Stray Fields and The Electron’s Electric Dipole Moment
byT. H. Wright
B.A., Amherst College, 2018
M.S., Yale University, 2020

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2024

Committee Members:
Eric Cornell, Chair
Jun YeJohn Bohn
Adam Kaufman
J. Mathias Weber

ii
Wright, T. H. (Ph.D., Physics)
Stray Fields and The Electron’s Electric Dipole Moment
Thesis directed by Prof. Eric Cornell

The universe is full of matter, and we cannot explain how it got there. According to ourmost accurate theory of particle physics, the Standard Model, the big bang created equal parts
matter and antimatter. In the billions of years since, matter and antimatter should have collided
and annihilated, leaving (almost) nothing behind. This obviously is not what happened; we live
inside of an entire universe made of matter. Despite this serious shortcoming, the Standard Modelis outrageously successful in predicting how particles will behave in experiments here on Earth. To
salvage the Standard Model, new theories tack on as-of-yet undiscovered particles and interactions
that violate the symmetry between matter and antimatter. A side effect of breaking this symmetryis that electrons should have a non-zero electric dipole moment (EDM). In this thesis, I present
the world’s most precise measurement of the electron EDM to date using electrons confined insidehafnium fluoride molecular ions (HfF+ ). We trap HfF+ in corotating electric and magnetic fields
and measure the electron EDM signal by performing Ramsey spectroscopy with coherence times up
to 3 seconds. Our result is consistent with an electron EDM of zero and improves on the previous
best upper limit by a factor of ∼ 2.4, further constraining proposed theories of particle physics.
I also worked towards a future measurement, hopefully 10× more precise, of the electron EDM
using thorium fluoride molecular ions. I discuss the systematic errors we uncovered in our HfF+
measurement that require our future experiment to be magnetically shielded.

Dedication

To everyone who reads my dissertation.

“As far as I’m concerned, if something is so complicated that you can’t explain it in ten seconds,
then it’s probably not worth knowing anyway.”- Bill Watterson, The Indispensable Calvin and Hobbes

iv

Acknowledgements

Thank you to my teachers, from those who taught me how to tie my shoes to the people stillteaching me how to work in a lab. Connie Isbell, Jeanne Jewell, Arthur Crumm and Mark Adams
are just a few of the incredible teachers I had growing up.
Thank you to the administrators who pushed me through hoops I did my best to stumble
through, particularly Krista Beck.
Thank you to anyone who spent time cleaning up after me.
Thank you to everyone who worked with me, particularly those who still think I know what
I am doing. Nate Clayburn, Claire Carlin, Olivier Grasdijk, Oskari Timgren, Jakob Kastelic, Andy
White, Guanchen Peng, Luke Caldwell, Tanya Roussy, Kia Boon Ng, Noah Schlossberger, Sun Yool
Park, Anzhou Wang, Addison Hartman and Patricia Hernandez have all been a joy to work with.
Thank you to Larry Hunter, Dave DeMille, Dave Kawall, Tanya Zelevinsky, Steve Lamoreaux,
Ben Sauer, Mike Tarbutt, Eric Cornell and Jun Ye. I learned an incredible amount working in theirlabs, from how to use a level to precision molecular spectroscopy.
Thank you to all of my friends and family who helped me find my way through two graduate
programs. My parents have been a wellspring of support my entire life, and I could not have madeit this far without them. My cousin Kate Berrigan gets a special mention because she adopted adog, letting me have a pet in graduate school without the responsibility.

v

Contents

Chapter
1 Introduction
1.1

1

Crash Course in Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

The Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.1.3

The Standard Model, Particles and Interactions . . . . . . . . . . . . . . . . .

9

1.1.4

The Standard Model, Discrete Symmetries . . . . . . . . . . . . . . . . . . . 15

1.2

The Baryon Asymmetry of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3

The Electron’s Electric Dipole Moment

1.4. . . . . . . . . . . . . . . . . . . . . . . . . 23

1.3.1

The eEDM Violates CP-Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.2

The eEDM is Background Free . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3.3

The eEDM is Sensitive to BSM Physics . . . . . . . . . . . . . . . . . . . . . 27

Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Generation 2 Overview

31

2.1

JILA eEDM Experimental Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2

How the JILA Generation Two eEDM Experiment Works . . . . . . . . . . . . . . . 40
2.2.1

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.2.2

Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.2.3

Effective Hamiltonian for Doublets . . . . . . . . . . . . . . . . . . . . . . . . 59

vi
3 Systematics

63

3.1

Systematics due to Off Diagonal Terms

3.2

Doublet-odd Frequency Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.3

3.4

3.5

3.6

. . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.1

Zeeman Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2.2

⃗2,0
Zeeman Shift from E⃗rot and B

3.2.3

Imperfect Doublet Overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

. . . . . . . . . . . . . . . . . . . . . . . . . 73

Systematics from Magnetic Shifts in f B and f DB . . . . . . . . . . . . . . . . . . . . 79
3.3.1

Analytic Approach to Zeeman Systematics . . . . . . . . . . . . . . . . . . . 79

3.3.2

Numerical Approach to Zeeman Systematics

3.3.3

Uniform Magnetic Field + Second Harmonic Systematic . . . . . . . . . . . . 87

3.3.4

Other Systematics from Zeeman Shifts . . . . . . . . . . . . . . . . . . . . . . 91

. . . . . . . . . . . . . . . . . . 81

Other Noteworthy Magnetic Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.4.1

Axial Magnetic Field and f R . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.4.2

Charging Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.4.3

Other Zeeman Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Berry’s Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.5.1

Berry’s Phase Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.5.2

Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

3.5.3

Ellipticity plus Axial Second Harmonic . . . . . . . . . . . . . . . . . . . . . . 103

3.5.4

nth Harmonic along z plus (n-1) or (n+1)th Harmonic in the xy-plane . . . . 105

3.5.5

Other Berry’s Phase Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Systematics Summary and Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4 Sign of the G-Factor

110

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.2

Possible Methods of Measuring the Sign of gF . . . . . . . . . . . . . . . . . . . . . . 112

4.3

The JILA eEDM Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

vii
4.3.1

Experimental Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.3.2

Experimental Switches and Frequency Channels . . . . . . . . . . . . . . . . 115

4.3.3

Berry’s Phase in JILA eEDM . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.4

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.5

Conclusion

4.6

Appendices for Sign of the G Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 125. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.6.1

Appendix 1: Berry’s Phase Derived

4.6.2

Appendix 2: Relating de to f DB . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.6.3

Appendix 3: Calculating Berry’s Phase from 2H and Ellipticity . . . . . . . . 134

4.6.4

Appendix 4: Erot Phases in HfF+ Experiment . . . . . . . . . . . . . . . . . . 135

4.6.5

Appendix 5: Defining the Frequency Channels . . . . . . . . . . . . . . . . . 137

4.6.6

Appendix 6: Table of Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.6.7

Appendix 7: The sign of δgF in HfF+ . . . . . . . . . . . . . . . . . . . . . . 140

5 Magnetic Shielding
5.1

5.2

5.3

. . . . . . . . . . . . . . . . . . . . . . . 125

144

Generation Three JILA eEDM Experiment . . . . . . . . . . . . . . . . . . . . . . . 145
5.1.1

Desired Statistical Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.1.2

One of Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.1.3

Methods of Generating f 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.1.4

Choosing a Method to Generate f 0 . . . . . . . . . . . . . . . . . . . . . . . . 152

Magnetic Systematics in Generation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.2.1

Known Magnetic Systematics from Generation 2 . . . . . . . . . . . . . . . . 155

5.2.2

Maxwell Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.2.3

Systematics from Spatial Gradients . . . . . . . . . . . . . . . . . . . . . . . . 162

5.2.4

Other Magnetic Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Magnetic Shielding Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.3.1

Diamagnetic, Paramagnetic and Ferromagnetic Materials . . . . . . . . . . . 167

viii
5.3.2

Cylindrical Shield – Transverse Shielding Factor . . . . . . . . . . . . . . . . 168

5.3.3

Cylindrical Shield – Axial Shielding Factor . . . . . . . . . . . . . . . . . . . 175

5.3.4

Layers of Magnetic Shields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.3.5

Shielding Induced Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

5.3.6

Seams and Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.3.7

Degaussing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.4

Magnetic Shield Design – Plan E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.5Magnetic Shield Design – Plan A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6 Conclusion

201

Bibliography

203

Appendix
A Bz and BzB Measurements

210

B The Harmonics of Erot

213

ix

TablesTable
2.1

Spectroscopic constants for 3 ∆1 state of HfF+ . . . . . . . . . . . . . . . . . . . . . . 41

2.2

Example experimental parameters for Generation 2 . . . . . . . . . . . . . . . . . . . 41

2.3

Lasers in our Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1

Magnetic Field Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2

f B Curavatures with and without Permanent Magnets . . . . . . . . . . . . . . . . . 77

3.3

Analytically Calculated Zeeman Shifts, Part 1 . . . . . . . . . . . . . . . . . . . . . . 82

3.4

Analytically Calculated Zeeman Shifts, Part 2 . . . . . . . . . . . . . . . . . . . . . . 83

3.5

Analytically Calculated Zeeman Shifts, Part 3 . . . . . . . . . . . . . . . . . . . . . . 84

3.6

Analytically Calculated Zeeman Shifts Grouped Together . . . . . . . . . . . . . . . 92

3.7

Berry’s Phase Shifts from Harmonics of Erot . . . . . . . . . . . . . . . . . . . . . . . 107

3.8

Uncertainty Budget

4.1

Signs for Calculating the g-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.1

Kia Boon’s Plans to Generate f 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.2⃗Maxwell . . . . . . . . . . . . . . . . . . 161
Analytically Calculated Zeeman Shifts from B

5.3

Known Magnetic Field Requirements for Generation Three . . . . . . . . . . . . . . 166

5.4

Coil Positions and Currents to generate uniform B2,0 ∼ 4 G/m, Part 1 . . . . . . . . 193

5.5

Coil Positions and Currents to generate uniform B2,0 ∼ 4 G/m, Part 2 . . . . . . . . 194. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

x
B.1 Constraints on Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

xi

FiguresFigure1.1Electron scattering, annihilation and pair production . . . . . . . . . . . . . . . . . .

2

1.2

Electron-muon scattering, annihilation and pair production . . . . . . . . . . . . . .

4

1.3

Pair production and annihilation Feynman diagrams . . . . . . . . . . . . . . . . . .

5

1.4Classical and QFT vacuums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.5

Electron g-factor Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.6Standard Model particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.7

Left- and right-handed particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.8

Muon decay and beta decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.9

eEDM symmetry violations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.10 Two- and Three-Loop SM Contributions to the eEDM . . . . . . . . . . . . . . . . . 26
1.11 Four-Loop SM Contributions to the eEDM . . . . . . . . . . . . . . . . . . . . . . . 27
1.12 Feynman diagram of an eEDM from BSM interactions . . . . . . . . . . . . . . . . . 28
2.1

EDM Measurement Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2

Energy levels in 3 ∆1 relevant to eEDM Measurement . . . . . . . . . . . . . . . . . . 36

2.3

Molecule Rotating in Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . 37

2.4

How Brot arises from Baxgrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5

Statistical Sensitivity of an eEDM Measurement . . . . . . . . . . . . . . . . . . . . 39

2.6

Schematic of Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . 42

xii
2.7

Transitions used during State Preperation . . . . . . . . . . . . . . . . . . . . . . . . 47

2.8

Ramsey spectroscopy in HfF+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.9

Ion-detection data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.10 Data cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.1

Off Diagonal Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2

Measurement of δgF /gF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3

f B vs z Data with Permanent Magnets . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4

Simulated Ion Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.5

Zeeman shift from Bx and E2hx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.6

f B vs Bx and By . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.7

f R vs Bz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.8

Radial Fin Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.9

Current in Radial Electrode vs Height . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.10

df BR
dz

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.11 Berry’s Phase on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.12 f B and f BR vs E2hz phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.13 Ellipticity Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1

Microwave spectroscopy diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.2

JILA eEDM experimental overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.3

Berry’s Phase on a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.4

f B and f BR vs 2nd Harmonic Frequency with θ =4.5

π. . . . . . . . . . . . . . . . . . 122
8πf B and f BR vs 2nd Harmonic Frequency with θ = − . . . . . . . . . . . . . . . . . 123
8

4.6

Sign of the g-factor data summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

4.7

Axes for Berry’s phase derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.8

Map of vectors in the 3 ∆1 state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.9

Schematic of the Generation 2 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . 136

xiii
4.10 f D vs f 0 data and residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.1

Bucket Brigade Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.2⃗rot Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
B

5.3

ThF+

5.4

Erot bump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.5Erot and Brot bumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.6

Mumetal B vs H Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.7

Infinite Length Transverse Magnetic Shielding . . . . . . . . . . . . . . . . . . . . . . 170

5.8

Infinite Length Transverse Magnetic Shielding Comsol Model . . . . . . . . . . . . . 173

5.9

Finite Length Transverse Magnetic Shielding . . . . . . . . . . . . . . . . . . . . . . 173

δgF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
gF

5.10 Axial Magnetic Shielding Comsol Model . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.11 Axial Magnetic Shielding vs R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.12 Axial Magnetic Shielding vs L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.13 Axial Magnetic Field with Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.14 Opening in the Shield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.15 Shielding Factors vs Opening Angle. . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5.16 Shielding Factors vs Shield Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.17 Axial and Transverse Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.18 Axial Magnetic Field with Plan E Shields . . . . . . . . . . . . . . . . . . . . . . . . 189
5.19 Magnetic Shield Seams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.20 Alternating Plan A Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.21 B2,0 Created by Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.22 Bz along ρ = 0 from Coils with and without Shielding . . . . . . . . . . . . . . . . . 198
5.23 Bz along ρ = 0.25 meters from Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
5.24 Bz along ρ = 0 from Coils with shields, with and without Background Bz = 100 mG 200
6.1

eEDM Results vs Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

xiv
B.1 Notch Filter Circuit Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
B.2 Throughput Data Both Switch Settings . . . . . . . . . . . . . . . . . . . . . . . . . 216
B.3 Throughput Data up to 17th Harmonic . . . . . . . . . . . . . . . . . . . . . . . . . 217

Chapter 1

Introduction“The universe is a big place, perhaps the biggest.”- Kilgore Trout, Venus on the Half-Shell

A survey of recent JILA electric dipole moment (EDM) theses shows that their introductions
focus on a few common subjects. These include the usual story about matter and antimatter [1,2,3],
an overview of how our measurement works [1, 3, 4], a history of past EDM measurements [2, 4]
and a survey of contemporary experiments [1, 2, 4].1 I will leave the latter three topics to the otherrecent theses, and dive into why even null results of the electron EDM (eEDM) are so interesting.

1.1

Crash Course in Quantum Field Theory
Nathaniel Craig, a professor at University of California Santa Barbara, is an expert in quan-tum field theory (QFT). He gave a virtual department colloquium at Boulder, titled “The Returnof the LHC” in April 2022, which began with the most succinct and understandable overview of
QFT that I have heard [5]. I will draw heavily from his talk to give a non-rigorous explanation of
QFT which will help to explain the motivation for the JILA EDM experiment. This is a notoriously
1

I particularly recommend the introduction of Tanya Roussy’s thesis [2] for those who would like to read moredetail about how the JILA eEDM experiment came to be. She’s a comedian.

2difficult subject, but I hope future EDM graduate students will be able to understand this section
even if they have never taken a QFT course.

1.1.1

Feynman Diagrams

Even though QFT is an abstract and mathematically dense subject, physical interactions are
represented pictorially in Feynman diagrams like the ones shown in Figure 1.1. The first Feynman
diagram, Figure 1.1a, depicts electron scattering. We know from classical physics that the electrons
moving towards one another will feel a repulsive force that grows as they get closer. In QFT, theelectromagnetic interaction is mediated by the “exchange of virtual photons.” This interaction isdepicted in Figure 1.1a by a squiggly red line between the two blue lines which represent electrons.
The photon is “virtual” because it is a particle that only exists during the interaction - it does notexist at the initial or final time depicted in the Feynman diagram. Particles that enter or leave
Feynman diagrams are called “real.” This interaction modifies the electrons’ momenta, indicatedby the electron lines going off in new directions.

(a) Electron scattering(b) Electron-positron annihilation and creation

Figure 1.1: A Feynman diagram (a) depicting electron scattering. Quantum field theories respect
the symmetries of special relativity so we can exchange the time and space axes of our Feynman
diagram, equivalent to rotating the diagram 90 degrees. This rotated diagram (b) shows electron
and positron annihilation followed by pair production.

There are a few more things worth pointing out about Figure 1.1a. The first is that theelectrons, like all fermions, are depicted by solid lines with an arrow. Lines with an arrow pointing

3in the direction the fermion is traveling represent matter particles, while lines with an arrow pointing
in the opposite direction depict antimatter particles. The arrows in Figure 1.1a all point in thesame direction the particles are moving, so they all represent electrons instead of positrons. The
other thing to know is that Feynman diagrams come with a set of mathematical rules to calculate
the depicted event’s probability amplitude [6]. I will not try to explain all of the math behind these
Feynman diagrams, but it is good to know that the usual rules of momentum, energy and charge
conservation apply to these calculations and can help guide our intuition.
Moving on to Figure 1.1b, it is clear that this is the same diagram as Figure 1.1a rotated
90 degrees. This is not a coincidence. QFT is a quantum theory that is consistent with andrespects the symmetries of special relativity, which include Lorentz transformations that convert
space into time and vice versa. This means Feynman diagrams which have been rotated describe
real, physical processes. In the case of electron scattering, the 90 degree rotated Feynman diagram
describes electron-positron annihilation which creates a virtual photon that then turns into a newelectron-positron pair.
It’s worth pausing to let this sink in – we can take a simple Feynman diagram like electron
scattering, rotate it on its side because QFT follows the rules of special relativity, and determine thatpair production and annihilation are really things that happen. This alone predicts the existenceof antimatter and that particle number is not a conserved quantity, both of which are observed
experimentally! Half of why I chose to write about QFT in more than the usual detail is to share
this point, I think this is so cool.
It is interesting to see what happens when we replace one of the electrons in Figure 1.1a withanother charged particle like a muon µ− , the 207 times heavier “cousin” of the electron, as shown in
Figure 1.2a. This diagram still represents particles scattering via the electromagnetic force, where
one particle is much heavier than the other. The interesting part comes when we rotate the diagramby 90 degrees, as shown in Figures 1.2b and 1.2c. In Figure 1.2b we have a muon and antimuon

4(a) Electron-muon scattering(b) Muon annihilation to electron (c) Electron annihilation to muon
pair production
pair production

Figure 1.2: A Feynman diagram of (a) electron-muon scattering which has been rotated (b)clockwise and (c) counter clockwise by 90 degrees. The second two diagrams depict annihilationand pair production interactions where the initial and final particles are different species.

annihilating which leads to electron-positron pair production. This means that we can start with
one pair of particles and end up with a completely new pair!
It is important to think about energy conservation in Feynman diagrams like Figure 1.2b.
P
Here the initial energy is given by the sum of the rest mass Erest = i mi c2 and kinetic energy.
Since muons are 207 times heavier than electrons, most of the muon’s and antimuon’s rest massenergy will be carried away as kinetic energy of the electron positron pair. Figure 1.2c shows
the same interaction happening in reverse. While this is a valid Feynman diagram, it can only
take place if the incoming electron and positron have enough kinetic energy produce a muon andantimuon.
There is a final point worth mentioning about Figure 1.1b that concerns momentum andenergy conservation. These conservation laws apply at every vertex. This means that the energy
and momentum of the initial electron/positron, the virtual photon and the final electron/positronpair must be the same. We can consider the Feynman diagram from the center of mass frame, where
ptot,initial = 0 and Etot,initial ≥ 2me c2 . This means the photon will have momentum pγ =

ℏω
=0
c

and energy Eγ = ℏω ≥ 2me c2 – which seems to be a contradiction! The resolution is that virtual
particles can be “off-shell,” meaning they have unusual values of energy and momentum which is5allowed since they only exist for a short amount of time and do not exit the Feynman diagram. All
particles that enter or exit the diagram must be “on-shell” and behave normally.2 In this case, thevirtual photon is off-shell and then produces two on-shell particles, an electron and a positron.
Annihilation and pair production can occur independently of one another, as shown in Figure
1.3. In order for energy and momentum to be conserved at the end of these processes they eachinvolve two real photons. Figure 1.3a shows an electron emitting a real photon and becoming an offshell particle, which can annihilate with a positron to produce a second real photon. Rotating thisFeynman diagram 180 degrees would depict two photons coming together to produce an electron
and a positron, an allowed but relatively rare two body process. The more common Feynman
diagram depicting pair production is shown in Figure 1.3b, where one energetic photon turns into
the particle and antiparticle pair directly.

(a) Electron-positron annihilation(b) Electron-positron pair production

Figure 1.3: Feynman diagrams of (a) annihilation and (b) pair production. Note that there is nopreference for the electron or positron to be the particle in the middle of each diagram.

1.1.2

The Vacuum

Every particle-antiparticle pair is associated with a multicomponent quantum field, which
has values at every point in spacetime.3 Electrons, for example, have a four component quantum
2

Particles that are on-shell have the usual dispersion relation E 2 = m2 c4 + p2 c2 . Particles which are off-shellhave energies and momenta which do not follow this relationship.
3

This is quite similar to electric fields, which three components Ex , Ey and Ez defined at each point in spacetime.

6field for the spin up/down components of electrons/positrons. When the electron quantum field
is in its ground state, there are no electrons or positrons at all. Excitations of the quantum fieldcorrespond to particles that exist at certain points in spacetime.
That means in QFT there are no truly empty regions of space. Regions of space where allof the quantum fields are in their ground state are considered the “vacuum,” but those fields canfluctuate. In quantum mechanics there is an uncertainty relationship between energy and time
which can be written as:∆E∆t ≥ℏ
2(1.1)

We can interpret this equation as saying that for short times ∆t, the energy of the quantum fields
can have fluctuations on the order ofℏ. If this energy is large compared to 2mc2 there is enough2∆tenergy to produce a short lived particle antiparticle pair. For example, an electron positron paircan be created for ∆t ≤ℏ
= 3 × 10−22 seconds. So what we normally consider to be the
4me c2“vacuum” includes these short lived particle pairs as shown in Figure 1.4.

(a) Classical vacuum, ℏ = 0(b) QFT vacuum, ℏ ̸= 0

Figure 1.4: Vacuum in (a) classical and (b) quantum field theories. In QFT, pair production andannihilation can occur due to energy fluctuations. Those virtual particles can interact with one
another, as shown in the upper right of (b).

The important point is that we never measure bare particles, we always measure particles

7that are interacting with the vacuum. Take the example of an electron with no orbital angularmomentum in a uniform external magnetic field.4 We know that the magnetic moment of the⃗S⃗ causing an energy shift:electron’s spin µ⃗ s = gs µB will interact with the magnetic field Bℏ⃗⃗ = −gs µB Ms B
⃗ = −gs µB S · B
HZeeman = −⃗µs · Bℏ(1.2)

Here gs is the electron’s g-factor, which encodes how strongly the electron’s states split in energy due
to the external magnetic field. If we were able to measure the bare electron, Paul Dirac predictedthat we would measure gs = −2. Dirac’s calculation was equivalent to evaluating the Feynman
diagram in Figure 1.5a, where the electron interacts with an external magnetic field represented bya real photon5 without any virtual particles popping in.
When we compare Dirac’s prediction of gs = −2 to the current state of the art measurement
gs = −2.00231930436(26), we see that he was only off by about 0.1 % [7]. This gap starts toclose when we calculate how virtual particles modify the size of the electron’s magnetic moment.
The simplest virtual particle we can consider is shown in Figure 1.5b where the electron emits andreabsorbs a virtual photon. This process changes the prediction of the electron’s g-factor from −2αto −(2 + ) = −2.0023228..., which is just two parts in a million different from the measured value.
π
We can keep drawing Feynman diagrams with more and more virtual particles, like the one
in Figure 1.5c, which will all contribute to the predicted value of gs . We fortunately do not need
4

Since I spend the entirety of Chapter 4 on the sign of the g-factor in HfF+ , I will make sure the signs of thequantities are clear here. The electron’s spin magnetic moment µ⃗ s has a positive magnitude µs = gs µB if the magnetic⃗ The Bohr magneton µB is defined to be a positive quantity, so the sign
moment is parallel to the electron’s spin S.
of the magnetic moment’s magnitude is determined by the sign of gs . Since we know that the electron’s magnetic
moment is antiparallel to its spin, gs ≈ −2.002 is a negative quantity.
5

In QFT, bosons (particles with integer spin) like photons are “force carriers”. Photons are the force carriers of

the electromagnetic force. Whenever a particle is interacting with an external electric or magnetic field, that shows
up in Feynman diagrams as a real photon.

8(a) g = −2(b) g = −(2 +α) ≈ -2.0022π(c) g = −(2 +α+ O(α2 ))2π

Figure 1.5: Feynman diagrams that describe the electron’s g-factor. The dominant term effectis given by (a), the electron interacting with an external magnetic field represented by a photon.
The next largest term is from (b), where a virtual photon emitted and then later absorbed by theelectron modifies the g-factor from −2 to −2.002. This is a one-loop Feynman diagram as the only
closed path we can draw is between the three electron-photon vertices. Higher order effects which
include more or heavier particles, like (c), contribute smaller perturbations to the g-factor. This isa two-loop Feynman diagram.

to calculate all of these Feynman diagrams to get a reasonably accurate result. As a general rule,the more complicated the Feynman diagram the smaller the effect it will have on a prediction.6
In fact, we can sort Feynman diagrams by the number of non-overlapping closed loops that they
contain. For example, Figure 1.5a has no closed loops and therefore is the zeroth order effect.
This makes sense because it is the only Feynman diagram with an incoming and outgoing electronthat interacts with the external electromagnetic field and no virtual particles. Figure 1.5b has one
closed loop – the triangle between the virtual photon and electron – so it provides the first order
perturbation.7 There are many Feynman diagrams we can draw with two loops, one of which is
Figure 1.5c. All Feynman diagrams with an incoming and outgoing electron interacting with a real
photon that have two loops are summed together to make up the second order perturbation to gs .
In order to match today’s experimental precision, theorists need to include all of the more
6

This is true for interactions of the electromagnetic and weak forces, but is not true for the strong force. This

makes strong force calculations very difficult.
7

There is another Feynman diagram we can draw with just one loop, where we replace the virtual photon witha virtual Z boson, see Subsection 1.1.3. Z bosons are massive and chargeless carriers of the weak force, and this2meFeynman diagram causes a≈ 3 × 10−10 times smaller change to gs than Figure 1.5b [8].
mZ

9than 10,000 unique Feynman diagrams out to five loops [7]. Amazingly, the predicted value agrees
with the experimental result even with its tiny uncertainty at the 0.1 part per trillion level. While
this level of agreement is a triumph of theoretical and experimental physics, people are working
hard to reduce the error bars on both sides. One motivation for this further work is to try anddiscover new particles and/or interactions which slightly modify the electron’s g-factor from theexpected theoretical value.
The important takeaway is this: particles interact with the vacuum which contains particles
popping into and out of existence. That interaction modifies the properties of all particles, such as
the electron’s magnetic dipole moment. If undiscovered particles exist that even indirectly interact
with electrons, the strength of the electron’s magnetic moment will be different than is predictedby our most accurate quantum field theory, the Standard Model (SM). So increasingly precise
measurements of quantities like the electron’s magnetic moment can signal the existence of newparticles and interactions if those measurements do not agree with what is predicted by the SM.
Alternatively, if more precise measurements continue to agree with the SM, the measurements canrule out proposed new theories which make different predictions than the SM.

1.1.3

The Standard Model, Particles and Interactions

Before jumping into the story about the electron’s electric dipole moment, it will help to havesome background on the Standard Model. The SM is the current state of the art quantum field
theory that best describes particle physics. It contains the 17 known particles, shown in Figure 1.6,
and how they interact via the electromagnetic, weak and strong forces. Notably, the only force itdoes not include is gravity, which is best explained by Einstein’s theory of General Relativity.
The blue and green particles in Figure 1.6 are fermions, paticles with half-integer spins whichmake up matter. All of these particles have associated antiparticles which have the same properties
but have the opposite signed charge.8 Fermions interact with one another via the three forces
8

Neutrinos have no charge, so their associated antimater particles are chargeless as well.

10

Figure 1.6: Particles in the Standard Model. Each particle’s mass relative to the electron’s massme = 0.511 MeV/c2 , charge and spin are listed. Blue and green particles are fermions which eachhave an associated antimatter particle with all the same properties except for the opposite signed
charge. Blue particles are leptons, green particles are quarks and yellow particles are force carrying
bosons. Interactions with the Higgs boson give particles their mass.

included in the Standard Model, which are mediated by the yellow bosons in Figure 1.6. The
electromagnetic force acts on all fermions with nonzero charge, and is mediated by photons asshown in Figure 1.1a. It attracts oppositely charged fermions together and repels fermions withthe same charge, as we would expect.
The strong force is mediated by gluons and only acts on quarks, which are the fermionsshown in green in Figure 1.6. In addition to their mass, charge and spin, quarks also have a “color”
which can be red, green or blue. The strong force is a powerful attractive force which is mediatedby gluons and binds quarks together to make “color-neutral” composite particles. A composite

11particle is color-neutral if it is made of a red, green and blue quark bound together, or if it is madeof a quark and antiquark of the same color. For example, a red up quark, a blue up quark and agreen down quark can be bound together by the strong force. A composite particle made of two
up quarks and a down quark, regardless which quark is which color, is called a proton. One up
quark can combine with two differently colored down quarks to make a neutron. Both protons andneutrons are examples of baryons, composite particles made of an odd number of quarks. Baryons
and antibaryons, particles made of an odd number of antiquarks, will play an important role in thestory about the eEDM. The strong force is notoriously difficult to calculate, so it is fortunate that
we do not need to dive into those details.
The weak force requires the longest explanation. It is mediated by the Z and W± bosons,and is the only force that interacts with all twelve fermions of the Standard Model. When fermionsinteract via the Z boson they exchange energy and momentum, which is similar to electromagnetic
interactions like the one shown in Figure 1.1a. As hinted at by its name, the forces mediated by
the Z boson are weak compared to electromagnetic ones. The more interesting weak interactions
are those mediated by the W+ and W− bosons, which cause fermions of one type to change toanother. These “flavor” or identity changing interactions are subject to many restrictions beyond
energy, momentum and charge conservation, which we will explore now.
One restriction comes from the “helicity” or “handedness” of particles. Particles are either“right-handed” or “left-handed” if their spin angular momentum is in the same or opposite direction as their motion, as shown in Figure 1.7. Most particles and antiparticles can be either leftor right-handed, occurring in nature with equal frequency. However, all neutrinos are left-handed
and antineutrinos are right-handed. Additionally, the W± bosons only interact with left-handed
fermions and right-handed antifermions. These facts together mean that the weak force acts differently on left- and right-handed particles, a peculiarity not shared by the electromagnetic or strong
forces.

12(a) Left-handed particles(b) Right-handed particles

Figure 1.7: Examples of (a) left-handed and (b) right-handed particles. Most particles and theirantiparticles, like electrons and positrons, can be left- or right-handed. However, all neutrinos areleft-handed and antineutrinos are right-handed.

Another important constraint on flavor changing weak interactions are that they cannot turnquarks into leptons or leptons into quarks. For interactions with leptons, W± bosons conserve a
property called “lepton number”. Lepton number is the total number of leptons minus the numberof antileptons, so this restriction means that the weak interaction cannot change a lepton into an
antilepton or vice versa. The final constraint for leptons is that W± bosons must interact withleptons of the same “generation”. Looking back at Figure 1.6, we see that each row contains
three fermions which have similar properties but vary in mass.9 For example, electrons are moreor less light versions of muons which in turn are light versions of taus. We call the leftmost andlightest column of particles the first generation, the middle column the second generation and theright column the third generation. Exactly why there are three generations of each fermion is
open question [9]. The restriction that the weak force cannot change lepton generation means thatelectrons and electron neutrinos can turn into one another, but they cannot turn into a muon.10
With these rules for lepton flavor changing in place, we can finally see an example of the W±boson changing the identity of a particle. In Figure 1.8a we see an incoming muon turn into a muon
neutrino by emitting a W− boson. We know the muon must be left-handed since it interacts with
9

Only upper limits have been measured for the neutrino masses, but they are expected to have different masses

as well.
10

Muons cannot directly turn into an electron, but they can indirectly as shown in Figure 1.8a13a W− boson, and it produces a left-handed muon neutrino. This interaction follows the rules ofcharge and lepton number conservation and it preserves lepton generation. However, this explains
only the first half of the Feynman diagram in Figure 1.8a. Muon neutrinos are stable particles thatcan leave the Feynman diagram, but the W− boson is not. W± and Z bosons are massive, unstableparticles with multiple decay channels and lifetimes of about 3 × 10−25 seconds [10]. The decay
mode shown in Figure 1.8a is the W− boson turning into an electron and antielectron neutrino.
These particles are both stable and the interaction complies with all of the rules listed above.

(a) Muon Decay(b) Beta Decay

Figure 1.8: Two Feynman diagrams showing decay due to the weak force. (a) A muon transforms
into a muon neutrino, emitting a W− boson. The unstable W− boson then decays into an electron
and antielectron neutrino. (b) A down quark inside of a neutron emits a W − boson and turns
into an up quark. This changes the composite baryon into a proton. The unstable W− boson thendecays into an electron and antielectron neutrino. The entire process is known as beta decay.

We can now turn to the weak interactions mediated by W± bosons between quarks. As
was the case for leptons, weak interactions cannot change quarks to antiquarks or vice versa.
This rule is normally expressed by stating that baryon number, the number of baryons minus
the number of antibaryons, must be conserved. However, this is where the similarities for weakinteractions between leptons and quarks ends. The weak force is allowed to transform quarks fromone generation into another, which it cannot do to leptons. The only additional restriction onquark flavor transformations is that the weak force always turns “up-type” quarks (the up, charm2
and top quarks with charge + ) into a “down-type” quarks (the down, strange and bottom quarks
3

14
1
with charge − ) or vice versa. This can be understood as a consequence of charge conservation,
3
since the W± bosons which mediate flavor change are themselves charged particles.
In total, there are nine weak interactions that turn one of the three down-type quarks into
one of the three up-type quarks. These nine interaction strengths are characterized by the 3x3
unitary CKM matrix:Vud Vus Vub 
VCKM =  Vcd Vcs Vcb Vtd Vts Vtb(1.3)

The entries in the top row of this matrix characterize the relative probability amplitudes of thedown, strange and bottom quarks transforming into the up quark. The next two rows parameterize
the relative probability amplitudes of down-type quark transitions into the charm and top quarks.
The relative probability amplitude of an up-type quark transforming into a down-type quark is
given by the complex conjugate of the relevant matrix element. This means that the CKM matrix
completely parameterizes all weak interactions that change quark flavor.
In general, all nine complex parameters of a 3x3 unitary matrix like the CKM matrix canbe described by just four real numbers, three rotation angles and one complex phase [11]. The
three rotation angles are the three “mixing angles” which parameterize the likelihood of each of
the nine quark transformations. The complex phase is a parameter which describes how much eachof these interactions violates time-reversal symmetry, which will be explained further in the next
subsection [6].11
A common example of quarks turning into one another is shown in Figure 1.8b, where a
11

Weak interactions that change lepton flavor cannot change generation, so these relative probability amplitudes

are described by the 3x3 identity matrix. This matrix obviously does not have any imaginary components, so weakinteractions that change lepton flavor do not violate time-reversal symmetry.

15
down quark inside a neutron turns into an up quark making the composite particle a proton. This
interaction is characterized by the Vud term in the CKM matrix.

1.1.4

The Standard Model, Discrete Symmetries

The last prerequisite we need to cover is discrete symmetry in the Standard Model. There
are three particularly important discrete symmetries – charge conjugation (C), parity (P) andtime-reversal (T). These three symmetries are defined by a particular transformation, and we saythat a system respects a given symmetry if the physical laws are the same before and after thetransformation.
It is easiest to see what that means with an example. The charge conjugation transformation
is to swap all of the matter particles in a system for their antiparticles, and vice versa.12 Consider
once again electron scattering, as shown in Figure 1.1a. The negatively charged electrons exchange aphoton resulting in a repulsive force. If we replace the electrons with positrons, the exact same thing
will happen – the two particles will approach one another, exchange a photon and feel a repulsive
force. Because the particles behave the same way whether or not the particles are swapped for
antiparticles, this interaction respects charge conjugation symmetry. In fact, all electromagnetic
and strong interactions are C-symmetric, though it is not the case for weak interactions. The
example of muon decay, shown in Figure 1.8a, involves a left-handed muon turning into a lefthanded muon neutrino while emitting a W− boson. This interaction is impossible if we swap theparticles for antiparticles because there are no left-handed antineutrinos, so it violates C-symmetry.
Note that weak interactions mediated by Z bosons do not necessarily involve neutrinos, so someweak interactions violate C-symmetry while others do not.
Parity’s transformation inverts spatial coordinates, (x, y, z) → (−x, −y, −z). This is equivalent to a mirror inversion (x, y, z) → (x, y, −z) followed by a 180 degree rotation (x, y, −z) →(−x, −y, −z) about the same axis. An 180 degree rotation is a standard Lorentz transformation
12

The C transformation also involves swapping the charge of W± bosons.

16
and is common fare, which is why parity symmetry is sometimes referred to as mirror symmetry.
Just as is the case for charge conjugation, the electromagnetic and strong forces respect parity while
the weak force does not. This is because the parity transformation flips the handedness of particles.
Consider again the left-handed muon decaying into a left-handed muon neutrino and emitting aW− boson. The parity inverted version of this decay would involve a right-handed muon, whichcannot interact with a W− boson, decaying into a right-handed muon neutrino, which does notexist. So weak interactions violate P as well as C symmetries.
The third symmetry is time-reversal symmetry, whose associated transformation is to reverse
the direction of time t → −t. While we can test various systems with matter and antimatter (C)and in different spatial orientations (P), it is a bit harder to set time running backwards in our
experiments. However, if we can infer that an interaction would behave differently if we reversed
time’s arrow, we say that it violates T-symmetry. The way time-reversal symmetry breaking is
described mathematically is by a Hamiltonian with imaginary components, like the CKM matrix
that describes quark flavor changing weak interactions [6]. This means the probability amplitudeof quark a turning into quark b is different than the probability amplitude of quark b turning intoquark a. These two processes are the same except the arrow of time has been reversed, so they
violate T-symmetry [12].
In addition to considering the three discrete symmetries on their own, they combine toform composite symmetries. The most important combined symmetry for the eEDM story is CPsymmetry, the symmetry whose operation is to 1) swap matter with anitmatter and 2) invert space.
If a process respects both of the joined symmetries, it is straightforward to see that it will respect the
combined symmetry. Electromagnetic interactions behave the same way under C and P symmetries
separately, so they will still behave the same way when both symmetry operations are applied. A
process will also conserve a combined symmetry if it violates both of the merged symmetries. In
Figure 1.8a we started with a left-handed muon that decays to a left-handed muon neutrino while
emitting a W− boson. Applying the C and P transformations, we get a right-handed antimuon that17
decays to a right-handed antimuon neutrino while emitting a W+ boson. All of these particles exist
and the interaction has the same probability amplitude as the the original interaction, so muon
decay respects CP-symmetry.
In addition to CP, CT and PT symmetries, all three symmetries can be combined to make
CPT-symmetry. If an interaction violates zero or two of the three individual symmetries it respects
the overall CPT symmetry, and if the interaction violates an odd number of the symmetries it
breaks CPT-symmetry. When QFT was first being developed in the early to mid 20th century it
was widely believed that all three of the symmetries are strictly conserved by nature [13]. This was
until the 1950s when it was found that beta decay, as shown in Figure 1.8b, violates parity [14].
Now it is believed that while the individual symmetries can be violated, the joint CPT-symmetry
will always be conserved. This is largely based on the CPT theorem, a mathematical proof thatshows that any QFT which is Lorentz invariant (plays nicely with special relativity) will conserve
CPT-symmetry [15].
If we assume on this basis that CPT-symmetry really is a good symmetry of nature, we
conclude that T-symmetry violation found in quark mixing implies that quark mixing also violates
CP-symmetry. While it is possible that CPT-symmetry is broken in nature, I will assume that it ispreserved and that CP- and T-symmetries are equivalent. Quark mixing is the only interaction in
the Standard Model that has been measured to violate CP-symmetry. Theorists have been puzzledthat the strong force is seemingly allowed to violate CP-symmetry [16], but every measurement of
the strong force so far indicates that it is CP-even [17].13
That covers the preliminaries we need to know about QFT and the Standard Model, so we
can begin the story that motivates our measurement of the eEDM.
13

A few recent papers claim that a more nuanced understanding of the Standard Model implies that the strong

force inherently respects CP-symmetry [18,19,20], though I believe that this viewpoint has not been widely accepted.

18

1.2

The Baryon Asymmetry of the Universe
14 billion years ago the universe began as we know it with the big bang. A particularlycolorful description of the big bang that I enjoy is that “at an early time in the standard cosmologicalmodel, the Universe began as a fireball, filling all space, with extremely high temperature and energy
density” [21]. The source of this quotation together with [22] provide a detailed and understandable
history of that fireball, which I will summarize parts of here.
Immediately after the fireball initially exploded, there was plenty of energy for all the fermionsand their antiparticles to pop into existence, depicted by Feynman diagrams like Figure 1.3b. The
particles that were produced had so much energy that the quarks, usually bound together by thestrong force, could roam freely. The big bang caused space itself to rapidly expand, spreading thefireball out and decreasing the energy density and temperature of the universe. After the first 20
microseconds or so of fun the temperature fell below the hadronization temperature Th ≈ 1.7 × 1012
K = 150 MeV/kB , and the strong force was able to overcome the quarks’ kinetic energy to wrangle
the individual quarks into stable composite particles. At this point there was no longer enoughenergy to form new protons and antiprotons or neutrons and antineutrons as 2mp c2 ≈ 2mn c2 ≈ 1.8
GeV, so the number of baryons and antibaryons in the universe had hit its maximum.
Shortly after the universe cooled to the hadronization temperature, the baryons and antibaryons found one another and annihilated in similar processes to the one shown in Figure 1.3a.
Since all of the interactions described in the Standard Model in Subsection 1.1.3 conserve baryon
number, we might expect that the mass annihilation event destroyed all of the baryons and antibaryons. But this is not what happened – while all the antibaryons disappeared, there were
baryons left over.14 That means there must be a process that can occur at energies large comparedto the hadronization temperature which can increase the baryon number, so that after annihilation
14

There is the possibility that matter and antimatter spread out in different directions, and we find ourselves in amatter dominated domain of the universe [21, 22]. This idea will be addressed later in this section.

