, as derived from the quantum theory of the

laser, goes from black-body statistics below threshold to Poissonian statistics farabove threshold. In between, when we are in the threshold region (and even abovethreshold as in the case, for example, of the helium neon laser), we have a newdistribution. We present a review of the laser photon statistical problem.

It has been said that the Bose–Einstein condensate is to atoms what the laseris to photons; even the concept of an atom laser has emerged. In such a case, onenaturally asks, “what is the statistical distribution of atoms in the condensate?”For example, let us first address the issue of an ideal gas of N atoms in contactwith a reservoir at temperature T . The condensate occupation distribution in theharmonic trap under these conditions at low enough temperatures is given by the


FIG. 1. (a) Mean value 〈n0〉 and (b) variance �n0 =√〈n2

0〉 − 〈n0〉2 of the number of condensedatoms as a function of temperature for N = 200 atoms in a harmonic trap calculated via the solutionof the condensate master equation (solid line). Large dots are the exact numerical results obtainedin the canonical ensemble. Dashed line for 〈n0〉 is a plot of N [1 − (T /Tc)

3] which is valid in thethermodynamic limit. Dashed line for �n0 is the grand canonical answer

√n0(n0 + 1) which gives

catastrophically large fluctuations below Tc .

BEC master equation analysis as

(7)pn0 = 1

ZN

[N(T/Tc)3]N−n0

(N − n0)! .

The mean number and variance obtained from the condensate master equationare in excellent agreement with computer simulation (computer experiment) asshown in Fig. 1. We will discuss this aspect of the fluctuation problem in somedetail and indicate how the fluctuations change when we go to the case of theinteracting Bose gas.

(5) The fascinating interface between superfluid He II and BEC in a dilute gaswas mapped out by the experiments of Reppy and coworkers [22]; and finite-sizeeffects were studied theoretically by M. Fisher and coworkers [23]. They carriedout experiments in which He II was placed in a porous glass medium which servesto keep the atoms well separated. These experiments are characterized by a dilutegas BEC of N atoms at temperature T .

Of course, it was the successful experimental demonstration of Bose–Einsteincondensation in the ultracold atomic alkali–metal [24–26], hydrogen [27] andhelium gases [22,28,29] that stimulated the renaissance in the theory of BEC.In less than a decade, many intriguing problems in the physics of BEC, that werenot studied, or understood before the 90s [30–36], were formulated and resolved.

(6) Finally, we turn on the interaction between atoms in the BEC and findexplicit expressions for the characteristic function and all cumulants of the prob-ability distribution of the number of atoms in the (bare) ground state of a trapfor the weakly interacting dilute Bose gas in equilibrium. The surprising result is


that the BEC statistics is not Gaussian, i.e., the ratio of higher cumulants to anappropriate power of the variance does not vanish, even in the thermodynamiclimit. We calculate explicitly the effect of Bogoliubov coupling between excitedatoms on the suppression of the BEC fluctuations in a box (“homogeneous gas”)at moderate temperatures and their enhancement at very low temperatures. Wefind that there is a strong pair-correlation effect in the occupation of the coupledatomic modes with the opposite wavevectors k and −k. This explains why theground-state occupation fluctuations remain anomalously large to the same ex-tent as in the noninteracting gas, except for a factor of 1/2 suppression. We findthat, roughly speaking, this is so because the atoms are strongly coupled in cor-related pairs such that the number of independent stochastic occupation variables(“degrees of freedom”) contributing to the fluctuations of the total number of ex-cited atoms is only 1/2 the atom number N . This is a particular feature of thewell-studied quantum optics phenomenon of two-mode squeezing (see, e.g., [37]and [18,19]). The squeezing is due to the quantum correlations that build up inthe bare excited modes via Bogoliubov coupling and is very similar to the noisesqueezing in a nondegenerate parametric amplifier.

Throughout the review, we will check main approximate analytical results(such as in Eqs. (162), (172), (223), (263), (271)) against the “exact” numericsbased on the recursion relations (79) and (80) which take into account exactly allmesoscopic effects near the critical temperature Tc. Unfortunately, the recursionrelations are known only for the ideal Bose gas. In the present review we discussthe BEC fluctuations mainly in the canonical ensemble, which cures misleadingpredictions of the grand canonical ensemble and, at the same time, does not haveany essential differences with the microcanonical ensemble for most physicallyinteresting quantities and situations. Moreover, as we discuss below, the canonicalpartition function can be used for an accurate calculation of the microcanonicalpartition function via the saddle-point method.

2. History of the Bose–Einstein Distribution

In late 1923, a certain Satyendranath Bose, reader in physics at the Universityof Dacca in East Bengal, submitted a paper on Planck’s law of blackbody radi-ation to the Philosophical Magazine. Six months later he was informed that thepaper had received a negative referee report, and consequently been rejected [38].While present authors may find consolation in the thought that the rejectionof a truly groundbreaking paper after an irresponsibly long refereeing processis not an invention of our times, few of their mistreated works will eventuallymeet with a recognition comparable to Bose’s. Not without a palpable amount ofself-confidence, Bose sent the rejected manuscript to Albert Einstein in Berlin,together with a handwritten cover letter dated June 4, 1924, beginning [39]:


Respected Sir:

I have ventured to send you the accompanying article for your perusal and opinion.I am anxious to know what you think of it. You will see that I have tried to deducethe coefficient 8πν2/c3 in Planck’s Law independent of the classical electrodynamics,only assuming that the ultimate elementary regions in the phase-space has the con-tent h3. I do not know sufficient German to translate the paper. If you think the paperworth publication I shall be grateful if you arrange its publication in Zeitschrift fürPhysik. Though a complete stranger to you, I do not hesitate in making such a request.Because we are all your pupils though profiting only from your teachings through yourwritings. . .

In hindsight, it appears curious that Bose drew Einstein’s attention only to hisderivation of the prefactor in Planck’s law. Wasn’t he aware of the fact that histruly singular achievement, an insight not even spelled out explicitly in Einstein’stranslation of his paper as it was received by the Zeitschrift für Physik on July 2,1924 [1,2], but contained implicitly in the mathematics, lay elsewhere?

2.1. WHAT BOSE DID

In the opening paragraph of his paper [1,2], Bose pounces on an issue which heconsiders unsatisfactory: When calculating the energy distribution of blackbodyradiation according to

(8)�ν dν = 8πν2 dν

c3Eν,

that is,

energy per volume of blackbody radiation with frequency between νand ν + dν

= number of modes contained in that frequency intervalof the radiation field per volume

× thermal energy Eν of a radiation mode with frequency ν,

the number of modes had previously been derived only with reference to classicalphysics. In his opinion, the logical foundation of such a recourse was not suffi-ciently secure, and he proposed an alternative derivation, based on the hypothesisof light quanta.

Considering radiation inside some cavity with volume V , he observed that thesquared momentum of such a light quantum is related to its frequency through

(9)p2 = h2ν2

c2,

where h denotes Planck’s constant, and c is the velocity of light. Dividing thefrequency axis into intervals of length dνs , such that the entire axis is covered


when the label s varies from s = 0 to s = ∞, the phase space volume associatedwith frequencies between ν and ν + dνs therefore is

(10)∫

dx dy dz dpx dpy dpz = V 4π

(hν

c

)2h dνs

c= V 4π

h3ν2

c3dνs.

It does not seem to have bothered Bose that the concept of phase space againbrings classical mechanics into play. Relying on the assumption that a singlequantum state occupies a cell of volume h3 in phase space, a notion which, inthe wake of the Bohr–Sommerfeld quantization rule, may have appeared naturalto a physicist in the early 1920s, and accounting for the two states of polariza-tion, the total number As of quantum cells belonging to frequencies between ν

and ν + dνs , corresponding to the number of radiation modes in that frequencyinterval, immediately follows:

(11)As = V8πν2

c3dνs.

That’s all, as far as the first factor on the r.h.s. of Eq. (8) is concerned. This iswhat Bose announced in his letter to Einstein, but this is, most emphatically, nothis main contribution towards the understanding of Planck’s law. The few lineswhich granted him immortality follow when he turns to the second factor. Back-translated from Einstein’s phrasing of his words [1,2]:

Now it is a simple task to calculate the thermodynamic probability of a (macroscop-ically defined) state. Let Ns be the number of quanta belonging to the frequencyinterval dνs . How many ways are there to distribute them over the cells belongingto dνs? Let ps0 be the number of vacant cells, ps1 the number of those containing onequantum, ps2 the number of cells which contain two quanta, and so on. The number ofpossible distributions then is

(12)As !

ps0!ps1! . . ., where As = V

8πν2

c3dνs ,

and where

Ns = 0 · ps0 + 1 · ps1 + 2 · ps2 . . .is the number of quanta belonging to dνs .

What is happening here? Bose is resorting to a fundamental principle of statisti-cal mechanics, according to which the probability of observing a state with certainmacroscopic properties—in short: a macrostate—is proportional to the number ofits microscopic realizations—microstates—compatible with the macroscopicallygiven restrictions. Let us, for example, consider a model phase space consistingof four cells only, and let there be four quanta. Let us then specify the macrostateby requiring that one cell remain empty, two cells contain one quantum each, andone cell be doubly occupied, i.e., p0 = 1, p1 = 2, p2 = 1, and pr = 0 for r � 3.


How many microstates are compatible with this specification? We may place twoquanta in one out of four cells, and then choose one out of the remaining threecells to be the empty one. After that there is no further choice left, since each ofthe two other cells now has to host one quantum. Hence, there are 4 × 3 = 12possible configurations, or microstates: 12 = 4!/(1! 2! 1!). In general, when thereare As cells belonging to dνs , they can be arranged in As ! ways. However, if acell pattern is obtained from another one merely by a rearrangement of those ps0cells containing no quantum, the configuration remains unchanged. Obviously,there are ps0! such “neutral” rearrangements which all correspond to the sameconfiguration. The same argument then applies, for any r � 1, to those psr cellscontaining r quanta: Each of the psr ! possibilities of arranging the cells with r

quanta leads to the same configuration. Thus, each configuration is realized byps0!ps1! . . . equivalent arrangements of cells, and the number of different configu-rations, or microstates, is given by the total number of arrangements divided bythe number of equivalent arrangements, that is, by Bose’s expression (12).

There is one proposition tacitly made in this way of counting microstates whichmight even appear self-evident, but which actually constitutes the very core ofBose’s breakthrough, and which deserves to be spelled out explicitly: When con-sidering equivalent arrangements as representatives of merely one microstate, it isimplied that the quanta are indistinguishable. It does not matter “which quantumoccupies which cell”; all that matters are the occupation numbers psr . Even more,the “which quantum”-question is rendered meaningless, since there is, as a matterof principle, no way of attaching some sort of label to individual quanta belongingto the same dνs , with the purpose of distinguishing them. This “indistinguishabil-ity in principle” does not occur in classical physics. Two classical particles mayhave the same mass, and identical other properties, but it is nevertheless taken forgranted that one can tell one from the other. Not so, according to Bose, with lightquanta.

The rest of Bose’s paper has become a standard exercise in statistical physics.Taking into account all frequency intervals dνs , the total number of microstatescorresponding to a pre-specified set {psr } of cell occupation numbers is

(13)W[{psr}] =∏

s

As !ps0!ps1! . . .

.

The logarithm of this functional yields the entropy associated with the consideredset {psr }. Since, according to the definition of psr ,

(14)As =∑r

psr for each s,

and assuming the statistically relevant psr to be large, Stirling’s approximationln n! ≈ n ln n− n gives


(15)lnW[{psr}] =∑

s

As lnAs −∑s

∑r

psr lnpsr .

The most probable macrostate now is the one with the maximum number of mi-crostates, characterized by that set of occupation numbers which maximizes thisexpression (15). Stipulating that the radiation field be thermally isolated, so thatits total energy

(16)E =∑s

Nshνs with Ns =∑r

rpsr

is fixed, the maximum is found by variation of the psr , subject to this con-straint (16). In addition, the constraints (14) have to be respected. IntroducingLagrangian multipliers λs for these “number-of-cells” constraints, and a furtherLagrangian multiplier β for the energy constraint, the maximum is singled out bythe condition

(17)δ

(lnW[{psr}]−∑

s

λs∑r

psr − β∑s

hνs∑r

rpsr

)= 0,

giving

(18)∑r,s

δpsr(lnpsr + 1 + λs

)+ β∑s

hνs∑r

rδpsr = 0.

Since the δpsr can now be taken as independent, the maximizing configuration{psr } obeys

(19)ln psr + 1 + λs + rβhνs = 0,

or

(20)psr = Bse−rβhνs ,with normalization constants Bs to be determined from the constraints (14):

(21)As =∑r

psr = Bs

1 − e−βhνs.

The total number of quanta for the maximizing configuration then is

(22)Ns =∑r

rpsr = As(1 − e−βhνs

)∑r

re−rβhνs = As

eβhνs − 1

.

Still, the physical meaning of the Lagrangian multiplier β has to be established.This can be done with the help of the entropy functional, since inserting the max-imizing configuration yields the thermodynamical equilibrium entropy:

(23)S = kB lnW[{psr}] = kB

[βE −

∑s

As ln(1 − e−βhνs

)],


where kB denotes Boltzmann’s constant. From the identity ∂S/∂E = 1/T onethen finds β = 1/(kBT ), the inverse energy equivalent of the temperature T .Hence, from Eqs. (22) and (11) Bose obtains the total energy of the radiationcontained in the volume V in the form

(24)E =∑s

Nshνs =∑s

8πhνs3

c3V

1

exp( hνs

kBT− 1)

dνs,

which is equivalent to Planck’s formula: With the indistinguishability of quanta,i.e., Bose’s enumeration (12) of microstates as key input, the principles of statis-tical mechanics immediately yield the thermodynamic properties of radiation.

2.2. WHAT EINSTEIN DID

Unlike that unfortunate referee of the Philosophical Magazine, Einstein immedi-ately realized the power of Bose’s approach. Estimating that it took the manuscriptthree weeks to travel from Dacca to Berlin, Einstein may have received it aroundJune 25 [8]. Only one week later, on July 2, his translation of the manuscript wasofficially received by the Zeitschrift für Physik. The author’s name was lacking itsinitials—the byline of the published paper [1] simply reads: By Bose (Dacca Uni-versity, India)—but otherwise Einstein was doing Bose fair justice: He even sentBose a handwritten postcard stating that he regarded his paper as a most impor-tant contribution; that postcard seems to have impressed the German Consulate inCalcutta to the extent that Bose’s visa was issued without requiring payment ofthe customary fee [38].

Within just a few days, Einstein then took a further step towards exploring theimplications of the “indistinguishability in principle” of quantum mechanical en-tities. At the end of the printed, German version of Bose’s paper [1], there appearsthe parenthetical remark “Translated by A. Einstein”, followed by an announce-ment:

Note added by the translator: Bose’s derivation of Planck’s formula constitutes, in myopinion, an important step forward. The method used here also yields the quantumtheory of the ideal gas, as I will explain in detail elsewhere.

“Elsewhere” in this case meant the Proceedings of the Prussian Academy ofSciences. In the session of the Academy on July 10, Einstein delivered a paperentitled “Quantum theory of the monoatomic ideal gas” [5]. In that paper, heconsidered nonrelativistic free particles of mass m, so that the energy-momentumrelation simply reads

(25)E = p2

2m,


and the phase-space volume for a particle with an energy not exceeding E is

(26)Φ = V4π

3(2mE)3/2.

Again relying on the notion that a single quantum state occupies a cell of volumeh3 in phase space, the number of such cells belonging to the energy interval fromE to E +�E is

(27)�s = 2πV

h3(2m)3/2E1/2�E.

Thus, for particles with nonzero rest mass �s is the analog of Bose’s As in-troduced in Eq. (11). Einstein then specified the cell occupation numbers byrequiring that, out of these �s cells, psr�s cells contain r particles, so that psris the probability of finding r particles in any one of these cells,

(28)∑r

psr = 1.

Now comes the decisive step. Without attempt of justification or even comment,Einstein adopts Bose’s way (12) of counting the number of corresponding mi-crostates. This is a far-reaching hypothesis, which implies that, unlike classicalparticles, atoms of the same species with energies in the same range �E are in-distinguishable: Interchanging two such atoms does not yield a new microstate; aswith photons, it does not matter “which atom occupies which cell”. Consequently,the number of microstates associated with a pre-specified set of occupation prob-abilities {psr } for the above �s cells is

(29)Ws = �s!∏∞r=0(p

sr�s)!

,

giving, with the help of Stirling’s formula,

(30)lnWs = −�s∑r

psr lnpsr .

Einstein then casts this result into a more attractive form. Stipulating that theindex s does no longer refer jointly to the cells within a certain energy interval,but rather labels individual cells, the above expression naturally generalizes to

(31)lnW[{psr}] = −

∑s

∑r

psr lnpsr ,

where the cell index s now runs over all cells, so that psr here is the probabilityof finding r particles in the sth cell. It is interesting to observe that this func-tional (31) has precisely the same form as the Shannon entropy introduced in 1948in an information-theoretical context [40].


Since∑

r rpsr gives the expectation value of the number of particles occupying

the cell labelled s, the total number of particles is

(32)N =∑s

∑r

rpsr ,

while the total energy of the gas reads

(33)E =∑s

Es∑r

rpsr ,

where Es is the energy of a particle in the sth cell. Since, according to Eq. (26),a cell’s number s is related to the energy Es through

(34)s = Φs

h3= V

h3

4π

3

(2mEs)3/2

,

one has

(35)Es = cs2/3

with

(36)c = h2

2m

(4πV

3

)−2/3

.

Considering an isolated system, with given, fixed particle number N and fixedenergy E, the macrostate realized in nature is characterized by that set {psr } whichmaximizes the entropy functional (31), subject to the constraints (32) and (33),together with the constraints (28) expressing normalization of the cell occupationprobabilities. Hence,

δ

(lnW[{psr}]−∑

s

λs∑r

psr − α∑s

∑r

rpsr − β∑s

Es∑r

rpsr

)(37)= 0,

so that, seen from the conceptual viewpoint, the only difference between Bose’svariational problem (17) and Einstein’s variational problem (37) is the appearanceof an additional Lagrangian multiplier α in the latter: In the case of radiation, thetotal number of light quanta adjusts itself in thermal equilibrium, instead of beingfixed beforehand; in the case of a gas of particles with nonzero rest mass, thetotal number of particles is conserved, requiring the introduction of the entailingmultiplier α. One then finds

(38)ln psr + 1 + λs + αr + βrEs = 0

or

(39)psr = Bse−r(α+βEs),


with normalization constants to be determined from the constraints (28):

(40)Bs = 1 − e−(α+βEs).

Here we deviate from the notation in Einstein’s paper [5], in order to be compat-ible with modern conventions. The expectation value for the occupation numberof the cell with energy Es then follows from an elementary calculation similar toBose’s reasoning (22):

(41)∑r

rpsr = 1

eα+βEs − 1.

Therefore, the total number of particles and the total energy of the gas can beexpressed as

(42)N =∑s

1

eα+βEs − 1,

(43)E =∑s

Es

eα+βEs − 1.

Inserting the maximizing set (39) into the functional (31) yields, after a briefcalculation, the equilibrium entropy of the gas in the form

(44)S = kB lnW[{psr}] = kB

[αN + βE −

∑s

ln(1 − e−(α+βEs)

)].

In order to identify the Lagrangian multiplier β, Einstein considered an infinites-imal heating of the system, assuming its volume and, hence, the cell energies Es

to remain fixed. This gives

dE = T dS = kBT

[N dα + β dE + E dβ −

∑s

d(α + βEs)

eα+βEs − 1

](45)= kBTβ dE,

requiring

(46)β = 1

kBT.

As in Bose’s case, the Lagrangian multiplier β accounting for the energy con-straint is the inverse energy equivalent of the temperature T . The other multi-plier α, guaranteeing particle number conservation, then is determined from theidentity (42).

In the following two sections of his paper [5], Einstein shows how the ther-modynamics of the classical ideal gas is recovered if one neglects unity againsteα+βEs

, and derives the virial expansion of the equation of state for the quantumgas obeying Eqs. (42) and (43).


2.3. WAS BOSE–EINSTEIN STATISTICS ARRIVED AT BY SERENDIPITY?

The title of this subsection is a literal quote from the title of a paper by M. Del-brück [8], who contents that Bose made an elementary mistake in statistics inthat he should have bothered “which quantum occupies which cell”, which wouldhave been the natural approach, and that Einstein first copied that mistake withoutpaying much attention to it. Indeed, such a suspicion does not seem to be en-tirely unfounded. In his letter to Einstein, Bose announces only his comparativelystraightforward derivation of the number (11) of radiation modes falling into thefrequency range from ν to ν + dνs , apparently being unaware that his revolution-ary deed was the implicit exploitation of the “indistinguishability in principle” ofquanta—a concept so far unheard of. In Einstein’s translation of his paper [1] thisnotion of indistinguishability does not appear in words, although it is what under-lies the breakthrough. Even more, it does not appear in the first paper [5] on theideal Bose gas—until the very last paragraph, where Einstein ponders over

. . .a paradox which I have been unable to resolve. There is no difficulty in treatingalso the case of a mixture of two different gases by the method explained here. In thiscase, each molecular species has its own “cells”. From this follows the additivity of theentropies of the mixture’s components. Therefore, with respect to molecular energy,pressure, and statistical distribution each component behaves as if it were the only onepresent. A mixture containing n1 and n2 molecules, with the molecules of the firstkind being distinguishable (in particular with respect to the molecular masses m1,m2)only by an arbitrarily small amount from that of the second, therefore yields, at a giventemperature, a pressure and a distribution of states which differs from that of a uniformgas with n1 + n2 molecules with practically the same molecular mass, occupying thesame volume. However, this appears to be as good as impossible.

Interestingly, Einstein here considers “distinguishability to some variable de-gree”, which can be continuously reduced to indistinguishability. But this notionis flawed: Either the molecules have some feature which allows us to tell onespecies from the other, in which case the different species can be distinguished, orthey have none at all, in which case they are indistinguishable in principle. Thus,at this point, about two weeks after the receipt of Bose’s manuscript and one weekafter sending its translation to the Zeitschrift für Physik, even Einstein may not yethave fully grasped the implications of Bose’s way (12) of counting microstates.

But there was more to come. In December 1924, Einstein submitted a secondmanuscript on the quantum theory of the ideal Bose gas [6,7], formally writtenas a continuation of the first one. He began that second paper by pointing out acuriosity implied by his equation of state of the ideal quantum gas: Given a cer-tain number of particles N and a temperature T , and considering a compressionof the volume V , there is a certain volume below which a segregation sets in.With decreasing volume, an increasing number of particles has to occupy the firstquantum cell, i.e., the state without kinetic energy, while the rest is distributedover the other cells according to Eq. (41), with eα = 1. Thus, Bose–Einstein con-densation was unveiled! But this discovery merely appears as a small addendum


to the previous paper [5], for Einstein then takes up a different, more fundamen-tal scent. He mentioned that Ehrenfest and other colleagues of his had criticizedthat in Bose’s and his own theory the quanta or particles had not been treated asstatistically independent entities, a fact which had not been properly emphasized.Einstein agrees, and then he sets out to put things straight. He abandons his pre-vious “single-cell” approach and again considers the collection of quantum cellswith energies between Eν and Eν +�Eν , the number of which is

(47)zν = 2πV

h3(2m)3/2E1/2

ν �Eν.

Then he juxtaposes in detail Bose’s way of counting microstates to what is donein classical statistics. Assuming that there are nν quantum particles falling into�Eν , Bose’s approach (12) implies that there are

(48)Wν = (nν + zν − 1)!nν !(zν − 1)!

possibilities of distributing the particles over the cells. This expression can easilybe visualized: Drawing the nν particles as a sequence of nν “dots” in a row, theycan be organized into a microstate with specific occupation numbers for zν cells—again assuming that it does not matter which particle occupies which cell—byinserting zν − 1 separating “lines” between them. Thus, there are nν + zν − 1positions carrying a symbol, nν of which are dots. The total number of microstatesthen equals the total number of possibilities to select the nν positions carrying a“dot” out of these nν + zν − 1 positions, which is just the binomial coefficientstated in Eq. (48). To give an example: Assuming that there are nν = 4 particlesand zν = 4 cells, Eq. (48) states that there are altogether

(4 + 4 − 1)!4! 3! = 7!

4! 3! =7 · 6 · 5

2 · 3= 35

microstates. On the other hand, there are several sets of occupation numbers whichallow one to distribute the particles over the cells:

Occupation numbers Number of microstates

p4 = 1, p0 = 3 4!1! 3! = 4

p3 = 1, p1 = 1, p0 = 2 4!1! 1! 2! = 12

p2 = 2, p0 = 2 4!2! 2! = 6

p2 = 1, p1 = 2, p0 = 1 4!1! 2! 1! = 12

p1 = 4 4!4! = 1


The right column of this table gives, for each set, the number of microstatesaccording to Bose’s formula (12); obviously, these numbers add up to the totalnumber 35 anticipated above. Thus, the binomial coefficient (48) convenientlyaccounts for all possible microstates, without the need to specify the occupationnumbers, according to the combinatorial identity

(49)∑ Z!

p0! . . . pN ! =(N + Z − 1

N

),

where the sum is restricted to those sets {p0, p1, . . . , pN } which comply withthe two conditions

∑r pr = Z and

∑r rpr = N , as in the example above.

In Appendix A we provide a proof of this identity. With this background, let usreturn to Einstein’s reasoning: When taking into account all energy intervals�Eν ,the total number of microstates is given by the product W = ∏ν Wν , providingthe entropy functional

lnW[{nν}] =∑

ν

[(nν + zν) ln(nν + zν)− nν ln nν − zν ln zν

](50)=

∑ν

[nν ln

(1 + zν

nν

)+ zν ln

(nν

zν+ 1

)].

The maximizing set {nν} now has to obey the two constraints

(51)∑ν

nν = N,

(52)∑ν

nνEν = E,

but there is no more need for the multipliers λs appearing in the previous Eqs. (17)and (37), since the constraints (14) or (28) are automatically respected when start-ing from the convenient expression (48). Hence, one has

(53)δ

(lnW[{nν}]− α

∑ν

nν − β∑ν

nνEν

)= 0

or

(54)∑ν

[ln

(1 + zν

nν

)− α − βEν

]δnν = 0,

leading immediately to

(55)nν = zν

eα+βEν − 1,

in agreement with the previous result (41).But what, Einstein asks, would have resulted had one not adopted Bose’s pre-

scription (12) and thus counted equivalent arrangements with equal population


numbers only once, but rather had treated the particles as classical, statisticallyindependent entities? Then there obviously are

(56)Wν = (zν)nν

possibilities of distributing the nν particles belonging to �Eν over the zν cells:Each particle simply is placed in one of the zν cells, regardless of the others.Now, when considering all intervals �Eν , with distinguishable particles it doesmatter how those nν particles going into the respective �Eν are selected fromall N particles; for this selection, there are N !/∏ν nν ! possibilities. Thus, takingclassical statistics seriously, there are

(57)W = N !∏ν

(zν)nν

nν !possible microstates, yielding

lnW[{nν}] = N lnN −N +

∑ν

[nν ln zν − nν ln nν + nν]

(58)= N lnN +∑ν

[nν ln

(zν

nν

)+ nν

].

This is a truly vexing expression, since it gives a thermodynamical entropy whichis not proportional to the total number of particles, i.e., no extensive quantity, be-cause of the first term on the r.h.s. Hence, already in the days before Bose andEinstein one had got used to ignoring the leading factor N ! in Eq. (57), withthe half-hearted concession that microstates which result from each other by amere permutation of the N particles should not be counted as different. Of course,this is an intrinsic inconsistency of the classical theory: Instead of accepting that,shouldn’t one abandon Eq. (57) straight away and accept the more systematicquantum approach, despite the apparently strange consequence of losing the par-ticles’ independence? And Einstein gives a further, strong argument in favor ofthe quantum theory: At zero temperature, all particles occupy the lowest cell, giv-ing n1 = N and nν = 0 for ν > 1. With z1 = 1, the quantum way of countingbased on Eq. (48) gives just one single microstate, which means zero entropy inagreement with Nernst’s theorem, whereas the classical expression (57) yields anincorrect entropy even if one ignores the disturbing N ! Finally, the variationalcalculation based on the classical functional (58) proceeds via

(59)∑ν

[ln

(zν

nν

)− α − βEν

]δnν = 0,

furnishing the Boltzmann-like distribution

(60)nν = zνe−α−βEν


for the maximizing set {nν}. In short, the quantum ideal gas of nonzero mass par-ticles with its distribution (55) deviates from the classical ideal gas in the samemanner as does Planck’s law of radiation from Wien’s law. This observation con-vinced Einstein, even in the lack of any clear experimental evidence, that Bose’sway of counting microstates had to be taken seriously, since, as he remarks in theintroduction to his second paper [6], “if it is justified to consider radiation as aquantum gas, the analogy between the quantum gas and the particle gas has to bea complete one”. This belief also enabled him to accept the sacrifice of the statis-tical independence of quantum particles implied by the formula (48), which, bythe end of 1924, he had clearly realized:

The formula therefore indirectly expresses a certain hypothesis about a mutual influ-ence of the molecules on each other which is of an entirely mysterious kind. . .

But what might be the physics behind that mysterious influence which noninter-acting particles appear to exert on each other? In a further section of his paper [6],Einstein’s reasoning takes an amazing direction: He considers the density fluctua-tions of the ideal quantum gas, and from this deduces the necessity to invoke wavemechanics! Whereas he had previously employed what is nowadays known as themicrocanonical ensemble, formally embodied through the constraints that the to-tal number of particles and the total energy be fixed, he now resorts to a grandcanonical framework and considers a gas within some finite volume V whichcommunicates with a gas of the same species contained in an infinitely large vol-ume. He then stipulates that both volumes be separated from each other by somekind of membrane which can be penetrated only by particles with an energy in acertain infinitesimal range �Eν , and quantifies the ensuing fluctuation �nν of thenumber of particles in V , not admitting energy exchange between particles in dif-ferent energy intervals. Writing nν = nν +�nν , the entropy of the gas within Vis expanded in the form

(61)Sgas(�nν) = Sgas + ∂Sgas

∂�nν�nν + 1

2

∂2Sgas

∂(�nν)2(�nν)

2,

whereas the entropy of the reservoir changes with the transferred particles accord-ing to

(62)S0(�nν) = S0 − ∂S0

∂�nν�nν.

In view of the assumed infinite size of the reservoir, the quadratic term is negligi-ble here. Since the equilibrium state is characterized by the requirement that thetotal entropy S = Sgas + S0 be maximum, one has

(63)∂S

∂�nν= 0,


so that

(64)S(�nν) = S + 1

2

∂2Sgas

∂(�nν)2(�nν)

2.

Hence, the probability distribution for finding a certain fluctuation �nν isGaussian,

P(�nν) = const · eS(�nν)/kB

(65)= const · exp

(1

2kB

∂2Sgas

∂(�nν)2(�nν)

2),

from which one reads off the mean square of the fluctuations,

(66)⟨(�nν)

2⟩ = kB

− ∂2Sgas

∂(�nν)2

.

Since, according to the previous Eq. (54), one has

(67)1

kB

∂Sgas

∂�nν= ln

(1 + zν

nν

),

one deduces

(68)1

kB

∂2Sgas

∂(�nν)2= −zν

n2ν + zνnν

,

resulting in

(69)⟨(�nν)

2⟩ = nν + n2ν

zν

or

(70)⟨(�nν/nν)

2⟩ = 1

nν+ 1

zν.

With zν = 1, this gives the familiar grand canonical expression for the fluctuationof the occupation number of a single quantum state. With a stroke of genius, Ein-stein now interprets this formula: Whereas the first term on the r.h.s. of Eq. (70)would also be present if the particles were statistically independent, the secondterm reminds him of interference fluctuations of a radiation field [6]:

One can interpret it even in the case of a gas in a corresponding manner, by associatingwith the gas a radiation process in a suitable manner, and calculating its interferencefluctuations. I will explain this in more detail, since I believe that this is more than aformal analogy.


He then refers to de Broglie’s idea of associating a wavelike process with singlematerial particles and argues that, if one associates a scalar wave field with agas of quantum particles, the term 1/zν in Eq. (70) describes the correspondingmean square fluctuation of the wave field. What an imagination—on the basis ofthe fluctuation formula (70) Einstein anticipates many-body matter waves, longbefore wave mechanics was officially enthroned! Indeed, it was this paper of hiswhich led to a decisive turn of events: From these speculations on the relevanceof matter waves Schrödinger learned about de Broglie’s thesis, acquired a copy ofit, and then formulated his wave mechanics.

Having recapitulated this history, let us once again turn to the title of thissubsection: Was Bose–Einstein statistics arrived at by serendipity? Delbrück’sopinion that it arose out of an elementary mistake in statistics that Bose madealmost certainly is too harsh. On the other hand, the important relation (49), seealso Appendix A, does not seem to have figured in Bose’s thinking. When writ-ing down the crucial expression (12), Bose definitely must have been aware thathe was counting the number of microstates by determining the number of differ-ent distributions of quanta over the available quantum cells, regardless of “whichquantum occupies which cell”. He may well have been fully aware that his wayof counting implied the indistinguishability of quanta occupying the same energyrange, but he did not reflect on this curious issue. On the other hand, he didn’t haveto, since his way of counting directly led to one of the most important formulas inphysics, and therefore simply had to be correct.

Many years later, Bose recalled [41]:

I had no idea that what I had done was really novel. . . I was not a statistician to theextent of really knowing that I was doing something which was really different fromwhat Boltzmann would have done, from Boltzmann statistics. Instead of thinking ofthe light-quantum just as a particle, I talked about these states.

By counting the number of ways to fill a number of photonic states (cells)Bose obtained Eq. (13) which is exactly the same form as Boltzmann’s Eq. (57)for zν = 1, but with new meanings attached to the new symbols: ns replacedby psr and N replaced by As . Bose’s formula leads to an entirely different newstatistics—the quantum statistics for indistinguishable particles, in contrast toBoltzmann’s distinguishable particles. It took a while for this to sink in even withEinstein, but that is the nature of research.

Contrary to Bose, Einstein had no experimental motivation when adaptingBose’s work to particles with nonzero rest mass. He seems to have been guidedby a deep-rooted belief in the essential simplicity of physics, so that he was quiteready to accept a complete analogy between the gas of light quanta and the idealgas of quantum particles, although he may not yet have seen the revolutionary im-plications of this concept when he submitted his first paper [5] on this matter. Butthe arrival at a deep truth on the basis of a well-reflected conviction can hardly


be called serendipity. His second paper [6] is, by all means, a singular intellec-tual achievement, combining daring intuition with almost prophetical insight. Andwho would blame Einstein for trying to apply, in another section of that secondpaper, his quantum theory of the ideal Bose gas to the electron gas in metals?

In view of the outstanding importance which Einstein’s fluctuation formula (70)has had for the becoming of wave mechanics, it appears remarkable that a puz-zling question has long remained unanswered: What happens if one faces, unlikeEinstein in his derivation of this relation (70), a closed system of Bose particleswhich does not communicate with some sort of particle reservoir? When the tem-perature T approaches zero, all N particles are forced into the system’s groundstate, so that the mean square 〈(�n0)

2〉 of the fluctuation of the ground-state oc-cupation number has to vanish for T → 0—but with z0 = 1, the grand canonicalEq. (69) gives 〈(�n0)

2〉 → N(N+1), clearly indicating that with respect to thesefluctuations the different statistical ensembles are no longer equivalent. What,then, would be the correct expression for the fluctuation of the ground-state oc-cupation number within the canonical ensemble, which excludes any exchangeof particles with the environment, but still allows for the exchange of energy?To what extent does the microcanonical ensemble, which applies when even theenergy is kept constant, differ from the canonical one? Various aspects of thisriddle have appeared in the literature over the years [13,15,42], mainly inspiredby academic curiosity, before it resurfaced in 1996 [43–46], this time triggered bythe experimental realization of mesoscopic Bose–Einstein condensates in isolatedmicrotraps. Since then, much insight into this surprisingly rich problem has beengained, and some of the answers to the above questions have been given by now.In the following sections of this article, these new developments will be reviewedin detail.

2.4. COMPARISON BETWEEN BOSE’S AND EINSTEIN’S COUNTING

OF THE NUMBER OF MICROSTATES W

In Bose’s original counting (12), he considered the numbers psr of cells occu-pied with r photons, so that the total number of cells is given by As = ∑r p

sr .

While Einstein still had adopted this way of counting in his first paper [5] onthe ideal Bose gas, as witnessed by his Eq. (29), he used, without comment,more economical means in his second paper [6], relying on the binomial coef-ficient (48). Figure 2 shows an example in which Ns = 2 particles are distributedover As = 3 cells, and visualizes that Bose’s formula for counting the num-ber of possible arrangements (or the number of microstates which give the samemacrostate) gives the same result as Einstein’s formula only after one has summedover all possible configurations, i.e., over those sets ps0, p

s1, . . . which simultane-


FIG. 2. A simple example showing Bose’s and Einstein’s (textbook) methods of counting thenumber of possible ways to put n = 2 particles in a level having A = g = 3 states.

ously obey the two conditions As = ∑r psr and Ns = ∑r rp

sr . In this example,

there are only two such sets; a general proof is given in Appendix A. Thus, it isactually possible to state the number of microstates without specifying the indi-vidual arrangements, by summing over all of them:(

As∑ps0=0

. . .

As∑psNs=0

As !ps0!ps1! . . . psNs

, As =Ns∑r=0

psr and Ns =Ns∑r=0

rpsr

)

(71)= (Ns + As − 1)!Ns !(As − 1)! .

3. Grand Canonical versus Canonical Statistics of BECFluctuations

The problem of BEC fluctuations in a Bose gas is well known [13,15]. However,as with many other problems which are well-posed physically and mathemati-cally, it is highly nontrivial and deep, especially for the interacting Bose gas.


3.1. RELATIONS BETWEEN STATISTICS OF BEC FLUCTUATIONS

IN THE GRAND CANONICAL, CANONICAL, AND MICROCANONICAL

ENSEMBLES

To set the stage, we briefly review here some basic notions and facts from the sta-tistical physics of BEC involving relations between statistics of BEC fluctuationsin the grand canonical, canonical, and microcanonical ensembles.

Specific experimental conditions determine which statistics should be appliedto describe a particular system. In view of the present experimental status thecanonical and microcanonical descriptions of the BEC are of primary importance.Recent BEC experiments on harmonically trapped atoms of dilute gases deal witha finite and well-defined number of particles. This number, even if it is not knownexactly, certainly does not fluctuate once the cooling process is over. Magneticor optical confinement suggests that the system is also thermally isolated and,hence, the theory of the trapped condensate based on a microcanonical ensem-ble is needed. In this microcanonical ensemble the total particle number and thetotal energy are both exactly conserved, i.e., the corresponding operators are con-strained to be the c-number constants, N = const and H = const. On the otherhand, in experiments with two (or many) component BECs, Bose–Fermi mixtures,and additional gas components, e.g., for sympathetic cooling, there is an energyexchange between the components. As a result, each of the components can bedescribed by the canonical ensemble that applies to systems with conserved par-ticle number while exchanging energy with a heat bath of a given temperature.Such a description is also appropriate for dilute 4He in a porous medium [22]. Inthe canonical ensemble only the total number of particles is constrained to be anexact, nonfluctuating constant,

(72)N = n0 +∑k=0

nk,

but the energy H has nonzero fluctuations, 〈(H − H )2〉 = 0, and only its averagevalue is a constant, 〈H 〉 = H = E = const, determined by a fixed temperature Tof the system.

However, the microcanonical and canonical descriptions of a many-body sys-tem are difficult because of the operator constraints imposed on the total energyand particle number. As a result, the standard textbook formulation of the BECproblem assumes either a grand canonical ensemble (describing a system whichis allowed to exchange both energy and particles with a reservoir at a given tem-perature T and chemical potential μ, which fix only the average total number ofparticles, 〈N〉 = const, and the average total energy 〈H 〉 = const) or some re-stricted ensemble that selects states so as to ensure a condensate wave functionwith an almost fixed phase and amplitude [30,32,47,48]. These standard formu-lations focus on and provide effective tools for the study of the thermodynamic


and hydrodynamic properties of the many-body Bose system at the expense ofan artificial modification of the condensate statistics and dynamics of BEC for-mation. While the textbook grand canonical prediction of the condensate meanoccupation agrees, in some sense, with the Bose–Einstein condensation of trappedatomic gases, this is not even approximately true as concerns the grand canonicalcounting statistics

(73)ρGCν (nν) = 1

1 + nν

(nν

1 + nν

)nν,

which gives the probability to find nν particles in a given single-particle state ν,where the mean occupation is nν . Below the Bose–Einstein condensation temper-ature, where the ground state mean-occupation number is macroscopic, n0 ∼ N ,the distribution ρGC

0 (n0) becomes extremely broad with the squared variance〈(n0 − n0)

2〉 ≈ n20 ∝ N2 even at T → 0 [13,49]. This prediction is surely at odds

with the isolated Bose gas, where for sufficiently low temperature all particles areexpected to occupy the ground state with no fluctuations left. It was argued byZiff et al. [13] that this unphysical behavior of the variance is just a mathemati-cal artefact of the standard grand canonical ensemble, which becomes unphysicalbelow the condensation point. Thus, the grand canonical ensemble is irrelevant toexperiment if not revised properly. Another extremity, namely, a complete fixationof the amplitude and phase of the condensate wave function, is unable to addressthe condensate formation and the fluctuation problems at all.

It was first realized by Fierz [42] that the canonical ensemble with an exactlyfixed total number of particles removes the pathologies of the grand canonicalensemble. He exploited the fact that the description of BEC in a homogeneousideal Bose gas is exactly equivalent to the spherical model of statistical physics,and that the condensate serves as a particle reservoir for the noncondensed phase.Recently BEC fluctuations were studied by a number of authors in different statis-tical ensembles, both in the ideal and interacting Bose gases. The microcanonicaltreatment of the ground-state fluctuations in a one-dimensional harmonic trap isclosely related to the combinatorics of partitioning an integer, opening up an in-teresting link to number theory [43,50,51].

It is worthwhile to compare the counting statistics (73) with the predictions ofother statistical ensembles [46]. For high temperatures, T > Tc, all three ensem-bles predict the same counting statistics. However, this is not the case for lowtemperatures, T < Tc. Here the broad distribution of the grand canonical statis-tics differs most dramatically from ρ0(n0) in the canonical and microcanonicalstatistics, which show a narrow single-peaked distribution around the conden-sate mean occupation number. In particular, the master equation approach [52,53] (Section 4) yields a finite negative binomial distribution for the probabilitydistribution of the ground-state occupation in the ideal Bose gas in the canonicalensemble (see Fig. 12 and Eqs. (154), (162), and (163)). The width of the peak


decreases with decreasing temperature. In fact, it is this sharply-peaked statisticaldistribution which one would naively expect for a Bose condensate.

