A new limit on the electron’s electric dipole moment
by
Tanya S. Roussy
BASc., Univeristy of British Columbia, 2014
MASc., University of British Columbia, 2016

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
2022

Committee Members:
Eric A. Cornell, Chair
Prof. Jun Ye
Prof. John Bohn
Prof. Keith Ulmer
Prof. Zach Berta-Thompson

ii
Roussy, Tanya S. (Ph.D., Physics)
A new limit on the electron’s electric dipole moment
Thesis directed by Prof. Eric A. Cornell

Physicists have been able to describe our entire universe using just two mathematical models: General Relativity, for gravity, and the Standard Model, for everything else. While theStandard Model has been tested to exquisite precision and is generally successful, it has someserious inconsistencies: it fails, for example, to explain why matter dominates over antimatter in
our universe. New models aiming to eliminate these inconsistencies often introduce additional
symmetry violation, which in turn results in symmetry-violating properties in fundamental particles like electrons. In particular, many new physics theories predict that the electron will have a
non-zero electric dipole moment (EDM) aligned with the spin axis, which violates time-reversal
symmetry. Placing limits on the magnitude of the electron EDM directly constrains new physics
theories at energies beyond the reach of the Large Hadron Collider. Here we present a newexperimental limit on the electron’s electric dipole moment using trapped HfF+ in rotating bias
fields: |de | < 4.4 × 10−30 e·cm. Our measurement is a factor of 2.5 better than the previous best
limit.

Dedication

To everyone fighting for a better world.

"It is our duty to fight for our freedom.
It is our duty to win.
We must love each other and support each other.
We have nothing to lose but our chains".
- Assata Shakur

iv

Acknowledgements

For my parents. Ok, ok, I’ll get a real job

Contents

Chapter12Introduction1

1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

JILA’s eEDM experiment(s): a brief history . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

A non-exhaustive list of other searches for the eEDM . . . . . . . . . . . . . . . . . .

9

1.4

Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

How the experiment works
2.1

2.2

2.3

11

Measuring the electron’s electric dipole moment: the basic recipe . . . . . . . . . . . 11
2.1.1

The basic recipe: put a dipole in a field - twice . . . . . . . . . . . . . . . . . . 11

2.1.2

Key ingredient 1: sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.3

Key ingredient 2: bias magnetic and electric fields . . . . . . . . . . . . . . . 14

2.1.4

Key ingredient 3: Ω doubling (aka clever use of molecular energy structure) 16

Measuring the electron’s electric dipole moment: theoretical details . . . . . . . . . 20
2.2.1

3 ∆ , v = 0, J = 1 Hamiltonian
1

2.2.2

Measurement of δgF /gF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.3

Molecular structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.4

Experimental noise vs the quantum projection noise limit . . . . . . . . . . . 23

. . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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2.3.2

Source chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.3

Ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.4

Computer control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.5

Applied magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.6

Applied electric fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.7

Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.8

Chops, switch states, blocks, and superblocks . . . . . . . . . . . . . . . . . . 38

2.3.9

Trap characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.10 Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4

3

Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.4.1

Hf counting and imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4.2

Spin asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.4.3

Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4.4

Frequency channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4.5

Blinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.4.6

Data cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Accuracy evaluation

60

3.1

Accuracy evaluation of the HfF+ electron electric dipole moment experiment . . . . 60

3.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.1

Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3.2

Molecular beam and ionisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.3

Ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.4

Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3.5

State preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3.6

Ramsey sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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3.3.7

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.3.8

Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3.9

Experimental protocol and switch states . . . . . . . . . . . . . . . . . . . . . 72

3.3.10 Images to determine doublet positions . . . . . . . . . . . . . . . . . . . . . . 74
3.4

Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4.1

Ion counting and asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.4.2

Least squares fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.4.3

Switch parity channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4.4

Blinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4.5

Data cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.5

Effective Hamiltonian for doublets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.6

Accuracy evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.7

Frequency shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.8

3.9

3.7.1

Magnetic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.7.2

Berry’s phase effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.7.3

Residual rotation induced doublet mixing . . . . . . . . . . . . . . . . . . . . 107

3.7.4

Oddities & miscellany . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Phase Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
3.8.1

State preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

3.8.2

Internal state measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.10 Supplemental Information / Appendices to the paper . . . . . . . . . . . . . . . . . 130
3.10.1 Overview of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.10.2 Overall comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.10.3 Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.10.4 Contrasts, and contrast decays . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.10.5 Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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3.10.6 Offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.11 Not Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.11.1 Shifts vs z position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.11.2 Ion number vs f 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3.11.3 f R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.11.4 Shifts vs x-y position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4

5

A new limit on the electron’s electric dipole moment

149

4.1

Summary of the eEDM result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4.2

Experimental Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.3

Accuracy evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.4

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

4.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

4.6

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.6.1

Bias fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.6.2

HfF+ 3 ∆1 details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.6.3

Quantum state preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4.6.4

Ramsey spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4.6.5

Final state readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

4.6.6

de signal extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

Dark Matter Exclusion

164

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

5.2

Experimental Constraint on Axionlike Particles over Seven Orders of Magnitudein Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

5.3

Experimental Constraint on Axionlike Particles over Seven Orders of Magnitudein Mass - Supplemental Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.3.1

Frequency Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

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5.3.2

Generating Least-Squares Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 181

5.3.3

Timestamp Shuffling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

5.3.4

Estimating the no-ALP distribution . . . . . . . . . . . . . . . . . . . . . . . . 186

5.3.5

Sources of Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5.3.6

Parameterizing the ALP-gluon coupling . . . . . . . . . . . . . . . . . . . . . 190

5.3.7

Aggregate Prior Update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Conclusion

194

Bibliography

198

6

Appendix
A Lasers

206

A.1 Neutral HfF production / ablation: “Abby" 532 nm . . . . . . . . . . . . . . . . . . . 206
A.2 Photionization 1+2 pump: “Mia" 532 nm . . . . . . . . . . . . . . . . . . . . . . . . . 207
A.3 Photoionization 1 seed: “Tuna" 1237.55598 nm . . . . . . . . . . . . . . . . . . . . . . 208
A.4 Photoionization 1 (first photon): “Harambe" 309.388 nm . . . . . . . . . . . . . . . . 209
A.5 Photoionization 2 (second photon): “Bertha" 735.464 nm . . . . . . . . . . . . . . . . 210
A.6 State transfer: “Toptica" 961.43495 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
A.7 Cleanup: “Vibational Cleanup" 818.372 nm . . . . . . . . . . . . . . . . . . . . . . . . 212
A.8 mF pumping: “DBR/1083" 1082.4137 nm . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.9 Depletion: “814/Depletion" 814.508 nm . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.10 Photodissociation 1 (first photon): “Cora" 736.989/2 nm or 736.986/2 nm . . . . . . 214
A.11 Photodissociation 2 (second photon) + Cora pump: “Olga" 266 nm . . . . . . . . . . 215
B One shot asymmetry
B.0.1

216

One-shot asymmetry and measured fringe contrast . . . . . . . . . . . . . . . 216

x
C A list of useful transitions

219

D An approximation to our shot noise limit

221

E All parity channel results

226

TablesTable
2.1

Magnetic field coil parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2

Photons used in our experimental sequence . . . . . . . . . . . . . . . . . . . . . . . 37

3.1

Constants used throughout this document. The total magnetic g-factor of J =1, F = 3/2 states gF ≡ ( G∥ + g N µ N /µ B )/3 results from the combination of nuclear
and electronic Zeeman effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2

Example experimental parameters and associated derived parameters from our2022 data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3

Photons used in our experimental sequence . . . . . . . . . . . . . . . . . . . . . . . 64

3.4

Magnetic-field expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5

Constraints on electric fields from higher-harmonic voltages on radial electrodes
and endcaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.6

Constraints on Berry’s phase frequency contributions to f B or f BR due to possiblecombinations of radial and axial field imperfections from harmonic distortion in
amplifiers used to drive radial electrodes. All entries are in µHz. . . . . . . . . . . . 106

3.7

Systematic error budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.1

Systematic error budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

A.1 Minilite II specs from manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

xii
A.2 Quanta-Ray-Pro-250-10 specs from manual . . . . . . . . . . . . . . . . . . . . . . . . 207
A.3 A few Tuna (Toptica TA Pro) specs from the data sheet . . . . . . . . . . . . . . . . . 208
A.4 Select Harambe (Radiant Narrow Amp) specs . . . . . . . . . . . . . . . . . . . . . . 210
A.5 Select Harambe (Radiant Narrow Amp) specs . . . . . . . . . . . . . . . . . . . . . . 211
A.6 A few Toptica/Transfer (Toptica TA Pro) specs . . . . . . . . . . . . . . . . . . . . . . 211
A.7 A few Vibrational Cleanup (Homebuilt ECDL) specs . . . . . . . . . . . . . . . . . . 212
A.8 A few Vibrational Cleanup (Homebuilt ECDL) specs . . . . . . . . . . . . . . . . . . 213
A.9 A few Depletion (Toptica TA Pro) specs . . . . . . . . . . . . . . . . . . . . . . . . . . 214
A.10 Select Cora (Syrah Cobra Stretch) specs . . . . . . . . . . . . . . . . . . . . . . . . . . 215
A.11 Quanta-Ray-Lab-170-10H specs from manual. Note the manual specified the pulse
width in nm but I’m pretty sure that’s a typo, especially given footnote 1 on table
3-2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
E.1 All contrast parity channels, disaggregated over distinct f 0 values and aggregatedover all 3 f 0 values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
E.2 All decay rate parity channels, disaggregated over distinct f 0 values and aggregated over all 3 f 0 values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
E.3 All frequency parity channels, disaggregated over distinct f 0 values and aggregated over all 3 f 0 values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
E.4 All phase parity channels, disaggregated over distinct f 0 values and aggregatedover all 3 f 0 values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
E.5 All offset parity channels, disaggregated over distinct f 0 values and aggregatedover all 3 f 0 values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
E.6 All frequency parity channels (but with 3 digits after the decimal), disaggregatedover distinct f 0 values and aggregated over all 3 f 0 values. . . . . . . . . . . . . . . . 232

Figures

Figure1.1

A comic I drew about the baryon asymmetry . . . . . . . . . . . . . . . . . . . . . . .

1.2

A cartoon depicting the time-reversal symmetry violation of an electron electric2dipole moment. The spin is indicated with two arrows and the electric dipole
moment is represented by a colour gradient and positive and negative ‘poles’.
When one flips the arrow of time, the spin flips but the electric dipole moment
does not. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3

3

A comparison between fringes taken in 2013 (upper left) and much more recently,
in 2020 (remaining panels). Note the time axis goes to 25 ms in 2013, while it goesto 1.5 seconds in 2020. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4

8A cartoon depicting the current status of eEDM searches. The pink region indicateswhere most new physics theories predict the eEDM should be, and the blue arrows
indicate recent exclusions (at the 90% confidence level). The exclusions measured
zero up to the uncertainty indicated, which means that they exclude eEDMs larger
than their uncertainty. The brown region indicates the standard model prediction
for the eEDM. One can see that the region we are probing is effectively ‘backgroundfree’ in terms of the standard model prediction, and that most new physics theories
predict EDMs right in the range we can measure - so our measurements directly
constrain those theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

xiv
2.1

Conceptual cartoon depicting the basic measurement principle. In each panel, theket indicates both the direction of the magnetic moment (spin) and the electric
dipole moment (EDM). In the first panel, a simple spin flip measurement is performed. In the second panel, an electric field is added, pushing the energy levels
apart for that particular relative orientation of the fields. In the third panel theelectric field is flipped, pushing the levels together. The difference between the last
two measurements cancels out the magnetic dipole contribution (to first order)
leaving only the electric dipole contribution. . . . . . . . . . . . . . . . . . . . . . . . 12

2.2

Conceptual cartoon depicting how three key parameters affect our sensitivity. In
the first panel, the separation between peaks scales linearly with the effective electric field, Eeff . In the second panel, having a higher count rate N reduces the√fractional error in the determination of the peak centers by 1/N. In the third
panel, the width of the peaks scales like the inverse coherence time, or 1/τ . . . . . 13

2.3

A cartoon depicting a polarized HfF+ molecule . . . . . . . . . . . . . . . . . . . . . 13

2.4

A cartoon depicting how a radial quadrupole magnetic field amounts to a static,quantization-axis aligned magnetic field in the rotating frame. In each panel, themagnetic field gradient is indicated with light blue arrows. A circular orbit for amolecule is indicated near the top, and pink arrows depict the vector decomposition of the magnetic field which the ions see as they are rotating. You can see thatas the molecule rotates, the vector decomposition of the magnetic field has twocomponents: one which is always pointing “up", and one which always aligns (oranti-aligns) with the molecular axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

xv
2.5

A cartoon depicting the vectors for Hund’s case a. The electronic spin and orbital angular momenta, S and L, are strongly coupled to the internuclear axis byspin-orbit and electrostatic interactions respectively. Hund’s case a arises whenspin-orbit interactions are much weaker than the electrostatic interactions which
split different electronic states. This means that widely separated electronic levels
are split into a set of spin-orbit manifolds, each of which contains many closely
spaced rotational levels. Λ = L · n is the projection of the electronic orbital angular momentum L on the internuclear axis n, S = |S| is the total electronic spin
angular momentum, Σ = S · n is the electron spin projection on the internuclear
axis, J = |J| = |L + S + R| is the electronic plus rotational angular momentum,Ω = Λ + Σ = J · n is the projection of the electronic angular momentum onto
the internuclear axis, I = |I| is the 19 F nuclear spin, F = |F| = |J + I| is the total
angular momentum of the molecule, and m F = F · ẑ is the projection of F on therotating quantization axis [1] [2]. We take the internuclear axis n to be directedfrom the 19 F nucleus to the 180 Hf nucleus [2].

2.6

. . . . . . . . . . . . . . . . . . . . . . 17

In HfF+ ’s 3 ∆1 J = 1 state we have two hyperfine states with total angular momentum F = 3/2 and F = 1/2. Each of these has two degenerate manifolds madeup of the different allowed projections m F = F · ẑ onto the quantization axis. The
two sets of degenerate states correspond to the eigenstates of the parity operator:√√|e⟩ = (|Ω = +1⟩ − |Ω = −1⟩)/ 2 and | f ⟩ = (|Ω = +1⟩ + |Ω = −1⟩)/ 2. These
degenerate manifolds are separated by the Ω-doublet splitting, which we call ωef . 18

xvi
2.7

Focusing only on the F=3/2 manifold of the 3 ∆1 J = 1 state of HfF+ . Σ = S · n
is the electron spin projection on the internuclear axis, and the internal electricfield experienced by the valence electron is indicated as ⃗Eeff . Left: an electric fieldpolarizes the molecule along or against the field, which we call the “upper" and“lower" “doublets". The energy levels without the added E field are indicatedwith dashes lines, and the energy levels with the added E field are indicated withsolid lines. Center: adding a B field further lifts the degeneracy, when the spinis aligned with the B field the level moves down, and when it is anti-aligned it
moves up. Again, the energy levels without the added B field are indicated withdashed lines and the energy levels with the added B field are indicated with solidlines. Right: finally, the addition of the eEDM itself further breaks the degeneracy,
we assume the eEDM points in the same direction as the spin; when the eEDM
is aligned with the internal electric field ⃗Eeff the energy level moves further down
and vice versa. As before, the energy levels without the effect of the eEDM are
indicated with dashed lines and the energy levels with the effect of the eEDM are
indicated with solid lines. Nothing in this cartoon is to scale. . . . . . . . . . . . . . 19

2.8

HfF+ ’s 3 ∆1 , J = 1, F = 3/2 state complete details. Σ = S · n is the electron spin
projection on the internuclear axis, and the internal electric field experienced by
the valence electron is indicated as ⃗Eeff . The energy levels without the effect of theeEDM are indicated with grey lines, while the ones with the effect of the eEDM
are indicated with black lines. Dashed lines are for visual aid only. The effect of
the eEDM has the opposite sign in the upper vs lower doublet. It is this energy
difference that we aim to measure with our experiment. . . . . . . . . . . . . . . . . 20

2.9

Measurement of δgF /gF by varying the mean fringe frequency f 0 , all data is taken
at Erot = 57.9V/cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.10 Measurement of δgF /gF by adding a non-reversing axial magnetic field gradient,leading to large f B . Note that f BD is still blinded in this measurement.

. . . . . . . 23

xvii
2.11 CAD rendering of the entire apparatus, excluding lasers. In purple are the X shimcoils, blue are the Y shim coils, and yellow are the Z shim coils. The orange coils
generate the axial magnetic gradient that we call Brot . . . . . . . . . . . . . . . . . . 29
2.12 Top down schematic of our experiment’s main chamber . . . . . . . . . . . . . . . . 29
2.13 CAD rendering of the main experimental chamber sliced in half for a view of theion trap. To give a sense of scale, you can fit a coffee cup between the electrodes. . 31
2.14 Confining and orienting field amplitudes throughout the experimental sequence,in Volts/cm, for a 5 ms Ramsey free-evolution time. The π/2 pulses are easily
visible in the Erot panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.15 Confining and orienting field amplitudes throughout the experimental sequence,in Volts, for a 5 ms Ramsey free-evolution time. The π/2 pulses are easily visible
in the Erot panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.16 The variation of the computed secular frequencies throughout the experimental
sequence for a 5 ms Ramsey free-evolution time. . . . . . . . . . . . . . . . . . . . . . 34
2.17 From Erot , computed values of Brot and rrot throughout the experimental sequence
for a 5 ms Ramsey free-evolution time. . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.18 Cartoon depicting the transitions used during our state preparation. . . . . . . . . . 37
2.19 Measurement of the ‘ambient’ f B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.20 Measurement of the ‘ambient’ f B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.21 The time of flight trace of the HfF+ cloud with a Gaussian overlaid . . . . . . . . . . 41
2.22 Using our HfF/voltage calibration, we compute the HfF in each slice of the TOF
trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.23 Constructing the approximate HfF ellipsoid from our imaging and TOF data. In
the Y-Z imaging data, the lines demarcate the approximate edge of the cloud. . . . 42
2.24 Approximate HfF density along X, and approximate Mean Field Parameter for ourcloud over a given temperature range . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

xviii2.25 Some typical images showing how the two doublets appear on our phosphor
screen: on the left, we depleted one doublet and imaged the one remaining. In
the center we did the same with the opposite doublet, and on the right we depleted neither (so we are imaging both doublets at the same time). The numbers
indicate pixels on the screen, the images are in false colour, and the images arean average of 20 separate runs of each experiment. The red line indicates ourprogrammatic determination of the ’center’ of the two blobs. . . . . . . . . . . . . . 48
2.26 An overlay of 20 images from our MCP assembly, where each doublet was fully
depleted 10 times. The center is indicated with a black dashed line. . . . . . . . . . 51
2.27 The integrated trace of Figure 2.26, showing the upper and lower doublets separately.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.28 Example Ramsey interferometry data . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1

CAD rendering of the entire apparatus, excluding lasers. On the left is the sourcechamber, where we produce neutral molecules. To the right of the source chamberis the differential pumping chamber, where the beam of neutral molecules is supersonically cooled to a rotational temperature of roughly ∼ 10K and collimatedby skimmers. Further right is the main experimental chamber, which contains our
8-electrode Paul trap. Surrounding the main chamber are the magnetic field coils:
the X shim coils in purple, the Y shim coils in blue, and the Z shim coils in yellow;⃗ are indicated in orange. . 63
the coils which generate the quadrupole magnetic field B
3.2

Top down schematic of our experiment’s main chamber. The neutral molecules
fly into the center of the trap where they are ionized and trapped. We apply the⃗rot and the quadrupole magnetic field B⃗ . The ionsrotating electric bias field E⃗rot , and theirundergo unifom circular motion, either aligned or anti-aligned with E⃗rot · B⃗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
unpaired spins precess about E
3.3

Cartoon depicting the transitions used during our state preparation. . . . . . . . . . 67

xix
3.4

Ramsey spectroscopy in HfF+ . Top, level structure in the eEDM-sensitive 3 ∆1 ,J = 1, F = 3/2 manifold in external electric Erot and magnetic Brot bias fields(not to scale). The oriented molecules are either aligned or anti-aligned with Erot ,which we call the lower and upper Stark doublets respectively. The component of
the energy splitting due to the eEDM, 2de Eeff , flips sign between doublets. Bottom,
example Ramsey fringes from our dataset. The fringes from the two doublets arecollected simultaneously. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5

Results as we shift the threshold for applying cuts to the data. In each plot, only
one cut is applied (the other cut is not applied). The cuts do not shift the resultant
value of f BD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.6

Measurement of gFF , by varying the mean fringe frequency f 0 and measuring theδgdifferential fringe frequency f D . Data taken at Erot ∼ 50 V/cm. . . . . . . . . . . . . 87
3.7

δg

Measurement of gFF , obtained by introducing a large δBnr and measuring the resultant values of f BD and f B . Note that f BD is still blinded in this measurement.
Data taken at Erot ∼ 50 V/cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.8

Example axial oscillation-induced phase modulation data, for a both large kick(color) and a small kick (grey). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.9

Phase modulation amplitude for various sizes of z kick. Upper doublet data is inblue and lower doublet data is in pink. Both fits have an intercept of −0.0127(0.0108). 99

3.10 Example data collected with E2 f Z ∼ 4 V m−1 and an ellipticity Eϵ /Erot ∼ 5 × 10−3 .
As we vary the phase of the axial modulation, f B and f BR vary sinusoidally witha π/2 phase offset, as expected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.11 Toy model to explain ϕ DBI . (i) uncoupled basis, (ii) coupled basis. . . . . . . . . . . 116
3.12 Example data. Plotting the slope of f D when the early-time data is taken at varying
offsets from the zero crossing of the difference fringe. The x-intercept gives us thetrue value of our imaging contrast, which we then use in our fitting program. . . . 122

xx
3.13 Varying the swatch width while correcting for the doublet contrast. We see nomeaningful shifts in our science channel and a small residual shift in f D . In thelower panel we have averaged over the 3 distinct values of f D that we operated at. . 124
3.14 The effect of moving the center of the swatch on our science channel. . . . . . . . . 126
3.15 Moving the center of the swatch has a measurable effect in several parity channels.
Note all panels show the entire aggregated dataset except for the f D panel, whichshows a subset of measurements performed at a single f 0 . . . . . . . . . . . . . . . 127
3.16 Imaging data collated over 25 blocks. The lower doublet data is in pink while theupper doublet data is in blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.18 Observed frequency shifts in z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.19 Observed frequency shifts in z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.20 Fitting f (z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.21 Comparing frequency shifts in z with the magnetic field shimming Z coil on or off 140
3.22 Comparing frequency shifts in z with the magnetic field shimming Z coil on or off 140
3.23 Comparing observed frequency shifts in z with 2 different values for the axial
confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.24 Comparing observed frequency shifts in z with 2 different values for the axial
confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.25 Fitting the measured frequency vs slosh amplitude in the non-displaced trap . . . . 145
3.26 Comparing the prediction for f BR from our model of the anharmonicity to theempirical results. We see that this model does not account for the observed shifts . 146
3.27 Averaging f R over our entire dataset, per primary fringe frequency f 0 . f R scaleswith f 0 with a slope of 0.00018, which is very close to 1/1024 = 0.00098 or a single
step in a 10-bit DAC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.28 Comparing observed frequency shifts with x translation . . . . . . . . . . . . . . . . 148
3.29 Comparing observed frequency shifts with x translation . . . . . . . . . . . . . . . . 148

xxi
4.1

Ramsey spectroscopy in HfF+ . Top, level structure in the eEDM-sensitive 3 ∆1 ,J = 1, F = 3/2 manifold in external electric Erot and magnetic Brot bias fields(not to scale). The oriented molecules are either aligned or anti-aligned with Erot ,which we call the lower and upper Stark doublets respectively. The component of
the energy splitting due to the eEDM, 2de Eeff , flips sign between doublets. Bottom,
example Ramsey fringes from our dataset. The fringes from the two doublets arecollected simultaneously. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.2

Top down schematic of our experiment’s main chamber. The neutral ions fly into
the center of the trap where they are ionized and trapped. We apply the rotating⃗rot and the quadrupole magnetic field gradient B⃗ . The ionselectric bias field E⃗rot , and theirundergo unifom circular motion, either aligned or anti-aligned with E⃗rot · B⃗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
unpaired spins precess about E
4.3

Systematic shift in the eEDM channel f BD due to non-reversing axial magnetic
field, which shows up in the f B channel. . . . . . . . . . . . . . . . . . . . . . . . . . 156

4.4

Results as we change the threshold for cutting blocks from our dataset. In eachplot, only one cut is applied (the other cut is not applied). Applying cuts does not

4.5

systematically shift our result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
p
Summary of our dataset after cuts have been applied and scaling δ f by χ2 to account for over-scatter. Top left, normal probability plot of f BD showing the data areconsistent with a normal distribution. Bottom left, histogram of normalized andcentered f BD data. Overlaid in blue is a normal distribution. Right, superblocks
taken under different experimental parameters compared to the overall averagevalue of f BD . Here N is the average number of trapped HfF+ ions per experimental trail. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4.6

Cartoon depicting the transitions used during our state preparation. . . . . . . . . . 161

5.1

Generation 1 EDM data vs time. Could there be an oscillatory signal buried in here?164

xxii
5.2

Least-squares spectral analysis of our EDM data. . . . . . . . . . . . . . . . . . . . . 172

5.3

Bayesian power measured analysis of the EDM data. The grayscale matrix corresponds to a sub-aggregation of the prior updates U over 100 logarithmicallyspaced frequency bins. Darker shading on the logarithmic color scale corresponds
to increased prior update, which can be interpreted as an increased belief in an
ALP of given frequency ν at ALP-gluon coupling CG /( f a m a ). We use a blue line
to indicate the 95% exclusion (which corresponds to the sub-aggregated update
dropping to 0.05 [3]) for sub-aggregation over each decade. The corresponding
greyscale matrix for this sub-aggregation is not shown. Finally, a dotted line corresponds to the aggregated exclusion over the full analysis range. The full unaggregated matrix of U values is available upon request. . . . . . . . . . . . . . . . . 176

5.4

Astrophysical and laboratory limits on the ALP-gluon interaction. All limits assume that ALPs saturate the local dark matter content and we have taken the mean
local dark matter density to be 0.4 GeV/cm3 [4]. All constraints correspond to a
95% confidence level. The purple region embodies the constraint that the virial deBroglie wavelength is smaller than the size of a dwarf galaxy [5]. The blue regionis the constraint derived from looking at neutron EDM oscillation [6]. The green
region represents constraints from big bang nucleosynthesis [7], and the yellow
region represents constraints from supernova energy loss bounds [8, 9]. The pinkregion is the constraint described in this work. . . . . . . . . . . . . . . . . . . . . . . 178

5.5(Top) Individual CS measurements plotted as a function of their acquisition timesince August 16 2016 (the temporal ‘center’ of our dataset). (Bottom) Our most sensitive CS measurements were collected over 11 days in 2017. For the low frequency
analysis we used the entire dataset (top), and for the high frequency analysis we
used only the most sensitive 11 days (bottom). . . . . . . . . . . . . . . . . . . . . . . 180

xxiii
5.6

Attenuation factor due to ALP field decoherence, calculated over the high frequency analysis range. Grey-filled circles are the calculated values, and the blue
curve is the fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.7

Attenuation due to finite sample time over the entire analysis range. The blue
curve shows the piecewise function generated by the fit to the grey circles. . . . . . 191

5.8

Aggregate prior update U (equivalently, exclusion) as a function of ALP-gluon
coupling CG /( f a m a ). The horizontal dotted line indicates 95% exclusion (where
the aggregate prior update drops to 0.05), while the vertical dotted line indicatesthe lower limit of CG /( f a m a ) which is excluded at 95% confidence over the entire
analysis band. Had our data presented us with strong affirmative evidence for the
existence of an ALP in our frequency range, the curve of U vs CG /( f a m a ) might
have peaked not at U = 2.5 but for instance at U = 1, 000 or higher. . . . . . . . . . 193

C.1 The infamous ’laser diagram’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Chapter 1
Introduction

1.1

Motivation
It’s honestly hilariously ambitious of physicists to imagine that the entire universe - fromthe dawn of time until its inevitable heat death, from our galactic core out to infinity - can be
described with a single mathematical equation. But that does seem to be the project for a sizeable
portion of the discipline, and this thesis is one very small contribution to that project. Physicists
have agreed for now that the language of our universe is mathematics, so the goal is to find theGrand Unified Theory of Everything, which is funnier if you read it in Douglas Adam’s voice.
In practice, this means taking a fairly hefty equation, like the Standard Model (SM), and testing
it and refining it and testing it and refining it and so on until we have pulled the blindfolds off
our eyes and can stare into the face of god as we walk backwards into hell1 .
The Standard Model, while a darling in most physics circles, is problematic2 . For example,
the SM does not account for dark matter or dark energy 3 , the unification of the SM with generalrelativity remains elusive, and there are other issues with opaque names like ‘the hierarchy
problem’ [11]. Our experiment is motivated by a problem called the ‘baryon asymmetry’, which
1 @dril wrote: “IF THE ZOO BANS ME FOR HOLLERING AT THE ANIMALS I WILL FACE GOD AND WALKBACKWARDS INTO HELL” Twitter, 2012-05-22
2 The Standard Model: our problematic fave.
3 How embarrassing - for example, dark matter makes up 84% of our universe’s cosmic matter density [10]2describes the preponderance of matter (baryons) over antimatter (antibaryons) in our universe.
This asymmetry explains why we actually exist, but this too is a problem in the SM. The SM
predicts that our universe would have been born with virtually equal parts matter and antimatter,
who would have met and (excitingly! explosively!) annihilated a long time ago. The result would
be a present-day universe made entirely of light4 , which sounds poetic - but a universe without
matter would be terribly boring. Our very existence, then, poses a problem for the “let’s describe
literally everything with an equation" people.
The baryon asymmtery can be explained by modifying the SM with the addition of someextra ‘asymmetry’ in the math, in particular some time-reversal (T) violation (and by the CPT
theorem, charge-parity (CP) violation). This was all worked out by a physicist named Andrei
Sakharov [12], whose papers I do not understand in the least. For a slightly more technicalexplanation of all this I’d recommend you read the introduction to Will Cairncross’ thesis [13],and for an even less technical explanation please enjoy this comic I drew (Figure 1.1).