19
a portion of the baryons can survive.
The asymmetry in the number of baryons and antibaryons just after baryon pair production
froze out is known as the baryon asymmetry of the universe (BAU).15 The BAU is defined as:

BAU =NB − NB
NB + NB T≳1GeV/kB(1.4)

Quantifying this asymmetry is an important step to solving the mystery of how we ended up in auniverse full of matter after the big bang. However, as almost all of the baryons and antibaryons thatexisted immediately after the big bang were annihilated, we cannot measure the quantities in thisexpression. Fortunately, we can relate the sum of baryons and antibaryons in the early universe to
the number of photons that exist currently. As shown in Figure 1.3a, when two particles annihilate
there will typically be two photons produced. Bravely ignoring the fact that leptons annihilate andproduce photons as well, we can say that the sum of baryons and antibaryons in the early universe
is something like the total number of photons in the universe now, which has a temperature of 3
Kelvin:BAU =NB − NB
NB − NBNB∼==ηNB + NB T≳1GeV/kBNγNγ T=3K
T=3K

(1.5)

We can use the observation16 that there is no antimatter currently in the universe to simplify thenumerator, and we find that the BAU can be estimated by the baryon to photon ratio η [22].
15

There is a similar story about leptons and the LAU [21]. Leptons on the whole are lighter than baryons, so

lepton pair production froze out when the universe was cooler, about a second after the big bang. The BAU gets
more attention because it can be measured quite precisely, especially compared to the LAU since it is challenging tomeasure the background neutrino level of the universe.
16

onAgain, the possibility of an inhomogeneous universe of matter and antimatter domains will be addressed later20
Astronomers have two methods of of determining the baryon to photon ratio. The first is bymeasuring the relative abundance of light nuclei (2 H, 3 He, 4 He and 7 Li) created a few minutes after
the big bang, when the baryon to photon ratio was stable but there was still enough latent energy(≳ 2 MeV) in the universe to trigger thermonuclear reactions [21,22]. This process is known as the
big bang nucleosynthesis (BBN). Contemporary research in plasma physics shows that the relative
abundances depend heavily on η, and astronomers find that η = 6.10 × 10−10 [22, 23].
The second method to determine η is by measuring the power fluctuations of the cosmic
microwave background (CMB). It took much longer, around 380,000 years, for most electronpositron pairs to annihilate [21]. At this point the energy of the universe was approximately 0.3
eV, and the remaining electrons bound with the nuclei to form atoms, primarily hydrogen. For the
first time the universe was no longer a plasma, so the photons emitted when the atoms were formed
could freely propagate. These photons were emitted nearly 14 billion years ago so they have been
substantially red-shifted and now appear as microwaves that approach Earth from every directionin the universe, hence the name cosmic microwave background. It turns out that the baryon tophoton ratio η, which has not varied since just seconds after the big bang, significantly affects thepower spectrum of the CMB [22]. This method gives η = 6.1 × 10−10 , in good agreement with the
BBN method.
Both of these results tell us that before the universe hit the hadronization temperature at
150 MeV just tens of microseconds after the big bang, the universe had approximately one billion
plus one baryons for every billion antibaryons. But where did the extra baryons come from? If
matter can only be created in ways that preserve baryon number, as described in Subsection 1.1.3,
this would be impossible.
In 1967 Andrei Sakhorov17 studied this problem and realized that there are three criteria that17

Not only did Sakhorov divine the three necessary conditions for a matter dominated universe, he came up with

the idea of a tokamak, the main experimental apparatus of modern nuclear physics. He did not win the Nobel prizein physics for either of these achievements, but he did win the Nobel peace prize in 1975 for his activisim against

21
a theory of particle physics must meet in order to have a positive baryon to photon ratio η [24]:(1) Baryon number violation. Since at the big bang the universe began with a baryon number
B = NB − NB = 0 − 0 = 0 and there is a positive baryon number now, there must be aprocess that changes baryon number.
(2) C- and CP-symmetry violation. If C is a good symmetry of nature, then baryon number
increasing interactions would be just as likely as baryon number decreasing interactions andthe total baryon number would remain zero. If C-symmetry was violated but CP-symmetry
was conserved, then interaction which produces right (left) handed baryons and left (right)
handed antibaryons would be preferred, but the total baryon number would still remain
zero [22].18(3) Deviation from thermal equilibrium. If the universe was in thermal equilibrium when thebaryon number violating interactions took place, then by definition the average baryon
number of the universe would not change.
So we live in a universe dominated by matter and Sakhorov has listed requirements of particle
physics for that to be the case. We have a theory of particle physics called the Standard Model,
which was loosely explained in Subsection 1.1.3. Does the Standard Model meet those requirements?
The Standard Model includes the non-perturbative19 sphaleron processes that violate baryon
and lepton number simultaneously, so it checks off the first box [22].20 As we saw before, the weakforce violates C-symmetry because all neutrinos are left-handed while all antineutrinos are rightnuclear proliferation and advocacy for human rights more broadly. Check out his wikipedia page, it’s a good read.
18

Note that there is no analogous requirement for CT or CPT-symmetry violation since the universe only runs

forwards in time and therefore whether or not these interactions are T-odd is irrelevant.
19“Non-perturbative” interactions cannot be explained via Feynman diagrams, and they are far too complicatedfor me to explain in Subsection 1.1.3.
20

One of the recent papers which claim that the strong force cannot violate CP-symmetry also claims that sphaleronprocesses do not occur [19], so it is possible that the Standard Model does not satisfy the first criterion.

22
handed. It also violates CP-symmetry when talking specifically about quark mixing (assuming the
CPT theorem holds), so the Standard Model meets the second criterion. The third requirement
is met as the universe was not in thermal equilibrium during its first few microseconds as it was
rapidly cooling and expanding [22].
So can the Standard Model explain the baryon to photon ratio η = 6.1 × 10−10 ? Despite
including processes that meet Sakhorov’s three requirements, the answer is a resounding no. The
SM can only generate η on the order of 10−20 , falling a full 10 orders of magnitude short of what
astronomers observe [22]. There are different ways the SM could be modified in order to bridge
this gap, but doing so requires much more CP-violation than is included in the SM [22].
The fact that the Standard Model cannot explain the BAU is just one of its most glaring
problems. Others include that it cannot explain gravity, it seems to omit most of the matterholding together galaxies (dark matter) and it cannot account for the accelerating expansion of
the universe (dark energy) [5]. But we should stay on topic. Because the Standard Model does
not contain enough CP-symmetry violation in order to explain the observed baryon asymmetry,
we should be able to find evidence of more CP-violating interactions than the Standard Model
predicts [17]. There are many new theories which predict the existence of more particles and/orinteractions in order to account for the deficiencies of the Standard Model, called Beyond theStandard Model (BSM) theories [17]. The goal of our experiment at JILA is to search for CPviolation that is not predicted by the SM in order to test these proposed BSM theories which can
explain why there is matter left over after the universe-scale particle annihilation which shortly
followed the big bang.
As a two paragraph aside, I will now address the possibility that the universe produced anequal number of baryons and antibaryons in the first tens of microseconds after the big bang. Thishypothetically could have happened if the big bang separated the universe into different spatial

23
domains, some filled with matter and others filled with antimatter.21 If this were the case, thedomain of matter we live in would have to be at least the size of the observable universe. If ourdomain was smaller than the observable universe astronomers would be able to see the domain
walls, the two dimensional planes where matter and antimatter meet. These domain walls would
be the location of many annihilation events which produce gamma rays (high energy photons) thatwould be easy for astronomers to spot.
So if there are domains of matter and antimatter in the universe, the observable universe
must exist entirely inside a matter dominated domain. The hypothesis that the universe is split
into domains is therefore untestable, we cannot rule it out. In this case Sakhorov’s conditions
would be relaxed – we no longer need baryon number violation (since there is as much matter as
antimatter). But we would still need plenty of CP-violation [19], so additional CP-violation beyond
what is predicted in the Standard Model is well motivated even if this untestable theory is correct.

1.3

The Electron’s Electric Dipole Moment
Our experiment searches for BSM CP-violating interactions by measuring the electron’s elec-

tric dipole moment. We think this is a sensible observable to measure for a few reasons. 1) A
nonzero eEDM would violate T-symmetry (and therefore CP-symmetry by the CPT theorem). 2)
The Standard Model predicts that the eEDM is almost negligibly small, so any nonzero measurement can be attributed to BSM physics. And 3) BSM theories which predict new instances of
CP-violation in order to explain the BAU generally induce an eEDM. In the next three subsections
I will explain each of these points in turn.
21

Another possibility, which can be relegated to a footnote, is that matter and antimatter were homogeneously

spread throughout the universe but did not annihilate. This is all but ruled out by galaxy collisions, which would
cause the regions of matter and antimatter in each galaxy to overlap and annihilate producing crazy amounts of highenergy photons that astronomers simply do not observe [22].

24
1.3.1

The eEDM Violates CP-Symmetry

Strictly speaking, the electric dipole moment of the electron is a T- and P-symmetry violating
effect.22 If we accept the CPT theorem as true, then T-symmetry violation is just as good as CPsymmetry violation as far as resolving the BAU goes. For a depiction showing that a nonzero valueof the eEDM would violate P- and T-symmetries, see Figure 1.9.

Figure 1.9: As electrons are spin-half particles, the Wigner-Eckart theorem implies that if electrons⃗ Suppose that the eEDM
have a nonzero EDM d⃗ it must be parallel or antiparallel to its spin S.
⃗⃗
is nonzero and d ∥ S, as shown on the left side of the figure. Its spin is indicated by the light
purple arrow indicating the rotation direction, and its electric dipole moment is depicted by thelump of extra negative charge at the bottom of the electron and the cavity of charge at the top.
Reversing the direction of time (top right) would cause the electron to spin in the opposite directionTT ⃗⃗−⃗ but would not change the orientation of the charge (d⃗ −
(S
→ −S)
→ d).
A parity inversion (bottomP⃗ but would not changeright) would cause the cavity and lump of charge to swap places (d⃗ −
→ −d)
P ⃗
⃗ −
the direction of the electron’s spin (S→ S).
In both cases, we start with parallel and end with⃗ and we know that all electrons are identical,
antiparallel vectors. Because we assumed that d⃗ ∥ S
the electrons on the right side of the above image do not exist in nature. Therefore the eEDM
violates T- and P-symmetries.
22

This is true of any fundamental fermion with an EDM, not just electrons. Electrons are a convenient choicesince they are stable, readily available and exist as free particles unlike quarks.

25
1.3.2

The eEDM is Background Free

A nonzero measurement of the eEDM would be a signal of CP-violating physics, but how do
we know if it comes from the Standard Model or from something more exciting? Fortunately, theSM predicts a value for the eEDM which can be calculated precisely via Feynman diagrams.
We saw before that the electron’s magnetic dipole moment is represented in Feynman diagrams where an electron interacts with a real photon as shown in Figure 1.5. Feynman diagrams
where an electron interacts with a real photon that also contain interactions that violate P- andT-symmetries instead generate the electron’s electric dipole moment, as the eEDM also violates
these symmetries [25].
The only T- and P-violating interactions in the Standard Model are weak interactions thatchange quark flavor. The simplest way to include those interactions in a Feynman diagram thatstarts and ends with an electron is to draw the two-loop diagram shown in Figure 1.10a. Herean electron is first converted into an electron neutrino and W− boson by the weak force. That
W− boson converts into a down-type and an antiup-type quark, which is described by a complex
entry of the CKM matrix that violates T-symmetry. The quarks then recombine into a W− boson,parameterized by the complex conjugate of the same CKM matrix element, and then the W−boson and electron neutrino turn back into an electron. While this two-loop diagram contains
interactions that violate T-symmetry, in the math of calculating the probability amplitude of thisFeynman diagram we multiply the CKM matrix element by its complex conjugate, making thetotal amplitude real and T-symmetry conserving. So this Feynman diagram does not induce aneEDM. Note that the external photon in Figure 1.10a is detached because it can interact with anyof the virtual charged particles in the Feynman diagram.
In order to find the Standard Model’s prediction of the eEDM, we must include more interactions. We can go to three-loop Feynman Diagrams, like the one shown in Figure 1.10b. Note that
this one diagram represents many diagrams with every permutation of up- and down-type quarks

26(a) Two-Loop SM Contribution to eEDM(b) Three-Loop SM Contribution to eEDM

Figure 1.10: (a) A two-loop Feynman diagram which connects a real electron to CP violating
physics in the Standard Model. ui and dj are any antiup-type and down-type quarks, and theexternal photon can couple to any of the charged virtual particles. While this Feynman diagram
contains CP-violating interactions at the vertices between the W− bosons, down- and antiup-type
quarks, the overall probability amplitude is CP-symmetric as it is proportional to Vij Vij∗ = |Vij |2 ,which is completely real. (b) A three-loop Feynman diagram that violates CP-symmetry and can
induce an eEDM. However, the sum of all of the probability amplitudes from three-loop diagramsof this type is real, meaning that even at three loops the eEDM predicted by the SM is zero.

included at the four quark lines, and the external photon interacting with any of the charged particles. These diagrams now include four vertices where quarks transform into one another via theweak force, so the probability amplitude is proportional to the product of four entries of the CKM
matrix. The probability amplitudes are in general complex numbers which violate T-symmetry
and therefore induce an eEDM. However, it turns out when you add together all of the possible
three-loop Feynman diagrams, the SM predicts an eEDM equal to zero [25].23
The Standard Model does predict a finite value of the eEDM at the four loop level. It
turns out that the four-loop diagrams which contain even more weak interactions contribute to theeEDM (Figure 1.11a), and so do four-loop diagrams that contain a strong interaction mediated by
a gluon (Figure 1.11b). These fourth order diagrams add together in a way that their probability
23

A satisfactory explanation that the three-loop diagrams add up to zero eEDM requires a deeper dive into themath, as explained in [25].

27
amplitudes have an imaginary component, which results in an eEDM of 5.8 × 10−40 e cm [26].

(a) SM eEDM, all weak interactions(b) SM eEDM, one strong interaction

Figure 1.11: Four-loop Feynman diagrams that give the electron an EDM. These diagrams are thesame as the one shown in Figure 1.10b with the addition of (a) a weak interaction that changes
quark flavor or (b) a strong interaction between quarks.

The Standard Model predicts a nonzero value of the eEDM, but it is orders of magnitude
smaller than the sensitivity of leading eEDM experiments [27, 28]. This means that if we canmeasure the eEDM with a smaller error bar than anyone before and we find that our measurement
does not agree with zero, it can only be explained by CP-violating physics beyond the Standard
Model (or, of course, a measurement error). This helps make the eEDM an appealing place to lookfor new CP-violating physics, but why should new CP-violating physics show up as eEDM at all?

1.3.3

The eEDM is Sensitive to BSM Physics

We saw in Subsection 1.1.3 that the strong force couples to quarks (half of the fermions),
the electromagnetic force couples to charged particles (75% of the fermions) and the weak force
couples to all fermions. How do we know which fermions will couple to Beyond the Standard Modelinteractions and therefore be the best probes of new physics? The answer is that we cannot know
in advance, we can only make an intelligent guess. Many of the most popular BSM theories haveCP-violating interactions that couple to the electron directly (via one-loop Feynman diagrams),

28
so the eEDM is a sensible place to look [17]. Supersymmetric theories are one example of a BSM
theory that has CP-violating interactions which couple to the electron inducing an eEDM, shownin Figure 1.12.

Figure 1.12: In supersymmetric BSM theories, every particle has a heavier supersymmetric partner.
The electron can turn into its supersymmetric parter, the selectron ẽ− , by emitting a photino γ̃, thesupersymmetric partner of the photon. The selectron interacts with the external electromagnetic
field, giving the electron an EDM. This interaction is CP-symmetry violating.

The magnitude of the eEDM from one-loop BSM theories, like the one shown in Figure 1.12,
is straightforward to calculate [29]:de ∼ea0 α g 2
m2
sin ϕCP e2
2 2π
M

(1.6)

The first fraction in the above equation gives the proper dimensions for the eEDM as ea0 is the
charge of the electron multiplied by the bohr radius. α is the fine structure constant, which isapproximately

1
g2. This is then multiplied by, where g is the interaction strength of each137
2πelectron-selectron vertex in Figure 1.12. As there are two such vertices, g 2 shows up in the magnitude of the eEDM. This is then multiplied by sin ϕCP , a number whose magnitude is between 0 and
1, which encodes how much this interaction violates CP-symmetry. If the interaction is completely
CP-symmetry violating sin ϕCP = 1. Naively, we expect sin ϕCP ∼ 1 since the new interactions

29
need to be CP-symmetry violating to explain the BAU. Finally, the eEDM is multiplied by

m2e
,
M2

the ratio of the electron’s mass divided by the characteristic mass of the new particles squared.
The heavier the BSM particles are the smaller the eEDM will be, which makes sense as according
to the Heisenburg uncertainty principle the heavier virtual particles will exist for a shorter timeand have a smaller effect on the electron.
It’s worth pointing out that Equation 1.6 tells us that we are at a particularly exciting time tomake precise measurements of the eEDM. If we assume that sin ϕCP = 1 and the BSM interactions
are as strong as electromagnetic interactions, i.e. g 2 = α ≈1, then we can directly relate limits of137

the eEDM magnitude to searches for new particles of a certain mass. The result of the experiment
I describe in my thesis is that, at the 90% confidence interval, |de | < 4.1×10−30 e cm [27]. Plugging
this relation into Equation 1.6 gives M ≳ 40 TeV/c2 .
This means that our experiment, which measured the eEDM and found a result consistent
with zero, was able to rule out the existence of particles which 1) have CP-symmetry violatinginteractions with the electron that 2) are as strong as the electromagnetic force and 3) have a mass
smaller than 40 TeV/c2 . We can compare this to the large hadron collider (LHC), the home of theworld’s leading high energy physics experiments, which can rule out BSM particles with masses lessthan about 3 TeV/c2 [30]. This means that our eEDM measurement, which is the combined effortof orders of magnitude fewer people than work at the LHC with a much smaller budget, was able
to search for a broad class of particles that are too massive to detect at the world’s best particle
collider. That’s pretty cool.

1.4

Thesis Outline
In this introductory chapter I have explained why we expect there to exist CP-symmetry

violation that is not predicted by the Standard Model and why measuring the eEDM is a good
way to search for these interactions. In chapter two, I will give a succinct description of the second
generation HfF+ eEDM measurement, which has already been described in these sources [2,27,31].

30
In chapter three I will fill in the gaps of what has not been written about the second generation
measurement, mostly discussing the systematic effects that I worked on. Chapter four discusses
how we determined the sign of the g-factor in the 3 ∆1 state of HfF+ , and chapter five is about
the magnetic shielding we will need for our third generation experiment to keep known systematiceffects small. I conclude my thesis in chapter six.

Chapter 2Generation 2 Overview“We all could have been killed - or worse, expelled. Now if you don’t mind, I’m going to bed.”- J. K. Rowling, Harry Potter and the Sorcerer’s Stone
The main accomplishment of my time in graduate school was my role in the second generation
measurement of the eEDM at JILA [27]. In addition to writing a paper on the result, we publisheda lengthy document that gives a complete description of the measurement and its uncertainty [31].
Tanya Roussy, who was the senior graduate student on the project when we took the data, wrote
an excellent thesis going into even more detail about the experiment [2]. As the only other graduate
student who worked full time on the experiment during the years leading up to the measurement,
my thesis will likely be the last document we write about the project.1 I do not see a great value
in rewriting how the experiment works in detail, so the bulk of this chapter will be a reproductionof the first half of our “Systematics” paper [31]. In Chapter 3 I will give an overview of thesystematic errors in our measurement, focusing on the ones I worked on. But first, I will give myown description of how we measured the eEDM.
1

Eric Cornell and Jun Ye presumably have better things to do at this point.

32

2.1

JILA eEDM Experimental Overview
A free electron with no orbital angular momentum in external electric and magnetic fields

will have an energy:⃗⃗⃗ − d⃗e · E⃗ = −gs µB S · B⃗ − de S · E⃗
He = −⃗µs · Bℏℏ(2.1)⃗ (i.e.
We know the electron has a magnetic dipole moment µ⃗ s that is antiparallel to its spin S⃗ causing an energy shift. While it has not
gs ≈ −2 < 0) which interacts with the magnetic field B
been measured, the electron may have a nonzero electric dipole moment d⃗e , which must be parallel⃗ by the Wigner-Eckart theorem [32]. In an electric(de > 0) or antiparallel (de < 0) to its spin S⃗
field E⃗ this would cause an additional, small energy shift −d⃗e · E.
⃗ = 0. In this case the only energy shift is
A naively appealing way to measure de is to set B
from the eEDM and we find:

He = −de⃗S· E⃗ = −de Ms E
ℏ

(2.2)

⃗ so E⃗ defines the quantization
Here the only energy shift is caused by the eEDM interacting with E,
axis. The two states Ms = ±1
are split by an energy de E which can be measured via a standard
2

spin-flip experiment.
While this approach is straightforward, it quickly runs into a few problems. The first isthat de is tiny; our current limit says that it would take an unreasonably large electric field of1015 V/cm or more to split a free electron’s spin up and down states just 1 Hz [27]. The mostambitious experiments I know of have electric fields on the order of 106 V/cm made from closely
spaced and highly charged electrodes [33]. Even in these fields the eEDM shift would be at thenHz scale or smaller, which means the eEDM measurement needs to be quite precise. Additionally,

33
the large electric field poses a new problem; the field will quickly accelerate the electron out of theexperimental apparatus.
Another problem is that there is no way of eliminating the magnetic field entirely. Earth’s
magnetic field is typically 0.5 Gauss, which would interact with the electron’s magnetic momentµs = 2µB /h = 2.8 MHz/G to cause a ∼ 1 MHz shift between the electron’s spin up and downstates. While a few orders of magnitude of magnetic shielding is achievable, the 15 orders ofmagnitude of shielding required to make the Zeeman effect as small as the eEDM signal are far
beyond what is possible. In addition to drowning out the eEDM signal, the background field willdefine the quantization axis.2 This would cause a host of problems, from reducing the magnitudeof the eEDM shift to making the spin-flip experiment nearly impossible.
We can solve some of these problems by intentionally applying a small magnetic field, as
shown in Figure 2.1. Per Equation 2.1, the magnetic field would define the quantization axis andcause the spin up and down states to split by 2.8 MHz/G. We could then apply an electric fieldparallel to the magnetic field. This would cause the energy splitting to slightly grow or shrink,depending on the sign of de . After measuring this energy difference in a spin-flip experiment, we
could reverse the direction of the electric field and repeat the measurement. As long as the Zeeman
shift is constant its magnitude will not matter – the difference between the two measurements willbe 2de E.
This new approach solves the problem of defining the quantization axis and allows us tomeasure the eEDM even though its shift is small compared to the magnetic effect. However,it does not solve the problem of the electric field kicking the electron out of our apparatus. It
also introduces the requirement that the noise in the magnetic field δB is small enough such thatgs µB δB < δde E, where δde is the desired precision of the eEDM measurement.
2

When one field causes a much larger energy shift than the other, the quantization axis is defined by the field

which causes the larger energy shift.

34(a) Spin up and down states in a (b) Parallel magnetic and electric (c) Antiparallel magnetic and elecmagnetic field
fieldstric fields

Figure 2.1: A simplified eEDM measurement protocol. In each subfigure the spin up |↑⟩ and down|↓⟩ states are split in a magnetic field, where the vertical axis represents energy. (a) The states
are split in energy by a magnetic field. To the right there is an inset that shows the frequencyν = ∆E/h which drives the spin flip. (b) The states are split by parallel electric and magnetic
fields. For this figure we assume that de is positive, meaning the splitting between the spin up anddown states decreases. This is shown by the red frequency trace in the inset while the blue traceshows the frequency from (a). (c) The states are split by antiparallel electric and magnetic fields
causing the energy difference to increase. The value of de E can be found by taking the differenceof energies measured in (b) and (c). Image adapted from [2].

Our experiment at JILA solves these problems with a method first suggested by Dave DeMille[34]; we measure the electron’s EDM by experimenting on electrons trapped inside of a heavy polar
molecule in the 3 ∆1 electronic state. These molecules have two valence electrons, one of which is
subject to a massive effective electric field Eeff on the order of 10 GV/cm [35] that is at least fourorders of magnitude larger than fields we can generate in labs. Our molecule of choice is HfF+
which has an effective electric field of + 23 GV/cm [36]. Even though one of the valence electrons
is subject to a substantial effective electric field it is trapped inside the molecule and does not fly
out of our experiment. Additionally, the magnetic sensitivity of HfF+ in the 3 ∆1 electronic stateis about three orders of magnitude smaller than a free electron [31]. This is because in a 3 ∆1electronic state, the two valence electrons’ spins are oriented in the opposite direction of two unitsof orbital angular momentum, largely cancelling out the molecule’s magnetic dipole moment.3
The upshot of using a heavy polar molecule in a 3 ∆1 electronic state is that the eEDM signal
might be as large as 10 µHz (four orders of magnitude larger than it can be from a lab electricfield) while the magnetic sensitivity is on the order of 1 kHz/G (three orders of magnitude smaller3

See Chapter 4 for a further discussion of this point.

35
than that of a free electron). This still implies that we need to control the magnetic field noise to
10 nG, a daunting task but one that is seven orders of magnitude less challenging than if we did
our experiment with a bare electron.
Fortunately, 3 ∆1 molecules have another trick up their sleeve. Their more complicated structure, which is explained fully in Chapter 2 of Will Cairncross’s thesis [1], contains two pairs of statescalled omega doublets, shown in Figure 2.2. These states have nearly the same g-factors, different
by only a part in 465(1) in HfF+ , but the opposite sign shift due to the eEDM [31]. Glossing overa few important details that will be addressed in Chapter 3, this means we can read out the eEDM
shift by subtracting the energy difference within the upper doublet from the energy difference inthe lower doublet. The crucial benefit this provides is that by measuring the upper and lower
energy differences simultaneously the magnetic field noise applies to both doublets and is naturally
cancelled.
This point about noise cancellation is important, so it is worth explaining how we implemented it in our experiment. We prepared a population of HfF+ molecules, half of which were in
the upper doublet of the 3 ∆1 electronic state and half in the lower doublet. These molecules were
held in an ion trap and overlapped spatially and temporally. They therefore experienced the samemagnetic field which caused Zeeman shifts in both doublets as shown in Figure 2.2.4 Any noise in
the magnetic field would change the magnitude of the Zeeman shifts in both the upper and lower
doublets simultaneously. Except for the slight difference in g-factors which we account for in [31],this noise cancels when we find de Eeff ∝ ∆Eupper − ∆Elower . Simultaneously measuring the energy
differences of both doublets to eliminate magnetic field noise has been incredibly useful.
We have one final scheme to reduce our sensitivity to magnetic fields. In addition to doing our
experiment in a 3 ∆1 state that has reduced magnetic sensitivity and doublets with similar g-factors
4This is true if we assume the molecules are in the exact same time and place. This is true to a very good

approximation, though we have a systematic error associated with this discussed in [31].

36

3Figure 2.2: Energy levels in the 3 ∆1 v = 0, J = 1, F = state of HfF+ where we perform the eEDM
2
measurement. The eight states indicated by the black horizontal lines are in the presence of anelectric field ∼ 60 V/cm and magnetic field ∼ 10 mG. The vertical axis denotes energy and is not toscale. We call the top two states the upper doublet and the lower two states the lower doublet. The
doublets are oriented the opposite direction and are split by a Stark splitting of approximately 100
MHz. Each doublet is split in energy by the Zeeman shift, about 100 Hz, and the eEDM, ≲ 40µHz.
The Zeeman shifts of the upper and lower doublet are not identical, indicated by their different
g-factors gu and gl , though as discussed in [31] and later in this thesis they are only different by apart in 465(1). The shift due to the eEDM has the opposite sign between the two doublets. If we
ignore for a moment that the Zeeman shifts are different, we can measure the eEDM by taking thedifference of the energy shifts of the upper and lower doublets. Because we measure molecules in
the upper and lower doublets simultaneously, magnetic field noise will affect both doublets and (tofirst order) not change our measurement of the eEDM. This is a good general description of ourmeasurement, though it contains a few loopholes which are addressed in Chapter 3 of this thesisand [31].

but opposite shifts due to the eEDM, we perform our experiment in a way that is to first-orderimmune to uniform magnetic fields. We hold the HfF+ molecules in a Paul trap and polarize themwith a rotating electric field, as shown in Figure 2.3a. The rotating electric field E⃗rot has magnitude
Erot = 58 V/cm and rotates with a frequency frot = 375 kHz. This causes the molecules to quickly
rotate in circles with radius rrot = 0.5 mm while the molecules slowly oscillate around the centerof the trap with a secular frequency fsec ∼ 1 kHz.

37(a) HfF+ rotating due to E⃗rot⃗axgrad(b) HfF+ rotating due to E⃗rot inside B

Figure 2.3: (a) A HfF+ molecule, with a red Hf nucleus and blue F nucleus, rotates in a circle
with rrot = 0.5 mm due to E⃗rot . We alternately take data with the molecules rotating clockwise
and counterclockwise. The molecule and instantaneous direction of E⃗rot is shown at eight distinct
times. The molecule undergoes many such rotations while it slowly oscillates around the originof the ion trap. (b) The molecule undergoes the same rotation in the ion trap our usual applied⃗axgrad = Baxgrad (2⃗z − ⃗x − ⃗y ). This figure is drawn with Baxgrad < 0, though notemagnetic field B
that we alternately take data with Baxgrad < 0 and Baxgrad > 0.

The quantization axis of our experiment is defined by E⃗rot as it causes the largest energy shift
to our 3 ∆1 state of ∼ 100 MHz. This rotating quantization axis, as shown in Figure 2.3a, points
in every direction in the xy-plane over a period Trot = 1/frot . Therefore any uniform magnetic
fields in the xy-plane average to zero in the direction of the quantization axis and do not cause
any Zeeman shifts.5 Uniform magnetic fields along the z-axis never point along E⃗rot so they cannot
cause any diagonal shifts, though they can cause off-diagonal couplings. These off-diagonal effects
are considered in Section 3.1, and the systematic effects they cause are manageably small.
In order to intentionally apply the small Zeeman shift necessary for our experiment, we apply⃗axgrad = Baxgrad (2⃗z − ⃗x − ⃗y ). Figure 2.3b shows a molecule rotatinga magnetic field gradient B⃗rot along the quantization
through this magnetic field. This results in an average magnetic field B
5

Systematic effects due to static magnetic fields will be discussed in Chapter 3 and are examined in [31]. These

effects will be important for our third generation experiment, which will be discussed in Chapter 5.

38axis, as described in Figure 2.4. How this works is, in my opinion, one of the more clever aspectsof the JILA EDM experiment.

⃗axgrad at four different time steps of
Figure 2.4: A molecule rotating due to E⃗rot in a magnetic field B
the rotation period Trot = 1/frot . Just as is shown in Figure 2.3b, the magnetic field at the location⃗axgrad canof the ion varies in direction and magnitude as the ion rotates. The total magnetic field B⃗static , which does not change in magnitude or direction, and B⃗rot ,
always be expressed as a sum of B
⃗rotwhich is constant in magnitude but always antiparallel to E⃗rot . For the opposite sign of Baxgrad , B⃗
is always parallel to Erot . The magnetic fields cause a Zeeman shift if they have a nonzero average⃗static does not cause a Zeeman shift as it
projection along the quantization axis defined by E⃗rot . B⃗⃗
is parallel as often as antiparallel to Erot , but Brot does cause a Zeeman shift.

So far I have discussed why the naturally large Zeeman shift makes it very difficult to measurea nonzero eEDM. We employ three main strategies at JILA to make our experiment possible: 1) theuse of a 3 ∆1 molecule with a large Eeff and small g-factor, 2) the simultaneous measurement of theupper and lower doublets which have very similar g-factors but opposite sensitivities to the eEDM,

39
and 3) the use of a rotating quantization axis that averages out uniform magnetic fields. The firststrategy is not unique to our experiment, the ACME collaboration has a comparable limit on theeEDM and uses the 3 ∆1 state of the ThO molecule [28]. While ACME does not measure their
doublets simultaneously, they rapidly switch back and forth between the two doublets. However,
we are the only eEDM experiment that currently takes advantage of the rotating quantization
axis. It is not a coincidence that we are the only recently published result without any magnetic
shielding.6 Systematic shifts due to magnetic effects are a leading error for EDM measurements
and are a serious candidate for what could limit progress in the field. All three of these advantages
may be key in advancing the precision of EDM measurements.

Figure 2.5: The x-axis is energy and the y-axis is number of measurements in an EDM experiment.
The orange and blue curves represent data from two datasets where Energy = Zeeman shift ±EDM shift. The three graphs represent the key ingredients to statistical sensitivity of an EDM
experiment. Left: the splitting between the two datasets increases with a larger Eeff . Center: thewidth of each dataset shrinks with the coherence time τ . Right:√ The center of each dataset can befound more reliably
with
more
data
pointsandincreases
with
N . An ideal EDM experiment has√large Eeff , τ and N .

In addition to making sure that the Zeeman shift or other systematic effects do not overwhelm
the eEDM shift, we want the measurement to be as precise as possible. As discussed above, wefind the eEDM by measuring the energy difference of the upper and lower doublets and thentaking the difference of those measurements. As we see in Figure 2.5, the central values of those
doublet measurements will be farther apart in the presence of a larger Eeff . The distributions ofour measurements will be narrower for data taken with a longer coherence time τ , and we will be
6

We will need shielding for our next generation measurement, see Chapter 5. However, our shielding requirements

will still be quite relaxed compared to competing experiments.

40
able to find the center of each distribution as we take more data points√

N [17]. All together, wefind that the statistical uncertainty of the measurement is given by:

δde ∼

ℏ
√
|Eeff |τ N

(2.3)

As discussed earlier, we have chosen a 3 ∆1 molecule with a large Eeff = 23 GV/cm. A key
advantage of our experiment is that we can hold our HfF+ molecules in an ion trap for arbitrarily
long times. Unfortunately the 3 ∆1 electronic state of HfF+ is metastable with a finite lifetime ofabout two seconds, limiting our coherence time τ ≈ 3 seconds at most.7 This is still orders ofmagnitude longer coherence time than competing experiments like ACME which can only maketheir measurements over a few milliseconds [28]. We perform our experiment on as many HfF+
molecules at a time as we can, which is typically a few hundred ions in each omega doublet. All
together this gave us a good enough statistical sensitivity to make the most precise measurementof the eEDM to date [27].

2.2

How the JILA Generation Two eEDM Experiment Works
Hopefully the previous section sufficiently introduced how to measure the eEDM in generaland our approach at JILA. Now I will give an explanation of how the experiment works in detail.
Because this was already written up in Tanya Roussy’s thesis and our “Systematics” paper, which
was written by Luke Caldwell, Tanya Roussy and myself, there’s not a great need for a third
independent write up [2, 31]. Below is the explanation of our experiment from the Systematics
paper, edited just a bit to fit smoothly into this thesis.

41

Table 2.1: Spectroscopic constants for 3 ∆1 state of HfF+ used throughout this document.
Constant
Be /h
A∥ /h
dmf /hωef /(2π)
gN
G∥gF|Eeff |/hValue
8.983(1) GHz−62.0(2) MHz
1.97(1) MHz V−1 cm
0.74(4) MHz
5.257 74(2)−0.0122(3)−0.0031(1)5.5 × 1024 Hz e−1 cm−1

Description
Rotational constantHyperfine constant
Molecule-frame electric dipole momentΩ-doubling constantNuclear magnetic g-factor of 19 F
Effective electronic g-factor
F = 3/2 state g-factor
Effective electric field

Reference[37][38][39, 40]
[37]
[41]
[39]
[39]
[42, 43]

The effective electronic g-factor given here is inferred from the measured gF , G∥ ≡ 3gF − gN µN /µB ,and suited to calculations including only states in 3 ∆1 . A slightly smaller G∥ must be used when
considering the effects of interactions with other electronic states [40, 44].

Table 2.2: Example experimental parameters and associated derived parameters from our 2022
data.
Parameter
Erot
π/2
Erot
ωrot
Brot
rev
B2,0
rrot
δgF
gF
∆0

∆D
VRF
ERF
ωRF
VDC
EDC
ωx
ωy
ωz

Value
58 V cm−1
7 V cm−1
2π × 375 kHz
10 mG (typ.)200 mG cm−1 (typ.)0.5 mm
−0.002 146(2)∼ 0.6 Hz∼ −0.9 Hz
23.5 V
0.5 V cm−1
2π × 50 kHz
3.7 V
10 mV cm−1
2π × 0.95 kHz
2π × 1.51 kHz
2π × 1.60 kHz

Description
Magnitude of rotating electric field during free evolution
Magnitude of rotating electric field during π/2 pulses
Angular frequency of Erot
Effective rotating magnetic field
Applied magnetic quadrupole gradientRadius of ion circular motion
Stark doublet-odd magnetic g-factor ratio (see Figure 3.2)
Rotation induced mF coupling
Doublet-odd correction to ∆RF radial confinement voltage during free evolution
RF electric field amplitude at typical ion during free evolution
Radial-confinement RF frequency
DC axial confinement voltage during free evolution
DC axial confinement electric field at typical ion
x secular frequency during free evolution
y secular frequency during free evolution
z secular frequency during free evolution

In the Systematics paper we report ∆ ∼ 1.0 Hz and ∆D ∼ −0.6 Hz. Recent numerical simulations by
Anzhou Wang indicate that we incorrectly swapped the magnitudes of these two values. Fortunately, as
each term has a magnitude of ∼ 1 Hz, this does not effect the analysis of our experiment.

42

Figure 2.6: Schematic of experimental apparatus. On the left is the source chamber, where we
produce neutral molecules. On the right is the main experimental chamber, containing the ion
trap. The two chambers are connected by a differential pumping chamber with two small aperturesat either end. The endcap electrodes of the ion trap have a hole in the center to allow optical accessalong the z direction (vertical in the lab). Inset shows fields applied during experimental sequence:
the rotating electric bias field E⃗rot , and the quadrupole magnetic field B⃗0 . The molecular axis ofeach of the ions is either aligned or anti-aligned with E⃗rot .

2.2.1

Experiment

Our experiment uses HfF+ ions, confined in an ion trap and prepared in the metastable 3 ∆1state. Relevant molecular properties are given in Table 2.1. In the 3 ∆1 state, one of the valence
electrons is subject to a large intramolecular effective electric field Eeff = 23 GV cm−1 [45], along
the internuclear axis of the molecule. We orient this molecular axis in the lab frame by applying
an external electric field which rotates to maintain confinement of the ions. We then prepare theelectron spin of the molecule in a coherent superposition of states, corresponding to the spin of
the eEDM-sensitive electron oriented either parallel or antiparallel to Eeff , and measure the energy
difference between them using Ramsey spectroscopy. The eEDM will give a contribution to thisenergy proportional to de Eeff . To reject other unwanted contributions, we perform this measurement
simultaneously on two spatially overlapping clouds of ions with their molecular axes aligned and7

In our third generation experiment we will switch to ThF+ molecules where 3 ∆1 is the ground electronic state,

hopefully allowing us to increase the coherence time τ to 20 seconds.

43
anti-aligned with the externally applied field. The difference between the measured energies in eachcase is our science signal.
This section describes the apparatus and each of the steps used in state preparation andmeasurement of the ions. A summary of typical experimental parameters is given in Table 2.2.

2.2.1.1

Lasers

The experiment uses a total of 9 lasers; 5 pulsed lasers used for ablation, ionization, and
818
1082
814
photodissociation, and 4 CW lasers—which we denote L961
trans , Lvc , Lop , Ldepl —used for state-

preparation and readout. A summary is given in Table 2.3 and Fig. 2.7, and each is describedin detail in the following sections. All lasers are locked to wavemeters using simple, ∼ 1 Hz servo
loops. The CW lasers are locked to within ∼ ±30 MHz, and the pulsed lasers to ∼ ±500 MHz.

2.2.1.2

Molecular beam and ionization

Our experiment begins with a pulsed beam of neutral molecules. We use a pulsed Nd:YAG
laser to ablate a solid Hf rod into a pulsed supersonic expansion of Ar, seeded with 1% SF6 . Chemical reactions between the Hf plasma and the SF6 produce neutral HfF which are entrained in the
supersonic expansion and rovibrationally cooled by collisions with the Ar atoms to a temperatureof ∼ 10 K. When they arrive in our main chamber, ∼ 50 cm away, a pair of pulsed UV lasers at
309 nm and 368 nm excite a two-photon transition to a Rydberg state 54 cm−1 above the ionizationthreshold, from which they autoionize [39, 46]. The molecular ions are created in the first few
rotational levels of 1 Σ+ (v = 0), the electronic and vibrational ground state of the molecule. The
ions are stopped at the center of our RF ion trap by pulsed voltages on the radial trap electrodes,
after which the confining potentials are immediately turned on. We typically trap ∼ 2 × 104 HfF+
ions with a lifetime8 of ∼ 5 s. The trap is described in detail in the next section.
8

We note that the trap lifetime is limited by slow heating of the ions and is strongly dependent on the trapping

parameters. The 5 s here is for the very shallow trap used during the Ramsey interrogation time.

Name
Ablation
Photoionization 1
Photoionization 2
TransfermF Pumping
mF Depletion
Vibrational Cleanup
Dissociation 1
Dissociation 2
L961
trans
L1082
op
L814
depl
L818
vc

SymbolTransition–Ω = 3/2 ← X 2 ∆3/2
Rydberg ← Ω = 3/2
P (1) 3 Π0+ ← X 1 Σ+
P (1) 3 Π0− ← 3 ∆1
Q(1)3 Σ−← 3 ∆10+−
P (1)3 Σ0+ ← 3 ∆1
Ω = 2 ← 3 ∆1?←Ω=2Power/Energy
10 mJ
30 µJ
1.3 mJ
600 mW
21 mW
550 mW
30 mW
1.6 mJ25 mJ

λ (nm)
532
309.388367.732
961.43495
1082.4137
814.508
818.37198
368.494
266

Polarization
linearlinearlinearlinearcircularcircularlinearcircularcircular

Pulse Width
10 ns
10 ns
10 ns
CW
CW, strobed
CW, strobed
CW
10 ns
10 ns

Table 2.3: Photons used in our experimental sequence. All lasers address v = 0 state of ground and excited levels, except L818
vc which
961
818
addresses v = 1. Ltrans and Lvc propagate along the trap z axis, all other lasers propagate in the x-y plane.

4445
2.2.1.3

Ion trap

Our linear Paul trap has 8 radial electrodes and 2 endcaps. The radial confinement is provided
by driving the radial electrodes in a quadrupole configuration producing a field,
VRF
E⃗RF (⃗r, t) = 2 cos (ωRF t)(⃗x − ⃗y ),
R0

(2.4)

where ωRF = 2π × 50 kHz, VRF is the voltage applied on each electrode, R0 ∼ 4.8 cm is the effective
radius the RF trap, and ⃗x, ⃗y are the radial position coordinates of the ions in the laboratory frame.
Axial confinement is provided by DC voltages VDC on a pair of endcaps, producing a field
VDC
E⃗DC (⃗r, t) = 2 (⃗x + ⃗y − 2⃗z),
Z0

(2.5)

where Z0 ∼ 17 cm is the effective height of the RF trap. We choose the values of VRF and VDC
immediately after ionization to best match the spatial mode of the initial ion cloud, giving trap
frequencies ∼ 5 kHz in all directions. We then linearly ramp the trapping voltages down over 10 ms
to expand and cool the ion cloud. The ramp takes the trap frequencies to ∼ 2.8 kHz and ∼ 2.0 kHz
in the radial and axial directions respectively.
In addition to the confinement fields, we also apply a rotating electric field E⃗rot ,hiE⃗rot (t) = Erot x̂ cos (ωrot t) + R̃ŷ sin (ωrot t) ,(2.6)

where ωrot = 2π × 375 kHz, R̃ = ±1 indicates the rotation direction and Erot = |E⃗rot | is typically∼ 58 V cm−1 . This field serves to orientate the molecular axis, and thus the effective electric field, of
the ions and we do our spectroscopy in this rotating frame. E⃗rot causes an additional micromotionof the ions,−e ⃗2 Erot = −rrot Êrot ,
mωrot

(2.7)

where rrot ∼ 0.5 mm. The shape of the radial electrodes is optimized to minimize inhomogeneitiesin E⃗rot across the ion cloud [1].

46
2.2.1.4

Magnetic fields

Measuring the electron EDM also requires orienting the electron spin of the molecules which
we do with an applied magnetic field B⃗0 . In order for the unpaired electrons to experience atime-averaged interaction with the intramolecular effective electric field, this magnetic field must
corotate with E⃗rot . We achieve this using a pair of coils in anti-helmholtz configuration aligned
along the axial direction, giving
revB⃗0 = B̃B2,0(2⃗z − ⃗x − ⃗y ).

(2.8)

rev is typically ∼ 200 mG cm−1 and B̃ = ±1 indicates the direction of the current in the
Here B2,0

coils, explained in more detail in Sec. 2.2.1.9. In the rotating and co-moving frame of the ions, thisquadrupole magnetic field appears as a time-averaged magnetic bias,
rev
B̃Brot = ⟨B⃗0 · E⃗rot ⟩ = B̃B2,0
rrot .

