A Test of CPT and Lorentz Symmetry Using a K-3He Co...

A Test of CPT and Lorentz Symmetry

Using a K-3He Co-magnetometer

Thomas Whitmore Kornack

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

ASTROPHYSICAL SCIENCES

NOVEMBER, 2005

Abstract

A K-3He co-magnetometer has been developed for a test of Lorentz and CPT sym-

metry. Polarized K vapor forms a spin-exchange relaxation-free (SERF) magneto-

meter that has record sensitivity of about 1 fT/√

Hz. The polarized 3He effectively

suppresses sensitivity to the magnetic fields and gradients. Together, the K-3He co-

magnetometer retains sensitivity to anomalous, CPT- and Lorentz-violating fields

that couple to electron and nuclear spins differently than a normal magnetic field.

Data over the course of 15 months provide upper limits on the coupling energy

of a CPT-violating field to neutron spin, bn < 1.4 × 10−31 GeV, to proton spin,

bp < 4.4× 10−30 GeV, and to electron spin, be < 1.0× 10−28 GeV. These limits are

consistent with the existing limits of bn < 1.1× 10−31 GeV (Bear et al., 2002) and

be < 3.0× 10−29 GeV (Heckel et al., 2000). The proton sensitivity is better than the

published limit of bp < 1.8× 10−27 GeV (Phillips et al., 2001). The long-term sen-

sitivity of the co-magnetometer was significantly limited by sources of systematic

noise.

The co-magnetometer provides a robust platform for precision measurements

primarily due to its inherent insensitivity to magnetic field drift and field gradients.

Detailed analytic and numerical modeling of the coupled spin ensemble dynamics

provides good agreement with steady state and transient response measurements.

iii

Elaborate procedures have been developed for running the system optimally and

minimizing the magnetic fields and lightshifts in the system.

The co-magnetometer also forms a sensitive gyroscope that inherits all the mag-

netic insensitivity features of the co-magnetometer and adds insensitivity to mag-

netic field fast transients. The sensitivity of this gyroscope is competitive with

existing compact gyroscope techniques.

iv

Dedicated to Jill Foley

Acknowledgments

First I’d like to thank my advisor Michael Romalis for providing me with this won-

derful opportunity. Mike’s enthusiasm for this work is infectious and his commit-

ment to finding the answers to these fundamental questions is inspiring. I am also

grateful for his hands-on style, nearly constant availability, and patience with my

development. I would not have been able to finish this dissertation in such a short

amount of time without his prompt reading. I also appreciate the efforts of my care-

ful readers Ernie Valeo and Will Happer, who suggested important improvements.

I’d like to thank my dissertation committee: Will Happer, Stewart Zweben, who

tirelessly answered my questions in his diagnostics classes, and John Krommes,

who taught an excellent and challenging class on irreversible processes.

There are many people who have directly helped me in this work: Ioannis Komi-

nis was a joy to work with on the initial magnetometer sensitivity measurements.

Igor Savukov has always been available and eager to discuss the finer points of

atomic theory and greatly helped with the first major renovation of the experiment.

Rajat Ghosh built most of the additions for the gyroscope measurements, which

worked very well on the first try. Saee Paliwal built a very useful wavelength feed-

back box. Tom Jackson wrote what turned out to be an absolutely essential viewing

program for the data.

vi

Mike Souza patiently made and re-made the spherical glass cells that were of

critical importance to the experiment. Charles Sule fabricated absolutely bomb-

proof, space-ready electronics. Dan Hoffman was a constant companion and was

always was free to help when help was most needed, especially with last-minute

machining. I am indebted to Mike Peloso for maintaining a most awesome, acces-

sible and efficient student machine shop. I also deeply appreciate all the pieces

made by Bill Dix, Laszlo Varga, Glenn Atkinson, Ted Lewis and everyone else

in the machine shop. Mary DeLorenzo, Ellen Webster, Claude Champagne and

Kathy Warren provided fantastic and friendly administrative and purchasing sup-

port. This work would not have been possible without the support from NASA,

NSF, a NIST Precision Measurement grant, and The Packard Foundation.

I could not have asked for a better officemate than Micah Ledbetter, who was

both a good friend and colleague. The same is true for Scott Seltzer, who has been

very supportive and helpful while I have been writing. I benefited tremendously

from the faithful companionship of Luis Delgado-Aparicio during our long studies

for the plasma physics general exams. I also deeply appreciate the support of the

prelims study group: Juan Burwell, Jack Laiho, Wei-Li Lee and Ben North.

Everyone in plasma physics has been extremely supportive. I am grateful

for the continuous support from Nat Fisch and the program in plasma physics.

Phil Efthimion offered tireless encouragement and guidance while I was finishing.

Sam Cohen was an inspiring, excellent teacher and oversaw a wonderful first-year

project. Barbara Sarfaty has been very supportive and helpful throughout my time

here and clearly worked hard to make sure that I never had to worry about admin-

istrative details.

vii

My love of physics was developed with the physics faculty at Swarthmore Col-

lege: John Boccio, Tom Donnelly, Peter Collings, Frank Moscatelli, Amy Bug and

others held me to very high standards. In particular, working with Michael Brown

on spheromak plasmas was an incredible experience that cemented my trajectory

into graduate school.

My interests in physics started with John Peterson in 7th grade science class and

continued with excellent teaching by Kathy Sweeney-Hammond, David Walker

and Jennifer Groppe. Jennifer, in particular, ran the Engineering Team, which I

loved, and the skills that I learned would become useful later in building physics

experiments. My general interest in academia became much deeper during an

amazing history class by Leonard King.

I’d like to thank my parents Mom and Dad for being so loving and supportive

of my interests throughout the years. They tirelessly encouraged me, enabled me

and gave me the confidence to pursue my dreams.

Though I met Jill Foley as a fellow student and lab mate, she has become so

much more to me and I am filled with joy that we will wed shortly hereafter. For

her trusted advice on all things, both in physics and on other aspects of life, for

providing healthy, tasty sustenance, and especially for her limitless, loving support

throughout this work, I dedicate this thesis to her.

viii

Contents

Abstract iii

Acknowledgments vi

1 Introduction 1

1.1 Lorentz Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 The CPT Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . 10

1.4 Experimental tests of Lorentz and CPT symmetry . . . . . . . . . . . 14

1.5 High sensitivity magnetometers . . . . . . . . . . . . . . . . . . . . . . 16

1.6 Dissertation structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Co-magnetometer theory 19

2.1 Alkali Metal Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.1 Pressure broadening . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.2 Light propagation . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 The K magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.1 Spin-exchange collisions . . . . . . . . . . . . . . . . . . . . . . 29

ix

2.2.2 Spin destruction collisions . . . . . . . . . . . . . . . . . . . . . 32

2.2.3 Spin-exchange efficiency . . . . . . . . . . . . . . . . . . . . . . 34

2.2.4 Spin diffusion relaxation . . . . . . . . . . . . . . . . . . . . . . 34

2.2.5 Total K relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.6 Magnetometer bandwidth . . . . . . . . . . . . . . . . . . . . . 37

2.2.7 Fundamental magnetometer sensitivity . . . . . . . . . . . . . 38

2.3 Optical Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4 Lightshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.5 Noble gas optical pumping . . . . . . . . . . . . . . . . . . . . . . . . 53

2.6 Coupled spin ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6.1 Transient response dynamics . . . . . . . . . . . . . . . . . . . 60

2.6.2 Oscillatory response dynamics . . . . . . . . . . . . . . . . . . 63

2.7 Steady state signal dependence . . . . . . . . . . . . . . . . . . . . . . 64

2.7.1 Steady state signal dependence refinements . . . . . . . . . . 66

2.7.2 Anomalous field dependence . . . . . . . . . . . . . . . . . . . 68

2.7.3 Rotation dependence . . . . . . . . . . . . . . . . . . . . . . . . 69

2.7.4 First order experimental imperfections . . . . . . . . . . . . . 70

2.7.5 Second order experimental imperfections . . . . . . . . . . . . 71

2.7.6 Signal pumping intensity dependence . . . . . . . . . . . . . . 76

3 Co-magnetometer implementation 78

3.1 Co-magnetometer setup . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.1.1 The pump laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.1.2 The probe laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.1.3 Probe beam steering optics . . . . . . . . . . . . . . . . . . . . 88

x

3.2 Co-magnetometer characterization . . . . . . . . . . . . . . . . . . . . 92

3.2.1 The potassium magnetometer . . . . . . . . . . . . . . . . . . . 92

3.2.2 Coupled spin ensembles . . . . . . . . . . . . . . . . . . . . . . 97

3.2.3 Intensity dependence . . . . . . . . . . . . . . . . . . . . . . . 108

3.2.4 Nonlinear dynamics . . . . . . . . . . . . . . . . . . . . . . . . 109

3.2.5 Relaxation rate measurement . . . . . . . . . . . . . . . . . . . 110

3.3 Zeroing fields and lightshifts . . . . . . . . . . . . . . . . . . . . . . . 112

3.3.1 Zeroing Bz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.3.2 Calibrating the magnetometer . . . . . . . . . . . . . . . . . . 117

3.3.3 Zeroing By . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.3.4 Zeroing Bx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.3.5 Zeroing lightshifts . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.3.6 Zeroing the pump-probe nonorthogonality . . . . . . . . . . . 127

3.3.7 Zeroing sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.3.8 Anomalous field dependence . . . . . . . . . . . . . . . . . . . 132

4 Signal analysis and systematic effects 134

4.1 Signal acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.1.1 Background subtraction . . . . . . . . . . . . . . . . . . . . . . 135

4.1.2 Zeroing schedule . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.2 Systematic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.2.1 Systematic effects with implemented controls . . . . . . . . . 141

4.2.2 Unresolved systematic effects . . . . . . . . . . . . . . . . . . . 145

4.2.3 Thermal sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.2.4 Systematic noise compensation . . . . . . . . . . . . . . . . . . 158

xi

4.2.5 Systematic error estimation . . . . . . . . . . . . . . . . . . . . 163

4.3 Anomalous field measurement . . . . . . . . . . . . . . . . . . . . . . 166

4.3.1 Sidereal amplitudes . . . . . . . . . . . . . . . . . . . . . . . . 167

4.3.2 Long term data analysis . . . . . . . . . . . . . . . . . . . . . . 171

4.4 Anomalous coupling energy and conversions . . . . . . . . . . . . . . 175

5 The co-magnetometer gyroscope 183

6 Conclusions 192

A Time and orientation conventions 195

Bibliography 199

xii

List of Figures

1.1 Solar system with anomalous field . . . . . . . . . . . . . . . . . . . . 14

2.1 Alkali magnetometer principle of operation . . . . . . . . . . . . . . . 19

2.2 Potassium level diagram with spin-orbit and magnetic structure . . . 21

2.3 Potassium level diagram illustrating optical pumping . . . . . . . . . 22

2.4 Potassium vapor linewidth measurements . . . . . . . . . . . . . . . . 25

2.5 Pump beam propagation and potassium polarization . . . . . . . . . 27

2.6 Alkali-alkali spin-exchange collisions . . . . . . . . . . . . . . . . . . 30

2.7 Hyperfine sublevel distributions . . . . . . . . . . . . . . . . . . . . . 31

2.8 Spin relaxation rate optimization . . . . . . . . . . . . . . . . . . . . . 36

2.9 Potassium level diagrams for D1 and D2 transitions . . . . . . . . . . 43

2.10 Optical rotation angle of the probe beam . . . . . . . . . . . . . . . . . 44

2.11 Probe beam spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.12 Pump beam lightshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.13 Probe beam pumping rate and lightshift . . . . . . . . . . . . . . . . . 52

2.14 Illustration of a K-3He spin-exchange collision . . . . . . . . . . . . . 54

2.15 Time dependent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.16 Co-magnetometer field compensation illustration . . . . . . . . . . . 65

xiii

2.17 Signal response to Bx, By and Bz scans . . . . . . . . . . . . . . . . . . 72

2.18 Signal dependence on lightshifts . . . . . . . . . . . . . . . . . . . . . 74

2.19 Intensity dependence of various parameters . . . . . . . . . . . . . . . 76

3.1 Experimental setup schematic . . . . . . . . . . . . . . . . . . . . . . . 80

3.2 Pictures of the cell, oven and cooling shield . . . . . . . . . . . . . . . 81

3.3 Pictures of the magnetic and thermal shields . . . . . . . . . . . . . . 82

3.4 Cell optics beam deviation . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.5 Cell optics raytracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.6 Probe beam lens configuration . . . . . . . . . . . . . . . . . . . . . . 89

3.7 Transverse resonance linewidth at 30 Hz synchronous pumping . . . 92

3.8 Transverse resonance linewidths for a range of frequencies . . . . . . 93

3.9 Potassium magnetometer noise . . . . . . . . . . . . . . . . . . . . . . 94

3.10 Magnetic field imaging and source location . . . . . . . . . . . . . . . 96

3.11 Potassium magnetometer bandwidth . . . . . . . . . . . . . . . . . . . 97

3.12 Uncoupled 3He T2 measurement . . . . . . . . . . . . . . . . . . . . . 98

3.13 Potassium polarization saturation curve . . . . . . . . . . . . . . . . . 99

3.14 Coupled spin ensembles transient response dynamics . . . . . . . . . 100

3.15 Coupled spin ensembles transient response dynamics, expanded . . 101

3.16 Coupled spin ensembles frequency and decay rate . . . . . . . . . . . 103

3.17 Co-magnetometer frequency response . . . . . . . . . . . . . . . . . . 104

3.18 Suppression of applied magnetic fields . . . . . . . . . . . . . . . . . . 105

3.19 Suppression of an external applied magnetic field . . . . . . . . . . . 106

3.20 Suppression of applied magnetic field gradients . . . . . . . . . . . . 106

3.21 Suppression of magnetic noise . . . . . . . . . . . . . . . . . . . . . . 107

xiv

3.22 Pump intensity profile with propagation model . . . . . . . . . . . . 108

3.23 Nonlinear response to a large tipping angle pulse . . . . . . . . . . . 109

3.24 Nonlinear spontaneous spin response to field reversal . . . . . . . . . 110

3.25 Determination of the co-magnetometer K spin destruction rate . . . . 111

3.26 Zeroing procedure example showing modulated raw signal . . . . . 113

3.27 Zeroing procedure example modulation response curve . . . . . . . . 114

3.28 Zeroing procedure for Bz . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.29 Single transient response simulation . . . . . . . . . . . . . . . . . . . 116

3.30 Zeroing procedure for By . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.31 Zeroing procedure for Bx . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.32 Signal response curves for zeroing-relevant modulations . . . . . . . 121

3.33 Signal response to Bz scans with no pump beam . . . . . . . . . . . . 125

3.34 Zeroing procedure for Lx . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.35 Zeroing procedure for pump-probe non-orthogonality . . . . . . . . 128

3.36 Magnetometer and misalignment sensitivities . . . . . . . . . . . . . 130

4.1 Raw data, second timescale . . . . . . . . . . . . . . . . . . . . . . . . 137

4.2 Raw data with zeroing gaps, minute timescale . . . . . . . . . . . . . 138

4.3 Raw data with zeroing gaps, hour timescale . . . . . . . . . . . . . . . 140

4.4 Cell wall dichroism and sweet spot illustration . . . . . . . . . . . . . 142

4.5 Vertical lightshift generated by back-reflection . . . . . . . . . . . . . 146

4.6 Relative sensitivity gradient through cell . . . . . . . . . . . . . . . . 148

4.7 Signal-temperature correlation example . . . . . . . . . . . . . . . . . 151

4.8 Thermal disequilibriation examples . . . . . . . . . . . . . . . . . . . 153

4.9 Signal-temperature correlation improvement . . . . . . . . . . . . . . 155

xv

4.10 Thermal expansion of table flap . . . . . . . . . . . . . . . . . . . . . . 156

4.11 Signal-pump position correlation example . . . . . . . . . . . . . . . . 159

4.12 Signal-probe position correlation example . . . . . . . . . . . . . . . . 160

4.13 Signal-pressure correlation and jump removal . . . . . . . . . . . . . 161

4.14 Smooth quadratic drift example . . . . . . . . . . . . . . . . . . . . . . 162

4.15 Measured systematic errors . . . . . . . . . . . . . . . . . . . . . . . . 164

4.16 Sidereal fit to a single data run . . . . . . . . . . . . . . . . . . . . . . 167

4.17 Long term frequency spectrum sample . . . . . . . . . . . . . . . . . . 169

4.18 Raw data over one year . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.19 Summary of data runs . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

4.20 Summary of days . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4.21 RMS cutoff dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4.22 Histogram of day sidereal amplitudes . . . . . . . . . . . . . . . . . . 177

4.23 Error analysis integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

5.1 Gyroscope experiment setup . . . . . . . . . . . . . . . . . . . . . . . . 184

5.2 Gyroscope experiment setup front view . . . . . . . . . . . . . . . . . 184

5.3 Gyroscope raw rotation signal . . . . . . . . . . . . . . . . . . . . . . . 185

5.4 Gyroscope noise spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.5 Gyroscope angle random walk . . . . . . . . . . . . . . . . . . . . . . 187

5.6 Co-magnetometer transient insensitivity . . . . . . . . . . . . . . . . . 188

A.1 Celestial coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

xvi

List of Tables

1.1 CPT transformation of electromagnetic fields . . . . . . . . . . . . . . 7

1.2 CPT transformation of Dirac fields . . . . . . . . . . . . . . . . . . . . 8

1.3 Existing limits on CPT violation parameters . . . . . . . . . . . . . . . 15

2.1 Alkali-metal spin-destruction cross sections . . . . . . . . . . . . . . . 33

2.2 Summary of timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.3 Typical experimental parameters . . . . . . . . . . . . . . . . . . . . . 67

4.1 Spin polarization reversal times . . . . . . . . . . . . . . . . . . . . . . 144

4.2 Survey of thermal sensitivity around the pump beam . . . . . . . . . 154

4.3 Survey of other thermal sensitivities . . . . . . . . . . . . . . . . . . . 158

4.4 Measured systematic errors . . . . . . . . . . . . . . . . . . . . . . . . 165

5.1 Gyroscope performance comparison . . . . . . . . . . . . . . . . . . . 190

xvii

Chapter 1

Introduction

GENERAL RELATIVITY AND QUANTUM THEORY have successfully led phys-

ics through most of the last century. General Relativity describes the behav-

ior of gravitation, space and time while Quantum Theory covers all of the other,

much stronger interactions in Nature. The success of this dichotomous system re-

lies on the vast differences in scale between the two theories; no experiment or

observation has ever probed Quantum Gravity, where both theories are simulta-

neously significant. The quest for a unification of Quantum Theory and General

Relativity in a theory of Quantum Gravity is primarily motivated by the notion

that Nature is a single entity.

For many decades, the experimental search for a signature of Quantum Grav-

ity was dismissed by the simplistic argument that gravitational effects would ap-

pear only at the experimentally unattainable Planck scale ∼ 1019 GeV. However,

in recent years, efforts to determine the measurable consequences of Quantum

1

2

Gravity have considerably intensified. In the mid-1980s two moderately success-

ful theories of Quantum Gravity were proposed: String Theory (Green et al., 1987)

and Loop Quantum Gravity (Rovelli, 1997). Not long afterward, Kostelecky and

Samuel (1989) showed that String Theory can lead to a spontaneous breakdown of

Lorentz and CPT symmetries in the Standard Model. CPT symmetry is closely re-

lated to Lorentz symmetry; it is a discrete symmetry of physical systems in which

a system is invariant upon charge conjugation (C), parity inversion (P), and time

reversal (T). Lorentz violation is not entirely unexpected, however, since almost

every other symmetry of Nature including charge, parity, time-reversal, CP sym-

metry, chiral symmetry, Supersymmetry, etc. has been found to be spontaneously

broken.

Most theories of Quantum Gravity have now been shown to violate Lorentz

symmetry in some way. Both Loop Quantum Gravity (Gambini and Pullin, 1999)

and String Theory (Ellis et al., 2001) have generally been shown to produce a mod-

ified dispersion relation for particle propagation that necessarily leads to Lorentz

symmetry breaking. The modifications stem from a discrete description of space-

time and typically cause photons with different energies to travel with different

velocities, v(E) 6= c, introducing small delays that would become detectable on

cosmological scales (Sudarsky et al., 2002). Such a modified dispersion relation

must be evaluated in a single, preferred reference frame, which clearly violates

Lorentz symmetry.

Conveniently, there is a preferred reference frame available: the frame at rest

with respect to the Cosmic Microwave Background. Other cosmological observa-

tions have also spurred interest in the possible violation of Lorentz symmetry: the

3

WMAP satellite and other cosmological observations have recently confirmed that

a majority of the energy in the universe consists of dark energy. Dark energy can

be associated with either a cosmological constant or a fifth “infrared” degree of

freedom that gives rise to a scalar field in a preferred frame, sometimes known as

quintessence (Bolokhov et al., 2005). This very low energy scalar field background

not only affects the expansion of the universe but may also interact with particles

in such a way that Lorentz-violating effects would appear.

It remains an open and compelling question whether an unambiguous sign of

Lorentz violation can be measured using present experimental techniques. The

experiment described in this dissertation searches for CPT- and Lorentz-violating

effects that are clearly forbidden by the Standard Model but may be allowed by a

more fundamental theory. In particular, this experiment compares the spin preces-

sion of electron and nuclear spins while the earth rotates in order to search for an

anomalous coupling along a particular preferred direction in spacetime. In search-

ing for a sort of New Aether wind, this experiment is a modern analogue of the

original Michelson-Morley experiment.

In general, there are many possible ways to search for Lorentz violation, in-

cluding (1) the violation of rotational symmetry (Aether), (2) the violation of boost

invariance, (3) anomalous dispersion in the vacuum, (4) violation of discrete sym-

metries such as CPT symmetry, and (5) the occurrence of forbidden processes

(Vucetich, 2005). There are currently two successful methods of experimentally

1.1. Lorentz Symmetry 4

testing Lorentz symmetry with sufficient sensitivity: astrophysical or cosmologi-

cal observations and “clock-comparison” experiments similar to the present exper-

iment. A null result from this experiment applies only to a narrow subset of pos-

sible Lorentz violation mechanisms; only in conjunction with many other types of

experiments and observations can the search for Lorentz violation be considered

thorough.

1.1 Lorentz Symmetry

Lorentz symmetry states that the physics of any system is symmetric through ro-

tations and boosts of the laboratory. The Lorentz group is formed by the set of

matrices that conserve the following spacetime quantity:

Invariant: s2 = c2t2 − x2i = x2

0 − x2i = xµgµνxν (1.1)

If all the terms in a particular theory are chosen from the Lorentz group, then the

theory is automatically Lorentz invariant. For such a Lorentz-invariant physical

system, transformations of both the observer and the system itself do not change

the physics of the system. An observer Lorentz transformation can be expressed

using the matrix Λµν for rotations and boosts and the vector aµ for translations:

x′µ = Λµνxν + aµ (1.2)

where the coordinate xν transforms into x′µ. A particle Lorentz transformation is

expressed using

U(Λ, a)ψ(x)U−1(Λ, a) = ψ(Λx + a) (1.3)

1.2. The CPT Theorem 5

where U is a transformation matrix containing both rotations, boosts and transla-

tions. These two types of transformation are equivalent under Lorentz invariance.

They are, however, distinct under any form of Lorentz violation. In a system with

Lorentz violation, observer transformations preserve Lorentz symmetry while par-

ticle transformations violate Lorentz symmetry. Under a particular scheme of

Lorentz violation, particles that are moving at different velocities would experi-

ence different physics. Moreover, in any given scheme of Lorentz violation, there

is a particular preferred frame in which the equations of motion are simplified.

Thus, the presence of a preferred frame leads to experimentally measurable conse-

quences in the local laboratory frame.

1.2 The CPT Theorem

CPT symmetry depends critically on Lorentz symmetry and is an exact symmetry

of the Standard Model. Under CPT symmetry, a physical system is invariant when

the charges are flipped (C), the spatial axes are reflected (P, parity) and time is

reversed (T):

C : e → −e

P : x → −x (1.4)

T : t → −t

The total transformation is defined as Θ ≡ CPT . This section contains a brief,

utilitarian tour of CPT transformations and a sketch of a proof of the CPT theo-

rem. The material in this section draws upon more detailed presentations by Kaku

(1993) and Colladay (1998).


It is instructive to consider the case of Maxwell’s equations to determine the

CPT transformation of the vector potential Aµ = (A0, A) and the source term jµ =

(ρ, j). Maxwell’s equations can be written as

∂µFµν = jν (1.5)

where

Fµν = ∂µ Aν − ∂ν Aµ =

0 E1 E2 E3

−E1 0 B3 −B2

−E2 −B3 0 B1

−E3 B2 −B1 0

(1.6)

Charge conjugation transforms a positive current into a negative one:

C jµC−1 = −jµ (1.7)

Under a parity transformation, the direction of current flow reverses (j → −j) but

the charge density stays the same:

P jµP−1 = jµ (1.8)

where the notation means jµ = (ρ,−j). Time reversal has the same effect:

T jµT −1 = jµ (1.9)

The derivative ∂µ transforms in a similarly straightforward manner:

C : ∂µ → ∂µ (1.10)

P : ∂µ → ∂µ (1.11)

T : ∂µ → −∂µ (1.12)


∂µ jµ Aµ

C ∂µ −jµ −Aµ

P ∂µ jµ Aµ

T −∂µ jµ Aµ

Θ −∂µ −jµ −Aµ

Table 1.1: CPT transformation of electromagnetic fields.

The transformation of the vector potential Aµ is derived from the transformation

of the source terms and Maxwell’s equations (Equation 1.5):

C : Aµ → −Aµ (1.13)

P : Aµ → Aµ (1.14)

T : Aµ → Aµ (1.15)

A summary of these transformations can be found in Table 1.1. Each term in

Maxwell’s Equation 1.5 gains a negative sign under a CPT transformation. Since

there are an even number of these terms appearing in Maxwell’s equation, it is

invariant and symmetric under CPT transformations.

A similar but significantly more involved derivation can be used to obtain the

transformation properties of a fermion ψ interacting with various fields. Fermions

of mass m obey the Dirac equation; a Lagrangian form is

L = ψ[−m + iγµ(∂µ + ieAµ)]ψ− 14

FµνFµν (1.16)

where ψ ≡ ψ†γ0 and the spin is defined using the notation,

ψ =

ψR

ψL

where ψL ≡1− γ5

2ψ and ψR ≡

1 + γ5

2ψ (1.17)


Scalar Pseudoscalar Vector Pseudovector Tensorψψ ψγ5ψ ψγµψ ψγµγ5ψ ψσµνψ

S P Vµ Bµ Tµν

C S P −Vµ −Bµ −Tµν

P S −P Vµ Bµ Tµν

T S −P Vµ Bµ −Tµν

Θ S P −Vµ −Bµ Tµν

Table 1.2: CPT transformation of Dirac fields.

For left-handed spin-1/2 fermions, the CPT operators transform according to (see,

for example, Kaku (1993))

CψL(x, t)C−1 = iηCγ2γ0ψ†R(x, t) (1.18)

PψL(x, t)P−1 = ηPγ0ψR(−x, t) (1.19)

T ψL(x, t)T −1 = iηTγ1γ3ψL(x,−t) (1.20)

ΘψL(x, t)Θ−1 = −iηCPTγ5γ0ψ†L(−x,−t) (1.21)

where the identity γ0γ1γ2γ3 = −iγ5 was used. Phase factors ηC, ηP, ηT, and

ηCPT arise from these transformations. A summary of the CPT transformation of

common Dirac bilinears can be found in Table 1.2. One can verify that using these

constructions, the Lagrangian in Equation 1.16 is indeed CPT invariant.

In general, any given Dirac bilinear can be represented by

O = ψΓψ where Γ = I, γµ, σµν, γ5 (1.22)

and Γ forms a complete basis set. Testing O for CPT symmetry can be generalized

according to

ΘOΘ−1 = (−1)kη1η2O†(−x) (1.23)


where k denotes the number of Lorentz indices (µ, ν) that appear in O. These sign

factors arise from the anticommutation of fermionic spinors. Phase factors η1,2

correspond to the CPT phase of fermion fields ψ1,2. Since the CPT operator Θ has

no eigenstates, the CPT phases are unobservable and can be set to unity (Colladay,

1998).

The treatment of bosons is somewhat more straightforward since the boson

fields commute; a bosonic field transforms by

ΘBµ(x)Θ−1 = (−1)kBµ(−x) (1.24)

Each of k gauge bosons in a Lagrangian term contributes a factor of −1 to the CPT-

transformed term. As with the fermions, this factor is equivalent to the number of

Lorentz indices appearing in each term.

The transformation of the total Lagrangian can thus be summarized according

to

ΘL(x)Θ−1 = (−1)kL†(−x) (1.25)

Now, two essential features of the Lagrangian are utilized: First, imposing Lorentz

covariance of the Lagrangian requires that even rank tensors transform into their

Hermitian conjugate while odd-rank tensors transform into the negative of their

Hermitian conjugate. In other words, all of the Lorentz indices must appear in

pairs and for a total of k Lorentz indices, (−1)k = 1. Second, the Lagrangian is

assumed to be hermitian; thus L†(x) = L(x). Under these conditions,

ΘL(x)Θ−1 = L(−x) (1.26)

As long as the Lagrangian is translationally invariant, the CPT-transformed L(−x)

is equivalent to the physics of L(x).

1.3. Spontaneous symmetry breaking 10

To complete the proof of CPT symmetry one must show that the action in the

Hamiltonian is also invariant. This proof is found in many texts such as Kaku

(1993). Invariance of the action requires the imposition of the usual spin-statistics

connection for bosons and fermions. That, together with a Hermitian, Lorentz-

invariant Lagrangian forms the core of the CPT theorem.

1.3 Spontaneous symmetry breaking

Theories of Quantum Gravity often lead to a spontaneous breakdown of Lorentz

and CPT symmetries. New fields can be added to the Standard Model Lagrangian

to account for the symmetry breaking. The particular type of fields that appear

depend strongly on the form of the fundamental theory.

The theoretical framework presented by Alan Kostelecky and colleagues pa-

rameterizes CPT and Lorentz violation effects in a Standard Model Extension. The

parameters are chosen to include all 4-dimensional terms that break Lorentz in-

variance for particles in the Standard Model. The following terms appear in the

Lagrangian for Dirac particles such as electrons (Colladay and Kostelecky, 1998):

L = −ψMψ +12

iψΓµ∂µψ (1.27)

where Γµ and M are given by

M = m + aµγµ + bµγ5γµ − 12

Hµνσµν (1.28)

Γµ = γµ + cµνγν + dµνγ5γν + eµ + i fµγ5 +12

igλνµσλν (1.29)

These alphabetical vector and tensor terms are constants that define the orienta-

tion in spacetime of the spontaneous symmetry breaking. The terms aµ, bµ, eµ, fµ,


and gλνµ are CPT-odd (they violate CPT symmetry) whereas the rest are CPT-even.

This can be readily verified using the results in Table 1.2. This parameterization of

Lorentz violation terms in a Standard Model Extension provides a standard frame-

work necessary to directly compare theoretical and experimental results. There

are other frameworks such as the one proposed by Myers and Pospelov (2004) that

accommodate up to dimension-5 Lorentz violation terms in the Lagrangian. The

latter becomes particularly useful for parameterizing certain modified dispersion

relations (Nibbelink and Pospelov, 2005).

The present experiment is sensitive to fields that couple to spins. For non-

relativistic electrons, the experiment couples to the following Lorentz violation

terms:

V = −bµψγ5γµψ = −bei σi

e = −be · S (1.30)

where an anomalous coupling to the electron be is given by the following combina-

tion of terms:

bei = be

i −medei0 − εijkHe

jk/2 (1.31)

Thus, Lorentz and CPT violation terms give rise to a vector field b that couples

to spins in the same way as a magnetic field. This anomalous field b points in a

fixed direction in spacetime. By carefully measuring the torque on spins as they

are rotated relative to a fixed axis, one is able to measure the Lorentz and CPT

violation terms in Equation 1.31.

The spontaneous symmetry breaking that is parameterized by the Standard

Model extension can be constrained by considering the attributes of specific fun-

damental theories. Although all of the Lorentz violation parameters aµ, bµ, cµν,

dµν, eν, fν, gλµν ... could conceivably have nonzero values, particular fundamental


theories place constraints on which of these are likely to appear. By measuring a

sufficiently wide selection of these parameters, it is possible to constrain or even

choose among possible formulations of quantum gravity.

The following two examples of specific predictions from fundamental theories

are provided to illustrate the utility of the Kostelecky parameterization. The par-

ticular details of the theories are not of great concern here; rather, the results are

presented to give a flavor of how this Standard Model Extension is useful for com-

parison.