Figure 1.1: A comic I drew about the baryon asymmetry

The tricky bit, as far as I understand it, is that we can’t just add some T violation willy-nilly
into the standard model, because there’s a multitude of so-called sectors it can be added to andwe need to figure out which one is the right one. Put a different way, there’s a lot of terms in
the SM which could be tweaked to accomodate the extra symmetry violation, and we need tofigure out which terms are the right ones to adjust. Our measurement aims to check the leptonic
4 no stuff, just vibes

3

Figure 1.2: A cartoon depicting the time-reversal symmetry violation of an electron electric dipole
moment. The spin is indicated with two arrows and the electric dipole moment is represented bya colour gradient and positive and negative ‘poles’. When one flips the arrow of time, the spin
flips but the electric dipole moment does not.

sector by measuring the permanent electric dipole moment (EDM) of the electron (eEDM). An
eEDM violates time-reversal symmetry because flipping the arrow of time flips the spin but not
the EDM (see Figure 1.2).
Finally, we can state the goal of this thesis: to measure the EDM of the electron (eEDM), if
it exists. We will cover the details of how one goes about measuring the eEDM in the following
chapters. For now, dear reader, please understand that the rest of this thesis is really about doinga spectroscopic experiment (with a quick detour into a dark matter search - see chapter 5), andnot quantum field theory or complicated calculations.
There is a second motivation for our particular experiment. The addition of more T violation in the SM is, by some strange magic I don’t understand, the same as adding new high energy
particles to the SM. Because of this, sorting out the details of the T violation is related to detecting new high-energy particles and measuring their properties. Traditionally this is what particle
accelerators are built for - but the new particles we are searching for are beyond the reach ofany existing accelerator, and (barring some incredible innovation) this fact is unlikely to change
for decades to come. Because of this, we really need to look outside the accelerator paradigm tocontinue to search for new physics. Luckily for the field, precision AMO (atomic and molecular
optics) experiments can do just that: our current experiment is probing new physics at a TeV

4range which is already beyond the reach of the Large Hadron Collider (LHC) [32].

1.2

JILA’s eEDM experiment(s): a brief history
Before Eric started this experiment, nobody had done an eEDM search with trapped ions.

To this day we are still the only trapped ion eEDM experiment in the whole world. The relative
uniqueness of this experiment is important to acknowledge off the top, because it means that thisexperiment was quite difficult to realize from scratch. Eric describes the early days of thinking
and planning5 :"There were several years there when I was thinking of how do a really good
AMO-meets-particle-physics experiment. I had BEC6 , which seemed like a goodstarting point. Then (as now) I was conscious of hailing from a line of AMO
or AMO-adjacent physicists who were interested in that sort of thing. I was in
Florence, Italy, in 2000 and went to an ICAP where Dave Demille had a poster on
his PbO, vapor cell, Ω = 1 vision. That was my first exposure to the really cool
possibilities of Ω = 1 molecular states for precision metrology, and it blew mymind. I’m glad that four years went by before I definitively set out to steal that
idea. And I’m glad that everything about my experiment except for Ω = 1 wasentirely different from Dave’s EDM vapor cell project. It can be a little rude (although not "illegal") to go to a talk or poster and then just say "cool experiment.
I am going to try to do it too, just do it faster." In the 2000s through summer 2003
I was thinking a lot about "what I could do for an encore to BEC". Starting in fall
2002 with Immanuel Bloch’s paper on the Mott transition [14] which basically set
everyone’s hair on fire, and increasingly for the next couple years, this idea of
doing quantum computing with individual neutral atoms in adjacent lattice sites
was like all anyone wanted to talk about. The whole scheme didn’t appeal to me,
or maybe it was just sour grapes that it was not my idea, so that pushed me tothink even more about precision metrology as an alternative to quantum information, and as a future direction for cool BEC experiments. Meanwhile, Debbie justdown the hall was doing these mind-blowing things with fermions, so I could
sort of enjoy cutting-edge ultra-cold physics vicariously through watching her
progress. Before too long I was so set on "doing a cool precision measurement"that I was no longer committed to "doing precision measurement using BEC". I
was pretty sure I didn’t want to do a beam-line experiment, ala Rabi then Ramsey then Kleppner then Commins. I wanted long coherence times, so trapped
or at least hobbled atoms or molecules. I spent time reading journal articles on
all sorts of crazy ideas, e.g. "trapping" or at least immersing molecules in liquid
5 Extensive quotes from E. Cornell extracted from a February 2022 email
6 Bose Einsten Condensate

5helium. I looked into exotic traps like acoustic-wave Paul traps and microwave
cavity traps."
At this point it’s helpful to point out that a key step in any EDM experiment is the additionof a very large electric field to polarize the species under investigation. And since Eric wanted
to do the experiment with molecules to realize the “Ω = 1 vision", but he also wanted longcoherence/interrogation times, he needed to figure out how to both trap and electrically polarize
molecules. At the time nobody had developed the technology to trap EDM-friendly neutral
molecules. In fact, considerable effort is still being expended on getting EDM-friendly neutrals
trapped, and the best neutral limits still come from beam experiments (which are severely limited
in interrogation time). Of course, the technology for trapping charged species was very mature,
but using trapped ions for an EDM experiment was widely considered impossible: in a normal
ion trap the application of a polarizing DC electric field would immediately kick the ions rightout of the trap. But here’s where Eric had a fantastic idea: use a rotating electric field to polarize
the molecules without kicking them out of the trap. Later chapters will discuss this in detail.
"The idea of trap with rotating field, I am not sure exactly when I came up with
it but it was much on my mind in summer of 2003, and by fall 2003 I decided I
wanted to try it. I had like 10 different molecules in mind, but HfF+ was not oneof them7 . The idea of using photodissociation as read out technique was there
from early on as well. I started trying to hire my first student in late fall of 2003.
Russ (Russel Stutz) joined in January 2004."
The first recorded proposal for the experiment that I could find comes from 2004, from the35th Annual Meeting of the APS AMO Division [15]. The abstract reads:
Search for the electron EDM using trapped molecular ions
The current limit on the electron electric dipole moment (de < 1.6 × 10−27 e·cm)
was set using an atomic beam of Tl [16]. We propose a new experiment using
molecular ions trapped in an RF quadrupole and cooled to cryogenic temperatures using a Helium buffer gas. This new experiment would benefit from thelarge effective electric fields experienced by an electron in polar molecules which
can be orders of magnitude larger than in heavy atoms [17]. We also hope to
7 Ultimately, HfF+ was the molecule we used in the experiment6
achieve much longer spin coherent times than previous experiments, resulting
in narrow magnetic resonance lines. Utilizing the near degenerate Λ-doublet
present in some molecular ions, we expect to achieve a highly polarized sample
of ions in a relatively weak (10’s of V/cm) rotating electric field. This will leadto high sensitivities to de while minimizing systematic effects due to laboratory
electric fields. Systematic effects should also be reduced by the low ion velocities
compared to atomic beam experiments.

It’s worth pausing for a moment here to give Russ the credit he deserves for being the very
first grad student to sign on to a never-been-done-before experiment as a new grad student. Eric
recalled a time when he gave a talk about the proposed experiment at NIST, with Russ in tow,
and afterwards one of the NIST scientists said, with Russ listening, “cool idea, but I’d hate to bea grad student working on that project!”By the spring of 2004 Laura Sinclair had also signed up as a grad student on the eEDM
project, and Aaron Leanhardt had agreed to be a postdoc. All 4 of them met at DAMOP in 2004
for the first group eEDM meeting, even though half the group hadn’t even been to Boulder yet!
Around this time, Eugene Commins’ experiment was the one to beat. He had published the leading paper on eEDM in 2002 [16] using an atomic beam of Thallium, and this limit stood for about
a decade before being bested in 2011 by the first eEDM experiment done with a paramagnetic
polar molecule: the YbF experiment from the Hinds group at Imperial College London [18].
By Fall of 2004 Russ, Laura, and Aaron were all in the lab, and John Bohn and his student
Ed Meyer were thinking hard about the experiment in general and molecular species in particular.
But just as things were getting off the ground, Eric got very, very sick with necrotizing fasciitis.
In his words:"It’s hard to describe how very very sick I was, how close to death, for twomonths, and how for at least a full 12 months I was either not around at all or
else around, but physically and emotionally kind of shattered, needing to workon physical therapy, pain issues, and something at least vaguely like ptsd. I had
definitely lost my swagger. In the meantime, Russ and co. plodded on, and byaround February 2005 Eric realized that "some combination of Aaron, John and7
Ed had identified a really promising molecule, HfF+. I have no idea how they
did that!!!!"
Obviously, settling on a species was a huge milestone for the experiment. After this theteam needed to figure out how to make it, trap it, map its energy landscape, control the internal
quantum state, do ramsey spectroscopy, and read out the final quantum state. A tall orderindeed! Eric describes the exploratory work that followed:"We wanted to do "practice photodissociation" on a molecule whose photodissociaton was well understood, and we wasted like four person-years on that effort.
Also it took us too long to figure out we really could not do survey spectroscopy
in our ion tap. We needed a neutral beam line, and we simply had to bite thebullet and learn to do our own survey spectroscopy. Enter Jun Ye!!!"Though the group did not publish a lot at this time, a metric tonne of work got done.
They figured out how to make and trap HfF+ , they did spectroscopy and saw their first of many
assignable rotation bands, they found excited states which coupled strongly to both 1 Σ+ and
3 ∆ , they measured the spectroscopic signature for ∆8 with no applied B field, and of course1they finally saw Ramsey fringes. A big deal for our experiment was when the team figuredout rotational-state-selective resonance-enhanced-multiphoton-dissociation (REMPD), which was
eventually published in 2014 [19]. This was so critical because the efficiency of the other readout
option, light induced fluoresence (LIF), was so low that it was clear that they would never get ahigh precision result with it. Eric said it was the “much better than expected REMPD” that madehim think “yes. We will by golly be able to beat Commins by a factor of ten. We may not do that
before ACME does, but we will eventually. This whole idea IS NOT CRAZY!!!!”
Eventually we published our first generation result [2], and we managed to almost tie
ACME’s world-leading result [20]. It was pretty amazing given we were counting about 5 ionson the side of a fringe at the time (our entire advantage was that we were measuring out to∼ 750ms compared to ACME’s ∼ 5ms). After that we tore down the first generation apparatus
8 Defined in Chapter 2

8and rebuilt everything while we re-invented almost every step of the experiment. We now count∼ 250 ions on the side of a fringe and we measure out to 1.5 − 2.5 seconds, allowing us to put a
new limit on the eEDM.
Looking back at all the theses and papers published over the last decade and a half it isremarkable what a labour of love our eEDM experiment is. I owe a huge debt of gratitude to all
the grad students, undergrads, post-docs, and PI’s who worked incredibly hard to made this notso-crazy idea actually come to life. I am exceedingly lucky to have walked into the experimentaround the time the first measurement was happening, and that I’ve been able to see the second
measurement through from beginning to end.

Figure 1.3: A comparison between fringes taken in 2013 (upper left) and much more recently, in2020 (remaining panels). Note the time axis goes to 25 ms in 2013, while it goes to 1.5 seconds in
2020.

9

1.3

A non-exhaustive list of other searches for the eEDM
So far I’ve only given a few examples of the larger eEDM landscape, but these measure-ments have been happening for 70 years so it’s a relatively mature field. Purcell and Ramsey were
the first to propose such a search in 1950 [21]. The first limits were set via scattering [22], Lamb
shift [23], and electron g-factor measurements in the late 1950’s and early 1960’s [24]. Limits werethen set with paramagnetic atoms such as Rb, Cs, and Tl (see, for example, [25],[26],[27]) in the
1960’s through 1970’s. To give you a sense of the progress, the 1964 Cs result [26] constrained|de | < 2 × 10−21 e·cm, while the measurement detailed in this thesis constrains |de | < 4.4 × 10−30
e·cm.
In the early 1960’s measurements were done in diamagnetic 129 Xe [28] and then later inthe late 1980’s with Hg [29]. From the late 1980’s through the early 2000’s Tl was consistently
setting the most stringent limits ([30][31][16]) - by 2002 the limit was |de | < 1.6 × 10−27 e·cm.
In the last 20 years, molecular experiments took over: since the first limit was set using YbF by
the Hinds group in 2011 [18] (|de | < 1.05 × 10−27 e·cm), all subsequent limits have been set usingparamagnetic polar molecules. In 2014 the ACME collaboration used a beam of ThO [20] toconstrain |de | < 8.7 × 10−29 e·cm, and in 2017 our group published a similar limit using trapped
HfF+ ions [2] (|de | < 1.3 × 10−28 e·cm). In 2018 ACME improved the limit by roughly an order ofmagnitude, again using a beam of ThO [32] (|de | < 1.1 × 10−29 e·cm). As mentioned above, themeasurement in this thesis constrains |de | < 4.4 × 10−30 e·cm.
At the time of this writing, we are still the only eEDM experiment which actually traps
molecules. There are a number of groups who are on the verge of trapping EDM-friendly neutrals, whether they have Ω-doubling structure [33] or not (which does not preclude sensitivity toto BSM physics but requires much larger laboratory fields, often results in incomplete polarization, and offers no internal co-magnetometry) [34][35][36][37], a technique has been suggested
for using already-trappable polar molecules [38], and some folks are even working on solid state

10
eEDM searches [39]. I even heard a rumor at DAMOP that another group is going to start using
rotating fields [40].

Figure 1.4: A cartoon depicting the current status of eEDM searches. The pink region indicateswhere most new physics theories predict the eEDM should be, and the blue arrows indicate
recent exclusions (at the 90% confidence level). The exclusions measured zero up to the uncertainty indicated, which means that they exclude eEDMs larger than their uncertainty. The brown
region indicates the standard model prediction for the eEDM. One can see that the region we areprobing is effectively ‘background free’ in terms of the standard model prediction, and that most
new physics theories predict EDMs right in the range we can measure - so our measurements
directly constrain those theories.

1.4

Overview of this thesis
This thesis is focused entirely on JILA’s eEDM search, and almost completely on the second

generation of the search, referred to as "Generation 2" or "Gen 2". Chapter 2 details how one
actually goes about measuring an eEDM and the specific details of our experiment, including
details about the data analysis and the mathematical and algorithmic specifics we use to generate
everything from Ramsey fringes (from raw ion counts) to linear combinations of data streams to
get "frequency channels". Chapter 3 is the systematics chapter, which is pretty self explanatory.
Chapter 4 contains the final results from the Generation 2 measurement. Chapter 5 is about thedark matter exclusion we did with Generation 1’s data. Chapter 6 contains my conclusions about
this work. Enjoy!

Chapter 2
How the experiment works

This chapter details how our experiment actually works, both theoretically and in practice.
We begin this chapter with a section providing a generic overview of how one might measure an
eEDM. Later sections will detail the theoretical foundations as well as the practical details of ourexperiment.

2.1

Measuring the electron’s electric dipole moment: the basic recipe
In the most general terms, the goal of this section is to answer the question: how do you

actually measure an electron EDM? We will cover the core concepts and basic ingredients here.

2.1.1

The basic recipe: put a dipole in a field - twice
Let’s begin with something we understand1 : a simple spin-flip experiment. Let’s say wehave a spin in a magnetic field, and we measure the energy splitting between the two states (spinaligned or spin anti-aligned with the field, see Figure 2.1, left). Now we can add in the EDM:
imagine it is pointing in the same direction as the spin, so the ket in Figure 2.1 represents both
the direction of the spin and the EDM. If we also add an electric field, then the energy levels will
get pushed apart for one relative orientation of the fields (see Figure 2.1, center). If we flip thedirection of the electric field, the levels get pushed together (see Figure 2.1, right). If we take the
1 hopefully

12difference between these two measurements, the magnetic term cancels out2 and we measure an
energy splitting directly proportional to the EDM. So you put a dipole in a field - twice.

Figure 2.1: Conceptual cartoon depicting the basic measurement principle. In each panel, theket indicates both the direction of the magnetic moment (spin) and the electric dipole moment(EDM). In the first panel, a simple spin flip measurement is performed. In the second panel,
an electric field is added, pushing the energy levels apart for that particular relative orientationof the fields. In the third panel the electric field is flipped, pushing the levels together. The
difference between the last two measurements cancels out the magnetic dipole contribution (tofirst order) leaving only the electric dipole contribution.

2.1.2

Key ingredient 1: sensitivity
To keep pushing the limit on the eEDM we must keep improving our sensitivity (i.e. re-ducing our measurement uncertainty). To do this we need to consider three primary things: theelectric field, the count rate, and the coherence/interrogation time. First, the larger the electricfield the larger the energy level splitting will be (see Figure 2.2, left). Second, a large count rate
will allow us to determine the centers of the peaks with more precision (see Figure 2.2, center).
Third, a long coherence will make the lines more narrow (see Figure 2.2, right). All together,our sensitivity improves as the product of the effective electric field, the coherence time, and thesquare root of the counts. To optimize our sensitivity (minimize our uncertainty) we want tomake all three parameters as large as possible.

δde ∼2 to first order1√| Eeff |τ N

(2.1)

13

Figure 2.2: Conceptual cartoon depicting how three key parameters affect our sensitivity. In thefirst panel, the separation between peaks scales linearly with the effective electric field, Eeff . In
the second panel, having√ a higher count rate N reduces the fractional error in the determinationof the peak centers by 1/N. In the third panel, the width of the peaks scales like the inverse
coherence time, or 1/τ

To maximize the effective electric field we use paramagnetic polar molecules: the unpaired
electron spin experiences a massively enhanced electric field inside the polarized molecule, requiring only modest laboratory fields to produce [41]. For example, in our experiment we apply
a field of ∼ 50 V/cm, which results in an effective electric field (experienced by the unpaired
electron) of ∼ 20 GV/cm. Of course, this comes with a catch: molecules have a much more
complex energy structure compared to atoms. This can be both a blessing and a curse: it’s ablessing because we are able to exploit certain molecular states to improve our measurement, but
it’s a curse because it makes the vast majority of molecules that would be sensitive to the eEDM
very difficult to trap and cool. Molecules being generically difficult to trap and cool3 means that
until recently all the leading eEDM measurements were done with molecular beams, which areconstrained to ∼ms coherence/interrogation times.

Figure 2.3: A cartoon depicting a polarized HfF+ molecule

To maximize the coherence time one needs to be able to trap the molecules - with a catch. To
3 Of course, there are molecules amenable to trapping and cooling, but so far none of those candidates appear

sensitive to the eEDM.

14
perform an eEDM experiment, you also need to orient the molecules in an external electric field.
Doing both at the same time is the difficult bit. Lots of groups are working on the technology totrap eEDM-sensitive neutral molecules, and once these are trapped orienting them is (nominally)quite easy. Trapping eEDM-friendly neutrals is not easy, however - being sensitive to the eEDM
usually means an energy level structure which is not very amenable to cooling and trapping.
The other option is to work with charged species, which are very easy to trap. But until our
experiment came along nobody could figure out how to orient trapped ions: any applicationof a DC electric field would push the ions right out of the trap. Eric’s insight was to apply arotating electric field in the x − y plane, Elab = Erot (cos(ωrot t) x̂ + sin(ωrot t)ŷ) which the ions canadiabatically follow while staying trapped. With the rotating field trick, we enjoy coherence times
three orders of magnitude larger than beam experiments [2] [32]. It’s a really brilliant idea, andwe are still the only people in the world using rotating fields to orient trapped ions.
Finally, to maximize the count rate you need to trap/cool as many molecules as possible,and detect them with maximum efficiency. This is where ion traps underperform compared tobeam experiments: we can only trap so many ions at a time due to coulomb repulsion, and theexperimental rep rate is limited by the interrogation time. In our experiment we count about 250
ions on the side of a Ramsey fringe (roughly per second), which seems really pathetic comparedto the ∼ 106 molecules per second counted in a beam experiment. But these numbers go undera square root, while the 3 orders of magnitude improvement we have in interrogation timecompared to beam experiments does not. This is one of the ways we stay competitive.

2.1.3

Key ingredient 2: bias magnetic and electric fields
To perform the spin-flip experiment described above we need to apply a polarizing electricfield as well as a bias magnetic field. The two fields need to be aligned then anti-aligned to get
the two core measurements we need. The previous section described our polarizing electric field.
The rotating field has two primary effects: first, it drives the ions in uniform circular motion,

15
eErot
where their orbits have radius r = mω(r ∼ 0.5 mm for our typical operating parameters).
2
rot

Second, the ions are accelerated into a rotating frame relative to the lab, which introduces a
coupling between near-degenerate m F levels, which we denote as ∆ for m F = ±3/2. ∆ D denotes
the difference in coupling between Stark doublets due to differing proximity to the F = 1/2
manifold.
The rotating electric field and resultant circular orbits of the molecules leaves us with one
more hurdle: how can we apply a bias magnetic field aligned (or anti-aligned) with a rotating
quantization axis?4 The answer is to use a quadrupole magnetic field. The circular orbits (in therotating frame) see a static magnetic field which is aligned or anti-aligned with the quantization
axis5 (see Figure 2.4). The primary component of the spin precession frequency that we measureis due to this so-called rotating magnetic field.

Figure 2.4: A cartoon depicting how a radial quadrupole magnetic field amounts to a static,quantization-axis aligned magnetic field in the rotating frame. In each panel, the magnetic field
gradient is indicated with light blue arrows. A circular orbit for a molecule is indicated near the
top, and pink arrows depict the vector decomposition of the magnetic field which the ions see
as they are rotating. You can see that as the molecule rotates, the vector decomposition of themagnetic field has two components: one which is always pointing “up", and one which alwaysaligns (or anti-aligns) with the molecular axis.
4 Why have a magnetic field at all? We want to bias away from an avoided crossing between the two energy levels

we measure, and to control the spin precession frequency.
5 Technically, the quantization axis is the vector sum of the electric and magnetic fields, but the Zeeman interaction

is so much smaller than the Stark interaction in our chosen states that we can, to an excellent approximation, neglect
the contribution of the magnetic field to the direction of the quantization axis.

16
2.1.4

Key ingredient 3: Ω doubling (aka clever use of molecular energy structure)
We have almost everything queued up for a great experiment. The main remaining problem

is that the shift we want to measure is either zero (up to our sensitivity) or very small, fractionally,
compared to the primary Zeeman-induced precession frequency we will measure. To measuresomething that is fractionally tiny without polluting the result is difficult. One must take care
to identify and, whenever possible, eliminate systematic shifts which might mimic the effect of areal eEDM. This is where Ω-doubling comes in! I’ll provide a brief review of Ω-doubling here aswell as a short discussion on why it is so helpful for systematic rejection. If you would like moredetails I’d recommend reading [42] and references therein, as well as [43]. The idea to exploitΩ-doubling to reduce systematics comes from Dave DeMille [44].
Ω-doubling is the name for a type of energetic substructure present in some molecules.
Ω-doublet substructure generally occurs in molecular states with internal electronic angular momentum J ≥ 1, and often in molecules with one heavy nucleus, two valence electrons in a triplet
state, and at least one valence electron in a σ or π1/2 orbital [42]. For the rest of this discussion
we will discuss the J = 1 manifold of the HfF+ 3 ∆1 state. HfF+ ’s 3 ∆1 state is well described by
Hund’s case a) basis states with coupled nuclear spin:|Λ = ±2, S = 1, Σ = ∓1, J, Ω = ±1, I = 1/2, F, MF ⟩ ,(2.2)

where Λ = L · n is the projection of the electronic orbital angular momentum L on theinternuclear axis n, S = |S| is the total electronic spin angular momentum, Σ = S · n is theelectron spin projection on the internuclear axis, J = |J| = |L + S + R| is the electronic plus
rotational angular momentum, Ω = Λ + Σ = J · n is the projection of the electronic angularmomentum onto the internuclear axis, I = |I| is the 19 F nuclear spin, F = |F| = |J + I| is thetotal angular momentum of the molecule, and m F = F · ẑ is the projection of F on the rotating
quantization axis [1] [2]. We take the internuclear axis n to be directed from the 19 F nucleus to

17
the 180 Hf nucleus [2]. The above might be easier to digest in pictoral form; if so see Figure 2.5.

Figure 2.5: A cartoon depicting the vectors for Hund’s case a. The electronic spin and orbital
angular momenta, S and L, are strongly coupled to the internuclear axis by spin-orbit and electrostatic interactions respectively. Hund’s case a arises when spin-orbit interactions are much
weaker than the electrostatic interactions which split different electronic states. This means thatwidely separated electronic levels are split into a set of spin-orbit manifolds, each of which contains many closely spaced rotational levels. Λ = L · n is the projection of the electronic orbital
angular momentum L on the internuclear axis n, S = |S| is the total electronic spin angular momentum, Σ = S · n is the electron spin projection on the internuclear axis, J = |J| = |L + S + R|
is the electronic plus rotational angular momentum, Ω = Λ + Σ = J · n is the projection of
the electronic angular momentum onto the internuclear axis, I = |I| is the 19 F nuclear spin,
F = |F| = |J + I| is the total angular momentum of the molecule, and m F = F · ẑ is the projectionof F on the rotating quantization axis [1] [2]. We take the internuclear axis n to be directed fromthe 19 F nucleus to the 180 Hf nucleus [2].

In the 3 ∆1 J = 1 state of HfF+ we have two hyperfine states with total angular momentum
F = 3/2 and F = 1/2. Each of these has a pair of manifolds made up of the two allowed
projections m F = F · ẑ onto the quantization axis. The two sets of states correspond to the√eigenstates of the parity operator: |e⟩ = (|Ω = +1⟩ − |Ω = −1⟩)/ 2 and | f ⟩ = (|Ω = +1⟩ +√|Ω = −1⟩)/ 2. The degeneracy of these two manifolds is lifted by the Ω-doublet splitting,which we call ωef (see Figure 2.6).

18

Figure 2.6: In HfF+ ’s 3 ∆1 J = 1 state we have two hyperfine states with total angular momentum
F = 3/2 and F = 1/2. Each of these has two degenerate manifolds made up of the different
allowed projections m F = F · ẑ onto the quantization axis. The two sets of degenerate√ statescorrespond to the eigenstates oftheparityoperator:
|
e
⟩
=
(|
Ω
=
+
1
⟩
−
|
Ω
=
−
1
⟩)
/
2 and
√
| f ⟩ = (|Ω = +1⟩ + |Ω = −1⟩)/ 2. These degenerate manifolds are separated by the Ω-doublet
splitting, which we call ωef

The molecule has a molecule frame dipole moment6 , and in the presence of an electricfield states of good parity are mixed to become states of good orientation. The orientation states
are either aligned or anti-aligned with the field, and we refer to these states as Stark doublets, or
doublets for short. We refer to the aligned orientation as the “lower doublet" and the anti-aligned
orientation as the “upper doublet" (see Figure 2.7, left). The addition of a magnetic field lifts thedegeneracy further, as the spin is either aligned or anti-aligned with the external field (see Figure
2.7, center). Finally the effect of the eEDM further shifts the levels, as the eEDM (which we
assume is aligned with the spin) is either aligned or anti-aligned with the internal electric fieldof the molecule (indicated with ⃗Eeff , see Figure 2.7, right). A summary of all the shifts discussed
above is depicted in Figure 2.8.
The effect of the eEDM has the opposite sign in the upper vs lower doublet, so this allows
us to make a differential measurement between doublets to measure the energy shift due to theeEDM. A small complication is the magnetic g-factor is different in the upper vs lower doublet(indicated as guf and glF ) due to electric-field-induced mixing with the F = 1/2 hyperfine level
6 The molecule frame dipole moment points from negative to positive charge

19Figure 2.7: Focusing only on the F=3/2 manifold of the 3 ∆1 J = 1 state of HfF+ . Σ = S · n is theelectron spin projection on the internuclear axis, and the internal electric field experienced by thevalence electron is indicated as ⃗Eeff . Left: an electric field polarizes the molecule along or against
the field, which we call the “upper" and “lower" “doublets". The energy levels without the added
E field are indicated with dashes lines, and the energy levels with the added E field are indicatedwith solid lines. Center: adding a B field further lifts the degeneracy, when the spin is alignedwith the B field the level moves down, and when it is anti-aligned it moves up. Again, the energy
levels without the added B field are indicated with dashed lines and the energy levels with theadded B field are indicated with solid lines. Right: finally, the addition of the eEDM itself further
breaks the degeneracy, we assume the eEDM points in the same direction as the spin; when theeEDM is aligned with the internal electric field ⃗Eeff the energy level moves further down andvice versa. As before, the energy levels without the effect of the eEDM are indicated with dashedlines and the energy levels with the effect of the eEDM are indicated with solid lines. Nothing inthis cartoon is to scale.

and with the J = 2 states.
Now here’s the clever bit: this energy level structure allows for robust systematic rejection due to misaligned fields: a transverse magnetic field, to first order, will couple m F andm′F = m F ± 1. Because we measure only the energy difference between stretched states, this firstorder shift won’t affect our measurement. To second order, the shift will affect the difference between the stretched states, but this shift is common-mode between doublets7 - so taking a differential
measurement between doublets eliminates this (up to time-variations, which can be dealt withby measuring both doublets simultaneously, as we do) [42].
7 Up to the diffference in g-factor between doublets, see section 2.2.1

20

Figure 2.8: HfF+ ’s 3 ∆1 , J = 1, F = 3/2 state complete details. Σ = S · n is the electron spin
projection on the internuclear axis, and the internal electric field experienced by the valence
electron is indicated as ⃗Eeff . The energy levels without the effect of the eEDM are indicated withgrey lines, while the ones with the effect of the eEDM are indicated with black lines. Dashedlines are for visual aid only. The effect of the eEDM has the opposite sign in the upper vs lower
doublet. It is this energy difference that we aim to measure with our experiment.