The coil pair is driven by a precision current source with 1 pA resolution,(2.9)
corresponding to

200 fG cm−1 . We refer to the pair of coils that produces this field as the B⃗0 -coils.
The apparatus also includes three pairs of coils setup along the lab frame x̂, ŷ, ẑ axes in
Helmholtz configuration for tuning the magnetic field at the position of the ions. The z coil isdriven by the second channel of the precision current supply used for the B⃗0 -coils, the x and y coils
are driven by a lower precision current supply. The magnetic field around the periphery of thetrap is measured by an array of eight, 3-axis fluxgate magnetometers bolted to the outside of themain experimental chamber. We use these measurements to infer the magnetic field at the centerof the trap. In contrast to other modern eEDM experiments [28, 47, 48], the apparatus includes
no magnetic shielding as we are principally only sensitive to magnetic fields rotating at ωrot , asdiscussed in detail in Chapter 3 of this document and Section VI A of the Systematics paper.

47

Figure 2.7: Cartoon depicting the transitions used during our state preparation. The 3 Π0+ and
1 +
3Π
3
3 −
0− states decay preferentially to ∆1 , while the Σ0 state decays preferentially to Σ .

2.2.1.5

State preparation

Immediately after ionization, the HfF+ ions are in the ground electronic and vibrational
state (1 Σ+ (v = 0)), primarily distributed over the lowest 4 rotational levels J = 0–3. We connect
these rotational levels using microwaves and perform incoherent transfer to the eEDM-sensitive
3 ∆ (v
1

3
1 += 0, J = 1) science state by using light from L961
trans to drive the Π0+ (J = 0) ← Σtransition, the excited state of which decays preferentially to 3 ∆1 . This light enters the chamber
along the z-axis and is on for 80 ms beginning immediately after ionization. The decay from 3 Π0+puts population in several vibrational levels in 3 ∆1 , which can decay into the v = 0 science state
if left untreated. We remove the population in higher vibrational levels by illuminating the cloud3
3 −with L818
vc light which connects Σ0+ (v = 1, J = 0) ← ∆1 (v = 1, J = 1) at ∼ 818 nm, preferentiallydecaying back to 1 Σ+ . The L818
vc laser also enters the chamber along the z axis and remains on forthe duration of the experiment. Potential systematics associated with this light are discussed in
Section VI B 2 of the Systematics paper.
After transferring the ions to the science state, we ramp on E⃗rot in 5 ms. Figure 2.8 shows

48
the structure of the science state at Erot = 58 V cm−1 . In this field, the stretched states of 3 ∆1 (J =1, F = 3/2) correspond to the molecule aligned or anti-aligned with the field. They form two pairsof levels—which we call the upper and lower doublet, highlighted in orange and blue respectively—
with their molecular dipole, and thus Eeff , either aligned or anti-aligned with Erot . Each doublet
consists of one state with mF = 3/2 and one with mF = −3/2.
We polarize the molecules in the rotating frame by optically pumping them using L1082
op light
addressing the 3 Π0− (v = 0, J = 0) ← 3 ∆1 (v = 0, J = 1) at 1082 nm. The light is circularly
polarized with its k-vector in the plane of E⃗rot . We use an AOM to strobe L1082
synchronously
opwith the rotation of E⃗rot on a 50% duty cycle9 such that it drives either σ + or σ − transitions to
an F ′ = 3/2 manifold in the excited state. This eventually leaves population only in either the
mF = 3/2 or mF = −3/2 states of 3 ∆1 (v = 0, J = 1). We define the preparation phase of theexperiment as the orientation of E⃗rot relative to the k-vector of the light when the light is on; inwhen E⃗rot is parallel, and anti when it is anti-parallel. This preparation phase can be changed byadjusting the timing of the strobing cycle. L1082
is on for a total of 80 ms, starting 40 ms after
op
trapping.
The final step of state preparation is applying L814
depl light at 814 nm. This laser is tuned to(v = 0, J = 0) ← 3 ∆1 (v = 0, J = 1), which preferentially decays to 1 Σ+ by aaddress the 3 Σ−0+
10:1 ratio, having weaker coupling to the 3 ∆1 state. L814
depl is circularly polarized with the same
handedness and k-vector as L1082
op . It is again strobed so as to only address and remove any residual
814
population left over in other mF states after L1082
op is turned off. Ldepl light is on for 7 ms, beginning

3 ms after L1082
is turned off.
op
These steps leave the population in an incoherent mixture of one of the stretched states of
the two doublets. The key difference from our previous measurement [38] is that the experiment9

We note that, although the light is on for 50% of each cycle, the micromotion-induced Doppler shifts mean it is

only resonant with the ions for less than 5%.

49
proceeds on both doublets simultaneously. Our detection scheme [49], described in Sec. 2.2.1.7,
allows us to read out each independently, enabling us to take advantage of common-mode noise
cancellation.

2.2.1.6

Ramsey sequence

Immediately prior to the Ramsey sequence, we ramp the radial confinement of the ions down
further, to trap frequencies of ∼ 1 kHz. This reduces the density of the cloud and improves thecoherence time due to mechanisms discussed in Section VI D 1 of the Systematics paper.
We apply a π/2 pulse to the ensemble of ions by temporarily ramping down the magnitudeof E⃗rot from ∼ 58 V cm−1 to ∼ 7 V cm−1 in 16 µs, holding it there for 1 ms and then ramping backup in a further 16 µs. Reducing Erot increases a rotation-induced coupling between mF = ±3/2
states in a doublet (see Sec. 2.2.3), causing the pure spin states in each doublet to evolve into a
coherent superposition. We allow this superposition to evolve for a variable amount of time tR —up
to 3 s—and then apply a second π/2 pulse to map the relative phase onto a population difference
between the two states in a doublet.

2.2.1.7

Measurement

We project the ions into their final state by applying L814
depl again to remove population fromone of the stretched states in each doublet. The readout phase is defined in the same way as the
preparation phase; in for E⃗rot is parallel with the k-vector of the light when it is on and anti for
antiparallel.
Finally, we detect and count the number of ions in the remaining stretched states via
resonance-enhanced multiphoton dissociation [50], driven by two pulsed UV lasers at 368 nm and266 nm. Immediately prior to the dissociation pulse, we ramp up both radial and axial confinement
to compress the cloud and improve the dissociation efficiency. The dissociation pulse is timed sothat E⃗rot is along I˜ŷ, parallel to the plane of a microchannel plate (MCP) and phosphor screen as-

50

Figure 2.8: Ramsey spectroscopy in HfF+ . Top: level structure of the eEDM-sensitive 3 ∆1 (v =0, J = 1) manifold in external electric Erot ∼ 58 V cm−1 . Solid (dashed) lines correspond to stateswith Ω = +1(−1). Gray lines correspond to states which asymptote to F = 1/2 at zero field,
all other states asymptote to F = 3/2. The upper (orange) and lower (blue) doublets used forthe measurement, corresponding to Eeff aligned and anti-aligned with the externally applied field
respectively, are separated by ∼ 100 MHz. The two states in each doublet are further split by
the Zeeman energy, not resolvable on this scale, and interaction of the eEDM with Eeff . Bottom:
example Ramsey fringes from our dataset. The fringes from the two doublets are collected simultaneously.

sembly. Here I˜ = ±1 and ŷ is defined by Fig. 2.6. Because the dissociation lasers enter at an angle
to ŷ, there is considerable Doppler shift from the micromotion of the ions at 45◦ to the k-vector of

51
the light. To account for this we adjust the frequency of the 368 nm light by ∼ ±2 GHz dependingon the product R̃I˜ which gives the sign of the Doppler shift.
Each of the lasers is circularly polarized to drive transitions which preserve the orientationof the molecules during dissociation [1]. In this way the resultant Hf+ ions from each doublet areejected in opposite directions. The handedness of the dissociation lasers P̃ is determined by a λ/2
waveplate which can be moved into or out of the beam path on a motorized mount. Immediately
after dissociation, we turn off the RF confinement and apply pulsed voltage on the radial electrodes
to kick the ions towards the MCP. The Hf+ ions from each doublet are imaged on opposite sides of˜ We time-gate thethe phosphor screen; the side each doublet is imaged on is set by the value of I.
phosphor screen such that we only image the dissociated Hf+ and not any remaining HfF+ ions.
We detect both Hf+ and HfF+ ions in time of flight. Technical details of our imaging and counting
system are described in [49, 51].
Of the ∼ 2 × 104 trapped HfF+ ions, we typically detect ∼ 550 Hf+ ions on each side of thescreen at early time (∼ 1100 total), and ∼ 120 (∼ 240 total) after tR ∼ 3 s. The latter is principally
limited by the finite lifetime of the 3 ∆1 state but with a contribution from ions being heated out
of our shallow trapping potential during tR .

2.2.1.8

Noise

Instability of the intensity of the pulsed lasers used for ablation, ionization and photodissociation means that the fluctuations in the number of Hf+ ions detected at the end of eachshot are ∼ 30%, roughly 3× the quantum projection noise limit for 120 ions. However thesesources of noise, and many others, are common mode; the exact same laser pulses address the ionswhich end up in the upper and lower doublets. If we measure the ion number when the Ramsey
oscillations of the two doublets are close to in phase with one another then we can take advantage of
excellent noise cancellation in the number difference [49] which we use to extract our science signal(see Sec 2.2.2). The two doublets oscillate at slightly different frequencies owing to a part in 230

52
difference in their magnetic moments and so we deliberately take our data at a beat; our early-timedata is taken when the two doublets are in phase10 and our late-time data ∼ 230 oscillations later
when they come back into phase again. The time of the second beat can be controlled by varying
the oscillation frequency via the B⃗0 -coils. In our final dataset, the noise in the science signal isroughly 30% above the shot noise limit.

2.2.1.9

Experimental protocol and switch states

In each shot of the experiment we can choose the preparation phase to be either in or anti.
˜ the direction of E⃗rot at the moment of dissociation, and P̃ , the handedness of
For a given choice of I,
the dissociation laser polarization, the readout phase is constrained by the need to drive stretchedto-stretched transitions which preserve molecule orientation. We label each shot of the experimentwith each of these phases. For example in-anti labels a shot where the cleanup (and opticalpumping) laser are parallel to E⃗rot during state preparation but anti-parallel during readout.
To record a Ramsey fringe, we repeat our measurement at different free evolution times. For
a given fringe, the phase of readout is kept fixed. At each Ramsey time, we take an even number ofshots with each pair consisting of one shot with each phase of state preparation. This set of shots
is called a point and a Ramsey fringe consists of 8 of these points taken at different tR ; we take 4
points at early Ramsey time, and 4 points at late Ramsey time, each consisting of two points on
the sides of fringes and one point each on the top and bottom as shown in Fig. 2.8. The points on
the sides of the fringes consist of 20 shots each, while the points on top and bottom consist of 8
shots each.
We record our data in ‘blocks’. Each block is constructed from a set of 23 = 8 fringes
recorded in each possible combination of 3 experimental switches. Each switch corresponds to an
10

Due to the finite length of the π/2 pulses, the doublets are already slightly out of phase at the earliest Ramsey

times accessible to us. Systematic effects associated with this imperfection are discussed in Section VII in the
Systematics paper.

53
experimental parameter whose sign can be reversed: B̃ the direction of the current in the B⃗0 -coils,R̃ the direction of rotation of E⃗rot , and I˜ the direction of E⃗rot relative to the y axis at the time ofdissociation, corresponding to which side of the phosphor screen each of the doublets are imaged
onto. Note that in our implementation of the I˜ switch, the direction of E⃗rot is reversed at all points
in time so that the opposite switch is prepared and read out. A fourth experimental switch, P̃ thepolarization of the dissociation light, is alternated every block.
˜ P̃ } = ±1 as a switch state. In each
We refer to each experimental configuration with {B̃, R̃, I,
block, the first Ramsey time is recorded for all switch states before moving onto the second Ramsey
˜ ={1,1,1},
time for each switch state etc. The order of the switch states at each point is {B̃, R̃, I}{-1,1,1}, {1,-1,1}, {-1,-1,1}, {1,1,-1}, {-1,1,-1}, {1,-1,-1}, {-1,-1,-1}. Every other block, the order ofswitch states is reversed. In each switch state, we simultaneously collect data for molecules in eachdoublets, corresponding to orientation of the molecular axis with respect to the applied electricfield which we represent by another switch D̃ = ±1. The Ramsey times for each switch state areadjusted independently based on the data from the previous block to ensure that the 20-shot points
are as close as possible to both the sides of the fringes, where our sensitivity is highest, and to thebeat, where our noise cancellation is best and where various systematic shifts are minimized.
For the eEDM dataset, we collected 1370 blocks or ∼ 600 hours of data over a ∼ 2 month
rev ,period of April–June 2022. During the data run, we took data with 3 different values of the B2,0

corresponding to fringe frequencies of ∼ 75, 105 and 151 Hz. About halfway through the dataset,814
we rotated the waveplates of L1082
op , Ldepl and the dissociation lasers to reverse the handedness of

the light from each.

2.2.1.10

Images to determine doublet positions

To determine where the dissociated Hf+ ions from each doublet fall on the phosphor screenin each switch state, we take a series of images with no Ramsey sequence. For these images we
prepare the stretched states as described in Section VII A of the Systematics paper but apply

54

Figure 2.9: Example ion-detection data for a single shot of the experiment. Swatch (solid lines),from which ion counts are discarded, is defined by region ±45 pixels from center line (dashed line).
Ions assigned to the upper and lower doublets are shown in orange and blue respectively.

no π/2 pulses before removing the population in one of the doublets using low-power L814
depl light,
tuned to resonance with the doublet to be depleted, and propagating along the z direction to avoid
micromotion-induced Doppler shifts. We take 3 types of image per switch state: one where wedeplete the lower Stark doublet, one where we deplete the upper Stark doublet, and one where thelaser is tuned between the doublets to deplete both symmetrically. We use these three images todetermine the center line between the two blobs for each switch state as described in Sec. 2.2.2.1.
We repeated this imaging routine roughly every 10 blocks during the dataset to protect againstalignment drifts. Potential effects of a systematic error in the determination of the center line are
discussed in Section VII B 2 of the Systematics paper.

2.2.2
2.2.2.1

Data Analysis
Ion counting and asymmetry

Our experimental signal is dissociated Hf+ ions read out via phosphorescence on an imaging
microchannel plate (MCP). The images are processed asynchronously and we save a file whichcontains the locations of each bright spot which was determined to be an ion according to our55smoothing and noise-reducing processing algorithm. The data from a tyical shot is shown in
Fig 2.9. The full eEDM dataset contains ∼ 108 ion detection events.
We use this same algorithm to analyze the test images described in Sec. 2.2.1.10 and finda center line for each switch state. We use this center line when analyzing the Ramsey data todefine a swatch which is a rectangular area, of fixed width, at the center of the image from which
ion counts are discarded, as shown in Fig. 2.9. We do this because the doublets are not entirely
separated on the screen, so in this area we cannot be sure that we will assign ions to the correct
doublet. The extent to which we are able to isolate the two doublets is given by the imaging
contrast CI . As the swatch width increases, the imaging contrast improves but total ion number N
decreases as we throw out more ions. For the final analysis of the dataset, we used a swatch of 90
pixels—to be compared with the total width covered by the detected ions, about 900 pixels—which√roughly maximizes CI N , proportional to our sensitivity. Potential systematics associated withthe swatch width and imaging contrast are discussed in Section VII B 1 of the Sysetmatics paper.
Once we have our center line for each switch state we can properly count Hf+ ions in ourRamsey data images and assign them to the correct doublet. For every image (which corresponds
to a single run of the experiment), we end up with a number of ions in the upper doublet N u , andnumber of ions in the lower doublet N l .
For every switch state, we take data in both the in-in and anti-in (or the anti-anti and in-anti)
combinations of preparation and readout phase. If the preparation and readout phase are the same(i.e. in-in and anti-anti), then the fringe formed as we vary tR will have a π phase shift from thefringe formed when they are different (anti-in or in-anti). We will refer to in-in and anti-anti as “In”
and in-anti and anti-in as “Anti”. Now for each pair of shots, we can form our spin asymmetry,
Au/l =

u/l

u/l

u/l

u/l

NIn − NAnti,(2.10)

NIn + NAnti

where the subscript refers to the preparation and readout phase combination. For each Ramsey
time and switch state we take n shots and so can form n/2 asymmetries for each of the upper and56
lower doublet. From these n/2 asymmetries we construct a mean Am
u/l , and from their scatter, astandard error on the mean δAu/l .
We then form two ‘meta’ asymmetries by taking the difference (D) and sum (S) of the upperand lower asymmetries,
AD = Au − Al ,(2.11)
AS = Au + Al ,mwith means Am
D , AS and standard errors δAD , δAS . The δAD are reduced compared to δAS (and

δAu/l ) due to common-mode cancellation of many sources of noise.

2.2.2.2

Least squares fitting

As mentioned previously, we perform our Ramsey experiment simultaneously on both doublets and use their opposing orientations at the time of dissociation to read them out on opposing
sides on the imaging MCP screen. Because the data are acquired simultaneously, the difference
asymmetry allows us to cancel much of the common-mode noise, leaving us with doublet-odd datawith less scatter than the raw data. Unfortunately, the doublets are not fully separated on thescreen, so we must be careful with how we fit our data.
For an ideal Ramsey fringe, with no leakage from the other doublet, we can define a functional
form for the asymmetry,hu/l (tR ) = Cu/l e−γu/l tR cos(2πfu/l tR + ϕu/l ) + Ou/l .

(2.12)

Here C is the initial fringe contrast, γ the contrast decay rate, f the fringe frequency, tR the freeevolution time, ϕ the initial phase, O the offset, and the subscripts indicate the upper or lower
Stark doublet. In our fitting routine, we initially fit each fringe separately to this function. From
the fit parameters we define the mean and difference parameters as
αu + αl,2αu − αl
αd =
,
2

αm =

(2.13)

57
with α ∈ {C, γ, f, ϕ, O}. Due to imperfect imaging contrast CI , in reality each doublet’s signal has
a contribution from the other doublet. In this case, the measured asymmetries are
1 + CI
1 − CI)hu + ()hl ,2
2
1 − CI
1 + CI
Al = ()hu + ()hl .
22

Au = ((2.14)

Now our sum and difference asymmetries are
AS = hu + hl
(2.15)AD = CI (hu − hl ),which we can express in terms of the mean and difference fitting parameters,

AS = (Cm + Cd )e−2γm tR cos(2π(fm + fd )tR + (ϕm + ϕd )) + (Om + Od )...
+ (Cm − Cd )e−2γd tR cos(2π(fm − fd )tR + (ϕm − ϕd )) + (Om − Od ),(2.16)AD = CI ((Cm + Cd )e−2γm tRcos(2π(fm + fd )tR + (ϕm + ϕd )) + (Om + Od )...

− (Cm − Cd )e−2γd tR cos(2π(fm − fd )tR + (ϕm − ϕd )) − (Om − Od )).
We use these two expressions to perform a simultaneous least-squares fit for the sum anddifference asymmetries in each experimental switch state. The value of CI , the imaging contrast, isfixed at 0.89 for this fit—determined as described in Section VII B 1 of the Systematics paper. The
parameter uncertainties are extracted based solely on the standard errors of the asymmetries usedin the fits. The resultant uncertainties on the fitted values of fd and ϕd are close to the shot noise
limit and much smaller than those on fm and ϕm thanks to our simultaneous data collection andfitting routine, which cancels most of the common-mode noise. The outputs of these simultaneous
fits are used for all further analysis.

2.2.2.3

Switch-parity channels

After fitting to each Ramsey fringe in a block to extract the 10 fitting parameters, we use
the resulting 8 values of each parameter to form 8 linear combinations which we call switch-parity

58
channels. The switch-parity channels for the mean and difference parameters αm and αd which are
odd under the product of switches [S̃a S̃b ...] are given by
αSa Sb ... =α

DSa Sb ...

1
8

1
=
8

X

˜
[S̃a S̃b ...]αm (B̃, R̃, I),˜
B̃,R̃,I=±1

X(2.17)˜[S̃a S̃b ...]αd (B̃, R̃, I).

˜
B̃,R̃,I=±1

So for example, C DBRI is formed from adding together the measured Cd in all switch states forwhich the product B̃ R̃I˜ = 1, subtracting all switch states for which the product B̃ R̃I˜ = −1 anddividing by the number of switch states. f BR is formed by adding together the fm measured inall switch states for which the product B̃ R̃ = 1, subtracting the fm measured in all switch states
for which the product B̃ R̃ = −1 and, again, dividing by the total number of switch states. To
avoid ambiguity, we label parity channels for the mean parameters αm which are even under all
switches with superscript 0; e.g. f 0 is the mean of the fm measured in all switch states. We note
that, because the measured fu/l are defined as positive quantities (see Sec. 2.2.3), the B̃ switch isanomalous in that frequency contributions which change sign with B̃ appear in B̃-even channels
while contributions which are independent of the B̃ switch appear in B̃-odd channels. In particular,
the absolute sign of the contribution from the eEDM, which appears in f DB , changes sign onlywith D̃. All other parity channels allow us to to diagnose experimental issues and identify sourcesof systematic error.

2.2.2.4

Blinding

We blinded the dataset by programming the fitting routine to add an unknown constant offset
to the f DB channel. This offset, drawn from a uniform distribution with a width of 10 mHz (∼9 × 10−28 e cm), was stored in an encrypted file and not removed until all statistical and systematic
checks on the dataset had been completed, and the uncertainties finalized.

59
2.2.2.5

Data cuts

After completing the dataset, we applied cuts to the blinded data based on non-EDM channels
indicating signal quality. Blocks with any individual fringe with late-time contrast below Clate = 0.2
were cut due to low signal to noise. By inspection of least-squares fits of individual fringes, thiscut served as a good proxy for pathological fitting results. Blocks were also cut if they contained afringe with a fitted difference frequency fd in any switch state which was more than 3.5σ different
from the mean fringe frequency for that switch state. The mean fringe frequencies were calculated
over all blocks not removed by the late-time-contrast cut and which had the same value of B⃗0 .
They were constructed from the linear combinations, including the blinded offset on f DB . This cuthelped remove blocks where an experiment malfunction, e.g. laser unlocking, affected just one or
two shots in a fringe. If our data were perfectly normally distributed, with no outliers, this would
be expected to remove ∼ 5 blocks, and decrease χ2 by ∼ 0.7%. Figure 2.10 shows the shift in thecenter value, and the error bar of the eEDM channel, as a function of each of these two cuts. The
first cut removed 26 blocks and the second a further 15, leaving 1329 blocks in the final dataset,with χ2 = 1.07(4) for f DB . Our final 1σ statistical error of 22.8 µHz has been relaxed by a factorpχ2 = 1.035.

2.2.3

Effective Hamiltonian for Doublets

To an excellent approximation, we can model the evolution of either of the Stark doublets,
shown in orange and blue in Fig. 2.8, as a two-level system. The effective Hamiltonian for eachpair can be parameterized
h f0 ∆ 
Heff = 
.
2
∆ −f0

(2.18)

60

Figure 2.10: Change in f DB over whole dataset as a function of cuts explained in text. (a) Latetime contrast cut. (b) Individual fringe outlier
p cut. In each plot, only one cut is applied. The errorDBbars on f
are corrected by a factor of χ2 . Dashed lines indicate cuts used in final analysis.

MF = +3/2 11This effective Hamiltonian acts on states . The diagonal components f0 contain
MF = −3/2
all terms which directly shift the energies of the two states relative to one another, while the offdiagonal components ∆ contain all terms which mix the two states. We can expand both f0 and∆ in terms of their leading contributions,11

f0 = B̃f00 + δF0 ,(2.19)∆ = R̃∆0 + R̃D̃∆D + δD.

(2.20)The order of the vectors in Equation 2.18 was not specified in the Systematics paper. The Systematics paper

implicitly uses the opposite convention and has a few sign errors and missing factors of h that will be highlighted infootnotes of this document.

61
Here the quantities with tildes are equal to ±1 and are determined by the experimental switch
state, discussed in Sec. 2.2.1.9. The remaining parameters are defined in the next two paragraphs.
The principal contribution to f0 is the Zeeman splitting f00 = −3gF µB Brot /h ∼ 100 Hz.12 The
off-diagonal component is dominated by two terms with similar magnitude, but different switch state
dependence; ∆0 and ∆D , which both have magnitudes of about 1 Hz, represent a slight mixing of
the two MF = ±3/2 states in each doublet and arise at fourth order in perturbation theory in thefull molecular Hamiltonian from the combined perturbations of rotation and Ω-doubling, breaking
the degeneracy of either Stark doublet at Brot = 0 [52, 53]. ∆0 and ∆D are given by
!


2
2
2
3ℏωef
ℏωrot 3 18A∥ − 19dmf Erot
0
h∆ =
,
2
2
dmf Erot
A2∥ − d2mf Erot!
!
2
3
9A2∥ − 8d2mf Erot3ℏωef
ℏ3 ωrot
D
h∆ =
,
2 A
2
2
d2mf Erot
A2∥ − d2mf Erot∥

(2.21)

where the various constants are defined in Table 2.1. These expressions are valid as long as
dmf Erot ≫ ℏωef and dmf Erot ≫ ℏωrot or, in other words, if the molecule is fully polarized. The
strong scaling of ∆ with Erot allows us to perform off-resonant π/2 pulses by modulating the magnitude of Erot as described in Sec. 2.2.1.6.
The additional perturbations are given by˜ D̃) =
δF0 (B̃, R̃, I,

X

S̃δf0s ,(2.22)

S̃δ∆s ,

(2.23)S̃∈W˜ D̃) =
δD(B̃, R̃, I,

X
S̃∈W

where both summations are over W, the set of all possible products of {B̃, R̃, I˜ D̃}, and the superscript s on the δf0s and δ∆s denote the switch state dependence of the perturbation relative to thelargest term in each matrix element, f00 and ∆0 respectively. For our purposes, the most important
12

Note that sign change and factor of h not included in the Systematics paper. See Subsection 4.6.2 for why

this shift has a negative sign. The lack of a negative sign in the Systematics paper is the cause of considerable signconfusion.

62
perturbation is that due to the eEDM which contributes a diagonal term, D̃δf0DB = 2D̃de Eeff /h13 .
Others which are important for our determination of de are discussed in detail in Section VI of the
Systematics paper.
˜ and doublet D̃,For each experimental switch state (B̃, R̃, I),

we measure a frequency

˜ D̃) corresponding to the energy difference between the two eigenstates, which we definef (B̃, R̃, I,
to be always positive. For typical experimental parameters, f00 is roughly two orders of magnitudelarger than any other term in the Hamiltonian, and so we can expand f about f00 ,

2

2

2
0
D
0 D
0
D
˜ D̃) = B̃f00 + δF0 + (∆ ) + (∆ ) + 2D̃∆ ∆ + 2R̃(∆ + D̃∆ )δD + δD
f (B̃, R̃, I,
2B̃f0022− δF0(∆0 ) + (∆D ) + 2D̃∆0 ∆D + 2R̃(∆0 + D̃∆D )δD + δD2
2f00
"

2

2

+ ...

2

(∆0 ) + (∆D ) + 2D̃∆0 ∆D + 2R̃(∆0 + D̃∆D )δD + δD2
2|f00 |#
2
2(∆0 ) + (∆D ) + 2D̃∆0 ∆D + 2R̃(∆0 + D̃∆D )δD + δD2+ ...
(2.24)− B̃δF0
2|f00 |2

= |f00 | + sgn(f00 )B̃δF0 +

Note the factor of sgn(f00 ) = sgn(−3gF µB Brot /h) = −sgn(gF ), multiplying all but the firstterm in this expression. As we see in Chapter 4, we have measured sgn(gF ) = −1 so sgn(f00 ) = +1.14

13

The factor of 2 in this expression arises because our definition for the frequency of each fringe corresponds to thefull energy difference between de aligned and anti-aligned with Eeff . The systematics paper also reports this shift witha positive sign, though this is due to 1) the opposite convention for the order of the MF = ±3/2 states in Equation
2.18 and 2) a sign error in the paper. See Subsection 4.6.2 for why the sign of the eEDM shift is positive. Also notethe inclusion of h, absent in the Systematics paper.
14

The systematics paper says sgn(f00 ) = −1 because it uses the definition f00 = +3gF µB Brot /h. I use the definition

with a negative sign the equation for f00 , and do so consistently throughout this thesis. Where there are discrepancies,
default to this thesis instead of the Systematics paper.

Chapter 3

Systematics“I’m not dumb. I just have a command of thoroughly useless information.”- Bill Watterson, The Indispensable Calvin and Hobbes
As discussed in the previous chapter, the eEDM shift is D̃-odd and shows up in the f DB
frequency channel which we have measured with a 1σ statistical error of 22.8 µHz. One of ourprimary goals for the experiment was to understand and reduce the size of the systematic errors
in our experiment so that they are each small compared to our statistical error bar. A complete
description of our systematic error budget is given in the Systematics paper [31]. In this chapter,
I will give an overview of our systematic errors and discuss some of those errors in more depth.
The systematic errors of our measurement can be nicely split into two camps. First are shifts
in the frequency difference within each doublet that we measure during the Ramsey experiment.
These shifts are the largest systematics in our experiment and will be the focus of this chapter.
The other set of systematic errors are due to phase shifts of the Ramsey fringe at either early orlate time. While these phase shifts change the measured frequency, the resulting frequency shifts
are suppressed because we measure the phase at the beginning and end of our long coherence time.
This is a key advantage we have over beam line experiments which cannot vary the coherence timeof their measurements. As the systematics from phase errors are relatively small, I leave their
discussion to Section VII of the Systematics paper [31].

64

3.1

Systematics due to Off Diagonal Terms
We know that the eEDM shift contributes to our measured frequency f DB , but it is just partof the full expression which we can find from Equations 2.22- 2.24:

f DB =

1
16

1
=
16

X

˜ D̃)
B̃ D̃f (B̃, R̃, I,˜ D̃=±1
B̃,R̃,I,"X

B̃ D̃|f00 | + D̃δF0

˜ D̃=±1
B̃,R̃,I,22(∆0 ) + (∆D ) + 2D̃∆0 ∆D + 2R̃(∆0 + D̃∆D )δD + δD2+ B̃ D̃
2|f00 |#
2
2(∆0 ) + (∆D ) + 2D̃∆0 ∆D + 2R̃(∆0 + D̃∆D )δD + δD2− D̃δF0+ ...
2|f00 |2

(3.1)

The only terms that survive the summation are the ones which are not multiplied by B̃, R̃, I˜ or D̃:

2

2

0 D
(∆0 ) + (∆D )∆0 δ∆DB + ∆D δ∆BB∆ ∆− δf0DB
−
δf
0
0
0
|f0 |
2|f0 |2|f00 |2"#X
1
δD2
2R̃(∆0 + D̃∆D )δD + δD2+− D̃δF0
+ ...
B̃ D̃
16
2|f00 |
2|f00 |2

f DB = δf0DB +(3.2)

˜ D̃=±1
B̃,R̃,I,The first term, δf0DB , contains the doublet-odd frequency shifts which show up directly inthe measured f DB channel.1 This includes the eEDM shift 2D̃de Eeff which we want to measure.
1

As defined in Subsection 2.2.3, frequency shifts are labeled with a superscript that denotes the switch state

dependence of the frequency shift relative to the largest frequency shift f00 . f00 is B̃-odd, so δf0DB is D̃-odd but
B̃-even. The same convention holds for the ∆ terms where the largest off-diagonal coupling ∆0 is R̃-odd. This
convention is confusing and should be abolished in Generation 3, but as it is deeply entrenched in the Generation 2
documents I use it in my thesis. Note that every term Equation 3.2 has the “net” superscript DB. For example, thesecond and third terms ∆0 δ∆DB and ∆D δ∆B each have one D and B in their superscripts, and therefore a “net”
superscript DB. If the same letter appears twice in the superscript it cancels out. This means that the δf0DB (∆D )2term has a “net” superscript DB as well.

65
All other terms in the above expression, including any other D̃-odd frequency shifts that are partof δf0DB , are systematic errors in our experiment.
All but the first term of Equation 3.2 are systematic shifts due to the nonzero off-diagonal
couplings ∆. The largest of these terms are the R̃-odd ∆0 ∼ 0.6 Hz and D̃R̃-odd ∆D ∼ −0.9 Hz. In
Subsection 2.2.3 it was explained that these couplings are calculated from fourth order perturbation
theory which gives the expressions in Equations 2.21. The important part of these expressions is3rotthat ∆0 is R̃-odd and is proportional to ℏωef dℏω, which is explained in Figure 3.1.
mf Erot

Figure 3.1: The level structure of 3 ∆1 v = 0, J = 1, F = 3/2 state in HfF+ in an external electricfield Erot ∼ 58 V/cm and magnetic field Brot ∼ 10 mG. The energies of the eight states are givenby the Stark shift, which splits the upper and lower doublets by ∼ 100 MHz (blue bracket), andthe Zeeman shift, which causes a ∼ 100 Hz shift between the MF = ±3/2 levels (green brackets).
The shifts are not shown to scale. Pairs of states with the same Ω and ∆MF = 1 are mixed by
the rotation matrix element ℏωrot (black lines) because the experiment is performed in a rotating
frame. Pairs of states with the same MF but different Ω values are mixed due to the Omega
doubling matrix element ℏωef (red lines). When we construct the two-level effective Hamiltonianin Subsection 2.2.3, ∆0 is the average off-diagonal coupling between the MF = ±3/2 states withineach doublet. These states are coupled by three rotation matrix elements and one Omega doubling
matrix element, spanning energies on the scale of the Stark shift dmf Erot . That is why fourth order3rot
perturbation theory gives the expression ∆0 ∝ ℏωef dℏω. Note that ∆0 is proportional to
mf Erot
three powers of the rotation coupling which changes sign when the molecules are rotating clockwise
vs counterclockwise. This explains why ∆0 is R̃-odd. This diagram does not include the F = 1/2
states, which as seen in Figure 2.8 are closer in energy to the upper doublet. These states complicate
the calculation and explain why the coupling has a large ∆D term.

The model explained in Figure 3.1 can also guide us towards the off-diagonal coupling terms

66
besides ∆0 and ∆D which are significant. While the rotation matrix element mixes pairs of stateswith the same value of Ω and ∆MF = 1, the same can be done by a magnetic field that isperpendicular to the quantization axis defined by E⃗rot .2 As E⃗rot rotates in the xy-plane, magnetic
fields along the z-axis of our experiment fit the bill.
There are two important sorts of uniform magnetic fields along the z-axis, those that reverse
rev (2⃗
with B̃ and those that do not. Recall that we apply a magnetic field gradient B⃗0 = B̃B2,0z −⃗x −⃗y ),rev ∼ 200 mG/cm, via a pair of anti-Helmholtz coils.3 B rev is the largest part of the total
where B2,0
2,0

⃗ Similar to the notation for f and ∆⃗ = B̃ B⃗0 + δB.
magnetic field experienced by the molecules, B
explained in Subsection 2.2.3, the switch state dependence of the magnetic field will be written as:

⃗=
δB

X

⃗s
S̃δ B

(3.3)

S̃∈W

˜ and the superscript s
The summation is over W, the set of all possible products of {B̃, D̃, R̃, I},
denotes the switch state dependence of the field relative to B⃗0 , which is B̃-odd. This means Bz
is the B̃-odd magnetic field along the z-axis, which can be generated if the anti-Helmholtz coils
are misaligned, while BzB is the B̃-even background field.4 See Appendix A for how we measured
Bz < 88 mG and BzB < 80 mG in our experiment.
These magnetic fields along the z-axis cause off-diagonal couplings between the same pairsof states coupled by the rotation matrix element. We can naively estimate the magnitude of ∆induced by Bz or BzB by replacing one of the three rotation matrix elements ℏωrot with gF µB Bz
or gF µB BzB . This would tell us that there are contributions to ∆ that are2

gF µB Bz
1
=
and
ℏωrot
982

See Chapter 2 of Willaim Cairncross’s thesis [1] for a detailed explanation of how magnetic and electric fields

cause diagonal and off-diagonal matrix elements.
3

See Subsection 2.3.5 of Tanya Roussy’s thesis [2] for a complete description of the coils used in our experiment.

4

As I made clear in a previous footnote, I do not like this confusing notation. I strongly encourage everyone stillworking on the JILA eEDM experiment to change this notation for the next measurement.

67
gF µB BzB
1=times ∼ 1 Hz, assuming that the magnetic fields are as large as our measurements
ℏωrot
1080
allow them to be. A more careful calculation by current JILA EDM graduate student Anzhou Wang
tells us that this misses a factor of three because the magnetic field matrix element can replace any
one of the three rotation matrix elements. This results in B̃-even and B̃-odd components of ∆ thatare ∼ 327 and 360 times smaller than 1 Hz. These ∆s are R̃-even because they now include only
two rotation matrix elements, and they can either be D̃-even or D̃-odd since we start with ∆0 and∆D of roughly the same magnitude. This means we should expect δ∆R , δ∆BR , δ∆DR and δ∆DBR
all to be roughly ∼ 3 mHz, given our limits on Bz and BzB .
The second term of Equation 3.2 contains δ∆DB and δ∆B . These δ∆s must be R̃ and B̃odd, which can be done if the fourth order perturbation is composed of one Omega doubling
matrix element, one B̃-even Zeeman matrix element, one B̃-odd Zeeman matrix element and
one rotation matrix element. These couplings, made from two Zeeman matrix elements, will be
1 ℏωrotℏωrot∼ 1.8 × 105 times smaller than ∆0 and ∆D . The factor of 6 comes from the six
6 gF µB Bz gF µB BzB
unique ways that gF µB Bz and gF µB BzB can replace the three ℏωrot matrix elements, meaning thatδ∆B and δ∆DB are both about 6 µHz.
Knowing the magnitude of the ∆s, together with the fact that |f00 | ∼ 100 Hz, we can evaluate
the terms in Equation 3.2 to see how large the systematic effect is from each term. The first term,
δf0DB , contains all of the D̃-odd shifts including the eEDM shift. This term will be discussed in
Section 3.2. The second and third terms in Equation 3.2 can be grouped with the largest terms in
B̃ D̃δD2 , where δD is the sum of all of the ∆ switch states except ∆0 and ∆D . These shifts together
are:

30gF2 µ2B Bz BzB ∆0 ∆D∆0 δ∆DB + ∆D δ∆B + δ∆DR δ∆BR + δ∆R δ∆DBR=
2 |f 0 ||f00 |
h2 frot
0(3.4)

On the right side of this equation we have used the approximate values of the six δ∆ terms de-68termined previously. Note that the four terms on the left hand side of the above equation are notindependent – their errors are correlated so we have taken the sum of their magnitudes instead of
the usual quadrature sum for uncorrelated errors. When we evaluate this term with the largest
allowed values of Bz and BzB we find that it has a magnitude of ∼ 150 nHz, comfortably smaller
than our statistical error bar and smaller than the 400 nHz we report in the Systematics paper.5
The next terms in Equation 3.2 are:

0 2
D 2
0 D
δf0DB
δf0B
DB (∆ ) + (∆ )
B∆ ∆
−δf0
−
δf
≈
−
+
0
104
104
2|f00 |2
|f00 |2(3.5)

The first term simply tells us that our measurement of f DB gives us 99.99% of the D̃-odd frequency
shifts instead of the entire shift, a small correction that we can safely ignore. The second term
tells us that we have a systematic from the frequency shifts that are even in all of our switches,
divided by 10,000. As we will see in Section 3.2.2, the root cause of many systematic errors in our
experiment is that a part in ∼ 460 of δf0B shifts leak into our measured f DB . The second term in
the above equation tells us that this leakage is about 5% larger than we would otherwise expect.
As we later constrain δf0B to be ∼ 3.5 mHz,6 we get an additional systematic shift on the order of∼ 210 nHz. This shift is not included in the Systematics paper, though we do mention that there
are effects at the ≲ 200 nHz level that we ignore.
The last term in Equation 3.2 is a sum over a term that is proportional to δF0 . As discussed
in Subsection 3.4.2, the largest frequency shift outside of f00 is δf0BR ≈ 210 mHz. This means thatthe largest systematic effects from the δF0 term are:5

There are multiple reasons for the discrepancy with the Systematics paper. The first is that we did not include

the factors of 3 and 6 found by Anzhou Wang discussed above. The second is that we did not properly find the limits
on Bz and BzB as discussed in Appendix A. We also did not include the terms proportional to ∆0 δ∆DB or ∆D δ∆B .
It is fortuitous that the magnitude of the systematic does not vary substantially when correcting these oversights.
6This number was found by adding in quadrature all of the shifts in f B in sections VI A and VI B of the

Systematics paper. Those shifts will be discussed later in this chapter.

69

−f BR

f BR 6gF µB Bz ∆0 ∆D∆0 δ∆DR + ∆D δ∆R
=
−
hfrot
|f00 |2|f00 |2(3.6)

On the right hand side we have replaced the δ∆ terms with the order of magnitude estimates wefound previously. We can evaluate this expression to find that f BR leaking into the eEDM channel
gives us a ∼ 70 nHz shift, smaller than the 200 nHz shift we report in the Systematics paper.
The updated values of the systematic effects in Equations 3.4 and 3.6 are not significant
enough to change the net systematic or total error bars of our experiment, but these more reliablecalculations may be helpful for when these effects are evaluated in our future measurement.

3.2

Doublet-odd Frequency Shifts
In Section 3.1 we saw that all of the terms in Equation 3.2 except for the first lead to sys-

tematics that are comfortably small compared to our statistical uncertainty. The first term, δf0DB ,contains all of the D̃-odd frequency shifts. This includes the T-symmetry violating eEDM shift
2D̃de Eeff that we want to measure. HfF+ is also sensitive to a T-symmetry violating pseudoscalarscalar electron-nucleon coupling, CS [54, 55]. This shows up as a D̃-odd frequency shift 2WS CS
where WS /h = −51 kHz in HfF+ [56]. While CS does not get the same attention as de , we are
equally interested in measuring these two T-symmetry violating effects.7
Other than T-symmetry violating effects, the only D̃-odd shifts are magnetic.8 Therefore
now is a good time to explain in detail the Zeeman effect in our experiment. This explanation will7

We actually measure a linear combination of de and CS . Our measurement alone cannot determine the value ofeither parameter. It is possible, if unlikely, that de and CS are large enough for us to measure but the effects cancel
in HfF+ . This is one reason why it is good that experiments to measure these quantities are done with different
molecules. I will only mention CS again briefly in Chapter 6.
8

Berry’s phase effects, which are discussed in Section 3.5, are inherently D̃-even. Berry’s phase, magnetic andT-symmetry violating effects are the only shifts that can create an energy difference between pairs of MF = ±F
states for which all other quantum numbers are the same.

70
follow the introduction to Section VI A of the Systematics paper quite closely.

3.2.1

Zeeman Shifts

The Zeeman Hamiltonian for the 3 ∆1 v = 1, J = 1, F = 3/2 state of HfF+ is given as:⃗
HZ = −(⃗µ0 + D̃⃗µD ) · B⃗= −MF µB (gF + D̃δgF )Êrot · B(3.7)

Here µ⃗ 0 is the part of the magnetic moment that is common to both doublets and µ⃗ D is thedifferential part. The magnetic moments of the upper and lower doublets are quite similar for ourtypical experimental parameters; their difference is usually only a part in ∼ 460. In the second line
we use the fact that both magnetic moments track the quantization axis defined by Êrot .
⃗ =As discussed in Section 3.1, we express the magnetic field as a sum of two parts, B⃗ 0 + δ B.
⃗ The first part is the idealized applied quadrupole magnetic field gradient which switchesB̃ B⃗ represents any additional magnetic field experienced byperfectly with B̃, and the second part δ B⃗ B , the magnetic field which does not switch sign
the molecules. We are particularly interested in δ Bwith B̃, causing shifts in f B and f DB .9 To first order10 these are:9⃗Bhδf B = −3gF µB Ê · δ B

(3.8)

⃗B
hδf DB = −3δgF µB Ê · δ B(3.9)

Note that a magnetic field with switch state dependence S will cause shifts in f S and f DS as given by Equations

3.8 and 3.9. With an exception discussed in Subsection 3.2.3, magnetic fields are necessarily D̃-even as the ions ineach doublet are overlapped in space and time.
10

There is a ≲ 5% correction to these shifts from the mixing within each doublet described in Section 2.2.3. These

effects are not included in our systematic uncertainty budget.

