SPIN EXCHANGE OPTICAL PUMPING OF NEON

AND ITS APPLICATIONS

RAJAT K. GHOSH

A DISSERTATION

PRESENTED TO THE FACULTY

OF PRINCETON UNIVERSITY

IN CANDIDACY FOR THE DEGREE

OF DOCTOR OF PHILOSOPHY

RECOMMENDED FOR ACCEPTANCE

BY THE DEPARTMENT OF

PHYSICS

ADVISER: MICHAEL V. ROMALIS

SEPTEMBER 2009

c© Copyright by Rajat K. Ghosh, 2009.

All Rights Reserved

Abstract

Hyperpolarized noble gases are used in a variety of applications including medi-

cal diagnostic lung imaging, tests of fundamental symmetries, spin filters, atomic

gyroscopes, and atomic magnetometers. Typically 3He is utilized because large

3He polarizations on the order of 80% can be achieved. This is accomplished by

optically pumping an alkali vapour which polarizes a noble gas nucleus via spin

exchange optical pumping.

One hyperpolarized noble gas application of particular importance is the K-

3He co-magnetometer. Here, the alkali atoms optically pump a diamagnetic noble

gas. The magnetic holding field for the alkali and noble gas is reduced until both

species are brought into hybrid magnetic resonance. The co-magnetometer ex-

hibits many useful attributes which make it ideal for tests of fundamental physics,

such as insensitivity to magnetic fields.

The co-magnetometer would demonstrate increased sensitivity by replacing

3He with polarized 21Ne gas. Tests of CPT violation using co-magnetometers

would be greatly improved if one utilizes polarized 21Ne gas. The sensitivity of

the nuclear spin gyroscope is inversely proportional to the gyromagnetic ratio of

the noble gas. Switching to neon would instigate an order of magnitude gain in

sensitivity over 3He.

In order to realize these applications the interaction parameters of 21Ne with

alkali metals must be measured. The spin-exchange cross section σse, and mag-

netic field enhancement factor κ0 are unknown, and have only been theoretically

calculated. There are no quantitative predictions of the neon-neon quadrupolar

relaxation rate Γquad.

In this thesis I test the application of a K-3He co-magnetometer as a navigational

gyroscope. I discuss the advantages of switching the buffer gas to 21Ne. I discuss

the feasibility of utilizing polarized 21Ne for operation in a co-magnetometer, and

iii

construct a prototype 21Ne co-magnetometer. I investigate polarizing 21Ne with

optical pumping via spin exchange collisions and measure the spin exchange rate

coefficient of K and Rb with Ne to be 2.9 × 10−20cm3/s and 0.81 × 10−19cm3/s.

We measure the magnetic field enhancement factor κ0 to be 30.8 ± 2.7, and 35.7 ±

3.7 for the K-Ne, and the Rb-Ne pair. We measure the quadrupolar relaxation

coefficient to be 214± 10 Amagat·s. Furthermore the spin destruction cross section

of Rb, and K with 21Ne is measured to be 1.9 × 10−23cm2 and 1.1 × 10−23cm2.

iv

Acknowledgements

I would first like to thank my advisor Michael Romalis for his help in the lab. His

passion to study fundamental physics is what drives the entire lab group. Without

his assistance and guidance this work would not have been possible. Of special

note is his availability to discuss the underlying physics so that one can gain a

deeper view of the atomic processes which we study.

The projects described herein have greatly benefited from the fruitful collabo-

ration with my colleagues in the Romalis lab. These projects could not have been

accomplished by me alone. Although I could never fully express my appreciation

and admiration for my fellow colleagues I would still like to take the time to thank

them each individually. First I would like to thank Tom Kornack. He built the

first generation CPT violation experiment. Without his help I would not have been

able to get any gyroscope data. I would also like to thank Γιωγos Bασιλακηs, and

Sylvia Smullin for their help with the scalar magnetometer experiment. I would

also like to thank Vishal Shah for his help with regards to the spin exchange rate

measurements. I really appreciate all the time you have taken to discuss all things

atomic physics.

This work has benefited from the support of the many staff members in the

physics department. I would like to thank Bill Dix for his help in machining the

parts of my experiment when I was unable to do so. I would also like to thank

Ted Lewis in the metal stockroom for all of his help. I would be remiss if I did not

thank Mike Peloso for all his help teaching me how to machine in the student shop

and always suggesting alternative designs, or easier ways in which to machine my

designs. I will miss our talks, or more accurately shouts, about guitars, and music

theory over the commercial lathe and drill press.

I would like to thank Mary Santay, Barbara Grunwerg, Kathy Warren, John

Washington, from the purchasing and receiving department. You really make it a

v

pleasure to come visit the A level. I would also like to thank Claude Champagne

of the purchasing department for always taking the time to try and make all of us

smile. I would also like to thank Regina Savadge, Ellen Webster Synakowski, and

Mary Delorenzo as the vanguards of the atomic physics group. I would also like to

thank Mike Souza for the glass cells which only he could so expertly craft. I would

like to give a special thanks to Laurel Lerner for all her help in all things, especially

near the end of my time here.

During my tenure here I have spent much of my time with my fellow grad stu-

dents in the deepest darkest Jadwin. You have been both great colleagues, and

even better friends. First I must thank Scott Seltzer for our many tea time conver-

sations about not only atomic physics, but politics, cinema, and the finer points

of life in general. I do miss our discussions about british comedy, and the beauty

of a proper Jewish Deli and the perfect matza ball soup. I would like to thank

the enthusiasm Justin Brown has shown over the years. If only we could all have

the energy you do I think the World wouldn’t need coffee. As for my good friend

Γιωγos Bασιλακηs I will miss your contagious love of science, and being able to

debate the biggest proponent of the Greek culture. My only hope is that they do

not one day find the work of my thesis had already been scribed on one of Aris-

totle’s tablets. I would also like to thank Ranjit Chima. Friends like you are very

rare. I think life in Princeton would have not been nearly as fun had you not been

here.

I would also like to acknowledge all the past members of the atomic physics

group from whom I have learnt a great deal over the years including Micah Ledbet-

ter, Hui Xia, Kiwoong Kim, Charles Sule, Andre Baranga, Oleg Polyakov, Parker

Meares, Scott Seltzer and Dan Hoffman. I will miss my conversations with Dan

about all things, and with Kiwoong about when it is appropriate to acknowledge

commoners.

vi

I would like to give thanks to my readers, and my committee members for tak-

ing the time to read my thesis and giving useful suggestions for the improvement

of this manuscript. I am also grateful to Scott Seltzer for his careful reading of the

first draft of my thesis.

Finally and most importantly I would like to thank all of my family. Without

you none of this would have been possible. I would like to thank my sister Sheila

for all her support. Your dedication to your own work encourages me to always

stay positive and persevere. You have always listened to me when I needed com-

forting, and cheered me when I need cheering. I don’t think I could have gone

through this without you. I have to give a special thanks to my parents. With-

out your guidance none of what I have accomplished would have been possible.

I would like to thank my mother for her constant encouragement, her support,

and her unwavering belief in me. Without you pushing me when I needed to be

pushed, and encouraging me when I needed to be encouraged I would not have

made it this far. I must also thank my father for his constant help in both dis-

cussing all thing physics, and in his advice. From the time I was little you have

always helped me in my education, and taught me a great deal. Your constant

encouragement has meant all the difference to me. To all of my family I owe you a

debt which I can never acknowledge enough, or ever justly repay. You inspire me.

vii

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Table of Contents x

List of Tables xi

List of Figures xiv

1 Introduction 1

2 Background 7

2.1 Optical Pumping background . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Atomic Energy Levels . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.3 Effects of Pumping Rate and resonance Lineshape on Optical

Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.4 Evolution of Alkali polarization due to Optical Pumping . . . 17

2.1.5 Dynamics of polarized alkali at low magnetic fields . . . . . . 20

2.2 Spin Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Spin Exchange Collisions . . . . . . . . . . . . . . . . . . . . . 24

2.2.2 Spin Destruction Collisions . . . . . . . . . . . . . . . . . . . . 29

2.2.3 Diffusion wall collisions, and Magnetic field Gradients . . . . 30

viii

2.3 Monitoring polarized Alkali . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.1 Optical Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3.2 Light Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 Coupled Spin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.1 Optical Pumping of Noble Gas . . . . . . . . . . . . . . . . . . 42

2.4.2 Interaction of polarized alkali with polarized noble gas . . . . 44

2.5 Manipulation of polarized noble gas spins, and Magnetic shielding . 48

2.5.1 Adiabatic Fast Passage . . . . . . . . . . . . . . . . . . . . . . . 49

2.5.2 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . 52

2.5.3 Magnetic Shielding . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 Nuclear Spin Gyroscope 59

3.1 Co-magnetometer Gyroscope Implementation and behaviour . . . . 60

3.2 Effect of Experimental Imperfections on Gyroscope Performance . . 66

3.3 Zeroing the Co-magnetometer Gyroscope . . . . . . . . . . . . . . . . 67

3.4 Co-magnetometer Gyroscope Sensitivity . . . . . . . . . . . . . . . . . 70

4 Initial tests of an alkali-Neon co-magnetometer 75

4.1 Magnetometer setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2 Neon Polarization Measurements and Preliminary Neon Co-Magnetometer

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3 Influence of Quadrupole collisions in Polarizing neon nuclei . . . . . 77

4.3.1 T1 measurement of Neon . . . . . . . . . . . . . . . . . . . . . 80

4.4 Improving Magnetometer Sensitivity . . . . . . . . . . . . . . . . . . . 84

4.4.1 Removing Birefringence and false Faraday Rotation signals . 84

4.4.2 Controlling and Monitoring the Laser stability . . . . . . . . . 85

4.4.3 Miniaturization of Gyroscope . . . . . . . . . . . . . . . . . . . 86

4.4.4 Alternate methods to heat Cell, and remove Convection noise 87

ix

5 Measurement of parameters for Polarizing Ne with K or Rb metal 91

5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2.1 NMR detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.2.2 Electron Paramagnetic Resonance Shift . . . . . . . . . . . . . 99

5.2.3 Alkali Polarization . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2.4 Alkali Polarization Decay Constant measurement . . . . . . . 103

5.2.5 Back Polarization measurement . . . . . . . . . . . . . . . . . . 104

5.2.6 Alkali density measurement . . . . . . . . . . . . . . . . . . . . 105

5.3 Fermi Contact interaction κ0 Results . . . . . . . . . . . . . . . . . . . 107

5.4 Results of neon quadrupolar relaxation Γquad measurement . . . . . . 107

5.5 Spin exchange Rate coefficient Results . . . . . . . . . . . . . . . . . . 108

5.6 Measurement of Spin destruction cross-sections of neon with Rb and K110

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6 Feasibility of utilizing 21Ne in a co-magnetometer 114

6.1 Effects of Light Propogation and alkali relaxation on Rb-Ne co-magnetometer

simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2 Simulation of Noble gas relaxation . . . . . . . . . . . . . . . . . . . . 117

6.3 Noise mechanisms in a Rb-Ne co-magnetometer . . . . . . . . . . . . 118

6.3.1 Spin Projection Noise . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3.2 Photon Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.4 Results of Rb-Ne co-magnetometer simulation . . . . . . . . . . . . . 122

7 Conclusions and future work 125

A Properties of Ne21 129

x

List of Tables

2.1 Line broadening and shift of K in various Gases . . . . . . . . . . . . 16

2.2 Spin Destruction Cross Sections . . . . . . . . . . . . . . . . . . . . . . 30

3.1 Gyroscope performance comparison . . . . . . . . . . . . . . . . . . . 74

5.1 K-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . . 110

5.2 Rb-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . 110

5.3 Fermi Contact Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.4 Rb-Ne and K-Ne spin destruction cross sections. . . . . . . . . . . . . 112

7.1 K-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . . 125

7.2 Rb-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . 126

7.3 Fermi Contact Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.4 Rb-Ne and K-Ne spin destruction cross sections. . . . . . . . . . . . . 126

A.1 Properties of Ne21 relevant for optical pumping . . . . . . . . . . . . 130

xi

List of Figures

1.1 Basic operation of an Atomic magnetometer . . . . . . . . . . . . . . . 3

2.1 Alkali metal energy level diagram . . . . . . . . . . . . . . . . . . . . 9

2.2 Optical pumping of the electron spin of an alkali atom . . . . . . . . . 12

2.3 Ground-state Zeeman level splitting . . . . . . . . . . . . . . . . . . . 26

2.4 Spin-exchange collisions can cause atoms to switch hyperfine levels . 26

2.5 Spin-temperature distribution . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 Principle of optical rotation . . . . . . . . . . . . . . . . . . . . . . . . 33

2.7 Branching ratios for the D1 and D2 transitions . . . . . . . . . . . . . 35

2.8 Methods for detecting optical rotation . . . . . . . . . . . . . . . . . . 36

2.9 Methods for detecting optical rotation . . . . . . . . . . . . . . . . . . 37

2.10 Polarized Noble gas screens transverse fields . . . . . . . . . . . . . . 48

2.11 Effective magnetic field in the rotating frame . . . . . . . . . . . . . . 51

2.12 NMR tip of atoms with spin ~S . . . . . . . . . . . . . . . . . . . . . . . 54

2.13 Affinity of Magnetic fields line for Magnetic Shields . . . . . . . . . . 57

3.1 Schematic of the co-magnetometer implemented as a gyroscope . . . 61

3.2 Side view of the gyroscope configuration for the co-magnetometer . 62

3.3 In-situ calibration of non-contact displacement sensors. . . . . . . . . 62

3.4 Comparison of co-magnetometer gyroscope signal to displacement

sensor signal with no free parameters . . . . . . . . . . . . . . . . . . 63

xii

3.5 noise spectrum of the comagnetometer gyroscope . . . . . . . . . . . 63

3.6 Suppression of an applied magnetic field gradient by the co-magnetometer

compared to that of a non-compensating magnetometer . . . . . . . . 64

3.7 The co-magnetometer suppressing magnetic fields . . . . . . . . . . . 65

3.8 Response of co-magnetometer to a magnetic field transient . . . . . . 65

3.9 Long term drift of gyroscope . . . . . . . . . . . . . . . . . . . . . . . 72

4.1 Experimental setup of Neon Magetometer . . . . . . . . . . . . . . . . 76

4.2 T2 time of ≈ 14minutes for Neon polarization when operating away

from the compensation point in the co-magnetometer configuration. 77

4.3 Compensation behaviour of the K-Ne comagnetometer to an exter-

nally applied magnetic transient field . . . . . . . . . . . . . . . . . . 78

4.4 T1 of 105 minutes for a 1.6atm cell of Ne at 170C. . . . . . . . . . . . 79

4.5 Theoritical simulation of the noble gas spin for positive gain κ . . . . 83

4.6 Theoritical simulation of the noble gas spin for negative gain κ . . . . 83

4.7 Magnetic field homogeneity for a 3cm×3cm region in magetic shields 87

4.8 Silvered oven holding Boron-Nitride housing for Pyrex cell . . . . . . 89

5.1 Experimental Setup for measuring spin exchange parameters of K,

and Rb with Ne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 Representative NMR signal of polarized 21Ne gas . . . . . . . . . . . 99

5.3 Representative EPR shifts after 2 hours of polarization . . . . . . . . . 101

5.4 Determination of Alkali polarization via RF sweep over Zeeman levels103

5.5 Potassium Polarization decay as a result of Pump beam being man-

ually chopped . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.6 Absolute Potassium Back polarization as Neon is flipped via Adia-

batic fast passage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.7 Neon Relaxation as a function of cell pressure. . . . . . . . . . . . . . 108

xiii

5.8 Absolute neon polarization as function of time for determination of

Spin Exchange Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.9 Scatter in Spin exchange rate measurements for K-Ne . . . . . . . . . 111

5.10 Scatter in Spin exchange rate measurements for Rb-Ne . . . . . . . . 111

6.1 Absolute Rb polarization as function of propagation distance through

cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

xiv

Chapter 1

Introduction

Since the time of the first compass Man has been intrigued by the properties of

magnetic fields. This fascination only increased with the birth of modern electron-

ics. However the reach of magnetism extends well beyond that of electronics. The

duality of the theory of magnetism and electricity proved to be a fundamental re-

alization when special relativity was utilized to unify them. Even light has been

shown to be a form of electromagnetic radiation. The properties of magnetic fields

have been used for navigational purposes, and as a means to unify seemingly dis-

similar forces in physics for over a century.

In the modern era one finds even more applications of magnetic fields. A few

of these include medical imaging, detection of explosives, generation of electricity,

tests of fundamental symmetries, and maglev trains. Therefore, it should be no

surprise that the ability to precisely and accurately measure magnetic fields with

as much sensitivity as possible is of prime importance.

During the past century scientists have continually sought to improve the sen-

sitivity of magnetic field measurements. This began with Hall probes, fluxgates,

and ultimately in superconducting quantum interference devices (SQUIDS). For

the past few decades these SQUIDS have been the state of the art with respect to

1

measuring magnetic fields. They exhibit sensitivity on the order of 1fT when oper-

ating in an environment with superconducting shields (Clarke & Braginski (2004)).

However this decade has seen the re-emergence of spin precession based atomic

magnetometers with improved sensitivity which now surpass that of SQUIDS.

In 1957 Dehmelt (Dehmelt (1957)) proposed observing the precession of alkali

spins in order to measure the strength of a magnetic field. This was carried out ex-

perimentally the same year by Bell and Bloom (Bell & Bloom (1957)). These works

were aided by the contribution of Kastler who created a technique to produce sig-

nificant population changes from free atoms in the ground state (Kastler (1957)).

For this work he was awarded the 1966 Nobel prize in physics.

Most modern day spin precession atomic magnetometers operate on the same

underlying principles as those used by Bell, Bloom, and Kastler. Spins are polar-

ized by optically pumping with a laser beam, and allowed to precess in a magnetic

field according to Larmor precession. See fig.1.1.

~ω = γ~B (1.1)

Here the frequency of precession is directly proportional to the strength of the

magnetic field. This precession can be detected via optical rotation of the plane of

polarization of a linearly polarized probe laser beam. The plane of polarization of

the probe laser becomes rotated as it interacts with a magnetically polarized sam-

ple. This leads to a simple way in which to measure the alkali precession frequency,

and hence the magnetic field.

Normally the atoms one uses to measure precession are alkali atoms. This is for

multiple reasons. The most important of which is that because alkali only have one

valence electron, they are simple spin systems. That is their total spin can be char-

acterized by the sum of the nuclear spin, and valence electron spin. Furthermore

2

F

B

ω

Pump Beam

Probe Beam

Figure 1.1: For operation of a magnetometer alkali atoms are polarized via circularpolarized pump beam. A probe beam measures the orientation of the alkali po-larization via Faraday rotation as the atoms precess due to a magnetic field. Thefrequency of precession can be used to determine the strength of the magnetic field.

3

the ground state is spherically symmetric. This leads to long lived ground state po-

larization lifetimes. Additionally the alkali atoms can be easily interrogated with

lasers via a strong optical transition. This enables us to pump the majority of the

alkali into specific states so that their magnetic moments are all parallel.

In order to pump the alkali metal in its gas state, we heat the metal sample

and rely on the saturated vapour pressure to supply the vapour to be optically

pumped. The sensitivity of the spin precession magnetometer is characterized by

the time for which the spins can precess coherently.

∆B =1

γT2(1.2)

Where T2 is the transverse coherence time of the optically pumped vapour. Quan-

tum mechanically we can interpret this spin precession as the measurement of the

splitting of the Zeeman levels of the alkali, in this case in its ground state. The

magnetic linewidth can be expressed as

∆B =∆ω

γ(1.3)

Thus to optimize the sensitivity of the magnetometer, we maximize the spin pre-

cession coherence time, and effectively minimize the linewidth of the Zeeman res-

onance line. Therefore, we can express the magnetometer sensitivity δB in terms

of the Zeeman level linewidth ∆B, and the signal to noise ratio of the Zeeman

resonance.

δB =∆B

SNR(1.4)

When the alkali atoms collide with the cell walls they depolarize. In order to in-

crease the polarization lifetime one normally introduces a buffer gas into the cell.

One can coat the cell walls with either parrafin, or silane coatings as these reduce

4

the alkali depolarization rate upon collision. The buffer gas densities are typically

105 − 106 times greater than the alkali density under normal operating conditions.

Typically inert noble gases are utilized for this purpose. In fact if the noble gas

utilized has a nonzero nuclear spin one can transfer some of the alkali electron

spin polarization to the noble gas nuclear spin via an interaction known as spin

exchange collisions. This can be used to construct co-magnetometers, which are a

type of magnetometer containing a polarized noble gas species.

The strong D1 transitions of the alkali can easily be accessed by modern solid

state diode lasers. These lasers can easily be tuned to pump the alkali vapour,

and are quite stable in frequency. The preferred diode laser is a DFB (distributed

feedback) laser. These have reflection gratings etched directly on the diode sur-

face which result in operation with very stable frequency. This frequency stability

makes them ideal for measuring the alkali polarization as a linearly polarized de-

tuned probe beam.

The sensitivity to polarization of the probe beam due to spin projection noise

varies inversely as the number of alkali in the cell. This effect will be discussed

later in this text.

∆Sx ∝1√N

(1.5)

One can increase this by either increasing the vapour pressure of the alkali, by in-

creasing the temperature, or by increasing the path length of the vapour cell. If

one increases the density of the cell the alkali interact with themselves via spin ex-

change collisions which broaden the Zeeman resonance. For years this had been

the limiting mechanism in improving atomic magnetometer sensitivity. Typically

cells had been constructed with large volume, with low density, at room temper-

ature in order to minimize the reduction in polarization lifetime due to spin ex-

change collisions.

In 2002 Allred et al. (2002) experimentally demonstrated a magnetometer which

5

operated with small measurement volume, near zero field which did not experi-

ence broadening of the Zeeman resonance due to spin exchange collisions. It does

so by operating in a spin exchange relaxation free regime (SERF) which was first

proposed by Happer & Tam (1977). Many of the magnetometers described in this

thesis are based on this effect.

In this thesis we describe the potential application of enriched neon as a buffer

gas. We also demonstrate the application of a potassium-helium co-magnetometer

as a sensitive gyroscope. We show that the sensitivity of this gyroscope can be

further improved if we switch the buffer gas from helium to neon.

This potassium-helium co-magnetometer was originally used for tests of Lorentz

violation, and CPT invariance. CPT (charge parity time reversal) is a discrete sym-

metry of the universe. Breaking of this symmetry would lead to a violation of

Lorentz symmetry. While these symmetries are conserved in the Standard Model

they may be broken in a more fundamental underlying theory. There exist theo-

ries of quantum gravity which have been shown to violate Lorentz symmetry in

some way. Switching to a neon buffer gas will also increase of the sensitivity of the

co-magnetometer to these effects.

However many of the parameters governing the interaction between potas-

sium, or rubidium with enriched 21Ne have not been measured. The spin exchange

rate between the transfer of polarization of the alkali to the noble gas nuclei has

not been measured. Neither has the Fermi contact interaction between these pairs

of atoms. In order to design and construct an optimized magnetometer one must

have a knowledge of these parameters. We measure them here.

We construct the first 21Ne co-magnetometer and study its behaviour. And

finally we discuss what future work must be carried out in order to use 21Ne for

tests of fundamental symmetries.