2.2
2.2.1

Measuring the electron’s electric dipole moment: theoretical details
3 ∆ , v = 0, J = 1 Hamiltonian1

We perform ramsey spectroscopy on the stretched (m F = ±3/2) states in HfF+ ’s 3 ∆1 , J = 1,
F = 3/2 manifold. As mentioned in 2.1.4, the 3 ∆1 state is well-described by Hund’s case a) basis
states with coupled nuclear spin. In the 3 ∆1 , J = 1, F = 3/2 manifold |Σ| = 1, |Λ| = 2,
and |Ω| = 1, and, again as mentioned in 2.1.4, both hyperfine (due to the fluorine nucleus’ spin
I = 1/2) and Ω-doubling structure are present. The hyperfine splitting is −46.6(1) MHz between
the F = 1/2 and F = 3/2 manifolds, and the Ω-doubling splitting is 740(40) kHz between Ω = 1
and Ω = −1 states. Treating each Stark doublet as a two-level system, we include the Zeeman,frame-rotation, Ω-doubling, and eEDM terms as primary contributors to the energy shift between
the stretched states [2]. In the F = 3/2, m F = ±3/2 basis, the two-level perturbative Hamiltonian
has the form:

21
1 −3( g F ± δg F )µ B (Brot + δBnr + δBi ) ± 2de |Eeff | − 3αh f rot + . . .
u/l
Heff
= 
2
h(∆ ± ∆ D )h(∆ ± ∆ D )3( g F ± δg F )µ B (Brot + δBnr + δBi ) ∓ 2de |Eeff | + 3αh f rot + . . .
(2.3)

where the u/l superscript and upper or lower signs correspond to the upper or lower Stark
doublet, α ≡ E Z /Erot is the tilt of the electric field out of the XY plane (in other words, theaxial tilt), δB represents imperfect switching of the rotating magnetic field, and ∆ is a a rotationinduced mixing of the angular momentum projections in the laboratory frame8 with a Stark
doublet-odd contribution ∆ D (due to different levels of perturbation from the F = 1/2 manifold,the upper doublet being closer and therefore more perturbed). The subscripts on δB , nr and i,
indicate an axial magnetic field gradient δBnr that does not switch with our applied switchingaxial field gradient (which ultimately comes down to the limit of our abilty with the shim coil),and an actually-rotating-in-the-lab frame δBi due to the charging current i in our 8-electrodetrap. Here the magnetic g-factor gF results from the combined nuclear and electronic magnetic
moments, which are coupled by the nuclear hyperfine Hamiltonian. There is a difference inmagnetic g-factors between the upper and lower Stark doublets, denoted as δgF , which is due
to Stark-induced mixing of J = 1 with J = 2 and E -field induced mixing of adjacent magnetic
sublevels at second order in perturbation theory. δgF has the form

δgF
3d Erot≈ − mf
gF
20Be



gN µN
1−
3gF µ B2
3h̄2 ωrot+
dmf Erot A∥



2g N µ N
1−
3gF µ B≈ −1 × 10−3 .

(2.4)

For our experimental parameters the first term, which is linear in Erot , dominates. The
coupling ∆ first arises at fourth order in perturbation theory from the combined perturbationsof rotation and Ω-doubling, and breaks the degeneracy of the |m F | = 3/2 states in either Stark
doublet at Brot = 0 [43, 45]. ∆ and ∆ D are given by
8 I think of it is like this: if there is a projection that is initially aligned with the quantization axis, then the axis

rotates, the projection goes from being, eg., +z to some mixture of ± z. So, with our rotating bias field, adjacent mF
levels become mixed.


,

22

3h̄ωe fh∆ =2h̄ωrotdmf Erot3

2
18A2∥ − 19d2mf Erot2A2∥ − d2mf Erot!

3h̄ωe f, h∆ D =23h̄3 ωrot
2 A
d2mf Erot∥!

2
9A2∥ − 8d2mf Erot2A2∥ − d2mf Erot

!
.

(2.5)
These expressions are valid as long as dmf Erot ≫ h̄ωe f and dmf Erot ≫ h̄ωrot or, in otherwords, if the molecule is fully polarized. The strong scaling of ∆ with Erot allows us to perform
off-resonant π/2 pulses by modulating the magnitude of Erot .
The geometric frequency shift 3α f rot is due to a non-zero axial electric field, E Z . WhenE Z is nonzero, the total electric field tilts out of the x − y plane, and it sweeps out a solid angle
different from 2π. See [46] for a detailed discussion of this effect. 2de |Eeff | is the EDM term, wherede is the electron edm and Eeff is the effective electric field experienced by the valence electron.
u/lHigher order contributions to Heff
, denoted by ellipses in Eq. (2.3), are due to higher-order

combinations of perturbing Hamiltonian components, and primarily result in small corrections
to the terms already discussed [2].

2.2.2

Measurement of δgF /gF
We re-measured the differential g-factor ratio δgF /gF for improved accuracy compared to

the Generation 1 measurement. There are two different ways to measure δgF /gF , one can lookat the slope in f D as one varies f 0 (by, for example, varying the magnitude of Brot ), or the slope
B.
in f BD as one purposely applies a non-reversing axial magnetic field gradient, generating an f nr

We did both, see Figures 3.6 and 2.10.

2.2.3

Molecular structure
Put bluntly, HfF+ is a complex molecule, and entire PHDs are done calculating various

molecular parameters. Luckily for me, that is not what this thesis is about. For a nice detailed
discussion of the energetic structure of HfF+ , I’d recommend reading Chapter 2 of Will’s thesis

23

Figure 2.9: Measurement of δgF /gF by varying the mean fringe frequency f 0 , all data is taken at
Erot = 57.9V/cm.

Figure 2.10: Measurement of δgF /gF by adding a non-reversing axial magnetic field gradient,leading to large f B . Note that f BD is still blinded in this measurement.

[13]. This section is here simply to point you to the right place if you are looking for HfF+ details
:)

2.2.4

Experimental noise vs the quantum projection noise limit
We’ve already seen a heuristic expression for the sensitivity in section 2.1.2. Now let’s dig

into the details, and try to figure out how far above the quantum projection noise limit we are.

24
The problem with really deriving the correct expression is that we never really know how
many ions each experimental run starts with. In certain specialized experiments, the total numberof spin-1/2 particles maybe known precisely (for instance, in chains of trapped ions, or in certain
optical lattice experiments, or in situations where many photons can be scattered from eachparticle.) Often times, however, the total number of spins Ntot contained within a sample can
1/2
be known to at best only ± N̄tot. Where N̄tot is the average value for Ntot observed over many

samples. One can imagine this limit arises because we have somehow randomly down-selecteda small fraction much larger sample (say of neutral molecules) to get our sample spins (perhaps
they are ionized molecules). In any case, when one knows the total number of spins only to the
1/2?shot-noise? on Ntot , N̄tot, and when one then performs some coherent spin manipulation on

the sample, and then measures a projection, say, the number of up spins N↑ , one finds that thebest one can do is to measure N↑ to a one-sigma uncertainty of N̄↑1/2 where N̄↑1/2 is the average
number of up spins measured over may samples. If one succeeds in realizing this precision, one
is at the shot-noise limit. This situation can be contrasted to the situation in which one knows
Ntot precisely, in which case the corresponding precision on measuring up-spins can be somewhat
better:σ = ( N̄↑ (1 − N̄↓ /Ntot ))1/2 .

(2.6)

Note that in this “known Ntot ” limit, variance vanished as N̄↑ goes to N̄tot whereas in theother limit the variance on N↑ continues to be the naive shot-noise limit of N̄↑1/2 even as N̄↑ goesto N̄tot .
We are very much in the “imprecisely known Ntot ” limit, and worse yet, we actually haveconsiderable technical noise in the production of our molecules, so noise from sample-to-sample
1/2
on Ntot is actually often factor of 4 or 5 worse even than N̄tot. We use a differential technique,measuring the N↑ population simultaneously in both stark doublets, so as to cancel the technicalnoise and thus on the side of the fringe we can measure the difference between the spin up

25
populations in the two stark doublets to the precision you would anticipate if each individual
spin-up population were measured to within its shot noise limit [47].
I derived the expected shot noise limit in the case where we do know Ntot , you can find it
in Appendix D.
For now, let’s not worry about the stark doublet and think of a single fringe and see what
we anticipate for the error in our spin asymmetry if we were able to measure individual spin
projections at the shot noise limit. During our Ramsey experiment the spin precesses, so the initial√state (|↑⟩ + |↓⟩)/ 2 evolves such that at a later time t R after a second π/2 pulse the state vector
can be written as ∼ cos(ωt R ) |↑⟩ + sin(ωt R ) |↓⟩. So now the probabilities of measuring a givenbasis vector become sinusoidally time-varying: p↑ = |c↑ |2 = | cos(ωt R )|2 , which goes right into
the binomial distribution of equation D.1 and the corresponding variance (eq D.2):
σ2 (t R ) = Ntot | cos(ωt R )|2 (1 − | cos(ωt R )|2 ).

(2.7)

This is the quantum projection noise limited uncertainty for any measurement we will make ofan ensemble of independent ions. In other words, this is the minimum uncertainty we can expect
in any measurement of the number of ions in a given stretched state. How will this uncertainty
affect our total uncertainty?
To compute the quantum projection noise limited total uncertainty, we need to go froman uncertainty in a single spin asymmetry measurement to an uncertainty in the precession
frequency, then from that to a measurement of de . I’ll sketch out a heuristic argument using only
the sides of the fringe, I expect the rigorous argument is more complex than we need. Let’s step
through a perfect, quantum-projection-noise-limited version of our experiment and keep track of
the uncertainties:√
First, we prepare an ensemble of ions Ntot in a superposition state (|↑⟩ + |↓⟩)/ 2, let it
evolve, apply a second π/2 pulse, then measure the number of ions in the up state, N↑ . The

26
probability of measuring a given value of N↑ (on the side of a fringe, we expect N↑ = Ntot /2) isgiven by the binomial distribution with variance σ↑2/↓ = Ntot p↑ (1 − p↑ ) = Ntot /4, since p↑ = 1/2
on the side of a fringe.
Then we repeat the same process to measure N↓ and again σ↑2/↓ = Ntot /2 since we are on
the side of a fringe (p↑ = p↓ = 1/2).
Now we take these two measurements and form the fringe asymmetry: A = ( N↓ −N↑ )/( N↓ + N↑ ). Combining the errors we have a new uncertainty: the numerator and denominator have the same variance δA2n/d = Ntot /2, and the quotient of that has varianceδ A2 =N↓ − N↑ 2
Ntot /2 
Ntot /2+
2
N↓ + N↑( N↓ − N↑ )( N↓ + N↑ )2=

Ntot /2( N↓ − N↑ )2
Ntot /2+( N↓ + N↑ )2( N↓ + N↑ )4

=

1
2Ntot

(2.8)

where in the last line we used N↓ + N↑ = Ntot
Now we repeat this procedure r times, and we get r/2 asymmetry measurements. We
average these measurements to get a better estimate of the spin asymmetry, and the sampling
distribution of the mean9 is usually approximated by a normal distribution, so the quantumprojection-noise-limited uncertainty on an asymmetry measurement is δA2mean = δA2 /r:
δA2QPN =1
Ntot f rep Ttot

,

(2.9)

f rep is the rep rate of the experiment, and Ttot is the total time the experiment was running, andr = f rep Ttot /2.
On the side (zero-crossing) of a sinusoisally-varying asymmetry measurement, the slope(in units 1/s) is equal to ω 10 , which we approximate as a line with zero offset: y = ωt. If we are
9 For N
tot >> 1
10 The side of the fringe cos( ωt ) is at ωt = π/2, the derivative is d/dt cos( ωt ) = − ω sin( ωt ), plug in ωt = π/2

27
fitting a line to a point with errorbar σ′ (and σ′ << 1), then the line can be moved up or down σ′ ,so the fitted line can range from y = ωt to y = ωt ± σ′ . The horizontal distance (in this case, thetime shift) between the "true line" y = ωt and the "line inferred due to the errorbar" y = ωt ± σ′
is given by finding the point on the x-axis where y = ωt = σ′ , so t = σ′ /ω. In a cycle, the time
for a single degree is tdeg = 1/360 f = T/360, where f is the frequency and T is the period. The
time uncertainty due to our errorbar divided by the time for a single degree is the uncertainty
in degrees: σ′ /ω/1/360 f = σ′ /2π f /1/360 f = σ′ (remembering that the 2π is radians and 360
is degrees). So our phase error, in radians on the side of a fringe, is the same as the errorbar:
σ′ . Now if we say that our frequency measurement is given by (ϕL − ϕE )/2πt R , where ϕL is thelate-time phase, ϕE is the early-time phase and t R is the ramsey free evolution time, then the√ ′2 ′2√ ′(σ ) +(σ )2σfrequency uncertainty is δ f ==2πt R
2πt R . This is all for a fringe with perfect contrast, for√′2σa non-unity contrast C we can scale it δ f = C2πt
. Finally, to get from a frequency uncertainty toR

a de uncertainty, we divide by −2|Eeff |sgn(gF )/h [2], where |Eeff | is the effective field experienced
by the valence electron and sgn(gF ) is the sign on the g-factor. Thus,

√

δde =

2σ′ h
2|Eeff |C2πt R(2.10)

is the uncertainty in de for a single frequency measurement.

Finally, we will take the mean of all our frequency measurements, and the distributionof frequency measurements is again normal (we go to some effort to quantify this, see Chapter
4.) So our new uncertainty is just equation D.6 divided by the square root of the number ofmeasurements:

√

δde =

2σ′ h
p
.
2|Eeff |C2πt R f rep Ttot /2(2.11)

Let’s clean this up by taking h̄ = h/2π and cleaning up the factors of 2:
δde =

σ′ h̄
p

|Eeff |Ct R

f rep Ttot.

(2.12)

28
Now all that remains is plug in the quantum-projection noise limited errorbar on a singlepfrequency measurement, σ′ = 1/ Ntot f rep Ttot . Finally, we have the quantum-projection-noiselimited expression for our uncertainty in de :
δde =

h̄

√
.
|Eeff |Ct R f rep Ttot Ntot

(2.13)The expression above assumes that in each experimental cycle we read out an identicalnumber of ions, Ntot . In reality, N varies in every cycle due to fluctuations in the spatial modeand power of the lasers that ablate, ionize, and dissociate, as well as other experimental noise
sources. We avoid most of that noise by performing a differential measurement between theupper and lower doublets: in a single shot of the experiment we measure both the lower andupper Stark doublets, and we perform our measurements when the fringes for both are in phase,so the proportional experimental noise is effectively cancelled out. Details of this scheme are in[47] and [48].

2.3

Experimental apparatus

2.3.1

Overview
Our experiment is a spectroscopy-on-trapped-ions experiment, so we have the usual stuff:

lasers, a vacuum chamber, a source chamber, some oxford commas, and an ion trap. The generalaesthetic is ’home built spaceship’.

2.3.2

Source chamber
Our source chamber is where we make neutral HfF. It contains our piezo valve, which

opens to pulse SF6 into the chamber. Details of the disk-piezo11 valve can be found in Kevin
Cossel’s thesis [49]. The source is connected to the main chamber via a differential pumping
chamber which contains two skimmers (detailed in Huanquian’s thesis [50]).
11 Physik Instrumente P-286.23

29

Figure 2.11: CAD rendering of the entire apparatus, excluding lasers. In purple are the X shimcoils, blue are the Y shim coils, and yellow are the Z shim coils. The orange coils generate theaxial magnetic gradient that we call Brot

λ/2

λ/2

λ/2

m

F

de

pl
et
io
pu
n
m
pi
ng

n
tio
ia
oc
ss
di

MCP

HfF+
HfF

B

Z

Y

Erot
X
trap
electrodes

MCP
phosphor
CCD

Figure 2.12: Top down schematic of our experiment’s main chamber

30
2.3.3

Ion trap
Our 10-electrode linear Paul trap has 8 radial electrodes and 2 axial electrodes. The 8 radial

electrodes are driven in a quadrupole configuration to provide RF radial confinement (50 kHz),
while they are operated independently to generate a rotating electric field (375 kHz) or DC shims.
The 2 endcap electrodes provide DC axial confinement.
Our second-generation ion trap benefitted enormously from lessons learned in the first
generation of the experiment [2]. The 8 radial electrodes have a shape carefully optimized to provide a drastically improved rotating field uniformity (over a several centimeter diameter area),increased trap depth, increased trap harmonicity, and highly suppressed electric field inhomogeneties. The result is a larger trapping volume which allows us to trap about 10 times moreions at the same or lower density. New electronics allowed us to generate about 2× larger Erot ,which directly suppresses collisional decoherence. The driving electronics were also improved tosuppress ground loops and high frequency noise, all of which can cause general ‘heating’. Our
trap is very unusual, in part for the rotating field but also because it is relatively big. To give youa sense of scale, you could put a coffee mug between the electrodes (see Figure 2.13). The detailsof the design of the Generation 2 ion trap and electronics can be found in Will Cairncross’ thesis[13].
The amplitudes of the confining, rotating, and shim fields vary throughout the experimental sequence, as pictured in Figures 2.17-2.16 for a 5 ms free evolution time Ramsey sequence.
To get from a voltage on the electrodes to a value of V/cm in the center of the trap, we
need to account for the geometry of the electrodes. For the radial electrodes: take the electrodevoltage and multiply by 0.1748 to get Erot or RF amplitudes in V/cm in the center of the trap,
and for the endcap (axial) electrodes multiply the amplitude in voltage by 0.0146.

31
axial
electrodeelectrical
feedthroughs
chamber

Y coil
X coilBrot coil
Z coilradialelectrodesZ
Y

Figure 2.13: CAD rendering of the main experimental chamber sliced in half for a view of theion trap. To give a sense of scale, you can fit a coffee cup between the electrodes.

2.3.4

Computer control
While almost every line of Matlab code got rewritten since Generation 1, and the LabView

state machine was largely reprogrammed, the structure of the computer control is still about thesame as detailed in Section 2.3 of Matt Grau’s thesis [51] and Section 4.4 of Will Cairncorss’ thesis
[13].

2.3.5

Applied magnetic fields
Our experiment has 4 pairs of magnetic field coils12 and no magnetic shielding. All coils

are on the outside of the main vacuum chamber, see Figure 3.1.
Three of the coil pairs are set up along our lab-frame X, Y, Z axes in Helmholtz configuration for either shimming out the earth’s magnetic field or for generating large transverse fields(for systematics studies). The magnetic field at the center of the trap is measured by an array
12 *Slaps top of experiment* this baby can fit so many field coils in it

32

Figure 2.14: Confining and orienting field amplitudes throughout the experimental sequence, in
Volts/cm, for a 5 ms Ramsey free-evolution time. The π/2 pulses are easily visible in the Erotpanel.

of fluxgate magnetometers13 bolted to the main experimental chamber. The magnetometers are
sensitive enough to detect whether the elevator across the hall is on the same floor as us or not.
The final coil pair, which we colloquially call “B axgrad coils” generate our so-called ‘rotating’ magnetic field. They are aligned with the lab-frame Z axis and produce the axial magnetic
13 From Bartington Instruments, custom-built Mag612U’s with a 500µT range and a ∼ 1kHz bandwidth

33

Figure 2.15: Confining and orienting field amplitudes throughout the experimental sequence, in
Volts, for a 5 ms Ramsey free-evolution time. The π/2 pulses are easily visible in the Erot panel.

gradient B = B ′ axgrad (2Z − X − Y ) (where, for example, |B ′ axgrad | ≈ 11.89 G/m for an applied current of 0.55A). In the rotating frame of the ions this appears as a magnetic bias field

Brot ≡ ⟨B ·E rot /Erot ⟩ ≃ B ′ axgrad rrot .
Physical coil parameters are detailed in Table 2.1.

34

Figure 2.16: The variation of the computed secular frequencies throughout the experimental
sequence for a 5 ms Ramsey free-evolution time.

2.3.6

Applied electric fields
As mentioned in 2.3.3, our 10-electrode ion trap has 8 radial electrodes and 2 axial elec-

trodes. We can apply the RF, Erot or DC shim/confining potentials to any electrode, but in ournormal operation we only apply a DC confining field to the endcap electrodes, and everythingelse is confined to the radial electrodes. In an idealized experiment, in the laboratory frame, wewould only have 3 applied fields:

35

Figure 2.17: From Erot , computed values of Brot and rrot throughout the experimental sequence
for a 5 ms Ramsey free-evolution time.

First, the radial RF confining fields:

E RF (R, t) =Vrf
cos (ωRF t)(X − Y ),R20

where ωRF ≡ 2π f rf = 50 kHz and R0 ∼ 2.4 cm is the effective radius the RF trap.

(2.14)

36
Table 2.1: Magnetic field coil parameters
Coil
X
Y
Z
Axgrad# turns
80
85
57
85

Radius [m]
0.217
0.198
0.171
0.135

Separation [m]
0.216
0.197
0.172
0.192

Second, the axial DC confining fie

E DC (R, t) =Vdc
(X + Y − 2Z )Z02

(2.15)

where Z0 ∼ 8.3 cm is the effective height of the RF trap.
And third, the radial rotating electric field:
E rot (R, t) = Erot X̂ cos (ωrot t) − R̃Ŷ sin (ωrot t) ,(2.16)

where ωrot ≡ 2π f rot and R̃ is the rotation switch sign.
In our actual experiment, we see higher harmonics of both time-varying fields, though it is
the higher harmonics of Erot which cause us trouble. We also see leakage of Erot and its higher
harmonics on to the endcap, which is itself problematic (we will discuss this in Chapter 3). The
second harmonic of Erot is of particular concern:

E 2h (R, t) = E2h!
X X̂ − YŶXŶ + Y X̂cos (2ωrot t + ϕ2h ) +sin (2ωrot t + ϕ2h ) ,R0
R0(2.17)

where R0 is the radius of the ion trap, and this is again in the laboratory frame [13]. These four
fields comprise the main electric fields we are concerned with in our experiment.
For the radial electrodes: take the electrode voltage and multiply by 0.1784 to get Erot or
RF amplitudes in V/cm in the center of the trap, and for the endcap (axial) electrodes multiply
the amplitude in voltage by 0.0146.

37
2.3.7

LasersBelow is a table detailing the laser beams used directly in our experiment. The generationof these beams takes a somewhat larger number of actual lasers. Complete details of all the lasers
used in our experiment can be found in Appendix A.
Table 2.2: Photons used in our experimental sequence
Name
Ablation
Photoionization 1
Photoionization 2
TransfermF Pumping
mF Depletion
Vibrational Cleanup
Dissociation 1
Dissociation 2

Transitionn/a2Ω = 3/2 ← X ∆3/2
Rydberg ← Ω = 3/2
P ( 1 ) 3 Π 0+ ← X 1 Σ +P ( 1 ) 3 Π 0− ← 3 ∆ 1
Q(1)3 Σ0−+ ←3 ∆1
P(1)3 Σ0−+ ←3 ∆1Ω = 2 ←3 ∆1
?←Ω=2

Power
10 mJ
30 µJ
1.3 mJ
600 mW
21 mW
550 mW
30 mW
1.6 mJ25 mJ

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

Polarization
linearlinearlinearlinearcircularcircularlinearcircularcircular

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

All the lasers used in our experiment are locked to wavemeters using a simple servo loop.
Note that for the first dissociation photon the wavelength is actually 368.4945nm or 368.493nm
depending on the direction of rotation, to adjust for doppler broadening.

Figure 2.18: Cartoon depicting the transitions used during our state preparation.

38
2.3.8

Chops, switch states, blocks, and superblocks
The core of this measurement is taking the difference between measured transition ener-

gies (Ramsey fringe frequencies) when the electric field flips direction relative to the electron
spin, as explained in Section 2.1.1. We also like to measure the difference and average of thefringe frequencies when we change other experimental parameters, like the Erot rotation direction. Ultimately, we measure the fringe frequency in 16 different experimental states, and weform linear combinations of those measurements to diagnose systematics and isolate our eEDM
signal, as explained in Section 2.4.4. In addition to that, we prepare one of 2 stretched states andread out one of 2 stretched states, and we chop between those possible preparation and readout
choices. Finally, we sometimes change a single parameter and take a set of measurements withthat new parameter. All of this is what we call switch states, blocks, chops, and superblocks. Let
me explain.
A chop is the fastest change we make to our experiment. Between single experimental runs,
we change which stretched state we prepare and which we readout. For example in one run
we might prepare m F = +3/2 and readout m F = −3/2, then in the next run we might preparem F = −3/2 and readout m F = −3/2. We do this by changing the circularly polarized pumping
and depletion lasers (see Section 2.3.7 and Section 2.3.10 ) to be σ+ or σ− in the ion’s frame. The
pumping and depletion lasers are circularly polarized by passing through λ/4 waveplates rightbefore they enter the vacuum chamber. These waveplates stay fixed (they are not rotated between
experimental chops). The pumping and depletion lasers are strobed at f rot on a 49% duty cycle,
and the relative direction of ⃗k and E⃗rot , plus the handedness in the lab frame, determines if thelight is σ+ or σ− . So, in practice, the chop consists strobing the lasers in phase or out of phase with

Erot . There are 4 possible combinations: in/in, in/out, out/in, and out/out.
A switch is the next fastest change we make to our experiment. When we take measurements we do them in one of 24 = 16 possible experimental states, which we call switch states.

39
We switch the sign of the magnetic field (in or out), the Erot rotation direction (again in or out),
the Stark doublet (upper or lower), and the side of the screen we read a given Stark doublet outon (left or right). Note, we measure both doublets simultaneously (so the Stark doublet switch
comes for ‘free’), but separately (on either side of the screen, so we still use it as a switch in theanalysis - see Section 2.4.4). We indicate which switch state we are in with the letters B̃ , R̃, D̃ , andĨ , i.e. B̃ = ±1, R̃ = ±1, D̃ = ±1, and Ĩ = ±1. Because the D̃ switch comes for free, the actual
number of switches we physically make is 23 = 8.
In practice, there’s a few extra details to keep track of here, but we don’t have specialnames for them. The dissociation transitions need to be between stretched states to ensure thedissociation products inherit the correct momentum and the doublets separate on the screen. So
if the chop is in/in or out/in (in both cases the readout state is “in”), then the dissociation beams,
which are circularly polarized, need to be fired when they are also ‘in phase’ with Erot - in otherwords, if the depletion transition is σ+ then the dissociation transition also need to be σ+ , andvice-versa. In addition to this, the first dissociation photon will be red or blue shifted dependingon the rotation direction, so we shift its wavelength depending on which combination of R̃ andσ+/− we have.
A Block is what we call a set of Ramsey fringe measurements in all possible switch states again since the D̃ switch comes for free this means a block is comprised of 8 ramsey fringes.
Finally, a superblock is when we make a change to the apparatus then measure a set ofblocks. For example, the λ/4 waveplates are all fixed in our normal operation, but they caneach be rotated (and the corresponding logic changed to ensure we always dissociate stretchedstretched etc) and we can collect a set of blocks with them in their new position. The example
given should have no effect on our science signal, but we do things like that just in case we dosee something - in which case we investigate thoroughly!

40
2.3.9

Trap characterization

2.3.9.1

Secular frequencies

Because we change the RF and axial confinement throughout the experimental sequence,the secular frequencies are not constant. Their computed values for the entire sequence are shownin Figure 2.16, but those computed values do not account for any experimental perfections. We
can easily measure the secular frequencies by giving the ions a little kick to start them sloshing,then sending them on to the MCPs as a function of trapping time. Measuring the secular frequencies during the free-evolution time, we find that f X = 1 kHz, f Y = 1.3 kHz, and f Z = 1.7
kHz.

2.3.9.2

Ambient fB measurement

We collected data with Iaxgrad set to 1.08 A in both the in and out direction and shifted theions in z to see if we could get a handle on the ’ambient’ f B . The data is in X:\2021\210901\systematic1
and X:\2021\210831\systematic1 X:\2021\210831\systematic2 . We observed quadratic shifts
in both the f B and f BR channels. We explore this in more detail in Chapter 3.

Figure 2.19: Measurement of the ‘ambient’ f B

41

Figure 2.20: Measurement of the ‘ambient’ f B

2.3.9.3

Mean field

We have approximately 20,000 ions in our trap on any given day. Coulomb repulsion causes
mean field effects that we can observe. For example, in time of flight a thermal cloud should fit
a Gaussian profile, but our cloud clearly deviates from this (see Figure 2.21).

Figure 2.21: The time of flight trace of the HfF+ cloud with a Gaussian overlaid

Generally, there is always going to be tension when we try to define things like temperatureand size for an ion cloud with mean field effects. In particular, a lot of the following analysis
relies on assumptions like expansion during time of flight is ballistic, so I just want to put adisclaimer for the rest of this section that there are some serious self-consistency issues. The partwhich does not rely on assumptions is the breathe, which we did measure, and which does show

42
the presence of mean field effects (see the very end of this section).
We can chop the trace in Fig 2.21 into slices and integrate each slice to compute the HfF
number in each slice (see Fig 2.22).

Figure 2.22: Using our HfF/voltage calibration, we compute the HfF in each slice of the TOF
trace

The TOF data along with the Y-Z image data allows us to construct an approximate ellipsoid which contains the majority of the ions (see Fig 2.23).

Figure 2.23: Constructing the approximate HfF ellipsoid from our imaging and TOF data. In the
Y-Z imaging data, the lines demarcate the approximate edge of the cloud.

Now we can use the area in each slice in the X direction along with how many ions are ineach slice to determine the HfF density:
Using this, we can compute the density weighted density (8.9 for this dataset) and the43

Figure 2.24: Approximate HfF density along X, and approximate Mean Field Parameter for ourcloud over a given temperature range

approximate mean field parameter (see Fig 2.24).
If we estimate that our cloud is about 10 K then our mean field parameter is roughly 6.
This means that we are definitely not a gas but clearly not in the coulomb crystal regime. The
ion cloud behaviour should be somehwere between a pure gas and a liquid.
We can also give the could kicks and measure the breathe and slosh frequencies. For athermal cloud the ratio of the breathe over the slosh frequency should be 2. For our cloud wemeasure ∼ 1.414 .

2.3.10Experimental sequence

2.3.10.1

Neutral HfF production (∼ 3 ms)

We make neutral HfF by ablating a Hafnium rod in the presence of SF6 . First, a disk-piezo15
based valve16 opens for ∼65 µs, and Ar(99%)/SF6 (1%) gas at ∼ 120 psi is injected into the sourcechamber. The gas exits through a 1mm diameter nozzle, just downstream of which a Hf rod
is positioned (such that it is on the edge of the expanding beam). About 1.21 µs after the start
14 Analysis and plotting scripts are in X:\2020\200403\analyze_cloud_density .
15 Physik Instrumente P-286.23
16 See Kevin Cossel’s thesis for details of the valve assembly [49]

44of the gas pulse, an Nd:YAG pulsed laser (Abby, 532 nm, 10ns, ∼10 mJ pulses) ablates the Hfrod. The Hf molecules from the ablation are entrained in the gas pulse and through the magic ofchemistry, HfF molecules are produced.
The jet of HfF molecules (at this point both neutral and charged) is supersonically expanded, which cools the molecular beam to rotational temperature of ∼10 K. As the beam traverses a differentially pumped chamber on its way to the main chamber it is is collimated with
two skimmers: the first (3 mm diameter) is located about 11 mm from the source and is biased at
200 V to remove any ions which were produced in the ablation; and the second (2 mm diameter)
is about 9.6 mm further downstream and is not biased.

