Learning about order from noise

Quantum noise studies of ultracold atoms

Eugene Demler Harvard University

Funded by NSF, Harvard-MIT CUA, AFOSR, DARPA, MURI

Collaborators:Ehud Altman, Robert Cherng, Adilet Imambekov, Vladimir Gritsev, Mikhail Lukin, Anatoli Polkovnikov

Introduction. Historical review

Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices:HBT experiments and beyond

Quantum noise in interference experiments with independent condensates

Outline

Adiabaticity of creating many-body fermionic states in optical lattices

Quantum noiseClassical measurement: collapse of the wavefunction into eigenstates of x

Histogram of measurements of x

Probabilistic nature of quantum mechanics

Bohr-Einstein debate on spooky action at a distance

Measuring spin of a particle in the left detectorinstantaneously determines its value in the right detector

Einstein-Podolsky-Rosen experiment

Aspect’s experiments:tests of Bell’s inequalities

SCorrelation function

Classical theories with hidden variable require

Quantum mechanics predicts B=2.7 for the appropriate choice of ‘s and the state

Experimentally measured value B=2.697. Phys. Rev. Let. 49:92 (1982)

+

-

+

-1 21 2

S

Hanburry-Brown-Twiss experimentsClassical theory of the second order coherence

Measurements of the angular diameter of SiriusProc. Roy. Soc. (London), A, 248, pp. 222-237

Hanbury Brown and Twiss, Proc. Roy. Soc. (London), A, 242, pp. 300-324

Quantum theory of HBT experiments

For bosons

For fermions

Glauber,Quantum Optics and Electronics (1965)

HBT experiments with matter

Experiments with 4He, 3HeWestbrook et al., Nature (2007)

Experiments with neutronsIanuzzi et al., Phys Rev Lett (2006)

Experiments with electronsKiesel et al., Nature (2002)

Experiments with ultracold atomsBloch et al., Nature (2005,2006)

Shot noise in electron transport

e- e-

When shot noise dominates over thermal noise

Spectral density of the current noise

Proposed by Schottky to measure the electron charge in 1918

Related to variance of transmitted charge

Poisson process of independent transmission of electrons

Shot noise in electron transport

Current noise for tunneling across a Hall bar on the 1/3 plateau of FQE

Etien et al. PRL 79:2526 (1997)see also Heiblum et al. Nature (1997)

Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices:Hanburry-Brown-Twiss experiments and beyond

Theory: Altman, Demler, Lukin, PRA 70:13603 (2004)

Experiment: Folling et al., Nature 434:481 (2005); Spielman et al., PRL 98:80404 (2007); Tom et al. Nature 444:733 (2006)

Atoms in optical lattices

Theory: Jaksch et al. PRL (1998)

Experiment: Kasevich et al., Science (2001); Greiner et al., Nature (2001); Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004); Ketterle et al., PRL (2006)

Bose Hubbard model

tunneling of atoms between neighboring wells

repulsion of atoms sitting in the same well

U

t

4

Bose Hubbard model

1n

U

02

0

M.P.A. Fisher et al.,PRB40:546 (1989)

MottN=1

N=2

N=3

Superfluid

Superfluid phase

Mott insulator phase

Weak interactions

Strong interactions

Mott

Mott

Superfluid to insulator transition in an optical lattice

M. Greiner et al., Nature 415 (2002)

U

1n

t/U

SuperfluidMott insulator

Why study ultracold atoms in optical lattices

t

U

t

Fermionic atoms in optical lattices

Experiments with fermions in optical lattice, Kohl et al., PRL 2005

Antiferromagnetic and superconducting Tc of the order of 100 K

Atoms in optical lattice

Antiferromagnetism and pairing at sub-micro Kelvin temperatures

Same microscopic model

Positive U Hubbard model

Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)

Antiferromagnetic insulator

D-wave superconductor

Atoms in optical lattice

Same microscopic model

Quantum simulations of strongly correlated electron systems using ultracold atoms

Detection?

Quantum noise analysis as a probe of many-body states of ultracold

atoms

Time of flight experiments

Quantum noise interferometry of atoms in an optical lattice

Second order coherence

Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment

Experiment: Folling et al., Nature 434:481 (2005)

Hanburry-Brown-Twiss stellar interferometer

Second order coherence in the insulating state of bosons

Bosons at quasimomentum expand as plane waves

with wavevectors

First order coherence:

Oscillations in density disappear after summing over

Second order coherence:

Correlation function acquires oscillations at reciprocal lattice vectors

Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment

Experiment: Folling et al., Nature 434:481 (2005)

Second order coherence in the insulating state of fermions.Hanburry-Brown-Twiss experiment

Experiment: Tom et al. Nature 444:733 (2006)

How to detect antiferromagnetism

Probing spin order in optical lattices

Correlation Function Measurements

Extra Braggpeaks appearin the secondorder correlationfunction in theAF phase

How to detect fermion pairing

Quantum noise analysis of TOF images is more than HBT interference

Second order interference from the BCS superfluid

)'()()',( rrrr nnn

n(r)

n(r’)

n(k)

k

0),( BCSn rr

BCS

BEC

kF

Theory: Altman et al., PRA 70:13603 (2004)

Momentum correlations in paired fermionsGreiner et al., PRL 94:110401 (2005)