Each statistical ensemble is described by a different partition function. Themicrocanonical partition function Ω(E,N) is equal to the number of N -particlemicrostates corresponding to a given total energy E. Interestingly, at low tem-peratures canonical and microcanonical fluctuations have been found to agree inthe large-N (thermodynamic) limit for one-dimensional harmonic trapping poten-tials, but to differ in the case of three-dimensional isotropic harmonic traps [54].More precisely, for the d-dimensional power-law traps characterized by an expo-nent σ , as considered later in Section 5.3, microcanonical and canonical fluctua-tions agree in the large-N limit when d/σ < 2, but the microcanonical fluctua-tions remain smaller than the canonical ones when d/σ > 2 [55]. Thus, in d = 3dimensions the homogeneous Bose gas falls into the first category, but the har-monically trapped one into the second. The difference between the fluctuations inthese two ensembles is expressed by a formula which is similar in spirit to the oneexpressing the familiar difference between the heat capacities at constant pressureand at constant volume [54,55].

The direct numerical computation of the microcanonical partition function be-comes very time consuming or not possible for N > 105. For N � 1, and forlarge numbers of occupied excited energy levels, one can invoke, e.g., the ap-proximate technique based on the saddle-point method, which is widely used instatistical physics [31]. When employing this method, one starts from the knowngrand canonical partition function and utilizes the saddle-point approximationfor extracting its required canonical and microcanonical counterparts, which thenyield all desired quantities by taking suitable derivatives. Recently, still anotherstatistical ensemble, the so called Maxwell’s demon ensemble, has been intro-duced [54]. Here, the system is divided into the condensate and the particlesoccupying excited states, so that only particle transfer (without energy exchange)between these two subsystems is allowed, an idea that had also been exploitedby Fierz [42] and by Politzer [44]. This ensemble has been used to obtain an ap-proximate analytical expression for the ground-state BEC fluctuations both in thecanonical and in the microcanonical ensemble. The Maxwell’s demon approxi-mation can be understood on the basis of the canonical-ensemble quasiparticleapproach [20,21], which is discussed in Section 5, and which is readily generaliz-able to the case of the interacting Bose gas (see Section 6). It also directly provesthat the higher statistical moments for a homogeneous Bose gas depend on theparticular boundary conditions imposed, even in the thermodynamic limit [20,21,56].

The canonical partition function ZN(T ) is defined as

(74)ZN(T ) =∞∑E

e−E/kBT Ω(E,N),


where the sum runs over all allowed energies, and kB is the Boltzmann con-stant. Eq. (74) can be used to calculate also the microcanonical partition functionΩ(E,N) by means of the inversion of this definition. Likewise, the canonicalpartition function ZN(T ) can be obtained by the inversion of the definition of thegrand canonical partition function Ξ(μ, T ):

(75)Ξ(μ, T ) =∞∑N=0

eμN/kBT ZN(T ).

Inserting Eq. (74) into Eq. (75), we obtain the following relation betweenΞ(μ, T )

and Ω(E,N):

(76)Ξ(μ, T ) =∞∑N=0

eμN/kBT

∞∑E

e−E/kBT Ω(E,N).

As an example, let us consider an isotropic 3-dimensional harmonic trap witheigenfrequency ω. In this case, a relatively compact expression for the grandcanonical partition function,

(77)Ξ(μ, T ) =∞∏E

(1

1 − e(μ−E)/kBT

)(E/hω+1)(E/hω+2)/2

,

allows us to find Ω(E,N) from Eq. (76) by an application of the saddle-pointapproximation to the contour integral

Ω(E,N) = 1

(2πi)2

∮γz

dz

∮γx

dxΞ(z, x)

zN+1xE/hω+1,

(78)x = e−hω/kBT , z = eμ/kBT ,

where the contours of integration γz and γx include z = 0 and x = 0, respec-tively. It is convenient to rewrite the function under the integral in Eq. (78) asexp[ϕ(z, x)], where

ϕ(z, x) = lnΞ(z, x)− (N + 1) ln z− (E/hω + 1) ln x.

Taking the contours through the extrema (saddle points) z0 and x0 of ϕ(z, x),and employing the usual Gaussian approximation, we get for N → ∞ andE/hω → ∞ the following asymptotic formula:

Ω(E,N) = [2πD(z0, x0)]−1/2

Ξ(z0, x0)/zN+10 x

E/hω+10 ,

where D(z0, x0) is the determinant of the second derivatives of the func-tion ϕ(z, x), evaluated at the saddle points [45]. For N → ∞ and E/hω → ∞the function exp[ϕ(z, x)] is sharply peaked at z0 and x0, which ensures goodaccuracy of the Gaussian approximation. However, the standard result becomes


incorrect for E < Nε1, where ε1 = hω1 is the energy of the first excited state(see [45,57]); in this case, a more refined version of the saddle-point method isrequired [58–60]. We discuss this improved version of the saddle-point method inAppendix F.

An accurate knowledge of the canonical partition function is helpful for thecalculation of the microcanonical condensate fluctuations by the saddle pointmethod, as has been demonstrated [57] by a numerical comparison with exactmicrocanonical simulations. In principle, the knowledge of the canonical parti-tion function allows us to calculate thermodynamic and statistical equilibriumproperties of the system in the standard way (see, e.g., [13,61]). An importantfact is that the usual thermodynamic quantities (average energy, work, pressure,heat capacities, etc.) and the average number of condensed atoms do not have anyinfrared-singular contributions and do not depend on a choice of the statisticalensemble in the thermodynamic (bulk) limit. However, the variance and highermoments of the BEC fluctuations do have the infrared anomalies and do dependon the statistical ensemble, so that their calculation is much more involved andsubtle.

3.2. EXACT RECURSION RELATION FOR THE STATISTICS OF THE NUMBER

OF CONDENSED ATOMS IN AN IDEAL BOSE GAS

It is worth noting that there is one useful reference result in the theory of BECfluctuations, namely, an exact recursion relation for the statistics of the numberof condensed atoms in an ideal Bose gas. Although it does not give any simpleanalytical answer or physical insight into the problem, it can be used for “exact”numerical simulations for traps containing a finite number of atoms. This is veryuseful as a reference to be compared against different approximate analytical for-mulas. This exact recursion relation for the ideal Bose gas had been known fora long time [61], and rederived independently by several authors [46,62–64]. Inthe canonical ensemble, the probability to find n0 particles occupying the single-particle ground state is given by

(79)ρ0(n0) = ZN−n0(T )− ZN−n0−1(T )

ZN(T ); Z−1 ≡ 0.

The recurrence relation for the ideal Bose gas then states [61,62]

ZN(T ) = 1

N

N∑k=1

Z1(T /k)ZN−k(T ),

(80)Z1(T ) =∞∑ν=0

e−εν/kBT , Z0(T ) = 1,


which enables one to numerically compute the entire counting statistics (79). Hereν stands for a set of quantum numbers which label a given single-particle state,εν is the associated energy, and the ground-state energy is taken as ε0 = 0 byconvention. For an isotropic, three-dimensional harmonic trap one has

Z1(T ) =∞∑n=0

1

2(n+ 2)(n+ 1)e−nhω/kBT = 1

(1 − e−hω/kBT )3,

where 12 (n+ 2)(n+ 1) is the degeneracy of the level εn = nhω.

In the microcanonical ensemble the ground-state occupation probability isgiven by a similar formula

ρMC0 (n0) = Ω(E,N − n0)−Ω(E,N − n0 − 1)

Ω(E,N);

(81)Ω(E,−1) ≡ 0,

where the microcanonical partition function obeys the recurrence relation

Ω(E,N) = 1

N

N∑k=1

∞∑ν=0

Ω(E − kεν,N − k),

(82)Ω(0, N) = 1, Ω(E > 0, 0) = 0.

For finite E the sum over ν is finite because of Ω(E < 0, N) = 0.

3.3. GRAND CANONICAL APPROACH

Here we discuss the grand canonical ensemble, and show that it loses its va-lidity for the ideal Bose gas in the condensed region. Nevertheless, reasonableapproximate results can be obtained if the canonical-ensemble constraint is prop-erly incorporated in the grand canonical approach, especially if we are not tooclose to Tc. In principle, the statistical properties of BECs can be probed withlight [65]. In particular, the variance of the number of scattered photons may dis-tinguish between the Poisson and microcanonical statistics.

3.3.1. Mean Number of Condensed Particles in an Ideal Bose Gas

The Bose–Einstein distribution can be easily derived from the density ma-trix approach. Consider a collection of particles with the Hamiltonian H =∑

k a†k ak(εk − μ), where μ is the chemical potential. The equilibrium state of

the system is described by

(83)ρ = 1

Zexp(−βH ),


where Z = Tr{exp(−βH )} = ∏k(1 − e−β(εk−μ))−1. Then the mean number ofparticles with energy εk is

〈nk〉 = Tr{ρa

†k ak} = (1 − e−β(εk−μ)

)∑nk

⟨nk∣∣a†k ake

−βa†k ak(εk−μ)∣∣nk ⟩

(84)= (1 − e−β(εk−μ))d(1 − e−β(εk−μ))−1

d(−β(εk − μ))= 1

exp[β(εk − μ)] − 1.

In the grand canonical ensemble the average condensate particle number n0 isdetermined from the equation for the total number of particles in the trap,

(85)N =∞∑k=0

nk =∞∑

k=0

1

expβ(εk − μ)− 1,

where εk is the energy spectrum of the trap. In particular, for the three dimen-sional (3D) isotropic harmonic trap we have εk = hΩ(kx + ky + kz). Forsimplicity, we set ε0 = 0.

For 3D and 1D traps with noninteracting atoms, Eq. (85) was studied byKetterle and van Druten [11], and by Grossmann and Holthaus [43,66]. Theycalculated the fraction of ground-state atoms versus temperature for various Nand found that BEC also exists in 1D traps, where the condensation phenomenonlooks very similar to the 3D case. Later Herzog and Olshanii [67] used the knownanalytical formula for the canonical partition function of bosons trapped in a 1Dharmonic potential [68,69] and showed that the discrepancy between the grandcanonical and the canonical predictions for the 1D condensate fraction becomesless than a few per cent for N > 104. The deviation decreases according to a1/ lnN scaling law for fixed T/Tc. In 3D the discrepancy is even less than inthe 1D system [64]. The ground state occupation number and other thermody-namic properties were studied by Chase, Mekjian and Zamick [64] in the grandcanonical, canonical and microcanonical ensembles by applying combinatorialtechniques developed earlier in statistical nuclear fragmentation models. In suchmodels there are also constraints, namely the conservation of proton and neutronnumber. The specific heat and the occupation of the ground state were found sub-stantially in agreement in all three ensembles. This confirms the essential validityof the use of the different ensembles even for small groups of particles as longas the usual thermodynamic quantities, which do not have any infrared singularcontributions, are calculated.

Let us demonstrate how to calculate the mean number of condensed particlesfor a particular example of a 3D isotropic harmonic trap. Following Eq. (85), wecan relate the chemical potential μ to the mean number of condensed particles n0as 1 + 1/n0 = exp(−βμ). Thus, we have

(86)N =∞∑

k=0

〈nk〉 =∞∑

k=0

1

(1 + 1/n0) expβεk − 1.


The standard approach is to consider the case N � 1 and separate off the groundstate so that Eq. (86) approximately yields

(87)N − n0 =∑k>0

1

exp(βεk)− 1.

For an isotropic harmonic trap with frequency Ω ,∑k>0

1

exp(βεk)− 1= 1

2

∞∑n=1

(n+ 2)(n+ 1)

exp(βnhΩ)− 1

(88)≈ 1

2

∞∫1

(x + 2)(x + 1)

exp(xβhΩ)− 1dx.

In the limit kBT � hΩ we obtain

(89)∑k>0

1

exp(βεk)− 1≈ 1

2

∞∫0

x2

exp(xβhΩ)− 1dx =(kBT

hΩ

)3

ζ(3).

Furthermore, when T = Tc we take n0 = 0. Then Eqs. (87) and (89) yield

(90)kBTc = hΩ

(N

ζ(3)

)1/3

and the temperature dependence of the mean condensate occupation with a cuspat T = Tc in the thermodynamic limit

(91)n0(T ) = N

(1 −(T

Tc

)3 ).

Figure 3 compares the numerical solution of Eq. (86) (solid line) for N = 200with the numerical calculation of n0(T ) from the exact recursion relations inEqs. (79) and (80) in the canonical ensemble (large dots). Small dots show theplot of the solution (91), which is valid only for a large number of particles, N .Obviously, more accurate solution of the equation for the mean number of con-densed particles (86) in a trap with a finite number of particles does not show thecusp. In Appendix B we derive an analytical solution of Eq. (86) for n0(T ) validfor n0(T ) � 1. One can see that for the average particle number both ensemblesyield very close answers. However, this is not the case for the BEC fluctuations.

3.3.2. Condensate Fluctuations in an Ideal Bose Gas

Condensate fluctuations are characterized by the central moments 〈(n0 − 〈n0〉)s〉=∑r

(sr

)〈nr0〉〈n0〉s−r . The first few of them are

(92)⟨(n0 − n0)

2⟩ = ⟨n20

⟩− 〈n0〉2,


FIG. 3. Mean number of condensed particles as a function of temperature for N = 200. The solidline is the plot of the numerical solution of Eq. (86). The exact numerical result for the canonicalensemble (Eqs. (79) and (80)) is plotted as large dots. The dashed line is obtained using Maxwell’sdemon ensemble approach, which yields n0 = N −∑∞

k>0 1/[exp(βεk)− 1].

(93)⟨(n0 − n0)

3⟩ = ⟨n30

⟩− 3⟨n2

0

⟩〈n0〉 + 2〈n0〉3,

(94)⟨(n0 − n0)

4⟩ = ⟨n40

⟩− 4⟨n3

0

⟩〈n0〉 + 6⟨n2

0

⟩〈n0〉2 − 3〈n0〉4.

At arbitrary temperatures, BEC fluctuations in the canonical ensemble can bedescribed via a stochastic variable n0 = N −∑k =0 nk that depends on and iscomplementary to the sum of the independently fluctuating numbers nk , k = 0,of the excited atoms. In essence, the canonical-ensemble constraint in Eq. (72)eliminates one degree of freedom (namely, the ground-state one) from the set ofall independent degrees of freedom of the original grand canonical ensemble, sothat only transitions between the ground and excited states remain independentlyfluctuating quantities. They describe the canonical-ensemble quasiparticles, or ex-citations, via the creation and annihilation operators β+ and β (see Sections 5and 6 below).

At temperatures higher than Tc the condensate fraction is small and one canapproximately treat the condensate as being in contact with a reservoir of non-condensate particles. The condensate exchanges particles with the big reservoir.Hence, the description of particle fluctuations in the grand canonical picture, as-suming that the number of atoms in the ground state fluctuates independently fromthe numbers of excited atoms, is accurate in this temperature range.

At temperatures close to or below Tc the situation becomes completely differ-ent. One can say that at low temperatures, T � Tc, the picture is opposite to thepicture of the Bose gas above the BEC phase transition at T > Tc. The canonical-ensemble quasiparticle approach, suggested in Refs. [20,21] and valid both for


the ideal and interacting Bose gases, states that at low temperatures the noncon-densate particles can be treated as being in contact with the big reservoir of the n0condensate particles. This idea had previously been spelled out by Fierz [42] andbeen used by Politzer [44], and was then employed for the construction of the“Maxwell’s demon ensemble” [54], named so since a permanent selection of theexcited (moving) atoms from the ground state (static) atoms is a problem resem-bling the famous Maxwell’s demon problem in statistical physics.

Note that although this novel statistical concept can be studied approximatelywith the help of the Bose–Einstein expression (84) for the mean number of excited(only excited, k = 0!) states with some new chemical potential μ, it describesthe canonical-ensemble fluctuations and is essentially different from the standardgrand canonical description of fluctuations in the Bose gas. Moreover, it is ap-proximately valid only if we are not too close to the critical temperature Tc, sinceotherwise the “particle reservoir” is emptied. Also, the fluctuations obtained fromthe outlined “grand” canonical approach (see Eqs. (95)–(102) and (C10)–(C12)),an approach complementary to the grand canonical one, provide an accurate de-scription in this temperature range in the thermodynamic limit, but do not take intoaccount all mesoscopic effects, especially near Tc (for more details, see Sections 4and 5). Thus, although the mean number of condensed atoms n0 can be foundfrom the grand canonical expression used in Eq. (85), we still need to invoke theconservation of the total particle number N = n0 + n in order to find the highermoments of condensate fluctuations, 〈nr0〉. Namely, we can use the following re-lation between the central moments of the mth order of the number of condensedatoms, and that of the noncondensed ones: 〈(n0 − n0)

m〉 = (−1)m〈(n − n)m〉.As a result, at low enough temperatures one can approximately write the centralmoments in the well-known canonical form via the cumulants in the ideal Bosegas (see Eqs. (220), (221), (219), (223) and Section 5 below for more details):

(95)⟨(n0 − n0)

2⟩ = κ2 = κ2 + κ1 ≈∑k>0

(n2k + nk

),

(96)⟨(n0 − n0)

3⟩ = −κ3 = −(κ3 − 3κ2 − κ1) ≈ −∑k>0

(2n3

k + 3n2k + nk

),

⟨(n0 − n0)

4⟩ = κ4 + 3κ22 = κ4 + 6κ3 + 7κ2 + κ1 + 3(κ2 + κ1)

2

(97)≈∑k>0

(6n4

k + 12n3k + 7n2

k + nk)+ 3

[∑k>0

(n2k + nk

)]2.

These same equations can also be derived by means of the straightforward calcu-lation explained in Appendix C.

Combining the hallmarks of the grand canonical approach, namely, the valueof the chemical potential μ = −β−1 ln(1 + 1/n0) and the mean noncondensateoccupation nk = {exp[β(Ek − μ)] − 1}−1, with the Eq. (95) describing the fluc-


tuations in the canonical ensemble, we find the BEC variance

�n20 ≡ ⟨(n0 − n0)

2⟩(98)=

∑k>0

{1

[exp(βEk)(1 + 1n0)− 1]2 + 1

exp(βEk)(1 + 1n0)− 1

}.

In the case kBT � hΩ , this Eq. (98) can be evaluated analytically, as is shown inAppendix D. The variance up to second order in 1/n0 from Eq. (D4) is

�n0 ≈√

N

ζ(3)

(T

Tc

)3 [π2

6− 1

n0(1 + ln n0)+ 1

n20

(1

2ln n0 − 1

4

)]1/2

.

(99)

The leading term in this expression yields Politzer’s result [44],

(100)�n0 ≈√ζ(2)N

ζ(3)

(T

Tc

)3

≈ 1.17

√N

(T

Tc

)3

,

plotted as a dashed line in Fig. 4, where ζ(2) = π2

6 ≈ 1.644 9 and ζ(3) ≈ 1.202 1(compare this with D. ter Haar’s [15] result �n0 ≈ N( T

Tc)3, which is missing

the square root). The same formula was obtained later by Navez et al. using theMaxwell’s demon ensemble [54]. For the microcanonical ensemble the Maxwell’sdemon approach yields [54]

(101)�n0 ≈√N

(ζ(2)

ζ(3)− 3ζ(3)

4ζ(4)

)(T

Tc

)3

≈ 0.73

√N

(T

Tc

)3

;

higher-order terms have been derived in Ref. [60]. The microcanonical fluctua-tions are smaller than the canonical ones due to the additional energy constraint.For 2D and 1D harmonic traps the Maxwell’s demon approach leads to [46,55]

(102)�n0 ∼ √NT

Tc

√lnTc

Tand �n0 ∼ T

TcN,

respectively.Figure 4 shows the BEC variance �n0(T ) as a function of temperature for a 3D

trap with the total particle number N = 200. The “grand” canonical curve refersto Eq. (98) and shows good agreement for T < Tc with the numerical result for�n0(T ) (large dots), obtained within the exact recursion relations (79) and (80)for the canonical ensemble. The plot of Politzer’s asymptotic formula (100)(dashed line) does not give good agreement, since the particle number consid-ered here is fairly low. We also plot

√n0(n0 + 1), which is the expression for the

condensate number fluctuations �n0 in the grand canonical ensemble; it workswell above Tc. Figure 5 shows the third central moment 〈(n0 − n0)

3〉 as a function


FIG. 4. Variance 〈(n0 − 〈n0〉)2〉1/2 of the condensate particle number for N = 200. The solidline is the “grand” canonical result obtained from Eq. (98) and the numerical solution of 〈n0〉 fromEq. (86). Exact numerical data for the canonical ensemble (Eqs. (79) and (80)) are shown as dots. Theasymptotic Politzer approximation [44], given by Eq. (100), is shown by the dashed line, while smalldots result from Eq. (D4). The dash-dotted line is obtained from the master equation approach (seeEq. (172) below).

of temperature for the total particle number N = 200, plotted using Eqs. (96)and (86). Dots are the exact numerical result obtained within the canonical en-semble. We also plot the standard grand canonical formula 2n3

0 +3n20 + n0, which

again works well only above Tc.At high temperatures the main point, of course, is the validity of the stan-

dard grand canonical approach where the average occupation of the ground statealone gives a correct description of the condensate fluctuations, since the excitedparticles constitute a valid “particle reservoir”. Condensate fluctuations obtainedfrom the “grand” canonical approach for the canonical-ensemble quasiparticlesand from the standard grand canonical ensemble provide an accurate descrip-tion of the canonical-ensemble fluctuations at temperatures not too close to thenarrow crossover region between low (Eq. (98)) and high (

√n0(n0 + 1)) tem-

perature regimes; i.e., in the region not too near Tc. In this crossover range bothapproximations fail, since the condensate and the noncondensate fractions havecomparable numbers of particles, and there is no any valid particle reservoir. Note,however, that a better description, that includes mesoscopic effects and works inthe whole temperature range, can be obtained using the condensate master equa-


FIG. 5. The third central moment 〈(n0 − n0)3〉 for N = 200, plotted using Eqs. (96) and (86).

Exact numerical data for the canonical ensemble (Eqs. (79) and (80)) are shown as dots. The dashedline is the result of master equation approach (see Eq. (174) below).

tion, as shown in Section 4; see, e.g., Fig. 13. Another (semi-analytic) technique,the saddle-point method, is discussed in Appendix F.

4. Dynamical Master Equation Approach and LaserPhase-Transition Analogy

One approach to the canonical statistics of ideal Bose gases, presented in [52] anddeveloped further in [53], consists in setting up a master equation for the con-densate and finding its equilibrium solution. This approach has the merit of beingtechnically lucid and physically illuminating. Furthermore, it reveals importantparallels to the quantum theory of the laser. In deriving that master equation, theapproximation of detailed balance in the excited states is used, in addition to theassumption that given an arbitrary number n0 of atoms in the condensate, theremaining N −n0 excited atoms are in an equilibrium state at the prescribed tem-perature T .

In Section 4.2 we summarize the master equation approach against the resultsprovided by independent techniques. In Section 4.1 we motivate our approach bysketching the quantum theory of the laser with special emphasis on the pointsrelevant to BEC.


4.1. QUANTUM THEORY OF THE LASER

In order to set the stage for the derivation of the BEC master equation, let us re-mind ourselves of the structure of the master equations for a few basic systemsthat have some connection with N particles undergoing Bose–Einstein condensa-tion while exchanging energy with a thermal reservoir.

4.1.1. Single Mode Thermal Field

The dissipative dynamics of a small system (s) coupled to a large reservoir (r)is described by the reduced density matrix equation up to second order in theinteraction Hamiltonian Vsr

(103)∂

∂tρs(t) = − 1

h2Trr

t∫0

[Vsr (t),

[Vsr (t

′), ρs(t)⊗ ρthr]]dt ′,

where ρthr is the density matrix of the thermal reservoir, and we take the Markov-ian approximation.

We consider the system as a single radiation mode cavity field (f ) of frequencyν coupled to a reservoir (r) of two-level thermal atoms, and show how the radia-tion field thermalizes. The multiatom Hamiltonian in the interaction picture is

(104)Vf r = h∑j

gj(σj a

†ei(ν−ωj )t + adj)

where σj = |bj 〉〈aj | is the atomic (spin) operator of the j th particle correspond-ing to the downward transition, a† is the creation operator for the single modefield and gj is the coupling constant. The reduced density matrix equation for thefield can be obtained from Eqs. (103) and (104). By using the density matrix forthe thermal ensemble of atoms

(105)ρthr =∏j

(|aj 〉〈aj |e−βEa,j + |bj 〉〈bj |e−βEb,j)/Zj ,

where Zj = e−βEa,j + e−βEb,j , we obtain

∂

∂tρf = −1

2

∑j

κj{Paj(aa†ρf − 2a†ρf a + ρf aa

†)(106)+ Pbj

(a†aρf − 2aρf a

† + ρf a†a)},

where Pxj = e−βExj /Zj with x = a, b and the dissipative constant is 1

2κj =Re{g2

j

∫ t0 e

i(ν−ωj )(t−t ′) dt ′}. Note that the same structure of the master equation isobtained for a phonon bath modelled as a collection of harmonic oscillators, asshown in Appendix E.


Taking the diagonal matrix elements ρn,n(t) = 〈n|ρf |n〉 of Eq. (106), we have

∂

∂tρn,n(t) = −κPa

{(n+ 1)ρn,n(t)− nρn−1,n−1(t)

}(107)− κPb

{nρn,n(t)− (n+ 1)ρn+1,n+1(t)

},

where κ is κj times a density of states factor and ρn,n.The steady state equation gives

(108)ρn,n ≡ pn = e−nβhωp0.

From∑∞

n=0 pn = 1 we obtain the thermal photon number distribution

(109)pn = e−nβhω(1 − e−βhω

)which is clearly an exponentially decaying photon number distribution.

4.1.2. Coherent State

Consider the interaction of a single mode field with a classical current J describedby

(110)Vcoh(t) =∫V

J(r, t) · A(r, t) d3r = h(j (t)a + j∗(t)a†),

where the complex time dependent coefficient is

j (t) = A0

h

∫V

J(r, t) · xei(k·r−νt) d3r

and A0 is the amplitude vector potential A(r, t) of the single mode field, as-sumed to be polarized along x axis. An example of such interaction is a klystron.Clearly, the unitary time evolution of Vcoh is in the form of a displacement op-erator exp(α∗(t)a − α(t)a†) associated with a coherent state when dissipation isneglected.

Thus, the density matrix equation for a klystron including coupling with a ther-mal bath is

∂ρf (t)

∂t= 1

ih

[Vcoh(t), ρf (t)

](111)− 1

h2Trr

t∫0

[Vf r (t),

[Vf r(t ′), ρf (t)⊗ ρthr

]]dt ′,

where Vf r is given by Eq. (E3). The second term of Eq. (111) describes the damp-ing of the single mode field given by Eq. (E4). By taking the matrix elements of


FIG. 6. Photon number distributions for (a) thermal photons plotted from Eq. (108) (dashed line),(b) coherent state (Poissonian) (thin solid line), and (c) He–Ne laser plotted using Eq. (121) (thicksolid line). Insert shows an atom making a radiation transition.

Eq. (111), after a bit of analysis, we have

∂ρn,n′(t)

∂t= −i(j (t)√n+ 1 ρn+1,n′ + j∗(t)

√n ρn−1,n′ − j (t)

√n′ ρn,n′−1

− j∗(t)√n′ + 1 ρn,n′+1

)− 1

2C[(n+ n′

)ρn,n′ − 2

√(n+ 1)

(n′ + 1)ρn+1,n′+1

](112)− 1

2D[(n+ 1 + n′ + 1

)ρn,n′ − 2

√nn′ ρn−1,n′−1

].

Clearly, the first line of Eq. (112) shows that the change in the photon numberis effected by the off-diagonal field density matrix or the coherence between twostates of different photon number. On the other hand, the damping mechanismonly causes a change in the photon number through the diagonal matrix elementor the population of the number state. This is depicted in Fig. 7.

It can be shown that the solution of Eq. (112) for D = 0 is the matrix elementof a coherent state |β〉 = e|β|2/2∑

nβn√n! |n〉, i.e.,

(113)ρn,n′(t) =⟨n∣∣{|β〉〈β|}∣∣n′⟩ = β(t)nβ∗(t)n′√

n!√n′! e−|β(t)|2 ,


FIG. 7. Diagonal (star) and off-diagonal (circle) density matrix elements that govern temporaldynamics in (a) klystron and (b) thermal field.

where β(t) = α(t) − 12C∫ t

0 α(t′) dt ′. This can be verified if we differentiate

Eq. (113),

dρn,n′

dt={dα(t)

dt− 1

2Cα(t)}√

n ρn−1,n′

+ √n′{dα∗(t)dt

− 1

2Cα∗(t)

}ρn,n′−1

−{dα(t)

dt− 1

2Cα(t)}√

n′ + 1 ρn,n′+1

(114)−{dα∗(t)dt

− 1

2Cα∗(t)

}√n+ 1 ρn+1,n′

and using√n ρn,n′ = ρn−1,n′ ,

√n′ ρn,n′ = ρn,n′−1,

(115)√n+ 1 ρn+1,n′+1 = ρn,n′+1,

√n′ + 1 ρn+1,n′+1 = ρn+1,n′ ,

where dα(t)dt

= −ij∗(t) is found by comparing Eq. (114) with Eq. (112).

4.1.3. Laser Master Equation

The photon number equation (107) for thermal field is linear in photon number, n,and it describes only the thermal damping and pumping due to the presence of athermal reservoir. Now, we introduce a laser pumping scheme to drive the single


FIG. 8. Typical setup of a laser showing an ensemble of atoms driving a single mode field. A com-petition between lasing and dissipation through cavity walls leads to a detailed balance.

mode field and show how the atom–field nonlinearity comes into the laser masterequation.

We consider a simple three level system where the cavity field couples level aand level b of lasing atoms in a molecular beam injected into a cavity in the excitedstate at rate r (see Fig. 8). The atoms undergo decay from level b to level c. Thepumping mechanism from level c up to level a can thus produce gain in the singlemode field. As shown in [19], we find(

∂ρn,n(t)

∂t

)gain

= −{A(n+ 1)− B(n+ 1)2}ρn,n

(116)+ {An− Bn2}ρn−1,n−1,

where A = 2rg2

γ 2 is the linear gain coefficient and B = 4g2

γ 2 A is the self-saturationcoefficient. Here g is the atom–field coupling constant and γ is the b → c decayrate. We take the damping of the field to be

(117)

(∂ρn,n(t)

∂t

)loss

= −Cnρn,n + C(n+ 1)ρn+1,n+1.

Thus, the overall master equation for the laser is

∂ρn,n(t)

∂t= −{A(n+ 1)− B(n+ 1)2

}ρn,n + {An− Bn2}ρn−1,n−1

(118)− Cnρn,n + C(n+ 1)ρn+1,n+1,


which is valid for small B/A � 1.We emphasize that the nonlinear process associated with B is a key physical

process in the laser (but not in a thermal field) because the laser field is so large.We proceed with detailed balance equation between level n− 1 and n,

(119)−{A(n+ 1)− B(n+ 1)2}pn + C(n+ 1)pn+1 = 0,

(120){An− Bn2}pn−1 − Cnpn = 0.

By iteration of pk = A−BkC pk−1, we have

(121)pn = p0

n∏k=1

A− BkC ,

where p0 = 1/(1 +∑∞n=1∏n

k=1A−Bk

C ) follows from∑∞

n=0 pn = 1. Eq. (121) isplotted in Fig. 6. There we clearly see that the photon statistics of, e.g., a He–Ne,laser is not Poissonian pn = 〈n〉ne−〈n〉/n!, as would be expected for a coherentstate.

4.2. LASER PHASE-TRANSITION ANALOGY

Bose–Einstein condensation of atoms in a trap has intriguing similarities withthe threshold behavior of a laser which also can be viewed as a kind of a phasetransition [70,71]. Spontaneous formation of a long range coherent-order para-meter, i.e., macroscopic wave function, in the course of BEC second-order phasetransition is similar to spontaneous generation of a macroscopic coherent fieldin the laser cavity in the course of lasing. In both systems stimulated processesare responsible for the appearance of the macroscopic-order parameter. The maindifference is that for the Bose gas in a trap there is also interaction between theatoms which is responsible for some processes, including stimulated effects inBEC. Whereas for the laser there are two subsystems, namely the laser field andthe active atomic medium. The crucial point for lasing is the interaction betweenthe field and the atomic medium which is relatively small and can be treated per-turbatively. Thus, the effects of different interactions in the laser system are easyto trace and relate to the observable characteristics of the system. This is not thecase in BEC and it is more difficult to separate different effects.

As is outlined in the previous subsection, in the quantum theory of laser, thedynamics of laser light is conveniently described by a master equation obtainedby treating the atomic (gain) media and cavity dissipation (loss) as reservoirswhich when “traced over” yield the coarse grained equation of motion for thereduced density matrix for laser radiation. In this way we arrive at the equation ofmotion for the probability of having n photons in the cavity given by Eq. (118).


From Eq. (121) we have the important result that partially coherent laser light hasa sharp photon distribution (with width several times Poissonian for a typical He–Ne laser) due to the presence of the saturation nonlinearity, B, in the laser masterequation. Thus, we see that the saturation nonlinearity in the radiation–matterinteraction is essential for laser coherence.

One naturally asks: is the corresponding nonlinearity in BEC due to atom–atom scattering? Or is there a nonlinearity present even in an ideal Bose gas? Themaster equation presented in this Section proves that the latter is the case.

More generally we pose the question: Is there a similar nonequilibrium ap-proach for BEC in a dilute atomic gas that helps us in understanding the under-lying physical mechanisms for the condensation, the critical behavior, and theassociated nonlinearities? The answer to this question is “yes” [52,53].

4.3. DERIVATION OF THE CONDENSATE MASTER EQUATION

We consider the usual model of a dilute gas of Bose atoms wherein interatomicscattering is neglected. This ideal Bose gas is confined inside a trap, so that thenumber of atoms,N , is fixed but the total energy,E, of the gas is not fixed. Instead,the Bose atoms exchange energy with a reservoir which has a fixed temperature T .As we shall see, this canonical-ensemble approach is a useful tool in studying thecurrent laser cooled dilute gas BEC experiments [24–27,88]. It is also directlyrelevant to the He-in-vycor BEC experiments [22].

This “ideal gas + reservoir” model allows us to demonstrate most clearly themaster equation approach to the analysis of dynamics and statistics of BEC, andin particular, the advantages and mathematical tools of the method. Its extensionfor the case of an interacting gas which includes usual many-body effects due tointeratomic scattering will be discussed elsewhere.

Thus, we are following the so-called canonical-ensemble approach. It de-scribes, of course, an intermediate situation as compared with the microcanonicalensemble and the grand canonical ensemble. In the microcanonical ensemble, thegas is completely isolated, E = const, N = const, so that there is no exchangeof energy or atoms with a reservoir. In the grand canonical ensemble, only theaverage energy per atom, i.e., the temperature T and the average number of atoms〈N〉 are fixed. In such a case there is an exchange of both energy and atoms withthe reservoir.

The “ideal gas + thermal reservoir” model provides the simplest descriptionof many qualitative and, in some cases, quantitative characteristics of the experi-mental BEC. In particular, it explains many features of the condensate dynamicsand fluctuations and allows us to obtain, for the first time, the atomic statistics ofthe BEC as discussed in the introduction and in the following.


FIG. 9. Simple harmonic oscillators as a thermal reservoir for the ideal Bose gas in a trap.

4.3.1. The “Ideal Gas + Thermal Reservoir” Model

For many problems a concrete realization of the reservoir system is not very im-portant if its energy spectrum is dense and flat enough. For example, one expects(and we find) that the equilibrium (steady state) properties of the BEC are largelyindependent of the details of the reservoir. For the sake of simplicity, we assumethat the reservoir is an ensemble of simple harmonic oscillators whose spectrumis dense and smooth, see Fig. 9. The interaction between the gas and the reservoiris described by the interaction picture Hamiltonian

(122)V =∑j

∑k>l

gj,klb†j aka

†l e

−i(ωj−νk+νl)t + h.c.,

where b†j is the creation operator for the reservoir j oscillator (“phonon”), and a†

k

and ak (k = 0) are the creation and annihilation operators for the Bose gas atomsin the kth level. Here hνk is the energy of the kth level of the trap, and gj,kl is thecoupling strength.

4.3.2. Bose Gas Master Equation

The motion of the total “gas + reservoir” system is governed by the equa-tion for the total density matrix in the interaction representation, ρtotal(t) =−i[V (t), ρtotal(t)]/h. Integrating the above equation for ρtotal, inserting it backinto the commutator in Eq. (123), and tracing over the reservoir, we obtain theexact equation of motion for the density matrix of the Bose-gas subsystem

(123)ρ(t) = − 1

h2

t∫0

dt ′ Trres[V (t),[V(t ′), ρtotal(t ′)]]

,

where Trres stands for the trace over the reservoir degrees of freedom.We assume that the reservoir is large and remains unchanged during the in-

teraction with the dynamical subsystem (Bose gas). As discussed in [53], the


density operator for the total system “gas + reservoir” can then be factored,i.e., ρtotal(t

′) ≈ ρ(t ′) ⊗ ρres, where ρres is the equilibrium density matrix of thereservoir. If the spectrum is smooth, we are justified in making the Markov ap-proximation, viz. ρ(t ′) → ρ(t). We then obtain the following equation for thereduced density operator of the Bose-gas subsystem,

ρ = −κ

2

∑k>l

(ηkl + 1)[a

†kala

†l akρ − 2a†

l akρa†kal + ρa

†kala

†l ak]

(124)− κ

2

∑k>l

ηkl[aka

†l ala

†kρ − 2ala

†kρaka

†l + ρaka

†l ala

†k

].

In deriving Eq. (124), we replaced the summation over reservoir modes by an inte-gration (with the density of reservoir modes D(ωkl)) and neglected the frequencydependence of the coefficient κ = 2πDg2/h2. Here

(125)ηkl = η(ωkl) = Trres b†(ωkl)b(ωkl) =

[exp(hωkl/T )− 1

]−1

is the average occupation number of the heat bath oscillators at frequency ωkl ≡νk − νl . Equation (124) is then the equation of motion for an N atom Bose gasdriven by a heat bath at temperature T .

4.3.3. Condensate Master Equation

What we are most interested in is the probability distribution

pn0 =∑{nk}n0

pn0,{nk}n0

for the number of condensed atoms n0, i.e., the number of atoms in the groundlevel of the trap. Let us introduce pn0,{nk}n0

= 〈n0, {nk}n0 |ρ|n0, {nk}n0〉 as adiagonal element of the density matrix in the canonical ensemble where n0 +∑

k>0 nk = N and |n0, {nk}n0〉 is an arbitrary state of N atoms with occupationnumbers of the trap’s energy levels, nk , subject to the condition that there are n0atoms in the ground state of the trap.

In order to get an equation of motion for the condensate probability distributionpn0 , we need to perform the summation over all possible occupations {nk}n0 of theexcited levels in the trap. The resulting equation of motion for pn0 , from Eq. (124),is

dpn0

dt= −κ

∑{nk}n0

∑k>l>0

{(ηkl + 1)

[(nl + 1)nkpn0,{nk}n0

− nl(nk + 1)pn0,{...,nl−1,...,nk+1,...}n0

]+ ηkl[nl(nk + 1)pn0,{nk}n0


− (nl + 1)nkpn0,{...,nl+1,...,nk−1,...}n0

]}− κ∑{nk}n0

∑k′>0

[(ηk′ + 1)(n0 + 1)nk′pn0,{nk}n0

− (ηk′ + 1)n0(nk′ + 1)pn0−1,{nk+δk,k′ }n0−1

+ ηk′n0(nk′ + 1)pn0,{nk}n0

(126)− ηk′(n0 + 1)nk′pn0+1,{nk−δk,k′ }n0+1

],

where ηk′ = η(νk′) is the mean number of thermal phonons of mode k′ and thesum∑

k′ runs over all excited levels.To simplify Eq. (126) we assume that the atoms in the excited levels with a

given number of condensed atoms n0 are in an equilibrium state at the tempera-ture T , i.e.,

(127)pn0,{nk}n0= pn0

exp(− hT

∑k>0 νknk)∑

{n′k}n0exp(− h

T

∑k>0 νkn

′k),

where∑

k>0 nk = N−n0, and we assume that the sum∑

k>0 runs over all energystates of the trap, including degenerate states whose occupations nk are treatedas different stochastic variables. Equation (127) implies that the sum

∑k>l>0 in

Eq. (126) is equal to zero, since as depicted in Fig. 10,

(ηkl + 1)pn0,{nk}n0= ηklpn0,{...,nl+1,...,nk−1,...}n0

,

(128)(ηkl + 1)pn0,{...,nl−1,...,nk+1,...}n0= ηklpn0,{nk}n0

.

Equation (128) is precisely the detailed balance condition. The average numberof atoms in an excited level, subject to the condition that there are n0 atoms in theground state, from Eq. (127), is

(129)〈nk′ 〉n0 =∑{nk}n0

nk′pn0,{nk}n0

pn0

.

Therefore, the equation of motion for pn0 can be rewritten in the symmetrical andtransparent form

d

dtpn0 = −κ{Kn0(n0 + 1)pn0 −Kn0−1n0pn0−1

(130)+Hn0n0pn0 −Hn0+1(n0 + 1)pn0+1},

where

(131)Kn0 =∑k′>0

(ηk′ + 1)〈nk′ 〉n0 , Hn0 =∑k′>0

ηk′(〈nk′ 〉n0 + 1

).

We can obtain the steady state distribution of the number of atoms condensedin the ground level of the trap from Eq. (130). The mean value and the variance of


FIG. 10. Detailed balance and the corresponding probability flow diagram. We call Kn0 cooling(kooling) rate since the laser loss rate is denote by “C”.

the number of condensed atoms can then be determined. It is clear from Eq. (130)that there are two processes, cooling and heating. The cooling process is repre-sented by the first two terms with the cooling coefficient Kn0 while the heatingby the third and fourth terms with the heating coefficient Hn0 . The detailed bal-ance condition yields the following expression for the number distribution of thecondensed atoms

(132)pn0 = p0

n0∏i=1

Ki−1

Hi

,

where the partition function

(133)ZN = 1

pN=

N∑n0=0

N∏i=n0+1

Hi

Ki−1

is determined from the normalization condition∑N

n0=0 pn0 = 1. The functionsHi

and Ki as given by Eq. (131), involve, along with ηk′ (Eq. (125)), the func-tion 〈nk′ 〉n0 (Eq. (129)). In the following sections, we shall derive closed formexpressions for these quantities under various approximations. The master equa-tion (130) for the distribution function for the condensed atoms is one of our mainresults. It yields explicit expressions for the statistics of the condensed atoms andthe canonical partition function. Physical interpretation of various coefficients inthe master equations is summarized in Fig. 11.

