The ThF+ eEDM experiment: concept, design, andcharacterization
by
Kia Boon Ng
B.A., National University of Singapore, 2015
M.S., University of Colorado, Boulder, 2018

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 A. Cornell, Chair
Jun YeJames Thompson
John Bohn
Marco Nicotra

ii
Ng, Kia Boon (Ph.D., Physics)
The ThF+ eEDM experiment: concept, design, and characterization
Thesis directed by Prof. Eric A. Cornell

The Standard Model of particle physics is one of the most successful models that we have to describe
the universe at a fundamental level. However, the Standard Model is known to be incomplete. For instance,
the Standard Model does not explain the degree of abundance of matter over anti-matter. When theorists
make extensions to the Standard Model through new physics models, their models usually make predictions
on the value of the electric dipole moment of the electron (eEDM). By measuring (or placing improved
limits on) the eEDM, we can place constraints on these new theories. The eEDM experiments at JILA use
molecular ions to measure the eEDM. Molecular ions are good for two main reasons: (i) our molecules give
us enhanced sensitivity to the eEDM, and (ii) trapped ions allow us to take advantage of the long coherence
times of the eEDM-sensitive states. The Generation II JILA eEDM experiment using HfF+ concluded in
November 2022 by setting the world’s best limit. In our Generation III project, we seek to further improveupon our limit by switching over to ThF+ , which promises enhanced sensitivity to the eEDM. Herein, I
present (i) spectroscopy work performed on ThF+ to allow us to control the molecule in both the external
and internal degrees of freedom, (ii) design and characterization of a prototype of a multiplexed ion trap.

iii

Acknowledgements

Rating: ó ó ó ó ñComment: 5-star experience in JILA and CU Boulder. Many thanks to the whole JILA community,
and staff in the Department of Physics. Special thanks to Bosses, lab mates, support staff, andcollaborators, who have made my learning journey fruitful and exciting. Special mention to all myfriends and family members who have kept me sane and emotionally healthy. One star deductedbecause of disappointing food choices in Boulder (within graduate student budget).
Recommendation: Would strongly recommend to friends and family.

Contents

Chapter

1 Introduction

1

1.1

Background of the eEDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

eEDM measurement with molecular ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.1

JILA eEDM experiment with HfF+ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.2

JILA eEDM experiment with ThF+ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.3

How an eEDM experiment is done at JILA . . . . . . . . . . . . . . . . . . . . . . . .

5Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

2 Preparing ThF+ ions

11

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.3

Sample of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.4

Spectroscopic constants, energy scales, and fitting form

. . . . . . . . . . . . . . . . . . . . .

15

2.5

Vibrational progression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.6

Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.6.1

Search for high angular momentum state . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.6.2

Perturbations in the deep ultraviolet region . . . . . . . . . . . . . . . . . . . . . . . .

22

2.6.3

Auto-ionization into the ground vibronic state of ThF+ . . . . . . . . . . . . . . . . .

23

2.6.4

Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

v
3 State readout of ThF+

27

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.2

Rule of thumb for two-state detection using angle-resolved photofragments . . . . . . . . . . .

30

3.2.1

Angular distribution of photofragments . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.2.2

Energy of ejected photofragments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Angle-resolved photofragments in ThF+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.3.1

The [32.09]+266 nm transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.3.2

Photon polarization combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.4.1

Other important transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.4.2

Possible future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.3

3.4

4 Quantum control of ThF+
4.1

4.2

4.3

55

Spectroscopic constants of ThF+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.1.1

State preparation and readout for eEDM-sensitive state spectroscopy . . . . . . . . . .

56

4.1.2

Microwave spectroscopy on the J=1 to J=2 transition; Hyperfine coupling constantand molecular electric dipole moment . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.1.3

Ramsey spectroscopy within the J=1 eEDM-sensitive state; Magnetic g-factor . . . . .

64

4.1.4

Summary of Results & Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67Blackbody Radiation Excitation and T1 Relaxation Time . . . . . . . . . . . . . . . . . . . .

68

4.2.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4.2.2

Measuring the v=1 spontaneous decay lifetime . . . . . . . . . . . . . . . . . . . . . .

70

4.2.3

Suppression of blackbody radiation excitation . . . . . . . . . . . . . . . . . . . . . . .

72

Experiment sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4.3.1

State preparation and detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4.3.2

Ramsey sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

4.3.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

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5 Interlude
5.1

97

Revisiting wisdom lost in time

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

5.1.1

Wisdom up till now . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

5.1.2

Other exotic ways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.2

Feasibility of using ThF+ in an eEDM measurement . . . . . . . . . . . . . . . . . . . . . . . 105

5.3

Bucket Brigade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6 Ion Cloud & Decoherence

110

6.1

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

6.2

Ion cloud dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3

6.4

6.5

6.6

6.2.1

Studying mean field effects with spherical ion clouds . . . . . . . . . . . . . . . . . . . 114

6.2.2

Simulating ion dynamics with aspherical ion clouds . . . . . . . . . . . . . . . . . . . . 119

Decoherence from spatial inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3.1

Frequency dependent decoherence rate (naı̈ve decoherence rate) . . . . . . . . . . . . . 123

6.3.2

Suppressed decoherence from collisional narrowing . . . . . . . . . . . . . . . . . . . . 125

6.3.3

Net decoherence rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Collisional narrowing in the ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4.1

Collisional narrowing in experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.4.2

Collisional narrowing in simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.4.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Collision rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.5.1

Collision rate ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.5.2

More collision-related rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.5.3

Comparison of various collision-related rates . . . . . . . . . . . . . . . . . . . . . . . . 139

6.5.4

Heating of ion clouds in Paul traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.5.5

Relevance to the Bucket Brigade and ThF+ . . . . . . . . . . . . . . . . . . . . . . . . 141

Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

vii
6.6.1

Collisions that could change mF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.6.2

Spatial sampling by an ion in an anharmonic trap . . . . . . . . . . . . . . . . . . . . 145

6.6.3

Difficulty in measuring ion cloud properties in the experiment . . . . . . . . . . . . . . 146

7 Design of the Bucket Brigade

148

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7.2

Rings: Axial confinement & Translation of ions . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7.3

7.4

7.5

7.6

7.2.1

“Continuous rings” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.2.2

Discrete rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.2.3

Translation of ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.2.4

Shielding effect from Rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Rods: Radial confinement & Rotating electric field . . . . . . . . . . . . . . . . . . . . . . . . 160
7.3.1

Radial confinement in a Paul trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7.3.2

Rotating electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

7.3.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

The baby Bucket Brigade (b.B.B.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.4.1

Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.4.2

Trap loading & Ion kick out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.4.3

Remarks on wisdom acquired . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Electronics design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.5.1

Rod driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

7.5.2

Ring driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Parametric excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.6.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.6.2

Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.6.3

Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

viii
8 Generating f0 in the Bucket Brigade

189

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.2

Plan A: Co-moving axial magnetic quadrupole . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.3

8.4

8.5

8.6

8.2.1

Working principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.2.2

In the Bucket Brigade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

8.2.3

RF trapping field-induced decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

8.2.4

Shimming away gradients in Baxgrad . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8.2.5

Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Plan B: Berry’s phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.3.1

Working principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.3.2

In the Bucket Brigade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.3.3

New switches for the experiment: S and T . . . . . . . . . . . . . . . . . . . . . . . . . 211

8.3.4

fDS channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

8.3.5

Q-factor & Resonant Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

8.3.6

Spatial homogeneity of the dithering axial electric field . . . . . . . . . . . . . . . . . . 217

8.3.7

Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

Plan C: Uniform ambient magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.4.1

Working principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

8.4.2

In the Bucket Brigade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

8.4.3

Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Plan D: Transverse magnetic quadrupole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.5.1

Working principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

8.5.2

In the Bucket Brigade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

8.5.3

Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Plan E: Rotating magnetic field with time-oscillating currents . . . . . . . . . . . . . . . . . . 227
8.6.1

Working Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

8.6.2

In the Bucket Brigade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

ix
8.6.3
8.7

Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

9 Control System

232

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

9.2

Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

9.3

9.4

9.2.1

Legacy system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

9.2.2

Current system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
9.3.1

ARTIQ for the uninitiated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

9.3.2

Important folders and files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

9.3.3

Communication with auxiliary control computers. . . . . . . . . . . . . . . . . . . . . 241

9.3.4

The JILA eEDM suite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

10 Cryogenics

250

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
10.2 Design in the context of the Bucket Brigade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
10.2.1 Heat load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
10.2.2 Spreading the cold with the Cold Jacket . . . . . . . . . . . . . . . . . . . . . . . . . . 255
10.2.3 Thermal contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
10.3 Current status in the baby Bucket Brigade. . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

10.3.1 Household insulation foam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
10.3.2 Cryocooler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
10.3.3 Effective temperature and lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
10.4 Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
10.4.1 Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
10.4.2 Liquid nitrogen, LN2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

x10.4.3 Raining cryogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
10.4.4 Cold shield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

11 Conclusion

282

Bibliography285

Appendix

A Paul Trap and Mathieu Equations

291

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
A.2 Definition of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
A.3 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
A.4 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

B Typical Parameters for the Bucket Brigade

296

B.1 Trap parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
B.2 Internal state and trapping field parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

C Hamiltonian298

C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
C.2 Terms in the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
C.2.1 Hyperfine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
C.2.2 Omega-doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
C.2.3 Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
C.2.4 Stark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
C.2.5 Zeeman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
C.2.6 Offset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
C.2.7 eEDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

xi
C.3 Solving the Schrödinger’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
C.3.1 Time-independent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
C.3.2 Time-dependent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
C.4 Two-level effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

Tables

Table

2.1

Comparison of molecular constants of the ground state of ThF in experiment and theory. . .

20

2.2

Comprehensive list of tentative assignments of vibrational progressions. . . . . . . . . . . . .

21

3.1

Relevant parameters of fitted transitions in the REMPD spectroscopy of ThF+ . . . . . . . . .

49

4.1

Measured spectroscopic structural constants for X 3 ∆1 . . . . . . . . . . . . . . . . . . . . . .

67

4.2

Prediction of blackbody radiation excitation lifetime (in seconds) out of the v = 0 vibrational
level in X 3 ∆1 , in a radiative environment of the indicated temperature. . . . . . . . . . . . .

72

5.1

Comparison of parameters of ThF+ and HfF+ . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2

Naı̈ve expected statistical sensitivity for ThF+. . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.3

Expected statistical sensitivity for ThF+ in the Bucket Brigade. . . . . . . . . . . . . . . . . 109

6.1

Coefficients of inhomogeneity expansion of f in the Gen. II setup. . . . . . . . . . . . . . . . . 131

6.2

Comparison of simulation results of different ion cloud parameters against predictions . . . . 139

7.1

Axial frequencies at various geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.2

Potential barrier of the wells along the trap axis. . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.3

Attenuation factors of various harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

10.1 Thermal contraction of a 3 m long setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
10.2 Some important parameters of setup used for the experiment to determine thermal conductivity of foam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

xiii
B.1 Table of typical parameters for the (Baby) Bucket Brigade. . . . . . . . . . . . . . . . . . . . 296
B.2 Table of typical parameters for the Gen. III experiment. . . . . . . . . . . . . . . . . . . . . . 297

Figures

Figure1.1Upper bounds on the eEDM set by experiments over the years. . . . . . . . . . . . . . . . . .

2

1.2

Energy landscape of the molecular ion ThF+ . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.3

Basic idea behind optical pumping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.4

Energy level diagrams of the eEDM-sensitive states. . . . . . . . . . . . . . . . . . . . . . . .

10

2.1

Schematic diagram of the LIF experimental setup. . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2

Compressed spectra of all bands found (blue trace) and fitted (red dots above traces) in our
LIF survey spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3

Samples of rotational bands obtained from the LIF survey spectroscopy (Hund’s case (c)). . .

14

2.4

Quick summary of Hund’s cases (a), (b), and (c). . . . . . . . . . . . . . . . . . . . . . . . . .

16

2.5

Sample of a Hund’s case (b) rotational band obtained from the LIF survey spectroscopy. . . .

18

2.6

Sample of a band with resolvable Lambda-doubling. . . . . . . . . . . . . . . . . . . . . . . .

19

2.7

Crossings of diabatic potential curves of different states distort the potential curves. . . . . .

23

2.8

Rotational distributions of ThF+ ions created from different intermediate rotational states of
ThF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.9

Typical traces of the REMPI TOF-MS experiments. . . . . . . . . . . . . . . . . . . . . . . .

25

3.1

Schematic diagram showing state readout using state-selective REMPD. . . . . . . . . . . . .

28

3.2

Semiclassical picture of molecule alignment with polarizing field. . . . . . . . . . . . . . . . .

31

3.3

Simulated angular distribution of photofragments I(θ, ϕ) for Ω′ = 1. . . . . . . . . . . . . . .

32

xv
3.4

Simulated angular distribution of photofragments I(θ, ϕ) for Ω′ = 2. . . . . . . . . . . . . . .

33

3.5

Schematic diagram of molecular potential curves. . . . . . . . . . . . . . . . . . . . . . . . . .

37

3.6

Schematic diagram comparing the effects of KEdissoc. w.r.t. KEtherm. . . . . . . . . . . . . . . .

38

3.7

Schematic diagram of the REMPD experimental setup.

. . . . . . . . . . . . . . . . . . . . .

40

3.8

Scatter plot of the Th+ ion positions at the ion detector. . . . . . . . . . . . . . . . . . . . . .

42

3.9

Histograms of the Th+ ion positions, integrated over the X direction, at the ion detector. . .

43

3.10 Histograms of the Hf+ ion positions, integrated over the X direction, at the ion detector. . .

44

3.11 Pathway taken by molecules to during dissociation with various combinations of photon polarizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.12 Distribution of ions along the Y axis on the ion detector with various combinations of photon
polarizations for the [32.09] intermediate state. . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.13 Distribution of ions along the Y axis on the ion detector with various combinations of photon
polarizations for the [32.50] intermediate state. . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.14 Various configurations of REMPD beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

4.1

State preparation sequence (not to scale). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

4.2

Energy landscape (not to scale) of the neighborhood of the eEDM-sensitive state X 3 ∆1 (v =0), showing only the first two rotational levels. . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.3

Microwave scan at zero net external electric field. . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.4

Microwave transitions between the X 3 ∆1 (v = 0, J = 1, mF = −3/2) and J = 2 states at an
applied electric field strength of 24 V/cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.5

Allowed transitions between J = 1 and J = 2 in X 3 ∆1 (v = 0). . . . . . . . . . . . . . . . . .

61

4.6

Heat map of the coefficients of the eigenstates expanded out in our chosen basis set. . . . . .

62

4.7

Energy level diagram of the 32-level system in X 3 ∆1 (v = 0, J = 1, 2) expressed in our chosen
the basis set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.8

Energy differences between the mF = ±3/2 states; measurement of the magnetic g-factors.

.

65

4.9

Measuring δg/g in ThF+ at Erot = 60 V/cm. . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

xvi
4.10 Energy level diagram of the lowest few vibronic states in ThF+ . . . . . . . . . . . . . . . . . .

69

4.11 State preparation for measurement of excited state lifetime. . . . . . . . . . . . . . . . . . . .

70

4.12 Vibrational population decay from v = 1 to v = 0 of the ground electronic state in ThF+ . . .

71

4.13 Effect of suppressing blackbody radiation excitations at room temperature. . . . . . . . . . .

74

4.14 Summary of the lasers (and recommended wavelengths in vacuum) used for state preparationand detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

4.15 Overlapping transitions of the J = 1 to J = 2 transition in the X 3 ∆1 (v = 0, 1, 2) manifolds. .

78

4.16 Relevant Franck-Condon and Hönl-London factors for transitions between X 3 ∆1 and Ω =0− [14.59]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.17 From the optical pumping model: distilling out the effects of spontaneous decay and BBR
excitation from the effect of Goat Q(1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.18 From the optical pumping model: population in the (v = 0, J = 1) state, which is a sum of
population in “D” and “B”, plotted as a function of time at which Goat is turned on. . . . .

86

4.19 Comparing predictions from the optical pumping model with experiment. . . . . . . . . . . .

86

4.20 Plots of “impurity” and population migration in and out of the J = 1 manifold as a functionof state preparation time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

4.21 Plot of |∆|. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

4.22 Plot of the special Erot vs frot .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

4.23 Rabi flopping of both doublets at various Erot . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

4.24 Summary of experiment sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

5.1

Transition frequencies of all transitions connecting the (J = 1, mF = ±1/2) states to the(J = 2, mF = ±1/2) states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2

The four states used in a typical eEDM experiment with co-magnetometry. . . . . . . . . . . 101

5.3

States bumping into each other through Stark shifts. . . . . . . . . . . . . . . . . . . . . . . . 103

5.4

Comparisons of energy levels between ThF+ and HfF+ . . . . . . . . . . . . . . . . . . . . . . 107

5.5

Schematic diagram of the Bucket Brigade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

xvii
6.1

Dependence of total ion number on initial condition of numerical solver. . . . . . . . . . . . . 115

6.2

Mean field energy and potential energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3

Ion number radial density vs total ion number. . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.4

Number density at ion cloud center vs temperature. . . . . . . . . . . . . . . . . . . . . . . . 117

6.5

Sample of results from a LAMMPS simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.6

Decoherence of an ensemble-averaged Ramsey fringe for a Gaussian frequency distribution. . 124

6.7

Simulated result of the spread of Ramsey phases in an ensemble of ions. . . . . . . . . . . . . 126

6.8

Decomposing spatial inhomogeneity into odd and even functions about trap center. . . . . . . 127

6.9

Decoherence rates seen in the Gen. II setup through spatial inhomogeneity of Berry’s phase. . 130

6.10 Collisional narrowing seen in a simulation of ions. . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.11 Schematic of a collision between two singly positively charged ions. . . . . . . . . . . . . . . . 133
6.12 Fits to the self correlation function of the kinetic energies. . . . . . . . . . . . . . . . . . . . . 137
6.13 Fit of Equation 6.15 to the decay rate of Equation 6.16. . . . . . . . . . . . . . . . . . . . . . 138
6.14 Density-weighted density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.15 Ion cloud heating rates predicted by Equation 6.18 and comparison to experiment. . . . . . . 141
6.16 Plot of the average fractional increase in energy of an ion per collision in a Paul trap. . . . . 142
6.17 Ion cloud heating rates predicted by Equation 6.18 for ThF+ in the Bucket Brigade at various
secular trap frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.18 Rate of collisional narrowing predicted by Equation 6.15. . . . . . . . . . . . . . . . . . . . . 143
6.19 Simulated trajectories of an ion in two different traps. . . . . . . . . . . . . . . . . . . . . . . 146

7.1

Spoiler: perspective view of a CAD drawing of the Bucket Brigade to visualize the relevant
relationship between various electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.2

Schematic comparison of waveform amplitude decay from the Rings with various k. . . . . . . 152

7.3

Graphical representation of summing a linear potential to quadratic and quartic potentials. . 156

7.4

Cross sectional view of the Bucket Brigade involving the Rods, Rings, and the vacuum chamber.158

7.5

Attenuation of potentials of Rings by Rods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

xviii7.6

RF electric potentials in the Bucket Brigade, evaluated along the x- and y-axes. . . . . . . . . 162

7.7

Mathieu stability plot for the Bucket Brigade. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.8

Plots of spatial homogeneity of rotating electric field. . . . . . . . . . . . . . . . . . . . . . . . 164

7.9

CAD drawing of the b.B.B. setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.10 Pictures of the b.B.B. setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.11 Picture of the b.B.B., with yours truly and an approving-looking advisor for scale. . . . . . . 170
7.12 Losing ions past 30 cm into the b.B.B.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.13 Picture of the microchannel plate assembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.14 Tank circuit used to drive voltages on the Rods. . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.15 T-model equivalent circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.16 Pictures of a prototype transformer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.17 Schematic of an overview of the Rod driver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.18 Schematic of an overview of the Ring driver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.19 Number of ions remaining in the trap after parametric excitation. . . . . . . . . . . . . . . . . 187
7.20 Using parametric excitation to diagnose the trap. . . . . . . . . . . . . . . . . . . . . . . . . . 188

8.1

Illustration of the coils and magnetic field configurations used to generate a Zeeman offset
frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.2

Plan A: using current-carrying coils to introduce an axial magnetic field gradient. . . . . . . . 194

8.3

Schematic of a coil with radius R in the x-y plane with current going clockwise. . . . . . . . . 195

8.4

Plan A: Temporal modulation of current in the mth coil. . . . . . . . . . . . . . . . . . . . . . 198

8.5

Plan A: Plots of Baxgrad along the trap axis for various bucket length. . . . . . . . . . . . . . 200

8.6

Plan A′ : Plots of Baxgrad along the trap axis for various bucket length. . . . . . . . . . . . . . 201

8.7

Plan B: using the Rings to introduce a time-varying axial electric field. . . . . . . . . . . . . . 210

8.8Resonance curves for Q = 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

8.9

Spatial inhomogeneity of the dithering electric field w.r.t. point at center of a bucket. . . . . . 218

xix
8.10 Lissajous figures traced out by the rotating electric field as seen by ions at various locations
within the trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8.11 Lissajous figure traced out by the rotating electric field as seen by ions in the Bucket Brigade. 223
8.12 Plan E: geometry of setup with charging currents. . . . . . . . . . . . . . . . . . . . . . . . . 228
8.13 Charging-current–induced Zeeman frequencies in the b.B.B.. . . . . . . . . . . . . . . . . . . . 230

9.1

List of things connected to the TTL and DAC channels. . . . . . . . . . . . . . . . . . . . . . 236

9.2

The ARTIQ dashboard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

9.3

Typical code structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

9.4

Example of having multiple non-“underscored” classes inheriting from the main non-“underscored”
class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

10.1 Number of blackbody radiation photons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
10.2 Energy level diagram for the lowest lying states for HfF+ and ThF+ . . . . . . . . . . . . . . . 251
10.3 Schematic diagram of a simplified setup for heat load analysis. . . . . . . . . . . . . . . . . . 253
10.4 Calculated conduction power through household foam. . . . . . . . . . . . . . . . . . . . . . . 254
10.5 Early plans for Cold Jacket. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10.6 Schematic of model used to study cool down time of cold plate. . . . . . . . . . . . . . . . . . 258
10.7 Equivalent mathematical model for the actual setup. . . . . . . . . . . . . . . . . . . . . . . . 259
10.8 Temperature profile along cold slab at various times. . . . . . . . . . . . . . . . . . . . . . . . 260
10.9 Plots of temperature drop at various positions with time. . . . . . . . . . . . . . . . . . . . . 260
10.10Plot of cooling power required versus time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
10.11First attempt at spraying foam into a container. . . . . . . . . . . . . . . . . . . . . . . . . . 265
10.12Photo montage of the process of making the foam blocks. . . . . . . . . . . . . . . . . . . . . 267
10.13Schematic diagrams and pictures of the setup used to test thermal conductivity of foam. . . . 268
10.14Working range for the Ametek CryoTel DS30 cryocooler. . . . . . . . . . . . . . . . . . . . . . 271
10.15Power resistors mounted onto the extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
10.16Temperature readings of RTDs in one experimental run. . . . . . . . . . . . . . . . . . . . . . 273

xx
10.17Testing the cryogenics out in the b.B.B. setup. . . . . . . . . . . . . . . . . . . . . . . . . . . 274
10.18Setup used to measure the magnetic fields from the cryocooler. . . . . . . . . . . . . . . . . . 276
10.19Plot of Baxgrad vs d, the distance between the Cold Head and the center of the magnetometercross. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
10.20Effective blackbody radiation excitation lifetime in the Bucket Brigade. . . . . . . . . . . . . 279

A.1 Comparing definition of Uec,0 between conventional linear Paul traps and the Bucket Brigade. 293

Chapter 1

Introduction

1.1

Background of the eEDM
An improved or non-zero measurement of the electron’s electric dipole moment (eEDM, de ) would

have huge implications for our understanding of the universe [1–4]. Our current understanding is nicely
encapsulated at a fundamental level by the Standard Model of particle physics, but many observations
hint that the Standard Model is incomplete. This problem is being addressed by numerous works on thetheoretical front to introduce new physics through extensions of Standard Model [5, 6]. These new physics
models come in many different classes, and each class makes a different prediction of the value of the eEDM
[7–12]. Hence, a measurement (or a null-measurement) of the eEDM would place constraints on these newtheories, and guide us towards a better understanding of the universe.
There are many ongoing works trying to measure the eEDM [13–20], and in recent years the effort
has been switched from atoms to molecules because of molecular enhancement factors and robust systematic
error rejection [21–27] that are available only to molecules because of their rich internal structures. There
are several proposals that seek to exploit the rich internal structures even further to realize exotic ways tomeasure the eEDM [28, 29]. Figure 1.1 is a compilation of published upper bounds of the eEDM.
The richness of the molecular internal structure provides additional tools for diagnostics and eEDM
sensitivity enhancements, but it proves to be a double-edged sword, as non-diagonal Franck-Condon factors
and rotational branching increase the complexity of compressing the Hilbert space into the states that we careabout. This additional complexity complicates state preparation, and also makes standard AMO practices

2Upper bound on eEDM (e cm)

10 13

Lamb shift
g-factore -4He scattering
Cs
Rb
Tl
Xe

10 16
10 19

Hg
TlF
YbF
PbO
ThO
HfF +

10 22
10 25
10 28

0

196

0

197

0

198

0
0
199
200
Publication date

0

201

0

202

Figure 1.1: Upper bounds on the eEDM set by experiments over the years. Adapted from Paul
Hess’s thesis from the Gabrielse Group. Bounds are plotted at 90% confidence level.

like laser cooling and state detection through cycling transitions less applicable to molecules. Incidentally,
there are groups trying to get around this limitation by working with polyatomic molecules [30–32] to marryfavorable photon cycling properties of the heavy metal atom in the molecule to the advantages that comewith the rich internal structures of a molecule. Although these groups are in the very early stages of theirspectroscopy ground work, much related work [33, 34] is showing promising signs for future work.
The eEDM experiment here at JILA uses diatomic molecular ions. The statistical sensitivity to theeEDM depends on three main factors in the experiment: (i) effective electric field strength that couples
to the eEDM, (ii) interrogation time of the eEDM-sensitive state, and (iii) total number of counts in theexperiment for statistics. The first two generations of the JILA eEDM experiment [14, 35, 36] take advantageof the long ion trapping times to tap the long coherence times of the eEDM-sensitive state. The following
sections give brief introductions to the JILA eEDM experiment with HfF+ and ThF+ .

3

1.2

eEDM measurement with molecular ions

1.2.1

JILA eEDM experiment with HfF+
The determination to measure the eEDM with molecular ions was first written in stone in the JILA

2011 paper [25], which addressed many concerns that skeptics in the AMO community had regarding performing a precision metrology experiment with trapped ions with a rotating electric field. Fast forward
to 2017, the first generation measurement (Gen. I) from JILA using trapped HfF+ convinced the last of
the skeptics by placing an upper bound on the eEDM at |de | < 1.3 × 10−28 e cm (90% confidence) [14],falling just slightly behind the world’s best record set by ACME at that time at |de | < 8.7 × 10−29 e cm
(90% confidence) [37]. Since then, there has been friendly competition between JILA, ACME, and Imperial
College [13] to see who can set the best limits on the eEDM, or even measure the eEDM itself. Soon after,
ACME published an improved limit at |de | < 9.6 × 10−30 e cm (90% confidence) [15] to spice up the game. In
the meantime, JILA has made improvements to various aspects of the experiment, culminating in a second
generation measurement (Gen. II, unblinded in November 2022) [35, 36], and is now the world’s best limit
at |de | < 4.1 × 10−30 e cm.
Some of the major improvements made in the Gen. II experiment include (i) better designed ion
trap for field homogeneity, (ii) trapping more ions for better statistics, and (iii) new state preparationand detection techniques to reduce measurement noise. The details of upgrades pertaining to the Gen. II
measurement can be found in numerous papers [38, 39] and theses from the JILA eEDM group (e.g. William
Cairncross and Tanya Roussy) and shall be omitted here to save the spotlight for ThF+ .

1.2.2

JILA eEDM experiment with ThF+At the time of this writing, JILA has the world’s best upper bound on the eEDM. The race is stillon, however, so we need ideas on how to improve on this result. Small epsilon upgrades typically take about
five years from conception to fruition; big overhauls can take up to fifteen years, but our competitors arecertainly not patient enough to wait fifteen years for our big overhaul before resuming the competition. To

4
get a head start for the third generation experiment, we used1 the time machine called foresight to go backin time to 2008 to start laying the foundations for a third generation experiment (Gen. III).
During the early days of the HfF+ experiment, the JILA eEDM team learnt about a new molecule
ThF+ that can potentially help them get/stay ahead in the race [40–42]. The level of sensitivity to theeEDM (δde ) depends on three main factors:
δde ∼Eeff τ

p

1,
Nsingle · frep. · Ttot.

(1.1)

Effective electric field, Eeff . Relativistic effects allow for a non-zero electric field to couple to the EDM of
the electron, despite having net zero time-averaged electric field inside the molecule. The effective
electric field is molecular species dependent, and it turns out that ThF+ has a slightly larger Eeff =35 GV/cm [22, 41] than HfF+ at Eeff = 23 GV/cm [24, 25]
Coherence time, τ . A longer coherence time is in general a good thing to have. In the context of the eEDM
precision metrology experiment, which is a Ramsey spectroscopy measurement, a long coherence timeallows for long Ramsey fringes to be taken. A longer Ramsey fringe translates to a more precise
determination of the Ramsey frequency. The coherence time can be affected by, say, less thanperfect control over the ions or fields (T2∗ type losses), and even if we have full control over the wholeexperiment, the coherence time of the eEDM sensitive states is bounded by the natural decay lifetimeof the states (T1 type losses). The eEDM sensitive states in HfF+ have natural decay lifetimes ofabout 2 s. However, the eEDM sensitive states in ThF+ are shown to be the ground rovibronic state[43], so the coherence times in ThF+ can be much longer than 2 s.
Number statistics,pNsingle · frep. · Ttot. , where Nsingle , frep. , and Ttot. are the total number of ionscounted in a single shot of the experiment, the repetition rate of the experiment, and the total
time for which the graduate students are willing to take data, respectively. In the Gen. II experiment, the dead time of the experiment is short compared to a single Ramsey integration time, which
in turn is limited by the coherence time. Hence the repetition rate is frep. ≈ 1/τ . As we shall see
1 It is not immediately obvious to the author, who does not major in the English language, what tense to use when time
travel is involved.

5
in later sections, we have tricks up our sleeves that allow us to work with frep. ≫ 1/τ that we will
implement in our eEDM experiment with ThF+ .
Despite the successes we have in measuring the eEDM with HfF+ [14, 35, 36], it is not immediately
obvious if the same experimental techniques (i.e. state preparation and readout) used on HfF+ can betransferred over to ThF+ . For instance, the laser transitions required to address states in ThF+ may beentirely different from those of HfF+ . Are the relevant transitions covered by lasers that are readily available?
Are the internal quantum states easy to understand given that thorium, a heavy metal that is even heavier
than hafnium, may break all the usual rules that we use for state assignments? How do we even create
thorium monofluoride ions? Someone has to turn knobs in the laboratory to answer these questions. That
someone, for the past eight years, happens to be me.

1.2.3

How an eEDM experiment is done at JILA
This thesis is a build up to a precision measurement of the eEDM in ThF+ , involving a lot of spec-

troscopy, design, and characterization of both the molecule and the setup. To understand why we do thesein certain ways, mostly guided by our experience with the experiment using HfF+ , we need to understandhow the eEDM experiment is done at JILA. This has already been covered in numerous theses before mine(e.g. Huanqian Loh, Kevin Cossel, Matt Grau, Daniel Gresh, William Cairncross, and Tanya Roussy), but
I will present a brief summary in the context of ThF+ (first time this is done) to make this thesis more
self-contained. The ideas in this section serve only as a “quick start guide”, and will be further developed
throughout this thesis.
The first step to understanding how we perform an eEDM experiment at JILA involves knowing theenergy landscape of the molecule. Figure 1.2 shows the most important features of the energy landscape of
ThF+ . A diatomic molecule like ThF+ has a very rich internal structure. Much like atoms, our molecules haveelectronic states. Unlike atoms, however, the electronic states are further split into vibrational manifolds,
which are then further split into rotational manifolds, and finally within each rotational manifold, we havethe hyperfine structure and parity doublets. The states that we choose to perform our eEDM measurementin are the X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = ±3/2) states. Much must be done to compress internal

6
population of our molecule into just these four states alone. We do this through optical (and microwave)
pumping.
The second step to understanding how we perform an eEDM experiment at JILA involves knowing
how optical pumping works in our molecule. Figure 1.3 shows the basic idea behind optical pumping. These
are standard techniques used in the AMO community. Further explanation in the captions.
The last piece of information we need to understand how we perform an eEDM experiment at JILA
is what we are measuring that tells us about the eEDM. Figure 1.4 shows the energy level diagrams forthe eEDM-sensitive states, and introduces the frequencies f0 (the frequency offset that we intentionallyintroduce), f u (frequency difference of the upper Stark doublet), and f l (frequency difference of the lower
Stark doublet). At the end of the day, all efforts discussed in this thesis are aimed at measuring f u and f lto the best of our abilities. Further explanation in the captions.

1.3

Thesis Layout
This thesis is a culmination of two main efforts: (i) various spectroscopy work on ThF and ThF+ , and(ii) design work on a prototypical multiplexed ion trap to take full advantage of the long coherence time of
ThF+ . It serves as a guide on how to use ThF+ to measure the eEDM, albeit an incomplete one, one thatis a work in progress. Correspondingly, this thesis is split into the two parts:(1) Part I: controlling the internal states of ThF+
(a) Preparing ThF+ ions (Chapter 2): how do we create ions state selectively;(b) State readout of ThF+ (Chapter 3): how do we use action spectroscopy to readout the quantum
states of our ions;(c) Quantum control of ThF+ (Chapter 4): how do we usher the ions into the desired quantum
state through optical and microwave pumping;(2) Part II: the JILA proposal for how we want to use ThF+(a) Interlude (Chapter 5): can we do something more with ThF+ than what we are already doingwith HfF+ , formally introducing a multiplexed ion trap (the Bucket Brigade);7(b) Ion cloud & decoherence (Chapter 6): the dynamics of an ion cloud and its consequences on
the design of the multiplex ion trap;(c) Electric fields in the Bucket Brigade (Chapter 7): the design of the electrodes of a multiplexed
ion trap;(d) Generating f0 in the Bucket Brigade (Chapter 8): exploration on various ways to turn bugs
(sources of systematic errors) into features (let them work for us);(e) Control system (Chapter 9): an introduction to our upgraded control system;(f) Cryogenics (Chapter 10): using cryogenics to take full advantage of the long coherence time in
ThF+ .
So turn over to the next chapter, and we shall begin the story of how to perform an eEDM experimentwith ThF+ starting with the first step: how to create ThF+ ions.

8
Energy..
.

|Ω| = 2 [32.09] (v = 0)32100 cm−1..
.
(v = 2)586 cm−1(v = 1)590 cm−1Ω = 0− [14.58] (v = 0)14600 cm−1..
.
(v = 2)619 cm−1(v = 1)623 cm−1Ω = 0+ [10.49] (v = 0)10500 cm−1..
.
(v = 2)651 cm−1(v = 1)655 cm−13∆3 (v = 0)3150 cm−1..
.

(v = 2)649 cm−1..
.

(v = 1)(v = 2)649 cm−1..
.

653 cm−1(v = 2)3(v = 1)653 cm−11+a Σ (v = 0)314 cm−1

649 cm−1(v = 1)653 cm−1
X 3 ∆1 (v = 0)

∆2 (v = 0)1050 cm−1...
J =3
43.6 GHz

J =2

F = 1/2

29.1 GHz

J =1
F = 3/2
mF = − 32

− 12

1
2

3
2

Figure 1.2: Energy landscape of a molecular ion like ThF+ . Not drawn to scale. 1 cm−1 ≈ 30 GHz.
The electronic ground state is the X 3 ∆1 . Within each electronic state, there are vibrational manifolds,
labelled by quantum number v. Within each vibrational level, there are rotational manifolds, labelled byquantum number J (Hund’s case (c)). Within each rotational state, there are hyperfine manifolds (labelledby quantum number F ), parity doublets (labelled by quantum number Ω; Ω = ±1 are color coded blackand blue, respectively), and Zeeman sublevels (labelled by quantum number mF ). This diagram shows acompilation of (i) the few lowest lying electronic states (3 ∆1 , 1 Σ+ , 3 ∆2 , and 3 ∆3 ), (ii) higher lying statesused for optical pumping (Ω = 0+ and Ω = 0− ), and a high lying state available for photodissociation(|Ω| = 2). The grey horizontal boxes indicate breaks in the vertical (energy) axis. The two pairs of stateshighlighted in pink are those used for the the eEDM measurement.

9

..
.
J =2
Ω = 0− [14.29] (v = 0)

J =1
J =0

..
.

Other electronic states

J =3
X 3 ∆1 (v = 1)

J =2
J =1

..
.
J =3
X 3 ∆1 (v = 0)

J =2
J =1

Figure 1.3: Basic idea behind optical pumping. Only the relevant states from Figure 1.2 are shown
here. We start with population distributed among several rotational states in the X 3 ∆1 (v = 0) vibronic
manifold (blue shaded circles). A laser incoherently pumps population out of the J = 2 rotational state intothe Ω = 0− [14.29] (v = 0, J = 1) state (solid black arrow).
Population pumped to the excited state then decays into lower lying states (grey dashed arrows). The decay
is subjected to selection rules (∆J = 0, ±1). Hence, molecules can fall back to the J = 1, 2 rotational states.
Population that fall back to the X 2 ∆1 (v = 0, J = 2) state can be pumped up again by the laser. Populationthat fall back to the X 2 ∆1 (v = 0, J = 1) state will remain dark to the laser, but this also happens to bewhere we want the molecules to be for the eEDM measurement. This basic pattern (using the resonanceselectivity of excitations plus the “selection rule” selectivity of spontaneous decay) is used (i) to deplete
population from where we do not want it, or (ii) to accumulate population where we do, or (iii) sometimes
both. The entropy associated with the internal state distribution is transferred to the dissipated photons
from the spontaneous emission.
Population in the X 2 ∆1 (v = 0, J = 3) state do not move in this partial scheme, but as we will see in
Chapter 4, we can use microwaves to couple the J = 2 and 3 to pump population from J = 3. Nondiagonal Franck-Condon factors (vibrational wavefunction overlap between the electronic states X 3 ∆1 andΩ = 0− [14.29] (v = 0)) allow for decay into higher vibrational manifolds in X 3 ∆1 . These molecules will
decay to the v = 0 manifold after about 200 ms. We could also speed up the process by using vibrational
repumps: lasers that connect to the Ω = 0− [14.29] state from the v > 0 vibrational manifolds.
In general, molecules in in excited states can decay to multiple electronic states. In ThF+ , however, thebranching ratio from Ω = 0− [14.29] back to X 3 ∆1 is very close to unity. Therefore, to close the opticalpumping cycle, we only need vibrational repump lasers.

10
(a)

(b)
n̂
−

∆

n̂

−
∼ dmf Erot

+

+

n̂

n̂
∆

(c) mF = − 32

EnergymF = 32(d)
f u ≈ f0 + 2de Eeff

∆
2

f=

p

∆2 + f02

Ω Σ

Ω Σ

Brot

Ω Σ
Ω Σ

−∆
2
mF = 32

f0 = 3gµB Brot

mF = − 32

f l ≈ f0 − 2de Eeff

Figure 1.4: Energy level diagrams of the eEDM-sensitive states. Only the X 3 ∆1 (v = 0, J = 1, F =3/2) states are shown. Refer to Figure 1.2 for the bigger overview.
(a) In the absence of external fields, the eigenstates are states of good parity. The molecule does not havea preferred orientation. We assume that the positive parity state has the lower eigenenergy in thisschematic. We did not determine the energy ordering of the parity states in our experiments.
(b) When we apply an external electric field, we get Stark shifts that scale with the molecular dipole moment
dmf and the applied electric field Erot . The parity states mix and produce states of good orientation.
The internuclear axis unit vector n̂ points from the lighter anion to the heavy cation. To polarize ourmolecular ions with an external electric field without ejecting them out of the ion trap, we apply arotating electric field, with rotation frequency frot that is fast compared to typical ion motion in thetrap, but adiabatic compared to typical energy scales in the molecule. As a result, we work in therotating frame of the molecules. In the absence of external magnetic fields, the mF = ±3/2 states mix
from the non-inertial-frame term in the rotating frame. The coupling strength is ∆.
(c) Without the non-inertial-frame term, the energy levels of the mF = ±3/2 states within either the upperor lower Stark doublet follow the dashed lines. The energy difference f0 is just the Zeeman splitting
between them. In the presence of the non-inertial-frame term, we get an avoided crossing with splitting∆. The net frequency difference between the mF = ±3/2 states is the quadratic sum of ∆ and f0 . Notethat f0 is of a Zeeman nature in this schematic diagram, but as we will see in Chapter 8, it can alsocome from Berry’s phase. The only purpose of f0 is to lift the degeneracy of the mF = ±3/2 states.
For our eEDM measurements, we park at an f0 ≫ ∆, where |mF = ±3/2⟩ are good eigenstates of thesystem. To perform our π/2 pulse to prepare a superposition of the mF = ±3/2 states for our Ramsey
sequence, we ramp our fields so that f0 ≪ ∆ and allow the new eigenstates (almost equal superpositionof mF = ±3/2) to evolve for a time τ , such that 2π · ∆ · τ = π/2, before restoring our fields to thef0 ≫ ∆ setting.
(d) We measure the eEDM in both the upper (red) and lower (blue) Stark doublets. The energy level of eachstate goes as E ∼ EStark +EZeeman +EEDM . We measure the energy difference within each Stark doublet,−→so the Stark term drops out. The eEDM term is EEDM = −de Eeff , where de ||Σ · n̂ and Eeff || − n̂. We
take the difference between f u and f l to extract EEDM . Higher order corrections like slight differencesin the g-factors and the ∆ coupling between the upper and lower Stark doublets are ignored here forsimplicity.

Chapter 2

Preparing ThF+ ions2.1

Introduction
Following the footsteps of the experiments with HfF+ , the first step in performing an eEDM exper-

iment with ThF+ involves creating neutral ThF. These neutral ThF will then be ionized state selectively
through resonance-enhanced multi-photon ionization (REMPI) to produce ThF+ in the eEDM-sensitive statewith high preparation efficiency. This process involves exciting ThF to a pre-ionization state, which is higher
in energy than the ground state of ThF+ , whereupon it sheds energy by giving up an electron and falls
into a lower energy state in ThF+ . To perform this state selective REMPI, we first need to perform survey
spectroscopy of the excited states of ThF to find the appropriate intermediate state for REMPI.
Previous work on ThF [44, 45] were laser-induced fluorescence (LIF) spectroscopy in a very limited
range slightly below 21500 cm−1 above the ground state X 2 ∆3/2 , which is less than half of the ionizationenergy (IE = 51585(50) cm−1 [44]). This is not the most optimum state for state selective REMPI becauseof the following:• The second photon of a 1 + 1′ REMPI usually requires more pulse energy than the first, so that thetransition from the intermediate state to the final state can be saturated.
• Since the laser pulse of the second photon is of higher energy, it is more likely to drive a one-color
multi-photon (off-resonant) transition than the first photon, because this process is a strong functionof the peak laser intensity. If the second photon has more energy than IE/2, then it could drive its
own 1′ + 1′ multi-photon ionization, and create ions with undesired internal states.

12• Therefore, it is better to have a first photon that is bluer in wavelength than the second, i.e. the first
photon should be more than IE/2. In ThF, the first photon should have a wavelength of less than
387 nm.
To find a state that fulfils all the above criteria for state-selective ion creation, we perform laserinduced fluorescence (LIF) spectroscopy on ThF. This chapter is structured as follows. We first begin withan introduction to the setup of the experiment in Section 2.2. Then we take a look at a sample of the resultsof the scan in Section 2.3. A simple explanation of how we extract the spectroscopic constants is shown in
Section 2.4 and 2.5. Finally we conclude with further discussions of the LIF results and their relation to theeEDM experiment with ThF+ in Section 2.6. The following sections will be paraphrases of our publication
Ref. [46].

2.2

Setup
A schematic of the setup is shown in Figure 2.1. The control system and data acquisition systems we

use are similar to those used in the HfF+ experiment, and shall be omitted here for brevity. The interested
reader can refer to theses of earlier members of the group (e.g. Huanqian Loh and Matt Grau) for moredetails.
We use a tunable pulsed dye laser (Sirah Cobra-Stretch, 1800 grooves/mm single grating, 0.06 cm−1
linewidth at 600 nm) to perform the LIF spectroscopy. The wavelength of the laser is monitored continuously
by a wavemeter (Bristol 871B) during the scan. We reference the wavemeter to a stabilized He-Ne laser with
∼50 MHz absolute accuracy. We note that there are vibronic bands that overlap with each other, such thatthey cannot be rotationally resolved by the pulsed dye laser, especially in the deep ultraviolet region. To
resolve these lines, we use a narrow-band pulsed laser system, which consists of a tunable cw-ring dye laser(Sirah, Matisse DR2) and a pulsed dye amplifier (Radiant Dye Amp, 150 MHz linewidth). The wavelengthof the seeding laser is monitored by a high resolution wavemeter (High Finesse, WS7), which is referenced
to an external-cavity diode laser locked to a 87 Rb transition at 384.227982 THz. Lastly, to reduce noise due
to laser scattering (e.g. from the ablation laser), a fast switch gates the PMT on 10 ns after the excitation

13
Parabolic mirror
Ablation laser

Intensity & lifetime analysis
V
LIF laser (out of page)

SkimmertCollimating lensTh coinFast gated PMT

Pulsed valve
Gas reservoir (Ne, 0.1% SF6 )Figure 2.1: Schematic diagram of the LIF experimental setup. The LIF spectroscopy consists of thefollowing steps:(1) A plume of Th plasma is created through the ablation of a Th metal with a 532 nm laser focuseddown to a beam waist of about 150 µm with about 2 mJ per 5 ns pulse from a Q-switched Nd:YAG
laser. The Th metal is mounted on an electronically controlled movable stage to give a fresh surface
for each ablation.
(2) The nozzle lets out a buffer gas of Ne at room temperature seeded with SF6 from a stagnant 100 PSI
reservoir. SF6 reacts with the Th plasma to give ThF, which gets entrained in the supersonic
expansion of Ne to form a molecular beam of ThF and cools ThF down to about 10 K in translationaland rotational degrees of freedom.
(3) A tunable laser intercepts the molecular beam in a cross-beam setup. If the laser frequency isresonant with a transition in ThF, the molecule is excited to the excited state, and relaxes throughfluorescence, which is collected with optics and a photomultiplier tube (PMT). The signal from the
PMT goes to an oscilloscope, where the amplitudes and decay time of the trace are recorded andanalyzed.
(4) Repeat the experiment with different excitation wavelengths.

laser pulse is fired to extract the long-lived fluorescence signals (> 50 ns).

2.3

Sample of results
We record survey scans over the ranges of 13000 to 16000 cm−1 , 18000 to 20000 cm−1 , and 26000

to 44000 cm−1 using multiple laser dyes (LDS 698, DCM, Coumarin 540A, Coumarin 503, Coumarin 480
and Coumarin 460), and as necessary, frequency-doubled with a BBO crystal. 345 previously unreported
vibronic bands of ThF are recorded. The tables of quantum numbers of all the 345 vibronic bands span 15
pages in the paper, and shall not be included in this thesis for brevity. I would refer the interested reader toour publication [46] for the fitted transitions. Specific vibronic bands that are interesting will be reproduced
here for discussion. A compressed plot of all the bands found and fitted in our spectroscopy is shown in14Int. (arb.)

Figure 2.2.
300
250
200
150
100
50
0

Fitted trans.

30 00035 00040 00045 000-1

Wavenumber (cm )Figure 2.2: Compressed spectra of all bands found (blue trace) and fitted (red dots above traces)in our LIF survey spectroscopy. The grey region in the center indicates the region that we did not scan.
The area of interest lies to the right of the grey region because of the “above IE/2” requirement mentioned
in the introduction of this chapter.

Plotting the LIF amplitudes against the frequencies of the laser gives rotational structures as shownin Figure 2.3.

(a) Ω′ = 1/2(b) Ω′ = 3/2(c) Ω′ = 5/2

Figure 2.3: Samples of rotational bands obtained from the LIF survey spectroscopy. The quantum
number Ω (Section 2.4) is listed for each of the three sample plots shown. The blue inverted traces are data
points from experiment, while the red traces are profiles of the simulated fits.

Interpretation of these bands and assignment of quantum numbers require knowledge of allowedtransitions through selection rules. We shall cover this in the next section.

15

2.4

Spectroscopic constants, energy scales, and fitting form
The Hamiltonian governing a diatomic molecule is highly non-trivial. I would refer the interested

reader to textbooks (e.g. Ref. [47]) for a complete treatment of the full Hamiltonian and the selection rulesthat come from it. Here, I shall only attempt to describe very specific cases, and lay out the spectroscopic
constants and quantum numbers relevant to selection rules that apply to most of the bands found in this
LIF survey spectroscopy:
Rotational constant, B. Inversely proportional to the moment of inertia of our diatomic molecule modelled as a rigid rotor. Usually reported in units of cm−1 .
Origin of vibronic band, ν0 . Usually reported in units of cm−1 .
Rotational quantum number, J. The quantum number J comes from the total angular momentum operator J = L + S + R = Ja + R, where L, S, Ja and R are the electronic orbital angular momentum,
electronic spin angular momentum, total electronic angular momentum, and rotational angular momentum of the nuclei, respectively. Selection rules for E1 (electric dipole; single photon) transitions
only allow for transitions between states with ∆J = −1, 0, +1, which, in spectroscopy literature,correspond to P, Q, and R transitions, respectively. The P, Q, and R transitions give rise to distinctive P, Q, and R branches, which can be seen from the marked ticks in Figure 2.3. The relative
intensities of the P, Q, and R branches are determined by Hönl-London factors, which are detailed
below.
Projection of total angular momentum onto internuclear axis, Ω. The quantum number Ω takes
values from −Ja to Ja (in integral steps). It is formally defined as Ja · n, where n is the internuclear axis vector pointing towards the cation. The quantum number Ω reported for the LIF
spectroscopy are the absolute values |Ω| since our setup is not sensitive to the sign of Ω. Selection
rules for E1 transitions only allow for transitions between states with ∆Ω = 0, ±1.
For completeness, the projections of L and S onto the internuclear axis are denoted as L·n =: Λ and S·n =: Σ.
Some combination of all the above quantum numbers are used in the molecular term symbol: 2S+1 ΛΩ to16label the molecular quantum states, much like the Russel-Saunders term symbol for atomic states. Theseterm symbols are strictly only valid for states that fall under Hund’s case (a), because quantum numbers
like Λ may no longer be good quantum numbers for states belonging to other Hund’s cases. While more
information on Hund’s cases can be found in many textbooks (e.g. Brown et al. [47]), Figure 2.4 provides aCase (a)

R?

L?

J

S

Λ

Σ

BJ(J + 1); J ≥ Ω

Ω

Hrot

Ω

Next fine structure
e.g. 3 ∆1 vs 3 ∆2

R?

L?

Ja

Ω

J

BJ(J + 1); J ≥ Ω

Ω

Hrot

= BR2 = B(J − L − S)2† Overwritten by J ≥ |N − S|e.g. N = 0 only has S states.

Λ
L?

R?

Λ

Case (c)S

J= BR2 = B(J − L − S)2∆Λ = 0, ±1; ∆S = 0; ∆Σ = 0; ∆Ω = 0, ±1; ∆J = 0, ±1

S?

Case (b)N

BN (N + 1); N ≥ Λ

Next fine structure
e.g. 3 ∆1 vs 3 ∆2

quick summary of three Hund’s cases relevant to this thesis.

Split into Js

2S + 1 states†

Hrot = BR2 = B(N − L)2∆Λ = 0, ±1; ∆S = 0; ∆Σ = 0; ∆Ω = 0, ±1; ∆N = 0, ±1

L, electronic orbital angular momentum
S, electronic spin angular momentum
Ja = L + S, total electronic angular momentum
R, nuclei rotational angular momentum
J = R + Ja , total angular momentum of system
N = R + L = J − S, total angular momentum sans spin
Nuclear spins I1 and I2 not included yet.
e.g. F1 = J + I1 and F2 = F1 + I2

∆S = 0; ∆Ω = 0, ±1; ∆J = 0, ±1

Figure 2.4: Quick summary of Hund’s cases (a), (b), and (c). Adapted from Ref. [47]. Hund’s cases(d) and (e) are not relevant to this thesis. Solid filled arrows indicate angular momenta or their projections
onto the internuclear axis. Unfilled arrows indicate couplings between angular momenta or the internuclear
axis; solid unfilled arrows imply a stronger coupling than dashed unfilled arrows. Quantum numbers with“?” indicate not-a-good quantum number. Selection rules are also indicated at the bottom of each box.
Examples of the relevant rotational energy level structure are also shown in each box.

The transitions between the ground state in ThF and excited states have energies of the followingform:ν = ν0 + E ′ (J ′ ) − E ′′ (J ′′ ),(2.1)

where E(J) can take on several forms depending on which Hund’s case the relevant states fall under. We
managed to fit our transitions to three main classes, and these are the forms of E(J) relevant to our LIF

17
spectroscopy:
Hund’s case (c). L and S couple strongly together, and their resultant angular momentum then couples tothe internuclear axis to give Ω. This situation arises in heavy molecules. We expect most electronic
states in ThF to fall under this category because of the heavy thorium atom. This is especially true
when J is small. This will give:
E(J) = BJ(J + 1).

(2.2a)The ground electronic state of ThF (X 2 ∆3/2 ) is well described by Hund’s case (c). Good quantum
numbers are J and Ω. The observed Ω-doubling is much smaller than the linewidth of our laser, andshall be neglected in our fitting model. Hönl-London factor is given by:
HLΩ′ ,J ′ ,Ω′′ ,J ′′ = (1 + δΩ′ ,0 + δΩ′′ ,0 − 2δΩ′ ,0 δΩ′′ ,0 ) (2J ′ + 1) (2J ′′ + 1) ×



J′
1
J ′′
−Ω′ (Ω′ −Ω′′ ) Ω′′2, (2.2b)where δi,j is the Kronecker delta function, and the last term is the Wigner 3j symbol. Single(double-) primed quantum numbers correspond to those of the excited (ground) state. Samples ofbands in Hund’s case (c) are shown in Figure 2.3.
Hund’s case (b). This is the case where Λ = 0, e.g. when L precesses around the internuclear axis,
but is fully orthogonal to it. In this case, spin-orbit coupling vanishes, and S decouples from theinternuclear axis. One consequence of this is that Ω is now ill-defined; the good quantum number
to use now is N , where N is the quantum number for the total angular momentum excluding spin
N = J − S. Another consequence is that there will multiple sets of PQR branches corresponding to
the multiplicity of the spin. For the case of S = 1/2, we have two sets of PQR branches with:γ
Ea/b (N ) = BN (N + 1) −
2



1±s
N+
2


,

(2.3a)

where {a, b} denote the doublets, γ is the spin-rotation constant, and s = ±1 corresponds to the a

18and b doublets, respectively. The PQR branches are determined by the following selection rules:Pa/b :

Qa/b :

Ra/b :

1
2



1
N ∓
2



1
2



N′ ∓
′

N′ ∓

− J ′′ = −1,− J ′′ = 0,
− J ′′ = 1,

where the upper (lower) sign corresponds to the a (b) state, and once again single- and doubleprimed variables denote the excited and ground states involved in the transitions, respectively. The
Hönl-London factors for a Hund’s case (c) (the ground state) to a Hund’s case (b) transition is:2

HLΛ′ J ′ N ′ ,Ω′′ J ′′ =X √2N ′ + 1 (−1)

′

′

p
J +Ω

HLΩ′ ,J ′ ,Ω′′ ,J ′′



S N′
′
′

′

J
Σ Λ −Ω′



,(2.3b)

Σ′ ,Ω′

where HLΩ′ ,J ′ ,Ω′′ ,J ′′ is the Hönl-London factor for a Hund’s case (c) to case (c) transition, as shownin equation 2.2b. A sample of a rotational band under Hund’s case (b) in the LIF spectroscopy isshown in Figure 2.5.

Figure 2.5: Sample of a Hund’s case (b) rotational band obtained from the LIF survey spectroscopy. The blue inverted traces are data points from experiment, while the red traces are profiles of thesimulated fits.

2

Π states The last class of transitions involve excited 2 Π states. These states have significant Λ-doubling.
Since we do not have enough spectroscopic resolution to resolve the e/f symmetry components of
the lower state of the transition, we cannot determine the absolute parities of the upper states.
Following the convention used in Loh et al. [48], we label related Λ-doubling pairs as a/b instead of

19e/f . The upper state can be modeled by:
1
1
Ea/b (J) = B ′ J(J + 1) ∓ (p + 2q) J +
,
2
2

(2.4)

where (p + 2q)/2 is the Λ-doubling constant. The upper (lower) sign corresponds to the a (b)symmetry component. An example of a band with resolvable Λ doubling is shown in Figure 2.6.

Figure 2.6: Sample of a band with resolvable Λ-doubling. Red arrows indicate strong thorium atomic
transitions (from NIST Atomic Spectra Database [49]). The blue inverted traces are data points fromexperiment, while the red traces are profiles of the simulated fits.

2.5

Vibrational progression
All of the ThF transitions that we have assigned seem to have the X 2 ∆3/2 ground electronic state as

the lower level. The first excited state of ThF is 2 ∆5/2 , which is 2500 cm−1 higher in energy and has not been
observed in our experiments [44]. However, there are finer structures within the ground electronic states,e.g. vibrational. During the molecule production, some of the molecules are created in excited vibrational
states within the ground electronic manifold. The vibrational spacing of the ground state is about 600 cm−1
(1 cm−1 ≈ 30 GHz). Given favorable Franck-Condon factors, different vibrational states from the groundelectronic manifold can couple to the same excited vibronic state, giving rise to a vibrational progression oftransitions separated by the vibrational spacing of the ground electronic manifold. Likewise, the same lower
vibronic state can couple to vibrational progressions of the excited electronic state.
Using a combination difference analysis, we tentatively group vibronic bands that share the same
upper and lower electronic states, but different vibrational levels, into sets of vibrational progressions. The

20
combination differences of the lower states are known from previous work [44, 45]. The criteria for extracting
combination differences for the upper states are: (i) vibronic bands that belong to the same electronic stateshould have similar rotational constants; and (ii) vibrational constants should agree with ab initio calculations
within a reasonable range (500 < ωe < 700 cm−1 , 0 < ωe χe < 10 cm−1 , similar radiative lifetime).
For transitions that we associate with a vibrational progression, we redo the fit to the vibronic bandusing the rotational constant (Bv′′ ) for the relevant vibrational quantum number (v ′′ ) of the lower state in
the transition. From these fits, we extract the molecular constants νe , Be , αe , ωe , and ωe χe . These constants
are related to the fitting constants (ν0 , Bv , and v, which are defined in previous sections) as such:



2



2
1
1
1
1
′
′
′
′′
′′
′′
′′
v +
− ωe χe v +− ωe v ++ ωe χe v +
,
2
2
2
2


1
Bv = Be − αe v +.
2
ν0 = νe + ωe′(2.5a)(2.5b)We extract the molecular constants of the ground state with global fit to all the transitions involvedin vibrational progressions, and the result is summarized in Table 2.1.

−1ωe (cm )ωe χe (cm−1 )Be (cm−1 )αe (cm−1 )re (Å)This work (exp.)

Barker et al. [44]Theory601.00(2)2.07(3)0.2339(2)0.0014(3)2.026(3)605(15)0.237(5)2.01(3)598.8
2.1
0.2325
2.032

Table 2.1: Comparison of molecular constants of the X 2 ∆3/2 ground state of ThF in experiment and theory.
90% confidence intervals are quoted in parentheses. The parameter re is the equilibrium internuclear distance,
and the other molecular constants are defined in equation 2.5. We collaborated with Prof. Lan Cheng fromJohns Hopkins University for the theoretical calculations [46].

A comprehensive list of the upper state vibrational progressions that we have tentatively assigned isshown in Table 2.2.
For progressions with only two different v ′ , e.g. v ′ = 0 and v ′ = 1, we conventionally set ωe χe tozero so that ωe is just the splitting between the relevant rotational band origins of the upper state. 90%
confidence intervals, which come from a convolution of fitting uncertainties and vibrational progression fitting
uncertainties, are quoted in parentheses in Table 2.2.
Since the bands in our data set have relatively low resolution and high state density, there is a risk that21
Ω = 3/2 [14.09]Ω = 3/2 [15.18]Ω = 3/2 [18.62]Ω = 3/2 [19.98]Ω = 3/2 [20.40]2
Π1/2 [27.98]Ω = 3/2 [28.37]2
Π1/2 [29.22]

νe

Be

αe

ωe

ωe χe

p+2q
2

14093.80(3)15180.19(1)18616.19(2)19978.19(3)20397.40(2)27977.94(4)28370.52(3)29223.19(6)0.2276(5)0.2221(9)0.2179(5)0.2172(4)0.2126(3)0.2173(8)0.2138(5)0.2175(3)0.0013(2)0.0007(3)-557.25(2)584.00(4)517.54(3)533.05(3)569.47(2)563.13(4)577.59(3)580.5(1)2.00(1)3.80(8)0.009(6)0.022(3)Table 2.2: Comprehensive list of tentative assignments of vibrational progressions. Vibrational
progressions are groups of transitions which share the same upper and lower electronic states, but different
vibrational levels. Units for all the above constants are quoted in cm−1 . 90% confidence intervals, which comefrom a convolution of spectroscopic fitting uncertainties and vibrational progression fitting uncertainties, arequoted in parentheses. Large fitting uncertainties for the rest of the vibrational progressions prevent us fromextracting their corresponding αe , so αe is set to zero for our fitting purposes.

transitions are incorrectly grouped, and that apparent progressions occur only due to coincidence. Hence,
more experiments are required to confirm these preliminary assignments, and we leave this as a challenge to
our fellow spectroscopists.
The aforementioned analysis allows us to group most of the vibronic bands found in the visible region
into vibrational progressions. In the ultraviolet region, however, we are unable to identify any definite
vibrational progressions. Missing transitions in vibrational progressions could be explained by perturbationsof these high energy levels, which will be discussed in Section 2.6.2.

2.6

Further discussion

2.6.1

Search for the s2 l state in ThF
The ground electronic state of ThF has an s2 d structure for its valence electrons. A one-photontransition can bring the configuration to spd, s2 p, or s2 f . We started the LIF experiment hoping to find an
s2 f Rydberg state, where we can excite it with more photons to bring it to a pre-ionization state with s2 l
configuration, where l ≥ 4. States with s2 l configurations are known as core-nonpenetrating Rydberg states,
where the orbital angular momentum of the Rydberg electron prevents the electron from having significant
spatial overlap with the rest of the molecule. This molecule can then autoionize by shedding energy fromthe s2 ion core to the Rydberg electron and preserve its s2 nature.

22
An ionization process like this would prepare all the ions in the J = 0 state. In principle, thiscan give a much higher state preparation efficiency as compared to the one we settled on (Section 2.6.3),
where the molecule autoionizes into a distribution of J states within the ground vibronic manifold of ThF+
X 3 ∆1 (v = 0, J = 1 to 5).
Working with our theorist collaborator Prof. Lan Cheng, we know that s2 l states are high-lying
states and have B constants that are close to that of ThF+ (0.243 cm−1 [43]) because of their similarities inelectronic structures. However, all the B constants of the transitions that we find are smaller than the groundstate (0.233 cm−1 ), which is consistent with the fact that excited states usually have longer internuclear2ℏ
separation of the nuclei, resulting in smaller B = 2µr
2 (where µ is the reduced mass). Disappointingly, theseestates with B smaller than that of the ground state are probably spd states.
Moreover, the spectra obtained at high energy levels (deep ultraviolet region) in our LIF spectroscopy
exhibit characteristics of strong perturbations (lack of regular rotational band structure), which make identifying the states very hard with the resolution that we have. Hence, we ended up with an extensive databaseof transitions but not the one we started out to search for. Nevertheless, we managed to find a suitable state
to use as our intermediate state for REMPI (Section 2.6.3), but before we move on to that we shall address
the problem with perturbations in the deep ultraviolet region in the next section for completeness.

2.6.2

Perturbations in the deep ultraviolet region
One explanation for the lack of regular rotational band structures in the deep ultraviolet region is

the result of perturbations between the s2 f states and spd states. In contrast to the region where only spd
states exist, i.e. low lying states, the diabatic potential curves of both types of states cross very frequently athigh energy levels (Figure 2.7), because of the significant difference in their internuclear distances (different
rotational constants of spd and s2 f ). In this region, interactions could give rise to many avoided crossings,
and profoundly distort the molecular potentials for both the s2 f and spd states. The crossings of thesepotential curves would redistribute the oscillator strength of a vibrational level in a potential curve to manyother vibrational levels of other potentials, hence making the transitions very weak and come with irregularvibrational patterns.

23

spd

Internuclear distance(a) Similar internuclear distances.

EnergyEnergy

s2 f
s2 f
spd

Internuclear distance(b) Different internuclear distances.

Figure 2.7: Crossings of diabatic potential curves of spd and s2 l states distort the potential
curves. The horizontal lines represent the vibrational levels. Left: low lying spd and s2 f states have similarinternuclear distances, so their potential curves do not cross each other. Right: high lying spd and s2 f states
have different internuclear distances, so their potential curves (dashed) cross each other and mix, giving rise
to very different adiabatic potential curves (solid).

2.6.3

Auto-ionization into the ground vibronic state of ThF+
We choose the Ω = 3/2 [32.85] state as the intermediate state for the creation of ThF+ in the

X 3 ∆1 (v = 0) state through REMPI. This choice is motivated by several considerations:(1) This chosen intermediate state lies above IE/2, and so non-resonant ionization from ground state of
ThF by the more intense second photon is greatly suppressed.
(2) The transition from the ground state to the Ω = 3/2 [32.85] state is a relatively strong transition ina wavelength region that is easily accessible.
(3) A convenient 532 nm photon (from the second harmonic of Nd:YAG laser) can be used as thesecond photon, which excites the neutral molecule into a broad autoionizing resonance that lies only
62 cm−1 above the ionization potential for decay into the X 3 ∆1 (v = 0) state in ThF+ . Since the next
vibronic state a 1 Σ+ (v = 0) lies 314 cm−1 above the ground state, energy considerations restrict allions formed to be in the ground vibronic state, which coincidentally is the preferred eEDM-sensitive
state.
We verify the formation of ions in the ThF+ ground vibronic state by state-selective resonance-

24
enhanced multi-photon dissociation (REMPD). The details of the dissociation technique will be described in
Chapter 3. Using the dissociation readout technique, we also show that the distribution of the ion population
across the different J + levels depends on the J ′ quantum number of the intermediate state used in the REMPI
process (the + superscript indicates quantum numbers of the ThF+ to distinguish it from those of ThF).
The J + -distributions of the ions created with different J ′ intermediate states are shown in Figure 2.8.

Figure 2.8: J + -distributions of ThF+ ions created from different intermediate J ′ states of ThF.
The rotational distribution of the ThF+ ions shifts toward higher J + states as the J ′ value of the intermediate
state is increased. For easy comparison purposes, the red and blue traces are offset upward from zero by 30
and 60, respectively. These traces are scans of the R branch of the dissociated intermediate state used in
the resonance-enhanced multi-photon dissociation spectrum.

Figure 2.8 shows that the rotational distribution of the ions shifts toward higher J + states when ahigher J ′ intermediate state is used. Since the electron that is lost during the ionization process carries
a fixed range of angular momenta, we expect to see a higher final angular momentum distribution for theions when an intermediate state of higher angular momentum is used, and vice versa [50]. Since the eEDM
sensitive state is the lowest rotational state (J + = 1), we use the J ′ = 3/2 state as our intermediate state
for the REMPI process, which results in 30% of the ions being produced in the desired J + = 1 state.
Before we move on to talk about spectroscopy experiments performed on ThF+ created with thisnew-found intermediate state, we shall have some closing remarks on the LIF experiment.