Fermion pairing in an optical lattice

Second Order InterferenceIn the TOF images

Normal State

Superfluid State

measures the Cooper pair wavefunction

One can identify unconventional pairing

Interference experimentswith cold atoms

Interference of independent condensates

Experiments: Andrews et al., Science 275:637 (1997)

Theory: Javanainen, Yoo, PRL 76:161 (1996)Cirac, Zoller, et al. PRA 54:R3714 (1996)Castin, Dalibard, PRA 55:4330 (1997)and many more

Nature 4877:255 (1963)

x

z

Time of

flight

Experiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 441:1118 (2006)

Experiments with 1D Bose gas S. Hofferberth et al. arXiv0710.1575

Interference of two independent condensates

1

2

r

r+d

d

r’

Clouds 1 and 2 do not have a well defined phase difference.However each individual measurement shows an interference pattern

x1

d

Amplitude of interference fringes,

Interference of fluctuating condensates

For identical condensates

Instantaneous correlation function

For independent condensates Afr is finite but is random

x2

Polkovnikov, Altman, Demler, PNAS 103:6125(2006)

Fluctuations in 1d BEC

Thermal fluctuations

Thermally energy of the superflow velocity

Quantum fluctuations

For impenetrable bosons and

Interference between Luttinger liquids

Luttinger liquid at T=0

K – Luttinger parameter

Finite temperature

Experiments: Hofferberth,Schumm, Schmiedmayer

For non-interacting bosons and

Distribution function of fringe amplitudes for interference of fluctuating condensates

L

is a quantum operator. The measured value of will fluctuate from shot to shot.

Higher moments reflect higher order correlation functions

Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006Imambekov, Gritsev, Demler, cond-mat/0612011

We need the full distribution function of

Distribution function of interference fringe contrastExperiments: Hofferberth et al., arXiv0710.1575Theory: Imambekov et al. , cond-mat/0612011

Comparison of theory and experiments: no free parametersHigher order correlation functions can be obtained

Quantum fluctuations dominate:asymetric Gumbel distribution(low temp. T or short length L)

Thermal fluctuations dominate:broad Poissonian distribution(high temp. T or long length L)

Intermediate regime:double peak structure

Interference of two dimensional condensates

Ly

Lx

Lx

Experiments: Hadzibabic et al. Nature (2006)

Probe beam parallel to the plane of the condensates

Gati et al., PRL (2006)

Interference of two dimensional condensates.Quasi long range order and the KT transition

Ly

Lx

Below KT transitionAbove KT transition

x

z

Time of

flight

low temperature higher temperature

Typical interference patterns

Experiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 441:1118 (2006)

integration

over x axis

Dx

z

z

integration

over x axisz

x

integration distance Dx

(pixels)

Contrast afterintegration

0.4

0.2

00 10 20 30

middle Tlow T

high T

integration

over x axis z

Experiments with 2D Bose gasHadzibabic et al., Nature 441:1118 (2006)

fit by:

integration distance Dx

Inte

grat

ed c

ontr

ast 0.4

0.2

00 10 20 30

low Tmiddle T

high T

if g1(r) decays exponentially with :

if g1(r) decays algebraically or exponentially with a large :

Exponent

central contrast

0.5

0 0.1 0.2 0.3

0.4

0.3high T low T

2

21

2 1~),0(

1~

x

D

x Ddxxg

DC

x

“Sudden” jump!?

Experiments with 2D Bose gasHadzibabic et al., Nature 441:1118 (2006)

Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al., Nature 441:1118 (2006)

The onset of proliferation coincides with shifting to 0.5!

Fraction of images showing at least one dislocation

0

10%

20%

30%

central contrast

0 0.1 0.2 0.3 0.4

high T low T

Adiabaticity of creating many-body fermionic states in optical lattices

Formation of molecules with increasing interaction strength

Strohmaier et al., arXiv:0707.314

Saturation in the number ofmolecules created is related to the finite rate of changinginteraction strength U(t)

Formation of molecules with increasing interaction strength

U

As U is increased, the excess energy of two unpaired atoms should be converted to the kinetic energy of bound pairs.

The kinetic energy of a single molecule is set by .When U>>t many particles will have to be involved in the relaxation process.

During adiabatic evolution with increasing attractive U, all single atomsshould be converted to pairs. Entropy is put into the kinetic energy of bound pairs.

Hubbard model with repulsion:dynamics of breaking up pairs

Energy of on-site repulsion Energy of spin domain walls

U

E

Hubbard model with repulsion:dynamics of breaking up pairs

Energy of on-site repulsion UEnergy of spin domain wall

Stringent requirements on the rate of change of the interaction strength to maintain adiabaticity at the level crossing

Hubbard model with repulsion:dynamics of breaking up pairs

Hubbard model with repulsion:dynamics of breaking up pairs

Dynamics of recombination: a moving pair pulls out a spin domain wall

High order perturbation theory

Hubbard model with repulsion:dynamics of breaking up pairs

N itself is a function of U/t :

U

E

Hubbard model with repulsion:dynamics of breaking up pairs

Probability of nonadiabatic transition

12 – Rabi frequency at crossing point

d – crossing time

Extra geometrical factor to account for different configurations of domain walls

Formation of molecules with increasing interaction strength

U

Value of U/t for which one finds saturation in the production of molecules

V0/ER=10, 7.5, 5.0, 2.5

Rey, Sensarma, Demler

Summary

Experiments with ultracold atoms provide a new perspective on the physics of strongly correlated many-body systems. Quantum noise is a powerful tool for analyzing many body states of ultracold atoms

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