6

Chapter 2

Background

In order to understand the experiments involving alkali metal-neon optical pump-

ing it would be prudent to first review the physical processes pertaining to optical

pumping. We begin by discussing the basics of polarizing alkali via optical pump-

ing. We discuss the properties of the energy levels relevant to optical pumping as

well as the effects of line broadening on optical pumping. We also briefly discuss

utilizing polarized alkali to measure magnetic fields.

The process of optical pumping does not polarize the alkali to unity. In the next

section we discuss the relaxation mechanisms which limit the alkali polarization.

Relaxation mechanisms of particular interest include spin relaxation due to spin

exchange collisions, spin destruction collisions, and magnetic field gradients.

Additionally we discuss the means by which the alkali polarization is inter-

rogated. This includes a number of schemes based on optical rotation utilizing

polarimeters. We also investigate spurious signals due to the effects of light shifts.

We discuss the consequences of adding diamagnetic noble gas to the vicinity of

polarized alkali atoms. We investigate the transfer of polarization from the alkali

spin to the noble gas nuclei via spin exchange optical pumping. We describe the

response of the coupled system and the dynamics of an atomic co-magnetometer.

7

Finally we discuss useful techniques with which to manipulate the polarized noble

gas spins, such as adiabatic fast passage, and nuclear magnetic resonance.

2.1 Optical Pumping background

Optical pumping is a technique which can be used to polarize the spins of an

atomic species through application of laser light. The most popular atomic species

to utilize during optical pumping experiments are the alkali atoms. Typically the

transitions utilized to optically pump the alkali are the D1 or D2 doublets. In this

section we discuss the basic principles of optical pumping. We discuss the prop-

erties which make alkali atoms the preferred species for many optical pumping

experiments. We investigate the properties of the alkali atoms energy levels which

are relevant for optical pumping and ways to manipulate these levels to produce a

long lived ground state alkali polarization. Finally we discuss the dynamics of op-

tical pumping alkali in terms of the optical pumping rate, and the spin relaxation

rate.

2.1.1 Atomic Energy Levels

Alkali metals are a convenient choice for optical pumping because they possess

a single unpaired valence electron. The spectroscopic properties are well approxi-

mated by ignoring the interaction of the filled electron sublevels and concentrating

on the valence electron, and its interaction with the atomic nucleus. As such the

wavefunctions describing the energy levels are well described by total angular mo-

mentum quantum numbers of the valence electron spin and the nuclear spin.

Consider the ground state S shell of the alkali atom. The spin of the valence

electron is S= 1/2, and the orbital angular momentum in this state is L= 0. Thus

the total electron angular momentum is J= 1/2. The first excited state is the P

8

s

p

D1 D2

Orbital

Structure

Fine

Structure

Hyperfine

Structure

2P3/2

2P1/2

2S1/2

F=I+1/2

F=I–1/2

F=I+1/2

F=I–1/2

F=I+1/2

F=I+3/2

F=I–1/2

F=I–3/2

Figure 2.1: Energy level splitting of the ground state and first excited state of analkali metal atom. The fine structure splits first excited state further into J=1/2 andJ=3/2 levels. The hyperfine structure further splits the J energy levels. Not drawnto scale.

shell with L= 1. Due to fine structure L·S coupling the P state splits into 2P3/2 and

2P1/2 states. Here we use the standard spectroscopic notation 2S+1LJ to describe

the energy levels. Historically the 2S1/2 → 2P1/2 and 2S1/2 → 2P3/2 transitions are

referred to as the D1 and D2 lines. See fig. 2.1.

The nuclear spin of the alkali metal I couples to the total electron spin J via the

hyperfine interaction to further split the energy levels into states with good quan-

tum number F. Here F is the total angular momentum F=I+J. It follows from the

Wigner-Eckart theorem (see Cohen-Tannoudji 1972) that the total electron angular

momentum J must be parallel to F. Thus when we probe the hyperfine manifolds

the orientation of the total atomic angular momentum vector we also determine

the total electron angular momentum vector direction.

Implementation of a magnetic field lifts the degeneracy between different Zee-

man sublevels of states with the same total angular momentum F, but different

projection m f along the quantization axis defined by the magnetic field. The result-

ing energy splitting between Zeeman levels is proportional to the field for small

9

field strengths, typically less than a Gauss. This splitting gives rise to Larmor

precession of the atoms between the energy levels with frequency ωL = ∆EL =

γ |B|. Here the gyromagnetic ratio γ for alkali atoms is given by γ ≈ ±2π ×

(2.8MHZ/G)/(2I+1), where I is the nuclear spin of the atomic nucleus and the

sign corresponds to the hyperfine manifold, ie. The F = I + 1/2 manifold yields a

+ sign in the gyromagnetic ratio.

For large magnetic fields however the Zeeman energy level splitting is non-

linear and is given by the Breit-Rabi splitting. We can calculate this by studying

the ground state Hamiltonian of the alkali atom.

The ground state Hamiltonian is of the form:

H = AJ I · J + gsµBS · B − gIµN I · B (2.1)

where µN is the nuclear magneton, AJ = 2hωhf/(2I + 1) is the hyperfine coupling

constant, gs≈2 is the electron g-factor, µB=9.274×10−24 J/T is the Bohr magneton,

and gI the nuclear g-factor. The hyperfine coupling constant is specific to the atom

under discussion. The energy spectrum can be calculated from the eigenvalues of

the Hamiltonian to be (Corney (1977)):

E(F = I ± 1/2, mF) = − hωhf

2(2I + 1)− gIµNBmF ±

hωhf

2

√x2 +

4xmF

2I + 1+ 1 (2.2)

Where

x ≡ 2(gsµB + gIµN)B

(2I + 1)AJ=

(gsµB + gIµN)B

hωh f(2.3)

Notice that the energy spacing as a function of the magnetic field is now non-

linear. In the low field adjacent sublevels would be separated by gIµNBmF. This

10

clearly scales with the magnetic field. In the high field regime the last term in eq.

2.2 causes the splitting to no longer be directly proportional to the magnetic field.

2.1.2 Optical Pumping

The experiments described in this thesis require a large source of polarized spin.

One could thermally polarize the sample using brute force by introducing the sam-

ple into a large magnetic field:

Pther = tanh

(12 gsµBB

kBT

)(2.4)

where gs≈2 is the electron g-factor and µB=9.274×10−24 J/T is the Bohr magneton.

The polarization can only be raised to 2% at room temperature if one implements

large magnetic fields on the order of 10T . For comparison the typical absolute

thermal polarization is only 1 × 10−7 at room temperature, in Earth’s field. The

more elegant technique of optical pumping can yield alkali polarization on the

order of unity.

To describe optical pumping we consider a toy model where the atom has no

nuclear spin. However, the following scheme is of a general nature and can be

extended to the case where I 6= 0. Typically one utilizes the ground state D1 tran-

sition of an alkali metal for optical pumping for a variety of reasons. First, the

alkali doublet have strong oscillator strength which leads to larger absorption by

the pump laser. Second, it is theoritically possible to polarize the ground state to

unity if the D1 is excited (Franzen & Emslie (1957)). In the case of D2 pumping the

maximum achievable polarization is limited to 1/2. This transition is not efficient

for optical pumping. It can be utilized under conditions where the gas density

is rare, and there is little collisional mixing in the excited P state. The details of

collisional mixing will be described shortly.

11

mJ = -1/2 m

J = +1/2

2S1/2

2P1/2

σ+

gnipmuP

gni

hc

ne

uQ

gni

hc

ne

uQ

Collisional Mixing

Spin Relaxation

Figure 2.2: Optical pumping of the electron spin of an alkali.

The experiments in this work rely on optically pumping alkali by tuning to

the D1 transition. To describe this process first consider the D1 2S1/2 → 2P1/2

transition of an atom in a magnetic field as shown in fig.2.2. Let us define the

quantization axis along the direction of the magnetic field. The ground and excited

state sublevel degeneracy is lifted in the presence of a magnetic field and can each

be resolved into states with magnetic quantum number mJ .

The goal of optical pumping is to increase the ground state population of one

of the magnetic sublevels, 2S1/2 (mJ = +1/2) in this example. To accomplish this

we first utilize σ+ photons to excite the 2S1/2 (mJ = −1/2) →2 P1/2 (mJ = +1/2)

transition. In the excited state two effects occur. The first is spontaneous emission

to both 2S1/2 magnetic sublevels, and the second is collisional mixing in the excited

state (Walker & Happer (1997)).

Due to the nonzero orbital angular momentum of the excited state the alkali

wavefunction is not spherically symmetric. This causes collisional mixing effects

to occur. Collisional mixing transfers the atom from the excited P mJ = +1/2 to

the mJ = −1/2 sublevel. Typically when alkali metal is optically pumped the al-

kali vapour is in the vicinity of other gases. These can include noble gas, and a

12

molecular quenching gas. The collision between alkali vapour, and noble buffer

gas leads to collisional mixing in the excited state. Physically these may be de-

scribed as the electron orbital angular momenta coupling to the rotational angular

momenta of the molecule temporarily created by the pair of colliding atoms (Wu

et al. (1985))(Walker & Happer (1997)).

In general the Hamiltonian of such an interaction can be expressed as (Wu et al.

(1985)):

Hcm = γ(r)S · N (2.5)

Here the S refers to the alkali spin, N to the rotational angular momentum of the

temporarily formed molecule, and γ(r) as the coupling between the two. This last

factor depends on the inter-atomic potential, the separation of the constituents of

the temporarily formed molecule, and the corresponding wavefunction density of

the pair.

During collision the degree to which the alkali wavefunction is perturbed de-

termines the collisional mixing. Atoms with large polarizability have greater col-

lisional mixing cross-sections. The ground state of the alkali is less susceptible to

wavefunction deformation than the P state sublevels because it is spherically sym-

metric (Corney (1977)).

If the pumping scheme is iterated the atoms eventually populate the 2S1/2mJ =

+1/2 state, as atoms in this state can no longer absorb the σ+ light. This represents

an ideal case. There exist mechanisms to prevent this. Fluorescent light from the

spontaneous decay of the excited P state can interact with other alkali in the sample

and excite atoms out of the 2S1/2mJ = +1/2 sublevel. This is referred to in the

literature as radiation trapping. This can be prevented by introducing a molecular

quenching gas. A typically quenching agent is N2. N2 molecules collide with the

excited alkali atoms and transfer them to the ground state without re-irradiation

of the alkali (Happer (1972)). This can occur because the N2 molecule has a large

13

number of closely spaced energy levels, due to its rotational, and vibrational level

structure. As the N2 molecules return to their ground state they may either transfer

the energy from the alkali atom to its rotational or vibrational modes to be re-

radiated at a frequency different from the D1 line. The excited alkali atoms become

non-radiatively quenched. Typically 100 Torr of Nitrogen is sufficient to prevent

radiation trapping in cells with alkali densities near 1014/cm3. There are also spin-

relaxation mechanisms which can transfer atoms between the two ground state

magnetic sublevels to depolarize the sample. These will be discussed in the next

section.

The alkali nucleus is strongly coupled to the total electron angular momen-

tum via the hyperfine interaction. Thus when we polarize the electrons we also

successfully polarize the alkali nuclei. This leads to some interesting effects. The

alkali electron spin polarization decay is slower than one would initially estimate.

This is due to the electron spins being re-polarized via the hyperfine interaction

with the polarized nucleus. This effect is termed the slowing down factor, and will

be quantitatively described later in this work.

2.1.3 Effects of Pumping Rate and resonance Lineshape on Opti-

cal Pumping

In describing optical pumping it is useful to define a quantity named the optical

pumping rate. The optical pumping rate is defined as the rate at which an unpo-

larized alkali atom absorbs photons from the pump laser.

Rop =∫

I(ν)

hνσ(ν)dν (2.6)

I(ν) is the light spectral density in units of Watts cm−2Hz−1, and σ(ν) is the photon

absorption cross section.

14

The photon absorption cross section is defined by the atomic response as a func-

tion of incident photon frequency. In general it is influenced by the natural lifetime

of the atomic state of interest, pressure broadening of the atomic resonance, and

Doppler broadening of the resonance due to thermal velocity distribution. The

absorption cross section can be related to the classical electron radius by:

∫ ∞

0σ(ν)dν = πrec f (2.7)

This is valid regardless of the spectral line shape. In eq.2.7 the oscillator strength

f is the quantum mechanical correction factor to the classical expression for the

relation between absorption cross section, and classical electron radius. It is a di-

mensionless number and depends on which transition we are pumping. For the

D1 line it is ≈ 1/3, and for D2 it is ≈ 2/3(Migdalek & Kim (1998)). The absorption

cross section is given by:

σ(ν) = πrec f Re[V(ν − ν0)] (2.8)

where V(ν − ν0) is the atomic lineshape. It is given by the Voight profile (Happer

& Mathur (1967)), and includes the effects of pressure broadening, natural lifetime,

and Doppler broadening.

V(ν − ν0) =∫ ∞

0L(ν − ν

′)G(ν

′ − ν0)dν′

(2.9)

Where

V(ν − ν0) =2√

ln 2/π

ΓGw

(2√

ln 2[(ν − ν0) + iΓL/2]

ΓG

)(2.10)

And the function w is given in terms of the complex error function as:

w(x) = e−x2(1 − erf(−ix)) (2.11)

15

Gas Energy Level Half-halfwidth ShiftHe P1/2 1.55 ± 0.03 +0.45 ± 0.04

He P3/2 2.06 ± 0.04 +0.24 ± 0.04Ne P1/2 0.85 ± 0.02 −0.41 ± 0.02Ne P3/2 1.16 ± 0.04 −0.62 ± 0.02Ar P1/2 2.45 ± 0.03 −2.31 ± 0.05Ar P3/2 1.98 ± 0.03 −1.52 ± 0.04Kr P1/2 2.31 ± 0.05 −1.65 ± 0.04Kr P3/2 2.31 ± 0.05 −1.16 ± 0.04Xe P1/2 2.75 ± 0.03 −1.79 ± 0.06Xe P3/2 2.75 ± 0.03 −1.79 ± 0.06N2 P1/2 2.45 ± 0.03 −1.83 ± 0.04N2 P3/2 2.45 ± 0.03 −1.32 ± 0.04

Table 2.1: Comparison of the line broadening and shift parameters for K in variousGases. All experimental measurements were made in the 400 − 420K temperatureregime. The line broadening parameter and shift parameter have units of 10−9 rads−1 atom −1 cm3. Data taken from Lwin & McCartan (1978)

er f denotes the standard error function. Here ΓG is the linewidth of the Gaussian

contribution to the lineshape due to Doppler broadening for an atom of mass M

G(ν − ν0) =2√

ln 2/π

ΓGexp

(−4 ln 2(ν − ν0)

2

Γ2G

)(2.12)

ΓG = 2ν0

c

√2kBT

Mln 2 (2.13)

And ΓL is the linewidth of the Lorentzian contribution to the Voight profile due to

pressure broadening.

L(ν − ν0) =ΓL/2π

(ν − ν0)2 + (ΓL/2)2(2.14)

The values for ΓL as a function of pressure is listed for K vapour in table 2.1.

16

2.1.4 Evolution of Alkali polarization due to Optical Pumping

The evolution of the polarization of the alkali atoms due to optical pumping can

calculated. In order to compute the polarization achieved via optical pumping one

must clarify whether the target cell contains any buffer gas. For certain applica-

tions it becomes advantageous to operate without a buffer gas (see Knappe et al.

(2006),Cates et al. (1988),and Pustelny et al. (2006) for more information), and uti-

lize a cell coating instead. However those applications will not be discussed in this

work.

Once the D1 2S1/2 → 2P1/2 transition is excited the electron will decay back

to both the mJ = 1/2, or mJ = −1/2 sublevels. For the case of no buffer gas the

branching ratios are determined by the Clebsch-Gordon coefficients to be 1/3, and

2/3 respectively (Budker et al. (2004)). However when buffer gas is introduced in

the cell, collisional mixing in the excited states alters the branching ratio to be 1/2

for both cases.

Under these conditions we can now calculate the time evolution of the alkali

polarization. Let us denote the populations in the mJ = +1/2 and mJ = −1/2

sublevels as N+, and N− respectively. The pumping rate is defined as the rate at

which an unpolarized atom absorbs a photon. The mJ = +1/2 state is unable to

absorb a photon if one uses σ+ light. Therefore the rate at which the mJ = −1/2

state absorbs a photon is at twice the optical pumping rate. This is because the

optical pumping rate is defined as per unpolarized atom. Using this and including

the 1/2 branching ratio we can describe the pumping process by realizing dN+dt =

− dN−dt . Thus:

dN+

dt=

1

2(2Rop)N− +

Rrel

2N− − Rrel

2N+ (2.15)

Where Rrel is the relaxation rate. By noting that the polarization can be expressed

17

as:

P = 2 < Sz >=N+ − N−N+ + N−

(2.16)

the time evolution of the polarization can be ascertained. Assuming P(0) = 0:

P(t) = Pequil(1 − e(Rop+Rrel)t) (2.17)

During equilibrium:

Pequil =Rop

Rop + Rrel(2.18)

This is only strictly true in the case of an atom with zero nuclear spin. The electron

spin and nuclear spin are coupled via the hyperfine interaction. As the electron

spin precesses it drags the nuclear spin along with it. This results in a slower

precession frequency than that of a free electron in a magnetic field (Budker et al.

(2004)). It is modified as:

γ =γe

2I + 1(2.19)

Where γe = gsµb/h is the gyromagnetic ratio for a free electron, and has magni-

tude 2π × 2.8MHz/G. The coupling between the valence electron and the nucleus

also alters the rate at which the valence electron is depolarized. Just as the hy-

perfine interaction can polarize the nucleus when we pump the valence electron,

the opposite can occur. When the electron spin is depolarized, the nuclear spin

can re-polarize the electron. Thus the actual rate at which the electron is either

depolarized by collisions, or polarized by a laser must be modified. To good ap-

proximation one can describe this effect by a linear slowing down factor ǫ (Walker

& Happer (1997)). For atoms in low magnetic field where γB << Rse the system

can be described accurately in terms of a two level system. The equation of motion

18

of the atom can now be described as:

dS

dt=

1

ǫ + 1

[γeB × S + Rop

(1

2sz − S

)− RrelS

](2.20)

This is often referred to as the Bloch equation in the literature. Here the first term

describes the precession of the spins in a magnetic field. The second term describes

the pumping of the spins to their equilibrium value. Here ǫ is given by:

ǫ(I, β) = (2I + 1)coth(β/2)coth(β[I + 1/2]) − coth2(β/2) (2.21)

β = ln

(1 + P

1 + P

)(2.22)

P = tanh(β/2) (2.23)

Here I is the nuclear spin, and β is the spin temperature of the system (Walker

& Happer (1997)), which can be related to the polarization P. A more complete

discussion on spin temperature will be described later in this work. But for now

it is sufficient to note the special limiting cases for ǫ for low and high polarization

respectively as:

ǫ(I, β << 1) = 4I(I + 1)/3 (2.24)

ǫ(I, β >> 1) = 2I (2.25)

Although this alters the time evolution of the electron spin polarization to:

P(t) = Pequil(1 − e(Rop+Rrel)t/(ǫ+1)) (2.26)

The equilibrium polarization reached remains the same eq 2.18.

19

2.1.5 Dynamics of polarized alkali at low magnetic fields

Optical pumping is used to polarize the spins of an alkali species. An application of

the polarized alkali spin of particular importance is the determination of magnetic

fields. This is accomplished by determination of the precession frequency of the

alkali by monitoring the spin dynamics and relating it to the magnetic field.

The alkali undergo Larmor precession due to the interaction of the spins with

the magnetic field:

H = γalkali h~B · ~S (2.27)

The dynamics of the alkali due to this interaction gives a response:

d

dt~S =

i

h[H,~S] (2.28)

Noting that the components of the spin follow the commutation relation:

[Sx, Sy] = iSz (2.29)

we can describe the evolution of Sz as

d

dt~Sz = iγalkali(−iBxSy + iBySx) (2.30)

One can describe the evolution of the Sx, and Sy components of the spin with sim-

ilar equations. Inclusion of the effects of optical pumping, and spin relaxation

modify the spin evolution given by eq.2.30 to that given by the phenomonological

equation eq.2.20.

The magnetometer bandwidth can be calculated utilizing a set of simplified

Bloch equations. Assume we impose an oscillating field of the form By = B0exp(−iωt).

Let us rewrite eq.2.20 in terms of the components of the alkali polarization Pxe , and

20

Pze . Then, for a magnetometer where Bx, and Bz are set to zero imposing a field By

gives the following behaviour:

dPex

dt=(γeByPe

z − RtotPex

) 1

ǫ + 1(2.31)

dPez

dt=(−γeByPe

x − RtotPez + Rop

) 1

ǫ + 1(2.32)

The first term represents precession about By, the second represents spin relax-

ation, and the third term represents optical pumping. Solution of the above equa-

tions yields:

Pex =

Pez γeB0

Rtot − iω(ǫ + 1)(2.33)

S = R(Pex) =

Pez γeB0

R2tot − ω2(ǫ + 1)2

(2.34)

ω0 is the frequency of the spin precession in the ambient magnetic field. For low

frequencies the signal can be simplified to obtain:

S =Pe

z γeB0

Rtot(2.35)

2.2 Spin Relaxation

In order to achieve optimal sensitivity and minimize the magnetic linewidth dur-

ing optical pumping experiments, one must maximize the spin polarization life-

time. Since magnetic linewidth is correlated to the lifetime of the Zeeman states,

we are effectively maximizing the spin polarization. In general we must consider

two cases, the longitudinal polarization lifetime T1, and the transverse polariza-

tion lifetime T2. Here T1 is the lifetime of the polarization parallel to the magnetic

holding field, and T2 is the lifetime of polarization in a perpendicular direction.

21

One can express the longitudinal lifetime T1 as:

1

T1=

1

ǫ + 1(Rsd + Rop + Rpr) + Rwall + Rinh (2.36)

Here the first term is due to spin destruction collisions. These collision may be

of several types, including collisions between alkali atoms, collisions between the

alkali atoms and the buffer gas, or collisions between the alkali atoms and the

quenching gas.

Rsd = Rsel fsd + R

bu f f ersd + R

quenchsd (2.37)

Each of the collisional spin destruction rate mechanisms can be described by:

Rsd = nσv (2.38)

where n is the density of the gas species in question which is colliding with the al-

kali, and σ is the spin destruction cross section. Calculation of the spin destruction

crpss-sections can be performed from a knowledge of the atomic wavefunction,

interaction potential and the interaction Hamiltonian. These calculations are nor-

mally accurate to 50% with the measured value, due to imprecise knowledge of

the specifics of the collisional interaction (Walker & Happer (1997)). The last term

v is the relative velocity of the colliding pair and is given by:

v =

√8κBT

πM(2.39)

and M is the reduced mass of the alkali and its colliding partner:

1

M=

1

m+

1

m′ (2.40)

The second term in eq (2.36) is the optical pumping rate. This affects T1 be-

22

cause the absorption of the pump beam alters the angular momentum state of the

alkali. The next term is similar, but due to absorption of the probe beam. These

relaxation mechanisms depolarize the valence electrons, while leaving the nucleus

unaffected. Thus the sum of the first three terms must be divided by the slow-

ing down factor because of the previously mentioned re-pumping of the electron

due to the polarized nucleus. The next term is due to collisions with the wall. In

uncoated cells collisions with the cell wall completely depolarize the alkali atoms.