25
2.6.4

Closing remarks

2.6.4.1

Contamination from other species and REMPI TOF-MS

The LIF experiment on ThF has been plagued by contamination from other species created during
the molecule creation stage. Performing a REMPI time-of-flight mass spectrometry (TOF-MS) on ourmolecular beam shows contamination by species like ThO and ThF2 . The TOF-MS is performed by kicking
the molecular beam with electric fields towards a microchannel plate (MCP) after they are ionized by the
REMPI lasers. The kick imparts the same momentum to all the ions (of the same charge), resulting in spatial
separation of species with different masses as they fly towards the MCP, giving rise to different arrival times
at the MCP. A sample of the trace from the REMPI TOF-MS is shown in Figure 2.9

Figure 2.9: Typical traces of the REMPI TOF-MS experiments. The separation of species by massin the REMPI TOF-MS trace allows us to distinguish the ion species to separate the ThF signals from therest. Our mass resolution is sufficiently high to separate ThO+ from ThF+ .

One possible reason for seeing ThO signals in our LIF is that ThO is created during the ablation stage
where residual oxygen from air that is mixed into our buffer gas reservoir reacts with the thorium plasma tocreate ThO.

26
2.6.4.2

Current status of ion creation

In Section 2.6.3, we saw that we can create ions with 30% of them in the J + = 1 rotational manifold
by addressing the Ω = 3/2 [32.85] (v ′ = 0, J ′ = 3/2) intermediate state. Having a lot of ions created in the
J + = 1 rotational manifold is desirable because that is the eEDM-sensitive manifold in our experiment, andions in the rest of the rotational manifolds only contribute to decoherence through ion-ion collisions in our
ion trap.
We can go through the Ω = 3/2 [32.85] (v ′ = 0, J ′ = 3/2) intermediate state by addressing the groundstate of our neutral ThF molecule with a laser tuned to the P(5/2) or a Q(3/2) line as the first photon of our
REMPI process. However, as we shall see in Chapter 4, we have the ability to move ion population in the
J + > 1 rotational manifolds into the J + = 1 manifold with optical and microwave pumping. Hence, there
is no pressing need for us to stick to the J ′ = 3/2 intermediate state to maximize our yield of useful ThF+
ions. In fact, we now illuminate our ions with our dye laser parked at the Q branch, addressing the Q(3/2),
Q(5/2), and Q(7/2) lines simultaneously (incoherently) with the broad linewidth of our pulsed dye laser.
The ions created no longer have a population peak at J + = 1, but we now have many more ions because theQ lines address the molecules in the X 2 ∆3/2 (v ′′ = 0, J ′′ = 3/2 to 7/2) vibronic manifold instead of only the
J ′′ = 3/2 with the Q(3/2) line only. Ions spread out across the rotational manifolds can then be ushered
into the J + = 1 rotational manifold with techniques to be introduced in Chapter 4.
Understanding how we usher ions to our desired quantum states involve knowing how to perform state
readout of our ions first. So we shall be patient and first look at how we read out the quantum states of ourions using photodissociation in the next chapter. Note that from here on, we will only be talking about theThF+ ion, so we shall drop the + superscript from the quantum numbers.

Chapter 3

State readout of ThF+

3.1

Introduction
The previous chapter taught us how to prepare ThF+ ions. Using techniques to be discussed in thischapter, we managed to verify that the ions are prepared in the ground vibronic state of ThF+ . Similar
to its ionization counterpart, REMPI, we use resonance-enhanced multi-photon dissociation (REMPD) todissociate our molecule state selectively to perform our quantum state readout. Our pulsed dye laser (Sirah
Cobra-Stretch) has a linewidth that is narrow enough (∼ 6 GHz after doubling with BBO) to distinguishbetween rotational lines (∼ 10 GHz), but not structures like hyperfine, Stark, parity doublets, and Zeeman
sublevels within each rotational manifold. To discern ion population within the finer structures, we use cw
lasers to shelf ions in higher rotational or vibrational states (see Chapter 4) before dissociating everythingthat is left in that rotational manifold. Upon dissociation, the molecules in the targeted rotational state
will turn into Th+ ions and neutral F atoms, and the ions (both the dissociated Th+ and the undissociated
ThF+ ) are ejected towards our microchannel plate (MCP) assembly for detection (see Figure 3.1). The timeof flight is long enough for the ions of different species to separate by their masses. This has been the basisof state readout in the Gen. I experiment. With this technique, we can read out the ion population in asingle selected rotational manifold with each shot of our experiment.
In the Gen. II experiment, in collaboration with Prof. Tanya Zelevinsky from Columbia University, wemanaged to squeeze more out of our dissociation technique to give us two-state detection, instead of just one
state, with each destructive measurement with our action spectroscopy [38]. In short, this upgrade makes

28

Right before detector

Detector

In the trap

Kick out direction

Figure 3.1: Schematic diagram showing state readout using state-selective REMPD. On the left
we have the ions right after dissociation; molecules in the targeted state are dissociated (indicated withdashed lines connecting the blue fluorine atom and the red thorium ion), and those in the wrong stateremain intact as molecules (indicated with solid lines). Right after dissociation, the ions are ejected out of
the trap (rightwards in the diagram) towards the ion detector. The Th+ ions are separated from the ThF+
ions by their difference in masses, and they reach the detector first. By counting the number of Th+ ions,
we can deduce how many ions are in the targeted state right before dissociation.

use of the fact that the photofragments can be ejected out of the molecule with an angular dependence on
the state of the molecule before dissociation. By choosing the intermediate state used for REMPD carefully,
we can have the photofragments ejected in largely opposite directions for the two different states that wewish to detect. This also relies on the fact that our molecules have well defined orientations in our trap
along the direction of the electric field (Erot ) that we use to polarize our molecules. The physics behind thisangle-resolved photofragments technique is quite involved, but it is mentioned in many sources (see Ref.
[38, 51–53], and William Cairncross’s thesis) so I shall only state the expression of the angular distribution
here:
I(θ, ϕ) ∝

=

X

X

Ωf

Jf ,(mJ )f

X

X

Ωf

Jf ,(mJ )fJf(ϕ, θ, 0)D(mJ )f ,Ωf∗

2µ · ϵ|ϕ′ ⟩⟨ϕcont.
Jf ,(mJ )f ,Ωf ,E |µp

2Jf + 1p∗Jf
2Jf + 1 D(m
(ϕ,
θ,
0)
J )f ,Ωf×qX(−1)p+(mJ )f −Ωf (2Jf + 1)(2J ′ + 1)
p,q
2

×



Jf
1 J′
−(mJ )f −p m′J



Jf 1 J ′−Ωf q Ω′



µ q,{Jf ,(mJ )f ,Ωf ,E}ϵ p ,(3.1)

where D is the Wigner D-matrix; µ q is the q th (spherical tensor) component of the transition dipole moment;

29
ϵ p is the pth (spherical tensor) component of the (second REMPD) photon polarization; primed variables
belong to the state that we are dissociating from (the intermediate state in our 1 + 1′ scheme); variables
with the subscript f indicate those for the predissociation state. By E1 selection rules (or properties of the
Wigner 3-j symbol), we demand that (mJ )f = m′J − p and Ωf = Ω′ + q. The sum is coherent in {Jf , (mJ )f }but incoherent in Ωf . Caveat: this necessarily assumes that predissociation states of different Ωf are well
separated in energies.
The question now is whether we can also implement this upgraded detection scheme on ThF+ . The
rest of this chapter will proceed as follows. We will begin with the rule of thumb (Section 3.2) for choosing
the intermediate state that will give us the best angle-resolved photofragment resolution. We will then lookat results from our spectroscopy (Section 3.3). Finally we will discuss how we can further optimize thistechnique on ThF+ (Section 3.4).

30

3.2

Rule of thumb for two-state detection using angle-resolved photofragments
There are two main things1 that we care about in performing two-state detection using angle-resolved

photofragments: (i) angular distribution of photofragments, and (ii) energy of ejected photofragments.

3.2.1

Angular distribution of photofragments
We want the angular distribution to be asymmetric about the equator, and as polar as possible. Itturns out that the angular distribution depends on both the intermediate and the predissociation states that
we address. There is no hard and fast rule on what dissociation pathway we should take to get the mostdirected angular distribution, a polar one in an ideal case, but here is a rule of thumb on how to choose theintermediate state.
Dissociate from states with Ω′ ̸= 0. States with Ω′ = 0 are not polarized, i.e. the molecule does not havea preferred orientation w.r.t. our polarizing Erot = 60 V/cm at the current field strength applied.
Our Erot is not strong to couple states of opposite parity for states with Ω′ = 0; the closest stateof opposite parity in an Ω′ = 0 state is the next rotational state, but the Stark coupling is small
compared to the rotational splitting (dmf Erot ∼ 100 MHz vs rotational splitting ∼ 30 GHz)2 . Sincestates with Ω′ = 0 are not polarized, the photofragments will be ejected equally likely in bothhemispheres, thus rendering the two-state detection through angular resolution of photofragments
ineffective.
Dissociate from states with m′J ̸= 0. Molecules in the m′J = 0 state have the J ′ vectors precessing on theplane perpendicular to the quantization axis. We do not expect highly polar angular distributions
when dissociating from such states.
The first REMPD photon: J ′ = 2. Our first REMPD photon takes us from our eEDM-sensitive state
X 3 ∆1 (v = 0, J = 1, mJ = 1, |Ω| = 1) to an intermediate state (J ′ , m′J , |Ω′ |).3

We choose to work

1 There is a third. We want to have high dissociation efficiency. We will talk about this more in Section 3.4.2.
2 For the |Ω′ | > 0 states, states of opposite parities are separated only by the Ω-doubling splitting ∼ 10 MHz
3 Our eEDM-sensitive states are in the m

F = ±3/2 stretched Zeeman states, so they correspond to the mJ = ±1 states.
We will be switching between these two languages for the bulk of this section.

31
with J ′ = 2 because the linewidth of our pulsed dye lasers are not narrow enough to resolve the Q
lines to dissociate only the X 3 ∆1 (v = 0, J = 1) rotational manifold, so we park on the R(1) line
instead. The quantum number m′J depends on the polarization of the photon used, e.g. m′J = 2 if
we use a σ + photon, and m′J = 1 if we use a π photon. Our molecule decides what |Ω′ | is at the
wavelength of the first photon used, but we get to decide which wavelength to use.
One could simulate the angular distributions of all possible dissociation pathways from our eEDMsensitive state, assuming that the transitions exist. One can then see what intermediate states are available
to us from our spectroscopy, and decide on a REMPD scheme that gives the most polar angular distribution.
All the constraints above limit us to only ten possible outcomes: four for |Ωf | = 1, and six for |Ωf | = 2.
There is no reason not to look at all of them, shown in Figures 3.3 and 3.4.
The first thing we noticed from Figure 3.3 is that the directedness of the photofragments do not look
impressively polar for the cases of |Ωf | = 1. One reason is that when Ω′ < J ′ for a fixed m′J , the internuclear
axis of the molecule is not as aligned with the polarization axis (see Figure 3.2).

(a)

(b)

(c)
q

q

Erot

m0J

J 0 (J 0 + 1)
Ω0

q

m0J

J 0 (J 0 + 1)

m0J

J 0 (J 0 + 1)
Ω0

Ω0Figure 3.2: Semiclassical picture of molecule alignment with polarizing field. The dashed line
indicates the direction of our polarizing axis defined by our applied electric field Erot . (a) The quantum
numbers m′J and Ω′ are the projections of J ′ onto the quantization axis and internuclear axis, respectively.
The internuclear axis precesses and nutates around the polarizing axis to preserve the quantum numbers.
The configuration in (a) shows the situation where |m′J | ≠ |Ω′ |: the internuclear axis is not well aligned with′′
the quantization axis. (b) Situation where J = |mJ | = |Ω′ |; the internuclear axis is better aligned with thepolarizing axis, giving more directed photofragments towards the pole. (c) Having a large J ′ = |m′J | = |Ω′ |makes the J ′ vector precess even closer to the internuclear axis as compared to (b), giving even more directed
photofragments towards the pole.

Furthermore, we only have access to |Ωf | = 0, 1, 2 if we start from the |Ω′ | = 1 intermediate state by
the E1 selection rules. Throwing Ωf = 0 out of our list of choices for aforementioned reasons, we are limited
to only four choices. Only the σ + plot in Figure 3.3b looks promising, but there is no promise that we will32
end up with an |Ωf | = 2 predissociation state. Fortunately, the directedness of the photofragments do look
much more polar if |Ω′ | = 2. We shall take a look at this next.
Intensity (arb. unit)

+

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0

45

90 135
(deg)

180

0.0

0

45

90 135
(deg)

180

0.0

+

0

45

90 135(deg)

180(a) From J ′ = 2, m′J = 1, Ω′ = 1 to Ωf = 1. Transition strength: (0.102, 0.158, 0.188)

Intensity (arb. unit)

+

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0

0

45

90 135
(deg)

180

0

45

90 135
(deg)

180

0.0

+

0

45

90 135(deg)180(b) From J ′ = 2, m′J = 1, Ω′ = 1 to Ωf = 2. Transition strength: (0.102, 0.145, 0.249)

Intensity (arb. unit)

+

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0

45

90 135
(deg)

180

0.0

0

45

90 135
(deg)

180

0.0

+

0

45

90 135(deg)180(c) From J ′ = 2, m′J = 2, Ω′ = 1 to Ωf = 1. Transition strength: (0.200, 0.116, 0.381)

Intensity (arb. unit)

+

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0

0

45

90 135
(deg)

180

0

45

90 135
(deg)

180

0.0

+

0

45

90 135(deg)

180(d) From J ′ = 2, m′J = 2, Ω′ = 1 to Ωf = 2. Transition strength: (0.0392, 0.201, 0.476)

Figure 3.3: Simulated angular distribution of photofragments I(θ, ϕ) for Ω′ = 1. Left column:
Newton sphere plots. The Newton sphere plots should be interpreted this way: (i) imagine the molecule
to be at the center of the sphere, oriented with the Th side pointing upwards; (ii) upon dissociation, theTh+ ion is ejected from the molecule, and it hits the inside surface of the sphere, painting it blue. To
effect a two-state detection with the anisotropic angular distribution of photofragments, we demand theblue (photofragments) to be as concentrated at the poles as possible, so that the overlap of photofragmentsfrom molecules with the opposite orientation is minimized for maximum distinguishability. In each subfigure,there are three Newton spheres, corresponding to the result of using different polarizations of the dissociating
photon (second REMPD photon in our case). Right column: number density along the polar angle (θ)
direction. In other words, the total number of photofragments
within a solid angle spanned by dθ dϕ sin θRat (θ, ϕ) is I(θ, ϕ) sin θ dθ dϕ, and I have plotted out I(θ, ϕ) dϕ. The dissociation
efficiencies are shown inR1order of the cases (σ − , π, σ + ). The strengths shown are evaluated from 4πI(θ, ϕ) dΩs.a. with µ in Equation
3.1 given only by the relevant Clebsch-Gordan coefficients, and Ωs.a. the solid angle, not to be confused withquantum number Ω. More discussion of the plots in main text.

33
Intensity (arb. unit)

+

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0

45

90 135
(deg)

180

0.0

0

45

90 135
(deg)

180

0.0

+

0

45

90 135(deg)

180(a) From J ′ = 2, m′J = 1, Ω′ = 2 to Ωf = 1. Transition strength: (0.175, 0.113, 0.0582)

Intensity (arb. unit)

+

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0

0

45

90 135
(deg)

180

0

45

90 135
(deg)

180

0.0

+

0

45

90 135(deg)180(b) From J ′ = 2, m′J = 1, Ω′ = 2 to Ωf = 2. Transition strength: (0.176, 0.0862, 0.180)

Intensity (arb. unit)

+

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0

45

90 135
(deg)

180

0.0

0

45

90 135
(deg)

180

0.0

+

0

45

90 135(deg)180(c) From J ′ = 2, m′J = 1, Ω′ = 2 to Ωf = 3. Transition strength: (0.0286, 0.203, 0.317)

Intensity (arb. unit)

+

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0

0

45

90 135
(deg)

180

0

45

90 135
(deg)

180

0.0

+

0

45

90 135(deg)

180(d) From J ′ = 2, m′J = 2, Ω′ = 2 to Ωf = 1. Transition strength: (0.397, 0.153, 0.0476)

Intensity (arb. unit)

+

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0

45

90 135
(deg)

180

0.0

0

45

90 135
(deg)

180

0.0

+

0

45

90 135(deg)180(e) From J ′ = 2, m′J = 2, Ω′ = 2 to Ωf = 2. Transition strength: (0.0751, 0.322, 0.238)

Intensity (arb. unit)

+

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0

45

90 135
(deg)

180

0.0

0

45

90 135
(deg)

180

0.0

+

0

45

90 135(deg)

180(f) From J ′ = 2, m′J = 2, Ω′ = 2 to Ωf = 3. Transition strength: (0.00317, 0.0794, 0.713)

Figure 3.4: Simulated angular distribution of photofragments I(θ, ϕ) for Ω′ = 2. Left column: Newton sphere plots. See caption of Figure 3.3 on the interpretation of the Newton sphere plots. Right column:
number density along the polar angle (θ) direction. In other words, the total number of photofragmentsRwithin a solid angle spanned by dθ dϕ sin θ at (θ, ϕ) is I(θ, ϕ) sin θ dθ dϕ, and I have plotted out I(θ, ϕ) dϕ.
The transition
strengths are shown in order of the cases (σ − , π, σ + ). The strengths shown are evaluatedR
1from 4π I(θ, ϕ) dΩs.a. with µ in Equation 3.1 given only by the relevant Clebsch-Gordan coefficients, andΩs.a. the solid angle, not to be confused with quantum number Ω. More discussion of the plots in main text.

34
We shall use the results in Figure 3.4 to make a few comments:
Choice of second REMPD photon polarization. Let me draw your attention to the last three rows ofFigure 3.4. These correspond to angular distributions of photofragments with the same intermediate
state (with a σ + -polarized first REMPD photon), but different |Ωf |. The thing to note is that
depending on what |Ωf | is, one has to choose the appropriate polarization of the second REMPD
photon to maximize the directedness of the angular distribution of photofragments, i.e. σ − -, π-, andσ + -polarized second REMPD photon for |Ωf | = 1, 2, and 3, respectively.
How can we know what |Ωf | is? We could just scan the whole photon polarization parameter space tofind out, but let me suggest an approach that is more guided from an experimental standpoint.
We still focus on the last three rows of Figure 3.4. We can compare the contrast of the angular
distribution of the case where the second REMPD photon is σ + against that of σ − . If they lookvery similar, chances are the predissociation state has a very strong |Ωf | = 2 character (see Figure3.4e).
To verify the |Ωf | = 2 character, one would have the switch either the first photon (Figure 3.4b) or
the second photon (Figure 3.4e) to π-polarization, and see an improvement in contrast.
If the contrasts look very different with the handedness of the circular polarization for the second
photon, one can switch both REMPD photon polarizations to π, and see if we end up with Figures
3.4a or 3.4c.
What should we do for the different cases of |Ωf |? If |Ωf | = 3, we just need to use a σ + -polarized
first photon, and a σ + -polarized second photon. This is as good as it gets for |Ωf | = 3.
If |Ωf | = 2, we have a choice. Comparing Figures 3.4b and 3.4f, one may think that the σ + plot informer gives a more polar angular distribution than the π plot in the latter, because the part above
the equator is more concentrated at the pole in the former. However, the former has some structureon the other side of the equator from the pole, while the latter decreases monotonically from thepole. The differences are small, but the latter is cleaner in principle. Nevertheless, the best doublet
angle-resolved scheme for an |Ωf | = 2 predissociation state will involve one σ + photon and one π

35
photon. This has to be a cross-beam setup, where the two photons will have to come in throughdifferent windows at the trap.
For the |Ωf | = 1 case, a similar comparison to the |Ωf | = 2 case holds for the π plot in Figure 3.4aand the σ − plot in Figure 3.4d. We either do a π first photon and π second photon (for the former),or a σ + first photon and a σ − second photon (for the latter). In either case, both photons can comein through the same window at the trap.
What should we make of the transition strengths reported in the figures? The net dissociation efficiency depends on a few factors: (i) the strength of the transitions, (ii) the intensities of the
dissociation lasers, and (iii) the potential energy landscape past the dissociation threshold.
For a weak transition, e.g. the σ − transition from Figure 3.4e, one can increase the overall dissociation
efficiency by overwhelming the transition with a very intense laser. Jumping ahead to data shown in
Figure 3.12, which we believe to be a case of Figure 3.4e, it is then interesting to see that the σ − andσ + transitions result in similar dissociated Th+ numbers. This suggests that the second dissociation
photon is saturating the transitions for both transitions. This could result in the two transitions
having the same overall dissociation efficiencies, despite having vastly different predicted transition
strengths. The puzzle now, then, is why would the overall dissociation efficiency be so low (around10%) for those transitions. Our guess right now is that the second photon could be connecting to a
predissociation curve that has non-trivial crossings with other predissociation curves much like thoseshown in Figure 2.7. It could also be that the first photon is not saturating the bound-to-boundtransition. However, we are seeing signs of power broadening of the first photon at the current laser
intensity that we are using, so this is likely not the case.
These rules are by no means rigorous, and it depends on the structure of the predissociation potential landscape. The plots in Figures 3.3 and 3.4 are generated assuming that (i) we connect to only one
predissociation state and that it comes with a definite Ωf , (ii) there are no potential crossings with otherpredissociation potential curves, and (iii) we can sum contributions from all allowed Jf coherently withamplitudes given by the Clebsch-Gordan coefficients. We have no reason a priori to believe that all these36
assumptions hold. We make these assumptions only to simplify the model for understanding. In fact, we seein the Gen. II eEDM experiment with HfF+ that the REMPD protocol used resulted in two radial bands ofphotofragments, corresponding to the dissociated ions going through two different pathways ending up withdifferent KEdissoc. (see William Cairncross’s and Tanya Roussy’s theses, and also the section below). Interpret the results of Figures 3.3 and 3.4 qualitatively and with a teaspoon of soy sauce; they only serve to guidethe general direction of the experiment design, not to claim full understanding of the angular distributions
seen in the experiments like those in Section 3.3.

37
3.2.2

Energy of ejected photofragments
Right before dissociation, the ThF+ molecules have a spatial extent in our ion trap. The molecules

also have a thermal velocity distribution. Upon dissociation, the Th+ ions inherit the positions and velocitiesof their parent molecules. In addition to the kinetic energy (KEtherm. ) that the Th+ ions inherit from their
ThF+ parents, they can also gain additional kinetic energy (KEdissoc. ) from the dissociation (see Figure 3.5).

KE<
dissoc.
Th+∗ + FKE>dissoc.
Th+ + FD0+

X 3 ∆1Figure 3.5: Schematic diagram of molecular potential curves. Not drawn to scale. The horizontal
axis indicates the internuclear distance of our molecule, and the vertical axis indicates potential energy. The
horizontal lines indicate various vibrational manifolds for each potential curve. D0+ indicates the minimum
energy required to dissociate our molecule from the ground state. In our 1+1′ REMPD scheme, the first
photon is a bound-to-bound transition bringing our molecules from X 3 ∆1 (in black) to an intermediate
state (in green), and the second photon brings our molecules into the predissociation state (indicated by thedashed lines). Upon dissociation, the photofragments can acquire an additional kinetic energy by moving
down the potential curves of the predissociation molecular potentials. One would naı̈vely think that we canimpart more kinetic energy to our photofragments through using REMPD photons of shorter wavelength sothat we end up with more energy above the dissociation threshold. This is wishful thinking, because KEdissoc.
depends on which predissociation potential the second photon connects to: the purple second photon is moreenergetic than the blue second photon in the figure, but connects to a predissociation potential curve that
relaxes to an excited state of Th+ , imparting an overall lower kinetic energy to the photofragment.

In the limit of KEdissoc. ≪ KEtherm. and in the velocity mapping regime (because we have a very longtime of flight compared to the ratio of our ion cloud size and thermal velocities of our ions), the angular38
distribution of ejected photofragments on our ion detector will be smeared out by the initial distribution of
ThF+ thermal velocities. This is unfavorable for two-state detection.
We want the other limit of KEdissoc. ≫ KEtherm. for us to see the effect of angular distribution of
the photofragments on our ion detector. Working with typical ion cloud temperatures of 10 to 200 K,
corresponding to ion speeds of 20 to 80 m/s, we need to find a REMPD process that will give our Th+ ionsadditional kinetic energy corresponding to more than 80 m/s in speeds. Figure 3.6 gives a visual comparisonof the two limits.
In the trap

At the ion detector

KEdissoc.  KEtherm.

KEdissoc.  KEtherm.

Figure 3.6: Schematic diagram comparing the effects of KEdissoc. w.r.t. KEtherm. . The left column
corresponds to the molecule positions right after dissociation, while the right column corresponds to thoseat the ion detector after the ions are kicked out of the trap. The orange arrows indicate thermal velocity.
The red and green arrows indicate additional kinetic energy gained from dissociation; they are color coded
red and green to guide the eye to the dissociation ejection direction. The top row shows the case where
KEdissoc. ≪ KEtherm. : one can see that the Th+ moving upwards are not well separated in position fromthose going downwards; the ion positions are dominated by the thermal velocities. The bottom row shows
the other limit of KEdissoc. ≫ KEtherm. : one can see that all the Th+ ions moving upwards are on the top
half of the detector, while those moving downwards are on the bottom half; the ion positions correlate verystrongly to the dissociation ejection directions, allowing us to effect a two-state detection with each shot of
the experiment.

39
Unfortunately, unlike choosing an intermediate state with the right quantum numbers, there is noknob for us to turn to control KEdissoc. directly. We can only scan the wavelength of our lasers to look forstrong transitions with the right quantum numbers, and hope that they have enough KEdissoc. . Fortunately,at least one such transition exists, as we shall see in the next section.

40

3.3

Angle-resolved photofragments in ThF+
The setup used to look for good REMPD intermediate states is similar to that used in the LIF

experiment, and a schematic diagram is shown in Figure 3.7.
Ion. & Dissoc. lasers
IMG MCP

Ablation laser
Diff. pumping chamber

ThF+

Source chamber
Th coinPulsed valve
Gas reservoir (Ne, 0.1% SF6 )Trap
TOF MCP

Figure 3.7: Schematic diagram of the REMPD experimental setup. The REMPD spectroscopy
consists of the following steps:(1) A molecular beam of ThF is created in a similar manner as the LIF experiment.
(2) The molecules are ionized into the ground vibronic state of ThF+ with REMPI lasers at the centerof ion trap, and the ions are loaded into the trap.
(3) We have an option of populating the X 3 ∆1 (v = 1) or the a 1 Σ+ (v = 0) vibronic states by depleting
the X 3 ∆1 (v = 0) states with a laser into an excited state found in previous work [43], and allowing
the states to decay back down to one of the aforementioned states through selection rules. As such,
we will see excited states of ThF+ that couple to the aforementioned states, allowing for state readoutof those states, which will prove to be very useful in later chapters.
(4) Another set of lasers consisting of (i) a tunable laser, and (ii) a second UV laser excites the ionsin the trap. The sum of the energies of the two photons is greater than the dissociation energy of
ThF+ . If the first photon is resonant with a transition between the ground vibronic state and an
excited state, the ions will be excited to the intermediate state where it gets further excited to a
pre-dissociation state by the second photon. The ion then dissociates into Th+ and F. To ensurethat the signal is real, we take four scans: (i) with no REMPD laser, (ii) with only the first photon,(iii) with only the second photon, and (iv) with both REMPD photons.
(5) The resulting mix of ThF+ and Th+ ions are kicked out towards one of the two MCPs, where we
can read out ion numbers of both species through TOF-MS (Section 2.6.4.1).
(6) Repeat the experiment with different excitation wavelengths.

The ion trap used in the setup is the 6-fin Paul trap from the setup used in the Gen. I experiment [14].
At some point, we switched the 6-fin Paul trap to the baby Bucket Brigade (which is also a Paul trap, see
Section 7.4), but the general workings of both traps are the same. The workings of the 6-fin Paul trap andtrap electronics are detailed in theses of previous group members (e.g. Huanqian Loh and Matt Grau) and41
shall be omitted here for brevity. A simple description of how a Paul trap works is summarized in Appendix
A.
We began on the spectroscopy to search for the intermediate state for REMPD before we knew aboutangle-resolved photofragments, so we did not pay attention to the polarization of the REMPD lasers during
the early days of the spectroscopy. Still feeling hurt from the disappointment in the results of our LIF
spectroscopy, we decided to start the REMPD spectroscopy using a convenient dye with the widest tuning
range: DCM in ethanol. This dye allows us access to 602 nm to 660 nm with a peak wavelength at 627 nm.
We doubled the output from our dye laser with a BBO and found a few transitions. The transition with thestrongest strength is the [32.09]+266 nm combination.

3.3.1

The [32.09]+266 nm transition
After we learnt about angle-resolved photofragments from Prof. Tanya Zelevinsky, and saw that it canbe used for two-state detection in HfF+ , we decided to implement it in ThF+ . We chased our tails for sometime in the optimization of our REMPD process,4 but to cut the chase short, Figures 3.8 and 3.9 show theexperimental proof of principle results of angle-resolved photofragments with ThF+ .
Figure 3.10 shows a typical two-state detection characterization plot on the Gen. II JILA eEDM
experiment setup using HfF+ . Comparing Figures 3.9 and 3.10 shows that the quality of our two-state
detection scheme for ThF+ is on par with that for HfF+ .
In Figures 3.8 and 3.9, we present the positions of the ions on the ion detector for two combinationsof the experimental parameters: keeping the polarizations of the REMPD photons fixed, we prepared ourions in the mF = 3/2 or −3/2 Zeeman states. The “relative sign” of the Zeeman state chosen during state
preparation and the helicities of the REMPD photons matter for the separation of the doublets on the ion
detector as we shall see in the next section.

4 For example, we first started to look for angle-resolved photofragments along the time-of-flight axis instead of the transverse
imaging plane [38] because we were limited by the relative orientations of our laser beams and ion detectors.

42

Figure 3.8: Scatter plot of the Th+ ion positions at the ion detector (same data as that usedin Figure 3.9). Each plot shows the ions from a sum of five shots of the experiments. The two rows
correspond to two slightly different experiment conditions: the polarizations of the REMPD photons were
kept the same across the two rows, but we prepare the molecules in the (J = 1, mJ = −1) state for the
top row, and (J = 1, mJ = 1) for the bottom row. The first REMPD photon increases the mJ quantum
number by 1, and brings the molecules in the top (bottom) row to m′J = 0 (m′J = 1). The horizontal lines
in each plot corresponds to the mean position of the ions (of the corresponding color), while the shaded area
corresponds to the 1σ spread of the ion locations in the Y direction. The middle column is an overlay of theleft and right columns. There is clear evidence of angle-resolved photofragments in the bottom row, where
the ions in the upper doublet end up on the upper side of the detector, while those in the lower doublet end
up on the lower side.

43

Figure 3.9: Histograms of the Th+ ion positions, integrated over the X direction, at the ion
detector (same data as that used in Figure 3.8). The two columns correspond to two slightly different
experiment conditions: the polarizations of the REMPD photons were kept the same across the two columns,
but we prepare the molecules in the (J = 1, mJ = −1) state for the left column, and (J = 1, mJ = 1) for theright column. The first REMPD photon increases the mJ quantum number by 1, and brings the molecules in
the left (right) column to m′J = 0 (m′J = 1). The top row shows the overlay of the ion distribution in the Y
direction from both doublets. The grey dashed line indicates the mean of the mean positions of ions in both
doublets. The bottom row takes the difference between the two doublets. The grey dashed lines indicate theregion of the “swatch” — the region where we do not count ions to either doublet for our two-state detection
protocol, and we associate ions to the left (right) of the swatch with the upper (lower) doublet.

44

Figure 3.10: Histograms of the Hf+ ion positions, integrated over the X direction, at the ion
detector. Courtesy of the group members who worked Gen. II experiment. Comparing this with the topright plot in Figure 3.9 indicates that the quality of our two-state detection for ThF+ is on par with thatfor HfF+ .

45
3.3.2

Photon polarization combination
Depending on the “relative sign” of the Zeeman state chosen during state preparation and the helicitiesof the REMPD photons, the molecules could take a different pathway to dissociation. Figure 3.11 shows thevarious pathways the molecules can take.
PBS(1,10 ) : (0, 0)◦

PBS(1,10 ) : (0, 90)◦

Good if |Ωf | = 3

Goes through m0J = 0, bad

Goes through m0J = 0, bad

Good if |Ωf | = 1

GG*Figure 3.11: Pathway taken by molecules to during dissociation with various combinations ofphoton polarizations. The three lower (five upper) horizontal lines correspond to the mJ sublevels in theJ = 1 from the eEDM-sensitive state (J ′ = 2 in the intermediate state). The arrows indicate the REMPD
photons. The REMPD photons nominally go through a PBS each, exiting horizontally polarized, and theygo through the same QWP (after being combined through a dichroic mirror) to have the same circularity
handedness. This configuration is represented by the “PBS(1,1′ ):(0,0)◦ ” notation, and it corresponds to thefirst column. We change the handedness of the first photon by rotating the PBS of the first photon by 90◦
to make it go through the QWP vertically polarized; this configuration corresponds to the second column.
The first row, denoted by “G” is where we prepare the ions in the mF = 3/2 Zeeman state, while “G*”corresponds to preparing the ions in the mF = −3/2 Zeeman state. Molecules that take pathways that go
through m′J = 0 lose their directedness, so they should be avoided. Only the top left and bottom rightpathways can be good, but the quality of each depends on the |Ωf | that we connect to with the second
photon in REMPD.

We went through the combinations and obtained results shown in Figure 3.12 for the [32.09] intermediate state, and Figure 3.13 for the [32.50] intermediate state. Common to both intermediate states, we seethat the combinations on the off-diagonal axis show poor doublet separation, agreeing with the model wehave in Figure 3.11. Using combinations along the diagonal axis, we also saw that the [32.50] intermediate
state exhibits inferior doublet separation than the [32.09] one. This could be because the [32.50]+266 nm
combination imparts the dissociated Th+ ions with less energy than the [32.09]+266 nm combination.

46(a) mF = 3/2, σ1+ , σ1+′ .

(b) mF = 3/2, σ1− , σ1+′ .

(c) mF = −3/2, σ1+ , σ1+′ .

(d) mF = −3/2, σ1− , σ1+′ .

Figure 3.12: Distribution of ions along the Y axis on the ion detector with various combinationsof photon polarizations for the [32.09] intermediate state. Top (bottom) row: we prepare the ionsin the mF = 3/2 (mF = −3/2) state. Left (right) column: the first REMPD photon has the same (opposite)
circular polarization as the second photon. Note that the combination used in the top left plot gives thebest separation of doublets. Having similar quality of doublet separation in (a) and (d) suggests that thesecond photon connects to a predissociation state that has a predominant |Ωf | = 2 nature (compare with
Figure 3.4e). Assuming that the predissociation state is predominantly |Ωf | = 2, the fact that we get similardissociated Th+ numbers in (a) and (d) suggests that our second REMPD photon is saturating (c.f. Figure3.4e). See the discussion on dissociation efficiencies on Page 35.

47(a) mF = 3/2, σ1+ , σ1+′ .

(b) mF = 3/2, σ1− , σ1+′ .

(c) mF = −3/2, σ1+ , σ1+′ .

(d) mF = −3/2, σ1− , σ1+′ .

Figure 3.13: Distribution of ions along the Y axis on the ion detector with various combinationsof photon polarizations for the [32.50] intermediate state. Top (bottom) row: we prepare the ionsin the mF = 3/2 (mF = −3/2) state. Left (right) column: the first REMPD photon has the same (opposite)
circular polarization as the second photon. Even though there is some separation of doublets in some of theplots shown here, they are not as good as those using [32.09] as the intermediate state (see Figure 3.12a),indicating that perhaps the [32.50]+266 nm combination imparts a lower dissociation kinetic energy to theTh+ ions than the [32.09]+266 nm combination.

48

3.4

Further discussion
In the previous sections, we have seen that we can perform two-state detection with each destructive

shot of our action spectroscopy on ThF+ . Before we close this chapter, we shall address a few points:(i) other important REMPD intermediate states, and (ii) possible future directions to make the two-state
detection scheme better.

3.4.1

Other important transitions
Table 3.1 shows a list of transitions that we found in our REMPD spectroscopy with doubled frequency

using the DCM dye. We performed the scan once with 266 nm as the second photon, and once with 355 nm
as the second photon. Given the same first photon, we noticed that the choice of the wavelength of thesecond photon can affect the degree of doublet separation. This is probably because the combination of thefirst and second photon energies address a different dissociation potential depending on which second photon
is used, resulting in different dissociation kinetic energies (see Section 3.2.2). Switching between 266 nm and355 nm is not an easy task — we have to change the optics within the Nd:YAG laser, and all the downstream
optics between the laser and the trap — this is not an easy knob to turn.
As of this writing, we typically perform dissocation from the following lines:
32097. Our main dissociation line for the X 3 ∆1 (v = 0) manifold. We characterized angle-resolved photofragments on this line. With 266 nm as our second photon, this puts us about 16500 cm−1 above thedissociation threshold.
31973. Our main dissociation line for the X 3 ∆1 (v = 1) manifold for diagnostics.
31201. Our main dissociation line for the X 3 ∆1 (v = 2) manifold for diagnostics.

49

X 3 ∆1 (v = 0)

X 3 ∆1 (v = 1)a 1 Σ+ (v = 0)

Ω

30799
30857
30934
31030 (*)
31073
31144
31210
31230
31242
31312
31355
31392
31450
31461
31572
31619
31662
31723
31751
31780
31825
31922
31957
31984
32097 (*)
32171
32179
32269
32449
32504 (*)
-

30796
30808
30919
30966
31009
31070
31097
31126
31171
31269
31304
31331
31518
31526
31616
31796
31851
31973 (*)

30759
30830
30896
30916
30998
31305
31409
31436
31511
31643
31670
31857
32135
-

2 (?)
1 (?)
0 (?)
1
0 (?)
0 (?)
1
2 (?)
1 (?)
2 (?)
0
0 (?)
1 (?)
1 (?)
0 (?)
1 (?)
2 (?)
1 (?)
1 (?)
2
2
2

Table 3.1: Relevant parameters of fitted transitions in the REMPD spectroscopy of ThF+ .
The first three columns indicate the transition frequencies in cm−1 connecting the same upper state to thecorresponding lower state stated for each column; dashed entries indicate transitions not found. The column
for Ω indicate the fitted values of the quantum number Ω for the transitions; entries marked with (?) indicate
a crude fit due to the low signal to noise ratio of the scans, while those without the mark have very goodfits; dashed entries indicate that Ω was not fit for that transition. The transitions marked with (*) are ones
that we frequently use. The scanning range of the DCM dye was not enough for us to classify bands into
vibrational progressions like that in Section 2.5. We associate certain bands together in this table through (i)similar B ′ for the upper state, and (ii) reasonable energy difference between the transitions, corresponding
to the vibrational spacing of the X 3 ∆1 electronic state.

50
3.4.2

Possible future directions
We note that 266 nm is more energetic than our first photon at around 311 nm. This means that

266 nm can induce a one-color two-photon dissociation by itself. For a resonant two-photon process, one
would rather have a more energetic first photon than the second photon, and in the ideal situation the first
photon should exceed half of the dissociation energy. This allows the first photon to be much weaker than
the second photon, so that the first photon would not induce its own one-color two-photon process, while
the stronger but less energetic second photon will not be able to induce a one-color two-photon dissociation
by energy considerations. A possible future direction will be to use a first photon that exceeds half of ourdissociation threshold (D0+ = 53230 cm−1 from Ref. [44]). This would require survey spectroscopy on the
Pyrromethene 597 or Coumarin 153 dyes; these dyes are chosen such that we can use the convenient 532 nm
laser as the second photon.
Things to look out for in the survey spectroscopy include:
Kinetic energy of photofragments upon dissociation. We want this to be as high as possible to overwhelm the original ion cloud temperature for good angle-resolved photofragments. We just need tomake sure that the photofragments do not have so much energy that they miss the ion detector, but
this can be easily solved by transverse focusing using some ion optics.
|Ω′ | of the intermediate state. We want |Ω′ | = 2 (see Section 3.2). The absence of a P(2) line in theband would indicate such a state.
B ′ of the intermediate state. The rotational constant of the intermediate state determines the spacing
between the R lines. We want this spacing to be large compared to our laser linewidth so that we
dissociate only one J manifold. We want a large B ′ .
|Ωf | of the predissociation state. This would determine what photon polarizations we want to use foreach of the two photons in the REMPD scheme (see Figure 3.4).
Unwanted one-color multi-photon dissociation in a 1 + 1′ scheme. We want the first photon to address only the bound-to-bound transition, and the second photon to saturate the bound-to-continuum

51
transition. We do not want the first photon to excite a bound-to-continuum transition; we also do
not want the second photon to address off-resonant bound-to-bound transitions. We can always
achieve this by reducing the energies of the laser pulses, but we also want to have as high a dissociation efficiency as possible, which requires laser pulses with high energies. Hence we want tofind an intermediate state such that there are no accidental resonances that could enhance one-color
multi-photon transitions.
One-color two-photon dissociation with good angle-resolved photofragments. If it turns out thatthe first photon can also act as a second photon, hitting a pre-dissociation state that gives us very
high KEdissoc. , there would be no reason not to use this 1+1 REMPD transition.
Dissociation efficiency. We have conveniently left out dissociation efficiency in all our previous discussions,but dissociation efficiency is also a very important factor. Only the dissociated Th+ ions contributeto our statistics, and the rest of the ions only contribute to decoherence in our ion trap. The[32.09]+266 nm scheme currently gives us 10% dissociation efficiency. We determined the efficiency
in two ways, both giving consistent results:(1) All our ions are in the ground vibronic state, spread across a few rotational states. With ourusual experiment parameters (i.e. laser powers and ion numbers), we scanned the R branch forthe intermediate state [32.09] to count the number of dissociated Th+ ions. We then compare
the total number (accounting for the Hönl-London factors for the R lines) to the total ThF+
number. The latter is inferred through a calibrated area under the curve on the capacitively
coupled output from our microchannel plate assembly.
(2) Alternatively, we could compare the areas under the curve for both Th+ and ThF+ from thecapacitively coupled output from our microchannel plate assembly. We would need to increase
our total ThF+ ion numbers for the Th+ ions to have an appreciable signal on the capacitively
coupled output.
In the Gen. II experiment, the dissociation efficiency is at least 20%. The lower bound comes fromnot having full control/knowledge of the Hilbert space spanned by all the HfF+ ions in the trap. It

52
would be great if we can find a new intermediate state (with the relevant second photon) that cangive us higher dissociation efficiency.
The above can be very tedious work with no promised returns, and the quality of our angle-resolved
photofragments is already good enough for most practical purposes. Let me suggest possible optimizationroutes and consequences, arranged in increasing order of tediousness.
Current configuration. We are currently using the [32.09]+266 nm scheme, both circularly polarized withthe same handedness. These photons go through the same windows on our vacuum chamber, andlies perpendicular to the axis of the molecular beam source. The vacuum chamber has eight flanges
arranged in an octagon. The molecular beam comes in through one of the flanges horizontally. The
REMPD beams go up windows vertically. The rest of the windows are used for other lasers andmicrowaves. See part (a) of Figure 3.14.
(a)

M.B.

(b)

M.B.

Erot

(c)

π

Erot

M.B.

σ+
σ

+

σ

+

σ

Erot

π

+

Figure 3.14: Various configurations of REMPD beams. M.B. stands for molecular beam. The grey
octagon represents our vacuum chamber with eight flanges (2.75′′ CF). The into-page and out-of-page directions are 10′′ CF flanges that are not meant for optics, see Figure 7.9. (a) Current configuration. (b)Naı̈ve implementation of Route 1. This does not work because the π photon will be hitting the skimmers
and molecular beam source. (c) One will have to move the beams around like this to implement Route 1.

Route 1: switching to π-polarized photon when |Ωf | = 2. We saw in Section 3.2 that we can increase
the directedness of the angular distribution of photofragments if |Ωf | = 2 by switching either of the
REMPD photons to π-polarization. This can be done in a cross-beam setup, where the REMPD
photons go into the trap perpendicularly. The polarization axis will be aligned with the k-vector of
the circularly polarized photon. We will have to move the beams around from the current configuration to avoid hitting the molecular beam source with our lasers (see parts (b) and (c) in Figure533.14). Using four windows instead of two for the high pulse energy lasers also means fewer windows
for the cw lasers for state preparation and detection. This may be a little tricky given the numberof cw lasers we need.
Furthermore, it turns out that in our current trap setup (see Figure 7.9 and Section 7.4), thepositions of our ions at our ion detector is velocity (position) dominated in the vertical (horizontal)
direction. This means that we may lose some contrast if the molecules are dissociated at 45◦ w.r.t.
the vertical (part (c) of Figure 3.14). We will need to understand the ion lensing effects during
the flight between the trap and the detector, and turn both vertical and horizontal directions to bevelocity map imaging.
Route 2: changing the second photon in the [32.09]+266 nm scheme. We can change the second
photon to 355 nm, or any arbitrary wavelength with a tunable dye laser. Doing so can change the|Ωf | of the predissociation state, and also the dissociation kinetic energy imparted to Th+ . We cankeep the current laser beam configuration if we find address an |Ωf | = 1, 3 predissociation state. We
would have to ensure that the dissociation efficiency and kinetic energy are still high enough. This
would involve (i) changing the downstream optics from those of 266 nm to 355 nm, or (ii) introducing
an additional dye laser and doubling optics.
Route 3: changing both photons to get a [?]+532 nm scheme. In this scheme, the second photon is
below half of the dissociation energy. This means that it will be much harder for the second photon
to induce one-color multi-photon dissociation. One will need to switch the dye used to generate thefirst photon, and perform survey spectroscopy. The group will have to look out for intermediate
states with all the properties mentioned at the start of this section.
Route 4: changing both photons to get a [?]+(≈532 nm) scheme. If the above is too easy, or if the[?]+532 nm scheme results in (i) lower dissociation efficiency, (ii) lower dissociation kinetic energy, or(iii) an undesirable |Ωf |, then one might want to tune the 532 nm photon. Route 4 is a combinationof Routes 2 and 3.
I shall leave the decision of whether to perform this survey spectroscopy in the good hands of the54group members that stay on after me. In the meantime, we have answered the question of how to detect theinternal states of our ions, so we shall now move on to discuss how we can perform quantum control of thestates inside ThF+ .

Chapter 4

Quantum control of ThF+

Now that we know how to create ThF+ ions (Chapter 2), and how we can perform state readout(Chapter 3), we are now ready to look at how we can perform quantum control of ThF+ . We intend to use
lasers and microwaves to do this, so we need to know the frequencies of these transitions required for state
preparation and detection. Fortunately, some of the laser part of the spectroscopy has already been done
before I joined the group (see Daniel Gresh’s thesis and Ref. [43]), but the microwaves part still needs to bedone, and we also need to decide on which of the transitions that Gresh et al. found is suitable for our needs.
Furthermore, we have been emphasizing how ThF+ can give us better statistical sensitivity than HfF+ , but
we also need to understand systematic effects in using ThF+ to measure the eEDM. This requires knowing
how the X 3 ∆1 electronic state responds to external electric and magnetic fields. We thus need to determine
certain spectroscopic constants in the effective Hamiltonian governing our system. Much of this chapter isdirectly lifted from our publication [54].
The layout of this chapter is as follows. We first show results of our spectroscopy on the eEDMsensitive states of ThF+ in Section 4.1. We then go on to discuss in Section 4.2 a possible mechanism
limiting coherence time in ThF+ despite having a ground state for the eEDM-sensitive state, and how we
intend to overcome it. Finally, we discuss an experiment sequence that we can use to obtain Ramsey fringes
in the eEDM-sensitive state in ThF+ in Section 4.3.

56

4.1

Spectroscopic constants of ThF+
The following sections detail the spectroscopy process to extract the spectroscopic constants crucial

to the eEDM experiment. We begin with an overview of the state preparation process in Section 4.1.1, then
we touch on the details of the measurement of the hyperfine coupling constant (Section 4.1.2), molecular
frame electric dipole moment (Section 4.1.2), and the magnetic g-factor (Section 4.1.3). Finally we present
a summary of the results in Section 4.1.4.

4.1.1

State preparation and readout for eEDM-sensitive state spectroscopy
The energy level of a diatomic molecule like ThF+ has nested progressively finer structure (see Figure1.2). The eEDM-sensitive states are labeled by the quantum numbers X 3 ∆1 (v = 0, J = 1, F = 3/2, mF =±3/2), which correspond to the electronic, vibrational, rotational, hyperfine, and Zeeman manifolds, respectively, with increasing fineness in their structures. We first prepare ThF+ in the X 3 ∆1 (v = 0) vibrational
manifold through REMPI of neutral ThF [46] (Chapter 2), and usher them into the finer energy structures with optical pumping via an excited electronic state Ω = 0− [14.59] (Figure 4.1). The Ω = 0−electronic state lies approximately 14600 cm−1 above X 3 ∆1 [43]. The full process to prepare our ions in
the X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = −3/2, Ω = 1) is illustrated in Figure 4.1, where Ω is the quantum
number for the Ω-doublets.
State preparation involves two pulsed lasers at 304 nm and 532 nm for REMPI, multiple cw lasers at
685 nm for optical pumping [Figures 4.1(a-c)], a cw repump laser at 717 nm [Figure 4.1(a)], and microwave
channels at 29.1 GHz and 43.6 GHz [Figure 4.1(b,d)].
We perform our state readout by dissociating our molecular ions state-selectively with methods usedin our previous work [38, 46, 55] and detailed in Chapter 3.

57(a) Overview(b) Rotational cooling

Re

pu
mp

p

m

pu
Re
v≥2

v=1

Optical pumpingΩ = 0−

X 3 ∆1 (v = 0)

Laser
µwave
Spon. em.

Ω = 0−
P(2)

J =1

2

≥4

3

(c) Zeeman sub-levelΩ = 0−(d) Doublet selectivity
J =2

Q(1), σ −mF Ω = − 32mF Ω = + 32

mF = − 32

− 12

1
2

3
2

Figure 4.1: State preparation sequence (not to scale). The state preparation process is shown as asequence of steps for clarity, but all the steps involved occur at the same time in our experiment.
(a) All our optical pumping lasers connect from the X 3 ∆1 (v = 0) vibronic manifold to the Ω = 0− (v = 0)
vibrational manifold [43]. We have vibrational repump lasers from the higher vibrational manifolds. For
some of the spectroscopy shown in the following sections, we only have the vibrational repump laserfrom the (v = 1) manifold. We added more vibrational repump lasers as time progressed.
(b) Within the X 3 ∆1 (v = 0) vibronic manifold, we perform rotational cooling by optically pumping theJ = 2 state through the Ω = 0− state. We use microwaves to couple the J = 2 and J = 3 states together
to transfer J = 3 population eventually into J = 1.
(c) Within the J = 1 manifold, we pump all the ions into a single Zeeman mF state using circularly polarized
light on the Q(1) line. We also introduce a magnetic field to prevent the Zeeman levels from mixing
through rotation coupling (refer to Section 4.1.3). We show only one lower state for each mF value (inthe grey box), but there are in fact two or four levels at each mF value; compare with (d).
(d) Ω-doubling gives rise to two closely spaced states with the same mF number. We can deplete one ofthese states by coupling it with microwaves to J = 2.

58
4.1.2

Microwave spectroscopy on the J = 1 to J = 2 transition; Hyperfine coupling constantand molecular electric dipole moment
A schematic diagram of the energy levels of the J = 1 and J = 2 rotational states in X 3 ∆1 (v = 0) is

shown in Figure 4.2. We use state preparation steps shown up to panel (b) in Figure 4.1 to prepare our ions
into X 3 ∆1 (v = 0, J = 1). We then perform microwave spectroscopy on the detailed structure of the J = 1
to J = 2 rotational transition. Our observable is the appearance of population in the J = 2 state, detectedby state-selective photodissociation. No external magnetic fields are applied for the microwave spectroscopy
experiments in this section.
At near-zero external electric field, selection rules and energy degeneracies result in just six distinct
resonant frequencies (refer to Figures 4.2 and 4.3), from which we perform a fit to the spectroscopic constants
A|| (hyperfine coupling constant) and ωef (Ω-doubling splitting constant). We are able to extract 4B =29.09733(4) GHz (see Equation 2.2a), which is roughly the J = 1 to J = 2 separation. This number isconsistent with 29.093(9) GHz obtained in our previous work [43]. Figure 4.2 illustrates how the spectroscopic
constants fit into the energy level structure.
By repeating the above experiment with a non-zero external electric field strength, we can see the
Stark shifts in the spectral lines (illustrated in Figure 4.2), shifts which depend on mF and dmf (molecular
frame electric dipole moment). To enhance signal-to-noise ratio for the extraction of dmf , we prepare all
the ions into X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = −3/2) with state preparation sequence up to panel (c)in Figure 4.1. This allows us to suppress all lines coming from the mF = ±1/2 states to obtain a muchcleaner spectrum to extract dmf . We search for the lines corresponding to the (L) and (U) transitions shownin Figure 4.2. The spectroscopy of these two lines are shown in Figure 4.4. These two lines were used forthe measurement of dmf because of their strong intensities. The (L) and (U) lines will be used to perform
doublet depletion [preparation step (d) of Figure 4.1] in subsequent sections.
Before we end off this section we shall look at some diagrams to give us a better understanding of
the energy level structure of ThF+ . Figure 4.5 shows transitions that are allowed between J = 1 and 2 in
v = 0, intensity-coded for the relative strengths. Figures 4.6 and 4.7 show the corresponding eigenstates andWithout an external electric field
Energy J = 2

3ωef
5
12

59

With an external electric field

With more realistic ordering

J =2

J =2

F = 3/2

F = 3/2

A||
(U)

3ωef

J =1

F = 5/2

(a)

(b)

J =1

ωef
3
4

J =1

F = 1/2

A||

F = 1/2(L)ωef
mF = − 52 − 32 − 12

F = 5/2

F = 3/2
1
2

3
2

(U)

(L)

F = 3/2

5
2

Figure 4.2: Energy landscape (not to scale) of the neighborhood of the eEDM-sensitive state
X 3 ∆1 (v = 0), showing only the first two rotational levels. See Figure 1.2 for the bigger picture
involving the other states.
In the absence of external electric field, the eigenstates are states of good parity. The energy levels with e(f ) parity are denoted by black (grey) lines in the energy level diagram on the left.
Selection rules for E1 transitions only allow for transitions connecting states of opposite parities, ∆F = 0, ±1,
and ∆mF = 0, ±1, thus resulting in only six distinct frequencies for resonant transitions connecting the J = 1
to J = 2 rotational manifold in the absence of an external magnetic field. For example, the transitions withthe highest and lowest frequencies are labelled (a) and (b), respectively, here and also in Figure 4.3.
In the presence of an external electric field, states of opposite parities mix. The parity selection rule is now
replaced by the ∆Ω = 0, ±1 selection rule. The red arrows in the middle panel labeled (L) and (U) correspondto the stretched-to-stretched transitions used to determine the Stark shift in our microwave spectroscopy.
The Stark shift depends on mF and dmf . The eEDM-sensitive states used for the eEDM measurement are
the upper (red) and lower (blue) doublets.
To reduce clutter in the left and middle diagrams, the states drawn do not reflect the true nature of thestates in three aspects: (i) the states are drawn with an artificially large A|| in comparison to ωef for well
separated hyperfine levels in the diagram; (ii) the Stark shifts in the diagram are portrayed proportionallysmaller than those in the actual experiment, where Stark shifts are large enough to allow some states in thelower hyperfine level to be more energetic than the upper; and (iii) F is no longer a good quantum number inthe presence of a strong external electric field, except for the most stretched Zeeman states in each rotational
manifold. Hence, in the presence of an external electric field, the only selection rule remaining on the J = 1
to J = 2 transition is ∆mF = 0, ±1. The panel on the right shows a more realistic ordering of the energy
levels.

energy level diagram to scale. In the figures, we make use of the result of the g-factor from the next section.

60

Th + number

40

Data
Predicted

(a)

30
(b)

20
10
29.08

29.09

29.10
Microwave frequency (GHz)

29.11

29.12

Transition strength (arb. unit)

Fractional depletion

Figure 4.3: Microwave scan at zero net external electric field. The blue peaks and yellow markers
correspond to the actual measurement and simulated position of the peaks, respectively. The unexpectedpeak at 29.11 GHz is most likely due to a spike in the experimental noise. Error bars from the simulation arepropagated from the values shown in Table 4.1. The uncertainty of the Th+ numbers in our data is typicallyaround 5 ions. The lines labelled (a) and (b) are the corresponding transitions labelled in Figure 4.2.

(U): f0 = 29.0942 , = 0.0003(L): f0 = 29.1038 , = 0.0003

0.4
0.2
0.0
29.0932

29.0942
Frequency (GHz)29.0952

0.4
0.2
0.0
0.2
29.102829.1038
Frequency (GHz)29.1048

0.25
0.20
0.15
0.10
0.05
0.00

29.06

29.08

29.10
Frequency (GHz)

29.12

29.14

Figure 4.4: Microwave transitions between the X 3 ∆1 (v = 0, J = 1, mF = −3/2) and J = 2 states at
an applied electric field strength of 24 V/cm. The top two plots show the transition lines corresponding
to (U) and (L) of Figure 4.2. Error bars show the estimated 1σ error in the signal. The bottom plotshows a simulation of the microwave spectrum across a wider frequency range. The left and right peaks
highlighted in yellow correspond to (U) and (L), respectively. The intensities are evaluated from ClebschGordan coefficients. In the limit of dmf E ≫ ωef , where E is the applied electric field, the splitting between(U) and (L) is given by dmf E/3. At E = 24 V/cm, dmf E is roughly 8 times larger than ωef .

61
29.200Allowed transition
Stretched to Any
Stretched to Stretched29.175Frequency (GHz)29.15029.12529.10029.07529.05029.02529.000
20

25

30

35

40

rot (V/cm)

45

50

55

60

45

50

55

60

(a) Zoomed out view.

29.120Allowed transition
Stretched to Any
Stretched to Stretched29.115Frequency (GHz)29.11029.10529.10029.09529.09029.08529.080

20

25

30

35

40

rot (V/cm)(b) Zoomed in view.

Figure 4.5: Allowed transitions between J = 1 and J = 2 in X 3 ∆1 (v = 0). The (L) and (U) transitions(stretched to stretched) are colored red. We want to operate at an Erot such that these red lines are notcrossing any other line. There is another pair of stretched to stretched transition (also colored red) that is
weakly allowed by Ω-doubling in the J = 2 manifold. Transitions connecting the stretched states, i.e. thedoublets, in the v = 0 manifold are colored yellow.

Basis states

62| J=1.0, =-1.0, F=0.5, mF=-0.5| J=1.0, =-1.0, F=0.5, mF=0.5| J=1.0, =-1.0, F=1.5, mF=-1.5| J=1.0, =-1.0, F=1.5, mF=-0.5| J=1.0, =-1.0, F=1.5, mF=0.5| J=1.0, =-1.0, F=1.5, mF=1.5| J=1.0, =1.0, F=0.5, mF=-0.5| J=1.0, =1.0, F=0.5, mF=0.5| J=1.0, =1.0, F=1.5, mF=-1.5| J=1.0, =1.0, F=1.5, mF=-0.5| J=1.0, =1.0, F=1.5, mF=0.5| J=1.0, =1.0, F=1.5, mF=1.5| J=2.0, =-1.0, F=1.5, mF=-1.5| J=2.0, =-1.0, F=1.5, mF=-0.5| J=2.0, =-1.0, F=1.5, mF=0.5| J=2.0, =-1.0, F=1.5, mF=1.5| J=2.0, =-1.0, F=2.5, mF=-2.5| J=2.0, =-1.0, F=2.5, mF=-1.5| J=2.0, =-1.0, F=2.5, mF=-0.5| J=2.0, =-1.0, F=2.5, mF=0.5| J=2.0, =-1.0, F=2.5, mF=1.5| J=2.0, =-1.0, F=2.5, mF=2.5| J=2.0, =1.0, F=1.5, mF=-1.5| J=2.0, =1.0, F=1.5, mF=-0.5| J=2.0, =1.0, F=1.5, mF=0.5| J=2.0, =1.0, F=1.5, mF=1.5| J=2.0, =1.0, F=2.5, mF=-2.5| J=2.0, =1.0, F=2.5, mF=-1.5| J=2.0, =1.0, F=2.5, mF=-0.5| J=2.0, =1.0, F=2.5, mF=0.5| J=2.0, =1.0, F=2.5, mF=1.5| J=2.0, =1.0, F=2.5, mF=2.5

Heat map of sgn( basis | eig ) | basis | eig |2

1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Eigenstates in order of increasing eigenenergy

1.00

Figure 4.6: Heat map of the coefficients of the eigenstates expanded out in our chosen basis
set. This heat map is obtained by diagonalizing the 32-level Hamiltonian (see Appendix C.3.1) of the
X 3 ∆1 (v = 0, J = 1, 2) manifold in our chosen basis set. The Hamiltonian is diagonalized at our usualexperiment parameters Erot = 60 V/cm, frot = P
147.5 kHz, and Brot = 800 µG corresponding to an f ≈ 50 Hz.
For an eigenstate expanded of the form |·⟩ = i ci |i⟩, where |i⟩ is the ith basis state, the heat map showssgn ci |ci |2 (information of the complex phase is lost in this representation; the sgn operator has a “branchcut” along the negative real axis). The eigenstates are ordered from least to most energetic.

63

ThF + : X 3 1(v = 0, J = 2) , rot = 60 V/cm , rot = 0.8 mG

29140
29130

[31]: |F = 1.5, = 1[30]: |F = 1.5, = 1[29]: |F = 2.5, = 1[28]: |F = 2.5, = 1

29120[26, 27]: |F = 1.5, = 1[26, 27]: |F = 1.5, = 1[25]: |F = 2.5, = 1[24]: |F = 2.5, = 1

Energy (MHz)29110[22, 23]: |F = 2.5, = 1[22, 23]: |F = 2.5, = 1[20, 21]: |F = 1.5, = 1[20, 21]: |F = 1.5, = 1[18, 19]: |F = 2.5, = 1[18, 19]: |F = 2.5, = 1

29100
29090
29080

[16, 17]: |F = 2.5, = 1[16, 17]: |F = 2.5, = 1

29070[15]: |F = 1.5, = 1[14]: |F = 1.5, = 1

29060[13]: |F = 2.5, = 1[12]: |F = 2.5, = 1

2.5

1.5

0.5

0.5

mF

1.5

2.5

(a) J = 2.

ThF + : X 3 1(v = 0, J = 1) , rot = 60 V/cm , rot = 0.8 mG

60[11]: |F = 0.5, = 1[10]: |F = 0.5, = 1[9]: |F = 1.5, = 1[8]: |F = 1.5, = 1

40Energy (MHz)20
0[6, 7]: |F = 1.5, = 1[4, 5]: |F = 1.5, = 1[6, 7]: |F = 1.5, = 1[4, 5]: |F = 1.5, = 1[2, 3]: |F = 0.5, = 1[2, 3]: |F = 0.5, = 1

0.5

0.5

20
40
60

[1]: |F = 1.5, = 1[0]: |F = 1.5, = 1

2.5

1.5

mF

1.5

2.5

(b) J = 1.

Figure 4.7: Energy level diagram of the 32-level system in X 3 ∆1 (v = 0, J = 1, 2) expressed in ourchosen the basis set. Drawn to scale (c.f. Figure 4.2). The presence of an electric field, F is no longera good quantum number except for the most stretched Zeeman states, but we still label our states with F
to track how the states shift from zero electric field. Each line corresponds to a basis state. The indicesprefixed to each basis state corresponds to the eigenstate that aligns most with that basis state (numbers on
the horizontal axis of Figure 4.6). For example, [0] is predominantly |J = 1, F = 3/2, mF = −3/2, Ω = −1⟩,
as one can check with Figure 4.6. Basis states with multiple indices indicate an equal superposition state. For
example, [4, 5] means that states [4] and [5] are equal superpositions of |J = 1, F = 3/2, mF = −1/2, Ω = 1⟩
and |J = 1, F = 3/2, mF = 1/2, Ω = −1⟩.

64
4.1.3

Ramsey spectroscopy within the J = 1 eEDM-sensitive state; Magnetic g-factor
The last spectroscopic constant that we determine is the magnetic g-factor for X 3 ∆1 . We use arotating electric field to polarize our molecular ions without ejecting them from our ion trap. The rotation
micromotion traced out by the ions couples to the applied quadrupole magnetic field gradient to give an
averaged net non-zero magnetic field along the instantaneous quantization axis in the frame of the ions, thus
resulting in Zeeman shifts of the molecular states. Details of the underlying mechanism can be found in
Section 4.11 of Ref. [25].
The rotation frequency is fast compared to trap secular frequencies, but slow compared to typicalenergy differences between quantum states within the molecular ions. As the ions follow the rotation of the
field adiabatically, there is a non-inertial-frame term in the Hamiltonian in the frame of the rotating ions.
This non-inertial-frame term introduces rotational coupling between states of ∆mF = ±1, and couples the
mF = ±3/2 states through a third order process. Restricting ourselves to the Hilbert space involving only
the mF = ±3/2 states, the good eigenstates of the system are |mF = +3/2⟩ ± |mF = −3/2⟩ at zero external
magnetic field, and approach |mF = ±3/2⟩ asymptotically as the external magnetic field strength increases(see part (c) of Figure 1.4). Thus, as the strength of the applied magnetic field is swept, the energy difference
between the mF = ±3/2 states traces out a hyperbola like that shown in Figure 4.8, where the vertical offset
is due to the avoided crossing introduced by the rotational coupling.
Following the procedure reported in Ref. [56], we map out the the energy differences between the
mF = ±3/2 states for both the upper and lower doublets by performing Ramsey spectroscopy on X 3 ∆1 ,which is prepared with sequence up to panel (d) in Figure 4.1. We repeat the experiment at various appliedmagnetic field strengths. The data and fits are shown in Figure 4.8.
Since we operate at dmf E ∼ A|| , the lower doublet is energetically closer to more mF = ±1/2 states
than the upper doublet. Thus, the lower doublet has a stronger rotational coupling than the upper doublet,
resulting in a much larger avoided crossing seen in the plot for the lower doublet than the upper doublet in
Figure 4.8.
By measuring the frequencies at long Ramsey times for each doublet individually, we can also measureFrequency (Hz)65

Upper doublet
150
100
50
0

Lower doublet

150
100
50

2 1 0 1 2
(mG)

0

2

1 0 1
(mG)

2

Figure 4.8: Energy differences between the mF = ±3/2 states; measurement of the magnetic
g-factors. The fits are performed with our model which takes into account the avoided crossing introduced
by going into the rotating frame of the molecules. The energy difference at the avoided crossing is fixed by
our ab initio calculations through diagonalizing the 12-level Hamiltonian (see Appendix C.3.1). The only fit
parameters are the asymptotic gradients (corresponding to 3gF =3/2 µB /h) and the horizontal offset due toambient fields. The error bars are 1σ error estimates extracted from a non-linear fit to each Ramsey fringe.

the difference in the g-factor between the two doublets. The ratio δg/g is measured to be −0.00255(6) (seeFigure 4.9a) at Erot = 60 V/cm. Diagonalizing the 32-level Hamiltonian1 involving the X 3 ∆1 (v = 0, J =1, 2) manifold gives a theoretical value of about −0.00223. The 15% discrepancy may come from coupling
between the X 3 ∆1 and the 3 ∆2 states that is not accounted for in the 32-level Hamiltonian; we saw a similarcorrection in HfF+ .

1 Although we are looking at the g-factors of the J = 1 manifold, coupling with the J = 2 states through the Stark
Hamiltonian modifies the g-factor significantly, so we need to use the 32-level Hamiltonian instead of the 12-level one.

66

(Normalized) Ion number

Files: 17/000050181-Ramsey.h5 ; 18/000050186-Ramsey.h5 ; 18/000050185-Ramsey.h5 ; 17/000050182-Ramsey.h5 ; 17/000050183-Ramsey.h5
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1600
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0.6

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(a) Ramsey fringes for the lower (left) and upper doublets (right). The top row indicates the dissociated Th+ numbers
from the mF = 3/2 (blue) and mF = −3/2 (yellow) states, while the middle row is the contrast obtained throughthe difference between the two chops. The left column corresponds to the lower doublet, while the right column
corresponds to the upper. The last row is the total ThF+ numbers, and it is used for normalization. An exponential
decay is fit to the results in the top row, giving us some hints on T1 decoherence. An exponentially decaying sine
is fit to the results in the middle row, with insets showing the zoomed in view comparing the fit to the raw data.
Through the fits, we obtain the Ramsey frequencies for the lower (f l ) and upper (f u ) doublets used in the plot inpart (b) below.