71⃗
In these equations Ê is the unit vector pointing along the total electric field E.
As will be calculated in Subsection 4.6.7, the differential magnetic moment µ⃗ D arises primarily
from the mixing of J = 1 and J = 2 levels induced by the electric and magnetic fields:δgF ≈ −⃗
dmf G∥ |E|
20B(3.10)The values of the molecular constants in this equation are given in Table 2.1. There are ∼ 10 %corrections due to the mixing of adjacent MF levels in the rotating frame and interactions withother electronic states, as explained by [57]. We can use this equation together with Equation 3.9
to find expressions for the B̃-even time averaged frequency shifts we measure in the experiment:⃗B ⟩h ⟨δf B ⟩ = −3gF µB ⟨Ê · δ B(3.11)δgF⃗B ⟩µ ⟨E⃗ · δ B⃗ B|E|(3.12)h ⟨δf DB ⟩ = −3

Appreciating the subtle differences in these equations is incredibly important for understanding the systematic shifts in our experiment. The Zeeman shift that is common to the two doublets⃗ B in the direction of the quantization
is proportional to the time average of the magnetic field δ Baxis Ê. On the other hand, the differential Zeeman shift is proportional to the time average of the⃗ B dotted into the the entire electric field E.
⃗ Note that δgF is independent of themagnetic field δ B⃗|E|electric field due to Equation 3.10 and is just a constant. This difference between whether Ê or E⃗
is included in the time average is the source of most of our magnetic systematic errors.
⃗ B as:Going forward it will be useful to expand δ B

72X⃗ B (⃗x, ⃗y , ⃗z) =δBlX⃗ l Yl,m
Bl,m ∇r(3.13)

l=1,2,... m=−l

Here ⃗x, ⃗y and ⃗z and the ion’s position relative to the trap center, Bl,m is the coefficient of eachcomponent and Yl,m are the seminormalized real spherical harmonics:√i (Y m − (−1)m Y −m ) if m < 0


l
2 l

r


4π
Yl,m =
× Y0
if m = 0 .
l2l + 1 






 √1 (Yl−m + (−1)m Ylm ) if m > 02(3.14)⃗ l Yl,m for l ≤ 3 are given in Table 3.1.
The ∇rTable 3.1: Magnetic Field Expansion. Estimated sizes of components are given in parentheses.
Coefficient⃗ l Yl,m∇r
Uniform fields (≲ 10 mG)11

B1,−1
B1,0
B1,1

ŷ
ẑ
x̂

First-order gradients (≲ 10 mG cm−1 ) 12√B2,−2√3(yx̂ + xŷ)B2,−13(z ŷ + yẑ)
B2,0
−xx̂
√ − y ŷ + 2z ẑ
B2,1√3(z x̂ + xẑ)B2,23(xx̂ − y ŷ)
Second-order gradients (≲ 10 mG cm−2 )
q
3
5
B3,−3
2
2 (2xyx̂ + (x − y)(x + y)ŷ)√B3,−215(yz x̂ + xz ŷ + xyẑ)
q
3
1
2
2
2
B3,−1
2 (−xyx̂ + 2 (−x − 3y + 4z )ŷ + 4yz ẑ
1 2
2
2
B3,0
q −3(xz x̂ + yz ŷ + 2 (x + y − 2z )ẑ)
B3,1

B3,2
B3,3

3 1
2
2
2
2 ( 2 (−3x − y + 4z )x̂ − xy ŷ + 4xz ẑ)
√
1
15(xz
q x̂ − yz ŷ + 2 (x − y)(x + y)ẑ)3522 ((x − y)(x + y)x̂ − 2xy ŷ)

73
⃗2,0
Zeeman Shift from E⃗rot and B

3.2.2

The largest electric field that the molecules experience is E⃗rot = Erot [cos (ωrot t)x̂+sin (ωrot t)ŷ].
e
2 , wheremωrote > 0 and m are the charge and mass of HfF+ respectively. As discussed in Section 2.1, there is noThis field causes the molecules to undergo a circular micromotion with radius rrot =⃗1,m . We can now
induced Zeeman shift if the molecules are rotating in a uniform magnetic field B⃗1,m ⟩ = ⟨Êrot · B⃗1,m ⟩ = 0 for all allowed values of m = ±1, 0. In fact,
see that this is because ⟨E⃗rot · B
the only magnetic field with l ≤ 3 which causes a nonzero time-averaged shift in any frequency⃗2,0 . This leads to frequency shifts:
channel is the intentionally applied B

rev
rev
⃗2,0
h ⟨f 0 ⟩ = −3gF µB ⟨Êrot · B⟩ = −3gF µB B2,0
rrot

h ⟨δf D ⟩ = −3

δgF
rev
rev
⃗2,0
µB ⟨E⃗rot · B⟩ = −3δgF µB B2,0
rrot
Erot

(3.15)(3.16)⃗ = ⟨E⃗ · B⟩
⃗ /|E|.
⃗ This gives us the “usual” case where the D̃-odd
Note that in this case that ⟨Ê · B⟩
shift isgF
∼ −460 times smaller than the D̃-even shift.13 We reliably see in our experiment that
δgF

when f 0 ≈ 100 Hz, we measure f D ≈ −210 mHz.
While this combination of ideal electric and magnetic fields does not give rise to D̃-odd
frequency shifts, it is never that easy in real life. There will inevitably be a magnetic field gradient⃗2,0 that does not reverse sign with B̃, B⃗ nr . This non-reversing field can arise from either an
B
2,0
11

We used an array of magnetometers around our apparatus to shim the magnetic fields to about 10 mG, though

we were unsure if we could trust the measurement of magnetometers outside of our vacuum chamber. We were able
to confirm that B1,1 and B1,−1 were less than about 10 mG at the location of the ions by intentionally applying alarge second harmonic of the electric field and measuring the resulting shifts from the Zeeman effect described in
Subsection 3.3.3. In Appendix A we find that Bz < 80 mG and BzB < 88 mG.
12See Section VI A 4 of the Systematics paper for how we approximated the first- and second-order magnetic field

gradients.
13

See Subsection 4.6.7 if you would like an explanation of why

gF
is negative.
δgF

74⃗ rev or from a backgroundimperfection in how we reverse the current in the coils that drives B
2,0magnetic field gradient. In either case we get the frequency shifts:

nr
nr
⃗2,0
h ⟨δf B ⟩ = −3gF µB ⟨Êrot · B⟩ = −3gF µB B2,0
rrot

h ⟨δf DB ⟩ = −3

δgF
nr
nr
⃗2,0
µB ⟨E⃗rot · B⟩ = −3δgF µB B2,0
rrot
Erot

(3.17)(3.18)The latter of these two shifts is a systematic error, caused by a non-reversing magnetic field gradientinteracting with the slight difference in g-factors between the two doublets. The magnitude of this
nr
nr
B2,0
B2,0
systematic shift in f DB will be ⟨δf D ⟩ rev ≈ 210 mHz × rev .
B2,0
B2,0
We go to great lengths to suppress this worrying systematic shift. First note that every time
we take a block of data, we measure all 16 frequency channels made from the B̃, D̃, R̃ and I˜
nr .
switches. This includes f B , which according to Equation 3.17 contains a shift proportional to B2,0

If we assume that this is the only shift in f B , an assumption that we carefully check,14 we can⃗ 0 coils in the subsequent
use our measurement of f B in order to shim the currents through the B
rev as close to zero as we can. This shimming procedure was quite successful; theblock to bring B2,0

magnitude of the largest f B measured in any block was 35 mHz while the median magnitude was
2.3 mHz. These correspond to non-reversing magnetic fields that give rise to systematics of 75 and5 µHz per block.
To remove the systematic effect more completely, we also can correct our measurement of
14This is a two part assumption. The first part is that Equation 3.17 is the only frequency shift in our experimentthat is even with respect to all of our switch states, which will be evaluated in Sections 3.3 and 3.5. The second
assumption is that our measurement of f B does not contain significant shift due to the mixing terms like the ones inEquation 3.2. Since ∆0 and ∆D are similar in magnitude, the shifts caused by the mixing terms in f B will be similar
in magnitude to the ones discussed in Section 3.1. Those showed up at the ∼ 100s nHz level, and are therefore smallenough to ignore.

75
f DB using our measured value of f B :DB
fcorrected= f DB −

δgF B
f
gF

(3.19)

Note that the two approaches, shimming the magnetic field and correcting the measurement, aremathematically equivalent approaches to removing the systematic. We do both. Because weshimmed the non-reversing magnetic field before we corrected f DB , the average correction over thewhole dataset was just 90 nHz.
There are two more comments to make about this systematic effect. The first is that, in orderfor our correction to be successful, we need to know

δgF
quite accurately. This is an experimentallygF

measured quantity with the data shown in Figure 3.2. The second point is that, while we havesuccessfully stomped out this systematic, we now have a host of new problems. Any shift δf B which
δgF
times smaller will cause a systematic error ingF⃗ =⃗ /|E|,⃗
our measurement of the eEDM. These shifts can be magnetic in cases where ⟨Ê · B⟩̸ ⟨E⃗ · B⟩
does not have an associated D̃-odd shift that iswhich will be addressed in Section 3.3. Additionally, Berry’s phase can cause shifts that are evenwith respect to all of switch states without any D̃-odd effect, which are discussed in Section 3.5.

3.2.3

Imperfect Doublet Overlap

Before moving onto Zeeman effects that cause shifts in f B and f DB that may not be cancelled
out perfectly, it is worth asking if there are any Zeeman shifts that are inherently D̃-odd. We assume
throughout the rest of this thesis that the cloud of upper doublet molecules is overlapped in space
and time with the cloud of lower doublet molecules. If this is the case the magnetic fields seen by
the ions are inherently D̃-even and the only way to cause D̃-odd Zeeman shifts is via the small
difference in g-factor δg.
But what if the clouds are in slightly different places ⟨ri ⟩D ? Or what if the doublet clouds
D

have different sizes ⟨ri2 ⟩ ? In this case B̃-even magnetic fields that cause first- and second-order

76

D
0
F
Figure 3.2: Measurement of δg
gF . (a) f vs f for various values of the applied quadrupole magnetic
0 D
field, B⃗0 . Data taken at Erot ∼ 58 V cm−1 . Fit is to f D = δgF f 0 + ∆ ∆0 , giving δgF = −0.002 149(3),∆0 ∆D = −0.39(4) Hz2 .

gF
gF
f
DB
B
(b) Change in f
vs f induced by introducing a large non-reversing
0

D

∆ ∆
0 D
F
B2,0 . Fitted gradient is equal to δg
gF − f 0 2 and when combined with value of ∆ ∆ from (a) gives
δgF
gF= −0.002 13(2), in excellent agreement. Note that this is the only effect we ever saw cause ashift in f DB . Under each plot we show the residuals after fits are subtracted.

gradients in f B would lead directly to D̃-odd systematic shifts:

δf DB =

X
i=x,y,z⟨ri ⟩D

2 B
∂f B
D ∂ f
+ ⟨ri2 ⟩∂ri∂ri2

We measured these gradients by moving the ions around in the trap and found(3.20)

∂f B
∼ 40
∂ri

∂2f B
∼ 10 mHz/cm2 . In order to keep the systematics shifts at least an order of∂ri2Dmagnitude smaller than our systematic error bar, we need ⟨ri ⟩D ≲ 1×10−4 cm and ⟨ri2 ⟩ ≲ 5×10−4
mHz/cm and77
Table 3.2: f B Curavatures with and without Permanent Magnets
Effect∂f B
without Magnets (mHz/cm)
∂ri
∂2f B
without Magnets (mHz/cm2 )
∂ri2
∂f B
with Magnets (mHz/cm)
∂ri
∂2f B
with Magnets (mHz/cm2 )
∂ri2

x

y

z

18(2)

47(2)

47(2)

-2(10)

9(10)

-4(4)

-1090(10)450(10)-1500(20)250(50)-170(30)-1210(20)cm2 .
We took two approaches to measure these D̃-odd moments of the doublet clouds. The first
approach, which I will discuss now, did not work. It was fun though, and I want to record it in mythesis. Our idea was to place permanent magnets – found in the hallways of JILA holding up posters– around our ion trap to induce B̃-even gradients in f B . This was after we completed our ∼ 650
hour long dataset, so we really hoped that this would not permanently magnetize our apparatus if
we needed to retake any data.15 With the magnets in place we repeated our measurements of

∂f B
∂ri

∂2f B
. An example of the gradients we measured along the z-axis is shown in Figure 3.3. The∂ri2
gradients created by this particular orientation of magnets is summarized in Table 3.2

andAfter measuring the curvatures we took a few hours of precision data with the magnets still in
15

Putting the magnets above the vacuum chamber was easy. Three magnets were stacked together and they sat

in a small cardboard box that rested on the vacuum chamber. The cardboard box was there so the magnets would
not sit directly on our permanent vacuum chamber and possibly permanently magnetize it. Getting the other set of
permanent magnets below the vacuum chamber was trickier, as it was very hard to reach into the space and place
anything precisely. I ended up stacking the cardboard box that contains the permanent magnets on top of a pile ofpaper towels. I could adjust the number of paper towels to change the magnets’ height, and attempted to have the
top of the cardboard box rest directly against the bottom of the vacuum chamber. I should clarify that this was done
when I was left alone in lab for a week or so, as I think Tanya Roussy and Luke Caldwell have no desire to claim thiswork as their own. I, on the other hand, made sure this brilliant stroke of engineering for a precision measurement
is permanently recorded in my thesis.

78

Figure 3.3: Three permanent magnets were placed above and below the ion trap, generating acurvature in f B via the Zeeman effect. We moved the center of the ion trap to different positions
along the z-axis and measured f B on two different days, the 7th and 9th of September 2022. On
∂2f B
∂f B
both days we saw∼ 1200 mHz/cm2 and∼ 1500 mHz/cm, both much larger values
2
∂z
∂z
B
∂2f B
2 and ∂fthan background values of∼5
mHz/cm∼ 47 mHz/cm. For a full comparison of∂z 2∂z
the curvatures with and without the magnets in all three directions, see Table 3.2.

place and the ions at the center of the trap. We measured f DB = 90(627) µHz, showing no evidenceof a shift in f DB from our central value -15(23) µHz. In principle, we can use this measurementD

to set a limit on a linear combination of the six D̃-odd sizes ⟨ri ⟩D and ⟨ri2 ⟩ . If we took more
precision data with different values of

∂f B
∂2f B
and
, and continued not to see shifts in f DB , we
∂ri
∂ri2
D

perhaps could set sufficiently small limits on ⟨ri ⟩D and ⟨ri2 ⟩ .
In practice, this was quite difficult. We were limited in terms of where we could physicallyplace the permanent magnets by our apparatus. As can be seen in Figure 2.6, the vacuum chamber
extends to both sides of the ion trap along the y-axis, meaning it was especially difficult to makelarge gradients

∂2f B
∂f B
and
. Additionally, we were nervous about using magnets that were too∂y∂y 2

large around our trap in case the apparatus became permanently magnetized. This could havebeen disastrous if for any reason we needed to take more precision EDM data. In the end we came79
up with a number of arguments for how ⟨ri ⟩D and ⟨ri2 ⟩Dcould arise in our experiment and why

they must be small. See Section VI D 2 of the Systematics paper for that discussion. Via thosearguments we conservatively constrain the net systematic shift to be 3.5 µHz at most.

Systematics from Magnetic Shifts in f B and f DB

3.3

The correction to f DB discussed in Subsection 3.2.2 relies on the assumption that all shifts
δgF
times smaller in f DB . However, we saw that this assumptiongF⃗ ̸= ⟨E⃗ · B⟩
⃗ /|E|.
⃗ In
fails for any Zeeman shifts caused by electric and magnetic fields where ⟨Ê · B⟩
in f B are magnetic and appear

this case we would have:DB
fcorrected= f DB −

δgF B
f
gF

⃗
⟨E⃗ · B⟩δgF
= 2de Eeff − 3δgF µB
−
⃗
gF
|E|

!
⃗
− 3gF µB ⟨Ê · B⟩⃗
⟨E⃗ · B⟩⃗
+ 3δgF µB ⟨Ê · B⟩⃗
|E|!
⃗ · B⟩⃗⟨
E⃗ −= 2de Eeff + 3δgF µB ⟨Ê · B⟩⃗|E|

= 2de Eeff − 3δgF µB(3.21)In order to minimize this systematic shift we need to find all possible combinations of electric⃗ =⃗ /|E|.
⃗ Luke Caldwell approached this problem analytand magnetic fields where ⟨Ê · B⟩̸ ⟨E⃗ · B⟩
ically while I simulated the shifts numerically. These methods will be discussed in the next two
subsections.

3.3.1

Analytic Approach to Zeeman Systematics

The ideal electric field in our experiment is E⃗ = E⃗rot + E⃗RF + E⃗DC , where the three electric fields
are given by Equations 2.4 – 2.6 and the largest of which is E⃗rot . As discussed in Subsection 2.2.1.3,
we create E⃗rot ∼ 58 V/cm by applying voltages ∼ 350 V to eight radial electrodes. Those voltages
which oscillate at 375 kHz are generated by DDS (direct digital synthesis) boards and amplified

80
by high-powered op-amps. The op-amps inevitably add harmonic distortions when generating thevoltages to apply E⃗rot , which we can write as:E⃗nh (t) = Enhx cos (nωrot t + ϕnx )x̂ + Enhy cos (nωrot t + ϕny )ŷ + Enhz cos (nωrot t + ϕnz )ẑ(3.22)The other predominant imperfection in our electric field is the ellipticity of E⃗rot . This canarise either because the voltages on the eight radial electrodes are unequal or the electrodes areslightly out of place. We can write the ellipticity as:E⃗ϵ (t) = Eϵ cos (2θ − ωrot t)x̂ + sin (2θ − ωrot t)ŷ(3.23)The strategy now is to analytically calculate the micromotion of the HfF+ molecules in the⃗ where δE⃗ represents all perturbations to the idealized rotating electric
electric field E⃗ = E⃗rot + κδE,
field. This runs into a chicken and egg problem because E⃗RF and E⃗DC both vary spatially, as seen in
Equations 2.4 and 2.5. We want to find the ion trajectories from the electric field, but we need theion trajectories to know the magnitude of E⃗RF and E⃗DC . To get around this obstacle we approximate
the effects of the spatial variance with time-varying fields:E⃗sec = Esecx cos (ωsecx t + ϕsecx )x̂ + Esecy cos (ωsecy t + ϕsecy )ŷ + Esecz cos (ωsecz t + ϕsecz )ẑ(3.24)

E⃗RFeff = ERFeffx cos (ωsecx t + ϕsecx ) cos (ωRF t + ϕRF )x̂+ ERFeffy cos (ωsecy t + ϕsecy ) cos (ωRF t + ϕRF )ŷ+ ERFeffz cos (ωsecz t + ϕsecz ) cos (ωRF t + ϕRF )ẑ(3.25)

81
Equation 3.24 is an electric field that causes the secular motion of the trapped ions. Equation 3.25
is meant to capture the RF micromotion – it is large when the RF field is large and the secularmotion is at its maximum. This captures most of the effects of our electric fields, though it does
not include pondermotive forces. We note that those forces are included in the numerical model
described in Subsection 3.3.2.
Once we have the particle trajectories, we consider the molecules’ motion through a magnetic⃗ where δB⃗ are the zero, first and second order magnetic field gradients listed⃗=B⃗2,0 + κδB,
field B
⃗2,0 . The calculated trajectory allows us to write down the time-dependent
in Table 3.1 other than B⃗ and ⟨E⃗ · B⟩,⃗ wheremagnetic field experienced by a HfF+ molecule. We can finally calculate ⟨Ê · B⟩
the time average is over an integer number of periods of all relevant frequencies. We keep terms up
to O(κ2 ) before setting κ = 1.
As mentioned before, the only frequency shift that is zeroth order in κ is the one from E⃗rot⃗2,0 . There are no frequency shifts first order in κ. This means that there are nointeracting with B⃗ or from one field in δE⃗ interacting withfrequency shifts from E⃗rot interacting with one field in δB⃗2,0 . There are a host of frequency shifts that are second order in κ, listed in Tables 3.3 – 3.5.
B
Note that these table only include up to the the n = 4th harmonic, a choice which will be justified
in Subsection 3.3.4.

3.3.2

Numerical Approach to Zeeman Systematics

In addition to the analytic approach of finding which combinations of electric and magnetic
fields generate Zeeman shifts, we wanted to approach the problem numerically as well. The idea
was to calculate the trajectory of HfF+ ions in the full electric and magnetic fields at small discrete
time steps. We then found the electric and magnetic fields at the ion locations, which allowed us to⃗ and E⃗ · B⃗ at each time step. The values were then averaged over the entire trajectory,
calculate Ê · B⃗ and ⟨E⃗ · B⟩.
⃗
giving us values for ⟨Ê · B⟩

82
Table 3.3: Analytically Calculated Zeeman Shifts, Part 1⃗⟨Ê · B⟩

Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14

15

16

17

18

B1,1 E2hx cos ϕ2x
4Erot
B1,1 E2hy sin ϕ2y4ErotB1,−1 E2hy cos ϕ2y4ErotB1,−1 E2hx sin ϕ2x
4Erot√
3B2,2 E3hx rrot cos ϕ3x
4Erot√
3B2,2 E3hy rrot sin ϕ3y
−
4Erot
√
3B2,−2 E3hy rrot cos ϕ3y
−
4Erot
√
3B2,−2 E3hy rrot sin ϕ3y−4Erot
5B2,0 E3hx Eϵ rrot cos (2θ + ϕ3x )−218Erot5B2,0 E3hy Eϵ rrot sin (2θ + ϕ3y )218Erot5B2,0 E2hx E4hx rrot cos (ϕ2x − ϕ4x )−2128Erot
5B2,0 E2hy E4hy rrot cos (ϕ2y − ϕ4y )2128Erot
5B2,0 E2hy E4hx rrot sin (ϕ2y − ϕ4x )−2128Erot
5B2,0 E2hx E4hy rrot sin (ϕ2x − ϕ4y )−2128Erotr
5
2 cos ϕB3,3 E4hx rrot
3
4x
2
−
8Erot
r
5
2 cos ϕ3
B3,−3 E4hy rrot
4y
2
8Erot
r
5
2 sin ϕ3
B3,−3 E4hx rrot
4x
2
8Erot
r
5
2 sin ϕ3
B3,3 E4hy rrot
4y
2
8Erot
−

⃗ /Erot⟨E⃗ · B⟩

⃗ /Erot⟨E⃗ · B⟩⃗⟨Ê · B⟩

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

83

Table 3.4: Analytically Calculated Zeeman Shifts, Part 2⃗⟨Ê · B⟩

⃗ /Erot⟨E⃗ · B⟩

2 r
B2,0 E2hz
rot
2
4Erot
2 r
B2,0 E3hz
rot
−
2
9Erot
2 r
B2,0 E4hz
rot−2
16Erot2 r ω2B2,0 Esecz
rot rot−22Erot ωsecz
2
2
B2,0 ERFeffz
rrot ωrot−
4Erot (ωRF − ωsecz )22
2
rrot ωrot
B2,0 ERFeffz−
4E (ω + ω )2
√ rot RF 2 secz
3 15B3,2 E2hz rrot cos ϕ2z
8Erot√
2 sin ϕ
3 15B3,−2 E2hz rrot
2z
−
8Erot
√
3 3B2,2 Eϵ rrot cos 2θ
−
2Erot
√
3 3B2,−2 Eϵ rrot sin 2θ
−
2Erot
r
5
2 cos ϕ3
B3,3 E2hx rrot
2x
2
4Erot
r
5
2 cos ϕ3
B3,−3 E2hy rrot
2y
2
4Erot
r
5
2 sin ϕ3
B3,−3 E2hx rrot
2x
2
−
4Erot
r
5
2 sin ϕ3
B3,3 E2hy rrot
2y
2
4Erot

2 r
B2,0 E2hz
rot
2
4Erot
2 r
B2,0 E3hz
rot
−
2
9Erot
2 r
B2,0 E4hz
rot−2
16Erot2 r ω2B2,0 Esecz
rot rot−22Erot ωsecz
2
2
B2,0 ERFeffz
rrot ωrot−
4Erot (ωRF − ωsecz )22
2
rrot ωrot
B2,0 ERFeffz−
4E (ω + ω )2
√ rot RF 2 secz
3 15B3,2 E2hz rrot cos ϕ2z
8Erot√
2 sin ϕ
3 15B3,−2 E2hz rrot
2z
−
8Erot
√
2 3B2,2 Eϵ rrot cos 2θ
−Erot√
2 3B2,−2 Eϵ rrot sin 2θ
−
Erot
r
5
2 cos ϕ9
B3,3 E2hx rrot
2x
2
8Erot
r
5
2 cos ϕ9B3,−3 E2hy rrot
2y
2
8Erot
r
5
2 sin ϕ9B3,−3 E2hx rrot
2x
2
−
8Erot
r
5
2 sin ϕ9
B3,3 E2hy rrot
2y
2
8Erot

Number
19
20
21
22
23
24
25
26
27
28

29

30

31

32

−

−

⃗ /Erot⟨E⃗ · B⟩⃗⟨Ê · B⟩
1
1
1
1
1
1
1
1
4
3
4
3
3
2
3
2
3
2
3
2

84
Table 3.5: Analytically Calculated Zeeman Shifts, Part 3
Number
33
34
35
36
37
38
39
40
41
42
43
44
45

⃗
⟨Ê · B⟩

⃗ /Erot⟨E⃗ · B⟩

2 r
B2,0 E2hx
rot
2
16Erot2 r
B2,0 E2hyrot

2 r
B2,0 E2hx
rot
2
8Erot
2 r
B2,0 E2hyrot

2
16Erot2
B2,0 E3hx rrot
236Erot2 r
B2,0 E3hy
rot

2
8Erot
2
B2,0 E3hx rrot
2
18Erot2 r
B2,0 E3hyrot

2
36Erot2
B2,0 E4hx rrot
264Erot2 r
B2,0 E4hy
rot

2
18Erot2
B2,0 E4hx rrot
2
32Erot2 r
B2,0 E4hyrot

2
64Erot2
B2,0 Eϵ rrot
22Erot
2 r ω2B2,0 Esecx
rot rot
2 ω2
4Erot
secx
2 r ω2B2,0 Esecy
rot rot224Erot ωsecy
2
2
rrot ωrot
B2,0 ERFeffx
2 (ω216ErotRF − ωsecx )22rrot ωrot
B2,0 ERFeffx
2 (ω216ErotRF + ωsecx )2
2
B2,0 ERFeffy rrot ωrot2
32Erot2
B2,0 Eϵ rrot
2Erot
2 r ω2B2,0 Esecx
rot rot
2 ω2
2Erot
secx
2 r ω2B2,0 Esecy
rot rot222Erot ωsecy
2
2
rrot ωrot
B2,0 ERFeffx
2 (ω
2
8Erot
RF − ωsecx )22rrot ωrot
B2,0 ERFeffx
2 (ω
2
8Erot
RF + ωsecx )2
2
B2,0 ERFeffy rrot ωrot2 (ω216ErotRF − ωsecy )22
B2,0 ERFeffy
rrot ωrot

2 (ω
2
8Erot
RF − ωsecy )22rrot ωrot
B2,0 ERFeffy

2 (ω216ErotRF + ωsecy )8E 2 (ω + ωsecy )2r rot RF3
2 cos ϕ3
B3,1 E2hx rrot
2x
2
−
8Erot
r
3
2 cos ϕ3
B3,−1 E2hy rrot
2y
2
8Erot
r
3
2 sin ϕ3
B3,−1 E2hx rrot
2x
2
8Erot
r
3
2 sin ϕ3
B3,1 E2hy rrot
2y
2
8Erot

46

0

47

0

48

0

49

0

⃗ /Erot⟨E⃗ · B⟩⃗⟨Ê · B⟩
2
2
2
2
2
2
2
2
2
2
2
2
2

∞

∞

∞

∞

85
The numerical simulations added a level of realism not included in the analytic approach.
The simulations always included E⃗rot and the full DC and RF electric fields which give rise topondermotive potentials. In many of the simulations the electric fields were written as the perfectly
uniform fields given by Equations 2.4 – 2.6. Some of the more detailed simulations instead calculated
the electric fields from voltages applied to our ten electrodes, with voltage to electric field conversion
factors given in Appendix C.4 of William Cairncross’s thesis [1]. The resulting electric fields wereless homogeneous and more accurately represented the fields experienced by our ions. We did
not see any significant difference between the two methods of simulating electric fields, which isconvenient because the simpler fields are faster to simulate and indicate that the analytic approach
does not miss anything important in this instance.
In addition to the three standard electric fields, a numerical simulation typically included
one additional electric field perturbation such as an ellipticity or harmonic along a particular axis.
rev and one more of the magnetic fields in TableThe simulation also included typically included B̃B2,0

3.1 that would not change sign with B̃. As these are numerical simulations, particular values forthe magnitudes and phases for each field were chosen.
After selecting the fields, a time step ∆t and integration time Tint were chosen. ∆t was
selected so it was small compared to the inverse of the largest frequency in the simulation. Tint waschosen to include many secular frequencies ∼ 1 kHz, as that was always the slowest characteristicfrequency of the system. As long as Nsteps =Tint≫ 1, it did not matter if Tint was an integer∆tnumber of every characteristic period of the simulation.
Next we specified how many individual ion trajectories were being calculated. This was
typically an integer multiple of 16, as the computer I was working on had 16 cores that could
calculate the trajectories in parallel. We split the ions into equal groups of four where (B̃, R̃) =(+1, +1), (+1, −1), (−1, +1) and (−1, −1). Each individual ion was given a random initial positionin three dimensions drawn from a normal distribution centered around zero with a width set to set

86
to give the ion cloud a 1σ radius of about 1 cm. The initial velocity was determined the same way
except that it had a R̃-odd offset so that at t = 0 the molecules were already moving in circles withE⃗rot .
The trajectories of the 16N ions were then individually calculated. These trajectories include
the force from the magnetic field but did not include ion-ion interactions. Ion-ion interactions werethe largest effect on the trajectories that were not considered by the numerical simulations. A two
dimensional cut of a standard ion trajectory is shown in Figure 3.4. After calculating the ions’⃗ Ê and B⃗ were found at each of those points in space and time. The
positions at each time step, E,⃗ and Ê · B⃗ were then found at each time step for every ion. Finally the averages
dot products E⃗ · B⃗ and ⟨Ê · B⟩
⃗ were found for each ion by summing the individual dot products and dividing⟨E⃗ · B⟩by Nsteps .
The average frequency f and D̃-odd frequency f D were calculated from each ion trajectory.
By taking appropriate linear combinations of the data taken with different values of B̃ and R̃, theD̃, B̃ and R̃-odd frequency channels were constructed. This let us identify the magnitude andswitch dependence of the Zeeman shifts present in the simulation.
With this method, we confirmed many of the shifts identified analytically in Tables 3.3 – 3.5.
Take, for example, the first shift listed in the tables. It says that a second harmonic of the electricfield along x will interact with a magnetic field along x to give a Zeeman shift that is entirely
doublet even.16 We confirmed this shift by applying the fields in question and seeing a shift in f B
that varied with the magnitudes B1,1 and E2hx , in addition to the phase of the second harmoniccos ϕ2x .
I checked many of the systematic effects in Table 3.3–3.5 this way and did not find any
surprises. These effects will be discussed in the next two subsections.
16

Recall that the shift is entirely doublet even up to the two ≲ 10% effects listed in Subsection 3.2.1. We should

expect the doublet-odd shift to be ∼ 0.1 ×δgF
times the size of the doublet-even shift.
gF

87

Figure 3.4: A two dimensional cut of a standard ion trajectory. At t=0 the ion’s position is givenby the black dot, and its trajectory follows the green trace. The molecule traces out many circles
with radius ≈ 0.5 mm due to E⃗rot . The molecule also undergoes slower oscillations in the xy-plane
as the ion undergoes secular motion, moving around x = y = 0 due to the ion trapping fields.

3.3.3

Uniform Magnetic Field + Second Harmonic Systematic

Our task now is to show that all of the systematic effects from the entries in Tables 3.3–3.5
are small compared to our statistical error bar 22.8 µHz. This is easy for entries 19 through 26⃗ = ⟨E⃗ · B⟩
⃗ /|E|⃗ and their D̃-odd shift is cancelled by our shimming/correction
as they have ⟨Ê · B⟩
procedure. It is worth noting that many of these shifts can be grouped. For example, the Zeeman
shifts numbered 1 through 4 are all caused by a uniform magnetic field in the xy-plane interactingwith a second harmonic of Erot in the xy-plane. This subsection will focus on this group of systematic
shifts in detail, and the next subsection will address the rest of the effects.
Consider the simplified case where the total electric field is equal to E⃗rot plus a second har-88
monic in the xy-plane:









cos (ωrot t) E2hx cos (2ωrot t + ϕ2hx )

 


 

E⃗ = Erot  sin (ωrot t)  + E2hy cos (2ωrot t + ϕ2hy )

 


 

0
0

(3.26)

We can now find Ê by expanding to first order in 1/Erot :









cos (ωrot t)
E2hx cos (2ωrot t + ϕ2hx )




1




Ê ≃  sin (ωrot t)  +E2hy cos (2ωrot t + ϕ2hy )
 2Erot 





0
0






− cos (4ωrot t + ϕ2hx ) − cos (ϕ2hx )
− sin (4ωrot t + ϕ2hy ) + sin (ϕ2hy )
 EE2hx 



2hy 
+
 − sin (4ωrot t + ϕ2hx ) + sin (ϕ2hx )  + cos (4ωrot t + ϕ2hy ) + cos (ϕ2hx )  (3.27)
 4Erot 
4Erot 




0
0

Note thatE⃗ ismade of terms that oscillate at ωrot and 2ωrot . When we take the dot product
Bx 
 

⃗=
of E⃗ and BBy , all of the terms still oscillate at ωrot or 2ωrot . These terms will go to zero
 
 
0
⃗ both have constantwhen time averaged, resulting in no D̃-odd Zeeman shift. However, Ê and B⃗ ̸= 0 and we will get the D̃-even Zeeman shifts in entries 1-4 of Table 3.3.
terms. This means ⟨Ê · B⟩
We confirmed that this shift exists experimentally by applying a magnetic field Bx ∼ ±0.2
Gauss and a second harmonic of the electric field with E2hy ∼ 100 V/m and ϕ2hy ∼π 17. As seen4

in Figure 3.5, we saw no significant shift in f DB but a ∼ ±8.5 Hz shift in f B .18
17

Getting a value for the phase ϕ2hy is a bit complicated because it is defined relative to Erot which has a R̃-odd

phase. See Subsection 4.6.4 for discussion, but ϕ2hy ∼πis a reasonably good approximation as the R̃-odd part of4

89

Figure 3.5: We applied a magnetic field Bx ∼ ±0.2 Gauss and a second harmonic of the electricπfield with E2hy ∼ 50 V/m and ϕ2hy ∼ . The plot shows the resulting f B and f DB in Hz, where4
f DB includes the 16.1 mHz blind. Error bars of f B are too small to resolve on this scale. Datawith Bx ∼ +0.2 Gauss is on the right with f B ∼ 8 Hz, and data with Bx ∼ −0.2 Gauss is on theleft with f B ∼ −9 Hz. This confirms the predicted D̃-even Zeeman shift.

The remaining challenge is to set a limit on the magnitude of this effect while we took theeEDM data set. See Appendix B for how we set limits on the harmonics of Erot in our apparatus.
We carefully shimmed the second harmonic voltage on each electrode and found that the residual
magnitude was ≲ 11 mV, about than 30,000 times smaller than the voltages we apply to generate
Erot . Since we do not know if those voltages constructively or destructively interfere to generate asecond harmonic in the xy-plane, we conservatively assume that the magnitdue in x and y is lessthan 2.5 mV/cm, 24,000 times smaller than Erot . In this case we would expect shifts of about 80
mHz/G along both the x and y-axes.
After our harmonic shimming procedure we deliberately applied large magnetic fields ∼ 2
Gauss along x and y and observed shifts in f B to be 103(1) mHz/G along x and 46(9) mHz/G
Erot ’s phase is small.
18

We took this data on January 27th 2022. At the time we thought that all Zeeman shifts effected the upper
δgF. This was quite a surprise. The magnitude of the shift in f B
gF
is about three times larger than predicted, but the magnitude and phase of the second harmonic were not carefully
and lower doublets differently by the same ratio

calibrated at the time.

90
along y, see Figure 3.6. These shifts indicate that the estimate of the second harmonic in Appendix
B is accurate. We were then able to apply a large second harmonic in the xy-plane in order tomeasure Bx and By . We used this measurement to shim the current in our x and y Helmholtz coils,
described in Section 2.3.5 of Tanya Roussy’s thesis [2], to keep these background fields below 10mG.

(a) f B (Hz) vs Bx (G)(b) f B (Hz) vs By (G)Figure 3.6: We took data with (a) Bx = ±1 G and (b) By = ±1 G. Note the difference in scale off B of the two graphs. We saw shifts in f B in both cases that are likely from the residual second
harmonic of Erot after it was shimmed away.

The background magnetic field and second harmonic can drift throughout our dataset, so
we redid these measurements about once a week while we were taking eEDM data. We did notmeasure any drifts outside of the aforementioned upper bounds. We therefore set a conservative
upper limit of the uniform magnetic field in the xy-plane + second harmonic in the xy-plane shift
in f B to be 100 mHz/G × 10 mG = 1 mHz. Because there is no corresponding shift in f DB ,this shift in f B will cause a slight error in our shimming/correction procedure. This results in asystematic shift with magnitude 1 mHz ×
statistical error bar 22.8 µHz.

δgF
= 2.2 µHz. This is comfortably smaller than our
gF

91
3.3.4

Other Systematics from Zeeman Shifts

The previous subsection addressed entries 1 through 4 of Tables 3.3–3.5. We know entries⃗ = ⟨E⃗ · B⟩
⃗ /|E|⃗ in those cases. That
19 through 26 will not cause a systematic shift because ⟨Ê · B⟩
leaves 37 Zeeman shifts to check. If I addressed those shifts in as much detail as I did 1 through4, this chapter would go on for quite a while. Instead I will group them together in Table 3.6 anddiscuss the 15 remaining entries briefly. The effects in this table have all sines and cosines removed
because we assume the phases are optimally bad. Electric fields along the x or y-axes have beengrouped together, as have magnetic fields Bl,±m .
Entries 5-8 are D̃-even Zeeman shifts from a third harmonic in the xy-plane interacting witha magnetic field gradient in the xy-plane. As discussed in Appendix B, we measured these third
harmonics to be ∼ 1.3 V/m, and as discussed in Section VI A 4 of the Systematics paper we foundB2,±2 ≲ 10 mG/cm. Plugging these values into the expression in Table 3.6, we find a shift in f Bwith magnitude 0.7 mHz corresponding to a systematic in f DB of 1.5 µHz.
Entries 9-10 are interesting because they are proportional to B2,0 . This must be a B̃-evenmagnetic field gradient in order to show up as a systematic, and from Subsection 3.2.2 we know
nr . Over the entire dataset the median B nr ≈ 0.8 µG/m. These entries are alsothat this is B2,02,0proportional to the ellipticity Eϵ , which we measure in Subsection 3.5.3 to be about 1.7 V/m. This
gives an absolutely tiny shift in f B of 1 × 10−7 µHz and systematic in f DB of about 2 × 10−10 µHz.
nr and are also tiny. They are in fact smaller because
Entries 11-14 are also proportional to B2,0of the larger number in their denominator. We calculate the shift in f B to be 9 × 10−10 µHz andthe associated systematic 2 × 10−12 µHz.
Entries 15-18 are larger than 9-14 but still are not particularly worrying. We calculate theshift in f B to be 22 µHz and the associated systematic 5 × 10−2 µHz.
⃗ = ⟨E⃗ · B⟩
⃗ /|E|.
⃗ Any shift caused
Entries 19-26 do not cause systematics as they have ⟨Ê · B⟩

92

Table 3.6: Analytically Calculated Zeeman Shifts Grouped Together
Number in
Tables 3.3–3.5
1-4
5-8
9-10
11-14

15-18
19-21
22
23-24
25-26
27-28

29-32
33-38
39
40-41
42-45

46-49

⃗
⟨Ê · B⟩

⃗ /Erot⟨E⃗ · B⟩

⃗ /Erot⟨E⃗ · B⟩⃗⟨Ê · B⟩

Systematic
(µHz)

0

0

2.2

0

0

1.5

0

0

2 × 10−10

00

2 × 10−12

0

0

5 × 10−2

B1,±1 E2hxy
4Erot√
3B2,±2 E3hxy rrot
4Erot
5B2,0 E3hxy Eϵ rrot
2
18Erot5B2,0 E2hxy E4hxy rrot
2128Erotr
52
B3,±3 E4hxy rrot
3
2
8Erot
2 r
B2,0 Enhz
rot−2
n2 Erot
2 r ω2B2,0 Esecz
rot rot−22Erot ωsecz
2
2
rrot ωrot
B2,0 ERFeffz−
4Erot (ωRF ± ωsecz )2√2
3 15B3,±2 E2hz rrot
8Erot√
3 3B2,±2 Eϵ rrot
−
2Erot
r
5
2
B3,±3 E2hxy rrot
3
2
4Erot
2
rrot
B2,0 Enhxy2 r
B2,0 Enhz
rot2
n2 Erot
2 r ω2B2,0 Esecz
rot rot−22Erot ωsecz
2
2
rrot ωrot
B2,0 ERFeffz−
4Erot (ωRF ± ωsecz )2√2
3 15B3,±2 E2hz rrot
8Erot√
2 3B2,±2 Eϵ rrot
−
Erot
r
5
2
B3,±3 E2hxy rrot
9
2
8Erot
2
rrot
B2,0 Enhxy2
4n2 Erot2
B2,0 Eϵ rrot
2
2Erot
2
2
B2,0 Esecxy rrot ωrot
2 ω2
4Erot
secxy
2
2
B2,0 ERFeffxyrrot ωrot2
2n2 Erot2
B2,0 Eϵ rrot
2
Erot
2
2
B2,0 Esecxy rrot ωrot
2 ω2
2Erot
secxy
2
2
B2,0 ERFeffxyrrot ωrot2 (ω216ErotRF ± ωsecxy )

2 (ω
8Erot
± ωsecxy )2
r RF
3
2
3
B3,±1 E2hxy rrot
2
8Erot

0

−

1
1
1
1
4
3

1.7

3
2

2 × 10−2

2

8 × 10−122

5 × 10−10

2

1 × 10−5

2

3 × 10−6∞

1 × 10−2

93
by these effects will be corrected as discussed in Subsection 3.2.2.
Entries 27 and 28 are unique because they are the only second order shifts in κ with⃗⃗
⟨E · B⟩ /Erot4= . This means the correction will account for 75% of the shift these effects cause in⃗3⟨Ê · B⟩f DB , but the rest of the shift leads to a systematic. By plugging in the limits for the parameters inthis shift, as we have been been doing for the last few paragraphs, we calculate the shift in f B tobe 5 mHz. However as we mention in Section VI A 5 of the Systematics paper, we measured thisshift with a five times larger ellipticity than is usually present in our experiment. We did not see
any shift in f B due to this effect and the 1σ upper limit was 12 mHz, so we set a more stringent
limit on the effect at 2.4 mHz. This corresponds to a systematic uncertainty in f DB of 1.7 µHz.
Entries 29 through 32 are interesting as well for being the only shifts with

⃗ /Erot3⟨E⃗ · B⟩= .
⃗2
⟨Ê · B⟩

These shifts, however, are naturally smaller than 27 and 28 because they are proportional to ahigher order gradient and the second harmonic that we shimmed particularly well. We calculate
the shift in f B to be 17 µHz, corresponding to a systematic error of about 20 nHz.
nr . Their
Entries 33 through 45 are all predictably small because they are proportional to B2,0

associated systematic shifts can be found in Table 3.6.19
Entries 46 through 49 are Zeeman shifts which are entirely D̃-odd. These effects show up
directly as systematics! Fortunately they are only 14 nHz, small enough to sweep under therug. If they were much larger this would be a difficult systematic to diagnose as we blinded ourmeasurements of f DB and they do not show up in any other frequency channel.
Note that our choice to only include up to the fourth harmonic in Erot is sensible. As seen in
Appendix B, higher harmonics generally have smaller amplitudes. In addition, we can see in Table
19

Note that the largest systematic shifts from entries 33 through 38 are from n=3 due to the third harmonic being

significantly larger than the shimmed second harmonic, see Appendix B. The shift due to n=3 is the systematic shift
reported in Table 3.6. The magnitude of the RF electric field reported in Table 2.2 was used for ERFeffxy . Esecxy was
found using dimensional analysis assuming a secular frequency of 1.3 kHz.