The last term is due to magnetic field inhomogeneity. The pumped alkali align

with the net magnetic holding field. If a gradient is present it can locally alter the

direction of the net magnetic field and change T1.

Consider the mean free path λ in the gas. Since the Larmor frequency is much

faster than the transit rate v/λ between atomic collisions the atomic polarization

follows the total magnetic field. This leads to relaxation when the atom changes

direction after a collision (Budker et al. (2004)). The atom experiences a small mag-

netic field, which varies slowly compared to the Larmor frequency. It is transverse

to the holding field and corresponds to a rotation of the total magnetic field vector

with a frequency ω ≈ δBvBholdingλ . It can be treated as the flipping probability of an

atom with spin oriented along a magnetic holding field, by the presence of a fluc-

tuating transverse magnetic field. Magnetic field gradients are normally not the

dominant alkali relaxation mechanism. Typical gradients found in the low field co-

magnetometer experiment are 10µG/cm, with a holding field of 1mG. This gives a

negligible alkali relaxation rate due to gradient relaxation.

Additionally magnetic field gradients cause the atoms to precess due to local

variation in the magnetic field and de-phase. This causes relaxation of the trans-

verse component of the spins. Additionally the mechanisms discussed earlier per-

taining to the T1 relaxation similarly affect the transverse polarization because they

23

also randomly orient the spin polarization. The T2 lifetime can thus be given as:

1

T2=

1

T1+

1

qseRse + Rgrad (2.41)

At high field where γB >> Rse Happer & Tam (1977) give 1qse

as:

1

qse=

2I(2I − 1)

3(2I + 1)2(2.42)

Whereas at low magnetic field where γB << Rse they show that 1qse

→ 0. Rse is

the spin exchange rate between alkali atoms, and Rgrad is the dephasing due to

nonuniform magnetic field across the sample cell. The significance of eq. (2.42)

and its physical description will be described in more detail in the next section.

In order to maximize the sensitivity of most polarization experiments one at-

tempts to maximize the T2 because it is the relevant parameter in the determination

of the precession frequency. One also attempts to maximize T1 in a sense because

it limits the maximum achievable value of T2. It would be prudent to review each

of the aforementioned pumping and relaxation mechanisms in more detail. In the

next section we do so.

2.2.1 Spin Exchange Collisions

When alkali atoms collide there is the possibility of their exchanging spin states

(Purcell & Field (1956)). This can be shown as:

|+〉A |−〉B → |−〉A |+〉B (2.43)

At large alkali density this has been shown to be the dominant relaxation mecha-

nism. The following treatment on spin exchange collisions follows the treatment

by Budker et al. (2004).

24

One can describe the spin exchange mechanism by realizing that the inter-

atomic potential during collision has a spin dependent contribution:

V(r) = V0(r) + SA · SBV1(r) (2.44)

Where V0 is a spin independent interaction term, and SA and SB are the respective

spins of the colliding alkali. The wavefunction of a free atom before collision can

be expressed in the |S, MS〉 basis in terms of the singlet |0, 0〉, and triplet |1, 0〉 states

as:

|ψ(0)〉 = |+〉A |−〉B =1√2(|1, 0〉 + |0, 0〉) (2.45)

During collision the singlet and triplet states acquire a relative phase

|ψ(0)〉 =1√2(|1, 0〉 + ei∆φ(t) |0, 0〉) (2.46)

where the relative phase acquired is:

∆φ(t) =2π

h

∫ t

0V1[r(t)]dt (2.47)

The atoms have a probability of undergoing a spin exchange collision when ∆Φ is

an odd multiple of π. In this case

|ψ(t)〉 → |−〉A |+〉B (2.48)

Though spin exchange collisions conserve total mF quantum number during

collision they can alter the F state of the alkali. These collisions are fast with re-

spect to the nuclear hyperfine interaction, thus they do not affect the nuclear spin

state. These collisions also cause decoherence of the transverse spins, and dephas-

ing because the two hyperfine manifold actually have gyromagnetic ratios which

25

-2-1

-10

+1

0+1

+2

F=2

F=1

+ωL

+ωL

+ωL

+ωL

−ωL

−ωL

Figure 2.3: Ground-state Zeeman sublevels for the case I=3/2. Sublevels are la-beled by their mF azimuthal quantum number. Note that the energy level splittingchanges sign for the different F manifolds. This causes both F manifold to havegyromagnetic ratios with opposite sign

Figure 2.4: During spin-exchange collisions the total angular momentum F1 + F2 isconserved, the atoms may switch between mF hyperfine sublevels. The hyperfinelevels F = I ± 1/2 are represented here by the colors red and blue. The atomsis different hyperfine levels have gyromagnetic ratios with different sign,t equalmagnitude. They will precess in opposite directions and decohere.

while having the same magnitude differ in sign. See fig.2.3 and fig.2.4.

This can readily be seen by solving the ground state alkali Hamiltonian eq.

(2.1)(Walker & Happer (1997)): and solving for the eigenvalues in the |F, mF〉 basis.

ωF=I+1/2 = −ωF=I−1/2 = 2πgsµB

(2I + 1)h(2.49)

This decoherence causes a broadening of the Zeeman sublevels which at high

26

field and low polarization can be expressed as (Happer & Tam (1977)):

1

qse=

2I(2I − 1)

2(2I + 1)2(2.50)

This expression is accurate in the high field regime where ωLarmor >> Rse. This

effect also contributes to the reduction of the T2 of the alkali atom precession. See

eq. (2.41).

In the low field limit where ωLarmor << Rse we find interesting behaviour as

spin exchange processes no longer contribute to the transverse decoherence of the

spins. This can be explained if we think in terms of the relative populations of the

different hyperfine states. When the spin exchange rate is much higher than both

the optical pumping and relaxation rates the atoms mix the m f sublevel popula-

tions. This condition applies at high alkali density when spin exchange collisions

occur frequently. Here the steady state m f sublevel population can be described

by a spin temperature distribution (Anderson et al. (1959),Anderson et al. (1960)).

See fig.2.5.

ρ(F, mF) =1

ZFeβmF (2.51)

where

ZF = ΣeβmF =sinh[β(F + 1/2)]

sinh(β/2(2.52)

Here the sublevel population possess a Boltzmann like distribution but with a spin

temperature β instead of the normal kinetic thermal temperature. We find that

because of the high spin exchange rate the atoms spend time in both hyperfine

manifolds. However they do so with a probability corresponding to the spin tem-

perature distribution. Since the spin exchange rate is much higher than the Larmor

precession rate the atoms motion can be thought of as a slower averaged preces-

sion due to the fact that the gyromagnetic ratios in each hyperfine manifold has

opposite sign but have unequal time spent in each manifold. This results in a net

27

F=2

F=1

mF=-2 m

F=-1 m

F=0 m

F=+1 m

F=+2

e-2β e-β 1 e+β

e-β 1 e+β

e+2β

Figure 2.5: When the rate of spin-exchange collisions is high, the Zeeman sublevelpopulations are given by a Boltzmann distribution which is characterized by aspin-temperature. The sublevel population in this case scales as eβmF . The case ofnuclear spin I= 3/2 is shown.

rotation of the spins in one direction.

The alkali experience spin exchange collisions and hop between the two hy-

perfine manifolds. The two hyperfine manifolds possess gyromagnetic ratios with

opposite sign. The atoms precess a small amount between collisions. However

the atoms spend a larger amount of time in the hyperfine manifold with larger F

value. This is determined by the spin temperature distribution. Thus even though

the atoms hops back and forth between the hyperfine manifolds it preferentially

spends a larger fraction of time in the hyperfine manifold with the larger F value.

This causes the atom to have a net precession in the orientation consistent with the

sign of the gyromagnetic ratio of the larger F valued hyperfine manifold. Since

this precession is coherent, spin exchange is effectively removed as a source of re-

laxation to first order. The spin exchange relaxation rate becomes (Allred et al.

(2002)):

Rse = ω2ser f Tse

(Q)2 − (2I + 1)2

2(2.53)

Q is described in eq. (2.55). In the equation above Tse is the time between spin ex-

change collisions, and depends on the specifics of the cell temperature and buffer

28

gas pressure. However under typical conditions of the experiments in this thesis

it is ≈ 15µS. Optical pumping schemes operating in this regime are called spin

exchange relaxation free (SERF). The rate of precession is modified from the tradi-

tional Larmor rate by (Allred et al. (2002)):

ωSERF =2πgsµBB

(Q)h(2.54)

Where

Q = 1 +I(I + 1)

S(S + 1)(2.55)

for high polarization. One finds the dependence of ǫ on polarization as (Savukov

& Romalis (2005)):

Q(P) =2I + 1

2 − 2I+13+P2

(2.56)

2.2.2 Spin Destruction Collisions

Spin destruction collisions are the next leading mechanism for spin relaxation. Pic-

torially they may be represented as:

|↑〉A + |↓〉B → |↓〉A + |↓〉B (2.57)

They occur through when the spin angular momentum becomes coupled to the

orbital angular momentum of the atom-atom collision. This may be between the

alkali atoms and any of the other alkali atoms, buffer gas atoms, or quenching gas

molecules. This angular momentum coupling is hypothesized to be due to a spin

axis interaction (Bhaskar et al. (1980)):

VSA =2

3λS · (3RR − 1) · S (2.58)

29

Alkali Metal σsel f σHe σNe σN2

K 1 × 10−18 cm2 8 × 10−25cm2 1 × 10−23cm2 −−Rb 9 × 10−18 cm2 9 × 10−24cm2 −− 1 × 10−22 cm2

Cs 2 × 10−16 cm2 3 × 10−23cm2 −− 6 × 10−22 cm2

Table 2.2: Spin destruction cross sections of alkali atoms with various gases.Adapted from Allred et al. (2002)

A list of spin destruction cross sections of alkali gases with typical buffer, and

quenching gases is listed in table 2.2.

2.2.3 Diffusion wall collisions, and Magnetic field Gradients

When alkali interact with the cell walls they normally become completely depo-

larized. Physically this process involves the alkali becoming adsorbed on the glass

surface. Though the adsorption time is typically between 10µ S and 100nS, the

atoms experience very large magnetic and electric field emanating from the glass

surface during this time. This is sufficient to depolarize both the spin of the elec-

tron and the nucleus. In order to minimize this effect one can either treat the cell

wall with an anti-relaxation coating or fill the cell with a high pressure of buffer

gas to decrease the alkali diffusion to the cell wall. In this work we consider the

second option.

If one assumes that the alkali becomes fully depolarized when it becomes ad-

sorbed on the wall then one can model the polarization in the cell by the diffusion

equation.

d

dtP = D∇2P (2.59)

Here D is the diffusion constant, where λ is the mean free path, and v the average

thermal velocity.

D =1

3λv (2.60)

30

The diffusion constant for K in neon is given by(Franz & Volk (1982)):

DK−Ne = 0.19cm2/s

(√1 + T/273.15K

pamagat

)(2.61)

Where the rate of K relaxation for the case of a spherical cell is found by solving

eq. (2.59):

Rdi f f = DK−Ne

(π

a

)2(2.62)

Here the temperature T is given in Celsius, and p is the buffer gas pressure in

amagat, and a is the cell radius. This is only strictly valid for a spherical cell.

Franzen (1959) gives the diffusion constant of Rb in Ne as:

DRb−Ne = 0.31cm2/s

(√1 + T/273.15K

pamagat

)(2.63)

The alkali may also experience relaxation due to traveling through a magnetic

field gradient. The specifics of this depend on cell geometry, diffusion constant,

and magnetic gradient orientation. The gradient relaxation rate is defined as:

Rgrad = D

(∇B

B

)2

(2.64)

Where δB is the magnetic field variation over the cell, and D is the diffusion con-

stant.

The variation in the magnetic field also causes broadening of the alkali preces-

sion frequency ≈ γ∇B. This local variation of the field also causes dephasing of

the spin precession and places a limit on T2, the transverse relaxation time. For a

formal treatment of this effect see Cates et al. (1988). We take a more simplified

approach to describe this effect.

We can compute this by imagining the cell partitioned into two halves each

differing in field by δB. In the regime where the atoms are free to move about the

31

cell unimpeded they dephase according to:

2γδBT2 ≈ 1 (2.65)

However in actuality the atoms travel across the cell and experience a δB given by

the time averaged field. This movement can be treated as a random walk across

the cell. Thus the difference in averaged field experienced by two alkali atoms is:

δBavg ≈ δB√N

(2.66)

Where N is the number of collisions before dephasing. N can be expressed in terms

of T2 as

N ≈ v

RT2 (2.67)

Here R is the cell dimension. Noting that one can generalize this argument from a

cell of dimension R to a buffer gas included cell of mean free path λ one can solve

this set of equations to find:

1

T2>

λ

v(2γδB)2 (2.68)

2.3 Monitoring polarized Alkali

2.3.1 Optical Rotation

In order to determine the orientation of the alkali polarization we utilize the tech-

nique of optical rotation. A linearly polarized beam, which is slightly detuned

from the optical transition frequency utilized for pumping, is employed as a probe

beam. One uses a weak probe beam so that the alkali vapour is not significantly

pumped along the probe direction. As the probe beam interacts with the magne-

tized sample in the ground state the plane of rotation on the probe beam rotates.

32

θ

Figure 2.6: When linearly polarized light passes through a medium which is polar-ized the axis of polarization of the light rotates. This rotation angle is proportionalto the projection of atomic spin along the propagation direction.

See fig.2.6. To understand this recall that the linear polarized beam is equivalent

to a composition of two equal beams which are circularly polarized with opposite

helicity. The rotation occurs when the index of refraction of the probing transition

differs for the transitions of different helicity.

To describe this effect we follow the approach of Mort et al. (1965), and Erick-

son (2000). To describe optical rotation mathematically let us first define the probe

beam in terms of electromagnetic waves. The electric field of such waves propa-

gating along the x direction can be described as:

E(0) =E0

2y + c.c. (2.69)

E(0) =E0

4(y + iz) +

E0

4(y − iz) + c.c. (2.70)

Where c.c is the complex conjugate. After propagating the length of the cell l =

tc/n(ν) the electric field along the y becomes:

E(l) =E0

4eiωn+(ν)l/c(y + iz) +

E0

4eiωn−(ν)l/c(y − iz) + c.c. (2.71)

Where n+, and n− are the indices of refraction for σ+ and σ− polarized light re-

33

spectively. It becomes convenient to define the quantities:

n(ν) = [n+(ν) + n−(ν)]/2 (2.72)

∆n(ν) = [n+(ν) − n−(ν)]/2 (2.73)

We can then substitute this into eq. (2.71) to obtain:

E(l) =E0

4eiωn(ν)l/ceiω∆n(ν)l/c(y + iz)+

E0

4eiωn(ν)l/ce−iω∆n(ν)l/c(y− iz)+ c.c. (2.74)

One can ignore the common phase factor exp(iωn(ν)l/c when determining the

optical rotation angle. The rotation angle is defined to be:

θ =πνl

cRe[n+(ν) − n−(ν)] (2.75)

We can substitute this into eq. (2.74) to find the electric field vector of the emerging

probe beam as:

E(l) = E0(cosθy − sinθz) (2.76)

One can show that an atomic vapour is a medium where through appropriate

choice of laser frequency n+(ν) 6= n−(ν). To validate this assertion let us con-

sider the following. First the index of refraction of light for the D1 transition is

given by:

n(ν) = 1 +nrec

2 f

4νIm[V(ν − ν0)] (2.77)

Where f is the familiar oscillator strength, and V(ν − ν0) is the Voight profile of the

transition. We notice that when the probe beam interacts with the ground state,

only transitions which obey the quantum selection rules ∆mj = ±1 will occur

depending on the helicity of the light. Each σ± component of the linearly polarized

probe beam then couples to the ground state depending on the population of the

34

mJ = +1/2m

J = -1/2

2S1/2

2P3/2

1 1

D1 Transition

mJ = +1/2 m

J = +3/2m

J = -3/2 m

J = -1/2

2S1/2

2P3/2

1/41/4

D2 Transition

3/4 3/41/4

Figure 2.7: Branching ratios for the D1 and D2 transitions.

ground state sublevel. Thus we can effectively write:

n+(ν) = 1 + 2ρ(+1/2)nrec

2 fD1

4νIm[V(ν − νD1)] (2.78)

n−(ν) = 1 + 2ρ(−1/2)nrec

2 fD1

4νIm[V(ν − νD1)] (2.79)

For an unpolarized vapour ρ(+1/2) = ρ(−1/2). However for a polarized vapour,

Px = ρ(+1/2)−ρ(−1/2)ρ(+1/2)+ρ(−1/2)

, we see that n+ 6= n−. Substituting this into equation 2.75,

and recalling that ρ(+1/2) + ρ(−1/2) = 1 we find:

θ =πlnrecPx

2(− fD1 Im[V(ν − νD1)]) (2.80)

One can generalize this result taking into account the effect of the D2 line on the

probe beam to obtain:

θ =πlnrecPx

2

(− fD1 Im[V(ν − νD1)] +

1

2Im[V(ν − νD2)]

)(2.81)

The negative sign and factor of 1/2 can be attributed to the different branching

ratios for the D2 transition. See fig.2.7.

To detect the rotation of the polarization angle one uses a polarizing beam split-

ter to split the probe beam into two orthogonally polarized beams. These are fed

into two photo-diodes whose output is fed through a subtracting photodiode am-

35

Probe LaserLinear

PolarizerCell

Polarizing

BeamsplitterPhotodiode

Photodiode

Figure 2.8: Optical detection with a linearly polarized probe beam passing througha polarized cell, then being split by a polarizing beam splitter into two sepa-rate photodiode detectors. The resultant signals from each photo-detector is fedthrough a photodiode amplifier and then into a subtraction circuit, giving the ro-tation.

plifier. See fig.2.8.

The plane of polarization of the probe beam is normally adjusted so that one

obtains equal signal in each photodiode in the case where there are no polarized

alkali. This corresponds to the point where the beam splitter is at 45 to the probe

beam polarization. Then the difference signal is directly proportional to the polar-

ization. In this case the signal on each photodiode is given by:

I1 = I0sin2(θ − π

4) (2.82)

I2 = I0cos2(θ − π

4) (2.83)

where I0 is the intensity of the probe beam, and I1 and I2 are the intensities on the

two photo-detectors. Solving for the rotation angle in radians we get:

Θ =1

2sin−1

(I1 − I2

I1 + I2

)(2.84)

Which is often useful to write as:

Θ = Φ

(1 +

2

3Φ3 +

6

5Φ5 +

20

7Φ7 + · · · · · · · · ·

)(2.85)

36

Probe LaserLinear

Polarizer

Faraday

ModulatorCell

Linear

PolarizerPhotodiode

Lock-In

Amplifier

Input Reference

Figure 2.9: Optical detection with a linearly polarized probe beam passing througha Faraday rotator, then a polarized cell, a second linear polarizer at 90degrees to thefirst, and finally a photodiode detector. The photo-detector signal is fed througha photodiode amplifier and using a Lockin amplifier referenced to the Faradaymodulator frequency. The in phase component of the Lockin amplifier gives therotation

where

Φ =I1 − I2

2(I1 + I2)(2.86)

For the case where Φ << 1 we have Θ = I1−I22(I1+I2)

. For sensitive polarimetry one

often alters the detection scheme to increase the sensitivity or decrease the 1/ f

noise of this measurement.

An example of this is to send the probe beam through a Faraday modulator

which modulates the plane of polarization of the probe beam by a small angle via

application of strong internal magnetic fields. See fig.2.9. In this arrangement one

first passes the probe beam through a plane polarizer, then through the Faraday

modulator, then the cell, and finally through a second polarizer set at 90 to the

first polarizer before finally terminating on a photodiode. The resultant signal on

the photodiode is:

I = I0sin2[θ + αsin(ωmodt)] (2.87)

Here α denotes the amplitude of modulation of the probe polarization axis, I0 de-

notes the light intensity transmitted through the cell and ωmod denotes the fre-

quency of modulation of the Faraday modulator. To low order one can Taylor

37

expand the eq. (2.87) to obtain:

I ≈ I0[θ2 + 2θαsin(ωmodt) + α2sin2(ωmodt)] (2.88)

One then references the detected signal to a lockin amplifier to detect only the

Fourier component of the signal at the Faraday modulation frequency. This method

greatly reduces 1/ f noise and gives a signal:

ILockin ≈ 2I0θα (2.89)

Another commonly used detection scheme is to employ a dichroic plate before

the polarizing beam splitter in the standard detection arrangement. The dichroic

plate is a type of poor polarizer which greatly attenuates but does not extinguish

one particular orientation of polarized light. These often have extinguish or atten-

uation ratios on the order of 100 : 1. The attenuating axis of the dichroic plate is

aligned with the polarization axis of the probe beam. This increases the magnitude

of the signal in one photo-detector relative to the other, and effectively increase the

rotation angle. This however does not alter the maximum sensitivity as the rela-

tive noise on both channels remains unaffected. This system also requires a careful

calibration by comparing the output signal as the probe polarization axis is varied

because the signal to angle conversions factor is a function of angle.

2.3.2 Light Shifts

When operating with an off resonant, or nearly resonant laser beam one must take

special precautions to avoid the effects of light shifts. Light shifts mimic the be-

haviour of a magnetic field and will alter both the precession frequency, and ori-

entation of free spins. Light shifts can arise from two different mechanisms. Our

38

description will follow that by Appelt et al. (1998).

The mechanism for light shifts typically encountered in precision frequency

measurements is due to the AC Stark effect of the electric field of the light beam

interacting with the atomic vapour. This interaction can be described by:

δH = ∆Eν −ih

2< Γ >= −E∗ · α(ν)E (2.90)

Here the energy shift is given by ∆Eν, α(ν) describes the complex atomic polar-

izability, and < Γ > is the average photon absorption rate. It can be related to the

photon flux Φ, and the absorption cross-section σ(ν) by:

< Γ >= σ(ν)Φ(1 − s · S) (2.91)

where S is the atomic spin, and s is the photon spin vector. This is given by:

s = iǫ × ǫ∗ (2.92)

where ǫ is the unit Jones polarization vector. Jones vectors are a convenient way to

describe polarization. In Jones terminology a σ+, σ−, and π polarized beams are

given by:

ǫσ+ =1√2

1

i

0

(2.93)

ǫσ− =1√2

1

−i

0

(2.94)

39

ǫπ =

1

0

0

(2.95)

Because the Hamiltonian is an analytic function, its real and imaginary parts are

related. One can apply the Kramers-Kronig relations to eq. (2.90) to find the energy

shift to be:

∆Eν =h

2πrec f Φ(1 − 2s · S)Im[V(ν − ν0)] (2.96)

One can readily see that this has the same form as the Zeeman interaction:

∆Eν = hγeBLightShi f t · S (2.97)

where the effective magnetic field the atom experiences is:

BLightShi f t =−πrec f Φ

γeIm[V(ν − ν0)]s (2.98)

Here we ignore the common offset given by eq. (2.96) and only retain the relevant

spin dependent energy shift. One can generalize this and include the effect of the

D2 line by replacing s in the above equations by s/2.

Since the light shift interaction appears as a fictitious magnetic field the atoms

respond by precessing around the total effective magnetic field B = B0 + BLightShi f t.