A*f0 + B/f0
A = -0.00255 ± 5.6e-05
B = -0.53 ± 0.098
0.10

fD (Hz)

0.15
0.20
0.25
0.30
0.35
0.40
0

20

40

60

80
100
f0 (Hz)

120

140

160

(b) Global fit to repeated instances of Figure 4.9a at various f 0 := (f u + f l )/2. The vertical axis corresponds
to f D := (f u − f l )/2. The quantities f u and f l are defined above in part (a). The fit is to the functional form
f D = Af 0 + B/f 0 , where A = δg/g, and B = ∆ ∆D (see Equations 8.24a and 8.24c). The fit value of B = −0.5(1)is consistent with our model predicting a value of −0.45.

Figure 4.9: Measuring δg/g in ThF+ at Erot = 60 V/cm. By performing the experiment at several Ramsey
frequencies, we can perform a global fit to extract the difference in g-factor between the two doublets.

67
4.1.4

Summary of Results & Remarks
Table 4.1 shows the measured and calculated spectroscopic structural constants for X 3 ∆1 .
Parameters

Exp.

TheoryPrevious work

Notes

A|| /2π (MHz)ωef /2π (MHz)dmf (D)|gF =3/2 |δgF =3/2 /gF =3/2

−20.1(1)5.29(5)3.37(9)0.0149(3)-0.00255(6)

−21.5
3.46
See main text.
-

5.21(4) [43]4.03 [41], 3.46 [22]-At Erot = 60 V/cm.

Table 4.1: Measured spectroscopic structural constants for X 3 ∆1 . A|| , ωef , dmf , gF =3/2 , and δgF =3/2
are the 19 F magnetic hyperfine coupling constant, Ω-doubling splitting constant, molecular frame electric
dipole moment of ThF+ , the average value of the magnetic g-factors of the F = 3/2 hyperfine level in X 3 ∆1
for the upper and lower doublets, and the difference in magnetic g-factors between the upper and lower
doublets, respectively. Theoretical calculations by our collaborator Prof. Lan Cheng from Johns Hopkins
University and previous work are also shown here for comparison. A|| and ωef have units of rad/s and forconvenience we divide by 2π and present our results in millions of cycles per second.

Our spectroscopy is not sensitive to the sign of the magnetic g-factor shown in Table 4.1. Neglecting
the rotational contribution to the g-factor, and converting |gF =3/2 | into G|| , we get −0.042(2) if gF =3/2 > 0
and 0.048(2) otherwise. The latter is not far away from the theoretical predictions of G|| = 0.034 [22] and0.035 calculated by our collaborator Prof. Lan Cheng from Johns Hopkins University. We do not have asystematic estimate for the error in the theoretical value of G|| , therefore there remains some ambiguity insign of the g-factor.
The spectroscopic constants measured in ThF+ are similar to those in HfF+ [56]. This means that wewill be operating in a familiar experimental parameter space. Therefore, the eEDM experimental complexity
will not increase with the planned molecule upgrade from HfF+ to ThF+ .

68

4.2

Blackbody Radiation Excitation and T1 Relaxation Time

4.2.1

BackgroundX 3 ∆1 has been shown to be the ground electronic state of ThF+ [43]. Hence the coherence time of

X 3 ∆1 is not subject to spontaneous decay. However, stray photons, e.g. blackbody radiation, can excite
ThF+ from X 3 ∆1 , and they subsequently either decay into other long-lived states where they no longer
contribute to the measurement statistics, or back into X 3 ∆1 with corresponding delayed decay in coherent
spectroscopy contrast.
The transition rate between two states 1 → 2 for incoherent excitation is given by:2

Γ(f, T, µ12 ) =|µ12 |2ρ(f, T ),6ϵ0 ℏ2(4.1)

where µ12 is the transition dipole moment, and ρ(f, T ) is the spectral energy density from Planck’s law:3
ρ(f, T ) =8πhf 31,
c3 e khf
T
B
−1

(4.2)

where the first term is the density of states, while the second term is the mean photon number. We cancalculate the 1/e lifetime by taking the inverse of the transition rate:τ (f, T, µ12 ) = 1/Γ(f, T, µ12 ).
A simple relation exists between frequency and wavenumbers (ν̃):
f [Hz] = 100 · c[m/s] · ν̃[cm−1 ].

All that remains is to determine µ12 . There are three types of possible pathways for blackbody radiation
excitation from X 3 ∆1 (v = 0, J = 1):
Electronic transitions. Electronic transitions are typically on the order of 1000 cm−1 . This sits on the
tail of the exponentially decaying mean photon number at 300 K. Hence, we are not concerned withelectronic transitions.4
2 See, for instance, Equation 9.47 of Ref. [57] or Equation 2 of Ref. [58].
3 See, for instance, Equation 5.113 of Ref. [57] or Equation 1 of Ref. [58].
4 The only exception is transition to the a 1 Σ+ state at just 314 cm−1 above the ground state.

highly forbidden by selection rules, so the excitation rate to a 1 Σ+ is also highly suppressed.

However, the transition is

69
Rotational transitions. Rotational transitions sit on the other side of the spectrum, where the transition
frequencies are so small (∼ 30 GHz) that the density of states (∼ f 3 ) is so small that the transition
rates are also highly suppressed.
Vibrational transitions. These transition frequencies are at the anti-Goldilocks range where they areneither so large as to sit on the exponentially decaying tail of the mean photon number, nor so small
as to have negligible density of states. The excitation to v = 1 accounts for the main source of
blackbody radiation excitation from X 3 ∆1 (v = 0). The expression for µ12 for vibration excitation
is:5rµv,v+1 =

v+1
2

r

ℏ
ddmf
,
2πmωe · 100c dR(4.3)

where ωe = 656.96(1) cm−1 from our previous spectroscopy work [43], m is the reduced mass ofmf
ThF+ , and dd
dR is something that we will determine in this section.

An energy level diagram of the lowest few vibronic states in ThF+ is shown in Figure 4.10.

653 cm−1
a 1 Σ+ (v = 0)314 cm−1

EnergyX 3 ∆1 (v = 1)

X 3 ∆1 (v = 0)458.2 GHz3
43.6 GHz
29.1 GHz2
J =1

Figure 4.10: Energy level diagram of the lowest few vibronic states in ThF+ . The transition
between X 3 ∆1 and a 1 Σ+ is forbidden by selection rules. A zoomed-in view of the rotational states within
the ground vibronic state is shown in the dashed box. 1 cm−1 ≈ 30 GHz.

The blackbody radiation excitation rate from X 3 ∆1 (v = 0) through X 3 ∆1 (v = 1) is related to thespontaneous decay lifetime of the v = 1 states. This relation comes through the transition dipole moment
between the v = 0 and v = 1 states. One should also note that the spontaneous decay rate (Einstein’s A
5 See, for instance, Equation 7 in Ref. [58].

70
coefficient) is: 6A=

16π 3 f 3 |µ12 |2.
3ϵ0 hc3

Using the above, we get the relationship between Γ (blackbody radiation excitation rate) and A (spontaneous
decay rate) to be:
A

Γ=
e

hf
kB T

.
−1

Hence, a measurement of the lifetime of the v = 1 state will allow us to predict the blackbody radiation
excitation rate from X 3 ∆1 .

4.2.2

Measuring the v = 1 spontaneous decay lifetime
To measure the spontaneous decay lifetime of the v = 1 state, we first prepare all our ions in the

ground vibronic state X 3 ∆1 (v = 0) selectively with REMPI, and optically excite all our ions to the Ω = 0−
excited state, allowing for the ions to decay back into the X 3 ∆1 (v > 0) states, as illustrated in Figure
4.11(a). We note that the branching ratio from the Ω = 0− state to the X 3 ∆1 manifold is very close
(a)

(b)

Ω = 0− (v = 0)

Ω = 0+ (v = 0)4

3 22 703 0. 0.

0.

0.5 0.32 0.08
87

Ω = 0+ (v = 0)

Ω = 0− (v = 0)

v=2
v=2

v=2
v=2

v=1
v=1
v=0

v=1
v=1

a 1 Σ+
3

X ∆1

v=0
v=0

a 1 Σ+

v=0

3

X ∆1

Figure 4.11: State preparation for measurement of excited state lifetime. (a) For measurement of
X 3 ∆1 (v = 1) state lifetime. (b) For measurement of a 1 Σ+ (v = 0) state lifetime. Solid line indicates laser
used for optical pumping. Dotted lines indicate spontaneous emission. Numbers attached to dotted lines
indicate Franck-Condon factors determined from our previous spectroscopy work [43].

to unity. Off-diagonal Franck-Condon factors allow the v ≥ 1 manifolds to be populated through opticalpumping. The whole optical pumping process takes about 100 ms. We then allow the ions to decay fromthe excited vibrational states to lower ones, and read out the population in each vibrational state with
6 See, for instance, Equation 9.56 from Ref. [57]71
resonance-enhanced–multi-photon dissociation [46, 55] much like how we have described in Section 4.1.1.
We dissociate through the R(1) line for both the v = 0 and v = 1 manifolds. Our result is shown in Figure

Dissociated Th +Dissociated Th +

4.12.

v=1
Fit with model
Data

10
5
0

50

100 150 200 250 300 350
v=0

20
10
0

50

Fit with modelData
100 150 200 250 300 350
Time (ms)Figure 4.12: Vibrational population decay from v = 1 to v = 0 of the ground electronic statein ThF+ . All the population was pumped from the ground vibrational manifold into the excited vibrational manifold through the Ω = 0− state [43]. Population in each vibrational manifold is read out withphotodissociation. The observed vibrational decay lifetime agrees with our model, which is used to predict
the blackbody radiation excitation lifetime from X 3 ∆1 . The total dissociated Th+ numbers do not appear
to be conserved in these two plots because the dissociation efficiencies for these two vibrational manifolds
are not the same.

To extract the spontaneous decay lifetime of the v = 1 state from Figure 4.12, we employ the following
model:(1) We allow for non-zero initial population in the v = 1 and v = 2 states. The v = 1 and v = 2 states
are populated by decay from the Ω = 0− (v ′ = 0) state, and the Franck-Condon factors are such that
we may approximate the initial populations of v ≥ 3 as zero.
(2) The decay rates from each vibrational manifold are governed by their respective Einstein’s A coefficients. We also assume that only ∆v = ±1 transitions are allowed.
(3) Fit parameters include (i) absolute scaling to account for different detection efficiencies for the v = 0
and v = 1 manifolds; (ii) number of background ions; (iii) ratio of initial populations in the v = 1
and v = 2 manifolds; and (iv) d(dmf )/dr, with this being held the same across the fits for v = 0 andv = 1.

72
Note that we can group (i) the effect of imperfect optical pumping from the v = 0 manifold, which results
in an initial non-zero population in the v = 0 manifold, and (ii) background ions detected for the v = 0
manifold together into a single v = 0 “background ion” fit parameter in the fitting model. With the above
model, we obtain d(dmf )/dr = 7(2) D/Å, in good agreement with Prof. Lan Cheng’s calculations. This value
corresponds to spontaneous decay lifetimes of 0.16(11) s and 0.08(6) s for v = 1 → v = 0 and v = 2 → v = 1,
respectively.
Using the value of the molecular dipole, we predict the effective lifetime (T1 relaxation time) of
X 3 ∆1 (v = 0, J = 1) to be about 3 s at room temperature (300 K), which is limited by blackbody radiation
excitation from X 3 ∆1 to the first excited vibronic state X 3 ∆1 (v = 1).

4.2.3

Suppression of blackbody radiation excitation
We can suppress blackbody radiation excitation from X 3 ∆1 (v = 0) by introducing cryogenics to lower

the temperature of the setup. The effective lifetime of X 3 ∆1 (v = 0) increases drastically with a decrease intemperature (Table 4.2), and we anticipate establishing a blackbody environment at 180 K, which will be aworkable balance between technical convenience and sufficiently long interrogation times.
77 K

120 K

150 K

180 K

200 K

300 K

To X ∆1 (v = 1)
To a 1 Σ+

32000
2200

400
270

84
130

29
80

17
64

3.5
33

Combined

2100

160

51

21

13

3.2

3

Table 4.2: Prediction of blackbody radiation excitation lifetime (in seconds) out of the v = 0
vibrational level in X 3 ∆1 , in a radiative environment of the indicated temperature.

We calculate blackbody radiation excitation across rotational levels to occur at a time scale of 103
seconds and above, even at 300 K. Since we plan to measure the eEDM with an interrogation time of about
20 s, which is only about 10 times longer than our current experiment using HfF+ , the effects of rotational
blackbody radiation excitation are small and shall be neglected for the rest of the discussion.
Usual selection rules forbid transition between X 3 ∆1 and a 1 Σ+ , but our molecular ion falls underHund’s case (c), so these states contain slight admixtures of states of other character. Hence, a transition
between X 3 ∆1 and a 1 Σ+ is not entirely forbidden. By using an Ω = 0+ state [43] that couples both to73
X 3 ∆1 and a 1 Σ+ for optical pumping, we populate the a 1 Σ+ (v = 0) manifold through a process similar to
the experiment for measuring the X 3 ∆1 (v = 1) spontaneous decay lifetime [refer to Figure 4.11(b)]. We
observe the spontaneous decay lifetime from a 1 Σ+ (v = 0) back to X 3 ∆1 (v = 0) to be about 6 s, which isabout 40 times longer than from X 3 ∆1 (v = 1). We calculate blackbody radiation excitation from X 3 ∆1 toa 1 Σ+ to be in excess of 30 seconds (Table 4.2).
The net effect of blackbody radiation excitations to a 1 Σ+ and X 3 ∆1 (v = 1) at 180 K gives an
expected lifetime of X 3 ∆1 to be about 20 s.
We tested the effects of the cryogenic suppression of blackbody radiation excitation in our new setup(see Figure 7.9) by comparing the rate of population loss from the X 3 ∆1 (v = 0, J = 1) manifold at 170 K
and 300 K. The results are summarized in Figure 4.13.

74

Parked at Octagon (SUMuDG)
Parked at Science (SUMuDG)

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File: 16/000050857-Ramsey.h5

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Parked at Science (SUMuDG)
0.0

0.5

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4.0

Figure 4.13: Effect of suppressing blackbody radiation excitations at room temperature. The top
plot shows the detected ion population in the X 3 ∆1 (v = 0, J = 1) manifold vs time held in the trap. The
blue (yellow) points correspond to holding the ions at 300 K (170 K). The bottom plot shows the total ThF+
ions detected, and serves as shot-to-shot normalization of the counted Th+ ions. The experiment sequenceis as follows.
(1) We create ions in the X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = −3/2) manifold with state preparation
steps up to (c) in Figure 4.1.
(2) We move our ions into the cryogenic part of the trap, with an ambient temperature of 170 K. At
that location, the ions can see about 0.5% of the full 4π steradians of the 300 K environment. See
Figure 7.9 for more details of this new setup. The ions are translated into the cold region in 200 ms.
(3) For the yellow points, we hold the ions here for a variable amount of time (horizontal axis in theplot, unit in seconds).
(4) We then move our ions back out into the warm in 200 ms.
(5) For the blue points, we hold the ions here for the same amount of time that the yellow points areheld in the cold.
(6) We then dissociate ions in the J = 1 rotational manifold.
The results show that we lose ions slower from the X 3 ∆1 (v = 0, J = 1) manifold when they are held in the
170 K environment than in the 300 K environment.

75

4.3

Experiment sequence
Now that we understand all that we need to of the quantum states of ThF+ , we can now design anexperiment sequence for performing an eEDM experiment.

4.3.1

State preparation and detection

4.3.1.1

Lasers and microwaves

Figure 4.14 summarizes the lasers (and recommended wavelengths) used for state preparation anddetection. We first prepare our ions in the ground vibronic manifold using REMPI (Chapter 2) throughthe [32.85] intermediate state. We address the bound-to-bound transition with Lion (LiopTec LiopStarN, linewidth < 0.04 cm−1 before doubling, 180 µJ of UV pulse energy at trap) that is pumped with a
532 nm pump beam from Horse (Spectra-Physics INDI) which also acts as the second photon (1.1 mJ of
532 nm pulse energy at trap). We then usher our ions within the ground electronic manifold using our cw
lasers and microwaves. Here are detailed descriptions of the cw lasers and microwaves used (each of thelasers/microwaves is prefixed with a code name in parenthesis; to be elaborated in Section 4.3.1.2):(S) Sheep P(2) (Precilaser, 76 mW at trap, 2σ diameter of 10 mm, retroreflected). Sheep is a P(2)transition connecting the X 3 ∆1 (v = 0, J = 2) to the Ω = 0− [14.59] (v = 0, J = 1) state (part (b) of
Figure 4.1). Upon exciting the ions in the X 3 ∆1 (v = 0, J = 2) manifold to the Ω = 0− [14.59] (v =0, J = 1) manifold, they can decay back into the X 3 ∆1 (v, J = 1, 2) manifolds, with Sheep turned on,
all the ions will ultimately end up in the X 3 ∆1 (v = 0, J = 1) dark manifold. Transitions between
X 3 ∆1 and Ω = 0− [14.59] turn out to have unity branching ratio, so we only need to close thevibrational decay channels. Relevant Franck-Condon factors are shown in Figure 4.16a.
(U) 43 GHz microwave. We use the 43 GHz7

microwave to couple the X 3 ∆1 (v = 0, J = 2) andX 3 ∆1 (v = 0, J = 3) states together so that we can also pump the X 3 ∆1 (v = 0, J = 3) states
into the X 3 ∆1 (v = 0, J = 1) manifold.
7 The author notes that we are treating microwaves differently from the optical wavelengths. At the point of this writing,the author is not aware if microwaves have feelings, but if they are proven to have emotions in the not so distant future, theauthor wishes to apologize in advance for not considering the feelings of microwaves. Deep down the author’s heart, “43 GHz”
is affectionally called “Lamb”.

76

Ω = 0#, 𝑣 = 0

1

𝐽=0

3

2

Pumping

Llama

!
Dog

Bad state
Monkey

Sheep

𝐽=1

3

Tiger

266nm
37600cm-1
Tallest: R(1) to Ω = 2
2x 623.086nm
32098.3cm-1

Hottest: R(1) to Ω = 2
2x 615.298nm
32504cm-1

NiceP: R(1) to Ω = 1
2x 644.511nm
31031.3cm-1
532nm
18800cm-1
Q(3/2) to Ω = 3/2
2x 608.745640nm
32854.4cm-1

355nm
28200cm-1

𝐽=1

Cow

ThF+: "Δ$, 𝑣 = 0

2

2

4

2

Other lasers
Ablation
532nm

Hamster
To ! Σ "(Standby)

ChickenSnake
IE: 51581 cm-1
D0+: 53230 cm-1

REMPI
(Standby)

Lion
RabbitThe JILA ThF+ eEDM Zoo
Lasers (nm):✓ Goat: 685.443527 Q(1) ,q685.443699 [+110 MHz]
✓ Sheep: 685.489059 P(2)q
✓ Llama: 685.38160 R(2)q✓ Monkey: 717.591233 P(1)q
✓ Dog: 752.658850 P(1)qq Chicken: 954.987809 P(1)Microwaves (GHz):

Ti:Sapph(Standby)

Horse

ThF: !Δ"/!, 𝑣 = 0

𝐽=1!Δ , 𝑣 = 0"

Goat

REMPI &
REMPD

!Δ , 𝑣 = 1"

Δ" , 𝑣 = 2

DragonTransition

20 V/cm

60 V/cm𝑣 = 0 1-2l-l

29.102357

29.115175

𝑣 = 0 1-2u-u

29.095664

29.082941

𝑣 = 0 2-3

43.647 ± 0.020 @ 500 Hz

𝑣 = 1&2

28.970 ± 0.030 @ 500 Hz
28.850 ± 0.030 @ 500 Hz

Last modified: 30 April 2023

Figure 4.14: Summary of the lasers (and recommended wavelengths in vacuum) used for state
preparation and detection. We have decided to name our lasers after animals. Lasers with functions
related to each other are named after animals related to the corresponding animal of the first laser with thatfunction named.

77(M) Monkey P(1) (Home-built ECDL, 15 mW at trap, 2σ diameter of 14 mm, retroreflected).
Monkey is a vibrational repump connecting the X 3 ∆1 (v = 1, J = 1) manifold to Ω = 0− [14.59] (v =0, J = 0).
(D) Dog P(1) (Toptica DL-Pro, 31 mW at trap, 2σ diameter of 7.4 mm). Like Monkey, Dog is a
vibrational repump, but connecting the X 3 ∆1 (v = 2, J = 1) manifold.
(u) v = 1, 2, J = 1 ↔ 2 microwave. We chirp a 29 GHz microwave across a range of about 60 MHz tocouple the J = 2 to the J = 1 in both the v = 1 and v = 2 vibrational manifolds of X 3 ∆1 sothat they can also be repumped with Monkey and Dog. We are able to address the J = 1 ↔ 2transitions in the v = 1, 2 manifolds but not the v = 0 because of vibrational dependence of therotation constant (Equation 2.5b). Figure 4.15 plots the transitions for the v = 0, 1, 2 manifolds.
The transition with the lowest frequency that couples to a doublet in the v = 0 manifold corresponds
to a transition between the (v = 0, J = 1, F = 3/2, mF = ±3/2, Ω = ∓1) to the (v = 0, J = 2, F =5/2, mF = ±5/2, Ω = ±1). This transition seems to be forbidden from ∆Ω = 0, ±1 selection rule, but
is weakly allowed because of a slight admixture of the (v = 0, J = 2, F = 5/2, mF = ±5/2, Ω = ∓1)states through Ω-doubling, i.e. the molecules in the J = 2 manifold are not fully polarized yet at
Erot = 60 V/cm. This transition happens at the frequency 29.013247 GHz (third blue blob to theright of the 29.0 GHz tick mark in Figure 4.15) at Erot = 60 V/cm. For the frequency chirp to couple
the J = 1 and J = 2 states in v = 1, 2, we ensured that the chirped frequency stays below thisfrequency.
(G or G*) Goat Q(1) (Precilaser, 0.48 mW with 0.5% duty cycle at trap, 2σ diameter of 9 mm).
Goat is a circularly polarized light that is strobed at frot at 0.5% duty cycle by using the (−1)st orderdiffracted beam of a gated AOM at 110 MHz. Goat moves the population within X 3 ∆1 (v = 0, J = 1)
to the stretched Zeeman states X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = 3/2) (part (c) of Figure 4.1
during state preparation). Goat also acts as our depletion laser to remove ions from one of thestretched mF states before we perform REMPD for our state detection. Goat is non-cycling enoughthat we still manage to deplete most of the ions in the unwanted stretched mF states for our stateTransition strength(arb. unit)

78

0.3

v=0
v=1
v=2

0.2
0.1
0.0

28.8

28.9

29.0
Frequency (GHz)

29.1

29.2

Figure 4.15: Overlapping transitions of the J = 1 to J = 2 transition in the X 3 ∆1 (v = 0, 1, 2)
manifolds. The transition frequencies are obtained by solving the 32-level Hamiltonian of the X 3 ∆1 (v =0, J = 1, 2) manifold, and looking for E1-allowed transitions between the eigenstates. We then scale thefrequencies by Bv /B0 to obtain the transition frequencies for the v = 1, 2 manifolds. The transition strengths
are determined from the Clebsch-Gordan coefficients. The two lines highlighted in yellow are the stretchedto-streched transitions (L) and (U) mentioned in Section 4.1.2 at Erot = 60 V/cm. Note that the strongtransitions (corresponding to the (L) and (U) transitions of the v = 0, 1, 2 manifolds) happen to stay clearof transitions belonging to the other vibration manifolds. This will not be true if we use Erot much larger
than 60 V/cm.

detection sequence; about 8% of the ions in the unwanted stretched mF states end up in the wanted
stretched mF states (see Section 4.3.1.3). In the ideal situation, we may want Goat to sit on theP(1) transition for state preparation for higher cycling efficiency (no chance of falling down to the
X 3 ∆1 (v = 0, J = 2) manifold), and use another laser (maybe Chicken) on the Q(1) transition to theΩ = 0+ [10.47] electronic state that also couples to the a 1 Σ+ state for cleaner and faster depletion.
A 0.5% duty cycle was chosen for Goat for two reasons. First, the strobe was required so that when
Goat is strobed on, Erot is quite precisely parallel or anti-parallel with the Goat laser beam to enforce
a good circular polarization on Goat. Second, our Doppler shift from rotation micromotion at about
4 GHz could become large enough to redshift Goat to a Q(2) transition (about 1 GHz away fromthe Q(1) transition) for longer duty cycle; this Doppler shift would not be a problem if we change
Goat from Q(1) to P(1) instead.
(L) Llama R(2) (Home-built ECDL + TA, 23 mW at trap, 2σ diameter of 9.7 mm). During our
experiment, ions get bumped to higher rotational states through ion-ion collisions, or get excited tohigher rotational states through blackbody radiation that takes ions out of the X 3 ∆1 (v = 0) into
the X 3 ∆1 (v = 1), whereupon they can decay back to the X 3 ∆1 (v = 0) manifold. Llama is turnedon during our state detection sequence to shelf ions from the X 3 ∆1 (v = 0, J = 2) so that our power
broadened REMPD lasers parked on the J = 1 dissociation line do not accidentally dissociate J = 2

79
ions.
For completeness, I shall also briefly mention the history/status of the other lasers and microwave
mentioned in Figure 4.14:
Hamster (Continuum Minilite, 3 mJ per pulse focused down to a diameter of 150 µm). Laser used
for ablation of the thorium metal. Hamster replaced Mouse (Litron Nano-S-130) after the latter decided to retire.
Chicken (Home-built ECDL), not currently in use. We used this laser during the early ThF spectroscopy days to populate the a 1 Σ+ through the Ω = 0+ [10.47] state for diagnostics after creating
our ions in the ground vibronic state.
Snake (Coherent MBR-110). A Ti:Sapph laser used in the spectroscopy work in Ref [43]. Snake canperform the same function as Chicken, and with even higher frequency tunability!
Rabbit (Sirah Matisse), not currently in use. A cw ring dye laser that we used to seed Dragon with.
The output is then doubled into UV for a very narrow linewidth laser pulse to address individual Q
lines for the first photon in our REMPI protocol. Mainly used for a proof of principle work. The ring
dye laser (and the pulsed dye amplifier downstream) is much more complicated to maintain than anormal pulsed dye laser, and we do not need to resolve the Q lines for REPMI (we could just pump
the whole Q branch), so we swapped Rabbit+Dragon for Lion.
Dragon (Radiant Dye Lasers Narrowamp), not currently in use. A pulsed dye amplifier. Seededby Rabbit. Used for the first photon for the REMPI process. Currently decommissioned andenjoying its rightful vacation.
(ˆ or v) v = 0, J = 1 ↔ 2 microwave. We use this to deplete one of the two doublets in the (v = 0, J = 1)
manifold for diagnostics, e.g. to characterize the state-selectivity of our two-state detection anisotropy
measure. We checked that the intensity of the microwaves is low enough that we do not accidentally
deplete the other doublet from power broadening; we also ensured that the frequencies used do not80
happen to drive other transitions through accidental degeneracy. When operating in a two-state
detection mode, we do not use this microwave frequency.

4.3.1.2

Code names

On a day to day basis, we run our experiments with various combinations of lasers and microwaves
that are turned on. It will be very cumbersome to label an experiment with names like “(J = 2 rotationalcooling) + (J = 2 ↔ 3 microwave) + (v = 1 vibrational repump) + (v = 2 vibrational repump) + (v = 1, 2
microwaves coupling J = 1 ↔ 2) + (mF strobe pumping)”. Naming our lasers and microwaves after animals
allows us to shorten the above into “Sheep + 43 GHz + Monkey + (v = 1, 2, J = 1 ↔ 2) + Dog + Goat”.
This is still too long for our liking. We go one step further and abbreviate them with their one-charactercode names introduced with each laser/microwave in Section 4.3.1.1, so the above will look like “SUMuDG”.
Short and sweet. Cute and concise. The order of the code names does not matter, for instance “SUMuDG”
means the same thing as “SMDGUu”.
Most of the abbreviated code names are just the first letter of the name of the laser (e.g. “S” for
Sheep).8

Goat has two code names: “G” and “G*”. The former refers to Goat with σ + polarization, andthe latter refers to σ − . The v = 0 J = 1 ↔ 2 microwave also has two code names: “ˆ” and “v”, referring to
the microwave frequencies for depleting the upper and lower doublets, respectively.
Using this convention presents two additional advantages. First, having this convention also allows
us to turn lasers and microwaves on or off very easily when we run our experiments simply by including or
excluding the code names from our experiment user interface (see Section 9.3.4.2). Second, these abbreviatedcode names appear on all the plots of our automatically processed data, allowing for easy forensics on whichlasers/microwaves were turned on during the experiment just by looking at the abbreviated code names.9
In some of the figures that follow, one may see code names in the legends for the plots. The author
has also taken the extra effort to annotate the legends with descriptions for readers who are uninitiated withthe code names for clarity.

8 The Latin alphabet only has 2 × 26 = 52 letters. The author is trying to advocate for expanding the set of usable charactersto include Chinese, Japanese, and Korean characters. The possibilities are (practically) endless!
9 The full names are too long and clutter the plots.

81
4.3.1.3

Optical pumping model

Our state preparation stage is usually 300 ms long, and our state detection stage is about 40 ms long.
The choices for these timings are informed by our optical pumping model, and are fine-tuned by scanninghow long it takes for the lasers and microwaves to do their job, and are limited by the laser intensities that
we currently operate at.
The optical pumping model involves solving differential equations of the form:

 

X
X
d
ni (t) = 
Γj→i nj (t) − 
Γi→j  ni (t),
dt
j
j

(4.4)

where Γi→j are rates at which population move from the i state to the j state. We assume that (i) the
lifetime of the Ω = 0− [14.59] excited state is short as compared to the laser/microwave excitation rates, and(ii) the branching ratio to X 3 ∆1 from Ω = 0− [14.59] is unity. Hence, the i and j states are states fromthe X 3 ∆1 electronic manifold. We also make use of the fact that rotational transitions are slow comparedto the typical rates that we are looking at, so we do not include rates connecting different rotational states
within the same vibronic manifold.
The first term on the R.H.S. of Equation 4.4 corresponds to population going into the i state, while thesecond term corresponds to population going out of the i state. One can then easily vectorize the equation,
assume that the evolution is time independent (i.e. Γ rates are not functions of time), and solve it as a simple
linear algebra differential equation:dn(t) = Γ n(t) =⇒ n(t) = eΓt n(0),
dt

(4.5)

where n(t) is a vector with the population of all the states that we are interested in, and Γ is a matrix withdimensions dim(n) × dim(n) governing all the rate constants, and it includes (i) laser pumping rates, (ii)microwave pumping rates, (iii) blackbody radiation excitation rates, and (iv) spontaneous decay rates. It
does not include rotational heating through ion-ion collisions, and it assumes that the Ω = 0− [14.59] stateis short-lived as compared to the fastest pumping rates. Expansions to this model for time dependent Γ(t)or to include the two aforementioned effects are straightforward.
The optical pumping model takes in as input (i) the measured depletion rates of individual lasers
from relevant states, and (ii) the calculated Franck-Condon and Hönl-London factors (see Figure 4.16). We

82(a) Franck-Condon factors for transitions between X 3 ∆1 and Ω = 0− [14.59].

(b) Hönl-London factors for Ω = 0− [14.59] → X 3 ∆1 .

(c) Hönl-London factors for Ω = 0− [14.59] ← X 3 ∆1 .

Figure 4.16: Relevant Franck-Condon and Hönl-London factors for transitions between X 3 ∆1and Ω = 0− [14.59]. Franck-Condon factors are calculated from the measured spectroscopic constants(Be , ωe ) using equations from Ref [59]. Hönl-London factors are calculated with equations from Ref [60],being careful with the state multiplicities. Here is an example of how to get the element in Γ that represents
the rate at which population moves from (v = 0, J = 2) to (v = 0, J = 1) from the effect of Sheep P(2). For
convenience, we call the rate at which Sheep moves population out of (v = 0, J = 2) to the Ω = 0− state
to be ASheep . The rate at which population falls back to (v = 0, J = 2) is ASheep FC0′ 0′′ HL1′ 2′′ , where thesingle- (double-) primed numbers indicate the upper (lower) state. Hence the net rate at which population ismoving out of (v = 0, J = 2) is ASheep (1 − FC0′ 0′′ HL1′ 2′′ ) =: ASheep,meas. , which is the rate that we measured
with our setup. The rate at which Sheep moves population into (v = 0, J = 1) is the product of the rate at
which population is moving out of (v = 0, J = 2) and the probability that it falls down to (v = 0, J = 1).
This is given by ASheep,meas. FC0′ 0′′ HL1′ 1′′ = ASheep,meas. × 0.576 × 0.5. The relevant Hönl-London factors
are the ones associated with decay (part (b)), but the factors for excitation are also included for convenience
to compare rates across Sheep P(2), Monkey P(1), and Goat Q(1).

83
neglect the Zeeman mF sublevels in all our states in the optical pumping model except for the (v = 0, J = 1)
rotational state, which is split into the “D” and “B” states. The “D” (“B”) state represents the states thatare dark (bright) to the circularly polarized Goat Q(1) laser. For instance, a σ + -polarized Goat Q(1) laser
can only address the mF = {−3/2, −1/2, 1/2} states but not that mF = 3/2 states. The former forms the“B” state, and the latter forms the “D” state. One can tell from Figure 4.7 that there are actually two (ten)states in the “D” (“B”) state. Spontaneous decay into the (v = 0, J = 1) manifold are split into a 1:5 ratio
between the “D”:“B” states. Of note, we also neglect the effect of polarization of all lasers except for Goat
Q(1).10
We verified the model by comparing its predictions to experiments, and we saw very good agreement.
Here we present an example of matching predictions from the model with results from experiment.
After our Ramsey sequence, we map the phase difference acquired during the Ramsey evolution into
population difference in the two mF = ±3/2 states. Since our state detection protocol using REMPD does
not discern between the two states, we have to deplete one of the mF states before we dissociate to readout population in the other state. We use circularly polarized Goat Q(1) to perform this depletion. It is
important to understand what Goat does to the ion population. Ideally we want Goat to deplete all the
population in one of the mF states and leave the other mF states untouched. Figure 4.17 looks at the effectof Goat Q(1) on moving population within our molecule. The time dependence shown in the figure only
cares about the pumping rates of the lasers, which can be changed by changing the beam intensity and mode
overlap of the laser beam and the ion cloud.
Without the effect of spontaneous decay and BBR excitation (or equivalently if Goat’s pumping rate
is much faster than the rates of spontaneous decay and BBR excitation), we see that we end up pumping
8% of “B” (states bright to Goat) into “D” (states dark to Goat). This means that in using Goat Q(1) todeplete the unwanted mF state at the end of our Ramsey sequence, we end up with 8% pollution in oursignal. At this point in time, this is still within acceptable range, but if one wishes to improve upon it, one
would use Chicken or Snake for depletion instead of Goat.
With the effect of spontaneous decay and BBR excitation, one can imagine that if we turn Goat
10 The rotation of the rotating electric field mixes the polarization of the lasers that are not strobed.

84
on past some amount of time, we do not win in terms of contrast, i.e. we will end up with unwanted ion
migration into the dark mF states through vibrational decay. In fact, Figure 4.18 shows the total populationin the (v = 0, J = 1) manifold (= “D” + “B”) as a function of time at which Goat is turned on, and we cansee that past a certain pumping time, we start to see growth in the ion numbers in “D” (+“B”) throughvibrational decay. This means that with our current laser intensities and mode overlap, we do not win byhaving our depletion Goat Q(1) on for more than 50 ms, i.e. there is no point in having a state detection
stage longer than 50 ms. In fact, we are parking ourselves at 40 ms.
Figure 4.19 shows a comparison of model prediction and experiment result. We expect a population
growth in “D” (corresponding to chop “Prep. + det. (same pol.)” in the figure) after a certain time, andwe indeed see it in the experiment. The optical pumping model is not perfect because it makes several
simplifications (e.g. state lifetime of the Ω = 0− [14.59] is infinitely short compared to typical pumping rates;
treats the rotational manifolds as one state without sub-structure, except for (v = 0, J = 1) with the “D”
and “B” sub-structure), and it certainly does not include all the effects seen by our ions (e.g. “D”-“B” mixing
effects through mF changing collision; ion-ion collisions causing rotational heating), but it is good enoughfor us to design our state preparation and detection scheme.

85With Goat_Q(1)1.0(v=0, J=1), D(v=0, J=1), B(v=0, J=2)
(v=0, J=3)
(v=0, J=4)
(v=0, J=5)
(v=1, J=1)
(v=1, J=2)
(v=1, J=3)
(v=1, J=4)
(v=1, J=5)
(v=2, J=1)
(v=2, J=2)
(v=2, J=3)
(v=2, J=4)
(v=2, J=5)
(v=3, J=1)
(v=3, J=2)
(v=3, J=3)
(v=3, J=4)
(v=3, J=5)
Elsewhere

Intensity (arb. unit)

0.8

0.6

0.4

0.2

0.0
0

50

100

150

200
Time (ms)

250

300

350

400

(a) The effect of having only Goat Q(1) on.

With Goat_Q(1) (spon. dec.) BBR @ 300 K(v=0, J=1), D(v=0, J=1), B(v=0, J=2)
(v=0, J=3)
(v=0, J=4)
(v=0, J=5)
(v=1, J=1)
(v=1, J=2)
(v=1, J=3)
(v=1, J=4)
(v=1, J=5)
(v=2, J=1)
(v=2, J=2)
(v=2, J=3)
(v=2, J=4)
(v=2, J=5)
(v=3, J=1)
(v=3, J=2)
(v=3, J=3)
(v=3, J=4)
(v=3, J=5)
Elsewhere

1.0

Intensity (arb. unit)

0.8

0.6

0.4

0.2

0.0
0

50

100

150

200
Time (ms)

250

300

350

400

(b) Spontaneous decay and BBR excitation are turned on.

Figure 4.17: From the optical pumping model: distilling out the effects of spontaneous decay
and BBR excitation from the effect of Goat Q(1). Here we consider all states with v ≤ 3 and J ≤ 5.
The (v = 0, J = 1) manifold is split into “D” and “B” representing the states that are Dark and Bright to thecircularly polarized Goat Q(1) laser, respectively. We start with population all in the “B” state, and we turnGoat Q(1) on. Note that the early time behavior is the same with and without the effects of spontaneous
decay and BBR excitation because the pumping rate of Goat is much faster than the rates of decay fromions shelved in the higher vibrational states by Goat and the BBR excitation rate out of the v = 0 manifold.
The effects of spontaneous decay become apparent at later times when ions shelved in higher vibrational
manifolds start to decay down, and if they fall back into the “B” state, they in turn can be addressed byGoat again. More discussion in the main text.

86

Goat Q(1) with starting population (D=0, B=1)

Population in (v=0, J=1)

100

Without spontaneous decay and BBR
With spontaneous decay and BBR

Population = 7.8% @ t = 48 ms
10 1
0

50

100

150

200 250
Time (ms)

300

350

400

Figure 4.18: From the optical pumping model: population in the (v = 0, J = 1) state, which is asum of population in “D” and “B”, plotted as a function of time at which Goat is turned on.
Without the effect of spontaneous decay and BBR excitation, the population in the (v = 0, J = 1) manifold
saturates at 6%, but with the aforementioned effects, we reach a lowest population fraction of about 8% ataround 50 ms of Goat on time. Thereafter we see significant unwanted population growth.

(Normalized) Ion numbers (count)

File: 17/000041826-Ramsey.h5
200

100
50
0.00

0.3
0.2
0.1
0.00

0.02

0.04 0.06
Time (s)

0.08

0.10

(a) Predictions from the optical pumping model.

0.02

0.04

0.06

0.08

0.10

7500
7000
6500
Do nothing (0)Prep. only (SUMuG+0)6000
Prep. + det. (same pol.) (SUMuG+G)
Prep. + det. (diff. pol.) (SUMuG+G*) 5500
5000
4500

0.15
0.14
0.13
0.12
0.11
0.10
0.09

Number of ionsPrep. only
Prep. + det. (same pol.)Prep. + det. (diff. pol.)Do nothing (0)Prep. only (SUMuG+0)
Prep. + det. (same pol.) (SUMuG+G)
Prep. + det. (diff. pol.) (SUMuG+G*)

150

0.4
Area under TOF curve (V µs)

Population in (v=0, J=1)

0.5

250

0.00

0.02

0.04
0.06
scan_parameter

0.08

0.10

(b) Checking predictions with experiment.

Figure 4.19: Comparing predictions from the optical pumping model with experiment. The top
plot on the right corresponds to the number of dissociated Th+ detected, while the bottom plot is a checkon the total ThF+ numbers in the trap; the lines in the bottom plot should look flat. We see that the
population in “B” (corresponding to the chop “Prep. + det. (diff. pol.)”, red line) decays at early time andincreases slightly after about 20 ms, both in the simulation on the left plot and in the experiment on the
top right plot. The population in “D” (corresponding to the chop “Prep. + det. (same pol.)”, green line)
decays with time in the experiment, but not in the simulation. We believe this is because the model does
not include the effect of “D”-“B” mixing through mechanisms like mF -changing collisions. We can model
this mixing by including an ad hoc decay rate of about 3 s−1 to make the two plots agree more.

87
4.3.1.4

Current scheme and possible future upgrades

With our current scheme using 300 ms of state preparation (with Goat Q(1), Sheep P(2), 43 GHz
microwave, Monkey P(1), Dog P(1), and the v = 1, 2 microwaves), we expect about 55% of all our ions to
be in the X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = 3/2) states and another 2% in the X 3 ∆1 (v = 0, J = 1, F =3/2, mF ̸= 3/2) states after state preparation. 25% of our ions are hiding in the X 3 ∆1 (v = 0, J = 4), withanother 10% in the X 3 ∆1 (v = 1, J > 1), and the rest (8%) spread out across the higher vibrational states.
We could leave our state preparation lasers and microwaves turned on for longer, but leaving them on formuch longer than 100 ms starts to give diminishing returns. Figure 4.20 shows two plots for visualizing thediminishing returns after 100 ms.
Starting population (J=1: 128, J=2: 237, J=3: 218, J=4: 193)
100

Starting population (J=1: 128, J=2: 237, J=3: 218, J=4: 193)

Without spontaneous decay and BBR
With spontaneous decay and BBR

0.6

10 1

(B+D)

B/(B+D)

0.5
0.4
0.3
Without spontaneous decayWith spontaneous decay

0.2

10 2
0

100

200

300
400
Time (ms)

500

600

(a) “Impurity” of the J = 1 manifold.

700

0

100

200

300
400
Time (ms)

500

600

700

(b) Fraction of ions in the J = 1 manifold.

Figure 4.20: Plots of “impurity” and population migration in and out of the J = 1 manifold as afunction of state preparation time. “Impurity” is defined as the fraction of the population within theJ = 1 manifold in the “B” state; we want “impurity” to be as close to zero by the end of our state preparation
sequence. The initial population numbers (shown on the title of each plot) used for the simulation come fromcounting dissociated Th+ ions by scanning the R branch of the first photon in the REMPD transition used,
and taking into account the Hönl-London factors. (a) With our current pumping rates (limited by the current
laser and microwave intensities), we expect to go below an “impurity” of 10% at 100 ms. Beyond 100 ms,
BBR excitation kicks in to excite population in the “D” state into the higher vibrational states, so the lasers
have to fight harder to get the ions back. (b) The sharp rise in the population in J = 1 (=“B”+“D”) at thestart comes from Sheep and the 43 GHz microwave moving population from the J = 2&3 into J = 1. The
decrease after that comes from Goat moving population into the mF = 3/2 states through the Ω = 0− [14.59]
state; this is an inefficient process because only about 6% end up in the mF = 3/2 states after the first pass.

We can make several upgrades to our system to make the state preparation sequence more efficient.
Firstly, we can recover the population in X 3 ∆1 (v = 0, J = 4) by including a repump through microwaves
connecting the J = 3 and J = 4. This will give us an additional 25% ions. Secondly, we can recover the

88
population in X 3 ∆1 (v = 1, J > 1) by using stronger microwaves coupling the X 3 ∆1 (v = 1, J = 1) andX 3 ∆1 (v = 1, J = 2), and also microwaves coupling to higher rotational states. Lastly, we can also make
Goat’s mF pumping a more efficient process by switching Goat from Q(1) to P(1); using P(1) would also
alleviate the problem of Doppler shifting Q(1) into Q(2) because the P lines are much more separated thanthe Q lines. A quick study on the pumping rates of the lasers indicate that we are still not in the saturation
regime yet, so we can still win more with more laser power (which will allow us to expand the beam more
for the same/higher intensity with a better overlap with the entire ion cloud). We did not do a full study
on the saturation curve for the microwaves because it turns out that the setup made it much harder to turnthe knob on microwave power, but it seems like we are still in the linear regime so there is still more to begained with higher microwave power.
For state detection, we turn the lasers (Goat Q(1) and Llama R(2) only) on for 40 ms. We havealready discussed limitations in the state detection scheme in the previous section. Goat Q(1) and Llama
R(2) couple to the Ω = 0− [14.59] excited state, with unity branching ratio back to X 3 ∆1 . Fortunately Goat
Q(1) only brings 6% of our population back into the dark state, i.e. Goat is ∼ 94% efficient in depletion.
Llama R(2) does not move population into the dark state through the R(2) transition, but in principle ionsR(2)could take the pathway (v = 0, J = 2) −→ (v = 2, J = 3)

spon. decay−→(v = 1, J = 2)

spon. decay−→(v = 0, J = 1)

to go back to the dark state. To make this process less cycling, we can change their wavelength to addressthe Ω = 0+ [10.47] excited state instead with a less than unity branching ratio back into the X 3 ∆1 manifold,shelving ions in the a 1 Σ+ manifold.

89
4.3.2

Ramsey sequence
Upon preparing the ions in the X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = 3/2) states, we wish to perform a

Ramsey experiment by performing a π/2 pulse to prepare a superposition of the mF = 3/2 and mF = −3/2
states (see part (c) of Figure 1.4). The previous eEDM experiments with HfF+ use a π/2 pulse sequence asfollows:(1) Ramp Erot down from 58 V/cm to 7 V/cm in 16 µs;(2) Hold for 1 ms;(3) Ramp Erot back up to 58 V/cm in 16 µs.
Since Brot ∝ Erot and the rotational coupling11 (∆) between the mF = ±3/2 states increases with decreasing
Erot to about ∆ ∼ 500 Hz in HfF+ with an frot = 375 kHz, the above π/2 sequence effects a “simultaneous
π/2 pulse” for both the upper and lower doublets; the “simultaneous” is in quotes because the upper andlower doublets actually have a differential ∆ of about 50 Hz at 7 V/cm, but it is small enough compared to500 Hz so that while they are actually π/2 ± ϵ pulses, the ϵ is small. Figure 4.21a shows a plot of the ∆s in
HfF+ .
We borrowed wisdom from the HfF+ experiment and tried to perform the same π/2 pulse on ThF+ ,but we have not been very successful. We notice that we lose a lot of ions from our desired quantum
states after each π/2 pulse. It turns out that the spectroscopic constants of ThF+ are sufficiently different
from HfF+ to make the HfF+ protocol hard to implement in ThF+ . However, the seemingly unfriendly
spectroscopic constants of ThF+ also opened a window of opportunity to realize a true simultaneous π/2pulse in ThF+ , which we shall see in the next few sections.

11 See Figure 1.4 for an illustrated explanation of the rotational coupling in the context of the eEDM-sensitive states. See
Figure 4.21 for the plot of ∆ vs Erot . See Section C.3.1 for the Hamiltonian that we diagonalize to extract ∆.

90
4.3.2.1

Working principle behind the old protocol

The energy splitting between the mF = ±3/2 states goes as12 (also see Figure 1.4):
f u,l =q2

2[∆u,l (Erot , frot )] + [f0 (Erot , frot )] ,(4.6)

where {u, l} correspond to the upper and lower doublets, respectively, and it is made explicit that ∆ andf0 (Erot , frot ) = 3gµB Brot (Erot , frot ) both depend on Erot and frot . The dependence of Brot on those variables
eErot
are Brot = Baxgrad · m(2πf2 , while those of ∆ are more complicated, but it suffices to say that ∆ increases
rot )very quickly with decreasing Erot and increasing frot . Representing the mF = ±3/2 states as the poles on
a Bloch sphere, the Bloch vectors on the sphere precess about a precession vector pointing in the (∆, 0, f0 )direction with frequency f as given in Equation 4.6. For perfect Rabi flopping, we want the Bloch vectors toprecess around (1, 0, 0). We can realize this in our system by ramping down Erot to reduce f0 and increase∆ so that f0 ≪ ∆ for our π/2 pulse. Except during the π/2 pulses, Erot is large enough that f0 ≫ ∆ so thatthe Bloch vectors precess around the poles. This is the working principle behind the π/2 pulses in the firsttwo generations of the JILA eEDM experiment using HfF+ .
The above picture is painted as a two-level system, but there are many more energy levels in ourmolecule. It turns out that the width of the hyperfine crossing “resonance” in ThF+ is so much broader
than HfF+ (see Figure 4.21b) that we can get population leakage from our two-level system into other states
when we sweep across the hyperfine crossing, e.g. unstretched Zeeman states through rotational coupling ofneighboring mF states and, worse still, into the other doublet through Ω-doubling coupling. Furthermore,
the two doublets have such different ∆s that it is not possible for us to perform a “simultaneous π/2 pulse”
for both doublets in ThF+ (∆u ≈ 100 Hz and ∆l ≈ 300 Hz at 7 V/cm with frot = 147.5 kHz). We could
perform a π/2 pulse for the upper doublet and a 3π/2 pulse for the lower doublet simultaneously, but thisstill does not solve the problem of population leakage when we cross the hyperfine crossing. Instead, we
implement a true simultaneous π/2 pulse protocol by always staying on the same side of the hyperfine
crossing and parking ourselves at a special Erot during the π/2 pulse.

12 The difference in g-factor for the upper and lower doublets are small for our discussion here, and are intentionally left out.

91

frot = 375 kHz

500

Lower doublet
Upper doublet

Hyperfine crossing at 34.6 V/cm
400

14
12
10
8
6
4
2
0

| | (Hz)

300
200
100
0

0

10

25

30

20

35

40

30

45

50

40

rot (V/cm)

55

50

60
60

(a) HfF+ with experiment parameters used in the second generation eEDM measurement.

frot = 147.5 kHz

500

Lower doublet
Upper doublet

Hyperfine crossing at 11.4 V/cm
400
120

| | (Hz)

300

100
80

200
60
100
0

40
12.6
0

10

20

12.8

13.0
30

rot (V/cm)

13.2

13.4

13.6

40

50

13.8

14.0
60

(b) ThF+ with current experiment parameters.

Figure 4.21: Plot of |∆|. The plots are obtained by diagonalizing the 12-level Hamiltonian of the X 3 ∆1 (v =0, J = 1) manifold in the molecular frame (with zero magnetic field, and at various electric field Erot ), andextracting the energy difference within the doublet for each of the upper and lower doublets. The Hamiltonianthat we diagonalized is shown in Section C.2. After diagonalizing the Hamiltonian, we obtain eigenenergies
like those shown in Figure 5.4. ∆u would correspond to the energy difference between the states labeled [8]and [9] after the hyperfine crossing (or [6] and [7] before the hyperfine crossing). ∆l corresponds to the energy
difference between the states labeled [0] and [1]. The part where ∆ blows up corresponds to the hyperfine
crossing, where the upper doublet crosses over the lower unstretched Zeeman states of the F = 1/2 manifold.
Note that the hyperfine crossing occurs at a much higher Erot in HfF+ than in ThF+ because of the muchlarger hyperfine coupling constant in the former (3 times). The differential ∆ between the doublets is muchlarger in ThF+ than in HfF+ because of the much larger Ω-doubling constant in the former (7 times). Hard
to see from this diagram is an interesting Erot ≈ 30 V/cm for ThF+ where ∆u = 0 Hz.

92
4.3.2.2

New protocol with the special Erot

It turns out that the values of ∆s for both doublets cross at a special Erot of about 13.35 V/cm in
ThF+ with frot = 147.5 kHz (see Figure 4.21b). It also turns out that 13.35 V/cm brings a few otherbenefits:(1) It is sufficiently large for the molecules to retain their polarized orientations (dmf Erot ≈ 23 MHz ≫ωef = 5.3 MHz).
(2) It is sufficiently small as compared to 60 V/cm. We intend to have an f0 = 50 Hz when we
nominally operate at 60 V/cm, and since a magnetic field quadrupole-typed Zeeman induced f0
scales linearly with Erot , f0 will be reduced to about 10 Hz at 13.35 V/cm. This is small comparedto ∆u (Erot = 13.35 V/cm) = ∆l (Erot = 13.35 V/cm) ≈ 65 Hz, allowing us to perform Rabi floppingthat goes almost from pole to pole on the Bloch sphere, retaining good contrast.
Hence, without needing to go through the hyperfine crossing, and with a π/2 pulse that is truly simultaneousfor both the upper and lower doublets, the π/2 pulse protocol for ThF+ is:(1) Ramp Erot down from 60 V/cm to 13.35 V/cm in 1.2 ms;(2) Hold for 4.1 ms;(3) Ramp Erot back up to 60 V/cm in 1.2 ms.
We note that while we ramp Erot down to 13.35 V/cm, ∆l is always higher than ∆u , and one mightnaı̈vely think that there would be an additional phase accumulated by the lower doublet than the upper
doublet after the procedure. We propose two reasons not to worry. Firstly, it turns out that the ramp isfast enough that the extra phase accumulation is much less than 1% of a radian. Secondly, We could always
compensate for the difference by parking at an Erot that is slightly smaller than 13.35 V/cm. Regardless, thevalue for the special Erot is something that can be scanned experimentally. In fact, as we shall see in Figure
4.23, we scanned for the special Erot , and found it to be about 13.7 V/cm. We believe the discrepancy with
13.35 V/cm comes from a combination of COMSOL simulation precision (to convert applied voltage on the93electrodes to electric field strength along the trap axis) and imperfections in the experiment setup. This
discrepancy is also consistent with our estimated error of dmf (see Table 4.1).
It is also interesting to note that the value of the special Erot varies very slowly with frot within ourusual operating range between 147.5 kHz and 400 kHz. Figure 4.22 shows a plot of the special Erot vs frot .
13.4

Special rot (V/cm)

13.2
13.0
12.8
12.6
12.4
12.2
12.0

0

500

1000 1500 2000 2500 3000 3500 4000
frot (kHz)3. Hence the value
Figure 4.22: Plot of the special Erot vs frot . As frot increases, ∆u,l increase as ∼ frotof the special Erot changes very little, until frot is starting to be comparable to the hyperfine splitting (≈15 MHz).

94

0.03

0.00

0.0

0.0
0.02

0.03

0.02

0.00

0.01

0.02

0.03

$SUMuDG*+GL:+$SUMuDG+GL:+

5000

0.10
0.005

0.010

0.015
0.020
piby2_hold

20
10
0.00

Diff. pol. (SUMuDG*+GL)
Same pol. (SUMuDG+GL)

20
0.01

0.02

0.03

10
0.00

0.01

0.02

0.03

0.02

0.03

0.5
0.0

0.0
0.5
0.00

0.01

0.02

0.03

0.5
0.00

0.01

$SUMuDG*+GL:+ 5500
$SUMuDG+GL:+
5000

0.11
0.10

4500
0.000

0.005

0.010

0.015
0.020
piby2_hold

0.025(b) Erot = 13.35 V/cm.

File: 13/000050062-Ramsey.h5
Doublet A
Doublet B

File: 11/000050027-Ramsey.h5
Doublet A
Doublet B

40

Diff. pol. (SUMuDG*+GL)
Same pol. (SUMuDG+GL)

30
20

10
0.00

10
0.00

0.01

0.02

0.03

0.5

0.5

0.0

0.0

Diff. pol. (SUMuDG*+GL)
Same pol. (SUMuDG+GL)

0.01

0.02

0.03

0.02

0.03

0.00

0.01

0.02

0.03

$SUMuDG*+GL:+ 5000
$SUMuDG+GL:+
4500

0.09
0.005

0.010

0.015
0.020
piby2_hold(c) Erot = 13.7 V/cm.

0.025

0.030

30
20

4000

Diff. pol. (SUMuDG*+GL)
Same pol. (SUMuDG+GL)

40

0.01

0.02

0.03

0.00

0.5

0.5

0.0

0.0

0.5
0.00

0.030

Diff. pol. (SUMuDG*+GL)
Same pol. (SUMuDG+GL)

20

10
0.00

0.5
0.01

0.10

0.000

0.030

30

(a) Erot = 12.6 V/cm.

20

0.5
0.00

0.025

40

Diff. pol. (SUMuDG*+GL)
Same pol. (SUMuDG+GL)

0.01

0.02

0.03

0.01

0.02

0.03

0.5
0.01

0.02

0.03

0.00

5000

Number of ions

30

4500

(Normalized)
Contrast

0.09
0.000

(Normalized)
Ion numbers(count)0.03

Number of ions

0.11

0.01

0.01

30

File: 11/000050024-Ramsey.h5
Doublet A
Doublet B

Number of ions

0.02

(Normalized)
Contrast

0.01

0.2

(Normalized)
Ion numbers(count)(Normalized)
Contrast

15

0.5

0.5
0.00

Diff. pol. (SUMuDG*+GL)
Same pol. (SUMuDG+GL)

20(Normalized)
Ion numbers(count)

10
0.00

25

Area under TOF curve(V µs)

20

40

Area under TOF curve(V µs)

Diff. pol. (SUMuDG*+GL)
Same pol. (SUMuDG+GL)

Area under TOF curve(V µs)

30

40

Number of ions

Area under TOF curve(V µs)(Normalized)
Contrast(Normalized)
Ion numbers(count)File: 11/000050025-Ramsey.h5
Doublet A
Doublet B

0.10

4500

0.09

$SUMuDG*+GL:+ 4000
$SUMuDG+GL:+

0.08
0.000

0.005

0.010

0.015
0.020
piby2_hold

0.025

0.030

(d) Erot = 14.0 V/cm.

Figure 4.23: Rabi flopping of both doublets at various Erot . The top row in each subfigure corresponds
to the measured Th+ ion numbers. The second row corresponds to the contrast, which is the difference ofthe measured numbers of the two chops, divided by the sum. The last row corresponds to the total ThF+
numbers detected, for normalization. The numbers in each doublet are extracted by using angular resolved
photofragments as demonstrated in Figures 3.8 and 3.9. In this set of runs, Doublet A (B) corresponds tothe ions in the lower (upper) doublet. From our model, and assuming that Erot in the trap is what we believe
it is, we expect both doublets to have the same Rabi frequency at Erot = 13.35 V/cm (see Figure 4.21b).
However, (b) shows that the upper doublet is decohering very quickly, and is at a slightly higher frequency
than the lower doublet. This is probably because Erot is slightly closer to the hyperfine crossing than thespecial Erot point, causing us to loss ions from the upper doublet into the unstretched Zeeman states at
the hyperfine crossing. Going to a lower Erot = 12.6 V/cm gives a higher lower doublet Rabi frequency (asexpected, see Figure 4.21b), and a totally decohered Rabi flopping for the upper doublet as it gets evencloser to the hyperfine crossing. Going to a higher Erot = 14.0 V/cm makes the Rabi frequency for both
doublets go down as expected. The sweet spot happens to be around Erot = 13.7 V/cm.

95
4.3.3

Summary
The experiment sequence in the baby Bucket Brigade (see Figure 7.9) is summarized in Figure 4.24.

3

4

6

9
10

5

2

12

11
7

8

1

Figure 4.24: Summary of experiment sequence. The channels 1 (yellow) and 2 (cyan) correspond to
pickup voltages on Rod 8 (radial electrode) and the Southeast Quadrant Ring (“end cap”) (see Figure 7.9).
(0, not shown on the scope trace) We pulse the nozzle and fire the ablation laser to create neutral ThF
entrained in supersonic expansion beam of neon buffer gas at room temperature (ultimate flow speed ofabout 800 m/s). The steps are detailed in the main text. Also compare with Figure 7.9.

(1) We turn on the radial trap, fire the REMPI lasers to ionize the neutral ThF into the ThF+ X 3 ∆1 (v =0) manifold, and apply a stopping voltage to stop the ions into a trap with secular frequencies ofabout 2 kHz in all directions.
(2) We move the ions to the center of the trap, while turning the radial trapping frequency up to about
4 kHz from 2 kHz.
(3) We ramp up Erot from zero to 60 V/cm.
(4) State preparation for 300 ms, where we send in all the state preparation lasers and microwaves toprepare the ions in the X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = 3/2) states.
(5) We translate our ions into the Science region by modulating the potentials on the Rings (see Figure
7.9).

96
(6) We apply our π/2 pulse to prepare our ions in a superposition of the X 3 ∆1 (v = 0, J = 1, F =3/2, mF = 3/2) and X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = −3/2) states by ramping down our Erot to
13.7 V/cm with the protocol mentioned in Section 4.3.2.2.
(7) Ramsey free evolution time, followed by another π/2 pulse labelled by (6) to map the phase difference
between the states into a population difference.
(8) We translate our ions out from the Science region back into the Octagon (see Figure 7.9).
(9) State detection at 40 ms, where we send in the lasers (and microwaves) to transfer population fromeither the mF = 3/2 of mF = −3/2 states to states outside the J = 1 manifold.
(10) We ramp Erot down to 20 V/cm to retain the polarization of our molecule, but to ensure that Erot issmall enough that the ions to do not miss the ion detector because of their huge transverse rotation
micromotion speeds.
(11) We ramp the radial trapping frequency back to about 2 kHz, and move the ions to overlap with the
REMPD lasers.
(12) We fire our REMPD lasers and kick our ions out of the trap towards our ion detector.

Chapter 5Interlude

5.1

Revisiting wisdom lost in time
Before we get too excited with our newfound tools using ThF+ , it serves to revisit some JILA eEDM

wisdom that have been either (i) lost in time, or (ii) taken for granted. We shall see if the wisdom (mostly
built upon HfF+ ) still holds for ThF+ . This section does not aim to shed any new insight on top of theancient wisdom, and mainly serves as a quantum repeater or a buffer op-amp for wisdom from previous
generations to remind the younger students in the group why we chose to use ThF+ the way it is being used
now. As an exercise, we shall also explore the feasibility of a new proposal on how to measure the eEDM
with ThF+ from the Hutzler group [29].

5.1.1

Wisdom up till now
The whole state preparation and detection scheme in ThF+ has been built around performing theeEDM experiment on the 3 ∆1 (v = 0, J = 1, F = 3/2, mF = ±3/2) states. We shall now revisit the wisdom
behind the choice for each of the quantum numbers.

5.1.1.1

Using the 3 ∆1 state

The 3 ∆1 electronic state is wonderful for an eEDM measurement for a few reasons. Summarizing thepoints in Ref. [25], we have:Ω-doubling. The 3 ∆1 electronic state has an Ω doublet structure with a very relatively small energy gap
between the parity doublets. This allows the molecule to be polarized very easily with very modest

98electric fields as compared to atoms. A high degree of polarization allows the ensemble of moleculesin our ion cloud to sing in unison like in a choir, instead of cancelling each other’s effects like in acanteen. We borrowed this idea liberally from Prof. David DeMille’s PbO experiment [61].
|Ω| = 1: small g factor. With an |Ω| = 1 structure and a |Λ| = 2 in a 3 ∆1 state, the projections of the spin
and angular momentum of the electron onto the internuclear axis are anti-parallel, i.e. L·n̂ =: Λ = ±2
when S · n̂ =: Σ = ∓1. Given that the g-factor for the spin of the electron is quite nearly two times
that of its orbital angular momentum, the net g-factor of the 3 ∆1 state can be highly reduced. The
reduced g-factor makes us less susceptible to drifts in external magnetic fields, reducing systematiceffects in our system.
Co-magnetometry. Having two doublets (from the Ω doublet structure) with very similar sensitivities
to external fields, but sensitivities of opposite sign to the physics that we are probing allows us toperform co-magnetometry to have excellent rejection of many systematic effects.

5.1.1.2

Using the (v = 0, J = 1) stateWithin the 3 ∆1 electronic state, we use the v = 0 vibrational manifold because it is the groundvibronic state, where the state lifetime is not limited to any form of spontaneous decay to lower energy
levels. Within the v = 0 vibrational manifold, we use the J = 1 rotational manifold because this is the
manifold which has the lowest multiplicities, making quantum control much easier.
Furthermore, much in the spirit of Figure 3.2, since we are in the 3 ∆1 electronic state with |Ω| = 1,
we want to be in the J = |Ω| = 1 rotational manifold for the electron spin to be most aligned with theinternuclear axis to have the best sensitivity to the eEDM.

5.1.1.3

Using the stretched Zeeman state (mF = ±3/2)

Within the 3 ∆1 (v = 0, J = 1) rovibronic manifold, we choose to use the stretched Zeeman states
because the eEDM is sensitive to the electron spin. The electron spin is fully aligned with the projectionof the net angular momentum onto the internuclear axis in the stretched Zeeman states (|F = 3/2, mF =

99
±3/2⟩ = |F = 3/2; mJ = ±1(Σ = ∓1, Λ = ±2), mI = ±1/2⟩), whereas the unstretched Zeeman states comewith a superposition of electron spin projections onto the internuclear axis (e.g. |F = 3/2, mF = 1/2⟩ =q
2√1 |F = 3/2; mJ = 1, mI = −1/2⟩ +3 |F = 3/2; mJ = 0, mI = 1/2⟩). Therefore, using the stretched3
Zeeman states gives us the best sensitivity to the eEDM.

5.1.2

Other exotic ways
In light of proposals to perform eEDM measurements in unconventional ways (but still using diatomic

molecules) [28, 29, 34], one wonders if we could win by breaking ourselves free from tradition.1

In fact,

Ref. [29] proposes to perform an eEDM measurement in ThF+ by using a first-order electric field-insensitivetransition at a magic2 electric field strength. The physics behind this protocol is that as we increase theelectric field strength, we go from the “quadratic Stark regime” where F is a relatively good quantum number
to the linear Stark regime where F is no longer a good quantum number, but mF still is. This results in nontrivial shifts in energy levels within the molecule, which can result in transition energies between states to havea zero first-order gradient w.r.t. the field strength. This is much like the magic trapping wavelength concept
for trapping neutral atoms [62] or magic magnetic fields for first-order field-insensitive qubit transitions [63].
The proposal from Ref. [29] involves a transition between a (J = 1, mF = 1/2) and a (J = 2, mF =1/2) state at an electric field strength of 18.6 V/cm. Figure 5.1 shows a plot of transition frequencies as afunction of applied electric field. We shall attempt to analyze the proposal from Ref. [29], and see how it
can or cannot apply to our experiment.

1 Also known as peer pressure from people who are no longer around.
2 Although not yet officially defined in textbooks, the author notes that “magic” is now a technical term in AMO physics
with a specific meaning: (adj.) used in AMO physics to refer to a specific value of an experiment parameter that allows thesystem to be insensitive to changes in that experiment parameter; e.g. “magic wavelength” [62].

100Frequency (GHz)

29.12

29.10
CP-FIT
29.08

29.06
12.5

15.0

17.5

20.0

(V/cm)

22.5

25.0

27.5

30.0

25.0

27.5

30.0

(a) All transitions.

29.092CP-FIT
29.090Frequency (GHz)29.08829.08629.08429.08229.080

12.5

15.0

17.5

20.0

(V/cm)

22.5(b) Zoomed in for better view of the curvature.

Figure 5.1: Transition frequencies of all transitions connecting the (J = 1, mF = ±1/2) states tothe (J = 2, mF = ±1/2) states. CP-FIT stands for CPV-sensitive field-insensitive transitions, as definedin Ref. [29]. These plots are obtained in a similar way as Figure 4.5, but only showing the mF = ±1/2transitions, and plotting the lines with the same intensities instead of being weighted by transition strength.
The CP-FIT are transitions that are insensitive to the applied electric field to first order, and they show up as
transition energies with zero gradients on the plots. The rotating frequency frot is set to zero in these plots.
The vertical lines indicate 18.6 V/cm and 21.3 V/cm, where the CP-FITs are. The lines for the CP-FITs
look flat on this scale (2 MHz per small division in the zoomed in plot), but if there is a 1% deviation in E,
it would correspond to a 200 Hz deviation on the CP-FIT frequencies.