94
3.6 that the third and fourth harmonics generally couple to higher order magnetic field gradients
2 , we can safely assume that magnetic
and larger powers of rrot . Since B1,m > B2,m rrot > B3,m rrot

systematics caused by the fifth harmonic or higher will be smaller than the largest effects in Table
3.6.

3.4

Other Noteworthy Magnetic Shifts
In Sections 3.2 and 3.3 I addressed the Zeeman shifts which are doublet odd and show up as

systematic effects or appear in the f B frequency channel, leading to systematics via our correction
on f DB . In this section I will discuss a few other magnetic shifts that do not cause systematic
errors in our experiment but are interesting nonetheless.

3.4.1

Axial Magnetic Field and f RE⃗rot causes the molecules to trace out circles in the xy-plane with radius rrot . A magnetic⃗ force:field along the z-axis will slightly modify the radius of that rotation due to the ⃗v × B

rrot =

eErot
evBz
eErot
e2 Erot Bz
+
R̃
≃
+
R̃
2
2
2
3
mωrot
mωrot
mωrot
m2 ωrot(3.28)⃗ force causes a R̃-odd change in the radius. The effect depends on the
Since ⃗v is R̃-odd, the ⃗v × B
velocity of the ions, which in turn depends on the radius because ωrot is fixed. This makes themath more complicated, but we can approximate the velocity as v = ωrot rrot ≃ ωrot
as long as Bz ≪mωrot
∼ 5 Tesla so it only causes a small perturbation to rrot .
e

eErot
eErot
=
2
mωrot
mωrot

rev :
The change in rrot causes a R̃-odd frequency shift as the molecules move through B2,0

rev
rev
hf = −3gF µB B2,0
rrot ≃ −3gF µB B2,0



eErot
e2 Erot Bz
+
R̃
2
3
mωrot
m2 ωrot(3.29)

rev ≈ 230 mG/cm, which generates f 0 ∼ 150 Hz, and B ∼ ±10
We measured this effect with B2,0z

95
rev we predict f R to have a slope ofGauss. The data is shown in Figure 3.7. With this value of B2,0

−3.1 mHz/G, which is consistent with our measurement −2.98(9) mHz/G. This is the first shiftthat we as a group could not initially explain that I understood before anyone else, so it gets apage in my thesis.

Figure 3.7: We applied a large non-reversing Bz of ±10 G. We observed a shift in f R of −2.98(9)mHz/G, consistent with the explanation that Bz modifies the size of rrot .

3.4.2

Charging Currents

The largest magnetic field we intentionally apply to our molecules is the quadrupole magnetic
field that reverses sign with B̃. The largest magnetic field that we unintentionally apply is createdby the currents which drive the oscillating voltages Vrot to create Erot . The voltages have amplitude
Vrot ∼ 350 V and oscillate at frot = 375 kHz. They are applied to our eight radial electrodes which
are shown in Figure 3.8.
The radial electrodes, which we sometimes call “fins” due to their peculiar shape, are numbered n = 1 through 8 in Figure 3.8. We apply R̃-odd voltages to the electrodes with a phase shiftofπ
between neighboring electrodes:20
420n
The actual phases of Vrot
are discussed in Subsection 4.6.4. These toy phases are simpler to consider and are

sufficient to model the Zeeman shift caused by the charging currents.

96

Figure 3.8: Key dimensions of our ion trap, all dimensions in millimeters. The radial electrodes,
two of which are shown in detail on the left, have a peculiar shape to make the E⃗rot as uniform as
possible at the center of the trap. Note that the electrodes have “tabs” at the top and bottom thatreach beyond the top (T) and bottom (B) endcaps. The electrodes can be charged and discharged
through either or both of these notches. A full description of the radial electrode design can befound in Chapter 4 of William Cairncross’s thesis [1], which is the source of this image.

n
Vrot= Vrot cos (R̃ωrot t −

nπ
+ π)
4

(3.30)

Ignoring the spatial inhomogeneities that are discussed in Chapter 4 of William Cairncross’s thesis⃗[1], these voltages applied to our radial electrodes creates an electric field Erot = Erot cos (ωrot t)x̂ +R̃ sin (ωrot t)ŷ as we would expect.
Originally we charged and discharged all eight of our electrodes from the top and bottom
n to both thesimultaneously. That is, we connected the output of the op-amp which generates Vrot

97
top and bottom tabs of fin n. The idea was to minimize the current flowing through the electrodes
as Vrot varied, therefore reducing the resulting magnetic field. However, due to differences in
the resistances of the electrical feedthroughs above and below each fin, this resulted in all eightfins generating an unpredictable magnetic field. These resistances were small but had substantial
fractional variations, which meant each electrode had a different proportion of current above andbelow them.
Rather than attempt to model the resulting magnetic field, we decided to connect the opamps of the odd numbered fins only to the top of the odd numbered electrodes. The even numberedfins were charged and discharged from the bottom. This meant we knew exactly which direction
the current was traveling as each fin charged and discharged. We can create a simple model of themagnetic field by assuming that the eight radial electrodes have a height z0 and are centered at
z = 0. We also assume the radial electrodes have capacitance Cfin which is constant as a functionof height so Cfin =

dCfin
dV
z0 . With these assumptions and the relationship I = C
we reach thedzdtfollowing expression for the currents on each fin:



dCfin
nπ
n
Irot
= R̃
Vrot ωrot sin (R̃ωrot t −
+ π)
dz

4

z0
z∓
2


(3.31)The ∓ symbol is minus for odd radial electrodes that are charged from the top and plus for evenradial electrodes that are charged from the bottom.
We can now calculate the magnetic field at the center of the trap. To keep the expression
simple we assume the radial electrodes are infinitely long thin wires that are a distance Rtrap from⃗cc created by the charging currents
the trap center. If this is true we find that the magnetic field B
is:

98


⃗cc =
B







dCfin R̃ cos (ωrot t)
R̃ cos (ωrot t)



z



dz 
 sin (ωrot t)  = Bcc z  sin (ωrot t) 
πRtrap002Vrot µ0 ωrot

(3.32)

According to this equation the magnetic field caused by the charging currents rotates with a frequency equal to frot and is parallel to E⃗rot for z R̃ > 0 and antiparallel for z R̃ < 0.21
⃗cc , we can calculate its amplitude more
While this model captures the general behavior of B
precisely. We used Comsol Multiphysics to simulate where the charge accumulates on the fins as afunction of height.22 We found that each fin has a total capacitance of about 9.7 pF and we foundthe current through each fin as a function of height, shown in Figure 3.9.
We then calculated the magnetic field from these current distributions assuming the fins werethin wires with finite length z0 . It was initially unclear what value to use as the distance between
the eight imaginary thin wires and the center of the trap. To be safe we did the calculation twicewith the radius set to 48 and 84 mm, two reasonable radii shown in the bottom left of Figure 3.8.
These calculations predicted the Bcc = 2.9 and 1.1 mG/m respectively. Because the magnetic fieldis R̃-odd and B̃-even, this corresponds to a slope

df BR
between 31.8 and 12.4 mHz/mm. Thisdz

compares very well with our observed data, shown in Figure 3.10, where we see a shift of 12.69(6)mHz/mm. This perhaps indicates that the majority of the charge moves along the back of theradial electrodes.
There are a few comments worth making about this shift before moving on. The first is thatwhile the shift ideally goes to zero at the center of the ion trap, we measure f BR ∼ 210 mHz. This
could easily be explained by differences in capacitances of the wiring inside the vacuum chamber
21

This magnetic field has a nonzero curl; see Section 5.1.2 for a discussion of this field and Maxwell’s equations.

22

These simulations used a realistic rendering of the entire ion trap. It was important to include the voltages

applied to all eight radial electrodes when calculating the capacitance as this effected how much charge needed toaccumulate on a given fin to reach a given voltage.

99

Figure 3.9: This graph shows the percentage of the current that drives Vrot through a radial
electrode that is plugged in from the bottom. For the fin to reach Vrot charge must be distributedroughly evenly in z. Therefore 100% of the current goes through the bottom layer of the fin,and there is less and less current as we move up the fin. The curves of the graph are explainedby different heights of the radial electrode having different capacitances which were calculated in
Comsol Multiphysics. The two discontinuities occur at points where where the fins suddenly jut
out toward the center of the trap, see Figure 3.8.

between the eight op-amps and fins.
This shift is entirely R̃-odd and shows up in f BR , so it does not contribute to our systematic
error budget. Additionally, because this magnetic field is parallel or antiparallel to E⃗rot , it has⃗ /|E|⟨E⃗ · B⟩
= 1. This means that there will be a ∼ 460 times smaller shift that appears in f DBR .
⃗⟨Ê · B⟩Even if somehow a small part of the shift is R̃-even and appears in f B , which could happen becauseof the R̃-odd Erot phases discussed in Subsection 4.6.4, it will not cause a systematic error in our
experiment because the effect is taken care of by our shimming/correction procedure.
There is the possibility that this frequency shift will be used intentionally in our third gen-

100

Figure 3.10: The y-axis reports f BR in Hz and the x-axis gives the position of the ion cloud along
df BR
z in mm. We observe a gradientof 12.69(6) mHz/mm which agrees well with our model.
dz

eration experiment to measure the eEDM. This will be addressed in Section 5.1.3.

3.4.3

Other Zeeman Shifts

For a discussion of other miscellaneous Zeeman shifts, see Section 3.11 of Tanya Roussy’s
thesis [2].

3.5

Berry’s Phase
Other than Zeeman shifts, the only effect that can cause a frequency shift that shows up

in f B is Berry’s phase.23 Berry’s phase shifts are almost on equal footing in importance with the
Zeeman effect in terms of our systematics, but I have relegated them to the fifth section of thischapter. This is an interesting choice on my part, as when I give talks about the experiment the
Berry’s phase shift described in Subsection 3.5.3 is usually the only systematic effect I mention.
23

I mention at the beginning of this chapter that I leave discussion of systematics due to phase shifts to the

Systematics paper. In this section I describe how we can continuously accumulate a Berry’s phase which shows up
in our experiment as a frequency shift that can cause systematic errors. Hopefully this clarifies any confusion.

101
Unlike Zeeman shifts, Berry’s phase is an inherently D̃-even effect so it is a bit less worrisome. It
can only causes systematic shifts by changing f B and spoiling our correction to f DB described in
Subsection 3.2.2. That said, Berry’s phase is an important effect that we spent many collective
months (years?) worrying about for the generation two measurement.
As a note, I wrote Chapter 4 to be an independent paper which does not reference any otherchapter in this thesis on the off chance we decide to publish it. It explains how we measure thesign of gF using Berry’s phase, so it contains an explanation of Berry’s phase I will describe Berry’s
phase from scratch in this section as well, please forgive the repetition. This explanation is alsosimilar to the one given in Section VI B of the Systematics paper in case you would like to read athird version.

3.5.1

Berry’s Phase Overview

Our quantization axis defined by E⃗rot completes frot = 375, 000 rotations per second in thexy-plane. As E⃗rot traces out closed loops, the quantum state accumulates a geometric phase of:ϕgeo = ∆MF Ωs(3.33)

During our spectroscopy the molecules are in a superposition of MF = ±1.5 states, so themolecules will accumulate a relative phase shift of 3Ωs . Here Ωs is the solid angle traced out by
the electric field, as shown in Figure 3.11, which is a signed quantity. If the electric field rotates
perfectly in the xy-plane, Ωs = ∓2π for R̃ = ±1. In this ideal case the difference of ϕgeo between
the two MF states is ∓6π, meaning that the Berry’s phase is not observable as it is an integer
number of 2π phase shifts. But if we intentionally tilt the electric field up out of the xy-plane by asmall angle α as shown in Figure 3.11, we induce a measurable phase shift that will be accumulated
every Trot = 1/frot . For small tilts, this will appear as a frequency shift for each MF state given
by:

102

Figure 3.11: Ideally, the quantization axis defined by E⃗rot rotates perfectly in the xy-plane andtraces out a solid angle of −2π R̃. If E⃗rot is tilted up by a small angle α, it will deviate from theideal solid angle by an amount Ω. This image has been reproduced from [31].

3frot
fBerry ≈ −2π

Z Trotα(t)ϕ̇(t)dt(3.34)

0

Here α(t) is the tilt of the quantization axis out of the xy-plane and ϕ̇(t) is the azimuthal
angular velocity. If neither parameter depends on time, we find that:

fBerry = −3frot αR̃

(3.35)

At first glance it is reassuring that fBerry is R̃-odd, which is the case because ϕ̇(t) is R̃-odd.
However we will soon see that our imperfections to the electric field, the ellipticity and various
harmonics of Erot , can themselves be R̃-odd. This results in R̃-even shifts which generally appearin f B .

3.5.2

Gravity

The only way E⃗rot can have a constant tilt α out of the xy-plane is due to gravity. This isbecause the ions sit at the point in the trap where the average net force is equal to zero. Since
gravity pulls the ions downward there is an average electric field pointing up to cancel its effect. Thisfield E⃗grav =mgẑ ∼ 20 µV/m. This causes a tilt in our total electric field α = arctan
e



103

Erot
∼
Egrav

3.3 × 10−7 , resulting in a shift f BR ∼ 4 mHz.24 This shift is completely R̃-odd so it does not causea systematic effect.
All other Berry’s phase shifts discussed in this section are caused by electric fields that
oscillate along the z-axis interacting with another electric field in the xy-plane which together
cause an average tilt ⟨α⟩.

3.5.3

Ellipticity plus Axial Second Harmonic

The simplest way to get a nonzero value of ⟨α⟩ is from a second harmonic of Erot along thez-axis and an ellipticity of Erot . Ignoring the trapping electric fields, this gives us a total electricfield:




7π
7π
π
π
 Erot cos (ωrot t + 12 + R̃ 4 ) + Eϵ cos (2θR̃ − ωrot t − 12 − R̃ 4 ) 


π
7π
π
7π 
⃗ =
E(t)
Erot R̃ sin (ωrot t ++ R̃ ) + Eϵ R̃ sin (2θR̃ − ωrot t −− R̃ )

12
4
12
4 



(3.36)

E2hz cos (2ωrot t + ϕ2f )The peculiar choice of Erot phase

7π
π
+ R̃
was used in our experiment to ensure that the ions of
12
4

the upper and lower doublet land on different sides of our MCP, as explained in Section 2.2. See
Subsection 4.6.4 for how we confirmed the values of these phases. Here there is an ellipticity Eϵalong an angle θ from the x-axis and a second harmonic along the z-axis with phase ϕ2f .
As we see in Subsection 4.6.3, we can combine Equations 3.34 and 3.36 to find frequency
shifts that show up in f B and f BR :25
24

It would be really cool if we could measure this and see gravity’s effect on molecular ions. 4 mHz is a large

enough shift for us to easily resolve, but it shows up in the same frequency channel as the ∼ 200 mHz shift causedby the charging currents described in Subsection 3.4.2. If only we could turn our apparatus upside down.
25

Note that we get a R̃-odd ⟨α⟩, and therefore a R̃-even frequency shift, even though we assume that E⃗2hz isR̃-even. If we relax this assumption we find that the resulting frequency shifts are split between f B and f BR as well.

104

fB = −

9E2hz Eϵ frotπcos (2θ) sin (ϕ2f − )
2
6
2Erot

(3.37)π
9E2hz Eϵ frotsin (2θ) sin (ϕ2f + )
2
3
2Erot

(3.38)

f BR = −While we are not particularly bothered by the shift in f BR , the shift in f B leads to a systematicδgF
times smaller. Assuming the worst cases for the phases, that the entire shift shows up
gF
9E2hz Eϵ frot δgF
in f B , we have a systematic shift with magnitude
.
2
gF
2Erot

that isWe know the magnitude of all of these terms, including E2hz from Appendix B, except forEϵ . To find Eϵ we initially applied E2hz ∼ 4 V/m and Eϵ /Erot ∼ 7 × 10−3 to make sure that we saw
shifts in f B and f BR . In the data in Figure 3.12 we fixed the angle of the ellipticity θ = −

π
and
8

scanned ϕ2f . We saw both frequencies vary sinusoidally with the magnitudes we expected.

Figure 3.12: Measurement of Berry’s phase shift in f B and f BR . We applied an ellipticity withπmagnitude Eϵ /Erot ∼ 7 × 10−3 and phase θ = − . We also applied a second harmonic along the8
z-axis with magnitude E2hz ∼ 4 V/m and scanned its phase. We see the sinusoidal frequency shifts
predicted in Equations 3.37 and 3.38.

We then repeated the measurement without intentionally applying an ellipticity and saw asmaller sinusoidal effect in both frequency channels, shown in Figure 3.13. We inferred from this

105
measurement that the natural scale of the ellipticity in our trap is Eϵ /Erot = 3 × 10−4 . This let us
put a limit on the shift in f B to be 0.8 mHz, corresponding to a systematic effect of 1.7 µHz.

Figure 3.13: To measure Eϵ we applied a second harmonic along the z-axis with magnitude E2hz ∼ 4
V/m and scanned its phase. We see sinusoidal shifts in f B and f BR indicating that there is anonzero Eϵ /Erot ∼ 3 × 10−4 .

3.5.4

nth Harmonic along z plus (n-1) or (n+1)th Harmonic in the xy-plane

Generally, we can get a Berry’s phase shift from the combination of two harmonics when one
is along the z-axis and the other is in the xy-plane:⃗ x,y = Enhx cos(nωrot t + ϕnhx )x̂ + Enhy cos(nωrot t + ϕnhy )ŷδE(3.39)⃗ z = Emhz cos(mωrot t + ϕmhz )ẑδE(3.40)In these equations m and n are both integers. The combination of these perturbations will result

106
in a non-zero time-averaged frequency shift at first order provided m and n differ by ±1. The shift
is given by:δf = ∓


3Emhz frot (n ± 2)
Enhx cos(ϕnhx − ϕmhz ) ± Enhy sin(ϕnhy − ϕmhz )2
4Erot(3.41)

Here the ± correspond to m = n ± 1. Depending on the R̃ dependence of the various phases, thisshift could appear in either f B or f BR . We note that the frequency shift corresponding to thefirst harmonic on the endcap and second harmonic radially, m = 1, n = 2, is zero, and so the next
largest shifts are expected to come from effects involving the third harmonic or higher.
We measured the voltages of up to the 17th harmonics and put constraints on the various
electric fields in Appendix B. We can plug the electric field measurements in Table B.1 into Equation
3.41 to find the magnitude of shifts that appear in f B or f BR . The results are shown in Table 3.7.
The quadrature sum of all shifts in this table is 1.4 mHz. Assuming the worst case, that the shift
is entirely in f B , we are left with a 3.0 µHz systematic error in f DB .

3.5.5

Other Berry’s Phase Effects

The systematics paper discusses two more sources of Berry’s phase. They are caused by 1) an
AC Stark shift from our only laser that is on during the Ramsey time (Section VI B 2) and 2) the
molecular cloud’s axial slosh (Section VI B 3). I do not have much to add about these effects otherthan to point out that I think Luke Caldwell was clever to realize that there could be a Berry’s
phase shift from our laser at all.

3.6

Systematics Summary and Result
To end the chapter I include Table 3.8 which summarizes the systematic errors in our experi-ment. This table does include the smaller value for the axial magnetic field systematic discussed in
Section 3.1. The section listed in the table below refers to the relevant section of the Systematics

107
Table 3.7: Constraints on Berry’s phase frequency contributions to f B or f BR due to possiblecombinations of radial and axial field imperfections from harmonic distortion in amplifiers used
to drive radial electrodes. All entries are in µHz. This table is an extension of Table VI in the
Systematics paper.
nz
nx,y

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17

0
11
0
0
0
0
0
0
0
0
0
0
0
0
0
0

17
0
22
0
0
0
0
0
0
0
0
0
0
0
0
0

0
95
0
75
0
0
0
0
0
0
0
0
0
0
0
0

0
0
87
0
31
0
0
0
0
0
0
0
0
0
0
0

0
0
0
83
0
139
0
0
0
0
0
0
0
0
0
0

0
0
0
0
38
0
13
0
0
0
0
0
0
0
0
0

0
0
0
0
0
947
0
545
0
0
0
0
0
0
0
0

0
0
0
0
0
0
24
0
16
0
0
0
0
0
0
0

0
0
0
0
0
0
0
556
0
90
0
0
0
0
0
0

0
0
0
0
0
0
0
0
35
0
33
0
0
0
0
0

0
0
0
0
0
0
0
0
0
200
0
143
0
0
0
0

0
0
0
0
0
0
0
0
0
0
30
0
37
0
0
0

0
0
0
0
0
0
0
0
0
0
0
107
0
144
0
0

0
0
0
0
0
0
0
0
0
0
0
0
41
0
9
0

0
0
0
0
0
0
0
0
0
0
0
0
0
155
0
105

0
0
0
0
0
0
0
0
0
0
0
0
0
0
11
0

paper.
When we removed the blind from our data we found that:f DB = −14.6 ± 22.8stat + 6.9syst µHz

We divide this result by −2Eeff(3.42)

sgn(gF )µHz≃ 1.11 × 1031
and we find a value for the eEDM:
h
e cm



de = 1.3 ± 2.0 ± 0.6 × 10−30 e cm

(3.43)

The combined statistical and systematic uncertainty σde = 2.1×10−30 e cm improves on the ACME
collaboration’s result, which was previously the world leading measurement, by a factor of ∼ 2.

Total systematic
Statistical
Total

Magnetic
Non-reversing B⃗0
Second harmonic of Erot and transverse magnetic field
Third harmonic of Erot and magnetic field gradients
Higher harmonics of Erot and higher-order magnetic field gradients
Ellipticity of Erot and magnetic field gradients
Berry’s phase
Phase modulation due to axial secular motion
Axial 2nd harmonic with ellipticity of Erot
Higher harmonics of Erot
Rotation induced mixing
Leaking of f BR
Axial magnetic fields
Other frequency
Imperfect overlap of spatial distributions
Phase
Improperly characterized imaging contrast
Swatch position

Effect

0.1

0.7
1.2

VII B 1
VII B 2

6.9
22.8
23.8

3.5

VI D 2

0.1

0.1
0.2

VI C 1
VI C 2

< 0.1
2.2
1.5
< 0.1
1.7

Uncertainty (µHz)

3.4
1.7
3.0

0.1

Correction (µHz)

VI B 3
VI B 4
VI B 5

VI A 1
VI A 3
VI A 4
VI A 4
VI A 5

SectionTable 3.8: Uncertainty Budget

108

109
We can translate our result to an upper bound on de with 90% confidence:|de | < 4.1 × 10−30 e cm

(3.44)Per Equation 1.6, this result rules out the existence of certain beyond the Standard Model particles
with masses up to 40 TeV, well beyond the direct reach of the Large Hadron Collider [30].

Chapter 4

Sign of the G-Factor“Absolute sign determinations are extremely difficult to perform; there are too many ways one canoverlook one minus sign out of the many which may be present.”- Larry A. Cohen, John H. Martin, and Norman F. Ramsey. “Signs of rotational g factors.” PhysicalReview A 19.2 (1979): 433.

4.1

Introduction
The imbalance between matter and anti-matter in the universe can only be explained if the

combined charge and parity (CP) symmetry is violated [24]. While the Standard Model does
violate CP symmetry in the quark sector, it is not enough to explain the predominance of matterover antimatter [58]. Many extensions of the Standard Model incorporate novel CP violationthat would induce a non-zero, and potentially measurable, electric dipole moment of the electron(eEDM) [29, 59, 60].
We recently made the most precise measurement of the eEDM to date using electrons confined
inside HfF+ molecular ions, subjected to a huge intra-molecular electric field, and evolving coherently for up to three seconds [27]. Like the other leading eEDM measurement made by the ACME
collaboration, we performed our measurement in the 3 ∆1 electronic state of a diatomic molecule, astate which is sensitive to the eEDM but relatively immune to magnetic effects [28]. The magnetic

111
g-factor gF is suppressed in 3 ∆1 states because two units of orbital angular momentum with g = 1
point opposite to one unit of spin angular momentum with g ≈ 2.002.
The small g-factor of 3 ∆1 is a powerful tool to suppress systematic errors in our measurements[31,61]. However, its small size makes the sign of gF difficult to predict, especially in HfF+ where themagnetic moment of the fluorine nucleus is comparable in size to the residual electronic magnetic
moment. It is also a tricky parameter to extract experimentally, as only absolute magnitudes ofenergy differences are directly observable in standard Ramsey precession experiments.1
In both experiments, the sign of gF is critical because it is necessary to find the sign of theeEDM. In the HfF+ experiment the eEDM shows up asde =−sgn(gF )hf DB
2Eeff

(4.1)

where de is the eEDM, f DB is a particular linear combination of frequencies that we measured (seeSubsection 4.3.2) and Eeff = +23 GV/cm is the effective electric field of HfF+ [36, 42, 43, 45].2
Once an experiment finds a non-zero value of de , knowing its sign will be crucial to comparewith other measurements to confirm the result. At JILA, we plan to make our next generation
measurement of the eEDM in the 3 ∆1 state of ThF+ molecules, while the ACME collaboration’s
upcoming measurement continues to use the 3 ∆1 state of ThO [62]. In order to accurately determine
1

Suppose you can apply pi/2 pulses around any axis on the Bloch sphere with states |a⟩ and |b⟩, with the initialquantum state |ψ(t = 0)⟩ = |a⟩. Consider an experiment where you first apply a pi/2 pulse to rotate about the +x
axis, wait for the state to pick up a phase shift of π/2, and then apply another pi/2 pulse around the +y axis. The
population will end up in |a⟩ if Ea > Eb or in |b⟩ if Ea < Eb . However, it is not clear that you can determine thedirection of the axis of rotation if you do not already know which state has more energy than the other. In any case,when we apply pi/2 pulses by reducing the magnitude of the rotating electric field, we cannot control the rotation
axis in the Bloch sphere.
2

See Appendix 4.6.2 for a discussion of the sign of Eeff .

112
the sign of our result, this chapter will detail how we experimentally determined the sign of gF of
the 3 ∆1 state in HfF+ . This method can be used in the near future to do the same for ThF+ .

4.2

Possible Methods of Measuring the Sign of gF
Determining the magnitude of a 3 ∆1 state’s g-factor can be done by measuring the magnitudeof the energy difference between ±MF states in the presence of a known magnetic field. That alone
will not provide information about the g-factor’s sign. A generic technique that can work for 3 ∆1states is microwave spectroscopy between neighboring rotational levels that are typically spaced
tens of GHz apart. Consider the J = 1 and J = 2 rotational levels of a 3 ∆1 state with nuclear spinI=1
as is the case in HfF+ and ThF+ and shown in Figure 4.1. The diagonal Stark and Zeeman
2shifts of the fully stretched states in a given rotational level are given by:

1
1
EStark+Zeeman (Ω, J, I = , F = J + , MF = ±F )22
= − D∥ Ez Ω + (3gF − gNmeMF)µB Bzmp
F (F + 0.5)

(4.2)

In this equation D∥ is the effective molecular dipole moment, gF is the g-factor of the 3 ∆1state, gN ≈ 5.25 is the magnetic moment of the fluorine nucleus, and me and mp are the massesof the electron and proton. Ez and Bz are the components of the electric and magnetic fields
respectively along the quantization axis that give rise to the diagonal shifts, which in our experimentare Erot and Brot . If we perform microwave spectroscopy between two stretched states of J = 1 andJ = 2 as shown in Figure 4.1, in addition to the energy offset due to the rotational and hyperfine
splittings, we find an energy difference of:113

Figure 4.1: A schematic of the J = 1 and J = 2 rotational energy levels of the 3 ∆1 state with
I = 0.5 in the presence of an electric field. This is in the limit where the hyperfine structure islarge compared to the Stark splitting, which is large compared to the omega doublet splitting. The
red lines show transitions between the lower (L) and upper (U) stretched states. Measuring eithertransition in the presence of a magnetic field can determine the sign of the state’s g-factor. This
image has been reproduced from [4].

114

1
5
5
1
3
3
E(Ω = −1, J = 2, I = , F = ,MF = − ) − E(Ω = −1, J = 1, I = , F = , MF = − )
2
2
2
2
2
2
µN
(3gF − gN)µB Bz − D∥ EzµB=+ Constant(4.3)
6
The above equation gives the energy difference of the transition labeled (L) in Figure 4.1. Fora fixed value of Ez , the energy difference becomes a constant plus two terms - one that’s proportional
to the sign of gF and another that depends on very accurately known constants. By measuring thistransition frequency with positive and negative values of Bz , the offset can be removed and the signof gF determined. A challenge of this method is that one needs to know which fully stretched MF
state is initially prepared. If the initial state is prepared via optical pumping, the absolute sign ofthe optical pumping laser’s helicity must be known.
While this method could be used in a number of instances, it is not feasible in our apparatus.
In order to isolate the sign of gF , the magnitude of the Zeeman shift must be larger than the noisein the shift due to the electric field. In the part of our dataset where we applied 151 Hz frequency
shifts, the measured frequencies had a 1σ spread of 143 mHz that can be attributed to noise in
Erot . This part in 1000, or 60 mV/cm, electric field noise together with D∥ = 1.97 MHz / (V/cm)
gives about 20 kHz of noise in Equation 4.3. In order for the Zeeman shift to have a comparable
magnitude, we would need a reversible Brot ∼ 10 Gauss, which is roughly three orders of magnitudelarger than we typically apply.
This method could be implemented at the ACME experiment where they apply static electricand magnetic fields that do not rotate and therefore can be more stable. With a reasonably large∼ 100 G magnetic field3 , they would expect to see a ±1.4/6 MHz ≈ ±230 kHz shift. This would
require electric field stability at the V/cm level or better to see this signal and determine the signof their g-factor, which seems attainable.4
31000 Amp*turns and 10 cm radius coils gives a 90 G field at the center of Helmholtz coils

4

Note that ACME’s molecule, ThO, does not have any hyperfine structure, simplifying Equation 4.3 as gN = 0.

115
Alternatively, the sign of the g-factor can be measured via Ramsey spectroscopy between any
two states that are separated in energy by both a magnetic and non-magnetic effect like Berry’s
phase. For example, suppose two states ±MF are separated by a Zeeman shift of ∼ −sgn(gF )100
Hz. If a non-magnetic then effect lifts the +MF state 10 Hz above the −MF state, the sign of theg-factor can be determined by observing if the magnitude of the shift becomes 110 or 90 Hz.
One downside of this approach is that not many experiments are set up to apply readilymeasurable Berry’s phase shifts. Fortunately, performing our spectroscopy in a bias field Erot thatrotates at 100s of kHz allows us to quite easily apply large Berry’s phase shifts. It can be difficult to
get the sign of a Berry’s phase shift correct, which is crucial for this measurement, so it is re-derived
in Appendix 4.6.1. In Section 4.3 we will introduce the JILA eEDM experiment. In Section 4.4 wewill explain the results of Berry’s phase measurements that reveal the sign of gF in HfF+ .

4.3

The JILA eEDM Experiment

4.3.1

Experimental Overview

We trap ∼ 20, 000 HfF+ molecular ions in a two-dimensional Paul trap. We populate thestretched states of the lowest rovibrational level of the 3 ∆1 electronic state in the presence of arotating bias field E⃗rot with magnitude Erot ≈ 60 V/cm, see Figure 4.2. E⃗rot rotates in the xy-planewith frequency frot = 375 kHz, causing the ions to move circularly with radius rrot ≈ 0.5 mm. E⃗rot
also polarizes the molecules and defines the quantization axis for our spectroscopy. A magnetic field
causes a Zeeman shift between the stretched MF = ±1.5 states with an absolute value of ∼ 100
Hz, which we measure via Ramsey spectroscopy. For details on how we used HfF+ in this platform
to measure the eEDM, see [2, 27, 31] and Chapter 2 of this thesis.

4.3.2

Experimental Switches and Frequency Channels

As is common in precision measurements, we greatly benefit from experimental “switches”.
By measuring Ramsey fringes in different configurations we can isolate the eEDM induced shift

116

Figure 4.2: Top: The level structure of 3 ∆1 v = 0, J = 1 state in HfF+ in an external electric field
Erot ∼ 58 V/cm. The energy levels are similar to the 3 ∆1 , v = 0, J = 1 state in ThF+ . Grey lines
1
correspond to states which asymptote to F =as Erot tends toward zero, and all other states
2
3
asymptote to F = . Solid (dashed) lines correspond to states with Ω = +1 (-1), see Appendix
2
3
4.6.2. The upper (orange) and lower (blue) doublets of the F = state are split by ∼ 100 MHz by
2
3
Erot . We can split the MF = ± states with a Zeeman shift and/or a Berry’s phase, typically on
2
the order of ∼ 100 Hz. If the eEDM is nonzero it will cause an energy splitting in the upper andlower doublets of the same size but opposite sign. Bottom: We simultaneously perform Ramsey
measurements within each doublet to measure the ∼ 100 Hz energy difference. The points are fromexample HfF+ Ramsey fringes; the curves are fits. This image has been reproduced from [31].

117from much larger spurious effects. Our recent measurement had five switches, but only three thatare relevant for determining the sign of the eEDM and the g-factor: D̃ = ±1, whether we take datain the upper or lower doublet of 3 ∆1 as shown in Figure 4.2; B̃ = ±1, whether the magnetic field
is parallel or anti-parallel to E⃗rot ; R̃ = ±1, whether the rotation of the electric field points up or
down via the right hand rule. For a description of the other switch states, see [2, 31].
Combining these three switches we make 23 = 8 Ramsey measurements of the energy difference between the stretch states shown in Figure 4.2. We fit the Ramsey fringes to sine waves, andwe label the eight fit frequencies as f D̃=±1,B̃=±1,R̃=±1 .
The ∼ 100 Hz energy difference that we measure is caused by the Zeeman shift between
MF = ±1.5 states and therefore changes sign when we reverse B̃. We do not know this energy
difference’s absolute sign unless we know sgn(gF ). We arbitrarily constrain all of the fit frequencies
to be positive. We then construct eight frequency channels:



f


1



 D 

 f 
1






 fB 
1






 DB 
 1 1
f



= 

 R  8
 f 
1






 f DR 
1






 BR 


f
1






f DBR
1


1
−1

1

1

1 −1

1 −1 −1
−1 −1
1
−1

1

1

1

1

1

1 −1

1

1

1 −1

1 −1 −1

1 −1 −1 −1

1 −1 −1

1 −1

1 −1 −1 −1 −1
−1 −1

1 −1

1

1
1



1 f +1,+1,+1 




−1 f −1,+1,+1 







+1,−1,+1−1 f  −1,−1,+1 


1

 f




−1 f +1,+1,−1 







−1,+1,−1
1 f



  +1,−1,−1 


1

 f


f −1,−1,−1
−1

(4.4)

The presence of D or R in the superscript of a frequency channel indicates that it is thedifference between the upper and lower doublet or between the two rotation directions. The absenceof superscript D or R conversely indicates that it is the average of the doublets or rotation directions.
However, because B̃ determines the sign of the energy difference and we have already taken the

118absolute value, the inclusion or absence of B has the opposite meaning. For example, f BR is theaverage of the B̃ and D̃ switches, but the difference between rotation up and rotation down.
As derived in Appendix 4.6.2, f DB , the difference frequency between the upper and lower
doublet and average of the other switches, tells us the value of the eEDM. It is clear from Equation
4.1 that in order to extract the sign of de we need to know the sign of gF . Intuitively, this is forthe same reason that we need to know the sign of the Berry’s phase effect to determine the sign of
gF - because we only measure the magnitude of the splitting between MF = ±3states, we need to2know the sign of the Zeeman effect to know the sign of the eEDM.

4.3.3

Berry’s Phase in JILA eEDM

During the Ramsey spectroscopy, the quantization axis defined by E⃗rot completes on the orderof 105 rotations in the xy-plane per second. As E⃗rot traces out closed loops, the quantum state
accumulates a geometric phase of:ϕgeo = MF Ωs(4.5)

During our spectroscopy the molecules are in a superposition of MF = ±1.5 states, so themolecules will accumulate a relative phase shift of 3Ωs . Here Ωs is the solid angle traced out by
the electric field, as shown in Figure 4.3, which is a signed quantity. If the electric field rotates
perfectly in the xy-plane, Ωs = ∓2π for R̃ = ±1. In this ideal case the difference of ϕgeo between
the two MF states is ∓6π, meaning that the Berry’s phase is not observable as it is an integer
number of 2π phase shifts. But if we intentionally tilt the electric field out of the xy-plane by asmall angle α, we induce a measurable phase shift that will be accumulated every Trot = 1/frot .
For small tilts, this will appear as a frequency shift for each MF state given by:

119

Figure 4.3: Ideally, the quantization axis defined by E⃗rot rotates perfectly in the xy-plane and traces
out a solid angle of −2π R̃. If E⃗rot is tilted up by a small angle α, it will deviate from the ideal solid
angle by an amount Ω. This image has been reproduced from [31].

fBerry ≈ −MF frot
2π

Z Trotα(t)ϕ̇(t)dt

(4.6)

0

Here α(t) is the tilt of the quantization axis out of the xy-plane and ϕ̇(t) is the azimuthal
angular velocity. If neither parameter depends on time, we find that:

fBerry = −MF frot αR̃

(4.7)

In order to measure the sign of the g-factor, we cannot simply apply a constant tilt to E⃗rot asshown in Figure 4.3 since the average force on our trapped ions must equal zero.5 Attempting totilt E⃗rot upwards would simply lift the center of the ion trap. However, we can generate a Berry’s
phase by applying an electric field along the z-axis that time averages to zero. A combinationof fields that gives a nonzero Berry’s phase is an electric field along z that oscillates at twice therotation frequency along with an elliptical rotating field:
5

There is always a small electric field pointing up to balance the force of gravity. This slightly tilts E⃗rot andgenerates a small Berry’s phase that we cannot tune, which is not particularly useful for measuring the sign of gF .
For further discussion of the effect of gravity on Berry’s phase, see [31].

120





 Erot cos (ωrot t + ϕR + R̃ϕ0 ) + Eϵ cos (2θR̃ − ωrot t − ϕR − R̃ϕ0 ) 



⃗ =
E(t)
Erot R̃ sin (ωrot t + ϕR + R̃ϕ0 ) + Eϵ R̃ sin (2θR̃ − ωrot t − ϕR − R̃ϕ0 )
E2hz cos (2ωrot t + ϕ2f )(4.8)

Eϵ is the magnitude of the ellipticity and E2hz is the magnitude second harmonic that oscillates
at 2ωrot along the z axis. Note that both magnitudes are defined to be positive. At t = 0, E⃗rot
points in the xy-plane at an angle ϕ0 + R̃ϕR from the positive x-axis. The long axis of the ellipticity
points at an angle θ in the xy-plane and the second harmonic has a phase of ϕ2f . By applying thiscombination of fields, we see in Appendix 4.6.3 that we generate a Berry’s phase that will appear
as a frequency shift in f B and f BR and depend on the sign of gF .

fB =9E2hz Eϵ frotsgn(gF ) sin (2θ − 2ϕ0 ) sin (ϕ2f − 2ϕR )
2
2Erot

(4.9)

9E2hz Eϵ frotsgn(gF ) cos (2θ − 2ϕ0 ) cos (ϕ2f − 2ϕR )2
2Erot(4.10)

f BR = −In order to measure the sign of the g-factor, we need to apply these fields with known phases
and see how f B and f BR change as we scan the phase of the second harmonic for a fixed valueof the ellipticity.6 We can then compare the two observed sine waves to the above equations forsgn(gF ) = ±1 to see which value is correct for each of our molecules.

4.4

Results
As discussed in Appendix 4.6.4, we take all of our data with ϕ0 =π 77π
and ϕR =, so the4
12

above equations simplify to:6

Alternatively, we could fix ϕ2f and scan θ, but it is easier to scan ϕ2f experimentally.

7

These seemingly arbitrary choices of ϕ0 and ϕR ensure that at the time of dissociation the molecules are oriented

to optimize our imaging contrast.

121

fB =

9E2hz Eϵ frotπsgn(gF ) cos (2θ) sin (ϕ2f − )
2
6
2Erot

(4.11)

f BR =

9E2hz Eϵ frotπsgn(gF ) sin (2θ) sin (ϕ2f + )
2
3
2Erot

(4.12)We took data with our usual values of Erot = 58 V/cm and frot = 375 kHz, and appliedEϵ ∼ 400 mV/cm at an angle of θ =π. We also applied a second harmonic along the z-axis
8

E2hz ∼ 40 mV/cm and scanned its phase ϕ2f . We therefore expect the data, which are shown in
Figure 4.4, to behave as follows:fB =9E2hz Eϵ frotπ
√ 2 sgn(gF ) sin (ϕ2f − )6
2 2Erot(4.13)

f BR =

9E2hz Eϵ frotπ
√ 2 sgn(gF ) sin (ϕ2f + )3
2 2Erot(4.14)We see that the data conform to a sine waves with amplitudes similar to those predicted by
Equations 4.13 and 4.14. The phase of the sine waves predicted by these equations depend on thesign of the g-factor, and it is obvious from both frequency channels in Figure 4.4 that sgn(gF ) = −1.
We can confirm our assignment by considering data where the ellipticity is rotated by 90
degrees, so θ = −π
as shown in Figure 4.5. While there is a slight phase offset between the8predicted and measured values of f B and f BR on the order of a few degrees, that is explained by
the imprecision in determining ϕ0 and ϕR as explained in Appendix 4.6.4. It is again clear thatsgn(gF ) = −1 fits the data.
The data in Figures 4.4 and 4.5 are conclusive, but the result is obscured by a haze of math.
Instead of splitting the data into frequency channels as described in Equation 4.4, we can focus

122

(a)

(b)

π
Figure 4.4: f B (a) and f BR (b) data taken with applied ellipticity at angle θ =and different
8phases of the second harmonic along the z-axis. The data points in both graphs have error bars on
the order of 10 mHz that are too small to resolve at this scale. Equations 4.13 (a) and 4.14 (b) areplotted with both signs of gF , there is no fitting. Both datasets clearly show that sgn(gF ) = −1.

on the absolute magnitude of the energy differences measured directly. Consider the eight energy
differences when B̃ = ±1, R̃ = ±1, and when the time-average of α, the angle of the total electricfield above the xy-plane, is positive or negative. To simplify the notation, we introduce the notationα̃ = ±1 for these two cases respectively.
We have α̃ = +1 when E⃗2hz points downward when the elliptical electric field points along its

123π
Figure 4.5: f B and f BR data taken with applied ellipticity at angle θ = − and different phases
8of the second harmonic along the z-axis. The data points in both graphs have error bars on theorder of 10 mHz that are too small to resolve at this scale. Equations 4.13 (a) and 4.14 (b) areπplotted with sgn(gF ) = −1, which we found from the data with θ = shown in Figure 4.4. This8
dataset also shows that sgn(gF ) = −1.

major axis, forming a small downward angle. When the elliptical E⃗rot rotates 90 degrees, E⃗2hz will
point upward as the E⃗rot points along its minor axis, forming a larger upward angle. Therefore thetime-averaged α is positive in this case.8
The sign of the Zeeman shift is −sgn(gF )B̃,9 while the sign of the Berry’s phase shift is −α̃R̃,as given by Equation 4.7.10 When the Zeeman and Berry’s phase shifts have the same sign, theabsolute energy difference will increase. Therefore, the entries in Figure 4.6 will be larger/smaller(∼ 159 or ∼ 143 Hz) when −sgn(gF )B̃ × −α̃R̃ = sgn(gF )α̃B̃ R̃ = ±1. The top left entry in Figure
4.6 is one of the smaller entries, with α̃B̃ R̃ = (+1)(+1)(+1) = +1, so it must be that sgn(gF ) = −1.
Checking any of the other seven entries gives the same result, all without the potential confusion
8

For our data in Figure 4.6, we use data where ϕ2f is 15 degrees away from the value where α is at its maximal or

minimal value. This marginally reduces the magnitude of the Berry’s phase shift, but does not change the analysis.
3
3
9
We assume that energy shifts are positive when the MF = state has more energy than the MF = − state,2
2
as we do elsewhere.
10

While in this case α and ϕ̇ depend on time so we have to use the more general Equation 4.6, Equation 4.7 gives

the correct sign if we replace α with the time-averaged α. This helps clarify the underlying physics.