2.3.10.2

Photoionization (10ms)

When the neutral HfF molecules enter the trapping region, still entrained in the Argonbuffer gas, they are are excited via an optical-optical double resonance to a Rydberg state 54
cm−1 above the ionization threshold, and they autoionize (the two ionization photons come fromHarambe, dye amplifier, 309.388 nm with ∼ 30µJ per pulse, and Bertha, dye laser, 368.351 nm
with ∼ 2 mJ per pulse, both pumped by Mia, YAG 532 nm, 10 ns, 10 Hz, and Harambe is seededby Tuna, CW ECDL, 60 mW, 2 × 309.388nm) 17 . The ions inherit the 600 m/s center-of-mass
velocity of the neutral molecules, and ’kicks’ (short pulses) are applied to the electrodes to stop
the ion beam in the center of the trap. The trap turns on immediately after the kicks are applied,
and this RF quadrupole trap confines the ions. At the same time, the rotating electric field isturned on. We usually trap around 20, 000 ions.
The ionization process is state-selective: the Rydberg state is energetically forced to decay
to a single 1 Σ0+ , ν = 0 vibronic ground state of HfF+ . By choosing the correct parity of theintermediate state, as much as 50% of the ions end up in the 1 Σ0+ , m J = 0, J = 0 state. Complete
17 Harambe provides the 1st ionization photon and Bertha provides the 2nd ionization photon. Harambe is seededby Tuna but both are pumped by Mia.

45
details of our ionization procedure can be found in Huanquian Loh’s thesis [50].
We take some care when stopping and catching the ion cloud to minimize secular motion,
which will ultimately dephase and result in a hotter cloud. We do this by applying small DC
shims to the trap to shift the center of the RF trap to best overlap with the cloud (we call this‘minimizing the slosh’). After the trap is turned on and the ions are trapped, we slowly (over10 ms) ramp the shim values down to zero, which we call ’centering the trap’. I’ve quoted thetiming for photoionization as 10 ms because I have included the time taken to center the trap.
Note that the linewidth of the first ionization laser (Harambe) is narrow enough to resolve transitions of the various HfF isotopologues, allowing us to address transitions of only the
199 Hf19 F isotopologue18 [50]. Also note that ionization itself only takes tens of nanoseconds, but

the YAG which pumps both ionization lasers operates at 10 Hz, so our experimental rep rate islimited to at most 10 Hz.

2.3.10.3

State transfer, cleanup, and m F pumping (120 ms)

At this point we have trapped HfF+ in the ground electronic and vibrational state (1 Σ+ ,v = 0), spread over the lowest 4 rotational levels. We connect J = 0 → 3 using microwaves andperform incoherent transfer to our ‘science’ state: 3 ∆1 , v = 0, J = 1. We turn the CW transfer
laser (Toptica, CW ECDL, 961.434 nm, 600 mW) on as soon as the ions are trapped and leave
it on for 120 ms. The transfer laser is resonant with 3 Π0+ , J = 0 which decays preferentially to3∆ .
1

You might notice that this seems to violate the transition selection rule that the total spincannot change, but the 3 Π0+ state has an admixture of singlet and triplet character, allowing us
to transfer from an S = 0 state to an S = 1 state.
Using a resonant transition and its resultant decay means that we are left with ions in 3 ∆1
18 The isotopologue of a chemical species has at least one atom which is a different isotope (has a different numberof neutrons).

46
spread over several vibrational levels, primarily (until vibrational decay) in the lowest rotational
level (thanks to coming from J = 0 upstairs). This presents two problems: first, ions which arein the lowest vibrational and rotational level are still spread over the zeeman sublevels, and weneed to prepare a pure spin state. And second, ions in the upper vibrational levels will decay
throughout our spin precession experiment, polluting any pure spin state that we might preparein the lowest vibrational level.
We deal with the first issue by illuminating the cloud with low power CW lasers which
resonantly connect 3 ∆1 , v = 1, J = 1 → 3 Σ0−+ , v = 1, J = 0 or 3 ∆1 , v = 1, J = 2 → 3 Σ0−+ , v = 1, J = 1
or 3 ∆1 , v = 2, J = 1 → 3 Σ0−+ , v = 2, J = 0. The 3 Σ0−+ preferentially decays back to 1 Σ+ . These CW
lasers are linearly polarized and are brought through the top of the trap, and have about 10 mWof power. We only actually use one laser at the moment (3 ∆1 , v = 1, J = 1 → 3 Σ0−+ , v = 1, J = 0,
which we call vibrational cleanup, home-built ECDL, 10 mW, 818.3719 nm) as this seems tomaximize the contrast improvement we observe. We still have poor fringe contrast overall (only50%) from non science state ions getting dissociated, we are not sure what the root cause is.
I have been suspicious that the skimmer is no longer attached to the electrical lead and is nolonger actually biased, so our background might be due to random state ions created during theablation process being co-trapped with our state-prepared ions.
We perform our Ramsey spectroscopy between the m F = ±3/2 levels in either the upperor lower Stark doublet, so we must prepare a pure spin state in anticipation of preparing a
superposition. 40 ms after transfer is initiated we turn on a strobed, sigma polarized, CW laser(1083, DBR laser, 1082.4137 nm, strobed at f rot , 49% duty cycle, 21 mW) which is brought infrom the side of the trap and connects 3 ∆1 , v = 0, J = 1 → 3 Π0− , v = 0, J = 0. The strobing is
synchronous with Erot such that the wavevector is either parallel or anti-parallel to E rot , which
drives a σ+/− transition to an F ′ = 3/2 manifold upstairs, eventually leaving only one stretchedstate populated in 3 ∆1 . This pumping laser is on for 80 ms.

47
2.3.10.4

First depletion (7 ms)

At this point we have nominally prepared a pure spin state, but just to be sure we illuminate
the ion cloud with our CW depletion laser (814, CW ECDL, 814.5077nm, 540 mW at 50% duty
cycle, strobed at f rot such that it is parallel/antiparallel to E rot ) which drives a σ+/− transition to3 Σ− .
0+

Our depletion laser connects 3 ∆1 , v = 0, J = 1 → 3 Σ0−+ , v = 0, J = 0, which preferentiallydecays to 1 Σ+ by a 10:1 ratio, having weaker coupling to the 3 ∆1 state. This additional step
ensures that any ions in the wrong stretched state due to imperfect pumping are eliminated fromthe 3 ∆1 manifold. Figure 4.6 depicts the lasers and transitions used so far (excluding ionization).

2.3.10.5

π/2 pulses and free evolution (varying timescale)Now we have a pure spin state. We apply a π/2 pulse to create a coherent superposition of
the m F = +3/2 and m F = −3/2 states. We apply the π/2 pulse by temporarily ramping down
the magnitude of Erot , which increases a rotation-induced coupling ∆u/l between m F = ±3/2
states. The pure spin state will evolve into a superposition in ∼ 1 ms [43] [50]. We ramp Erotback up, and allow the superposition to evolve for a variable amount of time. After the desired
precession time has elapsed, we apply a second π/2 pulse to map the relative phase onto a
population difference between the m F = ±3/2 states.

2.3.10.6

Final state projection (second depletion, 25ms)

To project the ions into their final state, we again apply the depletion laser we used in
Section 2.3.10.4, which selects which final state we will read out (m F = ±3/2).

2.3.10.7

Dissociation and imaging

To detect and count the number of ions in our science state, we employ resonance-enhanced
multiphoton dissociation [19] using two UV pulsed lasers (Dissociation 1, also known as Cora,a pulsed dye laser, fired such that its k-vector is parallel/antiparallel to E rot , and Dissociation

48
2, also known as 266, a pulsed YAG, fired such that its k-vector is parallel/antiparallel to E rot ).
The dissociation is partially state-selective, it should only dissociate molecules which are in the
3 ∆ , v = 0, J
1

= 1, 2 manifold (but we do have a large background of dissociated ions, we do

not know where they are from. I suspect the skimmer biasing is broken). Because our ions areoriented in the trap, we dissociate them when they are parallel to our detector, and as long as weobey the angular momentum requirements detailed in Will Cairncross’ thesis [13] (importantly,that both dissociation photons drive σ+/− transitions), the Hafnium atoms from the different
doublets will fly to opposite sides of the phosphor screen. Chapter 6 of Will Cairncross’ thesis
covers our oriented photodissociation procedure in detail [13].
As mentioned in Section 2.3.8, if the chop is in/in or out/in (in both cases the readout stateis “in”), then the dissociation beams need to be fired when they are also ‘in phase’ with Erot in other words, if the depletion transition is σ+ then the dissociation transitions also need to be
σ+ , and vice-versa. In addition to this, the first dissociation photon will be red or blue shifted
depending on the rotation direction, so we shift its wavelength depending on which combinationof R̃ and σ+/− we have.

Figure 2.25: Some typical images showing how the two doublets appear on our phosphor screen:on the left, we depleted one doublet and imaged the one remaining. In the center we did thesame with the opposite doublet, and on the right we depleted neither (so we are imaging both
doublets at the same time). The numbers indicate pixels on the screen, the images are in false
colour, and the images are an average of 20 separate runs of each experiment. The red line
indicates our programmatic determination of the ’center’ of the two blobs.

We gate the phosphor screen such that we only image the Hf+ atoms, the un-dissociated
HfF+ ‘spectator’ ions (ions which were presumably in the wrong state) would otherwise ruin the49
image. We detect both the Hf+ and the HfF+ spectator ions in time of flight. The HfF+ signal
allows to diagnose experimental issues. We do not normalize the counted Hf+ by the integrated
HfF+ signal, we found it was less noisy to perform a differential measurement between the simultaneously detected orientation states [47]. The technical details of our imaging and counting
system are detailed in [48] and [47]. The photons used in our experimental sequence are summarized in Table 3.3 and Figure 4.6.
For diagnostic purposes (see Section 2.4.1), we can also use the depletion laser tuned to be
on resonance with one Stark doublet only (and sent in, linearly polarized, through the top of thetrap) to deplete one Stark doublet at a time, and to image this doublet alone.

2.3.10.8

Data collection and basic processing

Our experimental signal is dissociated Hf+ ions read out via phosphoresence on an imaging microchannel plate (MCP), as well as the time of flight scope trace from the MCP. The image
is processed asynchronously such that instead of storing the whole image, which would takean enormous amount of disk space, we save a file which contains the locations and brightness of each bright spot which was determined to be an ion according to our smoothing andnoise-reducing processing code. At the same time, we store the entire experimental state (whichmeans, in practice, the state of every object in our LabView state machine). These two items, theion counts and locations as well as the state of the experiment, form our base data.
Once we have this file, we use our determined ’center’ of the image to count how manyions are on each side of the screen, and our switch state to determine which side of the screen
corresponds to which doublet. We also know which stretch state we are reading out, so ultimately
we end up with a count of the m F = ±3/2 population in each doublet.

50

2.4

Data analysis

2.4.1

Hf counting and imaging
As mentioned in the previous section, our experimental signal is dissociated Hf+ ions readout via phosphoresence on an imaging microchannel plate (MCP), and we store a list of thelocations and brightness of each ion in the image, as well as the precise state of the experiment.
Before we take any Ramsey fringes, we take a set of images in each switch state to determine where the center line between the two blobs of ions should be for each switch state (in case,say, the rotation direction shifts the center of the image). In total we take 3 types of image: one
where we depleted the lower Stark doublet, one where we depleted the upper Stark doublet, andone where we deplete neither (see 2.25). We use these three images to determine the center line
between the two blobs for each switch state. Then we define a swatch, which is a rectangulararea centered on the center line whose ion counts are discarded. We do this because the doublets
are not entirely separated on the screen, so in this area we can’t be sure that we will assign thecorrect bright spots to the correct doublet.
Using Figures 2.26 and 2.27, we can get a heuristic understanding of the doublet/imaging
contrast. The doublet contrast essentially quantifies the fraction of ions from a given doublet
which land on the wrong side of the screen. In our experimental data, we see a blob for eachdoublet (like in Figure 2.26). If we integrate along one dimension, we will get two gaussian-likefunctions (Figure 2.27). If those traces are completely separated, then we can say the doublet
contrast is 1. Say the 1D function resulting from integrating in one dimension is called f L for the
lower doublet and f U for the upper doublet. Define the center of the blobs as yc (indicated by
the dashed black line in Figure 2.27). We can define the doublet contrast as
Ry
R 1200
2 yc f U (y)dy + 0 c f L (y)dy
DC = 1 −R 1200f U (y) + f L (y)dy0(2.18)The doublet contrast defined above does not account for any background, meaning ions51

Figure 2.26: An overlay of 20 images from our MCP assembly, where each doublet was fully
depleted 10 times. The center is indicated with a black dashed line.

Figure 2.27: The integrated trace of Figure 2.26, showing the upper and lower doublets separately.

who were dissociated but who weren’t from the science state. Our experimental data has arelatively high background (order 50 out of 350 ions in each doublet), so it’s important to consider
the effect of a background signal. The background ions arrive as a blob in the center of the image(we verify this by depleting all of 3 ∆1 before dissociating), and call its integrated trace b(y). In
that case,52

DC = 1 −2

R 1200ycRy R 1200f U (y)dy + 0 c f L (y)dy + 0
b(y)dy)R 1200f U (y) + f L (y)dy + b(y)dy0(2.19)

R 1200Let’s re-write this extremely cursed equation to make it more readable. Let T = 0 ( f U +R 1200
R 1200Ry
f L )dy, B = 0b(y)dy and N = yc f U (y)dy + 0 c f L (y)dy then our measured doublet contrast is
DCapparent = 1 −2N
2B−
T+B T+B(2.20)

So background ions reduce our apparent doublet contrast (for example, in the image
above). However, these background ions get effectively subtracted away in our difference asymmetry (see section 2.4.2), so our effective doublet contrast is just
DCeffective = 1 −2N.
T(2.21)

In either case, now that we have our center line and swatch for each switch state we canproperly count Hf+ ions in our images and assign them to the correct doublet and spin polarization. For every image (which corresponds to a single run of the experiment), we end up with aspin polarization assignment (in-in or in-anti etc), number of ions in the upper doublet, numberof ions in the lower doublet, and complete experimental state including the switch state.

2.4.2

Spin asymmetry
For every switch state, we will take data in both the in-in and anti-in (or the anti-anti andin-anti) state-prep and readout combinations. If the preparation and readout phase are the same(ie in-in and anti-anti), then the fringe formed as we measure different free evolution times will
have a 180 degree phase shift from the fringe formed when they are different (anti-in or in-anti).
You can reason this through easily if you just imagine what happens on the Bloch sphere atzero time: if you prepare ’in’ (say, m F = +3/2) and readout ’in’ with zero time evolution, you53should measure the maximum Nin = Nmax , while if you prepare ’out’ (in this example this is
now m F = −3/2) and readout ’in’ you should measure the minimum Nin = 0.
Now let’s call in-in and anti-anti “In" and in-anti and anti-in "Anti"19 . Now we can form
our asymmetry:

Au/l =

NIn − NAnti,
NIn + NAnti(2.22)

where the subscript u/l indicates the upper or lower doublet, respectively. As you can see, ittakes at least two separate experimental runs to form a single asymmetry measurement. However, thanks to our imaging technique, we get an asymmetry measurement in both doublets
simultaneously.
An typical ramsey fringe looks like Figure 2.28. The fringes from the upper and lower
doublets have slightly different frequencies, mostly due to the difference in g-factor between the
doublets.

Figure 2.28: Example Ramsey interferometry data

We can also form two ‘meta’ asymmetries by taking the difference or the sum of the upper
19 Sorry

54
and lower asymmetries:

A D = Au − Al(2.23)

AS = Au + Al
2.4.3

Fitting
As mentioned previously, we perform our Ramsey experiment simultaneously on both

doublets and use their oppositely-signed angular momentum at the time of dissociation to readthem out of opposing sides on the imaging MCP screen (see section 2.3.10.7). Unfortunately, thedoublets are not fully separated on the screen, so we must be careful with how we fit our data.
Below is a summary of our fitting routine.
For a simple Ramsey fringe with no leakage from the other doublet, we can define a decaying sine wave for the asymmetry:gu/l (t R ) = Cu/l e−γu/l tR cos(2π f u/l t R + ϕu/l ) + Ou/l

(2.24)

where C is the initial fringe contrast, γ is the contrast decay rate, f is the fringe frequency, t R is
the free evolution time, ϕ is the initial phase, O is the offset, and the u/l subscript indicates theupper or lower Stark doublet. In our fitting routine, we initially fit each fringe separately to thisfunction. From the fit parameters we define the mean and difference parameters as
αu + αl2αu − αl
,
αd =
2

αm =

(2.25)

where α represents C, γ, f , ϕ, or O. We can redefine our upper and lower fit parameters in termsof the mean and difference parameters:
αu = αm + αd(2.26)

αl = αm − αd .
Due to imperfect imaging/doublet contrast, in reality each doublet’s sine wave has a contribution
from the other doublet, which leads to beating. If we let DC be our doublet contrast, then our

55measured asymmetries will go like
1 + DC
1 − DC) gu + () gl
2
2
1 − DC
1 + DCAl = () gu + () gl ,
2
2

Au = ((2.27)

Now our sum and difference asymmetries are
A u + A l = gu + gl

(2.28)

Au − Al = DC ( gu − gl ),which we can express in terms of the mean and difference fitting parameters:
Au + Al = (Cm + Cd )e−(γu +γl )tR cos(2π ( f m + f d )t R + (ϕm + ϕd )) + (Om + Od )...

+ (Cm − Cd )e−(γu −γl )tR cos(2π ( f m − f d )t R + (ϕm − ϕd )) + (Om − Od )Au − Al = DC ((Cm + Cd )e−(γu +γl )t R

(2.29)

cos(2π ( f m + f d )t R + (ϕm + ϕd )) + (Om + Od )...

− (Cm − Cd )e−(γu −γl )tR cos(2π ( f m − f d )t R + (ϕm − ϕd )) − (Om − Od )),We use the two functions above to simultaneously fit the sum and difference asymmetries.
The fit uncertainty on f d and ϕd are smaller than the fit uncertainty on f m and ϕm thanks to ourdifferential data collection and fitting routine.

2.4.4

Frequency channels
When we take measurements we do them in one of 16 possible experimental states, where

we switch the sign of the magnetic field (in or out), the rotation direction (again in or out), theStark doublet (upper or lower), and the side of the screen/image we read a given Stark doublet
out on (left or right). Note, we measure both doublets simultaneously, but separately (on eitherside of the screen). We will indicate which switch state we are in with the letters B̃ , R̃, D̃ , and Ĩ ,i.e. B̃ = ±1, R̃ = ±1, D̃ = ±1, and Ĩ = ±1. With our switch state notation we can re-write our
two-level Hamiltonian20 :20 The Ĩ switch has no physical mechanism by which I can reasonably insert it into the hamiltonian, it’s more of asanity check than anything.

56


u/l
Heff
=

1 −3( g F + D̃ δg F )µ B (B̃Brot + δBnr + R̃δBi ) + D̃ 2de |Eeff | − 3αhR̃ f rot + . . .
2h(∆ + D̃ ∆ D )h(∆ + D̃ ∆ D )3( g F + D̃ δg F )µ B (B̃Brot + δBnr + R̃δBi ) − D̃ 2de |Eeff | + 3αhR̃ f rot + . . .

.

(2.30)

This two-level system has the familiar form




∆ 
 Eα
H=
,
∆ − Eαwhich has eigenvalues E = ±1/2
∆Eu/l =qp(2.31)∆2 + Eα2 . The full expression for the energy of this system is[3( gF + D̃ δgF )µ B (B̃Brot + δBnr + R̃δBi ) + 3αhR̃ f rot − D̃ 2de |Eeff |]2 + [h(∆ + D̃ ∆ D )]2 .
(2.32)

To simpify this, we can think of it likep

( A + a)2 + x2 , where A >> a and A >> x since de |Eeff |is (obviously) tiny and (when Erot is large enough) ∆ ± ∆ D is too. We expand our expression inp2
2
a Taylor series:( A + a)2 + x2 ∼ | A + a| + 2| Ax+a| ∼ | A + a| + 2x| A| . Dividing by h we get themeasured frequency:f =1h(∆ + D̃ ∆ D )2
|3( gF + D̃ δgF )µ B (B̃Brot + δBnr + R̃δBi ) + 3αhR̃ f rot − D̃ 2de |Eeff || +. (2.33)h
2|3gF µ B Brot |You need to be careful when taking the absolute value: 3gF µ B Brot is much bigger than all theother terms, so taking the absolute value flips B ’s sign onto the other terms (it can help to draw
this out on the number line to convince yourself of this). So after taking the absolute value one
is left with the theoretical measured frequency:
1(3| gF |µ B Brot + 3(| gF |µ B (δBnr − R̃δBi )B̃ − 3δgF µ B Brot D̃ − 3δgF µ B (δBnr + R̃δBi )B̃ D̃
hh(∆ + D̃ ∆ D )2− 3αh f rot B̃ R̃ + 2de Eeff B̃ D̃ +.
6gF µ B Brotf (B̃ , D̃ , R̃) =(2.34)

We take the measurement in all possible switch states and we form linear combinations ofthat set of measurements to create what we call frequency channels or parity states. One measurement in all of the switch states is called a block. For example, (ignoring the Ĩ switch for now), we

57
form linear combinations of frequency measurements f ( B̃, D̃, R̃) to form the frequency channels
f B̃ D̃ R̃ via the transformation



0
 f 
+



 B 

 f 
+






 fD 
+






 BD 

 f

1


+

= 
 R 
8
 f 
+






 f BR 
+






 DR 

f

+






f BDR
+



f +++



+ + + + + + +   −++ − + − + − + −  f







+−+
+ − − + + − −  f  −−+ − − + + − − +  f,  ++− + + + − − − −  f







−+−
− + − − + − +  f  +−− + − − − − + +  f− − + − + + −f −−−(2.35)

where ± signs represent ±1. Combining the errors on each fringe we end up with one standard
error for the D-odd channels and one for the D-even channels, given by
1
δf =
8

s

∑

(δ f B̃ D̃ R̃ )2 ,(2.36)

B̃, D̃, R̃=±

where δ f B̃ D̃ R̃ are the standard error estimates obtained from the fitting routine described in
Section 2.4.3. Note: given the matrix above you might expect the demoninator in equation 2.36
should be 4 since we average the D-odd channels separate from the D-even channels, but in thematrix above I ignored the Ĩ switch, so the correct number is 8.
We follow the same prescription to obtain parity states for other measured quantities likethe contrast, phase, decoherence rate, and fringe offset.
To understand why we do this, consider the leading order terms of some of the frequency channels.

f 0 ∼ 3| gF |µ B Brot /h is the average fringe frequency. On the other hand,B
f B ∼ 3| gF |µ B δB /h ∼ f 0 ( Bδrot) measures how accurately the magnetic field is reversed, you canthink of this as the average fringe frequency suppressed by the magnetic field chop. f D is thedifference in frequency between the two Stark doublets: f D ∼ −3δ| gF |µ B Brot /h, the leading termis the fringe frequency suppressed by the difference in g-factors between the two doublets. If the

58
g-factor was identical for each doublet and the rotation-induced coupling terms were identical,
the leading order f D term would vanish. And so on for all the other channels. Listing the leading
terms for all frequency channels we have:f 0 ∼ 3gF µ B Brot /h +h ( ∆ + ∆ D )2
6gF µ B Brotf B ∼ 3gF µ B δBnr /hf D ∼ −3δgF µ B Brot /h +h∆∆ D
3| gF |µ B Brot

f BD ∼ 2de Eeff /h − 3δ| gF |µ B /h(δBnr + δBi ) + . . .

(2.37)

fR ∼ 0f BR ∼ −3⟨α⟩ f rot + 3gF µ B δBi /h
f DR ∼ 0
f BDR ∼ 3δgF µ B δBi /h
Where again the subscripts on δB , nr and i , indicate an axial magnetic field gradient δBnr
that does not switch with our applied switching axial field gradient (which ultimately comes
down to the limit of our abilties with the shim coil), and an actually-rotating-in-the-lab frameδBi due to the charging current i in our 8-electrode trap. The frequency channel we are mostinterested in is the one containing the eEDM: f BD . f BD is the antisymmetric combination of
chopping both the doublet and the magnetic field. In general, forming frequency channels allows
us to diagnose experimental issues and get a handle on most systematics.

2.4.5

Blinding
We performed our measurment blind: the frequency channel which contains de , f DB , wasblinded by the computer before we got to see it. In practice, this meant that as soon as thefrequency channels were computed, a random number was added to the f DB value. The random
number was chosen once from a uniform distribution between 5 and 20 mHz, and it is stored
in a coded file which the computer opens, reads the blind, adds to f DB , then closes. When we

59
unblinded we subtracted the blind from the mean value of f BD .

2.4.6

Data cuts
Please see chapter 4 for details of our data cuts.

Chapter 3
Accuracy evaluation

This chapter was co-written by Luke Caldwell and myself.
Welcome to the systematics chapter, comrade. This chapter begins with a reproductionof a manuscript in preparation, titled ’Accuracy evaluation of the HfF+ electron electric dipole
moment experiment’. After the manuscript draft follows a section titled ’Not Systematics’ whichcontains everything that explicitly did not lead to systematic shifts in our measurement channel
but which contributed to our understanding of our system.

The following sections are a reproduction of a manuscript in preparation
3.1

Accuracy evaluation of the HfF+ electron electric dipole moment experiment
We have completed a new precision measurement of the electron’s electric dipole
moment using trapped HfF+ in rotating bias fields. We report on the accuracy
evaluation of this measurement, describing the mechanisms behind our systematic shifts. Our systematic uncertainty is reduced by a factor of 33 compared to
the first generation of this measurement [2].

3.2Introduction
Symmetries are fundamental in physics and exploring them has been vital to revealing

much of what we understand about nature today. In 1967 Sakharov showed that violation of

61
combined charge and parity (CP) symmetry is required to explain the observed baryon asymmetry of the universe - an imbalance between the amount of matter and anti-matter. CP symmetry
is not a good symmetry of nature but is only weakly violated by the Standard Model. Explaining
the baryon asymmetry requires new physics with additional sources of CP violation and manyextensions to the Standard Model have been proposed to do this.
Electric dipole moments of fundamental particles like the electron violate time-reversal (T)symmetry, equivalent to CP-violation assuming CPT invariance. The Standard Model predicts
an electron EDM (eEDM) which is well below current experimental sensitivity. However many
proposed extensions predict eEDMs which are several orders of magnitude larger, bringing its
observation within experimental reach. Measurements of the eEDM thus constitute sensitive
probes for physics beyond the Standard Model.
We recently completed the most precise measurement of the eEDM to date, constraining
its size at |de | < 4.4 × 10−30 e·cm (90% confidence). This paper explains the details of ourmeasurement. Section 4.2 describes our apparatus and experimental sequence, Section 3.4 theprotocol used for data analysis, and Section 3.5 details the effective Hamiltonian we use to model
the results. In sections 3.6–3.8 we report the measures we have taken to identify, characterize andmitigate sources of systematic error.

3.3

Experiment
Our experiment uses HfF+ ions, confined in an ion trap and prepared in the metastable3∆

1 state. In this state, the valence electrons are subject to a large intramolecular effective electricfield Eeff = 23 GV cm−1 , along the internuclear axis of the molecule. We orient this molecularaxis in the lab frame by applying an external electric field which rotates to maintain confinementof the ions. We then prepare the electron spin of the molecule in a coherent superposition ofstates parallel and antiparallel to Eeff , and use Ramsey spectroscopy to measure the precession
frequency. The electron EDM will give a contribution to this precession frequency proportional

62

Table 3.1: Constants used throughout this document. The total magnetic g-factor of J = 1, F =3/2 states gF ≡ ( G∥ + g N µ N /µ B )/3 results from the combination of nuclear and electronic Zeeman effects.
Constant
Be /h
A∥ /h
dmf /hωe f /(2π )gF
gN|Eeff |/hValue
8.983(1) GHz−62.0(2) MHz
1.79(1) MHz V−1 cm
0.74(4) MHz−0.0031(1)5.257 74(2)5.63 × 1024 Hz e−1 cm−1

Description
Rotational constant of science state
Hyperfine constant
Molecule-frame electric dipole momentΩ-doubling constantF = 3/2 state g-factor
Nuclear magnetic g-factor of 19 F
Effective electric field

Reference
[52]
[2]
[2]
[52]
[50]
[53]
[54]

Table 3.2: Example experimental parameters and associated derived parameters from our 2022
data.
Parameter

Value

Erot
ωrot
Brot

58 V cm−1
2π × 375 kHz
11.9 mG⃗0||B

23.5 G m−1

rrot

0.5 mm
−0.002 146(2)∼ 1 Hz∼ −0.6 Hz
23.5 V
4.1 V cm−1
50 kHz3.7 V
0.05 V cm−1
2π × 0.95 kHz
2π × 1.51 kHz
2π × 1.60 kHz

δg F
gF

∆
∆D
RF
RF
RF
Axial confinement
Axial confinement
ωx
ωy
ωz

Description
Magnitude of rotating electric field during free evolution
Rotation rate of Erot
Computed from a measurement of Baxgrad and calculation of rrot
Measured using the magnetometers, for an Iaxgrad ofroughly 1.08 A
Radius of HfF molecule circular motion
Stark doublet-odd magnetic g-factor ratio (see Figure 3.6)
Electric field-induced m F coupling
Doublet-odd correction to ∆RF radial confinement amplitude during free evolution
RF radial confinement amplitude during free evolution
RF radial confinement frequency during free evolution
DC axial confinement amplitude during free evolution
DC axial confinement amplitude during free evolution
X secular frequency during free evolution
Y secular frequency during free evolution
Z secular frequency during free evolution

63

Figure 3.1: CAD rendering of the entire apparatus, excluding lasers. On the left is the sourcechamber, where we produce neutral molecules. To the right of the source chamber is the differential pumping chamber, where the beam of neutral molecules is supersonically cooled to arotational temperature of roughly ∼ 10K and collimated by skimmers. Further right is the main
experimental chamber, which contains our 8-electrode Paul trap. Surrounding the main chamber
are the magnetic field coils: the X shim coils in purple, the Y shim coils in blue, and the Z shim
⃗ are indicated in orange.
coils in yellow; the coils which generate the quadrupole magnetic field B

λ/2

λ/2

λ/2

m

F

de

pl
et
io
pu
n
m
pi
ng

n
tio
ia
oc
ss
di

MCP

HfF+
HfF

B

Z

Y

Erot
X
trap
electrodes

MCP
phosphor
CCD

Figure 3.2: Top down schematic of our experiment’s main chamber. The neutral molecules fly
into the center of the trap where they are ionized and trapped. We apply the rotating electric⃗rot and the quadrupole magnetic field B⃗ . The ions undergo unifom circular motion,
bias field E⃗rot , and their unpaired spins precess about E⃗rot · B⃗.
either aligned or anti-aligned with E

64
Table 3.3: Photons used in our experimental sequence
Name
Ablation
Photoionization 1
Photoionization 2
Transferm F Pumping
m F Depletion
Vibrational Cleanup
Dissociation 1
Dissociation 2

Symbol

L961
trans
L1082
op
L814
depl
L818
vc

TransitionPowerλ (nm)

Polarization

Pulse Widthn/aΩ = 3/2 ← X 2 ∆3/2
Rydberg ← Ω = 3/2
P ( 1 ) 3 Π 0+ ← X 1 Σ +P ( 1 ) 3 Π 0− ← 3 ∆ 1
Q(1)3 Σ0−+ ←3 ∆1

10 mJ
30 µJ
1.3 mJ
600 mW
21 mW
550 mW

532
309.388367.732
961.43495
1082.4137
814.508

linearlinearlinearlinearcircularcircular

10 ns
10 ns
10 ns
CW
CW, strobed
CW, strobed

30 mW
1.6 mJ
25 mJ

818.37198
368.494266

linearcircularcircular

CW
10 ns
10 ns

P(1)3 Σ0−+ ←3 ∆1Ω = 2 ←3 ∆1
n/a ← Ω = 2

to de Eeff . To reject other unwanted contributions, we perform this measurement simultaneously
on two spatially overlapping clouds of ions with their molecular axes aligned and anti-alignedwith the externally applied field. The difference between the precession frequencies measured ineach case is our science signal, proportional to 2de Eeff .
This section describes the apparatus, and each of the steps used in state preparation andmeasurement of the ions. A summary of typical parameters used in the experiment is given in
Table 3.2.