101
5.1.2.1

Co-magnetometry

In our eEDM experiment, there are four states involved (see Figure 5.2). We tend to measure the
New proposal

Traditional JILA method

mF = −3/2

mF Ω = −3/2

J =2

mF Ω = 3/2

J =1

3/2

mF = −1/2

1/2

Figure 5.2: The four states used in a typical eEDM experiment with co-magnetometry. In
our traditional JILA method, we measure the frequencies of the upper and lower Stark doublets. Ramsey
spectroscopy is performed “horizontally” across the doublets. The energy difference of the upper (lower)
Stark doublet is EZeeman + EEDM (EZeeman − EEDM ). Taking the difference in energies between the doublets
reveals EEDM . In the new proposal from Ref. [29], Ramsey spectroscopy will be performed “vertically” acrossthe rotational states. The energy difference of the mF = 1/2 (mF = −1/2) pair is Emol. rot. EStark + EEDM
(Emol. rot. EStark − EEDM ). Taking the difference in energies between the two mF pairs reveals EEDM .

frequency within each Stark doublet, and perform co-magnetometry across the Stark doublets. The new
proposal changes it to measuring the frequency within each mF manifold, and performing checks on systematics by comparing the frequencies measured for the different mF . This is a paradigm shift, but it certainly
sounds like we could do this with our current setup.

5.1.2.2

Ramsey frequency

Our Ramsey frequencies have been on the order of f ∼ 100 Hz. In the third generation experiment,
we are also aiming for 50 Hz. In contrast, the new proposal is looking at a Ramsey frequency of about
29 GHz. This is huge. To perform a Ramsey experiment on this transition, and aiming for a precision on theorder of 1 mHz, we would need to have a stable microwave source down to a part in 3 × 1013 . This sounds
challenging, but not entirely impossible.

5.1.2.3

State preparation

Our current state preparation protocol involves (i) creating our ions in the ground vibronic state of
ThF+ , (ii) using lasers and microwaves to move our ion population to X 3 ∆1 (v = 0, J = 1, F = 3/2, mF =

1023/2), and (iii) performing a π/2 pulse by ramping Erot down to prepare a superposition of the mF = 3/2
and mF = −3/2 states for Ramsey spectroscopy.
The new proposal involves preparing ions in the mF = 1/2 states. In our setup, this would probably
mean that we will need to prepare our ions in the X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = 3/2) states first
like what we have been doing, and then performing a coherent transfer to the X 3 ∆1 (v = 0, J = 1, F =3/2, mF = 1/2) state, probably through stimulated Raman adiabatic passage (STIRAP). One would needto find a good intermediate state to use for STIRAP, or some form of adiabatic rapid passage scheme thatwould allow for such a population transfer. Unfortunately, our current setup (and our future Bucket Brigade
setup, see Section 5.3) only allows lasers and microwaves to come in along the plane of the rotating polarizing
electric field. This means that our lasers and microwaves can only be σ ± or x polarized, not π. This would
certainly limit schemes available to us, for example, a naı̈ve two-photon process to move population frommF = 3/2 to mF = 1/2 would involve a σ − and a π photon, but this cannot be done with our current setup.
Furthermore, the new proposal does not take into account that we need to work in a rotating frame
to polarize our molecular ions with a rotating electric field without ejecting them out of the trap. There will
be strong rotational coupling between the mF = ±1/2 states on the order of a few 100 kHz. We can apply
a magnetic field to offset the states through Zeeman shifts. This requires an axial magnetic field gradient of
about 100 G/cm (about 30000 times of what we are applying now). This is getting tricky.
The new proposal also requires a different type of π/2 pulse. We would probably need to address the(J = 1, mF = 1/2) → (J = 2, mF = 1/2) transition directly with π-polarized microwaves. These microwaves
will have to come in along the trap axis (perpendicular to the plane of the rotating polarizing electric field
Erot , and they either have to be pulsed, or they can be circularly polarized, but we would need an frot of
29 GHz3 . This sounds hard.
An alternative to the π-polarized microwaves is to use a two-photon Raman process, using photonsof opposite helicities4 . We would need to do some spectroscopy to make this happen.
3 This would introduce additional complications because f
rot is now much larger than typical energy scales within the J = 1
manifold.
4 We could use two photons with π-polarization, but as mentioned earlier on, our current setup (and the Bucket Brigade)does not allow us to address our ions with π-polarized lasers or microwaves.

103
5.1.2.4

What about the mF = ±3/2 manifold?

Ref [29] calculated the case for mF = 1/2, but what about mF = 3/2? It would save us one step inthe state preparation of moving our ions coherently into the mF = 1/2 state, and it would also relieve usof the rotational coupling problem between mF = ±1/2 states. It turns out that such a CP-FIT for the
mF = ±3/2 manifold does not exist for ThF+ because of the relative sizes of the hyperfine splitting constantand the Ω doubling coupling constant.
The proposed scheme relies on the fact that the Stark shifts move in the direction determined by thesign of mF Ω, and that the hyperfine splitting is large compared to the Ω-doublet splitting. In this regime(hyperfine splitting more than Ω-doublet splitting, the lower Ω doublet of the more energetic hyperfine
manifold will be Stark shifted down in energy, while the upper Ω doublet of the less energetic hyperfine
manifold will be Stark shifted up, and if they have the same mF , they will try to repel each other throughan avoided crossing, resulting in non-trivial energy shifts (see Figure 5.3).
J =1

Naı̈ve J = 2

Actual J = 2

Figure 5.3: States bumping into each other through Stark shifts. The horizontal lines indicate
Zeeman levels. Energy levels colored blue belong to one hyperfine manifold, and those colored black belong
to another. The arrows indicate the Stark shifts (EStark ∼ mF Ω) when an electric field is applied. In theJ = 1 manifold, the mF = ±1/2 states from one hyperfine manifold will come close to those from theother manifold, and there will be a strong interaction between the states, resulting in non-trivial energy
shifts for these states. The same is the case for the naı̈ve J = 2 manifold. It is called “naı̈ve” because wenaı̈vely assume that the energy levels are ordered like the J = 1 manifold, i.e. the hyperfine splitting is large
compared to the Ω-doubling constant. This is true for HfF+ , but it is not true for ThF+ . For ThF+ , thehyperfine splitting is comparable to the Ω-doubling splitting in J = 2, resulting in an energy ordering of thestates as shown in the right panel. In this ordering of states, Stark shifts do not cause states to bump into
each other, giving rise to more trivial energy shifts of the states. A combination of the physics behind theleft and the right panels allow for first-order field-insensitive transitions between the J = 1 and J = 2 in the
mF = ±1/2 states but not the mF = ±3/2 states in ThF+ .

104
In ThF+ , the states in the J = 1 manifold are arranged such that the Ω-doubling is still a finer
structure than the hyperfine splitting, but Ω-doubling and hyperfine splitting are of the same sizes for thestates in the J = 2. There are no states to repel the mF = 3/2 states in the J = 1 because there is only one
pair of mF = 3/2 states. There are two pairs of mF = 3/2 states in the J = 2 (each pair corresponding to
the F = 3/2, 5/2 hyperfine manifolds), but they do not repel each other head on through Stark shifts.
In contrast, the mF = 1/2 scheme works for ThF+ because the mF = 1/2 states in the J = 1 bumpinto each other through Stark shifts. In general, it seems like for the scheme proposed in Ref. [29], we would
require the hyperfine splitting to be large compared to the parity doublet splitting (e.g. Ω- or Λ-doubling for
diatomic molecules, Coriolis interactions for polyatomic molecules).

5.1.2.5

Sensitivity to the eEDM

At 18.6 V/cm, the molecule is neither fully polarized in the J = 1 nor the J = 2 manifold. Hence theeigenstates will not be expected to have |⟨Ω⟩| = 1, but it would be some number less than that. In fact, thestates involved for the 18.6 V/cm CP-FIT are:√

0.67 J = 1, Ω = 1, F =1
1, mF =
2
2


−

√

3 1
0.23 1, 1, ,
2 2


+

√

3 1
0.1 1, −1, ,
2 2



⇕
√

3 1
0.61 2, 1, ,
2 2


+

√

5 1
0.01 2, 1, ,
2 2


−

√

3 1
0.31 2, −1, ,
2 2


+

√

5 1
0.07 2, −1, ,
2 2


.

This means that the transition is between a state of ⟨Ω⟩ = 0.8 and ⟨Ω⟩ = 0.24, in contrast to the traditional
JILA method that probes the transition between a state of ⟨Ω⟩ = 1 and ⟨Ω⟩ = −1. Therefore, our statisticalsensitivity to the eEDM will be reduced to 28% if we wish to use the CP-FIT to give us first-order fieldinsensitive transitions.
Alas, it is not immediately clear if there is a sure win with the new proposal, given the various
complications that come with it. It also seems like it would require a lot of work to realize this new proposal
given where we currently are at. We can now go back humbled to performing an eEDM experiment with
ThF+ the JILA style.

105

5.2

Feasibility of using ThF+ in an eEDM measurement
Things look promising for using ThF+ in an eEDM experiment. We know how to create ions (Chapter2). We learnt how to perform quantum control, and we have also have a good protocol for performing a
Ramsey experiment (Chapter 4). We know how to perform state readout (Chapter 3). Now we can ask thequestion of whether we actually win by using ThF+ .
Since the first two generations of the JILA EDM experiment showed that HfF+ is a good molecule
to use, we shall discuss the feasibility of using ThF+ by comparing it to HfF+ . Table 5.1 shows a simple
comparison of the spectroscopic constants and other relevant parameters of ThF+ and HfF+ .
ThF+

HfF+

251
−2π × 20.1
2π × 5.29
2π × 1.70
0.014935 [22, 41]

199−2π × 62.2
2π × 0.740
2π × 1.80
0.003
23 [24, 25]

Parameters
Mass (u)A|| (MHz)ωef (MHz)dmf (MHz/(V/cm))|gF |Eeff (GV/cm)

Table 5.1: Comparison of parameters of ThF+ and HfF+ . Uncertainties are not included in this tablefor brevity.

From Table 5.1, we can draw a few inferences:(1) The combination of the smaller hyperfine splitting constant (A|| ) and bigger Ω-doubling constant(ωef ) in ThF+ than HfF+ has a few consequences (refer to Figure 5.4):(a) Since the Ω-doubling splitting constant is bigger in ThF+ , we expect ∆u/l to be bigger in ThF+
than HfF+ at similar strengths of Erot .
(b) The hyperfine states cross each other at a much smaller Erot in ThF+ (11.4 V/cm) than HfF+
(34.6 V/cm).
(2) Since the masses and molecular electric dipole moment are similar across the two species, we expect tobe applying electric field strengths to ThF+ that are similar to those used in the first two generationsof the experiment. Hence, we do not expect significant changes to the technology surrounding thecircuitry involved in generating electric fields.

106(3) ThF+ is much more susceptible to magnetic field noise than HfF+ because of the much larger gfactor. This could mean that we may need some sort of magnetic shielding in the third generation
experiment, e.g. the use of µ-metal.
(4) The effective electric field (Eeff ) in ThF+ is only about 1.5 times that of HfF+ . Given conservative
experimental parameters shown in Table 5.2, we see that swapping out HfF+ for ThF+ in the setupof the Gen. II experiment does not let us win in terms of the eEDM sensitivity.
Parameters

ThF+

HfF+ (Gen. II)τ (s)Nsinglefrep. (Hz)Ttot. (hours)

20
50†
0.05
100

2
500
0.5
100

δde (e cm)

4.4 × 10−30

6.7 × 10−30

Table 5.2: Naı̈ve expected statistical sensitivity for ThF+ . The value of δde for ThF+ is extrapolated
from that of HfF+ using Equation 1.1. The actual sensitivity achieved in the Gen. II experiment [35, 36]
is slightly better than the numbers shown here, but the numbers chosen here are easier numbers for easycomparison. † We have demonstrated having 5 dissociated Th+ ions on the side of our Ramsey fringe (see, forinstance, Figure 4.9a). This corresponds to a dissociation efficiency of only 2%, as compared to at least 20%
in the Gen. II experiment. The dissociation efficiency is currently limited by the choice of the intermediate
state in our REMPD scheme (see Section 3.4.2). We remain optimistic that a more suitable choice of theintermediate state for the REMPD scheme can bring our dissociation efficiency of ThF+ to be on par with
HfF+ in Gen. II.

At this point, it may seem that ThF+ is not that much more advantageous than HfF+ in terms ofeEDM statistical sensitivity despite the much longer coherence time in ThF+ . In fact, the long coherence
times limits the repetition rate of the experiment, since the experiment can only be repeated once every
20 s. To get around this, we can multiplex the experiment such that the repetition rate is not limited by thecoherence time that we are trying to probe. With this, we shall introduce the Bucket Brigade.

107

ThF + : X 3 1(v = 0, J = 1) , rot = 3 V/cm , rot = 0.8 mG[10, 11]

Energy (MHz)10[8, 9]

5
0[6][4, 5]

5
[0]

10

1.5

[2, 3][10, 11][8, 9][8, 9][4, 5][4, 5][2, 3]

0.5[10, 11][2, 3]

mF[10, 11]30[8, 9]

20[7][4, 5][2, 3]

0.5[0, 1]

Energy (MHz)15

HfF + : 3 1(v = 0, J = 1) , rot = 3 V/cm , rot = 0.8 mG

60

Energy (MHz)40[7, 8][7, 8][6, 9][4, 5][0]

1.5[2, 3][10, 11][6, 9][6, 9][4, 5][4, 5][2, 3]

0.5[10, 11][2, 3]

mF[10, 11][6, 9][7, 8][7, 8][4, 5][2, 3]
[1]

0.5

40

[6, 7][6, 7][4, 5][4, 5]

20
40
[0]

1.5

[2, 3][2, 3]

0.5[2, 3]

mF

0.5

1.5

mF

HfF + : 3 1(v = 0, J = 1) , rot = 34.6 V/cm , rot = 0.8 mG[10, 11]

[8]

[1]

1.5

[6, 7][4, 5][2, 3][10, 11][10, 11][10, 11][6, 7][6, 7][4, 5][4, 5][2, 3][2, 3][6, 7][9][4, 5][2, 3]
[1]

[0]

0.5

mF

0.5

1.5

60
[9]

[8]

20

[6, 7]

0[6, 7][6, 7][5][6, 7][4]

20
40[2, 3]

0.5

0.5

HfF + : 3 1(v = 0, J = 1)[10], rot = 58 [11]V/cm , rot = 0.8 mG[9][6, 7][6, 7][4, 5][4, 5]

1.5

1.5

40

0

[7]
[1]

20

[11]

[8]

[4, 5][4, 5][2, 3][2, 3]

0

1.5

20

60

20

ThF + : X 3 1(v = 0, J = 1) , rot = 60 V/cm , rot = 0.8 mG[10][4, 5][4, 5][2, 3][2, 3]

40
Energy (MHz)[10, 11]60

Energy (MHz)

Energy (MHz)ThF + : X 3 1(v = 0, J = 1) , rot = 11.4 V/cm , rot = 0.8 mG
20
15
10
5
0
5
10
15
20

[6]
[0]

0

20

1.5

[10, 11][10, 11][8, 9][8, 9]

10

10[0, 1][10, 11][10, 11][8, 9][8, 9]

60[2, 3][2, 3][2, 3][2, 3]
[1]

[0]

1.5

0.5

mF

0.5

1.5

Figure 5.4: Comparisons of energy levels between ThF+ and HfF+ . The left (right) column
corresponds to plots for ThF+ (HfF+ ). The first row consists of plots of the energy levels in the
X 3 ∆1 (v = 0, J = 1) manifold at low Erot to show the differences in the energy structure across both
species. Like in Figure 4.7, the basis states are labelled by their most dominant eigenstate in ascending orderof eigenenergies. For the unstretched Zeeman states, rotational coupling breaks the degeneracy between the
mF = 1/2 and mF = −1/2 states, and the energy levels are indicated by the black and green lines for thetwo superposition states. The second row shows the energy levels at hyperfine crossing. The third row showsthe energy levels at at the nominal electric field strength where we perform our Ramsey evolution. The
states are labeled with numbers from the least to the most energetic.

108

5.3

Bucket Brigade
The main idea behind the Bucket Brigade is multiplexing. The Bucket Brigade is a first-in-first-out

conveyor belt of ion traps, each containing its own ion cloud. The conveyor belt is continuously loaded fromthe left, all the traps move together from left to right, and the ion clouds are removed from the Bucket
Brigade on the right. A schematic diagram of the Bucket Brigade is shown in Figure 5.5.

Neon, 0.1% SF6
Bucket Brigade

1

State Detection

Turbo

State Preparation

Ablation

JILA eEDM Gen. III

Direction of travel

Turbo

State Preparation

2

State Detection
Insulation
4

MCP
3

5

Direction of travel
Conveyor belt of Paul traps (Bucket Brigade)
Turbo

Turbo

Figure 5.5: Schematic diagram of the Bucket Brigade. Our molecules go through five main vacuum
chambers for an eEDM measurement. (1) We create a plume of neutral ThF by ablating a piece of metallic
thorium, and reacting it with SF6 . The neutral plume of ThF is entrained in a supersonic expansion beam ofneon. (2) The molecular beam goes through a differential pumping chamber. (3) The neutral ThF molecules
are ionized with REMPI lasers and loaded into the first bucket in the Bucket Brigade. We would thenperform state preparation and the first π/2 pulse of the Ramsey sequence on the ThF+ ions. (4) Once thestate preparation is completed, the ions will be ushered along the Bucket Brigade into the cryogenic area,
where the ions perform Ramsey free evolution in a field-free region. While this is happening, steps (1-3)
are cycled uninterrupted, i.e. new batches of ions are being prepared in the first bucket and ushered into
the Bucket Brigade. (5) Once the ions exit the cryogenic area, we perform our state detection on the ions.
The inset on the top right shows how the ushering is done: time-varying potentials will be applied to thering electrodes (colors indicate magnitude of the potentials applied) to provide trapping forces along theaxial direction (potential wells) that move from left to right. Not shown in the inset are the rods that run
axially to provide radial confinement through RF-potentials. In the first two generations of the JILA eEDM
experiment, steps (3-5) happen at the same place, i.e. that of step (3). Note that the Ramsey evolution
happens concurrently for all the buckets, but they do not all begin and end at the same time.

109
With the multiplexing capabilities of the Bucket Brigade at a conservative repetition rate of about
2 Hz (limited by our state preparation sequence), the value of the eEDM sensitivity in Table 5.2 is thenupdated to that of Table 5.3.
Parameters

ThF+

HfF+ (Gen. II)

Eeff (GV/cm)
τ (s)Nsinglefrep. (Hz)Ttot. (hours)

35 [22, 41]
20
50
2
100

23 [24, 25]
2
500
0.5
100

δde (e cm)

7.0 × 10−31

6.7 × 10−30

Table 5.3: Expected statistical sensitivity for ThF+ in the Bucket Brigade. The value for δde isextrapolated from that of HfF+ using Equation 1.1.

At a repetition rate of 2 Hz and interrogation time of 20 s, the total number of buckets required to
run the multiplexed experiment concurrently is 2 Hz × 20 s = 40. For a nominal bucket length of 8 cm (withhindsight from Chapter 7), this means a total length for the Bucket Brigade at slightly more than 3 m, whichis still a reasonably sized table top experiment.
The value shown in Table 5.3 promises an order of magnitude improvement over Gen. II. As Prof.
Holger Müller often says, and I paraphrase, “sometimes when you measure something an order of magnitude
further, you can make an important discovery.” This order of magnitude improvement that the Bucket
Brigade and ThF+ promises us is precious. The next few chapters will be dedicated to the design of the
Bucket Brigade. To understand what levels of field homogeneity are involved in the design, we first begin
by studying the ion cloud dynamics to understand (i) the ion cloud sizes, and (ii) collision rates that could
lead to heating, decoherence from ion-ion collisions, or even suppressing decoherence due to spatial field
inhomogeneity.

Chapter 6

Ion Cloud & Decoherence

6.1

Introduction
We introduced the Bucket Brigade, a multiplexed ion trap, in the previous chapter. The Bucket

Brigade allows us to perform multiple experiments concurrently to take full advantage of the long coherencetime of ThF+ . However, our critics are not convinced. They ask, “this setup is huge, how are you going
to control the fields within such a huge volume?” This chapter is our reply to them, “you are rightfully
concerned, and so are we, but allow us to show you some equations and plots to convince you that the Bucket
Brigade is not as crazy as it sounds.”
Many of our critics come from the quantum information sector. They need to ensure that theirqubits, each with a highly localized position, do not decohere with each other. Such decoherence could comefrom spatial inhomogeneity of fields, such that each qubit precesses at a slightly different frequency. People
employ tricks like spin echo [64] or dynamic decoupling [65] to deal with the aforementioned decoherence
mechanism. However, there are also temporal variations of fields that could come from inadequate controlof the generators of the field, or imperfect shielding of fields from time-varying ambient sources. It is not
immediately obvious how one would deal with such temporal sources of decoherence. Our fellow ion trappers
in the quantum information sector are working with ion traps on a chip, and they find themselves working
very hard to make the fields well behaved in a microscopic volume. They are rightfully concerned that we
intend to perform precision metrology with ions sampling fields in a volume with a size of two small adults.
To tackle the concerns, we need to understand fast how our ions decohere w.r.t. each other in an

111
environment with inhomogeneous fields. This will allow us to place limits on design constraints of the
Bucket Brigade. Hence, we need to understand a few things: (i) what is a typical spatial extent of our ion
cloud, (ii) how does the decoherence manifest in an ensemble of ions, (iii) what role does ion-ion collisions
play in the context of decoherence, and (iv) what other relevant mechanisms could affect us in the greatercontext of an eEDM experiment using molecular ions? We shall start with simplified models, and tackle
these questions with increasing order of complexity in our models with each proceeding section.
Despite the high level of detail and complexity of the content in this chapter, we will not be arriving
at grand conclusions pointing us to definite directions for the design of the Bucket Brigade. Instead, we
will be confirming many small pieces of intuition. The work in this chapter was mainly done at the start ofthe pandemic, which incidentally coincided with the transition from spectroscopy to Bucket Brigade design.
Notwithstanding the lack of a grand conclusion, we still include work done on ion cloud dynamics and its
consequence to decoherence here in this thesis, in a hope that future generations of students and post-docs
can build on this work to push our understanding/control of decoherence in our ion cloud even further. This
chapter proceeds as follows.
We first investigate what an ion cloud at dynamic equilibrium looks like when we put it in a trap,
and also allow for there to be ion-ion Coulomb repulsion. We shall look at the mean field effects of ionion repulsion, and when they turn on to understand the regime that we will be in with our typical trap
parameters (Section 6.2.1). We then proceed to simulate the dynamics of an ion cloud to understand thespatial extent of our ion cloud at dynamic equilibrium (Section 6.2.2).
At this point, we switch gears a little to look at how decoherence is introduced into our ensemble ofions. We first formally introduce the naı̈ve decoherence rate, which is the decoherence rate in our ensemble ofions that is naı̈vely based only on the spatial inhomogeneity of our fields (Section 6.3.1). We then introduce
a possible mechanism built into our trap that could suppress the naı̈ve decoherence (collisional narrowing
in Section 6.3.2). Subsequently we combine the two effects together to come up with a model for the net
decoherence rate (Section 6.3.2).
We then proceed to combine experiment results (Section 6.4.1) with the aforementioned tools (Section
6.4.2) to help us build confidence in our model. Subsequently, we attempt to arrive at a closed analytic form

112
for the ion-ion collision rate (Section 6.5.1) and relate it to the collisional narrowing rate (Sections 6.5.2 and6.5.3). We then briefly discuss heating of ion clouds in Paul traps (Section 6.5.4) before discussing all theabove results in the context of the Bucket Brigade.
Finally, we close off this chapter with a brief discussion of other possible mechanisms of decoherence
not included in the above model (Section 6.6.1), and a quick disclaimer on how understanding decoherence
mechanisms in the experiment is highly non-trivial (Section 6.6.3).

113

6.2

Ion cloud dynamics
We use a Paul trap to confine our ions. Axial confinement is provided by DC voltages on end caps,

while radial confinement is provided by RF voltages on radial electrodes. Our ions exhibit harmonic secularmotion in the trap from the harmonic axial trapping potential, and the harmonic radial ponderomotive
trapping potential. In addition, our ions also exhibit fast RF micromotion driven by the RF potential that
increases in amplitude with radial distance from the trap axis. We shall simplify the analysis of our ion
dynamics by averaging over the fast RF micromotion of the ions in our Paul trap, and model the trap as a
3D harmonic trap with angular trap frequencies {ωx , ωy , ωz }. The two equations that govern the size of theion cloud at thermal equilibrium across all three spatial axes are given by Gauss’s law and the Boltzmann
distribution:
e
n(r),
ε0



N
1
n(r) =
exp −β eΦ0 (r) + m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 ) ,V2∇2 Φ0 (r) = −where V =RRR

(6.1a)(6.1b)


exp −β eΦ0 (r) + 21 m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 ) dV is a normalization constant. The quantityn(r) is the ion number density of the cloud. The kinetic energies of the ions are taken care of by the
β := 1/(kB T ) term.
It turns out that solving the above equations for arbitrary N , ωx , ωy , and ωz is not easy. It also turns
out that simulating the motion of all the ions in the trap can be computationally expensive. In the following
sections, we shall employ the following strategy. First, we shall study Equations 6.1 with a spherical ion
cloud to simplify the equations through the symmetry of the problem (Section 6.2.1). This will give us some
intuitive understanding of the effects of mean field (i.e. when ion-ion repulsion becomes significant) and whenit comes into play. We will then dive into simulations of ion motion in the ion cloud with scaled parameters
to make it computationally less expensive, but nonetheless still giving us the physics that we want (Section
6.2.2).

114
6.2.1

Studying mean field effects with spherical ion clouds

6.2.1.1

MethodFor a trap with spherical symmetry, Equations 6.1 reduce to a non-linear second order ODE:



er2
1
2 2
∂r r ∂r Φ0 (r) = −exp −β eΦ0 (r) + mω r
,
ε0
2
2

which can be solved numerically with the shooting method (a method to solve a boundary value problem byreducing it into an initial value problem). We massage the second order ODE into a first order ODE of thefollowing form1 : Ψ0  
=
∂r 

 
Φ0

2
− er
ε0 exp




1
2 2
−β eΦ0 + 2 mω r

,

2
Ψ0 /r

where Ψ0 = r2 ∂r Φ0 (r) is a variable defined for convenience. The shooting method involves solving the ODE
by starting with initial values on the R.H.S. (i.e. values of p0 := βeΦ0 when r = 0) and “shooting” theODE to non-zero r. We impose ∂r Φ0 (r = 0) = 0, which is equivalent to saying that we are coarse graining
individual ions away in the ion density.
Of note, depending on the choice of the initial condition p0 , the solution could correspond to very
different total ion numbers N . As the total number of ions N increases, mean field effects start to play a hugerole, where the inter-ion Coulomb repulsion suppresses the increase in the ion number density at the centerof the ion cloud. Hence, the total ion number of a solution can be very sensitive to the initial conditions
given to the numerical solver, as can be seen from Figure 6.1. To solve for Φ0 (r) for a specific N , one would
guess a p0 , solve for Φ0 (r), deduce N , make a better guess for p0 , and repeat until N is close enough to thedesired value.

6.2.1.2

Results

From Figure 6.2, we can see that the mean field is strong enough to modify the harmonic potential
near the center to give a “flat-top” ion density sliced profile in ion clouds.
Figure 6.3 show the sizes of the ion clouds at various trap frequencies, from which we can draw a few
conclusions:
1 1D numerical solvers typically work with the form y ′ (x) = f (y, x).

115

T=3K
T=6K
T = 10 K
T = 20 K

Total number of ions

104
103
102
101
100

24

22

20
p0

18

16

e 0(r)

0
10
20
30

1000
2000
3000
4000
5000
6000
7000
8000
9000
10000

0.0 2.5 5.0 7.5 10.0 12.5 15.0
r (mm)

Mean potential energy (K)Mean field energy (K)Figure 6.1: Dependence of total ion number N on initial value p0 = βeΦ0 used in the shooting
method. The plot shows that N is very sensitive to p0 at high N in the strong mean field regime (low T ).
The plots showed here are for ThF+ in a spherical trap of 1.6 kHz trap frequency.

e 0(r) + 12 m 2r2
100
50
0

1000
2000
3000
4000
5000
6000
7000
8000
9000
10000

0.0 2.5 5.0 7.5 10.0 12.5 15.0
r (mm)

Figure 6.2: Mean field energy eΦ0 (r) and mean potential energy as calculated, converted tokelvins. We used ω = 2π × 1 kHz. The legend indicates total ion numbers. Note the for the plot on theright, having high ion numbers makes the potential energy curve flat at small r, before reverting back to aquadratic curve at larger r.

(1) The stronger the mean field effects, the more the ion density profile deviates from a Gaussian.
(2) The effect of ion-ion interaction decreases with increasing trap frequencies.
(3) The ion number density at r = 0 approaches a maximum density asymptotically as we ramp up
mean field effects.
The maximum ion number density for a given trap can be evaluated at zero temperature, by demanding

2.5

5.0

0.0

7.5 10.0 12.5 15.0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000

5
0

0.0

2.5

5.0

5.0
2.5
0.0

0.0

2.5

5.0

2
0

0.0

2.5

5.0

7.5 10.0 12.5 15.00n(r) (mm 3)n(r) (mm 3)

7.5
5.0
2.5
0.0

0

0.2
0.0

2.5

5.0

N=6000

15

3K
6K
10 K
20 K

5
0

0

10
5
0

0

10
r (mm)

0.4
0.2
0.0

2.5

5.0

7.5 10.0 12.5 15.0
r (mm)

2
0

0

3K
6K
10 K
20 K

10
r (mm)2r (mm)

3

2

3

4

3

4

3

4

3

4

6K

0.4
0.2
0

1

2

10K

0.8
0.6
0.4
0.2
0.0

4

0

1

2

20K

1.0
0.8

4

0.6
0.4
0.2
0.0

0

1

2
r (mm)(b) Trap frequency of ω = 2π × 2 kHz

3K
6K
10 K
20 K

10
5
0

10

5

1

1

1.0

1000
2000
3000
4000
5000
6000
7000
8000
9000
10000

4

0

0.6

0.0

4

3

6

N=3000N=7000

0

2

0.2

0.8

20K

0.6

10
0

1

8

5

15

3

10

7.5 10.0 12.5 15.0

3K
6K
10 K
20 K

0

2

0.4

1.0

1000
2000
3000
4000
5000
6000
7000
8000
9000
10000

20K

10

0

10

1

0.6

0.0

4
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000

15

N=2000
3K
6K
10 K
20 K

3

6K

(a) Trap frequency of ω = 2π × 1 kHz

N=1000

2

10K

0.8

7.5 10.0 12.5 15.0
r (mm)

1

10K

0.4

0.0

25
20
15
10
5
0

0

0.8
n(r) (norm.)n(r) (mm 3)
5.0

1.0n(r) (norm.)n(r) (mm 3)

4

2.5

0.6

0.0

20K
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000

0.2

0.8

7.5 10.0 12.5 15.0

6

7.5 10.0 12.5 15.0

0.4

0.0

10
0

6K

1.0
n(r) (norm.)n(r) (mm 3)

7.5

5.0

0.6

0.0

10K
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000

2.5

0.8

7.5 10.0 12.5 15.0

10.0

0.0

1.0
n(r) (norm.)n(r) (mm 3)

6K
10

0.2

20

n(r) (norm.)

0.0

0.4

30

116

3K

1.0

1000
2000
3000
4000
5000
6000
7000
8000
9000
10000

n(r) (norm.)

0

0.6n(r) (mm 3)50.8n(r) (mm 3)

103Kn(r) (mm 3)n(r) (mm 3)

15

3K

1.0
n(r) (norm.)

1000
2000
3000
4000
5000
6000
7000
8000
9000
10000

n(r) (norm.)

3K

15
10
5
0

0

N=8000
3K
6K
10 K
20 K

0

10
r (mm)

3K
6K
10 K
20 K

10
5
0

10

N=4000

15

15
10
5
0

0

N=9000
3K
6K
10 K
20 K

0

10
r (mm)

3K
6K
10 K
20 K

10
5
0

10

N=5000

15

15
10
5
0

0

10

N=10000
3K
6K
10 K
20 K

0

10
r (mm)(c) Same data as trap frequency of ω = 2π × 1 kHz, rearranged to show thermal effects.

Figure 6.3: Ion number radial density vs Total ion number. The odd columns in (a) and (b) are theactual values for the ion number radial density n(r), and the even columns are normalized to n(r = 0) = 1
to tease out the effects of the ion-ion interactions. The last two rows are just a rearrangement of the 1 kHz
plots to show thermal evolution.

117
that the inter-ion Coulomb repulsion forces balance the trapping force:
FCoulomb + Ftrap = 0 =⇒ ∇ · [FCoulomb (r) + Ftrap (r)] = 01−e∇2 Φ(r) = ∇2 m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 )2eρmax (r)/ϵ0 = m(ωx2 + ωy2 + ωz2 )
nmax (r) =mϵ0 2(ωx + ωy2 + ωz2 ).
e2

(6.2)

The above expression evaluates to 17 mm−3 (68 mm−3 ) for a trap frequency of 1 kHz (2 kHz). We cansee from Figure 6.3 that the number density at r = 0 approaches 17 mm−3 asymptotically as the total ion
number increases for the {3 K, 1 kHz trap frequency} case.
Figure 6.4 shows how the ion number density at the center of the ion cloud varies with temperature
for a fixed total ion number. Note that the number density at the center of the trap changes as a weak
function of temperature at low temperatures.

N = 15000, f = 1.6 kHz
Numerical solution
Density-weighted density
nmax

n(r = 0) (mm 3)

40
30
20
10
10

20
30
40
Temperature (K)50

60

Figure 6.4: Number density at ion cloud center vs temperature. Variation of n(r = 0) with T for N= 15000 at f = 1.6 kHz. The expression for the maximum total ion number permissible by an ion trap nmax
is given in Equation 6.2. Jumping ahead to Section 6.5.1, a plot of the density-weighted density is shown
here for comparison. Unlike the density at the center of the ion trap, the density-weighted density is a muchstronger function of temperature at low temperatures.

118
6.2.1.3

Summary

To summarize, we simplified the geometry of the ion cloud to cast the governing equations into a form
that we can solve numerically. With this numerical method, we explored mean field effects coming from
ion-ion repulsion by varying the total ion number and temperature of the ion cloud. We saw that mean field
effects manifest at low ion cloud temperatures and high ion numbers. We also saw that as mean field effects
becoming more and more dominant, two things happen: (i) the net trapping potential curve becomes flatterat the center of the ion cloud, and (ii) the ion number density at the center of the ion cloud saturates to avalue that depends on the trapping frequencies.
The results in this sections are not surprising. This simple exercise with a spherical ion cloud serves
as a warm up to build some intuition. We are now ready to study the dynamics of the more general case ofan aspherical ion cloud.

119
6.2.2

Simulating ion dynamics with aspherical ion clouds

6.2.2.1

Statement of problem

We can combine the Equations 6.1 to obtain defining PDE for a non-spherical cloud. The PDE canbe written in two equivalent forms, expressed either as an equation for electric potential Φ0 (r) or ion number
density n(r):



e N
1
exp −β eΦ0 (r) + m(ωx2 x2 + ωy2 y 2 + ωz2 z 2 ) ,ε0 V2∇2 [log n(r)] + mβ ωx2 + ωy2 + ωz2 = 4πκβ · n(r),∇2 Φ0 (r) = −(6.3a)(6.3b)

2e
where κ := 4πε. Note that N seems to be missing from Equation 6.3b, but it is hidden in the normalization0

criterion:Rn(r) dr = N . The above equations are hard to solve in either form with simple methods because

the equations are:(1) Non-linear partial differential equations (PDEs). The usual trick to tackle PDEs would be to use
separation of variables (i.e. Φ0 (r), n(r) = X(x)Y (y)Z(z), but the Φ0 (r) term in the exponent for the
first equation, and the log n(r) term in the second equation make the equation non-linear and void
this trick.
(2) Not in the small variable expansion regime. Both Φ0 (r) and n(r) are not small and cannot be
expanded out with a series expansion to linearize the PDEs to solve it with the usual iterative tricks.
As a reminder, we wish to study the ion dynamics and use the results to estimate (i) the ion cloudsizes, (ii) collision rates between ions, and (iii) the resulting effects on decoherence. To this end, we should
simulate the motion of ions in a trap, and come up with a model based on the simulation results to explain
the dynamics of the ion cloud. We can then compare the model predictions with the ions in the Gen. II
setup.2 However, the typical ion number in the Gen. II setup is N = 15000, which is expensive to simulate
because the number of ion-ion interaction terms scales as N 2 ; it took about 75 mins to simulate the dynamics
of N = 1000 ions for 50 ms evolution time with trapping frequencies ∼ 1 kHz, so simulating 15000 ions is
simply not practical. Hence, we need to do some scaling of parameters to mimic the effect of having large
ion numbers with only a small number of ions.
2 At the time of this work, the only fully functioning ion trap in the JILA eEDM group is the Gen. II setup.

120
6.2.2.2

Scaling of parameters for realistic computation time

We want to study the ion cloud dynamics in the context of decoherence through, say, ion-ion collision.
As alluded to in the previous section, we need to do some scaling of parameters. Firstly, we wish to maintainthe ion density distribution, so that we can use the results to evaluate the effects of field inhomogeneity
within the spatial extent of the ion cloud in the trap. Secondly, we want to keep the same the collision
rate as the unscaled case so that we can study collision-induced decoherence effects. Therefore, we need to
combine the equations governing the two pieces of physics, and scale them in a consistent way to keep thephysics that we want to study invariant.
The equation governing collision rate is:"#2[κβ(T )]
1
Γcol. (T ) ∼ n(T ) · pln
3 ,mβ(T )n(T ) [κβ(T )](6.4)

We also made explicit that β(T ) is a function of temperature T . The above form for collision rate is informed2eby work in Section 6.5.1. In the above, κ := 4πεis a measure of the Coulomb interaction strength. We see0

that mβ and κβ have been grouped together suggestively to show that there are only two free parameters
available for us to tweak.
We wish to reduce the number of ions in the simulation, but we want to keep the ion cloud size thesame. We can scale n → n/λ and boost the Coulomb interaction strength up with κβ → λ · κβ to keep the
R.H.S. of Equation 6.3b invariant. The newly introduced λ term on the L.H.S. drops out under the laplacian.
This scaling is equivalent to using λ times less ions for the simulation, but with√

λ times the original charge

on each ion. However the aforementioned transformation will change Equation 6.4. One possible workaroundwould be to promote mβ into a field, but it is not obvious that a solution for such a field exists that can keep
both the above equations invariant. Therefore, we employ a slightly modified battle strategy as follows:(1) Simulate the effect of N singly charged ions with N/λ ions to reduce cost of computation. Each ion
comes with a scaled up charge√

λ e to mimic the ion cloud size of the former (Section 6.2.2.3);(2) Determine the field inhomogeneity sampled by the ions with the cloud sizes obtained (Section 6.3.1);(3) Verify that there is good agreement between the simulated ion dynamics of the N/λ ions with the121scaled version of Equation 6.4 (Sections 6.5.1 and 6.5.2);(4) Scale the collision model back down to that of N singly charged ions through Equation 6.4 andcompare the predictions with experiment results from the Gen. II setup (Sections 6.4.1, 6.5.1, and6.5.2).
(5) If the collision model works both in the unscaled version (matched with experiment data) and scaled
version (matched with simulations), use it for the design of the Bucket Brigade (Section 6.5.5).

6.2.2.3

Simulation package & sample of simulation results

We used the LAMMPS simulation package from Sandia National Laboratory [66] to simulate thedynamics of 1000 ions. The cut-off range for pair-wise Coulomb interaction is set to 10 cm, which is larger
than any anticipated size of clouds, to simulate the full effects of mean field. A sample of the simulation
results is shown in Figure 6.5.
The final temperature of the ion cloud can be extracted from a Maxwell-Boltzmann distribution fit to
the kinetic energies of the ions. Deviation from a Maxwell-Boltzmann distribution hints to us that the ion
cloud may not have reached thermal equilibrium, and the dynamics will need to be simulated for longer.
In most of the discussions that follow, we will be looking at the simulation results of N = 1000 ionswith charge√

15e in a trap with frequencies {2.0, 2.3, 1.5} kHz and cloud temperature 4.3 K. This simulation

mimics the ion cloud size of N = 15000 ions, so we can use the results to infer the amount of sampled spatial
field inhomogeneity as a typical ion cloud in the Gen. II experiment (the Gen. II trap typically has about
15000 ions3 ).
Before we dive into the results of the simulations, we shall now take a look at how spatial Ramsey
frequency inhomogeneity can affect decoherence of the fringes to understand what we need to look out forin the simulations.

3 When this joint project started, the Gen. II trap had about 15000 ions, but we managed to push it up to an impressive
40000 ions as the experiment went on.

RMS radius (mm)122

Ensemble, (xRMS, yRMS, zRMS) : (2.55, 1.7, 3.75) mm(1.51 : 1 : 2.21) vs (1/fx, 1/fy, 1/fz) : (1.31, 1, 1.7)

x2
y2
z2

0.4
0.2
0

10

20

30

40

50

Ensemble, U/K (x,y,z,sum) : (5.76, 4.24, 7.19, 5.72)
Ux/Kx
Uy/Ky
Uz/Kz
U/K

U/K

40
20

Kinetic energy (J)Mean Energy (J)

0
6
4
2
0

0

10

20

1

40

50

30

40

50

30

40

50

Ensemble

1e 22

Total
Coul.E.
K.E.
Harm.E.

0

1020

Ensemble

1e 23
2

30

Kx
Ky
Kz

0

10

20

Time (ms)Figure 6.5: Sample of√ results from a LAMMPS simulation. This is a simulation of N = 1000
ions, each with charge 15e to simulate the effects of 15000 ions in a harmonic trap with trap frequencies{1.3, 1.7, 1.0} kHz, with a starting temperature of 0.5 K. A few points to note:(1) The initial ringing of the RMS sizes in the top most plot shows breathing of the ion cloud as theions were not initialized with positions and velocities that match the steady state solution. Ion cloudbreathing damped out by 20 ms in this simulation instance. The final aspect ratio of the ion cloudat {1.51 : 1 : 2.21} is bigger than what we would expect if there were no mean field effects, i.e.
aspect ratio given by only the trap frequencies at {1.31 : 1 : 1.7}.
(2) The second row plots the ratio of the potential energy U and the kinetic energy K. The componentsof the energies are defined as such: Ui := 21 mωi2 x2i and Ki := 12 mvi2 , where m, ωi , xi , and vi are
the ion mass, angular trap frequency along the i-axis, position along the i-axis, and velocity along
the i-axis, respectively. For a harmonic trap, virial theorem says that U/K = 1 (grey dashed line),
but the ratios we have are clearly greater than 1, indicating that significant mean field effects aremaking our ion cloud bigger than what its temperature allows for.
(3) The third plot is a sanity check to show that total energy is conserved, and this shows that thesize of the time step chosen is appropriate. The pair-wise Coulomb energy is much higher than theions’ kinetic and trap potential energies, which shows, once again, that mean field effects are notnegligible.
(4) The last plot showed that the ion cloud is in thermal equilibrium across all three spatial dimensions,because all the kinetic energies along the three spatial axes converge together. Depending on thesimulation parameters chosen, e.g. small number of ions or high ion cloud temperature, the ion cloud
may not reach thermal equilibrium even after 50 ms of evolution time, so the simulation has to beextended until thermal equilibrium is reached as inferred from the convergence of kinetic energies
along the three spatial axes.
In all the simulations, we allowed for the ion cloud to settle down before extracting (i) the sizes of the cloud,(ii) the ion number density profiles, and (iii) ion trajectories.

123

6.3

Decoherence from spatial inhomogeneity
In the previous sections, we introduced the tools that we will use to simulate the ion dynamics. We

shall now switch gears a little to introduce the other set of tools required to understand decoherence in ourion cloud: a model for decoherence. We shall begin this section with a theoretical treatment of decoherenceof Ramsey fringes of ions in an ion trap. We will add more realism with each proceeding section. We will
finish off by comparing experiment results with our simulation model mentioned in the previous sections.

6.3.1

Frequency dependent decoherence rate (naı̈ve decoherence rate), γlin f
Suppose the ions evolve with Ramsey frequencies that do not depend on time, but fall under anormalized distribution P(ω). The spread in frequencies could come from the spatial inhomogeneity of, say,the magnetic fields that each ion samples at its location. In the simplest scenario, this could happen if ourions are very cold, and well localized, but we allow the ion cloud to have some spatial extent, like in a stringof ions [67], or in a Coulomb crystal [68]. Our setup does not allow us to probe each individual ions fortheir Ramsey phases; we are only able to read out the ensemble-averaged Ramsey phases. The resulting
ensemble-averaged Ramsey fringe is given by:ZZcos(ωt) P(ω) dω = Re

e

iωt


P(ω) dω= Re {F [P(ω)]} ,(6.5)

where F indicates Fourier transform.
We can represent the complex function F [P(ω)] (t) as an Argand vector rotating on the Argand plane(complex plane) with time. Its projection onto the real axis at various times represents the ensemble-averaged
Ramsey phase at the corresponding time. The envelope of the ensemble-averaged Ramsey fringe is given by
the amplitude of the Argand vector:Zeiωt P(ω) dω = |F [P(ω)] (t)| .

(6.6)
ω−ω0

2

1
As an example, for a Gaussian frequency distribution where P(ω) = √2πσe− 2 ( σω ) , the expressions
1

ω

124
in Equations 6.5 and 6.6 evaluate to
1

2

1

2

Re {F [P(ω)]} = e− 2 (σω t) cos ω0 t,(6.7a)|F [P(ω)]| = e− 2 (σω t) ,(6.7b)

where ω0 is the mean angular frequency and σω is the standard deviation of the angular frequency distribu-

Norm. Pop. Diff.

tion. Figure 6.6 shows a plot of the above two equations.

1.0
0.5
0.0
0.5
1.0

ActualSingle frequency
Envelope

0.0

0.1

0.2

0.3
Time (s)

0.4

0.5

0.6

Figure 6.6: Decoherence of an ensemble-averaged Ramsey fringe for a Gaussian frequency distribution. The blue solid line and the green dashed lines are plots of Equations 6.7a and 6.7b, respectively,for ω0 = 2π × 50 Hz and σω = 2π × 1 Hz. The beige dotted line shows what the fringe would look like ifσω = 0, i.e. no decoherence.

For any arbitrary P(ω), its Fourier transform will be a function of σω t. Often times the spread in
Ramsey frequencies seen by the ions comes from spatial field inhomogeneity in the trap, e.g. imperfectcurrent-carrying coils that give inhomogeneous magnetic fields gradients for Zeeman splitting, or imperfectelectrodes that give inhomogeneous electric fields that define the quantization axis. Such field inhomogeneity
gives rise to a spread in Ramsey frequencies that depends on the mean frequency. Hence, we express σω ina more illuminating form:σω = 2πσf = 2πf σf /f ,where f is the mean Ramsey frequency of the ensemble, and σf /f is the spread of the fractional frequency
distribution P(f /f ) with a mean at f /f = 1.
We define a half-time t1/2 that is the time at which the decoherence envelope drops to half, i.e.

125
F [P(ω)] (σω t1/2 ) = 1/2, such that:−1

F [P(ω)](1/2) = 2πf σf /f t1/2

1/t1/2 =2πσf /f−1

F [P(ω)](1/2)

f= γlin f ,(6.8)

where γlin captures all the details of the spatial field inhomogeneity. For decoherence effects due to spatial
field inhomogeneity, we expect the rate of decoherence to be proportional to the mean Ramsey frequency.
Note that the above only captures decoherence from spatial field inhomogeneity from fields generated bycoils and electrodes. Equation 6.8 does not capture decoherence effects due to spatial field inhomogeneityof stray fields from, say, magnetic patches of the optical table around the setup. Decoherence from such
sources are grouped under a background decoherence rate to be introduced in Section 6.3.3.
We note that γlin f is the naı̈ve decoherence rate, which would be the actual decoherence rate (assuming
spatial field inhomogeneity to be the only source of decoherence) if the ions are held in place, i.e. stationary.
If this were the only decoherence related mechanism in our trap, we would expect the Ramsey phases of
our ions to spread out ballistically, i.e. ∼ t, because the spread of the phase goes as σϕ = 2πf σf /f t ∝ t.
However, we shall see in the next section that there can be a suppression of the ballistic expansion of σϕ in
our ion trap.

6.3.2

Suppressed decoherence from collisional narrowing, Γc.n.
Our thermal ions move around the trap, and they collide with each other to send each other into

different trajectories around the trap. Associated with this is a rate at which the ions “reset” their trajectories, resulting in a corresponding “reset” of their Ramsey frequencies to another random frequency in P(ω).
Correspondingly, the spread in the ensemble of Ramsey phases is sub-ballistic, and is more of a classicalrandom walk.
Using the ion simulation tools from Section 6.2.2 in combination with known field inhomogeneity in
the Gen. II trap, we see from the simulation results that the spread of the Ramsey phases of ions in the ion
cloud ensemble evolves diffusively, i.e. σϕ ∼√t. This result is shown in Figure 6.7.

Stdev. of phase spread (rad)126

N = 1000, e2

15e2, {2.0,2.3,1.5} kHz, f = 3 kHz, T = 4.3 K

Standard Deviation
Fit to at b: b = 0.558

1.5
1.0
0.5
0.0

0

10

20
30
Time (ms)40

50

Figure 6.7: Simulated result of the spread of√ Ramsey phases in an ensemble of ions. The ion
cloud has N = 1000 ions, each with a charge of 15e. The trap has frequencies {2.0, 2.3, 1.5} kHz, Theion cloud temperature is T = 4.3 K. The ions have a mean Ramsey frequency of 3 kHz. We introduced arealistic spatial dependence of the local Ramsey frequency based on known field inhomogeneity in the Gen. II
trap. For book keeping, the Ramsey frequency is dominated by Berry’s phase (instead of the usual Zeeman
offset). The Berry’s phase is an axial electric field effect. The electric field inhomogeneity is derived from a
COMSOL model of the fields generated by the electrodes within the trap. The fit parameter b = 0.558(9) ismuch closer to 0.5 (diffusive) than to 1 (ballistic) and show that the phases indeed spread out diffusively.

To quantify the suppressed decoherence, we introduce the following model. The total phase accumulated by any ion is given by:
Z tϕ(t) =

ω(t′ ) dt′ =: ω(t) t,

0

where we define the net Ramsey (angular) frequency ω(t) given by the expression above. If we demand thatthe spread in ϕ(t) goes as ∼√√t, then the spread in ω(t) must go as ∼ 1/ t. We shall guess it to be of theform:σω (t) := √σω,1 + 2Γc.n. t(6.9)

where Γc.n. is a fitting parameter that we introduce for “collisional narrowing” (narrowing of decoherence
because of momentum-changing ion-ion collisions), and σω is the naı̈ve spread in frequencies without collisional narrowing. Γc.n. has units of inverse-time in this ansatz, and it is related to the ion-ion collision ratein the trap.4
At this point, it is important to address what kind of decoherence can be suppressed by ion-ion
collisions. The ions are sloshing around within the ion trap. There are a few key points:
4 Ions have collision cross section area of infinity because of the 1/r 2 scaling of the field strengths, but here we only refer tocollisions that change the ions’ trajectories significantly.

127(1) In the absence of ion-ion collision, each ion has a well defined total energy along each principal axesof the ion trap.
(2) The ion spends equal amounts of time on either side of the center of the trap along each of theprincipal axes of the ion trap.
(3) Any spatial inhomogeneity of the fields can be expanded out about the trap center as a sum of evenand odd functions; see Figure 6.8.

=+

Figure 6.8: Decomposing spatial inhomogeneity into odd and even functions about trap center.
The horizontal axis indicates position in the ion trap along a principal axis. The red line indicates the local
Ramsey frequency sampled by the ions at that location. The red line on the left is decomposed into a sumof odd and even functions about the trap center. The blue line shows the harmonic potential of the ion trap.
The dashed vertical line indicates the extrema of the sloshing motion of an ion of a given total energy. The
effect of the odd component averages out to zero in one slosh cycle. The effect of the even component does
not average out to zero, and the net value depends on the total energy of the ion.

(4) Spatial inhomogeneity of the fields that goes as an odd function about the trap center is averagedout by the ion as it sloshes around the ion trap. This is true regardless of the total energy of theion.
(5) Spatial inhomogeneity of the fields that goes as an even function about the trap center is not averagedout by the ion as it sloshes around the ion trap. The net accumulated effect as seen by the ion also
depends on its total energy along the relevant axes.
(6) In the absence of ion-ion collisions, then, ions with higher total energies along a particular axis will
sample net frequencies that will be different from those with lower total energies.
(7) In the presence of ion-ion collisions, there can be energy and momentum exchange between the ions,sending the ions into a different trajectory with each collision, allowing the ions to sample the entire

128
spatial extent of the ion cloud “ergodically”. This reduces the net spread of net frequencies seen by
the ensemble of ions.
In the next section, we shall attempt to put γlin and Γc.n. together to come up with a model to describe
the net decoherence rate in our ion cloud.

6.3.3

Net decoherence rate, Γ1/2
Here we attempt to derive an expression for the net decoherence rate as a result of the spatial field

inhomogeneity (γlin f ) and suppression of the effects of such spatial inhomogeneity from ion-ion collisions(Γc.n. ). Some of the parts may seem ad hoc, but we shall see in Section 6.4.1 that this model describes what
we see in the Gen. II trap very well.
Suppose we have a Gaussian distribution5 of frequencies, we can combine Equations 6.7b and 6.9 toshow that the contrast of the ensemble-averaged of Ramsey fringe has the form:
1 (σ t)2
CG (t) = exp −.
2 1 + 2Γc.n. t
Here we make a few observations:(1) At short times, t ≪ 1/(2Γc.n. ), the function behaves like a Gaussian, hence we expect it to start off
with zero gradient, instead of an exponential decay.
(2) At long times, t ≫ 1/(2Γc.n. ), the function behaves like an exponentially decaying function.
(3) The time it takes to get to half contrast is:

t1/2 =

Γc.n. +p

Γ2c.n. + σ 2 /(2 log 2)
σ 2 /(2 log 2)

Together with Equation 6.8, we get:t1/2 =

Γc.n. +q
Γ2c.n. + (γlin f )2
(γlin f )2.

5 Invoking the central limit theorem on the field inhomogeneity sampled by our ions, barring any exotic forms of spatial
inhomogeneity that break this limit.

129
From the above, we can define the net decoherence rate6 as the inverse of t1/2 , which gives:
Γ1/2 :=

1
t1/2=(γ f )2q lin+ Γbg ,
Γc.n. + Γ2c.n. + (γlin f )2

(6.10a)where we added a background decoherence rate Γbg by hand to account for decoherence that come fromother sources of spatial inhomogeneity such as those discussed in Section 6.6.1. In the limit of f ≫ Γc.n. /γlin ,which is equivalent to saying that collisional narrowing is weak compared to the naı̈ve decoherence, the above
evaluates to:


1

Γc.n.
Γ1/2 ≈+
γlin f(γlin f )2

−1
+ Γbg .

(6.10b)

Although the above expressions are evaluated for ions with Gaussian frequency distributions, we shall use
see in the following section that the above form fits to our data surprisingly well.

6.4

Collisional narrowing in the ion trap
Now that we have a model for various rates that go into decoherence, we would like to test it against

experiments and simulations. The naı̈ve decoherence and background decoherence rates are easy to understand, these are just decoherence rates that either depend or do not depend on the Ramsey frequency.
The collisional narrowing mechanism is a little unexpected, so we shall first check for its existence withexperiments (Section 6.4.1) and simulations (Section 6.4.2).

6.4.1

Collisional narrowing in experiments
We worked with the Gen. II team7 to measure the effects of spatial field inhomogeneity and collisional

narrowing. The Gen. II trap is optimized to have very good spatial magnetic field homogeneity for theireEDM measurement. However, the axial electric fields are not optimized for spatial homogeneity, whichmeans that there will be a spatially inhomogeneous Berry’s phase (Chapter 8.3). The net frequency is asum of very spatially homogeneous Zeeman frequency and very spatially inhomogenous Berry’s frequency.
Figure 6.9 shows the Berry’s frequency part of the results of the experiment. We performed the following
for the experiment:
6 Rates are usually defined by the inverse of the time required to get to 1/e instead of 1/2, but we are defining it this way
for easy comparison with data from Gen. II.
7 Many thanks to Dr. Yuval Shagam and Tanya Roussy for the experiment data.

130{2.0, 2.3, 1.5} kHz{1.3, 1.7, 1.0} kHz

Fit
Just lin
Experimental results
1/2 (s 1)

101

Fit
Just lin
Experimental results

100

100
100

101
fBerry (Hz)

102

10 1

100

fBerry (Hz)

101

102

Figure 6.9: Decoherence rates, Γ1/2 = 1/t1/2 , seen in the Gen. II setup through spatial inhomogeneity of Berry’s phase. The experiments were performed with two trap configurations (trap frequencies
annotated on top of the plots). The fits are of the form in Equation 6.10b. The fit parameters for the tighttrap (left plot) are γlin = 0.050(5) (s · Hz)−1 , Γc.n. = 1.4(5) s−1 , and Γbg = 0.64(2) s−1 ; while that for theloose trap (right plot) are γlin = 0.094(6) (s · Hz)−1 , Γc.n. = 2.2(5) s−1 , and Γbg = 0.34(1) s−1 . Here is aqualitative understanding of how each of the fit parameters affect the shape of the fit: Γbg determines where
the decoherence rate saturates at low frequency; γlin determines the horizontal displacement of the slope athigh frequency away from the vertical axis; Γc.n. determines sharpness of the ankle. Also shown on the plots
are the green dashed lines, which are just plots of γlin fBerry . Regions where the yellow fit line lies below thegreen dashed lines are regions where collisional narrowing is working.

(1) Introduce Berry’s phase into the Ramsey fringes of ions in the Gen. II setup (method similar to thatin Chapter 8.3).
(2) Extract the ensemble-averaged Ramsey fringes at various values of Berry’s frequency.
(3) Determine the half-time t1/2 and hence the decoherence rate Γ1/2 = 1/t1/2 .
The fit uncertainties on Γc.n. are small, indicating that Γc.n. is required in the fitting form of Equation 6.10b
to make a good fit. One can also intentionally vary Γc.n. or set it to zero, and see that the curve does notmake a good fit with the data regardless of the choice for the other two fit parameters.

6.4.2

Collisional narrowing in simulations
We can also see the effects of collisional narrowing in simulations. Here we show the results ofsimulations of N = 1000 ions with charge√

15e in a trap with frequencies {2.0, 2.3, 1.5} kHz and cloud

temperature 4.3 K at various Berry’s frequency (Figure 6.10). In the simulations, we model the spatial
dependence of frequency due to the spatial inhomogeneity of electric fields in the Gen. II setup with the

131

N = 1000, e2
1000

15e2, {2.0, 2.3, 1.5} kHz, T = 4.3 K

Simulation
Func. fit, lin = 0.049 s 1/Hz , c. n. = 448 s 1

1/2 (s 1)

800
600
400
200
0

5

10

15
20
fBerry (kHz)25

30

Figure √6.10: Collisional narrowing seen in a simulation. The simulation uses N = 1000 ions withcharge 15e in a trap with frequencies {2.0, 2.3, 1.5} kHz and cloud temperature 4.3 K. We did not simulate
the effects of background decoherence, so Γbg = 0 in the above fit. The rest of the fit parameters are
γlin = 0.0490(4) (s · Hz)−1 and Γc.n. = 450(10) s−1 .

following expansion in powers of the cylindrical coordinates:f≈Xai,j z i ρj ,i,jwhere z and ρ are the axial and radial coordinates. We only keep expansion terms up to sixth order inposition coordinates:
ai,j

ρ0

ρ2

ρ4

ρ6

z0

1.4636(2)−0.08360(7)1.75(1)×10−3−2.27(4)×10−5z20.1714(1)−0.01551(1)5.112(6)×10−4-z4

4.62(3)×10−3−4.32(1)×10−4

-

-

z6

2.1(2)×10−5

-

-

-

Table 6.1: Coefficients of inhomogeneity expansion of f in the Gen. II setup. Coordinates ρ andz are expressed in centimeters. Uncertainties are the standard deviations of the fits of the parameters overa 3 cm × 3 cm × 3 cm region centered at the trap center.

132
6.4.3

Summary
Through Equation 6.10b, we have a model for decoherence in our ion cloud. Our model splits thedecoherence rate into a component that is fringe frequency dependent (γlin f ) and one that is frequency
independent (Γbg ). We also introduced a decoherence suppressing mechanism (Γc.n. ) for the frequency
dependent component. Using all the simulation tools that we have developed in the previous sections andalso the Gen. II setup, we saw a good agreement between our model and results from experiment andsimulations.
Our model has implications for the design of the Bucket Brigade. Our model is saying that we canget away with a little bit of imperfection in the field homogeneity, because collisional narrowing will makeour ions decohere less than what we would naı̈vely expect from the overlap of the spatial extent of the ion
cloud and the spatial inhomogeneity of the fields.
As scientists, we are entitled to be greedy (at the expense of working even harder). In the next section,
we shall try to understand the collisional narrowing rate even more, and see if we can make it work evenharder for us.

133

6.5

Collision rate
In the previous sections, we saw good agreement between our model through Equation 6.10b, andresults from experiment and simulations. Our model relies heavily on the fact that ion-ion collisions canchange the ions’ trajectories and “reset” their Ramsey frequencies, resulting in a diffusive spread in Ramsey
phases instead of a ballistic one. This rate of collisional narrowing has to be related to the rate of trajectorychanging ion-ion collisions. In the following sections, we shall try to understand the relationship between
the rates of collisional narrowing and ion-ion collisions more rigorously.

6.5.1

Collision rate ansatz
Since the Coulomb force has infinite collision cross sectional area, it takes some art to define whether

a collision has happened between two ions. We start this section off with a simple derivation of the genericform of collision rates involved between ions in an ion cloud to motivate an expression for the collision rate.
We shall borrow wisdom from the plasma physics literature. Figure 6.11 shows a schematic diagram of theapproach of two ions.

y

y

v

v = v/2
r

b

r
x
2

1
e
F = 4πε2.
0 r

Lab framexb = b/2
2

1
F = (e/2)
4πε0 r2 , r = r/2.

Center of mass frame

Figure 6.11: Schematic of a collision between two singly positively charged ions. In the lab frame,
one ion starts off at rest, while the other comes in with an impact parameter b and an incoming speed v.
The stationary “background ion” will receive a recoil from the incoming “test ion”, which will make thescattering cross section harder to analyze. One neat trick is to switch over to the center of mass frame, andintroduce an infinitely heavy “effective ion” at the origin of the center of mass., with an effective charge of
e/4 with the incoming ion still having a charge e. The center of mass frame is moving to the right at speed
v/2.

Working in the center of mass frame of the collision, we can use the results from Rutherford scattering(light incoming ion and heavy stationary ion) of relating the deflection angle Θ to the incoming speed v and134
impact parameter b:tanΘ
Z1 Z2 e2
2e2
2κ
=
=
=
2
2
2
4πε0 mv b
4πε0 mv b
mv 2 b

,

κ :=

e2
.
4πε0

In the center of mass frame, the velocity of the incoming ion far away after the collision would be:(vx , vy ) = (v, 0) → (v cos Θ, v sin Θ) =v
2

4κmv 2 b(mv 2 b)2 − 4κ2,22
2(mv b) + 4κ (mv 2 b)2 + 4κ2

.

Back in the lab frame, the change in the longitudinal velocity ∆v|| and transverse velocity ∆v⊥ arethen given by:∆v|| = −∆v⊥ =4(κβ)2· v,
4(κβ)2 + (3b)2

6κβb
· v,
4(κβ)2 + (3b)23where we used the equipartition theorem 21 mv 2 = 32 kB T = 2β
. From the above, we can obtain the (kinetic)

energy transfer coefficient, i.e. the fractional change in energy per collision:4(κβ)2∆KE=−.
KE
4(κβ)2 + (3b)2(6.11)

Now that we know how much the kinetic energy of an ion changes per collision, we need to know therate of collision. The number of ions that each ion zips past per unit time is given by8 :
R
n·

dV
= nv
∆t

Z

Z
dA = 2πnv

db b.

(6.12)

Putting Equations 6.11 and 6.12 together, the (fractional) rate of decay of (kinetic) energy is given by:
Z λc.o.
ΓKE = 2πnv
0

√
 2

4(κβ)2(κβ)29λc.o. + 4(κβ)24π 3√db b ·
·
n
·
log
=
,
4(κβ)2 + (3b)294(κβ)2mβ(6.13)

where λc.o. is the cut off length for the integration limit; the logarithmic term blows up if we do not use afinite cut off. In the two-component plasma literature, the cut off length used is the Debye screen lengthq
ε0 kB T1(λD == √4πnκβ). The Debye screening length is the distance between two ions such that thene2
Coulomb repulsion between these ions are screened by the other component in the plasma of opposite
charge. The net result is that the ions do not accelerate away from each other as much as they would have
without the second component.
8 This is an operator, because b is different for collisions with different ions.

135
We have something similar in our ion trap. In addition to Coulomb repulsion between ions, our ionsalso feel a confining force from the trap. This means that beyond a certain ion-ion separation (for instance,
when the ions are on opposite sides of the trap), their Coulomb repulsion no longer results in acceleration
away from each other. It is as though the trap “screens” the Coulomb repulsion between ions beyond acertain ion-ion separation distance. This happens at a cut off length where the trapping force balances the
Coulomb repulsion:mω 2 λc.o. =κ
λ2c.o.

.

For a thermal cloud, we know that:1
1mω 2 λ2therm. =,22βwhere λtherm. is the 1σ radius of the thermal cloud. We also know that the λtherm. is related to the ion
number density (n) through:nλ3therm. ∼ N,
where N is the total ion number. Combining the above three relations, we have
2/3
λ2c.o. ∼ κβn−2/3.
Substituting the above into Equation 6.13 gives


B
(κβ)2
log
ΓKE ∼ A · n · √,
n · (κβ)3mβ(6.14)


where A and B are constants. A simple dimensional analysis check gives [β] = [1/(N · m)], [κ] = N · m2 ,[n] = 1/(m3 ) , so that [κβ] = [m], and [mβ] = 1/(m · s−1 )2 .
The above form, together with many others in plasma literature [69, 70] and studies of ion clouddynamics [71] suggest the following ansatz for the ion-ion collision rate (this is such an important ansatz
that I shall name it the “AB ansatz”):[κβ(T )]2
B,
(6.15)
Γcol. (T ) = A · ñ(T ) · p· logñ(T )[κβ(T )]3mβ(T )
R
where A and B are dimensionless parameters to be fit, and ñ = N1 n(r) · n(r) dV is the density-weighted
density of the ion cloud. The density-weighted density depends on temperature through the temperaturedependence of the ion number density. Deriving Equation 6.15 from first principles will give different values

136
on A and B depending on the assumptions used. As an experimental physicist, I shall simulate the dynamics
of our ion cloud, perform a fit to A and B, verify the results with the experiments done on the Gen. II setup,and challenge interested theorists out there to derive the analytical form for A and B.

6.5.2

More collision-related rates
The next question is: what quantity should we compare Equation 6.15 to in our ion dynamics sim-

ulation? Ultimately, we wish to find an expression for Γc.n. , which is the rate at which the ions reset theirperceived Ramsey frequencies. In the context of our trapped ion experiment, frequency resets correspondto position resets, which corresponds to trajectory resets. By monitoring the rate at which the ion changes
its trajectory, we would be able to relate that to Γc.n. . A good quantity, then, is to measure the decay rateof the self correlation of the kinetic energy of the ions along, say, the x-direction, because this is the rate at
which the ions forget about their trajectories along the x-direction, i.e. changed their trajectories. We define
the self correlation function of the kinetic energy in the x-direction to be:(1)gKEx (τ ) :=[KEx,i (t) − KEx (t)] · [KEx,i (t + τ ) − KEx (t + τ )]KEx (t)2,

KEx :=

1
mv 2 ,
2 x(6.16)

where the overline represents averaging over all ions and all time. We fit the above expression to an exponential decaying function:(1)

gKEx (τ ) = a exp (−ΓKEx τ ) ,where we extract the decay rate ΓKEx , and a is just a normalization constant. Figure 6.12 shows the fits tothe self correlation function of the kinetic energy in the x-direction for the simulation involving N = 1000
ions with charge√

15e in a trap with frequencies {2.0, 2.3, 1.5} kHz and cloud temperature 4.3 K.

We repeat the above process with simulations of the same ion cloud at various ion cloud temperatures.
We then fit all the simulation results to the form in Equation 6.15 to obtain the fit parameters:

A = 0.41(1),(6.17a)B = 19(5).