124

Figure 4.6: Absolute energy differences measured with a Zeeman shift plus Berry’s phase, as afunction of B̃, R̃ and α̃. Blue frequencies in the top left of each box have B̃ = +1, red frequencies
in the bottom right have B̃ = −1. The ∼ 8 Hz Berry’s phase shift has the same/opposite sign as
the ∼ 151 Hz Zeeman shift when sgn(gF )α̃B̃ R̃ = ±1, giving absolute energy differences of 151 + 8= 159 Hz or 151 - 8 = 143 Hz. All eight of these entries imply that sgn(gF ) = −1.

of defining f B and f BR . This method still relies on knowing the rotation direction, magnetic field
direction and relative orientation of electric fields of each of our fringes, but it is hopefully moreintuitive.
We can now plug in the sign of gF into Equation 4.1 and find a sign-sensitive relationship in
our HfF+ experiment:

de =

hf DB
2Eeff

(4.15)h is positive and Eeff = +23 GV/cm in our molecule, so de has the same sign as f DB .
It is important that the signed quantities have the same signs in our code, which is addressedin Appendix 4.6.5. For a full list of the quantities whose sign we need to define correctly in order
to arrive at these two equations, see Appendix 4.6.6. Given that the sign of gF is negative, we

125
can also confirm the calculation from second order perturbation theory thatδgF
is positive. See
gFAppendix 4.6.7 for that discussion.

4.5

Conclusion
By applying a Zeeman shift in addition to an energy difference with a known sign between

the ±MF states via Berry’s phase, we were able to determine that the sign of the g-factor in the3∆+
1 state of HfF is negative. As Norman Ramsey said, there are many places to go wrong when

determining absolute energy differences, but we are confident in our assignment. This method will
be repeated for the 3 ∆1 state of ThF+ , and can be checked by anyone who has enough time on
their hands to inspect this thesis chapter diligently.

4.6

Appendices for Sign of the G Factor

4.6.1

Appendix 1: Berry’s Phase Derived

Our method of determining the sign of gF relies on us applying an energy shift of known
sign between the MF = ±3states via Berry’s phase. Simply using an equation like Equation 4.7
2

is tempting but possibly misleading because the sign of the solid angle can depend on convention.
To confirm the sign of the Berry’s phase shift, we will derive Berry’s phase from purely quantum
mechanical arguments. This treatment will largely follow this paper [63] by Meyer, Leanhardt,
Cornell and Bohn.
Consider a spin-1
system11 in the presence of a magnetic field with the time-dependant
2

Hamiltonian:⃗ = − gµB B ⃗σ · B̂(t) = ℏ ωL⃗σ · B̂(t)H(t) = −⃗µm · B(t)22
11

The authors of [63] treat both a spin-same in both cases.

(4.16)

1and a spin-j system, and they show that the sign of the effect is the2

126
We are assuming that the magnitude of the magnetic field does not change, but its direction
varies. This Hamiltonian introduces the Larmor frequency ωL = −gµB B. We define the lab-frameℏCartesian coordinates (ξ, η, ζ), and allow the magnetic field to rotate around the ζ axis at a constantangle θr . In a time t the magnetic field rotates an angle ϕ = ωr tR̃ in the lab frame, where ωr is an
unsigned (always positive) quantity defined by ω
⃗ r = R̃ωr ζ̂. The rotating magnetic field defines therotating-frame coordinates (x, y, z) as shown in Figure 4.7.

Figure 4.7: Image and caption are from [63].

We can write the magnetic field in the lab frame as:⃗ = (Bξ (t), Bη (t), Bζ (t)) = B(sin θr cos ωr t, R̃ sin θr sin ωr t, cos θr )B(t)(4.17)

We can find an explicit representation of the Hamiltonian in Equation 4.16 in the lab-frame byplugging in the magnetic field as written in Equation 4.17 and using the explicit forms of the Pauli1matrices. The Hamiltonian acts on the lab-frame basis vectors |mζ = ± ⟩, and is written as:2H(t) =ℏωL  cos θr2sin θr eiR̃ωr t



sin θr e−iR̃ωr t − cos θr

(4.18)The goal now is to find the eigenvalues of this Hamiltonian. We’ll start by writing a general

127
ansatz for the wave function in this basis:−iω+ tα(t)e|ψ(t)⟩ = β(t)e−iω− t(4.19)

ω± are the currently unspecified frequencies of the eigenvectors, which we’ll solve for shortly. We
can now plug this ansatz and the Hamiltonian in Equation 4.18 into the time-dependent Schrodinger
equation:


iℏ

∂ 

∂t

α(t)e−iω+ t
β(t)e





 ℏωL =2−iω− tcos θrsin θr e−iR̃ωr t  α(t)e−iω+ t 

sin θr eiR̃ωr t



− cos θrβ(t)e−iω− t

(4.20)

After a few lines of algebra we find that:


iℏ





ωL
cos θr − ω+2∂ α(t) = ℏ
ωL∂tβ(t)sin θr ei(R̃ωr +ω− −ω+ )t
2



ωL
sin θr e−i(R̃ωr +ω− −ω+ )t  α(t)
2


ωL
−
cos θr − ω−β(t)2(4.21)

This is progress because we now have something resembling the Schrodinger equation, although
the vectors and Hamiltonian both depend on time. We can simplify this if we chose ω± = ±R̃ωr,2

as this makes the Hamiltonian time-independent:





ωL
ωr
∂ α(t)
 2 cos θr − R̃ 2iℏ 
 = ℏ
ωL∂tβ(t)sin θr2

ωL
sin θr2 α(t)


ωL
ωr
−
cos θr + R̃
β(t)
2
2

(4.22)

Recall that this is in the lab-frame basis. It’s also clear that we can split this Hamiltonian
into Zeeman and rotating parts:

Hdressed,lab = HZ,lab + Hr,lab(4.23)

128



cos
θ
sin
θ
r
r 
ℏωL 
HZ,lab =2sin θr − cos θr

(4.24)ℏωr +1 0 1 0
Hr,lab = −R̃ = −R̃ℏωr mζ 

2
0 −1
0 1

(4.25)

Now that we have a Hamiltonian, it’s tempting to read off the energy shift due to the partof the Hamiltonian that arises from rotation.12 However, we do our spectroscopy in the rotating
frame so we need to change the Hamiltonian’s basis. We can immediately write the Zeeman shift
in the rotating basis, even though it’s not particularly interesting for us at the moment:ℏωL 1 0 
HZ,rot =


2
0 −1

(4.26)

1
This Hamiltonian acts on basis vectors |mz = ± ⟩. To write Hr in the rotating frame, we
2need to transform from the |smζ ⟩ basis to the |smz ⟩ basis. We can do this by rotating through the
Euler angles (0, θr , 0): 13

"



0
R(0, θr , 0) = e−iθR Jη = Exp − iθR 
i
2


 
 

θr
θr
i #
cos
−
sin
− 


2
2 =   
 2 

θr 
θr
0
cos
sin
2
2

(4.27)

We can then write:
12

It’s worth noting that this Hamiltonian is exact, there is no assumption that ωr ≪ ωL as is usually the casewhen discussing Berry’s phase geometrically.
13[63] uses Wigner matrices instead of the generators of rotations. I believe the approach I have taken here iseasier to follow.

129   
 
  
θr
θr
−
sin
cos
 +ζ 
+z 
+ζ  2 
 2    = R  = 
 

θr
θr 
−z
−ζ
sin
cos
−ζ
2
2




(4.28)

Note that because the matrix R is invertible, we can find the Hamiltonian in the rotating
frame:


Hr,rot = RHr,lab R−1 = R̃ℏωr − cos θr − sin θr 2
− sin θr cos θr(4.29)

We can read off the diagonal Berry’s phase shift as −R̃mF ωr cos θr . We are interested in evaluating
the Berry’s phase for θr =π
− α, where α is a small angle. When α is positive it denotes a small2

angle above the xy-plane. In this case we have:ωBerry = −R̃mF ωr cos (π− α) ≈ −R̃mF ωr α
2

In the last step we used the first term from the following Taylor expansion: cos ((4.30)π− α) = α −2

α3 /6 + α5 /120 + .... Since α is small we can keep just the first term, and this justifies the overall
sign in Equation 4.7.

4.6.2

Appendix 2: Relating de to f DB

We define the internuclear axis n̂ to point from the lighter fluorine towards the heavier nucleus
as shown in Figure 4.8.14 The electronic angular momentum projections Λ, Σ and Ω = Λ + Σ arepositive if they point along n̂. In the 3 ∆1 state where |Λ| = 2, |Σ| = 1 and |Ω| = 1, Λ and Σ must
point in the opposite directions. This gives rise to the Ω = ±1 states shown in Figure 4.8.
14This is a common convention, but it is not universal.

130

Figure 4.8: The two possible configurations of Quantum numbers in 3 ∆1 . Note that the (Hf nucleus+ core of electrons) has a positive charge and the (F nucleus + core of electrons) has a negative
charge. The directions of the molecular dipole moment and effective electric field therefore follow
from the above definitions.

We can now justify the assignment of Ω = ±1 in the level diagram shown in Figure 4.2. In
the absence of an external electric field, the states with well defined Ω shown in Figure 4.8 arefully mixed and the eigenstates have well defined parity. In the presence of E⃗rot , which defines ourquantization axis, |3 ∆1 , Ω, J, F, MF ⟩ states are well defined. F is the total angular momentum andis always positive, while MF is positive if it points along E⃗rot , the quantization axis.
3 3Consider the particular rotational ground state (R = 0 so J = Ω + R = Ω) |3 ∆1 , 1, 1, , ⟩.
2 2
1
Since F = J + I = Ω + we know that MF points along the same direction as Ω. We also know
2that Ω points along n̂ and that MF points along Êrot , so it must be that n̂ ∥ Êrot . As we can see
in Figure 4.8, n̂ is parallel to the molecular dipole moment d⃗mf so we have Êrot ∥ d⃗mf . This means
3 3
the |3 ∆1 , 1, 1, , ⟩ state has a negative Stark shift EStark = −d⃗mf · E⃗rot = −dmf Erot and we can
2 2
identify it as part of the lower doublet, as shown in Figure 4.2. We can repeat a similar process to
identify the value of Ω in the remainder of the states in the upper and lower doublets.
This procedure is important because the shift due to the eEDM depends on Ω. The shift isgiven by:

⟨Hedm ⟩ = −de Eeff Ω(4.31)

131
This relationship is given by [36]15 , but also intuitively makes sense if we naively imagine that there
is an actual electric field pointing from the positively charged hafnium nucleus to the negatively
charged fluorine nucleus as shown in Figure 4.8. We define the electron’s EDM de to be positiveif d⃗e ∥ ⃗s. In our molecule that means dˆe ∥ Σ̂ ∥ −Ω̂. We should expect there to be an energy shift⟨Hedm ⟩ = −d⃗e · E⃗eff = −de Eeff (−Ω̂) · (−n̂) = −de Eeff Ω as given by [36].
The above argument is appealing and lands at the right answer, but only fortuitously. Drawing an effective electric field from one nucleus to another in Figure 4.8 is useful but misleading.
The energy shift due to the eEDM only occurs when one of HfF+ ’s valence electrons is very close
to the hafnium nucleus, where the electric field points radially outward and causes the electron tomove at relativistic speeds [64,65]. Instead, the proper treatment is to calculate the magnitude andsign of the effective electric field, as is done for HfF+ in the following papers [36, 42, 43, 45].
While theorists are consistent in determining the magnitude of Eeff for HfF+ , approximately
23 GV/cm, they are maddeningly unclear when it comes to the sign. For example, one author has
published that our Eeff is both positive and negative [43,45]. The confusion only gets worse when we
look at review articles which compare the signs of Eeff between different molecules. We can compare
the signs of Eeff for HfF+ , ThO and YbF, the molecules employed by the three currently leading
eEDM experiments. While an overall sign change between review articles could be explained bydifferent conventions, relative sign changes abound. In one article the signs of Eeff in all threemolecules are the same [35], in another YbF has a different sign than the others [56], and in a third
HfF+ is the odd molecule out [17]. This spans the space of possibilities!
I have adopted the convention of Alexander Petrov who reports that Eeff = +23 GV/cm in
HfF+ [36]. His article is relatively clear when it comes to the sign conventions and explicitly gives
Equation 4.31 which we can consistently apply to see how the eEDM shows up in our experiment.
15

Note that I have written the above equation with a minus sign that does not appear in [36] because we define

the internuclear axis n̂ to point in opposite directions and the sign of Ω depends on n̂.

132This has the perk of agreeing with John Bohn’s sign convention, and he’s on my committee [35,66].
We measure the eEDM induced shift in the presence of a magnetic field while minimizing
Berry’s phase, which can contribute systematic errors to our measurement [31]. The magnetic fieldinteracts with our molecule’s magnetic dipole moment µ⃗ F , which we define such that µF = gF µB >0 when µ⃗ F is parallel to the molecule’s total angular momentum F . Note that the Bohr magnetonµB is defined to be a positive quantity, while the g-factor can be positive or negative. As thequantization axis in our experiment is defined by E⃗rot , we have µ⃗ F = gF µB MF Êrot .
⃗rot that is parallel or antiparallel (B̃ = ±1) to E⃗rot which causes
We apply a magnetic field B
a Zeeman shift of:⃗rot = −gF µB MF Êrot · B⃗rot = −gF µB MF Brot B̃
HZeeman = −⃗µF · B(4.32)

We can now create an effective Hamiltonian for the upper and lower doublets using Equations 4.31
and 4.32:Dh(∆ + ∆ D̃)1 −3(gF + δgF D̃)µB Brot B̃ + 2de Eeff D̃
Heff = 2Dh(∆ + ∆ D̃)3(gF + δgF D̃)µB Brot B̃ − 2de Eeff D̃(4.33)In addition to the expected Zeeman and eEDM shifts, we have also included a D̃ odd Zeeman
shift to account for the fact that the upper and lower doublets have slightly different values of gF .
g upper − gFlower
We define gF to be the average g-factor of the two doublets, while δgF = F. We define2∆ and ∆D similarly to account for a D̃-odd
the states within the two doublets.
 mixing betweenMF = +3/2
This effective Hamiltonian acts on states .
MF = −3/2
If we diagonalize this Hamiltonian and solve for the energy difference, we find that:133[h(∆ + ∆D D̃)]2 B̃
+ 2de Eeff D̃
2 (gF + δgF D̃)µB Brot

EMF =1.5 − EMF =−1.5 = −3(gF + δgF D̃)µB Brot B̃ − 3(4.34)

While finding this energy difference we have dropped a term that is proportional to (de Eeff )2 andused a binomial approximation where we assumed that:

2de Eeff,(gF ± δgF )µB Broth(∆ ± ∆D )32 (gF ± δgF )µB Brot

2
≪1

(4.35)

Our new result tells us that the numerator of the first term is smaller than 40 µHz, while thedenominator is ∼ 100 Hz [27]. The numerator of the second term is ∼ 1 Hz, making the entire
term ∼ 10−4 . The second term is larger than the first, but is still much smaller than 1 so theapproximation is a good one.
As explained in Subsection 4.3.2, we always record a positive value of the energy. Since theleading term in Equation 4.34 is the largest, we take the absolute value of the energy difference bymultiplying by the sign of the leading term −sgn(gF )B̃:|∆E| = 3|gF |µB Brot + 3δgF µB Brot sgn(gF )D̃∆2 + D̃∆∆D + (∆D )2+ h2 sgn(gF ) 3− 2de Eeff sgn(gF )B̃ D̃
2 (gF + D̃δgF )µB Brot

(4.36)

We can immediately see that the only B̃ D̃-odd shift is caused by the eEDM. From Equation
4.4, we find thathf DB = −2de Eeff sgn(gF )This is equivalent to Equation 4.1, which is what we wanted to show.

(4.37)

134
4.6.3

Appendix 3: Calculating Berry’s Phase from 2H and Ellipticity

We generate a Berry’s phase by having E⃗rot trace out an elliptical Lissajous figure in xy-plane
and apply also a 2nd harmonic of Erot in the axial direction, as described in Equation 4.8. To
calculate the expected Berry’s phase shift, we need expressions for α(t) and ϕ̇(t) to compute theintegral in Equation 4.6:



E2hz
Eϵ
=
α(t) = qcos (2ωrot t + ϕ2f ) 1 −cos (2R̃(θ − ϕ0 ) − 2(ϕR + ωrot t))
Erot
Erot
E 2 (t) + E 2 (t)Ez (t)

xy(4.38)

ϕ̇(t) =∂
arctan∂tEy (t)Ex (t)

= R̃ωrot −2R̃ωrot Eϵcos (2R̃(θ − ϕ0 ) − 2(ϕR + ωrot t))
Erot(4.39)

In both equations we assume that Erot is large compared to the other electric fields. We can thenplug these expressions into Equation 4.6 and integrate:

fBerry =3E2hz Eϵ frot MF R̃
cos (2R̃(θ − ϕ0 ) + ϕ2f − 2ϕR )2
2Erot(4.40)

We can now construct an effective Hamiltonian like in Appendix 4.6.2, except this time wewill include the Berry’s phase shift and ignore all D̃-odd effects:



h∆
Diag
2 
Heff = 

h∆
−Diag
2

(4.41)The diagonal term is expressed below so everything fits nicely between the margins of this page:3
9E2hz Eϵ frot
Diag = − gF µB Brot B̃ + hcos (2R̃(θ − ϕ0 ) + ϕ2f − 2ϕR )R̃
2
2
4Erot

(4.42)

135
MF = +3/2
Again, the effective Hamiltonian acts on states . Solving for the difference in
MF = −3/2
the diagonalized energies, we get:|∆E| = 3|gF |µB Brot −9hE2hz Eϵ frot
h2 ∆ 2(4.43)
cos
(2
R̃(θ
−
ϕ
)
+
ϕ
−
2ϕ
)
B̃
R̃sgn(g
)
+
0
R
F
2f
2
6|gF |µB Brot
2Erot

Note that the the Berry’s phase shift is B̃-odd, but has a complicated R̃ dependence. This means
we should expect the Berry’s phase to show up in both f B and f BR . We can use Equation 4.4 to
find:

fB =

9E2hz Eϵ frotsgn(gF ) sin (2θ − 2ϕ0 ) sin (ϕ2f − 2ϕR )2
2Erot(4.44)

9E2hz Eϵ frotsgn(gF ) cos (2θ − 2ϕ0 ) cos (ϕ2f − 2ϕR )2
2Erot(4.45)

f BR = −4.6.4

Appendix 4: Erot Phases in HfF+ Experiment

It is easier to compare the above equations to our experiment if we replace ϕ0 and ϕR with
ϕup and ϕdown :ϕup = ϕ0 + ϕR

(4.46)

ϕdown = ϕ0 − ϕR

(4.47)

Substituting these into Equations 4.44 and 4.45 gives us:

136

fB =

9E2hz Eϵ frotsgn(gF ) sin (2θ − ϕup − ϕdown ) sin (ϕ2f − ϕup + ϕdown )2
2Erot(4.48)

9E2hz Eϵ frotsgn(gF ) cos (2θ − ϕup − ϕdown ) cos (ϕ2f − ϕup + ϕdown )2
2Erot(4.49)

f BR = −In order to use these equations to find sgn(gF ), we need to know all four phases in the above
equations. The x and y axes are defined as shown in Figure 4.9, with the z-axis pointing out of thepage. θ, the angle which defines the strong axis of the ellipse, is equal to zero when pointing along+x̂ and increases as it rotates toward the +ŷ axis. We define ϕup and ϕdown , the direction of Erotat t=0 for each rotation direction, the same way. ϕ2f is the phase of the second harmonic at t=0.

Figure 4.9: The eight radial electrodes are numbered 1 through 8. There are two “endcap” electrodes in and out of the page to apply fields along the z-axis.

The equations for f B and f BR do not depend on the choice of t=0, so we chose 1 ms after
the firstπpulse in our Ramsey sequence as it is a convenient time to trigger our oscilloscopes that2

measure the voltages on our electrodes. We can measure ϕup and ϕdown by simply measuring thevoltages on all eight radial electrodes and finding the direction Erot points at this time. We findthat ϕup =

10π
7π
π
11π
and ϕdown =to within 5 degrees, which means ϕ0 =and ϕR = .
6
6
4
12

137
When we apply the second harmonic field along the z-direction, we can control its phase in
our software. We find that when we program the computer to generate the field with “zero” phase,
1 ms after the firstπpulse the electric field is at its maximum pointing upwards. We also verified
2

that inputting a 30 degree phase shift into our software is equivalent to a +30 degree phase shift.
This means that the phase in our control software is equivalent to the phase in Equation 4.8.
Finally, we can verify that we know θ by observing the voltages that generate Erot on our eightradial electrodes. For example, if we apply an ellpiticity with θ = 0 we expect that the voltages onelectrodes 1, 4, 5 and 8 in Figure 4.9 will now have slightly larger amplitudes, while the amplitudeof the voltages on the remaining electrodes will decrease. This procedure went as expected.
We can therefore use the fixed values of ϕup and ϕdown and choose values of θ and ϕ2f togenerate a Berry’s phase. By choosing a value of θ and scanning ϕ2f , we should expect to see sine
waves in f B and f BR with phases that are completely known except for the sign of the g-factor.

4.6.5

Appendix 5: Defining the Frequency Channels

We need to know how our code constructs f B , f BR and f DB in order to accurately interpret
their signs. This involves understanding the Matlab code generate lcoms common.m which creates
the frequency channels from the raw data.
We take data through our homemade Labview code in 24 experimental configurations, by
taking every combination of magnetic field (B), rotation direction (R), imaging chop (I) and depletion phase (P). In each of these configurations, we take data for both the upper and lower doublet(D), giving us data for 25 = 32 total Ramsey fringes. As in Subsection 4.3.2, we are not interested
in the I or P chops in order to find the sign of the g-factor or eEDM. We can narrow our focus to
the 8 data sets made by switching B, R and D.
Labview saves the eight data sets with labels f D̃=±1,B̃=±1,R̃=±1 , but it mislabels the therotation switch. That is, data taken with E⃗rot rotating up are labeled R̃ = −1 and data taken with

138
E⃗rot rotating down are labeled R̃ = +1. This is a bit awkward, but as I will describe this error islater corrected in the data analysis.
Matlab imports the eight data sets, analyzes each one and fits a positive frequency for eachswitch state, as discussed in Subsection 4.3.2. Matlab then creates a doublet even and doublet odd
frequency four the four experimental configurations B̃, R̃ = ±1:

B̃=±1,R̃=±1
fM=
L

f D̃=+1,B̃=±1,R̃=±1 + f D̃=−1,B̃=±1,R̃=±1
2(4.50)

f D̃=+1,B̃=±1,R̃=±1 − f D̃=−1,B̃=±1,R̃=±1
2(4.51)

D,B̃=±1,R̃=±1
fM=
L

This is done when the data is processed in a subroutine called fit fringe common v6.m, see where
it defines the variables b mean and b diff. At this point, Matlab still has the sign of R̃ backwards
as it has inherited Labview’s error.
Matlab now constructs the four doublet even linear combinations, f f B , f R and f BR , out
B̃=±1,R̃=±1. It does the same for the four doublet odd linear combinations outof its four values fM
L
D,B̃=±1,R̃=±1, all of which is done in generate lcoms common.m. If inspectedof the four values fM
Lclosely, the matrices of ones and minus ones in the code that generate the eight frequency channels
seem to have the wrong signs. However, these “errors” must have been intentional as they fix therotation sign error that is inherited from Labview.16
The result is that the frequency channels are defined with the proper signs as given in Equation 4.4. f is the average of the absolute fitted frequencies, which are all defined to be positive.
⃗rot minus the sum of the fitted frequencies
f B is the sum of the fitted frequencies with E⃗rot ∥ B
16This is done without any comment in the code or, as far as I can tell, the lab notebook. Why not just fix themistake in Labview? Or at least make the correction when the data is first imported?

139⃗rot . f R is the sum of the fitted frequencies with the rotation direction of E⃗rot up
with E⃗rot ∥ −B
by the right hand rule minus the sum of the fitted frequencies with E⃗rot rotating down. f D is thesum of the fitted frequencies of the upper doublet minus the sum of the fitted frequencies of thelower doublet. For a frequency channel with multiple superscript letters like f DB , we add up thefrequency channels with positive B̃ D̃ and subtract the frequency channels with negative B̃ D̃. In
all cases, we divide by the number of frequency channels in the linear combination.

4.6.6

Appendix 6: Table of Signs

Here I provide a list of definitions of signed quantities used in determining the sign of theg-factor, gathered here in one table. For the entries f B , f BR and f DB , the note points to therelevant Matlab code which creates the linear combination of frequencies.
Table 4.1: List of quantities whose signs matter for determining the sign of the g-factor
Quantity
B̃
R̃
D̃
Labview B Sign
Labview R Sign
Labview D Sign
sgn(f B )Description⃗rot ∥ E⃗rot , -1 for B⃗rot ∥ −E⃗rot+1 for B+1 for Up, -1 for Down
+1 for Upper, -1 for Lower+1-1+1
f B̃=+1 − f B̃=−1

Notes
Magnetic Field Chop
Rotation Chop
Doublet Chop
Labview knows B Sign
Labview is wrong about R sign
Labview knows D Sign
See generate lcoms common.msgn(f BR )f B̃=+1,R̃=+1 − f B̃=−1,R̃=+1−f B̃=+1,R̃=−1 + f B̃=−1,R̃=−1

See generate lcoms common.msgn(f DB )f D̃=+1,B̃=+1 − f D̃=−1,B̃=+1−f D̃=+1,B̃=−1 + f D̃=−1,B̃=−1

See fit fringe common v6.m

d⃗
Eeff
n̂
µ
⃗
µ
⃗F
fBerryϕup
ϕdown

points from -q to +q+ 23 GV/cm
points from F to Hf nuclei
points normal to current loop
µF = gF µB > 0 when µ⃗ F ||F⃗= −mF frot αR̃
330 degrees
300 degrees⃗
Implies Udip = −d⃗ · E
See Appendix 4.6.2
The internuclear axis⃗
Implies Udip = −⃗µ·B
This defines the sign of the gF
Sign of Berry’s phase shift
Initial E⃗rot phase when R̃ = +1
Initial E⃗rot phase when R̃ = −1

140
4.6.7

Appendix 7: The sign of δgF in HfF+

From Equation 4.36 we can read that f = 3|gF |µB Brot and f D = 3δgF µB Brot sgn(gF ). Therefore we have:fD
δgF
3δgF µB Brot sgn(gF )
δgF
=
=
=
f
3|gF |µB Brot
|gF |sgn(gF )gF(4.52)

We can compare this expression with the Figure 6a in the systematics paper [31], repeated here as
Figure 4.10.

Figure 4.10: Repetition from systematics paper caption: f D vs f 0 for various values of the appliedδgF 0 ∆0 ∆Dquadrupole magnetic field, B⃗0 . Data taken at Erot ∼ 58 V/cm. Fit is to f D =
f +
,
gF
f0
δgF
giving= −0.002149(3), ∆0 ∆D = −0.39(4)Hz2 .
gF

We’ve concluded in Section 4.4 that gF is negative, and as we can see from Figure 4.10

δgF
gF

is negative as well. Therefore δgF is a positive quantity. We can compare this experimentally
determined sign to theory by evaluating the leading order term of δgF from our Hamiltonian as
follows.
δgF primarily comes from a difference in the sign of the mixing terms between the upper
(2)

(2)

and lower doublets to J = 2. We can start by finding gupper and glower , where the superscript (2)denotes that these are calculated from 2nd order perturbation theory:

141

(2)

(2)
gupper==(2)∆Eupper,MF =3/2 − ∆Eupper,MF =−3/2
 X
n∈J=2

−3µB Brot| ⟨upper, MF = 1.5| Hstark + Hzeeman |n⟩ |2−4B

X | ⟨upper, MF = −1.5| Hstark + Hzeeman |n⟩ |2 −/(−3µB Brot )−4B(4.53)n∈J=2Note that we define the g-factor of the upper doublet to be the difference in energy of the two
stretched states divided by −3µB Brot , which is the term for a Zeeman shift sans gF . There’s a similar(2)

expression for glower . Also note that we’re approximating the energy difference as −4B because theonly relevant states |n⟩ are in the J = 2 rotational level and E(J = 1)−E(J = 2) = 2B−6B = −4B.
⃗ ·E
⃗ −µ
⃗
Now we need to evaluate the matrix elements, where Hstark + Hzeeman = −D
⃗ · B.
Here’s an equation from chapter 2 of Will Cairncross’s thesis, which I’ve checked with Brown andCarrington [1, 67]:

⟨J, F, MF , Ω|Hstark + Hzeeman |J ′ , F ′ , MF′ , Ω′ ⟩


F′



1


 F
= − D∥ Erot − G∥ Brot µB Ω (−1)F −MF ′−MF 0 MF




′
′

J F 0.5
pF ′ +J+1.5′
× (−1)(2F + 1)(2F + 1)


F J
1


!
′
1
X
J
1
J
p


×
(−1)J−Ω  (2J + 1)(2J ′ + 1)
q=−1−Ω q Ω′

Note that G∥ = 3gF − gN

(4.54)me. This expression only applies for the simplified case where the electricmp

and magnetic fields are along the quantization axis and only applies within the 3 ∆1 , v = 0 manifold.
For the full expression, see Equation 2.15 in Will Cairncross’s thesis [1].

142
The above equation does not mix terms with different values of MF or Ω, which means there
are only a few terms that appear in the sums in Equation 4.53. Writing those terms out we get:(2)=
gupper| ⟨1, 1.5, 1.5, −1| Hstark + Hzeeman |2, 1.5, 1.5, −1⟩ |2−4B(−3µB Brot )| ⟨1, 1.5, 1.5, −1| Hstark + Hzeeman |2, 2.5, 1.5, −1⟩ |2+−4B(−3µB Brot )| ⟨1, 1.5, −1.5, 1| Hstark + Hzeeman |2, 1.5, −1.5, 1⟩ |2−−4B(−3µB Brot )| ⟨1, 1.5, −1.5, 1| Hstark + Hzeeman |2, 2.5, −1.5, = 1⟩ |2−−4B(−3µB Brot )D∥ Erot G∥=−
20B(4.55)(2)

A similar calculation for glower gives:

glower =D∥ Erot G∥
20B

(2)

(2)

(2)

(4.56)(2)

We can put them together to find δgF :

(2)

δgF =

D∥ Erot G∥
gupper − glower
=−
2
20B

(4.57)At the start of this appendix, we found experimentally that δgF is positive. The relationshipG∥ = 3gF −gN

metells us that G∥ is positive as well. We can take the sign of the terms in Equation
mp4.57 and we find that:(2)
sgn(δgF ) = sgn

D∥ Erot G∥
−
20B


=−

sgn(D∥ )sgn(Erot )sgn(G∥ )(+1)(+1)(−1)=−= +1sgn(B)(+1)(4.58)

143(2)

So we expect the sign of δgF to be positive, which makes sense because we found δgF to be positive
experimentally.
(2)

Since we’ve made it this far, we may as well calculate

δgF
gF

from G∥ = 3gF − gN

me
:
mp

(2)
D∥ Erot G∥
3D∥ Erot
D∥ Erot
δgF
3
Erot
 ≈−

=−
=−
= −4.298 × 10−5
m
e
g
m
gF
20B G + gN
5.102B
V/cm
N e∥
20B 1 +
mp
G mp∥

(4.59)
Note that in the last step, I’ve plugged in experimental values as reported in [31]. Using the value
in generation 2 of Erot = 58 V/cm, we find that:

(2)

δgF
= −0.00249
gF(4.60)

This has the same sign as and is 16% larger than the measured value −0.002149(3) from
Figure 4.10. The discrepancy in magnitude arises at least in part from the fact that we haveignored smaller perturbations. A more complete calculation was done by Alexander Petrov [57]
who included perturbations due to the rotation Hamiltonian and higher order effects that couple
to the 3 ∆2 , 3 Π0+ and 3 Π0− electronic states. Petrov claims that his numerical results are within
5% of the analytical formulas we presented in our systematics paper [31], and he found a value of:

1
δgF
=−
≈ −0.002114
gF
473(4.61)Petrov’s result is much closer to the measured value −0.002149(3) than the second order perturbation result I calculated in this appendix.

Chapter 5

Magnetic Shielding“Science is magic that works.”- Kurt Vonnegut, Cat’s Cradle.
One of the main challenges of measuring the eEDM is measuring the energy shift de Eeff inthe face of the much larger Zeeman effect. As explained in Section 2.1, our experiment 1) uses
a molecule with a small value of gF , 2) makes a differential measurement between the upper andlower doublets, pairs of states with similar magnetic sensitivities but opposite eEDM shifts, and 3)
uses a rotating quantization axis to average out shifts caused by uniform magnetic fields. There isa fourth, and perhaps obvious, strategy to minimize the systematic errors caused by the Zeeman
effect. That is to surround the apparatus with magnetic shielding to passively reduce the magnitudeof the field and therefore all associated systematic shifts. Fortunately, the first three methods tominimize magnetic systematics were sufficient in our first and second generation measurements ofthe eEDM at JILA. As discussed in Chapter 3, the total magnetic systematic error in generation
two was just a few µHz, almost an order of magnitude smaller than our statistical error bar.
The absence of magnetic shielding in our previous measurements makes us unique amongmodern EDM experiments. Two of the leading electron EDM measurements, the ACME collaboration and the Imperial College eEDM experiment, have both required multiple layers of magnetic
shielding from their outset [47, 68]. Regrettably, our experiment will need magnetic shielding in its

145next iteration. We want our third generation measurement to have an order of magnitude smaller
statistical and systematic error bars, the latter of which we can no longer achieve without shielding
the magnetic field in our lab.

5.1

Generation Three JILA eEDM Experiment

5.1.1

Desired Statistical Sensitivity

The goal of the third generation measurement at JILA is to improve our sensitivity to theeEDM by another order of magnitude. As seen in Table 3.8, our leading source of error in generation
two was statistical. To reduce the statistical error we can refer to Equation 2.3, which indicates wewill need to increase a combination of the effective electric field Eeff , the coherence time τ and thenumber of molecules we measure N .
There is not a clear path forward to substantially improving any of these quantities if we stickwith HfF+ . Eeff is a fixed quantity and τ is limited to about 3 seconds due to the finite lifetime of
the 3 ∆1 state. It should be possible to increase N by improving the state preparation efficiency of
3∆

1 as described in Subsection 2.2.1.5; we only move ≳ 20% of the HfF+ molecules we trap fromthe electronic ground state to 3 ∆1 . If we improve the efficiency to ∼ 90% we would gain roughlya factor of four in ion number, doubling our statistical sensitivity. It is not exactly clear how wewould improve this efficiency but it should be possible. Alternatively, we could just take more hoursof data. The ∼ 650 hours of data that Tanya Roussy, Luke Caldwell and I took over ∼ 2.5 months
could be extended indefinitely, or at least until the equipment/grad students/postdocs break.
Our plan is to work smarter instead of longer hours. The third generation experiment will bemade with ThF+ molecular ions instead of HfF+ . This switch has involved a significant overhead inorder to learn how to produce, trap and manipulate the quantum states of this new molecule [3, 4].
However, we are optimistic that the payoff will be worth it.
The 3 ∆1 state of ThF+ has Eeff = 35 GV/cm, about 1.5 times larger than HfF+ [69,70]. That

146
means in order to measure the eEDM 10 times more precisely, we only need a ∼ 6.7 times smaller
error bar in our frequency measurement. The key advantage of ThF+ is that 3 ∆1 is its electronic
ground state, so τ will not be limited by a finite lifetime. We hope to achieve τ ∼ 20 seconds forour generation three measurement.
In the generation two apparatus a longer coherence time τ means that we have to spend moretime waiting for the experiment to run and therefore measure fewer ions N . If we assume the bestcase that the “dead time” between the end of one measurement and the beginning of the coherencetime of the subsequent clouds of ions is minimal, increasing our coherence time only improves our√
statistical sensitivity by τ . This means an increase of coherence time from 3 to 20 seconds wouldp
only net us a factor of 20/3 = 2.6 in sensitivity. In order to improve our statistical sensitivity
even further, we have completely redesigned our ion trap. Instead of trapping a single cloud ofions, we will trap many ion clouds and make semi-continuous measurements of the eEDM.
See Figure 5.1 which shows the design of the “Bucket Brigade” ion trap. The eight radial
electrodes in the generation two ion trap shown in Figure 3.8 are replaced with eight long rod
electrodes that run the length of the new trap, from the bottom left to bottom right of the Figure5.1. The two endcap electrodes from generation two are replaced with a series of ring electrodes
whose voltages can be set to create many ion traps along the length of the rods.
We create a neutral beam of ThF molecules which are ionized and trapped on one end of the
Bucket Brigade. We do all of the state preparation and begin the Ramsey evolution at this initial
location. We will then slowly move the center of the ion trap by varying the voltages on the ring
electrodes to the other end of the Bucket Brigade, where after the coherence time τ the Ramsey
measurement is completed. However, once the ions have moved far enough,1 another “bucket” ofions is prepared in the state preparation region. In this way many buckets of ions are prepared1

How far is far enough between buckets of ions? This is currently an open question, look to a future JILA EDM

thesis for the answer.

147
and shuttled from one end of the ion trap to the other in unison, and we can make measurements
semi-continuously. This will substantially increase the number of ions N we measure in a given
amount of time.

Figure 5.1: A conceptual drawing of the Bucket Brigade for our generation three eEDM measurement. We plan to trap many buckets of ions simultaneously, all moving from the state preparation
region on the left to the dissociation region on the right. While the ions move in parallel they are
undergoing their ∼ 20 second coherence time. This image was made by Kia Boon Ng [4].

For significantly more detailed explanations of our next EDM experiment, please see thetheses recently written by Kia Boon Ng [4] and Noah Scholssberger [3].

5.1.2

One of Maxwell’s Equations

Before jumping into more details about generation three, it will help to discuss one of
Maxwell’s equations:⃗⃗ ×B⃗ = µ0⃗j + 1 ∂ E
∇
c2 ∂t

(5.1)

148




In our experiment the largest electric field is E⃗rot = Erot cos(ωrot t)x̂ + R̃ sin(ωrot )ŷ . According to
the above equation this will lead to time-varying magnetic field gradients:






∂By
∂Bz
 ∂z − ∂y 
 − sin(ωrot t) 




 ∂B


 z − ∂Bx  = ωrot Erot 


R̃
cos(ω
t)
rot


2


c
∂z 
 ∂x


 ∂Bx ∂By 
0
−
∂y
∂x

(5.2)

Because our experiment relies on E⃗rot we cannot remove these magnetic field gradients from ourexperiment. Note that the above equation does not uniquely determine any of the six gradients,
they must be determined by the boundary conditions of our experiment.
We saw the effect of these gradients in generation two; they were discussed in Subsection⃗cc . In that section we said that:
3.4.2 under the name BR̃ cos (ωrot t)
⃗cc = Bcc z B sin (ωrot t) 




0

(5.3)

This is equivalent to Equation 5.2 with boundary conditions that set

∂By
∂Bz
∂Bz
∂Bx
=
=
=
=
∂y
∂x
∂y
∂x

0. We can see that by applying these boundary conditions to Equation 5.2 and integrate along z
to find that:









 sin(ωrot t) 
By 


 
ω
E
z

 
rot rot Bx  = − R̃ cos(ωrot t)
c2


 
0
0

(5.4)

Noting that the first and second entries of the vectors are flipped from their usual order, we cancompare to Equation 5.3 to find Bcc = −ωrot Erot. When we plug in the generation two values wec2

149find that |Bcc | = 1.6 mG/m, which is about 50% larger than the value 1.1 mG/m we measured in
Section 3.4.2. This indicates that there were magnetic field gradients

∂Bz
∂Bz
and
possibly as∂y∂xlarge as 0.5 mG/m that we did not consider in generation two. The systematics shifts created bythese magnetic field gradients will be discussed in Subsection 5.2.2.
Because we plan to keep Erot and ωrot roughly the same magnitude, a choice we discussin Subsection 5.2.1, we should expect similarly sized magnetic field gradients in generation three.
These gradients will be more problematic in generation three because instead of sitting at one point
the molecules will move up to three meters along the z-axis! These gradients will be referenced
multiple times during the rest of this chapter.

5.1.3

Methods of Generating f 0

The combined improvements of Eeff , τ and N should give us a factor of ∼ 10 improvementin the statistical sensitivity of our next eEDM measurement. The larger value of Eeff means we
can achieve this factor of 10 by reducing our statistical uncertainty by a factor of 10/1.5 = 6.7,
corresponding to an uncertainty of about 3.5 µHz. This new sensitivity is crucial, but only mattersif our systematic error bar shrinks as well. As seen in Table 3.8, our largest individual systematic
error was 3.5 µHz. To keep our total systematic error ∼ 3 times smaller than our statistical error,
we want our each of our individual systematic errors to be no larger than 500 nHz at most.2
As we saw in Chapter 3, the magnitude of a given systematic effect depends on our method ofgenerating f 0 ≈ 100 Hz.3 In generation two f 0 was produced by the B̃-odd Zeeman shift caused by⃗ ⃗⃗2,0 . This effect has ⟨E · B⟩ /|E| = 1, so it also
our molecules rotating in the magnetic field gradient B⃗⟨Ê · B⟩
2

Assuming that all systematic errors are uncorrelated, there need to be 49 individual 500 nHz systematics toresult in a total systematic error bar of 3.5 µHz, the size of our desired statistical error bar. We do not want the
total systematic error bar to be this large, but a few ∼ 500 nHz systematic errors are fine.
3 0

f is the name for the channel that measures the ∼ 100 Hz Zeeman shift. In other JILA EDM documents it has

been called f and f0 .

150
causes a D̃-odd shift that isδgF
times smaller. We saw in Subsection 3.2.2 that all other frequency
gF

shifts with the same ratio of f D /f 0 did not cause an appreciable systematic shift because of ourshimming/correction procedure. This means that the magnitude of systematic effects in generation
3 will depend on the method we choose for generating f 0 . For example, if we generate f 0 with
⃗ /|E|⟨E⃗ · B⟩
Berry’s phase then all Berry’s phase shifts and magnetic effects with
= 0 will not cause⃗⟨Ê · B⟩a systematic error.4
As a group we have put a substantial amount of thought into how we want to generate f 0 in
generation 3, evidenced by chapter 8 of Kia Boon Ng’s thesis which is titled “Generating f 0 in the
Bucket Brigade” [4]. He details five plans for generating f 0 which are listed in Table 5.1, focusing
on how to implement each plan while keeping the required fields homogeneous enough to maintainτ ≳ 20 seconds.
Table 5.1: Kia Boon’s Plans to Generate f 0
fD
value
f0

Plan

Description

A

Zeeman shift from B2,0 plus Erot

δgF
gF

B

Berry’s Phase from Eϵ plus E2hz

0C

Zeeman shift from B1,±1 plus E2hxy0

D

Zeeman shift from B2,±2 plus EϵE

Zeeman shift from Brot plus Erot

4 δgF
3 gF
δgF
gF

Plans A through D make use of frequency shifts already discussed in Chapter 3. Plan E⃗rot that rotates in the xy-plane at a frequency frot so that it is always
relies on a magnetic field B⃗cc which also rotates inparallel or anti-parallel to E⃗rot depending on B̃. This is a similar field to B
4

This is true for the magnetic effects up to the small corrections discussed in Subsection 3.2.1 and [57]. In practice,these shifts would cause negligible systematics unless things go horribly wrong.