3.3.1

Lasers
The experiment uses a total of 9 lasers1 ; 5 pulsed UV lasers used for ablation, ionisation and

818
1082
814
photo dissociation, and 4 CW lasers—which we denote L961
trans , Lvc , Lop , Ldepl —used for state-

preparation and readout. A summary is given in Table 3.3 and Fig. 4.6, and each is describedin detail in the following sections. All lasers are locked to wavemeters using simple servo loops
with a bandwidth of ∼ 1 Hz.
1 There’s actually more lasers than this; the UV lasers need pumping and seed lasers, and we calibrate our waveme-

ter using an additional laser locked to a Rb transition. Our actual experiment had 13 ± 2 lasers going at any given
time.

65
3.3.2

Molecular beam and ionisation
Our experiment begins with a pulsed beam of neutral molecules. We use a pulsed Nd:YAG

laser to ablate a solid Hf rod into a pulsed supersonic expansion of Ar and SF6 . Chemical
reactions between the Hf plasma and the SF6 produce neutral HfF which are entrained in the
supersonic expansion and rovibrationally cooled by collisions with the Ar to a temperature of∼ 10 K. When they arrive in our main chamber, ∼ 50 cm away, a pair of pulsed UV lasers at
309 nm and 368 nm excite a two-photon transition to a Rydberg state 54 cm−1 above the ionizationthreshold, from which they autoionize [55, 50]. The molecular ions are created in the first few
rotational levels of 1 Σ+ , the electronic and vibrational ground state of the molecule. The ions are
stopped at the center of our RF ion trap by pulsed voltages on the radial trap electrodes, after
which the confining potentials are immediately turned on. We typically trap ∼ 20, 000 HfF+ ionswith a lifetime of ∼ 10 seconds. The trap is described in detail in the next section.

3.3.3

Ion trap
Our linear Paul trap has 8 radial electrodes and 2 endcaps. The radial confinement isprovided by driving the radial electrodes in a quadrupole configuration producing a field,⃗RF (⃗r, t) = VRF cos (ωRF t)(⃗x − ⃗y),
E
R20

(3.1)

where ωRF = 2π × 50 kHz, VRF is the voltage applied on each electrodes, R0 ∼ 4.8cm is the effective radius the RF trap, and ⃗x, ⃗y are the radial position coordinates of the ions in the laboratory
frame. Axial confinement is provided by DC voltages VDC on a pair of endcaps, producing a field⃗DC (⃗r, t) = VDC (⃗x + ⃗y − 2⃗z)
E
Z02

(3.2)

where Z0 ∼ 16cm is the effective height of the RF trap. We choose the values of VRF and VDC
immediately after ionisation to best phase match with the size of the ion cloud, giving trap
frequencies of around 5 kHz in all directions. We then linearly ramp the trapping voltages downover 10 ms to expand and cool the ion cloud. The ramp takes the trap frequencies to ∼ 2.8 kHz

66
and ∼ 2.0 kHz in the radial and axial directions respectively.

⃗rot ,In addition to the RF confinement, we also apply a rotating electric field E⃗rot (t) = Erot x̂ cos (ωrot t) − R̃ŷ sin (ωrot t) ,
E

(3.3)

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

(3.4)

where rrot ∼ 0.5 mm. The shape of the radial electrodes are optimized to minimize inhomo-⃗rot across the ion cloud.
geneities in E

3.3.4

Magnetic fields
Measuring the electron EDM also requires orienting the electron spin of the molecules⃗ 0 . In order for the unpaired electrons to experience
which we do with an applied magnetic field B
a time-averaged interaction with the intramolecular effective electric field, this magnetic field⃗rot . We achieve this by using a pair of coils in anti-helmholtz configuration,must corotate with E
and aligned along the axial direction, giving⃗ 0 = B̃B2,0 (2⃗z − ⃗x − ⃗y),
B

(3.5)

where B2,0 is typically 230 mG cm−1 and B̃ = ±1 indicates the direction of the current in thecoils, explained in more detail in Sec. WHAT. In the rotating and co-moving frame of the ions,this quadrupole magnetic field appears as a time-averaged magnetic bias,⃗0 · E⃗rot ⟩ = B2,0 rrot .
Brot = ⟨B(3.6)

The coil pair is driven by a precision current source (Keysight B2902A). We refer to the pair of⃗ 0 -coils.
coils that produces this field as the B

67

Figure 3.3: Cartoon depicting the transitions used during our state preparation.

The apparatus also includes three pairs of coils setup along the lab frame x̂, ŷ, ẑ axes in
Helmholtz configuration for tuning the magnetic field at the position of the ions. The z coil

⃗ 0 -coils, the xis driven by the second channel of the precision current supply used for the B
and y coils are driven by a lower precision current supply (Keithley 2230). The magnetic fieldaround the periphery of the trap is measured by an array of 8 fluxgate magnetometers (Bartington
Instruments, custom-built Mag612U’s) bolted to the main experimental chamber. We use thesemeasurements to infer the magnetic field at the center of the trap. In contrast to other modern
eEDM experiments, the apparatus includes no magnetic shielding.

3.3.5

State preparation
Immediately after ionization, the HfF+ ions are in the ground electronic and vibrational

state (1 Σ+ (v = 0)), primarily distributed over the lowest 4 rotational levels J = 0–3. We connect
these rotational levels using microwaves and perform incoherent transfer to the eEDM-sensitive
3 ∆ ( v = 0, J = 1) state by using light from L961 to drive the 3 Π + ( J = 0) ← 1 Σ+ transition,
1
0
trans

the excited state of which decays preferentially to 3 ∆1 . This light enters the chamber along thez-axis and is on for 80 ms beginning immediately after ionisation. The decay from 3 Π0+ puts
population in several vibrational levels in 3 ∆1 , which can decay into the v = 0 science state if left
untreated. We remove the population in higher vibrational levels by illuminating the cloud with

68
3 −
3
L818
vc light which connects Σ0+ ( v = 1, J = 0) ← ∆1 ( v = 1, J = 1) at ∼ 818 nm, preferentiallydecaying back to 1 Σ+ . The L818
vc laser also enters the chamber along the z axis and remains on forthe duration of the experiment.

⃗rot in 5 ms. In this field, theAfter transferring the ions to the science state, we ramp on E
stretched states of 3 ∆1 ( J = 1, F = 3/2), can be considered states of good orientation, forming
two pairs of levels—which we call doublets—with their molecular dipole, and thus Eeff , eitheraligned or anti-aligned with Erot as shown in Fig. 4.1. Each doublet consists of one state withm F = 3/2 and one with m F = −3/2.
We polarize the molecules in the rotating frame by optically pumping them using L1082
oplight addressing the 3 Π0− (v = 0, J = 0) ← 3 ∆1 (v = 0, J = 1) at 1082 nm. The light is circularly⃗rot . We use an AOM to strobe L1082
polarized with its k-vector in the plane of Eop synchronously⃗rot on a 50% duty cycle such that it drives either σ+ or σ− transitions towith the rotation of E
an F ′ = 3/2 manifold in the excited state. This eventually leaves population only in either them F = 3/2 or m F = −3/2 states of 3 ∆1 (v = 0, J = 1). The preparation phase of the experiment refers⃗rot relative to the k-vector of the light when the light is on; in when E⃗rot isto the orientation of E
parallel, and anti when it is anti-parallel. This preparation phase can be changed by adjusting
the timing of the strobing cycle. L1082
op is turned on for 80 ms, starting 40 ms after trapping.
The final step of state preparation is applying L814
depl light at 814 nm. This laser is tuned toaddress the 3 Σ0−+ (v = 0, J = 0) ← 3 ∆1 (v = 0, J = 1), which preferentially decays to 1 Σ+ by a
10:1 ratio, having weaker coupling to the 3 ∆1 state. L814
depl is circularly polarized with the same
handedness and k-vector as L1082
op . It is again strobed so as to only address and remove any
814
residual population left over in other m F states after L1082
op is turned off. Ldepl light is on for 7 ms,

3 ms after L1082
op is turned off.
These steps leave the population in an incoherent mixture of the stretched states of the two
doublets. The key difference from our previous measurement [2] is that the experiment proceeds

69
on both doublets simultaneously. Our detection scheme [47], described in Sec. 3.3.7, allows us toread out each independently, allowing us to take advantage of common-mode noise cancellation.

3.3.6

Ramsey sequence
Immediately prior to the Ramsey sequence, we ramp the radial confinement of the ions

down further, to a trap frequency of ∼ 1 kHz. This reduces the density of the cloud and improves
the coherence time.
We apply a π/2 pulse to the ensemble of ions by temporarily ramping down the magnitude⃗rot from 58 V cm−1 to 7 V cm−1 in 16 µs, holding it there for 1 ms and then ramping back up inof E
a further 16 µs. Reducing Erot increases a rotation-induced coupling between m F = ±3/2 statesin a doublet (see Sec. 3.5), causing the pure spin states in each doublet to evolve into a coherent
superposition. We allow this superposition to evolve for a variable amount of time t R —up to
3 s—and then apply a second π/2 pulse to map the relative phase onto a population difference
between the two states in a doublet.

3.3.7

Measurement
We project the ions into their final state by applying L814
depl again to remove population fromone of the stretched states in each doublet. The readout phase is defined in the same way as the⃗rot is parallel with the k-vector of the light when it is on and anti forpreparation phase; in for E
antiparallel.
Finally, we detect and count the number of ions in the remaining stretched states via
resonance-enhanced multiphoton dissociation [19], driven by two pulsed UV lasers at 368 nm
and 266 nm. Immediately prior to the dissociation pulse, we ramp up both radial and axial confinement to compress the cloud and improve the dissociation efficiency. The dissociation pulse is⃗rot , is along Ĩ ŷ, in the plane of a microchannel plate (MCP) and phosphor screen
timed so that E
assembly. Here Ĩ = ±1 and ŷ defined by Fig. 4.2. Because the dissociation lasers enter at an

70

Figure 3.4: Ramsey spectroscopy in HfF+ . Top, level structure in the eEDM-sensitive 3 ∆1 , J = 1,
F = 3/2 manifold in external electric Erot and magnetic Brot bias fields (not to scale). The oriented
molecules are either aligned or anti-aligned with Erot , which we call the lower and upper Stark
doublets respectively. The component of the energy splitting due to the eEDM, 2de Eeff , flips sign
between doublets. Bottom, example Ramsey fringes from our dataset. The fringes from the two
doublets are collected simultaneously.

71
angle, there is considerable Doppler shift from the micromotion of the ions at 45◦ to the k-vectorof the light. To account for this we adjust the wavelength of the 368 nm light depending both on
the product R̃ Ĩ which gives the sign of the Doppler shift.
Each of the lasers is circularly polarized to drive transitions which preserve the orientationof the molecules during dissociation [13]. In this way the resultant Hf+ ions from each doublet
are ejected in opposite directions. The handedness of the dissociation lasers P̃ is determinedby a λ/2 waveplate which can be moved into or out of the beam path on a motorised mount.
Immediately after dissociation, we turn off the RF confinement and apply pulsed voltage on theradial electrodes to kick the ions towards the MCP. The Hf+ ions from each doublet are imaged
on opposite sides of the phosphor screen; which side each doublet is imaged on is set by thevalue of Ĩ. We time-gate the phosphor screen such that we only image the dissociated Hf+ andnot any remaining HfF+ ions. We detect both Hf+ and HfF+ ions in time of flight. Technicaldetails of our imaging and counting system are detailed in [48] and [47].
We typically detect ∼ 550 Hf+ ions on each side of the screen early time, and ∼ 120 aftera 3 s interrogation time. The later is principally limited by the finite lifetime of the 3 ∆1 state butwith a contribution from ions being heated out of our shallow trapping potential during t R .

3.3.8

Noise
Fluctuations in the power of the pulsed lasers used for ablation, ionization and photo-

dissociation means that the fluctuations in the number of Hf+ ions detected at the end of eachshot are ∼ 30%, roughly 3× the quantum projection noise limit for 120 ions. However thesesources of noise, and many others, are common mode; the exact same laser pulses address theions which end up in the upper and lower doublets. If we measure the ion number when the
Ramsey oscillations of the two doublets are in phase with one another (or close) then we can takeadvantage of excellent noise cancellation in the number difference which we use to extract ourscience signal (see Sec 3.4). The two doublets oscillate at slightly different frequencies owing to72
a small difference in their magnetic moments of about a part in 230 and so we deliberately takeour data at the ‘beat’, where the two doublets are in phase. We take our early time data when
when the two doublets are in phase2 and our late time data ∼ 230 oscillations later when theycome back into phase again. The time of the second beat can be controlled by varying oscillation⃗ 0 -coils. In our final dataset, the noise in the science signal is roughly 10%
frequency via the B
above shot noise.

3.3.9

Experimental protocol and switch states
In each shot of the experiment we can choose the preparation phase to be either in or⃗rot at the moment of dissociation, and P̃–theanti. For a given choice of Ĩ–the direction of E
handedness of the dissociation laser polarization, the readout phase is constrained by the needto drive stretched-to-stretched transitions which preserve molecule orientation. We label eachshot of the experiment with each of these phases. For example in-anti labels a shot where the⃗rot during state preparation but anti-parallelcleanup (and optical pumping) laser are parallel to E
during readout.
To record a Ramsey fringe, we repeat our measurement at different free evolution times.
For a given fringe, the phase of readout is kept fixed. At each Ramsey time, we take an evennumber of shots with each pair consisting of one shot with each phase of state preparation. Thisset of shots is called a point and a Ramsey fringe consists of 8 of these points taken at differentt R ; we take 4 points at early Ramsey time, and 4 points at late Ramsey time, each consisting of
two points on the sides of fringes and one point each on the top and bottom as shown in Fig. 4.1.
The points on the sides of the fringes consist of 20 shots, while the points on top and bottom
consist of 8 shots.
We record our data in ‘blocks’. Each block is constructed from a set of 23 = 8 fringes
2 Due to the finite length of the π/2 pulses, the doublets are already slightly out of phase at the earliest Ramsey

times accessible to us. Systematic effects associated with this imperfection are discussed in Sec 3.8.

73
recorded in each possible combination of 3 experimental switches. Each switch corresponds toan experimental parameter whose sign can be reversed: B̃ the direction of the current in the⃗ 0 -coils, R̃ the direction of rotation of E
⃗rot , and Ĩ the direction of E
⃗rot relative to the y axis at theBtime of dissociation, corresponding to which side of the phosphor screen each of the doublets are⃗rot is reversed atimaged onto. Note that in our implementation of the Ĩ switch, the direction of E
all points in time so that the opposite switch is prepared and read out. A fourth experimental
switch, P̃ the polarization of the dissociation light, is alternated every block.
We refer to each experimental configuration with { B̃, R̃, Ĩ, P̃} = ±1 as a switch state. In
each block, the first Ramsey time is recorded for all switch states before moving onto the second Ramsey time for each switch state etc. The order of the switch states at each point is{ B̃, R̃, Ĩ } ={1,1,1}, {-1,1,1}, {-1,1,-1}, {-1,-1,1}, {1,1,-1}, {-1,1,-1}, {1,-1,-1}, {-1,-1,-1}. Every other block,the order of switch states is reversed. In each switch state, we simultaneously collect data formolecules in each doublets, corresponding to orientation of the molecular axis with respect to theapplied electric field which we represent by another switch D̃ = ±1. The Ramsey times for eachswitch state are adjusted independently based on the data from the previous block to ensure thatthe 20 shot points are as close as possible to both the sides of the fringes, where our sensitivity ishighest, and to the beat, where our noise cancellation is best and where various systematic shifts
are minimized.
For the eEDM dataset, we collected 1370 blocks or ∼ 600 hours of data over a ∼ 2 month
period of April–June 2022. During the data run, we took data with 3 different values of the B2,0 ,corresponding to fringe frequencies of ∼ 75, 105 and 151 Hz. About halfway through the dataset,814
we changed the waveplates of L1082
op , Ldepl and the dissociation lasers to reverse the handednessof the light from each.

74
3.3.10

Images to determine doublet positions

To determine where each doublet falls on the phosphor screen in each switch state, we takea series of images with no Ramsey sequence. For these images we prepare the stretched states as
described in Sec. 3.8.1 and then remove the population in one of the doublets using low-power

L814
depl light, tuned to resonance with the doublet to be depleted, and propagating along the z
direction to avoid micromotion-induced Doppler shifts. We take 3 types of image per switch
state: one where we deplete the lower Stark doublet, one where we deplete the upper Stark
doublet, and one where we deplete neither. We use these three images to determine the centerline between the two blobs for each switch state as described in Section 3.4.1. We repeated thisimaging routine roughly every 20 blocks during the dataset.

3.4

Data Analysis

3.4.1

Ion counting and asymmetry
Our experimental signal is dissociated Hf+ ions read out via phosphorescence on an imag-

ing microchannel plate (MCP). The images are processed asynchronously and we save a filewhich contains the locations and brightness of each bright spot which was determined to be an
ion according to our smoothing and noise-reducing processing code.
We use this same code to analyze the test images described in Sec. 3.3.10 and find a centerline for each switch state. We use this center line when analyzing the Ramsey data to define aswatch which is a rectangular area, of fixed width, at the center of the image whose ions counts
are discarded. We do this because the doublets are not entirely separated on the screen, so inthis area we can’t be sure that we will assign ions to the correct doublet. The extent to which
we are able to isolate the two doublets is given by the imaging contrast C I . As the swatch width
increases, the imaging contrast improves but total ion number N decreases as we throw out moreions. For the final analysis of the dataset, we used a swatch of 90 pixels which roughly maximizes

75
CI

√

N.
Once we have our center line and swatch for each switch state we can properly count Hf+ions in our Ramsey data images and assign them to the correct doublet. For every image (whichcorresponds to a single run of the experiment), we end up with a number of ions in the upper
doublet N u , number of ions in the lower doublet N l , and complete experimental state including
the switch state, preparation phase and readout phase.
For every switch state, we take data in both the in-in and anti-in (or the anti-anti and inanti) combinations of preparation and readout phase. If the preparation and readout phase are
the same (i.e. in-in and anti-anti), then the fringe formed as we measure different free evolution
times will have a 180 degree phase shift from the fringe formed when they are different (anti-in
or in-anti). We will refer to in-in and anti-anti “In" and in-anti and anti-in “Anti". Now for eachpair of shots, we can form our spin asymmetry,

Au/l =u/l
u/lNIn− NAntiu/lu/lNIn+ NAnti,(3.7)

where the subscript refers to the preparation and readout phase combination. For each Ramsey
time and switch state we take n shots and so can form n/2 asymmetries for each of the upperand lower doublet. From these n/2 asymmetries we construct a mean Am
u/l and a standard error
δAu/l which we use for fitting.
We can also form two ‘meta’ asymmetries by taking the difference (D) and sum (S) of theupper and lower asymmetries,

A D = Au − Al ,(3.8)

A S = Au + Al ,mwith means Am
D , AS and standard errors δ A D , δ AS . The δ A D are reduced compared to δ AS due

to common-mode cancellation of many sources of noise.

76
3.4.2

Least squares fitting
As mentioned previously, we perform our Ramsey experiment simultaneously on both

doublets and use their opposing orientations at the time of dissociation to read them out of
opposing sides on the imaging MCP screen. Because the data are acquired simultaneously, thedifference asymmetry allows us to cancel much of the common-mode noise, leaving us withdoublet-odd data with less scatter than the raw data. Unfortunately, the doublets are not fully
separated on the screen, so we must be careful with how we fit our data.
For a simple Ramsey fringe with no leakage from the other doublet, we can define a decaying sine wave for the asymmetry,gu/l (t R ) = Cu/l e−γu/l tR cos(2π f u/l t R + ϕu/l ) + Ou/l .

(3.9)

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

αm =

(3.10)

with α ∈ {C, γ, f , ϕ, O}. Due to imperfect imaging contrast C I , in reality each doublet’s signal
has a contribution from the other doublet. In this case, the measured asymmetries are
1 − CI
1 + CI) gu + () gl
2
2
1 + CI
1 − CI
Al = () gu + () gl22

Au = ((3.11)

Now our sum and difference asymmetries are
A S = gu + gl
A D = C I ( gu − gl ) ,(3.12)

77
which we can express in terms of the mean and difference fitting parameters,
AS = (Cm + Cd )e−2γm tR cos(2π ( f m + f d )t R + (ϕm + ϕd )) + (Om + Od )...

+ (Cm − Cd )e−2γd tR cos(2π ( f m − f d )t R + (ϕm − ϕd )) + (Om − Od )A D = C I ((Cm + Cd )e−2γm t R(3.13)

cos(2π ( f m + f d )t R + (ϕm + ϕd )) + (Om + Od )...

− (Cm − Cd )e−2γd tR cos(2π ( f m − f d )t R + (ϕm − ϕd )) − (Om − Od )),We use these two expressions to perform a simultaneous least-squares fit for the sum and difference asymmetries in each experimental switch state. The value of C I , the imaging contrast, isfixed at 0.9 for this fit—determined as described in Sec. 3.8.2.1. The parameter uncertainties are
extracted based solely on the standard errors of the asymmetries used in the fits. The resultant
uncertainties on the fitted values of f d and ϕd are close to the shot noise limit and much smaller
than those on f m and ϕm thanks to our simultaneous data collection and fitting routine, whichcancels most of the common-mode noise. The outputs of these simultaneous fits are used for all
further analysis.

3.4.3

Switch parity channels
After fitting to each Ramsey fringe in a block to extract the 10 fitting parameters, we use theresulting 16 values of each parameter to form 16 linear combinations which we call switch-parity
channels. The switch-parity channel for the mean and difference parameters αm and αd which
are odd under the product of switches [S̃ a S̃b ...] are given by
αSa Sb ... =α

D S a Sb ...

1[S̃ a S̃b ...]αm ( B̃, R̃, Ĩ ),16 B̃,R̃,∑Ĩ =±1

1=[S̃ a S̃b ...]αd ( B̃, R̃, Ĩ ).
16 B̃,R̃,∑Ĩ =±1(3.14)

So for example, f BR is formed by adding together the f m measured in all switch states for which
the product B̃ R̃ is even, subtracting the f m measured in all switch states for which the product B̃ R̃
is odd and dividing by the total number of switch states. C DBRI is formed from adding together
the measured Cd in all switch states for which the product B̃ R̃ Ĩ is even, subtracting all switch

78
states for which the product B̃ R̃ Ĩ is odd and again, dividing by the number of switch states. The
science signal is f BD . All other parity channels allow us to to diagnose experimental issues andidentify sources of systematic error.

3.4.4

Blinding
We blinded the dataset by programming the fitting routine to add an unknown constantoffset to the f BD channel. This offset was stored in an encrypted file and was not removed until
all statistical and systematic checks on the dataset had been completed, and the uncertainties
finalised.

3.4.5

Data cuts
After completing the dataset, we applied cuts to the blinded data based on non-EDM

channels indicating signal quality. Blocks with late-time contrast below 0.2 were cut due to low
signal to noise. Figure 4.4 shows how the center value and error bar of the science channel varyas a function this cut. Blocks were also cut if they contained a fringe with a fitted difference
frequency f d in any switch state which was over 3.5σ different from the mean fringe frequency
for that switch state. The mean fringe frequencies were calculated over all blocks which had⃗ 0 and were constructed from the linear combinations, including the blinded
the same value of B
offset on f BD . In total, the two cuts removed 42 blocks leaving 1328 blocks in the final dataset.

3.5

Effective Hamiltonian for doublets
To an excellent approximation, we can model the evolution of each of the Stark doublets asa two-level system. The effective Hamiltonian for each pair can be parameterizedh  f0 ∆ 
Heff = .
2∆ − f0(3.15)The diagonal f 0 components contain all terms which directly shift the energies of the two states
relative to one another, whilst the off diagonal ∆ components contain all terms which mix the

79

Figure 3.5: Results as we shift the threshold for applying cuts to the data. In each plot, only one
cut is applied (the other cut is not applied). The cuts do not shift the resultant value of f BD .

80
two states. We can expand both f 0 and ∆ in terms of their leading contributions,f 0 = B̃ f 00 + δF0 ,(3.16)∆ = R̃∆0 + R̃ D̃∆ D + δD .

(3.17)

Here the quantities with tildes are equal to ±1 and are determined by the experimental switch
state, discussed in Sec. 3.3.9. The principle contribution to f 0 is the Zeeman splitting f 00 =3gF µ B Brot ∼ 100 Hz. The off-diagonal component is dominated by two terms with similar magnitude, but different switch state dependence; ∆0 and ∆ D ∼ 1 Hz arise at fourth order in perturbation theory in the full molecular Hamiltonian from the combined perturbations of rotation andΩ-doubling, breaking the degeneracy of the m F = ±3/2 states in either Stark doublet at Brot = 0[43, 45]. ∆ and ∆ D are given by3h̄ωe fh∆ =23h̄ωe f
h∆ =
2
D



h̄ωrotdmf Erot2
18A2∥ − 19d2mf Erot

3

!

2
A2∥ − d2mf Erot3h̄3 ωrot
2 A
d2mf Erot∥!

2
9A2∥ − 8d2mf Erot2A2∥ − d2mf Erot

(3.18)!
.

These expressions are valid as long as dmf Erot ≫ h̄ωe f and dmf Erot ≫ h̄ωrot or, in other words, if
the molecule is fully polarized. The strong scaling of ∆ with Erot allows us to perform off-resonant
π/2 pulses by modulating the magnitude of Erot as described in Sec. 3.3.6.
The additional perturbations are given by
δF0 = ∑ S̃δ f 0s ,(3.19)

δD = ∑ S̃δ∆s(3.20)S̃∈W

S̃∈W

where both summations are over W , the set of all possible products of { B̃, D̃, R̃, Ĩ }, and thesuperscript s on the δ f 0s and δ∆s denote the switch state dependence of the perturbation relative to the largest term in each matrix element, f 00 and ∆0 respectively. For our purposes, themost important perturbation is that due to the electron EDM which contributes a diagonal term,D̃δ f 0BD = 2 D̃de Eeff . Some of largest diagonal perturbations are listed in Table WHAT. Those
which are important for our determination of de are discussed in detail in Sec. 3.7.

81
For each experimental switch state and doublet, we measure a frequency f ( B̃, R̃, Ĩ, D̃ ) corresponding to the energy difference between the two eigenstates, which we define to be always
positive. For typical experimental parameters, f 00 is roughly two orders of magnitude larger thanany other term in the Hamiltonian, and so we can expand f about f 00 ,f ( B̃, R̃, Ĩ, D̃ ) =∆B̃ f 00 + δF0 +B̃ f 00
2

2− δ F00 2 + ∆ D 2 + 2 D̃∆0 ∆ D + 2 R̃ ( ∆0 + D̃∆ D ) δ D + δ D 2∆0 + ∆ D + 2 D̃∆0 ∆ D + 2 R̃(∆0 + D̃∆ D )δD + δD 2
f 00∆
= | f 00 | + B̃δF0 +2− B̃δF0

3.6

2

+ ...

0 2 + ∆ D 2 + 2 D̃∆0 ∆ D + 2 R̃ ( ∆0 + D̃∆ D ) δ D + δ D 2(3.21)| f 00 |2∆0 + ∆ D + 2 D̃∆0 ∆ D + 2 R̃(∆0 + D̃∆ D )δD + δD 2
+ ...
| f 00 |2

Accuracy evaluation
Accurate determination of the eEDM-induced energy shift in HfF+ requires a careful eval-uation of all possible systematic shifts. There are several methods one can employ to search
for and understand systematics. First, one can change experimental parameters in a controlled
way to look for shifts in the science channel. To see shifts with this method, one typically needs
to change the parameter in question by a factor of 100, which is not always possible. Section
3.7.1.1 is a good example of doing this. Second, one can “break” the data analysis. This can bevery precise since it can be performed on our entire dataset before unblinding, and it runs norisk of actually breaking the physical experiment. Section 3.8.2 contains several examples of thisapproach. Third, we can use other parity channels for insight into the eEDM channel as they
can at times magnify certain effects. Section 3.7.2.4 is a good example of this approach. Finally,
one can perform detailed analytical calculations or simulations to understand how experimental
parameters might affect our measurement. Section 3.7.1 is an example of this approach.
Our systematics themselves can be divided into 2 categories: shifts in the actual precession
frequency of the spin, and shifts in the phase at early or late free evolution time. Frequency shifts

82
are the dominant class of systematics in our experiment, and are covered in Section 3.7. Phase
shifts tend to be smaller as they are suppressed due to our long coherence times and our ability
to measure both the early-time and late-time phase (which is not possible in beam experiments).
Phase shifts are covered in Section 3.8.
We include effects as systematics if they meet one of three criteria: first, if we saw a nonzero correlation between the eEDM channel and some parameter; second, if we saw a signal wethought would be important to understand but could not understand; and third, if we saw a correlation with the non-eEDM channel and some parameter but we know of a mechanism for thatto leak into the eEDM channel. We set limits on systematics either by varying parameters overa large dynamic range and setting direct limits using the experiment, or by making conservative
estimates using simulations.

3.7

Frequency shifts

3.7.1

Magnetic effects
Many experiments to measure EDMs are plagued by the interaction of the atom or moleculewith stray magnetic fields. In our experiment, three features make us comparatively immune tothese effects. First, the 3 ∆1 state we use has an extremely small magnetic moment, roughly
300 times smaller than the magnetic moment of a bare electron. Second, our measurement is adifferential one taken simultaneously between two pairs of states with almost identical magnetic
moments—equal to roughly 1 part in 500. This allows us to measure magnetic effects by looking
at the f B channel whilst they cancel to a high degree in our measurement channel f BD . Finally,
the rotating quantisation axis in our experiment means that most effects considered in otherexperiments tend to average to zero over an integer number of rotation cycles. However, there
are still some important magnetic effects which can shift our measured value of f BD and we
consider those in this section.