(6.17b)

The fit is shown in Figure 6.13. For completeness, the density-weighted density used in the fits shown in

137= 299 s 1
Simulation
Fit

10 1

10 2

0

2

4

(ms)

6

8

SimulationFit

100(1) ( )gKEx(1)( )
gKE100= 392 s 1

10

10 1

10 2

0

2

4

(ms)

6

8

10

Figure 6.12: Fits to the self correlation function of the kinetic energies. Right: the self correlation
function of the kinetic energy in the x-direction as given by Equation 6.16. Left: the same thing as the rightplot, but evaluated for the total kinetic energy.

Figure 6.13 is shown in Figure 6.14.
We now have a few collision-related rates, and this was a long section, so here is a summary:Collisional narrowing, Γc.n. . This is the rate at which the naı̈ve decoherence is suppressed. We wish to
get an expression for this as a function of ion number density and temperature of the ion cloud. Thiscan be extracted by fitting Equation 6.10b to our ensemble-averaged Ramsey phases.
Decay rate of self correlation function of kinetic energy in x-direction, ΓKEx . This is the rate at
which the ions forget about their trajectories along the x-direction, corresponding to the rate at
which they change their trajectories.
Collision rate with the AB ansatz, Γcol. . This is given by Equation 6.15. The fit parameters A and B
are obtained by fitting ΓKEx to a series of simulations (using scaled parameters to save on computational resources), and performing a global fit using Equation 6.15 to the obtained values ofΓKEx .
The three collision-related rates above are extracted through different means, and what remains now is tosee if they are equivalent measures of each other. We shall extract these three rates from various instancesof simulations, and perform a comparison across them.

138

A = 0.285±0.015, B = 36.2±17

KEx (s 1)

103
KE (s 1)A = 0.407±0.013, B = 18.6±5.1
Simulation
Fit
Fit std. err.

102

103
102

0.0

2.5

5.0

7.5

10.0
T (K)

12.5

15.0

17.5

0.0

Scaled back to singly charged N = 15000 ions
102
101

2.5

5.0

7.5

10.0
T (K)

12.5

15.0

17.5

Scaled back to singly charged N = 15000 ions

103
KEx (s 1)

KE (s 1)

Simulation
Fit
Fit std. err.

102
101

0.0

2.5

5.0

7.5

10.0
T (K)

12.5

15.0

17.5

0.0

2.5

5.0

7.5

10.0
T (K)

12.5

15.0

17.5

Figure 6.13: Fit of Equation 6.15 to the decay rate of Equation 6.16 for the total kinetic energy(top left) and the kinetic√ energy in the x-direction (top right). We simulated an ion cloud of
N = 1000 ions with charge 15e in a trap with frequencies {2.0, 2.3, 1.5} kHz at various temperatures. The
shaded region indicates the region spanned by the standard error of the fit parameters. The bottom plots
are predictions of the decay rates of N = 15000 singly charged ions using the A and B parameters from thefit. There are a few points to note regarding the fit and scaling:
2

2

15e
(1) The expression for κ used in the fits are (Ze)
4πε0 = 4πε0 . When scaling back to singly charged ions, κ2e.
should go back to its original definition of 4πε0(2) The parameters A and B are dimensionless numbers independent of the ion cloud parameters ñ, m,and β, hence the values obtained for A and B from this fit should apply to all ion clouds of similar
aspect ratio regardless of their cloud parameters.
(3) All information on the trap frequencies are already embedded in ñ, so as long as ñ is correctly
evaluated, A and B should be good parameters to use regardless of trap frequencies.

n (mm 3)3.0
2.5

N = 1000, e2 15e2, {2.0, 2.3, 1.5} kHz
Simulation
Interpolation

2.0
1.5
1.0

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5
T (K)Figure6.14: Density-weighted density for an ion cloud. The ion cloud has N = 1000 ions with charge√
15e in a trap with frequencies {2.0, 2.3, 1.5} at various temperatures.

139
6.5.3

Comparison of various collision-related rates
Table 6.2 compares the predictions of Equations 6.15 and 6.17 against simulations of ion clouds withvarious cloud parameters. Since these are simulations of just N = 1000 ions, the collision rates are low, sothey take a long time to reach thermal equilibrium. Simulating just one case took about half a day of work.

Case

RMS sizes (mm)ΓKEx (s−1 )Γcol. (s−1 )Γc.n. (s−1 )γlin (s·Hz)−1

Loose A
Tight A
Tight B

(1.3, 0.94, 1.8)
(0.96, 0.87, 1.3)
(1.2, 1.0, 1.7)

60
40
21

50
55
10

43
82
10

0.016
0.011
0.016

Table 6.2: Comparison of simulation results of different ion cloud parameters against prediction
from Equations 6.15 and 6.17. The “loose” and “tight” cases corresponds to trap frequencies of {1.3,
1.7, 1.0} kHz and {2.0, 2.3, 1.5} kHz, respectively. The “A” cases correspond to a cloud temperature ofabout 2 K, while the “B” case correspond to a cloud temperature of about 5 K. All the above cases are
simulations of N = 1000 singly charged ions. ΓKEx is obtained from a fit to Equation 6.16. Γcol. is obtained
from the prediction of Equations 6.15 and 6.17. Γc.n. and γlin are obtained from fitting the corresponding
versions of Figure 6.10.

Note that all three rates ΓKEx , Γcol. , and Γc.n. agree with each other very well to within a factor of
two. This is very settling, and not surprising. The former two should agree with each other since Γcol. was
obtained through Equations 6.15 and 6.17, the latter of which was obtained through a fit to ΓKEx in Figure
6.13. Their stark agreement further shows that Equation 6.15 has captured all the physics related to the ion
cloud parameters. The first and the last rates should also agree because of the discussion above that led toEquation 6.16. At last, we now have a way to connect measurable quantities from simulations (ΓKEx ) to a
prediction model in closed analytical form (Γcol. ) to the real physical effect that we care about (Γc.n. ).
Towards the end of this study, the Gen. II experiment had more important things to do (e.g. measure
the eEDM) so we did not pursue this further with experiments. It is unfortunate that we could not do aclean 1-to-1 comparison of our model with experiments. Nevertheless, we are still equipped a self-consistentmodel that we can use in the design of the Bucket Brigade.
Before we move on to talk about the design of the Bucket Brigade, there remains one loose end. We
see how the temperature of the ion cloud goes into Equation 6.15, but it has been known for a long timethat ion clouds in Paul traps heat up [71–76]. The next section is devoted to understand the heating of ion

140
clouds in Paul traps.

6.5.4

Heating of ion clouds in Paul traps
When ion clouds heat up, they expand and can feel more of the spatial field inhomogeneity in thetrap, leading to faster decoherence for the ensemble of Ramsey fringes. The short explanation for heating inPaul traps is that ion-ion collisions in Paul traps interrupt the RF micromotion of the ions, and redistribute
some micromotion energy into the secular motion degree of freedom. The ions then launch into new Mathieu
trajectories after the collision. Since micromotion is a driven motion, Paul traps can do work on the ionsthrough ion-ion collisions and heat the ion cloud up. We shall attempt to model heating of ion cloud in ourPaul trap. Borrowing wisdom from Chen et al. [71] the heating rate is given as:Ṫ = Γcol. ϵ(T + ηTex ),(6.18)

where• Γcol. is the ion-ion collision rate;• ϵ ≈ 32 1 + q02.24 , for q0 ≤ 0.4, is the average fractional increase in the energy of an ion per collision;• q0 is the Mathieu q0 parameter (refer to Appendix A);• η ≈ 3/5, for small q0 , from equipartition theorem with 5 partitions in total: 3 in secular motion and
2 in RF micromotion;• Tex is the temperature of the excess micromotion [77], which we take to be zero in our ion cloud (noDC field displacing ions from RF null; no excess micromotion).
A big caveat in using Equation 6.18 is that it was obtained from experiments performed on ion crystals.
This is a region where the ions are cold enough that they condense into a crystalline structure. We operate
at much higher ion cloud temperatures, so our ions are closer to being a “gas” than a crystalline “solid”.
Nevertheless, we shall use Equation 6.18 and see how far we can go. Figure 6.15 compares the prediction of
Equation 6.18 with temperature measurements taken in the Gen. II setup, and shows a decent agreement
between model and experiments.

Gen. II, N = 40000
50
40

AB ansatz
Chen et al.
Gen. II

20
T (K)dT (K s 1)dt

25

AB ansatz
Chen et al.

60

141

Gen. II, N = 40000

15
10

30
5

10

15
T (K)

20

25

5
0.0

0.5

1.0

1.5 2.0
Time (s)

2.5

3.0

3.5

Figure 6.15: Ion cloud heating rates predicted by Equation 6.18 and comparison to experiment.
The yellow plots use the original equation in Chen et al. [71] with the Chandrasekhar-Spitzer collision rate,
while the blue plots correspond to using Γcol. (given by Equations 6.15 and 6.17) for the collision rate. Chenet al. used a slightly unconventional definition of ρmax in their paper that is a factor of 4 smaller than theusual definition of the maximum ion number density in a Paul trap. The yellow plots here are without thecorrection. Comparison with temperature measurements from the Gen. II experiment shows good support for
using Equation 6.15 for Γcol. in Equation 6.18. The total ion number in the experiment is about N = 40000.
The temperature measurements are performed with Doppler broadening measurements with a laser shining
along one axis of the ion cloud.

One way to reduce the heating rate would be to reduce ϵ in Equation 6.18. This can be done by
reducing the Mathieu q0 parameter subjected to the same secular frequencies. We can show with Equation
A.4 that
q0 =

2
Ω

q
2ωi2 + ωz2,i ∈ {x, y}(6.19)

where Ω is the trap RF angular frequency, and ω are the secular trap angular frequencies. This meansthat we can keep the secular frequencies the same, but have a smaller q0 by increasing Ω. Unfortunately,
substituting Equation 6.19 into the expression for ϵ shows that we do not win with increasing Ω as quickly
as we would like (Figure 6.16); it looks like we are stuck with the heating rate in our trap.
Now that we have all the ingredients required (from the discussion at the end of Section 6.2.2.2), we
can proceed to use the model for decoherence on the design of the Bucket Brigade.

6.5.5

Relevance to the Bucket Brigade and ThF+
The current plan for an eEDM experiment with ThF+ in the Bucket Brigade involves about 1500 ions

per bucket. We assume that we can recreate the same conditions as the Gen. II experiment and load 1500
ions into the Bucket Brigade at 5 K. Simulation results show that such an ion cloud has only weak mean

2 (1 + q 2.24)
3142
f = 1 kHzf = 2 kHz

9 × 10 1
8 × 10 1
7 × 10 1
101

Trap RF frequency, /(2 ) (kHz)102


Figure 6.16: Plot of ϵ ≈ 23 1 + q02.24 against Trap RF frequencies. We plan to operate at Ω =2π × 50 kHz. The plot shows that it does not help us much by going to higher RF frequencies. The grey
dashed line shows the lower bound of ϵ = 2/3.

field effects, and can be modelled as an ion cloud with a 3D Gaussian number density distribution:

nthermal (x, y, z) =




N
1 x2
y2
z2
exp
−
+
+
,
2 σx2
σy2
σz2
(2π)3/2 σx σy σz(6.20)

where N is the total ion number, and σi , i ∈ {x, y, z} are the widths of the cloud, which are related to thetemperature of the cloud through:1
1
kB T = mωi2 σi2
2
2

=⇒

σi =

2πfi

1
√

2 kHz
= 3 mm ×
×
fi
mβ

r

T
.
50 K

(6.21)

Using Equations 6.20 and 6.21, we can show that the density-weighted density for such a thermal cloud isgiven by:ñthermal = N fx fy fz (πmβ)3/2 .

(6.22)

Using Equation 6.22 in 6.18, we obtain the plots shown in Figure 6.17.
Now that we know how the ion cloud sizes change with time with the initial ion cloud and trap
parameters that we plan to use in the Bucket Brigade, we can use this information in Equation 6.15 to
estimate the rate of collision, and hence collisional narrowing, of our ions in the trap. The results are shownin Figure 6.18.

143

N = 1500, ThF +

101

20
T (K)

dT (K s 1)
dt

25

f = 1 kHzf = 2 kHzf = 3 kHz102

N = 1500, ThF +

15
10

100
10

20

30
T (K)

40

50

5

f = 1 kHzf = 2 kHzf = 3 kHz

0

5

10
15
Time (s)20

25

Figure 6.17: Ion cloud heating rates predicted by Equation 6.18 for ThF+ in the Bucket Brigade
at various secular trap frequencies. For simplicity, the plots assume spherical clouds, i.e. fx = fy =fz = f .

N = 1500, ThF +col. (s 1)102

f = 1 kHzf = 2 kHzf = 3 kHz

101
100
10 1

N = 1500, ThF +102

f = 1 kHzf = 2 kHzf = 3 kHz

101
100

5

10

15

20 25
T (K)

30

35

40

10 1

0

5

10
15
Time (s)20

25

Figure 6.18: Γcol. predicted by Equation 6.15. The plot on the left is a direct application of Equation 6.22
on 6.15, while the plot on the right (solid lines) accounts for the time-dependence of the temperature of the ion
cloud given in Figure 6.17. The dashed lines are the “time-averaged” Γcol. given by Γcol. (t) := 1tRt0Γcol. (t′ ) dt′

to give a rough sense of the effective collision rate.

6.6

Further discussion
We now have decent confidence in our decoherence model, having checked our model predictions withsimulations and experiment results on several fronts. We are already applying our model in preliminary
investigations of decoherence in the Bucket Brigade.
We understand how our ion cloud dynamics can suppress some form of decoherence through ion-ion
collisions. Ironically, ion-ion collisions can also turn out to be a source of decoherence through Berry’s phase[25]. Before we move on to discuss the design of the Bucket Brigade, it serves to cover some other points

144
related to decoherence.

6.6.1

Collisions that could change mF
In the above sections, we assumed that our molecules remain in the mF = ±3/2 subspace, even withcollisions. This is not strictly true. Handwavingly, when two ions come together, there is a characteristic
time scale of the collision given by the ratio of the characteristic collision distance and the characteristic
speed of the collision. The physics is as follows.
(1) The molecular ions are polarized by an electric field. The quantization axis is also determined by
the direction of the electric field.
(2) When another ion approaches, the net electric field direction as seen by the first ion will be modified
by the electric field of the second ion.
(3) If the approach is adiabatic compared to typical energy scales in the molecule, then the molecule cantrack its quantum state adiabatically throughout the approach. After the ions separate, the first ion
will still stay in its initial quantum state.
(4) However if the approach is not adiabatic compared to typical energy scales in the molecule, then thequantum state of the first ion will be projected diabatically onto the new polarization axis during
the approach. After the ions separate, the first ion will then be left in a state that can be very
different from its initial one.
(5) Here comes the handwaving part. Suppose the incoming ion has comparable velocity components inthe transverse and longitudinal directions w.r.t. the approach, such that far away from the first ion,2
we have 12 kB T = 12 m(v⊥+ v||2 ) with v⊥ ∼ v|| . At closest approach, all the longitudinal kinetic energy
2is converted into Coulomb repulsion potential energy 12 mv||2 = 4πϵe0 rmin . We can define a transit time
to be the time it takes for the incoming ion to move by 2rmin in the transverse direction, i.e. rmin in

145
the transverse direction on both sides of the closest longitudinal approach:
2rmin
1
1 1
,
mv 2 ∼ × kB T
v⊥
2 ⊥
2 2
√ 2√
2 2e m1=π
ϵ0 (kB T )3/2

ttrans. ∼

,

e2
1
2
∼ mv⊥
4πϵ0 rmin2≈ 3 × 10−5 · T −3/2 ,where T is the ion cloud temperature in kelvins, and tchar. in seconds.
In our experiment, our molecules are in the mF = ±3/2 subspace. The energy scale of interest is the Stark
splitting between the mF = ±3/2 states and the mF = ±1/2 states. From Figure 5.4, this energy scale in
ThF+ (HfF+ ) is 10 MHz at Erot = 60 V/cm (20 MHz at Erot = 58 V/cm). Demanding 1/ttrans. to be smaller
than the aforementioned energy scales, we demand the ion cloud temperature to be less than 50 K for ThF+
(80 K for HfF+ ). These are not crazy high temperatures. For ion clouds with temperatures higher thanthis, there is a significant chance that ion-ion collisions could drive our molecules out of the eEDM-sensitive
subspace, leading to T1 type losses. The ions could also come back to the mF = ±3/2 subspace after a few
collisions, mimicking a T2 loss signal.

6.6.2

Spatial sampling by an ion in an anharmonic trap
In all the above analysis, we assumed that our ion motions are highly non-ergodic in the absence of

ion-ion collisions, where the ion only samples a very limited volume of the ion cloud. We had to invoke
ion-ion collision to throw the ions into different trajectories, whereupon we introduce collisional narrowing,a mechanism that reduces the spread of the mean Ramsey frequencies of the ion ensemble. However, ion
motion in an anharmonic trap can sample more of the ion cloud than a harmonic one. We compare thetrajectories of an ion in two traps in Figure 6.19.
With an anharmonic trap, the ion naturally “resets” its trajectory and samples a larger space within
the ion cloud. This would give rise to a decoherence narrowing effect similar to that of collisional narrowing.
The total energy of the ion is still unchanged throughout the secular motion, though.
In our ion trap, sources of anharmonicities could come from (i) mean field effects from the ions, (ii)small size of the trap as compared to the size of our ion cloud, such that ions at the edge of the ion cloudWith "mean field"y (cm)

1.00

1.00

0.75

0.75

0.50

0.50

0.25

0.25

0.00

0.00

0.25

0.25

0.50

0.50

0.75

0.75

146

Normal harmonic trap

1.00

1.00
1.0

0.5

0.0
x (cm)

0.5

1.0

1.0

0.5

0.0
x (cm)

0.5

1.0

Figure 6.19: Simulated trajectories of an ion in two different traps. The ion has the same initialconditions in both the left and right plots: v(t = 0) = 10 ŷ m/s, and r(t = 0) = 1 x̂ cm.
mω 2

Right: the confining field has the form Ei = − e i ri , where ωi = 2π × 1 kHz and 2π × 1.1 kHz for the x andy directions, respectively. The area traced out by the Lissajous figure is rather small; the ion only samples
a very limited volume of the entire space.


mω 2
R3
Left: the confining field has the form Ei (r > R0 ) = e i −ri + r30 · ri , where R0 = 5 mm, and E (r ≤R0 ) = 0. This is a very simplified model of the field seen by a hot ion in a very cold ion cloud such thatmean field effects dominate (like those shown in Figure 6.2). The region where mean field effects null out
fields is indicated by the grey circle r ≤ R0 . The ion samples more of the volume of the entire space, and its
direction of motion also “precesses” around.

can sample the deviation of the trapping potentials from a true quadrupole, and (iii) other imperfections
within our trap.

6.6.3

Difficulty in measuring ion cloud properties in the experiment
To match experiment results to the model we developed in this chapter, we would like to be able tomeasure a few properties of the ion cloud: (i) temperature of the ion cloud along all three principal axes9 ,and (ii) ion cloud density. However, the tools at our disposal are rather limited:
Thermal Doppler broadening of laser transition. In the Gen. II experiment, a laser beam with alinewidth narrower than the thermal Doppler width goes down the axial direction of the trap toserve as a depletion laser for the doublets. We can infer the temperature of the ion cloud by mea9 In addition to serving as an input parameter into our model, measuring in all three axes tells us if the ion cloud is inthermal equilibrium in all three dimensions.

147
suring the width of the transition. However, we can only do this for the axial direction. We could
have a laser going in from the radial direction, but RF and rotation micromotion also broadens thetransition.
Velocity map imaging of ions on ion detector. With our long time-of-flight in the prototypical Bucket
Brigade (Section 7.9), we can perform velocity map imaging of ions, where the ion positions on ourion detector corresponds to their velocity right before ejection from the trap. However, stray charges
along the flight path contribute to lensing effects that only give us velocity mapping along one
transverse axis (y) and not the other (x). Spatial variation of field strengths of the voltages seen by
the ions during the ejection also contribute to axial focusing effects. Furthermore, there could also
be ion-ion repulsion during the flight after the trap is turned off. All these effects complicate theinterpretation of our measurements on the actual temperature and ion number density of our ion
cloud.
We need to sharpen our detection tools to push our understanding of our ion cloud up a notch. This
would involve serious planning and significant changes to the current design of the setup. While we work on
this at the back of our minds, let us proceed to look at the electrode design of the Bucket Brigade.

Chapter 7

Design of the Bucket Brigade

7.1

Introduction
The Bucket Brigade is the second of the two main upgrades from Gen. II to III. In contrast to the Paul

traps that we have been using in the Gens. I, II, and ThF/ThF+ spectroscopy setup, the Bucket Brigade(i) is much larger in scale, (ii) has the capability to translate ions across meter-scale distances along thetrap axis, and (iii) allows for near-continuous multiplexed operation. Furthermore, all the above has to bedone in an environment that is highly controlled and devoid of noise for precision measurements that aim tosurpass our current Gen. II record. We are requiring a lot out of the Bucket Brigade.
Translating ions within a multi-sectioned trap is not new. Multiplexing operations on ions at different
parts of the multi-sectioned trap is also not new. These were already envisioned in the quantum CCD
architecture [78] two decades ago. Midway through the pandemic, the first trapped ion quantum CCD
computer architecture was also demonstrated [79]. Not long after, news also came from the industry side
(Quantinuum) with their version of the quantum CCD ion chip design [80]. Needless to say, we are neither
the first nor the only ones working on a multiplexed ion trap based on the quantum CCD architecture. But,
ours is the biggest.1
We do not aim to design the perfect Bucket Brigade from the get go. We do not claim to have foresightat such a level. We understand the weight of this big undertaking. We also take seriously all the helpfulcomments from our critics. We shall take the design one step at a time. We will start to scratch the surface
1 One can argue that large scale experiments with storage rings count as multiplexed ion traps. However, the author wishes
to point out that ion confinement in storage rings is implemented rather differently (not of quantum CCD-type) from all theaforementioned traps, so storage rings should be treated as a separate class.

149of the design with this chapter.
This chapter is structured as follows. First we will look at performing axial confinement and ion
translation with Rings (Section 7.2). Next, we will look at radial confinement and introducing the rotating
electric field with Rods (Section 7.3). We then introduce the baby Bucket Brigade, a prototype of the Bucket
Brigade (Section 7.4). Thereafter we have an overview of the electrode drivers to see how we control voltages
on all our electrodes (Section 7.5). Finally, we introduce a new tool in our JILA eEDM kit, parametric
excitation of ions (Section 7.6).

7.2

Rings: Axial confinement & Translation of ions
Axial confinement in linear Paul traps are typically introduced with end caps: electrodes on bothends of the trap in the axial direction. These end caps are placed at a positive DC voltage w.r.t. the DC
ground on the radial electrodes to confine positively charged ions. On ion chips based on the quantum CCD
architecture, the “end caps” are displaced away from the trap axis, but the potentials added onto them stillallows for axial confinement of the ions. There are also many more such “end caps”, each at a different
potential. These potentials can be spatially modulated to create the various ion trap wells required of amultiplexed ion trap.
We employ a similar strategy. We shall have Rings to serve as our “end caps”. To understand ourdesign constraints, we will first study the fields in our cylindrically symmetric setup. We start with static
axial confinement first. We begin with the case where the Rings are “continuous” in the axial direction(Section 7.2.1) to understand how the axial trap secular frequency depends on the basic design parameters.
We will then work with the more realistic Rings that are discretized (Section 7.2.2) to see how discretizing
the Rings change the analysis. Subsequently, we explain how we translate ions by modulating the potentials
on the discretized Rings (Section 7.2.3). Finally, we end off with a brief discussion on the shielding effectsof the radial electrodes on the fields generated by the Rings (Section 7.2.4). Without spoiling too much,
Figure 7.1 shows a CAD drawing of the Bucket Brigade for a quick visualization of the critical components
involved and definitions of some of the coordinates to be used in this section.

150

Figure 7.1: Spoiler: perspective view of a CAD drawing of the Bucket Brigade to visualize therelevant relationship between various electrodes. The inner diameter of the Rings is 2ρ0 . Some ofthe structural support is hidden for a better view of the Rings. The radial distance from the trap axis isdefined to be ρ, and the axial coordinate towards the right along the trap axis is z. The Rings colored red
are the Special Rings; yellow Rings are the Quadrant Rings; green rings are the Translation Rings; all to be
discussed in more detail in Section 7.4.

7.2.1

“Continuous rings”

7.2.1.1

Analytic solution

We begin with a simple model, where we solve for the potential inside the volume of a long cylindricalelectrode, where we impose an arbitrary boundary condition on the inner surface of the long cylinder. Thisis equivalent to the limit of having infinitely many Rings that are infinitely thin and infinitely close to eachother. The potential applied on the electrode holds the form:
V (ρ0 , z) = V0 sin kz.

(7.1)

Using separation of variables on Laplace’s equation in cylindrical coordinates gives:11∂ρ2 + ∂ρ + 2 ∂ϕ2 + ∂z2 P (ρ)Φ(ϕ)Z(z) = 0.
ρ
ρ

(7.2)

Now we need to solve the above with the boundary condition given by Equation 7.1.
Cylindrical symmetry dictates that Φ(ϕ) = 1. Assuming that the trap is infinitely long, the sinusoidal
modulation of potential at the boundary implies that Z(z) = sin kz, where k = 2π
λ , with λ the periodicity of

151
the potential modulation in the axial direction. The solution now looks like:

V (ρ, z) = P (ρ) sin kz.

Now the partial differential equation for P (ρ) is:1∂ρ2 + ∂ρ P = k 2 P.
ρ

The solutions to the above are the modified Bessel functions.2

Further demanding that the potential atρ → 0 to be finite, the solution then holds the form:

V (ρ, z) = αJ0 (ikρ) sin kz,
where α is a constant of proportionality, and J0 (x) is the zeroth order Bessel function of the first kind.
Matching the above to the boundary condition at the electrode with Equation 7.1 gives:

V (ρ, z) = V0

J0 (ikρ)sin kz.
J0 (ikρ0 )(7.3)

To aid us in the later sections, we note a few approximate forms of the Bessel function in the two extreme
limits:J0 (ikρ) =



2
4

6

1 + (kρ)+ (kρ)
4
64 + O(ρ ) kρ ≪ 1,

h
 i


 √ekρ 1 + O 1
kρ
2πkρ

(7.4)

kρ ≫ 1.

The small kρ expression will be useful in evaluating the potential in the vicinity of the ion cloud, while thelarge kρ expression will be useful in simplifying the expression for the potential on the boundary.
It is useful to note that voltage waveforms on the Rings with larger k (smaller λ) will decay much
faster in amplitude as we approach ρ → 0. Figure 7.2 shows a quick comparison of the waveforms withvarious k and the relevant coordinate parameters.

7.2.1.2

Dependence of axial frequency on design parameters

We intend to work with an ion cloud that is small compared to the periodicity of the spatial modulationof the axial potential, i.e. ∆z ≪ λ = 2π
k (see Figure 7.2). Expanding the potential along the axis in z about
2 The interested reader can refer to Jackson’s Classical Electrodynamics 3rd edition, equations 3.100 and 3.101.

Large k152

Small kAt the Rings, ρ = ρ0

V (ρ, z)V0 sin kz
λ
∆z
(ikρ)
V0 sin kz JJ00(ikρ0)

zAlong trap axis, ρ = 0

Figure 7.2: Schematic comparison of waveform amplitude decay from the Rings with various
k. Red (blue) wiggles show potential waveforms with large (small) k. The upper trace shows V (z) as
applied onto the “continuous rings” electrode, and the lower trace shows how this potential would look likealong the trap axis, ρ = 0. The orange disk indicates the size of our ion cloud, which is small compared to
λ = 2π/k ≫ ∆z. Within the vicinity of the ion cloud center, the potential can be approximated as harmonic.

a potential minimum gives:
V (ρ, z) ≈ constant + V0

J0 (ikρ) (k · ∆z)2.
J0 (ikρ0 )2Plugging the above into the equation of motion:m ∂t2 (∆z) = −e ∂∆z V = −eV0 k 2

J0 (ikρ)· ∆z,
J0 (ikρ0 )implies that the oscillation frequency at ρ = 0 is:sω=peV0 k 21
= V0
m J0 (ikρ0 )sp
4π 2 e1∼ V0 ·
2
mλ J0 (ikρ0 )re  ρ0 1/4 − πρ0··e λ ,m
λ5(7.5)

where in the last step, we expanded the Bessel’s function out (kρ0 ≫ 1) for a more intuitive feel of how theoverall expression scales with the relevant parameters. Using the exact expression, we show values of axial
frequencies for some combination of the design parameters in Table 7.1.
If we wish to operate in a experiment parameter space familiar to us in the Gens. I and II experiment,
we want to aim for between 1 and 10 kHz. Fortunately, the numbers show that we can get these frequencies
with very modest voltages applied if we pick the right geometry design parameters.

7.2.1.3

Peak-to-peak voltage along trap axis

One important aspect of the Bucket Brigade is that ions need to stay within their bucket. We do notwant ions to spill over to the neighboring bucket. We want to know the “axial energy” of an ion at which it153ρ0 \ λ

2 cm

4 cm

6 cm

8 cm

10 cm

12 cm

1 cm
2 cm
3 cm
4 cm
5 cm
6 cm
7 cm
8 cm
9 cm
10 cm

13200
3320
767
172
37.8
8.22
1.78
0.382
0.08180.0175

11800
6620
3370
1660
802
383
182
85.8
40.3
18.9

9080
6620
4410
2830
1780
1110
682
419
256
156

7190
5910
4510
3310
2380
1690
1190
830
578
401

5910
5170
4270
3400
2650
2030
1550
1170
884
664

4990
4540
3940
3310
2720
2210
1770
1420
1120
890

ω
Table 7.1: Axial frequencies at various geometries. The numbers shown here are 2π√in units
V0of Hz. For example, for an electrode modulation peak-to-peak voltage at Vpp = 2V0 = 2 V, distance toelectrode√ from trap axis ρ0 = 7 cm, modulating wavelength λ = 8 cm, the axial angular frequency is
2π × 1 × 1190 = 2π × 1.19 kHz. Mass is assumed to be that of ThF+ .

spills over classically from one potential well to the next. We have, in the convenient unit of kelvins:

Tmax =

1 eVpp.
kB J0 (ikρ0 )The maximum energy of the ions that can be trapped within this potential well for typical values of ρ0 andλ is shown in Table 7.2.
Even though we want to work with design parameters such that our ions of a certain temperature do
not spill over to the neighboring buckets, the ions will be sampling the anharmonic part of the axial trap
well before they start spilling over. We do not want the ions to sample anharmonic parts of the trap for fearof systematic effects on our eEDM measurement (Section 7.2.2.2). The numbers above serve only as a hard
limit (instead of the recommended values), beyond which we have no hope of using the Bucket Brigade fora multiplexed ion trap experiment.
If the ions in the Bucket Brigade have similar thermal energies as compared to the Gens. I and II
experiments, we expect ion cloud temperatures between a few kelvins to a few hundred kelvins. Fortunately,
the numbers show that our ions will not be spilling over to neighboring buckets with very modest voltages
applied if we pick the right geometry design parameters.

154
ρ0 \ λ

2 cm

4 cm

6 cm

8 cm

10 cm

12 cm

1 cm
2 cm
3 cm
4 cm
5 cm
6 cm
7 cm
8 cm
9 cm
10 cm

4230
266
14.2
0.711
0.03440.001630.0000763
3.53 × 10−6
1.62 × 10−7
7.37 × 10−9

13500
4230
1100
266
62.2
14.2
3.19
0.711
0.157
0.0344

17900
9520
4230
1740
689
266
101
38.1
14.2
5.26

20000
13500
7860
4230
2190
1100
544
266
129
62.2

21100
16200
11000
6980
4230
2500
1450
831
472
266

21700
17900
13500
9520
6440
4230
2730
1740
1100
689

Table 7.2: Potential barrier of the wells along the trap axis. The numbers are expressed in temperature
for an electrode peak-to-peak voltage of Vpp = 2 V. For example, if the distance to the electrode from trap
axis is ρ0 = 7 cm, and the modulating wavelength is λ = 8 cm, the ions start spilling over classically once
their “axial energy” exceeds 544 K.

7.2.2

Discrete rings
Having infinitely many Rings would allow us to set voltages on all those Rings such that the ionsalong the trap axis will see a perfectly sinusoidal modulation of the potential along the axial direction. If wedesire, we could even change the sinusoidal potential to something that looks more like a piece-wise seriesof harmonic traps, allowing the ions to feel a much bigger volume of trap harmonicity.
However, both the former and the latter have problems. The problem with the former is that wecannot control an arbitrarily large number of Rings in the experiment, it is simply impractical. We have tosettle for a finite number of Rings, countable with about ten pairs of hands. This means that our “boundary
condition”3 will no longer be a single sinusoid with only one periodicity; it will be a sum of the various
harmonics. This may complicate the analysis.
The problem with the latter is that Laplace’s equation in cylindrical coordinates attenuates higher
harmonics of spatial variations very quickly, as we will see in this section. To realize arbitrary modulation,
like the piece-wise stitched harmonic traps, we will have to apply very high potentials on the Rings to fightagainst the attenuation. It looks like we have to settle for a simple sinusoidal modulation. However, it turns
out that we can use this to our advantage to tackle the problem of the previous paragraph.
3 The boundary conditions are strictly only introduced by the surfaces of the discrete Rings, and not the space between
Rings. Nevertheless, if we look at the potential at ρ < ρ0 , the spatial modulation of the potential along the axial direction canbe decomposed into the various harmonics of the fundamental periodicity.

155
7.2.2.1

Suppression of the higher harmonics

If we have an arbitrary cylindrically symmetric potential at ρ = ρ0 , we can decompose it into thevarious harmonics:XV (k, z, ρ0 ) = V0 +

Vn(s) sin nkz + Vn(c) cos nkz .

n=1

Equations 7.3 and 7.4 then say that the potential at ρ ≪ {ρ0 , λ} is given by:
V (k, z, ρ) ≈ V0 +

X
n=1


 
(nkρ)2(nkρ)4  enkρ0√Vn(s) sin nkz + Vn(c) cos nkz · 1 +.
+464
2πnkρ0(7.6)

We see that the attenuation factor increases exponentially with the order of harmonicity n. Hence, thewaveform of the potential along the trap axis is predominantly that of the first harmonic unless we amplify
the higher harmonics components at the source. The attenuation factors for various λ = 2π/k are shown in
Table 7.3.
λ\n
8 cm
16 cm
50 cm

1
60.8
4.67
1.23

2
1.54 × 10
60.8
2.11

3
4

4.52 × 10
937
4.25

4
6

9

1.41 × 10
1.54 × 104
9.28

5

6

7

−
2.61 × 105
21.1−
4.52 × 106
49.1−
7.95 × 107
116

Table 7.3: Attenuation factors of various harmonics. The values shown are J0 (inkρ0 ). They areevaluated at ρ0 = 7.5 cm. The dashes are numbers that are too big and meaningless to include. For areasonable bucket length of about λ = 8 cm, we see that any higher harmonics will be so quickly suppressedthat the trapping potential along the trap axis is essentially a pure sinusoid.

A consequence of the above is that we do not need to have too many Rings per bucket. As alluded
to earlier, having more Rings allows for more parameters for us to tweak to adjust the waveform of thepotential, but since the attenuation of higher harmonics is so strong, we would have to apply unrealistic
magnitudes for the higher harmonics to be able to take advantage of having more rings. Therefore, we havechosen to use a lucky4 number of four Rings per bucket.
4 Four is an unlucky number in Chinese numerology. It is homophonous with “death” in various Chinese languages like
Cantonese, Teochew, and Mandarin. However, we shall associate the “death” with “uncertainties” in the Bucket Brigade design.

156
7.2.2.2

Anharmonicity

Lastly, we shall take a look at the harmonicity of the axial trapping potential. Since the potential
takes the form of a sinusoid, expanding the potential about the potential minimum at zmin is: [k(z − zmin )]2[k(z − zmin )]4(kρ)2(kρ)4  ekρ0√
V (k, z, ρ) ≈ V0 −1 +
−
· 1+
+
.
2
24
4
64
2πkρ0
Evidently the first anharmonic term comes in as the fourth power of z − zmin . Demanding the ratio of the
fourth power term to the harmonic term to be less than 10% (1%) at 1 cm away gives λ > 6 cm (18 cm).
The anharmonicity does not hurt the trapping of the ions. However, it could hurt our precision
metrology through systematic effects. Suppose in addition to a potential of the above form, we have an
unwanted axial electric field from, say, imperfectly shimmed axial trapping voltages for the rings. The resultof this additional axial electric field is that the minimum of the quadratic component of the trapping field will
be displaced from that of the quartic component (see Figure 7.3 for a graphical explanation). For the ions
(a)
=

+
(b)
+

+

=

+

=

Figure 7.3: Graphical representation of summing a linear potential to quadratic and quartic
potentials. (a) Adding a linear potential (uniform axial electric field) to a quadratic potential (harmonic
axial trapping potential) results in a displaced quadratic potential. The dashed line indicates the the positionof the original axis of symmetry, and the dotted line indicates mean position of ions sloshing around in thesaid potential. (b) Adding a displaced quadratic to a quartic potential (anharmonic part of the axial trap)
results in a trap that is asymmetrical about the mean ion position (indicated by the dotted line).

to remain trapped axially, the time averaged axial electric field seen by the moving ions has to be zero, i.e.
Rz>0REz dt+ z<0 Ez dt = 0. However, since the potential is no longer symmetric, it has to spend more time on

one side of the equilibrium position than the other, and we may end up with a non-zero time-averaged higher
moment of the axial electric field

R
z>0

R
Ez3 dt + z<0 Ez3 dt ̸= 0. The result of having a non-zero time-averaged

higher moment of Ez is that it can manifest as a systematic effect on our precision metrology (see Equation

157
8.15 and Section 8.3).
We shall leave this problem for another day; systematic effects are not of our immediate concern now.
Future graduate students and post-docs may want to bookmark this section for when the time comes tostudy systematic effects (with ions) in the (prototypical) Bucket Brigade.

7.2.3

Translation of ions
Translating the ions is trivially simple, we modulate the potentials on the Rings in space and time:
V (zn , t) = V0 sin(kzn − 2πfrep t) ,(7.7)

where zn corresponds to the axial position of the nth ring, and frep is the repetition rate of the Bucket
Brigade experiment from Equation 1.1.

7.2.4

Shielding effect from Rods
In our analyses of the Rings, we have been neglecting the radial electrodes, the Rods. Our Rods

will have RF voltages on them for the radial confinement of ions. They will also have the voltages for therotating electric field that polarizes our molecular ions. We require the rotating electric field to be spatiallyhomogeneous, so we have chosen to have the Rings to sit outside the Rods. This is so that the fields
from the Rods at the center of the trap will experience a much less severe shielding effect from the Rings.
Consequentially, the fields from the Rings will be shielded by the Rods.
To investigate the level of shielding, we shall first look at realistic dimensions of Rings and Rods. The
Bucket Brigade should fit inside a vacuum chamber of 8′′ outer diameter (0.120′′ thick wall), so a reasonabledimension for the Rings will be 1 mm thick Rings with inner and outer diameters of 15 cm and 17 cm,
leaving about 1.4 cm worth of space between the outer circumference of the rings and the inner wall of thevacuum chamber. There will be eight Rods at 1 cm in diameter, sitting at 6.5 cm away from the trap center.
The geometry is summarized in Figure 7.4.
Using the above dimensions, we run simulations in COMSOL to look at the shielding effect of theRods on the Rings. A comparison of the results with and without Rods are shown in Figure 7.5.

158
17 cm
15 cm

3

2
0.1200 thick

1 cm
4

1
y
x

1 mm thick

13 cm
5

8
6

7

800

Figure 7.4: Cross sectional view of the Bucket Brigade involving the Rods, Rings, and thevacuum chamber. The rods are color coded to indicate the pairing of the rods for the RF-potentials. The
rods are numbered here for future reference.

The simulation results from Figure 7.5 show that the Rods attenuate the potentials from the Rings
by about a factor of two. To compensate for this attenuation, we only need to amplify the potentials that wewish to apply on the Rings by the same factor. Hence, for an axial trapping frequency of 1 kHz at λ = 8 cm,
we just need V0 = 2 V (refer to Table 7.1).

159

Figure 7.5: Attenuation of potentials of Rings by Rods. Top row: the left (right) column shows
simulation results without (with) Rods; the verticallines on theright plot represents the edges of the Rods.
√
Potentials applied on the Rings are V (z) = − 2 V × cos 82π
cm z , where z ∈ {−3, −1, 1, 3} cm. We imposed
periodic boundary conditions on the top and bottom faces of the simulated domain. The top row shows
see-through color maps of electric potentials; the trap axis is the vertical direction. Bottom row: the profileof the electric potential along the trap axis. The amplitude for the case without Rods is about 16 mV,
which agrees with the attenuation factor from Table 7.3. On the bottom-right plot, the amplitude is further
attenuated by a factor of about 2.7 to about 6 mV.

160

7.3

Rods: Radial confinement & Erot
We will be using Rods to supply radial potentials. These potentials include the RF radial confinement,

and the potentials to create a rotating electric field to polarize our molecular ions Vrot . The geometries of
the Rods have already been alluded to earlier on in Figure 7.4. We will be analyzing the radial confinement
properties of our trap with the established machinery of the Mathieu parameters (see Appendix A for moredetails). We shall now briefly investigate the RF trapping properties (Section 7.3.1) for us to understandwhat kind of RF voltages we need to apply for the radial secular frequencies we want. We then look at
simulation results of the homogeneity of the rotating field at various sections along the Bucket Brigade(Section 7.3.2) to justify the dimensions chosen for the Bucket Brigade.

7.3.1

Radial confinement in a Paul trap
The physics behind confinement of ions with time-oscillating fields can be found in many literature(see, for example, Ref. [81]). The basic idea goes as follows:(1) Maxwell’s equations forbid us from having an electric potential minimum in free space, so we cannot
trap ions (electric monopoles) with static electric fields only (one can however engineer static electricfields to trap electric dipoles, see, for example, Ref. [82]).
(2) We apply a time-oscillating quadrupolar field in the x-y plane, which has a field strength that is zeroat all times along what we call the trap axis (the z-axis).
(3) The time-oscillating quadrupolar field moves the ion along with it. We call this motion the micromotion of the ion. Since the frequency of the oscillation of the field is typically in the RF range, wecall the micromotion the RF micromotion.
(4) The phase of the motion of the ion is matched with the phase of the oscillation of the field insuch a way that the ion sees a time-averaged net restoring force towards the trap axis. This is theponderomotive force. An ion would feel a ponderomotive force in the presence of a time-oscillating

161
spatially inhomogeneous electric field. For a field that oscillates as:⃗ t) = E(r)⃗ cos ωrf t,
E(r,the ponderomotive force that the ion of charge q and mass m feels is given by:
F=−

q2
2
⃗
2 ∇ E(r) .
4mωrf

(7.8)

⃗
(5) For a quadrupolar field, the expression for E(r)is proportional to x − y. This means the ponderomotive force that an ion sees in a time-oscillating quadrupolar electric field is given by:
Fquad. ∝ −∇(x2 + y 2 ),which is just a harmonic trap in the x and y directions, i.e. the radial directions. The motion
associated with the time-averaged restoring force is called the secular motion of the ion in the trap.
The secular frequency is usually much lower than the RF frequency.
The summary above shows how an ion in a time-oscillating quadrupolar electric field can feel a restoring
force back to the trap center. However, we did not mention how the amplitude of the secular motion of
the ion could change with time. Depending on the trap parameters chosen (e.g. RF frequency, RF voltage
amplitudes, and charge-mass ratio of the ion), the amplitude of the secular motion could increase with time
like in a resonance. These trapping conditions are deemed unstable. The ponderomotive picture only holds
if (i) the micromotion is small compared to the secular motion (hence the term “micro”motion), and (ii)the spatial variation in the fields are slow compared to the amplitude of the micromotion. We want to workin this parameter space. Fortunately, some machinery has already been built to allow us to analyze theparameter space, and, say, the stability of the trap more quantitatively, as we shall see below.
For a more quantitative approach, the potentials involved in Paul traps are usually parameterized in
the form of Equations A.1 and A.2. The relevant variables are defined in Section A.2. Using these forms, we
can calculate the Mathieu parameters, and use Equation A.4 to determine the secular frequencies. Hencethe first step to understand our radial confinement is to extract the relevant parameters (equivalent radiusof radial electrodes R0 and equivalent axial distance of the end caps Z0 ) from our Bucket Brigade with the162help of COMSOL. The relevant dimensions, definitions of our axes, and Rod numbering are shown in Figure
7.4. Applying positive (negative) RF voltage Vrf,0 to the red (blue) rods in Figure 7.4 and evaluating theelectric potential along the x- and y-axes gives Figure 7.6.
0.4
Residual (mV)

Vrf (V)

0.2
0.0
Along x-axis
Along y-axis
Fit along x-axis
Fit along y-axis

0.2
0.4
6

4
2 0
2
4
Distance from trap axis (cm)

6

1.5
1.0
0.5
0.0
0.5
1.0
1.5

Along x-axis
Along y-axis

3

2
1
0
1
2
Distance from trap axis (cm)3

Figure 7.6: RF electric potentials in the Bucket Brigade, evaluated along the x- and y-axes. For
the purpose of the plots, we used Vrf,0 = 1 V. The fits are done within the region of [−3, 3] cm. The residuals
from the fit are shown in the plot on the right. The profiles look identical when the 1D slices are taken at
the axial positions (i) between two Rings and also (ii) on the plane of the rings within this plotting region.
This says that the Rings are not significantly modifying the fields from the Rods within the region shown in
the plot.

From Figure 7.6, we find that R0 = 6.838 cm, which is not that far away from the actual 6.5 cm radial
distance of the rods from the trap axis. A similar analysis on the axial confining potential using the resultsof Section 7.2.4 gives Z0 = 25.5 cm. Voltages required for secular frequencies of 1 kHz in all three directions,are Uec,0 = 3.35 V, and Vrf,0 = 20.8 V. The resulting Mathieu stability plot is shown in Figure 7.7.5
So far, all the voltages required look reasonable. The final values that we settled on for the prototypical
Bucket Brigade are shown in Appendix B.1. The values shown there are slightly different from those shown
above because we scaled the transverse dimension down a little to make space for structural support.

7.3.2

Rotating electric field, Erot
To produce a rotating electric field Erot , the potentials we apply on the nth rod (compare with Figure7.4) is given by:(2n − 1)πVn = Vrot cos ωrot t +,8
5 It is interesting to note that operating in the stability region does not guarantee stability in the presence of higher ordermultipole time-oscillating potentials in the trap [83]. Our trap certainly has higher order multipoles because our electrodes are
not shaped like hyperboloids. It seems, though, that the instabilities in Alheit et al. [83] only come in at q > 0.45. We typicallyoperate at q ≈ 0.14, so this should not be a problem for us.

163

0.010
0.005

ThF +
Th +
ThF2+

3

0.005

kHz

a

0.000

4

0.010

2

0.015
0.020
0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200
q01

Figure 7.7: Mathieu stability plot for the Bucket Brigade. The colored dots represent the positions of
the three species (ThF+ , Th+ , ThF+
2 ) that we care about in our trap. The grey area bound by the dashedlines indicate stable trapping region. The solid colored lines indicate contours of radial trapping frequency
for ThF+ . Please refer to Appendix A for more information regarding the Mathieu parameters a and q0 .

where ωrot is the rotational angular frequency of the rotating electric field, and the phases are chosen suchthat Erot points in the negative x direction at t = 0 and rotates clockwise with time. Using COMSOL toevaluate the fields resulting from potentials applied to the Rods gives results shown in Figure 7.8.
From the results in Figure 7.8, we see that the deviation crosses the 0.1% and 0.01% marks at 1.6 cm
and 1.1 cm, respectively. We want all the ions in the ion cloud to see the same Erot , so we want the deviation
to be as small as possible within the extent of the ion cloud. Expecting a typical ion cloud size of 4σ ∼ 1 cm,
and requiring a 0.01% homogeneity level6 , we cannot afford to shrink the dimensions of the Bucket Brigade
down by much. One can say that we were lucky to have chosen to work with the current set of design
parameters; any smaller and we would have to calculate everything again with a bigger dimension.
⃗ 2 (see Equation 7.8), and the ponderomotive energy associatedPonderomotive potentials go as |E|with Erot is about 93500 K at the trap axis. For an ion cloud temperature of about 30 K, we see that thedeviation of the ponderomotive potential exceeds that of the ion cloud temperature beyond 1.3 cm from thetrap axis. We certainly do not wish to lose ions to the Erot ponderomotive potential (but in fact we do, see
Figure 7.20), so we would want to have small cold ion clouds given the current electrode configuration.
From COMSOL, we further see that for every 1 V of Vrot , we get 10.655 V/m of Erot . This means
6 The 0.01% value comes from demanding the ions to decohere less than 10% of a cycle after 50 Hz × 20 s = 1000 cycles.

20

2.0

15

1.5

|Deviation| (%)|E| (V/m)

164

100°, between rings
22.5°, between rings
0°, on rings
22.5°, on rings

5
0

6

4

2

0
x (cm)

2

4

6

0°, between rings
22.5°, between rings
0°, on rings
22.5°, on rings

1.0
0.5
0.0

3

2

1

0
x (cm)

1

2

3

Figure 7.8: Plots of Erot spatial homogeneity. Top row: color maps of electric potential, with planes
chosen to be between the rings, or on the plane of the rings. Second row: plots of the magnitude of theresultant electric field; range of color map is chosen such that the colors saturate beyond 1% from the value
at the center (10.655 V/m). Last row: 1D slice of the magnitude of the resultant electric field along thex-axis. For the first two rows, the odd columns correspond to Erot pointing along the negative x-axis (0◦ ),while the even columns correspond to Erot making 22.5◦ with the negative x-axis. Color legends are also
removed for clarity; warmer (cooler) colors indicate more positive (negative) values. The grey dotted line in
the bottom right plot indicates 0.1%.

that for a nominal Erot = 60 V/cm, we need to apply Vrot = 563 V. This is still a reasonable voltage that
we can deal with.
Once again, the final values that we settled on for the prototypical Bucket Brigade are shown inAppendix B.1. The values shown there are slightly different from those shown above because we scaled thetransverse dimension down a little to make space for structural support.

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7.3.3

Summary
With the current geometry, we saw that we can introduce the fields that we want, both radially andaxially, with reasonable voltages. The field homogeneity requirements for the Rods and the harmonic axial
region for the Rings are also at a comfortable level for the ion cloud size that we expect to work with. In
principle, this could work. In practice, we have to find out. There could be many things that could affect theexperiment in ways that are easier found out by the method of doing it, instead of trying to search throughthe whole parameter space exhaustively. There remains a lot to be said for building a prototype.
The next section will introduce a prototype of the Bucket Brigade. We will be using the prototype totest out our design ideas, and to have a feel of the construction and assembling tolerances.

166

7.4

The baby Bucket Brigade (b.B.B.)
Having a trap design that works in principle is a good start, but we still need to test this out

experimentally. To do so, we have constructed a prototype of the Bucket Brigade with realistic diameterand operating voltages, but only 1/3 as long as what we anticipate for the full design. We affectionally callthis prototype the baby Bucket Brigade (b.B.B.). In addition to testing out the trap design on the b.B.B.,
we also aim to test out many other new ideas that we wish to implement in the Gen. III experiment (to be
discussed in the later chapters).

7.4.1

Setup
A CAD drawing of the setup is shown in Figure 7.9, and critical components are elaborated on in thecaption. The drawing is a bird’s eye view from the top, but we made some minor changes in the final design.
For example, the turbomolecular pump attached to the Six-way Cross was drawn to be lying on the horizontal
plane, but it is now on the top flange of the Six-way Cross to make access to other flanges friendlier to thelab users. All the electrical feedthroughs for the electrodes inside the Cold Tube are mounted on various
blanks on the unused ports of the Six-way Cross. Electrical feedthroughs for the microchannel plate and theAccelerating Rings are mounted on the zero length reducer that the microchannel plate assembly is mounted
on. A picture of the setup is shown in Figure 7.10.
An 80 L/s turbomolecular pump (Varian V81) maintains the vacuum in the differential pumping
chamber, while the Six-way Cross has a 500 L/s turbomolecular pump (Varian V551) and a 1000 L/s (forhydrogen) nonevaporable getter (SAES CapaciTorr Z1000). The source chamber also has a Varian V551
attached to it. The molecular pumps are backed up by scroll pumps (Agilent IDP-15). Typical vacuum
pressures read by the ion gauges are 10−9 Torr in the differential pumping chamber and in the Six-way
Cross.
The distance from the molecular beam source to the trap is about 21′′ . The length of the wholeb.B.B. is about 1 m long. The distance between the Octagon to the microchannel plate assembly is about
70 cm. The microchannel plate assembly consists of two 4 cm diameter microchannel plates in the chevron

167
Nonevaporable
getterMicrochannel
plate
assembly
Acceleratingrings

Six-waycross

OctagonCold tube

Baby Bucket Brigade

Differential pumping chamber
Gate valve
Householdinsulating foam

Cold sleeve
Source chamber
Turbomolecular pumps

Stirling cryocooler

Figure 7.9: CAD drawing of the b.B.B. setup. A typical experimental sequence proceeds as follows(also compare with Figure 4.24):(1) We ablate a piece of thorium metal in the source chamber to create a plume of thorium plasma thatis reacted with sulfur hexafluoride to create neutral thorium monofluoride.
(2) The neutral molecules are entrained in a supersonic expansion buffer gas of neon at room temperature, which then passes through a gate valve and a differential pumping chamber (separated fromthe neighboring vacuum chambers with a 3 mm diameter copper skimmer on each side) before it
reaches the Octagon. The first skimmer is charged to 30 V to reject charged particles.
(3) As the ions arrive at the center of the 10′′ diameter Octagon, we fire REMPI lasers through the 2.75′′
windows to ionize the neutral molecules state-selectively into the X 3 ∆1 (v = 0) vibronic manifold.
(4) More lasers go through other windows on the Octagon for state preparation into the eEDM-sensitive
X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = ±3/2) states.
(5) The ions are then ushered down into the Cold Tube, where we perform our π/2 pulses to do Ramsey
spectroscopy.
(6) At the end of the Ramsey sequence, the ions are ushered back into the Octagon where they are
addressed with lasers for state detection (e.g. state clean up, state-selective REMPD).
(7) The ions are then ejected through the Six-way Cross towards the microchannel plate assembly for
detection. The Accelerating Rings can be turned on to accelerate the ions even further, or to be
used as some crude form of ion optics.
Rings with special functions are highlighted with thick colored lines: yellow indicates Special Rings, whileorange indicate Quadrant Rings. See Figure 7.1 for a closer view on the Rings. More on these rings in
Section 7.4.2.

configuration, followed by a phosphor screen. The potential difference across the front and back sides of theplates are kept at slightly less than −2 kV, with the front plate floated to −2.6 kV for a higher ion impact
velocity. The phosphor screen is at 3 kV. We read out signals from the microchannel plates capacitively (see168
Ref [39] and Noah Schlossberger’s thesis), and we also use a camera to capture the phosphorescence fromthe phosphor screen.
Figure 7.11 shows the constructed b.B.B.. One can argue against using kapton tape close to the ionsin an ion trap experiment (and this comes back to bite us again and again when we observe patch charge
effects when we try to translate the ions along the b.B.B.), but that was the quickest solution to tidy theloose ends while keeping the electrodes insulated from one another. We find solace in the fact that the kapton
tapes are only used on the side of the b.B.B. that sits in the warm region of the setup, while the part thatsits inside the Cold Tube does not have the tapes.

169

7

6

8
4

1

2

3

5

Figure 7.10: Pictures of the b.B.B. setup. Top row: (1) The Octagon. The 2.75′′ windows mounted on
the Octagon are the ones we use to send in laser beams. The windows on the vertical axis are uncoated, andare used for the higher energy pulsed lasers. The rest of the windows are AR coated for the cw lasers. (2)The Cold Tube. A 8′′ diameter 316LN stainless steel tube that houses the b.B.B.. (3) The Cold Sleeve. An
aluminum cold jacket the hugs the Cold Tube, conducting the cold from the cryocooler. Made in-house at
JILA. (4) The “Science” location where the ions sit for BBR suppressed Ramsey evolution. The “Science”
region is deep enough inside the cold to suppress BBR related T1 type losses, and far enough from themagnetic cryocooler. (5) The cryocooler. (6) A graduate student for scale. (7) The Cradle. The Cold Tubeis suspended from the Cradle for structural support. (8) The Six-way Cross. One can see the turbomolecularpump attached on to the top flange. Some of the electrical feedthroughs for the electrodes can also be seenon the blank facing the camera. The full b.B.B. is way more congested than the picture shown here. This
picture was taken after we took out several components to bake the system. Not shown in this picture include(i) the vertical breadboard that sits between the Octagon and the Cradle for the optics into the Octagon, (ii)the household insulating foam and panels that hold the foam, and (iii) all the disconnected cables from thetrap drivers. Bottom row: views of the b.B.B. setup from the other side. One can also see the back side ofthe (black) vertical breadboard in these pictures. The gas canister on the right contains the neon buffer gas,seeded with sulfur hexafluoride. The vacuum chamber on the right is part of the Source Chamber. Bottom
left: without the foam. Bottom right: with the foam and panels that hold the foam.

170

Figure 7.11: Picture of the b.B.B., with yours truly and an approving-looking advisor for scale.
The outer most square rods are mechanical supports (affectionally called Spines). The Spine are terminatedon both ends with “End Plates”. The Spines have a square cross section with side lengths of 0.75′′ . Midway
through the b.B.B., where screwholes on the Spines can be seen, there is a thicker Ring that also acts asa structural support. The horizontally orientated rings are the Rings, and the Rods are lined up along
the Spines inside the Rings. Some kapton tape is used to keep the wiring tidy. Vacuum-compatible d-sub
connectors can be seen at the top of the picture. These connectors allow for error-free connection of thewires with the electronic feedthroughs at the vacuum flange, because we only need to verify the connection
once on the bench in a comfortable configuration, and just connect the d-sub connectors when we slide theb.B.B. into the confined space of the vacuum chamber on the busy optical tables. The whole assembly wasassembled (by graduate students Sun Yool Park and Anzhou Wang, with the help of JILA instrument makerCalvin Schwadron) inside a makeshift clean room with a HEPA filter for good vacuum practice. The top
end of the b.B.B. in this picture corresponds to the side facing the ion detector.

171
7.4.2

Trap loading & Ion kick out
We split two Rings right at the center of the Octagon into four quadrants each. The quadrants areoriented such that each Quadrant Ring (denoted as thick orange lines in Figure 7.9) has one quadrant in theup, down, away-from-Source-Chamber and towards-Source-Chamber directions. Using a voltage difference
across the away-towards pair, we can apply a stop pulse to stop the newly created ions that are moving at
800 m/s. We can use a combination of voltage differences across the up-down pair and the overall voltage
difference across the two Quadrant Rings to steer the ions during the stop pulse in the case of imperfectalignment of the electrodes w.r.t. molecular beam.
For the ion kick out, we apply a potential gradient along the axial direction with the Special Rings(denoted as thick yellow lines in Figure 7.9) and the Quadrant Rings. At the end of the kick out acceleration,
the ions travel at about 4 km/s towards the ion detector.
The rest of the Rings only serve to translate the ions in and out of the Cold Tube, and are connectedwith every fourth neighbor to reduce the number of independently controlled electrodes. Hence, we have a
total of 4 × 2 (Quadrant Rings) + 1 × 10 (Special Rings) + 1 × 4 (Translating Rings)7 = 22 independently
controlled Ring voltages.

7.4.3

Remarks on wisdom acquired
We started designing the b.B.B. at the start of the pandemic (circa April 2020). We cleared out somelab space and knocked down some walls to make space for the new setup a year later (circa April 2021). We
celebrated the “effective end of the pandemic” by painting the walls of our new lab space in July 2021, andstarted populating the new lab space with equipment in August 2021. We saw our first ions in the b.B.B. in
June 2022, one week after DAMOP. We have our first dissociation of ThF+ in the b.B.B. in October 2022.
Finally, we have our first Ramsey fringe in the b.B.B. in February 2023.
The b.B.B. design, construction, and fine tuning turned out to be a much bigger problem than
anticipated. I shall briefly mention some wisdom that we acquired along the way to inform/inspire future
generations of students and post-docs working on the design of the full Bucket Brigade.
7 Although we have only four independently controlled voltages for the Translating Rings, we have 41 such Rings in total.

172
7.4.3.1

Tolerance

The position tolerance on the Rings are very forgiving because of Laplace’s equation; any effect fromunintentional displacements of the Rings with spatially sinusoidal modulated voltages applied to them will
manifest as a higher spatial harmonic of the potential that is exponentially suppressed by the radial distancefrom the electrodes (see Section 7.2.2).
In comparison, the Rods have a much more stringent position tolerance. A simple COMSOL simulationsuggests that at if we demand a 0.1% Erot spatial homogeneity requirement along the trap center at the sizeof our ion cloud, we can only tolerate a 200 µm radial or 1 mm azimuthal displacement of the Rods.

7.4.3.2

Shielding effect of insulator spacers

The b.B.B. has a lot of macroscopically sized electrodes in close proximity. They need to be held
together for structural integrity, but insulated from one another for independent voltage control. One wouldnaı̈vely assume that using thin insulating spacers to, say, insulate Rods from the Spines would do little tofields seen by ions along the trap axis some centimeters away. However, fellow graduate student Anzhou
Wang quickly found out through COMSOL simulations that the dielectric insulators modify the field lines
sufficiently enough to change the field strength along the trap axis. We can compensate for this effect byintroducing a metallic spacer along with the insulating spacer. The dimensions chosen for the insulating andmetallic spacers were guided by COMSOL simulation.

7.4.3.3

Patch charges and baking

During the early days of moving ions up and down the b.B.B., we noticed that there was a bad patcharound 30 cm into the Cold Tube (see Figure 7.12). We could not move ions past that location without
ramping up the radial confinement of the trap. Initially we guessed that perhaps some of the electrodes atthat region were not connected correctly, but it turned out in the end that a light bake of the Cold Tube atabout 100 ◦ C over the weekend solved the problem. We do not claim to understand what happened fully,
but the story is that there were some patch charges in that region that got baked away.

173

Figure 7.12: Losing ions past 30 cm into the b.B.B.. Data from 7 September 2022, RID 20202. The
blue trace corresponds to moving the ions into various locations of the b.B.B. (in units of Ncycles, where
each Ncycle corresponds to translating the ions in by one bucket, or 8 cm) in 300 ms, holding them there for
10 ms, before moving them back out into the Octagon again in 300 ms. The vertical axis is the area under
the time-of-flight curve, corresponding to the number of ions we retain after the operation. The yellow tracecorresponds to holding the ions at the Octagon for the same amount of time as the blue trace to act as acontrol. Note that we lose ions completely by the time we move to Ncycle = 3.8, corresponding to 30 cm
into the b.B.B.. This problem was fixed by baking the setup.

7.4.3.4

Long time of flightWith a distance of about 70 cm between the trap and the ion detector, and a kick-out velocity of
about 4 km/s (20 eV of kinetic energy for Th+ ), any stray fields during the 180 µs of time of flight could
steer our ions in unexpected ways. For reasons not fully understood, we require a slight steering of the ionsalong the x direction (Eastward direction in the lab, towards the Source Chamber w.r.t. the setup) to let theions hit the detector. We also see that the positions of the ions on our detector are mapped from positions(velocities) of the ions in the trap along the x (y, vertical) directions. The kick out angle that the steering
makes w.r.t. the trap axis is about 1◦ , which could be explained by (i) slight misalignment of the b.B.B.
w.r.t. the ion detector, or (ii) small patch charges on the side of the chamber (e.g. window on the Octagon
on the other side of the molecular beam) that break azimuthal symmetry.

7.4.3.5

Recessed ion detector

Towards the end of the spectroscopy of ThF+ before we started work on the b.B.B., we obtained a
new microchannel plate assembly to replace the one that was broken. We wanted a bigger microchannel

174
plate (4 cm) than our broken one (2 cm), but to make the entire setup fit into a 4.5′′ flange (limited by our
setup back then), the assembly was about 1′′ recessed from the end of the flange to make wiring easier (seeFigure 7.13). This turned out to introduce a lot of focusing effects on our ion cloud. At the time of thisTop MCP
Bottom MCP

44 mm

1.14700

0.7500300

Cylindrical shield
Mesh (35 mesh points)

Figure 7.13: Picture of the microchannel plate assembly. The assembly (Beam Imaging Solutions
BOS-40) is housed inside a 4.5′′ flange, with the microchannel plates about 1′′ below the surface of the flange
to give access to electrical connections in this tight space. There is a mesh in front of the microchannel
plate. A cylindrical shield that is electrically connected to the mesh shields the incoming ions from thefields from the electrical feedthroughs. Top: Going clockwise from the left most feedthrough, the electricalconnections are made to (i) phosphor screen, (ii) back side of the back microchannel plate, (iii) where thetwo microchannel plates meet, (iv) front side of the front microchannel plate. The unused feedthrough isconnected to the mesh (and grid) after taking this photograph. Bottom: a cross section view from the sidewith the relevant dimensions. MCP stands for microchannel plate.

writing, there is a plan to redesign the microchannel plate assembly to reduce the aforementioned effect.

175

7.5

Electronics design
Now that we have seen how the b.B.B. setup looks like, it is time to discuss how we drive the voltages

on our electrodes. First we shall look at the Rods (Section 7.5.1), then the Rings (Section 7.5.2).

7.5.1

Rod driver
Studies on systematics on the Gen. II setup shed light on the importance of stomping on the higher

harmonic of Vrot [36]. Furthermore, our Rods have substantial capacitance, and op-amps do not have infinite
slew rates. Hence, we decide to design a tank circuit to drive the voltages on the Rods. As will be elaborated
on in the following sections, this tank circuit involves a transformer with weak coupling between the primaryand secondary coils. There will be four such tank circuits to drive four pairs of Rods. Each tank circuit
drives a pair of Rods that are on opposite sides of the setup, e.g. Rods 1 and 5 in Figure 7.4 form a pair. A
typical tank circuit will look like that shown in Figure 7.14.

VRF

Rp

CRod
Rs
Vo

Lp

Monitoring readout

Ls

M

CRod

Figure 7.14: Tank circuit used to drive voltages on the Rods. Voltage Vo will be sent into the primarycircuit with op-amps, with a current-limiting resistor Rp . The inductances of the primary and secondary coils
are Lp and Ls , respectively. The inductors are coupled through a mutual inductance M . A common-mode
RF voltage VRF will be sent to the rods via a center-tap on the secondary inductor. A series of potential
dividing resistors with total resistance Rs will be used to readout the voltage for monitoring purposes. The
Rods are modelled as capacitors with capacitance CRod .

We will first begin with a foreword on some of the design considerations that we have learned froman early prototype. We then discuss transformers with perfect coupling to borrow intuition from textbook
examples. This is followed by a discussion of transformers with imperfect coupling. Finally, we will briefly

176
discuss the final design.

7.5.1.1

Foreword on transformer design

Before we launch into detailed discussion of the transformer, we shall look at a piece of wisdom (or ashortcut) that we acquired from an early prototype of the transformer. We will also introduce a neat trick
to help with analyzing circuit diagrams with transformers.
Air core

We used a ferrite core to link the fluxes between the primary and secondary coils in an

early prototype of the transformer. However, there are two problems with it. First, saturation effects in theferrite core by the flux from the primary coil could result in distorted waveforms with higher harmonics on
the secondary. The saturation effects do not have to be severe for this to be a problem for our systematicstudies; a slightly saturated ferrite core could already give us measurable systematic effects that have got
to do with second harmonics in Erot (see Section 8.4). Second, the ferrite core heated up during operation,and it turned out that the permeability of the ferrite core has a substantial temperature dependence. This
prompted a need for a non-trivial temperature control system. We decided to avoid the problem totally byusing two concentric solenoids as our primary and secondary coils, with an air core coupling the two.
For a system like this, we have governing equations (in the Fourier domain):where j =√Vp = jωLp ip + jωM is(7.9a)

Vs = jωM ip + jωLs is ,(7.9b)−1, M is the mutual inductance, V is the voltage across each inductor, ω is the angularfrequency of the voltage signal, L is the inductance, i is the current going through the inductor, and thesubscripts ‘p’ and ‘s’ stand for the primary and secondary inductors, respectively.
T-model equivalent circuit

Equations 7.9 can be massaged into a T-model equivalent circuit

form, enabling easier analysis of the circuit:
Vp = jω(Lp − M )ip + jωM (ip + is )(7.10a)Vs = jωM (ip + is ) + jω(Ls − M )is .

(7.10b)The equivalent circuit is shown in Figure 7.15.