This effect can be quite large when B0 ≈ BLightShi f t. Thus we try to minimize

this effect whenever possible. Note that the eq. (2.98) becomes zero either when

Im[V(ν − ν0)] or s become zero. The first case corresponds to a laser beam exactly

on resonance, as Im[V(ν − ν0)] has a dispersive shape. One can easily verify using

the Jones vectors that the second case where s vanishes corresponds to the case of

linearly polarized light. To minimize the effects of light shifts we ensure that the

40

pump beam is tuned to resonance. We must also ensure that beams which are off

resonance, such as the probe, are linearly polarized.

There exists a second mechanism by which light shifts can occur in the optical

pumping system. During optical pumping atoms spend a small amount of time

out of the ground state and in the excited state. The gyromagnetic ratio of the

alkali is different in the excited state. Thus atoms which were not pumped into the

excited state will acquire a phase relative to atoms which remained in the ground

state the entire time. This effect is generally much smaller than the precession

frequency itself and is negligible in our systems.

2.4 Coupled Spin Dynamics

In a vast number of fundamental physics experiments such as electric dipole mo-

ment search (Romalis et al. (2001)), CPT violation (Kornack et al. (2008)), and pul-

monary imaging (Oros & Shah (2004)), it would be useful to polarize noble gas

nuclei. Due to spin dependent interactions the electron spin of the alkali can be

used to polarize the nuclear spin of a noble gas. This is called spin exchange op-

tical pumping (SEOP). This is the typical way in which noble gases are polarized,

with the exception of helium. In some cases it is advantageous to polarize helium

gas via a technique called metastable exchange optical pumping. That technique

will not be discussed here (Schearer (1968)). In the following sections we discuss

the mecahnisms for SEOP and the dynamics of the interaction between the polar-

ized alkali, and the polarized noble gas.

41

2.4.1 Optical Pumping of Noble Gas

The spin dependent interaction between the alkali and the noble gas during colli-

sion is:

V1(r) = γ(r)N · S + Ab(r)Ib · S (2.99)

Here the noble gas spin is denoted as Ib, the alkali spin by S, and the rotational

angular momentum by N. The spin-rotation interaction arises from the magnetic

fields created by the motion of the charges of the colliding atoms. The second term

arises from the hyperfine interaction of the nucleus of the noble gas, and the alkali.

Here the coupling coefficients are strong functions of the inter-atomic separation

r. The subscripts a refer to the alkali atom, while the subscript b to the noble gas

atoms.

The spin-rotation term in the interaction potential leads to alkali spin relax-

ation. The second term leads to spin exchange between the alkali electron spin

and the noble gas nuclear spin. In the spin temperature regime we can express

the rate equations governing the polarization of the spins by (Walker & Happer

(1997)):

d 〈Fz〉dt

= − Γa(γ) 〈Sz〉 −

Γa(Ab)[ǫ(Ib, βb) 〈Sz〉 − 〈Ibz〉] (2.100)

d 〈Ibz〉dt

= Γb(Ab)[ǫ(Ib, βb) 〈Sz〉 − 〈Ibz〉] (2.101)

Due to the principle of detailed balance we can say that:

nbΓb(Ab) = naΓa(Ab) (2.102)

where the density of the vapour species is given as ni. For the case of 21Ne, Ib is

42

3/2. Thus eq. (2.101) simplifies to:

d 〈Ibz〉dt

= Γb(Ab)[3 〈Sz〉 − 〈Ibz〉] (2.103)

Since ǫ is a function of spin temperature this is valid uner conditions of high spin

temperature where Pa ≈ 1. We can readily see that in steady state, if all relaxation

mechanism are suppressed, then the noble gas nuclear spin expectation value is

one third of the alkali spin expectation value. However the nuclear spin of 21Ne

is 3/2, while the electron spin is 1/2. Taking this into account we can re-write eq.

(2.103) in terms of the polarization of each species respectively as:

3

2Pb = 3

1

2Pa (2.104)

Or, simply Pa = Pb, under conditions of high spin temperature. We see that in

steady state and under the absence of relaxation mechanisms the noble gas po-

larization equilibrates with the alkali vapour polarization. This argument is valid

for an atom with any nuclear spin value. However in practice the noble gas po-

larization does not reach the same value as the alkali polarization due to strong

relaxation mechanisms. These include spin destruction collisions, gradient relax-

ation, and long range magnetic dipolar and quadrupolar field interaction. Typical

polarizations are on the order of 40 − 50% under conditions of high pumping rate.

The equilibrium nuclear polarization can be written as:

Pb = Pa Rabse

Rabse + 1/Tb

1

ǫ

Ibz/Sz(2.105)

Where the spin exchange rate Rabse = d〈Ibz〉

dt = Γb(Ab), and 1/Tb1 is the relaxation

rate for neon polarization.

43

2.4.2 Interaction of polarized alkali with polarized noble gas

The interaction of the polarized alkali and polarized noble gas yields interesting

behaviour. The magnetic field experienced by the alkali is due to both the classi-

cal magnetic holding field in which the atoms are present, and the magnetic field

created by the polarized noble gas. The field experienced by the alkali due to the

noble gas is not given by the classical expression of a magnetic dipole field. This

is because collisions between the alkali and noble gas deform the wavefunction of

the alkali and also overlap. The dominant interaction between the alkali electron

spin and the noble gas nuclear spin is described by eq. (2.99).

The hyperfine interaction coefficient arises from the Fermi-contact magnetic

fields of the two atoms (Herman (1965)):

Ab(r) =8πgsµb

3Ib|Ψb(0)|2 (2.106)

Due to the deformation of the wavefunction during collision it becomes:

Ab(r) =8πgsµb

3Ib|ηΦ(R)|2 (2.107)

The enhancement factor η is the ratio of the wavefunction at the noble gas nucleus

during collision to the unperturbed wavefunction in the absence of noble gas. This

isotropic hyperfine interaction also leads to the frequency shift of the precession

frequencies of both the alkali and the noble gas. The shift is described by the pa-

rameter κ0, where κ0 is the ratio of the magnetic field experienced by the alkali due

to the collision with the noble gas, and that which would arise from a Fermi contact

interaction with no η wavefunction enhancement factor (Schaefer et al. (1989a)).

κ0 =∫

4πR2 |ηΦ0(R)|2 e−V0(R)/kBT (2.108)

44

Where V0 is the spin independent interaction between the Noble gas and the alkali.

Normally this is fit to pseudo-potentials using data from scattering experiments.

The dominant interaction between the alkali nuclear spin and the alkali electron

spin is:

Hhyp = Aa Ia · S (2.109)

The Aa term in the isotropic interaction can produce a pressure shift in the hyper-

fine splitting. This is because the valence electron density is perturbed by the noble

gas resulting in a shift in the energy levels. For the most part optical pumping ex-

periments measure magnetic resonance of Zeeman levels, and are thus not very

sensitive to shifts in the Aa parameter. This is an important effect in the operation

of atomic clocks.

The effective magnetic field the alkali experience due to the polarized noble gas

for a spherical volume is:

B =8πκ0

3MP (2.110)

Where M is the magnetization density of a fully polarized noble gas, and P is the

noble nuclei polarization. The alkali atoms precess in the resultant field caused

by both the static magnetic holding field, and the magnetic field produced by the

noble gas nuclei. The motion of the alkali polarization can be described by the

phenomenological Bloch equations (Kornack & Romalis (2002)):

∂Pe

∂t=

γe

Q(Pe)

(B +

8πκ0

3MnoblePnoble + L

)× Pe + Ω × Pe

+(

Ropspump + +Rprobesprobe + Rabse Pn − RrelP

e)

/Q(Pe) (2.111)

∂Pn

∂t= γn

(B +

8πκ0

3MalkaliPe

)× Pn + Ω × Pn + Rab

se (Pe − Pn) − RrelPn (2.112)

γe is the gyromagnetic ratio of a free electron, Rrel is the alkali relaxation rate, Pn

is the nuclear polarization, and Rabse is the alkali-noble gas spin exchange rate. Rop

45

and Rprobe are the pump and probe beam pumping rates.

One can write this system of equations in complex form by multiplying the y

component by the imaginary number i, adding it to the x component, and solving

the resulting complex differential equation. For small non-equilibrium excitation

of the spins when there is no transverse magnetic field present we can solve this

set of equations. The linear approximations to the solutions is:

Pe′⊥(t) = (−iγe(Bn +

8πκ0

3MnoblePz

noble)Pe⊥ − RtotP

e⊥ + i

8πκ0

3MnoblePz

noble)/Q(Pe)

(2.113)

Pn′⊥ (t) = −iγn(Bn +

8πκ0

3MalkaliP

zalkali)Pn

⊥ − RtotPn⊥ + i

8πκ0

3MalkaliP

zalkali)Pn

⊥

(2.114)

Here the x, and y components correspond to the real and imaginary components

of the solution.

One can see that the magnetic fields experienced by the alkali and Noble gas

species differ. This is because in addition to the holding field each species experi-

ences the magnetic field produced by the magnetization of the other sample.

Bez = Bn +

8πκ0

3MnoblePz

noble (2.115)

Bnz = Bn +

8πκ0

3MalkaliP

zalkali (2.116)

The co-magnetometer experiments in this thesis operate near a noble-alkali hybrid

resonance described by the magnetic field compensation point:

Bncomp = −8πκ0

3MalkaliP

zalkali −

8πκ0

3MNoblePz

Noble (2.117)

Although the alkali is not at zero field, the field is low enough that the alkali re-

mains in the SERF regime. Operation at the compensation point leads to screening

46

of transverse magnetic fields. This will be discussed shortly.

One application of particular interest is the signal dependence of the co-magnetometer

on rotation. Solving the Bloch equations one finds the signal to be (Kornack et al.

(2005b)):

Srot =Pe

z γeΩy

γnRtot

(1 − γn

γeQ(Pe) − C − γ2

e

R2tot

B2z +

Bz

Bn− D + · · ·

)(2.118)

where

C =γePe

z Rnse

γnPnz Rtot

≈ 10−3 (2.119)

D =MeRe

se

MnRtot≈ 10−5 (2.120)

where Ωy is the angular rotation rate of the apparatus. Pez is the electron polariza-

tion, Rnse is the noble gas spin exchange rate, Pn

z is the noble gas polarization, Rtot

is the total alkali spin relaxation rate, and Rese is the alkali spin exchange rate. The

signal is defined to be S = R(Pex).

For a holding field set to the compensation point:

S(Ωy) =Pe

z γeΩy

γnRtot

(1 − γn

γeQ(Pe) − C − D + O(10−6)

)(2.121)

One can see that the system is sensitive to any rotations about the y axis. The

experimental implications of this will be discussed later, suffice it to say that this

allows one to operate the coupled system as a gyroscope.

The signals from rotation about the other two axes is much smaller. This is by

a factor of approximately 100 for the x and 105 for the z axes.

The coupled system also exhibits interesting shielding behaviour when set to

the compensation point. The signal becomes insensitive to small applied trans-

verse magnetic fields. When the system is tuned to the compensation point the

noble gas spins are aligned with the holding field. If a small transverse field in-

47

Bn

B xBn

(a) 21 Ne cancels the external field Bn (b) 21 Ne compensates for B x

IeN

M

B

S K

M K

K feels no change

M Ne Bn

S K

M K

K feels no field

21 Ne

21

21 Ne

21

II

Bn

B

Figure 2.10: When set to the compensation point the polarized noble gas screenstransverse magnetic fields.

teracts with the system the noble gas spins will adjust to align itself with the total

magnetic field. To first order the noble gas cancels out any transverse field. This

suppresses fields transverse to the holding field. Magnetic fields parallel to the

holding field do not tip the alkali since they will not adjust the coupled system.

2.5 Manipulation of polarized noble gas spins, and Mag-

netic shielding

In experiments where noble gas is polarized it is often necessary to manipulate

the noble gas orientation. Examples of this include performing adiabatic fast pas-

sage for determination of alkali noble gas spin exchange cross section (Chann &

Walker (2002)), and nuclear magnetic resonance for determination of κ0 (Stoner &

Walsworth (2002)), and pulmonary imaging (Oros & Shah (2004)). Many optical

pumping experiments benefit from operation in a low magnetic field environment

(Kornack et al. (2008)). This can be achieved by utilizing magnetic shielding. This

suppresses the magnetic field the atomic species experience by several orders of

48

magnitude. In this section we discuss each of these techniques for manipulation of

both the polarized noble gas, and shielding in greater detail.

2.5.1 Adiabatic Fast Passage

In optical pumping experiments it is often useful to flip the noble gas polarization

by 180. The most common technique for achieving this with relatively low loss

of polarization is Adiabatic Fast Passage (AFP). To achieve this one applies a mag-

netic field perpendicular to the magnetic holding field of the alkali. This field is

then swept in frequency through the magnetic resonance frequency of the alkali

due to the holding field. Alternatively the frequency of the perpendicular applied

field can be held constant, and the strength of the field can be varied. Under ap-

propriate conditions this process can result in the inverting of the alkali spins. To

explain this effect it is necessary to review some dynamics of operators viewed in

a rotating frame.

Consider the vector ~A in an initially inertial frame. If one transfers to a frame

which is rotating with frequency ~Ω then ~A transforms as:

(d~A

dt

)

rotating

=

(d~A

dt

)

inertial

− ~Ω × ~A (2.122)

(d~A

dt

)

inertial

=

(d~A

dt

)

rotating

+ ~Ω × ~A (2.123)

Now let us consider a spin ~S held with a field B0 in the inertial lab frame. It under-

goes Larmor precession according to:

(d~A

dt

)

inertial

= γ~S × ~B0 (2.124)

49

In the rotating frame this becomes:

(d~S

dt

)

rotating

= γ~S × ~B0 − ~Ω × ~S (2.125)

or, (d~S

dt

)

rotating

= γ~S ×(

~B0 +~Ω

γ

)(2.126)

The system behaves as if it experiences an effective magnetic field

~Be f f = ~B0 +Ω

γ(2.127)

in the rotating frame. The spins rotate about this effective field in the rotating

frame.

Let us consider the special case where the applied field ~B1 is rotating with ~Ω is

equal to the precession frequency ω0 = −γB0 of the alkali. Here we find:

~Be f f = ~B0 + ~B1 +ω

γz (2.128)

~Be f f = B0z + B1x +ω

γz (2.129)

or since ω0 = −γB0,

~Be f f = B1x (2.130)

When the frequency of the applied transverse field is exactly on resonance with

the precession frequency due to the holding field in the rotating frame, the spins

experience a static transverse field perpendicular to the holding field. This causes

the spins to rotate about the x and flip.

In the more general case where Ω 6= ω0 one does not completely cancel out

the effect of the holding field as described in the transition from equation 2.129

to eq. (2.130). The atoms still experience a static field B1x but also experience a

50

B

B

B

0

eff

1

θ

ωγ

Figure 2.11: Here one can see that if a transverse field is applied at a frequencyother than the resonant frequency it appears as having both transverse and axialcomponents in the rotating frame. The relative strength of these components is afunction of the detuning from resonance. The atoms precess around the effectivemagnetic field which lies at an angle θ to the holding field. As one sweeps thefrequency of the transverse field B1 from far below resonance to above it the angleθ goes from 0 to π. Thus we can flip the orientation of the spins in this manner.

field Ω−ω0γ z along the holding field axis. See fig.2.11. The alkali now rotate around

the general field ~Be f f in the rotating frame. As the frequency Ω is swept the field

along z changes. The angle ~Be f f makes with the holding field is also changed. If Ω

is varied slowly then the spins will follow and continue to precess around the field

Be f f . Using this knowledge we can sweep the field from far below the resonant

frequency to far above the resonant frequency to flip the spins by π radians.

To ensure that the alkali spins follow Be f f one must change the frequency of the

transverse field slowly. Physically this condition is satisfied when the precession

rate is very large compared to the rate at which the direction θ (t) of Be f f is chang-

ing (Powles (1958)). This condition is most stringent when the applied frequency

is equal to ω0 or geometrically when θ = π2 . So,

∣∣θ∣∣ =

1

B1

dBe f f (t)

dt(2.131)

51

or,

1

B1

1

γ

dΩ

dt<< γB1 (2.132)

Substituting the condition ω = γB1 which occurs at θ = π/2 we find:

dΩ

dt<< ω2

0 (2.133)

to ensure the spins adiabatically follow the Be f f .

One cannot sweep the magnetic field frequency arbitrarily slowly. When one

has a Be f f which is not parallel to the holding field the spins will dephase and lose

polarization after a time T1. We must sweep the spins faster than a time T1 to retain

spin polarization. Mathematically this corresponds to:

γB1

T1<<

dΩ

dt(2.134)

Thus to ensure one can flip the spin by sweeping the frequency of the applied field

one must satisfy:

γB1

T1<<

dΩ

dt<< ω2

0 (2.135)

2.5.2 Nuclear Magnetic Resonance

In order to determine the κ0 magnetic enhancement factor one must determine

noble gas polarization. To do this we utilize the techniques of nuclear magnetic

resonance. As this is quite a broad field only the fundamentals which were applied

in the experiments in this thesis will be reviewed.

Nuclear magnetic resonance (NMR) is the technique by which polarized spins

precessing in a magnetic field are manipulated and detected. These are tipped or

excited so that they possess a transverse polarization. As the transverse magnetiza-

tion rotates they produces an oscillating magnetic field which is typically detected

52

by inductive pickup coils. To tip the spins one excites them with a magnetic field

pulse which is at the same frequency as the precession frequency of the atoms. If

the excitation tipping field, along the x direction, is of strength B1 then the field

experienced in the frame rotating at the Larmor frequency of the nuclei is:

~Be f f = ~B0 + ~B1 +ω

γz (2.136)

~Be f f = B0z + B1x +ω

γz (2.137)

or since ω0 = −γB0,

~Be f f = B1x (2.138)

This argument follows directly from that in the previous section, where we assume

the holding field along the z direction. We can see that the nuclei will now precess

along the x direction. Typically the applied field however is of the form:

~B1 = B1 cos(ωt)x (2.139)

which is not equivalent to a field rotating at the precession frequency. However it

can be decomposed into two counter-rotating fields at the precession frequency:

~B1 =1

2B1 (cos(ωt) − sin(ωt)) σ+ +

1

2B1 (cos(ωt) + sin(ωt)) σ− (2.140)

The σ− component is located far enough off resonance that its effect is negligi-

ble in most cases. We will not discuss it but it can be of interest in certain co-

magnetometers operating in the SERF regime. If the duration of the magnetic field

pulse is for a time t then we see that the angle the spins rotate through an angle

given by:

Θ =1

2γBt (2.141)

53

B0

θ

S

ω

Β1 (ω t)cos

Figure 2.12: If an oscillating magnetic field B1 is applied at the atom precessionfrequency orthogonal to the holding field B0 then the atoms will become tipped offaxis.

See fig.2.12.

In order to detect the field produced by the precession of the tipped spin one

constructs an inductive pickup coil. To ensure that the signal is larger than the

various instrument noise one utilizes the pickup coil to create a resonant LC circuit.

Here a capicator is placed is parallel with the pickup coil. For an ideal coil of

inductance L, and resistance r the impedance of parallel RLC circuit is given by

(Fukushima & Roeder (1981)):

Ztank =r − iωL(1 − ω2CL − r2C/L)−1

r2 + ω2L2(2.142)

The resonant condition occurs when:

1 − ω2CL − r2C/L = 0 (2.143)

However for most systems the resistance of the pickup is quite small. Here we

54

get the resonant frequency to be:

ω =1√LC

(2.144)

The impedance at the resonant condition is given by:

Zresonant =r2 + ω2L2

r(2.145)

Typically the resonant frequency of the pickup is tuned by placing a variable capi-

cator in the RLC circuit, and adjusting it.

It is often convenient to define the quantity Q = ωL/r. This is referred to as

the quality factor of the pickup circuit. For an inductive pickup coil the Q factor

is the factor by which the pickup voltage is magnified at the resonance condition

as compared to the case of detecting the precessing spins directly with the pickup

coil, and not using a RLC circuit.

2.5.3 Magnetic Shielding

For sensitive magnetometry experiments it is often advantageous to operate in-

side magnetic shields. These reduce or eliminate magnetic field contributions due

to the Earth’s field, and laboratory power line 60Hz magnetic noise. This ensures

that the field in which the atoms are held is well known and uniform. These shields

typically are made from ferromagnetic materials, have high permeability, and are

easy to both magnetize and demagnetize. Typical commercial brands are made

from Mu-Metal. Other materials often used in magnetic shield manufacture are

Conetic alloys, and Moly Permalloy. Recently work has been done to observe the

effectiveness of utilizing ceramic ferrite as a shield (Kornack et al. (2007)) which

have lower thermal noise than the Johnson noise of Mu-Metal shields at room tem-

55

perature.

The standard arrangement for magnetic shielding is to make multiple concen-

tric cylinders which are nested within each other. If one performs the calculation

they find that the shielding factor in the transverse and axial direction differ for a

cylindrical shield. The transverse shielding factor is given by (Jackson (1999)):

ST =Bi

B0=

µt

2R(2.146)

where µ is the magnetic permeability of the shield, t is its thickness, and R its

radius. Here B0 is the uniform field outside the shields, and Bi the field inside the

shield. The axial shielding factor is given by (Khriplovich & Lamoreaux (1997)):

SA =Bi

B0≈ 2µtR1/2

L2/3(2.147)

where L is the length of the cylinder. See fig.2.13. This approximation is only valid

in the region where 4 < L/R < 80.

Normally one must place holes in magnetic shields for various electronic feed-

throughs, power cables, optical beam paths, etc. This decreases the shielding factor

slightly. For a hole of radius r the field perpendicular to the shield surface falls off

as (Khriplovich & Lamoreaux (1997)):

B(l) = Be−1.5l/r (2.148)

where l is the distance from the hole.

The shielding from nested cylinders is not simply the product of the individ-

ual shielding factors of each layer. This is because the internal shields affect the

boundary conditions used in the previous calculation of the shielding factors. The

56

Baxial

MagneticShields

Air

Figure 2.13: The magnetic fields lines have an affinity for magnetic shields. Theabove figure shows how magnetic fields lines from a previously homogeneousfield warp in the presence of magnetic shields. The solution of the Laplaceequation in cylindrical polar coordinates gives a magnetic scalar potential Φ =

(Ckρ + Dkρ )cosθ. The relevant boundary conditions for the magnetic field compo-

nents normal and tangential the surface are µkdΦkdρ |rk

= µk+1dΦk+1

dρ |rk, and 1

rk

dΦkdθ |rk

=

1rk

dΦk+1dθ |rk

The magnetic field pattern can be constructed by solution of the Laplace

equation, subject to application of the boundary conditions described above.

57

transverse shielding factor for nested cylinders is given by (Sumner et al. (1987)):

ST = SnT

n−1

∏i

SiT

[1 −

(Ri

Ri=1

)2]

(2.149)

where the script i refers to each individual layer. The 1 −(

RiRi=1

)2compression

of internal flux (volume loss) in the region between layers. The axial shielding is

given by:

SA = SnA

n−1

∏i

SiA

(1 − Li

Li+1

)(2.150)

Here the 1− LiLi+1

term reflects the reduction in volume or compression of flux along

the cylinder length.

Before use the magnetic shields must be de-gaussed. This necessitates reduc-

tion of the magnetization of the shields to zero. To accomplish this one passes a

high alternating current, normally 50 amp-turns, through the shields to saturate

them. The current in the shields is then slowly reduced to zero. The idea is that

when one passes a high current through the shields the saturation is symmetric on

current reversal. If one then reduces the current slowly, much slower than the al-

ternating current frequency, one can eliminate the magnetization when the current

becomes zero.