83⃗ is well described by the effective
The interaction of the science states with a magnetic field B
Zeeman Hamiltonian,⃗,⃗0 ± µ⃗D ) · B
HZ = ( µ(3.22)⃗,= m F µ B ( gF ± δgF )Êrot · B⃗0 is the part ofwhere the ± corresponds to the upper and lower doublet respectively. Here µ
the magnetic moment which is common to both doublets and µ⃗D is differential part, typically
500 times smaller for our choice of experimental values. In the second line, we have madeexplicit the fact that both magnetic moments track the quantisation axis defined by the unit vector⃗ 0 + δ⃗B
⃗ = B̃B
pointing along Erot . We choose to write the magnetic field as a sum of two parts, B
where the first part is due to the idealized applied quadrupole magnetic field gradient whichreverses perfectly with the B̃ switch, and the second part δ⃗B represents any additional magnetic
field experienced by the molecules. In general, δ⃗B can be composed of many components withdifferent dependence on the switch state and can be writtenδ⃗B = ∑ S̃δ⃗B s ,(3.23)

S̃∈W

where the summation is over W , the set of all possible products of { B̃, D̃, R̃, Ĩ }, and the super-⃗ 0 . For example, the largest
script s denotes the switch state dependence of the field relative to B
magnetic shifts measured in the experiment are caused by charging currents induced by the oscillating voltages on the radial electrodes. The charging currents produce a uniform magnetic⃗rot and whose sign depends on the rotation direction,
field which rotates with E
R̃δB⃗BR = R̃µ0 ωrot Ceff Vrot
Êrot .
2πReff(3.24)

Here Vrot is the amplitude of the oscillating voltages on each of the fins, Ceff is the effective
capacitance, and Reff the effective radius of the trap. In our experiment |δB⃗BR | ∼ 15 µG.
We will be particularly interested in B⃗B , the magnetic field which does not chop sign with84
B̃, causing shifts in f B and f BD . To first order3 these arehδ f B = 3gF µ B Ê · δ⃗B B ,
hδ f

BD

(3.25)= 3δgF µ B Ê · δ⃗B B ,⃗.
where Ê is the unit vector pointing along the total electric field E
The differential magnetic moment µ⃗D arises principally from two effects—mixing of neighbouring m F levels in the rotating frame, and mixing of the J = 1 levels with J = 2 induced by theelectric field. For the experimental parameters used throughout our dataset, the latter dominatesand so we neglect the former here. The differential g-factor δgF can then be calculated fromsecond-order perturbation theory as,⃗|
3D∥ G∥ |E
.
20B

δgF ≃

(3.26)We see that, as expected, the mixing is linear in the size of the electric field. We can use this, in
combination with Eq. 3.25 to give expressions for the time-averaged frequency shifts we measurein the experiment,h⟨δ f B ⟩ = 3gF µ B ⟨Ê · δ⃗B B ⟩,h⟨δ f BD ⟩ = 3

δgF⃗ · δ⃗B B ⟩.
µ B ⟨E
Erot(3.27)We see that the common mode part of the Zeeman interaction is proportional to ⟨Ê · δ⃗B B ⟩ whilst⃗ · δ⃗B B ⟩.
the differential part is proportional to ⟨E
Any possible B⃗B can be expanded asδ⃗B B (⃗x, ⃗y, ⃗z) =l∗⃗ r l Yl,m,∑ ∑ Bl,m ∇(3.28)

l =1,2,... m=−l3 We note that there are higher order corrections to these shifts arising from rotation-induced mixing of the twostates in a doublet. These effects are well understood but, under the experimental parameters used in the dataset,
contribute less than 10 % corrections to the shifts discussed and so are not included in our systematic uncertainty
budget.

85

Table 3.4: Magnetic-field expansion
Coefficient∗Yl,mUniform fields
B1,−1
B1,0
B1,1

B2,−2
B2,−1
B2,0
B2,1
B2,2

B3,−3
B3,−2
B3,−1
B3,0
B3,1
B3,2
B3,3

ŷ
ẑ
x̂
First-order gradients√√3(y x̂ + x ŷ)3(zŷ + yẑ)−√x x̂ − yŷ + 2zẑ
√ 3(z x̂ + x ẑ)3( x x̂ − yŷ)
Second-order gradientsq5
3
2
2 (2xy x̂ + ( x − y )( x + y ) ŷ )√15(yz x̂ + xzŷ + xyẑ)
q
3
1
2
2
2
2 (− xy x̂ + 2 (− x − 3y + 4z ) ŷ + 4yz ẑ
1 2
22
q −3( xz x̂ + yzŷ + 2 ( x + y − 2z )ẑ)
3 1
2
2
2
2 ( 2 (−3x − y + 4z ) x̂ − xy ŷ + 4xz ẑ )√15q
( xz x̂ − yzŷ + 21 ( x − y)( x + y)ẑ)3522 (( x − y )( x + y ) x̂ − 2xy ŷ )86where ⃗x, ⃗y, ⃗z are the ions position relative to the trap center, Bl,m is the coefficient of each compo∗ are the semi-normalized real spherical harmonics
nent and Yl,m√i (Y m − (−1)m Y −m )


l
l
2

r


4π
∗
Yl,m
=
× Y0
l
2l + 1 






 √1 (Y −m + (−1)m Y m )ll2

if m < 0
if m = 0 .

(3.29)

if m > 0∗ for l ≤ 3 are given in Table 3.4.
The Yl,m

3.7.1.1′
Non-reversing Baxgrad

In an idealised version of the experiment where the ions are subject only to a perfect⃗ =⃗rot = Erot (cos(ωt) x̂ + sin(ωt)⃗y), the micromotion of the ions is − e 2 E
rotating electric field Emω rot−rrot Êrot . In this case the only component of the magnetic field with l ≤ 3 which causes a
non-zero time-averaged shift in any frequency channel is the quadrupole magnetic field δ⃗B B =B2,0 (− x x̂ − yŷ + 2zẑ). As described in Section 4.2, we deliberately apply such a quadrupole′
magnetic field Baxgrad
whose sign reverses in the B̃ chop which is the main contribution to f .

However there can be an additional contribution which does not reverse sign with B̃, arisingeither from a background field present in the lab, or from imperfect reversal of the applied field.
We haveh⟨δ f B ⟩ = −3gF µ B B2,0 rrot ,
h⟨δ f

BD

(3.30)⟩ = −3δgF µ B B2,0 rrot .

Both shifts are proportional to B2,0 but with the D̃-even shift being ∼ 500 times larger. There
are two possible approaches to removing the effect of this systematic from our measurement, thefirst is to make a correction to the measured value of f BD based the measured value of f B ,BDδ f corr
= fB

δgF
.
gF

(3.31)The second is to shim out B2,0 by deliberately applying slightly different currents through the′Baxgradcoil in each direction. Both approaches are equivalent, we choose to shim f B to zero

87

δg F
g F , by varying the mean fringe frequency
Ddifferential fringe frequency f . Data taken at Erot ∼ 50 V/cm.

Figure 3.6: Measurement off 0 and measuring theon each block based on the measured value of f B in the previous block. We then correct themeasured value of f BD on each block by the measured value of f B on that block. The quantity
δg F
gF

in Eq. 3.31 is determined directly from experiment as shown in Figure 3.6. The largest

correction on any single block is 75 µHz, while the median correction size is 5 µHz. The averagecorrection over the whole dataset is just 0.2 µHz.
This approach is effective in reducing the contribution of this systematic shift to levels well
below our statistical error bar, but only holds if there are no other shifts in f B that scale withf BD with a constant of proportionality different from

δg F
g F —or even no shift in

f BD at all. The

remainder of this section explores and places limits on such shifts arising from stray magnetic
fields. Section 3.7.2 explores possible contributions from Berry’s phase effects.

3.7.1.2

General principles in determining other magnetic effects

Although no other magnetic field components couple directly to an idealised rotating electric field to give significant shifts, a number of potentially problematic shifts can arise when othercontributions to the electric field experienced by the ions are taken into account. In particular,when the magnitude of the total electric field experienced by the ions is not constant in time,

88

δg

Figure 3.7: Measurement of gFF , obtained by introducing a large δBnr and measuring the resultant
values of f BD and f B . Note that f BD is still blinded in this measurement. Data taken at Erot ∼ 50
V/cm.

⃗ · δ⃗B B ⟩ ̸= ⟨|E⃗ |⟩⟨Ê · δ⃗B B ⟩.
there can be effects for which ⟨E
To enumerate possible shifts in the experiment, we analytically calculate the micromotion⃗ =E⃗rot + κ δ⃗E where E⃗rot is the electric field of theof a charged particle subject to an electric field E
idealised rotating field, and δ⃗E represents perturbations to this, discussed in the sections below.
This micromotion allows us to write down the time dependent magnetic field experienced by amolecule moving through magnetic field imperfections κ δ⃗B . Finally we find the frequency shifts⃗ · δ⃗B⟩. Here the time average is over an integer⟨ f B ⟩ and ⟨ f BD ⟩ by calculating ⟨Ê · δ⃗B⟩ and ⟨E
number of periods of all relevant frequencies and we keep terms up to O(κ 2 ).
For those effects which give non-zero shifts in either channel, we constrain the maximum
size of the shifts by direct measurement using the ions, or by auxiliary measurements of the sizeof the imperfections. We note that this analytical approach, by necessity, does not include theeffects of spatially dependent electric fields and so misses the effects of ponderomotive forces
such as those from the RF confining fields. To validate this approximation, we performed comprehensive numerical simulations including the full motion of the ions in the trap whilst varying
the size of the imperfections listed over ranges well above their expected size in the experimentand observed no additional effects.

89
3.7.1.3

Second harmonic of Erot and transverse magnetic field

The rotating electric field in our experiment is generated by sinusoidally varying voltages
on each of the radial electrodes. These voltages are driven by op-amps which inevitably sufferfrom harmonic distortion, adding higher harmonics of the input signal. Consider the effect of anadditional electric field oscillating at the second-harmonic frequency. We have cos(ωt) E2hX cos(2ωt + ϕ2hX )

 

 

⃗ = Erot E sin(ωt)  + E2hY cos(2ωt + ϕ2hY )

 


 

0
0





(3.32)

E2hX cos(2ωt + ϕ2hX )
cos(ωt)




1




Ê ≃  sin(ωt)  +E2hY cos(2ωt + ϕ2hY )

 2Erot 




0
0




− cos(ϕ2hX )
 sin(ϕ2hY )  EE2hX 



2hY 
+
 sin(ϕ2hX )  +cos(ϕ2hY )
 4Erot 
4Erot 




0
0


− cos(4ωt + ϕ2hX )
E2hX +
 − sin(4ωt + ϕ2hX ) 
4Erot 
0
− sin(4ωt + ϕ2hY )

E2hY 
+ cos(4ωt + ϕ2hY ) 
4Erot 

0(3.33)

where Ê has been expanded to first order in 1/Erot . Alongside the components oscillating at
ω, and the corrections oscillating at 2ω, Ê also has terms which are time independent, and terms
that oscillate at 4ω. The time-independent terms can couple to a uniform transverse magnetic

90field δ⃗B B = B1,1 x̂ + B1,−1 ŷ to give time averaged shifts,h⟨δ f B ⟩ =3gF µ B
− B1,1 E2hX cos(ϕ2hX )4Erot+ B1,−1 E2hY cos(ϕ2hY )(3.34)+ B1,−1 E2hX sin(ϕ2hX )+ B1,1 E2hY sin(ϕ2hY ) ,h⟨δ f BD ⟩ = 0.

(3.35)

We note that depending on the details of the second harmonic, the phases ϕ2hX and ϕ2hY canchange sign under rotation. In this way some of this shift can (and usually does) show up in
f BR rather than f B . We have verified these shifts experimentally by intentionally applying alarge second harmonic to our electrodes along with large transverse magnetic fields. Due tothe correction applied described in Section 3.7.1.1, any shift in f B not due to a non-reversing
quadrupole magnetic field would cause a systematic δ f BD = gFF ⟨δ f B ⟩.
δg

During the datarun, we measure the transverse field using an array of magnetometers
positioned on the vacuum chamber and tune it to zero using three pairs of shim coils arranged
around the apparatus. The magnetometers can be subject to small offsets and are calibrated at
the beginning of the data run by intentionally applying large second harmonic electric fields
and measuring shifts in f B and f BR . We used the same technique at regular (roughly weekly)intervals throughout the dataset to check the size of the magnetic fields and guard against any
drifts in the magnetometer offsets. Based on these measurements, we conservatively estimate
the maximum size of the transverse fields to be < 10 mG in x and y, limited principally by theprecision of the current supplies used to drive the shim coils.
To reduce the size of the second harmonic electric fields present in the experiment, we
intentionally apply a secondary signal at 2ω with the opposite phase, cancelling any signal at 2ω
down to about a part in 40,000 of the signal at ω. In the worst case, where the relative phases of
the residual 2ω signal are such that the corresponding electric field at the position of the ions is

91
maximised, this corresponds to residual shifts in f B of about 80 mHz G−1 . After this procedure,
we deliberately apply large magnetic fields to the ions and observe the residual shifts in f B to
be 103(1) mHz G−1 and 46(9) mHz G−1 for fields along x and y respectively, in good agreementwith our direct voltage measurements. The amplitude of the second harmonic was checked
periodically throughout the dataset to guard against any drifts, for example in the characteristicsof the power operation amplifiers. Combined with the uncertainty on the transverse fields above,these measured gradients result in systematic uncertainty in f B of 1 mHz with a corresponding
systematic uncertainty in f BD of 2 µHz.

3.7.1.4

Higher-harmonics of Erot

We can generalize the arguments of the previous section to higher harmonics of Erot . For
the nth harmonic, the expansion of Ê in powers of 1/Erot will contain components which oscillate⃗rot -induced micromotion of the ions combined with an mth orderat (n − 2)ω and (n + 2)ω. The E
magnetic field gradient, produces a magnetic field in the frame of the ions which oscillates at
mω. So in general an nth harmonic of Erot can couple to (n − 2)th and (n + 2)th order magnetic
field gradients (l = n − 1 or n + 3) to give non-zero time-averaged frequency shifts. After thefeedforward is applied to reduce the second-harmonic as described above, the next largest shift
from harmonic distortion is due to 3ω, with an amplitude on each fin of about a part in 1500of the amplitude at ω. The 3rd harmonic can couple to first-order field gradients of the form√√δ⃗B B = B2,−2 3(y x̂ + x ŷ) + B2,2 3( x x̂ − yŷ) to give shifts,√
3 3gF µ B eBh⟨δ f ⟩ =B2,2 E3hX cos(ϕ3hX )4mω 2− B2,−2 E3hY cos(ϕ3hY )(3.36)− B2,−2 E3hX cos(ϕ3hX )− B2,2 E3hY cos(ϕ3hY ) ,h⟨δ f BD ⟩ = 0.

(3.37)

92
We intentionally applied a large third harmonic electric field with various phases and observed
shifts of ∼ 20 mHz. The applied third harmonic field was 28 times larger than that present in thedataset and so we conservatively set a limit on the maximum size of any possible shift in f B at
1 mHz, with a corresponding systematic uncertainty in f BD of 2 µHz.
Higher harmonics are comparable to, or lower in amplitude (the fourth harmonic is down
about a factor of 2/3 compared to the third, whilst the fifth is up a factor of about 4/3, sixthand higher harmonics are all smaller) and couple to higher-order magnetic field gradients. To
set limits on their possible contributions to shifts in f B we made measurements of the magnetic
field gradients. These measurements were made outside of, but within a few cm of, the vacuum
chamber. The electrodes in the experiment are machined from non-magnetic titanium and there
are no other sources of magnetic fields closer to the ions than the steel vacuum chamber whichis ∼ 10 cm away. Because of this, we expect that the field gradients outside the chamber are
similar to, or greater than, those inside the chamber. From these measurements we estimate themaximum size of first order field gradients to be 10 mG cm−1 and second-order gradients to be
10 mG cm−2 . To determine the size of the magnetic field oscillating at nω, the nth order magnetic
n where r
field gradient should be multiplied by rrot
rot = 0.05 cm. This means we expect any

higher-order harmonic contributions to shifts in f B to be reduced by a factor of ∼ 20 from thosefor the third harmonic, corresponding to systematic uncertainties in f BD < 100 nHz. Therefore,
we do not include any higher-order effects in our systematic uncertainty budget.

3.7.1.5

Ellipticity of Erot

Another possible electric field imperfection is an ellipticity of Erot (i.e. the oscillating voltage on different electrodes in the trap having slightly different amplitudes). A general rotating
electric field having an ellipticity with its major axis orientated at angle θ to the x axis can be93written









cos(2θ − ωt)cos(ωt)







⃗ =Erot E sin(ωt)  + Eϵ  sin(2θ − ωt)  .








0
0

(3.38)

The corresponding Ê is








cos(ωt)
cos(2θ − ωt)



Eϵ 




Ê =  sin(ωt)  + 2Erot  sin(2θ − ωt) 




0
0



(3.39)

− cos(2θ − 3ωt)

Eϵ 


+

.
2Erot  sin(2θ − 3ωt) 0This field can once again couple to first order magnetic field gradients of the form δ⃗B B =√√B2,−2 3(y x̂ + x ŷ) + B2,2 3( x x̂ − yŷ). After taking into account the modification to the ion micromotion caused by this perturbation, the resulting shifts are√
9 3gF µ B eEϵBh⟨δ f ⟩ = −B2,2 cos(2θ ) + B2,−2 sin(2θ ) ,2√ 2mω
6 3δgF µ B eEϵ
B2,2 cos(2θ )h⟨δ f BD ⟩ = −mω 2+ B2,−2 sin(2θ ) .

(3.40)(3.41)

Note that in contrast to the effects of higher-harmonics discussed above, the ellipticity does
cause a shift in f BD , with δ f BD /δ f B = 43 gFF . Due to the correction described in Sec. 3.7.1.1,
δgwhich assumes δ f BD /δ f B = gFF , the systematic associated with the shift described in this section
δgwill be 1/4 of the total shift induced in f BD . The size of the ellipticity in our experiment wasmeasured as described in Sec. 3.7.2. To set limits on the size of the shift, we applied an ellipticity
5 times larger than this, limited by our shallow ion trap during the Ramsey sequence, and varied
its angle. The one sigma upper limit on the shifts seen in this way was 12 mHz and so we set
an upper limit on the size of any shift in f B due to this effect of 2.4 mHz, corresponding to asystematic uncertainty in f BD of 1 µHz.

943.7.1.6

RF micromotion

In addition to the rotating electric field the ions are subject to the RF confining field oscillating at ωrf . This RF field induces its own micromotion of the ions and, through B2,0 , a magnetic
field oscillating at ωrf . The corresponding time-averaged shifts areh⟨δ f B ⟩ = 3gF µ B

eErf2 B2,0,
2
4Erot mωrf

(3.42)

eErf2 B2,0.
2
2Erot mωrf

(3.43)h⟨δ f BD ⟩ = 3δgF µ B

In this case we have ⟨δ f B ⟩/⟨δ f BD ⟩ = 2δgF /gF and so there is potential for a systematic, but both
shifts are proportional to B2,0 which we are shimming to zero. The ratio of the magnitude of the2 ω 2 ∼ 10−2 so that this shift will
shift caused by the RF field to that caused by Erot is ⟨Erf2 ⟩ω 2 /Erotrfjust provide a ∼ 1% correction to the—already very small—systematic uncertainty in Sec. 3.7.1.1.
In addition, because our measurement of δgF /gF was performed with the RF fields present, italready includes this correction. Therefore, we do not include any contribution from this effect inour systematic uncertainty budget. Other electric fields which oscillate at frequencies other thaninteger multiples of ω (e.g. time-averaged electric field due to secular motion) behave similarly
but with even smaller contributions.

3.7.1.7

Out of plane electric fields

All the electric field imperfections discussed thus far are additional fields contained within
the x, y-plane, the plane in which Erot rotates. However, it’s also possible for the ions to experienceoscillating fields in the z direction. This can occur, for example, due to thermal motion of theions or patch charges displacing the center of the ion cloud from the geometrical centre of thetrapping electrodes. The largest shifts from fields in the z direction are those due to B2,0 coupling
to their own micromotion. The shifts have δ f BD /δ f B = δgF /gF and so produce no systematiceffects not already accounted for by the approach described in Sec. 3.7.1.1.

95
3.7.2

Berry’s phase effects
The rotating electric bias field Erot defines our quantisation axis, and we do our spec-

troscopy in the rotating frame. Working in this rotating frame has two important effects: amixing of the eigenstates of the system, which depends on the rotation rate; and an additional
phase accumulation between eigenstates with different total angular momentum projections onto
the quantization axis, which does not. This latter part, a geometric or Berry’s phase—depending
only on the path traced out by the quantisation axis in time—is discussed in this section, whilst
the former is discussed in section 3.7.3.
As the electric field vector in the experiment rotates, it traces out closed loops in phasespace. In each loop, the two states in a doublet accrue a differential geometric phase given by
ϕgeo = ∆m F Ω = 3Ω,(3.44)

where ∆m F is the difference in the angular momentum projection of the two states and Ω isthe solid angle traced out by the electric field. In the idealised experiment, the electric fieldrotates strictly in the X, Y-plane, subtending a solid angle of 2π every Trot = 1/ f rot . Phase
shifts of integer multiples of 2π are indistinguishable from zero and so not observable in theexperiment. However effects which tilt the quantisation axis out the X, Y-plane can cause a
non-zero frequency shift. For small tilts, the shift is given by
3 f rotδf = −
2π

Z Trot0α(t)ϕ̇(t)dt(3.45)

where α(t) is the tilt of the quantization axis out of the equatorial plane, ϕ̇(t) is it’s azimuthal
angular velocity, and the integral is over one period of the rotating field.
This frequency shift affects both doublets identically but is independent of the B̃ chop andso appears in B̃-odd frequency channels. We note that the main contribution to ϕ̇(t) ∼ R̃ω isrotation odd and so the most natural channel for the shift to appear in is f BR , however rotationodd components of α(t) can cause effects in f B as well. Whilst the shifts do not appear directly

96
in our measurement channel, any shifts in f B will cause a systematic shift—suppressed by afactor of δg/g—due to the correction we make based on the measured value of f B , described inSec. 3.7.1.1.
The most straightforward way to tilt the quantisation axis is to add a time-independent
electric field in the axial direction. Our measurement using trapped ions is well protected fromthis possibility; any electric field in the axial direction which does not time average to zero, andwhich is not balanced by another force on the ions, will cause the mean position of the ions toshift to where the time-averaged electric field is zero. However, there are a number of mechanisms
which can cause non-zero time-averaged geometric phases in the experiment. In this section, we
discuss these mechanisms and place constraints on their possible size during our measurement.

3.7.2.1

Gravity

The gravitational force on the ions causes them to sit at a position in the trap where theaxial electric field is slightly different from zero. This field is Egrav = mg/e ∼ 20 µV m−1 , causinga shift δ f = 3 f rot Egrav /Erot ∼ 4 mHz. This shift appears rigorously in f BR which we do not use
for correction of f BD and so we do not include any contribution from this effect in our systematic
uncertainty budget.

3.7.2.2

AC Stark shift from vibrational cleanup light

The vibrational cleanup light is tuned to resonance with the 3 ∆1 (v = 1) → 3 Σ0+ (v = 0)transition to remove any population in 3 ∆1 (v = 1) and prevent it decaying into the science state.
The light is left on throughout the free evolution time and propagates in the axial direction
through the chamber. To estimate the effect of this light, we assume that the polarisabilityof the 3 ∆1 (v = 0) state is dominated by interaction with 3 Σ0+ (v = 0), from which the laser
is detuned δL ∼ 1.7 THz, and the nearby 3 Σ1 (v = 0), a further ∼ 30 THz away. Conservatively assuming transition matrix elements of dtrans ∼ 0.1 ea0 for each, we obtain a polarisabil-

97
ity of d2trans /(h2 ϵ0 cδL ) ∼ 2 × 10−4 Hz W−1 m2 . The light intensity at the position of the ions is∼ 300 W m−2 , giving an AC Stark shift of order ∼ 1 Hz.
The scalar AC Stark shift affects all states in 3 ∆1 (v = 0) equally and so has no effect on
our measurement. However, a gradient of the intensity could exert a force on the ions, pushingthem to a position where the electric field is non-zero in a similar way to the gravitational effectdescribed above. The light is roughly collimated through the chamber and we conservatively
constrain the intensity gradient to be < 500 W m−3 , corresponding to a force more than 10 ordersof magnitude smaller than that due to gravity discussed in section 3.7.2.1.
Effects from possible vector or tensor AC stark shifts depend on the polarization of thelight which is nominally linear in the X, Y-plane in our experiment. The tensor shift causes
an AC stark shift which is differential between doublets, but common mode within a doublet,
but which oscillates at 2 f rot , imitating an ellipticity of Erot . Conservatively assuming the tensor
polarisability is of similar size to the scalar, this effective ellipticity is h ∼ 1 Hz/Erot dmf ∼ 10−8 ,much smaller than our best limit on the true ellipticity in the trap.
For linearly polarised light, the vector Stark shift is zero but we can consider the effectof a residual handedness. In this case, the vector shift can tilt the quantisation axis out of theX, Y-plane. Conservatively assuming vector polarisability similar size to the scalar, and a 2 %residual handedness of the light, we find a geometric phase of ∼ 1 mHz. Like the gravitational
effect, this shift is in f BR and so has no systematic effect on our measurement. We took some
auxiliary blocks with the vibrational cleanup light turned off during the free evolution time andobserved no shifts, constraining any shift in f BR to less than 9 mHz.

3.7.2.3

Phase modulation due to axial secular motion

Ions displaced from the center of the trap experience a non-zero axial confining field. Any
axial motion due to the axial confining potential is harmonic, so the axial component of the

98electric field will average out to zero over many cycles. Instantaneously, however, the ions willaccrue a Berry’s phase which we could then alias, leading to a perceived phase shift.
We investigated by pulsing the endcap electrode to ‘kick’ the ion cloud in the axial direction
right before the first π/2 pulse; causing temporary coherent axial oscillations, then collecting a
Ramsey fringe. Example data is shown in Fig. 3.8. We repeated this procedure over a range ofkick amplitudes. For each Ramsey fringe we fit the data to the model
A cos(2π f t + ϕ + δ cos(2π f z t + ϕz )).

(3.46)

Figure 3.9 shows the amplitude of the phase modulation δ, plotted against the amplitude of theapplied kick. The data constrain δ < 0.02 radians at the 1σ confidence level. Assuming the worst
case scenario where we take our data at the maximum of the phase modulation, this corresponds
to an apparent frequency shift of < 1.4 mHz for the effective average late Ramsey time (∼ 2.3
seconds) used. This shift is likely to be in f BR but could leak into f B if the phase of the axial
oscillation were rotation odd. We did not monitor this during our dataset and so include ansystematic uncertainty of 3 µHz in our uncertainty budget.
While this phase shift is negligible at late time—we find any slosh decoheres in 7 ms—
the integral of the phase shift could be non-zero. We explored this numerically and found afrequency shift of f BD on the order of 100 pHz, which is completely negligible.

3.7.2.4

Axial 2nd harmonic + radial ellipticity

In addition to static tilts of the quantisation axis and aliasing of time-varying tilts, perturbations to the radial electric field can cause variations in the magnitude of the rotating field andits azimuthal angular velocity which can couple to axial displacements to give geometric phase
effects which do not time-average to zero. An important example of this type of effect is an ellipticity of Erot combined with a second-harmonic field in the axial direction. Such an axial field
could result either from leakage of the signals that drive the radial electrodes onto the endcaps,

99

Figure 3.8: Example axial oscillation-induced phase modulation data, for a both large kick (color)and a small kick (grey).

Figure 3.9: Phase modulation amplitude for various sizes of z kick. Upper doublet data is in blue
and lower doublet data is in pink. Both fits have an intercept of −0.0127(0.0108).

100
or from axial displacement of the ions from the geometric center of the trap combined with fields
generated by the radial electrodes themselves.
We can express a radial ellipticity in Erot as we did before (Eq. 3.38) but now explicitly
including rotational dependence, cos(ωt + ϕR + R̃ϕ0 ) ⃗ =Erot E R̃ sin(ωt + ϕR + R̃ϕ0 )




0




(3.47)

 cos(2 R̃θ − ωt − ϕR − R̃ϕ0 ) + Eϵ  R̃ sin(2 R̃θ − ωt − ϕR − R̃ϕ0 ) .




0
Here ϕ0 + R̃ϕR is the angle of the radial electric field to the X-axis at t = 0 and θ is the angle
between the major axis of the ellipse and the X-axis. An axial field oscillating at the second
harmonic is given by
⃗2hZ =E2hZ 
E





0
0
cos(2ωt + ϕ2hZ )




.



(3.48)

With these electric fields we find the tilt angle and azimuthal angular velocity areE2hZ
cos(2ωt + 2ϕ2 f Z )
Erot


Eϵ
cos(2ωt + 2ϕR + 2 R̃ϕ0 − 2 R̃θ ) ,
× 1−
Erot
2 R̃Eϵ ωϕ̇(t) ≃ R̃ω −cos(2ωt + 2ϕR + 2 R̃ϕ0 − 2 R̃θ ),Erotα(t) ≃(3.49)(3.50)

where in each case we have expanded to first order in Eϵ /Erot . Using Eq. 3.45, we findδf ≃9 R̃E2hZ Eϵ f rotcos(ϕ2 f Z + 2 R̃θ − 2 R̃ϕ0 − 2ϕR ).
2
2Erot(3.51)

Finally we can calculate the expected shifts in f B and f BR by taking the sum and difference of101

Figure 3.10: Example data collected with E2 f Z ∼ 4 V m−1 and an ellipticity Eϵ /Erot ∼ 5 × 10−3 .
As we vary the phase of the axial modulation, f B and f BR vary sinusoidally with a π/2 phaseoffset, as expected.

the shifts in the two rotation directions,
9E2hZ Eϵ f rotsin(2θ − 2ϕ0 ) sin(ϕ2 f Z − 2ϕR ),2
2Erot(3.52)

9E2hZ Eϵ f rotcos(2θ − 2ϕ0 ) cos(ϕ2 f Z − 2ϕR ).
2
2Erot(3.53)

δfB = −
δ f BR =We see that we expect frequency shifts which vary sinusoidally as a function of either θ or ϕ2 f Z ,with a π/2 phase shift between f B and f BR . Figure 3.10 shows example data for deliberately
applied ellipticity Eϵ /Erot ∼ 5 × 10−3 and an axial second-harmonic E2 f Z ∼ 4 V m−1 .
We used data like this to manually shim out any residual ellipticity by applying E2 f Z , andadjusting the Erot multiplying DAC on each electrode until we saw the effect was minimized. Wethen measured the residual ellipticity in the same way and found Eϵ /Erot = 3 × 10−4 , suppressedby a factor of 35. We constrain the axial second harmonic field by deliberately applying an
ellipticity and varying θ to constrain |E2 f Z | < 10 mV m−1 at the 1σ confidence level, suppressedby a factor 400. Combining these suppression factors allows us to place a limit on the maximum
size of any shift in f B , δ f B < 0.8 mHz. We include a corresponding contribution to our systematic
uncertainty of 2 µHz.