Vucetich (2005) provides a set of predictions for parameters in the Kostelecky

framework for a Loop Quantum Gravity system with a preferred frame. For a

laboratory experiment moving with a velocity Wµ with respect to the preferred

frame in a weave state with a small structure scale `P L λ such that spacetime

appears continuous on the macroscopic scale λ, the parameters in the Kostelecky

Lagrangian take the following form:

aµ = Hµν = dµν = eµ = fµ = 0 (1.32)

bµ =12

O4M2`PWµ (1.33)

cµν = O1M`P(gµν −WµWν) (1.34)

gαβγ = O2M`PεµαβγWµ (1.35)

where Oi are factors of order unity and M is the Planck mass.

Bolokhov et al. (2005) provides a specific prediction for a Lorentz-violating Su-

persymmetric QED system in terms of an explicit parity-breaking electron-posi-

tron mass difference ∆m2 = (m+s )2 − (m−

s )2 and real-space symmetry-breaking


unit vectors NµA,V :

aµ = − 1M

m2εNµ

V +α log(M/ms)

πM

(m2

s NµV +

∆m2

2Nµ

A −32

∆m2

2Nµ

)(1.36)

bµ =α log(M/ms)

πM

(m2

s NµA +

∆m2

2Nµ

V −32

m2s Nµ

)(1.37)

cµ =1M

(12

NµA − Nµ

), dµ =

1M

NµV

2, f µνρ =

2M

Tµνρ (1.38)

where α is the electromagnetic coupling constant, M is the Planck mass, and the

soft breaking mass can be chosen ms ≈ 1 TeV.

A scalar quintessence field φ that could account for the dark energy in the uni-

verse gives rise to the following vector coupling:

bµ =∇φ

Fa(1.39)

where Fa is a coupling constant (Pospelov and Romalis, 2004).

Regardless of the actual values of the Lorentz and CPT violation parameters,

these models offer divergent predictions for the form of the symmetry breaking.

In some theories, for example, aµ is present whereas in others it is absent. An

underlying theory’s particular prediction for these parameters forms a sort of fin-

gerprint for the underlying theory. By placing limits on Lorentz and CPT violation

parameters, these and other theories can be constrained. The measurement of a

non-zero value for one of these terms would allow one to begin to accurately dis-

cern between fundamental theories.

It is noteworthy that this experiment is dominantly sensitive to bµ, a term that

appears in all the theoretical predictions above and great majority of other funda-

mental theories. Thus, even though this experiment is dominantly sensitive to that

1.4. Experimental tests of Lorentz and CPT symmetry 14

Figure 1.1: This experiment rotates through a background anomalous field bµ as it movesaround the solar system.

one term, it is apparently well suited to detect Lorentz and CPT violation irrespec-

tive of the underlying theory.

1.4 Experimental tests of Lorentz and CPT symmetry

Experimental searches for Lorentz and CPT violation focus on the coupling of var-

ious Lorentz and CPT violation parameters to particular particles. Table 1.3 shows

the limits on a few of these parameters set by existing experiments. The terms that

couple to spins such as bµ have set the most stringent limits on CPT and Lorentz vi-

olation. An essential assumption in these experiments is that the anomalous field

bµ does not necessarily couple to spins proportional to their magnetic moments.

If bµ were to couple exactly according to the magnetic moments, then it would

be indistinguishable from a magnetic field. This difference in coupling allows a

co-magnetometer that uses two different spin species to distinguish between a nor-

mal magnetic field and an anomalous field coupling. The present experiment is

designed to be sensitive to the difference in coupling of bµ to electron and nuclear

1.4. Experimental tests of Lorentz and CPT symmetry 15

Experiment aµ bpµ bn

µ beµ cn

µν CitationK0-K0 10−20 Kostelecky (1999)Electron g− 2 10−24 Gabrielse et al. (1999)p-p 10−26 Bluhm et al. (1998)Cs-199Hg 10−27 10−30 10−27 Berglund et al. (1995)H Maser 10−27 10−27 Phillips et al. (2001)Polarized Solid 10−29 Heckel et al. (2000)3He-129Xe Maser 10−31 Bear et al. (2002)K-3He 10−30 10−31 10−28 (This work)

Table 1.3: Existing limits in GeV on CPT violation parameters are provided by a widerange of tabletop and high energy experiments. The last line indicates the performance ofthe present experiment.

(mostly neutron) spins, S ∼ be − bn. The design is insensitive to a normal magnetic

field which would appear as B = beµ = bn

µ.

These anomalous vector fields point in a certain direction in spacetime that is

constant on the scale of our solar system. Many experiments measure the ampli-

tude of a sidereal signal as the earth rotates and moves around the Sun through

such a constant background anomalous field as depicted in Figure 1.1. The detec-

tion of a sidereal signal, clearly distinct from a diurnal signal, is considered an

unambiguous indication of an anomalous field.

Heckel et al. (2000) use a torsional pendulum made of spin-polarized solids

to place a stringent upper limit on the electron coupling to the anomalous field of

be < 3.0× 10−29 GeV. The key to its high sensitivity is the use of two different types

of permanent magnet: one magnet that derives its magnetic moment entirely from

electron spin polarization and another that has a component of magnetization from

orbital angular momentum. These magnets were arranged in such a way that the

total magnetic moment of the pendulum was cancelled, rendering the experiment

1.5. High sensitivity magnetometers 16

largely insensitive to magnetic fields. The pendulum retains a total electron spin

moment which couples to the anomalous field bµ.

Bear et al. (2002) use a 3He-129Xe maser to place a stringent upper limit on the

neutron coupling to the anomalous field of bn < 1.1× 10−32 GeV. They use a co-

magnetometer setup that compares the precession of the 3He and 129Xe precession

frequencies to reject magnetic field sensitivity and retain anomalous field sensitiv-

ity.

The experiment described in this dissertation forms a co-magnetometer using

the electron spins of potassium (K) and the nuclear spins of Helium-3 (3He). This K-

3He co-magnetometer is sensitive to both nuclear and electron spin coupling and

has sufficient short-term sensitivity to set a new limit. The co-magnetometer can

be operated so that it is completely insensitive to magnetic fields and only retains

sensitivity to anomalous fields. However, as this dissertation will show at length,

the long-term sensitivity of this experiment is suppressed by systematic noise.

1.5 High sensitivity magnetometers

Advances in sensitive magnetometry have enabled the high projected sensitivity

of the present experiment. The co-magnetometer in this experiment is constructed

using a spin-exchange relaxation-free (SERF) magnetometer at its core with polar-

ized 3He buffer gas. The SERF magnetometer is a new type of ultra-sensitive alkali-

metal magnetometer that was developed for use in this experiment. Traditional

alkali metal magnetometers are fundamentally limited by spin-exchange collisions

between alkali atoms. Efforts to suppress spin-exchange led to designs with large

1.6. Dissertation structure 17

cells and very low densities of alkali metal (Alexandrov et al., 1996). Budker et al.

(2000) used evacuated cells with antirelaxation coatings and a nonlinear Faraday ro-

tation technique to obtain a projected shot-noise limited sensitivity of 0.3 fT/√

Hz.

These techniques were not very competitive with SQUID magnetometers and did

not find widespread use. The best low-frequency SQUID magnetometers achieve

1 fT/√

Hz and are fundamentally limited by Johnson noise in the shunt resistors

(Greenberg, 1998).

By operating an alkali-metal magnetometer at high density and in low mag-

netic field, the spin-exchange relaxation mechanism shuts down and the more

seldom spin-destruction collisions become the dominant relaxation rate. In this

regime, the measured sensitivity of a K vapor in a few atm 4He buffer gas is bet-

ter than 1 fT/√

Hz with a shot-noise limit of 20 aT/√

Hz. The fundamental shot

noise limit of a fully optimized spin-exchange relaxation-free (SERF) magnetome-

ter is 2 aT/√

Hz, almost three orders of magnitude better than the best SQUID. The

high sensitivity of this magnetometer is the basis of the co-magnetometer’s high

sensitivity to anomalous field coupling.

1.6 Dissertation structure

The first half of Chapter 2 contains a discussion of the essential components of a

SERF magnetometer. The second half of Chapter 2 discusses the behavior of an

alkali-metal-noble-gas co-magnetometer. The signal response is derived for tran-

sients, oscillations and in steady state. These analyses indicate how to operate the

co-magnetometer optimally while running the experiment.

1.6. Dissertation structure 18

Chapter 3 starts with the experimental verification of basic properties of the

SERF magnetometer. Since the co-magnetometer is insensitive to magnetic fields

in steady state, an elaborate “zeroing” mechanism has been developed to maintain

the magnetic field and other parameters near their optimal values. The zeroing

procedures presented in Section 3.3 exploit the second order sensitivity of the co-

magnetometer to various parameters of the system.

Chapter 4 contains all the technical details about how the data is gathered and

processed in such a way that the experiment is least sensitive to systematic errors

and drift. Many sources of systematic noise and ways to suppress each one are

presented. After all possible sources of systematic noise have been removed from

the data, Section 4.3 presents the methods used to search for the sidereal signature

of anomalous fields in the long-term data sets. The anomalous field measurements

derived from long-term data sets are given in Section 4.3.

The very high rotation sensitivity of the co-magnetometer was discovered dur-

ing its development. In Chapter 5 the measurements of the gyroscopic properties

of the co-magnetometer are presented. The performance of the gyroscope is com-

petitive with existing techniques such as fiber optic interferometer-based devices.

Future development of the co-magnetometer shows great promise for a very high

sensitivity, compact gyroscope.

Chapter 2

Co-magnetometer theory

THE K-3HE CO-MAGNETOMETER can be described in two parts: first, as a sensi-

tive, spin-exchange relaxation-free K magnetometer and second as a system

of interacting K and 3He spin ensembles. The alkali metal magnetometer operates

by accurately measuring the spin precession of spin-polarized alkali metal atoms

as illustrated in Figure 2.1. This chapter begins with a review of how alkali metal

spins are polarized by optical pumping using circularly polarized resonant light

(Section 2.1). The rate at which the spins are depolarized determines the funda-

mental limit on the sensitivity of the magnetometer (Section 2.2). At sufficiently

By

S

Probe Beam

Pump Beam

Figure 2.1: Principle of alkali magnetometer operation: a pump beam polarizes alkalimetal electron spins S and their precession in By is measured by a probe beam.

19

2.1. Alkali Metal Optical Pumping 20

high density and low ambient magnetic field, spin-exchange collisions do not de-

polarize the spins and the K vapor forms a very sensitive spin-exchange relaxation-

free (SERF) magnetometer. The orientation of the spins is measured using Faraday

rotation of a linearly polarized probe beam (Section 2.3). With lasers propagating

through the atomic vapor, light-induced shifts of the energy levels (lightshifts) can

arise that are indistinguishable from a magnetic field splitting whenever the lasers

are detuned from resonance and have some circular polarization (Section 2.4).

In the second half of the chapter, 3He is introduced to create a co-magnetometer

that is sensitive to anomalous fields yet is insensitive to magnetic fields. Spin-

exchange collisions between K and 3He atoms polarize the 3He spins (Section

2.5). The polarized K and 3He spin ensembles have coupled, resonant interac-

tions under certain conditions (Section 2.6). The same conditions render the co-

magnetometer insensitive to magnetic fields in steady state (Section 2.7). Small cor-

rections to the steady state signal dependence are calculated in order to understand

the degree to which the co-magnetometer can be rendered insensitive to magnetic

fields.

2.1 Alkali Metal Optical Pumping

Alkali metals are natural choices for magnetometry because the single outer shell

electron has an unpaired spin that is easy to utilize. Using an optical pumping

technique with circularly polarized light, the atomic spins can be polarized. The

first excited state of potassium, 4p, is split by the spin-orbit interaction into the

2P1/2 and 2P3/2 states. (The superscripted number denotes the 2S + 1 multiplicity


4s

Spin-OrbitOrbital MagneticIK = 3/2

4p2P1/2

2P3/2

2S1/2 –1/2

+1/2

–1/2

+1/2

–3/2–1/2+1/2

+3/2

mj

1

2

1

2

0123

F

D1, 770.1 nm

D2, 766.8 nm

Figure 2.2: Level splitting diagram for K showing the spin-orbit interaction splitting andthe magnetic interaction splitting of the 4s and 4p orbitals. D1 and D2 transitions corre-spond to excitation of 2P1/2 and 2P3/2 orbital angular momentum states. Not to scale.

of the state and the subscripted number denotes the j = l + s total electron angular

momentum.) D1 and D2 light denote the wavelengths necessary to excite to the

2P1/2 and 2P3/2 states, respectively. A good review of optical pumping is provided

by Walker and Happer (1997).

In pumping the D1 transition with circularly polarized light, the level diagram

can be redrawn according to Figure 2.3, with two ground state spin sublevels of

2S1/2 and two excited state sublevels of 2P1/2. The two sublevels in each state corre-

spond to the orientation of the electron spin. Starting with an unpolarized sample

of electrons in the ground state, distributed evenly between the ms = −1/2 and

ms = +1/2 sublevels of 2S1/2, a circularly polarized σ+ pumping laser will ulti-

mately place all the electrons in the ms = +1/2 sublevel: (1) The σ+ light promotes

electrons from ms = −1/2 to mj = +1/2 in the excited state. The electrons in the


4s 2S1/2

ms = –1/2 ms = +1/2

4p 2P1/2

Collisional Mixing

Spin relaxation

s + pumping la

ser

N2 Q

uenc

hing

N2 Q

uenc

hing

ee e e ee e eee e ee ee ee

eee e

Figure 2.3: Ground and first excited states of potassium. Circularly polarized pump beamphotons excite electrons with ms = −1/2 into the 4p, mj = +1/2 state. Collisions in theK vapor cause rapid mixing among the excited states and collisions with N2 de-excite theelectron. Spin relaxation processes cause spins to become disoriented and can repopulatems = −1/2. This diagram does not include the effects of nuclear spin.

ms = +1/2 state are not excited because the 2P1/2 excited state cannot accommo-

date more angular momentum. (2) In the excited state, collisions between K atoms

and noble-gas atoms cause very rapid mixing between mj = +1/2 and mj = −1/2.

(3) Sufficient N2 is added to quench the excited electrons, removing the orbital an-

gular angular momentum by exciting N2 rotational states without disturbing the

spin orientation (Happer, 1972). Without the N2 quenching, the excited electrons

radiate nearly unpolarized light that is reabsorbed by adjacent K atoms, thereby

depumping the atoms. Of the electrons that are excited only half are deposited in

the ms = +1/2 state. Since the σ+ light only depopulates the ms = −1/2 state,

over time that state would become completely depopulated. This process can be

seen as transferring the angular momentum of photons to the angular momentum


of electrons, thereby polarizing the electrons along the direction of light propaga-

tion.

The alkali nuclear spin is polarized because it is strongly coupled to the polar-

ized electron spin in the ground state. Since the total angular momentum F =

I + S + L is shared between electron and nuclear spins, continuous optical pump-

ing and spin-exchange collisions cause the atoms to become fully polarized in the

mF = 2 sublevel of the 2S1/2 (F = 2) ground state (in the absence of any relaxation).

In an ensemble of electrons, the polarization can be defined as the ensemble-

averaged expectation value of the orientation of the electronic spin, Pez ≡ 2〈Sz〉. For

potassium, fully polarized atoms are denoted by Pez = 〈Fz〉/2. Zero polarization

means that the spins in the ensemble are randomly oriented. Fully polarized atoms

along z are denoted by Pez = 1.

2.1.1 Pressure broadening

Noble gasses are commonly used as buffers to reduce diffusion to the walls where

spin coherence is rapidly lost. The presence of this gas broadens the spectral lines

and reduces the overall absorption of resonant light. In most cases the broadened

line shape is given by a Lorentzian:

L(ν) =Γ/2

(ν− ν0)2 + (Γ/2)2 (2.1)

where ν− ν0 is the detuning of light off resonance and Γ is the width (full width

half maximum) of the broadening. The cross section for absorption obeys the fol-

lowing sum rule (see, for example, Corney (1977)):∫σdν = πcre f (2.2)


where re is the radius of the electron and f is the oscillator strength. The resulting

absorption cross section is

σ(ν) = cre fL(ν) (2.3)

which on resonance is simply

σ(ν0) =2cre f

Γ(2.4)

This can be used to calculate the optical depth of the vapor as long as the density

n of the vapor is also known:

OD = nσ(ν)L (2.5)

where resonant light propagates a length L through the vapor. The light becomes

attenuated according to

S = S0 exp(−nσ(ν)L) (2.6)

The density n of potassium vapor as a function of temperature is well described

using the following empirical formula proposed by Killian (1926):

nK =1026.2682−(4453 K)/T

(1 K−1)Tcm−3 (2.7)

At a typical operating temperature of about 160C the density of potassium is

found to be nK = 2.2× 1013 cm−3.

The pressure broadening of K in a He environment is Γ ' 13.2 GHz/amg (Al-

lard and Kielkopf, 1982). The density of the helium buffer gas in the cell is ex-

pressed in amagats (amg); 1 amagat = 1 atm at 25C. Figure 2.4 shows the pressure

broadening in the two cells that are used in the experiment. The He pressure in

each cell is known from the cell filling process: the amount of gas in the cell is


0

0.5

1

1.5

Tra

nsm

itte

dS

ign

al(V

)

768.5 769 769.5 770 770.5 771 771.5

Wavelength (nm)

nK = 1.7× 1013 cm−3 , p = 7.0 amg (fixed)

nK = 4.6× 1013 cm−3 , p = 2.4 amg (fixed)

Figure 2.4: Absorption profiles showing pressure broadening for two co-magnetometercells filled with 2.4 amg and 7 amg 3He. A low intensity probe laser was tuned throughthe resonance and the transmission amplitude was measured on an arbitrary scale. Fitsusing Equation 2.6 give densities that are consistent with the density obtained knowingthe cell temperature and using Equation 2.7.

determined by measuring the pressure in the manifold before and after the cell

is pulled off. In fitting the absorption curve to Equation 2.6, the density is a free

parameter. The data for each cell was taken at different density and temperature;

the temperature of the 7.0 amg cell was 176C and the fit density implies a temper-

ature of 174C. The temperature of the 2.4 amg cell was 156C and the fit density

implies a temperature of 155C.1

Using these measurements for the 7.0 amg cell, one finds that there are 3.5 opti-

cal depths for resonant light in the 2.5 cm diameter cell. The attenuation of pump-

ing light as it propagates through the cell may be a significant issue considering

the implications of nonuniform pumping intensity and polarization in the cell.

1Note that the cell temperature is on average ∼ 4C less than the oven temperature set point,which was 180C and 160C for the 7.0 and the 2.4 amg cells, respectively.


2.1.2 Light propagation

The propagation of intense pumping light through an alkali vapor that is almost

fully polarized is not well described by the simple decaying exponential in Equa-

tion 2.6. As electrons are polarized by the pump light and the ms = −1/2 state is

depopulated, the absorption cross section decreases and the pumping light “burns

through” to polarize more atoms deeper in the cell. The polarization of atoms is

locally given by the balance of the pumping rate Rp, which is proportional to the

pumping light, and the relaxation rate Rtot according to

Pe =Rp

Rtot'

Rp

Rp + Rsd(2.8)

The relaxation rate Rtot has several contributions that will be discussed in greater

detail in a subsequent section; for now, it is sufficient to consider the relaxation to

be dominantly the sum of the pumping rate and a fixed “spin-destruction” rate, Rsd.

The propagation of circularly polarized pumping light through the cell is governed

by the amount of polarization according to

dRp

dx= −nσ(ν0)(1− Pe)Rp (2.9)

There is no attenuation for fully polarized portions of the vapor; dRp/dx = 0 for

Pe ' 1. The solution to these equations is the principal value of the Lambert W-

function (the inverse of the function f (W) = WeW):

Rp(x) = RsdW[

Rp(0)Rsd

exp(−xnσ(ν0) +

Rp(0)Rsd

)](2.10)

The solutions for normal and high pumping rate are shown in Figure 2.5. Under

typical conditions the K pumping rate is around 100 1/s and the polarization is


0

25

50

75

100

Po

lari

zati

on

(%)

0 0.5 1 1.5 2 2.5

Distance in Cell (cm)

20

50

100

200

5001000

Pu

mp

ing

Rate

(1/sec)

Nominal Rp ≃ 1.8Rsd

High Rp ≃ 14Rsd

Figure 2.5: At the nominal pump beam intensity, the pumping rate decreases as it prop-agates through the cell according to Equation 2.10 (solid line). The corresponding K po-larization (Equation 2.8) has a significant gradient through the cell. At sufficiently highpumping rate (dashed line), the pump beam “burns through” polarized K and creates amuch more uniform polarization.

around Pe ' 50% at the center of the cell. An effective optical depth can be gleaned

from these data; at the normal pump intensity ODeff ' 1.7, which is less than half

of the naıve calculation from Equation 2.5. Note that the polarization at high pump-

ing rate is barely attenuated on account of the pump laser almost fully polarizing

the cell. The strong polarization gradient at the nominal pumping rate is a cause

of some concern because it generates first order sensitivity to certain parameters of

the experiment. Such imperfections will be discussed in section 4.2.

2.2. The K magnetometer 28

2.2 The K magnetometer

An atomic magnetometer measures the precession of spins in a magnetic field. In

a given magnetic field, the spins will precess until they lose their coherence by, for

example, collisions with the cell wall. The signal of the magnetometer is generally

given by the ratio of the precession rate to the decoherence rate.

The precession of the atomic spins derives from the following Hamiltonian in

the presence of a magnetic field B and in the absence of collisions:

H = −2S · I− gsµBB · S + gNµNB · I (2.11)

The electron and nuclear spins interact through the I · S term. According to this

Hamiltonian, the electron and nuclear spins precess together with an angular fre-

quency

ωF=I±1/2 = ±ω0 = ±gsµBBQh

with Q = (2I + 1) (2.12)

This result is obtained for potassium atoms, with I = 3/2 and Q = 4. The two

potassium hyperfine states F = 1 and F = 2 precess in opposite directions. Q

represents a “slowing down” of the spin precession due to hyperfine coupling. It

is convenient to collect terms into a single gyromagnetic ratio γ:

ω = γB where γ = ±gsµB

Qh(2.13)

Here we have generalized Equation 2.12 using a vector representation; the plane

of precession is always perpendicular to the magnetic field.

A sensitive magnetometer is designed to allow the spins to precess coherently

with the longest possible lifetime, T2. There are several processes that destroy this


coherence:

Rtot =1T2

= Reese + Ren

se + Rcollisionssd + RD + Rp + Rm (2.14)

These terms are, in order, the relaxation rates due to alkali-alkali spin-exchange col-

lisions, alkali-noble gas spin-exchange collisions, spin-destruction collisions, diffu-

sion to the walls, pumping by the pump laser, and pumping by the probe laser.

The latter two pumping rates by the lasers are not fundamental and can be ad-

justed and reduced experimentally. The alkali-alkali spin-exchange term Reese is sup-

pressed at low field but modifies the precession frequency of the ensemble. The

spin-destruction and spin-diffusion relaxation rates constitute the most significant

contributions to the total relaxation and can be minimized by optimizing the com-

position of the cell (choice of alkali metal, cell temperature, buffer gas and gas

pressure). The following sections discuss each relaxation term in greater detail.

2.2.1 Spin-exchange collisions

Spin-exchange collisions between two alkali atoms preserve the total angular mo-

mentum projection mF1 + mF2 of the system but redistributes the angular momen-

tum among the hyperfine sublevels of the colliding atoms. Figure 2.6 shows how

the total angular momentum of the atom is redistributed after a collision. Tradi-

tional alkali metal magnetometers are limited by these collisions because they can

change the total angular momentum F of each atom without changing the projec-

tion mF1 + mF2 . After a spin-exchange collision the electron spins remain pointing

in the same direction but reside in different hyperfine states that precess with differ-

ent (opposite) ω. In time, the precession at different frequencies rapidly decoheres


KK

rr

|2,1〉 |2,1〉-2

mF

F = 1

F = 2-1 0 1 2

-1 0 1

mF

-2

F = 1

F = 2-1 0 1 2

-1 0 1

-2 -1 0 1 2

-1 0 1

-2 -1 0 1 2

-1 0 1

Figure 2.6: Spin-exchange collisions between alkali metal atoms. The total angular mo-mentum projection mF1 + mF2 is conserved but the individual mF level populations areredistributed. Adapted from Walker and Happer (1997).

and relaxes the system. Thus, the relaxation rate due to spin-exchange collisions

in a traditional atomic magnetometer is simply given by Reese ' 1/Tse where Tse is

the time between spin-exchange collisions.

For the past few decades spin-exchange interactions were considered the funda-

mental limiting factor in alkali metal magnetometers. However, Happer and Tang

(1973) discovered that at sufficiently low magnetic field and high spin exchange

rate such that Rse γB, this relaxation mechanism shuts down. Under these con-

ditions, each atom in the ensemble experiences spin-exchange collisions and hops

between hyperfine states F = 1 and F = 2, for which the precession frequencies are

−ω0 and +ω0, respectively. The atom precesses an infinitesimal amount between

collisions but has a net positive precession because F = 2 has higher statistical

weight (it has more sublevels) than F = 1. The advantage of operating in this

regime is that the atoms precess coherently albeit at a lower rate,

ω0 =gsµBB

Qhwith Q = 1 +

I(I + 1)S(S + 1)

(2.15)


-2

mF

w = -w0

w = +w0-1 0 1 2

-1 0 1

-2

mF

F = 1

F = 2-1 0 1 2

-1 0 1

-2

mF

-1 0 1 2

-1 0 1

Unpolarized, Q = 6 Partially Polarized, Q ~ 5 Polarized, Q = 4

Figure 2.7: Equilibrium hyperfine sublevel distributions for unpolarized, partially polar-ized and fully polarized ensembles of K atoms in the regime of strong spin-exchange.The partially polarized distribution is the steady-state “spin-temperature” solution thatis achieved after many spin-exchange collisions.

Since the precession is coherent, spin-exchange collisions no longer produce relax-

ation. For potassium, Q = 6, representing a further “slowing down” relative to

Equation 2.12 due to hopping between hyperfine levels. The value of Q depends

on the polarization of the ensemble, ranging from Q = 6 at low polarization to

the traditional Q = 4 from Equation 2.12 at high polarization. In a fully polarized

ensemble, spin-exchange collisions have no effect because all of the atoms reside

in the maximum angular momentum sublevel m f = 2 of the F = 2 hyperfine state

and F = 1 cannot accommodate that amount of angular momentum. The distribu-

tion of atoms among the hyperfine sublevels is illustrated in Figure 2.7. A partially

polarized ensemble achieves an equilibrium “spin-temperature” distribution (An-

derson et al., 1959) for which the slowing down factor is between the unpolarized

and fully polarized cases. Savukov and Romalis (2005) provide a simple expres-

sion for the dependence of Q on polarization:

Q(Pe) = 4(

2− 43 + Pe2

)−1

(2.16)


For typical polarization Pe ' 60%, one finds Q(Pe) ' 4.9. In the presence of

polarization gradients, atoms precess at different frequencies at different locations

in the cell.

In this regime of strong spin-exchange, the spin-exchange relaxation rate has

only a second order dependence on magnetic field that vanishes at zero field:

Reese = ω2

0TseQ2 − (2I + 1)2

2(2.17)

If ω0 Tse then the transverse relaxation time due to spin-exchange is much

longer than the time between spin-exchange collisions Tse. For typical conditions,

the spin-exchange rate is 1/Tse = 30, 000 1/s, which is much faster than the preces-

sion frequency of 0.44 1/s in 1 µG. A magnetometer utilizing this effect was first

presented in Allred et al. (2002); much of the data in that paper appears in this the-

sis. The suppression of spin-exchange relaxation in this manner is the key to the

high sensitivity of this magnetometer; magnetometers with these characteristics

are frequently called spin-exchange relaxation-free or SERF magnetometers.

2.2.2 Spin destruction collisions

In the absence of spin-exchange relaxation, spin destruction interactions between

potassium and the other elements in the cell become the dominant relaxation mech-

anisms. For this magnetometer, the contributions are

Rcollisionssd = σsd

HevnHe + σsdN2

vnN2 + σsdK vnK (2.18)


Alkali metal σsdSelf σsd

He σsdNe σsd

N2

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

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

Table 2.1: Alkali-metal spin-destruction cross sections reprinted from Allred et al. (2002).

In these interactions, the K electron spin is disoriented after collisions with K, N2

or He. The polarized spin angular momentum is ultimately converted to the trans-

lational or rotational degrees of freedom of the system. “Spin destruction” is some-

thing of a misnomer; spin rotation and spin disorientation also refer to these effects

in the literature.

Potassium was chosen among the alkali metals for the co-magnetometer be-

cause it has a lower spin-destruction cross section with the He buffer gas. Spin-

destruction cross-sections for various combinations of alkali metal and relevant

gasses can be found in Table 2.1. According to these cross sections, the spin-destr-

uction rates in a magnetometer with 7 amg 4He, 50 torr N2 at 160C are

Rensd = σsd

HevnHe = 29 1/s (2.19)

RK-N2sd = σsd

N2vnN2 = 0.1 1/s (2.20)

RK-Ksd = σsd

K vnK = 10 1/s (2.21)

The total of these spin-destruction rates is Rcollisionssd = 39 1/s.


2.2.3 Spin-exchange efficiency

In the co-magnetometer, the K atoms experience both spin-destruction and spin-

exchange collisions with the 3He buffer gas. Spin-exchange collisions transfer po-

larization between K atomic spin and 3He nuclear spin. It is useful to define the

alkali-metal-noble-gas spin-exchange efficiency η:

η ≡ Rense

Rense + Ren

sd(2.22)

corresponding to the fraction of the K-3He collisions that are spin-exchange colli-

sions. Baranga et al. (1998) measured η = 0.756− (0.00109 K−1)T for the K-3He

system. Although the spin-exchange efficiency drops slightly with increasing tem-

perature, the total spin exchange rate increases with increasing temperature and

potassium density. At the typical operating temperature of T = 160C, one finds

η = 0.28. This property can be used to determine the K spin-exchange rate Rense

knowing the spin-destruction rate Rensd of the K system:

Rense = Ren

sdη

1− η' 15 1/s (2.23)

where the result is obtained using the total spin-destruction rate from Equation

2.20. Since it is easy to measure the total relaxation rate of the potassium in the co-

magnetometer, it will be possible to use η to learn what fraction of the potassium

relaxation rate is spin-exchange relaxation.

2.2.4 Spin diffusion relaxation

The relaxation due to diffusion was modeled in Allred et al. (2002) assuming that

the K electron and nuclear spins completely depolarize on contact with the cell


walls. The diffusion constant for K in 3He buffer gas is (Franz and Volk, 1982)

DK-3He = 0.35 cm2/s

(√1 + T/(273.15 K)

pn/(1 amg)

)= 0.08 cm2/s (2.24)

where the result is for typical co-magnetometer conditions (7 amg 3He at 160C).

The diffusion rate of potassium atoms to the walls is approximated using the fun-

damental classical diffusion mode in spherical cell of radius a:

RD = Q(Pe)DK-3He

(π

a

)2= 2.5 1/s (2.25)

where a is the radius of the cell and the enhancement factor Q(Pe) accounts for the

destruction of both electron and nuclear spin polarizations at the cell wall.

2.2.5 Total K relaxation

The most significant contributions to the total potassium relaxation rate in the co-

magnetometer are

Resd = Ren

se + Rcollisionssd + RD ' 57 1/s (2.26)

At zero field, the K-K spin-exchange relaxation is negligible. The probe pumping

rate can be rendered negligible by running with very low intensity and at large

detuning.

In the pure K magnetometer using 4He instead of 3He, the noble-gas spin-exch-

ange term disappears, leaving just diffusion and spin destruction. Helium buffer

gas (4He) is added to reduce diffusion to the walls until the relaxation due to wall

diffusion is balanced by spin-destruction collisions with the buffer gas. In that

balance is the minimum total spin relaxation. The relaxation rate as a function


40

60

80

100

Rel

axat

ion

rate

RK−

He

sd(1

/s)

0 2 4 6 8 10

Buffer gas pressure (amg)

1

1.5

2

2.5

3

Lin

ewid

thR

K−

He

sd/

Q(P

e)(H

z)

T = 160C

T = 170C

Figure 2.8: Spin relaxation rate for a K magnetometer as a function of He buffer gas pres-sure.

of buffer gas pressure for a K magnetometer is plotted in Figure 2.8. In this case,

the optimum K magnetometer is created using 2 amg of buffer gas for which the

relaxation rate is

RK-Hesd = Rcollisions

sd + RD ' 30 1/s (2.27)

The relaxation rate is often quoted as RK-Hesd /[2πQ(Pe)] = 1.2 Hz, where Q(Pe) is

used to account for the relaxation of both the electron and nuclear spins. This very

low relaxation rate is an essential part of an ultra-sensitive magnetometer.