Under the above assumption of a thermal equilibrium for noncondensed atoms,we have

(134)〈nk′ 〉n0 =∑

{nk}n0nk′ exp(− h

T

∑k>0 νknk)∑

{n′′k }n0exp(− h

T

∑k>0 νkn

′′k)

.

In the next two sections we present different approximations that clarify the gen-eral result (132).

340V.V.K

ocharovskyetal.

[4

System coefficients Physics

Laser Gm = A[1 + BA(m+ 1)]−1 gain

Lm = C lossA: Linear stimulated emission gain

B: Nonlinear saturation

C: Loss “through” mirrors

BEC: Low temp limit (CNB I)Gm ⇒ Kn0 = N + 1 − (n0 + 1)

Lm ⇒ Hn0 =∑k[eβεk − 1]−1 ≡ H≈ N(T/Tc)

3 weak trap

N atom cooling coefficient due to spontaneous emission of phonons,adds atoms to condensate.

N atom heating coefficient due to phonon absorption from bath at tem-perature T removes atoms from condensate.

BEC: With cross excitations (CNB II)Gm ⇒ Kn0 = [N + 1 − (n0 + 1)][1 + η]Lm ⇒ Hn0 = H+ (N − n0)η

η = “cross-excitation” parameter

N atom cooling coefficient due to stimulated emission of phonon asenhanced by atoms.

N atom heating coefficient due to absorption of phonon. Absorptionrate is enhanced by (N −n0)η due to interaction with multiple phonons(stimulated absorption) and increased absorption due to presence ofatoms.

FIG. 11. Physical interpretation of various coefficients in the master equations


Short summary of this subsection is as follows. We introduce the probability ofhaving n0 atoms in the ground level and nk atoms in the kth level Pn0,n1,...,nk,....We assume that the atoms in the excited levels with a given number of condensedatoms n0 are in equilibrium state at the temperature T , then

(135)Pn0,n1,...,nk,... =1

ZN

exp[−β(E0n0 + E1n1 + · · · + Eknk+)

].

This equation yields

(136)PN ≡ Pn0=N,n1=0,...,nk=0,... = 1

ZN

exp[−βE0N ].

Assuming E0 = 0 we obtain the following expression for the partition function

(137)ZN = 1

PN.

We assume that Bose gas can exchange heat (but not particles) with a harmonic-oscillator thermal reservoir. The reservoir has a dense and smooth spectrum. Theaverage occupation number of the heat bath oscillator at a frequency ωq = cq is

(138)ηq = 1

exp(hωq/kBT )− 1.

The master equation for the distribution function of the condensed bosons pn0 ≡ρn0,n0 takes the form

pn0 = −∑k,q

κkq〈nk〉n0(ηq + 1)[(n0 + 1)pn0 − n0pn0−1

]Kool

(139)−∑k,q

κkq〈nk + 1〉n0ηq[n0pn0 − (n0 + 1)pn0+1

]Heat.

The factors κkq embody the spectral density of the bath and the coupling strengthof the bath oscillators to the gas particles, and determine the rate of the condensateevolution since there is no direct interaction between the particles of an ideal Bosegas. Since κkq = κ · δ(hΩk − hcq) the sum

∑k,q reduces to

∑k ,

1

κpn0 = −

∑k

〈nk〉n0(ηk + 1)[(n0 + 1)pn0 − n0pn0−1

]Kool

(140)−∑k

〈nk + 1〉n0ηk[n0pn0 − (n0 + 1)pn0+1

]Heat.

Particle number constraint comes in a simple way:∑

k〈nk〉n0 = N − n0.


4.4. LOW TEMPERATURE APPROXIMATION

At low enough temperatures, the average occupations in the reservoir are smalland ηk + 1 � 1 in Eq. (131). This suggests the simplest approximation for thecooling coefficient

(141)Kn0 �∑k

〈nk〉n0 = N − n0.

In addition, at very low temperatures the number of noncondensed atoms is alsovery small, we can therefore approximate 〈nk′ 〉n0 + 1 by 1 in Eq. (131). Thenthe heating coefficient is a constant equal to the total average number of thermalexcitations in the reservoir at all energies corresponding to the energy levels ofthe trap,

(142)Hn0 � H, H ≡∑k>0

ηk =∑k>0

(ehνk/T − 1

)−1.

Under these approximations, the condensate master equation (130) simplifiesconsiderably and contains only one nontrivial parameter H. We obtain

d

dtpn0 = −κ{(N − n0)(n0 + 1)pn0 − (N − n0 + 1)n0pn0−1

(143)+H[n0pn0 − (n0 + 1)pn0+1

]}.

It may be noted that Eq. (143) has the same form as Eq. (107) of motion for thephoton distribution function in a laser operating not too far above threshold. Theidentification is complete if we define the gain, saturation, and loss parameters inlaser master equation by κ(N + 1), κ , and κH, respectively. The mechanism forgain, saturation, and loss are however different in the present case.

A laser phase transition analogy exists via the P -representation [70,71]. Thesteady-state solution of the Fokker–Planck equation for laser near threshold is [19]

(144)P(α, α∗) = 1

N exp

[(A− CA

)|α|2 − B

2A |α|4]

which clearly indicates a formal similarity between

(145)lnP(α, α∗) = − lnN + (1 −H/(N + 1))n0 − (1/2(N + 1)

)n2

0

for the laser equation and the Ginzburg–Landau type free energy [19,70,71]

(146)G(n0) = lnpn0 ≈ const + a(T )n0 + b(T )n20,

where |α|2 = n0, a(T ) = −(N −H)/N and b(T ) = 1/(2N) for large N near Tc.The resulting steady state distribution for the number of condensed atoms is

given by

(147)pn0 = 1

ZN

HN−n0

(N − n0)! ,


where ZN = 1/pN is the partition function. It follows from the normalizationcondition

∑n0pn0 = 1 that

(148)ZN = eH#(N + 1,H)/N !,where #(α, x) = ∫∞

xtα−1e−t dt is an incomplete gamma-function.

The distribution (147) can be presented as a probability distribution for the totalnumber of noncondensed atoms, n = N − n0,

(149)Pn ≡ pN−n = e−HN !#(N + 1,H)

Hn

n! .

It looks somewhat like a Poisson distribution, however, due to the additional nor-malization factor, N !/#(N+1,H) = 1, and a finite number of admissible valuesof n = 0, 1, . . . , N , it is not Poissonian. The mean value and the variance can becalculated from the distribution (147) for an arbitrary finite number of atoms inthe Bose gas,

(150)〈n0〉 = N −H+HN+1/ZNN !,(151)�n2

0 ≡ ⟨n20

⟩− 〈n0〉2 = H(1 − (〈n0〉 + 1

)HN/ZNN !).

As we shall see from the extended treatment in the next section, the approx-imations (141), (142) and, therefore, the results (150), (151) are clearly valid atlow temperatures, i.e., in the weak trap limit, T � ε1, where ε1 is an energy gapbetween the first excited and the ground levels of a single-particle spectrum inthe trap. However, in the case of a harmonic trap the results (150), (151) showqualitatively correct behavior for all temperatures, including T � ε1 and T ∼ Tc[52].

In particular, for a harmonic trap we have from Eq. (142) that the heating rateis

H =∑k

〈ηk〉 =∑l,m,n

1

exp[βhΩ(l +m+ n)] − 1

(152)≈(kBT

hΩ

)3

ζ(3) = N

(T

Tc

)3

.

Thus, in the low temperature region the master equation (143) for the condensatein the harmonic trap becomes

1

κpn0 = −[(N + 1)(n0 + 1)− (n0 + 1)2

]pn0 + [(N + 1)n0 − n2

0

]pn0−1

(153)−N

(T

Tc

)3[n0pn0 − (n0 + 1)pn0+1

].


4.5. QUASITHERMAL APPROXIMATION FOR NONCONDENSATE

OCCUPATIONS

At arbitrary temperatures, a very reasonable approximation for the averagenoncondensate occupation numbers in the cooling and heating coefficients inEq. (131) is suggested by Eq. (134) in a quasithermal form,

(154)〈nk〉n0 = ηk∑k>0

〈nk〉n0

/∑k′

ηk′ = (N − n0)

(eεk/T − 1)H ,

where εk = hνk , ηk is given by Eq. (125) and H by Eq. (142). Equation (154)satisfies the canonical-ensemble constraint, N = n0 +∑k>0 nk , independently ofthe resulting distribution pn0 . This important property is based on the fact that aquasithermal distribution (154) provides the same relative average occupations inexcited levels of the trap as in the thermal reservoir, Eq. (125).

To arrive at the quasithermal approximation in Eq. (154) one can go along thefollowing logic. In the low temperature limit we assumed ηk � 1 and took∑

k

〈nk〉n0〈ηk + 1〉 ≈∑k

〈nk〉n0 = N − n0.

To go further, still in the low temperature limit, we can write

〈nk〉n0 ≈ (N − n0)

[exp(−βEk)∑k′ exp(−βEk′)

].

This is physically motivated since the thermal factor in [. . .] is the fraction of theexcited atoms in the state k, and N− n0 is the total number of excited atoms. Notethat

ηk = 1

exp(βEk)− 1�⇒ exp(−βEk) = ηk

1 + ηk.

Since we are at low temperature we take exp(−βEk) ≈ ηk and therefore

(155)〈nk〉n0 ≈ (N − n0)ηk∑k ηk

= (N − n0)

[exp(βEk)− 1]H ,

where

H =∑k

ηk.

Now this ansatz is good for arbitrary temperatures. As a result,

(156)∑k

〈nk〉n0〈ηk + 1〉 ≈ (N − n0)(1 + η),


where

η = 1

(N − n0)

∑k

〈nk〉n0ηk = 1

H∑k>0

η2k = 1

H∑k>0

1

[exp(βEk)− 1]2 .

Another line of thought is the following:

〈nk〉n0 ≈ (N − n0)nk∑k nk

.

But by detailed balance in the steady state

κ(n0 + 1)nk(ηk + 1) ≈ κn0(nk + 1)ηk

and if the ground level is macroscopically occupied then n0 ≈ n0 ± 1. Sinceeven at T = Tc one finds n0 ∼ √

N , one “always” has n0 � 1. Therefore,nk(ηk + 1) ≈ (nk + 1)ηk and, hence, nk ≈ ηk . As a result we again obtainEq. (155).

Calculation of the heating and cooling rates in this approximation is very sim-ple. For example, for the heating rate we have

(157)∑k

〈nk + 1〉n0〈ηk〉 =∑k

〈ηk〉 +∑k

〈nk〉n0〈ηk〉 ≈ H+ η(N − n0).

In summary, the cooling and heating coefficients (131) in the quasithermal ap-proximation of Eq. (154) are

(158)Kn0 = (N − n0)(1 + η), Hn0 = H+ (N − n0)η.

Compared with the low temperature approximation (141) and (142), these coef-ficients acquire an additional contribution (N − n0)η due to the cross-excitationparameter

(159)η = 1

N − n0

∑k>0

〈ηk〉〈nk〉n0 = 1

H∑k>0

1

(eεk/T − 1)2.

4.6. SOLUTION OF THE CONDENSATE MASTER EQUATION

Now, at arbitrary temperatures, the condensate master equation (130) containstwo nontrivial parameters, H and η,

dpn0

dt= −κ{(1 + η)

[(N − n0)(n0 + 1)pn0 − (N − n0 + 1)n0pn0−1

]+ [H+ (N − n0)η

]n0pn0

(160)− [H+ (N − n0 − 1)η](n0 + 1)pn0+1

}.


It can be rewritten also in the equivalent form

1

κ

dpn0

dt= −[(N + 1)(n0 + 1)− (n0 + 1)2

]pn0

+ [(N + 1)n0 − n02]pn0−1

(161)− (T /Tc)3N[n0pn0 − (n0 + 1)pn0+1

].

The steady-state solution of Eq. (160) is given by

pn0 = 1

ZN

(N − n0 +H/η − 1)!(H/η − 1)!(N − n0)!

(η

1 + η

)N−n0

(162)= 1

ZN

(N − n0 + H

η− 1

N − n0

)(η

1 + η

)N−n0

,

where the canonical partition function ZN = 1/pN is

(163)ZN =N∑

n0=0

(N − n0 +H/η − 1

N − n0

)(η

1 + η

)N−n0

.

It is worth noting that the explicit formula (162) satisfies exactly the general re-lation between the probability distribution of the number of atoms in the groundstate, pn0 , and the canonical partition function [46], Eq. (79).

The master equation (160) for pn0 , and the analytic approximate expres-sions (162) and (163) for the condensate distribution function pn0 and the partitionfunction ZN , respectively, are among the main results of the condensate masterequation approach. As we shall see later, they provide a very accurate descriptionof the Bose gas for a large range of parameters and for different trap potentials.Now we are able to present the key picture of the theory of BEC fluctuations,that is the probability distribution pn0 , Fig. 12. Analogy with the evolution of thephoton number distribution in a laser mode (from thermal to coherent, lasing) isobvious from a comparison of Fig. 12 and Fig. 6. With an increase of the numberof atoms in the trap, N , the picture of the ground-state occupation distributionremains qualitatively the same, just a relative width of all peaks becomes morenarrow.

The canonical partition function (163) allows us to calculate also the micro-canonical partition function Ω(E,N) by means of the inversion of the definitionin Eq. (74). Moreover, in principle, the knowledge of the canonical partitionfunction allows us to calculate all thermodynamic and statistical equilibrium prop-erties of the system in the standard way (see, e.g., [13,61] and discussion in theIntroduction).


FIG. 12. Probability distribution of the ground-state occupation, pn0 , at the temperature T =0.2Tc in an isotropic harmonic trap with N = 200 atoms as calculated from the solution of thecondensate master equation (130) in the quasithermal approximation, Eq. (162), (solid line) and fromthe exact recursion relations in Eqs. (79) and (80) (dots).

Previously, a closed-form expression for the canonical partition function (74)was known only for one-dimensional harmonic traps [68,69]

(164)ZN(T ) =N∏k=1

1

1 − e−khω/T.

In the general case, there exists only the recursion relation (80) that is quite com-plicated, and difficult for analysis [46,61–63].

The distribution (162) can also be presented as a probability distribution for thetotal number of noncondensed atoms, n = N − n0,

(165)Pn = pN−n = 1

ZN

(n+H/η − 1

n

)(η

1 + η

)n.

The distribution (165) can be named as a finite negative binomial distribution,since it has the form of the well-known negative binomial distribution [72],

(166)Pn =(n+M − 1

n

)qn(1 − q)M, n = 0, 1, 2, . . . ,∞,

that was so named due to a coincidence of the probabilities Pn with the terms inthe negative-power binomial formula

(167)1

(1 − q)M=

∞∑n=0

(n+M − 1

n

)qn.


It has a similar semantic origin as the well-known binomial distribution,

Pn =(M

n

)(1 − q)nqM−n,

which was named after a Newton’s binomial formula

[q + (1 − q)

]M =M∑n=0

(M

n

)(1 − q)nqM−n.

The finite negative binomial distribution (165) tends to the well-known distribu-tion (166) only in the limit N � (1 + η)H.

The average number of atoms condensed in the ground state of the trap is

(168)〈n0〉 ≡N∑

n0=0

n0pn0 .

It follows, on substituting for pn0 from Eq. (162), that

(169)〈n0〉 = N −H+ p0η(N +H/η).

The central moments of the mth order, m > 1, of the number-of-condensed-atom and number-of-noncondensed-atom fluctuations are equal to each other foreven orders and have opposite signs for odd orders,

(170)⟨(n0 − n0)

m⟩ = (−1)m

⟨(n− n)m

⟩.

The squared variance can be represented as

(171)�n20 = ⟨n2⟩− 〈n〉2 =

N∑n=0

n(n− 1)Pn + 〈n〉 − 〈n〉2

and calculated analytically. We obtain

�n20 = (1 + η)H− p0(ηN +H)(N −H+ 1 + η)

(172)− p20(ηN +H)2,

where

(173)p0 = 1

ZN

(N +H/η − 1)!N !(H/η − 1)!

(η

1 + η

)Nis the probability that there are no atoms in the condensate.

All higher central moments of the distribution Eq. (162) can be calculated ana-lytically using 〈n0

s〉 =∑Nn0=0 n

s0pn0 and Eqs. (162), (163). In particular, the third

central moment is


FIG. 13. The first four central moments for the ideal Bose gas in an isotropic harmonictrap with N = 200 atoms as calculated via the solution of the condensate master equation(solid lines—quasithermal approximation, Eq. (162); dashed lines—low temperature approximation,Eq. (147)) and via the exact recursion relations in Eqs. (79) and (80) (dots).

⟨(n0 − 〈n0〉

)3⟩ = −(1 + η)(1 + 2η)H+ p0(H+ ηN)

[1 + (H−N)2 + 2

(η2 +N(1 + η)

)+ 3(η −H(1 + η)

)]+ 3p2

0(H+ ηN)2(1 + η −H+N)

(174)+ 2p30(H+ ηN)3.

The first four central moments for the Bose gas in a harmonic trap with N =200 atoms are presented in Fig. 13 as the functions of temperature in differentapproximations.

For the “condensed phase” in the thermodynamic limit, the probability p0 van-ishes exponentially if the temperature is not very close to the critical temperature.In this case only the first term in Eq. (172) remains, resulting in

(175)�n20 = (1 + η)H ≡

∑k>0

(〈nk〉2 + 〈nk〉).


This result was obtained earlier by standard statistical methods (see [13] and ref-erences therein).

It is easy to see that the result (165) reduces to the simple approximation (149)in the formal limit η → 0, H/η → ∞, when

(176)#(N − n0 +H/η)

#(N +H/η)→(Hη

)−n0

.

The limit applies to only very low temperatures, T � ε1. However, due toEqs. (142) and (159), the parameter H/η tends to 1 as T → 0, but never toinfinity. Nevertheless, the results (169) and (172) agree with the low temperature

approximation results (150) and (151) for T � ε1. In this case the variance√�n2

0is determined mainly by a square root of the mean value 〈n〉 which is correctlyapproximated by Eq. (150) as 〈n〉 ≡ N − 〈n0〉 ≈ H.

4.7. RESULTS FOR BEC STATISTICS IN DIFFERENT TRAPS

As we have seen, the condensate fluctuations are governed mainly by two para-meters, the number of thermal excitations H and the cross-excitation parameter η.They are determined by a single-particle energy spectrum of the trap. We ex-plicitly present them below for arbitrary power-law trap. We discuss mainly thethree-dimensional case. A generalization to other dimensions is straightforwardand is given in the end of this subsection. First, we discuss briefly the case of theideal Bose gas in a harmonic trap. It is the simplest case since the quadratic en-ergy spectrum implies an absence of the infrared singularity in the variance of theBEC fluctuations. However, because of the same reason it is not robust relative toan introduction of a realistic weak interaction in the Bose gas as is discussed inSection 5.

4.7.1. Harmonic Trap

The potential in the harmonic trap has, in general, an asymmetrical profilein space, Vext(x, y, z) = m

2 (x2ω2

x + y2ω2y + z2ω2

z ), with eigenfrequencies{ωx, ωy, ωz} = ω, ωx � ωy � ωx > 0. Here m is the mass of the atom. Thesingle-particle energy spectrum of the trap,

(177)εk = hkω ≡ h(kxωx + kyωy + kzωz),

can be enumerated by three nonnegative integers {kx, ky, kz} = k, kx,y,z � 0. Wehave

(178)H =∑k>0

1

ehkω/T − 1, ηH =

∑k>0

1

(ehkω/T − 1)2.


The energy gap between the ground state and the first excited state in the trap isequal to ε1 = hωx .

If the sums can be replaced by the integrals (continuum approximation), i.e., ifhωx � T , the parameters H and ηH are equal to

(179)H = T 3

h3ωxωyωzζ(3) =

(T

Tc

)3

N,

(180)ηH = T 3

h3ωxωyωz

(ζ(2)− ζ(3)

) = ( TTc

)3

Nζ(2)− ζ(3)

ζ(3),

where a standard critical temperature is introduced as

(181)Tc = h

(ωxωyωzN

ζ(3)

)1/3

; ζ(3) = 1.202 . . . , ζ(2) = π2

6.

Therefore, the cross-excitation parameter η is a constant, independent of the tem-perature and the number of atoms, η = [ζ(2) − ζ(3)]/ζ(3) ≈ 0.37. The ratioH/η = N(T/Tc)

3[ζ(3)/(ζ(2) − ζ(3))] goes to infinity in the thermodynamiclimit proportionally to the number of atoms N .

In the opposite case of very low temperatures, T � hωx , we have

(182)H ≈ exp

(− hωx

T

)+ exp

(− hωy

T

)+ exp

(− hωz

T

),

(183)ηH ≈ exp

(−2hωx

T

)+ exp

(−2hωy

T

)+ exp

(−2hωz

T

)with an exponentially good accuracy. Now the cross-excitation parameter η de-pends exponentially on the temperature and, instead of the number 0.37, is expo-nentially small. The ratio

(184)Hη

= [exp(− hωxT)+ exp(− hωy

T)+ exp(− hωz

T)]2

exp(− 2hωxT

)+ exp(− 2hωyT

)+ exp(− 2hωzT

)∼ 1

becomes approximately a constant. The particular case of an isotropic harmonictrap is described by the same equations if we substitute ωx = ωy = ωz = ω.

4.7.2. Arbitrary Power-Law Trap

We now consider the general case of a d-dimensional trap with an arbitrary power-law single-particle energy spectrum [46,55,73]

(185)εk = h

d∑j=1

ωjkσj , k = {kj ; j = 1, 2, . . . , d},


where kj � 0 is a nonnegative integer and σ > 0 is an index of the energyspectrum. We assume 0 < ω1 � ω2 � · · · � ωd , so that the energy gap betweenthe ground state and the first excited state in the trap is ε1 = hω1. We then have

(186)H =∑k>0

1

eεk/T − 1, ηH =

∑k>0

1

(eεk/T − 1)2.

In the case ε1 � T , the sum can be replaced by the integral only for the para-meter H (Eq. (186)) if d > σ ,

(187)H = Aζ

(d

σ

)T d/σ =

(T

Tc

)d/σN, d > σ,

where the critical temperature is

(188)Tc =[

N

Aζ(d/σ)

]σ/d, A = [#( 1

σ+ 1)]d

(∏d

j=1 hωj )1/σ

.

The second parameter can be calculated by means of this continuum approxima-tion only if 0 < σ < d/2,

ηH = AT d/σ

(ζ

(d

σ− 1

)− ζ

(d

σ

))(189)=

(T

Tc

)d/σNζ( d

σ− 1)− ζ( d

σ)

ζ( dσ)

, 0 < σ < d/2.

If σ > d/2, it has a formal infrared divergence and should be calculated via adiscrete sum,

(190)ηH =(T

Tc

)2

N2σ/d aσ,d

[#( 1σ+ 1)]2σ [ζ( d

σ)]2σ/d , σ > d/2,

where

aσ,d =∑k>0

(∏d

j=1 hωj )2/d

ε2k

.

The traps with the dimension lower than the critical value, d � σ , can be analyzedon the basis of Eqs. (186) as well. We omit this analysis here since there is nophase transition in this case.

The cross-excitation parameter η has different dependence on the number ofatoms for high, d > 2σ , or low, d < 2σ , dimensions,

(191)η = ζ( dσ− 1)− ζ( d

σ)

ζ( dσ)

, d > 2σ > 0,


(192)η =(T

Tc

)2−d/σN2σ/d−1 aσ,d

[#( 1σ+ 1)]2σ [ζ( d

σ)]2σ/d , d < 2σ.

Therefore, the traps with small index of the energy spectrum, 0 < σ < d/2, aresimilar to the harmonic trap. The traps with larger index of the energy spectrum,σ > d/2, are similar to the box with “homogeneous” Bose gas. For the lattertraps, the cross-excitation parameter η goes to infinity in the thermodynamic limit,proportionally to N2σ/d−1. The ratio H/η goes to infinity in the thermodynamiclimit only for 0 < σ < d . In the opposite case, σ > d , it goes to zero. We obtain

(193)Hη

=(T

Tc

)d/σN

ζ( dσ)

ζ( dσ− 1)− ζ( d

σ), d > 2σ > 0,

Hη

=(T

Tc

)2(d/σ−1)

N2(1−σ/d)[#

(1

σ+ 1

)]2σ[ζ

(d

σ

)]2σ/da−1σ,d ,

(194)d < 2σ.

It is remarkable that BEC occurs only for those spatial dimensions, d > σ , forwhich H/η → ∞ at N → ∞. (We do not consider here the case of the critical di-mension d = σ , e.g., one-dimensional harmonic trap, where a quasi-condensationoccurs at a temperature Tc ∼ hω1N/ logN .) For spatial dimensions lower thanthe critical value, d < σ , BEC does not occur (see, e.g., [46]). Interestingly, evenfor the latter case there still exists a well-defined single peak in the probability dis-tribution pn0 at low enough temperatures. With the help of the explicit formulasin Section 3 we can describe this effect as well.

In the opposite case of very low temperatures, T � ε1, the parameters

(195)H ≈d∑

j=1

e−hωjT , ηH ≈

d∑j=1

e−2hωjT , η ∼ e−

ε1T

are exponentially small. The ratio Hη

∼∑dj=1 e

−(hωj−ε1)/T ∼ d becomes a con-stant.

Formulas (185)–(195) for the arbitrary power-law trap contain all particularformulas for the three-dimensional harmonic trap (d = 3, σ = 1) and the box,i.e., the “homogeneous gas” with d = 3 and σ = 2, as the particular cases.

In Fig. 13, numerical comparison of the results obtained from the exact recur-sion relations in Eqs. (79)–(80) and from our approximate explicit formulas fromSection 4 in the particular case of the ideal Bose gas in the three-dimensionalisotropic harmonic trap for various temperatures is demonstrated. The results in-dicate an excellent agreement between the exact results and the results based onquasithermal approximation, including the mean value 〈n0〉, the squared variance�n2

0 as well as the third and fourth central moments. The low temperature ap-proximation, Eq. (147), is good only at low temperatures. That is expected sinceit neglects by the cross-excitation parameter η.


4.8. CONDENSATE STATISTICS IN THE THERMODYNAMIC LIMIT

The thermodynamic, or bulk [13] limit implies an infinitely large number ofatoms, N → ∞, in an infinitely large trap under the condition of a fixed criti-cal temperature, i.e., Nωxωyωz = const in the harmonic trap, L3N = const inthe box, and Nσ

∏dj=1 ωj = const in an arbitrary d-dimensional power-law trap

with an energy spectrum index σ . Then, BEC takes place at the critical temper-ature Tc (for d > σ ) as a phase transition, and for some lower temperatures thefactor p0 is negligible. As a result, we have the following mean value and thevariance for the number of condensed atoms

(196)〈n0〉 = N −H ≡ N −∑k>0

1

eεk/T − 1,

(197)�n20 = (1 + η)H ≡

∑k>0

1

eεk/T − 1+∑k>0

1

(eεk/T − 1)2,

which agree with the results obtained for the ideal Bose gas for different traps inthe canonical ensemble by other authors [13,14,46,73–77]. In particular, we findthe following scaling of the fluctuations of the number of condensed atoms:

(198)�n20 ∼ C ×

(( TTc)d/σN, d > 2σ > 0

( TTc)2N2σ/d , d < 2σ

), ε1 � T < Tc,

�n20 ≈ 〈n〉 ≈

d∑i=1

exp

{− ωi

[∏dj=1 ωj ]1/d

[ζ

(d

σ

)]σ/d(199)×

[#

(1

σ+ 1

)]σTc

T Nσ/d

}, T � ε1,

where C is a constant. From Eq. (198), we see that in the high dimensional traps,d > 2σ , e.g., in the three-dimensional harmonic trap, fluctuations display theproper thermodynamic behavior, �n2

0 ∝ N . However, fluctuations become anom-alously large [46,55,74,78], �n2

0 ∝ N2σ/d � N , in the low dimensional traps,σ < d < 2σ . In the quantum regime, when the temperature is less than theenergy gap between the ground and the first excited level in the trap, it followsfrom Eq. (199) that condensate fluctuations become exponentially small. For alltemperatures, when BEC exists (d > σ), the root-mean-square fluctuations nor-malized to the mean number of condensed atoms vanishes in the thermodynamic

limit:√�n2

0/〈n0〉 → 0 as N → ∞.Another remarkable property of the distribution function obtained in Section 4

is that it yields the proper mean value and variance of the number of atoms in theground level of the trap even for temperatures higher than the critical temperature.In particular, it can be shown that its asymptotic for high temperatures, T � Tc,


yields the standard thermodynamic relation �n0 ≈ 〈n0〉 known from the analysisof the grand canonical ensemble [13]. This nice fact indicates that the presentmaster equation approach to the statistics of the cool Bose gas is valuable in thestudy of mesoscopic effects as well, both at T < Tc and T > Tc. Note that,in contrast, the validity of the Maxwell’s demon ensemble approach [54] to thestatistics of BEC remains restricted to temperatures well below the onset of BEC,T < Tc.

4.9. MESOSCOPIC AND DYNAMICAL EFFECTS IN BEC

In recent experiments on BEC in ultracold gases [24–29], the number of con-densed atoms in the trap is finite, i.e., mesoscopic rather than macroscopic,N ∼ 103–106. Therefore, it is interesting to analyze mesoscopic effects asso-ciated with the BEC statistics.

The mean number of atoms in the ground state of the trap with a finite numberof atoms is always finite, even at high temperatures. However, it becomes macro-scopically large only at temperatures lower than some critical temperature, Tc,that can be defined via the standard relation

(200)∑

ηk(Tc) ≡ H(Tc) = N.

This equation has an elementary physical meaning, namely it determines the tem-perature at which the total average number of thermal excitations at all energylevels of the trap becomes equal to the total number of atoms in the trap. The re-sults (162), (169), (172) shown in Fig. 13 explicitly describe a smooth transitionfrom a mesoscopic regime (finite number of atoms in the trap, N < ∞) to thethermodynamic limit (N = ∞) when the threshold of the BEC becomes verysharp so that we have a phase transition to the Bose–Einstein condensed stateat the critical temperature given by Eq. (200). This can be viewed as a specificdemonstration of the commonly accepted resolution to the Uhlenbeck dilemma inhis famous criticism of Einstein’s pioneering papers on BEC [6,9,10,12].

Although for systems containing a finite number of atoms there is no sharpcritical point, as is obvious from Figs. 3, 12, and 14, it is useful to define a criticalcharacteristic value of a temperature in such a case as well. It should coincide withthe standard definition (200) in the thermodynamic limit. Different definitions forTc were proposed and discussed in [11,66,79–85]. We follow a hint from laserphysics. There we know that fluctuations dominate near threshold. However, wedefine a threshold inversion as that for which gain (in photon number for the lasingmode) equals loss. Let us use a similar dynamical approach for BEC on the basisof the master equation, see also [86].

We note that, for a laser operating near the threshold where B/A � 1, theequation (118) of motion for the probability pn of having n photons in the cavity


implies the following rate of the change for the average photon number:

(201)d

dt〈n〉 = (A− C)〈n〉 − B

⟨(n+ 1)2

⟩+ A.

Here A, B, and C are the linear gain, nonlinear saturation, and linear loss coef-ficients, respectively. On neglecting the spontaneous emission term A and notingthat the saturation term B〈(n+1)2〉 is small compared to (A−C)〈n〉 near thresh-old, we define the threshold (critical) inversion to occur when the linear gain rateequals the linear loss rate, i.e., A = C.

Similar to laser physics, the condensate master equation (130) implies a cou-pled hierarchy of moment equations which are useful in the analysis of timeevolution. In the quasithermal approximation (160), we find

d〈nl0〉dt

= κ

l−1∑i=0

(l

i

){(1 + η)

[N(⟨ni0⟩+ ⟨ni+1

0

⟩)− ⟨ni+10

⟩− ⟨ni+20

⟩](202)+ (−1)l−i (H+ ηN)

⟨ni+1

0

⟩− (−1)l−iη⟨ni+2

0

⟩}.

Similar moment equations in the low-temperature approximation (143) followfrom Eq. (202) with η = 0,

d〈nl0〉dt

= κ

l−1∑i=0

(l

i

){N⟨ni0⟩+ (N − 1)

⟨ni+1

0

⟩− ⟨ni+20

⟩+ (−1)l−iH⟨ni+1

0

⟩}.

(203)

The dynamical equation for the first moment, as follows from Eq. (202), has thefollowing form:

(204)d〈n0〉dt

= κ{(1 + η)N + (N − 1 − η −H)〈n0〉 −

⟨n2

0

⟩}.

Near the critical temperature, T ≈ Tc, the mean number of the condensed atomsis small, 〈n0〉 � N , and it is reasonable to neglect the second moment 〈n2

0〉 com-pared toN〈n0〉 and the spontaneous cooling (spontaneous emission in lasers) termκ(1+ η)N compared to κN〈n0〉. In this way, neglecting fluctuations, we arrive ata simple equation for the competition between cooling and heating processes,

(205)d〈n0〉dt

≈ κ(N −H− η)〈n0〉.In analogy with the laser threshold we can define the critical temperature, T = Tc,as a point where cooling equals heating, i.e., d〈n0〉/dt = 0. This definition of thecritical temperature

(206)H(Tc)+ η(Tc) = N,


is valid even for mesoscopic systems and states that at T = Tc the rate of theremoval of atoms from the ground state equals to the rate of the addition, in theapproximation neglecting fluctuations. In the thermodynamic limit it correspondsto the standard definition, Eqs. (181) and (188). For a mesoscopic system, e.g., ofN = 103 atoms in a trap, the critical temperature as given by Eq. (206) is aboutfew per cent shifted from the thermodynamic-limit value given by Eqs. (181)and (188). Other definitions for Tc also describe the effect of an effective-Tcshift [11,66,79–85], which is clearly seen in Fig. 14, and agree qualitatively withour definition.

Note that precisely the same definition of the critical temperature follows froma statistical mechanics point of view, which in some sense is alternative to the dy-namical one. We may define the critical temperature as the temperature at whichthe mean number of condensed atoms in the steady-state solution to the mas-ter equation vanishes when neglecting fluctuations and spontaneous cooling. Wemake the replacement 〈n2

0〉 ≈ 〈n0〉2 in Eq. (204) and obtain the steady-state so-lution to this nonlinear equation, 〈n0〉 = N − H − η. Now we see that 〈n0〉vanishes at the same critical temperature (206). Finally, we remind again that aprecise definition of the critical temperature is not so important and meaningfulfor the mesoscopic systems as it is for the macroscopic systems in the thermody-namic limit since for the mesoscopic systems, of course, there is not any sharpphase transition and an onset of BEC is dispersed over a whole finite range oftemperatures around whatever Tc, as is clearly seen in Figs. 3, 12, and 14.

5. Quasiparticle Approach and Maxwell’s Demon Ensemble

In order to understand relations between various approximate schemes, we formu-late a systematic analysis of the equilibrium canonical-ensemble fluctuations ofthe Bose–Einstein condensate based on the particle number conserving operatorformalism of Girardeau and Arnowitt [87], and the concept of the canonical-ensemble quasiparticles [20,21]. The Girardeau–Arnowitt operators can be in-terpreted as the creation and annihilation operators of the canonical-ensemblequasiparticles which are essentially different from the standard quasiparticlesin the grand canonical ensemble. This is so because these operators create andannihilate particles in the properly reduced many-body Fock subspace. In thisway, we satisfy the N -particle constraint of the canonical-ensemble problem inEq. (72) from the very beginning. Furthermore, we do this while taking into ac-count all possible correlations in the N -boson system in addition to what one hasin the grand canonical ensemble. These canonical-ensemble quasiparticles fluc-tuate independently in the ideal Bose gas and form dressed canonical-ensemblequasiparticles in the dilute weakly interacting Bose gas due to Bogoliubov cou-pling (see Section 6 below).


Such an analysis was elaborated in [20,21] and resulted in the explicit expres-sions for the characteristic function and all cumulants of the ground-state occu-pation statistics both for the dilute weakly interacting and ideal Bose gases. Wepresent it here, including the analytical formulas for the moments of the ground-state occupation fluctuations in the ideal Bose gases in an arbitrary power-lawtrap, and, in particular, in a box (“homogeneous gas”) and in an arbitrary har-monic trap. In Section 6 we extend this analysis to the interacting Bose gas. Inparticular, we calculate the effect of Bogoliubov coupling between quasiparticleson suppression of the ground-state occupation fluctuations at moderate temper-atures and their enhancement at very low temperatures and clarify a crossoverbetween ideal-gas and weakly-interacting-gas statistics which is governed by apair-correlation, squeezing mechanism. The important conclusion is that in mostcases the ground-state occupation fluctuations are anomalously large and are notGaussian even in the thermodynamic limit.

Previous studies focused mainly on the mean value, n0, and squared variance,〈(n0 − n0)

2〉, of the number of condensed atoms.1 Higher statistical moments aremore difficult to calculate, and it was often assumed that the condensate fluctua-tions have vanishing higher cumulants (semi-invariants). That is, it was assumedthat the condensate fluctuations are essentially Gaussian with all central momentsdetermined by the mean value and the variance. We here show that this is nottrue even in the thermodynamic limit. In, particular, we prove that in the generalcase the third and higher cumulants normalized by the corresponding power ofthe variance do not vanish even in the thermodynamic limit.

The results of the canonical-ensemble quasiparticle approach are valid for tem-peratures a little lower than a critical temperature, namely, when the probabilityof having zero atoms in the ground state of the trap is negligibly small and thehigher-order effects of the interaction between quasiparticles are not important.We outline also the Maxwell’s demon ensemble approximation introduced andstudied for the ideal Bose gas in [42,44,46,54,55] and show that it can be justifiedon the basis of the method of the canonical-ensemble quasiparticles, and for thecase of the ideal Bose gas gives the same results.

This section is organized as follows: We start with the reduction of the Hilbertspace and the introduction of the canonical-ensemble quasiparticles appropri-ate to the canonical-ensemble problem in Section 5.1. Then, in Section 5.2, weanalytically calculate the characteristic function and all cumulants of the ground-state occupation distribution for the ideal Bose gas in a trap with an arbitrary

1 The only exception known to the authors is the paper [14] where the third moment of the ground-state occupation for the ideal gas in a harmonic trap was discussed in the Maxwell’s demon ensembleapproximation. Higher moments were also discussed on the basis of the master equation approach inRefs. [52,53].


single-particle energy spectrum. We also discuss the Maxwell’s demon ensem-ble approach and compare it with the canonical-ensemble quasiparticle approach.In Section 5.3 we apply these results to the case of an arbitrary d-dimensionalpower-law trap which includes a three-dimensional box with periodic boundaryconditions (“homogeneous gas”) and a three-dimensional asymmetric harmonictrap as the particular cases.

5.1. CANONICAL-ENSEMBLE QUASIPARTICLES IN THE REDUCED

HILBERT SPACE

In principle, to study the condensate fluctuations, we have to fix only the externalmacroscopical and global, topological parameters of the system, like the numberof particles, temperature, superfluid flow pattern (“domain” or vortex structure),boundary conditions, etc. We then proceed to find the condensate density matrixvia a solution of the von Neumann equation with general initial conditions admit-ting all possible quantum states of the condensate. In particular, this is a naturalway to approach the linewidth problem for the atom laser [88,89]. Obviously, thisis a complicated problem, especially for the interacting finite-temperature Bosegas; because of the need for an efficient technique to account for the additionalcorrelations introduced by the constraint in such realistic ensembles. The latteris the origin of the difficulties in the theory of the canonical or microcanoni-cal ensembles (see discussion in Section 3.1). According to [32], a calculationof equilibrium statistical properties using the grand canonical ensemble and aperturbation series will be impossible since the series will have zero radius ofconvergence.

One way out of this problem is to develop a technique which would allow us tomake calculations in the constrained many-body Hilbert space, e.g., on the basisof the master equation approach as discussed in Section 4. Another possibility isto solve for the constraint from the very beginning by a proper reduction of themany-body Hilbert space so that we can work with the new, already unconstrainedquasiparticles. This approach is demonstrated in the present Section. Working inthe canonical ensemble, we solve for the fluctuations of the number of atomsin the ground state in the ideal Bose gas in a trap (and similarly in the weaklyinteracting Bose gas with the Bogoliubov coupling between excited atoms, seeSection 6). More difficult problems involving phase fluctuations of the condensatewith an accurate account of the quasiparticle renormalization due to interaction atfinite temperatures and the dynamics of BEC will be discussed elsewhere.

We begin by defining an occupation number operator in the many-body Fockspace as usual,

(207)nk = a+k ak, nk∣∣ψ(n)

k

⟩ = n∣∣ψ(n)

k

⟩, a+k

∣∣ψ(n)k

⟩ = √n+ 1∣∣ψ(n+1)

k

⟩.


The particle number constraint (72) determines a canonical-ensemble (CE) sub-space of the Fock space. Again we would like to work with the particle-numberconserving creation and annihilation operators. The latter are given in the Gi-rardeau and Arnowitt paper [87],

(208)β+k = a+k β0, βk = β+

0 ak, β0 = (1 + n0)−1/2a0.

These operators for k = 0 can be interpreted as describing new canonical-ensemble quasiparticles which obey the Bose canonical commutation relationson the subspace n0 = 0,

(209)[βk, β

+k′] = δk,k′ .

We are interested in the properties of the fraction of atoms condensed in theground level of the trap, k = 0. We focus on the important situation when theground-state occupation distribution is relatively well peaked, i.e., its variance ismuch less than the mean occupation of the ground level of a trap,

(210)⟨(n0 − n0)

2⟩1/2 � n0.

In such a case, the relative role of the states with zero ground-state occupation,n0 = 0, is insignificant, so that we can approximate the canonical-ensemble sub-space HCE by the subspace HCE

n0 =0. Obviously, this approximation is valid onlyfor temperatures T < Tc.

The physical meaning of the canonical-ensemble quasiparticles, βk = β+0 ak,

is that they describe transitions between ground (k = 0) and excited (k = 0)states. All quantum properties of the condensed atoms have to be expressed viathe canonical-ensemble quasiparticle operators in Eq. (208). In particular, we havethe identity

(211)n0 = N −∑k=0

nk,

where the occupation operators of the excited states are

(212)nk = a+k ak = β+k βk.

Note that in Refs. [90,91] quasiparticle operators similar in spirit to those ofRef. [87] were introduced which, unlike βk, did not obey the Bose commutationrelations (209) exactly, if noncommutation of the ground-state occupation opera-tors a0 and a+0 is important. As was shown by Girardeau [92], this is importantbecause the commutation corrections can accumulate in a perturbation series forquantities like an S-matrix. Warning concerning a similar subtlety was stressedsome time ago [93].

We are interested in fluctuations in the number of atoms condensed in theground state of a trap, n0. This is equal to the difference between the total number


of atoms in a trap and the number of excited atoms, n0 = N − n. In principle, acondensed state can be defined via the bare trap states as their many-body mix-ture fixed by the interaction and external conditions. Hence, occupation statisticsof the ground as well as excited states of a trap is a very informative feature ofthe BEC fluctuations. Of course, there are other quantities that characterize BECfluctuations, e.g., occupations of collective, dressed or coherent excitations anddifferent phases.