58

Chapter 3

Nuclear Spin Gyroscope

Sensitive gyroscopes are utilized in applications ranging from inertial navigation,

and studies of the Earth’s rotation, to tests of general relativity (Stedman (1997)). A

variety of physical principles have been employed for rotation sensing. These in-

clude mechanical gyroscopes, the Sagnac effect for photons (Stedman (1997))(An-

dronova & Malykin (Andronova & Malykin)) and atoms (Gustavson et al. (1997))(Yver-

Leduc (Yver-Leduc)), the Josephson effect in superfluid 4He, and 3He (Avenel et al.

(2004)), and nuclear spin precession (Woodman, Franks & Richards (Woodman

et al.)). While mechanical gyroscopes operating in low gravity environments re-

main so far unchallenged (Buchman et al. (2000)) there is much competition in the

field of compact gyroscopes operating in Earth’s field.

We have developed a nuclear spin gyroscope (Kornack et al. (2005a)) based

on the co-magnetometer arrangement described earlier in this work (Allred et al.

(2002)). As mentioned earlier the signal from the atom co-magnetometer has a

dependence on the rotation of the apparatus, thus it can be utilized as a gyroscope.

Though the entire dynamics of the coupled alkali-noble gas system is complicated

it is useful to describe the system physically before delving into mathematics. The

work in this chapter can also be found in Kornack et al. (2005a).

59

3.1 Co-magnetometer Gyroscope Implementation and

behaviour

In the co-magnetometer arrangement the K alkali spins are polarized with a pump

laser. Through spin exchange collisions the nuclear spins of the 3He noble gas are

polarized parallel to this. Let us imagine the cell which contains noble gas to be in

an inertial reference frame. If we rotate the apparatus the alkali spins will quickly

be re-pumped along the new pump laser orientation since the T1 of the alkali is on

the order of 30ms. The noble gas then precess around the net magnetic field and

becomes aligned with the holding field in a time T2 of ≈ 100s. We will focus on

the interesting dynamics on a short time scale.

Initially if the co-magnetometer is properly zero-ed, and tuned to the compen-

sation point then in the steady state arrangement the noble gas nuclei experience

a net magnetic field parallel to the nuclear spin due to the magnetic compensation

field. When the apparatus is rotated the compensation field is now at an angle to

the nuclear spins causing them to precess about it. As the nuclear spins precess

the orientation of the net magnetic field which the alkali experience now changes.

This causes the potassium to rotate about the new orientation of the net magnetic

field. Consider the component of the magnetic field perpendicular to the pump

and probe beam optical axis which is produced by the noble gas. This field causes

the alkali to precess in the plane of the pump and probe beam. It is this alkali pre-

cession that we monitor with an off resonantly tuned probe beam. See eq. (2.121).

Using Green’s functions for the linearized Bloch equations one can show that

the integral of the co-magnetometer signal is proportional to the total rotation an-

gle about the y axis. This is independent of the time behaviour of Ωy. The angular

60

Hot AirCell

Magnetic Shields

Floating Optical Table

Position sensors

Polarizer

kc

atS

cirtc

ele

oz

eiP

kc

olB

elib

om

mI

Analyzing Polarizer

Photodiode

FaradayModulator

Field Coils

Lock-inAmplifier

y z

x

Pockel Cell

ma

eB

pm

uP

Probe Beam

?4

/

I 3HeM 3He

S K

B z

M K

Single Freq.Diode Laser

re

wo

P h

giH

res

aL

ed

oiD

Figure 3.1: A schematic of the co-magnetometer being implemented as a gyro-scope. Note the non-contact position sensors used to detect the rotation, and thepiezo stack used to force the apparatus to oscillate

frequency of the alkali can be related to the co-magnetometer signal by:

Ωy = γgS (3.1)

where S is the signal from the co-magnetometer in units of magnetic field and γg

is given by:

γg ≈(

1

γn− Q(Pe)

γe

)−1

(3.2)

The apparatus was driven to rotate by a piezo stack placed between the optical

table upon which the co-magnetometer sits and a heavy immovable concrete block.

This is depicted in fig.3.1, and fig.3.2.

Six non-contact displacement sensors were used to monitor the table orienta-

tion. They were mounted to the posts at the base of the optical table. They oper-

ate by measuring the capacitance between the sensor and a target metallic plate.

The target plates were mounted to the floating portion of the optical table. The

sensors was calibrated and found to give a linear response over displacement am-

61

Immobile

Block

Piezo Stack

Table rotation

z x

y

Photodiode

Cell

Spin polarization

Position

Sensors

Probe Beam

Pump Beam

zx

y

Figure 3.2: Alternate side view of the gyroscope configuration for the co-magnetometer

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Horizontal displacement from target (cm)

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Sig

nal

(vo

lts)

Figure 3.3: In-situ calibration of the non-contact displacement sensors for determi-nation of absolute rotation.

plitudes of a few centimeters. For comparison during operation the amplitude of

displacement was on the order of millimeters. The sensors were each individually

calibrated in-situ. See fig.3.3.

In order to convert the co-magnetometer signal to a gyroscope we must inte-

grate the co-magnetometer signal. Using eq. (3.1) we convert to angular units.

The resulting fit of the co-magnetometer gyroscope signal to the orientation signal

from the non-contact displacement sensors in shown in figure 3.4

The gyroscope signal agrees with the position measurement to within the 3%

calibration accuracy. The gyroscopic signal has also been shown to be insensitive

62

− 20

0

20

(n

oitat

oR

µ)c

es/

dar

0 2.5 5 7.5 10 12.5

Time (s)

− 100

− 50

0

50

100

)Tf

(dl

eiF

evi

tce f

fE

Figure 3.4: Comparison of co-magnetometer gyroscope signal to displacement sen-sor signal with no free parameters. The solid line depicts the co-magnetometersignal, and the dashed line the signal from the position sensors.

0

0.005

0.01

0.015

0.02ru

oh

/seerge

d(kl

aW

mo

dn

aR

elg

nA

1/

2)

0 200 400 600 800 1000

Frequency (hour − 1 )

0

5

10

15

20

25

zH

/Tf

(dl

eiF

1/

2)

Figure 3.5: Fourier transform of the noise spectrum of the comagnetometer gy-roscope. The discrete noise peaks are an artifact caused by the periodic zeroingroutines for the co-magnetometer

to the other two components of the angular velocity, and only depends on the

angular velocity vector perpendicular to the plane containing the pump and probe

beams.

Using the relation between the magnetic field measurement of the co-magnetometer

and the rotation measurement of the gyroscope one can calculate the noise spec-

trum of the gyroscope from previous magnetic noise measurements of the co-

magnetometer. As one can see from fig 3.5 the noise spectrum of the gyroscope

is nearly flat at 1.0ft/√

Hz, or translated to angular units 1.4× 10−5 rad/√

hour for

frequencies above 400 hour−1. At frequencies below this one sees a clear 1/ f noise

63

10− 4

10− 3

10− 2

10− 1

100

rotc

aF

noiss

erp

pu

Sdl

eiF

0.1 0.2 0.5 1 2 5 10

Frequency (Hz)

dBy/dx

dBy/dy

dBy/dz

dBx/dz

dBz/dz

Figure 3.6: Suppression of an applied magnetic field gradient by the co-magnetometer compared to that of a non-compensating magnetometer. Coloredpoints refer to measurements made with square wave modulation instead of sinu-soidal modulation.

dependence with a 1/ f noise knee at 0.05Hz. The noise of the gyroscope in mag-

netic units is much less than the magnetic Johnson contribution from the magnetic

shielding. This is because the co-magnetometer acts as a magnetic field suppressor.

This is discussed in more detail in section 2.6. The magnetic field is suppressed

even though the alkali metal and Noble gas have different spatial distributions.

(see fig 3.6). The reason for this behaviour is because the noble gas diffusion rate

is much lower than the nuclear spin precession rate. That is γnBn >> RD >> Rnsd,

where Rnsd is the nuclear spin destruction rate. The magnitude of the polarization

of the noble gas is constant in value, and orients itself parallel to the local magnetic

field. Thus the noble gas is able to cancel a non-uniform external field on a point

by point basis.

An oscillating magnetic field was applied and effectively shielded by the co-

magnetometer. This is shown in fig 3.7. In addition, using the linearized Bloch

equations one can show that the rotation angle created by a magnetic field transient

is zero as long as the spin polarizations are not rotated by a large amount during

the transient. Fig 3.8 shows the response of the gyroscope to a magnetic field tran-

sient. The co-magnetometer demonstrates a reduction of the spin rotation angle

by a factor of 400 as compared to that of a standard potassium co-magnetometer.

64

0.001

0.01

0.1

1

rotc

aF

noiss

erp

pu

Sdl

eiF

0.1 0.2 0.5 1 2 5 10

Frequency (Hz)

0.1

1

10)

Tp(

dlei

Fd

erus

ae

M

Bx

By

Bz

Figure 3.7: The co-magnetometer suppresses magnetic fields. Thus the contribu-tion from the Johnson noise of the magnetic shields is greatly reduced. The fieldsuppression in the x,y, and z directions are measured relative to that of a K mag-netometer. The solid line represents the theoretical prediction.

− 20

0

20

40

60

)T

p(dl

eiF

cite

ng

aM

0 0.1 0.2 0.3 0.4 0.5 0.6

Time (s)

− 0.1

0

0.1

0.2

0.3

)d

ar(

elg

nA

Figure 3.8: Response of co-magnetometer(dashed red line) to a magnetic field tran-sient(solid line). The total rotation angle(blue dashed line) is proportional to theintegral of the co-magnetometer signal. It is also much smaller than the rotationangle from a K magnetometer(dashed black line).

65

3.2 Effect of Experimental Imperfections on Gyroscope

Performance

Gyroscope performance can be affected by a number of experimental imperfec-

tions of the system. One such imperfection is that arising from not properly zero-

ing the magnetic field on the co-magnetometer. The only component of the mag-

netic field or light shift which contributes to the signal in first order is the Bx field.

It causes a false signal of (Kornack et al. (2005a)):

S(Bx) = BxPez (Ce

se + Cnse)/Bn (3.3)

where

Cese = (Re

sePnz )/(RtotP

ez ) (3.4)

is the electron spin exchange correction, and

Cnse = (γePe

z Rnse)/(γnPn

z Rtot) (3.5)

is the nuclear spin exchange correction factor which arise from solving eq. (2.111).

Substituting the measured values for Pez ,Pn

z , Rese, Rn

se, and Rtot we find Cese ≈ 10−2

and Cnse ≈ 10−3. Since both Ce

se and Cnse are small this field dependence is heavily

suppressed by a factor of ≈ 105. Pez is the alkali polarization,Pn

z the noble gas

polarization, Rese is the alkali metal-noble gas spin exchange rate for an alkali atom

and, Rnse is the same for a noble gas atom. Rtot = Rpump + Re

se + Resd + Rprobe

Misalignment of the pump and probe beam so that they are not orthogonal also

causes a systematic rotation signal. If they are misaligned an angle α away from

90 we measure a signal:

S = αRpump/Rtot (3.6)

66

Under typical operating conditions of the gyroscope a misalignment of 1µrad gives

a false signal of 10−8rad/s. However this can be corrected for because a true rota-

tion signal has no dependence on the pumping rate. One could correct for this by

varying the intensity of the pump beam, and aligning the probe beam until there is

no longer a gyroscope signal at the frequency at which the pump intensity is being

varied.

One can also measure a false signal if the probe beam polarization is not fully

linearly polarized, but has a small circular polarization component. This gives a

signal of:

S = smRm/Rtot (3.7)

where the circular polarization of the probe beam is given by sm, and Rm gives

the pumping rate of the probe beam. This pumping by the probe beam can be

eliminated by zeroing the probe beam light shift. This will be described further in

the next section.

Other experimental imperfections only contribute to the gyroscope signal in

second order. For comparison the signal from an improperly zeroed field By is

given by:

S =γeByPe

z

BnRtot(Bz − (Bz + Lz)Ce

se − (2Bz + Lz)Cnse) (3.8)

where Bz is the amount by which the z field is tuned away from the compensation

point. The Effects of imperfections in a Rb-Ne co-magnetometer are discussed in

Sec.6.4.

3.3 Zeroing the Co-magnetometer Gyroscope

In order for the co-magnetometer signal to be properly calibrated it must be zero-

ed. That is the net magnetic field along both the x, and y directions must be

67

zero. The light shifts must be removed, and the z field must be tuned to the

compensation point. To do this we employ a quasi-static modulation technique.

That is we modulate certain components of the field at very low frequency. That

is a frequency much lower than the resonance of the noble gas. In the 3He co-

magnetometer this corresponds to a frequency of roughly 7 Hz during normal op-

eration conditions. This frequency is determined by the field experienced by each

atom species when the co-magnetometer is brought to the compensation point.

The steady state solution to the coupled Bloch equation which govern the co-

magnetometer gives the co-magnetometer signal in the steady state regime in units

of magnetic field as:

S = Ly +Ωy

γn+

smRm + αRp

γePez

+ Bz

(By

Bn− Lx

γe

Rtot

)+

γe

Rtot

(BxBz(Bz + Lz)

Bn− LxLz

)

(3.9)

We shall consider the zero-ing of each component of the magnetic field or light

shift separately and in greater detail.

If one were to calculate the dependence of 3.9 on By they would find:

δS

δBy∝ Bz (3.10)

We see that the dependence of the signal is directly proportional to the Bz com-

ponent of the field. So if we modulate the By component of the field we observe

a modulation of the signal at the same frequency. Thus we can zero the Bz com-

ponent by modulating the By field, monitoring the signal, and varying Bz until

the modulation in the signal disappears. This corresponds to the point where the

compensation field cancels out the field contribution from the magnetization of the

atoms, and any residual field due to the magnetic shielding.

Once the field has been tuned to this compensation point one can use a similar

68

method to zero the By field. One sees that the signal dependence on Bz is:

δS

δBz∝

By

Bn+ Lx

γe

Rtot(3.11)

We ignore the contribution of the BxBz term here as it is the product of two small

numbers and is suppressed compared to the other terms in the above expression.

We see that the modulation of Bz and readjusting the By field will now zero a linear

combination of the By field and the lightshift Lx. We independently zero Lx as

well. This will be discussed later. Thus to zero By we iterate the modulation of Bz

followed by independent zeroing of Lx.

To zero Bx we take a slightly different approach. One can write the dependence

of the signal on Bz as:

δ2S

δB2z

∝ Bx (3.12)

Thus we can eliminate Bx if we modulate the second derivative of Bz. To do this

we asymmetrically modulate Bz between a zero, and non-zero value. If one were

to perform this modulation symmetrically one would simply be repeating the ear-

lier procedure used to zero By. Although this response would be different since

the BxBz term would now be of significant size compared to By since the latter

component has been zeroed. This function would also be dependent on Lx and By.

Once the magnetic field components have been zero-ed we zero the contribu-

tion of lightshifts. We consider the LxLz term of eq. (3.9). We see:

δS

δLz∝ Lx (3.13)

δS

δLx∝ Lz (3.14)

Thus we can modulate the appropriate component of the light shift to zero the

other. The light shift of a laser goes as the frequency of detuning, modified by the

69

Voight profile, and the degree of circular polarization. To modulate the light shift of

the pump beam Lz one modulates the frequency of the laser. This is because tuning

the pump to be directly on resonance maximizes pumping efficiency. To modulate

the lightshift of the probe beam Lx one varies the degree of circular polarization.

To do this we attempt to cancel any birefringence in the beam before it strikes the

cell. There are a few methods to do this including placing a Pockel cell in the beam

path or a plate of stressed glass. The stress of the glass can be varied by using a

piezo to compress it. This compression can be modulated.

One must also ensure that the pump and probe beam must be properly orthog-

onalized. The co-magnetometer signal does depend to first order on the product

of the angle misalignment α and the pumping rate Rpump. However in practice

one cannot make the pumping rate high enough to make it the only term which

contributes to the signal. The signal dependence on the pumping rate goes as:

δS

δRpump∝ αK

Rpump

Rtot+ γe

Ωy

γn

Rpump

R2tot

(3.15)

where K is a factor for calibrating the misalignment angle to signal units. To vary

the pumping rate we feed the pump beam on maximum intensity through two

crossed polarizers. In between these polarizers we place a liquid crystal waveplate.

By varying the retardance of the waveplate we are able to change the polarization

of the beam between the polarizers, and adjust the intensity of the pump upon

exiting the polarizers. Thus we can vary the angle α until the dependence on the

signal vanishes.

3.4 Co-magnetometer Gyroscope Sensitivity

The fundamental sensitivity of the gyroscope is accurately described by the spin

projection noise of the co-magnetometer. The measurement uncertainty of the co-

70

magnetometer is dominated by the noise from the alkali metal spins. In terms of

the angular frequency this can be expressed as:

δΩy =γn

γe

√Q(Pe)Rtot

nV(3.16)

where n is the alkali metal density, and V is the measurement volume.

Currently we operate the gyroscope with a K-3He co-magnetometer. The fun-

damental sensitivity for this is δΩy ≈ 1.2 × 10−8rad/s/√

Hz. This is roughly 50

times lower than the realized sensitivity of δΩy ≈ 5.0× 10−7rad/s/√

Hz. This dis-

crepancy can be attributed to the large amounts of angular noise contributed from

the probe beam laser.

By looking at eq. (3.16) we see that the fundamental sensitivity is a function of

the ratio of the noble gas and alkali gyromagnetic ratios. By switching to a K-21Ne

co-magnetometer should instigate an immediate improvement in the sensitivity

by a factor of roughly 10, since 21Ne has a gyromagnetic ratio roughly an order

of magnitude smaller than that of 3He. For a cell with 10cm3 volume, K density

of 1014cm−3, and 21Ne density of 6 × 1019cm−3 this would imply a fundamental

sensitivity of δΩy ≈ 2.0 × 10−10rad/s/√

Hz.

To be useful for applications such as navigation, a gyroscope must be small

and portable. The most widely used gyroscope for navigational purposes is the

fiber-optic gyroscope. After nearly two decades of improvement the sensitivity of

these gyroscopes have approached their fundamental sensitivity. Compact state

of the art fiber-optic gyroscopes have sensitivity of δΩy ≈ 2.0 × 10−8rad/s/√

Hz

(Sanders et al. (2000)). New compact interferometer gyroscopes using cold atoms

with a shot noise sensitivity of δΩy ≈ 1.4×10−7rad/s/√

Hz (Canuel et al. (2006))and

δΩy ≈ 5.0 × 10−9rad/s/√

Hz (Müller et al. (2007)). Another newly proposed

gyroscope is one which operates with MEMS technology. These hold promise if

71

− 0.6

− 0.4

− 0.2

0

0.2

0.4

0.6

)se

erg

ed(

elg

nA

0 1 2 3 4 5

Time (hours)

Constant drift of 0.1 deg/h

Figure 3.9: Long term drift of gyroscope

they continue to be developed in the future. Current designs have recently over-

come the extreme sensitivity to temperature drift past MEMS groscopes exhib-

ited.(Trusov et al. (2008)). However the current sensitivity of temperature insen-

sitive MEMS gyroscopes are δΩy ≈ 5.0 × 10−4rad/s/√

Hz (Trusov et al. (2008)).

This renders them unusable for navigational applications in the near future.

The compact co-magnetometer gyroscope is competitive with the methods men-

tioned above. The measurement volume is roughly 10 cm3, though the present

implementation occupies a square approximately 2m to a side. This setup can

be miniaturized. In fact many of the techniques found in the miniaturization

of atomic clocks can be used to minituarize the gyroscope (Knappe et al. (2006),

and Knappe (2004)). The lasers can be made more compact quite easily. In fact

miniaturizing the magnetic shields improves their performance. The dominant

source of long term drift in the co-magnetometer is due to temperature drift in the

system. Fig3.9 shows the long term drift of the gyroscope. This should also im-

prove dramatically upon minutuarization. The co-magnetometer gyroscope sen-

sitivity is competitive with larger commercial gyroscopes. For comparison gyro-

scopes based on the Sagnac effect (Stedman (1997)) have achieved sensitivities of

δΩy ≈ 2.0× 10−10rad/s/√

Hz using a ring laser with an enclosed area of 1m2, and

72

δΩy ≈ 6.0 × 10−10rad/s/√

Hz using an atom interferometer with path length of

2m (Gustavson et al. (2000)). One gyroscope which has a much greater sensitivity

is Gravity Probe B. However the wonderfully low drift which it demonstrates, sig-

nificantly decreases when operated under Earth’s gravity. This makes it much less

suitable for terrestrial based application.

73

Type Realized Projected Drift CitationSensitivity Sensitivity

rad/s/√

Hz rad/s/√

Hz rad/hour

Large Scale (∼2 m)

Ring Laser Gyro (CII) 2.2 × 10−10 — — Stedman (1997)Atom Interferometer (Yale) 6.0 × 10−10 2.0 × 10−10 1.3 × 10−4 Gustavson et al. (2000)

Intermediate Scale (∼50 cm)

Mechanical (Gravity Probe B) — — 3.0 × 10−14 Buchman et al. (2000)Superfluid 3He (Orsay) 1.4 × 10−7 3.0 × 10−10 2.1 × 10−5 Avenel et al. (2004)Atomic Interferometer (HYPER) — 2.0 × 10−9 — Jentsch et al. (2004)Atomic Fountain (Paris) — 3.0 × 10−8 — Yver-Leduc (Yver-Leduc)Atomic Spin ‘NMRG’ (Litton) 2.9 × 10−6 — 9.0 × 10−4 Woodman, Franks & Richards (Woodman et al.)

Small Scale (∼10 cm)

Fiber-optic Gyro (Honeywell) 2.3 × 10−8 — 1.7 × 10−6 Sanders et al. (2000)Atomic Spin (This work) 5.0 × 10−7 2.0 × 10−10 7.0 × 10−4

Miniature Scale (< 1 cm)

MEMS (CMU) 3.5 × 10−4 1.8 × 10−4 0.5 Xie & Fedder (2003)

Table 3.1: A survey of gyroscope performance.

74

Chapter 4

Initial tests of an alkali-Neon

co-magnetometer

The main objective of these experiments on 21Ne is to ultimately create a neon

co-magnetometer which can be utilized for experiments on tests of fundamental

symmetries, and for deployment as a sensitive gyroscope. We describe construc-

tion of a 21Ne co-magnetometer and investigate the practical problems limiting its

performance.

4.1 Magnetometer setup

The co-magnetometer operates with orthogonal pump and probe beams. These

are both distributed feedback lasers (DFB). These lasers have a reflection grating

etched on the diodes themselves providing a much more stable single frequency

emission. The pump beam has a power of 20 − 40mW, passes through the typical

λ/4 plate and is expanded to a cross section of 3cm2. It is tuned to the D1 resonance

of K. The probe beam is linearly polarized and passes through the cell and is split

by a polarizing beam splitter into two photodiode detectors. It has a power of

75

Figure 4.1: Experimental setup of Ne Magnetometer

10mW and is tuned 0.2nm away from the D1 resonance of K. The probe beam

cross section is 1.25 × 1.25cm2 defined by a mask. See fig. 4.1.

The cell is filled with a K, 1.6 atm 21Ne, and 60torr nitrogen mixture in a Pyrex

cell of volume 8.0cm3. It is heated to 180C. The cell is encased in a glass oven.

The cell is heated via a hot air line which feeds into the glass oven.