177
Lp − M
ip

Vp

ip

is

Lp

Ls

Vs

≡

(Lp − M )/2

Ls − M

Vp

ip

is

ip + isVs

M≡(Ls − M )/2

ip + is

Vp

is

Vs

M

M
(Lp − M )/2(Ls − M )/2

Figure 7.15: T-model equivalent circuit. A transformer (L.H.S.) can be modelled as a T-model equivalentcircuit (R.H.S.).

7.5.1.2

Transformer with perfect coupling

The physics behind a transformer with perfect coupling, i.e., M =p

Lp Ls , is well understood and isincluded in standard undergraduate courses on introduction to circuits. Defining the turns ratio n = Ns /Np ,we know that Vs = nVp from Faraday’s law of induction, and is = ip /n from conservation of energy. In fact,
we can show this with the T-model equivalent circuit. For a loaded transformer with load impedance ZL ,the total impedance seen by the source on the primary side is given by:
Ztot = jω(Lp − M ) +1
1
1jωM + jω(Ls −M )+ZL=

ω 2 M 2 − ω 2 Lp Ls + jZL ωLp.
ZL + jωLs(7.11)The voltage across the load on the secondary side is related to that on the primary by (through Ohm’s law):
1
1
1
Vs
ZL
M ZL
jωM + jω(Ls −M )+ZL==
Vpjω(Ls − M ) + ZL
Ztot
Lp ZL + jω(Lp Ls − M 2 )and we can see that Equation 7.12 reduces to Vs /Vp →p(7.12)

Ls /Lp = n in the case of perfect coupling, andwhere we use the relation8 Ls /Lp = n2 .
Equation 7.11 deserves some special attention before we move on to talk about transformers withimperfect coupling. At perfect coupling, and in the limit where |ωLs | ≫ |ZL |, we see that the expression
reduces to Ztot → ZL /n2 , where we make use of Ls /Lp = n2 . We can interpret the limit of |ωLs | ≫ |ZL | tobe the case where the load does not change the voltage/current of the circuit on the secondary side too much.
This means that for small loads on the secondary side, the circuit is equivalent to replacing the inductor on
the primary side with a load with impedance ZL /n2 , and ignoring the secondary side altogether. This may
8 This assumes that the inductors of both coils have geometries similar to each other, or if the coils share a ferrite core, suchthat the only difference between the two only come from the winding numbers.

178
be useful for estimating requirements on current flowing in the primary circuit. Also as a sanity check, inthe opposite limit |ωLs | ≪ |ZL |, where the secondary side is like an open circuit, we have Ztot → jωLp , as
expected.

7.5.1.3

Transformer with imperfect coupling

Going back to Equation 7.12, and introducing the general expression M = K

p

Lp Ls , where K ∈ [0, 1]

is the coupling constant, we have:
Vs
=
Vp

s

Ls
KZL
Lp ZL + jωLs (1 − K 2 )(7.13)We see that the denominator on the right hand side can be made small, which would make Vs large comparedto Vp , if the load is capacitive. Note that this can happen even if |ZL | is not small. In other words, for acarefully designed circuit, we can “step up” the voltage across a capacitive load on the secondary side withimperfectly coupled primary and secondary coils. Furthermore, this “stepping up” is frequency dependent
because of how ω goes into the denominator. This is essentially a tank circuit, which is just what we needto stomp on the second harmonic of Vrot that is plaguing the Gen. II systematics studies.
The optimum coupling that gives the maximum |Vs /Vp | for a fixed ω occurs at:9sp
|ZL |2 + ωLs [ωLs + 2Im(ZL )]K=≈ 0.25,ωLs(7.14)

where typical parameters in our setup are Ls = 10 mH, ω = 2π × 150 kHz, ZL = 1/ (1/Rs + jωC), with
Rs = 500 kΩ and C = CRod /2 = 120 pF. Hence, we see that we need to design bad coupling into thetransformer.
Designing imperfectness into a system is easy. We decide to have the following: a 400-turn secondary
coil in the form of a solenoid with diameter 4.5′′ and length about 14 cm using AWG 28 magnet wire, and a
400-turn primary coil in the form of a circular wire loop with average diameter 1.7′′ using AWG 30 magnet
wire. The primary coil sits within the secondary, and shares the same axis with the secondary. Estimatesof the inductances of the coils are about 10 mH each, and the mutual inductance near 2.5 mH. Pictures ofa prototype are shown in Figure 7.9.
9 As we will see in Equation 7.15, the resonant frequency has a K dependence. The fact that there is an optimal K for that

179

Figure 7.16: Pictures of a prototype transformer. Left: overview of the transformer, showing thesecondary coil, with keyboard and mouse for scale. Right: view of the primary coil inside the secondary coil.
The primary coil is mounted on a rod that can move along the axis of the transformer. This allows us toshim away the RF pickup on the primary coil from the center tap on the secondary. The primary coil has an
inner diameter of 1.5′′ and outer diameter of 1.9′′ . Measured primary, secondary, and mutual inductances
are 10 mH, 9.9 mH, and 2.0 mH, respectively.

Tests of the prototype indicate very good agreement with our model, and a second harmonic suppression of at least 34 dB. Of note, we have a few design guidelines:(1) There are two resonances: parallel and series. The parallel resonance happens at frequency ωparallel =√1/ Ls C, and it is where the impedance seen by the source at the primary side is the largest, i.e.,
smallest magnitude of driving current on the primary side. The series resonance happens ats
1 − 2RL2sC (1 − K 2 )s,(7.15)
ωseries =Ls C(1 − K 2 )and it is where the ratio |Vs /Vp | is the largest; we want to be operating here. This can be inferredfrom substituting M = K

p

Lp Ls into Equation 7.11 to get:
Ztot = jωLp

ZL + jωLs (1 − K 2 ),ZL + jωLs

and recognizing that ZL = 1/(1/Rs + jωC); parallel (series) resonance occurs when the denominator (numerator) is minimum. Evidently, for fixed Ls and K, we can fine-tune the series resonant
gives a maximum |Vs /Vp | comes from the resonance curve moving to higher frequencies with increasing K, and the optimal K
is where the resonance peak overlaps with the desired ω.

180frequency by adjusting C.
(2) At Rs ≫pp
Ls /C ≈ 9 kΩ, we have ωseries ≈ 1/ Ls C(1 − K 2 ), so changing the value of Rs onlychanges the Q-factor10 (Q ∼ ωseries Rs C), and not the series resonant frequency. Hence, we canfine-tune the Q-factor by adjusting Rs .
(3) All the design guidelines are based on analyses without the current-limiting resistor Rp , because theexpressions are much cleaner this way. The presence of Rp moves the resonant frequencies a little,
for instance, it moves the series resonance to a slightly lower frequency, and also reduces the Q-factor
by a bit. Vo = Vp + Rp ip (see Figures 7.14 and 7.15) will also have to be higher than Vp to drive
current through Rp . Since op-amps do not work with arbitrarily high voltage rails, and that they
can only draw finite current, there are some design constraints for Rp .
(4) To drive all eight Rods, we have four transformers to drive four pairs of Rods. In the testing of
the transformers, we discovered that there are capacitive couplings between neighboring Rods. This
results in capacitive couplings between transformers on the order of ∼ 7 pF. It is straightforward
to expand the above analyses to include the effects of coupled transformers using the principle ofsuperposition11 . However, the expressions are so complicated looking that they do no shed more lightthan the expressions for uncoupled transformers above, so we do not show them here. The generalfeatures of how the resonance depends on various parameters remain largely the same. In the designof the final system, we iterate between analytical simulations using our model (cross-checked with
LTspice) and testing the transformers in the actual setup.

10 There is a correction factor on the order of unity at the parameters that we are using.

culated power)/(Dissipated power), i.e., Q =The Q-factor is defined as (Cir-2
Lp Ls Rs K 2 ωseries
2
2
2
2 2 ≈
series +Rp [Rs (1−Ls Cωseries ) +ωs Ls ]Vs2 /[1/(ωseries C)]= ωseries Rs C ×
Vs2 /Rs +i2
L p L s Rs K 2 ω 2
p Rp

ωseries Rs C × 0.4. One can see that in the absence of the current limiting resistor Rp on the primary side, the correction factor
goes to 1.
11 Treating the voltage sources of “undriven” transformers as shorts.

181
7.5.1.4

Final design

It takes more than one person to construct and fine tune the transformers. Fellow graduate student
Noah Schlossberger and undergraduate Antonio Vigil also played huge roles. Here, I will just broadly lay out
the steps we took to get to the final product, and I would recommend interested readers to refer to Noah’s
thesis for the details.
We tested out the transformers individually and their performances matched very well with the modelthat we mentioned in the previous sections. However, as alluded on earlier, when the transformers are put
together the transformers talk to each other through capacitive coupling, resulting in output waveforms withmodified amplitudes and phases. As Noah details in his thesis, we overcome this problem by calibrating
a “transfer matrix” that takes in desired output voltage amplitudes and phases on the Rods, and spitout recommended input voltage amplitudes and phases from the op-amps. This is done by measuring thedifference voltage across each transformer though a precision monitor circuit box that sits right outside thevacuum flange with the electrical feedthroughs to the Rods. With this, we manage to shim the amplitudes
down to less than 0.1% and the phases down to less than 0.5◦ .
The transformers certainly do not know when to start driving the Rods, so they are only one set ofcomponents in a bigger ecosystem of what constitutes the Rod driver. Figure 7.17 shows an overview of thewhole Rod driver ecosystem.

182

Vacuum chamber
×4
DDS boxTransformer box

Monitor box

Oscilloscope
DAC

ARTIQ (Sinara) timing system

Control computer

Figure 7.17: Schematic of an overview of the Rod driver. A control computer programs (i) the DDS
box on the relevant (low voltage) waveforms to output for the RF frequencies (VRF ) and the rotating electricfield (Vrot ), (ii) the Sinara Zotino DAC for any DC shims we wish to apply to the Rods (VDC ), and (iii)the Sinara timing system (more details in Chapter 9) to synchronize the timing of the DDS and DAC. The
output from the DDS and DAC go into the transformer box, where VRF and Vrot are amplified with HV opamps PA94. VRF is summed onto Vrot on the secondary coil, whereupon VDC is added on through resistors.
The HV outputs are sent to the monitor box, where we have a voltage divider for us to readout the voltages
safely at low voltages on an oscilloscope; these can then be fed back into the system through our control
computer. The HV outputs then go from the monitor box to the Rods through the feedthroughs on thevacuum chamber. The flow of all communication signals and low voltages are indicated by unfilled arrows,
and high voltages are indicated with filled arrows. The schematic is drawn for one Rod-pair for brevity, butthere are four Rod-pairs for a total of eight Rods; we use four separate channels in the DDS and DAC boxes
to feed into four sets of transformer boxes, one for each Rod-pair. The DDSes are of the same design used in
the Gen. II JILA eEDM experiment. They are designed in-house by electronics shop staff Felix Vietmeyer,
and put together with the help of electronics shop staff James Fung-A-Fat. The PA94 HV op-amps in thetransformer box are cooled with processed chilled water.

183
7.5.2

Ring driver
As alluded to earlier on in Section 7.4.2, we use 22 independently controlled Ring voltages to (i) stop

our ions into our trap, (ii) move our ions within the trap transversely to overlap with the laser beams, (iii)move our ions within the b.B.B. axially in and out of the cold region, and (iv) kick our ions towards our ion
detector. All the voltage control is governed by our Ring driver. A schematic overview of our Ring driver isshown in Figure 7.18.

Special + Quadrant Rings

Consolidator box

Translating Rings···

HV op-amp box
Buffer op-amp
DAC

Vacuum chamber

ARTIQ (Sinara) timing system

Control computer

Figure 7.18: Schematic of an overview of the Ring driver. Control signal and low voltages are
represented by unfilled arrows, while the high voltages are indicated by filled arrows. A control computer
programs the FPGA based Sinara timing system, which then sets the voltages on the 64-channel DAC (Sinara
Zotino). Two sets of voltages are sent to a “consolidator box” through SCSI cables. The “consolidator box”
was designed in-house by electronics shop staff Terry Brown. It consolidates the voltages, sends them throughbuffer op-amps, and performs a simultaneous switch between the two sets of voltages. This allows for fastswitching between two sets of voltages for, say, ion stopping voltages and trapping voltages; the DACs are
not fast enough to do the switch on a ∼ 10 ns time scale. The Sinara timing system controls the timing of
the switch. The output from the “consolidator box” is amplified with PA94 HV op-amps before being sent
to individual Special and Quadrant Rings (see Figure 7.9), each Ring having its own electrical feedthroughon the vacuum flange. A third set of voltages are sent from the DACs into a set of buffer op-amps, before
going to the Translating Rings. In our current setup, the buffer op-amps live inside the “consolidator box”.
For the Translating Rings, there are only four electrical feedthroughs on the vacuum flange. We connect
every fourth Translation Ring to the same electrical bus for a total of four distinct buses. In our current
setup, we use the Spines as the buses, being careful to electrically insulate the Translating Rings from theSpines with the irrelevant voltages. The PA94 HV op-amps are cooled with processed chilled water.

184

7.6

Parametric excitation
Before we close off this chapter on the electric fields inside the b.B.B., let us look at a feature newly

introduced into the JILA eEDM toolkit: parametric excitation of the ion motion inside our trap. This idea
has been inspired by our fellow ion trappers, see for example Refs. [84, 85].

7.6.1

Motivation
Diagnosing the trap is a key part in ensuring success in our eEDM measurement. We have been able

to diagnose our ion trap very well in the first two generations of the JILA eEDM experiment. Here is a quicksummary of how we have been diagnosing our trap. We give our ions an intentional kick in the directionthat we are interested in. This will make the ions slosh around the trap at the respective secular frequency.
We then eject our ions out of the trap, and the position that they fall onto the ion detector will move aroundat the secular frequency of interest. We can then extract the frequencies of the trap by repeating this for all
three principle axes of the trap.
However, the above protocol may not work in Gen. III. We may want to diagnose parts deep inside ourb.B.B., where we do not have access to electrodes that can kick our ions onto the ion detector. Fortunately
we have an extra channel on our DDS that we can feed onto our Rods, giving us access to parametric
excitation at all locations of the b.B.B.. To diagnose any part of the b.B.B., we just need to (i) create ions in
the Octagon like we always do, (ii) move the ions to that desired location, (iii) apply parametric excitation,(iv) move the remaining ions back to the Octagon, (v) eject them towards the ion detector like we always do,and (vi) repeat by scanning the frequency of the parametric excitation, and look for dips in the remaining
ion numbers. We shall now see how parametric excitation works.

7.6.2

Mechanism
We introduce a time-varying quadrupole potential in the radial direction to trap our ions radially.

The frequency of this quadrupole potential is at 50 kHz. As a result of the ponderomotive potential, theions have a radial secular trap frequency of about 2 kHz. Suppose we add on top of the 50 kHz quadrupole
potential a 4 kHz quadrupole potential (twice the secular frequency), we can excite the radial mode of

185
motion resonantly, and amplify the secular motion of the ions. For generality, let us suppose that theexciting quadrupole potential is at an angle θ w.r.t. the principle axes of the secular motion, the equation of
motion is (limiting ourselves to the 2D x-y plane):F= − ωx2 x − ωy2 ym+ α cos Ωt [(x cos θ + y sin θ)x̂ + (x sin θ − y cos θ)ŷ] ,(7.16)

where ωx,y are the angular secular frequencies along the principle axes x and y, α is a measure of couplingstrength of the excitation to the ions, and Ω is the exciting angular frequency. The first line is just a particle
in a harmonic trap, and the second line is the additional quadrupole potential. We can rewrite the above as:

  
 



ẍ(t)  ωx2 = − 

  
ẍ(t)
0

0
cos θ
 + α cos Ωt 


ωy2
sin θ

sin θ  x(t)
 
.
 

− cos θy(t)(7.17)

We can solve the above perturbatively for small α using the method of series expansion in powers of α (weare solving for x(t) here, but the same applies for y(t) by symmetry):x(t) = x(0) (t) + αx(t) + O(α2 ),(7.18)

where the subscript (n) indicates the nth order term in the expansion series.
The zeroth order term is:x(0) (t) = X0 cos (ωx t + ϕx ) ,where X0 and ϕx are the amplitude and initial phase. This is just the solution to the ion motion along thex direction in a harmonic trap.
The equation governing the first order term is:ẍ(1) (t) = −ωx2 x(1) (t) + cos Ωt x(0) (t) cos θ + y(0) (t) sin θ
(


1
2
X0 cos θ cos (∆x t + ϕx ) + cos (Σx t + ϕx )= −ωx x(1) (t) +2)
+ Y0 sin θ cos (∆y t + ϕy ) + cos (Σy t + ϕy ) ,where ∆i := Ω − ωi and Σi := Ω + ωi . The equation is just an oscillator with a forced motion, and the186inhomogeneous solution to the above is:(1cos (∆x t + ϕx ) cos (Σx t + ϕx )x(1),inhomo. (t) =X0 cos θ+2Ω(Ω − 2ωx )Ω(Ω + 2ωx ))cos (Σy t + ϕy )cos (∆y t + ϕy ).
++ Y0 sin θ[Ω − (ωx + ωy )][Ω + (ωx − ωy )] [Ω + (ωx + ωy )][Ω − (ωx − ωy )](7.19)The homogeneous solution is grouped together with x(0) (t). Let us take a moment to analyze the above
solution.
The first term in the square brackets with the X0 term has a Ω − 2ωx term in the denominator.
This means that our ions will have huge amplitudes in the x direction when Ω = 2ωx , i.e. when the driving
frequency is twice the trap secular frequency. This is the resonant excitation.
The first term in the square brackets with the Y0 term has a Ω − (ωx + ωy ) term in the denominator.
This means that if Ω happens to be the sum of the two secular frequencies, our ions will also have motion
with a huge amplitude. This is the crossover term.
The picture above assumes that our trap is harmonic to an infinite range. In practice, our trap starts
to become anharmonic as we move a few centimeters away from the trap axis, and the ions will start to spillout of the trap because our trap depth is finite.
In summary, if we sweep the driving frequency Ω, we expect to see ion loss when Ω is either 2ωi orωx + ωy . If the secular frequency is different in x than in y (as is usually the case), we expect to see three
dips in ion numbers, two corresponding to the secular frequencies, and one corresponding to the crossover.
Figure 7.19 shows an example of data from such a scan.

7.6.3

Further discussion
As alluded to earlier on in this section, parametric excitation was developed as a diagnostic tool for

us to understand our trap at various locations inside the b.B.B. where we do not have access to electrodesthat can kick our ions. An example of how we can use parametric excitation to diagnose our b.B.B. is shownin Figure 7.20.
The dosage ( voltage on the Rods × time applied ) can be adjusted to pick out certain details. For

187

Figure 7.19: Number of ions remaining in the trap after parametric excitation. Data was taken
at the Octagon on 25 October 2022, RID 25916. The horizontal axis is the excitation frequency Ω/(2π),
and the vertical axis is the area under the curve from our ion detector, corresponding to number of ionsdetected. The ions are exposed to the parametric excitation for 50 ms, with a peak voltage of 150 mV on theRods (compared to 25.6 V for the radial confinement). The dips at 3.2 kHz and 3.8 kHz correspond to thesecular frequencies for the x and y directions. We confirmed this by kicking our ions intentionally (without
the parametric excitation) in the x and y directions within the Octagon, and ejecting them onto our ion
detector to observe their sloshing frequencies. The dip at 3.5 kHz is the crossover dip. We can employ either
the kicked ion method or parametric excitation to learn about our trap at the Octagon, but we can only use
parametric excitation at other locations inside the b.B.B..

example, one can choose to reduce the voltage if the parametric excitation is too saturated (dip is too broadand goes all the way down to zero), or to reduce the voltage but increase total excitation time to make thewidth of the dip narrower.
We now have a multiplexed ion trap, the b.B.B., and we have all the electronics and electrodes toperform the multiplexing. We are now in a good position to move on to the next chapter to investigate
various methods of using the b.B.B. to offset our Ramsey frequency away from zero. We shall also see why
we want to do this in the next chapter.

188

Figure 7.20: Using parametric excitation to diagnose the trap. Here we wanted to understand if
turning on Erot in the b.B.B. would introduce ponderomotive potentials that could change the trap secularfrequencies. We wanted to understand this both at the Octagon (chop named “Octagon”), and in the Cold
Tube where we intend to perform our long Ramsey interrogation (chop named “Science”). The first row
corresponds to a background experiment where we do not turn Erot and the parametric excitation on. The
last row is a control experiment where turn on parametric excitation but not Erot . There is only one broad
dip at around 8.2 kHz because we ramped our trap up to about 4 kHz. The second (third) row corresponds
to having parametric excitation on, and Erot rotating in the anti-clockwise (clockwise) direction as seen fromthe Cold Tube towards the ion detector. Note that the single broad dip from the last row has split into
three different dips with Erot turned on. This suggests that Erot is doing something to our trap frequencies
by modifying it through ponderomotive potentials. Also note that the second and third rows look different,
implying that having Erot rotating in one direction changes our trap differently than having Erot rotatingin the other direction. This set of data was taken during the early days (18 April 2023) of understanding
our trap. The system is much more well behaved now; this plot is just to show a possible application ofparametric excitation in our new setup.

Chapter 8Generating f0 in the Bucket Brigade

8.1

Introduction
To measure the eEDM, we need to measure the energy difference between the stretched Zeeman statesof the “upper Stark doublet” and the “lower Stark doublet” (see Figure 1.4). We apply a deliberate frequency
offset (f0 ) to the states within the doublets and subtract away this known additional energy difference fromthe measured quantity to obtain the contribution from the eEDM. We do this for two reasons.
First, we expect the eEDM to be very small, we expect a very small energy difference, corresponding
to a very small Ramsey frequency, which is hard to measure because of the 1/f noise (the power spectrum
density of noise in our setup typically scales with frequency as 1/f ). Hence, we bias the energy difference
between the stretched states deliberately to a non-zero value f0 .
Second, the polarization axis of our molecules rotates, so there is a non-inertial-frame coupling (∆)
in the molecules’ rotating frame that couples the Zeeman states within the doublets (see Figure 1.4). We
apply a frequency offset f0 ≫ ∆ so that each Zeeman state within the doublets remains a good eigenstate.
In the first two generations of the JILA eEDM experiment, we introduce the frequency offset between
the stretched Zeeman states within the doublets through Zeeman shifts with applied magnetic fields. Thisis certainly not the only way to introduce frequency offsets in our experiment. We will be investigating
various plans of introducing frequency offsets in the Bucket Brigade in this chapter. Many of these plans
were inspired by systematic effects that we saw in the first two generations of the JILA eEDM experiment;
we intend to turn these bugs into features. The feasibility of each of these plans will be judged based on the190following criteria:
Criterion 8.1 (Spatial homogeneity across Bucket Brigade) The Ramsey frequency, f0 , seen by an
ion packet should not change too much1 across the whole Bucket Brigade from shot to shot.

Criterion 8.2 (Spatial homogeneity within ion cloud) Our ion cloud has a non-zero spatial extent(Equation 6.21). Whatever mechanism we use to generate f0 , we care about its spatial variation within a±1 cm radius region (looking out to 2σ radius to include 95% of all ions) around the cloud center. For an f0of 50 Hz and a free evolution time of 20 s, we have 1000 such Ramsey oscillations. For a spread in frequencies
corresponding to 10% of an oscillation at the end of 20 s, we would need < 0.01% difference in f0 as seen by(i) the ensemble of ions in each ion cloud, and (ii) different ion packets from shot to shot.

We did not get creative with names this time for the various plans. The following is a quick summaryof the various plans for easy comparison.
Plan A Zeeman shift with an axial magnetic quadrupole, B = Baxgrad (x, y, −2z), and rotational micromotion; used in the first two generations of the JILA eEDM experiment.
Plan B Berry’s phase; an adaptation of a systematic effect seen in the first two generations of the JILA
eEDM experiment.
Plan C Zeeman shift with uniform ambient magnetic field and an oscillating uniform electric field thatis going as a second harmonic of Erot ; an adaptation of a systematic effect seen in the first twogenerations of the JILA eEDM experiment.
Plan D Zeeman shift with transverse magnetic quadrupole field, B = Btransgrad (x − y), and an elliptical Erot .
Plan E Zeeman shift with rotating magnetic field introduced by currents flowing along the electrodes.
We will be focusing more effort in analyzing Plans A and B in the context of the Bucket Brigade, using
them as classic examples of how we can introduce a frequency offset with the Zeeman shift and Berry’s phase,1 f can be allowed to change a little throughout the whole Bucket Brigade; if all the ion packets see the same variation0
as they go through the same stretch of Bucket Brigade, the variations in f0 will not result in additional uncertainties in themeasured Ramsey frequencies.

191
respectively. We will then introduce Plans C to E conceptually, which are ingenious ways of introducing
Zeeman frequency offsets differently from Plan A.

8.2

Plan A: Co-moving axial magnetic quadrupole

8.2.1

Working principle
In the first two generations of the JILA eEDM experiment, we used a Zeeman effect to give us afrequency offset between the two stretched Zeeman states in a doublet. The physics behind this effect has
been described in detail in the theses of previous members of the group who worked on the Gens. I and II
experiments. A brief summary is as follows.
Current-carrying coils in the anti-Helmholtz configuration generate a magnetic field of the form B =Baxgrad (x + y − 2z) (see Figure 8.1). An electric field rotating in the x-y plane polarizes our ions in theSide view

Top view
Erot

R

rrot
z
d

x

y

r
ρ0
y
z

y
x

z

x

Figure 8.1: Illustration of the coils and magnetic field configurations used to generate a Zeeman
offset frequency. In the side view, the blue arrows indicate the directions of the current flowing inside thetwo respective coils (for example, the current in the top coil is going clockwise looking from the top). The
dashed lines indicate the fields generated by the coils. The spacing
between the two coils is d, and the radius√of each coil is R. For an anti-Helmholtz configuration, d = 3R. In the top view, the arrows indicate theradial magnetic field along the x-y plane. The length of each arrow represents the strength of the field atthat position. In the right panel, r is the radial position vector of an ion from the trap center. This radial
position vector can ve decomposed into a constant vector ρ 0 and a rotating vector rrot . The rotating vector
rrot is anti-parallel with the direction of the rotating polarizing electric field Erot . As Erot rotates in the x-yplane, r traces out a circle indicated by the dotted line. In the rotating frame of our molecules (defined by
Erot ), rrot is non-rotating, and ρ 0 rotates.

eErot
rotating frame, and induces a rotational micromotion with radius rrot = mω2 . In the rotating frame of therotions performing rotational micromotion, they see a time-varying magnetic field of the form:
B(t) = −Baxgrad rrot n̂ + Baxgrad ρ0 (cos ωrot t x̂ + sin ωrot t ŷ),

192
where n̂ indicates the polarization axis, and ρ0 is the radial position of the ion within the trap.2

The

contribution of the time-varying part of the magnetic field averages out to zero in one cycle of the rotational
micromotion. Only the first term contributes to the net Zeeman offset. Therefore, there is a net Zeeman
offset between the stretched states of the form:
hf0 = 3gµB Baxgrad rrot =: 3gµB Brot .

(8.1)

We call this method of introducing a Zeeman frequency offset between the two Zeeman stretchedstates within each doublet: Plan A.

8.2.2

In the Bucket Brigade
As alluded to in Section 5.3, the Bucket Brigade consists of a conveyor belt of ion traps, where theradial confinement of ions is provided by Rods with RF-fields, and the axial confinement is provided by
the Rings, acting as “end caps” of the usual Paul traps. The ions are moved down the Bucket Brigade bymodulating the potentials on the Rings periodically with a fixed and well chosen phase difference between
neighboring Rings. We now try to implement a naı̈ve version of Plan A in the Bucket Brigade, where eachbucket of ions has its own “anti-Helmholtz” coils translating along with it.3

We do not intend to do thiswith anti-Helmholtz coils that move physically with the ions. Instead, we plan to have arrays of current
carrying coils, with currents that are modulated periodically in time and space with a well chosen phasedifference between neighboring coils.
There are many length related parameters in this analysis. Before we proceed further, we shallsummarize the definitions of some variables here:
Distance along the translation direction/trap axis, z. We define the z axis to be the direction along
which ions translate, with z = 0 at the center of the Bucket Brigade.
2 The ion could be performing secular motion within the trap, so ρ is actually a time-varying variable, but the time scale0

at which this is happening (1 kHz) is slow compared to the rotational micromotion (150 kHz), so ρ0 can be treated to betime-independent by separation of time scales.
3 Here we draw the distinction between anti-Helmholtz and “anti-Helmholtz” in quotes. The former refers to the classic
anti-Helmholtz√ coil pair (e.g. left panel of Figure 8.1) with a fixed ratio between the coil separation distance (d) and coil radius(R) at d = 3R. This fixed ratio gives ∂zk Bz (z = 0) = 0 for k ∈ {0, 2, 3, 4} for a pair of coils centered at z = 0 with theirsymmetry axis pointing along the z direction. The latter one in quotes refers to the combined geometry and current amplitudeconfiguration of a set of coils that gives the condition ∂zk Bz (z = 0) = 0 for k ∈ {0, 2, 3, 4}.

193
Axial position of the nth translating ion packet, Zn (t). For an ion packet that is translating at speed
vtrans , then we have Zn (t) = vtrans t. The n index is just a placeholder, but for concreteness, we caythat the first packet that we send in corresponds to n = 1.
Radius of the coils, R. We work with the simple case where all the coils have the same radius R.
Distance between the two coils in the classic anti-Helmholtz setup, d. In the classic anti-Helmholtz
setup, there are only two coils. These coils have counter-propagating currents, and they are separated
by the distance d (see Figure 8.1 for the configuration). In an anti-Helmholtz configuration, there isa special relation between the distance between the coils and the radius of the coils, d =√3R, where

the axial gradients (and higher order) of the axial magnetic field is zero up to and including thefourth order except for the first order gradient, i.e. ∂zk Bz (z) = 0 for k ∈ {0, 2, 3, 4}. Symmetry of the
setup makes all the even order derivatives zero. The d =√3R condition allows for the coils centeredaround z = 0 to give a zero third order gradient of the axial field ∂z3 Bz (z = 0) = 0. This means that
there is a substantial volume of region where there is a very homogeneous first order axial magnetic
field gradient; B = Baxgrad (x, y, −2z) is a good approximation within a large volume.4
Distance between two coils in the Plan A setup, l. In contrast to the classic anti-Helmholtz setup,
we will need an array of coils in the Bucket Brigade setup to generate field gradients throughout thewhole Bucket Brigade. The coils will have separation distance of l from each other. Note that l andd are not related to each other, they describe entirely different systems.
Bucket length or distance between neighboring sets of coils in the coil array, Λ. As alluded to earlier on, we plan to have an array of current-carrying coils, each with a time-modulated current toeffect an “anti-Helmholtz” magnetic field gradient that moves with the ion buckets. There will befour coils per bucket of ions. The separation between each set of four coils is Λ = 4l (see Figure 8.2).

We also note that the spacing between two neighboring buckets of ions is the bucket length: Zn (t) −Zn+1 (t) = Λ.
4 ∂z B(z) = −2Baxgrad

194
B̃ = −1B̃ = +1

l = Λ/4······
Λ

Four coils per one bucketR

Ion cloud translation direction

Figure 8.2: Plan A: using current-carrying coils to introduce an axial magnetic field gradient.
This is a cross-section view of a section of the Bucket Brigade. The ions move from left to right. The light
blue disks indicate the ion clouds in the Bucket Brigade. The dashed lines are visual aids to indicate theboundary of each bucket. The arrow head/tail symbols indicate the magnitudes and directions of the current
in the coils; the red dots indicate presence of coils with no current flowing in this snapshot in time. The
coils will sit outside the vacuum chamber, and are fixed in position. For better visualization, I have included
additional blue arrows to indicate the direction of the current in some of the coils coming out of the page.
The dashed arrows indicate the direction of the local magnetic fields generated by the current-carrying coils.
There are four coils per bucket, with each set of coils separated by Λ. The radius of the coils is R. To
introduce a non-zero Baxgrad , the currents in the coils have to flow in opposite directions on either side of
the ions. This gives rise to the configuration where the sign of Baxgrad alternates with each bucket. B̃ = +1(−1) indicates the bucket with the radial magnetic field going outwards (inwards) radially.

One accidental feature of Plan A is that neighboring buckets have Baxgrad of opposite signs, as can beseen from Figure 8.2. This alternating sign of Baxgrad corresponds to switching the sign of the Zeeman offset(the B̃-switch). Being able to switch B̃ frequently (once every other ion packet) allows us to understand andsuppress systematic effects associated with magnetic fields. This is not necessary for an eEDM experiment,
but is certainly very desirable to have. Plan A provides a very elegant solution to fast B̃-switching.
We shall now go into the details. First, we will determine the modulation on the currents required togive us as homogeneous a Baxgrad as possible at the ion locations (Section 8.2.2.1). Second, we will determine
how long each bucket needs to be so that Baxgrad is homogeneous enough throughout the spatial extent ofa typical ion cloud (Section 8.2.2.2). Finally, we discuss a slight variation of Plan A, which we call Plan A′(Section 8.2.2.3).

195
8.2.2.1

Determining constraints on currents in coils

For Plan A to work, we need a non-zero ∂z Bz . We also want many of the higher order derivatives
to be zero. Unlike the case of a single pair of anti-Helmholtz coils with a single ion trap (Section 8.2.1), wecannot do this by setting the ratio of the coil separation d and the coil radius R to be some magical number;this might work at a specific ion location w.r.t. the coil positions, but it will not work for the entire lengthof the Bucket Brigade where we will be translating the ions. We can mimic the anti-Helmholtz condition of
having zero higher order derivatives to the third order5 by modulating the currents in the coils in a specificway. To understand how we want to modulate the currents, we need to sum the contributions of the fields
from all the coils, and impose some constraints on them.
We first see that the analytical solution to Bz (x, 0, z) with a coil placed at z = 0 on the x-y plane anda current flowing counter-clockwise is (by Biot-Savart law, also see Figure 8.3):z(x, 0, z)yRIx

Figure 8.3: Schematic of a coil with radius R in the x-y plane with current going clockwise. We
are interested in the axial magnetic field Bz at an arbitrary location (x′ , y ′ , z ′ ). By the symmetry of the
setup, one can always rotate the coordinate system about the z axis
psuch that the point of interest lies above
the x axis such that the coordinates in the new system is (x = x′2 + y ′2 , 0, z = z ′ ). Hence, to evaluate
the field at an arbitrary point, we just evaluate it for the case of (x, 0, z), and perform the relevant rotationback to (x′ , y ′ , z ′ ).

µ0 IR
Bz (I, x, 0, z) =
4π

Z 2π
dϕ
0

R − x cos ϕ(R2 + x2 + z 2 − 2Rx cos ϕ)3/2


µ0 IR
=
R · I(α, β) − x · C(α, β)
4π

(8.2)
(8.3)

5 We only include up to the third order because we plan to have only four coils per bucket. Each coil has a current flowing
through it with a magnitude that we can use as a free tuning parameter. We want to keep ∂z Bz constant at the ion location
during translation, and this uses up one free parameter, leaving us with only three more free parameters to tweak; one for eachk ∈ {0, 2, 3} in ∂zk Bz .

196
where
√
I(α, β) := 2

 √



2β
2β
+ α + βE α+β
α − βE − α−βα2 − β 2
"





p
p
2β
2β
2
α
α−βE −C(α, β) := −+ α+βEβ α2 − β 2
α−β
α+β




 #
p
p
2β
2β
1
α+βK −+ α−βK−pα−βα+βα2 − β 2(8.4a)(8.4b)

α = R 2 + x2 + z 2 ,(8.4c)

β = −2Rx,(8.4d)

where E(x) and K(x) are the elliptic E- and K-integrals. When x → 0, the above expression reduces toµ0 IR2, as expected, where we use the limits:2(R2 +z 2 )3/2

lim E(x) =

x→0

π
,
2

lim K(x) =

x→0

π
,
2

lim

x→0

E(x) − K(x)
π
=− .
x
4

The Bucket Brigade consists of many such coils. These coils will be placed along the outside of thevacuum chamber, with spacing l = Λ/4 between neighboring coils, where Λ is the length of one bucket. The
mth coil (with current Im (t), where we conveniently define m = 0 to be the coil placed at the center of the
Bucket Brigade) placed on the x-y plane at z = mΛ/4 (see Figure 8.2) will produce an axial magnetic field(m)

Bz(Im , x, 0, z) given by:
Bz(m) (x, 0, z, t) = Bz (Im (t), x, 0, z − mΛ/4).

(8.5)

Using Equations 8.2 and 8.5, we sum the contributions from all the coils. We then solve for Im (t) by
demanding that Bz =(m)fulfills the following constraints:
m Bz

PConstant Baxgrad . We want ∂z Bz (Zn ) to be a constant at the (moving) ion location for the entirety of theion translation down the Bucket Brigade. This will give the ions a constant Zeeman frequency offset
throughout the translation. Note that this constraint is set by a moving point instead of a stationary
point.
Zero k th order gradients of Bz . Given that we have four coils per bucket, and we have already imposedon ourselves one constraint from the above, we only have three more degrees of freedom. With these197
three degrees of freedom, we demand that ∂zk Bz (Zn ) = 0 for k ∈ {0, 2, 3}. This will ensure that theion ensemble does not decohere through spatial inhomogeneity of the fields during the translation.
These three constraints must also be fulfilled at the (moving) ion location for the entirety of the iontranslation.
With the above constraints, we solve for Im (t) at various times during the ion translation within the Bucket
Brigade. We note that:
Im (t) = Im+8 (t),because the spatial periodicity happens every two buckets (see Figure 8.2), and there are four coils per
bucket. We also note that:
Im (t) = Im (t + 2Λ/vtrans. ),from the temporal periodicity of the setup, where, once again, Λ is the length of a bucket, and vtrans. is theconstant speed at which the buckets are translating down the Bucket Brigade. From the above, we can also
say that:Im (t) = Im′ t − (m′ − m) · Λ/(4vtrans. ) .

(8.6)

Making use of the periodic conditions above, we solve for Im (t) with various Λ, and the results are shown in
Figure 8.4.

0.0
0.5
1.0

0

5

10

1.0
0.5
0.0
0.5
1.0

5
0
5
10
Axial position of ion packet, Z(t) (cm)= 16 cm, R = 10 cm
Current, Im = 0(t) (normalized)

0.5

1.0
0.5
0.0
0.5
1.0

0

10

20

1.0

Current, Im = 1(t) (normalized)= 8 cm, R = 10 cm
Current, Im = 0(t) (normalized)

1.0

Current, Im = 1(t) (normalized)

Current, Im = 1(t) (normalized)

Current, Im = 0(t) (normalized)

198

0.5
0.0
0.5
1.0

10
0
10
20
Axial position of ion packet, Z(t) (cm)= 24 cm, R = 10 cm

1.0
0.5
0.0
0.5
1.0

10

0

10

20

30

1.0
0.5
0.0
0.5
1.0

20 10 0
10 20 30
Axial position of ion packet, Z(t) (cm)Figure 8.4: Plan A: Temporal modulation of current in the mth coil. The plots show Im (t) plottedagainst Z(t), where, once again, Z(t) is the axial position of the ion packet. For example, when the ion
packet is at the center of the Bucket Brigade, i.e. Z = 0, the current in the m = 0th coil (Im=0 ) is thethird blue point in the plots on the first row; I1 corresponds to the fourth blue point, and so on. These
modulations in Im (t) give a constant Baxgrad and ∂zk Bz = 0 for k ∈ {0, 2, 3}. The points indicate the valuesof t where we solve for Im (t), and the dashed lines are third order polynomial interpolation functions fromthe points. The blue (yellow) points correspond to instances where the ion packet is at a coil (between two
coils). The points are solved for the case where the coil radius is R = 10 cm, with about 20 buckets on eachside of the ion packet. To get Im′ (t) for any arbitrary m′ , perform the relevant graph transformation on theplots with Equation 8.6. Note that the curves are not just simple sinusoids, in contrast to the sinusoidal
voltages we send to Rings to translate the ions down the Bucket Brigade.

199
8.2.2.2

Spatial homogeneity

Using the values of currents in Figure 8.4, we can evaluate the spatial homogeneity of Baxgrad in the
Bucket Brigade. Figure 8.5 shows the results for various bucket lengths Λ.
We can see from Figure 8.5 that we need a bucket length of at least 16 cm for the spatial variationof Baxgrad to be less than 0.01% (Criterion 8.2) within an ion cloud of radius 1 cm. This means that for a
40-bucket Bucket Brigade, the setup will be at least 40 × 16 cm = 6.4 m long. This is longer than the 3 mlength that we are initially aiming for. If we limit ourselves to only 3 m, then we will lose half the numberof buckets and take a 30% hit in statistical sensitivity.
It looks like Plan A can be a little tricky to implement as is. We cannot have short buckets if we wanta region of spatial homogeneity large enough for our ion cloud. One thing to note is that we have limited
ourselves to four coils per bucket in the above analysis. We could introduce more coils per bucket for moredegrees of freedom to tweak the even higher order gradients of Bz to give us a greater region of homogeneity.
However, at four coils per bucket in a 40-bucket Bucket Brigade, this is already 160 coils that we need toconstruct and control. Having more parts would introduce more sources of errors; we would only want to go
down this road if necessary.
What if we give up on alternating B̃-switch with every bucket, and just have currents going throughcoils that do not change with time, but are allowed to vary according to the axial location of the coils? Could
the fields can be more spatially homogeneous that way? We shall look at this in the next section.

Baxgrad (normalized)

5.0
2.5
0.0
2.5
5.0

Baxgrad (normalized)

1.0
0.5
0.0
0.5
1.0

Baxgrad (normalized)

1.0
0.5
0.0
0.5
1.0

Baxgrad (normalized)

1.0
0.5
0.0
0.5
1.0

Baxgrad (normalized)

1.0
0.5
0.0
0.5
1.0

Baxgrad (normalized)

1.0
0.5
0.0
0.5
1.0

200

= 8 cm, R = 10 cm

8

20

10
0
10
Bucket number from center of Bucket Brigade

20

6

4
2
0
2
4
Axial distance from center of Bucket Brigade, z (cm)68

= 16 cm, R = 10 cm

20

15

10
0
10
Bucket number from center of Bucket Brigade

20

10
5
0
5
10
Axial distance from center of Bucket Brigade, z (cm)15

= 24 cm, R = 10 cm

20

20

10
0
10
Bucket number from center of Bucket Brigade

10010Axial distance from center of Bucket Brigade, z (cm)20

20

Figure 8.5: Plan A: Plots of Baxgrad along the trap axis for various bucket length. There is a coilat z = 0 (corresponding to the center of each plot) and there are coils going out to ±20 buckets in either
direction, with four coils per bucket, coil-coil spacing of Λ/4. The yellow circles indicate ion packets in the
Bucket Brigade. The plots are normalized to the value of Baxgrad at the center of the Bucket Brigade. Every
second plot is a zoomed in version of the first plot centered around the center of the Bucket Brigade. The
blue shaded region indicates the region where Baxgrad is within 0.01% of the value at the center of the ion
packet for this particular snapshot in time. The size of this region could change with the axial position of
the ion cloud as it translates down the Bucket Brigade. Having only four coils per bucket only allows us tosuppress higher order axial gradients in Bz up to the third order, and I did not investigate how the next
non-zero higher order gradient varies with axial position of the ion cloud. The grey dashed lines indicate±1 cm from the center of the ion packet. From the plots, one can see that the field starts to settle down a
few buckets into the Bucket Brigade, and it settles down much faster if each bucket is longer.

201
8.2.2.3

Plan A′ : Stationary axial magnetic quadrupole

Suppose now we have coils right outside the vacuum chamber with radius R = 10 cm, and the current
flowing through the mth coil is Im = m · i0 , where i0 is the difference in current between neighboring coils,
and, once again, the m = 0th coil sits right at the center of the Bucket Brigade. The value of i0 is directly
proportional to the Baxgrad that we will get along the Bucket Brigade. Figure 8.6 shows the profile of Baxgrad
along the trap axis of the Bucket Brigade for various inter-coil spacing l. Here we drop the constraint of

Baxgrad (normalized)

Baxgrad (normalized)

Baxgrad (normalized)

having four coils per bucket.

l = 4 cm, R = 10 cm
1
0
1
150

100
50
0
50
100
150
Axial distance from center of Bucket Brigade, z (cm)l = 6 cm, R = 10 cm
1
0
1
150

100
50
0
50
100
150
Axial distance from center of Bucket Brigade, z (cm)l = 8 cm, R = 10 cm
1
0
1
150

100
50
0
50
100
150
Axial distance from center of Bucket Brigade, z (cm)Figure 8.6: Plan A′ : Plots of Baxgrad along the trap axis for various bucket length. There is a coilat z = 0 (corresponding to the center of each plot), and there are coils going out in both directions withspacing l, all the way until z = ±160 cm. The yellow circles indicate locations of the ion packets with bucket
length Λ = 8 cm. The blue shaded region indicates where Baxgrad is within 0.01% of that at the center ofthe Bucket Brigade. When the inter-coil spacing is comparable to the coil radius R, there are noticeablefluctuations in Baxgrad .

One can tell from the plots in Figure 8.6 that the fields can be much more homogeneous. The edge

202
effects at the ends of the Bucket Brigade can be compensated for by use current magnitudes with nonlinear dependence of the coils’ axial positions. This would require some numerical modelling, but it isstraightforward nonetheless.
In addition to being able to have homogeneous fields with short buckets, another advantage that Plan
A′ has over Plan A is that the currents are do not vary in time in the former. This means easier control overthe experiment; one can just send in DC currents instead of AC currents with special waveforms.
With Plan A′ , switching the direction of the magnetic field will be a slow switch: we will have to clear
the whole Bucket Brigade of ions, switch the direction of the currents, and load the whole Bucket Brigade
again to perform the experiment with fields in the opposite direction. This is in contrast to Plan A, where
the fields switch every bucket. Once again, this is not necessary for an eEDM experiment, but being able toswitch the fields quickly allows for suppression of systematic effects in our precision measurement.
One consequence of using Plan A′ is that we will have large axial magnetic fields towards the ends of
the Bucket Brigade. We need Baxgrad = − 12 ∂z Bz = 3 mG/cm for an f0 = 50 Hz. Suppose Bz (z = 0) = 0 G
by the (anti-)symmetry of the current configuration, then at the end of the Bucket Brigade at about 1.5 m
away, we have an axial field of Bz (z = ±1.5 m) = ∓0.9 G. It is not immediately clear what this axial
magnetic field would do to our eEDM measurement. I shall leave this to the rest of the group to find out.
We have seen how we can implement Plan A (and its variant) that involves a Zeeman frequency offset
from an axial quadrupole magnetic field in the Bucket Brigade. Before we move on to the other Plans, it
serves to discuss (document (verb) in this thesis) one peculiar source of decoherence that could come fromthe combined effect of RF micromotion and axial quadrupole magnetic fields.

8.2.3

RF trapping field-induced decoherence
A form of decoherence within the ion cloud was brought up during one of our group meetings aspart of a discussion on systematics in the Gen. II experiment. This form of decoherence has avoided ourradar so far because its effects have not been seen directly in the Gen. I and II setups. Unlike the previous
sections, where we were talking about decoherence effects due to spatial inhomogeneities in Baxgrad , thephysics behind this new form of decoherence comes from the interplay between (a perfectly homogeneous)

203
Baxgrad and the RF trapping fields.
The physics behind Plan A involves the ions seeing an effective magnetic field in their rotating frameof the form (handwavingly; reproduced from Equation 8.1 for comparison with a latter equation):
Brot = Baxgrad rrot = Baxgrad

eErot
2 .
mωrot

In other words, Plan A works because of how the rotating polarization axis moves in phase with the rotation
micromotion, resulting in a net magnetic field as seen by ions in an axial quadrupole magnetic field.
In a spirit similar to the above, there is a small contribution to the net B-field to the above with theform (handwavingly):
Brf (r) = Baxgrad rrf (r) = Baxgrad

eErf (r)2 ,mωrf

where Erf (r) is the RF electric field at location r, ωrf is the angular frequency of the RF field. Note that thisis a position-dependent effect, because the RF field strength is a quadrupolar field.
The above handwaving expressions neglect the fact that the polarization axis is not rotating at aconstant angular velocity. To be more precise, the net magnetic field as seen by the ions in their rotating
frame has the form:
Brot = B(r) · Ê(r),(8.7a)

where the overline indicates time-averaged, andB(r) = Baxgrad (x, y, −2z),(8.7b)

2Vrf⃗
E(r)= Erot (cos ωrot t, sin ωrot t, 0) + 2 cos ωrf t (x, −y, 0),R0(8.7c)

where Vrf is the RF potential applied, and R0 is a trap geometry parameter defined in Equation A.1.
Splitting up the ion motion into its various components based on time scales (ωrot = 2π × 150 kHz ≫ωrf = 2π × 50 kHz), we have
Brot = Baxgrad r · Ê(r)= Baxgrad (r0 + rrot + rrf ) · Ê(r)= Baxgrad (0 + rrot + rrf ),

(8.8)

204
where we see that the r0 term (instantaneous secular position) evaluates to zero through the symmetry of the
fields in space and time, and the overlines on rrot (rotation micromotion amplitude) and rrf (RF micromotion
amplitude) are just symbols to remind us that these are time-averaged values. Evaluating the time-averaged
r’s up to first order in 1/Erot gives (being very careful with the signs): 2 2x + y2
eErot
e
Vrf
+
2
2
2
mωrot
mωrot R02Erot 2 22
x +ye
Vrf
rrf = −.
2
2mωrf R0
Erot

rrot = −(8.9a)(8.9b)

Combining Equations 8.8, 8.9a, and 6.21, we evaluate6 the expression at a 1σ ion cloud radius toget (substituting x2 + y 2 → σ 2 , and using the Mathieu parameters in Appendix A to convert Vrf into the
angular secular frequencies):
f0 = 3gµB |Brot |
"

 #


2
2
kB T mωrfo2
1 2kB T mωrot
o2
˜
1+
+
1+
,
= f0 1 rrot −(eErot )22
2 (eErot )2
2
rrot
rrf

(8.10)

eErot
where f˜0 = 3gµB |Baxgrad | mω
2 , o := ωz /ω{x,y} is the oblateness of the ion cloud, and ω{x,y,z} are the angularrot

secular frequencies of the ions in the trap. Typical values at typical operating parameters in Gen. III7 give
fractional decoherence of 0.0048% and 0.043% for the rrot and rrf terms, respectively. Those for Gen. II8give 0.0016% and 0.088%. The surprising factor of 1/2 in the expression for rrf comes from taking into
account the fact that the RF micromotion amplitude changes sinusoidally as the ion performs its secularmotion, so the average amplitude-squared is given by cos2 ωsec t = 1/2.
Equation 8.10 also says that the oblateness of the ion cloud contributes to decoherence. The physics
is easy to understand: a stronger confinement in the z direction necessitates a stronger Vrm to keep the ionsconfined radially, ultimately resulting in RF-induced decoherence. Moral of this paragraph: prolate good,oblate bad.
It is interesting why we are seeing decoherence much less than 0.088% in the Gen. II setup. One
possible guess is collisional narrowing (the topic of Section 6.5.2). Our ansatz Equation 6.15 tells us that the
6 The time averaging is done over a time domain t ∈ [0, T ] big enough that the rotation and RF micromotions come back tothe same phase together, i.e., T = 2πp/ωrot = 2πq/ωrf where p and q are very large integers.
7 Gen. III: T = 50 K, ω = 2π × 50 kHz, ω
rot = 2π × 150 kHz, Erot = 60 V/cm, o ≈ 1.
rf
8 Gen. II: T = 20 K, ω = 2π × 50 kHz, ω
rot = 2π × 375 kHz, Erot = 58 V/cm, o ≈ 1.
rf

205
rate of collisional narrowing decreases with increasing temperature and decreasing ion number density. We
expect to have a hotter and less dense ion cloud in Gen. III than in Gen. II, so it may not be wise to pin our
hopes of pulling the 0.2% decoherence down to 0.01% with collisional narrowing, especially when the effectis still not very well studied in the group at this point in time.
On the topic of Gen. II, Equation 8.10 also suggests that the mean frequency of the ion cloud shiftswith temperature. As the temperature of the ion cloud increases, the modal radius of the ion cloud also
increases, so the mean frequency of the ion cloud would increase. In fact, we saw that the ion cloud heats up
with time in Section 6.5.4 and Figure 6.15. This is a possible systematic, and I shall leave detailed studies
to my colleagues.
Plan A seems difficult: homogeneity requirements in Baxgrad demands either very long buckets or few
buckets. Furthermore, we still get RF-field induced decoherence even if Baxgrad is perfectly homogeneous.
But let us perform one last struggle: can we shim away/in gradients in Baxgrad ? Doing so allows us to tweak
some things:
Shim away the RF-induced decoherence. Since the decoherence effect discussed in this chapter behaves as though Baxgrad varies quadratically as we move away from the trap axis, we could try toshim this decoherence away by intentionally introducing a quadratic spatial variation of Baxgrad .
Shim away spatial inhomogeneity of Baxgrad . For decoherence through, say, axial variation of Baxgrad ,it would be nice if we can use shimming coils as extra degrees of freedom to shim the spatial variation
away. Subjected to constraints from Maxwell’s equations.
A combination of the above. Maxwell’s equations do not let us win for free. When we shim the curvatureof Baxgrad away along one axis, we have to pay by introducing curvature in another axis. We would
probably have to settle for a compromise between the two points mentioned above.
The next section will cover some basic ideas of what kinds of field gradient shimming are allowed by Maxwell.

206
8.2.4

Shimming away gradients in Baxgrad
Within our Bucket Brigade, we do not have sources of magnetic fields (ignoring the magnetic dipolesof our molecules). In a region absent of sources of magnetic fields, Maxwell’s equations allow us to express
the magnetic field as a gradient of a scalar potential field:
B(r) = −∇ΦB (r),where ΦB (r) is the scalar magnetic potential. Together with ∇ · B(r) = 0, we have:∇2 ΦB (r) = 0,
which, in turn, implies that ΦB (r) can be expanded out in terms of spherical harmonics Ylm (Ω):
X

ΦB (r) =l=0,−l≤m≤l



bl,m
l
al,m r + l+1 Ylm (Ω),
r

where Ω is a shorthand for the usual angular variables (θ, ϕ), l and m are the usual quantum numbers for total
angular momentum and projection of the angular momentum onto the quantization axis, respectively, anda and b are constants to be determined through boundary conditions. We can simplify the above expression
even further by noting that (i) the magnetic field is a field with real components, and (ii) the field does notblow up at r = 0. We define real spherical harmonics by:





√i

Ylm − (−1)m Yl−m
if m < 0,

2




Yl,m = Y 0
if m = 0,
l









 √1 Yl−m + (−1)m Ylm
if m > 0,2
and write the scalar magnetic potential as:XΦB (r) =al,m rl Yl,m (Ω).

(8.11)

l=0,−l≤m≤l

We further label components of the magnetic field obtained from taking the gradient of the corresponding
real spherical harmonic component of the scalar potential with the same quantum numbers, i.e.,Bl,m (r) := −∇ rl Yl,m (Ω) .
B(r) =

X
l=0,−l≤m≤lal,m Bl,m (r).

(8.12)(8.13)

207
There are many theses from our group that go in depth into the study of these spherical harmonics (forinstance, check out Noah Schlossberger’s upcoming thesis), and here I will just flash out some of the more
important spherical harmonics:
B1,{1,−1,0} Uniform field in the {x, y, z} directions.
B2,0 Quadrupole field of the form (x, y, −2z), i.e., term that gives a uniform ∂z Bz =: −2Baxgrad .
B2,2 Quadrupole field of the form (−x, y, 0).
B2,−2 Quadrupole field of the form (y, x, 0); like B2,2 , but rotated 45° along the z-axis.
B4,0 Term that gives a Baxgrad that varies with position as x2 + y 2 − 2z 2 .
B4,2 Term that gives a Baxgrad that varies with position as x2 − y 2 .
B4,−2 Term that gives a Baxgrad that varies with position as xy; like B4,2 , but rotated 45° along the z-axis.
Note that any gradients in Baxgrad that are odd in {x, y, z} will not result in an overall decoherence,
because any positive (negative) phase accumulation on one side of the trap will be nulled out by the negative(positive) phase accumulation on the other side of the trap when the ion sloshes around the trap. Hence,
the lowest order terms in the spatial variation of Baxgrad that contributes to decoherence are terms that goas {x, y, z}2 . Assuming that the secular trapping frequencies along the x, y, and z directions are incommensurate, or that trajectory-changing collisions happen frequently enough in the trap, terms like {xy, yz, zx}
also do not contribute to decoherence, so the only terms that contribute to decoherence are {x2 , y 2 , z 2 }. The
l = 4 terms are the lowest order terms that give spatial variations of Baxgrad that vary as {x2 , y 2 , z 2 }, andthe two l = 4 terms above are the only such terms at this order.
All the fields above can be engineered with a combination of coils, e.g., Helmholtz coils for B1,{1,−1,0} ,anti-Helmholtz coils for B2,0 , and it turns out that we can engineer a set of coils placed in strategic positions
to create B4,0 and B4,2 . The interested reader can find more details in Noah Schlossberger’s upcoming
thesis. The take home messages from evaluating these spherical harmonics and designing coil sets thatgenerate these harmonics are:

208
(1) We can shim away any imperfections in Baxgrad with our B4,0 and B4,2 coils to reduce decoherence
introduced by spatial inhomogeneity of Baxgrad that are even about the ion cloud center. This
shimming effect only works for a local region, though; it will require much more design effort todesign coil sets that shim away B4,0 and B4,2 along a line, say, down the entire Bucket Brigade.
(2) The B4,0 term gives a Baxgrad that varies with position as x2 + y 2 − 2z 2 . So for every two unitsof decoherence we shim away in the axial direction, we incur a cost of one unit of decoherence inthe radial direction. This sounds like a win. We could make our ion cloud shaped like a pencil
along the axial direction, and put all sources of decoherence into the radial modes with our B4,0
coils. We do not win forever with this method though. The suppression factor of decoherence
through a combination of (i) making our ion cloud longer and (ii) dumping decoherence from ouraxial dimension to the radial dimension with our B4,0 coils as compared to a spherical ion cloud isgiven by (see Equation 8.10):
Suppression factor =

1+

o2
2


1+

12
2


=

2 + o2,3where, once again, o = ωz /ω{x,y} is the oblateness of the ion cloud. Evidently, the best we can ever
do is to suppress decoherence to 2/3 its original value, where o → 0 is an ion cloud that is loose in
the axial direction and tight in the radial.

8.2.5

Final remarks
Working with Plan A would allow us to continue to stay in a familiar place with regards to under-

standing our systematic effects. There is a new systematic effect when we translate ions down the Bucket
Brigade, where the magnetic and electric fields conspire and vary in space such that it could mimic an eEDM
signal, but this is beyond the scope of this thesis.
Plan A does not seem easy, but it certainly is not impossible. However, there is nothing limiting us
from using other ways of introducing a frequency offset between the stretched states of the upper and lower
doublets. In the next section, we shall look at how we can introduce a frequency offset with Berry’s phase.

209

8.3

Plan B: Berry’s phase

8.3.1

Working principle
The quantization axis of our ions is given by the direction of the rotating electric field, Erot . As Erotrotates, the Erot vector traces out a solid angle. The solid angle that Erot traces out is directly related to the
Berry’s phase picked up by the quantum states [25]:ϕBerry = Ωs.a. mF ,(8.14)

where Ωs.a. is the solid angle, not to be confused with the quantum number Ω.
Defining α to be the instantaneous angle that Erot makes with the horizontal plane, and expressing
Equation 8.14 as the time-averaged energy difference between our stretched states (mF = ±3/2), we get:hf0 = 3hfrot ⟨sin α⟩,(8.15)

where ⟨·⟩ indicates time-averaging.
In the first two generations of the JILA eEDM experiment, we deliberately shim α to zero for all times
and characterize this as one of our systematic effects. In the Bucket Brigade, however, this can serve as ourPlan B for introducing a bias f0 into our system.
At first glance, one may think that to get a non-zero ⟨sin α⟩, one needs to have a non-zero timeaveraged electric field in the z-direction Ez , which will move our ions out of the trap. However, it is possible
to have a non-zero ⟨sin α⟩ AND a zero time-averaged electric field in the z-direction by modulating theamplitudes of Erot and Ez as we shall see in the next section.

8.3.2

In the Bucket Brigade
In addition to providing axial confinement of the ions, the Rings in the Bucket Brigade can also

provide a time-varying axial electric field Ez (t). An example of such a setup is shown in Figure 8.7.

210
S̃ = −1S̃ = +1······

Ion cloud translation direction

Figure 8.7: Plan B: using the Rings to introduce a time-varying axial electric field. This is across-section view of a section of the Bucket Brigade. The ions move from left to right. The blue disk
indicates the ion clouds in the Bucket Brigade. The dashed lines are visual aids to indicate the boundary ofeach bucket. The thick lines indicate cross-sections of the Rings. The colors of the thick lines indicate theamplitudes of the voltages applied on the Rings: warmer (cooler) colors indicate more positive (negative)
potentials. The straight arrows indicate the direction of the local Ez generated by the potentials on the
Rings. The direction of the axial electric field flips with each neighboring bucket. S̃ = + ± 1 indicate buckets
with the time-varying axial electric field pointing in directions that are π out of phase with each other.

Much like Plan A, Figure 8.2 also shows that Plan B allows for neighboring buckets to have Ez of
opposite signs, which provides a very elegant solution to fast S̃-switching (Berry’s phase analogue of B̃switching; to be introduced in Section 8.3.3) to suppress uncertainties and systematic effects associated withEz .
Suppose the axial electric field (ignoring the axial confinement fields here) goes as:Ez (t) = Edit cos ωdit t,(8.16a)where Edit is the amplitude of the dithering axial electric field and ωdit is the angular dithering frequency.
Also suppose that Erot is modulated with fractional amplitude a at angular modulation frequency ωdit :Erot (t) = Erot,0 (1 − a cos ωdit t)(8.16b)

211
Then the tangent that the axial electric field makes with Erot (t) is:tan α =

Edit cos ωdit t.
Erot,0 (1 − a cos ωdit t)(8.17)For Edit ≪ Erot,0 , we use the small angle approximation, and perform a time-average over one full cycle inthe dither with the notorious half-tangent substitution to get:⟨α⟩ =Edit 1
Erot,0 a


√




Edit
1
a 3a3
−1 =
+
+ O(a5 ) .
Erot,0 2
8
1 − a2(8.18)The ion accumulates the “carrier” Berry’s phase by accumulating a lot of it first when it is above theplane, then removing the accumulated phase incompletely below plane. The actual phase evolves as follows:∆ϕ(t) = ϕmF =+3/2 (t) − ϕmF =−3/2 (t)
sin 2ωdit ttEdit sin ωdit t++a+ O(a2 )= 3ωrot
Erot,0
ωdit
2
4ωdit
i
3 Editωrot Edit ha=ωrot a · t + ϕ0 (t) , ϕ0 (t) = 3
sin ωdit t + sin 2ωdit t + O(a2 ) .
2 Erot,0ωdit Erot,04
8.3.3(8.19)New switches for the experiment: {S̃, T̃ }There are many switches we can use to reverse the sign of Berry’s phase, which can be very helpfulin studying systematic effects in our setup. Inferring from Equations 8.17 and 8.18, we have three switches
associated with Berry’s phase: {R̃, S̃, T̃ }. The switches can each take on values ±1. The effect of the switches
are as follows:• R̃: Reversing the rotation direction of Erot .
• S̃: Reversing the sign of Edit .
• T̃ : Reversing the sign of amplitude modulation.
Formally, the effect of the switches on Berry’s phase is given by:

S̃
·
E
+
E
1
1
dit
nr
q
⟨α⟩R̃S̃ T̃ = R̃ ·Erot,0T̃ · a + anr
1 − a2 − a2 − T̃ · 2aanr≈ R̃ ·

S̃ · Edit + Enr T̃ · a + anr+ αnr ,
Erot,02

− 1 + αnr
nr(8.20)

212
where we introduce the subscript “nr” to indicate the component of the experiment parameter that does notflip with the sign of the switch through imperfections in our experiment like: imperfect potential shimming
for the Rings (Enr ), imperfect potential shimming for the Rods (anr ), any asymmetry involved in the switchingof the rotation direction of the rotation electric field (αnr ).
Note that {R̃, T̃ } are slow switches: the whole Bucket Brigade has to be unloaded before these switches
can be flipped. In contrast, S̃ is a fast switch that can have alternating signs for neighbouring buckets.
Once again, to measure the eEDM, we try to measure f u and f l very well, at various configurationsof the experiment parameters, and take a smart linear combination of the measured frequencies. Each of
the smart linear combinations reveal a certain aspect of our experiment, and one of them tells us about theeEDM. We now attempt to borrow language used in the first two generations of the JILA eEDM experiment(see theses of previous students) to have a glimpse of how using Plan B (Berry’s phase) could change theanalysis of our data. Each of the linear combinations are called “channels”. For the purpose of understanding
what the channels will look like in Plan B, we remove the slow switches from Equation 8.20 and keep only
the fast S̃ switch in the time-averaged α:⟨α⟩S̃ = S̃⟨α⟩ + αnr .

8.3.4

f D̃S̃ channels

8.3.4.1

Introduction(8.21)

With each repetition of our eEDM experiment, we have the freedom to change the signs of the switches.
In a simplified version of Plan B, we have the following switches:
Doublet: D̃. D̃ = +1 refers to the upper doublet f u , and D̃ = −1 refers to the lower doublet f l .
Sign of Edit : S̃. We have already introduced this at the start of the Plan B section.
These switches take on values ±1. We express the frequencies we measured with the various combinationsof switches as f D̃S̃ . As an example, frequency f D̃S̃ = f ++ corresponds to f u when the experiment was
performed with S̃ = +1, i.e. when Edit has a positive sign. As another example, frequency f D̃S̃ = f −+corresponds to f l with the same S̃ experiment condition as the above.

213
To get the frequency “channels”, we can then take a switch-weighted average of the above frequencies.
For instance:







0




D̃ S̃

++

=f 
 f 
+ + + + f










 fD 
+ + − − f D̃S̃ = f +− 





1
.
 := 







4
 fS 
+ − + − f D̃S̃ = f −+ 















D̃ S̃
−−
DS
f
=f
f
+ − − +
It turns out that each switch-weighted average frequency channel is sensitive to a different systematiceffect in the experiment, and we can use these frequencies to compare our experimental results with analyticalexpressions to learn about our systematic effects. To understand what each frequency channel is sensitive to,we work in a two-level effective Hamiltonian of our system (see Section C.4), and expand out the eigenenergies
of our Hamiltonian in powers of the small quantities with the following steps:(1) Measure the non-inertial-frame coupling term for the upper and lower doublets (∆u and ∆l ), andmeasure the g-factors for both doublets (g u and g l ). Massage them into the following form:∆ = (∆u + ∆l )/2∆D = (∆u − ∆l )/2

g = (g u + g l )/2

δg = (g u − g l )/2(2) Perform studies on the two-level Hamiltonian.
(a) Solve for eigenenergies of the two-level Hamiltonian. The frequency f D̃S̃ is just the differenceof the two eigenenergies.
p
A2 + B 2q= 3frot ⟨α⟩ 1 + 1̃(1 term) + D̃(D term) + S̃(S term) + D̃S̃(DS term).

f D̃S̃ =In the above, we identified 3frot ⟨α⟩ to be the biggest term in the square root, so we pull it
out. The “1 term” are terms small as compared to 3frot ⟨α⟩, but do not depend on any of theswitches. The rest of the terms are terms that depend on the relevant switches. For example,
the “D term” include all the terms that depend only on the D̃ switch.
(b) Recognize that the following are small compared to relevant quantities:214• {3gµB Bnr , ∆, ∆D , 2EEDM } ≪ 3frot ⟨α⟩• δg ≪ g• αnr ≪ ⟨α⟩and expand the terms out in the square root of the f D̃S̃ expression, being consistent in keeping
the small terms up to the correct order.