151⃗cc is parallel or anti-parallel to E⃗rot depending on
the xy-plane at frot . The differences are 1) that B⃗cc is a magnetic field gradient along the z-axis while B⃗rot is spatially uniformR̃ instead of B̃, 2) B
⃗cc is applied unintentionally due to the charging currents while
over the Bucket Brigade and 3) B⃗rot would be applied by intentionally sending additional current through the radial electrodes. In
B⃗rot we do not want B⃗cc interfering and causing theorder to apply a ∼ 100 Hz frequency shift with B⃗rot with minimal
frequency shift to be R̃-odd or having a spatial gradient. The implementation of B
⃗cc is depicted in Figure 5.2 and can be created with the following voltages and currents appliedB
to the eight radial electrodes:nVrot= Vrot cos (R̃ωrot t −

nπ
+ π)
4

(5.5)

n
Irot
= B̃Irot sin (R̃ωrot t −

nπ
+ π)
4

(5.6)

n are the intentionally applied currents that flow through the eight rods to create B⃗rot ,
Note that Irot
n . When these voltages andthey are not the currents that charge the rods and give rise to Vrot

currents are applied to the rods they create the following electric and magnetic fields:⃗Erot = Erot cos(ωrot t)x̂ + R̃ sin(ωrot t)ŷ

(5.7)



⃗
Brot = Brot B̃ cos(ωrot t)x̂ + R̃ sin(ωrot t)ŷ

(5.8)

Assuming that the rods are infinitely long,5 we find that Brot =2Irot µ0
, where R0 is the radius of
πR0

the Bucket Brigade. For R0 = 5.85 cm, the value of our current design [3], we need a reasonable
5

While this is a better approximation in generation three then in generation two, it should not be taken too

seriously. It is sufficient to find the approximate value of Irot .

152⃗rot without B⃗cc .
Figure 5.2: If we choose to generate f 0 via plan E this is our design to generate B
The eight rods in the Bucket Brigade, two of which are shown in blue, have finite capacitance. If⃗cc will arise as described in Subsection 3.4.2. We
they are charged from one side a magnetic field B
can minimize this field by charging the radial electrodes via the charging current Icc which connects
to many points spaced along the rods and is generated by the amplified Vrot . This minimizes the⃗cc . In order to create B⃗rot we can apply the currents
current traveling along the rods and therefore B
Irot in Equation 5.6 to run all the way though the electrodes. Since these currents do not need to⃗rot will
induce a voltage in the rods, they will have the same magnitude as a function of z and B
not have a spatial gradient.

Irot = 12 mA to apply Brot = 1.6 mG and a Zeeman shift of 100 Hz.

5.1.4

Choosing a Method to Generate f 0

Now that all five plans listed in Table 5.1 have been explained, we can see which would be
best in terms of limiting our systematic error. Plan B, where we generate f 0 ∼ 100 Hz via a second
harmonic and ellipticity to create a Berry’s phase shift, at first appears promising. Berry’s phase

153
does not cause a D̃-odd shift so there would be no need to worry about higher harmonics causing
additional small Berry’s phase shifts, and we would not need to apply a correction to our eEDM
frequency channel. Additionally, some of the Zeeman effects which are currently systematics have
⃗ /|E|⟨E⃗ · B⟩= 0 so they will not cause systematic errors either. The only systematic errors from⃗⟨Ê · B⟩frequency shifts that we need to mind are magnetic and the experiment could be done inside someserious magnetic shielding.
There are two problems with this approach. The first is that the Zeeman shift from our⃗2,0 would cause a systematic 3δgF µB B2,0 rrot .
molecules rotating inside the magnetic field gradient B
Unlike all of the magnetic systematics listed in Table 3.6, this systematic effect is not multiplied byan electric field imperfection divided by Erot . That will make this the most worrying Zeeman shift
and require B2,0 < 75 nG/cm for the effect to be 500 nHz or less. As we will soon see, this would
be the most stringent restriction on the allowed magnetic fields by far! The second problem is that,as described in Subsection 5.1.2, we cannot completely remove magnetic field gradients from ourapparatus. As discussed in Subsection 5.2.2, the largest Zeeman effects caused by these gradients⃗ /|E|⟨E⃗ · B⟩have
= 1 and would therefore cause systematic errors.
⃗⟨Ê · B⟩
While these considerations eliminate plan B, they have also pushed us away from plans C⃗ /|E|⟨E⃗ · B⟩and D. Plan C, to generate a Zeeman shift with= 0, suffers the same strict magnetic⃗⟨Ê · B⟩⃗ /|E|⟨E⃗ · B⟩
4shielding requirements. Plan D generates f 0 ∼ 100 Hz with= and could possibly⃗3⟨Ê · B⟩be simpler to apply over a two or three meter long region,6 but it still requires B2,0 < 225 nG/cm.
As we see in Subsection 5.2.1, this somewhat relaxed requirement is still three orders of magnitude⃗ /|E|⟨E⃗ · B⟩
more demanding then if we pick a plan with= 1. From this point on we assume that⃗⟨Ê · B⟩
we generate f 0 from either plan A or E, where one of B2,0 or Brot causes a f 0 ∼ 100 Hz Zeeman
shift via the interaction with Erot .
6

Plan D involves an ellipticity interacting with a magnetic field gradient B2,±2 which is entirely in the xy-plane.

The ellipticity could be intentionally applied by “displacing” one or more of the rod electrodes and the magnetic field
gradient could be created by a single current carrying wire that runs the length of the Bucket Brigade.

154

5.2

Magnetic Systematics in Generation 3
One of the downsides of ThF+ is that its g-factor |gF | = 0.0149(3) in the 3 ∆1 electronic

state is nearly five times larger than in HfF+ [62].7 Its differential g-factor is larger as well, with
|δgF | ∼ 5.7 times larger than in HfF+ , see Figure 5.3. This larger magnetic sensitivity and thefact that the molecules will move over a 2 or 3 meter long region during the coherence time makemagnetic systematics more worrying in our next measurement. I will first discuss how the magnetic
systematic effects that worried us in generation two translate to our new experiment in Subsection
5.2.1, and then discuss the new magnetic systematics we expect to encounter in Subsections 5.2.2
and 5.2.3.

Figure 5.3: A measurement of f D vs f 0 in ThF+ for various values of the applied magnetic field.
δgF
Data is taken with Erot = 60 V/cm. The data is fit to Af 0 + B/f 0 , where A =and B = ∆∆D .
gF
δgF
δgF
The parameter A is equal to= −0.00255(6), comparable toin HfF+ . This fit combined
gF
gF
with the measured value |gF | = 0.0149(3) gives |δgF | = 3.8(1) × 10−6 , about 5.7 times larger thanin HfF+ . This image was made by current JILA eEDM graudate student Sun Yool Park.
7

Note that the sign of gF in ThF+ is still unknown. This can be determined by repeating the method describedin Chapter 4.

155
5.2.1
5.2.1.1

Known Magnetic Systematics from Generation 2
Overview

The magnetic systematics in generation two, discussed in Section 3.3, concerned electric andmagnetic field imperfections in a small ∼ 1 cm3 region of space. In generation three our molecules
will move in a similarly sized ion cloud over a two to three meter long distance down the Bucket
Brigade. Fortunately, this does not complicate the systematics in Table 3.6 because they areproportional to the average electric and magnetic field imperfections experienced by the ions. Thatmeans the restrictions on these fields in generation three will be averaged over the entire length of
the Bucket Brigade, assuming the ions move at a uniform velocity.
As discussed earlier in this chapter we have a ∼ 5.7 times larger δgF and want to keepeach systematic smaller than 500 nHz. The largest magnetic systematics in generation two were
∼ 2 µHz. If we have the same field imperfections in our Bucket Brigade these will cause systematic
shifts greater than 10 µHz, over twenty times larger than we want to allow!
We would like the magnitude of these systematics to drop by at least a factor of ∼ 20,
which we can accomplish in one of two ways. The first method is to reduce the magnitude of thestray electric fields relative to Erot , which we plan to keep ∼ 60 V/cm. The ellipticity is largely
constrained by geometric imperfections that we expect to be similar in magnitude to our generation
two ion trap. We do have a plan to reduce the harmonics of Erot by replacing the op-amps which
generated Vrot in generation two with resonant transformer circuits. These circuits generate a large
voltage at their resonant frequency frot and suppress the off-resonant harmonics. The circuits havebeen built with a new resonant frequency frot ∼ 150 kHz and are currently in use, and they aredescribed in Chapter 4 of Noah Schlossberger’s thesis [3].
While the circuits have been successful in suppressing harmonics, the lower value of frot ∼ 150
eErot
2 by a factor of 5 relative to rrot in generation two. This will
4π 2 mfrot
l−1
cause the systematic effects due to magnetic field gradients to grow as they scale like Bl,m rrot.

kHz has increased rrot =156
The lower frequency has also caused larger pondermotive forces which have caused trouble in ourcurrent generation three ion trap. In order to solve both of these problems we plan on rebuilding
the resonant circuits with frot ∼ 375 kHz. This will result in rrot ∼ 0.4 mm, a slightly smaller value
than in generation two due to the higher mass of ThF+ . Because the new resonant circuit has notbeen built yet we do not know the values of the harmonic electric fields. To be overly cautious, I
assume that the harmonics will have the same values in generations two and three for the rest of
this section.
While we hope to have smaller electric field imperfections, the other option for reducing the
Zeeman systematics is to reduce the magnitude of the magnetic field and its gradients in the ion
trap. We can determine the required magnetic field limits by investigating the largest effects inTable 3.6 and the off diagonal effects in Subsection 3.1.

5.2.1.2

Uniform Magnetic Fields

The magnitude of the only systematic effect in Table 3.6 caused by a uniform magnetic field is
3δgF µBof

B1,±1 E2hxy, which caused a 2.2 µHz systematic in generation two. Assuming a similar value
4Erot

E2hxy
∼ 4.2 × 10−5 and accounting for the larger value of δgF , we require B1,±1 = Bxy < 0.25
Erot

mG. Magnetic fields along the z-axis in generation two caused systematics by creating off-diagonal
mixing terms ∆ that had switch state dependencies other than ∆0 and ∆D . In generation three
we need to consider all magnetic fields that are perpendicular to E⃗rot , not just B1,0 = Bz . If we⃗rot , there will inevitably be a portion ofchoose plan E and apply f 0 via a rotating magnetic field B
that field that is perpendicular to E⃗rot . We next consider the systematics caused by both of theseperpendicular magnetic fields which will be grouped together with the label B⊥ .
In general the systematics described in Section 3.1 due to ∆ will be larger in generation
three. This is because ∆0 and ∆D are both proportional to the Ω-doubling constant ωef , as shownin Equation 2.21, which is ∼ 7 times larger in ThF+ than it was in HfF+ . Assuming that frot = 375
kHz in generation three, ∆0 and ∆D will be ∼ 5 Hz. We can first consider the systematic effects

157
from the terms in Equation 3.4 which are:B ∆0 ∆D
30gF2 µ2B B⊥ B⊥∆0 δ∆DB + ∆D δ∆B + δ∆DR δ∆BR + δ∆R δ∆DBR=
2 |f 0 ||f00 |
h2 frot
0(5.9)

See Section 3.1 for how we arrived at the right hand side of this equation. Using ∆0 ∼ ∆D ∼ 5 Hz
as we expect in generation three, we find that the shift has a magnitude of 23

nHz
B.
B⊥ B⊥
mG2

B ≲ 20In order to keep this combined systematic effect smaller than 500 nHz, we require B⊥ B⊥⃗rot then we are only concerned with magnetic
mG2 . If we follow plan A and we do not apply B
fields along the z-axis. In this case we will intentionally apply a magnetic field gradient along z
B2,0 that flips with the magnetic field. Therefore the ions will experience an average magnetic
field B⊥ = B2,0 zoffset as they move through the Bucket Brigade. zoffset is the distance between thecenter of the Bucket Brigade and the location where the applied magnetic field along z goes tozero. Assuming that we reduce the background field BzB to ≲ 1 mG by magnetic shielding, thenwe require B2,0 zoffset < 20 mG.
If we choose plan E we also have to consider B⊥ due to imperfect alignment between E⃗rot⃗rot . As discussed in Subsection 5.1.3 we only need Brot ∼ 1.6 mG in order to create a ∼ 100
and B⃗rot so badly that B⊥ = B B = 1.6 mG, Equation 5.9
Hz frequency shift. Even if we messed up B⊥
would still only result in a ∼ 60 nHz systematic shift. Because this shift is small enough to not
be a problem and plan E does not involve applying large Bz gradients, this systematic shift would⃗rot . Limiting Bz ≲ 1 mG would be more than good
not be very concerning if we generate f 0 with B
enough to keep this systematic shift small.
The last systematic we need to consider from Bz is the shift proportional to f BR , which we
assume will still be our largest frequency channel outside of f 0 , given in Equation 3.6. We find that
we need f BR B⊥ ≲ 0.6 mG Hz. It is hard to predict how large f BR will be in generation three due tothe charging currents, so it will be important to check this requirement in the future. If we assume

158
f BR ∼ 200 mHz like in generation two, we need B⊥ ≲ 3 mG. Recall that this is the magnetic
field that is perpendicular to E⃗rot and switches sign with B̃. This should be straightforward if wechoose plan E, but if we choose plan A we would require B2,0 zoffset < 3 mG unless f BR is smaller
in generation three.

5.2.1.3

Magnetic Field Gradients

Table 3.6 has two more magnetic systematic effects that are a few µHz in size, both from√√
3B2,±2 E3hxy rrot
3B2,±2 Eϵ rrot
first order magnetic field gradients. They are 3δgF µB
and 3δgF µB
,
4Erot
2Erot
which caused systematics of 1.5 and 1.7 µHz respectively in generation two. Assuming the generation two values for E3hxy and Eϵ , we require B2,±2 < 0.25 mG/cm. This provides the strongest
restriction on first order magnetic field gradients as we will shim B2,0 to zero.8 To be safe, we would
like all first order gradients B2,m < 0.25 mG/cm.
Table 3.6 does include frequency shifts caused by second order magnetic field gradients B3,m ,though they only caused systematics on the order of 10s of nHz in generation two. In order to keepthese systematics less than 500 nHz in generation three, we require B3,m ≲ 25 mG/cm2 . Note that if
we apply magnetic shielding to keep the zero and first order gradients of the magnetic field less than
0.25 mG and 0.25 mG/cm respectively, it would be very odd if B3,m ≲ 25 mG/cm2 . We can makethe argument stronger by noting that the ions will always be more than 10 cm from the magnetic
shielding,9 so we would expect higher order gradients to fall at least as fast as
8

0.25 mG/cm
with(10 cm)nB
If we go with plan E and generate f 0 ∼ 100 Hz via Brot , shimming f B to zero by changing the value of Brot

B
B
will likely leave nonzero values of Brot
and B2,0. At first glance this appears to cause systematics due to effects 9-14
⃗ /|E|⟨E⃗ · B⟩and 33-45 in Table 3.6 which are proportional to B2,0 but have
̸= 1. However, the shimming procedure⃗⟨Ê · B⟩
B
B
will guarantee that Brot
= −B2,0rrot . Additionally, Table 3.6 does not include frequency shifts caused by Brot . Since

B2,0 rrot looks exactly like a rotating magnetic field to the ions (see Figure 2.4) Table 3.6 would include duplicates ofall entries proportional to B2,0 with B2,0 rrot replaced by Brot . Therefore the frequency shifts from B2,0 will cancel
with shifts caused by Brot and there will be no resulting systematics.
9

As discussed in Sections 5.4 and 5.5 we expect the innermost magnetic shield to have R1 = 25 cm, so 10 cm isconservative. The shield will be the nearest object to the ions that is magnetic.

159
higher orders of gradient n. All that is to say that if we constrain the zero and first order gradientsof the magnetic field to be < 0.25 mG and < 0.25 mG/cm respectively, and we do not increase
the size of rrot by using a lower value of frot , systematics from higher order gradients should be
negligible.

5.2.2

Maxwell Systematics

Figure 5.2 outlines a potential plan to control the magnetic field gradients guaranteed in our
experiment by Maxwell in Equation 5.1. Since we are not sure if we will implement this plan, andeven if we do there is no guarantee how well it will work, we now consider the systematic effects
caused by any of the magnetic gradients created by E⃗rot .
From Equation 5.2 we see that E⃗rot can lead to four10 magnetic field gradients, where thetotal “Maxwell” magnetic field is:⃗Maxwell = 
B





−R̃zBx∂z cos (ωrot t)−zBy∂z sin (ωrot t)R̃xBz∂x cos (ωrot t) + yBz∂y sin (ωrot t)(5.10)

While only the differences in these amplitudes is fixed by Equation 5.2, we can expect that they
are each ∼ωrot Erot= 1.6 mG/m.
c2What frequency shifts are created by this magnetic field gradient? Using E⃗rot as defined in⃗Maxwell ⟩ /h = −1.5gF µB (Bx∂z +Equation 5.7, we arrive at a D̃-even frequency shift of −3gF µB ⟨Êrot · B⃗ /|E|
3gF µB ωrot Erot⟨E⃗ · B⟩
By∂z )z R̃/h ∼ −
z
R̃,
which
has
a
magnitude
of
about
1
Hz/cm.
Since
=1
⃗
hc2
⟨Ê · B⟩δgF
. Fortunately this willin this case, there will also be a D̃-odd shift that is smaller by a factor ofgF
not cause a systematic if we choose plans A or E, but it does mean that our ions would have their
10

Because E⃗rot has no component along the z-axis, we assume that the two remaining magnetic field gradients areeach zero.

160
Ramsey frequency vary by ∼ 200 Hz over a 2 meter long Bucket Brigade! If we engineer so most of
the gradients are in Bz∂x or Bz∂y , we can make the gradient small. This may be a necessity as weneed the frequency to be large compared to ∆ for the entire coherence time or else the MF = ±1.5
stretched states will mix.
If we gloss over the problem of the frequency going through zero, when we average theeffect over the entire Bucket Brigade there will be a frequency shift of 1Hz
z0 R̃ where z0 is thecm

distance between the center of the Bucket Brigade and the position along the z-axis where theMaxwell magnetic field goes to zero. We think those locations should be near one another since theapparatus will be mostly symmetric, but a z0 of at least few centimeters would not be surprising
given that the length of the entire apparatus will be a few meters.
In order to see how this frequency gradient can cause systematic effects I have repeatedthe procedure outlined in Section 3.3. I numerically and analytically searched for frequency shifts⃗Maxwell and stray electric fields. Table 5.2 contains all of the shifts found by the analytic
caused by B
method up to O(κ2 ), which were verified numerically.11
Shifts number 1-4 in Table 5.2 cause a systematic shift because they cause a D̃-even but
Bx∂z − By∂z E3hx z0 cos ϕ3x
not D̃-odd shift. The four shifts can suggestively be rewritten as
and
8Erot

By∂z − Bx∂z E3hy z0 sin ϕ3y. In this form it is perhaps clear why we ignored this systematic in
8Erot
generation two. Assuming that our ion trap is rotationally symmetric about the z-axis, there areno major differences between x and y, Bx∂z and By∂z will have the same magnitude and sign causing
these frequency shifts to cancel. And even if the effect does not cancel and we suppose somehow thatBx∂z − By∂z ∼ωrot Erot= 1.6 mG/m, the systematic in generation two is still just 1.2 µHz/meterc2

times z0 . Per Figure 3.8 the entire generation two ion trap was 160 mm tall, meaning z0 should
be much smaller than 80 mm. Assuming the worst case where z0 = 80 mm, the systematic shift
11

For curious graduate students in the JILA EDM group, or in case these systematic effects are dramatically larger

than anticipated, I have recorded the shifts up to O(κ3 ) in the group meeting slides on November 13th 2023.

161⃗MaxwellTable 5.2: Analytically Calculated Zeeman Shifts from B
Number
1
2
3
4
5
6
7
8
9
10

⃗
⟨Ê · B⟩
Bx∂z E3hx z0 cos ϕ3x
8Erot
By∂z E3hx z0 cos ϕ3x−
8Erot
Bx∂z E3hy z0 sin ϕ3y−
8Erot
By∂z E3hy z0 sin ϕ3y
8Erot
Bx∂z E2hz rrot cos ϕ2z
16ErotBy∂z E2hz rrot cos ϕ2z−
16ErotBz∂x E2hz rrot cos ϕ2z−
4Erot
Bz∂y E2hz rrot cos ϕ2z
4Erot
Bx∂z Eϵ z0 cos 2θ
−
4Erot
By∂z Eϵ z0 cos 2θ
4Erot⃗ /Erot⟨E⃗ · B⟩

⃗ /Erot⟨E⃗ · B⟩⃗⟨Ê · B⟩

0

0

0

0

0

0

0

0

Bx∂z E2hz rrot cos ϕ2z
16ErotBy∂z E2hz rrot cos ϕ2z−
16ErotBz∂x E2hz rrot cos ϕ2z−
4Erot
Bz∂y E2hz rrot cos ϕ2z
4Erot
Bx∂z Eϵ z0 cos 2θ
−2Erot
By∂z Eϵ z0 cos 2θ
2Erot

1
1
1
1
2
2

would only be 100 nHz in generation two.
While we were safe to ignore these shifts in generation two, it is not so easy in generation
three. All magnetic effects are inherently 5.7 times larger, our desired uncertainty is 10 times
smaller and the ion trap will be multiple meters long. This time around the shift will be 6.7
ωrot ErotµHz/meter if Bx∂z − By∂z ∼= 1.6 mG/m. In order to keep this shift smaller than 500 nHz
c2
we must keep Bx∂z − By∂z z0 ≲ 0.1 mG.12 This should not be particularly challenging, but it willneed to be confirmed experimentally.
Shifts number 5-8 in Table 5.2 are interesting because they are proportional to rrot instead
12

This constraint assumes the generation two value for E3hxy . If this harmonic is suppressed in generation three,which it should be due to the resonant circuits discussed in Subsection 5.2.1, this restraint will be relaxed.

162
of z0 . Fortunately they will not cause systematic errors because they each have

⃗ /|E|⟨E⃗ · B⟩= 1.
⃗
⟨Ê · B⟩
By∂z − Bx∂z Eϵ z0 cos 2θ
. Like shifts 1
Shifts 9 and 10 in Table 5.2 can be combined to get
4Erot
through 4 this effect goes to zero if the ion trap is symmetric between x and y. Assuming this is not
the case, generation two would have had a systematic of 3.3 µHz/meter and is still small enough toignore. In the worst case described above, the systematic shift would be less than 300 nHz and islikely much smaller. For generation three I expect this systematic to be more worrying than shifts 1
through 4 because while we have a plan to reduce the third harmonic, the ellipticity will likely haveωrot Erotthe same magnitude as in generation two. If that is the case and if Bx∂z − By∂z ∼
= 1.6
c2

mG/m, this shift will be 18.8 µHz/meter times z0 . This will require Bx∂z − By∂z z0 ≲ 0.04 mG,
a more demanding constraint then before. This will require a bit of thought and effort to achieve,
but should not be an intractable obstacle.

5.2.3

Systematics from Spatial Gradients

The last category of magnetic systematics to consider are those that are caused by spatial
inhomogeneities in our electric and magnetic fields. In generation two our ion cloud sat at the same
⃗2,0 .
point in space, and we could verify the temporal stability of Vrot and the current that drove B⃗2,0 or B⃗rot
In generation three it will be inevitable that spatial inhomogeneities in E⃗rot and either B⃗2,0 depends on rrot
which can lead to systematic errors. Because the frequency shift caused by B
which is a function of Erot , we will start by considering the simpler case where our frequency shift⃗rot .
is caused by B
We start by considering the simplest case where Brot is uniform over the entire length of the
Bucket Brigade L but Erot is larger by Ebump for a small region of the Bucket Brigade LE , as shown⃗ · Ê⟩in Figure 5.4. This change in Erot magnitude will not change the D̃-even frequencies because ⟨B
will be the same size throughout the Bucket Brigade. On the other hand the D̃-odd frequencies⃗ · E⟩⃗ is not the same over the entire ion trap.13 Assuming that the ionswill change size because ⟨B
13

See Subsection 3.2.1 for an explanation of this difference between D̃-even and D̃-odd Zeeman shifts.

163
move with a uniform velocity, we measure:

LE Ebump 
fD
δgF
1+
=
0
f
gF
LErot

Note that this can cause an increase in
LE = L, this equation tells us that

(5.11)

fD
because Ebump is a signed quantity. In the case wheref0

δgF
is proportional to Erot , which we found in Equation 4.57.
gF

Figure 5.4: Brot is uniform over the L = 2 meter long Bucket Brigade, but Erot is larger by Ebump
for a region LE < L. This changes the ratio f D /f 0 .

On its own, Ebump is not problematic. It will change our measured value offD, but as longf0

as it is stable over time we will take all of our data with the same ratio.14 Spatial variations in themagnitude of Erot are most likely to come from geometric imperfections of the rods, i.e. the rods
are closer together on one end of the Bucket Brigade then the other, which will be stable over time.
Spatial variation in Erot and Brot at the same time is more worrying. Now we assume that themagnitude of Brot changes by Bbump for a region LB as shown in Figure 5.5. Let LEB be the lengthof the region where the two bumps overlap. In the case of Figure 5.5 LEB = 0, but in general the14

This modification of

Systematics paper [31].

fD
is similar to an effect of the trapping electric fields described in Section VI A 6 of the
f0

164
bumps can be centered anywhere in the region L and the two bumps can partially or completely
overlap. In this case the change in

Bbump
fD
is more complicated. Expanding in powers of
, we
0
f
Brot

find that:

fD
δgF
=
0
f
gF

Ebump
1+
Erot



Bbump LLEB − LE LB
Bbump 2
LE
+
+O
2
L
Brot
L
Brot

!
(5.12)

Note that this equation reduces to Equation 5.11 when Bbump = 0, as expected.

Figure 5.5: Erot is larger than its usual value by Ebump for a region LE < L and Brot is larger
than its usual value by Bbump for a region LB < L. This changes the ratio f D /f 0 and can cause asystematic shift.

B
The scary part of Equation 5.12 is that Bbump should be replaced by B̃Bbump + Bbump
as ingeneral the Brot non-uniformity will have B̃-even and B̃-odd components. The B̃-even component
B
Bbump
will cause a shift in f DB with magnitude:

Bδf DB = f 0

Note that we have dropped the factor of

δgF Ebump Bbump
gF Erot Brot

(5.13)LLEB − LE LB
from Equation 5.12 because the spatial
L2

inhomogeneities of the fields will not be perfect bumps and this order one number will be replaced

165
by another order one number. What makes this systematic effect worrying is that there will not
be an analogous shift in f B that occurs without the factor of

δgF. That is because this effectgF

is inherently D̃-odd like the D̃-odd cloud size effect described in Subsection 3.2.3 – the D̃-odd
frequency changes suddenly due to the bump in Erot while the D̃-even frequency is unaffected.
If there are inhomogeneities in Brot we do have a method to measure them. We can movethe ions to a certain point in the trap, perform the entire Ramsey sequence by applying the π/2pulses with the molecules in place, and then read out the frequency f 0 . This works because thefrequency depends only on Brot and not Erot in this case, and we can map out the variation in Brot .
Unfortunately, given the method for creating Brot in Figure 5.2, it may be difficult to correct forgradients in Brot . After mapping out Brot we could then measure variations in Erot by looking forvariations infD
as a function of ion position. Assuming we have f 0 = 100 Hz, we need to keepf0

BEbump Bbump< 2 × 10−6 in order to keep the systematic effect less than 500 nHz.
Erot Brot
As mentioned before, the story becomes more complicated if we use B2,0 instead of Brot togenerate f 0 ∼ 100 Hz because the frequency shift which is proportional to B2,0 rrot depends on themagnitude of Erot . However, this does not significantly change Equation 5.12. All it does is add
Ebump 2O
terms that are small compared to the shift we focus on in Equation 5.13.
Erot
The upside of B2,0 for this systematic is that we will likely be able to tune the magnetic fieldalong the Bucket Brigade quite easily. We envision applying B2,0 via a series of coils along theoutside of the vacuum chamber around the Bucket Brigade, which will be discussed in Subsection
5.5. We can change the value of the B̃-even and B̃-odd bumps in B2,0 by adjusting the currents
through these coils. Unfortunately, it will be challenging to measure the spatial profile of B2,0 with
the ions. Our frequency measurements in this case will give us information about B2,0 rrot ∝ B2,0 Erot ,so it will be difficult to disentangle the information about the two fields. Because these spatial
gradients do not oscillate they could be measured by inserting a magnetometer into the apparatus.

166
B

In the case that we apply B2,0 we also require

Ebump Bbump< 2 × 10−6 where Bbump is now
Erot B2,0

understood to be a bump in the magnetic field gradient.

5.2.4

Other Magnetic Systematics

There are more magnetic systematics to be concerned about for generation three. For example, we only limited the D̃-odd cloud size systematic discussed in Subsection 3.2.3 to 3.5 µHz
in generation two. This is the desired size of the statistical error bar in generation three, so thesystematic effect will need to be reduced by about an order of magnitude. For a summary of all
the known requirements for the magnetic fields in generation three, see Table 5.3.
Table 5.3: Known Magnetic Field Requirements for Generation Three. These limits assume frot =375 kHz and f 0 = 100 Hz.
Limit

Section in Thesis

Comments

B1,±1 < 0.25 mG

5.2.1.2

B1,±1 = Bx , By

B1,0 < 1 mG

5.2.1.2

B1,0 = Bz

B < 20 mG2
B⊥ B⊥

5.2.1.2

B⊥ is perpendicular to E⃗rot

f BR B⊥ < 0.6 mG Hz

5.2.1.2

B⊥ is perpendicular to E⃗rot

B2,m < 0.25 mG/cm

5.2.1.3

Assumes gen two values of E3hxy and Eϵ

B3,m < 25 mG/cm2

Bx∂z − By∂z z0 < 0.04 mG

5.2.1.3

Assumes gen two values of E3hxy and Eϵ

5.2.2

Assumes gen two values of E3hxy and Eϵ

B

Ebump Bbump< 2 × 10−6
Erot BrotD̃-odd Cloud Sizes

5.2.3
3.2.3

See Section VI D 2 of the Systematics paper

There will likely be new magnetic systematics we will discover when we take data with higher
statistical precision. We did not fully appreciate the distinction between the D̃-even and D̃-odd
Zeeman shifts until we saw evidence that Zeeman shifts could have different ratios offD
in Januaryf02022, months before we took the generation two dataset. It will be exciting to see what details
another order of magnitude or so in precision reveal about our experiment.

167
Of course there can always be systematic effects, magnetic or otherwise, that we do notpredict or discover in our data. This is equally true of the JILA eEDM experiment and every otherprecision measurement, but that doesn’t help me sleep at night.

5.3

Magnetic Shielding Overview
Our group is entirely new to magnetic shielding. Fortunately, there are groups here at CU andEDM experiments around the world that have extensive experience in the subject. In this section
I will compile the useful resources I have found which will hopefully serve as a useful jumping off
point for future JILA EDM students.

5.3.1

Diamagnetic, Paramagnetic and Ferromagnetic Materials⃗ in a vacuum. An external magnetic
To this point we have discussed the magnetic field B
field will induce a magnetic response in a solid that is parameterized by its dimensionless relative
magnetic permeability µR . The convention is to distinguish between the external magnetic field⃗ which has units of Amps/meter, and the induced magnetic field inside the material B.
⃗
intensity H,
These quantities are related by:⃗ = µR µ0 H⃗
B

(5.14)

Here µ0 = 4π × 10−7 T·m/A is the magnetic permeability of free space.
I have always found notation a bit confusing. In order to understand magnetic shielding weneed to know what magnetic field will be induced inside a material (in units of Gauss or Tesla)
depending on the external field in our lab (also in units of Gauss or Tesla). It is easy to imagine⃗ would be useful in the case where one material is enclosed in another, but
how the definition of H
⃗int = µR B
⃗ext . In the end it is lessfor our purposes it would be easier to simply use the equation B
work to use the notation in Equation 5.14 as it is ubiquitous in the literature.

168
Materials are classified magnetically by their value of µR . Diamagnetic materials are thosewith µR slightly less than one. An example is silver which has µR = 0.99998 [71]. That means inside
an external magnetic field, silver generates its own magnetic field that is 2 × 10−5 times smaller⃗ext . The net magnetic field inside silver still points inand pointing in the opposite direction as B
the same direction as the external field but it is slightly reduced. Paramagnetic materials haveµR slightly greater than one such as aluminum with µR = 1.00002 = 1 + 2 × 10−5 . The magnetic
field inside aluminum is therefore slightly greater than the external field. Ferromagnetic materials,
those with µR ≫ 1, will be the ones we use for magnetic shielding. Two commonly used materials
for shielding are steel (µR ∼ 2, 000) and mumetal (µR ∼ 80, 000). These materials significantly
amplify the external magnetic field.
An important point to make is the relative permeability µR of ferromagnetic materials is only
constant for small values of Bext . Figure 5.6 shows the B vs H and Bext vs Bint curves of mumetal,
and the slope of these curves gives the value of µR . For small values of Bext ≲ 0.2 G mumetal has
a constant µR ≈ 80, 000. After this value mumetal begins to saturate – the the dipole moments ofalmost all of the atoms in the mumetal have already aligned in the external magnetic field – andthe internal magnetic field no longer increases as quickly. Once Bext ≳ 2 G and all of the dipolesin the mumetal have been aligned, the field can no longer be further amplified and we say thatthe material has been saturated. Unless otherwise mentioned, the calculations in this section willassume that the ferromagnetic materials are unsaturated and µR can be treated as a constant.

5.3.2

Cylindrical Shield – Transverse Shielding Factor

We can now see how ferromagnetic materials like mumetal can be used to passively shielda volume of space from an external magnetic field. We are particularly interested in cylindricalshields as they will fit nicely around the Bucket Brigade and have conveniently calculable shielding
factors. The transverse shielding factor ST is defined as the magnitude of an external magnetic
field perpendicular to the length of the shield divided by the field’s magnitude inside the shield.

169

Figure 5.6: On the left is the B vs H curve for mumetal. The same information is shown, in myopinion more clearly, on the right in the Bext vs Bint plot. The data shows that mumetal is responds
linearly to an external field only up to about 0.2 Gauss. The data in these plots is from Comsol
Multiphysics and is a set of discrete points connected by lines. This B vs H curve is what is used
to simulate saturation effects later in this chapter.

The shield also has an axial shielding factor SA defined the same way for magnetic fields along
the shield’s z-axis. In this subsection we analytically calculate ST for an infinitely long cylindricaltshield with mean radius R and thickness t, so the outer and inner radii are R ± . The situation is2
shown in Figure 5.7, and the derivation follows Timothy Sumner’s 1979 dissertation on magnetic
shielding for a neutron EDM experiment [72].15
We are interested in finding ST at steady state without any currents moving around, so we⃗ = −∇ϕ.
⃗can define the magnetic potential ϕ where H
Because there are no magnetic monopoles so⃗ ·B⃗ = 0, and B⃗ = µH,⃗ we know that ϕ will satisfy Laplace’s equation ∇2 ϕ = 0. At the boundary∇of two materials with different values of µR , there are two relevant boundary conditions to calculate⃗ that is normal to the boundary is continuous, and thethe shielding factor. The component of B
⃗ is continuous.16 Because the external magnetic field is uniform, we cantangential component of H
write the potential in all three regions of space as:
15

While many such derivations assume that R ≫ t, Timothy Sumner’s derivation does not.
⃗ = 0, which we have already assumed. The general condition is
H is continuous as long as the surface current K

16 ⃗⃗2 − H⃗ 1 ) = K.
⃗n̂ × (H170

Figure 5.7: A two dimensional view of an infinitely long cylindrical magnetic shield. The space issplit into regions one through three, inside the shield, the shield itself and outside the shield. There⃗3 that is transverse to the shield. The magnetic field lines
is a uniform external magnetic field B⃗1 isbunch together inside the shield as its material has µR ≫ 1, and the internal magnetic field B
B
1⃗3 . The transverse shielding factor ST =small compared to B
is calculated in the text.
B3

ϕk = (Ck ρ +Dk) cos θ
ρ

(5.15)

Here k = 1, 2, 3 for the inner, shield and outer regions respectively, and Ck and Dk are yet to
be determined constants. The expression is written in cylindrical coordinates (ρ, θ, z). We canimmediately determine that D1 = 0 to prevent the potential from going to infinity at ρ = 0.
The boundary conditions set constraints on the potentials:

171

dϕk
dϕk+1
= µk+1dρ rk,k+1
dρ rk,k+1(5.16)1 dϕk
1 dϕk+1=
rk,k+1 dθ rk,k+1
rk,k+1 dθ rk,k+1(5.17)µk

Here rk,k+1 is the radius of the boundary between regions k and k + 1. These two boundary
conditions at each side of the shield provide four equations that relate C1 , C2 , C3 , D2 and D3 . The
system of equations can be solved to give the following expressions in terms of C1 :µR + 1
2µR(5.18)(µR − 1)(t − 2R)28µR

(5.19)(2RµR + t)(2R + tµR )µR (2R + t)2(5.20)

Rt(µ2R − 1)
2µR

(5.21)

C2 = C1D 2 = C1C3 = C1

D 3 = C1

Note that the magnetic field outside the shield goes like C3 −D3, so C3 on its own defines
ρ2

the external magnetic field far away from the shield. Therefore the shielding factor is given by:

ST =

C3
(2µR R + t)(2R + µR t)=
C1µR (2R + t)2

In the limit where R ≫ t the transverse shielding factor simplifies to:(5.22)

172

lim ST = 1 +R≫t(µR − 1)2 t
2RµR(5.23)

A good sanity check is that ST = 1 when t = 0, as there will be no shielding if the magnetic
shield is not there. This equation makes it clear why we want to make the magnetic shields out offerromagnetic material like mumetal with µR ≈ 80, 000. So long as the mumetal is not saturated,
we can take the limit where µR ≫ 1 to further simplify the expression:

ST ≈ 1 +µR t
2R(5.24)

Now we have arrived at the formula for ST that is most easily found online and in theliterature. We can compare the general result in Equation 5.22 to Comsol Multiphysics simulations,as shown in Figure 5.8. The Comsol simulations were run in 2D with µR = 80, 000 and t = 2 mm
for various values of R. The results were fit to a two parameter model ST =a
+ b and showsR

remarkable agreement with the analytically calculated shielding factor.
We can now consider how ST changes when the cylinder has a finite length L that stretchesfrom z = −LLto , as shown in Figure 5.9. For L ≫ R we should expect that the shielding factor22

at z = 0 is unchanged, while the transverse magnetic field leaks some distance into the shield.
Sumner shows this is the case by considering the general solution of the magnetic potential inside
the shield [72]. The potential is given by Bessel functions of the first kind:

ϕ=

X



kz
−kz
Jn (kρ) Ak e + Bk e
cos (nθ)(5.25)

k,n

As the external magnetic field is transverse to the cylindrical shield its angular dependence is given
by cos (θ). We should expect that the solution to ϕ has the same angular dependence so we canrestrict the sum to n = 1:173

Figure 5.8: Comsol Multiphysics was used to calculate ST for an infinitely long cylindrical shieldwith µR = 80, 000 and t = 2 mm for various values of R. The results are shown in green, which areathen fit to a two parameter model+ b shown in red. The prediction of Equation 5.22 is shownR
in blue which agrees quite nicely with the Comsol results.

Figure 5.9: Transverse magnetic shielding with a cylinder of finite length. At the ends of thecylinder the magnetic shielding w.

ϕ=

X
k



kz
−kz
J1 (kρ) Ak e + Bk e
cos (θ)(5.26)

174
We can further restrict ϕ by noticing that the axial magnetic field at z = 0 must vanish by symmetry.
That means that∂ϕ
= 0 which can only be achieved if Ak = Bk . We can therefore write ϕ as:∂z z=0ϕ=X

J1 (kρ)Ak cosh (kz) cos (θ)(5.27)k

For high permeability shielding the axial and azimuthal magnetic field components must go to zero
1 ∂ϕ
∂ϕ
=
= 0. This can only be true if J1 (kR) = 0, so we are∂z ρ=Rρ ∂θ ρ=R
3.83 5.14 6.38
restricted to k =,,etc. where the coefficients in the numerator are the zeros of J1 .
RRRat ρ = R, meaning that

With this restriction in place we can find what we were after originally, the transverse field within
the shielded region:Bρ = −µ0

X
∂ϕ
= −µ0
kJ1′ (kρ)Ak cosh (kz) cos (θ)∂ρ(5.28)k

We are interested in how the transverse magnetic field behaves as we move from z = ±L
2toward z = 0. The argument cosh (kz) in the above equation tells us that the field will exponentiallydecay as we enter the shield. The smallest allowed value of k, k =3.83, will quickly be the largestRterm in the summation as we move into the magnetic shield. If we let x be the distance we movefrom the edge of the shield towards its center, the magnetic field decays as e−3.83x/R . This result
derived by Sumner agrees reasonably well with another source that approximates the decay as
e−3.5x/R [73].
Note that the leaking transverse field Bρ will not decay all the way to zero as we move further
toward the center of the shield. Its value will asymptote toward the shielded value
In cases where L is not large enough for the field to decay below
transverse shielding factor at the center of the shield is given by:Bρ (z → ∞).
STBρ (z → ∞)at z = 0, the total
ST

175

1
1
=
+
tot
ST
ST

1

3.83L
cosh
2R


(5.29)

We can use this equation, as well as the analogous equation for the axial shielding factor in Subsection 5.3.3, to determine how far the shield should extend beyond the ends of the Bucket Brigade.

5.3.3

Cylindrical Shield – Axial Shielding Factor

The axial shielding factor SA for an open cylindrical shield is, perhaps unsurprisingly, worse
than its transverse shielding factor. Timothy Sumner derives this shielding factor as well, but I
found it much less illuminating than his derivation of ST . If we first ignore the effects of magnetic
fields leaking into the ends of the cylinder, or take the limit L ≫ R and only worry about theshielding factor at the origin, we find that:SA = 1 +

K=β−

2K
µR t
L αL2 2R
1+ +
R 3R2

R βR3
+
+ 2α(I1 − I2 )
L
4L3

L
+
R

r

r

R2 2R
−
L2
L


I1 = lnI2 = 2 1 +

1+

L2
R2

(5.30)(5.31)

(5.32)(5.33)

α = 0.85 and β = 1.83 are coefficients that Sumner determines in his thesis.
The contribution from the axial field that leaks into the ends of the shield can again be foundby consider the magnetic potential ϕ written in terms of Bessel functions in Equation 5.25. This

176
time the external field has no angular dependence so we restrict the sum to n = 0. A similarargument means that the first zero of J0 will determine the length scale for the exponential decay.
We find that the axial magnetic field decays as e−2.41x/R , which compares well to another sourcethat approximates the decay as e−2.25x/R . The total axial shielding factor at the center of the shieldis therefore:

1
1
+
=
tot
SA
SA

1

2.41L
cosh
2R


(5.34)

tot , the axial shielding factor at the origin, against the length of the shield in Figure
We plot SA

5.10. Here we use values of R = 0.25 m, t = 2 mm and µR = 80, 000. The figure also includes
Comsol simulations of the axial shielding factor with the same parameters, which confirms thegeneral behavior of Equations 5.30 and 5.34. It also verifies that for L ≳ 8R the axial shielding
scales like

1.
L

This is a surprising result! It makes sense that the axial shielding is poor for L ∼ R because
the shield is not long enough to attract the magnetic field lines away from the shielded region. But
as L becomes large compared to R the axial shielding factor gets worse again. We do not have an
intuitive explanation for this phenomenon but it is be important to remember when designing theshields. The axial shielding factor also gets worse with larger values of R, as shown in Figure 5.11,
but that is less surprising.

5.3.4

Layers of Magnetic Shields

We can extend the transverse and axial shielding factors ST and SA to the case where wehave multiple concentric magnetic shields with lengths Li and radii Ri , where i = 1 denotes theinnermost magnetic shield. To keep the expressions simple we assume they are made of the samematerial with magnetic permeability µR and thickness t ≪ R1 , and that Li ≫ Ri so we do nothave to worry about field lines that enter through the ends of the shields.