5.2. CUMULANTS OF BEC FLUCTUATIONS IN AN IDEAL BOSE GAS

Now we can use the reduced Hilbert space and the equilibrium canonical-ensemble density matrix ρ to conclude that the occupation numbers of thecanonical-ensemble quasiparticles, nk, k = 0, are independent stochastic vari-ables with the equilibrium distribution

(213)ρk(nk) = exp(−nkεk/T )(1 − exp(−εk/T )

).

The statistical distribution of the number of excited atoms, n = ∑k=0 nk, whichis equal, according to Eq. (211), to the number of noncondensed atoms, is a simple“mirror” image of the distribution of the number of condensed atoms,

(214)ρ(n) = ρ0(n0 = N − n).

A useful way to find and to describe it is via the characteristic function

(215)Θn(u) = Tr{eiunρ}.

Thus upon taking the Fourier transform of Θn(u) we obtain the probability distri-bution

(216)ρ(n) = 1

2π

π∫−π

e−iunΘn(u) du.

Taylor expansions of Θn(u) and logΘn(u) give explicitly initial (noncentral)moments and cumulants, or semi-invariants [72,94]:

(217)Θn(u) =∞∑m=0

αmum

m! , αm ≡ ⟨nm⟩ = dm

dumΘn(u)

∣∣∣∣u=0

,

logΘn(u) =∞∑m=1

κm(iu)m

m! , κm = dm

d(iu)mlogΘn(u)

∣∣∣∣u=0

,

(218)Θn(u = 0) = 1.

The cumulants κr , initial moments αm, and central moments μm ≡ 〈(n − n)m〉are related to each other by the simple binomial formulas [72,94] via the mean


number of the noncondensed atoms n = N − n0,

μr =r∑

k=0

(−1)k(r

k

)αr−knk, αr =

r∑k=0

(r

k

)μr−knk,

n = κ1,⟨(n− n)2

⟩ ≡ μ2 = κ2,⟨(n− n)3

⟩ ≡ μ3 = κ3,⟨(n− n)4

⟩ ≡ μ4 = κ4 + 3κ22 ,⟨(n− n)5

⟩ ≡ μ5 = κ5 + 10κ2κ3,

(219)⟨(n− n)6

⟩ ≡ μ6 = κ6 + 15κ2(κ4 + κ2

2

)+ 10κ23 , . . . .

Instead of calculation of the central moments, μm = 〈(n − n)m〉, it is moreconvenient, in particular so for the analysis of the non-Gaussian properties, tosolve for the cumulants κm, which are related to the moments by simple binomialexpressions. The first six are

κ1 = n, κ2 = μ2, κ3 = μ3, κ4 = μ4 − 3μ22,

(220)κ5 = μ5 − 10μ2μ3, κ6 = μ6 − 15μ2(μ4 − 2μ2

2

).

As discussed in detail below, the essence of the BEC fluctuations and the mostsimple formulas are given in terms of the “generating cumulants” κm which arerelated to the cumulants κm by the combinatorial formulas in Eq. (223),

κ1 = κ1, κ2 = κ2 + κ1, κ3 = κ3 + 3κ2 + κ1,

(221)κ4 = κ4 + 6κ3 + 7κ2 + κ1, . . . .

The main advantage of the cumulant analysis of the probability distributionρ(n) is the simple fact that the cumulant of a sum of independent stochasticvariables is equal to a sum of the partial cumulants, κr =∑k=0 κ

(k)r . This is a con-

sequence of the equalities logΘn(u) = log∏

k=0 Θnk(u) = ∑k=0 logΘnk(u).For each canonical-ensemble quasiparticle, the characteristic function can be eas-ily calculated from the equilibrium density matrix as follows:

Θnk(u) = Tr{eiunk ρk

} = Tr{eiunke−εknk/T

}(1 − e−εk/T

) = zk − 1

zk − z.

(222)

Here we introduced the exponential function of the single-particle energy spec-trum εk, namely zk = exp(εk/T ), and a variable z = exp(iu) which has thecharacter of a “fugacity”. As a result, we obtain an explicit formula for the char-acteristic function and all cumulants of the number of excited (and, according tothe equation n0 = N − n, condensed) atoms in the ideal Bose gas in an arbitrarytrap as follows:

logΘn(u) =∑k=0

log

(zk − 1

zk − z

)=

∞∑m=1

κm(eiu − 1)m

m! =∞∑r=1

κr(iu)r

r! ,


(223)κm = (m− 1)!∑k=0

(eεk/T − 1

)−m; κr =r∑

m=1

σ (m)r κm.

Here we use the Stirling numbers of the 2nd kind [72],

(224)σ (m)r = 1

m!m∑k=0

(−1)m−k(m

k

)kr ,

(ex − 1

)k = k!∞∑n=k

σ (k)n

xn

n! ,

that yield a simple expression for the cumulants κr via the generating cumu-lants κm. In particular, the first four cumulants are given in Eq. (221).

Thus, due to the standard relations (219), the result (223) yields all momentsof the condensate fluctuations. Except for the average value, all cumulants areindependent of the total number of atoms in the trap; they depend only on thetemperature and energy spectrum of the trap. This universal temperature depen-dence of the condensate fluctuations was observed and used in [42,44,46,54,57]to study the condensate fluctuations in the ideal Bose gas on the basis of the so-called Maxwell’s demon ensemble approximation. The method of the canonical-ensemble quasiparticles also agrees with and provides further justification for the“demon” approximation. The main point is that the statistics is determined bynumbers and fluctuations of the excited, noncondensed atoms which behave in-dependently of the total atom number N for temperatures well below the criticaltemperature since all “excess” atoms stay in the ground state of the trap. There-fore, one can calculate statistics in a formal limit as if we have an infinite numberof atoms in the condensate. That is we can say that the condensate plays the part ofan infinite reservoir for the excited atoms, in agreement with previous works [14,42,44,46,54,57].

Obviously, our approximation in Eq. (210) as well as the Maxwell’s demonensemble approximation does not describe all mesoscopic effects that can be im-portant very close to the critical temperature or for a very small number of atomsin the trap. However, it takes into account the effect of a finite size of a trap viathe discreteness of the energy levels εk, i.e., in this sense the approximation (210)describes not only the thermodynamic limit but also systems with a finite num-ber of atoms N . In addition, the mesoscopic effects can be partially taken intoaccount by a “grand” canonical approximation for the probability distribution ofthe canonical-ensemble quasiparticle occupation numbers

(225)ρk(nk) = exp(−nkεk/T )(1 − exp(−εk/T )

), εk = εk − μ,

where the chemical potential is related to the mean number of the condensedatoms n0 = 1/(1 − exp(−βμ)) and should be found self-consistently from thegrand-canonical equation

(226)N − n0 =∑k=0

(eεk/T − 1

)−1.


The canonical-ensemble quasiparticle result for all cumulants remains the sameas is given by Eq. (223) above, with the only difference that now all quasiparticleenergies are shifted by a negative chemical potential (εk = εk − μ),

κm = (m− 1)!∑k=0

(eεk/T − 1

)−m, m = 1, 2, . . . ;

(227)κr =r∑

m=1

σ (m)r κm.

The first, m = 1, equation in Eq. (227) is a self-consistency equation (226).The way it takes into account the mesoscopic effects (within this mean-number“grand” canonical approximation) is similar to the way in which the self-consistency equation (264) of the mean-field Popov approximation takes intoaccount the effects of weak atomic interaction. The results of this canonical-ensemble quasiparticle approach within the “grand” canonical approximation forthe quasiparticle occupations (225) were discussed in Section 3 for the case of theisotropic harmonic trap. Basically, the “grand” canonical approximation improvesonly the result for the mean number of condensed atoms n0(T ), but not for thefluctuations.

5.3. IDEAL GAS BEC STATISTICS IN ARBITRARY POWER-LAW TRAPS

The explicit formulas for the cumulants demonstrate that the BEC fluctuationsdepend universally and only on the single-particle energy spectrum of the trap, εk.There are three main parameters that enter this dependence, namely,

(a) the ratio of the energy gap between the ground level and the first excited levelin the trap to the temperature, ε1/T ,

(b) the exponent of the energy spectrum in the infrared limit, εk ∝ kσ at k → 0,and

(c) the dimension of the trap, d .

The result (223) allows to easily analyze the condensate fluctuations in a generalcase of a trap with an arbitrary dimension d � 1 of the space and with an arbitrarypower-law single-particle energy spectrum [46,55,73]

(228)εl = h

d∑j=1

ωj lσj , l = {lj ; j = 1, 2, . . . , d},

where lj � 0 is a nonnegative integer and σ > 0 is an index of the energyspectrum. The results for the particular traps with a trapping potential in the formof a box or harmonic potential well can be immediately deduced from the generalcase by setting the energy spectrum exponent to be equal to σ = 2 for a box and


σ = 1 for a harmonic trap. We assume that the eigenfrequencies of the trap areordered, 0 < ω1 � ω2 � · · · � ωd , so that the energy gap between the groundstate and the first excited state in the trap is ε1 = hω1. All cumulants (223) of thecondensate occupation fluctuations are given by the following formula:

κm = (m− 1)!∑

l≡(l1,...,ld )>0

[exp

(h

T

d∑j=1

ωj lσj

)− 1

]−m,

(229)κr =r∑

m=1

σ (m)r κm.

Let us consider first again the case of moderate temperatures larger than theenergy gap, ε1 � T < Tc. The first cumulant, i.e., the mean number of non-condensed atoms, can be calculated by means of a continuum approximation ofthe discrete sum by an integral if the space dimension of a trap is higher than acritical value, d > σ . Namely, one has an usual BEC phase transition with themean value

κ1 ≡ n ≡ N − n0 = Aζ

(d

σ

)T d/σ =

(T

Tc

)d/σN,

(230)d > σ, ε1 � T < Tc,

where the standard critical temperature is

(231)Tc =[

N

Aζ(d/σ)

]σ/d, A = [#( 1

σ+ 1)]d

(∏d

j=1 hωj )1/σ

.

The second-order generating cumulant can be calculated by means of this contin-uum approximation only if d > 2σ ,

κ2 = AT d/σ

(ζ

(d

σ− 1

)− ζ

(d

σ

))=(T

Tc

)d/σNζ( d

σ− 1)− ζ( d

σ)

ζ( dσ)

,

(232)d > 2σ.

In the opposite case it has to be calculated via a discrete sum because of a formalinfrared divergence of the integral. Keeping only the main term in the expansionof the exponent in Eq. (229), exp( h

T

∑dj=1 ωj l

σj )− 1 ≈ h

T

∑dj=1 ωj l

σj , we find

κ2 ={T

TcNσ/d

/[#

(1

σ+ 1

)]σ[ζ

(d

σ

)]σ/d}2

a(2)σ,d ,

(233)σ < d < 2σ,


where a(2)σ,d = ∑l>0(

∏dj=1 hωj )

2/d/ε2l . The ratio of the variance to the mean

number of noncondensed atoms is equal to√κ2/κ1 =

√κ−1

1 + κ2/κ21 , i.e.,√〈(n− n)2〉

n= N−1/2

(Tc

T

)d/(2σ)√ζ

(d

σ− 1

)/ζ

(d

σ

),

(234)d > 2σ,√〈(n− n)2〉n

=√√√√ 1

N

(Tc

T

)d/σ+N

2( σd−1)(Tc

T

)2( dσ −1)a(2)σ,d

[#

(1

σ+ 1

)]−2σ [ζ

(d

σ

)]−2σ/d,

(235)σ < d < 2σ.

We see that the traps with a relatively high dimension of the space, d > 2σ ,produce normal thermodynamic fluctuations (234) ∝ N−1/2 and behave simi-lar to the harmonic trap. However, the traps with a relatively low dimension ofthe space, σ < d < 2σ , produce anomalously large fluctuations (235) in thethermodynamic limit, ∝Nσ/d−1 � N−1/2 and behave similar to the box witha homogeneous Bose gas, where there is a formal infrared divergence in thecontinuum-approximation integral for the variance.

The third and higher-order central moments 〈(n− n)m〉, or the third and higher-order cumulants κm, provide further parameters to distinguish different traps withrespect to their fluctuation behavior. The mth-order generating cumulant can becalculated by means of the continuous approximation only if d > mσ ,

κm = AT d/σ

#( dσ)

∞∫0

tdσ−1

(et − 1)mdt =(T

Tc

)d/σN

#( dσ)ζ( d

σ)

∞∫0

tdσ−1

(et − 1)mdt,

(236)d > mσ.

In the opposite case we have to use a discrete sum because of a formal infrareddivergence of the integral. Again, keeping only the main term in the expansion ofthe exponent in Eq. (229), exp( h

T

∑dj=1 ωj l

σj )− 1 ≈ h

T

∑dj=1 ωj l

σj , we find

κm ={T

TcNσ/d

/[#

(1

σ+ 1

)]σ[ζ

(d

σ

)]σ/d}ma(m)σ,d ,

(237)σ < d < mσ,

where a(m)σ,d = ∑l>0(

∏dj=1 hωj )

m/d/εml . (For the sake of simplicity, as inEq. (248), we again omit here a discussion of an obvious logarithmic factor thatsuppresses the ultraviolet divergence in the latter sum

∑l>0 for the marginal case

d = mσ ; see, e.g., Eq. (250).)We conclude that all cumulants up to the order m < d/σ have normal behavior,

κm ∝ N , but for the higher orders, m > d/σ , they acquire an anomalous growthin the thermodynamic limit, κm � κm ∝ Nmσ/d . This result provides a simple


classification of the relative strengths of the higher-order fluctuation propertiesof the condensate in different traps. In particular, it makes it obvious that for allpower–law traps with 1 < d/σ < 2 the condensate fluctuations are not Gaussian,since

(238)κm

κm/22

∝ N0 → const = 0 at N → ∞, m � 3,

so that the asymmetry coefficient, γ1 ≡ 〈(n0 − n0)3〉/〈(n0 − n0)

2〉3/2 = 0, andthe excess coefficient, γ2 ≡ 〈(n0 − n0)

4〉/〈(n0 − n0)2〉2 − 3 = 0, are not zero.

Traps with d/σ > 2 show Gaussian condensate fluctuations, since all higher-order cumulant coefficients κm/κ

m/22 vanish, namely,

(239)κm

κm/22

∝ N1−m/2 → 0 at N → ∞ if 3 � m <d

σ,

(240)κm

κm/22

∝ Nm(σd− 1

2 ) → 0 at N → ∞ if m >d

σ.

(For the sake of simplicity, we omit here an analysis of the special cases whend/σ is an integer. It also can be done straightforwardly on the basis of the re-sult (229).) Very likely, a weak interaction also violates this nonrobust propertyand makes properties of the condensate fluctuations in the traps with a relativelyhigh dimension of the space, d/σ > 2, similar to that of the box with the homo-geneous Bose gas (see Section 6 below), as it is stated below for the particularcase of the harmonic traps.

For traps with a space dimension lower than the critical value, d < σ , it isknown that a BEC phase transition does not exist (see, e.g., [46]). Nevertheless,even in this case there still exists a well-peaked probability distribution ρ0(n0) atlow enough temperatures, so that the condition (210) is satisfied and our generalresult (223) describes this effect as well. In this case there is a formal infrareddivergence in the corresponding integrals for all cumulants (223), starting withthe mean value. Hence, all of them should be calculated as discrete sums. Formoderate temperatures we find approximately

κm � (m− 1)!(T

h

)m∑l>0

(d∑

j=1

ωj lσj

)−m

(241)∼ (m− 1)!d∑

j=1

(T

hωj

)m,


so that higher cumulants have larger values. In particular, the mean number ofnoncondensed atoms is of the same order as the variance,

κ1 ≡ n ≡ N − n0 � T

h

∑l>0

(d∑

j=1

ωj lσj

)−1

∼d∑

j=1

T

hωj

(242)∼√�n2

0 ∼

√√√√√ d∑j=1

(T

hωj

)2

, d < σ.

Therefore, until

n ∼√�n2

0 � N, i.e.,

(243)T � Tc = N/∑

l>0

(d∑

j=1

hωj lσj

)−1

∼ N/ d∑j=1

1

hωj,

there is a well-peaked condensate distribution,√�n2

0 � n0.The marginal case of a trap with the critical space dimension, d = σ , is also

described by our result (229), but we omit its discussion in the present paper.We mention only that there is also a formal infrared divergence, in this case alogarithmic divergence, and, at the same time, it is necessary to keep the wholeexponent in Eq. (229); because otherwise in an approximation like (241) thereappears an ultraviolet divergence. The physical result is that in such traps, e.g., ina one-dimensional harmonic trap, a quasicondensation of the ideal Bose gas takesplace at the critical temperature [43,46] Tc ∼ hω1N/ lnN .

For very low temperatures, T � ε1, the second and higher energy levels inthe trap are not thermally excited and atoms in the ideal Bose gas in any trapgo to the ground level with an exponential accuracy. This situation is also de-scribed by Eq. (229) that yields n0 � N and proves that all cumulants of thenumber-of-noncondensed-atom distribution become the same, since all higher-order generating cumulants exponentially vanish faster than κ1,

(244)κm � κ1 ≡ κ1, κm � (m− 1)!d∑

j=1

e−mhωj /T , T � ε1.

The conclusion is that for the ideal Bose gas in any trap the distribution of thenumber of noncondensed atoms becomes Poissonian at very low temperatures. Itmeans that the distribution of the number of condensed atoms is not Poissonian,but a “mirror” image of the Poisson’s distribution. We see, again, that the com-plementary number of noncondensed atoms, n = N − n0, is more convenient for


the characterization of the condensate statistics. A physical reason for this is thatthe noncondensed atoms in different excited levels fluctuate independently.

The formulas (229)–(244) for the power-law traps contain all correspondingformulas for a box (d = 3, σ = 2) and for a harmonic trap (d = 3, σ = 1) in a3-dimensional space as particular cases. It is worth to stress that BEC fluctuationsin the ideal gas for the latter two cases are very different. In the box, if the temper-ature is larger than the trap energy gap, ε1 � T < Tc, all cumulants, starting withthe variance, m � 2, are anomalously large and dominated by the lowest energymodes, i.e., formally infrared divergent,

(245)κm ≈ κm ∝ (T /Tc)mN2m/3, m � 2.

Only the mean number of condensed atoms,

n0 = N − n = N − κ1 = N(1 − (T /Tc)

3/2),(246)Tc = 2πh2

m

(N

L3ζ(3/2)

)2/3

,

can be calculated correctly via replacement of the discrete sum in Eq. (223) by anintegral

∫∞0 . . . d3k. The correct value of the squared variance,

κ2 ≡ ⟨(n0 − n0)2⟩ = N4/3

(T

Tc

)2s4

π2(ζ(3/2))4/3,

(247)s4 =∑l=0

1

l4= 16.53,

can be calculated from Eq. (223) only as a discrete sum. Thus, for the box the con-densate fluctuations are anomalous and non-Gaussian even in the thermodynamiclimit. To the contrary, for the harmonic trap with temperature much larger thanthe energy gap, ε1 � T < Tc, the condensate fluctuations are Gaussian in thethermodynamic limit. This is because, contrary to the case of the homogeneousgas, in the harmonic trap only the third and higher-order cumulants, m � 3, arelowest-energy-mode dominated, i.e., formally infrared divergent,

κ3 ≈ κ3 = 2∑l>0

(ehωl/T − 1

)−3,

(248)κm ≈ κm ≈ (m− 1)!(T

h

)m∑l>0

(ωl)−m ∝ Nm/3, m > 3,

and they are small compared with an appropriate power of the variance squared

(249)κ2 ≡ ⟨(n0 − n0)2⟩ = ζ(2)

ζ(3)

(T

Tc

)3

N.


The asymmetry coefficient behaves as

(250)γ1 ≡ κ3

κ3/22

≡ 〈(n0 − n0)3〉

〈(n0 − n0)2〉3/2∝ logN

N1/2→ 0,

and all higher normalized cumulants (m � 3) vanish in the thermodynamic limit,N → ∞, as follows:

(251)κm

κm/22

∝ 1

Nm/6→ 0.

It is important to realize that a weak interaction violates this nonrobust propertyand makes properties of the condensate fluctuations in a harmonic trap similar tothat of the homogeneous gas in the box (see Section 6 below). For the variance,the last fact was first pointed out in [78].

5.4. EQUIVALENT FORMULATION IN TERMS OF THE POLES

OF THE GENERALIZED ZETA FUNCTION

Cumulants of the BEC fluctuations in the ideal Bose gas, Eq. (223), can be writ-ten in an equivalent form which is quite interesting mathematically (see [95]and references therein). Namely, starting with the cumulant generating functionlnΞex(β, z), where β = 1/kBT and z = eβμ,

lnΞex(β, z) = −∞∑ν=1

ln(1 − z exp

[−β(εν − ε0)])

(252)=∞∑ν=1

∞∑n=1

zn exp[−nβ(εν − ε0)]n

,

we use the Mellin–Barnes transform

e−a = 1

2πi

τ+i∞∫τ−i∞

dt a−t#(t)

to write

lnΞex(β, z) =∞∑ν=1

∞∑n=1

zn

n

1

2πi

τ+i∞∫τ−i∞

dt #(t)1

[nβ(εν − ε0)]t

(253)= 1

2πi

τ+i∞∫τ−i∞

dt #(t)

∞∑ν=1

1

[β(εν − ε0)]t∞∑n=1

zn

nt+1.


Recalling the series representation of the Bose functions

gα(z) =∞∑n=1

zn/nα

and introducing the generalized, “spectral” Zeta function

Z(β, t) = [β(εν − ε0)]−t

,

we arrive at the convenient (and exact) integral representation

(254)lnΞex(β, z) = 1

2πi

τ+i∞∫τ−i∞

dt #(t)Z(β, t)gt+1(z).

Utilizing the well-known relations z ddzgα(z) = gα−1(z) and gα(1) = ζ(α),

where ζ(α) denotes the original Riemann Zeta function, Eq. (254) may be writtenin the appealing compact formula

κk(β) =(z∂

∂z

)klnΞex(β, z)

∣∣∣∣z=1

(255)= 1

2πi

τ+i∞∫τ−i∞

dt #(t)Z(β, t)ζ(t + 1 − k).

Thus, by means of the residue theorem, Eq. (255) links all cumulants of thecanonical distribution in the condensate regime to the poles of the generalizedZeta function Z(β, t), which embodies all the system’s properties, and to the poleof a system-independent Riemann Zeta function, the location of which depends onthe order k of the respective cumulant. The formula (255) provides a systematicasymptotic expansion of the cumulants κk(β) through the residues of the analyti-cally continued integrands, taken from right to left. The large-system behavior isextracted from the leading pole, finite-size corrections are encoded in the next-to-leading poles, and the non-Gaussian nature of the condensate fluctuations isdefinitely seen. The details and examples of such analysis can be found in [95].

Concluding the Section 5, it is worthwhile to mention that previously only firsttwo moments, κ1 and κ2, were analyzed for the ideal gas [13,46,53,57,73–77], andthe known results coincide with ours. Our explicit formulas provide a completeanswer to the problem of all higher moments of the condensate fluctuations in theideal gas. The canonical-ensemble quasiparticle approach, taken together with themaster equation approach gives a fairly complete picture of the central moments.

For the interacting Bose gas this problem becomes much more involved. Weaddress it in the next, last part of this review within a simple approximation thattakes into account one of the main effects of the interaction, namely, the Bogoli-ubov coupling.


6. Why Condensate Fluctuations in the Interacting Bose Gasare Anomalously Large, Non-Gaussian, and Governed by

Universal Infrared Singularities?

In this section, following the Refs. [20,21], the analytical formulas for the sta-tistics, in particular, for the characteristic function and all cumulants, of theBose–Einstein condensate in the dilute, weakly interacting gases in the canoni-cal ensemble are derived using the canonical-ensemble quasiparticle method. Weprove that the ground-state occupation statistics is not Gaussian even in the ther-modynamic limit. We calculate the effect of Bogoliubov coupling on suppressionof ground-state occupation fluctuations and show that they are governed by a pair-correlation, squeezing mechanism.

It is shown that the result of Giorgini, Pitaevskii and Stringari (GPS) [78] forthe variance of condensate fluctuations is correct, and the criticism of Idziaszekand others [96,97] is incorrect. A crossover between the interacting and idealBose gases is described. In particular, it is demonstrated that the squared varianceof the condensate fluctuations for the interacting Bose gas, Eq. (271), tends to ahalf of that for the ideal Bose gas, Eq. (247), because the atoms are coupled instrongly correlated pairs such that the number of independent degrees of freedomcontributing to the fluctuations of the total number of excited atoms is only 1/2 theatom number N . This pair correlation mechanism is a consequence of two-modesqueezing due to Bogoliubov coupling between k and −k modes. Hence, the factthat the fluctuation in the interacting Bose gas is 1/2 of that in the ideal Bose gasis not an accident, contrary to the conclusion of GPS. Thus, there is a deep (notaccidental) parallel between the fluctuations of ideal and interacting bosons.

Finally, physics and universality of the anomalies and infrared singularities ofthe order parameter fluctuations for different systems with a long range order be-low a critical temperature of a second-order phase transition, including stronglyinteracting superfluids and ferromagnets, is discussed. In particular, an effectivenonlinear σ model for the systems with a broken continuous symmetry is out-lined and the crucial role of the Goldstone modes fluctuations combined with aninevitable geometrical coupling between longitudinal and transverse order para-meter fluctuations and susceptibilities in the constrained systems is demonstrated.

6.1. CANONICAL-ENSEMBLE QUASIPARTICLES

IN THE ATOM-NUMBER-CONSERVING BOGOLIUBOV

APPROXIMATION

We consider a dilute homogeneous Bose gas with a weak interatomic scatteringdescribed by the well-known Hamiltonian [30]


(256)H =∑

k

h2k2

2Ma+k ak + 1

2V

∑{ki }

〈k3k4|U |k1k2〉a+k4a+k3

ak2 ak1,

where V = L3 is a volume of a box confining the gas with periodic boundaryconditions. The main effect of the weak interaction is the Bogoliubov couplingbetween bare canonical-ensemble quasiparticles, βk = β+

0 ak, via the condensate.It may be described, to a first approximation, by a quadratic part of the Hamil-tonian (256), i.e., by the atom-number-conserving Bogoliubov Hamiltonian [92]

HB = N(N − 1)U0

2V+∑k=0

(h2k2

2M+ (n0 + 1/2)Uk

V

)β+

k βk

(257)+ 1

2V

∑k=0

(Uk

√(1 + n0)(2 + n0) β

+k β

+−k + h.c.

),

where we will make an approximation n0 � n0 � 1, which is consistent with ourmain assumption (210) of the existence of a well-peaked condensate distributionfunction. Then, the Bogoliubov canonical transformation,

βk = ukbk + vkb+−k; uk = 1√

1 − A2k

, vk = Ak√1 − A2

k

,

(258)β+k = ukb

+k + vkb−k; Ak = V

n0Uk

(εk − h2k2

2M− n0Uk

V

),

describes the condensate canonical-ensemble quasiparticles which have a “gap-less” Bogoliubov energy spectrum and fluctuate independently in the approxima-tion (257), since

HB = E0 +∑k=0

εkb+k bk,

(259)εk =√(

h2k2

2M+ n0Uk

V

)2

−(n0Uk

V

)2

.

In other words, we again have an ideal Bose gas although now it consists ofthe dressed quasiparticles which are different both from the atoms and bare(canonical-ensemble) quasiparticles introduced in Section 5. Hence, the analysisof fluctuations can be carried out in a similar fashion to the case of the nonin-teracting, ideal Bose gas of atoms. This results in a physically transparent andanalytical theory of BEC fluctuations that was suggested and developed in [20,21].


The only difference with the ideal gas is that now the equilibrium density ma-trix,

(260)ρ =∏k=0

ρk, ρk = e−εkb+k bk/T(1 − e−εk/T

),

is not diagonal in the bare atomic occupation numbers, the statistics of which weare going to calculate. This feature results in the well-known quantum optics ef-fect of squeezing of the fluctuations. The number of atoms with coupled momentak and −k is determined by the Bogoliubov coupling coefficients according to thefollowing equation:

a+k ak + a+−ka−k = β+k βk + β+

−kβ−k

= (u2k + v2

k

)(b+k bk + b+−kb−k

)(261)+ 2ukvk

(b+k b

+−k + bkb−k

)+ 2v2k.

6.2. CHARACTERISTIC FUNCTION AND ALL CUMULANTS OF BECFLUCTUATIONS

The characteristic function for the total number of atoms in the two, k and −k,modes squeezed by Bogoliubov coupling is calculated in [20,21] as

Θ±k(u) ≡ Tr(eiu(β

+k βk+β+

−kβ−k)e−εk(b+k bk+b+−kb−k)/T

(1 − e−εk/T

)2 )= (z(Ak)− 1)(z(−Ak)− 1)

(z(Ak)− eiu)(z(−Ak)− eiu),

(262)z(Ak) = Ak − eεk/T

Akeεk/T − 1.

The term “squeezing” originates from the studies of a noise reduction in quantumoptics (see the discussion after Eq. (278)).

The characteristic function for the distribution of the total number of the ex-cited atoms is equal to the product of the coupled-mode characteristic functions,Θn(u) =∏k=0,mod{±k}Θ±k(u), since different pairs of (k,−k)-modes are inde-pendent to the first approximation (257). The product

∏runs over all different

pairs of (k,−k)-modes.It is worth noting that by doing all calculations via the canonical-ensemble

quasiparticles (Section 5) we automatically take into account all correlations in-troduced by the canonical-ensemble constraint. As a result, similar to the ideal gas(Eq. (223)), we obtain the explicit formula for all cumulants in the dilute weaklyinteracting Bose gas,

κm = 1

2(m− 1)!

∑k=0

[1

(z(Ak)− 1)m+ 1

(z(−Ak)− 1)m

],


(263)κr =r∑

m=1

σ (m)r κm.

In comparison with the ideal Bose gas, Eq. (223), we have effectively a mixtureof two species of atom pairs with z(±Ak) instead of exp(εk/T ).

It is important to emphasize that the first equation in (263), m = 1, is a nonlin-ear self-consistency equation,

(264)N − n0 = κ1(n0) ≡∑k=0

1 + A2ke

εk/T

(1 − A2k)(e

εk/T − 1),

to be solved for the mean number of ground-state atoms n0(T ), since the Bo-goliubov coupling coefficient (258), Ak, and the energy spectrum (259), εk, arethemselves functions of the mean value n0. Then, all the other equations in (263),m � 2, are nothing else but explicit expressions for all cumulants, m � 2, ifone substitutes the solution of the self-consistency equation (264) for the meanvalue n0. The Eq. (264), obtained here for the interacting Bose gas (257) inthe canonical-ensemble quasiparticle approach, coincides precisely with the self-consistency equation for the grand-canonical dilute gas in the so-called first-orderPopov approximation (see a review in [36]). The latter is well established as a rea-sonable first approximation for the analysis of the finite-temperature properties ofthe dilute Bose gas and is not valid only in a very small interval near Tc, givenby Tc − T < a(N/V )1/3Tc � Tc, where a = MU0/4πh2 is a usual s-wavescattering length. The analysis of the Eq. (264) shows that in the dilute gas theself-consistent value n0(T ) is close to that given by the ideal gas model, Eq. (246),and for very low temperatures goes smoothly to the value given by the standardBogoliubov theory [30,33,93] for a small condensate depletion, N − n0 � N .This is illustrated by Fig. 14a. (Of course, near the critical temperature Tc thenumber of excited quasiparticles is relatively large, so that along with the Bogoli-ubov coupling (257) other, higher-order effects of interaction should be taken intoaccount to get a complete theory.) Note that the effect of a weak interaction onthe condensate fluctuations is very significant (see Fig. 14b–d) even if the meannumber of condensed atoms changes by relatively small amount.

6.3. SURPRISES: BEC FLUCTUATIONS ARE ANOMALOUSLY LARGE AND

NON-GAUSSIAN EVEN IN THE THERMODYNAMIC LIMIT

According to the standard textbooks on statistical physics, e.g., [30,61,77,98],any extensive variable C of a thermodynamic system has vanishing relative root-mean-square fluctuations. Namely, in the thermodynamic limit, a relative squaredvariance 〈(C − C)2〉/C2 = 〈C2〉/〈C〉2 − 1 ∝ V −1 goes inversely proportional


FIG. 14. Temperature scaling of the first four cumulants, the mean value n0/N = N − κ1/N ,

the variance√κ2/N = 〈(n0 − n0)

2〉1/2/N1/2, the third central moment −κ1/33 /N1/2 =

〈(n0 − n0)3〉1/3/N1/2, the fourth cumulant |κ4|1/4/N1/2 = |〈(n0 − n0)

4〉 − 3κ22 |/N2, of the

ground-state occupation fluctuations for the dilute weakly interacting Bose gas (Eq. (263)), withU0N

1/3/ε1V = 0.05 (thick solid lines), as compared with Eq. (223) (thin solid lines) and with theexact recursion relation (80) (dot-dashed lines) for the ideal gas in the box; N = 1000. For the idealgas our results (thin solid lines) are almost indistinguishable from the exact recursion calculations(dot-dashed lines) in the condensed region, T < Tc(N). Temperature is normalized by the standardthermodynamic-limit critical value Tc(N = ∞) that differs from the finite-size value Tc(N), as isclearly seen in graphs.

to the system volume V , or total number of particles N . This fundamental prop-erty originates from the presence of a finite correlation length ξ that allows usto partition a large system into an extensive number V/ξ3 of statistically inde-pendent subvolumes, with a finite variance in each subvolume. As a result, thecentral limit theorem of probability theory yields a Gaussian distribution for the

variable C with a standard scaling for variance,√〈(C − C)2〉 ∼ V 1/2. Possible

deviations from this general rule are especially interesting. It turns out that BEC


in a trap is one of the examples of such peculiar systems. A physical reason for theanomalously large and non-Gaussian BEC fluctuations is the existence of the longrange order below the critical temperature of the second-order phase transition.

Let us show in detail that the result (263) implies, similar to the case of theideal homogeneous gas (Section 5), that the ground-state occupation fluctuationsin the weakly interacting gas are not Gaussian in the thermodynamic limit, andanomalously large. The main fact is that the anomalous contribution to the BECfluctuation cumulants comes from the modes which have the most negative Bo-goliubov coupling coefficient Ak ≈ −1 since in this case one has z(Ak) → 1, sothat this function produces a singularity in the first term in Eq. (263). (The secondterm in Eq. (263) never makes a singular contribution since always z(−Ak) < −1or z(−Ak) > exp(εk/T ).) These modes, with Ak ≈ −1, exist only if the inter-action energy g = n0Uk/V in the Bogoliubov Hamiltonian (257) is larger thanthe energy gap between the ground state and the first excited states in the trap,ε1 = ( 2πh

L)2 1

2M , i.e.,

(265)g

ε1≡ 2an0

πL� 1,

and are the infrared modes with the longest wavelength of the trap energy spec-trum. In terms of the scattering length for atom–atom collisions, a = MUk/4πh2,the latter condition (265) coincides with a familiar condition for the Thomas–Fermi regime, in which the interaction energy is much larger then the atom’skinetic energy.

Let us use a representation which is obvious from Eqs. (258) and (259),

(266)z(Ak)− 1 ≡ 2εk

g(1 − Ak)

[1 + Ak(1 − Ak)

1 + eεk/T

]≈ εk

gtanh

(εk

2T

),

where in the last, approximate equality we set Ak ≈ −1, and neglect the contri-bution from the second term in Eq. (263), assuming that the singular contributionfrom the modes with Ak ≈ −1 via the first term in Eq. (263) is dominant. Then,for all infrared-dominated cumulants of higher orders m � 2, the result (263)reduces to a very transparent form

(267)κm ≈ 1

2(m− 1)!

∑k=0

1

[ εkg

tanh( εk2T )]m

.

Finally, using the Bogoliubov energy spectrum in Eq. (259),

εk = ε1

√l4 + (2g/ε1)l2

with a set of integers l = (lx, ly, lz), we arrive to the following simple formulasfor the higher-order generating cumulants in the Thomas–Fermi regime (265):

(268)κm ≈ 1

2(m− 1)!

(T

ε1

)m∑l=0

1

l2m, g � 2T 2/ε1; ε1 � g � T ,


κm ≈ 1

2(m− 1)!

(g

2ε1

)m/2∑l=0

1

[|l| tanh(|l|√2gε1/T )]m ,

(269)2T 2/ε1 � g � T ; g � ε1,

(270)κm ≈ 1

2(m− 1)!

(g

2ε1

)m/2∑l=0

1

|l|m , g � 2T 2/ε1; g � ε1,

for high, moderate, and very low temperatures T , respectively, as compared to thegeometrical mean of the interaction and gap energies in the trap,

√gε1/2.

In particular, for the Thomas–Fermi regime (265) and relatively high tempera-tures, the squared variance, as given by Eq. (268),⟨

(n0 − n0)2⟩ = 1

2

∑k=0

[1

(z(Ak)− 1)2+ 1

(z(−Ak)− 1)2+ 1

z(Ak)− 1

+ 1

z(−Ak)− 1

](271)→ N4/3(T /Tc)

2s4

2π2(ζ(3/2))4/3

scales as 〈(n0 − n0)2〉 ∝ N4/3. Here the arrow indicates the limit of sufficiently

strong interaction, g � ε1 and T � g � ε1.The behavior (271), (245), and (275) is essentially different from that of the nor-

mal fluctuations of most extensive physical observables, which are Gaussian withthe squared variance proportional to N . The only exception for the Eqs. (267)–(270) is the low temperature limit of the variance that is not infrared-dominatedand should be calculated not via the Eq. (270), but directly from Eq. (263) us-ing the fact that all modes are very poorly occupied at low temperatures, i.e.,exp(−εk/T ) � 1, if g � 2T 2/ε1, ε1. Thus, we immediately find

κ2 = κ2 + κ1 ≈∑k=0

2A2k

(1 − A2k)

2= 1

2

∑k=0

g2

ε2k

≈ 2πg2

ε21

∞∫0

dr

r2 + 2g/ε1

(272)= π2

√2

(g

ε1

)3/2

= 2√π

(an0

L

)3/2

.

The above results extend and confirm the result of the pioneering paper [78]where only the second moment, 〈(n0 − n0)

2〉, was calculated. (The result of [78]was rederived by a different way in [99], and generalized in [100].) The higher-order cumulants κm, m > 2, given by Eqs. (263) and (267)–(270), are not zero,do not go to zero in the thermodynamic limit and, moreover, are relatively largecompared with the corresponding exponent of the variance (κm)

m/2 that provesand measures the non-Gaussian character of the condensate fluctuations. For the


Thomas–Fermi regime (265) and relatively high temperatures, the relative valuesof the higher-order cumulants are given by Eq. (268) as

(273)κm

(κ2)m/2≈ 2m/2−1(m− 1)! s2m

(s4)m/2, s2m =

∑l=0

1

l2m,

where l = (lx, ly, lz) are integers, s4 ≈ 16.53, s6 ≈ 8.40, and s2m ≈ 6 for m � 1.In particular, the asymmetry coefficient of the ground-state occupation probabilitydistribution

γ1 = ⟨(n0 − n0)3⟩/⟨(n0 − n0)

2⟩3/2

(274)= −κ3/κ3/22 ≈ 2

√2 s6/(s4)

3/2 ≈ 0.35

is not very small at all.This non-Gaussian statistics stems from an infrared singularity that exists for

the fluctuation cumulants κm, m � 2, for the weakly interacting gas in a boxdespite of the acoustic (i.e., linear, like for the ideal gas in the harmonic trap) be-havior of the Bogoliubov–Popov energy spectrum (259) in the infrared limit. Thereason is that the excited mode squeezing (i.e., linear mixing of atomic modes)via Bogoliubov coupling affects the BEC statistics (263) for the interacting gasalso directly (not only via a modification of the quasiparticle energy spectrum),namely, via the function (262), z(Ak), which is different from a simple exponentof the bare energy, exp(εk/T ), that enters into the corresponding formula for thenoninteracting gas (223).

6.4. CROSSOVER BETWEEN IDEAL AND INTERACTION-DOMINATED BEC:QUASIPARTICLES SQUEEZING AND PAIR CORRELATION

Now we can use the analytical formula (263) to describe explicitly a crossoverbetween the ideal-gas and interaction-dominated regimes of the BEC fluctuations.Obviously, if the interaction energy g = n0Uk/V is less than the energy gapbetween the ground state and the first excited state in the trap, g < ε1, the Bogo-liubov coupling (258) becomes small for all modes, |Ak| � 1, so that both termsin Eq. (263) give similar contributions and all fluctuation cumulants κm tend totheir ideal gas values in the limit of vanishing interaction, g � ε1. Namely, in thenear-ideal gas regime n0a/L � 1 the squared variance linearly decreases fromits ideal-gas value with an increase of the weak interaction as follows:⟨

(n0 − n0)2⟩ ≈ N4/3(T /Tc)

2s4

π2(ζ(3/2))4/3

[1 − π2 s6

s4

g

ε1

](275)= N4/3(T /Tc)

2s4

π2(ζ(3/2))4/3

[1 − 3.19

n0a

L

].


With a further increase of the interaction energy over the energy gap in the trap,g > ε1, the essential differences between the weakly interacting and ideal gasesappear. First, the energy gap is increased by the interaction, that is

(276)ε1 =√ε2

1 + 2ε1n0(T )U0/V > ε1 =(

2πh

L

)2 1

2M,

so that the border T ∼ ε1 between the moderate temperature and very low tem-perature regimes is shifted to a higher temperature, T ∼ √

gε1/2. Thus, theinteraction strength g determines also the temperature T ∼ √

gε1/2 above whichanother important effect of the weak interaction comes into play. Namely, accord-ing to Eqs. (267)–(270), the suppression of all condensate-fluctuation cumulantsby a factor of 1/2, compared with the ideal gas values (see Eq. (247)), takes placefor moderate temperatures, ε1 � T < Tc, when a strong coupling (Ak ≈ −1)contribution dominates in Eq. (263). The factor 1/2 comes from the fact that,according to Eq. (262),

z(Ak = −1) = 1, z(Ak = 1) = −1,

so that the first term in Eq. (263) is resonantly large but the second term is rela-tively small. In this case, the effective energy spectrum, which can be introducedfor the purpose of comparison with the ideal gas formula (223), is

(277)εeff

k = T ln(z(Ak)) � 1

2(1 + Ak)εk � 1

2ε2

kV/U0n0 � k2

2M.