The glass housing is placed inside a set of 4 concentric Mu-metal magnetic

shields, which sits upon a standard optical table. Multiple turns of wire run in-

side the magnetic shields for de-gaussing purposes. These are only connected and

run when the magnetometer in not under operation. Inside the magnetic shields

are sets of Helmholtz coils, and cosine windings wound around a G−7 frame for

magnetic field generation. The internal current used to drive these internal mag-

netic fields is created by a custom current source. It is based on a mercury battery

voltage reference with a FET input stage followed by an op-amp and transistor

output stage (Baracchino et al. (1997)).

76

).br

a( n

oitazir

alo

P n

oeN

Time (s)

0 100 200 300 400 500 600

Figure 4.2: T2 time of ≈ 14minutes for Neon polarization when operating awayfrom the compensation point in the co-magnetometer configuration.

4.2 Neon Polarization Measurements and Preliminary

Neon Co-Magnetometer data

Neon nuclei were polarized by optically pumping K vapour. The compensation

point for neon was approximately 250µG, which corresponds to a neon magneti-

zation of 8µG, or 0.08% polarization. We have also tipped the neon spins with a

uniform tipping field which was fed into cosine windings. This yielded a T2 time

of approximately 14minutes at low field. See fig.4.2. We have also been able to

show that the K-Ne co-magnetometer shares the same transient response, that is

to compensate for external magnetic transient fields, as the K-He comagnetometer.

See fig.4.2. In short we have demonstrated operation of a K-Ne co-magnetometer,

albeit one with low Ne polarization.

4.3 Influence of Quadrupole collisions in Polarizing

neon nuclei

We were also able to measure the T1 time of the spin exchange optically pumped

neon, as both a function of cell pressure and temperature. It seems that the dom-

77

0 50 100 150 200

Time (sec)

-2

-1

0

1

2

12

)V(

lan

gise

pocs

ory

gK-e

N

-1000

-500

0

500

1000

)T

p(dlei

Fcite

nga

Mesre

vsnar

T

Figure 4.3: Compensation behaviour of the K-Ne comagnetometer to an externallyapplied magnetic transient field.

inate limiting factor in achieving large neon polarization is due to quadrupolar

relaxation. These occur during neon-neon collisions when large electric field gra-

dients are created and couple to the quadrupole moments of the respective nuclei

causing depolarization. We believe this is the main cause of nuclear relaxation due

to the following reasons.

First, the T1 time of the longitudinal neon polarization is of the form:

T1 =1

Rse + Rrelax(4.1)

So, by knowing the Rse and Rrelax one should be able to reproduce the T1. Where,

Rrelax = Rgradient + σNe−Nesd [Ne]ν (4.2)

and

Rgradient = DNe

(∇B

B

)2

(4.3)

where

DNe = 0.79cm2/s

√1 + T/273.15

Pneon/1amagat(4.4)

78

0 100 200 300 400 5000

2

4

6

8

10

12

Po

la

ri

za

ti

on

(

ar

b.

u

ni

ts

)

Time (minutes)

Figure 4.4: T1 of 105 minutes for a 1.6atm cell of Ne at 170C.

For a 6.5atm cell of K-Ne we have a T1 time of 34minutes. Using the theoritical

value of the spin exchange rate and estimating the gradients to be 50µG/cm we

expect a T1 of approximately 140minutes. The gradient calculation was estimated

by modeling the cell as a uniformly polarized cube of magnetization 320µG.

Combining Rse, Rsd and the contribution due to magnetic gradients we still can-

not reproduce the observed T1 times. We attribute the discrepancy to a quadrupo-

lar relaxation mechanism. Furthermore we varied the spin exchange rate by mea-

suring T1 at various temperatures without observing a significant change in the T1

time. In fact by increasing the temperature of the 1.6atm cell from 170C to 190C

we only observe the T1 change from 105 to 97 minutes. The K density varies by

a factor of over 2 in this temperature range, which would have had a more dra-

matic effect on the T1 if this were the leading contribution to T1. It was difficult to

raise the temperature above 190 because the optically thick cell was not uniformly

polarized by the pump beam and caused a lower neon polarization. We are fur-

ther supported by the fact that cells with higher neon pressures have significantly

lower T1 times. For instance at 170C the 6.5atm cell has a T1 of 35.1 ± 1.9minutes

79

whereas the 1.6atm cell is 104.7 ± 0.7minutes. See fig.4.4.

The contribution from neon quadrupolar relaxation can be quantified by mea-

suring the T1 times of cells with different pressure. See fig 5.7. The slope of this fit

is 214 ± 10min.atm. This is discussed in more detail in sec.5.4. Grover (1983) has

also polarized neon and suggests that neon quadrupolar relaxation is the dominant

source of neon relaxation. His data suggests a relaxation of 240min.atm which is

consistent with our data. Although he performed his experiments with Rb instead

of K most relevant exchange cross sections do not vary dramatically between the

two alkali species. We can still attribute the dominant relaxation to be dependent

on the neon properties rather than that of the alkali.

Finally we investigated the effect of filling the cells with both 21Ne and 4He.

Originally this was done to broaden the optical resonance of the K line and de-

crease the optical thickness of the cell to ensure uniform polarization of the cell.

This was to ensure that the low T1 of the cell was not due to relaxation and inef-

ficient pumping caused by a non-uniformly polarized cell. We filled the cell with

1.73atm of neon and 3.24atm of helium. We measured a T1 of 40.7 ± 1.4 minutes.

This leads us to believe that the effect of Ne-Ne collisions may be comparable to

that of Ne-He collisions.

4.3.1 T1 measurement of Neon

The inference that quadrupolar relaxation is the dominant relaxation mechanism

for neon is dependent on the accurate measurement of the T1 time. To put this on

a firm footing it would be prudent to discuss the measurment of T1 in more detail.

Determining the spin dynamics of the co-magnetometer and determining the T1

data is not trivial. A problem arises because the T2 time is comparable to the T1

time for the measurements made inside magnetic shields.

The large T2 makes measuring the T1 time difficult. To measure the T1 we first

80

tipped the neon spins by a small angle using field coils wound in a cosine wind-

ing arrangement within the magnetic shields. As a result of tipping the spins the

Faraday rotation measured by the probe beam became modulated at the preces-

sional frequency. The signal from the photodiode subtracting amplifier was fed

into a Labview program which fit the data and calculated the precessional fre-

quency, amplitude, and decay constant. These can be utilized to calculate the T2

time, and polarization. However when the T2 time is large and comparable to T1

the cell would significantly increase in polarization during the decay of the trans-

verse polarization. This would not make it possible to accurately determine the T1

time. Instead we quenched the transverse oscillations to eliminate this potential

problem.

To quench the oscillations we first partially polarize the sample. Subsequently

the Faraday rotation signal from the detection system was inverted and used to

generate an oscillating magnetic field in a direction orthogonal to both the probe

and pump directions. This caused a nonlinear response which flipped the spins

to make them anti-aligned with the pumping direction. Subsequently a tipping

pulse was applied, and data was recorded for 60 seconds. Then the signal from the

detection system was again fed into the field coils. This then served to quench the

transverse oscillations.

Let us describe this process in more detail theoretically:

~S = a sin(φ(t)) [x cos(ωt) − y sin(ωt)] + a cos(φ(t))z (4.5)

where a is equal to the magnetization. The quenching field can be expressed as:

~Bquench = κ[~S · x

]y (4.6)

Here κ is a parameter with units of s−1 which is dependent on factors such as the

81

gain of the coils, and magnetic moments of the spins. So the total magnetic field

acting on the spins in the laboratory frame is:

~B = ~B0z + κ[~S · x

]y (4.7)

~B = ~B0z + κa sin(φ(t)) cos(ωt)y (4.8)

If we transfer to the frame rotating at the neon precessional frequency this trans-

forms to:

~Brotating = κa sin(φ(t)) cos(ωt)y (4.9)

Similarly the behaviour of our spins can be described as:

d~S

dt inertial= γ~S × ~Binertial (4.10)

d~S

dt rotating= γ~S × ~Brotating (4.11)

or,

d~S

dt rotating= γ~S × κa sin(φ(t)) cos(ωt)y (4.12)

We are interested in the behaviour of the z component of the spin. This becomes:

d~Sz

dt= −a sin(φ(t))

dφ

dt= γκa2 sin2(φ(t)) cos2(ωt)y (4.13)

Solving this we find the behaviour of the spins as a function of time:

φ(t) = 2 tan−1

[tan(φ0/2)exp

[−κ

(t

2+

sin(2ωt)

4ω

)]](4.14)

where φ0 is the initial tip angle. One can see that in the case of positive κ the

quenching field aligns the spins with the magnetic holding field during equilib-

82

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Time (s)A

ng

le (

de

gre

es)

Figure 4.5: Theoritical simulation of the noble gas spin. The quenching field elimi-nates transverse oscillations and aligns the spins with the holding field for positiveκ.

0 2 4 6 8 10 12 140

255075

100125150175

Time (s)

An

gle

(d

eg

ree

s)

Figure 4.6: Theoritical simulation of the noble gas spin. The quenching field flipsthe spins so that the are anti-aligned for negative κ.

rium. see fig 4.5. For negative values of κ we see that the spins align anti-parallel

to the magnetic holding field. That is the equilibrium value of the azimuthal angle

is π radians, regardless of the initial tip angle φ0. See fig 4.6

As time goes on and the spins are re-pumped along their original direction

the polarity of the quenching pulses must be flipped so as not to re-flip the spins

to their anti-aligned state again. A Labview program was utilized to control this

entire data taking process and monitor the spin orientation relative to the magnetic

holding field.

83

4.4 Improving Magnetometer Sensitivity

A number of technical schemes for improving the magnetometer noise have been

investigated before the final alkali-neon comagnetometer prototype is built. The

first we discuss is eliminating spurious signal from the gyroscopic signal.

4.4.1 Removing Birefringence and false Faraday Rotation signals

Birefringence in the probe beam can cause a false signal. If there is any degree of

circular polarization in the probe beam the system will additionally pump along

the probe beam axis and introduce light shifts into the system. To reduce this

effect we have utilized a cubic cell shape rather than the typical spherical cell. For

a spherical cell there is no dichroic effect produced for a diametrically traversing

beam. For a spherical cell the light experiences some dichroism as varying amount

of light are reflected off the cell wall for light polarized in the plane of reflection,

and perpendicular to the plane of reflection. Random walk of the beam off centre,

possibly due to convection currents of air, would induce some degree of circular

polarization. This is not an issue for a cubic cell since the surfaces are planar.

In a spherical cell different segments of the probe beam travel varying dis-

tances through the cell. Therefore segments of the probe beam experience different

amount of Faraday rotation. This effect is also eliminated for a cubic cell to first

order.

Circular birefringence can also be induced from reflections with mirrors, and

via transmission through various optical elements. This can particularly be the

case for stressed lenses, and mirrors. To eliminate this problem the probe beam

has been forced to travel through a variable waveplate. The variable waveplate

consists of a nematic liquid crystal with polar molecules. As an electric field is

applied the polar crystals orient themselves along the electric field axis. The popu-

84

lation which is oriented along the field is a function of the electric field magnitude.

When the probe beam enters the liquid it will experience different indices of re-

fraction along the direction of the electric field and perpendicular to it, inducing

a tunable phase lag. A square wave function generator to control this retardance

and to eliminate or cancel the circular polarization of the beam has been built.

It seems though that utilizing a stressed plate of glass to alter the birefringence

gives a better noise performance, and will most likely be used in the future neon

co-magnetometer.

4.4.2 Controlling and Monitoring the Laser stability

We were unable to obtain DFB lasers which operated at the Potassium D1 fre-

quency. Thus we had to cool commercially available Eagle Yard laser diodes to

−30C in order to tune it to the required frequency. To achieve this we attached

two thermoelectric coolers (TEC’s) in series to the laser diode baseplate. These

were in turn connected to a large heat sink which was cooled with a continuously

run chilled water supply cold plate. Originally the setup was cooled via a mechan-

ical fan, but the level of measurement noise induced by vibration of the air against

the heat sink was unacceptable.

There is the additional complication that the dew point in the lab is approxi-

mately 12C. Thus electronic shorting do to condensation of the electrical circuitry

was a major concern. To circumvent this problem we encased the laser and dual

TEC setup in an aluminum housing. Holes were drilled in this housing and air was

continuously flowed through the setup at positive pressure. The water vapour was

first removed from the air by flowing the air through a commercial dessicant called

Drierite. This setup was utilized for both the pump and probe laser arrangements.

With recent advances in DFB lasers one can now purchase lasers which are evacu-

ated around the actual diode. These can be cooled without danger below the dew

85

point of the laboratory.

Another concern in the setup was due to the laser frequency shifting with time.

To monitor this we constructed a constant pressure Fabry Perot etalon. It consists

of an Invar tube, which has low thermal expansivity, surrounded by Nichrome

wire, along with a thermocouple to measure temperature. Two confocal mirrors

are fitted to both ends of the hollow Invar tube, and attached to a piezo. The

Nichrome wire is utilized as a heater and used via negative feedback to keep the

temperature of the Invar constant. The entire setup enclosed inside three stainless

steel tubes and evacuated. The idea was to utilize this to look for correlations

between pressure, and temperature change of the lasing frequency and stabilize

them. This is currently being tested on a different setup.

4.4.3 Miniaturization of Gyroscope

In an effort to miniaturize the current setup one runs into difficulties creating ho-

mogeneous magnetic fields. With the standard Helmholtz arrangement the active

region, where the non linearity in the field is less than 1%, is small compared to

the radius of the coils. In order to miniaturize the system a dual pair of coils were

modeled analytically. The current ratio in the two pairs was taken to be a rational

integer. Using a numerical simulation we solved for the inter coil separation as a

function of the integer current ratio while imposing elimination of 4th order non-

linearity in the field dependence at the symmetric centre of the coils. Using this

process we found that the optimal compact arrangement occurred for a turn ratio

of 1 : 1. The coils are a distance 1.2cm and 2.2cm from the location of the cell along

the field axis, and have a diameter of 3.6cm.

The analytical solution is not strictly valid in the vicinity of magnetic shield-

ing. We used the simulation as a starting point to estimate the optimal placement

of theses coils within Mu metal shields. This was modeled using the commercial

86

Figure 4.7: Magnetic field homogeneity for a 3cm×3cm region. Here the coils arecarrying a 10mA current.

computer program Maxwell. See fig.4.7. An optimized magnetic field homogene-

ity was realized by simultaneously varying both the radii of the coils and their

positions independent of each other. This solution is independent to first order to

small asymmetric misplacing of the wires.

We were able to achieve a magnetic field homogeneity of 0.1% in a region 2cm

by 2cm which was produced by coils of diameter 4cm.

4.4.4 Alternate methods to heat Cell, and remove Convection noise

A source of error in the magnetometer signal is due to random walk of the laser

beam caused by the air convection used to heat the cell. In order to reduce this we

have enclosed the cell in an evacuated glass chamber. The cell is surrounded by

a 6 boron-nitride sheets. These are finally attached to a boron-nitride rod around

which is wrapped a coaxial mineral insulated heating cable. The entire enclosure

is cemented together utilizing aluminum nitride. A current is sent through the

coaxial cable, which is shorted at one end. The power dissipated through the cable

is used to heat the boron nitride walls, which heat the cell by conduction. The

87

entire enclosure is enclosed in a evacuated glass chamber which has been silvered

to reduce radiative loss of heat. The silver coating is scratched to prevent Johnson

noise from currents in the coating.

The co-axial coils carry a 10mA current and heat the cell via direct conduction.

Because of the coaxial arrangement the magnetic field due to the current is elim-

inated to first order. Boron-Nitride was chosen because it is has a large thermal

conductivity. (120 W/mK). Since it is not a metal it has limited magnetic prop-

erties. The fusing cement has a nominal conductivity of 110 W/mK. However

experiments were performed and indicate the conductivity of the cement form is

30 W/mK.

An arrangement utilizing Kapton heaters was tested to heat the cells. This sys-

tem was theoretically modeled. A temperature gradient exist across the boron-

nitride oven which can be explained by heat loss due to radiation. To eliminate

this problem we are attempting to redesign the boron nitride oven, to keep the

temperature gradient to a minimum. We also theoretically studied the radiation

exchange problem between the oven and magnetic shielding. We operate such that

the magnetic shielding is not raised above its Curie temperature. It was also shown

that one could simply use a twisted pair of heater coils to heat the cell. This is the

scheme which is currently being used. The advantage over Kapton heaters is that

the twisted pair can operate at higher temperature, whereas the Kapton heaters

melt at 200C. This is because the twisted wire contains a high temperature glass

sheath.

To reduce the noise further we oscillate the current in the twisted wire at a

frequency higher than the bandwidth of co-magnetometer. The bandwidth of the

co-magnetometer is ≈ 100Hz, and the current is oscillated at ≈ 25KHz. This also

ensures that the spin precess through a small angle during the period of current os-

cillation. This coaxial wire has an outer sheath of copper, and an inner sheath made

88

Figure 4.8: Silvered oven holding Boron-Nitride housing for Pyrex cell. These are all enclosed in a 4layer concentric Mu-metal magnetic shield system.

89

of Everdur 655. This is a higher resistance non magnetic metal. These were chosen

to reduce the inhomogeneous DC field produced from the wires themselves. Other

wires such as Nichrome have been experimented with. The oven is placed inside

a glass encasing which is then evacuated. This is then silvered to reduce heat loss

due to radiation leakage. The silver surface is scratched so as to reduce the effect

of eddy currents.

A version of the gyroscope where the Mu-metal shields have been replaced

with a smaller ceramic ferrite shield has been implemented for use as a new test of

CPT violation. It has a realized noise of 1fT/√

Hz, using a K-He co-magnetometer.

This is roughly 2.5 times more sensitive than the previous incarnation of the co-

magnetometer gyroscope which had a magnetic field sensitivity of 2.5fT/√

Hz,

and 5 × 10−7rad/s/√

Hz. In the current setup if one switched to a K-Ne co-

magnetometer cell the expected sensitivity would be 5 × 10−8rad/s/√

Hz due to

the 21Ne gyromagnetic ratio being a factor of ≈ 10 smaller than that of 3He. See

eq. (2.121).

90

Chapter 5

Measurement of parameters for

Polarizing Ne with K or Rb metal

Many of the applications currently using 3He, including the co-magnetometer,

would benefit, and realize increased sensitivity by substituting 3He with polarized

21Ne gas. In fact, tests of CPT violation using co-magnetometers would be greatly

improved if one utilizes polarized 21Ne gas (Kornack et al. (2008)). Additionally

the nuclear spin co-magnetometer gyroscope would realize an order of magnitude

gain in sensitivity (Kornack et al. (2005a)).

Very little is known about parameters which govern the spin-exchange polar-

ization of 21Ne. In order to realize these applications, and test the feasibility of a

21Ne co-magnetometer the interaction parameters of 21Ne with alkali metals must

be measured. The spin-exchange cross section σse, and magnetic field enhance-

ment factor κ0 have only been theoretically calculated (Walker (1989a)). Further-

more there are no quantitative predictions of the neon-neon quadrupole relaxation

rate Γquad. In this work we investigate polarizing 21Ne with optical pumping via

spin exchange collisions and measure the relevant spin exchange rate coefficient,

magnetic field enhancement factor, and quadrupolar relaxation coefficient. Fur-

91

thermore we measure the spin destruction cross section of Rb, and K with 21Ne,

and find agreement with the values found in the literature. Finally we discuss the

feasibility of utilizing polarized 21Ne for operation in a co-magnetometer.

5.1 Theory

In this work we refer to the spin exchange rate measurements of the K-21Ne system.

However the measurement technique is identical to that utilized for the Rb-21Ne

system.

21Ne becomes polarized via spin-exchange collisions with polarized alkali atoms.

During spin exchange collisions the electron wavefunction of the alkali atom over-

laps with the noble gas nuclei. They interact via a hyperfine Fermi contact interac-

tion of the form

Hse = α~In · ~Sa (5.1)

where ~In refers to the noble gas nuclear spin operator, and ~Sa the alkali electron

spin operator. During collision this interaction leads to an exchange of angular

momentum from the alkali valence electron to the noble gas nuclei. The collision

also brings both the alkali valence electron, and the noble gas nuclei into close

proximity. This causes them to both experience strong magnetic fields due to the

the interaction of their magnetic moments. This results in a change in the Larmor

precession frequency of both species. This frequency shift is described in terms of a

magnetic field enhancement factor κ0 (Walker & Happer (1997)). It is defined as the

ratio of the Larmor frequency shift of each species caused by the contact interaction

and to the shift caused by the classical magnetic field generated by each other gas

species. The value of κ0 is often much greater than unity. It can be measured by

comparing the precessional frequency shift of an alkali atom the noble gas is in

contact with, to the actual magnitude of the magnetic field produced by the noble

92

gas nuclei.

For light alkali atoms the dominant spin exchange mechanism for polarizing

noble gas involves binary collisions. Here the spin-exchange rate constant can be

expressed in terms of the familiar spin-exchange cross-section as Grover (1983):

κa = σsevK−Ne (5.2)

Where vK−Ne is the relative velocity between the potassium, and neon atoms.

The polarization of neon atoms via spin exchange collisions with potassium

atoms of number density [K] can be described by (Walker & Happer (1997),Appelt

et al. (1998)):

3

2

∂PNe

∂t= κa[K](ǫ(W, β)

1

2PK − 3

2PNe) − PNeΓquad (5.3)

Here the coefficient κa represents the spin-exchange rate constant. The terms PK

and PNe are related to the longitudinal spin polarization of the alkali atom of spin S,

and neon atom with nuclear spin K by PNe = 〈Wz〉 /W, and PK = 2 〈Sz〉. The neon

relaxation is dominated by the quadrupolar relaxation rate Γquad (Adrian (1965)).

One method to measure κa, suggested by Gentile (Gentile & McKeown (1993)),

is to simply measure the rate of rise of neon polarization at PNe = 0. This trans-

forms eq. (5.3) to

∂PNe

∂t=

5

3κa[K]PK (5.4)

We measure the buildup of neon polarization as a function of time extrapolated to

the slope at zero neon polarization to determine the alkali-neon spin exchange rate

κa. These measurements are made for times where the buildup of neon polariza-

tion remains linear.

In this work we additionally use a method based on re-polarization in the dark

93

to measure κa (Chann & Walker (2002),Kadlecek et al. (1998)). Here the polarized

neon spins repolarize the K spins in the absence of optical pumping. In this case

the total K spin, F = I + S, evolves as

∂Fz

∂t= D∇2Fz − ΓKSz + κa[Ne](Wz − ǫ(W, β)Sz) (5.5)

Here the first term represents the contribution due to diffusion of K through the

cell. The second term represents depolarization via spin destruction collisions. We

expect the contributions from diffusion to be small compared to the other terms.

The diffusion term is ≈ 2840 times smaller than the spin destruction rates. When

the neon has reached steady state polarization one can simplify eq. (5.5) as:

κa =12 ΓKPK0

(32 PNe0 − 1

2 ǫPK0)[Ne](5.6)

Each of these quantities can be measured independently. ΓK can be measured by

chopping the pump beam, and analyzing the resulting decay in the alkali polariza-

tion. [Ne] is directly measured as the vapour cells are filled. PNe0 can be computed

directly by performing NMR on the polarized sample. PK0 can be measured by

scaling the polarization when the pump beam illuminates the cell according to the

optical rotation signal in the light, and the dark. The individual measurements for

determining the spin exchange rate, spin destruction cross-sections, and κ0 values

are described in greater detail in the following sections.