2.2.6 Magnetometer bandwidth

The bandwidth of the magnetometer can be determined from a simplified set of

Bloch equations for the spin dynamics. If all the magnetic fields are properly ze-

roed, then the Bloch equations can be expressed as

dPex

dt=(+γeByPe

z − RtotPex) 1

Q(Pe)(2.28)

dPez

dt=(−γeByPe

x − RtotPez + Rp

) 1Q(Pe)

(2.29)

To the right of the equal signs, the first term causes spin precession in a magnetic

field By, the second term represents the spin relaxation rate and the third term

polarizes the atoms along z at the pumping rate Rp. Imposing an oscillating field

By = B0 exp(−iωt), one obtains

Pex =

Pez γeB0

Q(Pe)1

Rtot/Q(Pe)− iω(2.30)

S ≡ <(Pex) =

Pez γeB0

Q(Pe)2Rtot

[Rtot/Q(Pe)]2 + ω2 (2.31)

where the signal S is sensitive to the x projection of the polarization via optical

rotation of a probe beam. The bandwidth of the magnetometer has a Lorentzian

profile with half-width Rtot/Q(Pe) centered on zero frequency. This analysis can be

carried out in a similar fashion for a magnetometer in finite field (typically driven

by a synchronous pumping technique). One can move to the rotating frame and

retrieve a Lorentzian profile with nonzero frequency offset:

S =Pe

z γeB0

Q(Pe)2Rtot

[Rtot/Q(Pe)]2 + (ω−ω0)2 (2.32)


where ω0 is the frequency of the spin precession in the ambient magnetic field. On

resonance, ω0 = ω (or at very low frequency in zero field), the signal simplifies to

S =PeγeB0

Rtot(2.33)

A calibration constant κ is measured so that the magnetometer measures the ap-

plied field accurately; B0 = κS. The actual angle that spins achieve is

θ ' Px

Pez

=γeB0

Rtot' 2× 10−6 rad (2.34)

where the result is obtained for B0 = 10 fT and Rtot = 200 1/s. This very small

angle requires the very high precision detection scheme described in Sections 2.3

and 3.1.2.

2.2.7 Fundamental magnetometer sensitivity

Fundamentally, the magnetometer sensitivity is limited by the shot noise of the

atomic vapor that is being measured. The derivation in this section appears in

extremely abbreviated form in Savukov et al. (2005). The total angular momentum

F is governed by the uncertainty principle according to

δFxδFy ≥|Fz|

2(2.35)

where the angular momentum operators commute by [Fx, Fy] = iFz. Since the

states are not entangled or squeezed, the uncertainty in the transverse angular

momentum projections are equal; δFx = δFy. For an ensemble of N atoms, let

Fi → NFi and the uncertainty in the transverse component becomes

δFx =

√Fz

N(2.36)


In the magnetometer, probe beam photons make measurements of the spin com-

ponent Fx. If repeated measurements of a particular atom happen faster than the

relaxation time T2, then the measurements will be correlated and the measurement

is not improved. Gardner (1977, Equation 8.34) provides an expression for the

uncertainty in the measurement of Fx after continuous measurement for a time t:(〈δFx〉δFx

)2

=2t

∫ t

0(1− τ/t)K(τ)dτ (2.37)

where the spin time-correlation function is given by the relaxation time T2:

K(τ) = exp(−τ/T2) (2.38)

Since this experiment is primarily concerned with long term measurements, one

can take t T2 and perform this integral for t → ∞:

〈δFx〉 = δFx

√2T2

t(2.39)

To place this in terms of magnetic field sensitivity, use Equation 2.33, which can be

rewritten with S → Fx/2 and Pe → Fz/2 as

〈δFx〉 =〈δB〉γeFz

Rtot(2.40)

Although fully polarized potassium atoms are in the Fz = 2 hyperfine state, maxi-

mum sensitivity is found for lower pumping rates where the polarization is around

Pe ' 0.5 and Fz ' 1. The relaxation rate is identified as 1/T2 = Rtot/Q(Pe), where

the slowing down factor Q(Pe) accounts for the depolarization of both the elec-

tron and nuclear spins. Combining Equations 2.36, 2.39, and 2.40 and solving for

magnetic field, one obtains

δB =1γe

√2RtotQ(Pe)

FzNt(2.41)

2.3. Optical Rotation 40

The total number of atoms involved in the measurement, N = nV, reduces the

noise level according to√

N, as expected according to Poisson statistics. A probe

laser is used to measure spin precession using optical rotation (see next Section).

To achieve this noise limit, the probe laser must have sufficient photons such that

in a time T2 at least one photon interacts with each atom.

2.3 Optical Rotation

The average angle of the alkali metal spin is measured using optical rotation of a

linearly polarized probe beam. Linearly polarized, off-resonant light will experi-

ence rotation of the polarization axis as it propagates through a polarized medium

for which positive and negative helicity light experience different indices of refrac-

tion, n+(ω) and n−(ω), respectively. This section loosely follows the semi-classical

approach originally set forth by Mort et al. (1965) and refined by Erickson (2000).

First one must establish an expression for how much linearly polarized light is

rotated in a medium with n+(ω) 6= n−(ω). Start with the propagation of linearly

polarized light:

E(z = 0) =E0

2eiωty + c. c. =

E0

4eiωt [(y + ix) + (y− ix)] + c. c. (2.42)

where c. c. denotes the complex conjugate and in the last step the light was put

in terms of right- and left- circularly polarized components. (The terms positive

helicity, right-circular polarization and σ+ are used interchangeably throughout

this text.) A time t later, the fields will have propagated a distance l = tc/n(ω)

and the electric fields will become

E(z = l) =E0

4eiωln+(ω)/c(y + ix) +

E0

4eiωln−(ω)/c(y− ix) + c. c. (2.43)


It is useful to define the quantities

n(ω) = [n+(ω) + n−(ω)]/2 (2.44)

∆n(ω) = [n+(ω)− n−(ω)] (2.45)

One can now rewrite Equation 2.43 as

E(z = l) =E0

4eiωln(ω)/ceiωl∆n(ω)/c(y + ix) (2.46)

+E0

4eiωln(ω)/ce−iωl∆n(ω)/c(y− ix) + c. c. (2.47)

Ignoring the common phase factor due to n and defining the rotation angle,

θ =ωl2c<[n+(ω)− n−(ω)] (2.48)

the electric field becomes

E(z = l) = E0(y cos θ − x sin θ) (2.49)

This clearly shows that the polarization angle of light is rotated as it propagates

through a medium with n+(ω) 6= n−(ω).

The Kramers-Kronig relations allow one to derive the real part of the index of

refraction from the imaginary part. A derivation of the Kramers-Kronig relations

can be found in many texts such as Jackson (1998). The imaginary parts of the

refractive index are easily obtained by measuring the absorption of resonant light

propagating through the atomic vapor. To this end, the data in Figure 2.4 verified

that our alkali vapor is well described using a pressure-broadened, Lorentzian ab-

sorption profile. Converting Equation 2.3 to angular units, the cross section for

absorption is

σ(ω) = 2πcre fL(ω) where L(ω) ≡ γ/2(ω−ω0)2 + (γ/2)2 (2.50)


and where ω and γ are the angular versions of ν and Γ. Note that∫L(ω)dω = π.

The absorption coefficient α is defined as the characteristic decay length of the

intensity of decaying plane waves; I ∼ E2 ∼ E0e−2ωlni(ω)/c (note the factor of

two) where ni(ω) is the imaginary component of the index of refraction n(ω) =

nr(ω) + ini(ω). One can equate the absorption coefficient to the absorption cross

section simply using

a =2ωni(ω)

c= nσ(ω) (2.51)

where n is the number density in units of particles per volume. Here one must

be careful not to confuse the index of refraction ni(ω), nr(ω) with the density n.

Solving for ni(ω),

ni(ω) =πnc2re f

ωL(ω) (2.52)

It will be convenient to express the index of refraction in terms of the electrical sus-

ceptibility χe(ω). The electrical susceptibility, permittivity and index are related

by ε(ω) = χe(ω) + 1 = n2(ω). One can expand χe(ω) in terms of n(ω) since

[nr(ω)− 1] 1 and ni(ω) 1 are small:

χe(ω) = 2[nr(ω)− 1] + i2ni(ω) (2.53)

Since χe(ω) is a physical quantity that obeys causality in its temporal evolution,

one can employ the Kramers-Kronig relations to obtain expressions for the real

parts of χe(ω) and n(ω):

χr(ω) = − 1π

P∫ ∞

−∞

χi(ω1)ω−ω1

dω1 (2.54)

The solution to the principal value integral of a Lorentzian has a standard disper-

sion shape:

− 1π

P∫ ∞

−∞

L(ω1)ω−ω1

dω1 = D(ω), where D(ω) ≡ ω−ω0

(ω−ω0)2 + (γ/2)2 (2.55)


4s 2S1/2 ms = –1/2 ms = +1/2

4p 2P1/2

D1 Transition

D2 Transition

s + s–

11

4s 2S1/2 ms = –1/2 ms = +1/2

mj = –1/2mj = –3/2 mj = +3/2mj = +1/2

mj = –1/2 mj = +1/2

4p 2P3/2

s + s–

11

s–

3

s +

3

Figure 2.9: Level diagrams for K showing D1 and D2 transitions. Circled numbers indicatethe transition amplitudes.

The resulting real component of the index of refraction is

nr(ω)− 1 =πnc2re f

ωD(ω) (2.56)

Thus, the real part of the index of refraction was obtained with knowledge of the

imaginary part and by imposing causality through the Kramers-Kronig relations.

The specific indices of refraction for right- and left-circularly polarized light

interacting with the D1 transition can be obtained with small adjustments to Equa-

tion 2.56. Let Px = ρ+ − ρ− be the polarization of the alkali vapor, where ρ± is the

population of the ms = ±1/2 state. One can split the index of refraction into left-


−1

0

1

Sig

nal

(arb

.)

766 767 768 769 770 771

Wavelength (nm)

−4

−2

0

2

4

Ro

tati

on

An

gle

(rad

)

0.01

1

100O

pti

cal

Dep

th

p3He = 7 atm, T = 160 C

p3He = 2 atm, T = 170 C

Figure 2.10: The probe beam experiences optical rotation according to Equation 2.62 as itpasses through a typical 2.5 cm diameter cell with Px = 1. The top figure shows the opticaldepth of the probe light, which increases significantly around the D1 and D2 resonances.The middle plot shows the optical rotation angle experienced by probe beam photons. Thebottom plot shows the projected signal obtained by the product of the light transmissionand the rotation angle; near the resonances, the absorption of light attenuates the rotationsignal.


and right-circularly polarized components acting on half the population:

n+(ω)− 1 = ρ−2πnc2re fD1

ωDD1(ω) (2.57)

n−(ω)− 1 = ρ+2πnc2re fD1

ωDD1(ω) (2.58)

where due to selection rules σ± light can only interact with ρ∓. For unpolarized

atoms, ρ+ = ρ− = 1/2 and one retrieves n = n+ = n−. Finally, one can substitute

these expressions into Equation 2.48 to obtain a rotation angle of

θ = −12

lrec fD1nPxDD1(ν) (2.59)

The D2 transition is only slightly complicated by having more transitions that are

weighted by different amplitudes:

n+(ω)− 1 =

(ρ−Am=− 1

2→+ 12+ ρ+Am=+ 1

2→+ 32

Am=− 12→+ 1

2+Am=+ 1

2→+ 32

)2

πnc2re fD2

ωDD2(ω) (2.60)

n−(ω)− 1 =

(ρ+Am=+ 1

2→−12+ ρ−Am=− 1

2→−32

Am=+ 12→−

12+Am=− 1

2→−32

)2

πnc2re fD2

ωDD2(ω) (2.61)

where again one retrieves n = n+ = n− for ρ+ = ρ− = 1/2. Using the transition

amplitudes found in Figure 2.9 and noting that fD1 = fD2/2 = 1/3, the total angle

of rotation is equal in magnitude but opposite in sign to the rotation around the

D1 transition. In sum, the total rotation angle is

θ =12

lrec fD1nPx[−DD1(ν) +DD2(ν)] (2.62)

Optical rotation for typical parameters is plotted in Figure 2.10. Although maxi-

mum rotation is achieved within ∼ 0.1 nm of the resonance, the probe beam is

typically operated at 769.5 nm, between D1 and D2 where the signal has a small

first order dependence on wavelength. Away from resonance, fluctuations in the

wavelength do not cause significant fluctuations in the rotation signal.

2.4. Lightshifts 46

2.4 Lightshifts

When off-resonant, circularly-polarized light propagates through an atomic vapor,

the vapor can experience a Zeeman lightshift that makes the atoms respond as

though they were in a magnetic field pointing along the propagation of the laser. In

a sensitive magnetometer, this effect can be quite significant. This section contains

a semi-classical derivation of this effect following previous work by Appelt et al.

(1998) and Happer and Mathur (1967).

One seeks to determine the energy shift of atoms in a light field. This energy

shift can be expressed as a frequency shift or as an effective magnetic field. As with

the above derivation of optical rotation, knowledge of the absorption profile leads

to a complex polarizability, which can then be run through the Kramers-Kronig re-

lations to obtain the real part of the polarizability. The real part of the polarizability

gives energy shifts in the electric field of the light.

Consider a plane wave propagating through an alkali vapor for which the elec-

tric field is

E(z, t) =E0

2ei(kz−ωt) + c. c. (2.63)

Postulate that the propagation is governed by the wave equation:

dEdz

= 2πikn〈α〉E (2.64)

where the dielectric polarization tensor α is dependent on the average electron spin

orientation 〈S〉 of the alkali vapor and is given by

α = α(1− 2iS×) (2.65)

2.4. Lightshifts 47

Although it may not be immediately clear that this formulation is correct, one can

build trust in it by obtaining several sensible results along the way to deriving the

lightshift.

The first way to see that this expression is valid is to retrieve optical rotation

from the propagation of linearly polarized light. Substituting the polarizability,

Equation 2.65, into the wave equation, Equation 2.64, one obtains

dEx

dz= −(2πknαPz)Ey (2.66)

dEy

dz= +(2πknαPz)Ex (2.67)

where Pz = 2〈Sz〉. This represents optical rotation of the polarization that is pro-

portional to the polarization of the alkali vapor. For an appropriate choice of k and

α, this expression can be made to match the result derived in the previous section.

As light propagates through alkali vapor, the wave electric field induces an

oscillating electric dipole moment 〈p〉 = 〈α〉E. This interaction has an energy

δH = −p · E = −E∗ · αE = δE − ih2

δΓ (2.68)

where δE is the change in energy that leads to a lightshift and δΓ is the energy

associated with the absorption of the light. Find expressions for the real and imag-

inary parts of the polarizability α by equating the mean change in this energy,

d(δH)/dt = iωδH, with the absorption of optical power as follows:

−iωE∗ · αE + c. c. = 〈σα〉hνΦ(ν)/A (2.69)

where∫

Φ(ν) = P/hν is the total photon flux for a beam of power P incident upon

an area A and 〈σα〉 is the ensemble-average absorption cross section. Simplifying

2.4. Lightshifts 48

the left hand side,

E∗ · αE = E∗ · α(1− 2i〈S〉×)E (2.70)

= α(E∗ · E− 2iE · 〈S〉 × E) (2.71)

Circularly polarized light can be compactly expressed using

s =E∗ × E

iE2 (2.72)

where s is the photon spin. Rewriting Equation 2.71 using s and the identity E∗ ·

(〈S〉 × E) = −(E∗ × E) · 〈S〉, one finds

E∗ · αE = αE2(1− 2s · 〈S〉) (2.73)

Inserting this result into Equation 2.69,

〈σα〉hνΦ(ν)/A = −iωαE2(1− 2s · 〈S〉) + c. c. (2.74)

〈σα〉hνΦ(ν)/A = −i4πναE2(1− 2s · 〈S〉) (2.75)

〈σα〉 = σ(ν)(1− 2s · 〈S〉) (2.76)

where the absorption cross section is defined using the imaginary part of the polar-

izability,

σ(ν) =4παiE2A

hΦ(ν)= cre[ fD1LD1(ν) + fD2LD2(ν)] (2.77)

Note that L(ν) has the same Lorentzian form found in Equation 2.50, although this

instance does not use angular units. Now it is possible to solve for the complex

polarizability,

αi =hΦ(ν)σ(ν)

4πAE2 (2.78)

2.4. Lightshifts 49

0

0.5

1

1.5

Tra

nsm

itte

dS

ign

al(V

)

−500 0 500 1000 1500 2000

Frequency (MHz)

Fit σ = 23 MHz

Figure 2.11: The probe beam linewidth as measured by a 10 cm long, 1.5 GHz FSR Fabry-Perot cavity. In these data, imperfections in the alignment of the cavity are likely limitingthe resolution of the linewidth.

The imaginary part of the polarizability can be used to directly solve for the light

absorption operator δΓ from Equation 2.68 using the imaginary part of Equation

2.73:

δΓ = Rp(1− 2s · S), where Rp =∫ Φ(ν)σ(ν)

Adν (2.79)

is the pumping rate. This expression demonstrates correctly that the D1 absorption

is zero when the electrons are fully polarized (Sz = 1/2) along the helicity of the

pumping light (sz = 1). For D2 light, one must make the substitution 〈S〉 →

−〈S〉/2 everywhere in this analysis according to the D2 transition amplitudes that

were calculated explicitly in Equation 2.61.

To calculate the pumping rate, one must first find an expression for the photon

flux. In general, the photon flux from a single mode laser can be expressed as

Φ(ν) =Phν

(e(ν−νp)2/2∆ν2

p

∆νp√

2π

)(2.80)

2.4. Lightshifts 50

where the line shape is Gaussian with a half-width ∆νp. The linewidth of the probe

laser is measured in Figure 2.11 and the single mode pump beam is assumed to

have similar properties. As seen in Figure 2.4, the line broadening of K in 7 amg

3He is Γ = 13.2 GHz/amg × 7 amg = 91 GHz is sufficiently larger than the single

mode diode linewidths of the pump and probe beams that one can approximate

the diode line as a delta function:

Φ(ν) = Φ0δ(ν− νp) where Φ0 =Phν

(2.81)

where νp is the frequency of the laser. Using this expression for the photon flux the

pumping rate becomes

Rp =Φ0cre f

A

(Γ/2

(Γ/2)2 + (νp − ν0)2

)→ 2Pcre f

hν0AΓ(2.82)

Where in the last step the pump beam was tuned to the center of the resonance,

νp → ν0. For a pumping power of 1 mW over an area 2.5 cm2, one obtains a

pumping rate of 190 1/s.

The lightshift δE is obtained using the real part of Equation 2.73:

δE = −αrE2(1− 2s · S) (2.83)

The polarizability is a physical quantity that permits the use of the Kramers-Kro-

nig relations to obtain an expression for the real part of the polarizability from the

known imaginary part:

αr(ν) = − 1π

P∫ ∞

−∞

αi(ν1)ν− ν1

dν1 (2.84)

The lightshift becomes

δE = −hΦ(ν)cre f4πA

(1− 2s · S)(− 1

πP∫ ∞

−∞

L(ν1)ν− ν1

dν1

)(2.85)

= − hΦ(ν)cre f2A

(1− 2s · S)D(ν) (2.86)

2.4. Lightshifts 51

−20

−10

0

10

20

Lig

ht

Sh

ift

(µG

)

766 767 768 769 770 771

Wavelength (nm)

p3He = 7 atm, T = 160 C

p3He = 2 atm, T = 170 C

Figure 2.12: The lightshift according to Equation 2.88 for the pump beam at the typicalpumping rate of 180 1/s. The pump beam is operated at the zero crossing for the D1resonance at 770.1 nm to minimize the lightshift and maximize the pumping rate.

Finally, if one considers the energy shift as due to a magnetic-like field, 2µBL · S =

δE , then one obtains a lightshift:

L = −Φ(ν)cre fγe A

D(ν)s (2.87)

where γe ' 2µB/h. The constant energy term that gives rise to the scalar light shift

has been ignored. For D2 light, one must make the substitution 〈S〉 → −〈S〉/2

according to Equation 2.61 and let fD2 = 2 fD1. The total lightshift becomes

L =Φ(ν)cre fD1

γe A[−DD1(ν) +DD2(ν)]s (2.88)

Thus, the lightshift is a product of light’s degree of circular polarization and de-

tuning from resonance. The magnitude of the pump beam lightshift for typical

parameters can be found in Figure 2.12. The pump beam lightshift is minimized

by tuning the pump beam directly onto the resonance where the lightshift has a

zero crossing. The probe beam lightshift is shown in Figure 2.13 and is minimized

by operating the probe beam with minimum circular polarization.

2.4. Lightshifts 52

−50

−25

0

25

50

Lig

ht

Sh

ift

(nG

)

766 767 768 769 770 771

Wavelength (nm)

−0.004

−0.002

0

0.002

0.004

Po

lari

zati

on

Px

p3He = 7 atm, T = 160 C

p3He = 2 atm, T = 170 C

Figure 2.13: The probe beam is operated close to 769.5 nm, indicated here by the verticalline. For a probe pumping rate of 7 1/s and a circular polarization on the order of 0.005πretardation (1 V control voltage) gives the polarization Pe

x ' 8 × 10−6 and the lightshift3.7 nG. At the operating point the lightshift is suppressed by having no probe circularpolarization and has a very small first order dependence on wavelength drift.

2.5. Noble gas optical pumping 53

Small fluctuations in the wavelength of the pump beam generate significant

amounts of lightshift. Around resonance, the light shift is

dLdλ

= − 4Pre fD1chAγeΓ2λ

(2.89)

For the pump beam wavelength drifting on the order of a few picometers (pm)

around zero lightshift, the dependence is dLz/dλ = −1.4 µG/pm. By comparison,

the probe beam is less sensitive, generating only dLx/dλ = 3.0 pG/pm because of

its lower power, off-resonant wavelength and small ∼ 0.5% circular polarization.

Lightshifts behave exactly like magnetic fields while using a completely sepa-

rate mechanism to do so. The near field effect of the light propagating through

the vapor modifies the energies of the electronic states in a way that exactly mir-

rors a magnetic field. The interaction between the propagating light is not through

absorption but rather through the exchange of virtual photons. Polarized alkali

atoms precess due to a lightshift in the same way as in a magnetic field; the atoms

are subject to the slowing-down factor Q(Pe). In the co-magnetometer, a vertical

lightshift can be used to verify that the experiment retains sensitivity to an anoma-

lous field; a lightshift can be represented as an anomalous field with bn = 0 and

be = L.

2.5 Noble gas optical pumping

A co-magnetometer is formed using the polarized electrons of the K atoms and

the polarized nuclei of 3He atoms. The noble gas nuclear spin is polarized by spin-

exchange collisions with polarized K as depicted by Figure 2.14. For K-3He , these

collisions are dominantly binary collisions wherein the total angular momentum of


3He

3He

K

K

Figure 2.14: K-3He spin-exchange collisions preserve total angular momentum.

the system is conserved while the angular momenta of each atom can be exchanged

(Walker and Happer, 1997).

The equilibrium polarization of 3He is governed by the balance of the spin-

exchange polarization rate Rnese and the relaxation rate 1/Tn

1 :

Pn = Pe Rnese

Rnese + 1/Tn

1(2.90)

The spin-exchange rate experienced by 3He, Rnese , is proportional to the spin ex-

change rate experienced by the potassium, Rense , according to the ratio of their den-

sities:

Rnese = Ren

sene

nn' 1

150 hours(2.91)

This result is obtained using the value for Rense in Equation 2.23. Since there are

many more 3He buffer gas atoms than K atoms, the spin-exchange rate experienced

by 3He is much lower than the spin-exchange rate experienced by K.

In practice, the 3He polarization is ∼ 1% and is primarily limited by the relax-

ation due to diffusion of the polarized noble gas across magnetic field gradients.


The diffusion constant for 3He is taken to be

D3He = 1.2 cm2/s

(√1 + T/(273.15 K)

pn/(1 amg)

)' 0.28 cm2/s (2.92)

where the result is obtained for typical conditions (7 amg 3He at 160C). For a per-

fectly spherical, uniformly polarized cell volume, the field everywhere inside the

sphere is uniform. However, any asphericity in the cell creates first and higher

order gradients in the magnetic field. Spins precessing at different rates in a field

gradient across the cell decohere over time. The spin polarization relaxes as these

decoherent spins diffuse across the cell. Schearer and Walters (1965) originally

showed and Cates et al. (1988) confirmed that for sufficiently high buffer gas pres-

sures, the relaxation rate due to field inhomogeneities is

1Tn

1= D3He

|∇B⊥|2B2

z' 1

1 hour(2.93)

where ∇B⊥ is the gradient of the magnetic field transverse to the total field direc-

tion along z. The result is obtained for ∇B⊥ ∼ 70 µG/cm and B0 ∼ 2.2 mG. Since

the diffusion time in the co-magnetometer (a2/D3He ' 6 s) is much faster than

the nuclear spin relaxation rate Tn1 , the polarization of the nuclear spins is uniform

throughout the cell; the only source of field gradients must come from external

sources or the asphericity of the cell. It is likely that the largest source of asymme-

try in the cell is the cell pull-off stem, though this effect is mitigated by plugging

the stem with a drop of potassium metal.

The decay rate for transverse excitations of 3He spins due to diffusion across

gradients is1

Tn2

=8a4γ2

n|∇Bz|2175D3He

' 1210 s

(2.94)

2.6. Coupled spin ensembles 56

where a is the cell radius, gamma is the gyromagnetic ratio for 3He and ∇Bz ∼

5 µG/cm corresponds to the gradient in the field along the field direction.

Collisions between 3He atoms can also depolarize the ensemble. Newbury et al.

(1993) gives the dipolar relaxation rate:

1Tnn

1=

n3He744 amg

hour−1 ' 1106 hours

(2.95)

where n3He ' 7 amg is the density of 3He in amagats. In this experiment, dipolar

relaxation is much less significant than relaxation due to diffusion through gradi-

ents.

Having accounted for the 3He spin exchange and spin relaxation rates, the po-

larization of the nuclear spin is calculated to be:

Pn =(

1 +1

Rnese Tn

1

)Pe ' (3.9%)Pe (2.96)

For typical K polarization Pe ' 50%, the 3He polarization is about Pn ' 2%.

2.6 Coupled spin ensembles

The K-3He co-magnetometer exhibits striking coupled dynamics due to spin-ex-

change interactions. The co-magnetometer cell contains ensembles of polarized K

and 3He spins that are coupled to one another by both by dipolar field interactions

and by their spin-exchange contact interaction. The effective interaction field ex-

perienced by each spin species can be expressed as the field due to a uniformly

magnetized sphere of the other species,

B = λMP =8πκ0µ0

3MP (2.97)


Parameter Variable Typical ValueK-K spin-exchange rate Ree

se 30000 1/sK total relaxation rate Rtot = Ree

se + Rense + Re

sd + Rp + Rm 396 1/sK pumping rate Rp 100 1/sK total spin-destruction rate Re

sd = Rense + Rcollisions

sd + RD 57 1/sK collisional spin-destruction rate Rcollisions

sd = Rensd + RK-K

sd + RK-N2sd 39 1/s

3He precession frequency γnBc 33 1/s (5.2 Hz)K-3He spin-destruction rate Ren

sd = σsdHevnHe 29 1/s

K-3He spin-exchange rate Rense 15 1/s

K-K spin-destruction rate RK-Ksd = σsd

K vnK 10 1/sK probe beam pumping rate Rm 5 1/sK wall relaxation RD 2.5 1/sK-N2 spin-destruction rate RK-N2

sd = σsdN2

vnN2 10 s3He diffusion relaxation time Tn

2 ' 1/Rntot 210 s

3He diffusion relaxation time Tn1 1 hour

3He dipolar relaxation time Tnn2 106 hours

3He -K spin-exchange time 1/Rnese 150 hours

Table 2.2: Summary of expected co-magnetometer timescales.

where the κ0 is a spin-exchange enhancement factor due to the overlap of the K

electron wavefunction and the 3He nucleus. At a temperature of 170C, κ0 = 5.9

for K-3He spin-exchange (Baranga et al., 1998). The magnetization density M corre-

sponds to a fully polarized sample with atoms of magnetic moment µ and density

n:

M = µn (2.98)

Since the magnetometer cell is nearly spherical, the spin ensembles have negligible

self-interaction. The magnetic moments of the potassium and 3He are:

Potassium: µB = 9.274 009× 10−24 J/T =eh

2me(2.99)

Helium-3: µ3He = 1.074 552× 10−26 J/T = −1.158× 10−3µB (2.100)


For the alkali atoms, the magnetic moment is dominanted by the electron mag-

netic moment. The magnetic fields generated by the spin ensembles for typical

conditions are

Potassium: B(Me) ' 5 µG (2.101)

Helium-3: B(Mn) ' 2.2 mG (2.102)

Although the magnetic moment of the 3He is much smaller than the K moment,

the 3He magnetization is much larger than the K magnetization due to the high

density of the 3He gas.

The 3He and K spin ensembles are coupled through their magnetizations; the

3He precesses in the K field and the K precesses in the 3He field. The behavior

of the coupled spin ensembles can be robustly approximated by a set of Bloch

equations that couple the K ensemble polarization Pe with the 3He ensemble polar-

ization Pn:

∂Pe

∂t=

γe

Q(Pe)(B + λMnPn + L + be)× Pe + Ω× Pe

+ (Rpsp + Rense Pn + Rmsm − RtotPe)/Q(Pe)

∂Pn

∂t= γn(B + λMePe + bn)× Pn + Ω× Pn

+ Rnese (Pe − Pn)− Rn

totPn (2.103)

In order, terms in these equations are described as follows: Each spin species pre-

cesses in the sum of the ambient magnetic field B and the effective field due to

the polarization of the other species, λMP. Electron spins additionally precess in

a lightshift L and an anomalous field be. The spins experience non-inertial rota-

tion according to the rotation Ω of the surrounding system. The second line of


each equation contains the various pumping and relaxation rates. The electrons

relax at a rate Rtot = Resd + Ren

se + Rp + Rm. For a circularly polarized pump laser,

sp = 1 and the electron spins polarize according to (sp − Pe)Rp (including the con-

tribution from PeRtot). The electrons exchange spin polarization with 3He nuclei

according to (Pn − Pe)Rense ; this term usually depletes the K polarization, although

the 3He can polarize the K if Pn > Pe when the pump beam is shut off and the K po-

larization is low. The probe beam can pump K according to Rm(sm − Pe) whenever

the probe beam has some circular polarization sm 6= 0; this term is only significant

when the pump beam is off. The terms in the 3He evolution are a subset of the

those found in the K evolution. The 3He experiences a different anomalous field

coupling bn, does not experience any lightshift and the polarization is defined by

the balance of the spin-exchange pumping rate Rnese and the relaxation rate Rn

tot.

In the absence of interactions, the K and 3He spin ensembles precess according

to their gyromagnetic ratios:

ωe =γe

Q(Pe)=

gµB

hQ(Pe)=

2π × 2.8 MHz/GQ(Pe)

(2.104)

ωn = γn =µ3He

h= 2π × 3.244 kHz/G (2.105)

and decay according to their respective T2 times, which is ∼ 140 ms for K (due

to spin-destruction; see Equation 2.26) and ∼ 3 hours for 3He (due to diffusion;

see Equation 2.94). Dramatic and useful coupled dynamics arise, however, when

the spins are coupled such that each species precesses around the magnetization

of the other species. Transient, oscillatory and steady state solutions for the co-

magnetometer signal are carefully studied in the following sections.


2.6.1 Transient response dynamics

The time evolution of the coupled spin ensembles can be solved analytically with

somewhat simplified Bloch equations (Kornack and Romalis, 2002). Retaining only

the electron-nuclear coupling terms and a single relaxation term, the Bloch equa-

tions become

∂Pe

∂t=

γe

Q(Pe)(B + λMnPn)× Pe + Rtot(Pe

0 z− Pe)/Q(Pe) (2.106)

∂Pn

∂t= γn(B + λMePe)× Pn + Rn

tot(Pn0 z− Pn) (2.107)

Where Rtot and Rntot are generalized relaxation rates for potassium and 3He and

the system is designed to relax towards equilibrium polarizations Pn0 and Pe

0 point-

ing along z. For small transverse excitations of the spins, these equations can be

linearized and converted to complex notation as follows:

Pe⊥′(t) = (−iγe(Bn + λMnPn

z )Pe⊥ − RtotPe

⊥ + iλMnz Pn⊥)/Q(Pe) (2.108)

Pn⊥′(t) = −iγn(Bn + λMePe

z )Pn⊥ − Rn

totPn⊥ + iλMe

zPe⊥ (2.109)

where the real and imaginary components of the polarization P⊥ correspond to the

x and y components, respectively. The magnetic fields experienced by the nuclear

spins and the electron spins are different:

Bez = Bn + λMnPn

z (2.110)

Bnz = Bn + λMePe

z (2.111)

To discover strongly coupled dynamics, the external field Bn is set to the “compen-

sation point” where Bn largely cancels the nuclear magnetization λMnPnz . At the

compensation point, Bn is

Bncomp. = −λMnPn

z − λMePez (2.112)


−10

−5

0

5

10

Sig

nal

Pe x

(arb

.)

0 0.2 0.4 0.6 0.8 1

Time (s)

Bn< Bn

comp.

Bn∼ Bn

comp.

Figure 2.15: Transient responses according to the analytical solution to the Bloch equations2.109 for coupled spin ensembles. Far from the compensation point (solid), the electronand nuclear precession is distinct. Close to the compensation point (dashed), the coupleddynamics exhibit coherent oscillation and fast damping of the nuclear spin precession.