That is, the occupation of a pair of strongly coupled modes in the weakly interact-ing gas can be characterized by the same effective energy spectrum as that of a freeatom. It is necessary to emphasize that the effective energy ε

eff

k = T ln(z(Ak)),introduced in Eq. (277), describes only the occupation of a pair of bare atomexcitations k and −k (see Eqs. (260)–(262)) and, according to Eq. (263), theground-state occupation. That is, it would be wrong to reduce the analysis ofthe thermodynamics and, in particular, the entropy of the interacting gas to thiseffective energy. Thermodynamics is determined by the original energy spectrumof the dressed canonical-ensemble quasiparticles, Eq. (259).

This remarkable property explains why the ground-state occupation fluctua-tions in the interacting gas in this case are anomalously large to the same extentas in the noninteracting gas except factor of 1/2 suppression in the cumulants ofall orders. These facts were considered in [78] to be an accidental coincidence.We see now that, roughly speaking, this is so because the atoms are coupled instrongly correlated pairs such that the number of independent stochastic occupa-tion variables (“degrees of freedom”) contributing to the fluctuations of the totalnumber of excited atoms is only 1/2 the atom number N . This strong pair corre-lation effect is clearly seen in the probability distribution of the total number of


FIG. 15. The probability distribution (278) P2 as a function of the number of atoms in k and −kmodes, nk + n−k, for the interaction energy U0n0/V = 103εk and temperature T = εk. The paircorrelation effect due to Bogoliubov coupling in the weakly interacting Bose gas is clearly seen forlow occupation numbers, i.e., even occupation numbers nk + n−k are more probable than odd ones.

atoms in the two coupled k and −k modes,

P2(nk + n−k) = [z(Ak)− 1][z(−Ak)− 1]z(−Ak)− z(Ak)

[(1

z(Ak)

)nk+n−k+1

(278)−(

1

z(−Ak)

)nk+n−k+1 ](see Fig. 15). The latter formula follows from Eq. (262). Obviously, a higher prob-ability for even occupation numbers nk + n−k as compared to odd numbers atlow occupations means that the atoms in the k and −k modes have a tendencyto appear or disappear simultaneously, i.e., in pairs. This is a particular featureof the well-studied in quantum optics phenomenon of two-mode squeezing (see,e.g., [18,37]). This squeezing means a reduction in the fluctuations of the popula-tion difference nk − n−k, and of the relative phases or so-called quadrature-phaseamplitudes of an interacting state of two bare modes ak and a−k compared withtheir appropriate uncoupled state, e.g., coherent or vacuum state. The squeezingis due to the quantum correlations which build up in the bare excited modes viaBogoliubov coupling (258) and is very similar to the noise squeezing in a nonde-generate parametric amplifier studied in great details by many authors in quantumoptics in 80s [18,37]. We note that fluctuations of individual bare excited modesare not squeezed, but there is a high degree of correlation between occupationnumbers in each mode.

It is very likely that in the general case of an arbitrary power-law trap the inter-action also results in anomalously large fluctuations of the number of ground-state


atoms, and a formal infrared divergence due to excited mode squeezing via Bo-goliubov coupling and renormalization of the energy spectrum. In the particularcase of the isotropic harmonic trap this was demonstrated in [78] for the varianceof the condensate fluctuations. Therefore, the ideal gas model for traps with alow spectral index σ < d/2 (such as a three-dimensional harmonic trap whereσ = 1 < d/2 = 3/2), showing Gaussian, normal thermodynamic condensatefluctuations with the squared variance proportional to N instead of anomalouslylarge fluctuations (see Eqs. (232), (234) and (239), (240)), is not robust with re-spect to the introduction of a weak interatomic interaction.

At the same time, the ideal gas model for traps with a high spectral index σ >

d/2 (e.g., for a three-dimensional box with σ = 2 > d/2 = 3/2) exhibits non-Gaussian, anomalously large ground-state occupation fluctuations with a squaredvariance proportional toN2σ/d � N (see Eqs. (233), (234)) similar to those foundfor the interacting gas. Fluctuations in the ideal Bose gas and in the BogoliubovBose gas differ by a factor of the order of 1, which, of course, depends on thetrap potential and is equal to 1/2 in the particular case of the box, where �n2

0 ∝N4/3. We conclude that, contrary to the interpretation formulated in [78], similarbehavior of the condensate fluctuations in the ideal and interacting Bose gases ina box is not accidental, but is a general rule for all traps with a high spectral indexσ > d/2, or a relatively low dimension of space, d < 2σ .

As follows from Eq. (263), the interaction essentially modifies the condensatefluctuations also at very low temperatures, T � ε1 (see Fig. 14). Namely, in theinteracting Bose gas a temperature-independent quantum noise,

(279)κm → κm(T = 0) = 0, m � 2,

additional to vanishing (at T → 0) in the ideal Bose gas noise, appears due toquantum fluctuations of the excited atoms, which are forced by the interaction tooccupy the excited levels even at T = 0, so that nk(T = 0) = 0. Thus, in thelimit of very low temperatures the results of the ideal gas model (Section 5) areessentially modified by weak interaction and do not describe condensate statisticsin the realistic weakly interacting Bose gases.

The temperature scaling of the condensate fluctuations described above is de-picted in Fig. 14 both for the weakly interacting and ideal gases. A comparisonwith the corresponding quantities calculated numerically from the exact recur-sion relation in Eqs. (79) and (80) for the ideal gas in a box is also indicated. Itis in good agreement with our approximate analytical formula (263) for all tem-peratures in the condensed phase, T < Tc, except of a region near to the criticaltemperature, T ≈ Tc. It is worth stressing that the large deviations of the asymme-try coefficient, γ1 = 〈(n0 − n0)

3〉/〈(n0 − n0)2〉3/2, and of the excess coefficient,

γ2 = 〈(n0 − n0)4〉/〈(n0 − n0)

2〉2 − 3 from zero, which are of the order of 1 atT ∼ Tc/2 or even more at T ≈ 0 and T ≈ Tc, indicate how far the ground-stateoccupation fluctuations are from being Gaussian. (In the theory of turbulence,


the coefficients γ1 and γ2 are named as skewness and flatness, respectively.) Thisessentially non-Gaussian behavior of the ground-state occupation fluctuations re-mains even in the thermodynamic limit.

Mesoscopic effects near the critical temperature are also clearly seen in Fig. 14,and for the ideal gas are taken into account exactly by the recursion relations (79),(80). The analytical formulas (263) take them into account only via a finite-sizeeffect, L ∝ N−1/3, of the discreteness of the single-particle spectrum εk. Thisfinite-size (discreteness) effect produces, in particular, some shift of the character-istic BEC critical temperature compared to its thermodynamic-limit value, Tc. Forthe case shown in Fig. 14, it is increased by a few per cent. Similarly to the mean-number “grand” canonical approximation described at the end of Section 3.2 forthe case of the ideal gas, the canonical-ensemble quasiparticle approach can par-tially accommodates for the mesoscopic effects by means of the grand-canonicalshift of all quasiparticle energies by a chemical potential, εk = εk−μ. In this casethe self-consistency equation (264) acquires an additional nonlinear contributiondue to the relation exp(−βμ) = 1 + 1/n0. Obviously, this “grand” canonical ap-proximation works only for T < Tc, whereas at T > Tc we can use the standardgrand canonical approach, since the ground-state occupation is not macroscopicabove the critical temperature. However, the “grand” canonical approach takescare only of the mean number of condensed atoms n0, and does not improve theresults of the canonical-ensemble quasiparticle approach for BEC fluctuations.

6.5. UNIVERSAL ANOMALIES AND INFRARED SINGULARITIES

OF THE ORDER PARAMETER FLUCTUATIONS IN THE SYSTEMS

WITH A BROKEN CONTINUOUS SYMMETRY

The result that there are singularities in the central moments of the condensatefluctuations, emphasized in [20,21] and discussed above in detail for the BEC ina trap, can be generalized for other long-range ordered systems below the criti-cal temperature of a second-order phase transition, including strongly interactingsystems. That universality of the infrared singularities was discussed in [99,100]and can be traced back to a well-known property of an infrared singularity ina longitudinal susceptibility χ‖(k) of such systems [30,79,101]. The physics ofthese phenomena is essentially determined by long wavelength phase fluctua-tions, which describe the noncondensate statistics, and is intimately related to thefluctuation–dissipation theorem, Bogoliubov’s 1/k2 theorem for the static sus-ceptibility χϕϕ(k) of superfluids, and the presence of Goldstone modes, as will bedetailed in the following.


6.5.1. Long Wavelength Phase Fluctuations, Fluctuation–Dissipation Theorem,and Bogoliubov’s 1/k2 Theorem

First, let us refer to a microscopic derivation given in [102] that demonstrates thefact that the phase fluctuations alone dominate the low-energy physics. Also, letus assume that, as was shown by Feynman, the spectrum of excitations in theinfrared limit k → 0 is exhausted by phonon-like modes with a linear dispersionωk = ck, where c is an actual velocity of sound. Then, following a textbook [30],we can approximate an atomic field operator via a phase fluctuation operator asfollows:

Ψ (x) =√n0 e

iϕ(x),

(280)ϕ(x) = (mc/2V ntot)1/2∑k=0

(hk)−1/2(ckeikx + c+k e

−ikx),where n0 is the bare condensate density, ntot the mean particle density, m the par-ticle mass, ck and c+k are phonon annihilation and creation operators, respectively.An omission of the k = 0 term in the sum in Eq. (280) can be rigorously justi-fied on the basis of the canonical-ensemble quasiparticle approach (see Section 5)and is related to the fact that a global phase factor is irrelevant to any observablegauge-invariant quantity. In a homogeneous superfluid, the renormalized conden-sate density is determined by the long range order parameter,

(281)n0 = lim�x→∞

⟨Ψ+(�x)Ψ (0)

⟩ ≈ n0 exp(−⟨ϕ2(0)

⟩),

if we use the approximation (280) and neglect phonon interaction. Thus, the meannumber of condensate particles is depleted with an increase of temperature inaccordance with an increase of the variance of the phase fluctuations, n0(T ) ∝exp(−〈ϕ2(0)〉T ), which yields a standard formula [30] for the thermal depletion,n0(T )− n0(T = 0) = −n0(T = 0)m(kBT )

2/12ntotch3.

Similar to the mean value, higher moments of the condensate fluctuations of thehomogeneous superfluid in the canonical ensemble can be represented in terms ofthe phase fluctuations of the noncondensate via the operator

(282)n = N/V − n0 =∫

�Ψ+(x)�Ψ (x) d3x,

where

(283)�Ψ (x) = Ψ (x)−√n0(T ) =√n0(eiϕ(x)− e−〈ϕ2(0)〉/2).

In particular, the variance of the condensate occupation is determined by the cor-relation function of the phase fluctuations as follows [99]:

(284)⟨(n0 − n0)

2⟩ ≈ 2n20e

−2〈ϕ2(0)〉∫ ⟨

ϕ(x)ϕ(y)⟩2d3x d3y.


The last formula can be evaluated with the help of a classical form of thefluctuation–dissipation theorem,

(285)∫ ⟨

ϕ(x)ϕ(y)⟩2d3x d3y =

∑k=0

(kBT χϕϕ(k)

)2.

For a homogeneous superfluid, the static susceptibility in the infrared limit k → 0does not depend on the interaction strength and, in accordance with Bogoliubov’s1/k2 theorem for the static susceptibility of superfluids, is equal to [99]

(286)χϕϕ(k) = m/ntoth2k2, k → 0.

Then Eq. (284) immediately yields the squared variance of the condensate fluc-tuations in the superfluid with arbitrary strong interaction in exactly the sameform as is indicated by the arrow in Eq. (271), only with the additional factor(n0(0)/ntot)

2. The latter factor is almost 1 for the BEC in dilute Bose gases butcan be much less in liquids, for example in 4He superfluid it is about 0.1. Thus,indeed, the condensate fluctuations at low temperatures can be calculated via thelong wavelength phase fluctuations of the noncondensate.

6.5.2. Effective Nonlinear σ Model, Goldstone Modes, and Universalityof the Infrared Anomalies

Following [100], let us use an effective nonlinear σ model [103] to demon-strate that the infrared singularities and anomalies of the order parameter fluc-tuations exist in all systems with a broken continuous symmetry, independentlyon the interaction strength, and are similar to that of the BEC fluctuations in theBose gas. That model describes directional fluctuations of an order parameter Ψ (x) = m

(0)s

Ω(x)with a fixed magnitudem(0)s in terms of anNΩ -component unit

vector Ω(x). It is inspired by the classical theory of spontaneous magnetizationin ferromagnets. The constraint | Ω(x)| = 1 suggests a standard decomposition ofthe order parameter

Ω(x) = {Ω0(x),Ωi(x); i = 1, . . . , NΩ − 1},

(287)Ω0(x) =√√√√1 −

NΩ−1∑i=1

Ω2i (x),

into a longitudinal component Ω0(x) and NΩ − 1 transverse Goldstone fieldsΩi(x). The above σ model constraint, Ω2

0 (x) = 1 −∑i Ω2i (x), resembles the

particle-number constraint n0 = N −∑k=0 nk in Eq. (211) for the many-bodyatomic Bose gas in a trap. Although the former is a local, more stringent, con-straint and the latter is only a global, integral constraint, in both cases it resultsin the infrared singularities, anomalies, and non-Gaussian properties of the or-der parameter fluctuations. Obviously, in the particular case of a homogeneous


system the difference between the local and integral constraints disappears en-tirely. Within the σ model, a superfluid can be described as a particular case ofan NΩ = 2 system, with the superfluid (condensate) and normal (noncondensate)component.

At zero external field, the effective action for the fluctuations of the order para-meter is S[Ω] = (ρs/2T )

∫ [∇ Ω(x)]2 d3x, where the spin stiffness ρs is the onlyparameter. Below the critical temperature Tc, a continuous symmetry becomesbroken and there appears an intensive nonzero-order parameter with a mean value

(288)∫V

⟨ Ψ (x) · Ψ (0)⟩d3x = Vm2

L → Vm2s ,

which gives the spontaneous magnetization m2s of the infinite system in the ther-

modynamic limit V → ∞. The leading long distance behavior of the two-pointcorrelation function G(x) = 〈 Ψ (x) · Ψ (0)〉 may be obtained from a simpleGaussian spin wave calculation. Assuming low enough temperatures, we canneglect the spin wave interactions and consider the transverse Goldstone fieldsΩi(k) as the Gaussian random functions of the momentum k with the correlationfunction 〈Ωi(k)Ωi′(k′)〉 = δi,i′δk,−k′T/ρsk2. As a result, the zero external fieldcorrelation function below the critical temperature is split into longitudinal andtransverse parts G(x) = m2

s [1 +G‖(x)+ (NΩ − 1)G⊥(x)], where m2s = G(∞)

now is the renormalized value of the spontaneous magnetization. To the lowestnontrivial order in the small fluctuations of the Goldstone fields, the transversecorrelation function decays very slowly with a distance r , G⊥ ∝ T/ρsr , in ac-cordance with Bogoliubov’s 1/k2 theorem of the divergence of the transversesusceptibility in the infrared limit,

(289)χ⊥(k) = m2sG⊥(k)/T = m2

s /ρsk2.

The longitudinal correlation function is simply related to the transverse one [103],

(290)G‖(x) ≈ 1

4

⟨∑Ω2i (x)∑

Ω2i′(0)⟩c= 1

2(NΩ − 1)G2⊥(x),

and, hence, decays slowly with a 1/r2 power law. That means that contrary to thenaive mean field picture, where the longitudinal susceptibility χ‖(k → 0) belowthe critical temperature is finite, the Eq. (290) leads to an infrared singularity inthe longitudinal susceptibility and correlation function [101],

(291)χ‖(k → 0) ∼ T/ρ2s k; G‖ ∼ 1/r2.

Although the Eq. (290) is obtained by means of perturbation theory, the resultfor the slow power-law decay of the longitudinal correlation function, Eq. (291),holds for arbitrary temperatures T < Tc [103].


Knowing susceptibilities, it is immediately possible to find the variance of theoperator

(292)Ms = V −1∫V

∫V

Ψ (x) · Ψ (y) d3x d3y

that describes the fluctuations of the spontaneous magnetization in a finite systemat zero external field. Its mean value is given by Eq. (288) as 〈Ms〉 = Ms = Vm2

L.Its fluctuations are determined by the connected four-point correlation functionG4 = 〈 Ψ (x1) · Ψ (x2) Ψ (x3) · Ψ (x4)〉c. In the 4th-order of the perturbation theoryfor an infinite system, it may be expressed via the squared transverse susceptibil-ities as follows:

G4 = 1

2m4s (NΩ − 1)

[G⊥(x1 − x3)−G⊥(x1 − x4)−G⊥(x2 − x3)

(293)+G⊥(x2 − x4)]2.

For a finite system, it is more convenient to do similar calculations in the momen-tum representation and replace all integrals by the corresponding discrete sums,which yields the squared variance

(294)⟨(Ms − Ms

)2⟩ = 2m4s (NΩ − 1)T 2ρ−2

s

∑k=0

k−4 ∝ T 2V 4/3.

This expression has the same anomalously large scaling ∝V 4/3 in the thermo-dynamic limit V → ∞ (due to the same infrared singularity) as the variance ofthe BEC fluctuations in the ideal or interacting Bose gas in Eqs. (247) or (271),respectively. Again, although the Eq. (294) is obtained by means of perturbationtheory, the latter scaling is universal below the critical temperature, just like the1/k infrared singularity of the longitudinal susceptibility. Of course, very closeto Tc there is a crossover to the critical singularities, as discussed, e.g., in [104].Thus, the average fluctuation of the order parameter is still vanishing in the ther-

modynamic limit√〈(Ms − Ms)2〉/Ms ∝ V −1/3 → 0; that is, the order parameter

(e.g., spontaneous magnetization or macroscopic wave function) is still a well-defined self-averaging quantity. However, this self-averaging in systems with abroken continuous symmetry is much weaker than expected naively from the stan-dard Einstein theory of the Gaussian fluctuations in macroscopic thermodynam-ics. Note that, in accordance with the well-known Hohenberg–Mermin–Wagnertheorem, in systems with lower dimensions, d � 2, the strong fluctuations of thedirection of the magnetization completely destroy the long range order, and theself-averaging order parameter does not exist anymore. An important point also isthat the scaling result in Eq. (294) holds for any dynamics and temperature depen-dence of the average order parameter Ms(T ) although the function Ms(T ) is, ofcourse, different, say, for ferromagnets, anti-ferromagnets, or a BEC in differenttraps. It is the constraint, either | Ω| = 1 in the σ model or N = n0 +∑k=0 nk


in the BEC, that predetermines the anomalous scaling in Eq. (294). The tempera-ture dependence of the variance in Eq. (294) at low temperatures is also universal,since ρs → const at T → 0. The reason for this fact is that the dominant finitesize dependence is determined by the leading low energy constant in the effectivefield theory for fluctuations of the order parameter, which is precisely ρs in theeffective action S[Ω].

6.5.3. Universal Scaling of Condensate Fluctuations in Superfluids

In homogeneous superfluids the translational invariance requires the super-fluid density to be equal to the full density ntot, so that the associated stiff-ness ρs(T → 0) = h2ntot/m is independent of the interaction strength. Thus,Eq. (294) in accord with the Eq. (271) yields the remarkable conclusion that therelative variance of the ground-state occupation at low temperatures is a universalfunction of the density and the thermal wavelength λT = h/

√2πmT , as well as

the system size L = V 1/3 and the boundary conditions,

(295)√⟨(n0 − n0)2

⟩/n0 = B/

(ntotλ

2T L)2.

The boundary conditions determine the low-energy spectrum of quasiparti-cles in the trap in the infrared limit and, hence, the numerical prefactor B inthe infrared singularity of the variance, namely, the coefficient B in the sum∑

k=0 k−4 = BV 4/3/8π2. In particular, in accord with Eqs. (247) and (271),

one has B = 0.8375 for the box with periodic boundary conditions, andB = 8E3(2)/π2 = 0.501 for the box with Dirichlet boundary conditions. HereEd(t) = ∑∞

n1=1,...,nd=1(n21 + · · · + n2

d)−t is the generalized Epstein zeta func-

tion [95], convergent for d < 2t . Of course, in a finite trap at temperatures of theorder of or less than the energy of the first excited quasiparticle, T < ε1, the scal-ing law (294) is no longer valid and the condensate fluctuations acquire a differenttemperature scaling, due to the temperature-independent quantum noise producedby the excited atoms, which are forced by the interaction to occupy the excitedenergy levels even at zero temperature, as it was discussed for the Eqs. (279)and (244).

6.5.4. Constraint Mechanism of Anomalous Order Parameter Fluctuations andSusceptibilities versus Instability in the Systems with a Broken ContinuousSymmetry

It is important to realize that the anomalously large order parameter fluctuationsand susceptibilities have a simple geometrical nature, related to the fact that thedirection of the order parameter is only in a neutral, rather than in a stable equi-librium, and does not violate an overall stability of the system with a brokencontinuous symmetry at any given temperature below phase transition, T < Tc.


On one hand, on the basis of a well-known relation between the longitudinal sus-ceptibility and the variance �M of the order parameter fluctuations,

(296)χαα ≡ 1

N

∂Mα

∂Bα

= �2M

NkBT, �2

M = ⟨(Mα − Mα

)2⟩,

it is obvious that if the fluctuations are anomalous, i.e., go as �2M ∼ Nγ with

γ > 1 instead of the standard macroscopic thermodynamic scaling �2M ∼ N ,

then the longitudinal susceptibility diverges in the thermodynamic limit N → ∞as �2

M/N ∼ Nγ−1 → ∞. On the other hand, the susceptibilities in stable sys-tems should be finite, since otherwise any spontaneous perturbation will result inan infinite response and a transition to another phase. One could argue (as it wasdone in a recent series of papers [97]) that the anomalous fluctuations cannot existsince they break the stability condition and make the system unstable. However,such an argument is not correct.

First of all, the anomalously large transverse susceptibility in the infrared limitχ⊥ ∼ k−2 ∼ L2 in Bogoliubov’s 1/k2 theorem (289) originates from the obviousproperty of a system with a broken continuous symmetry that it is infinitesimallyeasy to change the direction of the order parameter and, of course, does not violatestability of the system. However, it implies anomalously large fluctuations in thedirection of the order parameter. Therefore, an anomalously large variance of thefluctuations comes from the fluctuations (M− M) ⊥ M, which are perpendicularto the mean value of the order parameter. This is the key issue. It means that alongitudinal external field B ‖ M, that easily rotates these transverse fluctuationstowards M when χ⊥B ∼ �M , will change the longitudinal order parameter by a

large increment, χ‖B ∼ |M| −√

M2 −�2M ≈ �2

M/2M . It is obvious that thispure geometrical rotation of the order parameter has nothing to do with an insta-bility of the system, but immediately reveals the anomalously large longitudinalsusceptibility and relates its value to the anomalous transverse susceptibility andvariance of the fluctuations,

(297)χ‖ ∼ �2M/2BM ∼ �Mχ⊥/2M.

This basically geometrical mechanism of an anomalous behavior of constrainedsystems constitutes an essence of the constraint mechanism of the infrared anom-alies in fluctuations and susceptibilities of the order parameter for all systemswith a broken continuous symmetry. The latter qualitative estimate yields theanomalous scaling of the longitudinal susceptibility discussed above, χ‖ ∼ L

with increase of the system size L, Eq. (291), since χ⊥ ∼ L2, M ∼ L3, and�M ∼ L2.

A different question is whether the Bose gas in a trap is unstable in the grandcanonical ensemble when an actual exchange of atoms with a reservoir is allowed,and only the mean number of atoms in the trap is fixed, N = const. In this case,


for example, the isothermal compressibility is determined by the variance of thenumber-of-atoms fluctuations κT ≡ −V −1(∂V/∂P )T = V 〈(N − N)2〉/N2kBT ,and diverges in the thermodynamic limit if fluctuations are anomalous. In par-ticular, the ideal Bose gas in the grand canonical ensemble does not have awell-defined condensate order parameter, since the variance is of the order of the

mean value,√〈(N − N)2〉 ∼ N , and it is unstable against a collapse [13,15,97].

In summary, one could naively expect that the order parameter fluctuationsbelow Tc are just like that of a standard thermodynamic variable, because thereis a finite restoring force for deviations from the equilibrium value. However,in all systems with a broken continuous symmetry the universal existence of in-frared singularities in the variance and higher moments ensures anomalously largeand non-Gaussian fluctuations of the order parameter. This effect is related to thelong-wavelength phase fluctuations and the infrared singularity of the longitudi-nal susceptibility originating from the inevitable geometrical coupling betweenlongitudinal and transverse order parameter fluctuations in constrained systems.

7. Conclusions

It is interesting to note that the first results for the average and variance of occupa-tion numbers in the ideal Bose gas in the canonical ensemble were obtained aboutfifty years ago by standard statistical methods [42,75,76] (see also [74,77] andthe review [13]). Only later, in the 60s, laser physics and its byproduct, the mas-ter equation approach, was developed (see, e.g., [16,17]). In this paper we haveshown that the latter approach provides very simple and effective tools to calcu-late statistical properties of an ideal Bose gas in contact with a thermal reservoir.In particular, the results (169) and (172) reduce to the mentioned old results in the“condensed region” in the thermodynamic limit.

However, the master equation approach gives even more. It yields simple an-alytical expressions for the distribution function of the number of condensedatoms (162) and for the canonical partition function (163). In terms of cumu-lants, or semi-invariants [72,94], for the stochastic variables n0 or n = N − n0, itwas shown [21] that the quasithermal approximation (154), with the results (169)and (172), gives correctly both the first and the second cumulants. The analysis ofthe higher-order cumulants is more complicated and includes, in principle, a com-parison with more accurate calculations of the conditioned average number ofnoncondensed atoms (129) as well as higher-order corrections to the second-ordermaster equation (124). It is clear that the master equation approach is capable ofgiving the correct answer for higher-order cumulants and, therefore, moments ofthe condensate fluctuations. Even without these complications, the approximateresult (162) reproduces the higher moments, calculated numerically via the exact


recursion relation (80), remarkably well for all temperatures T < Tc and T ∼ Tc(see Fig. 12).

As we demonstrated in Section 4, the simple formulas yielded by the masterequation approach allow us to study mesoscopic effects in BECs for a relativelysmall number of atoms that is typical for recent experiments [24–29]. Moreover,it is interesting in the study of the dynamics of BEC. This technique for study-ing statistics and dynamics of BEC shows surprisingly good results even withinthe simplest approximations. Thus, the analogy with phase transitions and quan-tum fluctuations in lasers (see, e.g., [19,52,70,71]) clarifies some problems inBEC. The equilibrium properties of the number-of-condensed-atom statistics inthe ideal Bose gas are relatively insensitive to the details of the model. The originof dynamical and coherent properties of the evaporatively cooling gas with an in-teratomic interaction is conceptually different from that in the present “ideal gas+ thermal reservoir” model. The present model is rather close to the dilute 4Hegas in porous gel experiments [22] in which phonons in the gel play the role ofthe external thermal reservoir. Nevertheless, the noncondensed atoms always playa part of some internal reservoir, and the condensate master equation probablycontains terms similar to those in Eq. (130) for any cooling mechanism.

For the ideal Bose gas in the canonical ensemble the statistics of the conden-sate fluctuations below Tc in the thermodynamic limit is essentially the statisticsof the sum of the noncondensed modes of a trap, nk, that fluctuate independently,n0 = N −∑k=0 nk. This is well understood, especially, due to the Maxwell’sdemon ensemble approximation elaborated in a series of papers [14,42,44,46,54,55], and is completed and justified to a certain extent in [20,21] by the explicitcalculation of the moments (cumulants) of all orders, and by the reformula-tion of the canonical-ensemble problem in the properly reduced subspace of theoriginal many-particle Fock space. The main result (263) of [20,21] explicitly de-scribes the non-Gaussian properties and the crossover between the ideal-gas andinteraction-dominated regimes of the BEC fluctuations.

The problem of dynamics and fluctuations of BEC for the interacting gas ismuch more involved. The master equation approach provides a powerful toolfor the solution of this problem as well. Of course, to take into account higher-order effects of interaction between atoms we have to go beyond the second-ordermaster equation, i.e., to iterate Eq. (123) more times and to proceed with thehigher-order master equation similarly to what we discussed above. It would beinteresting to show that the master equation approach could take into accountall higher-order effects in a way generalizing the well-known nonequilibriumKeldysh diagram technique [30,105,106]. As a result, the second-order masterequation analysis presented above can be justified rigorously, and higher-ordereffects in condensate fluctuations at equilibrium, as well as nonequilibrium stagesof cooling of both ideal and interacting Bose gases can be calculated.


The canonical-ensemble quasiparticle method, i.e., the reformulation of theproblem in terms of the proper canonical-ensemble quasiparticles, gives evenmore. Namely, it opens a way to an effective solution of the canonical-ensembleproblems for the statistics and nonequilibrium dynamics of the BEC in the in-teracting gas as well. The first step in this direction is done in Section 6, wherethe effect of the Bogoliubov coupling between excited atoms due to a weak in-teraction on the statistics of the fluctuations of the number of ground-state atomsin the canonical ensemble was analytically calculated for the moments (cumu-lants) of all orders. In this case, the BEC statistics is essentially the statisticsof the sum of the dressed quasiparticles that fluctuate independently. In particu-lar, a suppression of the condensate fluctuations at the moderate temperatures andtheir enhancement at very low temperatures immediately follow from this picture.

There is also the problem of the BEC statistics in the microcanonical ensem-ble, which is closely related to the canonical-ensemble problem. In particular,the equilibrium microcanonical statistics can be calculated from the canonicalstatistics by means of an inversion of a kind of Laplace transformation fromthe temperature to the energy as independent variable. Some results concerningthe BEC statistics in the microcanonical ensemble for the ideal Bose gas werepresented in [14,45,46,54,55,60]. We can calculate all moments of the micro-canonical fluctuations of the condensate from the canonical moments found in thepresent paper. Calculation of the microcanonical statistics starting from the grandcanonical ensemble and applying a saddle-point method twice, first, to obtain thecanonical statistics and, then, to get the microcanonical statistics [45], meets cer-tain difficulties since the standard saddle-point approximation is not always goodand explicit to restore the canonical statistics from the grand canonical one withsufficient accuracy [57]. The variant of the saddle-point method discussed in Ap-pendix F is not subject to these restrictions.

Another important problem is the study of mesoscopic effects due to a rela-tively small number of trapped atoms (N ∼ 103–106). The canonical-ensemblequasiparticle approach under the approximation (210), i.e., HCE ≈ HCE

n0 =0, takesinto account only a finite-size effect of the discreteness of the single particle spec-trum, but does not include all mesoscopic effects. Hence, other methods shouldbe used (see, e.g., [14,44,46,49,61–64,67]). In particular, the master equation ap-proach provides amazingly good results in the study of mesoscopic effects, as wasdemonstrated recently in [52,53].

The canonical-ensemble quasiparticle approach also makes it clear how to ex-tend the Bogoliubov and more advanced diagram methods for the solution ofthe canonical-ensemble BEC problems and ensure conservation of the numberof particles. The latter fact cancels the main arguments of Refs. [96,97] againstthe Giorgini–Pitaevskii–Stringari result [78] and shows that our result (263) and,in particular, the result of [78] for the variance of the number of ground-stateatoms in the dilute weakly interacting Bose gas, correctly take into account one


of the main effects of the interaction, namely, dressing of the excited atoms bythe macroscopic condensate via the Bogoliubov coupling. If one ignores this andother correlation effects, as it was done in [96,97], the result cannot be correct.This explains a sharp disagreement of the ground-state occupation variance sug-gested in [96] with the predictions of [78] and our results as well. Note alsothat the statement from [96] that “the phonon spectrum plays a crucial role inthe approach of [78]” should not be taken literally since the relative weights ofbare modes in the eigenmodes (quasiparticles) is, at least, no less important thaneigenenergies themselves. In other words, our derivation of Eq. (263) shows thatsqueezing of the excited states due to Bogoliubov coupling in the field of themacroscopic condensate is crucial for the correct calculation of the BEC fluctua-tions. Besides, the general conclusion that very long wavelength excitations havean acoustic, “gapless” spectrum (in the thermodynamic limit) is a cornerstone factof the many-body theory of superfluidity and BEC [93]. Contrary to a pessimisticpicture of a mess in the study of the condensate fluctuations in the interacting gaspresented in [96], we are convinced that the problem can be clearly formulatedand solved by a comparative analysis of the contributions of the main effects ofthe interaction in the tradition of many-body theory. In particular, the result (263)corresponds to a well-established first-order Popov approximation in the diagramtechnique for the condensed phase [36].

We emphasize here an important result of an analytical calculation of all highercumulants (moments) [21]. In most cases (except, e.g., for the ideal gas in theharmonic trap and similar high dimensional traps where d > 2σ ), both for theideal Bose gas and for the interacting Bose gas, the third and higher cumulantsof the number-of-condensed-atom fluctuations normalized to the correspondingpower of the variance do not tend to the Gaussian zero value in the thermodynamiclimit, e.g., 〈(n0 −〈n0〉)3〉/〈(n0 −〈n0〉)2〉3/2 does not vanish in the thermodynamiclimit.

Thus, fluctuations in BEC are not Gaussian, contrary to what is usually assumedfollowing the Einstein theory of fluctuations in the macroscopic thermodynamics.Moreover, BEC fluctuations are, in fact, anomalously large, i.e., they are not nor-mal at all. Both these remarkable features originate from the universal infraredanomalies in the order parameter fluctuations and susceptibilities in constrainedsystems with a broken continuous symmetry. The infrared anomalies come froma long range order in the phases below the critical temperature of a second-orderphase transition and have a clear geometrical nature, related to the fact that thedirection of the order parameter is only in a neutral, rather than in a stable equi-librium. Hence, the transverse susceptibility and fluctuations are anomalouslylarge and, through an inevitable geometrical coupling between longitudinal andtransverse order parameter fluctuations in constrained systems, produce the anom-alous order parameter fluctuations. In other words, the long wavelength phasefluctuations of the Goldstone modes, in accordance with the Bogoliubov 1/k2


theorem for the transverse susceptibility, generate anomalous longitudinal fluc-tuations in the order parameter of the systems below the critical temperature ofthe second-order phase transition. Obviously, this constraint mechanism of theinfrared anomalies in fluctuations and susceptibilities of the order parameter isuniversal for all systems with a broken continuous symmetry, including BEC inideal or weakly interacting gases as well as superfluids, ferromagnets and othersystems with strong interaction. It would be interesting to extend the analysis ofthe order parameter fluctuations presented in this review from the BEC in gasesto other systems.

The next step should be an inclusion of the effects of a finite renormalizationof the energy spectrum as well as the interaction of the canonical-ensemble qua-siparticles at finite temperatures on the statistics and dynamics of BEC. It canbe done on the level of the second-order Beliaev–Popov approximation, which isconsidered to be enough for the detailed account of most many-body effects (for areview, see [36]). A particularly interesting problem is the analysis of phase fluc-tuations of the condensate in the trap, or of the matter beam in the atom laser [88],because the interaction is crucial for the existence of the coherence in the conden-sate [21,30,32,33,48,93,107–109]. As far as the equilibrium or quasiequilibriumproperties are concerned, the problem can be solved effectively by applying eitherthe traditional methods of statistical physics to the canonical-ensemble quasipar-ticles, or the master equation approach, that works surprisingly well even withoutany explicit reduction of the many-particle Hilbert space [52,53]. For the dy-namical, nonequilibrium properties, the analysis can be based on an appropriatemodification of the well-known nonequilibrium Keldysh diagram technique [105,106,110,111] which incorporates both the standard statistical and master equationmethods.

Work in the directions mentioned above is in progress and will be presentedelsewhere. Clearly, the condensate and noncondensate fluctuations are cruciallyimportant for the process of the second-order phase transition, and for the overallphysics of the Bose–Einstein-condensed interacting gas as a many-particle sys-tem.

8. Acknowledgements

We would like to acknowledge the support of the Office of Naval Research(Award No. N00014-03-1-0385) and the Robert A. Welch Foundation (Grant No.A-1261). One of us (MOS) wishes to thank Micheal Fisher, Joel Lebowitz, ElliottLieb, Robert Seiringer for stimulating discussions and Leon Cohen for suggestingthis review article.


9. Appendices

Appendix A. Bose’s and Einstein’s Wayof Counting Microstates

When discussing Einstein’s 1925 paper [6], we referred to the identity (49),

(A1)∑′

{p0,p1,...,pN }

Z!p0! . . . pN ! =

(N + Z − 1

N

),

which expresses the number of ways to distribute N Bose particles over Z quan-tum cells in two different manners: On the left-hand side, which corresponds toBose’s way of counting microstates, numbers pr specify how many cells contain rquanta; of course, with only N quanta being available, one has pr = 0 for r > N .As symbolically indicated by the prime on the summation sign, the sum thus ex-tends only over those sets {p0, p1, . . . , pN } with comply with the conditions

(A2)N∑r=0

pr = Z,

stating that there are Z cells to accommodate the quanta, and

(A3)N∑r=0

rpr = N,

stating that the number of quanta be N . The right-hand side of Eq. (A1) gives thetotal number of microstates, taking into account all possible sets of occupationnumbers in the single expression already used by Einstein in his final deriva-tion [6] of the Bose–Einstein distribution which can still be found in today’stextbooks.

The validity of Eq. (A1) is clear for combinatorial reasons. Nonetheless, sincethis identity (A1) constitutes one of the less known relations in the theory of Bose–Einstein statistics, we give its explicit proof in this appendix.

As for most mathematical proofs, one needs tools, an idea, and a conjurer’strick. In the present case, the tools are two generalizations of the binomial theorem

(A4)(a + b)n =n∑

k=0

(n

k

)akbn−k,

where

(A5)

(n

k

)= n!

k!(n− k)!

396 V.V. Kocharovsky et al. [Appendix A

denotes the familiar binomial coefficients. The first such generalization is themultinomial theorem

(A6)(a1 + a2 + · · · + aN)n =∑∑

pr=n

n!p1!p2! . . . pN !a

p11 a

p22 . . . a

pNN ,

which is easily understood: When multiplying out the left-hand side, every prod-uct obtained contains one factor ai from each bracket (a1+a2+· · ·+aN). Hence,in every product ap1

1 ap22 . . . a

pNN the exponents add up to the number of brackets,

which is n. Therefore, for such a product there are n! permutations of the individ-ual factors ai . However, if identical factors ai are permuted among themselves,for which there are pi ! possibilities, one obtains the same value. Hence, the co-efficient of each product on the right-hand side in Eq. (A6) corresponds to thenumber of possible arrangements of its factors, divided by the number of equiva-lent arrangements. Note that the reasoning here is essentially the same as for thejustification of Bose’s expression (12), which is, of course, not accidental.

The second generalization of the binomial theorem (A4) required for the proofof the identity (A1) emerges when we replace the exponent n by a nonnaturalnumber γ : One then has

(A7)(a + b)γ =∞∑k=0

(γ

k

)akbγ−k,

with the definition

(A8)

(γ

k

)= γ (γ − 1)(γ − 2) · · · · · (γ − k + 1)

k! .

If γ is not a natural number, this series (A7) converges for any complex num-bers a, b, provided |a/b| < 1. This generalized binomial theorem (A7), whichis treated in introductory analysis courses, is useful, e.g., for writing down theTaylor expansion of (1 + x)γ .

Given these tools, the idea for proving the identity (A1) now consists in con-sidering the expression

(A9)PNZ(x) =(1 + x + x2 + x3 + · · · + xN

)Z,

where x is some variable which obeys |x| < 1, but need not be specified further.According to the multinomial theorem (A6), one has

(A10)PNZ(x) =∑

∑Nr=0 pr=Z

Z!p0!p1! . . . pN !

(x0)p0(x1)p1 . . .

(xN)pN .

Directing the attention then to a systematic ordering of this series with respect topowers of x, the coefficient of xN equals the sum of all coefficients encountered

Appendix B] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 397

here which accompany terms with∑

r rpr = N , which is precisely the left-handside of the desired equation (A1), keeping in mind the restrictions (A2) and (A3).

Now comes the conjurer’s trick: The coefficient of xN in this expression (A10),equaling the left-hand side of Eq. (A1), also equals the coefficient of xN in theexpression

(A11)P∞(x) = (1 + x + x2 + x3 + · · · + xN + xN+1 + · · ·)Z,since this differs from PNZ(x) only by powers of x higher than N . But this in-volves a geometric series, which is immediately summed:

(A12)P∞(x) =( ∞∑r=0

xr

)Z= (1 − x)−Z.

According to the generalized binomial theorem (A7), one has

(A13)(1 − x)−Z =∞∑k=0

(−Zk

)(−x)k,

so that, in view of the definition (A8), the coefficient of xN equals

(−1)N(−Z

N

)= (−1)N

(−Z)(−Z − 1) · · · · · (−Z −N + 1)

N != (Z +N − 1) · · · · · (Z + 1)Z

N !(A14)= (Z +N − 1)!

N !(Z − 1)! =(Z +N − 1

N

).

This is the right-hand side of the identity (A1), which completes the proof.

Appendix B. Analytical Expression for the Mean Numberof Condensed Atoms

We obtain an analytical expression for n0 from Eq. (86). One can reduce the triplesum into a single sum if we take into account the degeneracy g(E) of the levelwith energy E = hΩ(l+m+ n), which is equal to the number of ways to fill thelevel. This number can be calculated from the Einstein’s complexion equation,i.e.,

(B1)g(E) = {(l +m+ n)+ (3 − 1)}!(l +m+ n)!(3 − 1)! = 1

2

(E

hΩ+ 2

)(E

hΩ+ 1

).

398 V.V. Kocharovsky et al. [Appendix B

Now, we have reduced three variables l, m, n to only one. By letting E = shΩ

where s is integer, one can write

(B2)N =∞∑s=0

12 (s + 2)(s + 1)

( 1n0

+ 1)esβhΩ − 1� n0 + n0

(1 + n0)S,

where for n0 � 1,

(B3)S �∞∑s=1

12 (s + 2)(s + 1)

esβhΩ − 1.

The root of the quadratic Eq. (B2) yields

(B4)n0 = −1

2

[(1 + S −N)−

√(1 + S −N)2 + 4N

].

In order to find an analytical expression for S, we write

(B5)S � 1

2

∞∑s=0

s2 + 3s

esβhΩ − 1+

∞∑s=1

1

esβhΩ − 1.

Converting the summation into integration by replacing x = sβhΩ yields

S � 1

a

∞∫a

dx

ex − 1+ 1

2a3

∞∫0

x2 dx

ex − 1+ 3

2a2

∞∫0

x dx

ex − 1

(B6)= 1 − 1

aln(ea − 1

)+ ζ(3)

a3+(π

2a

)2

,

where a = βhΩ = hΩ/kBT = (ζ(3)/N)1/3Tc/T .Thus, Eq. (B3) gives the following analytical expression for S:

(1 + S −N) = −N{

1 −(T

Tc

)3 }+ π2

4

(N

ζ(3)

)2/3(T

Tc

)2

(B7)−(

N

ζ(3)

)1/3T

Tcln(e(ζ(3)/N)1/3Tc/T − 1

)+ 2.