1023.7.2.5

Effects of higher harmonics

We can generalize the discussion of the previous section to a pair of perturbations δ⃗E X,Yand δ⃗E Z which act in radial and axial direction respectively, and which are both phase locked toErot . We can write general expressions for these pertubations,δ⃗E X,Y = EnhX cos(nωt + ϕnhX ) X̂

(3.54)+ EnhY cos(nωt + ϕnhY )Ŷ,δ⃗E Z = EmhZ cos(mωt + ϕmhZ )(3.55)

where m, n are both integer. The combination of these perturbations will result in a non-zerotime-averaged frequency shift at first order provided m and n differ by ±1. The shift is given byδf = ∓

3EmhZ f rot (n ± 2)
EnhX cos(ϕnhX − ϕmhZ )
2
4Erot

± EnhY sin(ϕnhY − ϕmhZ ) ,(3.56)

where the ± correspond to m = n ± 1. Depending on the R̃ dependence of the various phases,
this shift could appear in either f B or f BR . We note that the frequency shift corresponding to thefirst harmonic on the endcap and second harmonic radially, m = 1, n = 2, is zero, and so thenext largest shifts are expected to come from effects involving the third harmonic or higher.
To place constraints on the possible size of these effects, we measured the Fourier transformof the signal on each of the radial electrodes and the two endcaps. The measured amplitudes ofeach harmonic are shown in the second column of Table 3.5. We first consider the possible radial
electric fields caused by these signals. The radial electric fields at the center of the trap generated
by an nω voltage signal on each electrode are
8

Vnhk
π
cos((9 − 2k ) ) cos(nωt + ϕnhk ),R
8
k=1 X,YδE Xnh = − ∑(3.57)

8Vnhk
πsin((9 − 2k) ) cos(nωt + ϕnhk ),R8
k =1 X,YδEYnh = ∑(3.58)

where Vnhk and ϕnhk are amplitude and phase of the voltage on each of the 8 electrodes and
R X,Y ≃ 23 cm is constant relating to the geometry of the trap. The amplitude of these fields is103
given by"

|δE Xnh | =8Vπcos((9 − 2k ) ) cos(ϕnhk )∑ Rnhk8
k =1 X,Y!2!2 #1/2πVnhk
cos((9 − 2k ) ) sin(ϕnhk )
,
+ ∑
R
8
k=1 X,Y
"
!2
8
Vnhk
π
|δEYnh | =∑ R sin((9 − 2k) 8 ) cos(ϕnhk )k =1 X,Y!2 #1/2
8
π
Vnhk
+ ∑
sin((9 − 2k) ) sin(ϕnhk ).
R
8
k =1 X,Y(3.59)

8(3.60)

The amplifiers driving each of the electrodes are nominally identical and so the harmonic distortion is approximately equal on each electrode. As a first order approximation, we takethe amplitude of the signal on each electrode to be equal, Vnhk = Vnh , and the phases to belocked to the fundamental with a small offset that can be different on each electrode, ϕnhk =ϕnh0 + n(9 − 2k) π8 + δϕnhk .
We first consider δϕnhk = 0. In this case all the radial fields from all electrodes cancel one
another unless n differs from a multiple of 8 by ±1. When n is one greater than a multiple of8, the relative phases on neighbouring electrodes are the same as for the fundamental and so
the harmonic field corotates with Erot . For n one less than a multiple of 8, the relative phaseson neighbouring electrodes are flipped compared to the fundamental and the resulting fieldcounterrotates. In both cases the amplitude is 4Vnh /R X,Y and we use this as an estimate of the
field size for these values of n in the third column of Table 3.5. For the other harmonics we
can estimate their size by looking at the variance of their amplitude due to the variance of theamplifier phases,
8⟨|δE Xnh | ⟩ = ∑
2

k =1
8 

⟨|δEYnh | ⟩ = ∑
2



k =1

∂|δE Xnh |∂δϕnhk2∂|δEYnh |∂δϕnhk22⟨δϕnhk
⟩,(3.61)

2⟨δϕnhk
⟩,(3.62)

where the derivatives in brackets are evaluated at δϕnhk = 0. Since in these cases the phaserelationship between the fields is random, we will be interested in their quadrature sum. After

104
Table 3.5: Constraints on electric fields from higher-harmonic voltages on radial electrodes andendcaps.
Amplitudes (mV)

Fields (mV m−1 )nRadialAxial|δE X,Ynh ||δE Znh |

2
3
4
5
6
7
8
9
10
11
12

11
310
190
410
100
230
45
170
37
140
42

0.5
0.3
0.1
0.2
0.4
0.5
0.7
1.0
1.4
1.8
2.1

250
1296
649
1715
341
4000
154
2957
126
586
143

1.0
2.1
1.8
2.7
0.8
1.7
3.1
1.8
2.0
2.7
3.1

some lengthy algebra, we find
1/2⟨|δE X,Ynh |2 ⟩1/2 = ⟨|δE Xnh |2 ⟩ + ⟨|δEYnh |2 ⟩" nπ 
√Vnh=
4 + (−1)n 2 cos
R X,Y
4

 #1/2
√
3nπ
n
2 1/2
− (−1) 2 cos
⟨δϕnh
⟩ ,
4

(3.63)

2 ⟩ = ⟨ δϕ2 ⟩.
where we have set the variances of the phases on all amplifiers to be equal, ⟨δϕnhk
nh
2 ⟩ = ( π/8)2 .
From measurements of the signals on the electrodes, we conservatively estimate ⟨δϕnh

The third column of Table 3.5 shows the estimated size of the radial electric fields from higher
harmonics calculated using this expression. Note that, because we shim the 2nd harmonic, we
can make no such claims about relative phases on electrodes and so we assume the worst caseof all electrodes on one side of the trap in phase and exactly out of phase with the electrodes onq
the other side, giving δE X,Y2h = 4 1 + √1 V2h /R X,Y .
2

We can take a similar approach to estimating the possible higher harmonic fields in the
Z direction. The most direct way of generating axial electric fields with angular frequency nω
is for them to appear directly on the endcaps of our trap where they generate a field VnhZ /R ax
Z ,

105
where R ax
Z ≃ 137 cm. Our measurements of possible higher-harmonic signals onto the endcaps
are shown in the third column of Table 3.5; for n > 3 these are upper limits, constrained by
the frequency-dependent noise floor of our FFT measurements. We find that in most cases thefields generated by these voltages are dominated by another contribution, signals on the radial
electrodes combined with axial displacement of the ions from the center of the trap. We can write
the axial electric field due to nω signals on the radial electrodes at axial displacement Z0 as
8

Vnhk Z0cos(nωt + ϕnhk ),
R2Z
k =1

δE Znh = ∑(3.64)

with amplitude
8

Z0|δE Znh | = 2
RZ∑ Vnhk cos(ϕnhk )

k =1
8

+

!2

∑ Vnhk sin(ϕnhk )(3.65)!2 !1/2.

k =1

The constant R Z ≃ 27 cm is fixed by the trap geometry. We measure the axial displacement by
deliberately applying an ellipticity plus a large second harmonic in phase to all radial electrodes
and find Z0 = 0.6 mm.
Again, we make the substitutions Vnhk = Vnh and ϕnhk = ϕnh0 + n(9 − 2k ) π8 + δϕnhk andstart by assuming δϕnhk = 0,
δE Znh =Vnh Z0 sin(nπ ) cos(nωt + ϕnh0 ).
R2Z sin nπ8(3.66)

The contribution of the 8 electrodes cancels out except for when n is a multiple of 8 when they are
all in phase with one another and we have |δE Znh | = 8Vnh Z0 /R2Z . For these values of n, we use
this as an estimate of the axial field amplitude from the radial electrodes. For the other possible
harmonics we can estimate their amplitudes in the same way we did for the radial fields. The
variance of the amplitude of the field is given by8⟨|δE Znh |2 ⟩ = ∑

k =1



∂|δE Znh |∂δϕnhk22⟨δϕnhk
⟩.

(3.67)

106
Table 3.6: Constraints on Berry’s phase frequency contributions to f B or f BR due to possiblecombinations of radial and axial field imperfections from harmonic distortion in amplifiers used
to drive radial electrodes. All entries are in µHz.
nZ
n X,Y

2

3

4

5

6

7

8

9

10

11

12

2
3
4
5
6
7
8
9
10
11
12

0
11
0
0
0
0
0
0
0
0
0

17
0
22
0
0
0
0
0
0
0
0

0
95
0
75
0
0
0
0
0
0
0

0
0
87
0
31
0
0
0
0
0
0

0
0
0
83
0
139
0
0
0
0
0

0
0
0
0
38
0
13
0
0
0
0

0
0
0
0
0
947
0
545
0
0
0

0
0
0
0
0
0
24
0
16
0
0

0
0
0
0
0
0
0
556
0
90
0

0
0
0
0
0
0
0
0
35
0
33

0
0
0
0
0
0
0
0
0
200
0

where again, the derivative in brackets is evaluated at δϕnhk = 0. After some more lengthy
algebra, we find2 1/2⟨|δE Znh | ⟩

V Z0
= nh2
RZ

sin(nπ )
4−
sin nπ
4

!1/2
2 1/2
⟨δϕnh
⟩ .

(3.68)

2 ⟩1/2 = π/8 to calculate the field estimates for all n which are
We use this expression with ⟨δϕnh

not a multiple of 8. Note that, again due to our shimming of the second harmonic, we can’t makeany claims about relative phases on the electrodes and so we assume the worst case of all in phasegiving |δE Z2h | = 8V2h Z0 /R2Z . The fourth column of Table 3.5 shows the sum in quadrature of ourestimates for the axial field from the radial electrodes with those from the endcaps.
In Table 3.6, we combine these estimates of the radial and axial field amplitudes with
Eq. 3.56 to set limits on the Berry’s phase from each possible combination of higher-harmonics.
Summing all the contributions in quadrature we get an uncertainty on the Berry’s phase of
1.4 mHz, corresponding to a systematic uncertainty in f BD of 3 µHz which we include in oursystematic uncertainty budget.

107
3.7.3

Residual rotation induced doublet mixing
The systematic effects we have considered so far are all concerned with the second term in

Eq. 3.21, various sources of δ f 0s . However, the third and fourth terms in the expression, containing
the off-diagonal components of the effective Hamiltonian, can also potentially cause systematic
shifts. The third term concerns shifts directly generated by the off-diagonal mixing components,
while the fourth term concerns the leaking of frequency shifts into other channels due to thismixing. In this section we discuss possible contributions to each and place constraints on theirsize during the measurement.

3.7.3.1

Leaking of f BR

The largest leak of any frequency channel into f BD is expected to come from the frequency
channel with the largest signal. This is f BR where a signal of ∼ 200 mHz is caused by the charging
currents in our electrodes. This could potentially leak into f BD when combined with a non-zerovalues of δ∆ DR or δ∆ R ,δ f BD = − f BR2∆0 δ∆ DR + 2∆ D δ∆ R.
| f 00 |2(3.69)

Non-zero values of δ∆ DR and δ∆ R can come from an axial magnetic field through a 4th ordercoupling similar to ∆ and ∆ D but where one of the rotation matrix elements is replaced with amagnetic field matrix element. Any effect would also result in a contribution to f DR ,
δ f DR =2∆0 δ∆ DR + 2∆ D δ∆ R.
| f 00 |(3.70)

So we can rewrite Eq. 3.69 as
δf

BD

f BR
≃− 0
f


f

DR


δgF R
−
f
.
gF

(3.71)

Our data places a limit on this contribution to f BD of < 170 nHz and so we include no contribution in our systematic uncertainty budget.
On similar grounds, we could also expect non-zero δ∆ DR or δ∆ R to cause f BR to leak into

108
f B,
δ f B = − f BR2∆0 δ∆ R + 2∆ D δ∆ DR,| f 00 |2(3.72)

causing a systematic via the correction described in Sec. 3.7.1.1. Since this expression involves
the same terms as Eq. 3.69, barring unlikely cancellations, the contribution can be expected to beof similar size. The corresponding systematic in f BD is suppressed by a further factor of δgF /gF
and so we again include no contribution in our systematic uncertainty budget.

3.7.3.2

Axial magnetic fields

Analogously to how a static axial magnetic field can generate δ∆ DR and δ∆ R , a B̃-odd axial′
magnetic field B ZB can cause δ∆ BDR and δ∆ BR . Such a field could be generated from the Baxgrad

coils if the ions are situated slightly away from their geometric center. The combination of these
two effects can cause a shift in f BD via the third term in Eq. 3.21,
δ f BD =2δ∆ DR δ∆ BR + 2δ∆ R δ∆ BDR.
| f 00 |2(3.73)An order of magnitude estimate for these R̃-odd mixing elements can be obtained from taking
the measured values of ∆ and ∆ D and multiplying them by the ratio of the magnetic-field matrix
element to the rotational matrix element, gF µ B B Z /h̄ωrot ∼ 0.01 G−1 . Since ∆ and ∆ D are ∼ 1 Hz,
we can expect effects of 10 mHz G−1 . We took data with large axial magnetic fields and saw
shifts in f DR of 170(30) µHz G−1 for f 0 = 151 Hz, confirming the order of magnitude of thisapproximation. Applying these approximations to Eq. 3.73, we can then expect a systematic shift
in f BD of ∼ 2 µHz G−2 × B ZB B Z . Looking at f DR − gFF f R and f BDR − gFF f BR for our dataset puts
δgδg

limits on the fields of B Z < 0.5 G and B ZB < 0.4 G. We note that the same magnetometers used
to shim out B Z are able to shim B X and BY to ∼ 10 mG but since we have not confirmed thisfor B Z , we use this more conservative limit. Finally we constrain any systematic shift in f BD to< 400 nHz.

109
3.7.4

Oddities & miscellany

3.7.4.1

Ion-ion interactions

The dominant ion-ion interaction is from the monopole charge on each ion. If we qualitatively model the cloud as 20,000 singly charged ions uniformly distributed across a sphere 1 cm⃗rot or thein radius, we find that the resulting mean-field electric field is much smaller than either E⃗RF , but comparable to the time-averaged value of E
⃗RF . That is to say, the mean-field selfpeak E
repulsion of the cloud is only modestly smaller than the effective electric fields providing secularconfinement. We see evidence for this for instance in the frequencies of the breathing modes of
the cloud. The ions therefore experience a net confinement that is somewhat anharmonic. Noneof the arguments for the limits of the size of systematic errors on de hinge on the confinement
being particularly harmonic. Each trapped ion must perforce experience a time-averaged axial
electric total field (whether external or from other ions) extremely close to zero, and thus to a⃗rot causes no Berry’s phase frequency shift. It is the case, however,high precision the rotation of E
that the mean-field repulsion causes the ion cloud to be larger than it otherwise would be at agiven temperature. In the presence of various field inhomogeneities, changes in the radius of thecloud can change both the decoherence rate and the average fringe frequency f 0 . We see shifts inthe mean Ramsey frequency f 0 that correlate with the number of ions in the trap. These do notappear in the D-odd frequency channels. The mean-field coulombic repulsion does not break thevarious symmetries that keep frequency shifts out of the science signal.
On a microscopic level, the coulomb potential between two atoms is roughly 10−3 kT, andthus ions are far from the ion-crystal regime. When small-impact-parameter ion-ion collisions
occur, the ion-ion electric field can briefly spike to a magnitude which is not infinitesimal com-⃗rot to briefly wobble in a way that encloses solidpared to Erot . This can cause the tip of the E
angle, and there can be a resulting random Berry’s phase shift that degrades the coherence between the m F = 3/2 and m F = −3/2 states, but does not bias the central value of the Ramsey

110fringe frequency. Adiabatic relaxation of the confining potential of the trap dramatically reduces
this source of decoherence, at the cost of increasing the decoherence from spatial inhomoegeneity
mentioned in this section. We empirically reoptimize the compromise value of the ramped-down
confinement several times during the course of a long data collection run.
Fields from the molecule-frame electric dipole moment are down from the monopolegenerated fields by a factor or more.
As for magnetic interactions, each ion carries a magnetic dipole moment (3/2) gF µ B . Approximating the distribution of 20,000 ions just as described above, the field arising from a uniform magnetization within the ball of ions could cause frequency shifts of order pHz, and is thus
neglected.

3.7.4.2

Effect of ion-cloud spatial distributions in which the Stark doublets are not perfectly overlapping

Much of the ultimate accuracy and precision our measurement of f BD ∼ de Eeff comes fromthe fact that we are measuring a resonance in two samples of ions which overlap perfectly inspace and in time, but for which Eeff points in opposite directions. If for some reason the two
samples do not perfectly overlap, then spatial variation in B̃-odd frequencies can cause systematic
shifts in our science signal. Generically, time-averaged D̃-odd displacements ⟨ri ⟩ D or sizes of theion cloud ⟨ri 2 ⟩ D can couple to first- and second-order gradients in f B to give systematic shifts,
δ f BD =∂fB∂2 f B∑ ⟨ri ⟩D ∂ri + ⟨ri 2 ⟩D ∂ri 2 .

(3.74)

i = x,y,z∂fB∂2 f BBy moving the ions around in the trap, we measure ∂ri ∼ 40 mHz cm−1 and ∂r2 ∼ 10 mHz cm−2iand so we are potentially interested in ⟨ri ⟩ D ∼ 1 × 10−4 cm and ⟨ri2 ⟩ D ∼ 5 × 10−4 cm2 .
We have good a priori reason to believe that the spatial distribution of ions in the two
doublets are identical to very high precision. The two Stark doublets are initially populated by

111
optical pumping from the 1 Σ ground state via a 3 Π0 state, both with Ω = 0. To an excellent
approximation, both these states have well defined parity in the modest electric fields used inour experiment and so the pumping process is completely independent of doublet. Any D̃-odd
spatial distributions must be imprinted subsequently, either by the lasers used to prepare or
readout the states of the molecules, or by D̃-odd forces on the ions.
We first discuss possible effects of the lasers used for state-preparation. The next laser tointeract with the ions is the 1082 nm which polarizes the ions by optically pumping them into
stretched m F states. Although this process again proceeds via an excited state with Ω = 0, and soeach photon scatter is equally likely to populate either doublet, in this case the laser excites out of
the 3 ∆1 state, and so can potentially cause a D̃-odd population difference. However, we operate
in the strongly saturated regime where all ions have sufficient time to interact with the laser andso any effects of spatial intensity variation are strongly suppressed. Due to this suppression, we
expect any effects to be smaller than those discussed in the next paragraph.
The other CW laser addressing molecules in the science state is the 814 nm laser used toremove population in unwanted m F states. This laser is on for 7 ms before the Ramsey sequence,
to clean up any population not successfully optically pumped, and for 25 ms after the Ramsey
sequence to remove population in one stretched state. Using measurements of ion number as afunction of laser interaction time, we estimate the time taken to remove all molecules in unwanted
m F levels is about 1 ms and so both interaction periods are strongly saturated, again greatly
suppressing any effects of spatial variation. However for this laser, there is a second relevant timescale; because the electric field is not perfectly aligned with the k-vector of the light, moleculesin the desired stretched m F state can also occasionally scatter photons, removing a fraction β dep
from the state we detect. This effect is largest for the time after the Ramsey sequence, whereinteraction time is longest and we conservatively estimate that β dep < 0.1. Although this step
takes place after the Ramsey sequence, if the ions removed have some D̃-odd spatial dependence,that will result in similar dependence being imprinted onto the remaining ions which we detect.

112⃗rot causes the ions to move in small circles at speeds of ∼ 1000 m s−1 . This motion
The rotating E
in the X-Y plane causes a sinusoidally varying Doppler shift oscillating at ωrot with amplitude∆Dopp ∼ 1000 MHz. This Doppler shift, combined with the Stark splitting of ∆St ∼ (dmf Erot −A∥ /2)/h ∼ 140 MHz between the two doublets means that the laser comes into resonance witheach at a slightly different time on each rotation. We have identified two possible mechanisms
related to this effect which can produce D̃-odd spatial structure in the cloud.
The first is that the interaction with each doublet takes place at slightly different spatial
locations, separated by δxdep ∼ 60 µm. This can cause an initial offset between the two clouds,
D ≲ β
ri0
dep δxdep ∼ 6 µm. Initial offsets in the center position of the two doublets can cause a

non-zero ⟨ri ⟩ D but, given that both clouds oscillate about the trap center with trap frequencies
ωsec ∼ 2π × 1 kHz, the time-averaged effect is heavily suppressed by the long Ramsey times,t R ∼ 2 s used in our experiment, ⟨ri ⟩ D ∼ ⟨ri0 ⟩ωsec /2πt R ∼ 30 nm. This effect is D̃ R̃-odd and so
produces a systematic when coupled to gradients in f BR . The f BR gradients we measure in theradial directions are < 20 mHz cm−1 , giving a systematic shift of < 100 nHz. In reality we expect
any effect to be significantly smaller still due to the depletion laser being on for a time which is
D is greatly reduced.
long compared to one trap period so that any ri0

If the laser beam has non-zero displacement from the center of the micromotion, xdep , theinteraction can selectively remove more of the hotter ions from one doublet than from the other,producing an effective ⟨ri2 ⟩ D in those ions remaining in 3 ∆1 . In the worst-case limit where thedepletion laser is much smaller than the ion cloud, the mean square position of the removed ionsis

( xdep +δxdep 2
δxdep 2) − ( xdep −) = 2xdep δxdep .
22(3.75)The ions oscillate backwards and forward in the trap and so, in the limit of no ion-ion collisions,
their time-averaged mean square position during the Ramsey time is reduced by a factor of 2.
Simulations of more realistic ratios of laser to cloud size reduce this by a further factor of ∼ 3. We

113estimate xdep < 2 mm and so estimate the difference in mean square position of the remaining
ions as ⟨ri 2 ⟩ D ∼ β dep xdep δxdep /3 ∼ 4 × 10−5 cm2 . The effect is again D̃ R̃-odd and so can produce
a systematic when accompanied by curvature in f BR . The largest such curvature we measureis 70 mHz cm−2 , producing a systematic of ∼ 3 µHz. In reality, ion-ion collisions redistributethe velocities and positions of ions in the trap so that any initial difference in the size of theclouds is rapidly scrambled. To set an upper limit on the timescale of this thermalisation, wenote that we measure a curvature in f of 1200 mHz cm−2 , caused by the spatial variation of the⃗ 0 coils. For a 1 cm gaussian cloud with no ion-ion collisions, this would
magnetic field from the B
cause decoherence of ∼ 20% of the contrast in ∼ 50 ms. This is roughly equal to our total lossof contrast over a 3 s Ramsey time and so ∼ 50 ms sets a rough upper limit for the timescaleof thermalisation. For the shortest Ramsey times used in the dataset of ∼ 1.5 s, the possible
systematic shift is reduced by a factor of 50 ms/1.5 s ∼ 30. As such we estimate the size of any
residual systematic to be < 100 nHz and so do not include its contribution in our systematic error
budget.
The second mechanism is that the electric field vector defining the quantisation axis is, on
average, pointing in slightly different directions when the laser interacts with each doublet. These
angles are given by θu = θ0 /2 + δθ, θl = −θ0 /2 + δθ where θ0 ∼ ∆St /∆Dopp ∼ 0.14 is the difference in the angle of Erot when it addresses each of the two doublets and δθ ∼ δdepl /∆Doppis half the difference in the magnitude of those angles caused by imperfect laser detuningδdepl ≲ 20 MHz. In the unsaturated limit, the scattering rate with which each doublet interacts with the laser is then different by a factor cos θu − cos θl ∼ −2∆St δdepl /∆2Dopp . Again, thelaser interacts with a fraction ∼ 0.1 of the molecules and so we estimate the possible difference inmean square position as ⟨ri 2 ⟩ D ∼ ⟨ri2 ⟩0.1(2∆St δdepl /∆2Dopp ) ∼ 6 × 10−4 cm2 . This effect is R̃-evenand so can produce a systematic when accompanied by curvature in f B . The largest such curvature we measure is 10 mHz cm−2 , producing a systematic of < 6 µHz. The same thermalizationarguments discussed in the previous paragraph reduce this to below 200 nHz and so we include

114
no contribution in our systematic error budget.
Finally, we consider the first dissociation laser. The photodissociation beam is a doubled,
pulsed dye laser with a 10 ns pulse width. The central frequency of the laser pulse is stabilized
to a wave meter, with the set point chosen to be a compromise between the peaks of the twotransitions for dissociating the upper and lower doublets. Over the course of the entire data
run we find that on average we have see D̃-odd term of 2% in the number of Hafnium ions
we create. A residual differences in efficiency, in combination with intensity-driven saturationand spatial structure on the beam can generate a D-odd contribution to the mean-square spatial
extent of the molecules that participate in the Ramsey fringes. This in turn can combine withcurvature in f B , which can be as large as 10 mHz cm−2 , to give a systematic error on f BD . Aswith possible D̃-odd effects arising from the optical pumping beams, the size of the resulting
systematic error is strongly suppressed by ion-ion thermalization that occurs over the courseof the t R and washes out size differences. The pulsed lasers width in frequency space is large
compared to the transform-limited value for a 10 ns pulse width, and frequency structure within
the pulse is not well characterized, which makes modeling the effect of the laser beam a little
uncertain. Just before entering the chamber, however, the dissociation beam passes through aniris with 1 cm diameter. We can make the simplifying and very pessimistic assumption that theextra 2% D̃-odd change in Hafnium ions is due entirely to molecules dissociated just at the outeredge of the laser beam. In this way we can set a conservative limit that the size of this effect on
f BD must be less than 3.5 µHz.
In addition to these effects from the lasers, we have also considered possible effects of D̃odd forces on the ions due to electric field gradients, and D̃-odd heating due to photon scattering.
We find each to be significantly smaller than our limits on those from the lasers.

115

3.8

Phase Shifts
In this section we explore possible systematic effects caused by shifts in the phase of theions, or our measurement of that phase. Using trapped ions for our measurement makes us
relatively insensitive to these effects when compared to other similar experiments using beams
of atoms or molecules. The first reason is that we can vary the free evolution time of our measurement; each block of our dataset consists of data taken at short and long Ramsey times. Most
phase shifts caused by state-preparation or measurement are common mode and so cancel out
in our science signal which is only sensitive to the differential phase evolution between the earlyand late time data. The second reason is that any residual shifts—from phase effects which are
different for the early and late time data—get divided by the difference in free evolution time
between the two measurements. The coherence time of the ions in our experiment is roughly
three orders of magnitude longer than comparable experiments using beams and so any effects
are reduced by a similar factor. In Sec. 3.8.2, we consider errors caused by imperfections in our
measurement of those accumulated phases, and in Sec. 3.8.1, we consider phase errors caused byimperfections in state-preparation.

3.8.1

State preparation

3.8.1.1

phases due to π/2 pulses

During preliminary data runs, we measured a small, and unexpected, contribution to theinitial phase (roughly 10 mrad) which is odd under D̃, B̃ and Ĩ. We studied the dependence of thiseffect on many experimental parameters. In particular we found: (a) the magnitude of the effectdepended strongly on the parameters used for the π/2 pulses–we could greatly suppress theeffect by implementing π/2 pulses with higher values of Erot during them and compensating byincreasing their duration; (b) the sign of the effect changed when we projected into the opposite
doublet; and (c) contrary to what we had anticipated for an effect which is odd under Ĩ, there isno dependence on the side of the phosphor screen that each doublet is imaged onto.

116

Figure 3.11: Toy model to explain ϕ DBI . (i) uncoupled basis, (ii) coupled basis.

We have settled on a probable physical mechanism for the effect which qualitatively matches
our observations by including the effects of other states 3 ∆1 (v = 0, J = 1) outside the two
doublets used for spectroscopy. The important features of this mechanism can be illustrated
with the toy model system shown in Fig. 3.11(i). The model includes just 4 of the 12 levels in
3 ∆ ( J = 1): the two F = 3/2, | m | = 3/2 states of a given doublet, labelled | a ⟩ and | b ⟩, and theF1

two F = 3/2, |m F | = 1/2 levels which lie closest to them in energy, labelled |c⟩ and |d⟩. In anelectric field, and in the absence of any magnetic field or rotation, the two pairs of states with|m F | = 3/2 and |m F | = 1/2 are each degenerate. The Stark splitting between the two pairs is κ.
When rotation is included, there is a direct coupling mixing | a⟩ and |c⟩, and |b⟩ and |d⟩. In thefull 12-level model there is also a second-order coupling between |c⟩ and |d⟩— involving a single
matrix element associated with each of rotation and Ω-doubling—and a fourth-order coupling
between | a⟩ and |b⟩— involving three matrix elements associated with rotation and a matrix element associated with Ω-doubling. Although these couplings are much smaller than the direct
coupling, they are important because they act in degenerate subspaces so we include their effects
in our toy model through couplings ∆ and ∆2 . We will not include the effect of a magnetic field
because we are interested in what happens during a π/2-pulse where the other effects are all117much larger. The Hamiltonian for our toy system in this basis is then




0 ∆ 
 κ δ




 δ 0 ∆2 0 
,
H=


 0 ∆2 0 δ 




∆ 0
δ κ

(3.76)

where the states are in order of m F : | a⟩ , |c⟩ , |d⟩ , |b⟩.
We can simplify our thinking by changing to a different basis. Consider the case where wesomehow independently tune δ to zero; we now have two uncoupled systems, | a⟩ and |b⟩, and |c⟩and |d⟩. We can diagonalize each system to give eigenstates which are fully mixed as shown in
Fig. 3.11(ii): |3/2, +⟩ = √1 (| a⟩ + |b⟩) and |3/2, −⟩ = √1 (| a⟩ − |b⟩), and |1/2, +⟩ = √1 (|c⟩ + |d⟩)2

2

2

and |1/2, −⟩ = √1 (|c⟩ − |d⟩). Rewriting the full toy-model Hamiltonian in this basis we have
2





δ
0
0 
 ∆2




0
0 
 δ κ+∆

,
H=

 0
0
− ∆2
δ 




0
0
δ
κ−∆

(3.77)

where the states are in the order |1/2, +⟩ , |3/2, +⟩ , |1/2, −⟩ , |3/2, −⟩. The Hamiltonian is now
block diagonal. During the second π/2 pulse, the mixing between | a⟩ and |b⟩ causes population
to oscillate between them (as intended), but it also causes a weak oscillation between the |m F | =3/2 states and the |m F | = 1/2 states. This oscillation depends on the initial phase difference
between the | a⟩ and |b⟩ states. The two most important phases to consider are the sides of thefringe; where we are most sensitive to phase shifts, and thus where we take most of our data. On
the side of the fringe, the state immediately before the second π/2 pulse is |3/2, +⟩ or |3/2, −⟩.
It can be seen from Eq. 3.77 that when δ = 0, these states are eigenstates of the Hamiltonian
and the π/2-pulse does nothing. When δ is non-zero, the coupling causes population to betransferred from |3/2, +⟩ to |1/2, +⟩ and from |3/2, −⟩ to |1/2, −⟩ with Rabi frequencies Ω+118and Ω− respectively,Ω+ =q

4δ2 + (κ + ∆ − ∆2 )2 ∼ κ + ∆ − ∆2 ,(3.78)Ω− =q

4δ2 + (κ − ∆ + ∆2 )2 ∼ κ − ∆ + ∆2 .