The dominant source of relaxation in neon arises from the quadrupolar relax-

ation rate Γquad. According to Adrian (1965) the relaxation rate is proportional to

the neon filling pressure. One can measure this contribution by measuring the T1

of neon cells with different filling pressure.

The dominant source of alkali relaxation is due to spin destruction collisions.

94

In the absence of optical pumping the spin destruction rate can be expressed as:

Rsd = nKσK−KVK−K + nNeσK−NeVK−K + nN2σK−N2

VK−N2(5.7)

In order to measure σK−Ne, one measures the decay of alkali polarization to de-

termine the total spin destruction rate and fit to eq. (5.7). One must modify the

measured relaxation rate by accounting for the paramagnetic slowing factor. This

is described in further detail in the section regarding measurement of the alkali

polarization decay.

5.2 Experimental Setup

The 21Ne gas sample is contained in a cubic pyrex cell of side length 20mm. It is

fully illuminated by a high power Sacher littrow diode laser which is tuned to the

K D1 line and outputs 100mW of power. This beam passes through a λ/4 wave-

plate before illuminating and polarizing the K atoms. A Toptica Dl-100 diode laser

is utilized as a probe beam. It passes through a linear polarizer and propagates

parallel to the pump beam. The pyrex cell contains a K droplet, ≈ 100 torr of nitro-

gen for quenching and 6.2 atm of 90% isotopically enriched21Ne gas. It is heated

to 180C via a hot air line which heats a glass oven. The oven is placed in the

middle of two large 34 inch diameter Helmholtz coils which create a holding field

of approximately 16.7 gauss. The Helmholtz coils are powered by a 100V Bipolar

amplifier. Perpendicular to these coils are a pair of NMR tip coils, also placed in a

Helmholtz configuration, and a pair of RF modulation coils. Along the y axis, out

of the plane of the optical table, (see fig 5.1) a NMR pickup coil is placed. It has a

Q of ≈ 14, and is tuned to the neon resonant frequency. For the Rb measurements

the pump laser is a 2W Coherent 19 diode laser array, and the probe is a 10mW

Nanoplus DFB diode.

95

Subtraction

Cell

OvenProbe Laser

Linear

Polarizer

Polarizing

Beamsplitter

Photo-

diodes

Pump Laser

λ/4

yx

Feedbackz

(Circular Polarizer)

Lock-In

Amplifier

Fluxgate

Current

Source

Modulation(Larmor Frequency)Reference

Magnetometer Signal

Compensating Coil

Feedback AFP frequency

Modulation

AFP Amplitude

Modulation

NMR tip Coils

NMR Pick-up

Figure 5.1: Experimental Setup

κ0 is measured with a sequence of EPR-AFP flips, followed by NMR tips. κ0 is

calculated by computing the ratio of the magnetic field shift experienced by the K

(EPR-AFP) and the magnetic field produced by the neon, as determined from the

NMR.

To determine the spin exchange constant using the repolarization method a

series of EPR-AFP flips was used to determine the K frequency shift. PNe0 was

calculated using the measured value for κ0 and the neon pressure in the cell which

was measured during filling. [Ne] is known from the cell filling pressure.

To determine PK0 a two step process was used. First the RF source was swept

over the K Zeeman levels to determine the K polarization with the pump on. The

pump beam was then blocked and the repolarization of K by Ne was measured by

monitoring the optical rotation. PK0 was calculated by calibrating the optical ro-

tation obtained during back polarization with the optical rotation measured while

the cell was illuminated. The zero optical rotation point when the pump beam is

blocked was found by flipping the neon spins with AFP and monitoring the sub-

sequent change in the K optical rotation in the dark. ΓK is determined by manually

96

chopping the pump beam, and measuring the resulting decay in the optical rota-

tion.

For the rate of rise measurements the average alkali polarization in the cell PK

must be measured while the cell is illuminated. To account for the variation in

K polarization across the cell, the probe beam was swept across the cell and the

optical rotation was measured. At the centre of the cell RF modulation was used

to measure the K polarization. By scaling the polarization as compared to the

optical rotation across the cell an estimate of the average polarization of the cell

was obtained.

By comparing the T1 data from the manufactured cells, and those constructed

by others the contribution of neon-neon quadrupolar collisions to overall neon

spin relaxation can be estimated. The T1 times were calculated by utilizing EPR

lock-AFP flips to calculate the neon polarization.

A measurement of the spin destruction rate of rubidium in the rubidium-neon

cell and subsequent calculation of the Rb-Ne spin destruction cross section was

carried out. The spin destruction rate was measured via the rubidium relaxation

in the dark measurement which was previously described. The spin destruction

rate was extrapolated to zero probe beam intensity by making multiple spin de-

struction rate measurements at different probe beam intensities. The variation in

probe beam intensity was achieved by placing neutral density filters of different

values directly in the beam path before the probe entered the cell.

In the following subsections we describe each of the individual measurements

in greater detail.

5.2.1 NMR detection

In order to determine the spin-exchange rate constant, several different quanti-

ties for each rate constant must be measured. PNe0 is measured using NMR. To

97

accomplish this the neon NMR was obtained by tipping the 21Ne using two cal-

ibrated 9 inch diameter coils in the Helmholtz configuration. Typical tip angles

are approximately 30. The resulting neon NMR is detected with a 4 coil pickup

with 780 turns per coil. This consists of two pairs of pickup coils with opposite

winding orientation so that the combined pickup has zero dipole moment. This is

done to minimize coupling of the pickup coil to external fields which are not due

to the 21Ne gas. Each pair of coils are located symmetrically about the cell, and

are wound on one Teflon rod. Each coil has 780 turns of enameled copper magnet

wire. A numerical model was written to determine optimum coil placement in

order to maximize pickup sensitivity.

In order to minimize systematic errors the tip coils were calibrated using two

independent methods. First the pyrex cell is replaced with a small pickup coil of

similiar dimension to the pyrex cell, while the tip coils are operated. The result-

ing signal is fed into a lockin amplifier and used to calibrate the magnetic field

strength. Care is taken to properly align the axis of the pickup coil with that of

the tipping coils. Second, the current entering the tipping coils is measured with

a clamp on ammeter. The magnetic field produced from the tipping coils is cal-

culated from a knowledge of this current and the coil separation, and diameter.

The two calibration schemes agree to 2%, which is within the compound preci-

sion of the coil geometry measurements, alignment of pickup coils, and ammeter

precision.

For NMR detection the pickup signal is fed into a low noise pre-amp, and moni-

tored via computer. A typical neon NMR signal is shown in fig.5.2. The pickup coil

is also calibrated using two methods. The first requires a dummy source of known

size to create a magnetic field. The magnetic field strength is calculated theoret-

ically based on the dummy coil geometry. The resulting pickup is measured to

produce a calibration. The pickup coils are also operated in reverse to produce a

98

0.8

0.6

0.4

0.2

0.

0.2

0.4

0.6

0 0.05 0.10 0.15 0.20

0.0

0.2

0.4

0.6

-0.2

-0.4

-0.6

-0.8

Time (s)

NMR

Pickup

gain 1e4

(volts)

Figure 5.2: Representative NMR signal of polarized 21Ne gas

magnetic field at the cell location. A small pickup coil is placed here. Using reci-

procity arguments (Jackson (1999)) the two signals are compared and agree to 6%.

The uncertainty of the Q of the resonant coils is 4%.

5.2.2 Electron Paramagnetic Resonance Shift

In order to calculate the κ0 value, the effective magnetic field the K atoms experi-

ence must be measured simultaneously with the NMR signal. This effective mag-

netic field is measured by monitoring the shift in the frequency in the electron

paramagnetic resonance(EPR)upon reversal of the neon spins. (Romalis & Cates

(1998),Schaefer et al. (1989b),Newbury et al. (1993),Barton et al. (1994)). A double

feedback scheme is utilized to accomplish this. The first feedback loop locks the

output of the holding field to minimize magnetic field fluctuations from external

sources. To accomplish this a fluxgate magnetometer is placed approximately 5

inches from the pyrex cell outside the glass oven. In order to prevent the fluxgate

from saturating, from the 16.7gauss field produced from the large 34 inch diameter

Helmholtz coils, a solenoid was wound around the fluxgate. This compensating

solenoid produces a field which cancels the holding field and results in zero field

at the fluxgate. The compensating solenoid is powered by a custom built voltage

99

controlled current supply. The output of the fluxgate is fed through PID feedback

electronics to the input of the voltage controlled power supply which powers the

large Helmholtz coil holding field.

The second feedback loop locks the magnetic holding field to the K resonance.

Thus when the K resonance is shifted the holding field for the atoms is also shifted

so that the K resonance frequency remains constant. In short the two feedback

loops are summed. The first feedback controls the current which supply the hold-

ing field for the atoms, while the second feedback give an offset to this field in

order to maintain a constant K precession frequency.

In order to operate the second feedback loop a small RF coil is used to produce

a magnetic field at 12.4957MHz, with a sweep width of 15KHz, and a sweep rate

of 340Hz. The optical rotation of the probe beam is measured as the RF coils sweep

in frequency across the K magnetic resonance. The derivative of this signal is pro-

duced taking the out of phase component after feeding the signal into the lockin

amplifier which has been referenced to the RF sweep rate. The out of phase compo-

nent is fed into an integral feedback box and used to control the voltage controlled

current source which powers the compensating solenoid. In this manner we can

lock to the K resonance.

The output of the integral feedback box is monitored and calibrated by intro-

ducing known frequency shifts in the base RF field. That is the RF field frequency

is shifted by 100Hz, and the resulting feedback box voltage is monitored. This re-

lation is linear, and is used to calibrate the EPR frequency shifts. This technique

is utilized to lock to the end resonance of the F = 2 manifold of the K spectrum.

Operating at high field causes the K resonance to be non-linearly spaced due to

the Breit-Rabi splitting. The neon NMR frequency is approximately 5500Hz while

the system is locked to the K end state. A numerical program is used to calcu-

late the effective gyromagnetic ratio of the end state due to the Breit-Rabi splitting

100

0 5 10 15 20 25

Time (s)

-600

-400

-200

0

200

400

Fre

qu

ency

Sh

ift

(Hz)

Figure 5.3: Representative EPR shifts after 2 hours of polarization. The spikes inthe data are due to temporary loss of lock while the AFP coils are operated.

(Woodgate (1989)). This coupled with the NMR data allows the determination of

κ0 (Romalis & Cates (1998)).

∆ν =8π

3

dν(F, M)

dBκ0µK[Ne]PNe (5.8)

In order to measure the magnetic field enhancement factor κ0 the technique

of Adiabatic fast passage (Abragam (1961)) is used to flip the orientation of the

polarized neon spins by 180, while keeping locked to the EPR frequency and

monitoring the frequency shift. See fig5.3. The peak to peak amplitude of the

frequency shift seen using EPR is twice the value of the frequency shift induced by

the polarized neon. In order to flip the spin orientation a magnetic field is swept

in frequency across the neon NMR resonance. This is accomplished by produc-

ing a field with the small set of Helmholtz coils while satisfying the adiabatic fast

passage (AFP)conditions (Abragam (1961)):

‖∇Bz‖2

B21

<<γω

B1<< γB1 (5.9)

101

To sweep the frequency of the AFP flipping field a voltage controlled oscillator

is fed through a multiplier circuit, and a ramping voltage from a NI-Daq analog

output. The resulting signal is then amplified through a 100V Bipolar amplifier to

supply the AFP field. The neon spins are flipped every 5 seconds. The frequency is

swept from 2500Hz to 7500Hz in 4s. The AFP field strength is ≈ 4Gauss. The flip

efficiency is 99.7% In order to optimize flip efficiency it was necessary to slowly

ramp up the AFP field strength before the AFP field frequency was swept. The

AFP field was raised from 0 to 4 Gauss in 0.7s.

5.2.3 Alkali Polarization

To determine the alkali polarization when the cell is fully illuminated, a RF field

is swept across the K Zeeman levels, while the optical rotation of the longitudinal

probe beam is measured (Chann & Walker (2002)). The transverse RF field causes

a depopulation of the end state, and a lower polarization. This in turn reduces

the optical rotation of the probe laser. By comparing the area under the peaks of

the different transitions of the F = 2 manifold and using eq. (5.11) one is able to

determine the K polarization. See fig5.4. The area under a transition peak (F, m) →

(F, m − 1) depends on the state population ρFm, and the RF field Br f as (Chann &

Walker (2002)):

AFm ∝ B2r f [F(F + 1) − m(m − 1)](ρFm − ρFm−1) (5.10)

The spin exchange rate measurement experiment operates in the regime where

the magnetic sublevel populations can be described by a spin temperature dis-

tribution (Walker & Happer (1997)). Substituting the spin temperature condition

ρFm ∝ exp(βm), and noting that the spin polarization is given by PK = tanh(β/2)

results in

102

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

Time (s)

-1

0

1

2

3

4

Op

tica

lR

ota

tio

n(a

rb.

un

its)

Figure 5.4: Sweep over the A22, A21and A20 transitions at reduced Pump beampower.

PK =

(7r − 3

7r + 3

)(5.11)

where

r =A22

A21 + A11(5.12)

where the state is designated with the indices AFm. Under normal operating con-

ditions the spin polarization is very close to 1. Only the end state transition A22 is

visible under these conditions. However one can view the entire spectrum if the

pump beam power is sufficiently attenuated.

5.2.4 Alkali Polarization Decay Constant measurement

In order to measure the K decay constant the optical rotation due to the K is mon-

itored, as the pump beam is manually chopped. See fig5.5. The decay constant is

then determined by fitting the decay curve on a log plot. One must account for

the fact that to obtain the true time constant the measured constant must be multi-

plied by the paramagnetic coefficient or ’slowing down factor’ (Walker & Happer

(1997),Chann & Walker (2002),Appelt et al. (1998)).

103

-0.1 0.0 0.1 0.2 0.3 0.4

Time (s)

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Ab

solu

teK

Po

lari

zati

on

Figure 5.5: Potassium Polarization decays as a result of Pump beam being manu-ally chopped.

For spin K, and spin temperature β the slowing down factor is given by eq.

(2.21): For β << 1 this reduces to 4I(I + 1)/3 see eq. (2.24)(Walker & Happer

(1997)). Only decay data from the low polarization portion of the data set were fit

to the β << 1 simplified expression.

5.2.5 Back Polarization measurement

In order to measure the K polarization in the dark, the optical rotation due to Ne-K

back polarization was measured. The optical rotation signal is routed through a

low noise Pre-amp and fed into a NI-DAQ for acquisition. To measure the zero

polarization level the AFP Helmholtz coils are run to flip the neon polarization.

See fig5.6.

To determine the back polarization in the dark the gain adjusted optical rota-

tion to the optical rotation signal with the pump beam is illuminating the cell is

compared to that when the pump beam is blocked. Since the alkali polarization

was measured when the pump beam was unblocked one can scale the optical ro-

tation signals to the polarization. During operation of the pump beam large (≈ 1)

radian optical rotations were observed. In this regime one can no longer use the

104

0 5 10 15 20 25

Time (s)

-6

-4

-2

0

2

4

6x10-4

KB

ack

Po

lari

zati

on

Figure 5.6: Absolute Potassium Back polarization as Neon is flipped via Adiabaticfast passage

standard small angle optical rotation formula

Φ =I1 − I2

2(I1 + I2)(5.13)

Where I1and I2 are the voltages on the individual channels of the balanced po-

larimeter. Instead one can use:

Φ =1

2sin−1

(I1 − I2

I1 + I2

)(5.14)

5.2.6 Alkali density measurement

In order to calculate κa using the rate of rise method, the alkali density must be

independently measured. This is accomplished by measuring both the optical ro-

tation of the longitudinal probe beam and its detuning. This enables calculation of

the alkali density, while accounting for the fact that the alkali is not polarized to

1. The detuning is measured by monitoring the optical rotation as the probe beam

frequency is swept. The probe beam frequency is tuned by varying the tempera-

ture, so as to minimize variation in the probe beam intensity. The probe beam is

105

detuned ≈ 0.5nm from the centre of the absorption peak. The pump beam is tuned

to resonance by varying the current until the optical rotation experienced by the

probe beam is a maximum. There is sufficient pump beam power, 100mW so that

the cell is uniformly polarized.

To determine the alkali density the optical rotation is measured and fit to:

Θ =π

2lnrePx

(−1

3Im [L(ν − νD1)] +

1

3Im [L(ν − νD2)]

)(5.15)

where the Lorentzian line shape is given by:

L(ν − ν0) =Γl/2π + i(ν − ν0)/π

(ν − ν0)2 + (Γl/2)2(5.16)

Here n is the density of K in the cell, l is the path length which the probe beam

propagates through the cell, and re is the classical electron radius. The contribu-

tion to the optical rotation from the D2 transition is on the order of a few percent.

The density is 4.7 × 1013/cm3 which is a factor of 4 less than that predicted by the

empirical formula which relates the vapour pressure to the cell temperature. How-

ever there is often a discrepancy in the value calculated from the saturated vapour

pressure and the actual vapour density by at least a factor of 2 (Chann & Walker

(2002)). This comes from the fact that the cell may not be of uniform temperature,

or from substantial heating due to the pump beam. This effect can also be caused

by absorption, or a reduced alkali vapour pressure due to interactions with the

glass walls of the cell. This has been observed for pyrex glass cells, which is the

same glass employed in these measurements.

106

5.3 Fermi Contact interaction κ0 Results

κ0 for the interaction of K with 21Ne is 30.8 ± 2.7. This is approximately 10% lower

than the value of 34 as predicted by Walker Walker (1989b). Although the values

do not agree with the predictions it should be noted that the general trend among

the predicted values, and experimentally measured κ0 for other noble gas-alkali

pairs is for the predicted values to be 10 − 20% higher than those obtained exper-

imentally. When the experimentally measured values are arranged by magnitude

the gas mixtures show the same order as those of the theoretically predicted val-

ues. In a Rb-Ne system Walsworth has measured κ0 to be 32.0 ± 2.9 (Stoner &

Walsworth (2002)), while the value predicted by Walker is 38 (Walker (1989b)).

The Fermi-contact interaction was also measured for the rubidium-neon pair.

It was found to be 35.7± 3.7 which is in agreement with the value found by Stoner

& Walsworth (2002) of 32.0 ± 2.9. Both of these values are below that predicted

by Walker (1989a) of 38. However both of the experimental measurements fol-

low the expected trend of being larger than the Fermi-contact interaction for the

potassium-neon pair. See Table 5.3

5.4 Results of neon quadrupolar relaxation Γquad mea-

surement

By comparing the T1 data from the manufactured cells, and those constructed by

others the contribution of neon-neon quadrupolar collisions to overall neon spin

relaxation can be estimated. In general one expects that for noble gases with a

quadrupole moment, the dominant form of relaxation is due to nuclear electric

quadrupole interaction (Adrian (1965)). As a consequence of this one expects the

spin-lattice relaxation time to vary inversely with the density of the buffer gas. In

107

0 1 2 3 4 5 6 70

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Cell Pressure (Amagat)

1/L

ife

tim

e1

/min

Quadrupolar Relaxation due to Neon Collisions

()

Figure 5.7: Neon relaxation as a function of cell pressure. Here the data satisfiesthe relationship Pressure×T1 = 214 ± 10Amagat×min. The fact that the T1 is in-versely proportional to the cell filling pressure seems to indicate that the dominantrelaxation mechanism is due to nuclear electric quadrupole collisions. This is con-sistent with what one would expect from a buffer gas with spin greater than 1/2.

fig 5.7 we see that the cells follow this linear relationship. The data from the cell

with the lowest pressure is from Grover (Grover (1983)) whereas the data from the

other cells is from this work. Both Grover’s cell and 3.34 atm cell is filled with a

Rb-Ne mixture, whereas the others are filled with a K-Ne mixture. The pressure

in the Rb-Ne cell is determined by measuring the broadening of the optical D1

transition in Rb. The reason for the large uncertainty in the 3.34 atm cell’s pressure

is due to the uncertainty in the literature of the broadening parameters for the D1

transition of Rb in neon gas (Ottinger et al. (1975)).

5.5 Spin exchange Rate coefficient Results

The K-21Ne spin exchange rate, measured with the repolarization method, is 3.36±

0.42 × 10−20cm3/s. The value obtained by using the theoretically predicted spin

exchange cross section by Walker is 7.5 × 10−20cm3/s. See Table5.1.

Data for the rate of rise method was taken in the limit of low neon polarization.

108

0 1 2 3 4 5

Time (minutes)

0

1

2

3

4

5

6

7x10-4

Neo

nP

ola

riza

tio

n

Figure 5.8: Absolute neon polarization as function of time to determine spin ex-change rate constant by rate of rise method.

Here the rate of rise of neon polarization remained linear. See fig5.8. The rate of

rise method gives a value of 2.34± 0.29× 10−20cm3/s for the K-21Ne spin exchange

rate.

Additionally neon was polarized using spin exchange optical pumping with Rb

metal. The Rubidium-Neon spin exchange rate was also measured and found to be

0.80 ± 0.12 × 10−19cm3/s. Walker (1989a) predicts a value of 1.66 × 10−19cm3/s.

See Table 5.2. However this value is not in agreement with the previously mea-

sured value by Chupp & Coulter (1985) of 4.66 × 10−19cm3/s. However Chupp

& Coulter (1985) determined their rubidium density by directly applying the sat-

urated vapour pressure calculated by the empirical formula given by Alcock et al.

(1984). This formula can be in disagreement with the actual alkali density by as

much as a factor of two (Chann & Walker (2002)). Chupp fits his data directly

through the origin implying that the contribution of neon quadrupolar relaxation

collisions to the T1 is negligible. We have shown in this work that this is incorrect,

and that the dominant contribution to the neon T1 is in fact due to quadrupolar

collisions. This coupled with the fact that the more precise measurements carried

out by Chann & Walker (2002) indicate a statistical variation in the rate constants of

109

Spin Exchange Rate Method cm3/s

Repolarization 3.36 ± 0.42 × 10−20

Theoretical Prediction 7.5 × 10−20

Rate of Rise 2.34 ± 0.29 × 10−20 cm2

Table 5.1: Spin exchange parameter of a potassium-neon system

Spin Exchange Rate Method cm3/s

Repolarization 0.80 ± 0.12 × 10−19

Theoretical Prediction 1.66 × 10−19

Rate of Rise 0.82 ± 0.18 × 10−19

Table 5.2: Spin exchange parameter of a rubidium-neon system

25% upon subsequent measurements implies that this disagreement is not entirely

unexpected.

5.6 Measurement of Spin destruction cross-sections of

neon with Rb and K

Utilizing the data from the potassium relaxation in the dark measurement one can

compare the measured spin destruction rate to that predicted by using the known

Species Fermi Contact InteractionK-Ne (this work) 30.8 ± 2.7K-Ne (prediction) Walker (1989a) 34Rb-Ne (this work) 35.7 ± 3.7Rb-Ne (Walsworth) Stoner & Walsworth (2002) 32.0 ± 2.7Rb-Ne (prediction) Walker (1989a) 38

Table 5.3: Fermi contact interaction measurements for the K-Ne, and Rb-Ne sys-tems in this work, and compared to both predictions ans measurements by othergroups.