8.3.4.2

Results of the expansion

With the above steps, we get, up to first non-zero correction:
1 ∆2 + (∆D )2
f 0 = 3frot ⟨α⟩ 1 +
2 (3frot ⟨α⟩)2
"

2 #
∆ ∆D
1 ∆2 + (∆D )2
3gµB Bnr + 3frot αnr
D
f =
1−
+
3frot ⟨α⟩
2 (3frot ⟨α⟩)23frot ⟨α⟩
∆2 + (∆D )2 3gµB Bnr + 3frot αnr
2EEDM ∆ ∆D+
+ 2EEDM(3frot ⟨α⟩)3(3frot ⟨α⟩)23frot ⟨α⟩2D 2
2EEDM ∆ ∆D
1 ∆ + (∆ )−f S = (3gµB Bnr + 3frot αnr ) 1 −
2 (3frot ⟨α⟩)2(3frot ⟨α⟩)21 ∆2 + (∆D )2∆ ∆D 3gµB Bnr + 3frot αnr
f DS = 2EEDM 1 −+ 3 δgµB Bnr −22 (3frot ⟨α⟩)3frot ⟨α⟩3frot ⟨α⟩(8.22a)(8.22b)(8.22c)(8.22d)A simple check shows that f 0 ± f D = f u/l with g u/l = g ± δg and ∆u/l = ∆ ± ∆D . For future reference, thefirst non-zero term in the f 0 channel for small Erot amplitude modulation (a ≪ 1) and small dither electricfield (Edit ≪ Erot,0 ) is:
f 0 = 3frot ⟨α⟩ =8.3.4.3

3 Editfrot a.
2 Erot,0(8.23)

Comparison with Zeeman-dominated f0 in Plan A

For reference, the channels for a Zeeman-dominated f0 , e.g. in Plan A, can be evaluated in a way
similar to the above. The relevant switches are:Sign of Baxgrad : B̃. The sign can be switched by switching the direction of the current in the coils.
Doublet: D̃. D̃ = +1 refers to the upper doublet f u , and D̃ = −1 refers to the lower doublet f l .
Rotation direction of Erot : R̃. We conveniently left this out for the Berry’s phase f D̃s̃ suite, but we shall
include this here for comparison with previous theses and papers.

215
The frequency channels are given as follows, to the first non-zero correction term (except for f R and f DR ,where we also include the next non-zero correction term):
1 ∆2 + (∆D )2
f = 3gµB Brot 1 +,2 (3gµB Brot )2
1 ∆2 + (∆D )2
2EEDM ∆ ∆D
f B = (3gµB Bnr + 3frot αnr ) 1 −−,2
2 (3gµB Brot )(3gµB Brot )2
0

∆ ∆D
,
3gµB Brot
3frot ⟨α⟩  2(∆ + (∆D )2 )(3gµB Bnr + 3frot αnr ) + 2EEDM · 2∆ ∆D ,fR =3
(3gµB Brot )∆ ∆D 3gµB Bnr + 3frot αnr
1 ∆2 + (∆D )2
BD
+
3
δgµ
B
−
f
,
= 2EEDM 1 −Bnr
2 (3gµB Brot )23gµB Brot
3gµB Brot
1 ∆2 + (∆D )2
f BR = 3frot ⟨α⟩ 1 −,
2 (3gµB Brot )2
f D = 3 δgµB Brot +
3frot ⟨α⟩ 
2∆ ∆D (3gµB Bnr + 3frot αnr ) + 2EEDM (∆2 + (∆D )2 ) ,3(3gµB Brot )
3frot ⟨α⟩
δg
BDR
D
2
D 2
f
=
−∆ ∆ +∆ + (∆ ).
(3gµB Brot )2g
f DR =(8.24a)(8.24b)(8.24c)(8.24d)(8.24e)(8.24f)(8.24g)(8.24h)These are the expressions more familiar to the first two generations of the JILA eEDM experiments. All theterms above have already been introduced in earlier parts of this section.
Here are some observations from comparing f D̃S̃ and f B̃ D̃R̃ :(1) f 0 and the eEDM channels {f DS , f BD } agree across the Berry’s phase f D̃S̃ and the Zeeman f B̃ D̃R̃
up to an expansion of this order, where we identify 3frot ⟨α⟩ with 3gµB Brot .
(2) Since there is no Brot in f D̃S̃ , the leading term in f D is the first correction term of that in f BDR .
D
D
Therefore, we see that fBerry< fZeeman, especially at large f 0 . In Gen. II, we expect f D to increase

with f 0 , but in the Berry’s phase picture, we expect f D to decrease with f 0 instead.
We shall now move on to the details of how we can implement Plan B in the experiment.

8.3.5

Q-factor & Resonant Circuit
One important ingredient in Plan B is the modulation of the amplitude of Erot . We plan to charge theRods with a resonant circuit. One advantage of the resonant circuit is that it stomps on the second harmonicof Erot that plagues the systematic effects seen in the first two generations of the JILA eEDM experiment.

216
One disadvantage, however, is that it stomps on all Fourier components that are not the resonant frequency,
thereby suppressing the effects of amplitude modulation. The degree of suppression of the Fourier frequencies
in the waveform depends on the Q-factor of the resonant circuit.
For an frot = 150 kHz, we want to be able to send in sidebands at about 15 kHz away, so we do notwant to work with too high a Q-factor. Hence, we aim for a Q-factor of about 20. Figure 8.8 shows theresonance plots at Q = 20.

1.34

1/|Z| (1/ )arctan(X/R) (rad)

2

0

100

120

Attenuation factor (%)

2

140 160
frot (kHz)

180

0.015
0.00434

0.010

0.0188

0.005

200

100

120

140 160
frot (kHz)

23

24

180

200

26
23%

24
22
20
17

18

19

20

21
Q

22

25

Figure 8.8: Resonance curves for Q = 20. Top plots: Resonance curves with resonance at 150 kHz. Fora V (t) going into the system, the response of I(t) will have a phase shift given by the left plot, and an
attenuation given by the right plot. Dashed lines show location of the 15 kHz sidebands. The sidebands
are at 23.0% of the maximum on the right plot. Bottom plot: variation of the attenuation factor of the
sidebands with a varying Q. The range of Q used corresponds to a change in R = 53.0 Ω of ±10 Ω.

217
We want to have a time modulating Erot that looks like:cos ωrot t





Erot (t) = Erot,0 (1 − a cos ωdit t)  sin ωrot t 





0




a
cos ωrot t − 2 [cos(ωrot + ωdit )t + cos(ωrot − ωdit )t]






a
= Erot,0  sin ωrot t − [sin(ωrot + ωdit )t + sin(ωrot − ωdit )t] 

2




0
so we reconstruct the input waveform using information in Figure 8.8, and they have the following expression:hErot (t) = Erot,0 1 −8.3.6

cos ωrot t


i


a

cos(ωdit t + 1.34)  sin ωrot t 
.
23.0%




0

Spatial homogeneity of the dithering axial electric field
The other crucial ingredient for Plan B is the dithering axial electric field. We shall see if Plan B

fulfils Criteria 8.1 and 8.2. Once again, we go back to Equation 7.6 and Table 7.3 and adapt the results forPlan B.
Given the attenuation factors in Table 7.3, it is not practical to work with harmonics of order higher
than n = 3. Including only up to n = 3, the waveform of a potential V (z, ρ) that minimizes spatial
inhomogeneity of Edit is:1
V (3k, z, ρ),
27




8k
k5
4
2 2
4
7
Edit = −∂z V (z, ρ) = −V0+−15ρ + 120ρ z − 40z + O {ρ, z}.
9120

V (z, ρ) = V (k, z, ρ) −Note, once again, that k = 2π/λ, where λ is the wavelength of the fundamental waveform on the Rings.
In the case of Plan B, λ = 2Λ. Evidently, the inhomogeneity comes in starting on the fourth power of thesize of the ion cloud 2σ ∼ {ρ, z}. From the coefficients of each inhomogeneity term, we see that the axial
inhomogeneity is more stringent than the radial.
The axial inhomogeneity in Edit comes in through the first order correction term involving (kz)4 . Toqfulfil Criterion 8.2, we demand that 83 (kz4σ )4 < 0.01%, which means that λ > 2π 4 30000
8 z4σ ≈ 49.2z4σ . For

218
an ion cloud with a 4σ radial extent of 1 cm, we see that each bucket needs to be about 25 cm in length. A
spatial plot of such an inhomogeneity is shown in Figure 8.9.

0.08
0.06

1

0.02

0

0.00

%

(cm)

0.04

0.02
1

0.04
1

0
z (cm)

1

0.06

Figure 8.9: Spatial inhomogeneity of Edit w.r.t. point at center of a bucket. The coordinates ρ andz represent the radial and axial directions, respectively. This plot is generated with Λ = 50 cm. The whitedashed circle indicates a 1 cm radius, corresponding to a 2σ ion cloud radius. The dotted lines indicate
contour lines of 0.01%, in line with Criterion 8.2.

8.3.7

Final remarks
Plan B started out looking rather promising, much like Plan A, until we saw the constraints that

Criteria 8.1 and 8.2 place on the dimensions of the Bucket Brigade. To this we say, “we should not be sadbecause it doesn’t work, we should be happy because we tried.” However, instead of alternating S̃ once
every bucket, we could alternate it once every few buckets. This will allow us to have short buckets andhomogeneous Edit at the same time. This relaxes a lot of design constraints, e.g. λ can even be longer than
50 cm, which improves spatial homogeneity. Having longer λ makes the attenuation factors work less hard(in our favor) so we can add more higher harmonics into the waveform to the potential to make it moretriangular. I shall leave the analysis to future generations of students and post-docs.
Much like Plan A, Plan B looks hard, but not impossible. With this, we have looked at examples of
how we can introduce a Zeeman frequency offset with Plan A, and Berry’s frequency offset with Plan B.
The rest of the chapter explores other ingenious ways of introducing Zeeman frequency offsets that do not219
require axial quadrupole magnetic fields.

220

8.4

Plan C: Uniform ambient magnetic field

8.4.1

Working principle

8.4.1.1

History: systematic effect in the Gen. I experiment

Before we talk about Plan C proper, it serves to take a trip back down memory lane to the firsttwo generations of the JILA eEDM experiments, where the inspiration for Plan C is drawn. One of thesystematic effects seen in the Gen. I experiment involves the following [14]:f B ∝ ϵ (Bx X0 + By Y0 ) ,(8.25a)f BR ∝ ϵ (Bx Y0 − By X0 ) ,(8.25b)where f B and f BR are the frequency channels defined in Equation 8.24; ϵ = V2f /Vrot is the ratio of theamplitude of the second harmonic of Erot to the fundamental; Bx,y is the ambient magnetic field strength in
the x and y directions, respectively; X0 and Y0 are the positions of the ion cloud w.r.t. the trap center inthe x and y directions, respectively. The physics behind the above systematic effects is as follows:(1) The electric potential for Erot in the Gen. I experiment was provided by six radial electrodes thatare powered by an op-amp each.
(2) Despite receiving a first harmonic low voltage input, the high voltage output from each op-amp also
has a component of the second harmonic that is at some phase w.r.t. the first harmonic. This results
h 


i
2nπ
in an output voltage on the nth electrode of the form: Vn (R̃) = Vrot cos ωrot t + R̃ nπ
+
ϵ
cos
2ω
t
+
R̃
+
ϕ
,
rot
3
3
where ϕ is the relative phase between the first and the second harmonic, and is determined by thenature of the op-amp; and R̃ is the rotation switch, e.g. +1 for Erot rotating counter-clockwise, and−1 for clockwise.
−−→(3) Summing the potentials from all the fins, we see that the Lissajous figures that Erot traces out asseen by ions at various locations within the ion trap is shown in Figure 8.10.

221

−−→
Figure 8.10: Lissajous figures traced out by Erot as seen by ions at various locations within
the trap. X0 and Y0 are the mean ion cloud positions w.r.t. the center of the trap, and are expressedin fractions of the radial distance of the electrodes away from the trap center (i.e. ions will be touching
the electrodes at, say, X0 = 1). The green dashed circle indicates the Lissajous figure traced out by the−−→
Erot vector without the second harmonic component as it rotates along the plane. The blue (yellow) circle
indicates the corresponding Lissajous figures with the effect of the second harmonic, and when R̃ is +1 (−1).

(4) From Figure 8.10, we note the following:(a) Moving out along the i-axis (where i ∈ {X0 , Y0 }) displaces the centroid of the Lissajous figure
along the i-direction.

222(b) Flipping the sign of R̃ flips the Lissajous figure about the radial axis.
(5) If there is an ambient magnetic field along the plane of the rotating electric field B = Bx x̂ + By ŷ,the ions can see a non-zero time-averaged9 magnetic field in the frame of the rotating electric field:
2⟨Bx x̂ · Ed
rot ⟩ ∝ Bx · ϵ · X0 cos ϕ − R̃ Y0 sin ϕ + O({ϵ, X0 , Y0 } ),(8.26a)
2⟨By ŷ · Ed
rot ⟩ ∝ By · ϵ · Y0 cos ϕ + R̃ X0 sin ϕ + O({ϵ, X0 , Y0 } ).

(8.26b)(6) Together with the fact that the ambient magnetic field flips direction w.r.t. the fields introduced by
Baxgrad under a sign flip of the B̃-switch, we see that:f B ∝ ϵ (Bx X0 + By Y0 ) cos ϕ + O({ϵ, X0 , Y0 }2 ),(8.27a)f BR ∝ ϵ (Bx Y0 − By X0 ) sin ϕ + O({ϵ, X0 , Y0 }2 ),(8.27b)which agrees with Equations 8.25 when the phase between the first and second harmonic ϕ is chosencorrectly. Evidently, this is a Zeeman effect.

8.4.1.2

Adaptation of the physics behind the systematic effect

From the previous section, we see that we can introduce a bias f0 if (i) we can disrupt the Lissajous
figure of Erot such that ⟨B · Ed̸ 0, and (ii) introduce a uniform ambient magnetic field parallel to therot ⟩ =
direction of the displacement. The effect that we saw in the previous section only turns on with a non-zeroion cloud displacement from the trap center, but we do not wish to displace our ions from the trap center
because this would introduce additional RF-micromotion and heat up our ion cloud at a much higher rate.
Hence, we need to engineer an Erot Lissajous figure such that Erot such that ⟨B · Ed
̸ 0 even when the ionrot ⟩ =
cloud is nominally at the trap center. We shall see how this can be done in the next section.

8.4.2

In the Bucket Brigade
It turns out that having an oscillating uniform electric field in the, say, x direction that oscillates atthe second harmonic of Erot would give Erot such that ⟨B · Ed
̸ 0. The analytical form for such an electricrot ⟩ =
9 Important: the averaging must be done over the time domain and not the spatial domain.

223
field is:

Erot cos ωrot t + E2harm cos(2ωrot t + ϕ)




,

E(t) = Erot sin ωrot t





0

(8.28)

where ϕ is the phase between the first and second harmonic, and E2harm is the amplitude of the second
harmonic. An example of an Lissajous figure traced out by such a field is shown in Figure 8.11.

yrot
rot + 2harmx−−→Figure 8.11: Lissajous figure traced out by Erot as seen by ions in the Bucket Brigade. The dashedline indicates the Lissajous figure in the absence of E2harm . For illustrative purposes, ϵ = E2harm /Erot = 0.2,
an unrealistically large value, to exaggerate the features.

The Plan C equivalent of Equations 8.26 are:⟨Bx x̂ · Ed
rot ⟩ ∝ −Bx · ϵ · cos ϕ,(8.29a)⟨By ŷ · Ed
rot ⟩ ∝ By · ϵ · R̃ sin ϕ,(8.29b)f 0 ∝ ϵ Bx cos ϕ,(8.30a)f R ∝ −ϵ By sin ϕ.

(8.30b)which necessarily implies10 :In fact, up to second order in ϵ = E2harm /Erot , we have a frequency offset:
f0 = 3gµB Beff,

Beff = Bxhϵ4i+ O(ϵ3 ) cos ϕ.

(8.31)

10 Even though “the proportional sign” is used, and frequency is a positive definite value, the relative sign difference between
the two expressions shown is included explicitly for future reference.

224
For typical f0 = 50 Hz and ϵ = 0.01, we just need Bx ≈ 0.3 G, which is a doable field that we can apply.
The uniform magnetic field will be produced by very wide “Helmholtz coils” running the entire length of thewhole Bucket Brigade.

8.4.3

Final remarks
One might think that since Beff ∝ Bx ϵ, as long as we choose the values such that they give the

Beff we want things will be fine. This is not true. Roughly speaking, what the second harmonic does is it
accumulates some Zeeman phase for one half of the Erot cycle, and removes some of it in the second half toresult in a net phase gain. The effect of this is frequency modulation of f0 . From our simulations, when ϵ istoo small, the frequency modulation becomes apparent. Since frot = 150 kHz and f0 = 50 Hz nominally, if
we want the modulation index to be less than 0.1, then ϵ ≳ 0.003.
One final thing to note is that the rods used to generate Erot will be driven by a resonant circuit,which does not like to have a second harmonic component. We will have to drive the second harmonic hard,
and we will also need to pay special attention to the stability of the amplitudes of the second harmonic on
all the Rods to ensure that we have a homogeneous uniform oscillating second harmonic electric field. Thissounds like a hassle, but not impossible. Still, in our quest to get away with being lazy, we came up with
Plan D.

225

8.5

Plan D: Transverse magnetic quadrupole

8.5.1

Working principle
The working principle behind Plan D is similar in spirit to Plans B and C: we allow for an asymmetricphase build-up and subtraction at various parts of the Erot cycle. In both Plans B and C, the asymmetry is
introduced by modulating the strength of Erot at a frequency that is commensurate with the phase-inducing
physics (Berry’s phase for Plan B, and Zeeman shift for Plan C). Plan D seeks to overcome the problem inPlan C where we will have to drive the second harmonic into the resonant circuit for the rods very hard. To
do this, we use the fields of the following form:(1 + ε) cos ωrot t




−−→

Erot = Erot  (1 − ε) sin ωrot t 





0

,





 cos 2ϕ

B = Btransgrad − sin 2ϕ0

sin 2ϕ0  x 
 
 
 
cos 2ϕ 0 −y  ,
 
 
0
1
0

(8.32)

where ϕ is the angle between the x-axis and the principal axis of the transverse magnetic quadrupole that−−→would nominally be aligned with the x-axis; and ε is the ellipticity. Such an Erot will give a slightly modified
rotational micromotion:(1 + ε) cos ωrot t





r(t) = rsec (t) − rrot  (1 − ε) sin ωrot t  ,




0

(8.33)

where rsec is the secular position of the ion; evidently the rotational micromotion is now elliptical.
Employing the same trick used in Plan C, i.e. evaluating ⟨B · Ed
rot ⟩, the resulting f0 holds the form:
f0 = 3gµB Beff

,

 0 




3
3

Beff = ε · Btransgrad rrot  sin 2ϕ  + O(ε ),
2




cos 2ϕ(8.34)

where the z-component of Beff is the one that goes into f0 , while the y-component is included there to remindourselves that a non-zero ϕ can induce additional coupling between states of neighboring mF through the
Zeeman term in the Hamiltonian. Fortunately, this additional coupling term (∼ f0 = 50 Hz) is small
compared to the rotation term in the Hamiltonian (∼ frot = 150 kHz), so modifications to ∆u/l should besmall.

226
8.5.2

In the Bucket Brigade
The transverse magnetic field gradient will be produced by very wide “anti-Helmholtz coils” runningthe entire length of the whole Bucket Brigade. For wires at radial distance R ≳ 4′′ away from the trap axis,
the magnetic field within the vicinity of the trap axis is:
"

5 #
2µ0 I  xy {x, y}
B=,,− ,0 + O
πR
R R
R
and for typical ion cloud radius of lsim1 cm, we see that the transverse field will be very homogeneous in
the region of interest.
The resonant circuits driving the Rods will be driven only at the resonant frequency. The amplitudes
will be tuned such that four of the Rods will operate at (1 + ε) the nominal amplitude, while the other fourat (1 − ε). This is easy.
However, to ensure that Criterion 8.1 is fulfilled, we need to ensure that we can maintain the amplitudesof Vrot on all the rods down to 0.01% of the desired ellipticity of Erot . To put things into perspective, thenumber of significant digits involved for an ε = 0.01 is:

1 + ε = 1.010000,
1 − ε = 0.990000,

which is a demand of stability down to 1 ppm on our resonant circuit.

8.5.3

Final remarks
As of the time of writing, we do not know if the above demand is realistic. We currently have a 0.1%

amplitude precision for the transformers in the b.B.B., and it is not immediately obvious how we can makethis better. To this, our advisors say, “work harder.”

227

8.6

Plan E: Rotating magnetic field with time-oscillating currents

8.6.1

Working Principle
The field generated by a current flowing down a straight infinitely-long wire with current i is givenµ0 i
by B(r) = 2πrϕ̂, as one can easily check with Ampère’s law. Suppose in addition to driving the radial

electrodes with the RF-trapping potentials and the rotating electric field that polarizes our ions, we drive
the electrodes with oscillating currents with phases matched to the rotating electric field. This would give
us a net magnetic field along the trap axis that rotates with the electric field — a magnetic field that is truly
co-rotating with the polarizing electric field.

8.6.2

In the Bucket Brigade

8.6.2.1

Intentional currents

Suppose we have eight radial electrodes that we number from 1 to 8, these radial electrodes can belong rods that stretch the whole length of the Bucket Brigade. The naı̈ve picture would involve sending
oscillating currents intentionally and voltages down the rods numbered n with phases like:
2π(n − 1)in (t) = −i0 sin ωrot t +,82π(n − 1)
Vn (t) = Vrot cos ωrot t +.
8(8.35)(8.36)

This would introduce a Brot that is anti-parallel to Erot . See Figure 8.12 for the same geometry of the setup,
but using charging currents instead of intentional currents.

8.6.2.2

Charging currents

The rods have non-zero capacitance, so charging the rods up to Vrot would require charging currents
to run through the rods. See Figure 8.12 for a schematic of the geometry involved.
The charging currents are to be distinguished from the currents that we intentionally want to add
to give a rotating Brot . If the Rods are only being fed the charging currents from one end, there will be a

Vrot cos (ωrot t)

Vrot cos ωrot t + 7π
4

Vrot cos ωrot t + 3π
2

Vrot cos ωrot t + π4

228



Vrot cos ωrot t + π2

Vrot cos ωrot t + 5π
4

Vrot cos ωrot t + 3π
4
Vrot cos (ωrot t + π)

Figure 8.12: Plan E: geometry of setup with charging currents. Grey disks represent a cross sectionof the Rods. The Rods are charged into the page. The red arrow indicates the direction of Erot at thisinstance in time. The blue arrows indicate magnetic field lines generated by the charging currents running
through the respective Rods. The arrow head/tail symbols indicate the direction of the current flowing in
those Rods. Only the charging currents and magnetic fields for two Rods are shown to prevent clutter. The
sum of all the magnetic field contributions give rise to a Brot,charging that is anti-parallel to Erot .

(linear) gradient of charging currents as we move along the Rods. Through the capacitor equation, we have:
2π(n − 1)
icharging,n (t) = −CRod Vrot ωrot sin ωrot t +,8(8.37)

where CRod is the capacitance of each Rod. The above expression incidentally has the same phase as theintentional currents that we wish to send. This would be great — we could just use the charging currents as
the intentional currents — except the gradient of charging currents along the rods would introduce a gradientin the magnitude of Brot along the Bucket Brigade. This calls for segmented rods or multiple charging points
along the rods11 to reduce charging current gradients.

8.6.3

Final remarks
Assuming that only the intentional current is flowing down the rods12 , the magnetic field generated

by Plan E can be surprisingly uniform in the vicinity of the trap axis with our eight-rod setup. The first
11 This method would require a current bus along the length of the rod, acting as a highway for current to be fed to multiple
points on the rods like highway exits. One would need to shield the fields generated by the currents flowing up and down thebus.
12 One can also use a separate set of long continuous rods or wires that carry the intentional current parallel to the rods
supplying the rotating electric field, as this would decouple the wiring for the intentional current from the segmented rods.

229
non-trivial deviation from uniformity comes in as the sixth order in radial coordinates, i.e., {x, y}6 .
Plan E looks promising in terms of uniformity. The trade off comes in the construction and assemblyof parts, especially if we are going to use the segmented rods. Inhomogeneity can come in through theimperfections of the assembly, e.g., if a rod segment is not fully aligned with the rest of the setup, it could
introduce deviations in the fields at the homogeneity levels that we care about.

8.7

Further discussion
There are uncertainties associated with all the aforementioned Plans. Spatial field homogeneity re-quirements also place hard limits on size and geometry of the setup, giving rise to several hard-to-go conclusions. Before we make the big decision to redesign the Bucket Brigade and tailor fit it to a particular Plan,
it would be wise to test out some of the Plans in the b.B.B.. This is something that will happen after I am
gone from the group, unfortunately.
As of this writing, we are using Plan A in the b.B.B.. There is a pair of circular anti-Helmholtz coils(8′′ diameter, 10 turns, giving 0.65 Hz/mA at Erot = 60 V/cm with frot = 147.5 kHz) at the Octagon for
diagnostics experiments at the Octagon. The wire was wrapped around the Six-way Cross and the Cold
Tube on either side of the Octagon. There is another pair of square anti-Helmholtz coils13 (56 cm each side,
3 turns, giving 21 Hz/A with the same conditions as before) at the Cold Tube where the ions sit, the Science
region. When we run experiments at the Science region, we only keep the square anti-Helmholtz coils turned
on, and we keep the circular anti-Helmholtz coils at the Octagon turned off.
When we charge up the Rods to give us the rotating electric field, there is enough capacitance perunit length of the Rods with the rest of the world to induce charging currents along the rods. In the current
b.B.B. configuration, we charge up the Rods from the warm end (the side closest to the ion detector). At
this end of the Rod, the magnitude of the charging current is:

Icharging = CRod Vrot ωrot ,where CRod is the total capacitance of the Rod-World system, Vrot and ωrot are the voltage amplitudes and
13 The anti-Helmholtz condition for a square set of coils is for the separation d to be 0.946 times that of the length of eachside of the square L, i.e. d ≈ 0.946L.

230
angular frequency of the voltage that we send to the Rods to generate the rotating electric fields, respectively.
In the same spirit as Plan E (Section 8.6), these charging currents generate a Brot,charging aligned with Erot ,giving rise to a net Zeeman shift in our molecules. The magnetic field induced by the charging current
decreases in magnitude as we move from the charged (warm) end of the Rods to the unconnected (cold) endof the Rods. Hence, there is a net gradient of Zeeman frequency as we move from the Octagon to the Science
region. Figure 8.13 shows a scan of the Zeeman frequency from the charging currents along the length of
the b.B.B..

9080
f R (Hz)

70
60
50
40
30
20

0

2

4
6
Number of buckets in8

Figure 8.13: Charging-current–induced Zeeman frequencies in the b.B.B.. Data was taken on 30
April 2023. Steps taken to produce this plot are as follow.
(1) Perform a Ramsey experiment at various locations within the b.B.B. (horizontal axis indicates location into the Cold Tube from the Octagon in units of buckets travelled; one bucket length ≡ 8 cm)
and extract the Ramsey frequency. Perform all these Ramsey experiments with the rotating electricfield rotating in one direction, say, clockwise, corresponding to rotation switch R̃ = +1.
(2) Repeat the experiments with the electric field rotating in the other direction, corresponding torotation switch R̃ = −1.
(3) Take the difference between these two frequencies (vertical axis, f R = (f R̃ = f + ) − (f R̃ = f − ) /2),
and plot them out for various locations within the b.B.B. (horizontal axis).
At each position, the Zeeman shift measured is a sum of two contributions: (i) ambient axial magnetic field
gradient (physics from Plan A), and (ii) magnetic fields from charging currents (Plan E). Reversing therotation direction of the electric field flips the sign of the Zeeman shift from the charging currents, but keeps
the sign of the Zeeman shift from the ambient magnetic field gradients invariant. By taking the differencein frequencies for the two rotation directions, we can extract the component of the Zeeman shift that comesfrom the charging currents only. We fit two lines to the points. These two lines have different gradients,
with a kink connecting them at about three buckets in. This suggests that the capacitance of the Rods perunit length has a sudden change at about three buckets in. It is not immediately clear why the kink happensat three buckets in, but we believe it to come from a combination of extra stray capacitance from the Rods
with additional structural support and wiring in the region before three buckets in (see Figure 7.11).

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The gradient of Zeeman shift from charging currents is bad for precision metrology with ions in
multiple buckets at different axial location within the b.B.B., but it is helpful in preserving the definition of
the Zeeman states as we move the ions (that have already gone through state preparation at the Octagon
into the X 3 ∆1 (v = 0, J = 1, F = 3/2, mF = 3/2) states) from the Octagon to the Science region.
In the future, there will be additional sets of coils that will suppress the axial gradient introduced by
the charging currents so that we can try out multiplexing protocols in the b.B.B. where we will have threeor four ion clouds in neighboring buckets inside the b.B.B. for a simultaneous Ramsey spectroscopy.
In the last few chapters, we laid out our ambition to perform a multiplexed ion trap experiment. Withso many electrodes to control, so many other instruments to coordinate, and so many ideas we wish to try
out, this calls for a robust and scalable control system to govern the operations of the experiment. In thenext chapter, we shall look at the brain that holds everything together.

Chapter 9

Control System

9.1

Introduction
We aim to perform a 20 s experiment with nanosecond timing precision on some of our operations,so this requires serious coordination of various components of our hardware. We also need to be able torun our experiment in a programmable and automated way with a user friendly interface, preferably witha system that does not break with operating system or software version updates. The systems used by the
first two generations of the JILA eEDM experiments and the ThF spectroscopy experiment use hardware(e.g. ViewPoint DIO64) that are no longer available on the market. It is also not obvious how to implementa pseudo-continuous operation mode that is required for the Bucket Brigade using the old control systems.
We decided to take this opportunity to develop a new control system and test it out on the baby Bucket
Brigade.
We had a sneak peak at how we will be driving the electrodes with the Rod driver (Section 7.5.1.4) andRing driver (Section 7.5.2). However, we also need to coordinate the timing of the voltages with the pulses
controlling when, say, the nozzle opens, or when the laser fires to perform our eEDM experiment (Section
4.3.3 and Figure 7.9). This requires a control system that knows and controls everything. At some point,
we will want to test out some of the Plans that we discussed in Chapter 8. This would require a control
system that is scalable and modular. The following sections will briefly touch on the various components ofour control system built upon the ARTIQ Sinara systems from M-Labs.

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9.2

HardwareWe built upon the wisdom from the Gen. I JILA eEDM experiment to set up our control system.

Much of the instruments used in the b.B.B. setup is also inherited from the ThF/ThF+ spectroscopy setup.
There are a few main components:
Master clock (SRS DS345). A master clock at 10 MHz is distributed to all our timing critical components
in the experiment. The clock is distributed via a clock distributor (SRS FS735).
Digital delay generator (SRS DG645). A digital delay generator (DDG) generates pulses at 10 Hz
repetition rate for our pulsed Nd:YAG lasers. These lasers need to be fired continuously at 10 Hz
for optimal performance. The DDG sends out four pulses with well defined delays between them.
Two of these pulses go to our SpectraPhysics Lab-190 Nd:YAG laser, and the other two to ourSpectraPhysics INDI Nd:YAG laser. The two pulses trigger the flashlamp and the Q-switch of thelasers. The DDG also sends a start pulse to our FPGA controller at the start of each burst of a setof four pulses to synchronize the timing of the DDG with the TTL generator.
FPGA controller. An independent processor that receives commands from the main control computer onwhen to set off the TTL generator, DACs, and DDSes to run an experiment. Uses the external clock
signal from the master clock.
TTL generators. An on-off square pulse generator (referred to as TTL generators, despite not necessarily
conforming to TTL standards) triggers the rest of our components, e.g. laser shutters, nozzles,
oscilloscope, microwave switches, camera, AOM.
Digital-to-analog converters (DAC). To generate arbitrary voltages that change at low frequencies.
This is usually used to add shimming voltages to our electrodes or to act as analog inputs to HVpower supplies.
Direct digital synthesizers (DDS). To generate arbitrary voltages that change at higher frequencies.
This is used to generate our RF trapping voltages, our rotating electric fields, and our parametric
excitation voltages. Uses the external clock signal from the master clock.

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Function generators. To strobe our AOMs or to chirp our microwave frequencies. They use the external
clock signal from the master clock.
Main control computer. A computer dedicated to talk to the FPGA controller directly.
Auxiliary control computers. Computers throughout the lab that run drivers for other satellite control
systems, e.g. wavemeter, fiberswitch, doubling optics servo system, asynchronous camera system,
asynchronous oscilloscope system, magnetic field coils driver, and cryocooler control system. These
control computers can receive commands from the main control computer, and they can also send
information to the main control computer regarding statuses of the systems. Communication is
performed with TCP/IP over an intranet system set up in the lab.
In the early days of the ThF spectroscopy, we used “spare parts” from the Gen. I JILA eEDM experiment to
set up our control system (Section 9.2.1), but we upgraded to a new system during the pandemic when we
switched gears from ThF/ThF+ spectroscopy to the design and construction of the b.B.B. (Section 9.2.2).

9.2.1

Legacy system
The DDSes used were the part of the trap driver from the Gen. I JILA eEDM experiment. The detailsof the trap driver can be found in Huanqian Loh’s thesis.
The DACs used were from National Instruments. We combined two sets of DACs together in ahome-built “consolidator box” to switch between the two sets of voltages quickly.
The main brain of the legacy system uses the ViewPoint DIO64 card, which served both as the FPGA
controller and the TTL generator. It boasted 64 TTL output channels, way more than any of its competitors,
and it came at a reasonable price. Unfortunately, good things never last, and they were no longer available
even at the start of the ThF spectroscopy experiments. We barely made it through the spectroscopy before
the DIO64 card started to show signs of failing. This was the main push to search for a new system.

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9.2.2

Current system
We were introduced to the ARTIQ control system by our fellow ion trappers from the Ion Storage

Group down the street at NIST. ARTIQ was first developed at NIST, and is commercialized by M-Labs,
based in Hong Kong. It is a system that is currently used by ion trappers at NIST, Duke, Oxford, and many
more. The Thompson Group at JILA is also porting their control system to ARTIQ. Alternatives to ARTIQ
includes Quantum Machines (based in Israel) or sticking to the National Instruments suite. We settled for
ARTIQ in the end.
We bought two ARTIQ crates (the “cu3 variant”) that come with (the datasheets can be found on
the M-Labs website, but some of the more critical specifications are listed here):
FPGA controller, the core device (Sinara 1124 Carrier, Kasli). Communicates with the main control computer through ethernet; this is an upgrade over the ViewPoint DIO64 that communicatesthrough GPIB.
64 TTL channels (Sinara 2118 BNC-TTL). There are four channels that can be switched between
input and output, and the rest of the sixty channels are output only. We use one of the input
channels to receive the start pulse from the DDG to synchronize the TTL outputs with the four
pulses for the pulsed Nd:YAG lasers. Outputs a high of 2 V on a 50 Ω load, with a timing jitter of
350 ps. Minimum pulse width of 3 ns.
64 DAC channels (Sinara 5432 Zotino). There are two Sinaro Zotino cards, each with 32 DAC channels. Outputs ±10 V with a 1 MSPS update rate shared across all channels (within each Zotinocard). Has a 3 dB bandwidth of 75 kHz, with a 1-99% rise/fall time of about 30 µs.
Figure 9.1 shows a list of how the channels are being used in the current setup.

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Figure 9.1: List of things connected to the TTL and DAC channels. A picture of the hardware isalso shown here for a feel of the density of cable connections.

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9.3

Software
One of the features of ARTIQ that appealed to us is that it comes with software with built-in graphics

user interface (GUI). Despite being students who have had programming experience, programming a GUI is
non-trivial, and there is no reason not to use a control system that comes with an in-built GUI. Figure 9.2
shows the ARTIQ dashboard, serving as the GUI frontend.

Figure 9.2: The ARTIQ dashboard. The panels on the left and right are applets that we can write (using
PyQtGraph) to show live data streaming in. For example, the top left panel shows the image captured byour camera; each bright spot corresponds to a Th+ ion hitting our ion detector. The top right panel shows
the progress of the experiment that is currently running. The panel in the center allows us to set parameters
for an experiment that we wish to use or scan. The bottom left panel shows experiments that are queued
to run, and the bottom right panel shows the log. The user is free to move the panels around to suit theirtastes.

Upon successful execution of the experiment, the experiment parameters given as user arguments
through the dashboard are saved together with the data collected into an HDF5 file on our local server.
These saved experimental parameters can be loaded onto the dashboard to rerun the experiment with theexact same conditions as the one saved. The Gens. I and II JILA eEDM experiments’ equivalent to this is
the *.ubin file (see Matt Grau’s thesis).

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We built our day-to-day experiment runs upon the ARTIQ software platform. I shall go into details
for some of the structure that we have built upon the ARTIQ software platform in the following sections.

9.3.1

ARTIQ for the uninitiated
These section is mainly a summary of some key points about the ARTIQ system for the uninitiated.

Interested readers can read their more extensive manuals on their website.
The artiq_master main control loop handles job scheduling and sends out commands to the FPGA
to start the experiment sequence. On the user side, one typically writes code with the typical structure asshown in Figure 9.3. The code is written in python.

Figure 9.3: Typical code structure. One would have to write a class for each experiment. The class inherits
from EnvExperiment, which is defined by ARTIQ, with methods build, prepare, run, and analyze. The
user then overloads the methods by defining their own version within the user-defined class. In this case, theuser defined class is Ramsey. The build method associates devices and user input arguments as attributes
to the class. The prepare method runs part of the code that is independent of the actual experiment, e.g.
calculating some variables from user specified arguments. The run part is where the actual sequences are
written. The analyze part (not shown in the figure because we do not use it) is where we can perform somepost-analysis of the data collected.

Users typically submit jobs to the queue through the ARTIQ dashboard, but can also submit jobs
through a command line interface command artiq_run or artiq_client submit.
Upon a power cycle (from, say, power surges during the summer at JILA), we would need to run
the artiq_master command on the main control computer once to let the main control loop run. On a
day-to-day basis, we would run the artiq_dashboard command on our workstations, which will then open
up the ARTIQ dashboard GUI, and communicate with the main control computer through our intranet
regarding job submission and information.
As alluded to earlier on, the only hardware that the main control computer is connected to is the Sinara

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FPGA. The main control computer communicates with the other instruments through remote procedure
calls (RPCs). When this happens, the main control computer talks to the instruments through the auxiliary
control computers. A list of devices is stored in the device_db.py file.

9.3.2

Important folders and files
The folder structure is as follows:
artiq_master/
dataset_db.pyon, last_rid.pyon
repository/
device_db.py
analyses/eedm.py, analyze.py, h5_unpack.py
applets/
controllers/
experiments/
Commissioned/
Personal_sandboxes/
Utilities/
helper/gen3_macro.py
ARTIQ_DAC/driver.py
TTL_helper/driver.py
results/<YYYY-MM-DD>/<HH>/<RID>_<experiment_name>.h5
First we have the home directory artiq_master/, not to be confused with the program artiq_mastermentioned above. It contains two Python JSON files (*.pyon). The dataset_db.pyon stores a list of240
persistent datasets that the users broadcast to the entire system. We use this to broadcast common variables
that could change on a day-to-day basis to every connected workstation, e.g. trap shimming voltages. The
last_rid.pyon file stores the Run ID number of the last run experiment. Within artiq_master/ there are
two folders: repository/ and results/.
Within the repository/ folder, we have a few important things. The device_db.py file stores alist of devices that the main control computer can talk to through the auxiliary control computers, and theaddresses and ports of the auxiliary control computers.
The analyses/ folder contains the suite of analysis tools that we have developed for the experiment(Section 9.3.4.3). analyze.py is the main workhorse of the data analysis package. eedm.py is the tool thatgives all the optical pumping, PQR branch simulations, and ThF+ Hamiltonian related tools. h5_unpack.py
is a macro we wrote to unpack the HDF5 files.
The folder applets/ contain the code for the applets that we saw on the ARTIQ dashboard (Figure9.2). The folder controllers/ contains all the drivers that we wrote for our devices. The drivers usuallycontain a list of important commands in the “programmer’s manual” that comes with the device. We try touse VISA communication with the devices whenever possible.
The experiments/ folder contains all the code with structure mentioned in Figure 9.3 that tells
ARTIQ how to run the experiment. We split it up into three main folders: Commissioned/ for all the codethat have been proven to work, Personal_sandboxes/ for code that each individual would like to messaround with before commissioning, and Utilities/ for macros to execute small things on devices that are
not directly related to an experiment per se, e.g. ramping the microchannel plate assembly to the rightvoltages.
The folder helper/ consists of macros that we wrote to make our lives easier. gen3_macro.py isa dictionary of names that we assign to TTL channels (see Figure 9.1). The ARTIQ_DAC and TTL_helper
driver files parses high-level commands used by the experimentalists into timing and voltage commands thatARTIQ understands. For example, it parses the following:<TTL_helper>.add_pulse(`Trap loading', `Consolidator TTL',\

241
pulse_duration = self.pulse_pulser,\delay_duration = 0)
into a series of <coredevice>.core.seconds_to_mu() commands to set the time for the next event, and<coredevice>.ttl.TTLOut.on() and <coredevice>.ttl.TTLOut.off() to turn the TTLs on or off at thecorresponding times.
Finally, we have the results/ folder, which contains all the HDF5 files with the data and user input
arguments for all the experiment. This folder is further sorted into the YYYY-MM-DD/HH/ subfolders basedon when the experiment was run.

9.3.3

Communication with auxiliary control computers.
Communication with devices from the main control computer is done through the auxiliary control

computers. We used to write our own TCP/IP communication VIs when we were using the LabVIEW
system. ARTIQ comes with the Simple Python Communication (SiPyCo) library1 that allows us to do justthat, but in a “pythonic” form that is also fully compatible with ARTIQ.
The communication comes in two parts: (i) the drivers with the relevant commands from the devices’“programmer’s manual”, and (ii) a controller server loop that opens a connection with the device throughthe drivers.
Device drivers.

The device drivers declare one python class for each device. Upon initiation of

the class, communication is established by the auxiliary control computer with the device if the device isconnected. Users can call on the methods of the class to ask the device to perform certain tasks. We show
an example of a code for a driver in Listing 1.
Controller server loop.

The server loop runs on the auxiliary control computer. It runs thedrivers and establishes direct communication with the devices. Within the loop, it listens for RPCs fromother control computers and runs the relevant RPC commands (involving the methods written in the device
drivers). An example of a code for a controller server loop is shown in in Listing 2.
1 SiPyCo used to be part of ARTIQ, but was split out so that one can use the former without having to install the rest of
ARTIQ.

"A snippet of the DS30 cryocooler driver"
1
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3
4
5
6
7
8
9
10
11
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13

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class ds30_cryocooler(_visa_comm): # inherit methods from `_visa_comm`""" DS30 Cryocooler """def __init__(self, visa_resource, baud_rate=None, read_termination=None,\write_termination=None, silent_bool=None):### Inherit `__init__` from `_visa_comm`if write_termination==None:
write_termination = '\r'
if baud_rate==None:baud_rate = 9600
_visa_comm.__init__(self, visa_resource, baud_rate=baud_rate,\
read_termination=read_termination,\
write_termination=write_termination,\silent_bool=silent_bool)

14
15
16
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def on(self):""" Turns on cryocooler. """self.WRITE('COOLER=ON')

18
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def off(self):""" Turns off cryocooler. """self.WRITE('COOLER=OFF')

22
23
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def set_TTARGET(self,TTARGET):""" Sets TTARGET (target temperature). Takes in argument `TTARGET`. """self.WRITE('TTARGET={}'.format(TTARGET))

Listing 1: A snippet of the DS30 cryocooler driver. We communicate with the DS30 cryocooler
through VISA. Three methods are shown here: <ds30 cryocooler>.on() to turn the cryocooler on,<ds30 cryocooler>.off() to turn the cryocooler off, and <ds30 cryocooler>.set TTARGET(TTARGET) to
set the target temperature to TTARGET.

One can talk to a particular device (e.g. to turn the DS30 cryocooler) on a particular auxiliary control
computer connected to the device (e.g. IP address 172.31.1.207, port 7000) from any workstation connectedto the intranet with a SiPyCo command (in the terminal or command prompt) of the form:
sipyco_rpctool 172.31.1.207 7000 call -t ds30 on
where the -t <device_name> flag is to indicate which device to talk to in case the auxiliary control computer
is connected to multiple devices. I recommend the learning graduate student or post-doc to check the manualon their website (https://m-labs.hk/artiq/sipyco-manual/#remote-procedure-call-tool) for more
information on SiPyCo commands.
In general, it is bad practice to connect all the devices to one auxiliary control computer. Doing so

"An example of a controller server loop"
1
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3
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5

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import argparse, sys, asyncio
from camera_vimba.driver import mcp_camera
from current_control.driver import keysight_current
from sipyco.pc_rpc import simple_server_loop
from sipyco import common_args

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### To parse argument for controller
def get_argparser():""" Argument parser """

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### Define argument parser object
parser = argparse.ArgumentParser()

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### Boilerplatecommon_args.simple_network_args(parser, 7000) # (parser, default_port)
common_args.verbosity_args(parser)
return parser

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### Main codedef main():### Boilerplateargs = get_argparser().parse_args() # retrieves dict of arguments
common_args.init_logger_from_args(args)

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### Define devices
connected_dev = {}try:connected_dev['camera_mako'] = mcp_camera('DEV_000F315C339E')
try:
connected_dev['Baxgrad_supply'] = keysight_current()

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### Open server in a looptry:simple_server_loop(connected_dev, common_args.bind_address_from_args(args),\
args.port)

Listing 2: An example of a controller server loop. The gist of the code is as follows. We import theclasses from the relevant driver files. We then create a dict of connected devices, with names of the devices
as keys and the relevant class as the values. Then we execute the server loop with simple server loop.

will reduce the bandwidth of the computer to work with all the devices simultaneously. In cases where there
needs to be simultaneous transfer of information from multiple devices, the computer may make unhappy
noises instead.

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9.3.4

The JILA eEDM suite
The ARTIQ software framework is very versatile, and in principle one can write code to run any

experiment of their liking. We can easily scan experiment parameters by using the for loop in python.
However, we quickly realized in the early days of getting familiar with ARTIQ that this is not scalable. We
would have to write a different for loop each time we wish to scan a different parameter or over a different
range. Depending on the order at which we change settings on our devices, changing the order of the forloops could end up being a total disaster. We need a scalable framework.
To address our needs, we introduce a particular structure to our code.2

We do not claim that thisis the best way of doing it. We also encourage users of the code to come up with creative ways to make thecode more efficient and more streamlined. The code that we use was written in the land of the free, so free
it shall be: free to modify and distribute. This section and the next only serve as a form of documentationof the code structure that we currently have in our lab. The target audience includes incoming students andpost-docs, and, of course, people who wish to adopt something similar in their labs.
Like all projects that involve coding, one does not hope to be an expert just from reading the documentation. I certainly do not intend for the content in this section to be fully self explanatory. I highly
encourage new students and post-docs to learn by using our system with the guidance of those who havealready been using it for some time. Hopefully through a combination of hands-on approach and the summary that I provide here, one would be able to pick up the essence of the JILA eEDM suite, and suit it totheir purposes.

9.3.4.1

Experiment code structure

As alluded to in Figure 9.3, to run an experiment, one writes a class with methods build, prepare,run, and analyze. Out of the four, only build and run are the absolutely necessary. The former declares
the devices, and the latter is the actual sequence to run for the experiment.
In the JILA eEDM suite, we introduce two types of classes in our attempt to make our code more
2 We learnt about ndscan (see https://github.com/OxfordIonTrapGroup/ndscan) a little too late for us to implement itwith minimal effort in our setup.

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modular:
The “underscored” class: class <experiment>( edmScan). We write the “underscored” class to be as
general as possible within the framework of a type of experiment. For example, we have a class called_Ramsey(edmScan) that performs a Ramsey experiment: (i) create ions and trap them, (ii) perform
state preparation, (iii) translate ions into the Science region, (iv) perform the Ramsey sequence, (v)translate ions back into the Octagon, (vi) perform the state detection sequence, and (vii) kick ionsout onto detector. Within each stage, all experiment parameters are set to be user input arguments.
These user input arguments can be set through ARTIQ dashboard; they also have default values if
the user does not wish to change them.
We have a separate “underscored” class for each type of experiment. For example, we also have aclass called _Slosh_Ions(_edmScan) that kicks the ions in the trap to induce slosh: (i) create ionsand trap them, (ii) kick them in a certain direction, (iii) wait for some time, and (iv) kick ions out
onto detector. Evidently this type of experiment is very different from that catered by _Ramsey.
The “underscored” class inherits from class _edmScan(HasEnvironment) that defines all the commonly used user arguments under its build method. These user arguments can be shared acrossmultiple types of experiments, e.g. _Slosh_Ions and _Ramsey. The class _edmScan also has a methodthat consolidates all the user input arguments into a list that can be accessed as an attribute of theclass. This list will be used in the non-“underscored” class.
The non-“underscored” class: class <experiment>(EnvExperiment). The non-“underscored” class isa wrapper around the “underscored” one, where it sets the user input arguments programmatically.
It assigns the “underscored” class as an attribute to itself, and invokes its methods whenever weneed to communicate with our devices. The non-“underscored” class allows the user to choose a user
argument from the list generated by _edmScan, and perform a scan by looping through multiple callsof the “underscored” class. The non-“underscored” class also allows the user to input “chop names”
that can be parsed by the “underscored” class to switch between various states of the experiment.
This allows lab users to run different variants of the same type of experiment without modifying the246
code each time.
We leverage on the class inheritance mechanism to define non-“underscored” classes that inheritfrom the base non-“underscored” classes to perform a fixed set of routine. For example, we can haveclass R_branch(Ramsey) inheriting from the non-“underscored” Ramsey. See Figure 9.4) for more
information.

Figure 9.4: Example of having multiple non-“underscored” classes inheriting from the main
non-“underscored” class. The class Pulse Phosphor is used to scan the timings of when to gate thephosphor off to distinguish phosphorescence signal between Th+ and ThF+ . The class R branch is used toscan the R branch of the REMPD intermediate state.

9.3.4.2

Parsing the code names

As mentioned in Section 4.3.1.2, we use abbreviated one-character code names to represent each laser
and microwave. The JILA eEDM suite turns all the laser and microwaves off by default, but it turns themon instead if it receives the corresponding code name. For example, it the software receives a string “SU”, itwill turn Sheep and the 43 GHz microwaves on (see Section 4.3.1.1 for the corresponding code names). The
user inputs this under the Chop section of the ARTIQ dashboard interface.
The code that implements this is a parser written in the “underscored” class. It is just a series of if
clauses of the form:
if 'S' in chop_name:<code to turn Sheep's shutter on>
if 'U' in chop_name:<code to turn 43~GHz microwave on>

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...

To differentiate between having the lasers/microwaves turned on during the state preparation andstate detection stages, we use a “+” as a divider. For example, “SU+GL” will turn only Sheep and the
43 GHz microwave on for the state preparation stage, and only Goat and Llama for the state detection stage.
This implementation is just the (upgraded) code version of clicking checkboxes on the LabVIEW GUI
in the Gens. I and II experiments to choose which set of lasers/microwaves to turn on. One can easily scale
up this protocol to include other “switches” (e.g. whether to translate the ions into the Science region) byencoding the relevant code names into the parser in the JILA eEDM suite.

9.3.4.3

Analysis package

All experiment data and user input variables are saved into HDF5 files after each run. Our analysis
suite is contained in the analyze.py file mentioned in Section 9.3.2. It consists of class get_moments()
that takes in an HDF5 file object and an optional dict of keyword arguments. Upon initialization of the class,it extracts (i) the capacitively coupled output from the oscilloscopes, and (ii) phosphorescence as captured
by our camera. From here, it massages the data into structured dicts sorted by “chop names” and ascendingorder of the scan parameter, if any. This forms the basis of our analysis package.
Within get_moments, there are methods that can take the structured dicts and perform further
analysis, e.g. segregate the detected Th+ ions into those of the two different doublets depending on whichside of the swatch they sit on, and plot out Ramsey fringes for both doublets. One usually writes suchanalysis code in their personal sandboxes, and only shares it with others by commissioning it into a methodin get_moments on the demand of others.
There is a wrapper function edm_analyze(fn) in analyze.py. This wrapper function takes in afilename (of the HDF5 file), analyzes the type of the experiment that was run (saved as metadata in the
HDF5 file), and calls the appropriate methods from get_moments to generate the plots associated with thattype of the experiment.

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9.4

Further discussion
We saw how we borrowed wisdom from the Gen. I experiment in setting up our control system. We

also listed some of the more critical components of our hardware system. We then discussed how we upgraded
our old system to a new one based on the ARTIQ control system. It is mainly an upgrade in the hardware,
but the software comes for free. As with all experiments, the control system is mutable. The control systemevolves with each stage of the experiment. It is up to the lab users to adapt the tools available to them totheir needs.
I led the team in building the control system that we have now. I tried to design it to be as scalable
and modular as possible, guided by my limited foresight of the kind of experiments that we wish to performwith the setup. Nevertheless, our experiment is heading in the direction of pseudo-continuous operation withthe multiplexed ion trap. This is something that has never been tried in the JILA eEDM labs. Turning anexperiment setup that runs one shot of the experiment at a time into one that can coordinate a pseudocontinuous operation is certainly non-trivial. Furthermore, it can also be tricky to correlate the stored HDF5
files with the corresponding bucket of ions. Here, I lay out some extensions to our current system that thegroup may wish to adopt in their endeavors to get to a fully mature Bucket Brigade experiment.
Idle kernel. The idle kernel is a piece of code that is stored on the core device. It runs whenever the coredevice is idle. One can possibly use this like a DDG, spitting out pulses continuously with welldefined delays between them. I imagine using a version of the idle kernel (or another SRS DG645)
to coordinate the pseudo-continuous operations in the Bucket Brigade setup.
Distributed Real Time Input/Output (DRTIO). Verbatim from the ARTIQ website:
DRTIO is a time and data transfer system that allows ARTIQ RTIO channels to be
distributed among several satellite devices synchronized and controlled by a central coredevice.
Much like how we have on-board FPGAs that control our DDSes in our Rod drivers, we could lessen
the burden on our core device and distribute it among several satellite devices. This may come inuseful in the future when we find ourselves exceeding 64 TTL channels, or when the core device

249
cannot keep up with the number of events happening concurrently.
Upgrading to ARTIQ 7+. New versions always come with new features. One of the features in ARTIQ
7 is that one can now configure the number of FIFO lanes in the scalable events dispatcher (SED)
directly. One of the problems that we see in our experiment is that when we try to schedule too
many events too close in timing, the FIFO lanes (eight of them, by default) fill up and the rest of theevents are jammed. This results in events being skipped, and the experiment does not work. When
the FIFO lanes are full, sometimes it returns an RTIO underflow error (which is already irritatingby itself) but sometimes it does not even return an error (a known bug in our system). It turns outthat configuring the number of FIFO lanes in the SED in our current ARTIQ 6 system is a real pain.
There is a saying that goes, “if it works, do not change it,” because there is always a non-zero chancethat the thing will break with the change. One will need to have great determination to upgrade
our currently working ARTIQ 6 framework into ARTIQ 7+ to get the aforementioned feature. I
shall leave this big decision to the diligent, intelligent, and responsible people in the group after I
am gone.

Chapter 10

Cryogenics

10.1

Introduction
The Ramsey interrogation time in the HfF+ experiments are limited by the radiative decay of 3 ∆1 →X 1 Σ+ . The radiative lifetime was estimated to be about 2 s [86]. This is sufficient for a Ramsey interrogation
time of about 3 s in the Gen. II experiment. As alluded to earlier on, the 3 ∆1 electronic state is the groundstate in ThF+ . This means that we will not be limited by decay from X 3 ∆1 in ThF+ to any lower lying
states. One limiting factor for our state lifetime (T1 ) is the rate at which molecules are excited out of
the X 3 ∆1 (v = 0) manifold through blackbody radiation. We have estimated this to be dominated by the(v = 1) ← (v = 0) channel with a lifetime of about 4 s at 300 K (see Table 4.2). We mentioned in Section
4.2.3 that we can suppress this loss by working with cryogenic temperatures. Figure 10.1 shows how quickly# of photons per mode

we can suppress the blackbody radiation photon numbers by going down to lower temperatures. The relevant101

X 3 1(v = 0) X 3 1(v = 1)

10 1
10 3
10 5

77 K
120 K
150 K
200 K
300 K

10 7
10 9
0

200

X3 1 a1 +
400
600
800
Frequency (cm 1)

1000

1200

Figure 10.1: Number of blackbody radiation photons. Plotted as a function of wavenumbers for easycomparison with Figure 10.2.

251energy scales for both molecules are shown in Figure 10.2 for comparison.
HfF+
3

ThF+
3

∆2

∆2

−1

2150 cm

1050 cm−1v=1755 cm−1v=1

36 GHz
3

∆1

v=0

J =2

653 cm−1

a 1 Σ+

J =1

977 cm−1

314 cm−1

X 3 ∆129 GHz

v=0

J =2
J =1

X 1 Σ+

Figure 10.2: Energy level diagram for the lowest lying states for HfF+ and ThF+ . 1 cm−1 ≈30 GHz. Blackbody radiation excitation from the Science state in HfF+ was never a major concern because
the lifetime of the Science state is limited by the radiative decay 3 ∆1 → X 1 Σ+ .

In this chapter we shall see how we introduce cryogenics into our setup. We will first look at thedesign of our cryogenics setup in the context of the Bucket Brigade (Section 10.2). We will then look at theperformance of the cryogenics system on our b.B.B. (Seciton 10.3). Finally, we will conclude this chapter
with a discussion on possible future upgrades (Section 10.4).

252

10.2

Design in the context of the Bucket Brigade
The cryogenic temperature that we are aiming for is only about 150 K. This is not serious cryogenics.

Here is a list of some common cold stuff for comparison.
• Coldest natural temperature recorded in Boulder: 237 K (17 January 1930, from NOAA)• Carbon dioxide, sublimation point (dry ice): 195 K• Coldest natural temperature recorded on Earth: 184 K (21 July 1983, Antarctica [87])• Bucket Brigade: 150 K• Methane, boiling point: 112 K• Oxygen, boiling point: 90 K
• Nitrogen, boiling point (LN2): 77 K
• Helium-4, boiling point: 4.2 K
There are many tried-and-tested solutions to cryogenics (e.g. multi-layer insulation1 , cold shields2 ,and vacuum jacket3 ), many of these are developed for setup involving serious cryogenics where they want
to minimize heat load from the ambient 300 K environment. Although the aforementioned tried-and-tested
solutions are established techniques that have been around for some time, they come with intricate problemsof their own, which can differ from setup to setup. We explored several options with engineers4 from NASA,
many of those involve using LN2, but the size of our setup makes LN2 an expensive option. In the end
we were advised by our in-house Dr. Eric Cornell to try a combination of household insulation foam and amodest cryocooler to get us down to 150 K.
1 MLIs: multiple layers of thin sheets to suppress radiative heat losses. Commonly used in spacecrafts and cryogenic setup.
This was recommended to us by Dr. Eric Silk from NASA.
2 Cold shields act as heat sinks.
3 Vacuum jackets are vacuum chambers surrounding the main chamber at cryogenic temperature. The vacuum suppresses
conductive and convective heat transfer, so the only mode of heat transport from the cryogenic main chamber to the ambientroom temperature environment will be through radiation. Vacuum jackets are typically used in conjunction with MLIs tosuppress radiative heat transfer.
4 Much thanks to Dr. Eric Silk and Dr. Richard Ottens from NASA.

253
This section proceeds as follows. We will estimate the heat load in the Bucket Brigade at steady state
operation (Section 10.2.1). We then look at the effects of (differential) thermal contraction of objects in the
Bucket Brigade which can be made of different materials (Section 10.2.3). Finally, we discuss how we wantto spread the cold from the cold source (Section 10.2.2).

10.2.1Heat load

The cryogenic part of the Bucket Brigade setup is shown as the blue part of Figure 5.5. For the
purpose of the analysis, we shall work with the schematic diagram shown in Figure 10.3.
Cold Head
Octagon for state prep.

Octagon for state det.

≈ 30 cm

0.1200800

Foam thickness ≈ 700

300 K

Thermal insulation (foam)

150 K
Cold Jacket

Figure 10.3: Schematic diagram of a simplified setup for heat load analysis. There is one Octagon
on each side for optics access. These are not temperature controlled, and are nominally at 300 K. A longnipple connects the two Octagon. The dimensions of the nipple are informed by a combination of field
homogeneity simulations (Section 7.3) and what can be bought, say, from Kurt Lesker. A Cold Jacket isclamped onto the nipple to spread the cold from the Cold Head. The Cold Head is situated at the centerof the setup. The Cold Jacket ends at about 15 cm from each end of the nipple. This provides a natural
temperature gradient from the Octagon at 300 K to the Cold Jacket at 150 K. The nipple and the Cold
Jacket are enclosed by thermal insulation. The Bucket Brigade is not drawn in the schematic diagram toprevent overclutter.

Heat load in the system comes mainly through three channels: (i) conduction through the nipple,(ii) radiation/conduction from the Bucket Brigade, and (iii) conduction through the thermal insulation. We
shall first look at the former.
The rate of heat flow per unit time associated with thermal conduction is given by:

P = κ(T )AdT=⇒ P = Adx

R ∆Tκ(T ) dT.
∆x(10.1)

254
Evaluating the above for the nipple with dimensions stated in Figure 10.3, we get a heat load of about 13
and 5.9 W coming in from each side of the nipple for the cases where the nipple is made of 304 SS and
Ti-6Al-4V, respectively.5
For the Bucket Brigade, we expect to work with eight stainless steel Rods, each 0.5′′ in diameter, 1 mm
thick. We assume that the Rods come in thermal equilibrium with the nipple along the axial direction. The
thermal equilibrium comes from radiative heat transfer along the radial direction.6 This gives 250 mW per
Rod per side, totalling to 4 W. Using titanium would half this amount.
For conduction through the thermal insulation, we estimate it for the case where the foam is cylindrically symmetric, with the outer surface at 300 K, inner surface at 150 K, total length of 3 m, with aconductivity of 60 mW/(m K).7 A plot of the conduction power at various foam thickness is shown in
Figure 10.4.

80
Pcond (W)

70
60
50
40
30
2

3

4

5
6
7
Foam thickness (in)

8

9

10

Figure 10.4: Calculated conduction power through household foam. Foam thickness is defined in
Figure 10.3. The foam is assumed to be cylindrically symmetric, with inner diameter the same as the outer
diameter of the nipple. The foam is set to 3 m long, with the outer surface at 300 K and inner surface at
150 K.

We will also have some structural support that will run down the Bucket Brigade. We do not expect
to use structures that have huge cross sectional area, or much higher thermal conductivity than stainlesssteel. The total heat load at steady state from the nipple, Rods, and structural support is estimated to beabout 40 W if all of them are made from stainless steel. Using 7′′ thick foam will give an additional heat
5 Thermal conductance of 304 SS and Ti-6Al-4V are obtained from Table 1A (NIST).
6 There will be some smoothing effect along the axial direction, but we will be ignoring this in our estimation for the heatload budget.
7 The foam was advertised to have an R-value per inch of 5.48, corresponding to 26 mW/(m K), but we ran tests and foundthat the net conductivity between a surface at 300 K and the other at 150 K is about 60 mW/(m K); see Section 10.3.1.

255
load of about 35 W, bringing the total to about 75 W.
Blackbody radiation from the warm ends down the Bucket Brigade contribute to about 30 W (summed
from both sides), bringing the total to about 105 W.
Resistive heating from charging currents running up and down the Rods contribute negligibly to theheat load (∼ 100 µW). We do not expect any other significant heat load in the Bucket Brigade. However,one may need to add additional sources of heat from, say, currents flowing in magnet wires used to generatemagnetic fields. These could go up to a few watts. They would contribute to the heat load significantly ifplaced in direct contact with the cold.
105 W is a reasonable heat load at 150 K. As we shall see in a later section, we can buy an off-theshelf Stirling cryocooler that can operate at around half that heat load, so we just need two of those. It
certainly looks like our modest cryogenic temperature allows us to get away with using a simple cryocooler
and household insulating foam. However, the above only holds if we can spread the cold effectively. We shalllook at how we do this in the next section.

10.2.2

Spreading the cold with the Cold Jacket

The Cold Head is a local heat sink, and we need to spread the cold to the rest of the Bucket Brigade
setup. This requires a Cold Jacket made from a material with high thermal conductivity, like copper oraluminum. We went through a few iterations for this design, including (see Figure 10.5):
Slab + copper braids. Cold Head attached to the middle of an aluminum slab on the top of the nipple tospread the cold axially along the top of the nipple. Copper braids wrap around the nipple to spreadthe cold azimuthally.
Backbone + ribs. A rib-like structure that hugs the nipple. It comes in two halves, one for the top half,and one for the bottom. The “backbone” of the rib-like structure will be on the top part of the nipple,
and the Cold Head will be at the center of the backbone. The ribs spread the cold azimuthally, andreplace the copper braids in the previous design.
Clamshells. Two clamshells of half-cylinders. This spreads cold in the same way as the rib-like struture,256
except that we fill in the gaps of the ribs to simplify the geometry.
(a)

Octagon for state prep.

Cold Head
Octagon for state det.

NippleCopper braid

300 K

150 K

Thermal insulation(b)

Backbone

Ribs

(c)

Top clamshell

Bottom clamshell

Figure 10.5: Early plans for Cold Jacket. (a) Slab + copper braids. (b) Backbone + ribs. (c) Clamshells.

Some design constraints include:• Being able to fit the Cold Jacket onto the nipple. The copper braid design just involves wrapping
the braids around. The other two designs involve clamping the two halves onto the nipple. Since wewant good thermal contact between the Cold Jacket and the nipple, it could be a little tricky to get

257
the machining tolerances right for the latter two.
• Having high thermal conductivity. This means that we would want to use copper or aluminum.
Aluminum is lighter (and cheaper) than copper, so the former is the preferred choice.
To get efficient thermal conduction, though, we would need to have big cross sectional areas. We firststumbled upon the rib design to save on some materials (and thermal mass). However, we realized that itcomplicated the machining process and did not save us that much time and money, so we ended up goingwith the clamshell of half-cylinders. Having a big cross sectional area means high thermal mass for the Cold
Jacket. The question now is: will the high thermal mass result in a longer cool down time?
Here I present a simple model for us to understand how the cool down time constant depends on thedesign parameters of the Cold Jacket. We make a few simplifying assumptions:(1) We assume that the thermal mass of the Bucket Brigade system is dominated by the thermal massof the Cold Jacket. This is true for a Cold Jacket with large cross section area. This ensures thatthe main route of axial thermal transfer is through the Cold Jacket.
(2) We assume that there is no additional heat load. We just want to know how long it takes for us to
extract all the thermal energy from the Cold Jacket.
(3) We simplify the geometry and model the Cold Jacket as a long slab with cross-sectional area A andlength L = 3 m.
This slab is being cooled from the center (x = 0) with a cold thermal bath held at Tcold (skip ahead to
Figure 10.7 for the geometry of this toy model). We discretize the slab into chunks of length ∆x to study thetime required to cool the slab down to some temperature of interest from room temperature. A schematicis shown in Figure 10.6.
In a short time ∆t, the net energy going into the discrete chunk at x is given by:
T (x − ∆x) − T (x)T (x) − T (x + ∆x)(Pin − Pout ) ∆t = κA
− κA
∆t
∆x
∆x
= κA ∆t

T (x + ∆x) − 2T (x) + T (x − ∆x).
∆x(10.2)

258(x)
κA T (x−∆x)−T∆x(x+∆x)
κA T (x)−T∆x

Ax − ∆xx

x + ∆xFigure 10.6: Schematic of model used to study cool down time of cold plate.

The change in the thermal energy stored in the chunk at x in this time is given by:
U (t + ∆t) − U (t) = mc[T (t + ∆t) − T (t)] = ρ ∆xAc[T (t + ∆t) − T (t)].
We equate the two equations above and work in the limits {∆x → 0, ∆t → 0} to get:∂x2 T (x, t) =ρc∂t T (x, t),κ(10.3)

where T (x, t) is the time-varying temperature profile of the slab, ρ is the mass density of the slab, c is the
specific heat capacity of the slab, and κ is the conductivity of the slab. Note that the cross sectional area
drops out.
Using the separation of variables, we demand that T (x, t) = X(x)T(t) to get:
1 2
1 ρc
∂x X =∂t T = −K 2 ,X
T κ
where K is some real positive constant to be determined. The solution to the above is as follows:

X(x) = AK sin Kx + BK cos Kx,
K2 κT(t) = e− ρc ·t ,(10.4a)(10.4b)where we conveniently define T(0) = 1. To determine the constants {AK , BK }, first we note by symmetryof the setup about x = 0, X(x) has to be an even function, so AK = 0. Next, we note that there are twoboundary conditions:(1) From the cryocooler: T (x = 0, t) = Tcold .
(2) There is no heat flow into the ends (by assumption), so ∂x T (x = ±L/2, t) = 0.