In this notation, Bn always refers to the externally applied field along z and must

not be confused with the variable Bz, which is used elsewhere in this text to mean

the sum, usually small or zero, of Bn and the magnetizations in the cell. Near

the compensation point, even though the electron and nuclear spin gyromagnetic

ratios are separated by three orders of magnitude, their precession frequencies can

be brought into resonance because they experience different fields Bez Bn

z .

One can verify that the following solution satisfies the linearized Bloch equa-

tions:

Pex(t) = <

[P1e−(Ae+An+F)t/2 + P2e−(Ae+An−F)t/2

](2.113)

Ae = (iγe(Bn + λMnPnz ) + Rtot)/Q(Pe) (2.114)

An = iγn(Bn + λMePez ) + Rn

tot (2.115)

F =√

(An − Ae)2 − 4γnγeλ2MeMnPez Pn

z /Q(Pe) (2.116)


The solutions contain two separate oscillators with hybrid precession frequencies.

Far from the compensation point these two oscillators are clearly identified as the

uncoupled precession of the electron and nuclear spins. The nuclear and electron

spin response can be identified by the first and second terms in Equation 2.113, re-

spectively. The analytical solution is plotted in Figure 2.15 using K and 3He under

typical experimental conditions. Close to resonance, the precession frequencies are

matched and the entire system exhibits a hybrid response where the contribution

from nuclear and electron spins becomes impossible to discern. At the compensa-

tion point the nuclear spin precession is damped much faster than in the uncou-

pled case because the nuclear spin motion is strongly coupled to the electron mo-

tion and, thus, is strongly damped by the significantly higher relaxation rate of the

electrons. The electron precession, in turn, experiences a longer effective lifetime

because it is driven by the nuclear motion.

The dominant co-magnetometer signal comes from of the slower-oscillating

and longer-lived of the two oscillators in Equation 2.113:

iωn + Γn = (Ae + An + F)/2 (2.117)

At large detunings from the compensation point, the frequency and decay rate of

observed oscillations are described by this expression, which can be identified as

the dynamics of the nuclear spin. Solving for the decay rate at the compensation

point, one finds

Γn =γnγeλ

2MeMnPez Pn

z Rtot

(γnλMnPnQ(Pe))2 + R2tot

(2.118)

Tuned to the compensation point for typical experimental conditions, excitations of

the K-3He coupled system decays at Γ ∼ 5 1/s, which is significantly shorter than


the uncoupled nuclear spin relaxation rate of ∼ 100 s. This dramatic foreshorten-

ing of the nuclear spin transverse relaxation rate, without a corresponding degrada-

tion of the nuclear spin polarization, is an essential aspect of the co-magnetometer.

Since the co-magnetometer reaches equilibrium in less than a second, the exper-

iment can run through a variety of complex calibration and zeroing procedures,

each requiring many quasi-static co-magnetometer response measurements, in a

finite amount of time. The quasi-static and equilibrium responses are discussed in

great detail in section 2.7.

2.6.2 Oscillatory response dynamics

The response to an applied oscillating magnetic field Bx = B0 cos(ωt) can be also

be solved using the linearized Bloch equations. Plugging in this oscillatory field,

asserting oscillatory solutions Pe⊥ = Pe

0ei(ωt+φe) and Pn⊥ = Pn

0 ei(ωt+φn), and solving

for Pe0 , one obtains

Pe0 =

Pez γeB0ω sin(ωt)

γnλMnPnz (iω + Rtot) + ω(iω + iγeλMePe

z + Rtot)(2.119)

To study the co-magnetometer response to slowly varying fields, take the real part

of this expression and expand the result in ω. The measured oscillating signal

amplitude to first order is

S(ω) =B0γePe

z ω sin(ωt)γnBnRtot

+O(ω2) (2.120)

Terms with higher orders of ω have been neglected because they do not signifi-

cantly contribute at low frequency. Thus, the suppression of an oscillating Bx field

is linear in frequency ω.

2.7. Steady state signal dependence 64

2.7 Steady state signal dependence

The long-term systematic effects of drifting magnetic fields and lightshifts are the

primary concerns in a test of CPT symmetry. Solving the Bloch equations (Equa-

tion 2.103) for the potassium magnetization Pex in steady state and keeping leading

order terms for each magnetic field and lightshift, one obtains the following expres-

sion:

S =Pe

z γeRtot

R2tot + γ2

e (Bz + Lz)2

×[

bny − be

y + Ly +Ωy

γn+

smRm + αRp

γePez

+Bz

(bey + By

Bn − Lxγe

Rtot

)+

γe

Rtot

(BxBz(Bz + Lz)

Bn − LxLz

)](2.121)

where the following notation has been introduced:

Bz = Bn + λMnPnz + λMePe

z (The compensation point) (2.122)

which is near zero when the external field Bn cancels the 3He and (much smaller)

K magnetizations. In the case that all quantities B and L are zero, Equation 2.121

simplifies to

S =Pe

z γe

Rtot(bn

y − bey) (2.123)

Hence, the co-magnetometer signal is proportional to the difference in field cou-

pling to the two spin species. For a regular magnetic field, the coupling to electron

and nuclear spins is equal, bny = be

y, and the co-magnetometer signal vanishes. Not

only does the signal vanish but also sensitivity to small changes in each quantity

is suppressed by at least one small factor.

Figure 2.16 illustrates how the 3He cancels an applied transverse field and main-

tains the K in a nearly zero field environment. In steady state, the 3He polarization


Bn

BxBn

(a) 3He cancels the external field Bn (b) 3He compensates for Bx

I3He

M3He

B

SK

MK

K feels no change

I3He

M3He Bn

SK

MK

K feels no field

Figure 2.16: The co-magnetometer is insensitive to applied magnetic fields: (a) the 3Hemagnetization is cancelled by an externally applied Bn z. (b) In response to a slowly chang-ing transverse field Bx, the 3He spins adiabatically follow the total field and to first orderthe 3He magnetization cancels the applied field.

is aligned with the ambient magnetic field. The ambient magnetic field Bn is ad-

justed to exactly cancel the 3He magnetization, M3He (= Mn), which points oppo-

site to its spin, thereby satisfying the compensation condition in Equation 2.122. If

a small transverse field Bx Bz is introduced, the 3He polarization will settle to lie

along the total magnetic field and, to first order, the 3He magnetization will cancel

the applied transverse field. As long as the transverse field does not significantly

alter the total magnitude of the magnetic field, the 3He spin is able to align with

the total field and its magnetization will cancel the total field.

Note that this compensation behavior only works for transverse fields Bx and

By; a longitudinal field, Bz is not compensated. That is not a problem because

the K magnetometer is generally insensitive to longitudinal fields because they are

parallel to the spins and do not cause precession. This behavior is useful for sup-

pressing low-frequency Johnson noise from the magnetic shields or other drifts in


the ambient magnetic fields. By virtue of keeping the K spins in zero field, the 3He

maintains the low field requirement of the spin-exchange relaxation-free operation

of the K magnetometer.

2.7.1 Steady state signal dependence refinements

The exact solution to the Bloch Equations 2.103 contains a large number of terms, a

great many of which are insignificant for small perturbations around the nominal

configuration. The most significant corrections to the co-magnetometer signal ex-

pressions in Equations 2.121 and 2.123 come from the inclusion of spin-exchange

and spin-destruction collision effects as well as other nonlinear dynamics. Includ-

ing these terms significantly proliferates the quantity of terms in the solution. One

must assign typical experimental values to each parameter and retain terms up to

the desired significance. The particular values chosen for the analysis in this sec-

tion can be found in Table 2.3 and reflect the nominal operating conditions of the

experiment.

It is convenient to place all terms in the solution in terms of a common denomi-

nator and analyze the significance of terms in the numerator and denominator sep-

arately. The denominator in Equation 2.121, R2tot + γ2

e (Bz + Lz)2, can be extended

to include higher order terms:

DSR2tot = R2

tot

(1 +

γ2e

R2tot

(Bz + Lz)2 + 2Cnse − 2

Bz

Bn + 2Dese +O(10−6)

), (2.124)


Parameter Variable Typical Value3He gyromagnetic ratio γn 3.24 kHz/GElectron gyromagnetic ratio γe 2.8 MHz/GK slowing down factor Q(Pe) 5.0Temperature T 160CK number density ne 2.2× 1013 cm−3

3He pressure p 7 amgK polarization field factor λMe 10 µG3He polarization field factor λMn 170 mGK pumping rate Rp 180 1/sK probe beam pumping rate Rm 6.6 1/sK spin-exchange rate Ren

se 26 1/sK spin-destruction rate Re

sd 79 1/sK total relaxation rate Rtot 396 1/s3He spin-exchange rate Rne

se 1.5× 10−5 1/s3He spin-destruction rate Rn

tot 0.001 1/sPump photon spin polarizations sp

z 0.99sp

x , spy 10−4

Probe photon spin polarizations smx 10−4

smy , sm

z 0K polarization Pe

z 54%3He polarization Pn

z 0.93%3He compensation field Bc

z 1.6 mGMagnetic field offset B 10−7 GHorizontal lightshifts Lx, Lz 10−7 GVertical lightshift Ly 0Rotation (earth) Ω 2.58, 5.16, 4.47 × 10−5 rad/secAnomalous fields be, bn 10−8 G

Table 2.3: Typical experimental parameters used to determine the relative significanceof terms in the solution to the Bloch equations. Wherever these values deviate from thetheoretical calculations earlier in this chapter, the new values are informed by direct mea-surements that are detailed in Chapter 3.


where Equation 2.122 was used wherever possible to simplify the expression and

where the following notation was introduced:

Cnse ≡

γePez Rn

seγnPn

z Rtot∼ 10−3, and (2.125)

Dese ≡

MeRese

MnRtot∼ 10−5. (2.126)

The terms immediately to the right and left of 2Cnse in the denominator are also on

the order of 10−3 and omitted terms are entirely higher order corrections.

Analysis of the terms in the numerator of the full solution is broken into sec-

tions for each term of interest. The signal dependence of anomalous fields, rotation,

magnetic fields and lightshifts can analyzed separately, but one must not attempt

to add them all together to obtain a complete expression for the signal dependence

because many of the terms are shared and would be over-counted.

2.7.2 Anomalous field dependence

The leading order corrections to the anomalous field coupling are

S(bey, bn

y) =Pe

z γebny

RtotDS

(1 + Cn

se −Bz

Bn + Dese +O(10−6)

)−

Pez γebe

y

RtotDS

(1 + Cn

se − 2Bz

Bn + Dese +O(10−6)

)(2.127)

Inserting the expression for the denominator in Equation 2.124 and utilizing the

approximation (1 + ε)−1 ' (1− ε) for small ε, one finds

S(bey, bn

y) =Pe

z γebny

Rtot

(1− Cn

se −γ2

e

R2tot

(Bz + Lz)2 +Bz

Bn − Dese +O(10−6)

)−

Pez γebe

y

Rtot

(1− Cn

se −γ2

e

R2tot

(Bz + Lz)2 − Dese +O(10−6)

)(2.128)


When the z magnetic field and lightshift are properly zeroed this expression sim-

plifies to

S(bey, bn

y) =Pe

z γe

Rtot(bn

y − bey)(1− Cn

se − Dese +O(10−6)) (2.129)

Thus, sensitivity to anomalous fields is only slightly reduced by spin-exchange

effects (a reduction of 10−3).

2.7.3 Rotation dependence

The co-magnetometer constitutes a sensitive gyroscope. Rotation Ω of the system

provides an effect analogous to the anomalous fields. The dominant dependence

on rotation, including terms from the denominator, is

Srot =Pe

z γeΩy

γnRtot

(1− γn

γeQ(Pe)− Cn

se −γ2

e

R2tot

(Bz + Lz)2 +Bz

Bn − Dese +O(10−6)

)(2.130)

With Bz and Lz properly zeroed the rotation signal becomes

S(Ωy) =Pe

z γeΩy

γnRtot

(1− γn

γeQ(Pe)− Cn

se − Dese +O(10−6)

)(2.131)

Thus, the co-magnetometer is sensitive to rotation around the y axis and is en-

hanced by a factor γe/γn 1. The slowing-down factor Q(Pe) provides an im-

portant 1% correction. It is convenient to define an effective gyromagnetic ratio to

convert between magnetic and rotation signals:

Ωy = γgBeffy where γg '

(1

γn− Q(Pe)

γe

)−1

. (2.132)

This equivalence allows one to calibrate a co-magnetometer gyroscope using mag-

netic fields instead of rotation.


Rotation around the other axes is suppressed compared to Ωy:

S(Ωz) =Pe

z Ωz

RtotDS

(−γeLxQ(Pe)

Rtot+

γeBy

γnBn +γ2

e BeLx

γnBnRtot+O(10−4)

)(2.133)

S(Ωx) =Pe

z Ωx

RtotDS

(γ2

e (Bz + Lz)γnRtot

− γe(Bz + Lz)Rtot

+γ2

e BeBz

γnBnRtot+O(10−4)

)(2.134)

Although Ωz is suppressed by a factor of 105, Ωx is suppressed by only a factor of

100 compared to Ωy for these parameters.

The rotation of the earth provides a significant offset in the co-magnetometer

signal. The gyroscopic properties of the experiment are experimentally investi-

gated in Chapter 5.

2.7.4 First order experimental imperfections

The pump and probe beams are nominally orthogonal, but small misalignments

can give significant signals. If the beams are an angle α away from orthogonal,

then the signal dependence is

S(α) = αRp

Rtot(2.135)

The signal due to non-orthogonality can be distinguished from other signals by the

dependence on pumping intensity. Whereas signals due to most misalignments

are suppressed at high pumping intensity, this misalignment generates a signal

that increases towards an asymptote with increasing pumping intensity.

The signal also has first order dependence on the degree of circular polarization

of the probe beam:

S(sm) = smRm

Rtot(2.136)


This effect is usually manageable because the probe beam intensity is very low and

the probe circular polarization is periodically zeroed. This effect becomes domi-

nant when the pump beam is blocked and the only remaining sources of K spin

pumping come from the probe beam and through spin-exchange with polarized

3He.

The only first order dependence on the magnetic fields and lightshifts comes

from the Bx field:

S(Bx) =Pe

z Bx

BnDS

(Ce

se + Cnse +

γ2e Bz(Bz + Lz)

R2tot

+O(10−6))

(2.137)

where the last term is order 10−5 and the following notation was introduced:

Cese ≡

Pnz Re

sePe

z Rtot∼ 10−2 (2.138)

The first order sensitivity to Bx is a non-negligible effect that may place a significant

limit on the field suppression capabilities of the co-magnetometer. The first order

sensitivity can be seen in Figure 2.17.

2.7.5 Second order experimental imperfections

Although the co-magnetometer is operated with lightshifts and magnetic fields set

close to zero, it is necessary to periodically introduce nonzero magnetic fields to

calibrate and re-align all the fields to compensate for drifts. Second and higher or-

der corrections to the signal response in Equation 2.121 can have significant effects

on calibrations and zeroing procedures, especially when the applied fields are par-

ticularly large. As with the rotation and anomalous field corrections, most of the

corrections are small factors due to spin-exchange and spin-destruction collisions.


−500

0

500

Sig

nal

(fT

)

−40 −20 0 20 40

Bz (µG)

Ω = Ω⊕

Bx = 500 nG

By = 500 nG

−4000

−2000

0

2000

4000

Sig

nal

(fT

)

−40 −20 0 20 40

By (µG)

−50

−25

0

25

50S

ign

al(f

T)

−40 −20 0 20 40

Bx (µG)

Figure 2.17: Signal response to Bx, By and Bz scans according to numerical simulations ofthe full Bloch equations (Equation 2.103). Top: The small first order dependence on Bx isdue to the spin-exchange terms found in Equation 2.137. Middle: To first order the systemis insensitive to By. Bottom: The earth’s rotation rate gives a slight first order dependenceon Bz around zero.


In zeroing Bz, the term S ∝ ByBz is typically utilized. For applied By field with

nonzero Bz field, the signal is

S(By, Bz) =Pe

z γeByBz

BnRtot

×(

1 +Ωz

γnBn −Bz + Lz

Bn Cese −

2Bz + Lz

Bn Cnse −

γ2e

R2tot

(Bz + Lz)2 +O(10−5))

(2.139)

The Ωz term is order 10−2. For vanishing Bz, this form simplifies to

S(By) =Pe

z γeBy

Rtot

(Ωz

γnBn −Lz

Bn (Cese + Cn

se) +O(10−8))

(2.140)

where the first term in the parentheses is order 10−5 and the second term is order

10−7.

When calibrating the magnetometer, one modulates By and measures the signal

response as a function of Bz. The slope of this response also has spin-exchange

corrections:

∂2S∂By∂Bz

=γePe

zBnRtot

(1− Cese − 2Cn

se +O(10−4)) ' γePez

|Bn|Rtot(2.141)

where the last step represents the usual approximations that are made in calculat-

ing the calibration. These additional terms provide a significant 1% correction to

the calibration.

For applied Bz field, the signal is

S(Bz) =Pe

z γeBz

RtotDS

(−γeLx

Rtot+

γeΩx

γnRtot+

Rpspy

Pez Rtot

+By

Bn +γe

Rtot+O(10−6)

)(2.142)

Note the strong, order 10−2, BzLx term in this expression; for Lx ∼ By, the signal

due to the Lx term is 103 times larger than the By term because γeBn Rtot. This


−500

−250

0

250

500

Sig

nal

(fT

)

−40 −20 0 20 40

Lz (µG)

Ω = Ω⊕

Lx = 190 nG

Bz = 10 µG, Bx = 1 µG

−10

−5

0

5

10

Sig

nal

(fT

)

−40 −20 0 20 40

Lx (µG)

Probe pumping

Lz = 10 µG

Figure 2.18: Signal response to lightshifts according to numerical simulations of the fullBloch equations. Top: The probe beam lightshift is dominantly a polarization pumpingeffect corresponding to the first order dependence in Equation 2.136; by comparison theterms in Equation 2.143 are insignificant. Bottom: The signal dependence on the pumpbeam lightshift showing the significance of the most important terms in Equation 2.144.

term is of particular concern because Bz is constantly drifting on account of drifting

3He polarization. Furthermore, the probe beam lightshift Lx is difficult to zero

everywhere in the cell because of potential non-uniform birefringence of cell walls.

Note, also, the dependence on the earth’s rotation Ωx of order 10−4. The rotation

sensitivity in both Ωx and Ωz provides the curious Bz dependence shown in Figure

2.17. The vertical pumping term Rpspy is on the order of 10−5.


For applied Lx, the signal is

S(Lx) =Pe

z γ2e Lx

R2totDS

×(−Lz − Bz − Bz

λMePez

λMnPnz

+ 2BzLz + B2

zBn − Ωz

γeQ(Pe) +O(10−12 G)

)(2.143)

Terms in the parentheses are order 10−7, 10−7, 10−10, 10−11 and 10−11 Gauss. If

the probe laser is detuned from resonance, the lightshift effects described here are

typically much smaller than the probe beam pumping term given by Equation

2.136. Indeed, in the simulations plotted in Figure 2.18, the lightshift only provides

a small perturbation on the signal due to pumping by the probe beam.

For applied Lz, the signal is

S(Lz) =Pe

z γ2e Lz

R2totDS

(−Lx +

Ωx

γg+

Rpspy

γePez

+2BzLx + BzBx

Bn +O(10−13 G)

)(2.144)

Terms in the parentheses are order 10−7, 10−9, 10−12, and 10−12 Gauss. The LxLz

term provides a clean method to zero the lightshifts. The dependence of the sig-

nal on Lz is plotted in Figure 2.18. The nonzero slope around Lz = 0 is due to

the earth’s rotation rate. The lightshift drift according to Equation 2.89 is about

dLz/dλ = 1.4µG/pm (140 pT/pm). According to Equation 2.144, with everything

perfectly zeroed, drifting wavelength provides a signal due to the earth’s rotation

of dS/dλ = 1.3 fT/pm. Thus, it is important to have a very stable pump laser and

a robust zeroing routine that maintains Lz at zero.


−0.25

0

0.25

0.5

0.75

1

Sig

nal

(fT

)

0 200 400 600 800

Pumping rate (1/sec)

bey = 8.16 pG

Ω = 26.3 nrad/s

Bx = 159 nG

By = 11 nG

Lx = 184 pG

α = 75 nrad

β = 5 µrad, Lz = 1 µG

Nominal operating point

Figure 2.19: Leading order contributions to the co-magnetometer signal are plotted as afunction of pumping intensity. Each contribution term is given a magnitude sufficient togenerate a maximum effective 1 fT signal. At the nominal operating point, contributionsfrom Bx, By, and Lx are explicitly zeroed out by zeroing procedures and are suppressedby at least one small factor. Note that the potassium polarization contribution to the totalBz field varies with pumping rate, which is what ultimately brings out the dependence ofmost of these terms.

2.7.6 Signal pumping intensity dependence

For the purpose of making measurements, the co-magnetometer is operated with

the pump beam intensity set to maximize the sensitivity. There are several in-

stances, however, when the intensity is increased or decreased for calibration or

diagnostic purposes. The signal due to various misalignments as a function of

pump beam intensity can be found in Figure 2.19. This plot is based primarily on

the leading order dependence of each term found in Equation 2.121. There is one

value of pumping intensity at the nominal operating point for which the sum of the

K and 3He polarizations are exactly compensated and Bz = 0. At other intensities,

the potassium polarization contribution to the total Bz field varies with pumping


rate, which elicits the dependence of most of these terms. There are only two ef-

fects that do not approach zero at high pumping rate: the first is the pump-probe

non-orthogonality angle α and the second effect arises from generating a vertical

lightshift Ly by steering the pump beam lightshift Lz by an angle β into the vertical

direction. The latter effect is brought to zero when the magnetic fields are aligned

using zeroing procedures. The non-orthogonality is explicitly zeroed out through

zeroing procedures. After everything else is properly zeroed, what remains is the

signal due to rotation of the earth corresponding to the black curve in this plot.

At very low pumping intensity many imperfections are strongly amplified. Sen-

sitivity to imperfections in Bx, By and Lx peak around Rp ' 10 1/s. When the

pump beam is blocked, the pumping rate does not reach zero but rather decreases

only to the level of the 3He spin-exchange pumping. A convenient way to suppress

this sensitivity involves applying a strong Bz field whenever the pump is blocked

for a low intensity background measurement. The large Bz primarily suppresses

the signal by making the denominator, DS ' 1 + B2z γ2

e /R2tot + · · · , very large (Equa-

tion 2.124).

Chapter 3

Co-magnetometer implementation

THE CO-MAGNETOMETER SEARCHES for a sidereal signal as the earth rotates

through a background anomalous field b. The previous chapter provided

a solid theoretical design for a co-magnetometer that is sensitive to anomalous

fields and largely insensitive to other imperfections. This chapter describes the im-

plementation of the co-magnetometer and verifies that the magnetometer behaves

as described in the previous chapter. The various measurements throughout this

chapter are not always directly comparable to each other and to the theoretical

expectations because they were taken over the course of three years under differ-

ent operating conditions. The objective of the characterization measurements pre-

sented in this chapter is to provide at least a qualitative verification of the most

important aspects of the co-magnetometer.

This chapter begins with a description of the experimental apparatus in Sec-

tion 3.1. The characterization measurements are contained in Section 3.2. Section

78

3.1. Co-magnetometer setup 79

3.3 discusses the methods that were developed to minimize imperfections in the

experiment such as misaligned laser beams and non-zero magnetic fields.

3.1 Co-magnetometer setup

The K-3He co-magnetometer is created using an optical pump-probe setup shown

in Figure 3.1. K vapor is held in a spherical, 2.5 cm diameter GE180 aluminosilicate

glass cell with thin walls. The walls of the cell, seen in Figure 3.2 are sufficiently

thin at about 6 mil (0.15 mm) that the light is not significantly deflected except at

the edges. Also in the cell are 50 torr N2 gas to quench the excited state and 7 amg

3He gas. The 3He has a few important roles: it provides a nuclear spin, it is an

effective buffer gas to reduce diffusion, and it broadens the K lines to increase the

optical depth of the vapor.

A double-walled oven shown in Figure 3.2 heats the cell to create a K vapor

density of n ∼ 1013 cm−3 by flowing 160C hot air between the walls that surround

the cell. The oven is constructed using G7, a woven fiberglass-epoxy composite,

and is insulated with a uniform 2.5 cm layer of fiberglass insulation. The seams of

the oven are sealed using Fomblin, a high temperature paste made of a Teflon-like

fluoropolymer. The double-walled design keeps turbulent hot air out of the optical

path and uniformly heats the cell. The insulated oven is placed in a water-cooled

box made of high thermal conductivity epoxy. The cell-oven-cooling-box assembly

resides inside a rigid G10 tube, shown in Figure 3.3, around which magnetic field

and gradient coils are wrapped. Five layers of high permeability metal magnetic

shields surround this assembly. Between each magnetic shield, white melamine


Oven

Probe BeamOptical Table

Optical TableLower Section

Encl

osur

es

Mag

netic

Shi

elds

Pump Beam

Pola

rizer Position Detector

DeviatorPinhole

Piezomirror

IsolatorTranslating Lens

Collimating Lens

(Analyzing) Polarizer

Photodiode

Position Detector

Probe BeamDiode Laser

Fabr

y-Pe

rot

Fara

day

Mod

ulat

or

Field Coils

l/2

y z

x

Pock

el C

ell

PinholePo

lariz

er

l/4

l/2VariableWaveplate

Polarizer

Shutter

Lenses

Final Mirror

Translation Stage

Position Detector

Spectrometer

Pump Beam Diode Laser

Cell

Hot Air FlowCollimator

InsulationCooling Jacket

Position feedback

Figure 3.1: Schematic of the experiment indicating: pump and probe lasers with surround-ing enclosures, frequency and intensity feedbacks, beam steering optics, the double-walledoven and magnetic shielding. The relative position of each component in this figure resem-bles the configuration in the actual experiment.


Figure 3.2: Left: The co-magnetometer cell is held by its stem. The wired device is a thin-film 10 kΩ RTD temperature sensor. Note that the potassium has migrated to a cold spoton the cell. Middle: The double-walled oven has an inner volume for the cell and an airgap between the two walls through which hot air flows. Right: Chilled water cools theexterior of the cooling shield, which is made of a thermally conductive epoxy.

acoustic absorption foam is installed to reduce acoustic noise. Everything is held

in place with a Ti frame and radial Ti rods that penetrate the magnetic shields

and directly support the G10 tube coil form. Evacuated glass tubes bring the laser

beams in from the outside through the magnetic shielding and thermal shielding

directly to the inner oven where the cell resides. The evacuated tubes eliminate

turbulent convection driven by the large thermal gradient between the oven and

the room.

Care was taken to minimize metal and other sources of magnetic noise inside

the magnetic shields of the experiment. The only metals and currents come from

the temperature sensors distributed around the oven and they are only operated

using high frequency AC modulation or they are momentarily operated with DC

excitation when data is not being taken. The AC excitation frequency is beyond the

bandwidth of the magnetometer. The magnetic shields are electrically insulated

from the support structure so as to avoid creating Ti-Fe thermocouple junctions.


Figure 3.3: Above: The five layers of high-permeability metal suppress external fields bya factor of 106. A Ti frame supports the shields and firmly holds the green G10 piece in thecenter. The oven assembly rests inside the G10 coil form. The white layers are melaminesound absorbing foam. Below: View of the entire experiment, showing the open thermalshields surrounding the experiment and Lexan enclosures around the optics. The magneticshields are covered by fiberglass insulation.


Great effort was expended controlling the temperature of the experiment. From

the beginning of the experiment, thermal drift of various components has con-

tributed to the signal in a variety of ways. Section 4.2.3 discusses thermal sensi-

tivity in greater detail. To improve temperature stability a number of parts were

added to the experiment. The pump and probe lasers have independent 10 mK-

stable analog temperature controllers that control the temperature of the laser mou-

nts by adjusting the temperature of cooling water. In Figure 3.3, fiberglass insu-

lation covers the magnetic shielding and a high performance thermal insulation

foam board forms an insulating shield around the entire experiment. In a more

recent improvement, an air conditioner in an adjacent room now pumps cold, dry

air through a bucking heater and blows temperature-controlled air into the thermal

shields. The positive pressure on the thermal shields prevents room temperature

air from leaking through cracks in the thermal shields.

3.1.1 The pump laser

The pump beam is formed using a diode laser tuned onto the K D1 resonance at

770 nm. Three types of laser diode have been used to create the pump beam. At

first several brands of 1 W broad area laser diode were used in an external grating

cavity feedback configuration. These diodes featured a broader linewidth that had

a longer effective optical depth and could be tuned in a reasonably smooth fashion

by adjusting the feedback grating. Their spectrum was significantly smoothed by

dithering or modulating the laser driver current. However, the broad area diode

lasers suffered from having short lifetimes, high mode noise and significant drift.

Half way through the data set in this thesis (at around 1800 sd), the experiment


started using a single mode, tapered amplifier diode laser for the pump beam. The

single mode laser greatly improved the mode noise and drift but was unable to

tune the wavelength smoothly over a wide range, making routine zeroing of the

pump lightshift impossible. Wavelength stability was achieved by carefully con-

trolling the temperature of the entire laser assembly to < 10 mK. A further up-

grade (at around 2060 sd) to a distributed feedback (DFB) laser is planned in order

to achieve both single mode operation and smooth wavelength tuning. Since DFB

lasers do not use an external cavity, they are less sensitive to ambient air tempera-

ture.

Glass wedges pick off a few percent of the pump beam in order to measure

intensity and wavelength. Intensity stabilization for the pump laser is achieved us-

ing feedback to a liquid crystal variable waveplate. While running with the broad

area diode lasers, the pump wavelength was stabilized using a spectrometer with

a two-segment photodiode at the outlet and a feedback to the laser external cav-

ity grating angle. Although wavelength feedback is not possible with the tapered

amplifier pump laser due to its tendency to hop modes, wavelength feedback and

adjustment is possible using a DFB laser.

The spatial profile of the single mode pump beam is significantly improved by

focusing the pump beam through a pinhole. After the pinhole, the pump beam

has a smoother, Gaussian intensity profile. A set of expansion optics creates a 3

cm wide by 1 cm high rectangular beam shape. The spherical aberration of short

focal length lenses is utilized to make the intensity profile slightly more uniform.

The beam is made uniform in intensity and wider than the cell so that small move-

ments in the pump beam do not significantly change which parts of the cell are


pumped. The position of the pump beam is measured using a four-segment pho-

todiode, positioned such that the position measurement reflects the position at the

cell. Finally, the pump beam is circularly polarized using a λ/4 waveplate before

it enters the glass tube, goes into the cell and pumps the K vapor in the cell.

3.1.2 The probe laser

The probe beam passes perpendicularly to the pump beam through the cell and

measures the spin polarization along the direction of propagation. It is created by

a single frequency diode laser with a tapered amplifier. Intensity stabilization is

accomplished by employing a feedback to the tapered amplifier current. A Fabry-

Perot interferometer can be used to stabilize the frequency of the probe beam by

modulating the cavity length and locking the frequency. However, the first version

of the interferometer was sensitive to barometric pressure changing the effective

path length of the cavity. A second, evacuated version of the interferometer was

sensitive to its temperature despite the use of an Invar tube. Without a solid wave-

length measurement, the wavelength of the probe beam is stabilized by careful

< 10 mK temperature control of the laser assembly. An upgrade to a DFB laser

may improve the wavelength tunability and reduce the broadband spontaneous

emission from the tapered amplifier. The probe beam passes through a protective

optical isolator and a pinhole to create a Gaussian spatial profile. After the glass

wedge pickoff for the intensity and wavelength feedbacks, the probe beam is atten-

uated by a pair of high contrast Glan-Laser calcite crystal polarizers. The probe

laser next passes through a Faraday modulator to modulate the angle of polariza-

tion, a Pockel cell to adjust the total circular polarization of the beam and a series


of beam expansion and steering optics. The expanded probe beam is about 1 cm

square and passes unimpeded through evacuated glass tubes and the cell in the

middle. The evacuated glass tubes are made using wedged windows on both ends

to reduce any interference effects. Four-segment position detectors measure the

position of the beam where it passes through the cell (at an equivalent but external

location) and after the cell.

The probe beam measures the component of the K polarization parallel to the

beam using optical rotation. To obtain a very high precision measurement of the

probe beam polarization angle, the polarization angle is modulated so that the sig-

nal is separated from 1/ f noise. The Faraday modulator uses a Tb-doped glass

Faraday rod surrounded by magnetic driver coil to modulate the angle of polar-

ization by α = 5 at a frequency ωm = 2π × 4.8 kHz before passing through the

cell. The probe beam picks up an angle φ due to optical rotation in the cell before

passing through a final, analyzing calcite polarizer set to extinction. The measured

photodiode signal amplitude after the final polarizer is given by

I = I0 sin2[α sin(ωmt) + φ] (3.1)

' I0[α2 sin2(ωmt) + 2φα sin(ωmt)] (3.2)

A lock-in amplifier measures the first harmonic component of the dominantly sec-

ond harmonic signal:

S ∝ Iωm ' 2I0φα (3.3)

If there is no optical rotation φ then the first harmonic has zero amplitude. Using

this modulation technique, angular sensitivity of a few 10−8 rad/√

Hz has been

achieved, which is sufficient to measure below 1 fT/√

Hz in this experiment.