Figure 16 compares different approximations for calculating n0 within thegrand canonical ensemble.

Appendix C] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 399

FIG. 16. Grand canonical result for 〈n0〉 as a function of temperature for N = 200, computedfrom analytical expressions, Eqs. (B4) and (B7) (line with circles); semi-analytical expressions givenby Eqs. (B4) and (B3) (solid line); and exact numerical solution of Eq. (86) (dots). Expanded viewsshow: (b) exact agreement of the semi-analytical approach at low temperature and (c) a small deviationnear Tc . Also, the analytical and the semi-analytical results agree quite well.

Appendix C. Formulas for the Central Momentsof Condensate Fluctuations

By using n0 = N −∑k =0 nk and 〈n0〉 = N −∑k =0〈nk〉, we have

(C1)⟨(n0 − 〈n0〉

)s ⟩ = (−1)s⟨[∑

k =0

(nk − 〈nk〉

)]s ⟩,

which shows that the fluctuations of the condensate particles are proportional tothe fluctuations of the noncondensate particles.

400 V.V. Kocharovsky et al. [Appendix C

As an example we show how to evaluate the fluctuation for the second-ordermoment, or variance,

(C2)⟨(n0 − 〈n0〉

)2⟩ = ∑j,k =0

(〈njnk〉 − 〈nj 〉〈nk〉).

The numbers of particles in different levels are statistically independent, since〈njnk =j 〉 = Tr{a†

j aj a†k akρ} = Tr{a†

j aj ρj }Tr{a†k akρk} = 〈nj 〉〈nk〉. Thus, we

find

(C3)⟨(n0 − 〈n0〉

)2⟩ =∑k =0

[⟨n2k

⟩− 〈nk〉2],(C4)

⟨(n0 − 〈n0〉

)3⟩ =∑k =0

[−⟨n3k

⟩+ 3⟨n2k

⟩〈nk〉 − 2〈nk〉3],(C5)

⟨(n0 − 〈n0〉

)4⟩ =∑k =0

[⟨n4k

⟩− 4⟨n3k

⟩〈nk〉 + 6⟨n2k

⟩〈nk〉2 − 3〈nk〉4].In the grand canonical approach, 〈nsk〉 can be evaluated using

(C6)⟨nsk⟩ = 1

Zk

∑nk

nske−β(εk−μ)nk ,

where Zk = (1 − e−β(εk−μ))−1. An alternative way is to use the formula 〈nsk〉 =dsΘk

d(iu)s|u=0 derived in Section 5.2. In particular, one can show that in this approach

(C7)⟨n2k

⟩ = 2〈nk〉2 + 〈nk〉,(C8)

⟨n3k

⟩ = 6〈nk〉3 + 6〈nk〉2 + 〈nk〉,(C9)

⟨n4k

⟩ = 24〈nk〉4 + 36〈nk〉3 + 14〈nk〉2 + 〈nk〉.Using Eqs. (C7)–(C9) we obtain

(C10)⟨(n0 − 〈n0〉

)2⟩ =∑k =0

[〈nk〉2 + 〈nk〉],

(C11)⟨(n0 − 〈n0〉

)3⟩ = −∑k =0

[2〈nk〉3 + 3〈nk〉2 + 〈nk〉

],

(C12)⟨(n0 − 〈n0〉

)4⟩ =∑k =0

[9〈nk〉4 + 18〈nk〉3 + 10〈nk〉2 + 〈nk〉

].

Appendix D] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 401

Appendix D. Analytical Expression for the Varianceof Condensate Fluctuations

In a spherically symmetric harmonic trap with trap frequency Ω one can convertthe triple sums in Eq. (98) into a single sum by using Eq. (B1), and then dointegration upon replacing

∑∞x=0 . . . → 1

βhΩ

∫∞0 dx . . . and βE → x, giving

�n20 =

∞∑E =0

1

2

(E

hΩ+ 2

)(E

hΩ+ 1

)

×{

1

[exp(βE)(1 + 1n0)− 1]2 + 1

exp(βE)(1 + 1n0)− 1

}

= 1

2βhΩ

∞∫a

dx

[(x

βhΩ

)2

+ 3x

βhΩ+ 2

]

×{

1

[exp(x)(1 + 1n0)− 1]2 + 1

exp(x)(1 + 1n0)− 1

}

(D1)= 1

2a

∞∫a

dx

[(x

a

)2

+ 3x

a+ 2

] exp(x)(1 + 1n0)

[exp(x)(1 + 1n0)− 1]2 ,

where a = βhΩ = hΩ/kBT = (Tc/T )(ζ(3)/N)1/3 and the density of statescan be written as ρ(E) = 1

2hΩ [( EhΩ

)2+ 3EhΩ

+2]. Then we integrate by parts usingthe identity

exp(x)(1 + 1n0)

[exp(x)(1 + 1n0)− 1]2 = − ∂

∂x

1

[exp(x)(1 + 1n0)− 1] ,

arriving at

(D2)�n20 = 3

a[ea(1 + 1n0)− 1] +

1

2a2

∞∫a

dx( 2xa

+ 3)

[exp(x)(1 + 1n0)− 1] .

The integral in Eq. (D2) can be calculated analytically, using

∞∫a

x dx

[exp(x)A− 1] = π2

6− 1

2ln2 A+ ln

(Aea − 1

)lnA

+ di log(Aea)+ a2

2,

402 V.V. Kocharovsky et al. [Appendix E

∞∫a

dx

[exp(x)A− 1] = ln(1 + α)− ln(Aea − 1

)+ a,

where

di log(x) =x∫

1

ln(t)

1 − tdt.

As a result, we get

�n20 = 1

a3

[π2

6+ di log

[ea(

1 + 1

n0

)]− 1

2ln2(

1 + 1

n0

)+ ln

[ea(

1 + 1

n0

)− 1

]ln

(1 + 1

n0

)(D3)+ 3

2a ln

(n0 + 1

ea(n0 + 1)− n0

)]+ 3

a[ea(1 + 1n0)− 1] +

2

a.

Taking into account that a = (Tc/T )(ζ(3)/N)1/3, we finally obtain

�n20 =(T

Tc

)3N

ζ(3)

[π2

6+ di log

[exp[(Tc/T )(ζ(3)/N)1/3](1 + 1

n0

)]− 1

2ln2(

1 + 1

n0

)+ ln

[exp[(Tc/T )(ζ(3)/N)1/3](1 + 1

n0

)− 1

]ln

(1 + 1

n0

)]+ 3

2

(T

Tc

)2(N

ζ(3)

)2/3

ln

[n0 + 1

exp[(Tc/T )(ζ(3)/N)1/3](n0 + 1)− n0

]+ T

Tc

(N

ζ(3)

)1/3[ 3

exp[(Tc/T )(ζ(3)/N)1/3](1 + 1n0)− 1

+ 2

].

(D4)

Appendix E. Single Mode Coupled to a Reservoir of Oscillators

The derivation of the damping Liouvillean proceeds from the Liouville–von Neu-mann equation

(E1)∂

∂tρ(t) = 1

ih

[Vsr , ρ(t)

],

Appendix E] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 403

where Vsr is the Hamiltonian in the interaction picture for the system (s) coupledto a reservoir (r).

We can derive a closed form of the dynamical equation for the reduced densityoperator for the system, ρs(t) = Trr{ρ(t)} by tracing out the reservoir. This is ac-complished by first integrating Eq. (E1) for ρ(t) = ρ(0)+ 1

ih

∫ t0 [Vsr (t ′), ρ(t ′)] dt ′

and then substituting it back into the right hand side of Eq. (E1), giving

∂

∂tρs(t) = 1

ihTrr[Vsr , ρ(0)

](E2)+ 1

(ih)2Trr

t∫0

[Vsr (t),

[Vsr (t

′), ρ(t ′)]dt ′].

We may repeat this indefinitely, but owing to the weaknesses of the system-reservoir interaction, it is possible to ignore terms higher than 2nd order in Vsr .Furthermore, we assume the system and reservoir are approximately uncorrelatedin the past and the reservoir is so large that it remains practically in thermal equi-librium ρthr , so ρ(t ′) � ρs(t

′)⊗ ρthr and ρ(0) = ρs(0)⊗ ρthr .For a single mode field (f ) coupled to a reservoir of oscillators, one has

(E3)Vf r = h∑

k

gk(bka

†ei(ν−νk)t + b†kae

−i(ν−νk)t).

Since Trr{a†ρthr } = 0 and Trr{aρthr } = 0, the first term vanishes and by us-ing Eq. (E3) we have 16 terms. But secular approximation reduces the numberof terms by half. We now perform the Markov approximation, ρf (t ′) � ρf (t),stating that the dynamics of the system is independent of the states in the past.

The thermal average of the radiation operators is

∑k

g2k

t∫0

e−i(ν−νk)(t−t ′) Trr{ρthr b

†kbk}dt ′

�∑

k

g2kπδ(ν − νk)n(νk) = n(ν)G(ν)/2,

where Trr{ρthr b†kbk} = n(νk) = (eβhνk − 1)−1. Thus, we have

∂

∂tρf (t) = −1

2C[a†aρf (t)− 2aρf (t)a

† + ρf (t)a†a]

(E4)− 1

2D[aa†ρf (t)− 2a†ρf (t)a + ρf (t)aa

†],where D = Gn and C = G(n+ 1).

404 V.V. Kocharovsky et al. [Appendix F

Appendix F. The Saddle-Point Method for Condensed BoseGases

The saddle-point method is one of the most essential tools in statistical physics.Yet, the conventional form of this approximation fails in the case of condensedideal Bose gases [13,58]. The point is that in the condensate regime the saddle-point of the grand canonical partition function approaches the ground-state sin-gularity at z = exp(βε0), which is a hallmark of BEC. However, the customaryGaussian approximation requires that intervals around the saddle-point stay clearof singularities. Following the original suggestion by Dingle [59], Holthaus andKalinowski [60] worked out a natural solution to this problem: One should ex-empt the ground-state factor of the grand canonical partition function from theGaussian expansion and treat that factor exactly, but proceed as usual otherwise.The success of this refined saddle-point method hinges on the fact that the emerg-ing integrals with singular integrands can be done exactly; they lead directly toparabolic cylinder functions. Here we discuss the refined saddle-point method insome detail.

We start from the grand canonical partition function

(F1)Ξ(β, z) =∞∏ν=0

1

1 − z exp(−βεν) ,

where εν are single-particle energies, β = 1/kBT and z = exp(βμ). The grandcanonical partition function Ξ(β, z) generates the canonical partition functionsZN(β) by means of the expansion

(F2)Ξ(β, z) =∞∑N=0

zNZN(β).

Then we treat z as a complex variable and using Cauchy’s theorem representZN(β) by a contour integral,

(F3)ZN(β) = 1

2πi

∮dz

Ξ(β, z)

zN+1= 1

2πi

∮dz exp

(−F(z)),where the path of integration encircles the origin counter-clockwise, and

F(z) = (N + 1) ln z− lnΞ(β, z) = (N + 1) ln z

(F4)+∞∑ν=0

ln[1 − z exp(−βεν)

].

The saddle-point z0 is determined by the requirement that F(z) becomes station-ary,

(F5)∂F (z)

∂z

∣∣∣∣z=z0

= 0,

Appendix F] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 405

giving

(F6)N + 1 =∞∑ν=0

1

exp(βεν)/z0 − 1.

This is just the grand canonical relation between particle number N and fugacityz0 (apart from the appearance of one extra particle on the left-hand side). Appen-dix C discusses an approximate analytic solution of such equations.

In the conventional saddle-point method, the whole function F(z) is taken inthe saddle-point approximation

(F7)F(z) ≈ F(z0)+ 1

2F ′′(z0)(z− z0)

2.

Doing the remaining Gaussian integral yields

(F8)ZN(β) ≈ exp(−F(z0))√−2πF ′′(z0).

The canonical occupation number of the ground state, and its mean-square fluc-tuations, are obtained by differentiating the canonical partition function:

(F9)n0 = ∂ lnZN(β)

∂(−βε0)= 1

ZN(β)

1

2πi

∮dz

1

zN+1

∂Ξ(β, z)

∂(−βε0),

(F10)�n0 = ∂2 lnZN(β)

∂(−βε0)2= −n2

0 + 1

ZN(β)

1

2πi

∮dz

1

zN+1

∂2Ξ(β, z)

∂(−βε0)2.

The saddle-point approximation is then applied to the integrands of Eqs. (F9) and(F10). Figs. 17 and 18 (dashed curves) show results for n0 and �n0 obtained bythe conventional saddle-point method for a Bose gas with N = 200 atoms in aharmonic isotropic trap. In the condensate regime there is a substantial deviationof the saddle-point curves from the “exact” numerical answer obtained by solutionof the recursion equations for the canonical statistics (dots).

The reason for this inaccuracy is that in the condensate region the saddle-pointz0 lies close to the singular point z = exp(βε0) of the function F(z). As a result,the approximation (F7) becomes invalid in the condensate region. To improve themethod, Dingle [59] proposed to treat the potentially dangerous term in (F4) as itis, and represent ZN(β) as

(F11)ZN(β) = 1

2πi

∮dz

exp(−F1(z))

1 − z exp(−βε0),

where

(F12)F1(z) = (N + 1) ln z+∞∑ν=1

ln[1 − z exp(−βεν)

]

406 V.V. Kocharovsky et al. [Appendix F

FIG. 17. Canonical occupation number of the ground state as a function of temperature forN = 200 atoms in a harmonic isotropic trap. Dashed and solid curves are obtained by the conventionaland the refined saddle-point method, respectively. Dots are “exact” numerical answers obtained for thecanonical ensemble.

FIG. 18. Variance in the condensate particle number as a function of temperature for N = 200atoms in a harmonic isotropic trap. Dashed and solid curves are obtained by the conventional andthe refined saddle-point method, respectively. Dots are “exact” numerical answers in the canonicalensemble.

Appendix F] FLUCTUATIONS IN BOSE–EINSTEIN CONDENSATES 407

has no singularity at z = exp(βε0). The singular point to be watched now is theone at z = exp(βε1). Since z0 < exp(βε0), the saddle-point remains separatedfrom that singularity by at least the N -independent gap exp(βε1) − exp(βε0) �hω/kBT . This guarantees that the amputated function F1(z) remains singularity-free in the required interval around z0 for sufficiently large N . Then the Gaussianapproximation to exp(−F1(z)) is safe. The subsequently emerging saddle-pointintegral for the canonical partition function can be done exactly, yielding [60]

(F13)ZN(β) ≈ 1

2exp[βε0 − F1(z0)− 1 + η2/2

]erfc

(η1√

2

),

where erfc(z) = 2/√π∫∞z

exp(−t2) dt is the complementary error function, η =(exp(βε0)− z0)

√−F ′′

1 (z0), and η1 = η − 1/η.

Calculation of occupation numbers and their fluctuations deals with integralsfrom derivatives of Ξ(β, z) with respect to −βε0. In such expressions the factorssingular at z = exp(βε0) should be taken exactly. This leads to the integrals ofthe following form:

1

2πi

∮dz

exp[−f1(z)− (σ − 1)βε0](1 − z exp(−βε0))σ

≈ 1√2π

(−f ′′1 (z∗))(σ−1)/2

(F14)× exp(βε0 − f1(z∗)− σ + η2/2 − η2

1/4)D−σ (η1),

where η = (exp(βε0)− z∗)√−f ′′

1 (z∗), η1 = η− σ/η, and z∗ is a saddle-point of

the function

(F15)f (z) = f1(z)+ (σ − 1)βε0 + σ ln(1 − z exp(−βε0)

);D−σ (z) is a parabolic cylinder function, which can be expressed in terms of hy-pergeometric functions as

Ds(z) = 2s/2e−z2/4√π

[1F1(−s/2, 1/2, z2/2)

#[(1 − s)/2]

(F16)−√

2z ·1 F1((1 − s)/2, 3/2, z2/2)

#[−s/2]].

Figures 17 and 18 (solid curves) show n0(T ) and �n0(T ) obtained by the refinedsaddle-point method. These results are in remarkable agreement with the exactdots. Figure 19 demonstrates that this refined method also provides good accuracyfor the third central moment of the number-of-condensed-atoms fluctuations.


FIG. 19. The third central moment 〈(n0 − n0)3〉 for N = 200 atoms in a harmonic isotropic trap

obtained using the refined saddle-point method (solid curve). Exact numerical results for the canonicalensemble are shown by dots.

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LIDAR-MONITORING OF THE AIR WITHFEMTOSECOND PLASMA CHANNELS

LUDGER WÖSTE1, STEFFEN FREY2 AND JEAN-PIERRE WOLF3

1Physics Department, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany2MIT, Department of Earth, Atmospheric, and Planetary Sciences, 77 Massachusetts Avenue,

Cambridge, MA 02139, USA3GAP-Biophotonics, University of Geneva, 20, rue de l’Ecole de Médecine,

1211 Geneva 4, Switzerland

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4132. Conventional LIDAR Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4153. The Femtosecond-LIDAR Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4194. Nonlinear Propagation of Ultra-Intense Laser Pulses . . . . . . . . . . . . . . . . . . . . . 4215. White Light Femtosecond LIDAR Measurements . . . . . . . . . . . . . . . . . . . . . . . 4276. Nonlinear Interactions with Aerosols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4337. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4378. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4389. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

AbstractLIDAR (Light Detection and Ranging) is the only remote sensing technique thatis capable to provide 3-dimensional range resolved measurements of atmosphericconstituents like pollutants, humidity or the aerosol, and of atmospheric parameterslike temperature or wind. Further perspectives arise from the advent of ultra-fasthigh-power laser sources which are capable to generate extended plasma channels inthe air, so called filaments. Their extraordinary properties, like backwards-enhancedwhite light emission, plasma generation along their trajectories, and their electricalconductivity provide further fascinating perspectives for applications in atmosphericresearch and beyond. Examples are remote multi-component analyses of the air andthe aerosol, bio-aerosol detection, hard target analysis and even lightning control.

1. Introduction

Observing and controlling the earth atmosphere is the most important issue forresearch. The total population on earth has reached a density where air pollu-


DOI 10.1016/S1049-250X(06)53011-3

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414 L. Wöste et al. [1

tion on local and global scales has severe consequences for mankind, aggravatingcatastrophes like hurricanes, floods and smog. The destruction of the ozone layerand the global warming are two prominent examples for anthropogenic changesthat occurred to the complex dynamic system of the atmosphere during the in-dustrial age. Since the change occurs fast, its consequences for the weather, theclimate and the air composition cannot be predicted at the required precision.This makes the problem so dangerous. Large congested areas, so called mega-cities, are growing all around the globe and lead into a spiral of more pollutionassociated with a deterioration of living conditions and health for hundreds ofmillion of people. Nowadays fear of the release of deadly toxic substances byindustry, research, military, and terrorists provides further reasons for the devel-opment of reliable observation techniques. Precise and reliable measurements are,last but not least, essential for improved weather prediction and support of trafficand aviation.

Optical remote sensing techniques have a long tradition when it comes to at-mospheric observations. Sky observation by naked eye may be the oldest methodfor weather prediction. Some vital discoveries, among them the surprising de-tection of the ozone hole in the stratosphere, are results obtained with opticalremote sensing instruments. Its discovery is historically of special interest, be-cause it demonstrates the importance of reliably operating observation platforms.Satellites for global ozone monitoring where already in place (Stolarski et al.,1986), but the unexpected catastrophe was discovered with the proven Brewer-spectrometer (Farman et al., 1985). The observation was confirmed later by are-evaluation of the existing satellite data. Prior to that the scientists did not yetsufficiently trust the accuracy of these data and might have ignored the ozone holecompletely.

Modern optical remote sensing instruments can be divided in active and pas-sive devices. The former have their own light source whereas the latter utilizenaturally existent sources like the sun, the moon, the earth, or stars. Passive de-vices are much easier to build and to operate. Prominent examples of this speciesare ground based, air borne, and space borne spectrometer or photometer. Themeasurement is an integral over the whole light path and therefore the instrumentcannot provide range resolution. As a result, only models can emphasize correc-tions for errors resulting from inhomogeneous distributions. These instrumentscommonly use lamps as a light source, which need to be placed at a suitabledistance from the detector (bi-static system). Alternatively the emitted light canbe reflected back to the place of origin (mono-static system). Depending on thebandwidth of the light source another distinction of optical remote sensing in-struments can be made. Long path differential absorption techniques, like DOAS(differential optical absorption spectrometer) and FTIR (Fourier transform in-frared spectrometer) use broadband sources. They are able to quantify a largenumber of atmospheric compounds simultaneously with a very high sensitivity

2] LIDAR-MONITORING OF THE AIR 415

(Platt et al., 1979). Mounted on space or airborne platforms they provide mea-surements on regional and global scales. Those advantages explain the widespreaduse of long-path absorption techniques for studies in atmospheric chemistry andglobal long term monitoring. Again, however, the instrumentation is only capa-ble to measure atmospheric constituents along a defined optical path. It does notprovide 3-dimensionally resolved concentration distributions, which are most im-portant in order to understand the dynamics of atmospheric reactions and transportprocesses.

2. Conventional LIDAR Measurements

3-dimensional range-resolved optical measurements can only be achieved by em-ploying the LIDAR techniques (Light Detection and Ranging). In such systemsa pulsed laser is commonly emitted into the atmosphere. There—after somedistance—the light is scattered, e.g., by means of Rayleigh- or Mie-scattering.A portion of the backscattered light is then re-collected at the site of the emitterwith a telescope. A time-resolved detection scheme allows to determine the lengthof the light path from the emitter to the scatter site and back to the receiver. Thesecond and third dimension of the measurement can be obtained by horizontallyor vertically turning either the entire system around, or by deflecting the emittedand received beams with appropriate mirrors. In order to discriminate particularatmospheric pollutants, the Differential Absorption LIDAR technique (DIAL) iscommonly used: It employs a set of two (or more) not too different wavelengths,which are collinearly emitted into the air. If one of those wavelengths is more ab-sorbed by a specific pollutant than the other, a data analysis allows to retrieve theconcentration distribution of this pollutant along the light path. A pioneer of thismethod is Herbert Walther, who—in the early days—established one of the firstDIAL systems, which was used for realistic pollution monitoring purposes. Oneof the results is depicted in Fig. 1 (Rothe et al., 1974); it shows an example of aremote obtained concentration map of NO2 in the area of a chemical plant. Theresult clearly indicates the location of the NO2 emission source.

Results as shown in Fig. 1 convincingly demonstrated the great monitor-ing power of the differential absorption LIDAR techniques. So we decided toconstruct—supported by the Swiss National Science Foundation—a similar sys-tem which was mobile. This allowed us to perform DIAL measurements at siteswhere most relevant results could be expected. In the Rhone valley we measured,for example, the pollutant distributions of NO2 and SO2. The results convinc-ingly demonstrated—especially during inversion weather situations—the envi-ronmental damage caused by industrial emissions in such mountainous regions(Beniston et al., 1990). We also performed DIAL measurements of traffic-causedNO-concentrations over urban areas like Lyon and Geneva (Kölsch et al., 1992).


FIG. 1. Measurement of the distribution of NO2 emission from a chemical plant by the differentialabsorption technique (Rothe et al., 1974).

The results clearly indicated the architectural need of sufficient ventilation spaceto avoid intolerable traffic smog formation. At that time one city was—due to ismultiple gradient situation—of greatest interest: the still divided city of Berlin:One side of the city was mainly heated by local brown coal, the other one byoil; one side was mainly motorized by two-stroke vehicles, the other one byfour-stroke engines, etc. We therefore brought our Swiss-licensed LIDAR sys-tem across the transit way to West Berlin and performed there in winter SO2-measurements across the city. One of the obtained results is presented in Fig. 2.It shows an example of unpurified emission from “Kraftwerk Mitte” which—atthat time still—burnt brown coal. From there the plume spread—unhindered bythe Berlin wall—over the entire city, contributing to the creation of dangerouswinter-smog (Kölsch et al., 1994).


FIG. 2. DIAL-measurement of a SO2-plume from a power station in former “East”-Berlin takenin winter 1988.

Today, fortunately, the winter-smog situation in Berlin has much improved,because the unpurified emission of industrially burnt local brown coal, which isstrongly contaminated with sulphur, is not allowed any longer. Another air pol-lution problem, however, the so-called photo-dynamical, or summer-smog, stillremains; due to the constantly increasing traffic density it even increases. Thecause of ozone formation results from sunlight photo-dissociation of NO2 to NOand O. Then the radical oxygen reacts with O2 forming the ozone. In the boundarylayer near ground ozone is—due to its high toxicity—the most important ingredi-ent of summer-smog. In order to monitor the development, we established on thetop of the centrally located Charité building in the middle of Berlin a LIDAR ob-servation station. A typical ozone-concentration map taken over Berlin on a sum-mer day at 100 m above ground is shown in Fig. 3. Surprisingly the distributionreaches its highest values in the recreational area of the “Tiergarten”, where notraffic passes (Stein et al., 1996). This clearly indicates the subtle equilibrium ofozone-formation and -destruction mechanisms due to the Leighton-relationship.

Even more threatening than the ozone increase in the troposphere is its de-pletion in the stratosphere: The O3 amount of the undisturbed stratosphere is sohigh that no summer smog- related increase can compensate the O3 loss in thestratosphere and its consequences with regard to the UV-radiation shield. AgainHerbert Walther was among the first, who performed from the mountain station“Zugspitze” DIAL measurements of stratospheric ozone concentrations (Werneret al., 1983), see Fig. 4. Responsible for the ozone depletion process is anthro-pogenic CFC, which after being released into the air, slowly migrates over yearsto the stratosphere. There it gets photo-dissociated by the hard UV-radiation form-ing active Cl. This destroys in a cyclic photo-catalytic process, e.g., on the particlesurface in natural polar stratospheric clouds (PSC) huge amounts of the ozone(Crutzen and Arnold, 1986). The presence of such PSC’s, is therefore most im-portant with regard to the stratospheric O3 depletion process. For this reason we


FIG. 3. Horizontal distribution of the ozone concentration over Berlin taken 100 m above groundin summer 1997.

routinely performed PSC-LIDAR measurements in Sodankyla in North Finland,which is well located for observing the rim of the arctic vortex. Occasionally weeven detected PSC’s from the Charité station above Berlin. For that purpose wehad developed a 4-wavelength backscatter LIDAR, which was capable to providealtitude profiles of aerosol abundances, and their size distribution. By performingdepolarization measurements we were even able to distinguish between liquid andsolid particles (Stefanutti et al., 1992).

The application of the above-mentioned “conventional” optical remote sensingmethods allowed us also to spot their limits: DOAS and FTIR detect simulta-neously a large variety of air constituents and pollutants, but only along a fixedoptical path. Further, they do not provide detailed information about the aerosol.DIAL provides up to 3-dimensional-resolved concentration maps of individualpollutants. The amount of simultaneously detectable substances, however, is lim-ited; usually it is just one substance at a time. Moreover in the IR-region themethod is not very sensitive, mainly because of the rather poor backscatter cross-section there. The multiple-wavelength backscatter LIDAR provides aerosol dis-tributions at a high spatial resolution. Particle compositions, however, cannot bedetermined by this method. Therefore a method was required, which does notexhibit these disadvantages.


FIG. 4. Monthly averages of stratospheric ozone concentrations for February 1983 (lower curve)and June 1983 (upper curve) (Werner et al., 1983).

3. The Femtosecond-LIDAR Experiment

New perspectives for optical remote sensing arose from the development of ultra-fast laser sources. They open an intensity regime, in which entirely new opticalphenomena occur, and where the classical laws of optics are generally no longervalid. Already 10 years ago our group performed encouraging laboratory exper-iments for the characterization of aerosol particles like water micro-droplets bymeans of nonlinear femtosecond (fs) laser spectroscopy (Kasparian et al., 1997).Then, with amplified fs-laser systems a power regime was made available, whichseemed to allow the realization of an old dream: A white light emitting artificialstar, which could freely be moved across the sky. In the laboratory this can sim-ply be achieved by tightly focusing (centimeters) a powerful nanosecond laserinto air, which then creates the well-known plasma focus. Performing the exper-iment at larger distances (kilometers) is more demanding, because it requires asignificantly higher power. With the advent of high-power fs-laser sources such


FIG. 5. Experimental setup of the femtosecond LIDAR.

a power level, could be reached even at large focusing lengths, provided that thegroup velocity dispersion (GVD) of the spectrally broad fs-pulse was appropri-ately compensated.

First experiments were carried out in 1996 at the laboratory of R. Sauerbrey inJena, where a 100 fs-laser with 4 terawatt peak power had just been installed. Forthe first experiments the laser was slightly focused with a 30 m focusing lens. Therequired compensation of the group velocity dispersion of the spectrally broadlaser pulse was achieved by negatively pre-chirping the time structure of the laserpulse, so that its differently fast propagating spectral components converged after30 m propagation in air at the site of the focus (temporal lens). As Fig. 5 shows,this was achieved by accordingly detuning the pulse compressor of the laser be-hind its last amplification stage. Then the slightly pre-focused and pre-chirpedlaser beam was fired out of the laboratory into the night time sky. Identical to thesetup of a conventional LIDAR system a telescope was directed along the emittedlaser beam in order to collect the backscattered light and feed it into a spectrom-eter. This allowed a spectrally and temporally resolved detection of the returnsignals. Surprisingly, however, the expected artificial star did not appear. Instead


FIG. 6. Photo of the first fs-LIDAR experiment in Jena: A clearly visible, white plasma channelemerges from the emitted IR-laser beam.

of that a clearly visible, extended white light channel emerged—as shown in thephotography of Fig. 6—from the almost invisible titanium–sapphire laser pulse(λ = 800 nm) (Wöste et al., 1997). Time resolved measurements indicated whitelight signals coming from altitudes up to 12 kilometers. Their spectrum exceededthe entire visible range. Later it was shown in the laboratory, that these amazingplasma channels were even electrically conductive (Schillinger and Sauerbrey,1999).

4. Nonlinear Propagation of Ultra-Intense Laser Pulses

The same effect of white light plasma-channel formation in high-intensity fs-laserbeams had briefly before also been seen in the laboratory (Braun et al., 1995).Soon after the underlying physical principles were also described: High powerlaser pulses propagating in transparent media—like air—undergo nonlinear prop-agation. For high intensities I , the refractive index n of the air is modified by theKerr effect (Kelley, 1965):

(1)n(I) = n0 + n2 · I,


FIG. 7. Kerr self-focusing (a) and plasma defocusing (b) of a high power laser beam.

where n2 is the nonlinear refractive index of the air (n2 = 3.10–19 cm2/W). Asthe intensity in a cross-section of the laser beam is not uniform, the refractiveindex increase in the center of the beam is larger than on the edge (Fig. 7a).This induces a radial refractive index gradient equivalent to a lens (called ‘Kerrlens’) of focal length f (I). The beam is focused by this lens, which leads to anintensity increase resulting in turn to a shorter focal length lens, so the wholebeam collapses on itself. Kerr self-focusing should therefore prevent propagationof high power lasers in air. The Kerr effect becomes significant when the self-focusing effect is larger than natural diffraction, i.e. over a critical power Pcrit:

(2)Pcrit = λ2

4π · n2.

If the laser pulse intensity reaches 1013–1014 W/cm2, higher order nonlinearprocesses occur such as multi-photon ionization (MPI). At 800 nm, 8 to 10 pho-tons are needed to ionize N2 and O2 molecules giving rise to plasma (Talebpouret al., 1999). The ionization process can involve tunneling as well, because of thevery high electric field carried by the laser pulse. However, following Keldysh’s


FIG. 8. Filamentation of a high power laser beam as it propagates in air (Berge and Couairon,2001).

theory (Keldysh, 1965), MPI dominates for intensities lower than 1014 W/cm2.In contrast to longer pulses, fs-pulses combine high ionization efficiency dueto their very high intensity, with a limited overall energy, so that the generatedelectron densities (1016–1017 cm−3) are far from saturation. Losses by inverseBremsstrahlung are therefore negligible, in contrast to a plasma generated by ananosecond laser pulse. However, the electron density ρ induces a negative varia-tion of the refractive index and, because of the radial intensity profile of the laserbeam, a negative refractive index gradient. This acts as a negative lens, whichdefocuses the laser beam, as schematically shown in figure Fig. 7b.

Kerr self-focusing and plasma defocusing should thus prevent long distancepropagation of high power laser beams. However, a remarkable behavior oc-curs in air, where both effects exactly compensate and give rise to a self-guidedquasi-solitonic propagation (Berge and Couairon, 2001). The laser beam is firstself-focused by the Kerr effect. This focusing then increases the beam intensityand generates a plasma by MPI, which in turns defocuses the beam. The inten-sity then decreases and plasma generation stops, which allows Kerr re-focusingto take over again. This dynamic balance between Kerr effect and plasma gener-ation leads to the formation of stable structures called “filaments” (Fig. 8). Theselight filaments have remarkable properties. In particular, they can propagate overseveral hundreds of meters, although their diameter is only 100–200 µm, whichwidely beats the normal diffraction limit. Their intensity (∼1014 W/cm2), theirenergy (∼1 mJ), their diameter, and their electron density (1016–1017cm−3) arealmost constant.

The laser pulse propagation is governed by the Maxwell wave equation:

(3)∇2E − 1

c2· ∂

2E

∂t2= μ0 · σ · ∂E

∂t+ μ0 · ∂

2P

∂t2,

where σ is the conductivity and accounting for the losses, and P is the polariza-tion of the medium. In contrast to the linear wave propagation equation, P nowcontains a self-induced nonlinear contribution corresponding to Kerr focusing and


plasma generation:

(4)P = PL + PNL = ε0 · (χL + χNL) · E,where χL and χNL are the linear and nonlinear susceptibilities, respectively. Con-sidering a radially symmetric pulse propagating along the z-axis in a referenceframe moving at the group velocity vg yields the following nonlinear Schrödingerequation (NLSE) (Berge and Couairon, 2001):

(5)∇2⊥ε + 2i

(k∂ε

∂z

)+ 2k2n2 · |ε|2 · ε − k2 ρ

ρc· ε = 0,

where ε = ε(r, z, t) is the pulse envelope of the electric field and ρc the criticalelectron density (1.8 × 1021 cm−3 at 800 nm). ε is assumed to vary slowly ascompared to the carrier oscillation and to have a smooth radial decrease. In thisfirst order treatment, group velocity dispersion (GVD) and losses due to multi-photon and plasma absorption are neglected (σ = 0). In (5), the Laplacian modelswave diffraction in the transverse plane, while the two last terms are the nonlinearcontributions: Kerr focusing and plasma defocusing (notice the opposite signs).The electronic density ρ(r, z, t) is computed using the rate equation (6) in a self-consistent way with (5):

(6)∂ρ

∂t− γ |ε|2α (ρn − ρ) = 0,

where ρn is the neutral molecular concentration in air, γ the MPI efficiency, and αthe number of photons needed to ionize an air molecule (typ. α = 10 (Talebpouret al., 1999)). Numerically solving the NLSE equation leads to the evolution ofthe pulse intensity

I = |ε|2as a function of propagation distance, as shown in Fig. 8. Initial Kerr lens self-focusing and subsequent stabilization by the MPI-generated plasma are well re-produced by these simulations. Notice that the filamentary structure of the beam,although only 100 µm in diameter, is sustained over 60 m. Numerical instabil-ity related to the high nonlinearity of the NLSE prevents simulations over longerdistances.

For higher laser powers P � Pcrit the beam breaks up into several localizedfilaments. The intensity in each filament is indeed clamped at 1013–1014 W/cm2

corresponding to a few mJ, so that an increase in power leads to the formationof more filaments. Figure 9 shows a cross-section of laser beams undergoingmono-filamentation (Fig. 9a, 5 mJ) and multi-filamentation (Fig. 9b, 400 mJ).The stability of this quasi-solitonic structure is remarkable: filaments have beenobserved to propagate over more than 300 m. However no direct measurementscould be performed on longer distances yet, because of experimental constraints.


FIG. 9. (a) Mono-filamentation (5 mJ) and (b) multi-filamentation (400 mJ) of high power laserbeams. Scale: The size of the filament in panel (a) (200 µm) is similar to one of the many filaments onpanel (b). The rings around the filaments are colorful and correspond to conical emission.

The spectral content of the emitted light is of particular importance for LIDARapplications. Nonlinear propagation of high intensity laser pulses not only pro-vides self-guiding of the light but also an extraordinary broad continuum spanningfrom the UV to the IR. This super-continuum is generated by self-phase modula-tion as the high intensity pulse propagates. As depicted above, Kerr effect leads,because of the spatial intensity gradient, to self-focusing of the laser beam. How-ever, the intensity also varies with time, and the instantaneous refractive index ofthe air is modified as:

(7)n(t) = n0 + n2 · I (t).This gives rise to a time dependent phase shift dφ = −n2I (t)ω0z/c, where ω0is the carrier frequency, which generates additional frequencies ω in the spectrumω = ω0 + dφ/dt . The smooth temporal envelope of the pulse induces thus astrong spectral broadening of the pulse around ω0. Figure 10 shows the spectrumemitted by filaments that were created by the propagation of a 2 TW pulse in thelaboratory. The super-continuum spans from 400 nm to over 4 µm, which coversabsorption bands of many trace gases in the atmosphere (methane, VOCs, CO2,NO2, H2O, etc.). Recent measurements showed that the super-continuum extendsin the UV down to 230 nm (see below), due to efficient third harmonic generation(THG) and frequency mixing (Akozbek et al., 2002; Yang et al., 2003). Theseresults open further multi-spectral LIDAR applications for detecting aromatics,SO2, and ozone.

Most of the filamentation studies showed that white light was generated in thefilamentary structure, and leaking due to coupling with the plasma in form ofa narrow cone in the forward direction (called “conical emission”, see Fig. 9a)(Kosareva et al., 1997; Nibbering et al., 1996). This cone spans from the longerwavelengths in the center to the shorter wavelengths at the edge, with a typicalhalf-angle of 0.12◦.


FIG. 10. Super-continuum generation in air: the dots are measurements in the laboratory aftera few meters propagation while the lines exhibit further broadening after having propagated acrosskilometer distances (Mejean et al., 2003).

However, an important aspect for LIDAR-applications is the angular distrib-ution of the white light continuum in the near backward direction. Already inthe first fs-LIDAR experiments, a pronounced backscattering component of theemitted white light was observed (see Fig. 6). For this reason angular resolvedscattering experiments have been performed. The emission close to the backwarddirection of the super-continuum from light filaments was found to be signifi-cantly enhanced as compared to linear Rayleigh–Mie scattering (Yu et al., 2001).Figure 11 shows the comparison of the linearly backscattered light (Rayleigh–Mie) from a weak laser beam and the nonlinear emission from a filament, forboth s- (left part) and p- (right part) polarizations. At 179◦ the backward enhance-ment extends an order of magnitude. An even greater enhancement is expected at180◦ (limited by the experimental apparatus). The enhancement may qualitativelybe attributed to a copropagating, self-generated longitudinal index gradient dueto plasma generation, inducing a back-reflection. Combined with self-guiding,which drastically reduces beam divergence, this aspect is extremely important forLIDAR experiments: Unlike backscatter-based, conventional LIDAR-systems asignificantly larger portion of the emitted white light is, therefore, collected bythe fs-LIDAR receiver.

To summarize, nonlinear propagation of TW-laser pulses exhibit outstandingproperties for multi-spectral LIDAR measurements: extremely broadband coher-


FIG. 11. Angular emission of the emitted white light compared to linearly scattered light.

ent light emission (“white light laser”), confined in a self-guided beam, and anincreased back-reflection to the emitter as the laser pulse propagates.

5. White Light Femtosecond LIDAR Measurements

Since filamentation counteracts diffraction over long distances, it allows to de-liver high laser intensities at high altitudes and over long ranges. This contrastswith linear propagation, in which the laser intensity is always decreasing whilepropagating away from the source, unless focusing optics such as large-aperturetelescopes with adaptative optics are used to reach focal lengths of the order ofhundreds of meters.

The distance R0 at which high powers are reached and thus where filamentationstarts, can be controlled by the following laser parameters: the initial laser diame-ter and divergence, and the pulse duration and chirp. The geometrical parametersare set by the transmitting telescope while the temporal parameters, leading to“temporal focusing” are determined by the grating compressor. These parametersare used to control the power and the intensity of the beam while propagating.A particular aspect is temporal focusing using an initial chirp, as it can be usedtogether with the air GVD, to obtain the shortest pulse duration and thus the on-set of filamentation at the desired location R0. The compressor is then aligned in


such a way that a negatively chirped pulse is launched into the atmosphere, i.e. theblue component of the broad laser spectrum precedes its red component. In thenear infrared, air is normally dispersive, and the red components of the laser spec-trum propagate faster than the blue ones. Therefore, while propagating, the pulseshortens temporally and its intensity increases. At the pre-selected altitude R0, thefilamentation process starts and white light is generated.

The extraordinary properties of the white light emitting plasma channels con-vinced our funding agencies DFG and CNRS to establish the French/GermanCooperation project “Teramobile”, which allowed us to construct a mobile fem-tosecond LIDAR-system (Wille et al., 2002). Its basic setup is according to theone shown in Fig. 5: the fs-laser pulse—after passing the compressor set as achirp generator—passes an emission telescope and then vertically sent into theatmosphere. The back-scattered portion of the white light generated in the at-mosphere is then collected and spectrally resolved by the LIDAR receiver. Thesystem was installed in a mobile, self-contained standard container, so it couldeasily be moved. This allowed us to perform fs-white-light LIDAR measurementsat different, relevant sites. Most rewarding in this regard was a campaign, whichwe performed at the Thüringer Landessternwarte Tautenburg (Germany). Therewe could make use of the detection power of its 2 m diameter telescope, whichwe operated at the high-resolution imaging mode. During these experiments thelaser was launched into the atmosphere, and the backscattered light was imagedthrough the telescope of the observatory.

Figure 12a shows a typical image at the fundamental wavelength of the laserpulse (λ = 800 nm), over an altitude range from 3 to 20 km. In this picture, strongscattering is observed from a haze layer at an altitude of 9 km and a weaker onearound 4 km. In some cases, scattered signal could be detected from distances upto 20 km. Tuning the same observation to the blue–green band (385–485 nm), i.e.observing the white light super-continuum, leads to the images shown in Fig. 12band c. As observed, filamentation and white light generation strongly depends onthe initial chirp of the laser pulse, i.e. white light signal can only be observedfor adequate GVD pre-compensation (Fig. 12b). With optimal chirp parameters,the white light channel could be imaged over more than 9 km. It should also bepointed out that, as presented above, the angular distribution of the emitted whitelight from filaments is strongly peaked in the backward direction, and that mostof the light is not collected in this imaging configuration.