259
Region of interest

x = −L/2

x = L/2

x=0

x = L/2

x=0

Actual setup

x=L

Equivalent setup
by symmetry

Figure 10.7: Equivalent mathematical model for the actual setup. Blue and grey areas denote thecryocooler and the cold-transmitting slab, respectively.

By the symmetry of the second boundary condition above, we recast the problem into an equivalent setup
shown in Figure 10.7.
Also important to note is that we can always add or subtract a constant from T (x, t) from Equation
10.3, so the quantity that matters is the temperature difference. Hence, we demand the following:(1) X(x = 0) = X(x = L) = 0 =⇒ BK = 0 and K = nπ/L.
(2) X(x) is to be an even function about x = L/2.
(3) X(x = L/2) = Tstart − Tcold .
The above gives:
An = (Tstart − Tcold ) ·4 sin2 (nπ/2)
nπ,Bn = 0 ,n ∈ Z+ ,and so our final solution is
T (x, t) = Tcold +∞X(Tstart − Tcold ) ·n=1

2 π2 κ
nπx − nρcL
4 sin2 (nπ/2)
2 ·t
· sin
·e
.
nπ
L

(10.5)One important thing to note from the above analysis is that we assumed that κ and c have no temperaturedependence. In the following results, we use values κ = min[κ(T = 150 K), κ(T = 300 K)] and c =max[c(T = 150 K), c(T = 300 K)] to get the upper bound on cooling time required. The plots below assume
Tstart = 300 K and Tcold = 120 K. We also show plots for the cases where the slab is made of aluminum andcopper.
Time constantWe can define the time constant as the following:
τ :=

ρcL2
,
π2 κ

(10.6)

260
2

which is the decay time constant for the n = 1 mode. Since the higher harmonics decay at rates ∼ en fasterthan the fundamental, this time constant gives us a good idea of how fast the system cools down. Figure
10.8 shows the temperature profile of the cold slab at various times.

Aluminium ( = 5.42 hr)

OFHC ( = 2.2 hr)

T (K)

300

300
t=0
t = /2
t=
t=2

250
200

t=0
t = /2
t=
t=2

250
200

150

150
0.00 0.25 0.50 0.75 1.00 1.25 1.50
x (m)0.00 0.25 0.50 0.75 1.00 1.25 1.50
x (m)Figure 10.8: Temperature profile along cold slab at various times. Ringing in the t = 0 plot is
attributed to taking a finite sum of Equation 10.5 to n = 50 for the numerical plot.

Not surprisingly, copper (OFHC) gives a shorter cooldown time constant than aluminum for the same
dimensions of the slab. Nevertheless, the time constant of about 5 hours for the case of aluminum is alreadya very friendly time scale to work with.
Temperature dropFigure 10.9 shows the temperature drop at various parts of the cold slab withtime. We see that the whole setup is cooled down to 150 K in about 10 hours. This means that one can turn

Aluminium ( = 5.42 hr)

300

x=0
x = L/4
x = L/2

T (K)

250

200

150

150
2

4

6
t (hr)

810x=0x = L/4
x = L/2

250

200

0

OFHC ( = 2.2 hr)

300

0

2

4

6
t (hr)

8

10

Figure 10.9: Plots of temperature drop at various positions with time.

the cryocooler on before they leave for the day, and come back the next morning to a nice and cool setup to
begin experiments with.
Cooling power

Figure 10.10 shows the cooling power required for the above cooling process. Notethat the powers plotted are those required to cool one half of the slab. This requires us to have two cryocoolers
with cooling power at 80 W at 150 K (or actually 120 K at the Cold Head) placed on both sides of the slab.

Aluminium ( = 5.42 hr)

Pcool (W)

102

1 mm
5 mm
1 cm

101

102
101
100

100
0

2

4

6
t (hr)

8261

OFHC ( = 2.2 hr)

10

10 1

1 mm
5 mm
1 cm

0

2

4

6
t (hr)

8

10

Figure 10.10: Plot of cooling power required versus time. The cooling power here only accounts forthe thermal mass of the slab, and neglects that of the Bucket Brigade. Width of slab is assumed to be 10 cm.
Grey dashed line shows the 80 W mark. Legend shows the thickness of the slab. We need more coolingpower to cool off the extra thermal mass of the cooling slab. Note that the thermal mass of the nipple (andthe Bucket Brigade) is assumed to be small compared to the slab in this (overly) simplified model.

One is actually enough if we are willing to wait a little while more. There are a few places to get cryocoolers
with the cooling power that we need: RIX, Cryomech, and Sunpower.
In this section we simplified the geometry of the cylindrical Cold Jacket to a planar slab. We also
ignored the thermal mass of the nipple and the Bucket Brigade. One can easily show with COMSOL
simulations that making the geometry and thermal masses more faithful only increases the time constantsby order unity. Equation 10.6 still captures the characteristic time scale involved. The time scales arereasonable time scales that one can be comfortable working with. The cooling power at steady state is also
manageable by cryocoolers that we can buy off the shelf. We believe we have spreading of the cold undercontrol. In the next section, we shall look at what happens to the system during the cool down.

10.2.3

Thermal contraction

We learn in kindergarten that objects contract when cooled. Even though we are not ambitious in
our cryogenic temperatures, thermal contraction will still be a noticeable effect in our setup. This effectcomes in twice: (i) change in the length of the setup at steady state, and (ii) differential changes in lengthfor different parts of the setup during the cooldown. We shall look at the former first.
Thermal contraction at steady state.

Using temperature-dependent thermal expansion coeffi-

cients from NIST, the absolute changes in length (of the 300-cm apparatus) during the temperature change
from 300 K to 150 K and 300 K to 120 K were calculated for 304 SS and Ti (Table 10.1). For a setup made

262
∆L 304 SS (cm)
∆L Titanium (cm)

150 K

120 K

-0.648-0.348-0.756-0.393

Table 10.1: Thermal contraction of a 3 m long setup. We calculate for stainless steel because it is verystrong (and cheap), so having the Bucket Brigade made from stainless steel will be strong. Titanium is alsostrong. Furthermore, it is non-magnetic. However, it is much more expensive than steel. Our current setup
uses 304 SS and the “non-magnetic” 316LN SS, but we have plans to upgrade to titanium in the future to
suppress spatial magnetic field noise coming from the material.

fully from stainless steel, we expect the whole setup to contract by slightly more than 5 mm. This implies a
few things:(1) We will need to install expansion tubes on at least one side of the nipple so that the thermal
contraction does not result in strain on the structural support of our setup.
(2) If the Bucket Brigade is one continuous structure, it will not end up aligned both with the Octagon
for state preparation and the other Octagon for state detection after cooling down. We will either
have to (i) break up the Bucket Brigade into smaller sections so that the thermal contraction does
not manifest all on one end, or (ii) be comfortable with realigning optics and reoptimizing the ion
ejection protocol every time we warm up or cool down the Bucket Brigade setup. We may have field
homogeneity concerns if we do the former. We have to ensure that the graduate students are well
fed if we want to go with the latter.
Differential thermal contraction during cool down.

In the above, we discussed the problem

at steady states, i.e. comparing the setup when it is warm and when it is cool. There could also be problems
during the cool down. Different parts of the Bucket Brigade setup will not be at the same temperatureduring the cool down. The parts closer (thermally) to the cryocooler will be colder than the rest, and sothey will thermally contract first. The Bucket Brigade sits in the nipple, and is supported by the structural
support which makes points of contact with the nipple. When the cool down happens, the Bucket Brigade
will cool down slower than the nipple, and the nipple will contract first. This will result in strain between
the nipple and the contact points of the Bucket Brigade. This strain could deform the Bucket Brigade, albeit
so slightly, and change the positioning of our electrodes. Given that we care about field inhomogeneity down

263
to a level of 0.01%, and that position tolerances of the Rods are only on the order of 100 µm, we will needto ensure that the structural support is strong enough to withstand strains from the differential thermal
contraction.
Caterpillar action from differential thermal contraction during cool down.

Furthermore,even if the structural support is strong enough to ensure that our electrodes do not move, there could be
another problem. If the differential thermal contraction between the nipple and the Bucket Brigade is large
enough to cause the contact points to slip and stick at a new place, the Bucket Brigade may end up moving
along the nipple like a caterpillar after a few warm-up–cool-down cycles unless we secure the Bucket Brigade.
Securing one structure is still relatively easy, but if we wish to break up the Bucket Brigade to ensure thatthe thermal contraction does not manifest all on one end, then we will have to secure all those structures
to the nipple. It is not immediately obvious at the point of this writing how we want to do this inside thenipple.
We are testing the cryogenics out in the b.B.B., and we are learning as we go along. We are not seeing
any major issues with this 1/3-scale prototype yet, but we are cautious in our design of the full scale Bucket
Brigade. Let us now look at the state of cryogenics in the b.B.B..

10.3

Current status in the baby Bucket Brigade
The short version of the current status of the cryogenic system in the b.B.B. is that it works. We havealready seen in Figure 4.13 all the way back in Chapter 4 showing the effect of cryogenics on increasing ourT1 time. We shall go into a little bit more detail in the next few sections, partly to serve as documentation.8
We will first look at characterization of the properties of the household insulation foam (Section 10.3.1).
We then look at the specifications of our cryocooler (Section 10.3.2). Finally, we shall discuss the effective
temperature that the ions see at the Science region (Section 10.3.3).
8 Or, looking ahead into Section 10.4, an obituary of using household insulation foam for insulation in the Bucket Brigade
setup, documenting all the fun times we had with foam. Spoiler alert: we have plans to move away from household insulation
foam.

264
10.3.1
10.3.1.1

Household insulation foam
Foam versus aerogel

Before we dive into the fun we had with foam proper, we should first address the debate the group had
early on in the cryogenics design. Having given up on vacuum jackets and MLIs, we were still presented withtwo choices for thermal insulation: household insulation foam and aerogel. There were many uncertainties
back then, and I will list some of them here:
Foam.

• How does foam perform at 150 K? Will it crumble at low temperatures?
• Can we spray it evenly enough that there are no air pockets?
• Are the foam kits from Home Depot going to work? If not, where should we get them?
• How does foam react to condensation and frost?
• Is it safe for us to spray it ourselves?

Aerogel.

• The datasheet (for Cryogel Z) includes 150 K, but how much can we trust it, especially given
our inexperienced skills in wrapping it?
• How hard is it to wrap it? How do we wrap it?
• Aerogel feels powdery to the touch between the layers. This is bad for optics. How bad will it
be for optics in the long run?
• How does aerogel react to condensation and frost?
• How bad will it be for us when we cut and wrap it? Is it going to be like asbestos?

Just looking at the datasheets, we see that aerogel boasts a thermal conductivity of only 14 mW/(mK) at 150 K. Aerogel is also the tried-and-tested insulating material used in many cryogenic applications.
Aerogel can also withstand baking temperatures of up to 125 ◦ C, whereas foam starts to degrade at around70 ◦ C. A lot can be said for using aerogel. However, we know for a fact that aerogel is bad for optics. We
will have to wrap the nipple up with aerogel with all the surrounding optics covered. It can be a pain if weneed to remove or replace the aerogel when all the optics are in place. Once again, advised by our in-house

265
Dr. Eric Cornell, “where’s the fun if we just use standard techniques,” he asked, so we decided to give foam
a try.

10.3.1.2

Recipe for working with household insulation foam

Now that we are committed to foam, we ordered some spray foam kits from Home Depot and started
spraying. We made a container with a 1/30-scale Bucket Brigade (just a half-cylinder) and filled it withspray foam. Figure 10.11 shows the disaster from our first try.

(a) We thought the foam has cured.

(b) It had not.

Figure 10.11: First attempt at spraying foam into a container. A total disaster. We sprayed the foam
in a fume hood, and let it cure for about an hour. The top layer has cured completely, but this forms aseal preventing the lower layers from curing. The mess was absolutely disgusting. It was very hard to clean.
The uncured part was sticky and left stains. The stains either had to be scrapped off through mechanicalmeans, or dissolved with acetone (recommended by the internet) which left even stickier stains if not done
thoroughly.

We got smarter with each attempt, and eventually we came up with a recipe:• Keep the spray foam at room temperature before using it. If not, it will not cure well.
• Spray foam at a distance of about half a meter away at least.
• With a sweeping action, spray a thin layer of foam each time. Allow the newly sprayed foam to
expand and fill gaps naturally. When the foam has stopped expanding, apply the next layer if
desired.

266• Aim for the corners first when spraying, otherwise air pockets will form around the corners.
• When making the mold, line the surfaces with aluminum foil. The foam will stick to the foil, but
one can easily slide the foam that is set out of the mold by detaching the aluminum foil from themold.
• After the foam has cured, surfaces that are exposed to air or the aluminum foil will have a nicefinish. However, if one cuts the foam (to shape it into something more convenient), the cut surface
will turn powdery. This is bad for optics. Paint these surfaces with acrylic paint to stop the powder
from coming off the surfaces.
• Outgassing from the spray foam stinks. Do it in a ventilated area.
A photo montage of the whole process is shown in Figure 10.12. This whole process happened at a random
garage that was available to us.9
Before we end this section, we have some closing remarks on foam. The manual says that the thermal
conductivity of the foam will degrade after about 90 days. We have not seen significant degradation of
insulating properties. The current set of foam on the b.B.B. setup is still those that we made on 25 May
2021 (from the photo montage in Figure 10.12). We tried several brands of foam, and we find that HandiFoam
has a better after-feel: less powdery.
Now that we know how to make foam blocks, it is time to characterize the performance of the foam.

10.3.1.3

Characterizing the thermal conductivity

The thermal conductivity of household insulation foam is typically reported as an R-value.10

The

R-value is related to the thermal conductivity through:κ = ∆x/R-value,
9 Prior to this, the author has never seen a garage before. Having grown up in Singapore reading books written by American
authors who write about elves, forests, and garages, the author is happy to have finally seen at least two out of the three during
his Ph.D..
10 There is a closely related quantity called the RSI-value, and it is the SI version of the R-value, but people in the industry
call the RSI-value the R-value nonetheless. The units have to be determined from context. This is the thermal insulation
version of kilocalories, Calories, and calories. The R-value (inch-Pound system) is expressed in units of ◦ F · ft2 · h/BTU. The
RSI-value is expressed in units of K · m2 /W. The conversion factor between the two is given by: RSI-value[K · m2 /W] ≈0.176 × R-value[◦ F · ft2 · h/BTU]. You are welcome.

267

Figure 10.12: Photo montage of the process of making the foam blocks. Top row: (i) mold for one
half of the foam, lined with aluminum sheets; (ii) applied first layer of foam; (iii) end product after spraying.
Bottom row: (i) painting the surfaces of the foam blocks with acrylic paint; (ii) cutting the top surface ofthe foam block, because it is easier to clamp the two halves together with flat surfaces; (iii) end products.
Shown in this picture are also fellow graduate students Sun Yool Park and Noah Schlossberger; one can also
make out undergraduate Gustavo Santaella cleaning up in the garage; the extra pair of hands holding on to
the foam being cut belongs to JILA instrument maker James Uhrich. James helped us make the big saw to
cut the foam, also shown in that picture.

where ∆x is the thickness of the sample used to determine the R-value. One needs to be careful to ensure
consistency of units (see Footnote 10 below).
Thermal conductivity of foam could be very different at 150 K than at room temperature or normal
winter temperatures. We constructed a simple setup that is a 1/20-scale version of the Bucket Brigade totest the insulation properties of the foam. Figure 10.13 shows some pictures of the setup.
The idea behind the experiment is simple. We want to measure the thermal conductivity of the foam.
To do this, we need to know (i) the temperature on the inner and outer surfaces of the foam, and (ii) theheat flux flowing through it. We can measure the temperatures with thermometers. We can also deduce theheat flux through the foam with conservation of energy flow:(1) Referring to the top right diagram in Figure 10.13, we see that heat is flowing from the warm outer268

Figure 10.13: Schematic diagrams and pictures of the setup used to test thermal conductivityof foam. Upper left: setup viewed from an angle. Upper right: parts of the setup. Lower left: the setup
with half of the foam block removed. Lower right: setup with viewing angle of upper left. The teal colored
goop is the sealant used to seal the gap between the two halves of the foam blocks. The tube is hollowed
out in the center to allow access for screws that secure the extension, thermal resistor, and the tube to theCold Head. Several holes for resistance temperature devices (RTDs) can also be seen in the left figure. The
extension and tube are made from aluminum, while the thermal resistor is made from Teflon.

surfaces of the foam to the cold inner surfaces of the foam (in contact with the tube).
(2) All the heat flux going through the foam between the thermal resistor and the far right end of thefoam has to go into the tube.
(3) This heat flux has to go through the thermal resistor to get to the heat sink on the Cold Head. This
results in a temperature difference across the thermal resistor.
(4) By using a thermal resistor with a well characterized thermal conductivity, and measuring thetemperature difference across the thermal resistor, we can deduce the heat flux through the foam.

269
As a result, there are several key aspects to this setup: (i) foam blocks, (ii) thermal resistor, (iii)
resistance temperature devices (RTDs), (iv) extension,(v) tube, (vi) power resistors, and (vii) sealant. A
table of dimensions and important parameters of the above are summarized in Table 10.2. Before we divePart

Parameters

Foam

Product: Handi-Foam model #P10625
Cross sectional dimensions: 21 cm by 21 cm
Bottom thickness: 2′′ thickness on far end from Cold HeadTop thickness: 3 cm thickness on for cold neck side
Inner diameter (tube part): 76 mm
Inner diameter (cold neck part): 30 mm

Thermal resistor

Material: Teflon (McMaster item #9266K82)
Thickness: 1.62 mm (measured)
Diameter: 76 mm
Thermal conductivity at 150 K: 26 mW/(m K)

RTD

Model: PT100, DigiKey #1528-2090-ND

Thermal grease

Model: Apiezon N Cryogenic Heat-Transfer Greases

ExtensionMaterial: aluminum
Length: 30 mm
Diameter: 76 mm

Tube

Material: aluminum
Length: 15 cm
Outer diameter: 76 mm
Inner diameter: 51 mm
Base thickness: 1 cm
Table 10.2: Some important parameters of setup used.

into the results, we shall cover a little bit more detail of the setup. We have already seen our recipe formaking foam blocks in Section 10.3.1.2. We shall go through the remainder on the list in the following
paragraphs.
Thermal resistor

Using a thermal resistor with known thermal conductivity at 150 K, we candeduce the amount of heat flowing through it by measuring the temperature difference across the thermal
resistor. We then attribute the heat flowing through the thermal resistor at steady state with the conduction
heat flux through the foam into the tube.
We use Teflon for the thermal resistor because it has a well known thermal conductivity11 at 150 K
of 260 mW/(m K), which allows for a measurable temperature drop (∼10 K) across a piece of reasonable
11 https://trc.nist.gov/cryogenics/materials/Teflon/Teflon_rev.htm

270
thickness (∼2 mm) for an expected conduction heat flux through the foam at steady state (∼10 W) simulatedwith COMSOL.
Resistance temperature devices (RTDs)

The RTDs used in this experiment are PT100 (DigiKey#1528-2090-ND). These RTDs have a temperature dependent resistance of the form:R(T ) = R0 1 + A · T + B · T 2 + C · T 3 (T − 100) ,where T is temperature in Celsius, A = 3.9083 × 10−3 , B = −5.775 × 10−7 and C = −4.138 × 10−12 if T < 0and C = 0 otherwise, and R0 = 100 Ω for the PT100.
We send in about 2 mA of current into each RTD12 and read out the potential difference across eachof them to deduce the temperature of the environment that the RTDs are exposed to. The RTDs are slotted
into machined holes on the extention and the tube of very similar dimensions to the RTDs, and thermal
grease (Apiezon N Cryogenic Heat-Transfer Grease) is applied to the RTDs to ensure good thermal contact
with the surroundings.
Extension

The extension is a cylindrical block of aluminum in direct contact with the Cold Head.

Its purposes are (i) to spread the temperature of the Cold Head to a bigger cross-sectional surface area, and(ii) to act as a holder for an RTD to measure the temperature on one side of the thermal resistor.
Tube

The tube is a cylindrical block of aluminum that conducts the conduction heat flux from thefoam to the Cold Head. It has two holes for RTDs on both of its ends. The RTD on the side of the thermal
resistor is used to measure the temperature difference across the thermal resistor, while the RTD on the far
side is there to ensure that things are making sense. We bored the tube out so that we can screw the wholeassembly to the Cold Head from the far end.
Power resistors

It turns out that the cryocooler that we have has a minimum cooling power13 .

The working range of the cryocooler is shown in Figure 10.14.
At steady state, our COMSOL simulation suggests a heat load of about 10 W at 150 K. Hence, weneed to introduce active heating elements into the setup to bring the total heat load into the grey region(about 35 W in total at 150 K).
12 Each RTD is in series with a 2 kΩ resistor, and a total potential of 5 V is applied across the setup.
13 The compressors use air bearing sliders to reduce friction between moving parts, so there is a minimum power that has tobe sent into the compressors to maintain the air flow, which results in a minimum cooling power.

DS30

100
Cooling power (W)

271

80
60
40
20
125

150

175 200 225
Temperature (K)

250

275

Figure 10.14: Working range for the Ametek CryoTel DS30 cryocooler. The blue points are thequoted minimum cooling power, while the orange points are the maximum cooling power that the cryocooler
can deliver. The grey region is a visual aid to the working region of the cryocooler.

The power resistors we use are Ohmite HS10 100R, which are 100 Ω resistors rated for 10 W each.
Three such power resistors are mounted on the Cold Head side of the extension as shown in Figure 10.15.
The power resistors are connected in series, and hooked up to a Variac. The nominal voltage applied on themat steady state is Vrms = 72.5 V, corresponding to about 17.5 W of additional heat load into the system.
Sealant

The sealant’s only function is to seal the small gaps between the two halves of the foam

blocks to prevent convection. We tried two types of sealant: (i) Molykote Valve Lubricant and Sealant, and(ii) Gasoila Soft-Set Thread Sealant with PTFE.
We noticed that the Gasoila sealant freezes at low temperatures, and it also dries up when exposed
to air in Boulder and turns into a strong glue that holds the two halves together. It is also quite messy todeal with, so our preferred choice is the Molykote sealant for areas close to delicate parts.
Discussion and Results

The results of an instance of an experimental run is shown in Figure

10.16. We performed the experiment with two different runs using two different sets of foam blocks, and also
an additional run to estimate additional heat load from the RTD cables14 . The final result is an effective
conductivity of the foam at about 60 mW/(m K), which is not far away from that of closed-cell foam atroom temperature as quoted in literature15 .
14 This run involves having additional RTDs attached to the tube, and having their cables come out between the two halvesof the foam blocks like the RTDs that are used to measure the temperatures. Each cable brings in an additional 1 W.
15 COMSOL has data on closed-cell foam between 293 K and 373 K. The conductivity quoted at 293 K is 49 mW/(m K).

272

Figure 10.15: Power resistors mounted onto the extension.

Note that the value of the measured conductivity is an effective thermal conductivity, because thecalculations assume that the foam blocks have a isotropic thermal conductivity throughout. However, there
is a huge temperature gradient along the thickness of the foam, and we expect the thermal conductivity
to change along the thickness of the foam because of the temperature gradient. So this effective thermal
conductivity is a weighted average of a case with a specific boundary condition. We put the words above
into an equation: approximating the foam blocks as cylinders instead of cuboids, we have for the conduction
power through the foam:

P = κ(T )A(r)

dT
dr

Z R2
=⇒

P
R1

1
dr =
2πLr

Z T2
κ(T ) dT
T1

P
R2
log
= κ̄T1 ,T2 (T2 − T1 )2πL
R1
P = 2πL ·κ̄T1 ,T2 (T2 − T1 ),log(R2 /R1 )where L is the length of the setup, {R1 , T1 } are the inner radius of the foam and the temperature there,{R2 , T2 } are the outer radius and the temperature there, and κ̄T1 ,T2 is the effective thermal conductivity.
We see that as long as T1 and T2 do not change, the effective thermal conductivity will remain unchanged
even if there are changes to the radii.

273

Figure 10.16: Temperature readings of RTDs in one experimental run. Points of interest are notedin the figure. The lower plot of power vs time is obtained from the difference between the values of the blue
and orange lines in the upper plot, i.e., the temperature difference across the thermal resistor.

10.3.1.4

In the baby Bucket Brigade

We now understand the thermal properties of the foam. It is time to scale up the experiment and test
this on the b.B.B.. Figure 10.17 shows the results of using foam as thermal insulation in the b.B.B..
The test shows that the design is a great success. The cold can be maintained by the foam, the cool
down time is acceptable, and the cryocooler has enough cooling power to sustain the cold. Subsequently
with the full setup, including the b.B.B. trap (and ThF+ ions), we demonstrated no problem maintaining
170 K at where the ions sit with a Cold Head temperature of 150 K. The b.B.B. trap also stomped on the“smoothing out” of the hot and cold sides of the Cold Tube seen in part (d) of Figure 10.17 by restricting
the reach of blackbody radiation from the hotter surfaces. This works using the same principle behind how
MLIs work.

10.3.2

Cryocooler

We now know how to make foam blocks. We also know its thermal insulation properties. It is time tomove on to the next big thing in our cryogenic system: the cryocooler. There are two main things to discuss
here: (i) cooling power of the cryocooler, and (ii) magnetic fields from the cryocooler.

274(a) Simulation of the steady state temperature profile. Numbers shown are temperature contour lines inkelvins.

(c) Test setup.

(b) Time evolution of the simulated temperature profile
along the inner surface of the Cold Tube.

(d) Measured temperature profile with time.

Figure 10.17: Testing the cryogenics out in the b.B.B. setup. In all the simulations and experiment, we
did not include the effect of the b.B.B. trap, i.e. the Cold Tube is empty inside (except for the thermometers).
(a) The Cold Head is set to 120 K, while the exterior surfaces of the foam and the exposed parts of the Cold
Tube are set to 300 K. The Cold Jacket starts at the 40 cm mark.
(b) The Cold Head is set to 150 K for this simulation. The Cold Jacket (40 cm mark) falls below 200 K in
about 14 hours. Bounds are not set on the cooling power in the simulation. Thermal radiation within
the Cold Tube is not included in the simulation.
(c) One half of the foam block is removed to show the insides. The white goop is the Molykote sealantused to break convective flow along the axial direction of the Cold Tube between the Cold Tube andthe foam. One can also see the Cold Jacket for the b.B.B. here. The Cold Jacket was machined by the
JILA instrument makers.
(d) We placed RTDs along the bottom of the Cold Tube. The Cold Tube was pumped down to about
1 mTorr with a scroll pump. The time constant is longer than those in (b), but it is still within an
acceptable range. The temperature profiles have smoother kinks than in (b) probably because (i) thetemperature on the surfaces are “smoothed out” by blackbody radiation within the inner surface, and(ii) there is imperfect thermal contact between the Cold Jacket and the Cold Tube.

275
10.3.2.1

Cooling power

We had a sneak peak at the cooling power of our cryocooler in Figure 10.14. It has a minimum coolingpower to run smoothly. The consequence is that if the heat load is smaller than the minimum cooling power,
the cryocooler will cool the sample to temperatures below the set point. This is only a problem if the setup
has a small heat load. We expect a heat load of about 80 W on the Bucket Brigade, so this is not a problem.
From Figure 10.14, we also see that the total cooling power at 150 K is also slightly lower than 80 W.
If we find ourselves liking the cryocooler, we could buy another one for a total of two in the Bucket Brigade
setup.

10.3.2.2

Magnetic fields

The cryocooler we have is a Stirling refrigerator. It has two counter-moving magnetic pistons thatare moved by the 60 Hz AC current from the wall. The counter-movement reduces vibration in the setup by
making making use of the symmetry to keep the center of mass stationary.
The magnetic fields from the pistons decay with distance away from the cryocooler. Associated withthe decaying field is a magnetic field with gradients. We saw in Section 8.2 that having an axial magnetic
field gradient gives rise to a non-zero f0 . We also saw in Section 6.3 that any spatial inhomogeneity of theaxial magnetic field gradient within the spatial extent of the ion cloud gives rise to decoherence. We canmitigate these effects by parking our ions far away from the cryocooler. However, the cooling power of thecryocooler decreases with distance because of the additional heat load that comes with the distance. We
want to park our ions far enough that the fields are homogeneous enough, but close enough that the ions
are still bathed in a 150 K thermal environment. The question now is where should we park our ions?
We measure the magnetic fields of the cryocooler with a setup shown in Figure 10.18. The cryocoolersits on a movable platform some distance away from an array of magnetometers. The Cold Head points
towards the magnetometer array. The magnetometer array consists of seven magnetometers, one on each endof the (3D) cross, and one at the center. The magnetometers are Adafruit LSM303 three-axes magnetometers,connected to an Arduino Uno with I2C communication. The arms of the cross are 9 cm in length, and theaxes are defined to be {x, y, z} in the directions { up, towards the measuring tape, the direction that the276

Figure 10.18: Setup used to measure the magnetic fields from the cryocooler. An array of magnetometers arranged in the shape of a cross is shown in the bottom right of the picture.

Cold Head is facing } in a right-handed convention, respectively.
We measure the magnetic field strengths at varying distances between the center of the magnetometercross and the Cold Head. We first do this with the cryocooler turned off, i.e. no moving parts in thecryocooler.
Axial direction

We assume that the local magnetic field can be expressed as an expansion up to

first order derivatives:

B(x, y, z) = B0 + Baxgrad

 x 
y
−2z

+ Btransgrad

 x 
−y
0

+ B1

y
x
0

+ B2

z
0
x

+ B3

0
z
y

.

(10.7)

Using the parameters that we got from performing a least squares fit of Equation 10.7 onto the data points
acquired, we obtain the variation of Baxgrad with axial distance from the Cold Head, as shown in Figure
10.19. The fit of the form f (d) = ae−bd + c, where a, b, and c are fit parameters, seems to work very well,

277
8

10

16

18

20

50

Data
Fit to aexp( bd) + c 3.0

40

2.5

30

2.0

|Baxgrad| (mG/cm)

f (Hz)6d (in.)
12 14

1.5

20

1.0

10
0
10

0.5
20

30

d (cm)

40

50

0.0

Figure 10.19: Plot of Baxgrad vs d, the distance between the Cold Head and the center of themagnetometer cross. The fit of the form ae−bd + c, where a, b, and c are fit parameters, seems to work
very well. A Baxgrad of 3.07 mG/cm corresponds to an f0 of 50 Hz for ThF+ at an frot = 150 kHz andErot = 60 V/cm.

with d in units of cm, and a = 206(2), b = 0.1116(8), c = −0.33(8). The fit is done after converting the
Baxgrad values to the Ramsey frequency as seen by ThF+ at an frot = 150 kHz and Erot = 60 V/cm. Taylor
expanding f (d) = a exp(−b d) + c about d = d0 , we get:1f (d) = f (d0 ) − b f (d0 ) · (d − d0 ) + b2 f (d0 ) · (d − d0 )2 + O[(d − d0 )3 ].
2
The constant term is the base frequency as seen by all ions. The linear term is averaged out by ions intheir secular motion in the trap. Evidently, the third term is the first non-trivial term that is responsiblefor decoherence within the ion cloud. We see that it scales with the distance between the Cold Head andthe center of the magnetometer cross (d0 ) as a decreasing exponential, and it scales with the size of the ion
cloud (d − d0 ) with the second power. For an ion cloud of 1 cm radius at 40 cm away from the Cold Head,this term corresponds to ∆f = 0.015 Hz, which is 0.03% of f = 50 Hz, not that far away from our desired
0.01%.
Radial direction

In the radial direction, the variations are much smaller than the radial direction

at d0 = 40 cm. We did not measure anything significant down to the precision of our magnetometers within
the spatial extent of ion cloud.
Cryocooler turned on

Turning on the cryocooler and repeating the experiment gives similar

results down to the precision of our magnetometers.

278
Summary

To suppress decoherence effects coming from gradients of axial magnetic field gradientsfrom the cryocooler, we just need to park our ions at least 40 cm away from the Cold Head.

10.3.3

Effective temperature and lifetime

Cryogenics is only applied to part of the b.B.B. setup. This means that the ions will see a non-zerosolid angle of 300 K blackbody radiation photons coming at them. However, as we shall see later on, thissolid angle decreases very quickly as the ions translate into the cold region. The effective rate is then therate given by Equation 4.1 weighted by the solid angle of the temperature involved:

Γb.B.B. (f, µ12 ) =|µv,v+1 |2
6ϵ0 ℏ2Zρ f, T (Ω) dΩ,

where µv,v+1 is given by Equation 4.3.
We estimate the effective lifetime for the full 3 m long Bucket Brigade, with both ends open to 300 K,
and Cold Tube cooled to 150 K.16 The results shown in Figure 10.20 are very reassuring. We do not haveto be worried about the ends exposed to 300 K.

10.4

Further discussion

10.4.1

VacuumOne can see from our design (see, for instance, Figures 10.5 and 10.17) that we do not have accessports in the cryogenic part of the Cold Tube to install vacuum pumps. Our cryogenic temperature is coldenough to suppress some outgassing, but it is not cold enough to act as a cryopump, except for water vaporand some organic molecules. Even without the cryopump action on water and organic molecules, the partialpressures are already very low in our vacuum chamber after a bake at around 150 ◦ C for a few days.
There are currently no vacuum pumps inside the Cold Tube in the b.B.B. setup. The pressure readingfrom our ion gauge at the Six-way Cross is at 1 × 10−9 Torr. The pressure is maintained by a 500 L/sturbomolecular pump and a non-evaporable getter pump at the Six-way Cross (see Figure 7.9). We are
currently limited by T1 losses from ion-ion collisions. We are not seeing signs that we are being limited
16 The b.B.B. has one end open to 300 K, so the estimate for the blackbody radiation excitation lifetime for the Bucket
Brigade will be a lower bound for the b.B.B..

279
102

Log scale

0.4
0.3

10 1

0.2

10 2
10 3

0.1
0.0

150

100

100

0

100

50

0
z (cm)

50

BBR excitation lifetime (s)Fraction of full solid angle

0.5

100

(a) Solid angle of 300 K seen by the ion packet.

150

101
150

100

50

0
z (cm)

50

100

150

(b) Effective blackbody radiation excitation lifetime.

Figure 10.20: Effective blackbody radiation excitation lifetime in the Bucket Brigade. We simplified the geometry such the walls of the Bucket Brigade are at 150 K within −150 cm ≤ z ≥ 150 cm and
300 K beyond that. We assume that the radial direction is bound by the inner radii of the Rings at 7.5 cm.
(a) The inset shows the same plot, but in log scale. The yellow dashed lines indicate one bucket in (8 cm).
One can see that the solid angle of 300 K drops to below 10% just one bucket into the cold region.
(b) The lifetime is evaluated for the transition (v = 0) → (v = 1). The vertical dashed lines indicate one
bucket in (8 cm). The horizontal dashed line indicates 20 s. One can see that once we are one bucket
into the Bucket Brigade, blackbody radiation excitation limits our T1 lifetime to only about 20 s.

by ion-neutral collisions, yet. We believe the vacuum in the b.B.B. is good enough for now. If we need to
improve the vacuum, we could put some getters inside the Cold Tube at strategically chosen locations to act
as passive pumps.

10.4.2Liquid nitrogen, LN2

Liquid nitrogen (LN2) costs about $160 per dewar (240 L) at JILA (circa 2020). LN2 has a latent heatof vaporization of about 1.6 × 106 J/L. This means LN2 costs about $0.40 per watt-day. One would naı̈vely
think that with 80 W of heat load (Section 10.2.1), this would total to only $32 per day. However, if we bathe
the Bucket Brigade setup in LN2, the temperature would be lower (77 K instead of 150 K), and there would
be additional heat load. So the correct cost would be around $50 per day, still cheap compared to graduate
students17 (accounting only for additional heat load from the increased temperature gradient between thecold region at 77 K and warm at 300 K). Furthermore, dewars can be magnetic, so we want to put them far
away from the Bucket Brigade, but in doing so the LN2 needs to travel a longer path before getting to the
17 And we know graduate students are only slight more expensive than fresh air in Boulder.

280
cold region. Having a longer path means more heat load. Depending on how we do the thermal insulationand LN2 transport, using LN2 on our setup could easily be just as expensive as a graduate student.
Instead of bathing the Bucket Brigade setup in LN2, we could instead use a temperature regulated
thermostat with LN2 as the cryogen. This has already been demonstrated in the Ye group downstairs, so wejust need to copy the technology over and scale it up. This is highly non-trivial though, but if the magnetic
fields from our cryocooler turns out to be a huge source of decoherence, we have to seek alternative sourcesof cold.

10.4.3Raining cryogen

An alternative cooling method that is being pursued in the group right now is to bridge the cryocooler
and the Bucket Brigade with a cryogen. Much of the cooling power is lost in the long thermal conduction
path between the b.B.B. setup and the Cold Head in the current setup. We could place the cryocooler
far above the b.B.B. setup, where its magnetic fields are small at the region where ions are. We then use
the cryocooler to cool a gaseous cryogen, whereupon it condenses (at the temperature we want) and rains
upon the b.B.B. setup. The cryogen then absorbs heat from the b.B.B. setup and evaporates back up to
the cryocooler. The condensation point of the cryogen can be controlled by choosing a suitable cryogen
and working pressure. This work was started by undergraduate Gustavo Santaella, and is currently being
pursued by graduate student Trevor Wright. The source of the cold need not be the cryocooler, it can be apool of LN2. I am excited as to how this will turn out.

10.4.4

Cold shield

One disadvantage of using foam for thermal insulation is that it is bulky. We learnt from the systematics studies in the Gen. II experiment that magnetic field gradients can be a huge source of systematic errors.
Furthermore, our ions are going to be translating inside the Bucket Brigade, so we do not have the option
to park them at a spot where the fields are the most homogeneous; we need the fields to be homogeneous
and well controlled throughout the Bucket Brigade. This requires some form of magnetic shielding.
Magnetic shielding with µ-metal (or other metals with high permeability) works by concentrating the281
field lines into itself. A nicely sealed enclosure made from µ-metal can give a very nice field-free region insideit. However, our Bucket Brigade extends in the axial direction to Octagons and Six-way Crosses, so there isno easy way for us to enclose only the Cold Tube inside the magnetic shielding. Fortunately, if the shield is along cylinder, the fields inside the cylinder (with distance inside z greater than the diameter of the cylinderd, i.e. z ≳ d) can still be suppressed.
The above would work if we can put magnetic shielding right outside the Cold Tube. Leaving thecryogenic behaviors of the shield aside, we see that the bulky foam is in the way of the shield. The shield
will have to be big to enclose the foam, but it would also lose its effectiveness in shielding external magnetic
fields. The solution is to (i) make the whole cryogenic setup smaller by moving away from foam, or (ii) comeup with a design where the magnetic shield is sandwiched between the Cold Tube and foam blocks, or even
within an inner and outer layer of foam.
Whatever we end up doing in the full Bucket Brigade setup will likely look quite different from what
we have now. Much of what I have described in this chapter will become outdated. I seek solace in that thischapter has been fun, with many pretty looking pictures of the setup.

Chapter 11

Conclusion

This thesis has been a rather extensive investigation on the prospects of performing a multiplexed ion
trap eEDM experiment with ThF+ . The group is currently working on suppressing ion-ion collisions hoping
to demonstrate 20 s coherence time. These collisions can be reduced by working with a weaker confining
trap, but we first need to reduce the spatial gradients of the ponderomotive potentials from the rotating
electric fields; the trap was ramped up to retain ions with the deconfining ponderomotive potentials.
The group is also branching out into many exciting new directions, and I shall list some here:
REMPD 1+(532 nm) (Section 3.4.2). I foresee in the near term performing the survey spectroscopy
will be a now-or-never experiment. We would like to settle down on the beam directions andpolarizations so that we can design the rest of the setup around the constraints imposed upon
us by the two-state detection protocol.
New ion detector (Figure 7.13). We believe that the current setup introduces substantial lensing effects
because the microchannel plates are recessed from the flange into the assembly. We could upgrade
to a better designed ion detector system after we settle on the REMPD protocol. If we found a
protocol such that KEdissoc. is much larger than what we have now, we may need to get a larger ion
detector to capture all the ions; or we need to introduce ion optics to focus them down (still in thevelocity mapping regime).
Using Ω = 0+ for depletion (Section 4.3.1.1). This would allow for a cleaner depletion from X 3 ∆1 (v =0, J = 1) instead of using Goat.

283
Multiple buckets. What is the point of having a prototype of a multiplexed ion trap if we are not using
it multiplexed? Multiplexing the experiment in the b.B.B. will be slightly different from that in the
Bucket Brigade in the sense that operations in the b.B.B. will not be pseudo-continuous, but will
still be a single shot experiment. Nevertheless, it will still be meaningful to try preparing multiple
buckets of ions and performing a simultaneous Ramsey experiment on all of them, first-in-last-out.
Magnetic shielding (Section 10.4). The JILA eEDM group has not experimented with magnetic shielding yet. We managed to get away with this because of our rotating quantization axis behaves likea lock-in detector, making us very insensitive to uniform ambient fields. Unfortunately, the g-factor
in ThF+ is larger than HfF+ , and the rotation micromotion is also larger because we are using asmaller frot in Gen. III than II. These two combine to make us about thirty times more sensitive
to magnetic field effects than the Gen. II experiment. Aiming to break our Gen. II record, we needmagnetic shielding.
New cryogenic scheme (Section 10.4). This will help with the design for magnetic shielding by making
the cryogenic setup more compact. It will also move the magnetic cryocooler further away from theions (or remove it totally).
Catching ions with Erot . We ramp Erot up after we caught our ions in the trap in the b.B.B.. However,the Bucket Brigade requires pseudo-continuous operation. It would be great if we can catch ourions with Erot always turned on. We would match the phase of Erot with when the molecular beam
enters the trap. Since the molecular beam goes into the trap at a speed lower than the rotation
micromotion, we will have to accelerate the ions instead of stopping them.
More DAC channels (Section 7.5.2). We now have two sets of DAC channels that allow us to switch
the voltages on the Rings between a short pulse (e.g. stop pulse to stop the ions) and a steady state(e.g. usual trapping conditions). Much can be said for having a third channel so that we can, say,switch from stop pulse (to stop the ions) to kick pulse (to eject the ions from the trap) to ground (sothat the ions see no fields for the rest of the flight to the ion detector). This will involve upgrades
to the current Ring driver.

284
Testing out various f0 -generating Plans (Chapter 8). As we learn about the decohering mechanisms
and possible new systematic effects in the b.B.B. we may want to explore various plans of introducing
f0 .
New π/2 pulse. Depending on the Plan we decide to use in the end, there could be more novel π/2 pulsesthat we can use. The current π/2 pulse involves ramping Erot , but this would be bad for ions inneighboring buckets in the Bucket Brigade. We could, instead, sweep through f0 = 0 in some kindof an adiabatic rapid passage in some way that would only affect this local bucket and not theneighboring ones.
Travelator. The state preparation and detection stages of the experiment are quite destructive to thequantum states: they move population around the Hilbert space. When we perform these operations
to the relevant buckets, we do not want ions in the neighboring buckets to be affected. One way
to do this is to keep the buckets undergoing these operations well separated in space from the next
bucket. Once state preparation is done, we move the bucket quickly into the cold region and matchthe translation speed with the rest of the buckets. On the far end, we also pull it out quickly to do
the state detection sequence. We call this problem the travelator: it is like matching one’s walking
speed when hopping on or off a travelator at the Denver International Airport. One could try thisout in the b.B.B..
Fighting against aging flashlamps, thinking about whether vacuum is broken during summer whenpower surges at JILA are common, and with so many things to try, the JILA eEDM group will stay busy
for quite some time. With that, I present you an incomplete guide on using ThF+ for an eEDM experiment,
the first of its series, with many more to follow.

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[87] John Turner, Phil Anderson, Tom Lachlan-Cope, Steve Colwell, Tony Phillips, Amélie Kirchgaessner,
Gareth J. Marshall, John C. King, Tom Bracegirdle, David G. Vaughan, et al. Record low surface airtemperature at vostok station, antarctica. Journal of Geophysical Research: Atmospheres, 114(D24),
2009.
[88] Dmitry Budker, Derek Kimball, Derek F. Kimball, and David P. DeMille. Atomic physics: an exploration
through problems and solutions. Oxford University Press, USA, 2004.
[89] A. N. Petrov, L. V. Skripnikov, and A. V. Titov. Zeeman interaction in the 3 δ1 state of HfF+ to search
for the electron electric dipole moment. Physical Review A, 96(2):022508, 2017.
[90] J. Robert Johansson, P. D. Nation, and Franco Nori. Qutip: An open-source python framework for the
dynamics of open quantum systems. Computer Physics Communications, 183(8):1760–1772, 2012.

Appendix A

Paul Trap and Mathieu Equations

A.1

Introduction
Paul traps come in various geometries, and I will just list two here:1

3D Paul trap. The original Paul trap. The electrodes consist of a ring shaped like a hyperboloid, and two
hyperbolic cylindrically symmetric caps placed on either side of the ring. An electric field oscillating
at an RF frequency is applied between the ring and caps for radial confinement. A DC potential
is applied on the caps w.r.t. the ring for axial confinement. See, for example, Figure 6 of Ref. [81].
The electrodes are shaped like hyperboloids because the equipotential surfaces for a quadrupolar
potential are hyperboloids.
2D/linear Paul trap. Originally used as a quadrupole mass filter; only charged particles with the rightcharge-mass ratio can stay within the “trap” (see Figure 4 of Ref. [81]). The electrodes consist of
two sets of radial electrodes, and end caps on either end of the radial electrodes. An electric fieldoscillating at an RF frequency is applied between the two sets of radial electrodes for radial confinement. A DC potential is applied on the end caps w.r.t. the radial electrodes for axial confinement.
See, for example, Figure 1 of Ref. [77]. The electrodes are sometimes shaped like hyperboloids, but
usually they are of the shape of more convenient geometries, e.g. rods for radial electrodes, plates
for end caps. The potentials generated by these electrodes will not be of a pure quadrupolar form,
but the quadrupolar potential is the most dominant one at the trap center.
1 Check out R. Bradford Blakestad’s thesis from the Wineland group for a compilation of common geometries not mentioned

here (e.g. surface trap, wafer traps, wire trap).

292
The equations of motion have the form of a Mathieu’s differential equations (see Section A.4), so the motionof the ions in a Paul trap are described by the Mathieu functions. Hence, the working of Paul traps areusually parameterized in terms of the Mathieu parameters (a, q) to be defined below. One thing to note isthat the relation between the Mathieu parameters to the trap configuration depends on whether it is a 3D
or 2D Paul trap. Here, I summarize the expressions for a 2D Paul trap, our Paul trap.

A.2

Definition of variables
There are quite a number of variables used throughout this appendix, so here I present a list of all

the parameters.
Spatial coordinates, x, y, z. The spatial coordinates in the Cartesian system. The trap center is defined
to be at x = y = z = 0. The z axis is defined to be along the axial direction.
RF voltage amplitude, Vrf,0 . The amplitude (peak) of the RF voltage on the radial electrodes.
Effective radial distance, R0 . The effective radial distance of the radial electrodes from the trap axis.
This would be the radial distance of the hyperboloid surface of the radial electrode from the trap
axis if it were shaped like a hyperboloid.
RF angular frequency, ωrf . The angular frequency of the RF oscillation.
DC axial voltage, Uec,0 . The potential difference between the z = 0 plane and the end cap. Usually this isjust the voltage applied on the end caps, because the radial electrodes are at DC ground. However, in
our Bucket Brigade, we apply a sinusoidal spatial modulation on the voltages on the Rings, so Uec,0would correspond to the peak-to-peak voltage applied on the Rings. See Figure A.1 for a comparison
between the definitions for linear Paul traps and the Bucket Brigade.
Effective axial distance, Z0 . The effective axial distance of the axial electrodes from the z = 0 plane.
This would be the axial distance of the hyperboloid surface of the caps from the z = 0 plane if itwere shaped like a hyperboloid. In the Bucket Brigade, this parameter is just there to parameterize..
.

U = Uec,0293
U = Uec,0 /2
U =0

U =0

U = −Uec,0 /2

z

U =0

U = Uec,0Linear Paul trap..
.
Bucket Brigade

U = Uec,0 /2

Figure A.1: Comparing definition of Uec,0 between conventional linear Paul traps and the Bucket
Brigade. The red blob represents the ion cloud. Left: a linear Paul trap. The red boxes indicate end caps,
while the blue fins indicate radial electrodes. The radial electrodes are held at DC ground. Right: the Bucket
Brigade with Rings and Rods. Rods are colored blue. The Rings are represented by colored horizontal lines.

the curvature of the axial trapping potentials as a “length”; it does not correspond to the distancefrom the ion cloud center to any Ring.
DC radial quadrupole voltage, VDC,0 . Half of the DC potential difference between the two sets of radial
electrodes, defined to be positive for the set of radial electrodes along the x direction. This DC
voltage will introduce a DC quadrupole potential in the radial direction, making the trap more
confining in x and deconfining in y for a positive VDC,0 . We turn this on sometimes to filter away
the heavier species from our trap, e.g. ThF+
2 . Some non-zero effective VDC,0 may be present due topatch charges on Rods.
Mathieu a parameter. Contains all the information of the DC voltages. Typically a ≈ −0.0027.
Mathieu q parameter. Contains all the information of the RF voltages. Typically q ≈ 0.14.
Angular secular trap frequency, Ω. To flash out the secular frequencies from the rest of the frequencies
in this appendix, we use Ω instead of ω to represent angular secular trap frequencies for the rest of
this appendix.
Electric charge, e. The charge of our singly charged ion.
Ion mass, m. The mass of our ion.
The rest of the sections will just list the equations involved with minimal elaboration.

294

A.3

Potentials
RF potential
 2

x − y2cos ωrf t.
R02

(A.1)



x2 + y 2z2 −, where Uec,0 is the peak-to-peak voltage on Rings.
2

(A.2)

Vrf = Vrf,0DC end caps

Uec =

Uec,0
Z02

DC electrodes
VDC =VDC,0 2(x − y 2 ).
R02

Total electric potential


 


 
VDC,0
Vrf,0
Uec,0
VDC,0
Vrf,0
Uec,0 2Uec,02−
+ 2 cos ωrf t x + −+− 2 cos ωrf t y 2 +z .
Vtot = −
2Z02
R02
R0
2Z02
R02
R0
Z02

A.4

Equations of motion
Total force

 



h 
i

Vrf,0
Uec,0
VDC,0
−2e
+ R2 cos ωrf t m xẍ  − 2Z02 − R02
0
  

i

  h ÿ  =  − Uec,0 + VDC,0 − Vrf,0 cos ω t −2e y  .
  

2
2
2
rf
m
2Z
R
R
0
0
0
  

  

Uec,0 −2e
z̈
z
Z2 m

(A.3)

0

Form of Mathieu equationsλ̈(τ ) = (−a + 2q cos 2τ )λ(τ ).

Our Mathieu equations

Performing the transformations 2τ := ωrf t and ∂t = ω2rf ∂τ on Equation(A.3) gives:
 





i
h 
Vrf,0
VDC,0
Uec,0
−8e
+ R2 cos 2τ mω2 x
ẍ  − 2Z02 − R02
0
rf
  


i
  h ÿ  =  − Uec,0 + VDC,0 − Vrf,0 cos 2τ −8e y  ,
  
2
2
2
2
2Z0
R0
R0
mωrf 
  

  

Uec,0 −8e
z̈
z
2
2
Z
mω
0

rf

295
where we can define the following:
8e
2
mωrf



ax = −

8e
2
mωrf



ay = −
az =

Uec,0
VDC,0
−
2Z02
R02



Uec,0
VDC,0
+
2
2Z0
R02



qx = −qy =8e Uec,0
2 Z2
mωrf
0

4e Vrf,0
2 R2
mωrf
0

4e Vrf,0
2 R2
mωrf
0

qz = 0,and relate to the more familiar forms:
4e Uec,02 Z2 < 0mωrf0ax = −|a0 | + a′qx = −q0a′ =8e VDC,0
>0
2
mωrf
R02

ay = −|a0 | − a′qy = q0

q0 =4e Vrf,0
2 R2 > 0mωrf0az = 2|a0 |qz = 0a0 = −

Secular frequencies

Ωz =

ωrf √
az
2

,

Ωx,y =

ωrf
2

qax,y + q02 /2.

(A.4)

Appendix B

Typical Parameters for the Bucket Brigade

In this part of the appendix, we present (i) the typical trap parameters of the baby Bucket Brigade forquick reference to ion dynamic quantities, and (ii) typical experiment and molecular parameters that could
go into the study of possible systematic effects in Gen. III.

B.1

Trap parameters
Parameter

Value

Comments

Ring inner radius
Ring outer radiusInter-ring spacing
Rod outer diameter
Rod radial distance

7.5 cm ×90%
8.5 cm ×90%
2 cm
0.5′′
6.5 cm ×90%

Scaled down to make space for structural support.
Scaled down to make space for structural support.
4 Rings per bucket.
Stock dimension.
Center-to-center. Scaled down to make space for structural support.

Erot per Vrot
Vrot
R0
Vrf,0
Z0
Uec,0
ωrf

14.0 (V/m)/V
430 V
5.36 cm
25.6 V33.8 cm
19.9 Vpp
2π × 50 kHz

From COMSOL simulations.
For Erot = 60 V/cm.
From COMSOL. Effective radial distance of radial electrodes.
RF voltage for radial secular frequency of 2 kHz, Vrf,0 ∼ fsec,radial .
From COMSOL. Effective axial distance of “end caps”.
2“End cap” voltage for axial secular frequency of 2 kHz, Uec,0 ∝ fsec,axial.
RF frequency.

Table B.1: Table of typical parameters for the (Baby) Bucket Brigade. Refer to Appendix A forthe definition of the variables. R0 is obtained from a fit to the potential within 2 cm of the trap axis.

297

B.2

Internal state and trapping field parameters
Parameter

Value (to 3 s.f.)

Notes

de Eeff
A||
Ehf
ωef
dmf
Erot
Ecapture
ωrot
Erot
rrot
rcapturedmf Erot3γ
dmf Erot
F=3/2
2
gF =3/2
gF =1/2
BaxgradBrot
3gF µB Brot
δgF /gF∆u∆lB⊥
ωx,y,z
r1σ
T
Ez
Erf
ωrf
⟨Erf ⟩
Erf
Ttrap,zTtrap,ρ

2π × 0.851 µHz
2π × −20.1 MHz
2π × −15.1 MHz
2π × 5.29 MHz
3.37 D
60 V/cm
18.8 V/cm
2π × 147.5 kHz
93500 K2.69 mm0.842 mm
2π × 102 MHz
2π × 50.9 MHz−0.0149
2.97 mG/cm
799 µG
2π × 50 Hz−2.55 × 10−3
2π × 0.511 Hz
2π × 1.43 Hz
2π × 2 kHz
0.251 cm
30 K
20.6 mV/cm
0.447 V/cm
2π × 50 kHz
15.5 mV/cm
22.6 K
1000 K
2100 K

Assuming de = 10−31 e · cm.
Hyperfine splitting constant.
Hyperfine splitting.
Ω-doubling splitting constant.
2π × 1.70 MHz/(V/cm).
.
Supersonic expansion speed for Neon is 780 m/s.
.
Kinetic energy of rotational micromotion.
Radius of rotational circular motion.
Radius of capture circular motion.
.
Stark shift of mF = ±3/2 states in J = 1.
Sign not measured.
Zeeman splitting.
At Erot == 60 V/cm.
.
.
Trap secular frequencies.
1σ radius of ion cloud.
Temperature of ion cloud.
Axial electric field for confinement at 2σ radius.
RF field strength at 1σ radius.
RF frequency
Averaged over one Paul cycle.
Kinetic energy of RF micromotion over one Paul cycle.
Trap depth along the z direction.
Trap depth along the radial direction.

Table B.2: Table of typical parameters for the Gen. III experiment. Additional notes:• Ez = mωz2 z/e.
• Erf = Vrf,pp r1σ /R02 , where R0 is given in Table B.1.
eE 2

• ⟨Erf ⟩ = 2mω2rfr1σ (Huanqian’s thesis Equation 4.7) and is lightly larger than Ez because of deconfining
rf
potential from rings.
e2 E 2• Erf = 4mωrf2 (Huanqian’s thesis Equation 4.8).
rfAppendix C

Hamiltonian

C.1

Introduction
Writing out the full Hamiltonian is highly non-trivial. The details of the Hamiltonian are scatteredacross many papers [14, 25] and theses of previous group members. This chapter seeks to (i) consolidate
information pertaining to ThF+ , (ii) address common pitfalls in understanding1 , and (iii) serve as a reference
for the previous chapters.
Before we go into the details of the Hamiltonian, it serves to address the first common pitfall inunderstanding: frames of reference. Two frames of reference are used in the evaluation of the Hamiltonian:(i) frame defining the quantization axis, and (ii) frame of the molecule. The former is defined to be a frameof reference that is rotating with Erot , and we define quantum numbers like F , mF , and J in this frame ofreference. The latter has quantization axis defined along the internuclear axis, pointing towards thorium,
and we define the quantum number Ω in this frame of reference.

C.2

Terms in the Hamiltonian
The Hamiltonian has this simple form:

H = Hhyperfine + HΩ-doub. + Hrot. + HStark + HZeeman + Hoffset (v, J) + HeEDM .

(C.1)

The following sections go into the details of each term in order.
1 Mainly to do justice to the insane amount of time I put into understanding and writing legible code to solve the Schrödinger’s
equation, and to lessen the burden on whoever the new Hamiltonian custodian is.

299
C.2.1

HyperfineThe hyperfine splitting arises from the interaction between the total angular momentum J and thetotal nuclear spin I. Since 232 Th has no nuclear spin, the nuclear spin comes solely from the spin-1/2 19 F
nucleus. Since the angular momenta precesses around the internuclear axis n, the interaction energy has theform: Ehyperfine ∼ (I · n) (J · n). Using the projection theorem (e.g. Sakurai Equation 3.10.40), we have:
Hhyperfine = A||F2 − I 2 − J 2
,
2J2

(C.2)

where A|| is the hyperfine splitting constant; F = I + J, I, and J are the hyperfine states, nuclear spin, andtotal angular momentum, respectively.
For ThF+ in the X 3 ∆1 (v = 0) manifold where I = 1/2, we have the ratio∆Ehyperfine,J=1 /∆Ehyperfine,J=2 = 9/5,
which says that the hyperfine splitting in the J = 1 state is larger than that in J = 2.

C.2.2Ω-doubling

The Ω-doubling splitting arises from the interaction between electronic and rotational motions thatsplits the energy degeneracy of Ω = J · n of opposite signs2 . This Hamiltonian only couples |Ω⟩ to | − Ω⟩with the same J, F , and mF . In our group, we defined it as:HΩ-doub. =M (−1)J
J≥Ω

where

L

2

ℏωef Ω(J)
x ,

(C.3)
(J)

is the direct sum, ωef is the energy splitting between the two Ω levels for J = 1, and Ωx

states of Ω with opposite signs within the same J subspace:
Ω(J)
x =



J(J + 1)
|J, Ω = +1⟩⟨J, Ω = −1| + |J, Ω = −1⟩⟨J, Ω = +1| .
2

The Ω states are connected to the parity states |±⟩ through:|Ω = ±1⟩ =|+⟩ ± |−⟩√.
2

The factor (−1)J is there to account for the alternating parity with J.
2 For more details see Chapter 7.9 of Dave DeMille’s book [88].

connects

300
C.2.3

Rotation

Since we are operating in a rotating reference frame defined by Erot , there is a non-inertial-frame
term in the Hamiltonian. This is the part of the Hamiltonian that is responsible for Berry’s phase. This
Hamiltonian couples states with ∆mF = ±1 with the same F , J, and Ω. We define the z-axis to be pointing
in the direction of Erot , and the x-axis to be anti-parallel to the (counter-clockwise) rotation vector. Hence,
we have:
Hrot. = ℏωrot. Fx ,−where Fx = F+ +F, F± |F, mF ⟩ =
2

p

(C.4)

(F ∓ mF )(F ± mF + 1)|F, mF ± 1⟩, and ωrot. is the angular frequencyof Erot .

C.2.4

Stark

In the presence of external electric fields, states of opposite parities are mixed to give us states ofgood Ω. The mixing happens across states of different J and F , but share the same Ω and mF . Hence,states between J = 1 and J = 2 can couple to each other, and depending on the exact energy difference
between the coupling states, properties like the magnetic g-factor can be different between the upper andlower doublets. Expressing the Hamiltonian in element-form, we have:



 J F
X
′
F +J +1+I
HStark = −dmf n · E = −dmf
Ep × (−1)
p∈{0,±1}
 F ′ J′




I 


1 



′

′

′

× (−1)F −mF

× (−1)

J ′ −Ω′

p

 F(2F + 1)(2F ′ + 1) 

−m′F


p
 J′
′
(2J + 1)(2J + 1) 

−Ω′

1
p
1

q=0

F 


mF

J
.

Ω

(C.5)

We have chosen q = 0 because the dipole is aligned along the internuclear axis, and p = 0, ±1, whichcorrespond to the spherical tensor components. The derivation is long, so I recommend interested readers
to start from page 167 in Brown and Carrington [47].

301
C.2.5

ZeemanIn the presence of external magnetic fields, states of different F are mixed. Since the magnetic g-factor
depends on F , we get different extents of mixing of gF depending on the exact energy difference between
the coupling states. This is the other reason for a differential g-factor δg. In the presence of weak external
magnetic fields, J couples more strongly to the internuclear axis than the external magnetic field, and the
Hamiltonian can be expressed in the following form:
HZeeman = −G|| µB (J · n) (n · B) − gN µN I · B,

(C.6)

where gN is the nuclear g-factor of 19 F, and G|| is related to the hyperfine g-factor through the following[89]:
F (F + 1) + J(J + 1) − 3/4µN F (F + 1) − J(J + 1) + 3/4+ gN,
2F (F + 1)J(J + 1)
µB
2F (F + 1)which evaluates to 31 −G|| + gN µµNB for F = 3/2, J = 1. Evaluating the dot products in the HamiltoniangF = −G||

give:

X

(J · n) (n · B) = Ω′

Bp × (−1)F +J +I+1

p∈{0,±1}




 J

F




I 



 F′

J′



1 



′

′

′

× (−1)F −mF

× (−1)

I·B =

X

′

J ′ −Ω′′

Bp × (−1)F +J +I+1

p∈{0,±1}′′

× (−1)F −mF

p

p

 F
(2F + 1)(2F ′ + 1) 

−m′F


p
 J′
′
(2J + 1)(2J + 1) 

−Ω′

p

I(I + 1)(2I + 1)




 I

F′

1
p

F 


mF


1
q=0




J′ 



 F


I



1 


 F′
(2F + 1)(2F ′ + 1) 

−m′F

1

F 
.

mF

p

J
.

Ω

The astute reader will recognize that the first form above is similar to Equation (C.5). Once again, thederivation is long, so I recommend interested readers to start from page 606 in Brown and Carrington [47].

302
C.2.6

OffsetWhile all the Hamiltonians above deal with energy scales ≲ 2π × 100 MHz, this last term is on theorder of 10 GHz and above. For the numbers relevant to ThF+ [43], the energy levels within the same
electronic level hold the approximate form:

Hoffset = G(v) + E(v, J),



2
1
1
G(v) = ℏωe v +,− ℏωe χe v +
2
2


1
2
E(v, J) = Be J(J + 1) − D [J(J + 1)] − αe v +J(J + 1).
2
2

= Bv J(J + 1) − D [J(J + 1)] .

(C.7a)(C.7b)(C.7c)(C.7d)

G(v) accounts for vibrational levels, and the energy spacing is on the order of about 600 cm−1 (1 cm−1 ≈30 GHz), while E(v, J) accounts for the rotational levels, with energy spacing on the order of 15 GHz. The
spectroscopic constants are the vibrational constant ωe , anharmonicity constant ωe χe , equilibrium rotational
constant Be , centrifugal distortion constant De , and rotational-vibrational coupling constant αe . The terms
involving Be and αe are often grouped together to give a v-dependent B = Be − αe (v + 1/2).

C.2.7

eEDM

For completeness, the effective Hamiltonian responsible for the shift in the energy levels due to theeEDM is [25]3 :HeEDM = −de Eeff

Ω
,
|Ω|

(C.8)

where Eeff ≈ 35 GV/cm is the effective electric field within ThF+ that is seen by the unpaired electron in
the X 3 ∆1 state [22, 40, 41].
Since the Stark shifts go as mF Ω while the eEDM term goes as Ω, we see that states that areenergetically degenerate under Stark shifts (e.g. |mF = +3/2, Ω = +1⟩ and |mF = −3/2, Ω = −1⟩) haveeEDM shifts of different signs.
3 There is an extra factor of 1/2 in the cited paper [25], which uses a different convention for how E
eff is defined in papers
that calculate the value [22, 40, 41]. The factor of 1/2 is removed for consistency with the cited value of Eeff .

303
In principle, all we need to do to determine the eEDM is to measure the energy difference between
degenerate Stark pairs. In practice, we choose to do so with the outer-most stretched Stark pairs, because
these stretched states are the easiest to prepare. There are two such pairs, and we call the pair that is more(less) energetic the “upper doublet” (“lower doublet”).

C.3

Solving the Schrödinger’s Equation

C.3.1

Time-independent

Solving the time-independent Schrödinger’s Equation is as easy as diagonalizing the full 12- (for J = 1
only) or 32-state (for J = 1&2) Hamiltonian. In general, the eigenstates are superposition states of our basis
set {|J, Ω, F, mF ⟩} because of the mixing induced by Hrot. , HStark , and HZeeman . Nevertheless, one can still
distill out the dominant component of the basis set from each eigenstate, and draw meaningful descriptionsof the system. Figure 4.7 shows an example of diagonalizing the 32-state Hamiltonian at Erot = 60 V/cm
for ThF+ .

C.3.2

Time-dependent

Propagating the quantum state with time is not cheap computationally. Unless one wishes to investigate the effects of the J = 2 states coupling to the J = 1 states, it is highly advisable to use the 12-state
Hamiltonian instead of the 32-state Hamiltonian. Unfortunately, many systematics involve contribution
from coupling between the J = 1&2 states. We can speed up the computation by propagating the quantum
state of different R̃ and B̃ switches in parallel. The python package QuTip [90] allows for such parallel
computation, and has been used for simulation works in this thesis.

304

C.4

Two-level effective Hamiltonian
For a more intuitive understanding of the relevant parameters involved in studies of systematics, it is

helpful to work with a two-level effective Hamiltonian that captures all the physics in newly defined effective
parameters. There are two of such two-level Hamiltonians, corresponding to the upper (states 8 and 9 inFigure 4.7) and the lower doublets (states 0 and 1 in Figure 4.7). The effective parameters are defined assuch:Magnetic g-factor, g, δg. The mean magnetic g-factor of the upper and lower doublets is defined as g =(g u + g l )/2, while the difference in magnetic g-factor between the upper and lower doublets is definedas δg = (g u − g l )/2, where “u” and “l” refer to the upper and lower doublets, respectively. Typicallyδg ≪ g.
Non-reversing magnetic field, Bnr . In the eEDM experiments with HfF+ , we switch the direction of theapplied magnetic field to search for systematics. In the event of an imperfect switch, the magnetic
field can be decomposed into a part that reverses, and a part that does not, i.e. B(B̃) = B̃B + Bnr ,where B̃ = ±1 is the sign of the switch. In the Bucket Brigade experiment, we do not plan to apply
Zeeman shifts to our eEDM sensitive states, so B = 0 and Bnr corresponds to the ambient magnetic
field.
Berry’s frequency, 3frot (S̃⟨α⟩ + αnr ). This term (integrated with time) is the Berry’s phase accumulatedby the states. Much like the case for magnetic fields in the second generation experiment, we canalso switch the sign of Berry’s phase in our experiment by switching the direction of the relevant
fields. In the case of an imperfect switch, Berry’s phase can also be decomposed into a reversing
part ⟨α⟩ and a non-reversing part αnr . We aim for αnr ≪ ⟨α⟩. The switch is defined as S̃ = ±1.
Energy splitting at avoided crossing, ∆, ∆D . Rotation coupling from Equation C.4 splits the degeneracy of the eEDM sensitive states within each doublet into those with avoided crossings ∆u and∆l for the upper and lower doublets, respectively. The mean and the difference are defined as∆ = (∆u + ∆l )/2 and ∆D = (∆u − ∆l )/2, respectively. Typically ∆D ≈ ∆. We also define a305
convenient “doublet switch” D̃ such that ∆u,l = ∆ + D̃ ∆D , D̃ = ±1.
Using the eEDM experiment with ThF+ in the Bucket Brigade using Berry’s phase as the dominant frequency
as an example, the steps involved are detailed in Section 8.3.4.