−50

0

50

Dev

iati

on

(mra

d)

−1 −0.5 0 0.5 1

Impact parameter (cm)

Thin probe beam

5 mm probe beam

Figure 3.4: Light is defocused by the cell walls and is worst at the edges. The effects aresomewhat mitigated by using a broad beam.

The Faraday rotator is a large assembly that generates a lot of heat and mechan-

ical noise. To reduce drift due to temperature variation, an optimized driver coil

was designed and fabricated with temperature-controlled chilled water flowing

around the inside surface of the coil. To reduce mechanical drifts, the Faraday rod

is mounted in a form-fitting phenolic tube that passes through the center of the

coils without touching the coil assembly at all and is separately held by posts on

the optical table. A µ-metal magnetic shield surrounds the coils to prevent ambient

magnetic noise from generating rotation and also to prevent the driver field from

affecting the surrounding electronics.

The pump and probe beams are enclosed by 0.5 in thick Lexan boxes with rub-

ber seals to reduce air currents in the room flowing through the path of the beam

and introducing extra noise. These enclosures significantly improve the short-term

noise of the signal and may also contribute to the long-term stability.


Untitled-2 3

Figure 3.5: Light is defocused by the cell walls and is worst at the edges. A single plano-convex lens can correct for the defocusing of the majority of the rays. The cell thicknesshas been greatly exaggerated for illustration purposes.

With ∼ 0.006 thick walls, the co-magnetometer cell is a weak lens. To maintain

a clean probe beam spot all the way through the experiment, one must compensate

for the slight lensing effect of the cell. Figure 3.4 shows a calculation for the angle

that a beam deviates as a function of the impact parameter on the cell. Figure 3.5

demonstrates that a simple plano-convex lens is sufficient to correct for the vast

majority of the cell defocusing. For the cell presently used in the experiment, an

f = 500 mm lens works well.

3.1.3 Probe beam steering optics

A set of three beam expansion and steering lenses shown in Figure 3.6 are used to

find the “sweet spot” where the beam passes through the center of the cell. The

significance of the sweet spot will be discussed in greater detail in Section 4.2.4.

The three components are a deviator, which steers the beam by adjusting a vari-

able angle glass wedge, a lens mounted on a 3-axis translation stage and a fixed

collimating lens. The deviator is constructed using matching concave and convex


Figure 3.6: Probe beam lens configuration for independent control of beam translation andangle at the cell.

lenses and an index-matching fluid that allows the two lenses to smoothly slip past

each other. The positions of the steering lenses are designed such that the deviator

and the lens on the translation stage independently change the angle of the beam

at the cell and the position of the beam at the cell, respectively. The exact positions

of these lenses were calculated using ABCD-type matrix operators representing

lenses and beam propagation (Tovar and Casperson, 1995):x

θ

1

= Mtot

x0

θ0

1

(3.4)

where x is the offset from an axis formed by a beam in the absence of any lenses, θ

is the angle of the beam relative to that axis and the total transformation matrix is

given by

Mtot = Md(d3)M f ( f2, x2)Md(d2)M f ( f1, x1)Md(d1) (3.5)


The propagation of light a distance d is given by the matrix operator,

Md(d) =

1 0 0

d 1 0

0 0 1

(3.6)

and the matrix operator for a lens of focal length f and offset x is

M f ( f , x) =

1 0 0

−1/ f 1 x/ f

0 0 1

(3.7)

The solution for independent position and angle adjustment of the beam is deter-

mined by the set of distances d1, d2, d3 between the lenses for which the position

offset at the cell is zero, x = 0, for arbitrary θ0. The total length of the beam path

is constrained by dtot = d1 + d2 + d3 and the distance between the two lenses is

constrained by being properly collimated: d2 = f1 + f2. The solution is

d1 =f1[( f1 + f2)2 − dtot f1]

f 22 − f 2

1(3.8)

d2 = f1 + f2 (3.9)

d3 = dtot − d2 (3.10)

With a fixed second lens, x2 = 0, the orientation of the probe beam at the cell is

x =dtot f1 f2 − f2( f1 + f2)2

f 31 − f1 f 2

2x1 (3.11)

θ = − f1θ0 + x1

f2(3.12)

Thus, the deviator angle θ0 adjusts the angle with which the beam passes through

the cell θ without moving the beam off the center of the cell. The position and


angle are not in fact completely independent; position adjustment x1 does change

the angle at the cell θ. Currently in use are lenses f1 = 100 mm and f2 = 250 mm

at positions d1 = 49 mm, d2 = 350 mm, d3 = 568 mm for dtot = 967 mm. The beam

is expanded by a factor of 2.5.

The independence of the angle adjustment can be verified by monitoring the

position of the probe beam on the four-segment position detector as the deviator

modulates the angle of the beam. The position detector is placed at a position

equivalent to the position of the cell. The positions of the lenses are tweaked until

the deviator modulation does not produce any translation at the detector; only

modifying the angle with which the beam hits the detector. This setup is not only

important for finding the sweet spot at the center of the cell (see Section 4.2.1),

but is also essential in zeroing the pump-probe orthogonality without moving the

probe beam off the center of the cell.

The position of the probe beam is fixed on the cell using the position detector

and a feedback to a piezo-controlled mirror as shown in Figure 3.1. Together with

zeroing of the pump-probe nonorthogonality, the horizontal angle, the horizontal

position, and the vertical position of the probe beam are maintained under feed-

back.

3.2. Co-magnetometer characterization 92

−0.2

−0.1

0

0.1

0.2

0.3

Lo

ck-i

nS

ign

al(V

rms)

10 20 30 40 50

Chopper Frequency (Hz)

1.2 Hz

Figure 3.7: The transverse resonance of the K magnetometer in response to chopped pumplight. The in-phase amplitudes fit a Lorentzian with a half-width of 1.2 Hz.

3.2 Co-magnetometer characterization

The co-magnetometer is evaluated first by considering the performance of a bare,

potassium magnetometer and second by probing the dynamics of a coupled K-3He

spin system.

3.2.1 The potassium magnetometer

Momentarily ignoring the 3He spins, the K atoms form a very sensitive SERF mag-

netometer. The SERF magnetometer was originally developed for this experiment

and the findings were published in Allred et al. (2002) and Kominis et al. (2003).

Several of the important experimental results are reproduced here.

The high sensitivity of a SERF magnetometer derives from the very low relax-

ation rate of the spin. This behavior can be fully appreciated only if other sources

of relaxation are minimized. The pumping rate at typical operating intensities is a


0

2

4

6

Res

on

ance

hal

f-w

idth

(Hz)

0 50 100 150 200 250

Chopper Frequency (Hz)

3 atm 3He Cell

∆ν = (79 µs)ν2 + (1.08 Hz)

1.08 Hz

Figure 3.8: Transverse resonance linewidths for a range of resonant synchronous pumpingfrequencies. In the limit of low frequency, this magnetometer has a linewidth of 1.1 Hz.The quadratic magnetic field dependence according to Equation 2.17 provides a fit of Tse =8 µs.

dominant source of relaxation. A synchronous pumping technique, whereby the

pump beam is chopped at the precession frequency of the atoms, allows the atoms

to precess without pumping-induced relaxation for a significant fraction of time.

In Figure 3.7, the chopper frequency is scanned across the resonant frequency of

atoms precessing in a finite magnetic field. The frequency response is accurately

given by Equation 2.32 as a Lorentzian centered on the free precession frequency.

The linewidth of Γ = 1.2 Hz is much smaller than the spin-exchange rate of 20 kHz,

indicating that the magnetometer is indeed in the SERF regime and the linewidth

is determined by the much slower spin-destruction rate. For these measurements

the chopper was modified to pass light with a duty cycle of less than 20% so as to

minimize pumping relaxation.

A series of these synchronously pumped linewidth measurements were made

for a range of applied magnetic fields. According to Equation 2.17 the relaxation


Frequency of applied By (Hz)

Mag

netic

Fie

ld (f

T/H

z1/2 )

0.1

1

10

100

140120100806040200

Figure 3.9: Noise levels in a single channel (dashed) and a differential measurement be-tween two adjacent segments of the probe beam, 3 mm apart. Noise levels were limited bythe magnetic Johnson noise generated by the magnetic shields. For frequencies < 20 Hz,the noise was limited by 1/ f noise and other technical sources.

rate due to spin-exchange collisions increases quadratically with applied field. In-

deed, the data in Figure 3.8 confirms this dependence and allows one to infer

the fundamental relaxation rate R = 1.1 1/s for this cell in the limit of no spin-

exchange relaxation at zero field. The quadratic increase in relaxation rate with

magnetic field agrees with Equation 2.17. From the fit to these data, one finds a

spin-exchange time of Tse = 8 µs, which is in agreement with other measurements

by Ressler et al. (1969).

This fundamental relaxation rate determines the fundamental magnetic field

sensitivity of the magnetometer. Using the expression in Equation 2.41 for the

shot-noise limited sensitivity of this magnetometer, one obtains 20 aT/√

Hz, using

density N = 3× 1013 cm−3 and volume V = 3 cm3, corresponding to the overlap

of the pump and probe beams. The measured noise of the magnetometer, shown in


Figure 3.9 is in practice limited by magnetic noise generated by Johnson noise cur-

rents flowing in the conducting magnetic shielding. The shield noise of 7 fT/√

Hz

is observed over a wide range of frequencies from 20 to 100 Hz. Above 100 Hz,

the bandwidth of the magnetometer decreases and below 20 Hz sources of 1/ f

drift dominate the noise. The amount of 1/ f drift was later significantly reduced.

A differential measurement between two adjacent sections of the probe beam im-

aged onto a two-segment photo diode is capable of eliminating the common-mode

noise between the two channels. The differential measurement gives a lower noise

level of about 500 aT/√

Hz, which is limited by magnetic field gradient noise.

The probe beam can be further partitioned and imaged onto a segmented photo-

diode to measure the magnetic field structure throughout the cell. For this applica-

tion, a T-shaped cell with three flat windows allows the undistorted transmission

of the probe beam through the cell. This setup allows more accurate measurements

of the magnetic fields near the edges of the cell, which are closer to the sources be-

ing measured. By measuring higher order field gradients one can localize and

characterize certain sources of magnetic field. The measurements in Figure 3.10

demonstrate this ability for a small, 1 cm diameter current loop placed 5.3 cm

above the center of the magnetometer cell. The measurement of the characteris-

tic 1/r3 dependence of the magnetic field uniquely determines of the location of

the current loop. The spatial resolution of this magnetometer is limited by the dif-

fusion length of K atoms within one coherence time. For this magnetometer, the

spatial resolution limit is estimated to be about 2 mm.


Position y (cm)

Mag

netic

Fie

ld (f

T)M

agne

tic F

ield

(fT)

Linear Photodiode Array Channel Number

Applied dBy/dy Gradient

Applied External Current Loop

-1.0 -0.5 0.0 0.5 1.0

-200

-100

100

200

300

100

120

140

160

180

200

1 2 3 4 5 6 7

0

yz

PumpBeam

Probe Beam

5.3 cm

Current Loop

Cell

Figure 3.10: An applied magnetic field gradient (top) and a external current loop (bottom)are imaged by projecting the probe beam onto a 7-channel segmented photodiode. Themagnetic field gradient dBy/dy = 315 fT/cm was applied with a 25 Hz modulation. Themeasurements are consistent with a straight line to within measurement error. The outer-most, hollow points were ignored because they had low signal. The external current loopgenerated a dipole µ = 1.25 µA/cm−2 oscillating at 25 Hz. The fit to a dipolar field locatesthe current loop at 5.3± 0.1 cm, which is accurate to within the measurement error. Thelarge error bar indicates the noise in a single channel. The smaller error bars indicate thenoise level with common mode noise removed.


Frequency of applied By (Hz)

Freq

uenc

y Re

spon

se (a

rb.) 20

15

10

5

140120100806040200

Figure 3.11: Bandwidth of the potassium magnetometer, broadened from the 1 Hz limitby relaxation due to spin pumping. The fit line is a Lorentzian centered on zero Hz witha half width of 20 Hz. Points chosen for this measurement avoided sources of noise at, forexample, 60 Hz.

The bandwidth and frequency response of the magnetometer at zero field is

given by a Lorentzian line shape according to Equation 2.31. The frequency re-

sponse curve is measured in Figure 3.11 and the shape of the curve is found to

agree with theory to within measurement error. The bandwidth of the magneto-

meter is found to be 20 Hz (126 1/s), which is consistent with the pumping rate

of the atoms being the dominant relaxation rate. This is much larger than the

linewidth in the synchronous pumping case for which the linewidth is limited by

spin-destruction collisions.

3.2.2 Coupled spin ensembles

In the co-magnetometer, spin-exchange collisions transfer polarization from the K

electron spins to the 3He nuclear spins. In general, the two spin ensembles interact


−1

−0.5

0

0.5

1

Am

pli

tud

e(a

rb.)

0 100 200 300 400 500 600

Time (s)

T2 = 210 s

Figure 3.12: Uncoupled T2 measured “in the dark” with the pump beam blocked for all butthree short measurement periods. The jumbles of points represent the rapidly oscillating3He polarization when the pump beam was momentarily on.

with each other wherein the K and 3He spin ensembles precess around the magneti-

zation of the other. In Figure 3.12 the 3He spins are freely precessing at a frequency

ω = γnB with a lifetime of about T2 = 210 s. The pump beam is only momentar-

ily on so that the 3He is mostly precessing in the absence of any coherent interac-

tions with K. If the pump beam is now turned on, the K are polarized and their

magnetization alters the total effective field that the 3He experience by an amount

∆B = λMePe. In Figure 3.13, the shift in precession frequency according to the

K magnetization is plotted for a range of pumping intensity. For each data point,

the K polarization was flipped by alternating the pumping light between left- and

right-circular polarizations. From the magnetization saturation level, BK = 45.1

µG, one infers a K density of NK = 5.3× 1013 cm−3 using Equation 2.97. This is

consistent with the density NK = 6.0× 1013 cm−3 inferred from a 180C cell. The

relaxation rate of 62 1/s is consistent with the prediction in Equation 2.26.


0

10

20

30

40

50

Po

tass

ium

Mag

net

izat

ion

Fie

ld(µ

G)

0 200 400 600 800 1000

Pumping Rate (1/s)

BK = (45.1 µG)Rp/(Rp + 61.85 1/s)

Figure 3.13: Potassium polarization measured by the 3He frequency shift for alternatingleft- and right-circularly polarized pumping light for a range of intensities. The polariza-tion fit is correctly predicted by Equation 2.8.

As discussed above in Section 2.6.1, the K and 3He dynamics become strongly

coupled when their precession frequencies approach resonance. Whereas the cou-

pling in Figure 3.13 amounted to a 1% effect, at resonance the K and 3He dynamics

merge into a single hybrid response that has an entirely distinct frequency and de-

cay rate. Figures 3.14 and 3.15 show the transient response of the co-magnetometer

as the external field is tuned through the compensation point. The dynamics are ac-

curately predicted by the analytical solutions in Equation 2.116. The complete set

of transient responses was simultaneously fit to the analytical response equations

with the only free parameters being the K and 3He polarizations and relaxation

rates, the tipping field, and an overall magnetic field offset, which were common

to all data. The externally applied fields Bn for each transient were accurately

recorded and used as fixed parameters with only an overall offset as a fit variable.


1

0 0.2 0.4 0.6 0.8 1

Time (s)

Bn

= −0.750 mG

1B

n= −0.975 mG

1 Bn

= −1.038 mG

1 Bn

= −1.076 mG

1B

n= −1.094 mG

1B

n= −1.107 mG

1B

n= −1.119 mG

1 Bn

= −1.132 mG

1 Bn

= −1.151 mG

1B

n= −1.219 mG

1

Bn

= −1.378 mG

Co

-mag

net

om

eter

Sig

nal

(arb

.)

Figure 3.14: The transient response to a transverse excitation. With Bz 6= 0, the K and 3Heprecession becomes decoupled and the signal clearly exhibits separate oscillatory contribu-tions. The analytical transient response function in Equation 2.116 fit the data.


1

0 0.05 0.1 0.15 0.2

Time (s)

Bn

= −0.750 mG

1B

n= −0.975 mG

1 Bn

= −1.038 mG

1 Bn

= −1.076 mG

1B

n= −1.094 mG

1B

n= −1.107 mG

1B

n= −1.119 mG

1 Bn

= −1.132 mG

1 Bn

= −1.151 mG

1B

n= −1.219 mG

1

Bn

= −1.378 mG

Co

-mag

net

om

eter

Sig

nal

(arb

.)

Figure 3.15: The transient response to a transverse excitation. Same data and fit as inFigure 3.14, with the first 0.2 s expanded.


The fit lines agree with the transients remarkably well, providing reasonably accu-

rate estimates of the system parameters: the K magnetization is 16 µG, the 3He

magnetization is 1.07 mG, the K relaxation rate is Rtot = 226 1/s and the 3He re-

laxation rate is Tn2 = 202 s. The 3He relaxation rate is consistent with the data

in Figure 3.12. The K relaxation rate is dominated by pumping relaxation. The

fast K response at far detuning from the compensation point is suppressed in both

the experimental measurements and the analytical response by a low-pass filter.

The differences between the filters in the fit model (single pole) and measurement

(multi-pole) may introduce the observed phase lag in the high frequency compo-

nents of the signal.

The analytical form of the transient response solutions has two oscillators corre-

sponding to the K and 3He atoms. Although on resonance the K and 3He responses

are indistinguishable, the separate dynamics are particularly clear for Bn far from

the compensation point. The lower frequency and longer decay time oscillator cor-

responds to the precession of the 3He atoms. The higher frequency K precession

decays quickly and is filtered out by the data acquisition equipment. Thus, suffi-

ciently far from the compensation point the entire transient response signal is well

described by a single decaying oscillator. The frequency and decay rate of a single

decaying oscillator fit to the transient data are plotted in Figure 3.16. The frequency

and decay rate data were simultaneously fit to Equations 2.117 and 2.118. The fits,

however, give parameters that are not entirely consistent with other measurements:

the 3He magnetization is 1.10 mG, the K magnetization is 18 µG, the K decay rate

is 205 1/s and the 3He decay time is 240 s. These data are reasonably consistent

previous measurements and expectations.


0.001

0.01

0.1

1

10

Dec

ayR

ate

(1/

s)

0.8 1 1.2 1.4

Magnetic Field (mG)

0

2

4

6

8

Res

po

nse

Fre

qu

ency

(Hz) Uncoupled K

Uncoupled 3He

Figure 3.16: The effective frequency and decay time of the transient response as a functionof Bz detuning. The fit lines are defined by Equations 2.117 and 2.118.


0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

Sig

nal

(V)

0 10 20 30 40 50

Frequency (Hz)

Bn

= 0; “Tuned”

Bn∼ 0; “Nearly tuned”

Bn

> 0; “Detuned”

Bn

< 0; “Detuned”

Hybrid Resonance

3He Resonance

K Resonance

Figure 3.17: Chirped By fields were applied to the co-magnetometer to obtain these fre-quency response curves for several values of the external field tuned through the compen-sation point.

The frequency response of the co-magnetometer shown in Figure 3.17 provides

another perspective on the behavior of the coupled spin system. When the external

field is detuned from the compensation point, the separate, uncoupled K and 3He

spin resonances are clearly visible. As the compensating field is tuned towards

the compensation point, the spin resonances merge into a single hybrid resonance

peak. At the compensation point, the frequency response below 6 Hz (the 3He

frequency) drops significantly faster than in the other cases. This indicates that the

co-magnetometer is properly compensating for applied fields.

The low frequency suppression of applied magnetic fields is further investi-

gated in Figure 3.18. The sensitivity to the applied Bx field is suppressed linearly

with the frequency, in accordance with Equation 2.120. The sensitivity to By is

unsuppressed at the resonance but drops off faster than Bx for the first decade in


0.001

0.01

0.1

1

Fie

ldS

up

pre

ssio

nF

acto

r

0.1 0.2 0.5 1 2 5 10

Frequency (Hz)

0.1

1

10

Measu

redF

ield(p

T)

Bx

By

Bz

Figure 3.18: Applied Bx, By, and Bz fields were suppressed according to Equations 2.120.At the 3He resonance the Bx and By field components are completely unsuppressed.

lower frequency. The magnetometer is insensitive to Bz because it is parallel to the

spins.

The entire experiment was subjected to a By magnetic field and the frequency

response is plotted in Figure 3.19. The magnetic shields provide a factor of about

106 suppression of the applied field and the co-magnetometer supplies the rest of

the suppression at low frequencies. At 100 s timescales, the total external field

suppression factor is 10−9. For modulations with periods longer than 100 s, the

suppression is still operative but long term drifts dominate the signal. Magnetic

storms can change the direction and magnitude of the earth’s magnetic field by

as much as 10%, which after 100 s would result in a 2 fT signal and after 1000

s would be just 0.2 fT. Normal diurnal distortion of the magnetosphere is much

less significant. Thus, external field fluctuations are not expected to be a source of

systematic noise and have never been directly observed in the experiment.


10−10

10−9

10−8

10−7

10−6

Su

pp

ress

ion

Fac

tor

0.001 0.01 0.1 1 10 100 1000

Frequency (Hz)

10−16

10−15

10−14

10−13

10−12

10−11M

easured

Field

(T)

Figure 3.19: Coils wrapped around the entire experiment generated a By magnetic fieldacross the entire experiment.

10−4

10−3

10−2

10−1

100

Fie

ldS

up

pre

ssio

nF

acto

r

0.1 0.2 0.5 1 2 5 10

Frequency (Hz)

dBy/dx

dBy/dy

dBy/dz

dBx/dz

dBz/dz

Figure 3.20: The measured co-magnetometer suppression of gradient magnetic fields com-pared to the signal one would obtain from a constant By field of the same magnitude mea-sured by a non-compensating magnetometer. Colored points on the axis correspond tothe same measurements in the quasi-static limit, with square-wave modulation instead ofsinusoidal modulation.


1

2

5

10

20

50

100

Am

pli

tud

e(f

T/√

Hz)

0.5 1 2 5 10 20

Frequency (Hz)

Signal

Background

Figure 3.21: The measured co-magnetometer suppression of low-frequency magnetic noisegenerated by the magnetic shields.

Magnetic field gradients are also suppressed in the co-magnetometer. The fre-

quency response due to applied magnetic field gradients is shown in Figure 3.20.

The co-magnetometer compensation works locally everywhere in the cell to can-

cel the local magnetic field. After settling for 10 s, the magnetic field gradients

are suppressed by factors of 500 to 5000, depending on the gradient. The smallest

suppression factor comes from the Bx gradient, which is understood by recalling

that Bx fields, together with a small Bz offset, generate signal according to BxB2z .

The largest suppression factor comes from the Bz gradient because the spins are

insensitive to Bz.

Magnetic noise is suppressed by the co-magnetometer just as well as any mag-

netic field. Figure 3.21 shows the noise of the co-magnetometer dropping for fre-

quencies below the 3He resonance at 3 Hz. This allows the co-magnetometer to

have sensitivity to anomalous fields below the magnetic noise generated by the

shields.


−100

0

100

200

300

400S

ign

al(f

T)

0 1 2 3 4 5 6

Pumping intensity (V)

0 500 1000 1500

Pumping intensity Rp (1/s)

Measurement

Bloch Equations

Propagation model

Model with Ly = 170 fT

Figure 3.22: Co-magnetometer signal as a function of pump intensity. The dash-dottedline comes from the earth’s rotation (Equation 2.121), the dashed line adds the propagationmodel from Equation 2.10 and the solid line adds sufficient vertical lightshift to the rotationsignal to match the data.

3.2.3 Intensity dependence

The intensity dependence of the signal is plotted in Figure 3.22. There are three

indistinguishable possible sources of this profile: an anomalous field, a vertical

lightshift and the earth’s rotation. The model curve given by Equation 2.121 for

the earth’s rotation does not properly describe the data in this graph. The model

is greatly improved by adding the pumping rate attenuation due to beam propa-

gation through the cell from Equation 2.10. The magnitude of the measurements

cannot be fully explained by the earth’s rotation; an additional vertical lightshift

can be added to the model to fit the data.


K M

agne

tizat

ion

Mxe (a

rb.)

Time (s)0 1 2 3 4 5

-4

-3

-2

-1

0

1

2

3

Figure 3.23: Transient response to a large 14 tip of the 3He magnetization.

3.2.4 Nonlinear dynamics

All of the dynamics discussed above involve small transverse excitations for which

linear approximations are valid. For large tip angle excitations, nonlinear behavior

arises from the complete set of Bloch equations. Figure 3.23 shows the signal re-

sponse to a large tipping angle pulse. Initially, for t < 3 s the large transverse

component of the 3He magnetization violates the low-field requirement for SERF

operation and significantly reduces sensitivity. As the magnetization returns to the

small-angle regime, the magnetometer regains sensitivity and the signal begins to

increase until the system resembles the linear case. For t > 4.6 s the decay resem-

bles the linear coupled dynamics shown in Figure 3.14.

When the spins are in equilibrium and the K polarization is suddenly reversed,

the 3He spins experience a dynamic instability wherein they spontaneously reverse

direction by executing the transient shown in Figure 3.24. The K polarization is re-

versed by switching between left- and right-circularly polarized light. The initial


0 500 1000 1500Time (s)

0 2 4 6Time (s)

0

2

-2

6

-6

4

-4

Tran

sver

se P

olar

izat

ion P x

e (arb

.)

Figure 3.24: Nonlinear co-magnetometer response to field reversal.

spike in signal corresponds to the spontaneous excitation of 3He precession and

the subsequent shutdown of the SERF sensitivity as soon as the system becomes

nonlinear. The remaining excitation time profile shows how the 3He slowly reori-

ents itself to be aligned with the K spins.

3.2.5 Relaxation rate measurement

The signal response to By modulation as a function of Bz has a standard dispersion

shape and can be used for two purposes: the asymmetry of the dispersion curve

can be used to detect Lz lightshifts and the width of the dispersion curve is a func-

tion of the relaxation rate Rtot. Figure 3.25 shows several of these response curves,

measured for a range of pumping intensities. The dispersion curve observed here

can be derived using the full expression for the signal, Equation 2.121. The relevant

portion of that equation for By modulation is

∂S∂By

=Pe

z Rtot

γe

(Bz

(Rtot/γe)2 + (Bz + Lz)2

)(3.13)


−40

−20

0

20

40

Mo

du

lati

on

Res

po

nse

(mV

)

−50 −25 0 25 50

Compensation Field Bc (µG)

Pp = 0.3 V

Pp = 0.7 V

Pp = 2.0 V

Pp = 5.0 V

0

250

500

750

1000

Res

po

nse

wid

th(1

/se

c)

0 1 2 3 4 5

Pump power Pp (V)

R = (207 1/sV)Pp + (79 1/s)

Figure 3.25: The signal response to By modulation as a function of Bz. In the limit of lowpumping rate, the width of the By modulation response curve gives the spin destructionrate. The line shape is given by Equation 3.13.

The dispersion half-width δB, obtained from fits to a dispersion curve, are con-

verted into a measure of the relaxation rate using Rtot = γ∆B. Figure 3.25 also

shows the measured relaxation rates plotted as a function of pumping rate. The

spin destruction rate can be inferred from the measured relaxation rates in the limit

of zero pumping rate. For these data, the spin-destruction rate was 79 1/s or 12 Hz.

This is higher than, but reasonably consistent with the prediction of Rtot = 57 1/s

from Equation 2.26.

3.3. Zeroing fields and lightshifts 112

3.3 Zeroing fields and lightshifts

It is critical to the operation of the co-magnetometer and the validity of Equation

2.123 that the magnetic field B and lightshift L be maintained at zero. This is partic-

ularly challenging because the co-magnetometer is insensitive to these quantities

when they are all near zero. A modulation technique has been developed to reli-

ably find the zeros of these and other important parameters. The modulation fre-

quency is quasi-steady-state, far enough away from the ∼ 7 Hz resonance of 3He

to validate the steady state solution in Equation 2.121. That steady state solution

for the signal is reprinted here in simplified form for convenience:

S ∼ bny − be

y + Ly +Ωy

γn+

smRm + αRp

γePez

+ Bz

(bey + By

Bn − Lxγe

Rtot

)+

γe

Rtot

(BxBz(Bz + Lz)

Bn − LxLz

)(3.14)

All higher order corrections to this equation have been excluded from this expres-

sion because they do not contribute significantly when the fields are near zero.

3.3.1 Zeroing Bz

Consider modulating By and measuring the signal response. Equation 3.14 simpli-

fies to

Zero Bz:∂S∂By

∝ Bz → 0 (3.15)

Here the signal response, ∂S/∂By, is proportional to Bz and no other fields or light-

shifts. It is possible, therefore, to employ a zero-finding routine that adjusts Bz

until the signal response and Bz are exactly zero. At that zero point, the applied


(a) Bz < 0

DS < 0

DS = 0 DS > 0

(b) Bz = 0 (c) Bz > 0Si

gnal

(arb

.)B y

Time (s)0 5 10 15

Figure 3.26: Procedure for zeroing Bz: A square wave modulation of By creates a modu-lation of the co-magnetometer signal. The steady-state response is indicated by the differ-ence between the dashed lines for Bz 6= 0. Note that the signal changes sign as Bz passesthrough zero, allowing the application of a zero-finding procedure. In case (b), when theapplied Bz exactly cancels the 3He and K magnetizations, the signal does not show anysteady state modulation. This illustrates the insensitivity of the co-magnetometer to mag-netic field drift.

Bn exactly cancels all the magnetizations of the atoms in the system as well as any

ambient field generated by the magnetic shields.

The zeroing process is illustrated by the data in Figure 3.26. In the first third

of the sequence, Bz is decreased by 5 µG and the 2 µG By modulation is applied

using a smoothed square wave modulation. After each signal transient decays and

the signal reaches equilibrium, a measurement is made. The measured sections are

averaged and subtracted from one another to obtain a signal amplitude indicated

by the dashed lines. For the first third of Figure 3.26, the modulation produces

a negative signal response. In the last third of the sequence, Bz is increased by

5 µG and the same modulation and measurements give a positive signal response.

There is a point between the positive response and the negative response for which

the applied By modulation produces no signal in steady state. At that point, shown

in the middle of Figure 3.26, Bz is zeroed.


Zero

dS/dB

y (a

rb.)

-4

-3

-2

-1

1

2

3

0

-2.0 -1.0 0.0 1.0 2.0Bz (arb.)

(a)(b)

(c)

Figure 3.27: Summary of the data collected in Figure 3.26 including data that extendsbeyond the linear regime described by Equation 3.15. A zero-finding routine samples thiscurve to find the point where the signal response to By modulation vanishes so that Bz = 0.

Similar modulation measurements were made for a wide range of Bz and are

plotted in Figure 3.27. Near the zero point the modulation response is linear in

Bz, in agreement with Equation 3.15. After measuring two data points around

zero, one can fit a line through them and find the value of Bz at the zero intercept,

corresponding to the value of Bz for which the co-magnetometer is insensitive to

changes in By in steady state. This zeroing routine must be executed frequently to

compensate for drifts in the 3He magnetization. When the magnetometer is prop-

erly zeroed, the insensitivity of the magnetometer in steady state is indicative of

the 3He magnetization properly canceling the applied field. In practice, if the first

two measurements are not close to the zero crossing, the zeroing procedure must

methodically map out the curve in Figure 3.27 until it discovers the zero crossing.

Additional information, such as the slope of the curve around zero, can inform the


−20

0

20

Sig

nal

(pT

)

3 4 5 6 7 12 13 14 15 16 21 22 23 24 25

Time (s)

Measure

δBy = 2 µG

∆Bz = 5 µG

Figure 3.28: Zeroing procedure for Bz.

zeroing routine about which way to continue its search for a zero crossing and can

help determine whether a zero crossing represents a valid zero.

A numerical simulation of the Bz zeroing routine using the full Bloch equations

is found in Figure 3.28. The modulation responses are measured for positive and

negative Bz detuning in the first two sections. For the last section of data, Bz was set

at the zero intercept of the first two modulation measurements. At that point, the

simulation verifies that the co-magnetometer equilibrium response is insensitive

to applied By.