110

0 1 2 3 4 5 62.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0x10-20

Sp

inE

xch

ang

eR

ate

(cm

/s)

3

Figure 5.9: Scatter in Spin exchange rate measurements for K-Ne

0 1 2 3 4 50.5

0.6

0.7

0.8

0.9

1.0x10-19

Sp

inE

xch

ang

eR

ate

(cm

/s)

3

Figure 5.10: Scatter in Spin exchange rate measurements for Rb-Ne

111

Species Group 10−23cm2

K-Ne This work 1.1 ± 0.1K-Ne Franz Franz & Volk (1982) 1.41 ± 0.14K-Ne Walker (prediction) Walker (1989a) 1.6Rb-Ne This work 1.9 ± 0.2Rb-Ne Franz and Volk Franz & Volk (1976) 1.9Rb-Ne Franzen Franzen (1959) 5.2Rb-Ne Walker (prediction) Walker (1989a) 1.8

Table 5.4: Spin destruction cross sections for K-Ne, and Rb-Ne as compared to boththeory and other measurements

spin destruction cross sections, and gas densities in the cells using eq. (5.7).

For the case of potassium with neon the predicted spin destruction rate is 8%

smaller in the 1.6atm cell, and 8%greater in the 6.2atm cell than the measured

value. This is reasonable agreement since the relevant spin destruction cross sec-

tions have only been measured to one significant figure. The spin destruction rates

are approximately 2840 times larger than the contribution due to gradient relax-

ation. It is larger than the rate of diffusion to the cell wall by a factor of 40.

A measurement of the spin destruction rate of rubidium in the rubidium-neon

cell and subsequent calculation of the Rb-Ne spin destruction cross section was

carried out. The spin destruction rate was measured via the rubidium relaxation

in the dark measurement which was previously described. The spin destruction

rate was extrapolated to zero probe beam intensity by making multiple spin de-

struction rate measurements at different probe beam intensities. The variation in

probe beam intensity was achieved by placing neutral density filters of different

values directly in the beam path before the probe entered the cell. The alkali-Ne

spin destruction cross sections are listed in Table 5.4.

112

One can accurately predict the polarization of the neon gas by using eq. (2.105).

PNe = Palkaliǫ

W/S

Rse

Rse + 1/T1(5.17)

Here the measured spin exchange rate for the Rse was substituted into eq. (5.17)

and T1 was calculated assuming that the quadrupolar relaxation mechanism is the

dominant relaxation mechanism. Making these assumptions, while taking into

account the non-uniform alkali polarization profile across the cell,a polarization of

0.55% at 140C was expected. A value of 0.8 ± 0.13% was measured using EPR.

5.7 Conclusion

The spin exchange rate coefficient for both the Rb-21Ne, and K-21Ne systems have

been measured using two different techniques. The values from the rate of rise

and repolarization techniques are consistent. The spin exchange rate coefficient

for Rb-21Ne does not agree with the previous value in the literature. We claim this

discrepency is caused by the exclusion of relaxation due to the neon quadrupolar

relaxation from previous measurements. The Fermi contact interaction for both

the Rb-21Ne, and K-21Ne pairs have been measured. There is agreement with the

previously measured value for the Rb-21Ne system. The neon quadrupolar relax-

ation has been measured. Also the various alkali spins destruction cross-sections

with 21Ne have been measured. These agree with the previous values quoted in

the literature. We have modeled the neon polarization as a function of alkali den-

sity and have show that the polarization dynamics are well described by assuming

quadrupolar relaxation is the dominant form of relaxation, and spin exchange the

dominant polarizing interaction.

113

Chapter 6

Feasibility of utilizing 21Ne in a

co-magnetometer

The main objective of these experiments on 21Ne is to ultimately create a neon co-

magnetometer which can be used for experiments on tests of fundamental symme-

tries, and for deployment as a sensitive gyroscope. However application of 21Ne

in a co-magnetometer requires polarization higher than that observed for the set

of spin-exchange rate measurements at 140C.

One can increase the neon polarization by increasing the alkali density. How-

ever this requires additional laser power to ensure the optically thick cell remains

uniformly polarized. Experiments were carried out at higher density at 180C and

resulted in ≈ 8% neon polarization.

A Mathematica model was used to calculate the sensitivity of the co-magnetometer

at normal operating temperatures by optimizing the laser power. Here the sources

of noise included were the effects of spin projection noise, and photon shot noise.

The first effect is due to the Heisenberg uncertainty principle as applied to the

transverse components of the spins and the fact that they do not commute. The

second source of noise is the noise of the rotation signal of the probe beam for a

114

balanced polarimeter setup. We ignore the effects of light shift noise because we

presume to operate with the pump and probe beams orthogonal to each other. In

this orientation the magnetometer signal is sensitive to the axis perpendicular to

the plane which contains both laser beams. Thus it is in-sensitive to light shift

noise caused by fluctuations in the ellipticity of the probe beam to first order. Let

us describe the Rb-Ne comagnetometer simulation in greater depth.

6.1 Effects of Light Propogation and alkali relaxation

on Rb-Ne co-magnetometer simulation

The propagation of the pump and probe beams must be modeled as the beams can

be strongly absorbed and result in non-uniform polarization through the cell. The

propagation of light through the cell is a function of the alkali polarization in the

cell which is given by eq. (2.18). The propagation dynamics can be described by:

dRop

dx= −nσ(ν0)(1 − Pequil)Rop (6.1)

As one can see the attenuationdRop

dx vanishes if the alkali is fully polarized. How-

ever this is never achieved in practice due to spin relaxation due to spin destruction

collisions. These can be described by:

Rrel = nRbσsdRb−RbVRb + nNeσ

sdRb−NeVNe + Rpr + (ǫ + 1)Rwall + Rse

Rb−NenRb (6.2)

Rwall is given by eq. (2.62), Vi−j is given by eq. (2.39), ni refers to the density per

cm−3 of species i, and ǫ is given by eq. (2.56). Rpr is the pumping rate due to the

115

probe beam and is given by:

Rpr = σRb−xs(λprobe) × Pprobe

λprobe

wlhc×

Exp

[−σRb−xs(λprobe)nRb

1

2

](6.3)

w refers to the width of the cell, and l its length. h is Planck’s constant, c is the

speed of light, and Pprobe is the probe beam power. The photon absorption cross

section σRb−xs is given by:

σRb−xs(ν) = rec ( fD1V(ν − νD1) + fD2V(ν − νD2)) (6.4)

Solution of eq. (6.1) gives a pumping rate profile across the cell as:

Rop(x) = RrelW

Rop−int

Exp[Rop−int

Rrel− σRb−xs(λD1)nRbx]

Rrel

(6.5)

Here Ppump is the pump beam power, and x is the propagation distance through

the cell. The function W is the principal value of the Lambert W-function. This is

defined as the inverse of the function f (W) = WeW . It is also refer

Rop−int = σRb−xs(λD1)PpumpλD11

Ahc(6.6)

where A is the cross sectional area of the cell, and λD1 is the wavelength of the D1

line.

An alkali polarization profile for a pancake cell of dimensions 6× 15mm under

condition of high pumping rate (≈ 7800s−1) is shown in fig.6.1

116

0.1 0.2 0.3 0.4 0.5 0.6

0.92

0.94

0.96

0.98

Distance Across cell (cm)

Ab

solu

te P

ola

riz

ati

on

Figure 6.1: Absolute Rb polarization as function of propagation distance throughcell, for pancake cell of dimensions 6 × 15 × 15mm and a pumping rate of ≈7800s−1.

6.2 Simulation of Noble gas relaxation

We are able to relate the noble gas polarization in the steady state to the alkali po-

larization using eq. (2.105). However eq. (2.105) indicates that the steady state

polarization of the noble gas not equilibrate with the alkali polarization due to

strong noble gas relaxation mechanisms. In this work we model the effects of both

quadrupolar relaxation, and relaxation due to diffusion of the noble gas in a mag-

netic field gradient.

The quadrupolar relaxation rate of neon is taken from the measured relaxation

rate as a function of cell pressure. This is depicted in fig.5.7.

The magnetic field gradient in the cell can be evaluated utilizing the technique

of magnetic vector potential(Jackson (1999)). The field inside the cell can be mod-

eled as a uniformly polarized mass. The magnetic field at an arbitrary point inside

the field can be evaluated by replacing the polarized mass with a surface current

of density equal to the cell’s magnetization (Jackson (1999)). This enables calcula-

tion of the relaxation rate given by eq. (2.64) on a point by point basis, which is

averaged over the cell. This is calculated numerically on a lattice. The noble gas T1

117

then becomes:

1

T1=

Pneon

12840s+∫

volumeDNe−Ne

(∇B(Pneon)

B(Pneon)

)2

dV (6.7)

In the above equation Pneon represents the pressure of neon in the cell in ama-

gat. The denominator of the first term describes the relaxation of neon due to

quadrupolar relaxation. The Pneon12840 is the quadrupolar relaxation rate given in s−1.

It is equal to 214Amagat·mins. DNe−Ne is the self diffusion coefficient of neon, and

∇B is the magnetic field gradient taken at every point in the cell. The eq. (6.7)

gives 1T1

in units of s−1. For a cell with pressure of 3 amagat the relaxation rate due

to diffusion is ≈ 30% as large as the quadrupolar relaxation rate.

6.3 Noise mechanisms in a Rb-Ne co-magnetometer

We describe the noise contributions to the magnetometer sensitivity in more detail.

The total noise in the magnetic field measurement has two major contributions.

The first is the spin projection noise due to the quantum uncertainty in the spin’s

orientation. The second contribution is due to the photon shot noise. To obtain the

total noise in the magnetic field measurement these contributions must be added

in quadrature.

δB =√

δB2spn + δB2

psn (6.8)

The descriptions of the uncertainty mechanisms below follow that of Savukov et al.

(2005).

118

6.3.1 Spin Projection Noise

To determine the spin projection noise of the co-magnetometer let us first consider

the commutation relation between the transverse components of the alkali spin.

[Fx, Fy] = iFz (6.9)

This leads to the uncertainty relation

δFxδFy ≥ |Fz|2

(6.10)

Let us operate under conditions of full polarization since this minimizes the uncer-

tainty relation. Due to symmetry δFx = δFy for spins which are not in a squeezed

state. We can describe the signal from N alkali atoms as making N uncorrelated

measurements. In such a case the uncertainty in the spin of the transverse compo-

nents becomes:

δFx =

√〈Fz〉2N

(6.11)

It is important to note that only uncorrelated measurements improve the sensitiv-

ity of the measurement. In the atomic magnetometer setup the probe beam con-

tinuously measures the spin projection. Correlated measurements from the same

atoms do not improve sensitivity. To take this into consideration let us define a

quantity χ(τ) as the degree of loss of spin coherence from a measurement at time

t = 0, and the time t = τ.

χ(τ) = e−τ/T2 (6.12)

Gardner (1990) gives the total uncertainty in a continuous measurement as:

δ 〈Fx〉 = δFx

[2

τ

∫ t

0

(1 − τ

t

)χ(τ)dτ

]1/2

(6.13)

119

δ 〈Fx〉 = δFx

[2T2

τ+

2T22 (e−t/T2 − 1)

t2

]1/2

(6.14)

Since typical measurement time for the magnetometer system satisfies t >> T2we

can combine eq. (6.11), and eq. (6.14) to obtain the total uncertainty:

δ 〈Fx〉 =

√2FzT2BW

N(6.15)

where BW= 1/2t is the bandwidth of the measurement. Here N describes the

total number of alkali spin that the probe beam interacts with. In units of root

mean square noise per root Hz we can rewrite the total uncertainty as:

δ 〈Fx〉rms =

√2FzT2

N(6.16)

This can be written in terms of magnetic noise by using eq. (2.35) and by setting

S → Fx/2 and Pez → Fz/2 to find:

δ 〈B〉rms =δ 〈Fx〉 Rtot

γeFz(6.17)

6.3.2 Photon Shot Noise

We will describe the photon shot noise associated with detection of a probe beam

with a balanced polarimeter setup. It is convenient to define the total photon flux

as:

Φ′ =∫

AΦdA (6.18)

where Φ is the photon flux per unit area, and Φ is the total photon flux taken over

the area of the probe beam. In terms of the photon flux in each photodiode we can

write the optical rotation as:

θ =Φ′

1 − Φ′2

2(Φ′1 + Φ′

2)(6.19)

120

If we assume the rotation angle θ is small we can state Φ′1 ≈ Φ′

2. In this case we

can write the fluctuation in the photon flux as:

δΦ′1 = δΦ′

2 =

√Φ′

2(6.20)

This leads to a fluctuation in the optical rotation angle given by:

δ 〈θ〉 =

√√√√2BW

[(δθ

δΦ′1

δΦ′1

)2

+

(δθ

δΦ′1

δΦ′1

)2]

(6.21)

δ 〈θ〉 =

√BW

2Φ′ (6.22)

In units of root-mean square noise per root Hz this becomes:

δ 〈θ〉 =

√1

2Φ′ (6.23)

For the magnetometer it is more useful to relate the noise in terms of the rotation

angle to the atomic polarization. Consider a probe beam detuned from the D1

resonance. We can calculate the optical rotation due to the D1 resonance if we

combine eq. (6.23) and eq. (2.81) to obtain:

δ 〈Px〉rms =2

πlnrec f√

Φ′Im[V(ν − ν0)](6.24)

Where Φ′ is given by:

Φ′ =ηPprobeλprobe

hcExp

[−nrblσRb−xs(λprobe)

](6.25)

121

Here η is the quantum efficiency

η =(Resp)hc

λprobee(6.26)

where Resp is the responsivity of the photodiode in A/W, and e is the charge of the

electron. The noise in angular units can be converted to magnetic units by using

eq. (2.35) and noting that S = Re(Pex) to obtain:

Px

Pz=

γeB0

Rtot(6.27)

or,

δ 〈B〉rms =δ 〈Px〉rms Rtot

γePez

(6.28)

6.4 Results of Rb-Ne co-magnetometer simulation

For a cell which is cubic with an edge of 5mm at 180C with 15mW pump beam

one finds the sensitivity averaged over the cell to be 0.77fT/√

Hz and compensa-

tion point of 4.7Hz. At 200C with 15mW pump beam one finds the sensitivity

averaged over the cell to be 0.31fT/√

Hzand compensation point of 4.7Hz. These

sensitivities are both below the 1fT/√

Hz noise floor caused by technical sources.

Thus we could operate a co-magnetometer with these settings. In both of these

cases the probe beam was optimized to 10mW and detuned 0.39nm lower than the

resonant wavelength.

For the situation where we operate at 240C we find that pumping with 75mW

gives a sensitivity of 0.67fT/√

Hz. This is the sensitivity averaged over the entire

cell. That is with a probe beam expanded over the entire cell.

If instead we go to a pancake geometry one can reach 7aT/√

Hz for a cell of

dimensions 6× 15× 15mm with 3atm of neon. This is a reasonable since the pyrex

122

glass we have available has wall thickness of 1mm, which corresponds to being

safely able to encase 3atm nominally. Also the cell temperature is 180C which

is near the upper temperature a pyrex cell be operated at without discoloration.

Thus it should be safe to operate at this temperature for a pyrex cell. The neon

frequency at the compensation point is 0.8Hz. This is not high enough that the co-

magnetometer zero-ing routines can be operated in a reasonable time. The pump

laser power is 40mW, and the probe beam is 10mW and detuned 0.4nm below the

D1 resonance wavelength. Typically one utilizes ultra stable single frequency DFB

laser diodes for precision experiments. These DFB diodes are normally available

with power rating up to 100mW. Thus it should be easy to obtain laser for pumping

and probing with the required power rating. The angular noise corresponding

to the atomic shot noise is 5nRad/√

Hz in this scheme. Thus it seems feasible

to create a rubidium-neon co-magnetometer with fundamental noise approaching

comparable to the state of the art atomic co-magnetometers.

We can use the experimentally measured values of the alkali-neon spin ex-

change rate to predict the feasibility of a alkali-neon co-magnetometer. One can

accurately predict the polarization of the neon gas by using eq. (2.105). Here we

substitute the measured spin exchange rate for the Rabse and calculate the T1 assum-

ing that the quadrupolar relaxation mechanism is the dominant relaxation mecha-

nism. Making these assumptions while taking into account the non-uniform alkali

polarization profile across the cell we expect a polarization of 0.55% at 140C. A

value of 0.8% which was measured using EPR.

Ultimately the quantity of interest when constructing a co-magnetometer is the

frequency of the noble gas when operating at the compensation point. This is

because the frequency of the noble gas sets the time scale for the co-magnetometer.

When processes are performed adiabatically for the co-magnetometer we really

mean that they are slow compared to the noble gas precession frequency at the

123

compensation point. Thus by increasing this frequency we are able to perform

adiabatic tasks, such as zeroing, faster. For practical purposes this is of great use

when running the co-magnetometer, and gyroscope experiments.

By predicting the noble gas polarization and utilizing the noble gas gyromag-

netic ratio we can predict the noble gas precession frequency at the compensation

point. Typically for operation of the co-magnetometer the noble gas frequency

must be set to at least 7Hz. Again utilizing eq. (2.105) we calculate the frequency

at the compensation point to be 2.30Hz at 180C, 3.3Hz at 190C, and 4.7Hz at

200C for a 3atm. At temperatures higher than 190C pyrex cells can often brown

because the alkali metal reacts with the glass in the cell wall. We can eliminate this

effect by employing aluminosilicate glass instead.

We have also investigated the influence of imperfections in the gyroscope for

a Rb-Ne co-magnetometer. The spin exchange correction factors Cese, and Cn

se from

eq. (3.4), and eq. (3.5) can be estimated. For operation with the pancake cell

Cese ≈ 8 × 10−5, and Cn

se ≈ 3 × 10−4. We predict a false signal due to misalignment

of the pump-probe orthogonality by 1µrad of ≈ 9 × 10−7rad/s. The rotational

sensitivity of the gyroscope is 1 × 10−9rad/s/√

Hz.

124

Chapter 7

Conclusions and future work

We have been able to measure many of the parameters necessary to construct an

optimized enriched neon co-magnetometer. The neon-neon quadrupolar relax-

ation rate was found to be 214 ± 10min×amagat. The neon-potassium spin ex-

change coefficient has measured and is listed tables 7.1. The neon-rubidium spin

exchange rate coefficient has been measured and is listed in Table 7.2. The Fermi

contact interaction κ0 for neon with both species is listed in table 7.3. The spin

destruction cross-sections for each alkali with neon is listed in Table 7.4.

We have demonstrated operation of an co-magnetometer gyroscope with mea-

sured sensitivity of ∆Ω ≈ 5.0 × 10−7rad/s/√

Hz. The fundamental sensitivity of

such a gyroscope using the current K-He mixture is ∆Ω ≈ 1.0 × 10−8rad/s/√

Hz

and is limited by angular noise of the probe beam. We have shown that switching

to 21Ne would increase the sensitivity of the detector to ∆Ω ≈ 1.0×10−9rad/s/√

Hz.

Spin Exchange Rate Method cm3/s

Repolarization 3.36 ± 0.42 × 10−20

Theoretical Prediction 7.5 × 10−20

Rate of Rise 2.34 ± 0.29 × 10−20 cm2

Table 7.1: Spin exchange parameter of a potassium-neon system

125

Spin Exchange Rate Method cm3/s

Repolarization 0.80 ± 0.12 × 10−19

Theoretical Prediction 1.66 × 10−19

Rate of Rise 0.82 ± 0.18 × 10−19

Table 7.2: Spin exchange parameter of a rubidium-neon system

Species Fermi Contact InteractionK-Ne 30.8 ± 2.7K-Ne (prediction) 34Rb-Ne 35.7 ± 3.7Rb-Ne (Walsworth) 32.0 ± 2.7Rb-Ne (prediction) 38

Table 7.3: Fermi contact interaction measurements for the K-Ne, and Rb-Ne sys-tems in this work, and compared to both predictions ans measurements by othergroups.

Species Group 10−23cm2

K-Ne This work 1.1 ± 0.1K-Ne Franz 1.41 ± 0.14K-Ne Walker (prediction) 1.6Rb-Ne This work 1.9 ± 0.2Rb-Ne Franz and Volk 1.9Rb-Ne Franzen 5.2Rb-Ne Walker (prediction) 1.8

Table 7.4: Spin destruction cross sections for K-Ne, and Rb-Ne as compared to boththeory and other measurements

126

Switching to 21Ne will also enable stricter limits on tests of Lorentz and CPT vio-

lation.

In future when using 21Ne to construct an improved co-magnetometer one

must further study the effects of quadrupole interactions and cell geometry which

differ from those of 3He. Since the dominant source of relaxation in a neon cell is

due to quadrupolar relaxation, and not due to long range dipolar fields one could

use a cubic geometry for the measurement cell. This will reduce effects of cell bire-

fringence and beam distortion of the lasers on the magnetometer noise. However

in this case the Bloch equations which describe the magnetometer interaction must

be modified to include a neon self interacting term due to the non-uniform mag-

netic field it would experience. Also the effect of quadrupolar splitting shifts the

Zeeman levels for the neon gas. This splitting in not uniform for each of the Zee-

man levels. To counteract this one needs to with operate with a cell which has zero

quadrupole moment. We have found that in order to generate high neon polariza-

tion one requires a uniform alkali polarization. One could also imagine creating a

thin pancake shaped cell so that the cell will be uniformly polarized.

The addition of a quadrupole moment allows one to test more parameters of

the standard model than helium. One could test Lorentz invariance by search-

ing for a quadrupole splitting in the Zeeman levels of the 21Ne which varies at a

frequency of twice a sidereal day. One could imagine constructing a vapour cell

with both 21Ne and 3He. The 21Ne could be used as a magnetometer and used

to correlate any systematic effects experienced by the 3He . This would give a

test for the local Lorentz invariance by measuring the mass anisotropy of 21Ne

(Chupp et al. (1989)). One could also measure the coherence relaxation between

the Zeeman levels of 21Ne. This has implications, and gives a test of the linearity

of quantum mechanics (Chupp & Hoare (1991)). The quadrupole moment of 21Ne

interacts with electric field gradients. Thus it could have many potential applica-

127

tions in pulmonary imaging. When the 21Ne interacts with pulmonary cell walls it

could be used as a probe of the surface dynamics. In fact another Noble gas with

a quadrupole moment 83Kr has been shown to successfully detect the difference

between hydrophilic and hydrophobic surfaces (Raftery (2006)). Since many lung

disease are dependent on the surface to volume ratio of lung tissue this could be

useful in diagnosis of pulmonary disease. As one can see the future of 21Ne optical

pumping is bright.

128

Appendix A

Properties of Ne21

129

Property Magnitude UnitsNatural Abundance 0.27 %Gyromagnetic ratio −336.10 Hz/GQuadrupole moment 0.10155 10−24cm2

Spin 3/2 –Nuclear Magnetic Moment −0.661796 µN

Diffusion Constant coefficient K in Ne 0.19

(√1+T/(273.15K)

pn/(1Amg)

)cm2/s

Diffusion Constant coefficient Rb in Ne 0.235

(√1+T/(273.15K)

pn/(1Amg)

)cm2/s

Self Diffusion Constant coefficient Ne 0.79

(√1+T/(273.15K)

pn/(1Amg)

)cm2/s

Pressure broadening 9.2 GHz/Amg

Table A.1: Properties of Ne21 relevant for optical pumping. The diffusion constantcoefficients are taken from Franz & Volk (1982)for K in Ne, Franz & Volk (1976) forRb in Ne, and Weissman (1973) for Ne in Ne

130

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