177

Figure 5.10: Comsol Multiphysics was used to calculate SA for an cylindrical shield with R = 0.25
meters, t = 2 mm and µR = 80, 000 for various values of L. The results of the Comsol simulationsashown in green. Comsol results with L ≥ 4 meters are then fit to a two parameter model+ b.
L
The prediction of Equation 5.30 is shown in blue, which slightly overestimates the Comsol results
but has the same general behavior.

Figure 5.11: The prediction SA vs R given by Equation 5.30. The calculation assumes L = 4
meters, t = 2 mm and µR = 80, 000.

178
The method in Subsection 5.3.2 where we calculated ST via a system of equations derived
from boundary conditions can be extended to the case where there are many cylindrical shields. We
define SiT =µR t
to be the transverse shielding factor of a cylindrical shield, ignoring the addition
2Riof one that gives the correct answer in the limit t → 0. Starting with the case of two nested shields
we have:

2 !
R
1
S2T nested = 1 + S1T + S2T + S1T S2T 1 −R2
 2 !
µ2R t2
R1
µR t
µR t
1−
=1+
+
+
2R1 2R2 4R1 R2
R2


(5.35)The total transverse magnetic shielding factor is equal to 1 plus sum of the individual shielding
factors plus another term that is proportional to the product of the two shielding factors. As seenin Figure 5.8 the transverse shielding values for the individual shields with reasonable parameters
will be on the order of 100, so the final term in Equation 5.35 will dominate. Importantly, this term
tells us that by using two nested shields we get to multiply the individual shielding factors. If we
had instead just doubled the thickness of one of our shields, by Equation 5.24 we would have only
R2 √doubled the shielding factor.17 Note that the shielding factor is maximized when
= 3 [73].
R1
We can extend the expression further to a system of n cylindrically nested shields:

SnT nested =1 +

n
X
i=1

+

SiT +

n−1
n
XX


1−

i=1 j>i

n−2
n
X n−1
XX
i=1 j>i k>j17

SiT SjT

SiT SjT SkT


1−

Rj−1
Rj

Rj−1
Rj

2 !

2 !


1−

Rk−1
Rk

2 !
+ ...

(5.36)

See this article [74] for an interesting discussion of modeling magnetic shielding as an electrical circuit, which isinteresting in its own right and explains why better shielding is achieved through multiple layers instead of thicker
shielding.

The final term in this expression is equal to

Qn−1

T T
i=1 Si Sn


1−

179
2 !
Ri
. This term is typicallyRi+1

the largest as it is proportional to the product of all the individual shielding factors.
We can also write down the axial shielding factor for 2 and n nested cylindrical shields.
i to be equal to the right hand side of Equation 5.30 without the +1 and, following
We define SA

Sumner [72], we find:

S2Anested = 1 + S1A + S2A + S1A S2A

SnA nested =1 +

n
X
i=1

+

SiA +

n−1
n
XX

SiA SjA

i=1 j>i

n−2
n
X n−1
XX

SiT SjT SkT

i=1 j>i k>j





L1
1−
L2

Lj−1
1−
Lj

(5.37)






Lj−1
Lk−1
1−
1−
+ ...
Lj
Lk

(5.38)For the transverse shielding factors we are guaranteed to get terms that are proportional to productsof the individual shielding factors because Ri+1 > Ri by definition. On the other hand, if thecylindrical shields all have the same length the net axial shielding factor will only be the sum of
the individual axial shielding factors, as can be seen in Figure 5.12. This tells us that we want ournested shields to grow in both radius and length as we move from the innermost shield outward.

5.3.5

Shielding Induced Gradients

Up to this point we have calculated shielding factors for axial and transverse magnetic fields.
These are important because they will inform us what sort of shields allow us to reduce Earth’s 500
mG field to ≲ 0.25 mG which we require, as discussed in Subsection 5.2.1. It is equally importantthat we keep the magnetic field gradients to be ≲ 0.25 mG/cm.
We can use Comsol to simulate the magnetic field gradients experienced by the ions due toshielded uniform transverse and axial magnetic fields. We define these “gradient shielding factors”

180

Figure 5.12: SA when there are two layers of magnetic shielding. The inner shield √has R1 = 0.25m, t = 2 mm, µR = 80, 000 and L1 is varied. The outer shield has R2 = 0.25 × 3 m, L2 = 4m, t = 2 mm and µR = 80, 000. The plot shows the prediction of Equation 5.37 in blue and theresults from Comsol simulations in green. Comsol verifies that the shielding factor grows as L1 gets
shorter than L2 .

Byext
Bzext
Bxext
grad
, which have units of distance. We are
and
S
=
=
T
∂Bzint∂Bxint∂Byint∂z∂x∂yinterested in calculating these gradient shielding factors far away from the edges of the shields as
grad
to be equal to SA=

we have already seen how transverse and axial fields decay as they enter the openings of the shields
in Subsections 5.3.2 and 5.3.3 respectively.
A single layer of mumetal shielding with R = 25 cm, L = 4 m, t = 2 mm and µ0 = 80, 000 has
a axial shielding factor of ∼ 30. Figure 5.13 shows the magnetic field as a function of z at ρ = 0.
We can see that the magnetic field gradient is ∼ 0.1 mG/cm over the central meter of the region
gradwith an external field of 100 mG, giving us SA∼ 1000 cm. Because the axial shielding function

in Equation 5.30 is fairly complicated this relationship should be checked for different parameters
grad
R, L and t, but SA∼ 1000 cm should provide a reasonable order of magnitude estimate.

With the finest mesh and highest precision I was able to manage, Comsol was still unable
to resolve the residual magnetic field gradients for transverse fields in an infinitely long magnetic
shield. We could conclude that STgrad ≫ 1000 cm, but in practice the dominant transverse gradients181(a) Full Plot(b) Zoomed In

Figure 5.13: A Comsol simulation of the magnetic field vs z along the center (ρ = 0) of a single
mumetal shield with R = 25 cm, L = 4 m, t = 2 mm and µ0 = 80, 000. (a) The shield is centeredat z = 0 and is in a background magnetic field of 100 mG along the z-axis. The blue trace shows
how the magnetic field has a slight increase from 100 mG and then falls as it enters the magnetic
shield at z = ±2. (b) The axial magnetic field has decayed to ≲ 5 mG over the region betweenz = ±1 where the ions travel, and the gradient is ≲ 0.25 mG/cm over this region as well.

182
will likely be caused by magnetized regions of the innermost magnetic shield. The consequences ofthese regions and how to minimize them are discussed in Subsection 5.3.7.
Fortunately, both external transverse and axial magnetic fields do not directly induce worryingly large magnetic field gradients inside the magnetic shielding. In the case where there is anexternal magnetic field gradient, the transverse and axial shielding factors are higher than thosefor uniform fields [75].

5.3.6

Seams and Holes

Up until now we have assumed that our magnetic shields will be perfect cylinders. This will
not be the case for two reasons. The first is that our cylinders will need holes to let laser light andvarious other parts of the apparatus through. The second is that mumetal cylinders with a R ∼ 0.5
meters and L ∼ 4 meters are hard to make, ship and install around our apparatus. Instead eachcylinder will be made of two half cylinders, or “clam-shells,” that attach together at a seam. In
this subsection I will address each of these imperfections.
We can approximate a hole in a magnetic shield as a magnetic dipole, so long as its diameterd is small compared to the shield’s characteristic lengths L and R [73].18 We can estimate themagnitude of the dipole to be µhole ≈ Bext

4π 3
d . This will cause a magnetic field and magnetic
3µ0

field gradient at the center of the cylinder, where the ions live. From dimensional analysis, themagnitudes of the magnetic field and its gradient will approximately beµ0 12µhole
d3≈4
Bext respectively.
4π R4
R4

µ0 3µholed3≈
Bext and
4π R3
R3

If we just had one shield with R = 25 cm and did not reduce the background magnetic fieldfrom 500 mG, we would require any holes to have d < 2.0 cm and d < 3.6 in order to keep themagnetic field and its gradient less than 0.25 mG and 0.25 mG/cm, respectively. As discussed in18

We can think of the shield as a bunch of magnetic dipoles that conspire together to reduce the magnetic field inthe shielded volume. A hole in the shield is equivalent to adding a magnetic dipoles pointing in the opposite directionin the same position as the hole.

183
Sections 5.4 and 5.5 we plan on using coils to reduce the background field to ≲ 100 mG and will
have multiple layers of shielding. With these precautions in place, holes in the shields will likely not
be a problem. Even so, it would be good to check this via simulations before building a finalized
design.19

Figure 5.14: It is possible that there is an opening with angle θ between the two halves of ourmagnetic shields. This could reduce the parallel and perpendicular shielding factors for the two
types of transverse magnetic fields shown in the image.

The other problem, that our cylinders will be made of two halves clamped together, could
be more significant.20 Suppose that the two halves of the shields do not quite meet each other sothere is an opening with angle θ on both sides of the shield, as shown in Figure 5.14. This opening
breaks the symmetry between magnetic fields along the x and y axes, which so far we have lumped
together when calculating ST . We now need to find both the parallel and perpendicular ST values
for the transverse magnetic field that are parallel and perpendicular to the opening.
We can see the calculated parallel and perpendicular transverse shielding factors as a function
19

If holes prove to be a problem, the penetrating magnetic fields can be minimized with cylindrical ferromagnetic

collars mounted over the holes [73]. See the cited article for more details.
20

The Imperial College eEDM team ran into this issue when they first built their magnetic shields. They havesince fixed the problem, and are good people to talk to if we have further questions.

184of θ in Figure 5.15. Shields with an opening are still reasonably good at shielding transverse
magnetic fields that are perpendicular to the gap for small values of θ, but are horrendous atshielding parallel transverse magnetic fields. We can mitigate this problem by 1) having the seamof subsequent layers of magnetic shielding be rotated and 2) ensuring the shield halves overlap one
another instead of just meeting at the seems.

Figure 5.15: Transverse shielding factors calculated in Comsol for an infinitely long single shieldwith R = 0.25 m, t = 2 mm, and µR = 80, 000. The opening angle, defined in Figure 5.14, isvaried between 0 and 20 degrees. Perpendicular magnetic fields are still shielded reasonably wellwith small opening angles, but ST drops to about 3 for parallel magnetic fields at an opening angleof just 0.25◦ .

Say that the two halves of the shield overlap one another by 10 degrees but there is a small
gap between the pieces of mumetal. How will the shielding factor change as a function of distance
between the two shields? Figure 5.16 shows that, again, the parallel shielding factor falls rapidly
according to simulations. While parallel ST is over 300 for our expected parameters when the shield
halves touch, it drops to about 40 if the distance is just 0.5 mm.
The discussion so far about gaps between the shields has avoided magnetic field gradients.
As a general rule, when the parallel ST is far below its intended value ∼ 300 in Figures 5.15 and185

Figure 5.16: Transverse shielding factors calculated in Comsol for an infinitely long single shieldwith R = 0.25 m, t = 2 mm, and µR = 80, 000. The shields are overlapped by θ = 10◦ . The gap
between the shields is varied between 0 and 10 mm. Surprisingly, perpendicular magnetic fields areshielded better with a small gap between the shields. However, parallel magnetic fields can findtheir way through a small gap even if the shields are overlapped.

5.16, simulations show that the magnetic field gradients that leak into the center of the shield areunacceptably large as well. Because the gradients are linked to whether or not ST is acceptably
large along both the parallel and perpendicular axes, this does not need to be treated as a separate
issue.

5.3.7

Degaussing

As mentioned in Subsection 5.3.5, improperly magnetized parts of the innermost shield cancause magnetic fields and gradients at the position of the ions. Highly ferromagnetic materials
like mumetal are magnetically fragile, standard handling and assembly of the shields can resultin magnetized regions. Fortunately, there is a standard procedure known as degaussing that can
reduce the residual magnetic fields to well below our required sensitivity [73].
The idea of degaussing is to apply an AC magnetic field that is originally large enough to186saturate the shield (∼ 1 G for mumetal) and then ramp the AC field to zero. If done properly, themagnetization of the shield will follow the applied magnetic field to zero, leaving the shield in its
best state to have the theoretically predicted shielding factors.
The theory behind degaussing is explained well in the first two sections of [76],21 and Zach
Lasner gives a detailed explanation of how it was implemented in the ACME II experiment insection 5.2 his thesis [77]. ACME II required five(!) layers of mumetal magnetic shielding for theireEDM result. While we have not yet degaussed magnetic shields in the JILA EDM experiment,
fortunately we do not need to reinvent the wheel.22

5.4

Magnetic Shield Design – Plan E
Now that I have introduced the generation three experiment (Section 5.1), covered its mag-netic systematics (Section 5.2) and described magnetic shielding (Section 5.3), I will now outline
our designs for the magnetic shields. The shielding requirements will depend on if we choose plan
A or E. Shielding is more complicated if we choose plan A which requires B2,0 ∼ 4 G/m to generatethe ∼ 100 Hz f 0 . This magnetic field would likely saturate the inner layer of magnetic shielding,complicating matters. Magnetic shielding with plan A will be discussed in Section 5.5.
Plan E, where we generate f 0 ∼ 100 Hz by applying a 1.6 mG magnetic field that rotates at
frot , poses no risk of saturating the shields. With this plan we only have to worry about the lab
magnetic field saturating the shields. Using the B vs H curve in Figure 5.6, we can use Comsol to
calculate the transverse and axial shielding factors for a single shield as a function of the external
magnetic field. The results of these simulations are shown in Figure 5.17, which show that the SA
starts to drop at ∼ 0.5 Gauss while ST remains constant until almost 5 Gauss. The important
conclusion to draw is that the calculated shielding factors are on the edge of being effected by
21

The paper [76] explains that degaussing was invented out of necessity – magnetized submarines made easy targets

in WWII.
22

Future JILA EDM graduate students should be able to find all of the information they need in the three citationsof this subsection.

187
saturation effects from Earth’s magnetic field ∼ 0.5 G.

Figure 5.17: Comsol simulations of axial and transverse shielding factors vs applied external
magnetic fields. The shield has R = 0.25 m, t = 2 mm and the B vs H curve defined in Figure 5.6.
SA is calculated with L = 4 m, while ST is calculated assuming L ≫ R.

In order to be confident that our outermost magnetic shield is not saturated we will build
pairs of lab-scale x, y and z coils to reduce the magnetic field around our apparatus. This will
be similar to the x, y and z coils that surrounded the generation two ion trap which brought themagnetic field to ∼ 10 mG in all directions. In generation three these coils, and the volume of space
they enclose, will be much larger. This sort of lab-scale shimming has been done at Nick Hutzler’s
lab at Caltech, where 6 ft x 7 ft x 8 ft coils reduce the ∼ 420 mG background field to ≲ 10 mG
within ∼ 15 cm from the center of the coils [78].
Our shields will a few meters long to enclose the ∼ 2 meter long bucket brigade, so we want
to reduce the magnetic field over length scales that are an order of magnitude larger. This would
be more challenging, except that we only want the field to be ≲ 100 mG. This is enough to preventthe mumetal from saturating and will relax our required shielding factors by a factor of ∼ 5. Withan ambient magnetic field is ∼ 100 mG, we need SA > 100 and ST > 400 to meet the requirements
spelled out in Table 5.3. To be conservative, and because magnetic shielding is notoriously finicky
to implement properly, we will design the shields to have SA > 1000 and ST > 4000. This will
also make imperfections from seams and holes in the shields discussed in Subsection 5.3.6 not as188
worrisome.
There are a number of constraints on the parameters of our shields. First, the curtained off
section of our lab that contains the apparatus and lasers is about 6.2 meters long. To leave room
for students and postdocs to walk around the magnetic shields we would like the length of ouroutermost and longest shield to be no more than 5 meters. The radius of our outermost shieldwould ideally be smaller than one meter so there is room for the optics, electronics and everythingelse that needs to be around the apparatus. On the other hand, we cannot make our innermost
magnetic shield too small. The innermost shield must be longer than the two or three meter long
Bucket Brigade or else there will be large magnetic gradients where the ions are coherently evolving.
Additionally, the innermost shield’s radius must fit around the apparatus and must be large enoughthat holes and imperfectly degaussed regions of the shield are not problematic.
To satisfy all of these constraints, I propose that we have three layers of mumetal shielding√
with t = 2 mm.23 They will have R1 = 0.25 m, R2 = 0.25 × 3 = 0.433 m, R3 = 0.75 m, L1 = 3.5m, L2 = 4 m and L3 = 4.5 m. These magnetic shields meet our practical constraints and shielding
requirements. Figure 5.18 shows the axial magnetic field along ρ = 0 in an external magnetic field
Bz = 100 mG. The magnetic field over the central two meters is ∼ 0.1 mG meaning that SA ∼ 1000.
Additionally, the magnetic field gradient∂Bz≲ 0.01 mG/cm for the central 2 meters, comfortably∂zbelow the required 0.25 mG/cm limit.
The same simulation that produced Figure 5.18 tells us that the magnitude of the gradientpeaks at about 3 mG/cm at z = ±2 meters, at the openings of the intermediate shield. The
magnitude of the magnetic field peaks close to 200 mG at z = ±2.5 meters, just outside theopening of the largest shield. While the axial magnetic field and its gradient are comfortably small
at z = ±1 meter, they are both too large at z = ±1.5 meters. This means that if we want to23This is a standard thickness for mumetal shields. Thicker shields will be more expensive and heavier, but will

improve SA and ST which are proportional to t.

189(a) Full Plot(b) Zoomed In

Figure 5.18: The axial magnetic field and its gradient in three layers of magnetic shielding designed
for plan E as described in the text. The external magnetic field is Bz = 100 mG, and we cansee that SA > 1000. The axial magnetic gradient is much smaller than the 0.25 mG/cm that we
require.

190
lengthen the Bucket Brigade, we will need shields with lengths approaching the 6.2 meters of space
we currently have.
This configuration of three shields works for the axial field, but what about the transverse
field? According to Comsol, ST = 3.1 × 106 , so a 100 mG external transverse magnetic field will
result in a negligible ∼ 33 nG magnetic field far from the openings of the shields. As discussed in
Subsection 5.3.2, transverse magnetic fields decay faster than axial magnetic fields as a functionof distance from the ends of the shield. If the shields work as advertised, transverse fields andgradients will not be a problem.
What if we cannot completely remove the issue of transverse magnetic fields leaking throughthe seams? We can combat this issue by rotating the seam locations in each shielding layer by
45 degrees, as shown in Figure 5.19. We assume that the layers of the shield are θ = 10 degrees
overlapped but have a 2 mm gap between layers. Note that Figure 5.16 tells us that the shielding
factor for magnetic fields parallel to the seam is only 12 in this case instead of ∼ 320 if there were
no seam. The net ST in this configuration depends on the orientation of the external transverse
magnetic field, but ST ≳ 5, 000 in all cases. This strategy of rotating the seams for each layer ofmagnetic shielding should mitigate the issue.
We have not settled on the number or size of holes needed in the magnetic shields. Whenthat has been decided, these simulations should be revisited to determine if the holes will cause anissue for the magnetic shielding. As long as the holes are not a problem, this geometry should give
shielding factors that are 10 times larger than we need and stomp out the magnetic field gradientsto a comfortable level.

5.5

Magnetic Shield Design – Plan A
Kia Boon Ng discusses two ways to implement plan A in his thesis [4]. The first option is to

apply a magnetic field gradient B2,0 ∼ 4 G/m that has the opposite sign for every bucket of ions, as191

Figure 5.19: A schematic of the seams in three layers of concentric magnetic shielding. Eachmagnetic shield will be made of two half cylinders that are attached together. The gaps visible inthe shields indicate the location of the seams, but the layers will have ∼ 10 degree overlap at thesespots. This diagram shows the optimal location of their seams to keep ST ≳ 5000 for magnetic
fields along x and y, given the shielding parameters proposed in Section 5.4, 10 degrees of layer
overlap and 2 mm gaps between the layers. ST will be larger if the gaps are smaller than 2mm.

shown in Figure 5.20. The main difficulty with this approach is that, in order to have a coherence⃗2,0∂2Btime τ ≳ 20 seconds, we have the magnetic uniformity requirement || ≲ 12.5 G/m3 .24 The∂z 2buckets of ions will likely be spaced ∼ 30 cm apart from one another, and the magnetic field gradient⃗2,0∂2B
needs to flip sign over that distance. Doing so while keeping≲ 12.5 G/m3 at the bucket∂z 2
locations with a few coils seems challenging, which is discussed in Section 8.2.2.2 of Kia Boon Ng’s
thesis [4]. The magnetic shielding for this version of plan A is the same as plan E as discussed in
the previous section.
24

This requirement assumes that ions maintain their secular amplitude for their entire coherence time. If this were⃗2,0
∂2B
the case and̸= 0 G/m3 , molecules with large secular amplitude would have a larger frequency than molecules∂z 2with a small secular amplitude. If the gradient is larger than 12.5 G/m3 , the two sets of molecules will dephase after
20 seconds. This assumption will not hold if ion-ion collisions change the secular amplitude of the ions involved.
These collisions do take place but are difficult to characterize, so the 12.5 G/m3 requirement is conservative.

192

Figure 5.20: A schematic depicting how we can implement plan A so B̃ changes sign between every
bucket. Clouds of ions are shown in blue, which are spaced by a distance Λ. The vertical dotslines are visual aids which indicate the boundary between buckets. There are four coils with radius
R for every bucket with currents that drive the magnetic field gradients B2,0 . The magnetic field
is indicated by the dashed lines that end in arrows. As the ions move left to right, the currents
through each coil change to keep the magnetic field gradient a constant magnitude at the bucket
locations. This image is from Kia Boon Ng’s thesis [4].

The simpler option to meet the magnetic field uniformity requirement is to apply B2,0 ∼ 4⃗2,0 = B2,0 − ⃗x −G/m over the entire Bucket Brigade. The magnetic field gradient is defined as B⃗y + 2⃗z , so this corresponds to a ∼ 8 G/m magnetic field gradient along the 2 meter long z-axis.
While it will be easier to make the magnetic gradient uniform, the challenge is that this field will
saturate the innermost magnetic shield.
We can generate B2,0 ∼ 4 G/m with

⃗2,0
∂2B
∼ 8.3 G/m3 with 32 coil pairs of radius 5.25
∂z 2

inches, with locations and currents shown in Tables 5.4 and 5.5. These coil positions and currents⃗2,0∂2B
were found numerically to minimize
and keep B2,0 ∼ 4 G/m over the two meter long bucket∂z 2brigade, with the constraint that the coils must be evenly spaced.25 The magnetic field created bythese coils along ρ = 0, as well as its first and third order gradients, are plotted in Figure 5.21. We
could generate a magnetic field that meets these requirements with fewer coil pairs if we increase
the coils’ radii. However, we want the coils to be as small as possible to minimize the magnetic
25

Kia Boon Ng mentions in Section 8.2.2.3 of his thesis that the homogeneity of B2,0 can be controlled with “somenumerical modeling” [4]. This result is one possible solution.

193
field at the innermost shield. The vacuum chamber containing the Bucket Brigade will have a 5
inch radius, so a radius of 5.25 inches is as small as the coils can be unless we put them in thevacuum chamber which is already crowded with electronics [3].
Table 5.4: Coil Positions and Currents to generate uniform B2,0 ∼ 4 G/m, Part 1
Coil Pair Number

Position (cm)Current (Amp Turns)

1±2.698±0.9372±8.095±2.8103±13.492±4.6834±18.889±6.5575±24.286±8.4316±29.623±10.3067±35.079±12.1818±40.476±14.0579±45.873±15.93310±51.270±17.81311±56.667±19.69012±62.064±21.57513±67.460±23.45314±72.857±25.34715±78.254±27.22316±83.651±29.136

The magnetic field generated by these coils along ρ = 0 is shown in Figure 5.22 with andwithout the three layers of magnetic shielding discussed in Section 5.4. We can see that the shields

194
Table 5.5: Coil Positions and Currents to generate uniform B2,0 ∼ 4 G/m, Part 2
Coil Pair Number

Position (cm)Current (Amp Turns)

17±89.048±30.99618±94.444±32.97419±99.841±34.71320±105.238±36.96221±110.635±38.87922±116.032±40.60323±121.429±41.86424±126.825±42.81225±132.222±46.335

26±137.619±46.56727±143.016±49.68028±148.413±50.58529±153.810±52.76030±159.206±53.62131±164.603±156.33332±170.000±91.189slightly increase the slope from 8.04 G/m to 8.11 G/m, a phenomenon explained in [75]. Since theprecise values of B2,0 and f 0 are not particularly important for our experiment, this 0.9% change
will not be a problem.
Figure 5.23 shows the magnetic field without magnetic shielding at ρ = 0.25 meters, the195radius of the innermost magnetic shield discussed in Section 5.4. This plot clearly shows thatthe innermost shield, which is 3.5 meters long and stretches between z = ±1.75 meters in the
figure, will saturated towards its ends due to the magnetic field generated by the coils. This shouldmake the shield less effective at reducing the background magnetic field and potentially allow large
systematics.
The safest approach is to assume the innermost shield does not shield the external field atall. Its role would be to protect the remaining shields from the magnetic field generated by thecoils. As we see in Table 5.5, the outermost coils have the largest number of amp turns and are∼ ±1.65 meters from the center of the Bucket Brigade. If we keep the radius of the inner shield
the same at R1 = 0.25 meters and increase its length slightly to L1 = 4 meters to better enclosethe coils, the magnetic field just outside this shield at ρ = 0.3 meters will peak at about 60 mG.
This means that we could have similar total magnetic shielding to plan E with four layers. These√layers would all have t = 2 mm, with R1 = 0.25 meters, R2 = 0.3 meters, R3 = 0.3 × 3 = 0.520
meters, R4 = 0.9 meters, L1 = 4 meters, L2 = 4 meters, L3 = 4.5 meters and L4 = 5 meters.
To see if another layer of shielding is necessary, we can look at the magnetic field along ρ = 0
with the coils and plan E’s shielding, with and without an external axial magnetic field ∼ 100
mG.26 These two magnetic field profiles are shown in Figure 5.24a. The difference in these twosimulations, displayed in Figure 5.24b, shows the axial shielding while the inner shield is partly
saturated. We can compare this magnetic field profile with Figure 5.18, which shows the axial
shielding without the magnetic coils. Remarkably, Comsol says that these magnetic field profiles
agree to within 10−8 mG at all points along ρ = 0, indicating that the partial saturation of themagnetic shields due to the coils will not make the shields less effective.
The takeaway is that the magnetic shields described in Section 5.4 should work for both26

SA is more important to check because SA < ST generally and axial magnetic fields leak farther into the shields

as we saw in Subsection 5.3.3. Conveniently, the coils, finite length shields and external axial fields can all be simulated
in 2D axially-symmetric models that are much faster to simulate than 3D models.

196
plans A and E. According to Comsol, we should not need an additional layer of shielding to prevent⃗2,0 from harming the magnetic shielding of the inner layer. It is important to note that theseB⃗2,0 many times over when we flip
simulations do not include the effect of us reversing the sign of B
B̃. To be safe with the magnetic shielding in plan A, an additional layer of shielding at ρ = 0.3
meters would be enough to keep the shielding factors large enough if the inner shield is not as
effective as Comsol suggests.
To summarize, we saw that magnetic systematic effects and magnetic shielding would bothbe more manageable if we choose plan E. Applying a uniform magnetic field of ∼ 1.6 mG thatrotates at frot is not something we have done at JILA before, but we expect that we can overcome
the new engineering challenge. If that fails we can still apply f 0 ∼ 100 Hz with plan A which wehave used in the previous two generations of the JILA eEDM experiment.

197(a) Bz

(b)

∂Bz
∂z

(c)

∂ 3 Bz∂z 3

Figure 5.21: The magnetic field and two of its gradients along ρ = 0 created by the coils with aradius of 5.25 inches and positions and currents given by Tables 5.4 and 5.5. The red vertical lines
indicate the Bucket Brigade region between z = ±1. There are no magnetic shields around the coilsin these calculations. The average first and third order gradients between z = ±1 are 8.04 G/m
and 8.3 G/m3 . This will create f 0 ∼ 100 Hz while keeping the coherence time above 20 seconds
even in the absence of ion-ion collisions. The peak magnetic field along ρ = 0 is ∼ ±15 Gauss andoccurs at z ∼ ±1.7 meters. Ideally the magnetic field would drop to zero outside of z = ±1, but itneeds to continue increasing to keep the third order gradient in the Bucket Brigade region small.
Further numerical optimization may be be able to decrease this peak field closer to ∼ ±10 Gauss.

198

(a)

(b)

Figure 5.22: (a) Comsol simulations of Bz vs z along ρ = 0 from magnetic field created by the coilswith parameters given in Tables 5.4 and 5.5. Blue and red dots are values of the field with andwithout magnetic shields. (b) The difference between the two magnetic fields in (a). The difference
between the two fields is caused by the magnetic shields changing the field made by the coils.

199

Figure 5.23: The magnetic field generated by the coils in Tables 5.4 and 5.5 along ρ = 0.25 meters
without magnetic shields. There total magnetic field is shown in blue, its radial component in green
and its axial component in red. The innermost mumetal shield is centered at zero and stretches
between z = ±1.75 meters, and is saturated at the peak ∼ 1.5 Gauss magnetic field (see Figure5.6). This will decrease the shielding efficacy of the innermost shield.

200

(a)

(b)

Figure 5.24: (a) The magnetic field Bz created by the coils in Tables 5.4 and 5.5 with the magnetic
shielding for plan E described in Section 5.4. The red field also has a 100 mG external magnetic
field along the z-axis. (b) The difference between the two magnetic fields in (a) which shows how
well the external magnetic field is shielded with the innermost shield partly saturated.

Chapter 6

Conclusion“I open at the close.”- J. K. Rowling, Harry Potter and the Deathly Hallows
This chapter is mostly an excuse to get in another Harry Potter quote, which I have alreadyaccomplished. I would like to wish everyone working on generation three of the JILA EDM experiment the best of luck. I think we don’t get enough credit for how complicated this experiment is,
especially given how few people work on it. Most AMO experiments are simpler, get results fasterand publish more frequently. I think the experiment is incredibly cool and worth working on, butthere really are a lot of steps between producing ions and making a particle physics measurement.
I’d like to conclude by presenting a plot that I’ve made, shown in Figure 6.1, which closely
resembles a plot made by Dave DeMille. He was my advisor for my first two years in graduate school,
so I think I can get away with it. It shows the relatively recent history of eEDM measurements
and how their results compare to predictions made by beyond the Standard Model theories.
I have been incredibly lucky to work with/for many of the professors responsible for theseexperiments. The Amherst measurement was led by Larry Hunter, my undergraduate advisor, ata small liberal arts school with little funding and no graduate students to do all the work. Despite
these constraints he reduced the limit on the eEDM by an astonishing factor of 21 [79]. Two of202

Figure 6.1: The black vertical lines are 90% confidence upper limits on the eEDM set by experiments
dating back to 1989. That is to say, each experiment is 90% sure that the true value of the eEDM
lies to the left of the black line. The experiments are labeled by the institution/collaboration
which made the measurement and the year the result was published. The height of the line is
proportional to the publication year, in some funny units. In the background are boxes whichvery roughly indicate the proposed value of the eEDM by different theories. The Standard Model
prediction is in a blue box,1 the beyond the Standard Model theories are in green boxes. The roughvalues of the eEDM predictions made by these theories were taken from Dave DeMille. The JILA
generation two result is in blue and in a larger font because I was involved in that measurement
and I made this plot.

the professors behind the Imperial College result, Ben Sauer and Mike Tarbutt, were kind enoughto invite me to work in their lab in London for the summer of 2023 [47]. Dave DeMille is one of
the three professors in charge of the ACME collaboration [28, 68]. I never met Eugene Commins,
the professor who led of the experiment at Berkeley, but he advised both Larry Hunter and Dave
DeMille who I learned much from [80, 81, 82]. And of course Eric Cornell and Jun Ye are myadvisors at JILA, who have done their best to make me a qualified AMO physicist [27, 38].
I find it incredibly meaningful my work in graduate school resulted in a new measurementon this plot. Thank you to everyone who made my journey through graduate school possible.
1

In Subsection 1.3.2 I state that the Standard Model prediction of the eEDM is 5.8 × 10−40 e cm, a value muchsmaller than is indicated in Figure 6.1. This larger value indicates the sensitivity required to measure the SM predictedvalue of the T-violating CS term discussed briefly in Section 3.2, which will be measurable before the eEDM [83].

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Appendix A
Bz and BzB Measurements

In Section 3.1 we saw that the off diagonal terms in our effective Hamiltonian gave us thefollowing systematic errors:δf DB =

0
DR + ∆D δ∆R∆0 δ∆DB + ∆D δ∆B + δ∆DR δ∆BR + δ∆R δ∆DBR
BR ∆ δ∆+f|f00 ||f00 |2(A.1)

Later in that section we estimated the size of the δ∆ terms in this expression, which depend on
Bz and BzB . We were able to measure the magnetic field along the x and y axes by using the
Zeeman shift described in Subsection 3.3.3, but we have no analogous effect that can easily probe
the magnetic field along the z axis.
Instead we can find upper limits on Bz and BzB by carefully investigating some of our otherfrequency channels. We can write out the full expression for f DR analogously to how we wrote
Equation 3.2 for f DB :

2

2

0
D
0 D
∆0 δ∆DR + ∆D δ∆RDR (∆ ) + (∆ )
R∆ ∆
−
δf
−
δf
0
0
|f00 |
2|f00 |2
|f00 |2"#X
1
δD2
2R̃(∆0 + D̃∆D )δD + δD2+
D̃R̃ 0 − D̃B̃ R̃δF0+ ...
16
2|f0 |
2|f00 |2

f DR = δf0DR +˜ D̃=±1
B̃,R̃,I,

(A.2)

211
Keeping only the largest terms as discussed in Section 3.1 we find that:f DR = δf0DR ++ f BR∆0 δ∆DR + ∆D δ∆R + δ∆B δ∆DBR + δ∆DB δ∆BR
|f00 |∆0 δ∆DB + ∆D δ∆B + δ∆BR δ∆DR + δ∆DBR δ∆R|f00 |2

We now assume that the only frequency shifts included in δf0DR are Zeeman effects with

(A.3)

⃗ /|E|⟨E⃗ · B⟩=⃗⟨Ê · B⟩

1 so they show up ∼ 460 times larger in δf0R .1 In that case we can write:

f DR −

δgF R ∆0 δ∆DR + ∆D δ∆R + δ∆B δ∆DBR + δ∆DB δ∆BR
f =gF
|f00 |+ f BR∆0 δ∆DB + ∆D δ∆B + δ∆BR δ∆DR + δ∆DBR δ∆R|f00 |2(A.4)

Note that we are ignoring the ∆ contributions to f R . While these shifts have similar magnitude to
the ones on the right hand side of the above equation, they are suppressed by a factor of

δgF
and
gF

can be ignored.
Plugging in the values of δ∆R , δ∆B , δ∆BR , δ∆DR , δ∆DB and δ∆DBR found in Section 3.1
into Equation A.4 we find that:f DR −

B 2 2
0 D
36Bz (BzB )2 gF3 µ3B ∆0 ∆D
δgF R 6Bz gF µB ∆0 ∆D
BR 30Bz Bz gF µB ∆ ∆
f =
+
+
f
3
2
gF
h|f00 |frot
h3 |f00 |frot
h2 |f00 |2 frot

(A.5)

⃗ /|E|⟨E⃗ · B⟩
= 1 and Berry’s phase shifts that are entirely D̃-even, the⃗⟨Ê · B⟩⃗ /|E|⟨E⃗ · B⟩largest shift by far in our experiment is a Zeeman shift with= 1. This shift ideally does not show up in⃗⟨Ê · B⟩f R but it can due to the digitization of how we apply Vrot , see Appendix B 3 of our Systematics paper. We expect⃗ /|E|⟨E⃗ · B⟩
this shift to overwhelm other Zeeman shifts with
̸= 1 and Berry’s phase shifts which naturally show up⃗⟨Ê · B⟩in our B̃-odd frequency channels.
1

While there are Zeeman effects with

212
The second and third terms in this equation are much smaller than the first for reasonable valuesof Bz and BzB ≲ 1 Gauss. We can therefore drop the latter two terms and use our measured
values of f DR = 0.07(2) mHz and f R = −4.23(9) mHz to find that Bz = 80 mG in our generation
two experiment. We expect that this is an overestimate of the magnetic field as we measure
Bx ∼ By ∼ 10 mG before using our ions to further reduce this field, but we use Bz = 80 mG tocalculate the systematic shifts in Section 3.1.
We can go through a similar procedure for f DBR and f BR , in which case we find:f DBR −

0 D
36Bz2 BzB gF3 µ3B ∆0 ∆D
δgF BR 6BzB gF µB ∆0 ∆D
BR ∆ ∆
+
f
+
f
=
3
gF
h|f00 |frot
h3 |f00 |frot|f00 |2(A.6)

Unsurprisingly, the B̃-odd version of Equation A.5 allows us to set a limit on the magnetic field withthe opposite B̃ dependence, BzB . In this equation the term proportional to f BR is not negligible.
This is because the product of f BR , ∆0 and ∆D , the largest frequency channel other than f 0 andlargest ∆s, happens to have the correct switch state dependence to show up in f DBR . While wecannot neglect this term we do know its value, so it can be included when finding BzB . In this case
we also allow for a 5 mHz contribution to f BR from Berry’s phase2 that is not otherwise accounted
for which inflates the maximum possible value of BzB . Even with this all taken into account we findthat BzB < 88 mG.

2

As discussed in Section 3.5, Berry’s phase shifts can naturally show up in f B or f BR . While analyzing systematics

we found that we expect that a 5 mHz shift is split between these two frequency channels which we assumed showed
up entirely in f B as that would cause the largest systematic. Here we assume the effect is entirely in f BR .

Appendix B
The Harmonics of Erot

“Is it over now?”- Taylor Swift, Is It Over Now? (Taylor’s Version)
In order to apply Erot we have eight op-amps each amplify a 375 kHz sine wave. The output
voltage Vrot of each op-amp is sent to the top or bottom of one of our eight radial electrodes as
described in Subsection 3.4.2. These voltages result in our rotating electric field E⃗rot that we needto perform our EDM measurement. Our op-amps are imperfect and in addition to outputting Vrot
the op-amps also produce harmonics that oscillate at n × 375 kHz where n is a positive integer.
These harmonic voltages produce electric fields that contribute to Zeeman and Berry’s phase shifts
which cause systematic shifts as explained in Chapter 3.
In order to quantify these systematic shifts we measured the magnitude of the harmonic
voltages on each of our radial electrodes. The challenge of this measurement is that there is a
∼ 350 Volt fundamental signal at 375 kHz on each of the radial electrodes that is, by design, muchlarger than the harmonics we want to measure. If our measurement device is nonlinear and causes
any harmonic distortion itself it will be difficult to tell if the measured harmonics are really on thefins or artifacts of our measurement.
To that end I built a very linear custom filter designed to notch the fundamental voltage at

214
375 kHz, as well as the smaller RF voltage ∼ 23.5 V at 50 kHz. This filter was also designed tohave a relative maximum of transmission at 750 kHz, the frequency of the second harmonic. This isbecause the second harmonic was a particularly worrying source of systematics – it interacts withmagnetic fields in the xy-plane to create a Zeeman shift or ellipticity in Erot to induce a Berry’s
phase shift, both of which are relatively large systematics in our experiment.
The circuit diagram of the home built filter, attached to a radial electrode and our fancy
oscilloscope, can be seen in Figure B.1. The input voltage comes from one of our eight op-amps
that creates Vrot . This voltage is connected directly to the radial electrode, denoted in this circuit
diagram by the fin capacitor. The input voltage also goes to a system of four resistors which, if
unattached from the rest of the circuit, act like a voltage divider. The voltage division is either afactor of 7 or 14 depending on if the switch is attached high or low in Figure B.1. We made eightsmall identical boxes that contain the voltage divider and switch, one for each of the fins. They
were electrically connected to their op-amp and fin during the data collection.

Figure B.1: Circuit diagram of the electronics used to measure harmonics. The components aredescribed in the text.

We made one circuit containing the two notch filters in Figure B.1. These inductors are madeusing hand wound coils with an air core to prevent harmonic distortion, each with an inductance1=
LCωrot , ωRF . These components together have very low impedance at their resonant frequencies,of about 200 µH. The inductors were connected to capacitors which were chosen such that √causing Vrot and Vrf to be grounded and not reach the output port of our circuit. In order to makesure the resonance of the filters was very close to ωrot and ωRF , the main capacitors were connected215
to variable capacitors that could vary from ∼ 0 to 50 pF. These two notch filters were held ina separate grounded box that had an input (connected to a voltage divider box) and an output(connected to the scope).
The two boxes and scope were connected by standard BNC cables which had a total capacitance of about 337 pF. The cabling connected the notch filter box to our oscilloscope which had
an input impedance of 100 MΩ. The complete circuit had one additional resonance that allowed
a relatively large amount of the signal to pass through. This was engineered to appear near 2ωrot ,the frequency of the second harmonic. We added a third variable capacitor in parallel with thecabling (which we put in the same box as the notch filters) that tuned this final resonance. The
measured throughput of the entire circuit is shown in Figure B.2.
Figure B.2 only shows the throughput of the first and second harmonics of Erot . We measured
the throughput all the way out to the 17th harmonic for the “High V” or “switch up” setting of
our circuit, shown in Figure B.3.
This allowed us to measure the voltages on each of our radial electrodes and two endcaps(electrodes at the top and bottom of our ion trap we used to apply the DC trapping potential) up to
the 17th harmonic, as shown in Table B.1. Because we were particularly worried about the second
harmonic, we used our initial measurements of its amplitude to apply a “feedforward” signal, also
at 2ωrot to minimize the second harmonic as much as possible. The data shown in Table B.1 iswith the feedforward signal applied. Note that we saw a signal that stood out from the noise floor
for all 17 harmonics on the radial electrodes. This was only the case up to the third harmonic on
the axial endcaps; for the higher harmonics on the endcaps we conservatively report the noise floorof our reading modulo the frequency dependent conversion factor in Figure B.3 as the magnitude.
Table B.1 also includes the magnitude of the electric fields generated by the harmonic voltages
we measured. There are two pages of fairly dry math in Section VI B 5 of the Systematics paperthat explain how we perform this conversion.

216

Figure B.2: Throughput data from home built circuit in Figure B.1. It was collected by the JILA
electronics shop’s signal analyzer which also had a 100 MΩ input impedance. The blue “High V”
data was taken with the switch in Figure B.1 “up”, and the red data was taken with the switch“down”. The thin blue vertical lines indicate the three desired resonance frequencies ωRF , ωrot and2ωrot . The power and frequencies are given at the local maxima/minima recorded by the signal
analyzer, which reports the throughput power at discrete frequencies.

All I will say here is to comment on why the electric fields in the xy plane for the 7th, 9th,
15th and 17th harmonics are particularly large given the radial voltages. This is because when ndiffers from a multiple of 8 by ±1 is a special case. Suppose that the nth harmonic on all eight
electrodes has the same phase difference ϕn from Vrot . In this case the voltages for when n differsfrom a multiple of 8 by ±1 constructively interfere to create the electric field Ex,ynh . For theseharmonics we assume this worst case which results in comparatively large electric field magnitudes.
This effect explains why we measured all the way out to the 17th harmonic.

217

Figure B.3: This is the recorded power throughput of our circuit immediately before measuring thevoltages on our ten electrodes, as reported in Table B.1.

218

Table B.1: Constraints on electric fields from higher-harmonic voltages on radial electrodes andendcaps. For reference, the amplitude of the fundamental on the radial electrodes is ∼ 350 V. This
table is an extension of Table V in the Systematics paper.
Amplitudes (mV)

Fields (mV m−1 )nRadialAxial|δEx,ynh ||δEznh |

2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17

11
310
190
410
100
230
45
170
37
140
42
120
60
45
16
34

0.5
0.3
0.1
0.2
0.4
0.5
0.7
1.0
1.4
1.8
2.1
2.2
2.2
2.1
1.9
1.9

250
1296
649
1715
341
4000
154
2957
126
586
143
501
206
777
53
596

1.0
2.1
1.8
2.7
0.8
1.7
3.1
1.8
2.0
2.7
3.1
1.8
1.7
1.5
1.4
1.4