One must use a smoothed square wave modulation of By at a sufficiently low

frequency that the spin precession transients can decay completely before a mea-

surement of the signal is made. The transient part of the square wave modulation

is not perfectly sharp to reduce excitation of spin precession. The modulation used

in the zeroing routines is given by

B(t) = B0 tanh(s sin(ωt)) (3.16)


−5

−2.5

0

2.5

5

Sig

nal

(pT

)

0 0.5 1 1.5 2

Time (s)

−2

−1

0

1

2

By

(µG

)By

Signal

Figure 3.29: Response to a step in the By field after having zeroed Bz.

where s represents the “sharpness” of the modulation and can be adjusted to be suf-

ficiently fast to make expedient measurements but not so fast that the modulation

induces significant transverse excitation. Figure 3.29 shows a simulated response

to a typical modulation profile. The primary advantage of this modulation form

is that all its time derivatives are continuous and it moves between the maximum

and minimum in finite time. The continuity of the derivatives helps reduce unnec-

essary transverse spin excitation, which is important for the timely measurement

of a steady-state effect. It is important for the modulation to move between levels

in finite time because any measurements must be made while the applied fields

are constant.

The magnetic field modulations are created using a 16-bit National Instruments

analog output board in the computer. The ±10 V full range of the output card is

attenuated by a 100 kΩ resistor (20 kΩ for Bnz) and directly drives the magnetic

coils inside the experiment. An ultra-stable, three-channel current source with a


mercury battery reference is added to the computer output and is used to find

rough zeros for the magnetic fields, leaving the computer to make refinements. At

the sensitivity level of this magnetometer, the noise level of the output card is sig-

nificant. A switchable time constant filter system is used to filter out the computer

output noise. The filters on each magnetic field control line switch between a very

long > 50 s time constant and a short 100 Hz time constant. The filters are de-

signed to change between slow and fast modes without any disruption to the filter

output levels. The computer switches these filters to the short time constant for

the duration of any magnetic field adjustment. In the modulation measurements

in Figure 3.26, the filters would switch to the fast time constant whenever the mod-

ulation level was moving and quickly switch to the long time constant during the

settling and measurement periods. The filters reduced the noise in the zeroing

routine measurements.

3.3.2 Calibrating the magnetometer

The co-magnetometer signal b is defined to be the magnitude of an anomalous

field in Tesla that couples exclusively to either K or 3He. One would obtain the

same signal from a magnetic field of the same magnitude measured by an identical

magnetometer with the 3He replaced by 4He. Define a calibration constant κ that

converts the K spin response expressed in Equation 2.123 into the magnetic field

quantity b as follows:

b = κS = bny − be

y where κ =Rtot

Pez γe

(3.17)


Since an anomalous field cannot be directly applied to calibrate the experiment, κ

is determined by measuring the slope of the line obtained while zeroing Bz. That

line is formed around the zero intercept in Figure 3.27. Combined with the known

amplitudes of Bn and the applied ∆By modulation, the calibration constant can be

expressed as

κ =(

Bn

∆By

d∆SdBz

)−1

(3.18)

where d∆S/dBz is the slope of the response obtained while zeroing Bz. In practice,

the measured signal S is in Volts from the lock-in amplifier and the calibration

accounts for the optical rotation and conversions through the photodiode amplifier

and lock-in amplifier. Although b and S are often used interchangeably for the co-

magnetometer signal in this work, one must multiply the expressions for S by κ to

obtain a result in magnetic field units.

3.3.3 Zeroing By

Once the Bz zero has been found, the remaining fields can be zeroed. By apply-

ing a Bz modulation around it’s newfound zero point (modulating Bn around the

compensation field), one obtains the following response:

Zero By:∂S∂Bz

∝ (bey + By)

1Bn + Lx

γe

Rtot→ 0 (3.19)

Here the quantity being zeroed is the sum of terms with By, Lx and the anomalous

field bey. Note that the BxBz term is suppressed by being the product of two small

factors. The Lx term is zeroed independently at a later stage and the whole proce-

dure is iterated several times. Zeroing (bey + By) does not render the magnetometer


−1000

0

1000

Sig

nal

(fT

)

3 4 5 6 7 12 13 14 15 16 21 22 23 24 25

Time (s)

Measure

δBz = 10 µG

∆By = 1 µG

Figure 3.30: Zeroing procedure for By.

insensitive to anomalous fields because it is always sensitive to the difference in

anomalous field coupling as shown in Equation 2.123.

The zeroing procedure for By is simulated in Figure 3.30 using the full Bloch

equations (Equations 2.103) under typical experimental conditions. In the first two

thirds of Figure 3.30, Bz is modulated for two values of By: one above the zero and

one below the zero. The linear fit to these two measurements accurately sets By to

zero; the last third of the plot demonstrates that the steady state response is con-

stant when By is properly zeroed. The total offset to the signal in these simulations

is due to the earth’s rotation. Note how the measurement periods are delayed from

the transients to allow the system to reach equilibrium.


0

500

1000

1500

Sig

nal

(fT

)

3 4 5 6 7 12 13 14 15 16 21 22 23 24 25

Time (s)

Measure

δBz = 10 µG

∆Bx = 1 µG

Figure 3.31: Zeroing procedure for Bx.

3.3.4 Zeroing Bx

To isolate the co-magnetometer response to Bx, one must measure the signal re-

sponse to the second derivative of Bz:

Zero Bx:∂2S∂B2

z∝ Bx → 0 (3.20)

This is accomplished by modulating Bz between zero and a non-zero value. If

one were to modulate Bz symmetrically around zero, one would retrieve the same

measurement that was just performed for zeroing By, with the more complicated re-

sponse that is sensitive to Lz and By in addition to Bx. The asymmetric modulation

excludes these terms and isolates the Bx dependence. The By zeroing removes the

linear component of the Bz modulation dependence and the Bx zeroing removes

the quadratic component of the Bz modulation dependence.

The only drawback to the asymmetric modulation measurement is that the mod-

ulation is somewhat sensitive to how well zeroed Bz is. If the 3He magnetization


−1

0

1

∂S

/∂

By

(pT

/20

µG

)

−100 −50 0 50 100

Bz (µG)

Lz = 0

Lz = 10 µG

−50

−25

0

25

50

∂S

/∂

Bz

(pT

/10

µG

)

−200 −100 0 100 200

By (µG)

−5

−2.5

0

2.5

5

∂2S

/∂

B2 z

(pT

/∆

Bz)

−100 −50 0 50 100

Bx (µG)

Bz = 1.5 µG, ∆Bz = 5 µG

Bz = 1.5 µG, ∆Bz = 1 µG, S× 10

Figure 3.32: Top: An asymmetric modulation of Bz measures the second derivative ofthe signal response to Bz modulation, which is sensitive to nonzero Bx. For the dashedline, a small offset in the Bz zero causes a slope reversal for sufficiently small modulationamplitude. Middle: A symmetric Bz modulation is sensitive to nonzero By. Bottom: Asymmetric By modulation is sensitive to nonzero Bz. The asymmetry of the response curveis proportional to the pump lightshift Lz.


drifts during the zeroing procedures then the asymmetric modulation may not be

starting from Bz = 0, which could lead to finding incorrect zeros for Bx. Figure 3.32

shows the modulation response curves relevant to zeroing each magnetic field com-

ponent. For each point on these curves, the full numerical Bloch equations (Equa-

tions 2.103) were solved. The top plot in Figure 3.32 shows the curve that is being

traversed during Bx zeroing. If the asymmetric Bz modulation is comparable to the

drift in the Bz zero due to 3He magnetization drift, the slope of the zero crossing

can change sign. If the modulation response changes sign due to Bz drift (Figure

3.32, Top, dashed line), then a total of three zeros appear, two of which have the

correct slope but are not the real zero. Thus, it is important to zero Bz immediately

before zeroing Bx and modulate Bz with an amplitude that is significantly greater

than the 3He magnetization drift.

3.3.5 Zeroing lightshifts

With the magnetic fields terms zeroed, the pump and probe beam lightshifts can

be zeroed straightforwardly by exploiting the S ∼ LxLz term. Modulating one

lightshift while zeroing the other allows the zeroing of both:

Zero Lx:∂S∂Ly

∝ Lx → 0 (3.21)

Zero Ly:∂S∂Lx

∝ Ly → 0 (3.22)

The pump lightshift is adjusted by slightly changing the wavelength of the laser

diode. The probe lightshift is zeroed using a Pockel cell to zero the degree of circu-

lar polarization and cancel the birefringence in the beam path before the cell. As

seen previously in Figure 2.18 and Equation 2.136, there is a significant first order


signal dependence on the adjustment of the probe circular polarization due to its

retardance and its ability to pump and polarize the spins along the measurement

direction. One must implement a method for measuring and subtracting the “back-

ground signal” of the Pockel cell. In practice, the pump beam is shuttered and the

signal due to the Pockel cell modulation is measured directly. This background sig-

nal is then subtracted from any further measurements involving that same Pockel

cell modulation.

There are two parts to the Pockel cell background signal: the optical rotation

due to probe pumping and the rotation due to retardance. The latter can be de-

scribed by using the following elements of a Jones algebra (Jones, 1941):

Mrot(θ) =

cos θ sin θ

− sin θ cos θ

, Mret(β) =

eiβ 0

0 1

, Mprojy =

0 0

0 1

(3.23)

where Mrot generates polarization rotation by an amount θ, Mret generates retar-

dance of x polarized light by an amount β, and Mprojy takes the projection of the

polarization along y as through a polarizer. A waveplate can be constructed from

these elements:

Mwp(θ, β) ≡ Mrot(−θ)Mret(β)Mrot(θ) (3.24)

The probe beam signal can be modeled including the Pockel cell and the Faraday

modulator using

I =

(

0 1

)Mproj

y︸︷︷︸y Polarizer

Mrot(φ)︸︷︷︸Optical Rotation

Mwp(θ, β)︸︷︷︸Pockel Cell

Mrot(α sin(ωmt))︸︷︷︸Faraday Modulator

1

0

︸︷︷︸

x Polarized

× c. c.

(3.25)


With the Pockel cell at an angle θ = φ/4, the first harmonic signal is, to leading

order,

S ∝ Iωm ' 2αφ

(1− β2

2

)(3.26)

Here the Pockel cell β modulation only appears in second order. First order de-

pendence appears, however, in the general case where additional birefringence of

retardance β2 at a different angle θ2 = π/8 is introduced in the path of the probe

beam:

S ∝ Iωm ' 2αφ

(1− β2

2

)+

−√2αββ2︸︷︷︸Linear in β

−αβ22

2+

αβ2β22

4

(12

+ φ

)+ · · · (3.27)

where terms up to second order have been included; higher order terms are in-

significant for small β, β2, α, and φ. The small angle approximations in this anal-

ysis are valid because typical Faraday modulation is α ' 5 and typical Pockel

cell modulation is β ' π/200. Thus, modulating the Pockel cell retardance β gen-

erates a background signal modulation that is linear in β and β2. This first order

background signal must be measured and subtracted from measurements involv-

ing Pockel cell modulation.

There is, however, an alternative procedure for zeroing the probe lightshift that

relies entirely on zeroing the pumping rate of the probe beam light. This scheme

works because the probe circular polarization, the probe lightshift and the K po-

larization due to probe beam optical pumping are all directly related; if one of

these is zeroed, they are all zeroed. In variables, the previous sentence translates

as Pex ∝ sm ∝ Lx. Slightly left- or right-circularly polarized light optically pumps

the K atoms parallel or anti-parallel to the probe beam. This polarization induces

optical rotation of the probe beam, which provides a signal. This signal, plotted


−250

−200

−150

−50

0

50

Sig

nal

(fT

)

−40 −20 0 20 40

Bz (µG)

By = 20 µG

Bx = 20 µG

−500

0

500

1000

1500

2000

Sig

nal

(fT

)

Lx = 2.3 pG

S× 100; Ω = Ω⊕

Figure 3.33: Signal response to Bz scans with Bx, By and sm offsets with no pumping light.The probe beam pumping becomes the dominant pumping effect and gives a strong signalat zero field.


−500

0

500

1000

1500

Sig

nal

(fT

)

3 4 5 6 7 12 13 14 15 16 21 22 23 24 25

Time (s)

Measure

δBz = −50 µG

Rp = 0

∆Lx = 2.3 pG

Figure 3.34: Zeroing procedure for Lx.

as a function of Bz in the top plot in Figure 3.33 has a strong peak around Bz = 0

whose amplitude is proportional to the polarization. The relevant portion of the

signal comes from Equation 2.121:

S(sm, Bz) =smRmRtot

R2tot + γ2

e B2z' sm

Rm

Rtot

(1− γ2

e B2z

R2tot

)(3.28)

where the last step is only valid for small Bz; for large Bz, the signal vanishes. The

zeroing procedure, shown in Figure 3.34, involves tuning the Pockel cell to adjust

sm while comparing the amplitude of this peak to a background point at large

nonzero Bz > 50 µG where the signal is strongly suppressed. When the amplitude

of the peak is equal to the background level, the probe polarization and, thus, the

probe lightshift have vanished. The term being zeroed can be expressed as

Zero Lx:∂2S∂B2

z

∣∣∣∣Rp=0

∝ sm ∝ Lx → 0 (3.29)


where the second derivative, in practice, indicates an asymmetric modulation. The

main advantage of this method is that it works with the pump beam shut off and

provides a clear signal with an unambiguous zero that is largely independent of

any other nonzero parameters. The main disadvantage of this measurement is

that it does not directly zero the LxLz signal due to the overlapping parts of the

pump and probe beams. If there are spatial nonuniformities in the lightshift of the

probe beam, then the polarization method will zero the entire probe beam average

lightshift whereas the zeroing Lx using the LxLz method will zero just those parts

of the probe beam that actually participate in the measurement.

3.3.6 Zeroing the pump-probe nonorthogonality

The co-magnetometer signal has first order sensitivity to the pump-probe beam

nonorthogonality angle α (Equation 2.135). The nonorthogonality can be zeroed

by considering the intensity dependence of terms contributing to the signal: ac-

cording to Figure 2.19, in the limit of high pumping rate, nonzero α generates a

signal while virtually all other imperfections are suppressed. The vertical light-

shift could also contribute at high intensity but it should be suppressed because

all the incident light dominantly propagates in the horizontal plane with zeroed

lightshift. By modulating the pump beam intensity between zero and high pump-

ing rates, the amplitude of the modulated signal is proportional to α. The concept

of this zeroing routine is to turn the K polarization on and off and adjust the rela-

tive angle of the pump and probe beams until the presence or absence of K spin

polarization has no effect on the probe beam, at which point the pump beam and

its polarization is orthogonal to the probe beam.


−200

0

200

Sig

nal

(fT

)

3 4 5 6 7 12 13 14 15 16 21 22 23 24 25

Time (s)

Measure

δRp = 0 ↔ 1300 1/sec

δBz = −100 µG

∆θm = 10 µrad

Figure 3.35: Zeroing procedure for pump-probe non-orthogonality.

In practice, however, the pump beam cannot achieve the limit of high pumping

rate where all the other terms do not contribute to the signal. Thus, the zeroing

routine is sensitive to terms that are not fully suppressed at high pumping rate.

The relevant terms that contribute to the zeroing modulation are

Zero α:∂S

∂Rp

∣∣∣∣Rp=Rbackp

Rp=0∝ αK

Rp

Rtot+ (bn

y − bey + Ωy/γn)

Rp

R2tot→ 0 (3.30)

where K is a calibration factor that converts pump-probe misalignment into signal

units. The signal due to anomalous fields and the earth’s rotation are suppressed

by a factor of 1/Rp in the limit of high pumping rate. They are nonetheless signifi-

cant for finite pumping rate in this zeroing routine.

The zeroing procedure is simulated in Figure 3.35. In an ideal configuration,

the signal at zero pumping rate would not have dependence on any parameters.


However, the pumping rate does not actually reach zero because of 3He spin ex-

change pumping; with the pump beam blocked, most parameters retain some sig-

nal dependence. To avoid this dependence when the pump beam is modulated

off, a strong Bz field of 100 µG is applied to suppress the sensitivity to the other

parameters. Thus, at the Rp = 0 side of the modulation, the sensitivity to system

parameters is negligible. At the high pumping rate side of the modulation, the sys-

tem is sensitive to the parameters in Equation 3.30. By steering the pump or probe

beam, the modulated signal can be zeroed. The carefully designed set of beam

attitude control lenses for the probe beam allows the probe beam to be steered us-

ing the variable wedge (the deviator) without translation on the cell (see Section

3.1.3). This setup is ideally suited for zeroing the pump-probe non-orthogonality.

If there were no rotation or anomalous fields, then this zeroing routine would set α

exactly to zero. In practice, however, α is set to cancel the suppressed but nonzero

contributions from anomalous fields and rotation.

Figure 3.36 illustrates how this zeroing routine suppresses sensitivity to anoma-

lous fields. In a perfectly aligned system, an anomalous field (or, equivalently, the

earth’s rotation), would create the signal profile given by the solid line. In the

absence of an anomalous field, nonzero α generates a signal profile given by the

dashed line. The zeroing routine compares the zero signal level at Rp = 0 to the sig-

nal level at high pumping rate and sets α such that those signal levels are the same.

With an anomalous field in the system, finding this zero sets α such that it cancels

the anomalous field signal contribution at high pumping rate. These contributions

are smaller by about 1/3 than their contributions at the peak sensitivity. Although


0

0.25

0.5

0.75

1

Sig

nal

(arb

.)

0 500 1000 1500 2000

Pumping rate (1/sec)

CPT signal

α sensitivity

CPT signal, α zeroed

Figure 3.36: In this simulation, the co-magnetometer sensitivity peaks at Rp = 240 1/s.The sensitivity to pump-probe nonorthogonality is maximized at high pumping rate. Thedotted vertical lines at Rp = 350 and Rp = 2000 represent the normal and backgroundpumping rates. This plot is a subset of Figure 2.19.

the entire sensitivity to anomalous fields is not removed, the sensitivity is nonethe-

less suppressed. After zeroing the modulation response, the resulting sensitivity

to anomalous field is given by the dash-dotted line which passes through zero at

the high pumping rate point where it was zeroed.

To quantify the sensitivity suppression due to this zeroing routine, consider a

zeroed system for which the pump-probe nonorthogonality α compensates for an

anomalous field b. The offset in α is

α = − bKRback

tot(3.31)

Compared to the signal S in the perfectly aligned α = 0 case, the signal SZα for α at

this offset is suppressed by a factor ηα:

ηα =S

SZα=

b Rp

R2tot

αK RpRtot

+ b Rp

R2tot

=1

1− RtotRback

tot

(3.32)


The ratio of the total relaxation rates for the background (high intensity) and nor-

mal pumping rates, Rtot/Rbacktot , can be obtained by measuring the sensitivity cali-

bration constant κback at high intensity in the same way that the normal calibration

constant is measured. The definition of the calibration constant is

κ =Rtot

Pez γe

=R2

totRpγe

(3.33)

where the last step used the expression for the polarization in Equation 2.8. Then

the ratio of the relaxation rates is

Rtot

Rbacktot

=

√κ

κback

Rp

Rbackp

(3.34)

Although the exact values of the normal and background pumping rates are not

known, the ratio Rp/Rbackp is well known. Thus, the suppression factor is

ηα =

(1−

√κ

κback

Rp

Rbackp

)−1

(3.35)

For a majority of the data since high intensity background zeroing started, the

background intensity is exactly Rp/Rbackp = 8 times larger than the nominal inten-

sity. The experiment typically obtains κback/κ ' 3, making the signal suppression

factor ηα ' 1.25. The experiment normally measures the suppressed signal SZα,

so one must multiply this signal by ηα to obtain the actual anomalous field. It is

convenient to define a new calibration factor κ′ = ηακ for the anomalous signal

obtained during long term data acquisition.

3.3.7 Zeroing sequence

The zeroing routines discussed in this section are run periodically during data ac-

quisition to maintain the zero of all relevant parameters. The 3He polarization is


very sensitive to experimental conditions and is subject to drift. The probe beam

steering also drifts faster than other parameters. Thus, every 100 s, this zeroing

sequence is executed:

Minor zeroing sequence: Bz α (3.36)

wherein the Bz magnetic field is zeroed followed by the pump-probe non-ortho-

gonality. Every 30 minutes a more elaborate zeroing routine is executed to zero all

the other parameters. Since Bz multiplies most of the terms in Equation 2.121, Bz

must be re-zeroed frequently, especially before zeroing other terms. The complete

zeroing sequence is currently

Major zeroing sequence: Bz By Bz Lx Bz Bx Bz α Bz (3.37)

The major zeroing sequence is repeated several times and the zeroing results are

averaged to reduce noise. It is particularly important to iterate over the By and Lx

zeroings because the former is dependent on the zero of the latter.

3.3.8 Anomalous field dependence

The sensitivity to anomalous fields is quite robust and is not eliminated by the

magnetic shields or the magnetic field zeroing procedures. The basic mechanism

for the sensitivity to anomalous fields is the differential precession of the K and

3He spins. If an anomalous field couples to K and not 3He (pure be), then the K

will precess independent of the 3He. If an anomalous field couples to 3He and not

K, then the 3He will move to cancel the anomalous field, but will introduce a real

transverse magnetic field that the K, seeing no anomalous field, will respond to.


Zeroing “By” always actually zeroes the sum (bey + By). This does not render

the magnetometer insensitive to anomalous fields because the signal is always sen-

sitive to the difference of the anomalous field couplings, S ∼ bny − be

y (Equation

2.123), regardless of the other magnetic fields involved. Even if the By field is set at

a finite value to cancel the anomalous field bey, the By sensitivity is suppressed by a

small Bz term. The steady state equation is still valid, and the signal, proportional

to the difference in nuclear and electron couplings, is unattenuated.

The magnetic shields operate using electron spin. A regular magnetic field

aligns the spins in the magnetic shield and the electron magnetization, which is

anti-parallel to the spin, cancels some of the incident field. These shielding elec-

trons will perceive the electron coupling of the anomalous field bey and react with

a regular magnetic field that cancels that component of the anomalous field. The

co-magnetometer, however, perceives the sum of the anomalous field and the op-

posing magnetic shield field. Since the co-magnetometer is only sensitive to the

difference in coupling of the anomalous field, the co-magnetometer will suppress

the shield field (3He will move to cancel it) but will remain fully sensitive to the

anomalous field.

Chapter 4

Signal analysis and systematic effects

THE SUPPRESSION of long-term systematic noise in the co-magnetometer sig-

nal is essential to the success of this experiment. The approach to reducing

systematic effects has two prongs: (1) a source of signal drift is accurately character-

ized and measured so that it can be subtracted from the signal during data analysis,

and (2) the source of signal drift is controlled or renovated until it is no longer a

dominant source of drift. In practice, these two prongs progress in parallel and

inform each other. Diagnostic measurements that correlate with the signal help

locate sources of systematic noise, which, in turn, lead to better measurements of

the sources of noise and more successful experimental solutions to minimize the

noise.

The zeroing routines described in the previous Chapter render the experiment

insensitive to field, lightshift and pump-probe angle drift. The techniques de-

scribed in this Chapter minimize sensitivity to a variety of other effects including

134

4.1. Signal acquisition 135

birefringence in the optics, dichroism of the cell, cell temperature, nonuniformities

in the cell, and imperfections amplified by the zeroing routines.

4.1 Signal acquisition

The co-magnetometer signal is measured continuously over the course of several

days to determine the amplitude of several sidereal periods. A computer reads the

output of the lock-in amplifier that measures the optical rotation of the probe light

through the co-magnetometer cell. The computer simultaneously reads in about 50

other channels of data from various sources such as temperature sensors, intensity

monitors and position monitors. These additional data are used to analyze the

systematic effects appearing in the signal.

4.1.1 Background subtraction

During data acquisition of the co-magnetometer signal, the pump beam intensity

is periodically changed to make background measurements. The background mea-

surements are designed to remove systematic noise from the experiment by chang-

ing the intensity from the normal operating intensity to an intensity—higher or

lower—for which the co-magnetometer is not sensitive to the anomalous field. In

earlier data sets, before 1843 sd, the background measurements were taken with

the pump beam shut off. In an ideal case with no pumping or K polarization, the

signal is simply a function of the probe beam polarization. Drift in the polariza-

tion angle due to the optics moving slightly or the table twisting is measured as


the background signal. These background measurements remained sensitive, how-

ever, to various parameters because of the small yet significant 3He spin-exchange

pumping rate. It is again useful to consult Figure 2.19 and note that small imperfec-

tions in almost every parameter have very strong dependence on the signal close

to zero pumping rate.

More recently the background measurements were changed to increase the

pump beam intensity by a factor of 8 to 10 over the normal intensity. In the

limit of high pumping rate, the K polarization is pegged along the direction of

the pump beam and is insensitive to any other parameters. Thus, the high inten-

sity background measurement includes both the pump-probe nonorthogonality

drift and the probe polarization drift. The only drawback to this method is that

the pump beam cannot practically reach the intensity necessary to render the co-

magnetometer insensitive to all parameters. Indeed, at just 8 times higher than

the normal operating intensity, the magnetometer signal is only about 2.5 times

less sensitive to anomalous fields (see Section 3.3.6). Thus, subtracting off the high

intensity background measurements from the signal tends to reduce sensitivity to

anomalous fields. However, if the pump-probe orthogonality zeroing routine has

already found the point of zero signal at high pumping rate, then the background

measurement will not remove any additional signal. Thus, the only sensitivity

correction that needs to be made while running the experiment is due to the pump-

probe nonorthogonality zeroing routine.

Figure 4.1 shows the raw co-magnetometer data with the periodic background

measurements. The normal signal is typically recorded for 10 seconds, followed


0 10 20 30 40 50 60

Time (s)

−1000

0

1000

Sig

nal

(fT

)

1981.001 1981.0012 1981.0014 1981.0016

Time (Sidereal Days)

Raw signal

Signal background subtracted

Figure 4.1: 66 s of raw data (solid) coming from the magnetometer, showing backgroundperiods for a duty cycle of ∼ 33%. Background periods are individually averaged anda background spline curve is subtracted from the raw signal to give the background-subtracted signal (dotted).

by a few seconds of background measurement. Each period of background mea-

surement is averaged down to a single point in signal and time. These background

points are fit to a spline curve and that spline is subtracted from the signal. The

background data average excludes the first 0.5 s of transient decay. Likewise, the

background-subtracted signal excludes the entire background plus the following

0.5 s of transient.

By varying the fraction of the time the pump beam is on during the background

measurement, one can control the equilibrium 3He magnetization. A PID feed-

back system (including proportional, integral and differential terms) takes mea-

surements of the 3He polarization from the Bz zeroing procedure and makes adjust-

ments to the background measurement time, thereby fixing the 3He polarization

to a desired value. Although the experiment nominally works at any polarization,


0 2 4 6 8

Time (min)

−1000

0

1000

Sig

nal

(fT

)

1981.001 1981.002 1981.003 1981.004 1981.005 1981.006


Raw signal

Signal background subtracted

Figure 4.2: Several minutes of raw data show gaps wherein zeroing procedures are per-formed to (1) keep Bz at zero and adjust for drifting 3He polarization and (2) main-tain pump-probe orthogonality. Note that after each zeroing period, the duty cycle isadjusted—in this case rather dramatically—to maintain a set 3He polarization. This sec-tion of data was also chosen to show that fluctuating background levels can be removedfrom the signal.

changes in polarization can contribute to the signal through various imperfections

and non-orthogonalities. Furthermore, there is clear empirical evidence that drift-

ing polarization directly correlates with the co-magnetometer signal even when

the changing calibration is correctly applied to the data.

4.1.2 Zeroing schedule

Every 100 seconds, the data acquisition pauses to zero the parameters that drift

the fastest and that are the most critical to the signal stability (the minor zeroing

routines). The Bz zeroing routine is always executed during this pause because

the 3He polarization is constantly drifting. This zeroing routine also provides the


sensitivity calibration constant and sets the background duty cycle through the

background PID algorithm. More recently, the pump-probe non-orthogonality has

also been included here to maintain the pump-probe orthogonality. Both pump

and probe beams sometimes drift on 10 minute timescales, so it is important to

include them here. Figure 4.2 shows a sample of the raw signal across several of

these minor zeroing routines. This selection of data shows how the duty cycle was

dramatically adjusted in an effort to maintain the 3He polarization. There is also

a significant drift in the raw signal that is removed by the background subtraction.

In this case, the horizontal position of the pump laser was wandering across the

cell.

The background periods continue unabated throughout the zeroing routines.

After each modulation measurement the pumping intensity is set to the backgr-

ound level for a period to maintain Bz throughout the zeroing routines. A timer

measures the time since the last background and calculates the appropriate amount

of background time to maintain the appropriate duty cycle. The PID algorithm is

normally allowed to set the background duty cycle between 15% and 60%. The

lower limit makes sure that sufficient data is taken during the background measure-

ment. The duty cycle is typically set to somewhere near the middle of this range.

Certain zeroing routines, however, require compensation for spending significant

fractions of their time with the pump beam off or at high intensity. Empirically

determined compensation factors are assigned to each zeroing routine to increase

or decrease the duty cycle. These compensation factors are allowed to reduce the

duty cycle above the upper limit and below 0%, in which case the pump beam is

turned off to kill polarization (if the backgrounds are normally at high intensity).


0 5 10 15 20 25

Time (hours)

−50

−25

0

25

50

Sig

nal

(fT

)

1915 1915.2 1915.4 1915.6 1915.8


Sinusoidal fit: 1.2 ± 0.3 fT, 0.66 ± 0.04 sd

Figure 4.3: One full day of raw data shows gaps wherein periodic, full zeroing proceduresare performed to zero all the fields, lightshifts, and laser beam orthogonality.

After 30 minutes (and previously up to a few hours) of alternating between

data acquisition and the minor zeroing routines, the complete, major zeroing rou-

tines are executed. These zeroing routines find the zero of all possible parameters

and are repeated several times. Figure 4.3 shows the gaps where this major zeroing

takes place. The anomalous field would appear in this data as a sinusoidal varia-

tion with a period of 1 sidereal day. For this run alone, one might infer that the

anomalous field is b ' 1.2± 0.3 fT; however, there are many sources of systematic

noise that could contribute to a false signal. In the following section, various ob-

served systematic drifts will be discussed. Further details regarding the analysis

can be found in Section 4.3.

4.2. Systematic Noise 141

4.2 Systematic Noise

The dominant sources of systematic noise in this experiment involve imperfections

associated with the pump and probe beams. In particular, pump and probe mo-

tion generate systematic noise whenever their relative angle drifts or whenever

their overlap changes. These and other significant sources of systematic noise that

have been identified are presented in this section. Various solutions to reduce the

sensitivity to each source of systematic noise are discussed. The section is broken

into systematic effects that have been addressed and systematic effects that remain

unresolved.

4.2.1 Systematic effects with implemented controls

Birefringence

Slight changes in the birefringence of the optics along the probe beam path, as

temperature or beam position drift, can cause significant systematic error. The bire-

fringence of the optics causes the probe beam to be slightly circularly polarized. In

the absence of any atoms, this birefringence has a rotation component that directly

contributes to the signal. The background measurements every 10 s, whether at

zero or high pumping rate, do an excellent job of removing the polarization off-

set. The birefringence also generates a probe lightshift Lx and significant probe

pumping. Although Lx is nominally suppressed to first order, the probe pump-

ing contributes to the signal in first order. The Lx zeroing routine explicitly zeroes

these contributions to the signal.


Probe Beam

PolarizationRotation

Cell

Sweet Spot

Figure 4.4: An off-axis probe beam passing through the cell experiences linear dichroismdepending on the angle at which it hits the cell wall, thereby rotating the angle of polariza-tion of the light that passes through. The dichroism vanishes for the probe beam passingthrough the center of the cell. This is called the “sweet spot.”

Cell dichroism

The co-magnetometer cell has two effects on the probe beam. First, the cell walls

possess birefringence due to stresses induced by the high pressure 3He buffer gas,

temperature gradients and built-in stresses from fabrication. This birefringence,

the same as any birefringence, is eliminated via the backgrounding and Lx zeroing

techniques. Second, if the probe beam passes through the cell off-center, motion of

the probe beam can translate into polarization rotation. As depicted in Figure 4.4,

the transmitted probe beam experiences some linear dichroism due to the different

amounts of reflection off the cell wall for light polarized in the plane of reflection

and perpendicular to the place of reflection. The transmitted light polarization an-

gle is effectively rotated as the beam passes through different parts of the cell. As

such, random probe beam motion across the cell, caused for example by air tur-

bulence, translates into angular noise. However, if the probe beam passes exactly

through the center of the cell, this effect vanishes to first order in the probe beam


motion. In that alignment, the beam passes through the so-called “sweet-spot” of

the cell. The set of optics before the cell for the independent rotation and transla-

tion of the probe beam (see Section 3.1.3) was designed specifically to more easily

locate the sweet spot.

Pump-probe angle drift

If the angle between the pump and the probe beams changes then the K polariza-

tion projection onto the probe beam would change accordingly. This effect is hard

to distinguish from precession under the influence of an anomalous field. The ori-

entation of the pump and probe beams is measured by four segment photodiodes

and can be directly removed from the signal as demonstrated in Figure 4.12. When

the pump-probe nonorthogonality zeroing routine was implemented, the sensitiv-

ity to beam drift was significantly suppressed. The remaining sensitivity may be

attributed to nonuniformities that are discussed below.