Under some initial laser parameter settings, conical emission due to leakage outof the plasma channel could also be imaged on a haze layer, as shown in Fig. 12d.Since conical emission is emitted side-wards over the whole channel length, thevisible rings indicate that under these experimental conditions the channel wasrestricted to a shorter length at low altitude.

Conventional LIDAR systems are based on the use of backscatter processeslike Rayleigh-, fluorescence or Mie-scattering, which only return a small fraction


FIG. 12. Long-distance filamentation and control of nonlinear optical processes in the atmosphere.Pictures of the beam propagating vertically, imaged by the CCD camera of a 2-m telescope.(a) Fundamental wavelength, visible up to 25 km through 2 aerosol layers. (b)–(d) Super-continuum(390–490 nm band) generated by 600-fs pulses with respectively negative (GVD pre-compensating),positive, and slightly negative chirp. On picture (d), conical emission appears as a ring on thehigh-altitude haze layer.

of the emitted light back to the observer. This leads necessarily to an unfavorable1/R2-dependency of the received light, where R is the distance from the scatterlocation to the observer. When spectrally dispersed, this usually leaves too smallsignals on the receiver, as arc-lamp-based LIDAR experiments have shown in thepast (Strong and Jones, 1995). Unlike these linear processes, the femtosecondwhite light plasma channel presents an almost ideal source for LIDAR applica-tions: Its strong backward emission allows high spectral resolution of the observedsignals, even at large distances. As a result, highly resolved spectral fingerprintsof atmospheric absorbers along the light path can be retrieved.

Figure 13a shows examples of spectrally filtered white light LIDAR returns inthree different spectral regions (visible at 600 nm and UV around 300 and 270 nm,1000 shots average). These profiles of in situ generated white light show scatterfeatures of the planetary boundary layer. The much faster decrease of the UVsignal is due to the stronger Rayleigh scattering at shorter wavelengths, and, theabsorption at 270 nm (compared to 300 nm) due to high ozone concentration. Thewhite light spectrum generated over long distances in the atmosphere showed sig-


FIG. 13. (a) White light LIDAR returns. The O3 absorption at 270 nm is clearly visible; (b) whitelight LIDAR resolved spectrum returned from 4.5 km altitude.


nificant differences with respect to the spectrum of Fig. 10, previously recorded inthe laboratory (Kasparian et al., 2000). Figure 13b displays the white light spec-trum backscattered from an altitude of 4.5 km (Mejean et al., 2003). The infraredpart of the spectrum (recorded with filters) is significantly stronger (full line, typ. 2orders of magnitude higher) than in the laboratory, which is very encouraging forfuture multi-VOC detection. A quantitative explanation of this IR-enhancementrequires the precise knowledge of the nonlinear propagation of the terawatt laserpulse, which cannot be simulated with the present numerical codes. However,it qualitatively indicates that the pulse shortens and/or splits while propagating,causing broader spectral components.

On the other end of the spectrum (not shown) it was observed that the super-continuum extends continuously down to 230 nm (limited by the spectrometer).This UV-part of the super-continuum is the result of efficient third harmonic gen-eration in air (Akozbek et al., 2002; Yang et al., 2003) and mixing with differentcomponents of the Vis-IR part of the spectrum. This opens very attractive ap-plications, such as multi-aromatics (Benzene, Toluene, Xylene, etc.) detectionwithout interference, NOx and SO2 multi-DIAL detection, and O3 measurements,for which the aerosol interference can be subtracted due to the available broad-band UV detection.

Very rich features arise from the white light backscattered signal, when it isrecorded across a high-resolution spectrometer, as shown in Fig. 13b. The spec-trum, which was detected from an altitude of 4.5 km with an intensified chargecoupled device (ICCD), shows a wealth of atmospheric absorption lines at highresolution (0.01 cm−1). The excellent signal to noise ratio (2000 shots average)demonstrates the advantages of using a high-brightness white light channel formulti-component LIDAR detection. The well-known water vapor bands around720 nm, 830 nm and 930 nm are observed simultaneously. Depending on the alti-tude (i.e. the water vapor concentration), the use of stronger or weaker absorptionbands can be selected. Figure 14 (upper) shows a higher resolution picture ofthe spectrum around 815 nm of the water vapor ((000) → (211) transition) andFig. 14 (lower) of the X → A transition of molecular oxygen. A fit using theHITRAN-database is shown in both cases. It leads to a mean water vapor concen-tration of 0.4%. Notice the excellent agreement with the database, demonstratingthat no nonlinear effects or saturation are perturbing the absorption spectrum.This is explained by the fact that the white light returned to the LIDAR receiveris not intense enough to induce saturation, and that the volume occupied by thefilaments (the white light sources) is very small compared to the investigated vol-ume.

The spectrum used to retrieve the water vapor concentration contains about700 data points. The use of so many wavelengths allows a major improve-ment in sensitivity as compared to the usual 2-wavelengths-DIAL. A systematic


FIG. 14. Upper: The water (000) → (211) transition. Lower: high-resolution spectrum of theX → A transition of molecular oxygen. The comparison with calculated results using the HI-TRAN-database shows excellent agreement in both cases. It leads to a mean water vapor concentrationof 0.4%.

study is in progress to quantify this gain, connected to the obtained signal-to-noise-ratio in each spectral element. Using a gated ICCD, the spectrum of theatmosphere can be recorded at different altitudes, yielding range resolved mea-surements.

Information about atmospheric temperature (and/or pressure) could be re-trieved from the line-shapes of the absorption lines. Another possibility is tomeasure the intensities of the hot bands, in order to address the ground state pop-ulation. As the molecular oxygen spectrum is very well known, O2 is particularlywell suited for this purpose. The access of the whole spectrum should again al-low to obtain significantly better precision than in former DIAL investigations(Megie and Menzies, 1980). The spectrum covered by the white light in Fig. 13bgives access to many bands to measure the H2O concentration, and to 2 bandsof O2 ((0)′′ → (0)′ and (0)′′ → (1)′ sequences of the X → A transition, around760 nm and 690 nm, respectively) to determine the atmospheric temperature. Thecombination of both information with good precision could allow to construct anefficient “relative humidity LIDAR profiler” in the future.


6. Nonlinear Interactions with Aerosols

Particles are present in the atmosphere as a broad distribution of size (from 10 nmto 100 µm), shape (spherical, fractal, crystals, aggregates, etc.), and composi-tion (water, soot, mineral, bio-agents, e.g., bacteria or viruses, etc.). The LIDARtechnique has shown remarkable capabilities in fast 3D-mapping of aerosols, butmainly qualitatively through the measurement of statistical average backscatter-ing and extinction coefficients. The most advanced methods use several wave-lengths, usually provided by the fixed outputs of standard lasers. The set ofLIDAR equations derived from the obtained multi-wavelength LIDAR data issubsequently inverted using sophisticated algorithms or multi-parametric fits ofpre-defined size distributions with some assumptions about the size range andcomplex refractive indices. In order to obtain quantitative mappings of aerosols,complementary local data (obtained with, e.g., laser particle counters, or multi-stage impactors to identify the sizes and composition) are often used togetherwith the LIDAR measurements. The determination of the size distribution andcomposition using standard methods must, however, be taken cautiously as com-plementary data, because of its local character in both, time and space. Nonlinearspectroscopy using ultra-short laser pulses appears as a promising new techniquefor a quantitative aerosol detection. In this section, we describe important non-linear interactions that exhibit a real potential to simultaneously measure sizeand composition of aerosol mixtures, and to identify a particular type of parti-cle within an ensemble.

The first approach of using ultra-intense laser pulses in a LIDAR arrangementto characterize aerosol particles is a direct extension of the multi-wavelength scat-tering technique. The extraordinary broadness of the super-continuum, spanningfrom the UV to the mid-IR, opens new perspectives in this respect. Instead ofsome discrete wavelengths, backscattered and extinction coefficients can now beobtained range-resolved over the whole continuum. This is particularly advanta-geous for mixtures of unidentified particles, where a wide range of size parametersx = 2πr/λ has to be addressed. The data inversion can follow the stream ofthe already developed sophisticated multi-wavelength algorithms, which will bevery powerful using such a wide and continuous spectrum. Besides the exper-iments performed with the Teramobile system (Kasparian et al., 2003), somerecent LIDAR measurements of aerosols using the femtosecond super-continuumgenerated in rare gas before transmitting were reported (Galvez et al., 2002). Partof the spectrum might also be analyzed at higher resolution, in order to detect vari-ations in the imaginary part of the refractive index, characterizing the absorptionprocess and thus provide some further insight on the particles composition.

A key parameter in these femtosecond LIDAR experiments will be the locationof the onset and the end of filaments. In particular, if the pulses are shaped in sucha way (negative chirp, see above) that filamentation occurs at short distances and


lasts only for some hundreds of meters, the light scattered back from longer dis-tances can be considered as linearly scattered. The data inversion can thus safelybe performed with the usual linear LIDAR algorithms. Conversely, if the laserpulses are initially shaped in a way that high intensity and filaments are presentin the investigated volume, nonlinear effects are induced directly in the aerosols,and new inversion algorithms have to be developed (Faye et al., 2001; Kasparianand Wolf, 1998). Examples of these nonlinear processes induced in the aerosolparticles are presented below.

Femtosecond laser pulses are able to provide very high pulse intensity at lowenergy, which allows to induce nonlinear processes in particles without thermaldeformation effects. The most prominent feature of nonlinear processes in aerosolparticles, and in particular in spherical micro-droplets, is the strong localizationof the emitting molecules within the particle, and the subsequent backward en-hancement of the emitted light. This unexpected behavior is extremely attractivefor LIDAR applications. For homogeneous spherical micro-particles, moleculesin certain regions are indeed more excited than others because of the focusingproperties of the spherical micro-resonator. Further localization is achieved by thenonlinear processes, which typically involve the nth power of the internal inten-sity In(r) (r denotes the position inside the droplet). Because the droplet acts asa spherical lens, the re-emission from these internal focal points occurs predomi-nantly in the backward direction. The backward enhancement can be explained bythe reciprocity principle (Boutou et al., 2002; Hill et al., 2000): Reemission fromregions of high I (r) tends to return toward the illuminating source by essentiallyretracing the direction of the incident beam that gave rise to the focal point.

We investigated, both theoretically and experimentally, incoherent multi-photon processes involving n = 1 to 5 photons (Boutou et al., 2002; Favre et al.,2002; Hill et al., 2000). For n = 1, 2, 3, we focused on multi-photon-excited flu-orescence (MPEF) of fluorophors- or amino acids-containing droplets. For n = 5(or more) photons we examined laser-induced breakdown (LIB) in water micro-droplets, initiated by multi-photon ionization (MPI). The ionization potential ofwater molecules is Eion = 6.5 eV (Noack and Vogel, 1999; Williams et al.,1976), so that 5 photons are required at a laser wavelength of 800 nm to initiatethe process of plasma formation. The growth of the plasma is also a nonlinearfunction of I (r). We showed that both localization and backward enhancementstrongly increases with the order n of the multi-photon process. Both MPEF andLIB have the potential of providing information about the aerosol composition.

The strongly anisotropic spontaneous emission of MPEF in a micro-dropletwas demonstrated on Coumarin 510 doped ethanol (Hill et al., 2000) droplets withsizes ranging from 10 to 50 µm. The directionality of the emission is dependenton the increase of n, because the excitation process involves the nth power of theintensity In(r). The ratio Rf = U(180◦)/U(90◦) increases from 1.8 to 9 whenn changes from 1 to 3. For 3-photon MPEF (3PEF), fluorescence from aerosol


micro-particles is, therefore, mainly backwards emitted, which is ideal for LIDARexperiments.

The backward enhancement also depends on the particle relative refractive in-dex m: the higher m the higher Rf . When excited by one photon at 266 nm,Rf from dye-doped polystyrene latex (PSL) micro-spheres (m = 1.59, typicaldiameter = 22.1 µm, fluorescence peaked at 375 nm) reaches 3.2 instead of1.8 for Coumarin 510 doped ethanol droplet. Such enhancement effect is alsoobserved for nonspherical transparent particles such as clusters of small (diame-ter smaller than 2 µm) PSL-spheres (Yong-Le et al., 2002). Another remarkableproperty is that the backward enhancement is insensitive to the size if the dropletdiameter exceeds some micrometers. This was shown by both the calculationsfor liquid spherical droplets and by experiments on clusters of PSL-spheres, forwhich the equivalent diameter was changed from 2 to 10 µm. However, althoughthe Rf ratio is not sensitive to the particle shape at least for a one-photon ex-citation process, the high resolution 2D-angular pattern in the near backwarddirection might be specific of its morphology.

Laser-induced breakdown (LIB) experiments were performed in pure andsaline water droplets. The white light (500±35 nm) angular distribution was mea-sured in the scattering plane for an incident intensity of 1.8 × 1012 W/cm2. Theobserved far-field emission is strongly enhanced in the backward direction, andexhibits a secondary narrow lobe near 150◦. The agreement between the experi-mental results and our Lorentz–Mie calculations (Favre et al., 2002) is excellent.LIB then takes place only at the internal hot spot of the droplet, and generatesa plasma of nanometric dimensions because of the I 5(r) dependency of multi-photon ionization (MPI). The white light emitted by the nanoplasma has a ratioRp = Up(180◦)/Up(90◦) that exceeds 35, i.e. 3× higher than for 3PEF.

The spectrum and the related plasma temperature have been measured by usingan optical multi-channel analyzer (OMA). The broadband visible emission wasrecorded in the backward direction from pure and saline droplets with variousincident intensities. In Fig. 15a we show that in the case of saline droplets andfor an incident intensity Iinc = 1.6 × 1012 W/cm2, the spectrum can be fittedby a Maxwell–Planck law, in agreement with laser heated plasma emission. Also,when the incident intensity is gradually increased to 1013 W/cm2 (curves (b)and (c)), the emission spectrum shifts towards the blue consistent with an increaseof the plasma temperature from 5000 to 7000 K. Similar behavior has been ob-served for pure water droplets (Fig. 15d) but unexpected and unidentified atomicor molecular lines appear in the spectrum. The shift of the emission maximum iscorrelated to the change in the angular distribution.

Dye-doped micro-droplets, because of their high multi-photon absorptioncross-sections, are good test cases to demonstrate the advantage of combiningMPEF and LIDAR techniques in order to identify the presence of fluorescingaerosols. An attractive application of the combined techniques is bio-aerosol de-


FIG. 15. Broadband emission spectra of saline water droplets (a), (b), (c) irradiated at increasingintensities from 1.6×1012 W/cm2 (curve (a)) to 1013 W/cm2 (curve (c)). The spectra exhibit besidesthe Na D lines continua that can be fitted to the Maxwell–Planck law, yielding plasma temperaturesfrom 5000 to 7000 K. Pure water droplets (d) show a similar behavior.

tection in the atmosphere. For this purpose, we performed the first multi-photonexcited fluorescence LIDAR detection of biological aerosols. The particles, con-sisting of water droplets containing 0.03 g/l riboflavin (a characteristic tracer ofbio-agents (Hill et al., 2001; Pan et al., 2001), were generated at a distance of 50 mfrom the Teramobile system. The size distribution peaked around 1 µm, a typicalsize of airborne bacteria. Riboflavin was excited with two photons at 800 nm andemitted a broad fluorescence around 540 nm. This experiment is the first demon-stration of remote detection of bio-aerosols using a 2PEF-femtosecond LIDAR(Fig. 16) (Kasparian et al., 2003). The broad fluorescence signature is clearly ob-served from the particle cloud (typ. 104 p/cm3), with a range resolution of a fewmeters. As a comparison, droplets of pure water did not exhibit any parasitic fluo-rescence in this spectral range. However, a background is observed for both typesof particles, arising from the scattering of white light generated by the filamentsin air. Competition between super-continuum generation before the laser reaches


FIG. 16. Two-photon excited fluorescence (2PEF)-LIDAR detection of bio-aerosols compared towater-aerosols.

the particles and 2PEF within the particles appeared critical. A possible solu-tion to this problem is to adapt the initial pulse duration, chirp, and geometricalcharacteristics of the laser such that the needed high intensity is reached exactlyat the target location. The use of tailored pulses is under investigation to solvethis problem; they will also be used to investigate possible simultaneous size andcomposition measurements in a pump–probe frame.

MPEF might be advantageous as compared to linear laser-induced fluorescence(LIF) for the following reasons: (1) MPEF is enhanced in the backward directionand (2) the transmission of the atmosphere is much higher for longer wavelengths.For example, if we consider the detection of tryptophan (another typical bio-tracerthat can be excited with 3 photons of 810 nm), the transmission of the atmosphereis typically 0.6 km−1 at 270 nm, whereas it is 3 × 10−3 km−1 at 810 nm (for aclear atmosphere, depending on the background ozone concentration). This mightcompensate the lower 3-PEF cross-section compared to the 1-PEF cross-sectionat longer distances. The most attractive feature is however the possibility of usingpump–probe techniques to measure both, composition and size.

7. Conclusion

The nonlinear propagation of ultra-short ultra-intense laser pulses providesunique features for LIDAR applications: a coherent white light emitting super-


continuum, which is self-guided and back-reflected towards the source. Back-ward enhancement also occurs for multi-photon-excited fluorescence (MPEF)and laser-induced breakdown (LIB) processes in aerosol particles. These char-acteristics open new perspectives for LIDAR measurements in the atmosphere:multi-component detection, reduced spectral interference, better precision dueto more absorption lines, improved IR-LIDAR measurements in aerosol-freeatmospheres, and remote measurement of aerosols size distributions and com-positions. The wide spread of the technique needs further characterization of thepropagation of laser pulses, in order to foresee the onset and the length of thefilaments, and to better control the intensity at each location along the laser path.

Beyond LIDAR applications the observed plasma channels exhibit fascinatingfurther perspectives; one of them is remote laser-induced breakdown spectroscopy(LIBS): By irradiating, for example, copper and iron plates with fs-plasma chan-nels, we have generated and identified their plasma lines over distances of 100 me-ters. Another exciting application is in the field of lightning control. The filamentsare electrically conductive; they may therefore be used as a laser lightning rod. Infirst exploratory experiments, which we performed at a high-voltage facility inBerlin, we could show that high-voltage discharges could indeed be triggered andguided—so far still over distances of some meters (Kasparian et al., 2003). An-other application concerns the triggered nucleation of water droplets. In a super-saturated atmosphere the laser-induced charges act—like in fog chambers—ascondensation germs for droplet formation. The phenomenon allows the remotedetection of super-saturation in the atmosphere.

As presented above, the potential applications of femtosecond plasma channelsin air are exciting, numerous and wide spread. It all resulted from an unsuccessfulattempt:

The creation of an artificial star!

8. Acknowledgements

The authors wish to thank the entire Teramobile team for the numerous hard work-ing nights, during which the results presented here were achieved. We owe partic-ular thanks to Roland Sauerbrey, André Mysyrowicz, Jerome Kasparian, EstelleSalmon, Jin Yu, Miguel Rodriguez, Holger Wille, Yves Bernard Andre, MichelFranco, Bernard Prade, Stelios Tzortzakis, Guillaume Mejean, Didier Mondelainand Riad Bourayou. Also we want to acknowledge the financial support of CNRSand DFG. One of the authors (L.W.) wishes to thank Professor Herbert Waltherfor having filled him with enthusiasm for spectroscopy, lasers and LIDAR.


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Index

Abbe’s theory of microscope, 106– and wavefront coding, 111–15Ablation, 238–9– from silicon, 239–40– – recoil pressure, 240–2– – surface morphology, 242–6– see also Coulomb explosion; Phase explo-

sionAblation regime, 234Ablation threshold, 234Acousto-optic modulators (AOMs), 95, 96Aerosols, 433–7Ambiguity functions, 110Amplitude object transmission function, 112Ancilla modes, nature, 144–7Anderson localization, 46Annihilation operators, 259, 261, 360Anti-bunching, 7, 14, 255Arbitrary power-law trap, 351–3– cross-excitation parameter, 352–3– ideal gas BEC statistics in, 364–70– with interacting Bose gas, 381–2– single-particle energy spectrum, 351, 364– see also Box; Harmonic trap; Isotropic har-

monic trapArtificial atoms see Quantum dotsAspheres, manufacture, 129–30Asymmetry coefficient, 382Atmosphere– parameters measurement, 432– super-saturation detection, 438Atom-field interaction, 191–4Atom laser– linewidth problem, 359– phase fluctuations of matter beam, 394Atom Trap Trace Analysis, 79Atomic conductance, fluctuations, 42–6Atomic photo cross sections, Ericson

fluctuations in, 49–51Atomic quasimomentum, 66Atomic transport, directed, due to interaction-

induced quantum chaos, 55–9ATTA method, 79

Autocorrelation function, 17– of average detection probability, 276–7, 285– of pupil function, 113Avalanche processes, 238Average detection probability, 259–60, 271,

273–4, 279–80

Barcode reading system, 109–10, 111BB84 protocol, 23, 26–7BCS state, momentum distribution of mole-

cules from, 180–1Beam splitters, 146, 147, 261–2BEC– counting statistics of molecules, 168–70– discovery, 307–8– experimental demonstration, 297– momentum distribution of molecules,

177–8– spinor-BECs, in spin-dependent optical lat-

tices, 223– wave-packet motion, 223BEC-BCS crossover, 152, 173BEC fluctuations– future research, 394– ground-state occupation distribution, 346–7– in ideal Bose gas, 323–8, 399–400, 401–2– in interacting Bose gas see Interacting Bose

gas– relations between statistics in various en-

sembles, 316–20– see also Condensate master equation ap-

proach; Quasiparticle approach; Sys-tems with broken continuous symme-try

Beliaev–Popov approximation, 394Benand–Marangoni instability, 247Berlin, 416–17Biexcitons, 5, 16Bifurcations, 246Bilinear transfer function, 115, 116Binomial distribution, 348Binomial theorem, generalized, 396Bio-aerosol detection, 435–7

443

444 Index

Blackbody radiation, 299Bloch oscillations, fermionic, 57Bloch vector model, 87Blood oxygenization determination, 123–5Bloodflow measurement, 125–7Blue-detuned lattices see Gray latticesBogoliubov canonical transformation, 373Bogoliubov coupling, 373–4Bogoliubov–Popov energy spectrum, 373,

379Bogoliubov’s 1/k2 theorem, 385, 389Bohr–Sommerfeld quantization rule, 300Born–Oppenheimer approximation, 213Bose, Satyendranath, 293–4, 298–9Bose–Einstein condensation see BECBose–Einstein distribution– history, 298–315– – analysis, 306–14– – Bose contribution, 299–303– – comparison between microstate counting

ways, 314–15, 395–7– – Einstein contribution, 303–6Bose–Hubbard Hamiltonian, 36, 55–6Boson–fermion model Hamiltonian, 163–4Bosonic bath, 56–9Bosonic commutation relation, 140Box– BEC fluctuations in ideal gas in, 369–70– temperature scaling of BEC fluctuations,

376, 382–3Breathing-mode wave-packets, 197, 200Broad resonance, 176

Caesium atom, 77–8, 80–1, 84–6, 88–9, 254– see also Neutral atoms; Single atomsCahn–Hilliard type equation, 247Calcium ion, single photon generation, 254Canonical ensemble, 335– counting statistics, 317– N -particle constraint, 316, 357– quasiparticle approach see Quasiparticle

approachCanonical partition functions, 318, 346, 404– contour integral representation, 404– for one-dimensional harmonic trap, 347Cataract surgery, 127Cavity-QED, 77, 84, 94, 95, 99Central limit theorem, 376Central retinal artery, 125, 126CFC, 417

Chaotic cavities, photonic transport in, 51–5Choroid, 125, 126, 127Chromium atom, feedback scheme in MOT,

82Clebsch–Gordan coefficients, 192Coherence function, 111–12– second-order, 7, 17Coherence theory, 175Coherence time, 255–6, 286Coherent backscattering, 55Coherent pumping, 156Coincidence probability see Joint detection

probabilityCold atom physics– quantum optics and, 152– see also Ultra-cold moleculesCold collisions, 81, 84, 100–1Computer numerical controlled (CNC)

machines, 129, 130Condensate master equation approach, 297– BEC statistics, 350–5– derivation of equation, 335–41– laser phase-transition analogy, 334–5, 342– low temperature approximation, 342–3– mesoscopic and dynamical effects in BEC,

355–7– physical interpretation of coefficients, 340– quasithermal approximation for non-

condensate occupations, 344–5– solution of equation, 345–9Conical emission, 425, 428, 429Cooling coefficient, 338–9, 340– low temperature approximation, 342– quasithermal approximation, 345Cooper instability, 172Cooper pairs, 180– coherent conversion, 173– detection, 174–5Corkscrew cooling, 206, 207Corneal reflex, 128Correlation function– connected four-point, 387– with jitter, 270, 272, 274– longitudinal, 386– second-order, 263, 264–6– transverse, 386Corrugation, variation of, 248Coulomb explosion, 238, 239, 241, 242, 246Counting statistics, 168–73, 317Coupled atom–molecule system, 155–8

Index 445

– see also Molecular micromaserCoupled fermion–molecule system, 175–7Creation operators, 257, 259, 261, 360Critical power, 422Critical temperature, 356–7, 383Cross-correlation function, 16, 18, 20Cubic phaseplates, 107–8, 117–20Cycling transition, 192

Decoherence, suppression, 86Degenerate approximation, 164–5Delbrück, M., 307, 313Density operator– photon pairs with jitter, 270– single-photon light field, 258Dephasing, suppression, 86Depth of focus (DOF), 106Depth resolution, lateral resolution and, 106Desorption dynamics, 234–8Diabatic potentials, 193DIAL, 415, 431, 432Dicke superradiance, 152, 164Differential optical absorption spectrometer

(DOAS), 414, 418Diffusive dynamics, 80–1Dipole trap (DT)– controlling atoms’ positions, 94–9– loading multiple atoms into, 89–91– optical conveyor belt, 95, 96– preparing single atoms in, 82–4DOAS, 414, 418Dressed atom approach, 228Drift-mode ToF spectra, 238–9Dynamical localization, 46Dynamical master equation see Condensate

master equation

Effective nonlinear σ model, 385–8Einstein, Albert, 294, 303, 307Ekert protocol, 26Electro-optic feedforward amplifier, 148Electro-optic modulators (EOMs), 27Elliptic islands, 59–61Emission-time jitter, 273–5, 285–6, 287Energy coupling, 229–33, 239Energy gap, in trap, 377, 379, 380Energy spectrum, effective, 380Entanglement schemes, four-photon, 100Equilibrium entropy, of ideal gas, 306Ericson fluctuations, 49–51

Ethanol, 434–5Excess coefficient, 382Excitons, 5, 16Eye, length measurement, 127–9

Fermi gas with superfluid component, 171–3Fermions, interaction with bosonic bath,

56–9Filamentation, control of start, 427–8Filaments, 423Finite negative binomial distribution, 347–8Flatness, 383Floquet–Bloch operator, 36–8Floquet operator, 38–40Fluctuation–dissipation theorem, 53, 385Flux operator, 258, 259Fock regime, 161–2Fokker–Planck equation, 342Fourier transform infrared spectrometer

(FTIR), 414, 418Free carrier absorption, 233, 239Frequency jitter, 271–3, 285–6, 287Frozen planet configuration, 64–5FTIR, 414, 418Fundus– bloodflow measurement, 125–7– imaging systems, 123– – numerical aperture, 106Fundus reflex, 128–9

Gamow factor, 198–9Gauge potential, 213Generalized binomial theorem, 396Generalized Zeta functions, 371, 388Generating cumulants, 362Ginzburg–Landau type free energy, 342Giorgini–Pitaevskii–Stringari result, defense,

392–3Girardeau–Arnowitt operators, 357Glauber coherent field, 169Glaucoma, 123Goldstone fields, 385, 386Good quantum numbers, 35Grand canonical ensemble, 321–8, 335,

389–90– chemical potential, 325– condensate fluctuations in ideal Bose gas,

323–8, 399–400, 401–2– condensate order parameter for ideal Bose

gas, 390

446 Index

– counting statistics, 317– mean condensate particle number in ideal

Bose gas, 321–3, 397–9– mean noncondensate occupation, 325Grand canonical partition function, 318, 404Gravitational constant, high precision mea-

surements, 67Gray lattices, 200–4– influence of magnetic fields on tunneling,

213–19– periodic well-to-well tunneling in, 208–13– see also Sloshing-type wave-packetsGroup velocity dispersion (GVD), 424

Hadamard gates, 94Hanbury Brown–Twiss setup, 8, 14–15, 17Harmonic trap, 350–1, 369– see also Isotropic harmonic trapHe II superfluid, 297Heating coefficient, 338–9, 340– low temperature approximation, 342– quasithermal approximation, 345Helium, liquid, Lambda point, 295Helium atom, 63–5Helium spectrum, semiclassical elucidation,

35Hemoglobin, absorption coefficients, 123Hohenberg–Mermin–Wagner theorem, 387Homogeneous broadening limit, 165Husimi phase space projections, 46, 48Hydrogen atom, microwave driven, 41, 44–6

Ideal Bose gas– central moments, 348–9– condensate fluctuations, 323–8, 399–400,

401–2– exact recursion relation for number of con-

densed atoms, 320–1– master equation, 336–7– mean condensate particle number, 321–3– – analytical expression for, 397–9Ideal gas + thermal reservoir model, 336Image intensity distribution, partial coherent

illumination, 115, 116Impact ionization, 233Incubation, 229–30, 236Inhomogeneously broadened Tavis–

Cummings model, 164, 170, 171InP quantum dots, 11–13Intensity modulation amplifier, 147

Interacting Bose gas– BEC fluctuations as anomalously large and

non-Gaussian, 375–9– – cumulants, 374–5– canonical-ensemble quasiparticles in

Bogoliubov approximation, 372–4– characteristic function for total number of

atoms, 374– crossover between ideal and interaction-

dominated BEC, 379–83– mesoscopic effects, 383– pair correlation effect, 380–1– see also Systems with broken continuous

symmetryInteraction energy, 377, 379Interaction volume, 242Interaction-induced, quantum chaos, 55–9Interferogram, spatial modulation frequency,

114–15Ion beam erosion, 247Ion etching configurations, 246Ion traps, selective addressing in, 91Ionization yield, of one electron Rydberg

states under microwave driving, 42–4Irregular level dynamics, 35–6Isotropic harmonic trap– condensate particle number, 322–3, 405–8– cross-excitation parameter, 351– with interactive Bose gas, 382

Jaynes–Cummings interaction, 156, 157Jitter, 270–7, 285–7Joint coherence operators, 165Joint detection probability– with emission-time jitter, 274–5– with frequency jitter, 272, 273– interference without time resolution, 264,

266–7– for perpendicular polarized photon pairs,

276, 277– time-resolved interference, 264, 267–70,

280–5Josephson regime, 161, 162

KAM theorem, 40, 60–1Kerr effect, 421–2Kerr self-focusing, 423, 424Kicked cold atoms, 66–7Kicked harmonic oscillator– Floquet operator, 38–40

Index 447

– mean energy, 47– web-assisted transport in, 46–9Klystron, 330–2Kolmogorov–Arnold–Moser (KAM)

theorem, 40, 60–1KPZ type equation, 247–8Kuramoto–Sivashinsky type equation, 247

Lamb–Dicke effect, 196, 200Lamb–Dicke parameter, 40, 46, 200Lambda point, in liquid helium, 295Landau–Zener transitions, 193Laser(s)– in chaotic resonators, 52– emission properties, 54– quantum theory of, 329–34, 402–3– see also Random lasersLaser cooling, types, 190–1, 206–8Laser-induced breakdown (LIB), 434, 435Laser-induced breakdown spectroscopy

(LIBS), 438Laser-induced fluorescence (LIF), linear, 437Laser interaction with solid surfaces, 227–49– discussion, 246–9– energy coupling, 229–33, 239– secondary processes, 233–46– – see also Ablation; Desorption dynamicsLaser master equation, 332–4Laser pulse propagation, 423Laserscanning microscope, 112–13Lateral resolution, depth resolution and, 106Lateral resolution limit, 106LIB, 434, 435LIBS, 438LIDAR (Light Detection and Ranging)– applications, 425, 438– conventional, 415–19, 428–9– Differential Absorption (DIAL), 415, 431,

432– femtosecond, experimental setup, 419–21– nonlinear interactions with aerosols, 433–7– nonlinear propagation of ultra-intense laser

pulses, 421–7– tailored pulses, 437– white light femtosecond, 427–32Light fields, 190–1Light quanta– indistinguishability, 301, 303, 307– momentum, 299Lightning control, 438

Lin-perp-lin configuration, 197Linear optical coherence tomography

(LOCT), 120, 130–3Linear optical device, general linear input–

output transformation, 140–1Linear optical quantum computation

(LOQC), 29, 254, 255Linear optical system theory, 106LOCT, 120, 130–3Long wavelength phase fluctuations, 384–5Longitudinal correlation function, 386Longitudinal susceptibility, 386, 389LOQC, 29, 254, 255Lorentz–Mie calculations, 435Luggage identification, 110, 111Lyapunov exponent, 50, 51

Macular degeneration, 123Magnetic field gradients method, 92–3Magnetic-field-induced laser cooling

(MILC), 204Magnetic-field-induced lattices, 204–6Magnetization, 385, 386, 387Magneto-optical trap see MOTMandel Q-parameter, 159Many-body Hamiltonian, excitation spec-

trum, 56Matter waves, many-body, 313Maxwell–Planck law, 435Maxwell wave equation, 423Maxwell’s demon ensemble, 324, 325, 326,

363Mean-field Popov approximation, 364Metal–Organic Vapor Phase Epitaxy

(MOVPE), 6, 11Metrology, industrial, 106Michelson add/drop filter, 22–6– application to quantum key distribution,

26–9Michelson interferometer, 14–15Microcanonical ensemble, 311, 335– counting statistics, 317– energy conservation, 316– ground-state fluctuations in one-

dimensional harmonic trap, 317– ground-state occupation probability, 321– particle number conservation, 316– partition function, 321Microcanonical partition function, 318, 346Micro-photoluminescence, 9–10

448 Index

Microscope– laserscanning, 112–13– theory of, 106, 111–15Microstate counting– Bose way, 301, 303, 307, 308– classical way, 310– comparison between Bose and Einstein

ways, 314–15, 395–7MILC-type lattices, 204–6Mode function, 257– see also JitterModulation transfer function (MTF), 107– of focused diffraction limited system, 108,

109– of laserscanning microscope, 113Molecular Beam Epitaxy (MBE), 6Molecular damping, 156Molecular fields, counting statistics, 168–73Molecular formation, passage time statistics,

163–8Molecular micromaser, 153–63– model, 154–8– results, 158–63– see also Coupled atom–molecule systemMono-filamentation, 424, 425MOT, 77– single atoms in, 77–82– transfer of atoms to dipole trap, 82–4– – modified procedure, 90–1MOVPE, 6, 11MPEF, 434, 435, 437MPI see Multi-photon ionizationMTF see Modulation transfer functionMulti-filamentation, 424, 425Multinomial theorem, 396Multi-photon cascades, 16–22Multi-photon-excited fluorescence (MPEF),

434, 435, 437Multi-photon ionization (MPI), 231–3,

422–3, 424, 434, 435Multiplexed quantum cryptography, 22–9Multiplexing, 22Multi-wavelength algorithms, 433

Nanocrystals, 3–4Narrow resonance, 176Nernst’s theorem, 310Neutral atoms, 76–7– Bose–Einstein condensation with, 76– entanglement, 99–101

– localization in space, 76– position control, 94–9– single see Single atoms– see also Dipole trapNo-cloning theorem, 23, 26NO2 emission distribution, 415–16Nondegenerate Tavis–Cummings model,

164, 170, 171Nondispersive wave packets, 60–1– in kicked cold atoms, 66–7– in one particle dynamics, 61–3– in three body Coulomb problem, 63–5Nonequilibrium Keldysh diagram technique,

391, 394Nonlinear Schrödinger equation (NLSE), 424Normal Fermi gas (NFG)– counting statistics of molecules from,

170–1– momentum distribution of molecules from,

179Numerical aperture (NA), 106

Occupation number operator, 359OCT see Optical coherence tomographyOne-dimensional lattice configurations, 196–

208Optical amplifiers, 139–48Optical attenuators, 140, 141, 147Optical cavity-QED, 77, 84, 94, 95, 99Optical coherence tomography (OCT), 107,

120–36Optical communication, 148Optical conveyor belt, 95, 96Optical high-finesse resonator, 99–100Optical lattices, 187–223– applications, 188– atom-field interaction, 191–4– future research, 223– light fields, 190–1– quantum Monte-Carlo wave-function simu-

lations see QMCWF simulations– see also Gray lattices; One-dimensional

lattice configurations; Sloshing-typewave-packets

Optical manufacturing, 129–30Optical multi-channel analyzer (OMA), 435Optical pumping, 80Optical remote sensing instruments, 414–15Optical transfer function (OTF), 107, 113,

117

Index 449

– inverse, 108Optical tweezers, 98Ozone, 414, 417–18, 437

P -representation, 342Parametric amplifiers, 140, 144Partial coherent illumination, 115–17Partial coherent imaging, theory of, 115Passage time statistics, of molecular forma-

tion, 163–8Paul trap, 76Pauli Exclusion Principle, 152Pegg–Barnett phase states, 162Perturbation depth, 247Perturbation Theory, 232Petermann factor, 53Phase explosion, 239, 242Phase insensitive amplifiers, 139–40, 141–3– multimode, 143–4Phase insensitivity, 142Phase sensitive amplifiers, 140Phase space structure, mixed regular chaotic,

40–1Phaseplates, 107–8, 117–20Photo-association, 155, 163–8Photo-dissociation, 166–8Photography, wavefront coding in, 136Photon detection, theory of, 177Photon ensemble, temporal envelope, 255Photonic localization length, 44π -pulses, 86Planck’s law of radiation, 311Plasma channels, 421, 438Plasma defocusing, 423, 424Point spread function (PSF), 108, 110, 116Polar stratospheric clouds (PSC), 417–18Polishing robot, 119, 129Politzer asymptotic approximation, 326, 327Polystyrene latex (PSL), 435Porter–Thomas distribution, 54Posterior ciliary artery, 125, 126Power exchange, 209, 220PSF see Point spread functionPupil function, 107, 113Purcell effect, 22

QMCWF simulations, 194–6, 199, 211– influence of magnetic fields on tunneling,

214, 215, 216

– sloshing-type wave-packet motion, 220,221

Quadrature-phase amplitudes, 381Quantum accelerator modes, 66–7Quantum chaos, 34–68– applications, 34, 68– control through, 59–67– – see also Nondispersive wave packets– interaction-induced, directed atomic trans-

port due to, 55–9– spectral properties, 34–41– see also Quantum transportQuantum cloning, 140Quantum cryptography see Multiplexed

quantum cryptographyQuantum dots, 4–7, 11–13Quantum gates, 84, 89, 99–101Quantum information processing, 2, 68, 84,

188– see also LOQC; Quantum gates; Quantum

registersQuantum key distribution, 26–9Quantum memory, 68Quantum Monte-Carlo wave-function

(QMCWF) simulations, 194–6, 199,211

– influence of magnetic fields on tunneling,214, 215, 216

– sloshing-type wave-packet motion, 220,221

Quantum numbers, good, 35Quantum optics, 34– cold atom physics and, 152– of ultra-cold molecules see Ultra-cold

moleculesQuantum phase gate, 100Quantum registers, 91–4Quantum resonances, 66–7Quantum states control– by rapid adiabatic passage, 93–4– magnetic field gradients method, 92–3Quantum transport, 41–59– atomic conductance fluctuations, 42–6– directed atomic transport due to interaction-

induced quantum chaos, 55–9– Ericson fluctuations in atomic photo cross

sections, 49–51– photonic transport in chaotic cavities, 51–5– web-assisted, in kicked harmonic oscillator,

46–9

450 Index

Quantum walks, 100Quasiparticle approach, 357–71– in atom-number-conserving Bogoliubov

approximation, 372–4– cumulants of BEC fluctuations in ideal

Bose gas, 361–4– – equivalent formulation in terms of poles

of generalized Zeta function, 370–1– grand canonical approximation for quasi-

particle occupations, 363– ideal gas BEC statistics in arbitrary power-

law traps, 364–70– in reduced Hilbert space, 359–61– see also Interacting Bose gasQubits, 84

Rabi oscillations, 80, 88–9Rabi regime, 161, 162Radiation field, interference fluctuations, 312Radiative escape processes, 81Raman amplifier, 148Raman photo-association, two-photon, 155Raman–Nath approximation, 165Ramsey spectroscopy, 87–8Random lasers, 51–5Rayleigh scattering, 429Reciprocity principle, 434Recoil pressure, 240–2Red-detuned lattices, 197–200Regular level dynamics, 35Relative humidity LIDAR profiler, 432Relaxation times, of atoms in dipole trap,

87–9Resonances, 49, 176Retina– bloodflow measurement, 125–7– tomographic imaging, 123–7Retinal reflex, 128–9Riboflavin, 436Riemann Zeta functions, 371Ripples, 242–6– orientation of structures, 248Rubidium– one-dimensional lattice structures see One-

dimensional lattice configurations– Rydberg states, 50, 51– single photon generation, 254Rydberg electrons, continuum decay, 49–51

Saddle-point method

– conventional, 404, 405, 406– refined, 328, 404–8Sag function, 107Saturation photon numbers, 54SDOCT see Spectral domain optical coher-

ence tomographySecond-order coherence function, 7, 17Semiclassical limit, 34–5Semiconductor superlattices, 49Sensor technology, 136Shannon entropy, 304Shift invariant function, 112Silicon– ablation from, 239–40– – recoil pressure, 240–2– – surface morphology, 242–6Single atoms– in MOT, 77–82– position control, 94–5– preparing in dipole trap, 82–4– quantum state detection, 85–6– quantum state preparation, 84–5– Stern–Gerlach experiment, 100– superposition states, 86–9– see also Dipole trapSingle photon(s)– add/drop filter, 22–6– – application to quantum key distribution,

26–9– characterization using two-photon interfer-

ence see two-photon interference– detection, 259–60– duration– – lower limit, 286– – measurement, 255– generation, 7–13– light fields, 257–8– as particle and wave, 13–16– source realization, 254Sisyphus cooling, 190, 197, 199, 200Skewness, 383Sloshing-type wave-packets, 197, 200,

219–22SO2 emission distribution, 415–17Spatial correlations, molecules as probes of,

173–81Spectral domain optical coherence tomogra-

phy (SDOCT), 133–6– noise comparison with TDOCT, 134–6Spin stiffness, 386

Index 451

Spontaneous emission, 141Spontaneous magnetization, 385, 386, 387Squeezing– noise, 298, 381– two-mode, 298, 374, 381Stark manifold, 61Stern–Gerlach experiment, single atom, 100Stimulated emission amplifiers, 144–7STIRAP, 254, 278Stop function, 107Stranski–Krastanov growth, 5Strong localization, 46Super-continuum generation, 425, 426Superfluid(s)– description in effective nonlinear σ model,