Zero pumping rate measurements

In addition to the the normal and high (background) pumping rate modes, the

experiment occasionally entirely shuts off the pump beam to make certain mea-

surements. This is currently used to zero the probe lightshift. In the likely chance

that the probe beam has nonuniform circular polarization and lightshift, zeroing

Lmx in the dark only provides the unweighted average zero probe lightshift. The

probe lightshift that participates in the co-magnetometer signal, Lx, is given by

the lightshift only within the pump-probe overlapping volume and is weighted ac-

cording to the nonuniform sensitivity curves discussed later with Figure 4.6. There


Time Spin Orientationsd sign(Pz)

< 1818 +1818 −1857 +1886 −1900 +1934 −1961 +

Table 4.1: Times indicating the reversal of the spin polarization.

are a variety of conditions under which Lx 6= Lmx . In this light, it would be ideal

to zero the probe lightshift using pump lightshift Lz modulation. Although the

pump lightshift cannot be reliably modulated with the single mode pump laser, it

will become possible in the future with a DFB pump laser.

In general, it is desirable to avoid using different pumping intensities to per-

form zeroing and background measurements since they always measure some-

what different volumes and are thereby sensitive to nonuniformities.

Diurnal variations

Systematic noise and drift is largely driven by environmental cycles linked to the

solar day. The barometric pressure and temperature both experience daily cycles

that can affect the experiment. Details on thermal sensitivity are discussed below

in Section 4.2.3. One way to differentiate diurnal and sidereal signals is to wait

several months for the two signals two separate. A far more expedient method is

to reverse the polarization of the spins. Reversing the spins changes the sign of the

co-magnetometer signal while many other systematic effects remain unaffected. In


Table 4.1, the times of these spin reversals is recorded. When the spins are reversed,

the calibration constant gains a negative sign to account for the opposite signals so

that all data is directly comparable in long-term data analysis.

4.2.2 Unresolved systematic effects

Cell temperature fluctuations

Cell temperature fluctuations of about 100 mK are sometimes observed. If the

density of the vapor is too high, then small changes in the density can create mea-

surable shifts in the signal: The pump beam has a relatively short optical depth

of < 1 cm and fluctuations in the K density would cause parts of the cell to be

pumped more or less. The probe beam experiences fluctuating absorption and

rotation signals with fluctuating temperature. The latter is particularly problem-

atic if the probe beam has a large steady-state optical rotation signal. In principle

these changes would be compensated at every point in time by the calibration. In

practice, however, the calibration does not compensate for the signal fluctuations

because the fluctuations happen faster than an accurate calibration can be estab-

lished. These correlations were observed with 1.6 pT/C while running at 170C

(Run 1789.82 sd); they were sufficiently reduced by operating at 160C.

Interference

Optical interference in the probe beam can attenuate the probe beam intensity and

thereby attenuate any nonzero signal. Furthermore, an interference pattern across

the beam profile causes different parts of the beam to be attenuated and thereby


Net vertical lightshift back-reflection

No net vertical lightshift

Figure 4.5: Vertical misalignment of the pump or probe beam could cause first order sensi-tivity to pump lightshift. In the second case depicted, the net back-reflection of the beamhas a significant vertical component.

causes shifting, nonuniform sampling of the cell. Interference is sensitive to wave-

length, beam motion and, in general, thermal expansion and thermal displacement

of optical elements. As temperature drifts, interference fringes are sometimes ob-

served as a sinusoidal modulation of the signal on one hour timescales. One pos-

sible source of interference is the Faraday rod, which has two flat, coated surfaces.

Even if the coating was 1% reflective, a 1% modulation of the signal could be sig-

nificant. The current solution for the Faraday rod is to cut, re-polish and re-coat

one end at a small angle. In general, elements that generate interference would

be highly sensitive to temperature. As with the cell temperature and density, this

effect is difficult to compensate using the sensitivity calibration alone because the

calibration measurement is usually rather noisy on the timescales that interference

appears.

Vertical lightshift and back-reflection

The vertical misalignment of pump and probe beams may contribute to the signal

drift by generating first-order sensitivity to pump and probe lightshifts. The probe

beam produces lightshift of about dLx/dλ ∼ 0.3 fT/pm for 0.5% circular polariza-

tion. At that level, the vertical component of the lightshift due to misalignments


is insignificant. The lightshift due to vertically steering the pump beam is elim-

inated when the magnetic fields are realigned so that they’re pointing along the

direction of the pump beam. However, the pump beam can reflect off the back of

the cell as shown in Figure 4.5. According to Figure 2.5 the pump beam intensity

is reduced by more than a factor of 10 through the cell and perhaps only 1% of that

gets reflected back into the cell. Even if the pump intensity is reduced by a factor

of 1000 for the back-reflected light, there is still sufficient light to provide a vertical

lightshift of about dLy/dλ ∼ −12 fT/pm (see Equation 2.89).

Nonuniform sensitivity

According to the models, with consistent zeroing of all the fields and α, the signal

should not be sensitive to any aspect of the pump or probe beams to first order.

Those aspects include their wavelengths, their positions and relative angle. The

vertical angle of the probe beam should have no effect on the signal measurement

whereas the vertical angle of the pump beam would temporarily create Ly until

the zeroing routines had a chance to realign the magnetic fields along the K polar-

ization direction. Thus, the system is designed to be perfectly insensitive to pump

and probe motion. Despite all this, the experiment remains somewhat sensitive to

pump beam motion and to a lesser extent probe beam motion.

Nonuniformities can explain the residual sensitivity to the many parameters

that are supposedly suppressed. Whenever the volume being measured changes

during zeroing or data acquisition, systematic signals can appear. There are two

ways that the effective measurement volume can change: the first occurs when the

pump and probe beam overlap changes or moves around due to beam drift (see


0.2

0.4

0.6

0.8

1

1.2

Sen

siti

vit

y∝

Pe/

Rto

t(a

rb.)

0 0.5 1 1.5 2 2.5

Distance in Cell (cm)

Low Rp = 0.5Rsd

Nominal Rp = 1.8Rsd

Background Rp = 14Rsd

Figure 4.6: Magnetometer sensitivity is given by the ratio of the alkali polarization to thetotal pumping rate. Using the data for pumping rate and polarization in Figure 2.5 (orfrom Equations 2.8 and 2.10), one can obtain this relative measure of the magnetometersensitivity along the pumping direction z through the cell.

the next section) and the second occurs when changing the pump beam intensity

shifts the area of maximum sensitivity around.

To understand how the sensitivity can move around as a function of pump-

ing intensity, first refer back to Figure 2.5 to see that polarization drops by 50%

through the cell. Over the width of the probe beam (' 1 cm), the nominal pumping

rate drops by more than a factor of 1.8. The decreasing polarization and pumping

rates along z through the cell cause the magnetometer sensitivity to vary somewhat

across the cell according to Figure 4.6. Because both the polarization and pumping

rate decrease with depth into the cell, the magnetometer sensitivity varies much

less than either alone. Across a 1 cm wide probe beam centered on the cell, the

sensitivity does not change more than 10% at the normal or high (background)

pumping rates. At the nominal pumping rate, a maximum in the sensitivity ap-

pears in the middle of the cell, corresponding to the optimum pumping rate. At


high pumping rate, the maximum sensitivity is near the rear of the cell; the sensitiv-

ity increases as the pump beam is attenuated through the cell and approaches the

optimum pumping rate towards the rear of the cell. Thus, the presence of nonuni-

formities in the cell could become a significant issue since the nominal and high

(background) pumping rate measurements emphasize the center and back edge of

the cell, respectively.

Not only does the pump beam intensity change the z location of the sensitivity,

it can also dramatically change the sensitivity profile along x. Pump light propa-

gating through the cell is attenuated at the edges due to lensing by the cell walls

and in the center where the pump beam has the greatest distance to travel through

the absorptive atomic vapor. This creates two areas of peak sensitivity along x; ap-

pearing on both sides of the center. By increasing the incident pump beam intensity,

the region of sensitivity is expanded towards the edges and towards the center. For

the areas near the edges of the cell where the pumping rate is particularly low due

to lensing, the signal is a strong function of various parameters.

There are two major consequences of nonuniform and shifting sensitivity: First,

any time intensity modulation is used in a zeroing routine, the average volume

being zeroed changes. Since the volume being zeroed is not the same as the volume

being measured during normal operation, the magnetometer may not be properly

zeroed for normal operation. Second, since the regions of sensitivity are different

at normal and high (background) pumping rates, background subtraction can only

partially compensate for the changes in the pump and probe beam misalignments

and intensity profiles.


Nonuniform K polarization

The K polarization drops by more than 50% across the cell due to the pump beam

absorption as it propagates through the cell. This gradient in the K polarization

makes it impossible for Bz to be zero everywhere in the cell. Thus, the co-magneto-

meter has some (small) first order sensitivity to many parameters. Indeed, the

zeroing routines find the zero of the spatial-average signal due to the application

of uniform fields. However, the co-magnetometer with the K polarization gradient

would retain first order sensitivity to nonuniform and gradient fields that are in

any way spatially distinct from the applied uniform fields used in zeroing. One

possible way to reduce this sensitivity would be to apply an appropriate gradient

field dBz/dz to do a better job of zeroing Bz throughout the cell. The total amount

of polarization gradient can also be reduced if the vapor were less optically thick.

Despite the first order sensitivity, the zeroing routines should be able to find the

average zero field—even with applied gradients—such that the long-term signal

does not respond to an applied gradient. There is no evidence that this effect has

contributed to the noise so far.

Nonuniform pump-probe overlap

Another aspect of spatial nonuniformity is the possibility for the active, overlap-

ping volume of the pump and probe beams to change significantly over time. If

the pump and probe beams are only partially overlapping, then the drift in the posi-

tions of the pump and probe beams will cause the total amount of signal to change

over time. That drift should be accurately reflected in the calibration, although the

calibration is too noisy to be used for corrections on timescales faster than a few


−400

−200

0

200

400

Sig

nal

(fT

)

1966 1968 1970 1972 1974


26.75

27

27.25

27.5

27.75

28

Tem

peratu

re(C

)

Signal

Pump pinhole temperature

Pump horizontal angle

Figure 4.7: Temperature in the room is strongly correlated with the signal and the pumpbeam position. The correlation is about 600 fT/C. This inspired the thorough investigationinto the temperature dependence of various components in the experiment.

hours. Thus, it is important to expand the pump beam sufficiently so that small

changes in orientation do not significantly alter the sensitivity in the volume of the

probe beam passing through the cell.

4.2.3 Thermal sensitivity

Many sources of long term drift are ultimately due to thermal drift of temperature-

sensitive components. Although it is easy to see the correlation between ambient

temperature and signal, it can be very difficult to pinpoint which component is

dominantly sensitive to temperature. In Figure 4.7 the room temperature clearly

correlates with the pump horizontal position and the co-magnetometer signal. Fit-

ting the temperature data to the signal gives a thermal sensitivity of about 600

fT/C. This very large sensitivity, coupled with typical temperature drifts that are,

at best, δT ∼ 0.2C, swamps any signals of interest at the 1 fT level. Furthermore,


these temperature drifts often follow the diurnal cycle of the building air condi-

tioning and thus would significantly contribute to the sidereal component of the

signal.

The signal correlation with the pump horizontal position in Figure 4.7 narrows

the possible sources down to about 15 possible optical elements in the path of the

pump beam. Beyond that, it is impossible to tell which element(s) contribute the

correlation because all of the components are at the same temperature and expe-

rience the same drifts. One way to learn which element is most sensitive to tem-

perature drift is to individually test the thermal response of each component. A

heater in the form of a 10 W power resistor is strapped or taped to the component

of interest and a temperature sensor is placed nearby to monitor the component

temperature. Then, while the experiment is running normally, sufficient current

is run through the heater to increase the temperature of the component by several

degrees. The changes in the signal and the relevant laser beam position measure-

ments are recorded and a temperature sensitivity measurement in fT/C is calcu-

lated. The raw data from one such test involving eight different components is

found in Figure 4.8. The signal change is often obscured by noise and drift, so in

many cases one must place a limit on the temperature sensitivity in lieu of a direct

measurement. The signal shifts are actually easier to discern than this plot suggests;

considerable detail is lost on this scale.

The complete set of thermal sensitivity measurements pertaining to the pump

beam components, including the results from Figure 4.8, is found in Table 4.2.

The largest single element sensitivities are the pump table flap near the cylindri-

cal lenses, the variable waveplate, and the optical isolator. Every element has


−100

−50

0

50

100

150

Sig

nal

(fT

)

1990 1991 1992 1993 1994


Signal

−10

0

10

Po

siti

on

(mm

)

Pump horizontal position ×10

Pump vertical position

0

2

4

6

Tem

per

atu

re∆

T(C

)

1. Upper case of pump laser

2. Pump laser first telecope lens

3. Probe final telescope lens

4. Faraday rotator cooling water

5. Pump table flap near cylindrical lens

6. Cell tube

7. Pump table near isolator – loose forks

8. Pump isolator

Figure 4.8: Heaters spread around the experiment are sequentially turned on for a fewhours each to pinpoint areas of the experiment that are particularly sensitive to tempera-ture and are in need of refinement.


Element ∆T ∆S dS/dT ∆xp ∆yp(C) (fT) (fT/C) (µm) (µm)

Room Temperature—loose forks +0.7 +400 +600 −450 —Pump Table, Cyl. Lens Inside +3.0 −104 −35 +350 −110Pump First Mirror +5.7 ≤ +39 ≤ +7 +130 —Pump Translation Stage +3.9 ≤ +13 ≤ +3 −270 —Pump Isolator +2.6 < +72 < +27 +100 —

Obscured by probe mode hop.Pump fixed polarizer +5.5 +26 +5 −160 −30Pump pinhole assembly +5.3 +78 +15 +540 +23Pump variable waveplate +5.4 −150 −28 +550 +45Pump laser upper casing +1.7 < +13 < +26 — —

Measurement ∆T limited.Pump first telescope lens +10.0 +13 +1 −140 +32Pump Table, Cyl. Lens Outside +2.8 −52 −19 — −609Pump table front isolator +3.8 +60 +16 +310 +18Pump table isolator front—tight forks +3.8 +43 +11 +380 +30Pump table isolator back—tight forks +6.2 +29 +5 −20 −9Entire Table—tight forks +0.8 −57 −71 — 21Room Temperature—after improvements +0.4 +75 +189 — —

Table 4.2: A survey of the thermal sensitivity of various components involved with thepump beam.

some sensitivity. Those elements with high sensitivity have particularly large ver-

tical position shift. Although vertical alignment does not contribute to the co-

magnetometer signal according to the models, it may contribute due to various

nonuniformities. In particular, if the pump beam moves a few 100 µm vertically,

the intersection of the pump and probe beams can be significantly altered or re-

duced. To reduce this effect, the pump beam was made even more uniform by

the careful adjustment of the cylindrical lenses and was expanded so that small

changes in the pump beam position should not significantly alter or reduce the

intersection of the pump and probe beams.


−100

−50

0

50

100

Sig

nal

(fT

)

2007 2008 2009 2010 2011 2012


28

28.5

29

29.5

30

Tem

peratu

re(C

)

Signal

Room Temperature

Figure 4.9: After several improvements, the temperature-signal correlation is reduced to189 fT/C, a factor of 3 improvement over the data in Figure 4.7.

Several measurements at the beginning of this survey indicated that the optical

isolator had thermal sensitivity in excess of 100 fT/C. Those initial measurements

used a heater attached directly to the side of the isolator and introduced a large ther-

mal gradient through the Faraday rod in the isolator, which significantly steered

the beam vertically. In subsequent measurements of thermal sensitivity of the opti-

cal isolator, now carefully avoiding applying a thermal gradient, the strongest sen-

sitivity came from heating the table next to the isolator. A modest improvement

in thermal sensitivity (from 16 to 11 ft/C) was achieved by simply repositioning

and tightening the optical pedestal forks that clamp the pedestals to the optical

table. After this minor success, screws were tightened all over the experiment.

After expanding the pump beam and tightening the forks, a long-term data run

measured the thermal sensitivity of the entire experiment to room temperature.

The resulting thermal sensitivity derived from the data in Figure 4.9 was +189

fT/C, more than 3 times better than the initial sensitivity.


Figure 4.10: Thermal expansion of the table flap induced by differential heating of the tableflap causes the beam to move vertically.

Thermal expansion of the pump table flap may fully explain the thermal sensi-

tivity due to heating the pump table flap near the cylindrical lens mount. The table

flap, shown in Figure 4.10 is a two inch thick optical breadboard that spans the gap

between the two main wings of the optical table. The top surface of the table flap

is enclosed by a lexan box and the air temperature in the box is cooled by chilled

water flowing through the pump beam. The bottom surface is more exposed to the

fluctuating room temperature. The differential expansion of the top and bottom of

this flap could cause it to warp like a bimetal strip.

One can estimate the vertical deflection of the table flap for the differential heat-

ing of this table flap. In particular, the heaters placed near the cylindrical lenses

(indicated by the single lens on the flap in Figure 4.10) created a particularly large

vertical beam motion of 38 µm/C. Nonmagnetic stainless steel used in the table

is categorized as anamorphic stainless steel, for which the thermal expansion is

ζ = 1.7 × 10−5 mm/mm/C. The deflection is modeled by considering the top


sheet of stainless as expanding relative to a fixed bottom sheet. The differential

expansion causes a vertical deflection ∆y of the table flap, where ∆y is given by

∆y =lxTζ∆T

w(4.1)

where a spot of length xT, heated up by an amount ∆T, deflects the table flap of

thickness w relative to the anchor point a length l away. For a heater affecting a

spot 20 cm in dimension around the cylindrical lens mount, one estimates a verti-

cal deflection of−35 µm/C. A Technical Manufacturing Corporation white paper

gives a vertical deformation of ∼ 28 µm/C for uniform heating of a 12 in thick

table. Although not directly comparable, our model gives the same order of mag-

nitude. If an f = 100 mm spherical lens next to the cylindrical lens were to move

vertically by the estimated amount, then the beam would be deflected vertically

by −240 µm/C at the cell. The measured deflection while heating the cylindrical

lens mount is −218 µm/C. Thus, the thermal expansion of the table flap due to

air thermal fluctuations may significantly contribute to the signal drift.

Two measures have been implemented to decrease systematic noise due to ver-

tical movement: First, the temperature of the entire experiment inside the thermal

shield is controlled to better than 0.1C by blowing temperature-controlled air into

the box. Second, the pump beam was expanded vertically a bit to prevent vertical

beam position drift from changing the pump-probe intersection too significantly.

Thermal sensitivity measurements were also made for optical elements in the

probe beam and for various other parts of the experiment. The results in Table 4.3

indicate that the Faraday rotator rod and the probe beam collimating lens have the

highest thermal sensitivity. The Faraday rod sensitivity may be due to interference

effects between the two surfaces. As a temporary solution, the rod was tilted at


Element ∆T ∆S dS/dT Comments(C) (fT) (fT/C)

Probe Faraday rotator rod +1.3 +78 +58 Heated input waterProbe Pockel cell +5.1 +13 +3 Large background shift

was fully compensated;slight probe steering.

Probe table—Left (near Isolator) +1.9 ≤ +42 ≤ +22 Obscured by roomtemperature change.

Probe fixed polarizer +2.7 +26 +10Probe translating lens +6.7 +86 +13 ∆ym = −0.13 VProbe position pickoff +7.5 +22 +3Magnetic shield frame ∼ +2.0 < +15 < +7Outer magnetic shield ∼ +2.0 < +15 < +7Probe final collimating lens +2.1 −39 −18Probe table—Right +7.4 < +13 < +2 ∆xm = +0.26 VCell holder tube +12.6 < +12 < +1

Table 4.3: A survey of thermal sensitivities over the whole experiment.

an angle sufficient to eliminate interference. The final solution will include cutting

and polishing a slight angle on one end of the rod and coating both ends.

4.2.4 Systematic noise compensation

The previous sections have discussed many sources of systematic drift, some of

which were observed during long-term data runs. After efforts to suppress those

sources of systematic noise, remaining systematic noise can be accurately charac-

terized and explicitly removed from the signal. There are four identifiable types

of systematic drift that can be explicitly removed: (1) any correlation between the

signal and another, separate measurement, (2) purely linear or, more rarely, second-

order polynomial drifts, (3) sudden jumps in the signal associated with mode hops

of the lasers, and (4) temporary excursions from a nominal trend. These corrections


−200

0

200

Sig

nal

(fT

)

1966 1968 1970 1972 1974


Signal

Fit pump horizontal angle and linear

Uncorrelated Signal

Figure 4.11: Pump horizontal position strongly correlates with the signal. The pump po-sition fits nicely to the data and can be subtracted to give the data for the ’UncorrelatedSignal’ trace.

may be made only if they do not risk reducing sensitivity to sidereal variation. This

section contains examples of these systematic effects and some details about their

causes and solutions.

Correlated measurement removal

In general, the clearest and most common systematic effects are due to correlation

between the signal and another measurement. These sources of systematic noise in-

clude drift in the angle and position of the pump and probe beams and the temper-

ature of various highly sensitive components. Figure 4.11 shows an archetypical

signal-pump beam position correlation. In these data, the optical elements used

for the pump beam were moving with changes in temperature. The moving opti-

cal elements caused the pump beam to move both horizontally and vertically. To

determine the degree of correlation, the pump position is fit to the signal using a


−200

−100

0

100

200S

ign

al(f

T)

1663 1664 1665 1666 1667 1668


Signal

Probe horizontal position

Figure 4.12: Position of the probe beam (dashed line) is strongly correlated with the signal(solid line).

nonlinear least-squares method. Since the signal and the pump position are sam-

pled at different rates and times, the pump position data is fit to a spline curve and

the fit uses that spline function. The signal and position traces in Figure 4.11 show

the result of this fitting process and indicate a high degree of correlation. This fit-

ting procedure was used for various other measurements and the most correlated

measurement is chosen for subtraction from the signal. The splined pump position

is subtracted from the signal to obtain the uncorrelated signal in Figure 4.11. This

correlation inspired the survey of thermal sensitivity found in Section 4.2.3 and

was ultimately reduced by tightening the forks that hold the pump beam optical

mounts to the table and expanding the pump beam.

Figure 4.12 contains another signal correlation example. The probe beam was

moving significantly over the course of several days. During this data set, the

pump-probe nonorthogonality was not zeroed. Thus, the motion of the probe


−1000

0

1000

2000

Sig

nal

(fT

)

1816.5 1817 1817.5 1818


Signal with probe mode hops

Signal with jumps removed

Scaled barometric pressure

Figure 4.13: The jump removal procedure eliminates signal jumps due to probe beammode hops. The remaining signal is correlated with the barometric pressure.

beam contributed directly to the signal. This correlation was eventually suppress-

ed by zeroing the pump-probe nonorthogonality and by replacing the glue hold-

ing a mirror in the probe laser. That the glue expanded significantly with humidity

was made eminently clear when a cooling water leak formed in the probe beam

area.

Jump removal

The short section of data in Figure 4.13 demonstrates the removal of sudden laser-

mode-hop jumps. One must identify real mode hops by looking at the intensity

and wavelength of the lasers to obtain the exact times of significant mode hops.

The jump removal procedure looks at the average signal level 10 minutes before

and 10 minutes after the jump, excluding data for 5 minutes around the jump. The

procedure then adds an appropriate constant correction to all the data on the right

hand side of the jump. Sometimes the probe laser becomes multi-mode when it


−200

−100

0

100

200

300S

ign

al(f

T)

1694 1695 1696 1697


Signal with probe mode hops

Signal with hops removed

Scaled quadratic curve

Figure 4.14: The signal fits reasonably well with a second-order polynomial after tempo-rary excursions have been removed.

is in the process of hopping modes. During multi-mode operation, the signal can

become quite noisy. It is therefore important to exclude from the jump removal

procedure the period of time—sometimes up to an hour—during which the probe

beam is multi-mode around a mode hop. After the mode hops were removed from

the data in Figure 4.13, the data were strongly correlated with the ambient baromet-

ric pressure. For this run, the probe laser wavelength was locked to a Fabry-Perot

cavity. The changes in barometric pressure caused the index of refraction of the air

in the cavity to change and, thus, the effective path length of the cavity. Simply

switching off the wavelength feedback stopped this correlation while a vacuum

Fabry-Perot cavity was constructed (which suffered from temperature sensitivity

despite the use of Invar).


Slow polynomial drift removal

The data in Figure 4.14 exhibit temporary mode hops and an overall second-order

dependence. The probe mode hops indicated by the dashed line are only tem-

porary and in all cases the probe beam returns to the same dominant mode and

wavelength. Thus, it is more accurate to simply clip out the data during the mode

hop instead of removing the beginning and ending mode jumps. After the mode

excursions are removed, the signal exhibits a clear second-order polynomial depen-

dence. Removing this quadratic curve is valid because it spans several days and

does not remove significant sidereal amplitude.

4.2.5 Systematic error estimation

Systematic effects dominate the long-term co-magnetometer signal. The correla-

tion between measured parameters and the signal can provide an estimate of each

parameters’ contribution to the systematic noise observed in the experiment. For

Figure 4.15, each parameter’s time series is fit to the signal and thereby placed

in units of femtoTesla. The correlation coefficient R between each parameter and

the signal is calculated; a correlation constant of 1 would imply that the signal is

entirely due to the correlated parameter. The systematic error due to each parame-

ter is given by the sidereal amplitude of the parameter, Cx,y, times the correlation

coefficient.

These systematic contributions δAx and δAy to the sidereal amplitudes Ax and

Ay are shown in Table 4.4. Before this correlation analysis, the pump and probe

correlations were removed from the signal to the extent possible. Although the


0

250

500

750

1000

1250

1500

femtoTesla

1950

1952

1954

1956

1958

1960

Sid

eralD

ays

Sig

nal

Pro

be

hori

zonta

lpos.

Pro

be

ver

tica

lpos.

Pro

be

pre

-cel

lhori

zonta

lpos.

Pro

be

pre

-cel

lver

tica

lpos.

Pum

phori

zonta

lpos.

Magnet

icShie

ld1

Tem

p.

Cel

lTem

per

atu

re

Room

Tem

per

atu

re

Pro

be

Osc

illa

tor

Inte

nsi

ty

Pum

pW

avel

ength

Bx

zero

corr

ecti

on

By

zero

corr

ecti

on

Bz

zero

corr

ecti

on

Pock

elce

llL

xze

roco

rrec

tion

Dev

iato

rα

zero

corr

ecti

on

Figu

re4.

15:

Para

met

erda

taar

eno

rmal

ized

tovi

sual

ize

corr

elat

ions

wit

hth

esi

gnal

.Th

eac

tual

corr

elat

ion

anal

ysis

isfo

und

inTa

ble

4.4.


Parameter Cx Cy R δAx δAy(fT) (fT) (fT) (fT)

Probe horizontal pos. −209.6 −284.0 0.036 −7.56 −10.24Probe vertical pos. 8.8 44.7 0.207 1.83 9.24Probe pre-cell horizontal pos. −20.9 22.9 −0.189 3.94 −4.32Probe pre-cell vertical pos. 17.0 11.4 0.418 7.11 4.77Pump horizontal pos. 4.2 18.6 −0.470 −1.95 −8.73Magnetic Shield 1 Temp. −21.4 −41.1 −0.179 3.83 7.38Cell Temperature 1.7 −24.8 −0.216 −0.36 5.35Room Temperature 56.1 −143.0 −0.081 −4.54 11.58Probe Oscillator Intensity 108.2 −81.6 −0.060 −6.53 4.92Pump Wavelength −7.2 −18.7 −0.378 2.70 7.05Bx zero correction 20.5 −58.0 −0.264 −5.40 15.33By zero correction 73.9 33.1 −0.185 −13.67 −6.11Bz zero correction −41.3 89.5 −0.055 2.28 −4.94Pockel cell Lx zero correction 110.3 205.2 −0.071 −7.81 −14.53Deviator α zero correction −108.7 −61.9 −0.099 10.72 6.10

Table 4.4: Measured systematic errors derived from the data in Figure 4.15. The correlationconstant R between each parameter and the signal is multiplied by the sidereal amplitudeof the parameters Cx,y to obtain the contributions δAx and δAy to the systematic noise inthe anomalous field measurement. Note that the signal had been compensated for pumpand probe position correlations before this analysis.

long-term drifts due to pump and probe motion were properly removed, the decor-

relation may have introduced short-term, correlated, and systematic noise that cre-

ate the relatively large correlation coefficients. The room temperature correlation is

suppressed relative to the data in the section on thermal sensitivity (Section 4.2.3)

because the temperature couples to the pump and probe positions, which were ex-

plicitly removed. The pump wavelength is strongly correlated with the signal but

the sidereal component of the drift appears to be small. A few of the top contribu-

tions to systematic error in A may merit further study: Bx, By, the Pockel cell and

the Deviator appear to have marginally larger contributions to the systematic error.

The Pockel cell sensitivity can be understood in terms of generating lightshifts Lx

4.3. Anomalous field measurement 166

while Bx may be simply compensating for changing Lx. The deviator sensitivity

can be understood in terms of the position correlations.

These systematic errors are likely to be highly correlated and cannot be sim-

ply added up. Temperature drift, for example, contributes to the beam position

measurements and perhaps even to the Pockel cell voltage (since the Pockel cell

is sensitive to beam angle). To determine the systematic error, one can attempt

to combine the systematic errors from a few sources that are considered indepen-

dent. For example, the systematic errors due to probe beam motion and the pump

wavelength may be sufficiently independent. In this case, the total systematic error

would be δA ' 15 fT. This correlation analysis only applies to the 12 days shown in

Figure 4.15 but similar results are obtained for other long data runs. Shorter runs

of 1 or 2 days give inconsistent results and are not well suited to correlation anal-

ysis. At this level of systematic error, the experiment searching for < 1 fT fields

is completely swamped by systematic noise. Indeed, as will be shown in the next

section, it took a very long time to average away this systematic noise to achieve

sub-femtoTesla sensitivity.

4.3 Anomalous field measurement

After all possible correlations and systematic drifts have been removed, the result-

ing signal is analyzed to search for the presence of any signals due to an anomalous

field. The signal due to an anomalous field would appear as a sinusoidal variation

in the signal with a period of exactly one sidereal day. Sidereal time measures the

rotation of the earth relative to a fixed, celestial coordinate system wherein one


−100

−50

0

50

100S

ign

al(f

T)

1966 1968 1970 1972 1974


Uncorrelated Signal

Sinusoidal fit: 7.7 ± 0.7 fT, 0.42 ± 0.01 sd

Figure 4.16: Sidereal signal fit to a data run after all correlation has been removed.

sidereal day is defined as exactly one 360 rotation of the earth in this celestial

frame. The anomalous fields of interest to this experiment are considered fixed in

this celestial frame, so any signal due to the earth rotation should appear in the

lab as a signal modulated with a period of one sidereal day (the direct gyroscopic

sensitivity of the earth’s rotation generates a constant offset with no Sidereal com-

ponent). The solar day is slower than the sidereal day because in one solar day

the earth must rotate an extra 4 minutes to compensate for the fact that the earth is

moving around the sun. All the data in this thesis are referenced in sidereal days

from the standard J2000 epoch. For more details on the J2000 epoch and relevant

conversions, see Appendix A.

4.3.1 Sidereal amplitudes

There are several possible ways to obtain the sidereal amplitude from a section

of data. One method is to find the best fit sinusoid with a fixed frequency of


Ω⊕ = 1/sd. Figure 4.16 shows a sample of data with the best fit sidereal sinusoid.

Here, a nonlinear least-squares fitting algorithm is used to find the best fit ampli-

tude, phase and offset. Since the fitting routine cannot directly find the global best

fit, the real and imaginary parts of the 1/sd frequency component of fast Fourier

transform of the signal provide an accurate estimate and initial guess of the ampli-

tude and phase. The fit of

κS = A sin(Ω⊕t + φ) (4.2)

= Ax cos(Ω⊕t) + Ay sin(Ω⊕t) (4.3)

can be expressed either in polar A, φ coordinates, or in Cartesian Ax, Ay coordi-

nates. The latter case is preferable because Ax and Ay are independent. Here κ con-

verts between raw signal units and fT. This nonlinear least-squares method is good

for analyzing data with significant gaps. Error estimates for the fit parameters can

be obtained from the best fit covariance matrix. However, the fitting routine and

the error estimates are only reliable for Gaussian distributed data, which is clearly

not the case in Figure 4.16 or in the majority of the data.

A lock-in or Fourier analysis technique is a superior method for highly noise-

compromised data. The data are multiplied by sine and cosine references with

unit amplitude and periods of 1 sidereal day. The multiplied data are averaged

over an integral number of periods to provide an accurate measure of Ax and Ay

(if the appropriate factors of 2 are included). This technique is a type of Fourier

transformation that obtains the amplitude of the Fourier component with sidereal

frequency. The uncertainty can be estimated if the noise spectrum is well known.


Osc

illat

ion

Am

plitu

de (f

T)

Frequency (1/sd)

10-4

.001

.01

0.1

1

10

100

.01 1.1 10 1000 105 107100 104 106

Figure 4.17: Lomb periodogram of 6 days of continuous data recording as a function offrequency. The CPT signal is expected to appear at the frequency of 1/sd shown by the ver-tical line. Peaks at higher frequencies correspond to periodic zeroing of the magnetometerfields.

However, since the exact form of the 1/ f noise is not well known, this error estima-

tion is not used. Instead, the uncertainty in the amplitude can be reliably obtained

from the scatter of repeated amplitude measurements.

The uncertainty of the sidereal amplitude is taken to be the upper limit on the

magnitude of the anomalous field. If the amplitude is greater than the uncertainty

in the amplitude, then either anomalous fields are coupling into the experiment or

systematic drift with a large sidereal frequency component dominates the signal.

One way to differentiate between these two possibilities with only a few days of

data is to consider their frequency spectra. The frequency spectrum of typical data

is found in Figure 4.17. A Lomb Periodogram (Press et al., 1992) was used to obtain

a frequency spectrum for nonuniformly sampled signal data. On short timescales


−100

−50

0

50

100Sig

nal

(fT

)

1700 1750 1800 1850 1900 1950 2000


Figure 4.18: Data runs taken over the course of a year. Gaps in the data represent timeswhen the experiment was being upgraded or when the data had too much drift.

at around 6 Hz (a bit shy of 106 1/sd), the noise peaks due to the resonance of the

3He spins. At progressively lower frequencies from that peak, the noise first drops

off a bit, indicative of the 3He canceling magnetic noise, before the magnetome-

ter becomes dominated by drift and the noise level rises. Systematic drift has no

discernible peak at frequency Ω⊕ = 1/sd; a drift frequency spectrum generally in-

creases for decreasing frequency, which is why it is often called 1/ f noise. Anoma-

lous field coupling would, in contrast, create a clear peak at f = 1/sd. Because

diurnal and sidereal signals are so similar on the timescale of a typical several-day

run, the only way to clearly verify anomalous field coupling at f = 1/sd is to

compare the phase of any sidereal variation between two data sets several months

apart. After 6 months, diurnal variation experiences a full 180 phase lag with

respect to sidereal variation due to the earth moving around the sun.


4.3.2 Long term data analysis

From all the signal data S recorded over 15 months, one must obtain a single pair

of sidereal amplitudes Ax and Ay. Figure 4.18 shows all of the data taken over

the course of a year that meet basic minimum requirements. In this plot, the data

were filtered by eliminating data runs that were shorter than 1 sidereal day and by

limiting the noise with a conservative cutoff of Arms < 100 fT; where Arms is the

RMS (root mean square) or standard deviation of the data:

Arms = κ

√√√√ 1NS

NS

∑i=0

(Si − 〈S〉)2 (4.4)

where 〈S〉 is the mean of the data and NS is the number of samples in the data set.

Figure 4.19 shows the sidereal amplitude of each run with a cutoff of Arms < 100

fT. Although the order of the data is preserved, the actual times have been dis-

carded. The error bars for each run are defined by the RMS of the data in the run

and the length NS of the run according to σi = Arms/√

NS. In this way, longer

runs have greater weight. The RMS uncertainties are conservative estimates of the

amount of noise contributing to the sidereal amplitudes. They are, however, accu-

rate representations of the relative noise level between the runs. One can justifiably

scale all the uncertainties σi by a multiplicative factor such that the resulting fit

provides χ2 = 1, indicating that the error bars are consistent with the scatter of the

data. For reference,

χ2 ≡ 1NA − 1

NA

∑i=0

(Ai − 〈Ai〉)2

σ2i

(4.5)


−50

0

50

Ay

(fT

)

0 5 10 15 20 25 30 35 40

Run number

Ay = (0.355± 0.941) fT

−50

0

50

Ax

(fT

)

Ax = (−0.694± 0.634) fT

Figure 4.19: Summary of the sidereal amplitudes of each run. These runs were selectedby limiting the noise level of each run to Arms < 100 fT. The error bars set by Arms/

√NS

where NS is the length of the run were scaled by a common factor to be consistent with thescatter of the data.


Here NA is the number of runs and 〈A〉 is the weighted mean of the daily ampli-

tudes given by

〈A〉 =∑NA

i=0Aiσ2

i

∑NAi=0

1σ2

i

(4.6)

To properly estimate the error for the data in Figure 4.19 such that it is consistent

with the scatter of the data (making χ2 ' 1), the RMS amplitudes were uniformly

scaled by a factor of order unity. The resulting weighted average amplitudes are

indicated in the figure. These results, however, may be skewed because of gaps in

the data during the runs.

occurrence In order to eliminate gaps in the runs, one can use a more fine-

grained approach wherein the runs are split up into individual sidereal day seg-

ments that avoid any data gaps in the run. To obtain the data and results in Figure

4.20, there are several discernible steps to describe: (1) First, the data are broken

down into short, equal 1 sidereal day segments that avoid any gaps. (2) The days

are assigned error bars or weights σi according to the standard deviation or RMS,

Arms, of the signal in each day. (3) Without introducing any bias or strongly af-

fecting the results, the noisiest data can be filtered out by setting an upper limit

Arms < 100 fT on the RMS of the signal. (4) The sidereal amplitudes Aix,y of the

data in each day are obtained using the lock-in technique described in Section 4.3.1.

(5) All the days of data with their amplitudes and uncertainties are combined us-

ing a weighted average to obtain a single pair of sidereal amplitudes Ax and Ay for

the entire data set. (6) If the data set has χ2 6= 1, the error bars may be uniformly

scaled by a factor of order unity in order to provide χ2 = 1 for which the scatter

of the data is consistent with the uncertainties. For Figure 4.20, the error bars were


−100

−50

0

50

100

Ay

(fT

)

0 10 20 30 40 50 60 70 80 90 100 110

Day number

Ay = (0.591± 0.807) fT

−100

−50

0

50

100

Ax

(fT

)

Ax = (−0.760± 0.740) fT

Figure 4.20: Summary of sidereal amplitudes of each day. These days were selected bylimiting the noise level of each day to Arms < 100 fT. The error bars set by Arms were scaledby a factor of 0.7 to be consistent with the scatter of the data. The indicated results for Axand Ay are weighted averages of these data.

4.4. Anomalous coupling energy and conversions 175

set to σi = 0.7Airms. The reduction of the error bars using this method indicates that

the RMS value overestimates the uncertainty in the sidereal amplitudes.

The only arbitrary element in this analysis was the cutoff of Arms < 100 fT.

Figure 4.21 shows how the results change for a range of different cutoff values.

As the cutoff increases, more days of data are included, the uncertainty decreases

towards an asymptote and the mean values of Ax and Ay become constant. Thus,

it is clear that the cutoff Arms < 100 fT discards insignificant data. The resulting

weighted average of the data in Figure 4.20 gives

Ax = (−0.76± 0.74) fT

Ay = (+0.59± 0.81) fT (4.7)

In the next section, these amplitudes are used to obtain a limit on anomalous fields.

Figure 4.22 shows a histogram of the sidereal amplitudes of each day. One

can see significant deviation from a Gaussian distribution, an indication that one

must avoid analysis techniques such as least-squares fitting that require Gaussian-

distributed data. The analysis technique presented here does indeed avoid this

requirement.

4.4 Anomalous coupling energy and conversions

The co-magnetometer is sensitive to the difference between the 3He nuclear cou-

pling and the K electron coupling to an anomalous field (Equation 2.123):

A = κS = bn − be (4.8)


−5

−2.5

0

2.5

5

Ay

(fT

)

0 20 40 60 80 100

Cutoff (fT)

−5

−2.5

0

2.5

5

Ax

(fT

)

Mean

Error

0

1

2

χ2

χ2x

χ2y

0

1

2

Un

cert

ain

ty(f

T)

δAx

δAy

0

50

100

150D

ays

of

Dat

a

Figure 4.21: Dependence of the mean value and uncertainty on the signal RMS cutoff.Adding days with larger RMS noise (above 40 fT) has little effect on the results. Theweights on each day were scaled by 0.7 for all parts of this analysis.


0

0.01

0.02

0.03

0.04

Occ

urr

ence

Fre

qu

ency

-38.

0-3

4.0

-30.

0-2

6.0

-22.

0-1

8.0

-14.

0-1

0.0

-6.0

-2.0 2.0

6.0

10.0

14.0

18.0

22.0

26.0

30.0

34.0

38.0

Ay (fT)

0

0.01

0.02

0.03

0.04

0.05

Occ

urr

ence

Fre

qu

ency

-38.

0-3

4.0

-30.

0-2

6.0

-22.

0-1

8.0

-14.

0-1

0.0

-6.0

-2.0 2.0

6.0

10.0

14.0

18.0

22.0

26.0

30.0

34.0

38.0

Ax (fT)

Figure 4.22: Distribution of the sidereal amplitudes of each day of data. These data reflectthe amplitudes found in Figure 4.20.


The couplings to K and 3He spins can be approximated to first order as couplings

to electron and neutron spins, respectively. Although the 41K nuclear spin I = 3/2

is due to a valence proton, the K coupling is dominated by the valence electron

coupling, which is a thousand times stronger for comparable be ∼ bp. 3He nuclear

spin can be approximated by a neutron spin because the 3He nucleus has two pro-

tons in a closed shell with a valence neutron. However, Friar et al. (1990) show

that in a polarized sample of 3He, 87% of the neutrons are polarized while 2.7% of

the protons are also polarized in the opposite direction. Thus, the co-magnetomter

is sensitive to the following linear combination of electron, neutron and proton

couplings:

A = κS = 0.87bn − 0.027bp − be (4.9)

A few percent of the spin coupling of 3He is due to proton spin while the great

majority can be attributed to neutron spin. In this way, the experiment can set

limits on the couplings to all three spins.

The coupling strength of the K electron and 3He nuclear spins to an anomalous

b field is typically given in units of energy. The amplitudes Ax,y of the anomalous

field in Tesla can be expressed in terms of energy using a magnetic moment:

bni = µ3HeAi/ f n µ3He = −6.707× 10−17 GeV/T f n = +0.87

bpi = µ3HeAi/ f p where µ3He = −6.707× 10−17 GeV/T , f p = −0.027

bei = µe Ai/ f e µe = −5.795× 10−14 GeV/T f e = −1 (4.10)

where the fractional contribution to the 3He spin polarization is f n from neutron

spin and f p from proton spin, µe is the magnetic moment associated with the K


electron (a Bohr magneton) and µ3He is the magnetic moments associated with the

neutron and proton components of the 3He nuclear spin.

The experimental measurements summarized in Equation 4.7 can be expressed

in energy terms using Equation 4.10:

bnx = (+5.9± 5.7)× 10−32 GeV bn

y = (−4.6± 6.2)× 10−32 GeV

bpx = (−1.9± 1.8)× 10−30 GeV bp

y = (+1.5± 2.0)× 10−30 GeV

bex = (−4.4± 4.3)× 10−29 GeV be

y = (+3.4± 4.3)× 10−29 GeV (4.11)

Conventions and conversions for anomalous field measurements are detailed in

Appendix A. It is standard practice to express the measured anomalous field cou-

plings in terms of components along the principal axes of the celestial sphere. The-

se conversions amount to a rotation in the x-y plane to correct for the Local Sidereal

Time and a division by cos(40.3449) to account for the latitude of the laboratory:

bnX = (−3.7± 8.1)× 10−32 GeV bn

Y = (−9.0± 7.5)× 10−32 GeV

bpX = (+1.2± 2.6)× 10−30 GeV bp

Y = (+2.9± 2.4)× 10−30 GeV

beX = (+2.8± 6.1)× 10−29 GeV be

Y = (+6.8± 6.1)× 10−29 GeV (4.12)

These couplings are dominated by uncertainty, indicating that there is no anoma-

lous effect at this level of sensitivity.

A simple bound on the level of these anomalous field couplings can be obtained

from these data. The bound b is set such that a measurement of bx and by has a


b

x

y

dbx

dby

Figure 4.23: Error analysis integral: The uncertainties in the measurements of bx and by areassumed to be Gaussian distributions. The bound b is set such that a measurement of bxand by has a 68.3% (1-σ) chance of being smaller than b.

68.3% (1-σ) chance of being smaller than b. Assuming normally distributed mea-

surements, this bound can be obtained by solving the following equation for b:

0.683 =∫ x=+b

x=−b

∫ y=+√

b2−b2x

y=−√

b2−b2x

1πδbxδby

exp(− (x− bx)2

δb2x

)exp

(−

(y− by)2

δb2y

)dy dx

(4.13)

This integral can be visualized using Figure 4.23. Using this technique, one can set

the following 68.3% confidence limits on the couplings of the CPT- and Lorentz-

violation term:

bn < 1.4× 10−31 GeV

bp < 4.4× 10−30 GeV

be < 1.0× 10−28 GeV (4.14)

Sensitivity to the nuclear spin coupling is comparable to the existing limit set

by the Harvard-Smithsonian noble gas maser (Bear et al., 2002). They quote a limit

of bnXY = (6.4± 5.4)× 10−32 GeV. The data tabulated in their paper can provide

the following measurements along the celestial axes:

3He-129Xe Maser: bnX = (−0.3± 5.6)× 10−32 GeV

bnY = (+8.8± 5.3)× 10−32 GeV (4.15)


From these measurements, the limit derived using Equation 4.13 is

3He-129Xe Maser: bn < 1.1× 10−31 GeV (4.16)

It appears that Bear et al. (2000) assume that the 3He nucleus is entirely composed

of neutron spin. The data from Friar et al. (1990) should be used to adjust the

neutron limit and can also be used to provide a proton limit as well. The proton

limit that can be derived from the 3He-129Xe maser should be comparable to the

limit that the present experiment achieves. However, since no numbers on proton

sensitivity have been published from this experiment, the following experiment

provides the existing limit on the proton coupling.

The proton spin coupling was directly measured using a hydrogen maser by

Phillips et al. (2001). The data in their paper provides the following measurements:

H Maser: bpα = (+0.1± 9.5)× 10−32 GeV

bpβ = (−8.4± 9.6)× 10−32 GeV (4.17)

Direct comparison to the X and Y celestial axes is not possible using these data, but

the limit is given by

H Maser: bp < 1.8× 10−27 GeV (4.18)

which is in agreement with the limit provided in their paper. The present experi-

ment provides a factor of 100 times improvement over this measurement.

The coupling to the electron spin is somewhat less sensitive than the Eot-Wash

torsional pendulum (Heckel et al., 2000), with measurements of

Torsional pendulum: beX = (0.1± 2.1)× 10−29 GeV

beY = (1.7± 2.3)× 10−29 GeV (4.19)


These measurements produce a limit of

Torsional pendulum: be < 3.0× 10−29 GeV (4.20)

In light of these previous measurements, this experiment does not establish

new limits on anomalous field couplings. Nevertheless, it provides an indepen-

dent verification using new techniques that anomalous field couplings are not ob-

served at the present sensitivity.

Chapter 5

The co-magnetometer gyroscope

THE CO-MAGNETOMETER is a very sensitive, compact gyroscope. Rotation of

the apparatus gives a co-magnetometer signal in much the same way that

anomalous fields and lightshifts do: all of these couple to the electron and nuclear

spins differently than a magnetic field and thus generate a co-magnetometer sig-

nal. In the most naıve approximation, the co-magnetometer apparatus is rotating

around spins that are stationary in an inertial frame. The lasers on the apparatus

perceive the spins rotating due to a non-inertial torque that rotates the spins with

respect to the apparatus. The information in this chapter is published in Kornack

et al. (2005).

A more accurate description of the co-magnetometer gyroscope is somewhat

more complex. The electron spins are re-pumped quickly along the rotating pump

laser while the nuclear spins lag behind. The nuclear spins lag behind the rotat-

ing compensation field and precess into the vertical axis, perpendicular to the

pump and probe beams. The vertical field generated by these vertical nuclear spins

183

184

Hot AirCell

Magnetic Shields

Floating Optical Table

Position sensors

Polarizer

Pie

zoel

ectr

ic S

tack

Imm

obile

Blo

ck

Analyzing Polarizer

Photodiode

FaradayModulator

Field Coils

Lock-inAmplifier

y z

x

Pockel Cell

Pum

p B

eam

Probe Beam

λ/4

I3HeM3He

SK

Bz

MK

Single Freq.Diode Laser

Hig

h P

ower

D

iode

Las

er

Figure 5.1: A diagram of the gyroscope experiment showing the non-contact position sen-sors and the piezo driver.

ImmobileBlock

Piezo Stack

Table rotation

z x

y

Photodiode

CellSpin polarization

PositionSensors

Probe BeamPump Beam

zx

y

Figure 5.2: A front view of the gyroscope experiment.

185

−20

0

20R

ota

tio

n(µ

rad

/se

c)

0 2.5 5 7.5 10 12.5

Time (s)

−100

−50

0

50

100 Effe

ctive

Fie

ld(fT

)

Figure 5.3: Angular velocity data from position sensors (dashed line) and co-magnetome-ter signal (solid line) are plotted with no free parameters.

causes the potassium spins to precess, which is what is ultimately being measured.

These dynamics are captured by Equation 2.132, reprinted here for reference:

Ωy = γgbeffy where γg '

(1

γn− Q(Pe)

γe

)−1

(2.132)

where beffy is the calibrated signal from the co-magnetometer. One can use this

equivalence between magnetic field and rotation to calibrate the gyroscope using

magnetic fields. Conversely, it is possible to check sensitivity to CPT-violating

terms in our experiment by rotating the table.

The optical table with the K-3He co-magnetometer cell at its center was rotated

using a piezo stack as shown in Figures 5.1 and 5.2. Six non-contact position sen-

sors measure the orientation of the table as it rotates. In Figure 5.3 the rotation

signal from the position sensors and the rotation signal from the co-magnetometer

agree to within the noise of the co-magnetometer. In this plot there is no fitting

or adjustment; the calibration for the magnetometer signals was obtained using

magnetic excitation and converted to angular units using Equation 2.132. Since

186

0

0.005

0.01

0.015

0.02

Ang

leR

ando

mW

alk

(deg

rees

/hou

r1/2)

0 200 400 600 800 1000

Frequency (hour−1)

0

5

10

15

20

25Field

(fT/H

z1/2)

Figure 5.4: Noise spectrum for the co-magnetometer gyroscope.

the gyroscope directly measures the rotation angle Ωy, the measured angular fre-

quency must be integrated to obtain the angle.

The equivalence between magnetic field and rotation allows the gyroscope to

inherit all of the measured magnetic field properties of the co-magnetometer. The

noise of the co-magnetometer, shown in Figure 5.4 is nearly flat at 1 fT/√

Hz or

1.4 × 10−5 rad/√

hour for frequencies greater than 400 1/hour. A clear 1/ f de-

pendence is visible at lower frequencies. The magnetic noise is lower than the

noise due to magnetic shields because the co-magnetometer compensates for that

magnetic field noise. The gyroscope also has the same field gradient suppression

shown in Figure 3.20.

The long term stability of the absolute orientation of a gyroscope is critical in

navigation applications. In Figure 5.5 a stationary gyroscope signal was measured

for several hours. The drifting signal had its linear drift removed and was then

187

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

An

gle

(deg

rees

)

0 1 2 3 4 5

Time (hours)

Constant drift of 0.1 deg/h

Figure 5.5: Integrated angle due to long term gyroscope signal drift.

integrated over time to produce the wandering angle measurement in this plot.

The drift is at most 0.1 deg/hour.

The gyroscope angular frequency measurement is usually integrated over time

to obtain the rotation angle for use in navigation applications. The disadvantage

to the integral is that small signal offsets linearly grow the integrated angle in

time. There is, however, one compelling advantage to the integral: the gyroscope

responds to fast transients of both Ω and magnetic field gracefully. It can be shown

that the integral of the signal accurately gives the rotation angle regardless of the

time dependence of Ω. Indeed, as long as the transverse spin excitation is small

and the linearized Bloch equations are correct, transients in Ω with any time de-

pendence integrate to give the correct angle. Furthermore, the integrated signal

after an arbitrary magnetic field transient is zero. To illustrate this behavior, in

Figure 5.6 a large magnetic field transient is applied to the gyroscope and the sig-

nal responses were measured and integrated. If this were a bare K magnetometer,

188

−20

0

20

40

60

Mag

net

icF

ield

(pT

)

0 0.1 0.2 0.3 0.4 0.5 0.6

Time (s)

−0.1

0

0.1

0.2

0.3

An

gle

(rad)

Figure 5.6: Response of the gyroscope in magnetic units (dashed line) to a short magneticfield transient (solid line) yields zero integrated angle (dash-dot line) in comparison to theangle achieved by an uncoupled K spin (dotted line).

the integral of the applied transient would be large, given here by the dotted line.

The co-magnetometer response given by the dashed lines wiggles around zero in

such a way that its integral given by the dash-dotted line is equal to zero after the

excitations decay. In this example, the sensitivity to magnetic field transients is

suppressed by more than a factor of 400.

The fundamental sensitivity of the gyroscope is given by the shot-noise of the

magnetometer, Equation 2.41, converted to angular units using Equation 2.132:

δΩy =γn

γe

√Q(Pe)Rtot

NVt(5.1)

For the present configuration, the sensitivity limit is δΩ ' 1.2× 10−8 rad/s/√

Hz.

This is significantly better than the measured value, δΩ = 5.0× 10−7 rad/s/√

Hz,

which is higher due angular noise in the probe beam setup. The fundamental

sensitivity can be significantly improved by using 21Ne, which has a nuclear spin

3/2 with a gyromagnetic ratio almost 10 times lower than that for 3He. For the

189

same conditions, 21Ne should give an immediate improvement in sensitivity by a

factor of 10. In an ideal scenario, for a 10 cc Ne detector the sensitivity limit would

be δΩ = 2.0× 10−10 rad/s/√

Hz.

This gyroscope is competitive with existing gyroscope techniques because its

active volume is comparatively compact, its sensitivity is quite good, and it comes

with the various appealing properties discussed above. Table 5.1 contains a sur-

vey of existing gyroscope techniques with as much performance data as possible

converted into units that can be used for direct comparison. This gyroscope is

categorized as being a relatively compact 10 cm device. Although the current im-

plementation is closer to 2 m on a side, the active region is just 2.5 cm diameter

and the magnetic shields and lasers can be made much more compactly. In fact,

the magnetic shielding performance improves as it gets smaller. The direct com-

petitor at this scale is the widely-used fiber-optic gyroscope. After more than two

decades of refinement, the fiber optic gyroscopes have reached their practical limits.

The present co-magnetometer gyroscope is just over an order of magnitude away

in noise from the best published fiber optic gyroscope. The use of 21Ne should be

able to surpass the performance of the fiber-optic gyroscope in the near term. The

drift of the co-magnetometer gyroscope will significantly improve as it is made

more compact. In a compact device, temperature drifts will become easier to con-

trol and the evacuation of the entire device could reduce noise due to convection

currents.

The co-magnetometer gyroscope is also competitive with larger gyroscopes us-

ing different techniques. Sagnac-based gyroscopes gain sensitivity with a larger en-

closed area. Thus, the very large ring laser gyroscopes and the atom interferometer

190

Type

Rea

lized

Proj

ecte

dD

rift

Cit

atio

nSe

nsit

ivit

ySe

nsit

ivit

yra

d/s/√

Hz

rad/

s/√

Hz

rad/

hour

Larg

eSc

ale

(∼2

m)

Rin

gLa

ser

Gyr

o(C

II)

2.2×

10−

10—

—St

edm

anet

al.(

2003

)A

tom

Inte

rfer

omet

er(Y

ale)

6.0×

10−

102.

0×

10−

101.

3×

10−

4G

usta

vson

etal

.(20

00)

Inte

rmed

iate

Scal

e(∼

50cm

)M

echa

nica

l(G

ravi

tyPr

obe

B)—

—3.

0×

10−

14Bu

chm

anet

al.(

1996

)Su

perfl

uid

3 He

(Ors

ay)

1.4×

10−

73.

0×

10−

102.

1×

10−

5A

vene

leta

l.(2

004)

Ato

mic

Inte

rfer

omet

er(H

YPE

R)

—2.

0×

10−

9—

Jent

sch

etal

.(20

04)

Ato

mic

Foun

tain

(Par

is)

—3.

0×

10−

8—

Yver

-Led

ucet

al.(

2003

)A

tom

icSp

in‘N

MR

G’(

Litt

on)

2.9×

10−

6—

9.0×

10−

4W

oodm

anet

al.(

1987

)Sm

allS

cale

(∼10

cm)

Fibe

r-op

tic

Gyr

o(H

oney

wel

l)2.

3×

10−

8—

1.7×

10−

6Sa

nder

set

al.(

2002

)A

tom

icSp

in(P

rinc

eton

)5.

0×

10−

72.

0×

10−

107.

0×

10−

4

Min

iatu

reSc

ale

(<1

cm)

MEM

S(C

MU

)3.

5×

10−

41.

8×

10−

40.

5X

iean

dFe

dder

(200

3)

Tabl

e5.

1:A

surv

eyof

gyro

scop

epe

rfor

man

ce.

191

gyroscopes currently have the best reported sensitivity. Both of these are not suffi-

ciently compact for practical applications. The co-magnetometer gyroscope is com-

petitive with devices in the intermediate ∼ 50 cm scale; the projected sensitivities

of these gyroscopes are all comparable or worse than the smaller co-magnetometer

gyroscope. The exception is the mechanical gyroscopes on board Gravity Probe B

that feature very low drift operating in low gravity but are significantly degraded

in the gravitational field on the surface of the earth. The even smaller MEMS gy-

roscope holds promise in future development, although the current designs are so

sensitive to temperature drift that they are useless for long term navigation.

Chapter 6

Conclusions

THE K-3HE CO-MAGNETOMETER forms an ideal detector for anomalous fields

because it has high short-term sensitivity and is insensitive to magnetic field

drift. Based on data over a 15 month period, the co-magnetometer coupling to an

anomalous field is quoted as follows:

This experiment (Previous measurements)

bn < 1.4× 10−31 GeV (bn < 1.1× 10−31 GeV, 3He-129Xe maser)

bp < 4.4× 10−30 GeV (bp < 1.8× 10−27 GeV, H maser)

be < 1.0× 10−28 GeV (be < 3.0× 10−29 GeV, torsional pendulum) (6.1)

These measurements indicate that there is no anomalous effect at this level of sen-

sitivity. Sensitivity to the nuclear spin coupling is comparable to the existing limit

set by Harvard-Smithsonian noble gas maser (Bear et al., 2002). The proton cou-

pling is a factor of 100 better than the existing published limit set by the Harvard-

Smithsonian hydrogen maser (Phillips et al., 2001). The coupling to the electron

spin is an factor of 3 less sensitive than the Eot-Wash torsional pendulum (Heckel

192

193

et al., 2000). This measurement represents an important verification of existing re-

sults using a new method. These results should be considered preliminary due to

the relatively low sample size and the high systematic noise found in the majority

of the data.

Significant improvements in the sensitivity of the co-magnetometer can be ach-

ieved by reducing systematic noise. Although the co-magnetometer can achieve

1 fT sensitivity (equivalent to the existing limit on neutron coupling strength) in

1 second, the poor long-term stability of the experiment significantly reduces the

sidereal sensitivity. Study of systematic noise and long-term drift represents the

bulk of the labor associated with this dissertation. Increased thermal and mechani-

cal stability of all aspects of the experiment directly improves the co-magnetometer

sensitivity. Work will continue on the stability of the experiment and procedural

changes can address systematic noise introduced by zeroing routines. A second

generation of this experiment should be designed from the ground up to eliminate

temperature sensitivity and mechanical instability. This can be most effectively

achieved by making the entire experiment as compact as possible and placing it

inside a well insulated, temperature-controlled, and evacuated enclosure.

In the pursuit of a test of CPT and Lorentz violation, four major results have

been obtained: (1) A new SERF magnetometer with unprecedented, < 1 fT/√

Hz

sensitivity has been developed. This technology has many applications wherever

SQUIDs are currently used and has the advantage of obviating the use of cryogens.

The localization of brain activity is an excellent application of SERF magnetome-

ters because it utilizes both the high sensitivity and high spatial resolution. (2) A

co-magnetometer that is insensitive to magnetic field drift has been developed for

194

this anomalous field measurement. The same co-magnetometer can be used to de-

tect any non-magnetic effect that couples to spin such as spin-mass and spin-spin

couplings. (3) Strongly coupled dynamics of electron and nuclear spin ensembles

has been observed for the first time. (4) A compact, high-sensitivity gyroscope has

been developed that is competitive with similarly compact gyroscopes based on

different techniques.

Appendix A

Time and orientation conventions

Following the convention set by Kostelecky and Lane (1999), the magnitude of

the anomalous field coupling is quoted in terms of a fixed coordinate system with

Z along the rotation axis of the earth. The X and Y axes are defined according

to celestial coordinates: X has both zero declination and zero right ascension, as

shown in Figure A.1. Thus, X points along the intersection of the ecliptic (solar

orbit) and equatorial planes. Greenwich Sidereal Time, GST time, the unit of time

used in all of the data here, is based on the movement of Greenwich, England

relative to the celestial sphere. At midnight GST, the position of Greenwich is

aligned with X. Since Princeton, NJ is at longitude θ = −74.6520, the apparent

Local Sidereal Time (LST) has an offset from GST of 1 sd × −74.6520/360 =

−0.207 sd. LST is defined such that at every time LST hits 0 hours (an exact integer

number of sidereal days), X is directly overhead.

195

196

Z Z

YX X

Wt Wt

Wc

c

nUp

nEast

nSouth

Right ascension

Celestial North Pole

Celestial Equator

Celestial FrameLab Frame

Vernal Equinox

Ecliptic

Earth

Declination

Figure A.1: Left: Celestial and lab axes standardized by Kostelecky and Lane (1999). Theaxes have been drawn with a common origin for clarity. Right: Standard definitions ofright ascension and declination

A series of coordinate system transformations are necessary to convert between

the celestial coordinates and the lab coordinates. Kostelecky and Lane (1999) pro-

vides the following standard transformation between celestial coordinates and lo-

cal directions:nSouth

nEast

nUp

=

cos χ cos Ωt cos χ sin Ωt − sin χ

− sin Ωt cos Ωt 0

sin χ cos Ωt sin χ sin Ωt cos χ

X

Y

Z

(A.1)

where χ is the complement of the declination; in Princeton, NJ the latitude is

φ = 40.3449, giving χ = 90 − φ = 49.6551. The experiment z axis is oriented

southeast at an angle ψ = 30 from south. That transformation is expressed byz

x

y

=

cos ψ sin ψ 0

− sin ψ cos ψ 0

0 0 1

nSouth

nEast

nUp

(A.2)

Since the experiment is only sensitive to anomalous fields along y, this transforma-

tion is trivial.

197

It is convenient to express the sidereal variation in terms of independent in-

phase and out-of-phase components by making the substitution cos Ωt → ALx and

sin Ωt → ALy , where the superscript L corresponds to LST. The amplitude AL

x corre-

sponds to a sinusoid with its maximum at midnight LST; the orthogonal sinusoid

reaches the maximum of ALy a time 0.25 sd later. To convert between the measured

Ax pegged to GST and the useful ALx in LST, one can use the following transforma-

tion: ALx

ALy

=

cos θ − sin θ

sin θ cos θ

Ax

Ay

(A.3)

where, again, θ is longitude. In retrospect, it would have been better to use LST as

the standard time in this experiment. With all of these transformation, it is possible

to obtain an expression for the anomalous field b projections along the celestial axes

in terms of the measured GST Ax and Ay amplitudes:

bX = +Axcos θ

sin χ− Ay

sin θ

sin χ(A.4)

bY = +Aycos θ

sin χ+ Ax

sin θ

sin χ(A.5)

This experiment does not have any sensitivity to the constant bZ term since it is

constant in the experiment’s frame.

It is an accepted standard to plot data against the Greenwich Mean Sidereal

Time (GST) in units of sidereal days (sd) since J2000. Meeus (1998) provides accu-

rate algorithms for calculating the present GST. To obtain sidereal time, one must

first calculate the number of solar days since the beginning of the J2000 epoch:

TdJ2000 =367× year− floor[7(year + floor[(mon + 9]/12)/4] + floor[275×mon/9]

+ day + (hour + mins/60 + secs/3600)/24− 730531.5 (A.6)

198

where the time to use here is Universal Time (UT) and the units of the result are

in solar days (d). This is, unfortunately, cumbersome to calculate on the computer.

The computer conveniently provides a standard UNIX epoch time defined as the

number of seconds since the time 00:00:00 UT on January 1, 1970. One can carefully

convert Equation A.6 into the UNIX epoch as follows:

TdJ2000 =

tUNIX + 3, 029, 572, 800 s86400 s/d

(A.7)

With a solid expression for the number of (fractional) solar days since J2000, one

can use Meeus (1998) to find the GST sidereal time:

TsdJ2000 =

280.46061837 + (360.98564736629/d)× TdJ2000

360/sd(A.8)

Note that the following ratio accurately relates the length of the solar day to the

length of a sidereal day:

1 d1 sd

=360.98564736629/d

360/d(A.9)

Whenever minutes and seconds are indicated in this text, they always refer to the

usual divisions of the solar day; use of sidereal time is always explicitly labeled.

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