386– static susceptibility, 385– universal scaling of condensate fluctuations

in, 388Superfluid-Mott insulator phase transition,

163Superfluidity, detection, 168, 174Superradiance, 163– Dicke, 152, 164Surface morphology, 242–6Surface roughening, 247Surface smoothening, 247Systems with broken continuous symmetry,

383–90

Tavis–Cummings Hamiltonian, 165Tavis–Cummings model, inhomogeneously

broadened, 164, 170, 171TDOCT see Time domain optical coherence

tomographyTemporal focusing, 427Teramobile system, 428, 433, 436Thomas–Fermi approximation, 177Thomas–Fermi regime, 377–9Threshold inversion, 355–6Time domain optical coherence tomography

(TDOCT), 120–30– blood oxygenization determination with,

123–5– bloodflow measurement and, 125–7– drawbacks, 130–1– eye length measurement with, 127–9– of fundus of eye, 122– interference signal, 121–2– noise comparison with SDOCT, 134–6

– in optical manufacturing, 129–30Time evolution operator, 42Time-of-Flight (ToF) spectroscopy, 236,

238–9Transverse correlation function, 386Transverse susceptibility, 386, 389Trapping states, 159Triexcitons, 16, 17–18, 19Tryptophan, 437Tunneling– influence of magnetic fields on, 213–19– periodic well-to-well, 208–13Tunneling ionization, 233Two-mode noiseless amplifier, 142Two-photon interference, 260–70– correlation function, 263, 264–6– experimental investigation, 277–86– with jitter see Jitter– principle, 262–3– quantum description of beam splitter,

261–2– single photon duration, 286– temporal aspects, 263–4– time-resolved, 267–8, 280–5– without time resolution, 266–7Two-photon Raman photo-association, 155

Uhlenbeck, George, 294Uhlenbeck dilemma, 294, 355Ultra-cold molecules, 151–82– counting statistics of molecular fields,

168–73– passage time statistics of molecule forma-

tion, 163–8– – see also Coupled atom–molecule system;

Molecular micromaser; Spatial corre-lations

van Zittert–Zernike theorem, 111Velocity-selective coherent population trap-

ping (VSCPT), 206–8Virial theorem, 199

Waiting time distribution, 9Water– droplets, 435, 436, 438– ionization potential of molecules, 434– vapor bands, 431–2Wave mechanics, formulation, 313Wave–particle duality, 13

452 Index

Wavefront, laserscanning microscope, 114Wavefront coding, 107–20– Abbe’s theory of microscope and, 111–15– limiting factors, 136– new applications, 136– partial coherent illumination and, 115–17– with variable phaseplates, 117–20

Web-assisted transport, in kicked harmonicoscillator, 46–9

White light laser, 427

Wien’s law of radiation, 311

Zeta functions, 371, 388

CONTENTS OF VOLUMES INTHIS SERIAL

Volume 1Molecular Orbital Theory of the Spin

Properties of Conjugated Molecules,G.G. Hall and A.T. Amos

Electron Affinities of Atoms andMolecules, B.L. Moiseiwitsch

Atomic Rearrangement Collisions,B.H. Bransden

The Production of Rotational andVibrational Transitions in Encountersbetween Molecules, K. Takayanagi

The Study of Intermolecular Potentialswith Molecular Beams at ThermalEnergies, H. Pauly and J.P. Toennies

High-Intensity and High-Energy MolecularBeams, J.B. Anderson, R.P. Anders andJ.B. Fen

Volume 2The Calculation of van der Waals

Interactions, A. Dalgarno andW.D. Davison

Thermal Diffusion in Gases, E.A. Mason,R.J. Munn and Francis J. Smith

Spectroscopy in the Vacuum Ultraviolet,W.R.S. Garton

The Measurement of the PhotoionizationCross Sections of the Atomic Gases,James A.R. Samson

The Theory of Electron–Atom Collisions,R. Peterkop and V. Veldre

Experimental Studies of Excitation inCollisions between Atomic and IonicSystems, F.J. de Heer

Mass Spectrometry of Free Radicals,S.N. Foner

Volume 3The Quantal Calculation of Photoionization

Cross Sections, A.L. StewartRadiofrequency Spectroscopy of Stored

Ions I: Storage, H.G. DehmeltOptical Pumping Methods in Atomic

Spectroscopy, B. BudickEnergy Transfer in Organic Molecular

Crystals: A Survey of Experiments,H.C. Wolf

Atomic and Molecular Scattering fromSolid Surfaces, Robert E. Stickney

Quantum, Mechanics in GasCrystal-Surface van der WaalsScattering, E. Chanoch Beder

Reactive Collisions between Gas andSurface Atoms, Henry Wise and BernardJ. Wood

Volume 4H.S.W. Massey—A Sixtieth Birthday

Tribute, E.H.S. BurhopElectronic Eigenenergies of the Hydrogen

Molecular Ion, D.R. Bates andR.H.G. Reid

Applications of Quantum Theory to theViscosity of Dilute Gases,R.A. Buckingham and E. Gal

Positrons and Positronium in Gases,P.A. Fraser

Classical Theory of Atomic Scattering,A. Burgess and I.C. Percival

Born Expansions, A.R. Holt andB.L. Moiseiwitsch

Resonances in Electron Scattering byAtoms and Molecules, P.G. Burke

453

454 CONTENTS OF VOLUMES IN THIS SERIAL

Relativistic Inner Shell Ionizations,C.B.O. Mohr

Recent Measurements on Charge Transfer,J.B. Hasted

Measurements of Electron ExcitationFunctions, D.W.O. Heddle andR.G.W. Keesing

Some New Experimental Methods inCollision Physics, R.F. Stebbings

Atomic Collision Processes in GaseousNebulae, M.J. Seaton

Collisions in the Ionosphere, A. DalgarnoThe Direct Study of Ionization in Space,

R.L.F. Boyd

Volume 5Flowing Afterglow Measurements of

Ion-Neutral Reactions, E.E. Ferguson,F.C. Fehsenfeld and A.L. Schmeltekopf

Experiments with Merging Beams, RoyH. Neynaber

Radiofrequency Spectroscopy of StoredIons II: Spectroscopy, H.G. Dehmelt

The Spectra of Molecular Solids,O. Schnepp

The Meaning of Collision Broadening ofSpectral Lines: The Classical OscillatorAnalog, A. Ben-Reuven

The Calculation of Atomic TransitionProbabilities, R.J.S. Crossley

Tables of One- and Two-ParticleCoefficients of Fractional Parentage forConfigurations sλstupq ,C.D.H. Chisholm, A. Dalgarno andF.R. Innes

Relativistic Z-Dependent Corrections toAtomic Energy Levels, Holly ThomisDoyle

Volume 6Dissociative Recombination, J.N. Bardsley

and M.A. BiondiAnalysis of the Velocity Field in Plasmas

from the Doppler Broadening of SpectralEmission Lines, A.S. Kaufman

The Rotational Excitation of Molecules bySlow Electrons, Kazuo Takayanagi andYukikazu Itikawa

The Diffusion of Atoms and Molecules,E.A. Mason and T.R. Marrero

Theory and Application of SturmianFunctions, Manuel Rotenberg

Use of Classical Mechanics in theTreatment of Collisions betweenMassive Systems, D.R. Bates andA.E. Kingston

Volume 7Physics of the Hydrogen Maser, C. Audoin,

J.P. Schermann and P. GrivetMolecular Wave Functions: Calculations

and Use in Atomic and MolecularProcess, J.C. Browne

Localized Molecular Orbitals, HarelWeinstein, Ruben Pauncz and MauriceCohen

General Theory of Spin-Coupled WaveFunctions for Atoms and Molecules,J. Gerratt

Diabatic States of Molecules—Quasi-Stationary Electronic States, Thomas F.O’Malley

Selection Rules within Atomic Shells,B.R. Judd

Green’s Function Technique in Atomic andMolecular Physics, Gy. Csanak,H.S. Taylor and Robert Yaris

A Review of Pseudo-Potentials withEmphasis on Their Application to LiquidMetals, Nathan Wiser andA.J. Greenfield

Volume 8Interstellar Molecules: Their Formation

and Destruction, D. McNallyMonte Carlo Trajectory Calculations of

Atomic and Molecular Excitation inThermal Systems, James C. Keck

CONTENTS OF VOLUMES IN THIS SERIAL 455

Nonrelativistic Off-Shell Two-BodyCoulomb Amplitudes, Joseph C.Y. Chenand Augustine C. Chen

Photoionization with Molecular Beams,R.B. Cairns, Halstead Harrison andR.I. Schoen

The Auger Effect, E.H.S. Burhop andW.N. Asaad

Volume 9Correlation in Excited States of Atoms,

A.W. WeissThe Calculation of Electron–Atom

Excitation Cross Section, M.R.H. RudgeCollision-Induced Transitions between

Rotational Levels, Takeshi OkaThe Differential Cross Section of

Low-Energy Electron–Atom Collisions,D. Andrick

Molecular Beam Electric ResonanceSpectroscopy, Jens C. Zorn and ThomasC. English

Atomic and Molecular Processes in theMartian Atmosphere, MichaelB. McElroy

Volume 10Relativistic Effects in the Many-Electron

Atom, Lloyd Armstrong Jr. and SergeFeneuille

The First Born Approximation, K.L. Belland A.E. Kingston

Photoelectron Spectroscopy, W.C. PriceDye Lasers in Atomic Spectroscopy,

W. Lange, J. Luther and A. SteudelRecent Progress in the Classification of the

Spectra of Highly Ionized Atoms,B.C. Fawcett

A Review of Jovian Ionospheric Chemistry,Wesley T. Huntress Jr.

Volume 11The Theory of Collisions between Charged

Particles and Highly Excited Atoms,I.C. Percival and D. Richards

Electron Impact Excitation of PositiveIons, M.J. Seaton

The R-Matrix Theory of Atomic Process,P.G. Burke and W.D. Robb

Role of Energy in Reactive MolecularScattering: An Information-TheoreticApproach, R.B. Bernstein andR.D. Levine

Inner Shell Ionization by Incident Nuclei,Johannes M. Hansteen

Stark Broadening, Hans R. GriemChemiluminescence in Gases, M.F. Golde

and B.A. Thrush

Volume 12Nonadiabatic Transitions between Ionic

and Covalent States, R.K. JanevRecent Progress in the Theory of Atomic

Isotope Shift, J. Bauche andR.-J. Champeau

Topics on Multiphoton Processes in Atoms,P. Lambropoulos

Optical Pumping of Molecules, M. Broyer,G. Goudedard, J.C. Lehmann andJ. Vigué

Highly Ionized Ions, Ivan A. SellinTime-of-Flight Scattering Spectroscopy,

Wilhelm RaithIon Chemistry in the D Region, George

C. Reid

Volume 13Atomic and Molecular Polarizabilities—

Review of Recent Advances, ThomasM. Miller and Benjamin Bederson

Study of Collisions by Laser Spectroscopy,Paul R. Berman

Collision Experiments with Laser-ExcitedAtoms in Crossed Beams, I.V. Hertel andW. Stoll

Scattering Studies of Rotational andVibrational Excitation of Molecules,Manfred Faubel and J. Peter Toennies


Low-Energy Electron Scattering byComplex Atoms: Theory andCalculations, R.K. Nesbet

Microwave Transitions of InterstellarAtoms and Molecules, W.B. Somerville

Volume 14Resonances in Electron Atom and

Molecule Scattering, D.E. GoldenThe Accurate Calculation of Atomic

Properties by Numerical Methods, BrainC. Webster, Michael J. Jamieson andRonald F. Stewart

(e, 2e) Collisions, Erich Weigold and IanE. McCarthy

Forbidden Transitions in One- andTwo-Electron Atoms, Richard Marrusand Peter J. Mohr

Semiclassical Effects in Heavy-ParticleCollisions, M.S. Child

Atomic Physics Tests of the BasicConcepts in Quantum Mechanics,Francies M. Pipkin

Quasi-Molecular Interference Effects inIon–Atom Collisions, S.V. Bobashev

Rydberg Atoms, S.A. Edelstein andT.F. Gallagher

UV and X-Ray Spectroscopy inAstrophysics, A.K. Dupree

Volume 15Negative Ions, H.S.W. MasseyAtomic Physics from Atmospheric and

Astrophysical, A. DalgarnoCollisions of Highly Excited Atoms,

R.F. StebbingsTheoretical Aspects of Positron Collisions

in Gases, J.W. HumberstonExperimental Aspects of Positron

Collisions in Gases, T.C. GriffithReactive Scattering: Recent Advances in

Theory and Experiment, RichardB. Bernstein

Ion–Atom Charge Transfer Collisions atLow Energies, J.B. Hasted

Aspects of Recombination, D.R. BatesThe Theory of Fast Heavy Particle

Collisions, B.H. BransdenAtomic Collision Processes in Controlled

Thermonuclear Fusion Research,H.B. Gilbody

Inner-Shell Ionization, E.H.S. BurhopExcitation of Atoms by Electron Impact,

D.W.O. HeddleCoherence and Correlation in Atomic

Collisions, H. KleinpoppenTheory of Low Energy Electron–Molecule

Collisions, P.O. Burke

Volume 16Atomic Hartree–Fock Theory, M. Cohen

and R.P. McEachranExperiments and Model Calculations to

Determine Interatomic Potentials,R. Düren

Sources of Polarized Electrons,R.J. Celotta and D.T. Pierce

Theory of Atomic Processes in StrongResonant Electromagnetic Fields,S. Swain

Spectroscopy of Laser-Produced Plasmas,M.H. Key and R.J. Hutcheon

Relativistic Effects in Atomic CollisionsTheory, B.L. Moiseiwitsch

Parity Nonconservation in Atoms: Status ofTheory and Experiment, E.N. Fortsonand L. Wilets

Volume 17Collective Effects in Photoionization of

Atoms, M.Ya. AmusiaNonadiabatic Charge Transfer,

D.S.F. CrothersAtomic Rydberg States, Serge Feneuille

and Pierre JacquinotSuperfluorescence, M.F.H. Schuurmans,

Q.H.F. Vrehen, D. Polder andH.M. Gibbs

Applications of Resonance IonizationSpectroscopy in Atomic and Molecular


Physics, M.G. Payne, C.H. Chen,G.S. Hurst and G.W. Foltz

Inner-Shell Vacancy Production inIon–Atom Collisions, C.D. Lin andPatrick Richard

Atomic Processes in the Sun, P.L. Duftonand A.E. Kingston

Volume 18Theory of Electron–Atom Scattering in a

Radiation Field, Leonard RosenbergPositron–Gas Scattering Experiments,

Talbert S. Stein and Walter E. KauppliaNonresonant Multiphoton Ionization of

Atoms, J. Morellec, D. Normand andG. Petite

Classical and Semiclassical Methods inInelastic Heavy-Particle Collisions,A.S. Dickinson and D. Richards

Recent Computational Developments in theUse of Complex Scaling in ResonancePhenomena, B.R. Junker

Direct Excitation in Atomic Collisions:Studies of Quasi-One-Electron Systems,N. Andersen and S.E. Nielsen

Model Potentials in Atomic Structure,A. Hibbert

Recent Developments in the Theory ofElectron Scattering by Highly PolarMolecules, D.W. Norcross andL.A. Collins

Quantum Electrodynamic Effects inFew-Electron Atomic Systems,G.W.F. Drake

Volume 19Electron Capture in Collisions of Hydrogen

Atoms with Fully Stripped Ions,B.H. Bransden and R.K. Janev

Interactions of Simple Ion Atom Systems,J.T. Park

High-Resolution Spectroscopy of StoredIons, D.J. Wineland, Wayne M. Itano andR.S. Van Dyck Jr.

Spin-Dependent Phenomena in InelasticElectron–Atom Collisions, K. Blum andH. Kleinpoppen

The Reduced Potential Curve Method forDiatomic Molecules and ItsApplications, F. Jenc

The Vibrational Excitation of Molecules byElectron Impact, D.G. Thompson

Vibrational and Rotational Excitation inMolecular Collisions, Manfred Faubel

Spin Polarization of Atomic and MolecularPhotoelectrons, N.A. Cherepkov

Volume 20Ion–Ion Recombination in an Ambient

Gas, D.R. BatesAtomic Charges within Molecules,

G.G. HallExperimental Studies on Cluster Ions,

T.D. Mark and A.W. Castleman Jr.Nuclear Reaction Effects on Atomic

Inner-Shell Ionization, W.E. Meyerhofand J.-F. Chemin

Numerical Calculations on Electron-ImpactIonization, Christopher Bottcher

Electron and Ion Mobilities, GordonR. Freeman and David A. Armstrong

On the Problem of Extreme UV and X-RayLasers, I.I. Sobel’man andA.V. Vinogradov

Radiative Properties of Rydberg States inResonant Cavities, S. Haroche andJ.M. Raimond

Rydberg Atoms: High-ResolutionSpectroscopy and RadiationInteraction—Rydberg Molecules,J.A.C. Gallas, G. Leuchs, H. Walther,and H. Figger

Volume 21Subnatural Linewidths in Atomic

Spectroscopy, Dennis P. O’Brien, PierreMeystre and Herbert Walther

Molecular Applications of Quantum DefectTheory, Chris H. Greene and Ch. Jungen


Theory of Dielectronic Recombination,Yukap Hahn

Recent Developments in SemiclassicalFloquet Theories for Intense-FieldMultiphoton Processes, Shih-I Chu

Scattering in Strong Magnetic Fields,M.R.C. McDowell and M. Zarcone

Pressure Ionization, Resonances and theContinuity of Bound and Free States,R.M. More

Volume 22Positronium—Its Formation and

Interaction with Simple Systems,J.W. Humberston

Experimental Aspects of Positron andPositronium Physics, T.C. Griffith

Doubly Excited States, Including NewClassification Schemes, C.D. Lin

Measurements of Charge Transfer andIonization in Collisions InvolvingHydrogen Atoms, H.B. Gilbody

Electron Ion and Ion–Ion Collisions withIntersecting Beams, K. Dolder andB. Peart

Electron Capture by Simple Ions, EdwardPollack and Yukap Hahn

Relativistic Heavy-Ion–Atom Collisions,R. Anholt and Harvey Gould

Continued-Fraction Methods in AtomicPhysics, S. Swain

Volume 23Vacuum Ultraviolet Laser Spectroscopy of

Small Molecules, C.R. VidalFoundations of the Relativistic Theory of

Atomic and Molecular Structure, IanP. Grant and Harry M. Quiney

Point-Charge Models for MoleculesDerived from Least-Squares Fitting ofthe Electric Potential, D.E. Williams andJi-Min Yan

Transition Arrays in the Spectra of IonizedAtoms, J. Bauche, C. Bauche-Arnoultand M. Klapisch

Photoionization and Collisional Ionizationof Excited Atoms Using Synchrotronand Laser Radiation, F.J. Wuilleumier,D.L. Ederer and J.L. Picqué

Volume 24The Selected Ion Flow Tube (SIDT):

Studies of Ion-Neutral Reactions,D. Smith and N.G. Adams

Near-Threshold Electron–MoleculeScattering, Michael A. Morrison

Angular Correlation in MultiphotonIonization of Atoms, S.J. Smith andG. Leuchs

Optical Pumping and Spin Exchange inGas Cells, R.J. Knize, Z. Wu andW. Happer

Correlations in Electron–Atom Scattering,A. Crowe

Volume 25Alexander Dalgarno: Life and Personality,

David R. Bates and George A. VictorAlexander Dalgarno: Contributions to

Atomic and Molecular Physics, NealLane

Alexander Dalgarno: Contributions toAeronomy, Michael B. McElroy

Alexander Dalgarno: Contributions toAstrophysics, David A. Williams

Dipole Polarizability Measurements,Thomas M. Miller and BenjaminBederson

Flow Tube Studies of Ion–MoleculeReactions, Eldon Ferguson

Differential Scattering in He–He andHe+–He Collisions at keV Energies,R.F. Stebbings

Atomic Excitation in Dense Plasmas, JonC. Weisheit

Pressure Broadening and Laser-InducedSpectral Line Shapes, Kenneth M. Sandoand Shih-I. Chu

Model-Potential Methods, C. Laughlin andG.A. Victor


Z-Expansion Methods, M. CohenSchwinger Variational Methods, Deborah

Kay WatsonFine-Structure Transitions in Proton–Ion

Collisions, R.H.G. ReidElectron Impact Excitation, R.J.W. Henry

and A.E. KingstonRecent Advances in the Numerical

Calculation of Ionization Amplitudes,Christopher Bottcher

The Numerical Solution of the Equationsof Molecular Scattering, A.C. Allison

High Energy Charge Transfer,B.H. Bransden and D.P. Dewangan

Relativistic Random-Phase Approximation,W.R. Johnson

Relativistic Sturmian and Finite Basis SetMethods in Atomic Physics,G.W.F. Drake and S.P. Goldman

Dissociation Dynamics of PolyatomicMolecules, T. Uzer

Photodissociation Processes in DiatomicMolecules of Astrophysical Interest,Kate P. Kirby and Ewine F. van Dishoeck

The Abundances and Excitation ofInterstellar Molecules, John H. Black

Volume 26Comparisons of Positrons and Electron

Scattering by Gases, Walter E. Kauppilaand Talbert S. Stein

Electron Capture at Relativistic Energies,B.L. Moiseiwitsch

The Low-Energy, Heavy ParticleCollisions—A Close-CouplingTreatment, Mineo Kimura and NealF. Lane

Vibronic Phenomena in Collisions ofAtomic and Molecular Species, V. Sidis

Associative Ionization: Experiments,Potentials and Dynamics, John WeinerFrançoise Masnou-Seeuws and AnnickGiusti-Suzor

On the β Decay of 187Re: An Interface ofAtomic and Nuclear Physics and

Cosmochronology, Zonghau Chen,Leonard Rosenberg and Larry Spruch

Progress in Low Pressure Mercury-RareGas Discharge Research, J. Maya andR. Lagushenko

Volume 27Negative Ions: Structure and Spectra,

David R. BatesElectron Polarization Phenomena in

Electron–Atom Collisions, JoachimKessler

Electron–Atom Scattering, I.E. McCarthyand E. Weigold

Electron–Atom Ionization, I.E. McCarthyand E. Weigold

Role of Autoionizing States in MultiphotonIonization of Complex Atoms,V.I. Lengyel and M.I. Haysak

Multiphoton Ionization of AtomicHydrogen Using Perturbation Theory,E. Karule

Volume 28The Theory of Fast Ion–Atom Collisions,

J.S. Briggs and J.H. MacekSome Recent Developments in the

Fundamental Theory of Light, PeterW. Milonni and Surendra Singh

Squeezed States of the Radiation Field,Khalid Zaheer and M. Suhail Zubairy

Cavity Quantum Electrodynamics,E.A. Hinds

Volume 29Studies of Electron Excitation of Rare-Gas

Atoms into and out of Metastable LevelsUsing Optical and Laser Techniques,Chun C. Lin and L.W. Anderson

Cross Sections for Direct MultiphotonIonization of Atoms, M.V. Ammosov,N.B. Delone, M.Yu. Ivanov, I.I. Bandarand A.V. Masalov

Collision-Induced Coherences in OpticalPhysics, G.S. Agarwal


Muon-Catalyzed Fusion, Johann Rafelskiand Helga E. Rafelski

Cooperative Effects in Atomic Physics,J.P. Connerade

Multiple Electron Excitation, Ionization,and Transfer in High-Velocity Atomicand Molecular Collisions, J.H. McGuire

Volume 30Differential Cross Sections for Excitation

of Helium Atoms and Helium-Like Ionsby Electron Impact, Shinobu Nakazaki

Cross-Section Measurements for ElectronImpact on Excited Atomic Species,S. Trajmar and J.C. Nickel

The Dissociative Ionization of SimpleMolecules by Fast Ions, Colin J. Latimer

Theory of Collisions between Laser CooledAtoms, P.S. Julienne, A.M. Smith andK. Burnett

Light-Induced Drift, E.R. ElielContinuum Distorted Wave Methods in

Ion–Atom Collisions, DerrickS.F. Crothers and Louis J. Dube

Volume 31Energies and Asymptotic Analysis for

Helium Rydberg States, G.W.F. DrakeSpectroscopy of Trapped Ions,

R.C. ThompsonPhase Transitions of Stored Laser-Cooled

Ions, H. WaltherSelection of Electronic States in Atomic

Beams with Lasers, Jacques Baudon,Rudalf Dülren and Jacques Robert

Atomic Physics and Non-MaxwellianPlasmas, Michèle Lamoureux

Volume 32Photoionization of Atomic Oxygen and

Atomic Nitrogen, K.L. Bell andA.E. Kingston

Positronium Formation by Positron Impacton Atoms at Intermediate Energies,B.H. Bransden and C.J. Noble

Electron–Atom Scattering Theory andCalculations, P.G. Burke

Terrestrial and Extraterrestrial H+3 ,

Alexander DalgarnoIndirect Ionization of Positive Atomic Ions,

K. DolderQuantum Defect Theory and Analysis of

High-Precision Helium Term Energies,G.W.F. Drake

Electron–Ion and Ion–Ion RecombinationProcesses, M.R. Flannery

Studies of State-Selective Electron Capturein Atomic Hydrogen by TranslationalEnergy Spectroscopy, H.B. Gilbody

Relativistic Electronic Structure of Atomsand Molecules, I.P. Grant

The Chemistry of Stellar Environments,D.A. Howe, J.M.C. Rawlings andD.A. Williams

Positron and Positronium Scattering atLow Energies, J.W. Humberston

How Perfect are Complete AtomicCollision Experiments?, H. Kleinpoppenand H. Handy

Adiabatic Expansions and NonadiabaticEffects, R. McCarroll andD.S.F. Crothers

Electron Capture to the Continuum,B.L. Moiseiwitsch

How Opaque Is a Star?, M.T. SeatonStudies of Electron Attachment at Thermal

Energies Using the FlowingAfterglow–Langmuir Technique, DavidSmith and Patrik Španel

Exact and Approximate Rate Equations inAtom–Field Interactions, S. Swain

Atoms in Cavities and Traps, H. WaltherSome Recent Advances in Electron-Impact

Excitation of n = 3 States of AtomicHydrogen and Helium, J.F. Williams andJ.B. Wang

Volume 33Principles and Methods for Measurement

of Electron Impact Excitation Cross


Sections for Atoms and Molecules byOptical Techniques, A.R. Filippelli,Chun C. Lin, L.W. Andersen andJ.W. McConkey

Benchmark Measurements of CrossSections for Electron Collisions:Analysis of Scattered Electrons,S. Trajmar and J.W. McConkey

Benchmark Measurements of CrossSections for Electron Collisions:Electron Swarm Methods,R.W. Crompton

Some Benchmark Measurements of CrossSections for Collisions of Simple HeavyParticles, H.B. Gilbody

The Role of Theory in the Evaluation andInterpretation of Cross-Section Data,Barry I. Schneider

Analytic Representation of Cross-SectionData, Mitio Inokuti, Mineo Kimura,M.A. Dillon, Isao Shimamura

Electron Collisions with N2, O2 and O:What We Do and Do Not Know,Yukikazu Itikawa

Need for Cross Sections in Fusion PlasmaResearch, Hugh P. Summers

Need for Cross Sections in PlasmaChemistry, M. Capitelli, R. Celibertoand M. Cacciatore

Guide for Users of Data Resources, JeanW. Gallagher

Guide to Bibliographies, Books, Reviewsand Compendia of Data on AtomicCollisions, E.W. McDaniel andE.J. Mansky

Volume 34Atom Interferometry, C.S. Adams,

O. Carnal and J. MlynekOptical Tests of Quantum Mechanics,

R.Y. Chiao, P.G. Kwiat andA.M. Steinberg

Classical and Quantum Chaos in AtomicSystems, Dominique Delande andAndreas Buchleitner

Measurements of Collisions betweenLaser-Cooled Atoms, Thad Walker andPaul Feng

The Measurement and Analysis of ElectricFields in Glow Discharge Plasmas,J.E. Lawler and D.A. Doughty

Polarization and Orientation Phenomena inPhotoionization of Molecules,N.A. Cherepkov

Role of Two-Center Electron–ElectronInteraction in Projectile ElectronExcitation and Loss, E.C. Montenegro,W.E. Meyerhof and J.H. McGuire

Indirect Processes in Electron ImpactIonization of Positive Ions, D.L. Mooresand K.J. Reed

Dissociative Recombination: Crossing andTunneling Modes, David R. Bates

Volume 35Laser Manipulation of Atoms,

K. Sengstock and W. ErtmerAdvances in Ultracold Collisions:

Experiment and Theory, J. WeinerIonization Dynamics in Strong Laser

Fields, L.F. DiMauro and P. AgostiniInfrared Spectroscopy of Size Selected

Molecular Clusters, U. BuckFermosecond Spectroscopy of Molecules

and Clusters, T. Baumer and G. GerberCalculation of Electron Scattering on

Hydrogenic Targets, I. Bray andA.T. Stelbovics

Relativistic Calculations of TransitionAmplitudes in the Helium IsoelectronicSequence, W.R. Johnson, D.R. Planteand J. Sapirstein

Rotational Energy Transfer in SmallPolyatomic Molecules, H.O. Everitt andF.C. De Lucia

Volume 36Complete Experiments in Electron–Atom

Collisions, Nils Overgaard Andersen andKlaus Bartschat


Stimulated Rayleigh Resonances andRecoil-Induced Effects, J.-Y. Courtoisand G. Grynberg

Precision Laser Spectroscopy UsingAcousto-Optic Modulators, W.A. vanMijngaanden

Highly Parallel Computational Techniquesfor Electron–Molecule Collisions, CarlWinstead and Vincent McKoy

Quantum Field Theory of Atoms andPhotons, Maciej Lewenstein and Li You

Volume 37Evanescent Light-Wave Atom Mirrors,

Resonators, Waveguides, and Traps,Jonathan P. Dowling and JulioGea-Banacloche

Optical Lattices, P.S. Jessen andI.H. Deutsch

Channeling Heavy Ions through CrystallineLattices, Herbert F. Krause and SheldonDatz

Evaporative Cooling of Trapped Atoms,Wolfgang Ketterle and N.J. van Druten

Nonclassical States of Motion in Ion Traps,J.I. Cirac, A.S. Parkins, R. Blatt andP. Zoller

The Physics of Highly-Charged Heavy IonsRevealed by Storage/Cooler Rings,P.H. Mokler and Th. Stöhlker

Volume 38Electronic Wavepackets, Robert R. Jones

and L.D. NoordamChiral Effects in Electron Scattering by

Molecules, K. Blum and D.G. ThompsonOptical and Magneto-Optical Spectroscopy

of Point Defects in Condensed Helium,Serguei I. Kanorsky and Antoine Weis

Rydberg Ionization: From Field to Photon,G.M. Lankhuijzen and L.D. Noordam

Studies of Negative Ions in Storage Rings,L.H. Andersen, T. Andersen andP. Hvelplund

Single-Molecule Spectroscopy andQuantum Optics in Solids, W.E. Moerner,R.M. Dickson and D.J. Norris

Volume 39Author and Subject Cumulative Index

Volumes 1–38Author IndexSubject IndexAppendix: Tables of Contents of Volumes

1–38 and Supplements

Volume 40Electric Dipole Moments of Leptons,

Eugene D. ComminsHigh-Precision Calculations for the

Ground and Excited States of theLithium Atom, Frederick W. King

Storage Ring Laser Spectroscopy, ThomasU. Kühl

Laser Cooling of Solids, Carl E. Manganand Timothy R. Gosnell

Optical Pattern Formation, L.A. Lugiato,M. Brambilla and A. Gatti

Volume 41Two-Photon Entanglement and Quantum

Reality, Yanhua ShihQuantum Chaos with Cold Atoms, Mark

G. RaizenStudy of the Spatial and Temporal

Coherence of High-Order Harmonics,Pascal Salières, Ann L’Huillier, PhilippeAntoine and Maciej Lewenstein

Atom Optics in Quantized Light Fields,Matthias Freyburger, AloisM. Herkommer, Daniel S. Krähmer,Erwin Mayr and Wolfgang P. Schleich

Atom Waveguides, Victor I. BalykinAtomic Matter Wave Amplification by

Optical Pumping, Ulf Janicke andMartin Wikens


Volume 42Fundamental Tests of Quantum Mechanics,

Edward S. Fry and Thomas WaltherWave-Particle Duality in an Atom

Interferometer, Stephan Dürr andGerhard Rempe

Atom Holography, Fujio ShimizuOptical Dipole Traps for Neutral Atoms,

Rudolf Grimm, Matthias Weidemüllerand Yurii B. Ovchinnikov

Formation of Cold (T ≤ 1 K) Molecules,J.T. Bahns, P.L. Gould and W.C. Stwalley

High-Intensity Laser-Atom Physics,C.J. Joachain, M. Dorr and N.J. Kylstra

Coherent Control of Atomic, Molecularand Electronic Processes, Moshe Shapiroand Paul Brumer

Resonant Nonlinear Optics in PhaseCoherent Media, M.D. Lukin, P. Hemmerand M.O. Scully

The Characterization of Liquid and SolidSurfaces with Metastable Helium Atoms,H. Morgner

Quantum Communication with EntangledPhotons, Herald Weinfurter

Volume 43Plasma Processing of Materials and

Atomic, Molecular, and Optical Physics:An Introduction, Hiroshi Tanaka andMitio Inokuti

The Boltzmann Equation and TransportCoefficients of Electrons in WeaklyIonized Plasmas, R. Winkler

Electron Collision Data for PlasmaChemistry Modeling, W.L. Morgan

Electron–Molecule Collisions inLow-Temperature Plasmas: The Role ofTheory, Carl Winstead and VincentMcKoy

Electron Impact Ionization of OrganicSilicon Compounds, Ralf Basner, KurtBecker, Hans Deutsch and MartinSchmidt

Kinetic Energy Dependence ofIon–Molecule Reactions Related toPlasma Chemistry, P.B. Armentrout

Physicochemical Aspects of Atomic andMolecular Processes in ReactivePlasmas, Yoshihiko Hatano

Ion–Molecule Reactions, WernerLindinger, Armin Hansel and ZdenekHerman

Uses of High-Sensitivity White-LightAbsorption Spectroscopy in ChemicalVapor Deposition and PlasmaProcessing, L.W. Anderson, A.N. Goyetteand J.E. Lawler

Fundamental Processes of Plasma–SurfaceInteractions, Rainer Hippler

Recent Applications of GaseousDischarges: Dusty Plasmas andUpward-Directed Lightning, AraChutjian

Opportunities and Challenges for Atomic,Molecular and Optical Physics in PlasmaChemistry, Kurl Becker Hans Deutschand Mitio Inokuti

Volume 44Mechanisms of Electron Transport in

Electrical Discharges and ElectronCollision Cross Sections, Hiroshi Tanakaand Osamu Sueoka

Theoretical Consideration ofPlasma-Processing Processes, MineoKimura

Electron Collision Data forPlasma-Processing Gases, Loucas G.Christophorou and James K. Olthoff

Radical Measurements in PlasmaProcessing, Toshio Goto

Radio-Frequency Plasma Modeling forLow-Temperature Processing, ToshiakiMakabe

Electron Interactions with Excited Atomsand Molecules, Loucas G.Christophorou and James K. Olthoff


Volume 45Comparing the Antiproton and Proton, and

Opening the Way to Cold Antihydrogen,G. Gabrielse

Medical Imaging with Laser-PolarizedNoble Gases, Timothy Chupp and ScottSwanson

Polarization and Coherence Analysis of theOptical Two-Photon Radiation from theMetastable 22Si1/2 State of AtomicHydrogen, Alan J. Duncan, HansKleinpoppen and Marian O. Scully

Laser Spectroscopy of Small Molecules,W. Demtröder, M. Keil and H. Wenz

Coulomb Explosion Imaging of Molecules,Z. Vager

Volume 46Femtosecond Quantum Control, T. Brixner,

N.H. Damrauer and G. GerberCoherent Manipulation of Atoms and

Molecules by Sequential Laser Pulses,N.V. Vitanov, M. Fleischhauer,B.W. Shore and K. Bergmann

Slow, Ultraslow, Stored, and Frozen Light,Andrey B. Matsko, Olga Kocharovskaya,Yuri Rostovtsev George R. Welch,Alexander S. Zibrov and MarlanO. Scully

Longitudinal Interferometry with AtomicBeams, S. Gupta, D.A. Kokorowski,R.A. Rubenstein, and W.W. Smith

Volume 47Nonlinear Optics of de Broglie Waves,

P. MeystreFormation of Ultracold Molecules

(T ≤ 200 μK) via Photoassociation in aGas of Laser-Cooled Atoms, FrançoiseMasnou-Seeuws and Pierre Pillet

Molecular Emissions from theAtmospheres of Giant Planets andComets: Needs for Spectroscopic andCollision Data, Yukikazu Itikawa, SangJoon Kim, Yong Ha Kim and Y.C. Minh

Studies of Electron-Excited Targets UsingRecoil Momentum Spectroscopy withLaser Probing of the Excited State,Andrew James Murray and PeterHammond

Quantum Noise of Small Lasers,J.P. Woerdman, N.J. van Druten andM.P. van Exter

Volume 48Multiple Ionization in Strong Laser Fields,

R. Dörner Th. Weber, M. Weckenbrock,A. Staudte, M. Hattass, R. Moshammer,J. Ullrich and H. Schmidt-Böcking

Above-Threshold Ionization: FromClassical Features to Quantum Effects,W. Becker, F. Grasbon, R. Kapold,D.B. Miloševic, G.G. Paulus andH. Walther

Dark Optical Traps for Cold Atoms, NirFriedman, Ariel Kaplan and NirDavidson

Manipulation of Cold Atoms in HollowLaser Beams, Heung-Ryoul Noh, XenyeXu and Wonho Jhe

Continuous Stern–Gerlach Effect onAtomic Ions, Günther Werth, HartmutHaffner and Wolfgang Quint

The Chirality of Biomolecules, RobertN. Compton and Richard M. Pagni

Microscopic Atom Optics: From Wires toan Atom Chip, Ron Folman, PeterKrüger, Jörg Schmiedmayer, JohannesDenschlag and Carsten Henkel

Methods of Measuring Electron–AtomCollision Cross Sections with an AtomTrap, R.S. Schappe, M.L. Keeler,T.A. Zimmerman, M. Larsen, P. Feng,R.C. Nesnidal, J.B. Boffard, T.G. Walker,L.W. Anderson and C.C. Lin

Volume 49Applications of Optical Cavities in Modern

Atomic, Molecular, and Optical Physics,Jun Ye and Theresa W. Lynn


Resonance and Threshold Phenomena inLow-Energy Electron Collisions withMolecules and Clusters, H. Hotop,M.-W. Ruf, M. Allan and I.I. Fabrikant

Coherence Analysis and TensorPolarization Parameters of (γ, eγ )Photoionization Processes in AtomicCoincidence Measurements,B. Lohmann, B. Zimmermann,H. Kleinpoppen and U. Becker

Quantum Measurements and NewConcepts for Experiments with TrappedIons, Ch. Wunderlich and Ch. Balzer

Scattering and Reaction Processes inPowerful Laser Fields, DejanB. Miloševic and Fritz Ehlotzky

Hot Atoms in the Terrestrial Atmosphere,Vijay Kumar and E. Krishnakumar

Volume 50Assessment of the Ozone Isotope Effect,

K. Mauersberger, D. Krankowsky,C. Janssen and R. Schinke

Atom Optics, Guided Atoms, and AtomInterferometry, J. Arlt, G. Birkl, E. Raseland W. Ertmet

Atom–Wall Interaction, D. Bloch andM. Ducloy

Atoms Made Entirely of Antimatter: TwoMethods Produce Slow Antihydrogen,G. Gabrielse

Ultrafast Excitation, Ionization, andFragmentation of C60, I.V. Hertel,T. Laarmann and C.P. Schulz

Volume 51Introduction, Henry H. StrokeAppreciation of Ben Bederson as Editor of

Advances in Atomic, Molecular, andOptical Physics

Benjamin Bederson Curriculum VitaeResearch Publications of Benjamin

BedersonA Proper Homage to Our Ben, H. LustigBenjamin Bederson in the Army, World

War II, Val L. Fitch

Physics Needs Heroes Too, C. Duncan RiceTwo Civic Scientists—Benjamin Bederson

and the other Benjamin, Neal LaneAn Editor Par Excellence, Eugen

MerzbacherBen as APS Editor, Bernd CrasemannBen Bederson: Physicist–Historian,

Roger H. StuewerPedagogical Notes on Classical Casimir

Effects, Larry SpruchPolarizabilities of 3P Atoms and van der

Waals Coefficients for Their Interactionwith Helium Atoms, X. Chu andA. Dalgarno

The Two Electron Molecular BondsRevisited: From Bohr Orbits toTwo-Center Orbitals, Goong Chen,Siu A. Chin, Yusheng Dou, Kishore T.Kapale, Moochan Kim, Anatoly A.Svidzinsky, Kerim Urtekin, Han Xiongand Marlan O. Scully

Resonance Fluorescence of Two-LevelAtoms, H. Walther

Atomic Physics with Radioactive Atoms,Jacques Pinard and H. Henry Stroke

Thermal Electron Attachment andDetachment in Gases, Thomas M. Miller

Recent Developments in the Measurementof Static Electric Dipole Polarizabilities,Harvey Gould and Thomas M. Miller

Trapping and Moving Atoms on Surfaces,Robert J. Celotta and Joseph A. Stroscio

Electron-Impact Excitation Cross Sectionsof Sodium, Chun C. Lin and John B.Boffard

Atomic and Ionic Collisions, EdwardPollack

Atomic Interactions in Weakly IonizedGas: Ionizing Shock Waves in Neon,Leposava Vuškovic and Svetozar Popovic

Approaches to Perfect/Complete ScatteringExperiments in Atomic and MolecularPhysics, H. Kleinpoppen, B. Lohmann,A. Grum-Grzhimailo and U. Becker

Reflections on Teaching, Richard E.Collins


Volume 52Exploring Quantum Matter with Ultracold

Atoms in Optical Lattices, ImmanuelBloch and Markus Greiner

The Kicked Rydberg Atom, F.B. Dunning,J.C. Lancaster, C.O. Reinhold, S.Yoshida and J. Burgdörfer

Photonic State Tomography, J.B. Altepeter,E.R. Jeffrey and P.G. Kwiat

Fine Structure in High-L Rydberg States:A Path to Properties of Positive Ions,Stephen R. Lundeen

A Storage Ring for Neutral Molecules,Floris M.H. Crompvoets, Hendrick L.Bethlem and Gerard Meijer

Nonadiabatic Alignment by Intense Pulses.Concepts, Theory, and Directions, TamarSeideman and Edward Hamilton

Relativistic Nonlinear Optics, DonaldUmstadter, Scott Sepke and ShouyuanChen

Coupled-State Treatment of ChargeTransfer, Thomas G. Winter

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