(3.79)The two Rabi frequencies are different because the energy gap is different in each case. The
amplitudes of the oscillations in each case are then
δ2
δ2
∼
,
4δ2 + (κ + ∆ − ∆2 )2κ2
δ2δ2A− = 2∼.
4δ + (κ − ∆ + ∆2 )2
κ2A+ =(3.80)(3.81)

The approximate expressions in both cases assume κ ≫ δ, ∆, ∆2 . This difference in Rabi frequencies results in a different amount of the population being transferred to |c⟩ and |d⟩, dependent on
the initial phase. We measure only the population in either | a⟩ or |b⟩ and never the populationin |c⟩ or |d⟩, and so this population transfer appears as an apparent phase shift. The population
oscillations are happening very fast—the frequencies are of order the energy gap between the
3/2 states and the 1/2 states, κ ∼ MHz—and so the size of the apparent phase shift can dependvery sensitively on the exact parameters of the π/2-pulse. The maximum size of the effect goesas 1/κ 2 and so depends strongly on the size of Erot during the π/2 pulse.
We now examine the dependence of this effect on the experimental switch state:(i) Ĩ: in our implementation of the Ĩ switch, the m F state we project into, and read out of, isĨ odd. This corresponds to measuring, for example, | a⟩ instead of |b⟩. On the side of thefringe where the population in | a⟩ is decreasing with Ramsey time, the population in |b⟩ isincreasing with Ramsey time and as a result the apparent phase shift—a relative change in
population which has same sign for both states—is Ĩ odd.

⃗ 0 and so is B̃-odd.
(ii) B̃: the phase shift is completely independent B(iii) D̃: in the opposing doublet, the sign of κ is flipped; the |m F | = 1/2 states are above the119|m F | = 3/2 states in energy. This means that the relative rate of transfer between the twosides of the fringe are flipped and the effect is D̃ odd.
(iv) R̃: δ, ∆ and ∆2 all involve odd numbers of matrix elements associated with rotation and so
flip sign with rotation direction. This flips the sign of the effect so that the relative rates
of population transfer out of the two states are reversed. However, the flipping sign of∆ means that the initial superposition produced by the first π/2 pulse is π out of phasecompared with the opposite rotation direction so the phase shift at a given Ramsey timeends up being R̃ even.
In the experiment it is very difficult to control the parameters of the π/2 pulses at the required
level to quantitatively verify this mechanism is at play. In addition small differences in the electricfield experienced by different ions, and coupling to other states in the full Hamiltonian, cause thepopulation oscillations to dephase rapidly, and so we were never able to observe them. However,
we believe the qualitative agreement with what we observe to be convincing.
We took three steps to mitigate the effects of ϕ DBI . We reduced the size of the effectdramatically by increasing the value of Erot that we ramp down to during the π/2-pulse. We
also added the P̃ chop to our experimental sequence for the eEDM dataset; in every other block,
we read out of the opposite state, so that any remaining effect would change sign. Finally,
midway through the dataset, we reversed the polarization of all the optical pumping, depletionand dissociation lasers, again flipping the sign of any effect. We can still check for any residual
effect by looking at the difference in ϕ DBIP before and after the waveplate change; in the eEDM
dataset its average value was 1.1(1) mrad. Such phase shifts could potentially leak into frequency
channels if, for example, the second π/2 pulse depended on the Ramsey time, perhaps becauseof heating in the amplifiers or similar. We note that during our exploration of the effect we wereable to increase its size to 170 mrad, and saw no shift in f DBI at the 1σ confidence level. Using
this data we can set a limit on any shift caused by the phase at −3(4) µHz mrad−1 , increasing

120
confidence in our ability to reject systematic effects associated with phases caused by the π/2pulses. In the eEDM dataset, none of the other D̃-odd phases (except ϕ D ) exceed 750 µrad.

3.8.2Internal state measurement

3.8.2.1

Improperly characterized imaging contrast

Our imaging process allows us to differentiate between Hf ions originating from the upperand lower doublets by projecting them onto different sides of the imaging MCP. However, due tothe initial size of the cloud and it’s finite temperature the two clouds are not perfectly separated
and they overlap slightly at the center of the screen. This overlap can cause our measurementof the difference frequency to be pulled and, if improperly characterised, can potentially cause asystematic error. To see how this works, consider how the number of ions detected on each sideof the MCP, and in each phase, depends on the phase evolution of the upper and lower doubletsΦu and Φl ,NN
(1 − sin(Φu ))(1 − ϵ) + (1 − sin(Φl ))(ϵ)
22NNNuin = (1 + sin(Φu ))(1 − ϵ) + (1 + sin(Φl ))(ϵ)
22NNNlanti = (1 − sin(Φl ))(1 − ϵ) + (1 − sin(Φu ))(ϵ)
22NNNlin = (1 + sin(Φl ))(1 − ϵ) + (1 + sin(Φu ))(ϵ),
22Nuanti =(3.82)

where N is the mean number of ions measured on each side of the MCP, ϵ is a parameter whichcharacterises the amount of leakage of each cloud onto the other side of the MCP screen, and wehave assumed perfect contrast. We can now form asymmetries as we do in our analysis,
Au =Nuin − NuantiNuin + Nuanti= (1 − ϵ) sin Φu + ϵ sin Φl ,N in − Nlanti
Al = lin
Nl + Nlanti= (1 − ϵ) sin Φu + ϵ sin Φl .

(3.83)

121
We can simplify our thinking by assuming that we are taking data close to the side of the fringe
such that Φu ≃ 2nπ + δu and Φl ≃ 2mπ + δl for n and m integer. In this limit we have
Au ≃ δu − ϵ(δu − δl )(3.84)Al ≃ δl + ϵ(δu − δl ).
The apparent phase of each of the doublets are pulled towards each other by an amount that
depends on the leakage ϵ and their difference in true phase δu − δl . We account for this in our
analysis by including a parameter C I , the imaging contrast. By comparing Eq. 3.83 with Eq. 3.11
we can identify ϵ = (1 − C I )/2. Now suppose we mischaracterize the imaging contrast C I ,assigning it an incorrect value C ′I . The resultant fitted values for the phases of the upper andlower doublet δu′ and δl′ will then satisfy the following expressions
1 − C ′I ′1 − CI(δu − δl′ ) = δu −(δu − δl ),22
1 − C ′I ′1 − CIδl′ +(δu − δl′ ) = δl +(δu − δl ).
2
2

δu′ −

(3.85)

We find that the fitted difference phase ∆′ = δu′ − δl′ is different from the true difference phase∆ = δu − δl by an amount∆′ − ∆ =(C I − C ′I )∆.
C ′I(3.86)

Both the early and late time data in our Ramsey are fringes are potentially susceptible to thisphase pulling effect and so it is important that we properly characterize and understand C I . To
do so, we took fringes where the data was collected deliberately offset from the early- or latetime zero crossings of the difference fringe. Taking the late time data removed from the beat by
a number of periods n causes the fitted f D to be pulled by an amountδfD =n (C I − C ′I ) f D∆′ − ∆
=
.
2πt R
tR
C ′I
f

(3.87)

Moving the early time data has the same size effect but with the opposite sign. For a given C ′I
and swatch size, we can fit the extracted f D vs the number of periods we are offset from the zerocrossing of the difference fringe. Figure 3.12 shows how this slope varies with the value of C ′I√
for the swatch size of 90 pixels used in our measurement, chosen to maximize C N. The correct

122

Figure 3.12: Example data. Plotting the slope of f D when the early-time data is taken at varying
offsets from the zero crossing of the difference fringe. The x-intercept gives us the true value of
our imaging contrast, which we then use in our fitting program.

value of C ′I is the one for which∂fD∂n= 0, here C ′I = 0.90(1), corresponding to ϵ = 0.050(5),which we use for analysis of our whole dataset. Whilst we can measure C I very precisely, itsvalue can wander slightly over time. We have repeated this measurement many times over thecourse a year and find that the early and late time C I are consistent with one another, andbetween measurements, to ±0.05. We conservatively estimate the largest possible deviation in
C I , averaged over the dataset to be δC I ∼ 0.05.
For this effect to leak into f BD and cause a systematic shift in our science signal requires
either the amount we miss the beat by to B̃-odd, or the imaging contrast itself to be B̃-odd. We
consider the B̃-odd phase first. Because the mean value of ϕ D is negative, the early time beathappens before zero, and so we are forced to take data with nonzero ∆. In this case a non-zerovalue of ϕ BD will mean that the early time data is taken at a different ∆ depending on the B̃switch. The value of ϕ BD over the dataset is −30(140) µrad. Combining this with the uncertainty
ϕ DB δC

in C I above leads to a systematic uncertainty 2πtR CII ∼ 0.7 µHz.
A systematic shift can also potentially be caused by ϕ D combined with a B̃-odd imaging
contrast C IB . In order for this shift to reach the 5 µHz level, would require C IB ∼ 0.005, or
equivalently ϵ B ∼ 2.5 × 10−3 , or about 5% of the total leakage. We know of no mechanism which123
can cause such an effect and so include no contribution in our systematic uncertainty budget.

3.8.2.2

Swatch position

As described in Section 4.2, the position of the center line and corresponding swatch foreach switch state is determined by an algorithm based on images of the cloud taken without
any π/2-pulses. In this section we explore the consequences of a possible systematic error in thecenter position dividing line between the two doublets.
We reanalysed the dataset, with the center line in each image displaced by different numberof pixels left and right. The largest observed shifts in the frequency and phase channels are shownin Fig. 3.15. We note that the signal to noise in these shifts is much better than indicated by theerror bars shown because we are reanalysing the same data. We can explain the observed shifts
using ideas from the previous section. Let’s use Eq. 3.85 as before but this time with the phasepulling on each of the doublets separated,(δu − δl )(ϵ − ϵ′ ),2ϵ′ − 1(δu − δl )(ϵ − ϵ′ )δl′ − δl = −.
2ϵ′ − 1δu′ − δu =(3.88)(3.89)

We can combine both of these equations as
δϕ = D̃∆(ϵ − ϵ′ )2ϵ′ − 1(3.90)

where δϕ is the phase pulling of a single fringe and ∆ = δu − δl is the amount we are missing
the beat by. We can express the effect of moving the swatch center around by modifying ϵ, theamount each doublet leaks into the otherϵ = ϵ0 + κ D̃ ĨYc − ν( D̃ ĨYc )2 .

(3.91)

This equation expresses the fact that as we move the swatch over to one side, there is moreleakage into one of the doublets and less into the other. Which way around they are depends
on the I switch. The first order effect, ∝ κ, is linear but for larger swatch displacements, the124

Figure 3.13: Varying the swatch width while correcting for the doublet contrast. We see nomeaningful shifts in our science channel and a small residual shift in f D . In the lower panel wehave averaged over the 3 distinct values of f D that we operated at.

125
second order effect, ∝ ν, starts to become important too. Substituting this back into Eq. 3.90 andassuming for simplicity that we have done a good job of picking the doublet contrast in the firstplace so that ϵ′ = ϵ0 , we getδϕ =∆2ϵ′ − 1( ĨκYc − D̃νYc2 ).

(3.92)The last factor we need to include to explain the data is to make the approximation that the mainplace we are missing the beat is at early time (due to the beat being before T = 0), so that ∆ ∼ ϕ D .
Here we have included the largest (by far) contribution to the D-odd phase only. Finally we haveδϕ =12ϵ′ − 1( Ĩκϕ D Yc − D̃νϕ D Yc2 ).

(3.93)

So the principle effects we expect are a linear shift in ϕ I and a smaller quadratic effects in ϕ D .
Because (almost) all of the pulling is happening at early time, each of these could be expected
to have an echo in the corresponding f channel, with opposite sign and smaller by a factor of
2π × T ∼ 15 s rad. Each of these can be seen in Fig. 3.15 from which we infer κ = −1.89(6) × 10−3
per pixel and ν = −7.0(2) × 10−6 per pixel squared. The shifts seen in ϕ DI and f DI (about a factorof 5 smaller than those in ϕ I and f I ) could be caused by a Ĩ-odd systematic error in the swatch
ϕD2

position Y I . Including this in Eq. 3.93 yields an additional contribution 2ϵ′ −1 (κY I − D̃νY I −2 D̃ ĨνYc Y I ). Comparing the term linear in Yc to the observed gradient in ϕ DI gives Y I = 25(1)pixels. Other possible systematic shifts in swatch position can be constrained by the data in thesame way to less than 5 pixels.
The largest gradient in a frequency channel seen in this analysis is ∂ f I /∂Yc ∼ 2 µHz per
pixel. It is important to realize that the swatch displacement that is carried out here is just oneof many possible ways of moving the swatch. Because we moved the swatch the same directionin each switch state, Yc shows up in Eq. 3.91 with D̃ Ĩ. There are 7 other possible ways to movethe swatch, always D̃-odd but with all other possible switch-state dependence. We can infer
the effects of these possible systematic effects from those shown here. We are interested in any
possible shifts in ϕ DB and f DB which can only show up quadratically in any systematic error

126

Figure 3.14: The effect of moving the center of the swatch on our science channel.

in the swatch position. To see an effect requires either missing the beat by a B̃-odd amount,or a combination of two systematic shifts in swatch position, one B̃-odd and on B̃-even. As
discussed in the previous section, the former can happen due to a combination of our non-zero
ϕ D and a non-zero ϕ DB . In our dataset, ϕ DB = −30(140) µrad which, combined with Y I gives
ϕ DB νY I

2

a systematic | 2πT (1−eϵ′ ) | < 70 nHz. The largest contribution to the latter is from Y I combinedwith Y BI , constrained by looking at the gradient of ϕ DBI in our analysis to be 1(2) pixels. This
2ϕ D νY I Y BI

gives a systematic of | 2πT (1−eϵ′ ) | < 1.2 µHz. We include the quadrature sum of these two in oursystematic uncertainty budget.

3.8.2.3

Counting saturation

Our imaging system has a limited capacity, and if that were somehow magnetic field anddoublet odd it could be a concern. It’s hard to conceive of a method for this to be both magnetic
field and doublet odd, however, because even if one side of the screen saturated faster than theother, that asymmetry is heavily suppressed by the fact that the Ĩ chop swaps which side of thescreen we read each doublet out on.
Collating the imaging data over 25 blocks we can see that our imaging does indeed saturate(the image intensity does not scale linearly with the number of Hf+ ions we count, see Figure

127

Figure 3.15: Moving the center of the swatch has a measurable effect in several parity channels.
Note all panels show the entire aggregated dataset except for the f D panel, which shows a subsetof measurements performed at a single f 0

128

Figure 3.16: Imaging data collated over 25 blocks. The lower doublet data is in pink while theupper doublet data is in blue.

3.16).
++
We can fit the data in Figure 3.16 to the function I = αHfcounted+ fi(Hfcounted)2 . For small++
Hfcounted, the image intensity should scale linearly with Hfcounted. Therefore, we can apply a+++saturation correction to our data: Hfreal= Hfcounted(1 + fi/ffHfcounted). We then re-analyzed all

our data with several saturation corrections applied: none/zero, two incorrect ones, and the“true” value of β/α = 8.52 × 10−4 . We see no observable shift in our science channel and so we
do not include this effect in our systematic error budget.

3.8.2.4

Time-varying offsets

Our asymmetry fringes can be offset from zero, characterised by the parameters Ou/l from
Eq. 3.9. It is possible that these offsets might decay as a function of Ramsey time, for example
if collisions slowly redistribute population between the two states in a doublet. Since we fit ourfringes with a single offset for all Ramsey times, this could potentially lead to a systematic shift
in the relative phases. To investigate this possibility, we refit our entire dataset with a fixed
differential offset between the early and late time asymmetry data points and looked for changes
in f BD . We observe no shifts at the nHz level and so include no contribution in our systematic
uncertainty budget.

129

Figure 3.17

Table 3.7: Systematic error budget
EffectMagnetic⃗0
Non-reversing B
Second harmonic of Erot and transverse magnetic field
Third harmonic of Erot and magnetic field gradients
Higher harmonics of Erot and higher order magnetic field gradients
Ellipticity of Erot and magnetic field gradients
Berry’s phase
Phase modulation due to axial secular motion
Axial 2nd harmonic with ellipticity of Erot
Higher harmonics of Erot
Rotation induced mixing
Leaking of f BR
Axial magnetic fields
Other frequency
Imperfect overlap of spatial distributions
Phase
Improperly characterized imaging contrast
Swatch position
Total

SectionCorrection (µHz)

Uncertainty (µHz)

3.7.1.1
3.7.1.3
3.7.1.4
3.7.1.4
3.7.1.5

0.2

< 0.1
2
2
< 0.1
1

3.7.2.3
3.7.2.4
3.7.2.53
2
3

3.7.3.1
3.7.3.2< 0.2
0.4

3.7.4.2

3.5

3.8.2.1
3.8.2.2

0.7
1.2
0.2

7

130

3.9

Interpretation
Table 4.1 presents our error budget. We have demonstrated a significant advance in thecharacterization of our experiment resulting in a lower systematic uncertainty and a measurement which is statistics dominated. The use of rotating bias fields and trapped molecules is a
powerful technique for suppressing systematics; the rotation means that most stationary fieldinduced systematics average away over a rotation cycle, and the trapped species allows us tomeasure both early- and late-time phase which eliminates a large class of state-preparation systematics one might otherwise have to worry about. As eEDM sensitivity is pushed into the next
decade of accuracy, these advantages may prove essential.

3.10

Supplemental Information / Appendices to the paper

3.10.1

Overview of data

The data blocks can be divided into three main categories, based on the approximate valueof f 0 , corresponding to roughly 77, 105, and 151 Hz. f D ≃ δg/g f 0 , thus came in respectively
at -171, -230 and −327 mHz, and by choosing late-time Ramsey fringes centered at 1/2 f D or
respectively roughly 2.9. 2.2 and 1.5 s, the late-time Ramsey fringes were taken at the re-phasingof the fringes for the upper and lower doublets. In the tables below, we present averages taken
separately over the three ranges of f 0 , plus an average taken over all the blocks. In each casethe quoted one-σ errors are based on combining the estimated error on all the relevant blocks,
pand then relaxing that value by multiplying by χ2 to correct for overscatter. For reference
we include the relevant value of χ2 as calculated before the error bars were relaxed. The χ2
associated with the relaxed error bars displayed here are all by construction equal to 1.
The tables under discussion in this section are presented in E.

131
3.10.2Overall comments

The parameters which are even in all switch states (superscript 0) are the noisiest, as they
are unprotected from all the drifts in the experimental conditions. These drifts arise from manysources, for instance (i) changes in the temperature of the lab, or of the water cooling the power
op-amps that drive the ion-trap electrodes. (ii) changes in the ambient magnetic field and its
gradients, arising e.g. from a distant freight elevator, a less-distant optical table, or a still-closer
weld in an only nominally nonmagnetic UHV chamber (iii) variation in the number, temperature,⃗ 0 coils. Cloud sizeand density of the ion cloud (iv) drifts in the current supply that drives the B
and thus ion density and trap temperature were not well-characterized parameters. Ion number
in particular drifts from block to block, and due to mean-field coulomb repulsion within thecloud the ion number is coupled to cloud size, and from there to f 0 which is an average overspatial field inhomogeneities. The strength of radial and axial trap confinement were treatedas parameters to be tweaked to maximize the precision per block of data. Ion production was
frequently re-optimized as well, with resulting jumps in f 0 between blocks of tens of milliherz
or more.
All P̃-odd values are in a category by themselves, in that they were not collected as partof a rigorously implemented intra-block chop. Instead, we reverse the direction of the relevant
polarizer every other block. On some days, we took an odd number of blocks. In other instances
we vetoed an entire block of data which left the next block of data unpaired with a block of
opposite P switch. For these reasons, from our entire run of ∼ 1300 blocks we do not try to
generate ∼ 650 sets of P̃-odd fit values based on pairs of matched, sequentially collected blocks.
Instead, for each of the f 0 superblocks, we divide the blocks into the P̃ = 1 and P̃ = −1 piles,
calculate weighted mean values for each fitting parameter and each of the { D̃, B̃, R̃, Ĩ } parity
channels, with error bars corrected for scatter. Then, element-wise across the large table ofvalues, we take either a sum or difference between the P̃ = 1 and P̃ = −1 to create the overallP̃-even or P̃-odd parity channels. In this method, there is no particular meaning to χ2 for the132P̃-odd parity channels, so no such value presented in the tables below.
After each block we attempt to servo the value of f B back to zero for the subsequent block.
Thus the mean value of f B is very low even though f B channel is susceptible to drifts in ambientconditions. These drifts can be tracked because we record the value of the current in the coils
used to servo f B .
Looking across the 480 parity-channel values associated with the three frequency superblocks,
and the 160 fully aggregated values, we see the overwhelming majority of the values are quite
close to zero. There are a few numbers which are dramatically different from zero, for goodreason:(i) f 0 itself of course, the absolute value of the energy difference between m F = ±3/2, which
we apply a bias magnetic field to set to a value between 75 and 155 Hz, depending on the
superblock.
(ii) f D comes in a −225 mHz, smaller than the mean value of f 0 by a factor of about −1/450,
which is the fractional difference in g-factor between the upper and lower states, plus a few
smaller correction factors.
(iii) f BR , at 212 mHz, comes from a magnetic field, rotating at 375 kHz, generated by oscillating
currents that arise from charging the fins to the oscillating voltages that create the rotating
electric bias field.
(iv) f BDR , at roughly 410 µHz, is f BR echoed in corresponding D̃-odd channel by the same factorof δg/g ∼ −1/450. After a correction for finite f BR is applied, f BDR is within 1.7 σ of zero.
(v) f BD is proportional to the electron’s electric dipole moment de and its value in principle
could be quite large. If we take the ACME 90% confidence limit, de < 1.1 × 10−29 e cm e-cm
at face value, we would have similar confidence that our measured value will lie between

133−120 µHz < f BD < 120 µHz. For all but the last few weeks of the experimental effort
described here, the blinding procedure built into our analysis software made f BD appear tohave the arbitrary value near 16.1 mHz.
(vi) γ0 The mean decay rate of coherence is about 0.1 s−1 . Arising from ion-ion collisions andfrom spatial inhomogeneities, gamma is a noisy number that depends sensitively on numberof trapped ions and fine details of the trap shimming.
(vii) ϕ0 , fairly large at tens of milliradians, is measured to be proportional to f 0 and is consistent
with ∼ 150 µs (CK) offset in where we define t R = 0. This is not unexpected given that theduring the π/2 pulses the rabi frequency varies during the 1 ms pulse duration.
(viii) O0 , O D , C D , ϕ D . These nonzero terms all arise predominantly from the same basic cause.
Due to the presence of the F = 1/2 manifold intermingled with the F = 3/2 levels, the coupling procedure we use to drive nominal π/2 pulses is characterized by a slightly different
effective Rabi frequency for the upper and lower levels. The duration of our coupling pulses
is chosen as a compromise, and results in a state mixing of slightly larger than π/2 for one
doublet and slightly smaller than π/2 for the other. This is the origin of the fringe offset
O0 and this compromise also contributes to the deviation of fringe contrast from unity. Our
compromise π/2 pulse duration was in retrospect chosen imperfectly, such that the deviation from perfect π/2 duration was not equal and opposite for the upper and lower levels.
This led to finite values of O D and C D . With the π/2 pulses not applied at exactly zerodetuning, a ϕ D term also results.
Beyond these large (and largely understood) nonzero fit values, in a perfect version of theexperiment we would like to see the remainder of the 480 values be very small, and ideally within
measurement error of zero.

134
3.10.3Frequencies

Of the remaining 27 frequency channels, there are five that differ from zero by more than
4σ, i.e. by more than twice their scatter-adjusted estimated error.
The four largest are f R , f RI , f I , and f BI with frequencies of -4.2, -2.2, -0.8 and −0.42 mHz
respectively. What f R , f RI , and f I have in common is that they correspond to switches which
involve changing the inputs to our direct digital synthesis (DDS) boards. Each fin is driven by a⃗rot , the R̃-switch , we change the relative
distinct DDS board. To change the sign of rotation of E
digital phases we load into the different DDS boards. To effect the Ĩ switch, which has to do with⃗rot electric field at the instant we do the photodissociation pulse, we
the direction of the the E
switch the sign of the amplitude input into all 8 boards. The digital math performed within the
DDS boards must always include at least implicitly a truncation to the least significant bit (LSB) of
the digital-to-analog converter, and this process is repeatable but subject to tiny discontinuities
from the round-off error. Our fringe frequency is directly proportional to the rotating electricfield amplitude. The commanded voltage on any fin is sinusoidal in time. A shift of 4 mHz⃗rot corresponding to much less than 1 LSB of
could result in a change in the magnitude of E
commanded voltage. In this context, our nonzero values of f R , f RI and f I seem unsurprising. f R
is additionally affected by small drifts in the equilibrium location of the center of the ion cloud,and by drifts in the magnitude of residual second harmonic contamination in Vr ot.
After the discussion above, we are left with 10 D̃-even and 14 D̃-odd channels which
we believe should be “quiet”, near zero, after a “δg/gcorrection” for the nonzero value in thecorresponding D̃-even channel, (the correction leaves f DR at 3.0 σ from zero, and f BDR at 1.6 σfrom zero). We have 24 “quiet” channels, and an additional channel f BD , which we believe should
be quiet except for any nonzero value of de , the electron’s EDM. We don’t have an independentmeasure of the correct value of de , but the 24 channels offer a chance for test of our accuracy andprecision claims. We take each quiet frequency channel, divide it by its corresponding sigma, and135
then we can ask what is the rms amount by which the items in the ensemble differ from zero?
If we omit the three-sigma value, f DR , and look at only the remaining 23 values, we get a very
pleasing answer - the frequencies differ from zero by an rms amount of 0.98. If instead we include
f DR , the rms normalized difference from zero is 1.3. This is not a catastrophically large number
for an ensemble of 24 independent points, but is a larger than one might hope. Put another way,
if you have 24 normally distributed points, there is about an 8% chance that at least one will
be as far away from its “correct” value as 3 σ. It is a thing that happens, sometimes. We haveno explanation for why f DR should be nonzero, and we can’t rule out that an uncharacterized
systematic error contributed to its nonzero value. We put a lot more effort into thinking about
and characterizing systematic errors that will move f BD around compared to errors in f DR , soeven if one in 24 measured frequencies does have an uncharacterized systematic error, this is not
a very worrisome impeachment of f BD . In the end we have chosen to comment on this slightly
aberrant observation but to make no corresponding relaxation in our error budget for our main
measurement.

3.10.4Contrasts, and contrast decays

We’ve already mention C0 and C D . The fit value C, the initial contrast in the Ramsey
fringes, is affected by parameters of the ion cloud exactly at the moment of photodissociation.
The cloud mean position, its mean velocity, and the direction in which Hafnium ions is ejected
after photodissociation, all impact where the atomic Hafnium ions eventually impact on theimaging ion detector. Different regions of the multi-channel ion detector have different sensitivity
and different propensity to saturate. We have a procedure which region of the detector receives
ions ejecting from upper doublet and which from lower doublet, but this procedure can be subject
to biases. All this means that changes in contrast can appear in multiple channels. In particular
C DI corresponds to a different contrast detected on the right or on the left side of the detector.
This is a particularly large term. C RI corresponds to the direction of the HfF+ velocity, along
the direction towards the detector, at the moment of photodissociation. This in turn affects the

136mean time-of-flight for the Hf+ ions and the HfF+ ions. We impose a hardware gate to cause
the image of ions to display only the Hf+ ions and not the heavier, slower, later-arriving HfF+
ions. The timing of this gate is designed to maximize count rate from Hf+ ions but suppress thecontrast-destroying HfF+ ions. Changing the molecule velocity at the moment of dissociation
changes the optimum time for the gate, but we do not make any adjustment to the detection
circuitry. Therefore a substantial value of C RI comes as no surprise. C R , C DR and C DRI likely
arise from similar but smaller effects. The entire point of including the contrast C as a fitting
term is to prevent the nonlinear fitting routine from interpreting changes in contrasts as changes
in frequencies. Frequencies are further isolated from misinterpreted changes in contrast (as for
instance, if the detector loses more sensitivity due to large-count rate saturation) by ensuring
we choose Ramsey times separately for data each switch state so that we take points as close as
possible to the exactly halfway up the sides of fringes.
What we call γ is really just a measure of the ratio of the contrast in the early-time fringe
to that in the late-time fringe. We don’t routinely collect over the intermediate times between thefirst, “early-time” sinusoidal cycle of the Ramsey fringe and the last “last-time” fringe. When
we have filled in some of the intermediate times, the resulting full fringe does not fit well to apure exponential decay in contrast. The various effects that limit contrast can’t be combined ina purely multiplicative way, and thus our approach to fitting (Eq. 3.13) means that any channel
with a distinctly nonzero contrast, say C RI , will have a nonzero decay rate, ie γ RI .

3.10.5PhasesThe largest observed nonzero channels in phase and offset (O, O D , ϕ, ϕ D ) are well understood. Phase errors typically arise from small imperfections in the π/2 pulses. The observed
value of ϕ BR is for instance consistent with the presence of an otherwise imperceptibly small axial⃗rot . Other channels of ϕ differ from zero by statistically significant
gradient in the magnitude of E
amounts, but all less than 3 mrad. With exception of ϕ D , no D̃-odd channel differs from zero by

137
more than 1 mrad, and the all important ϕ DB is measured to be 0.03(14) mrad. The frequencies
we measure are isolated from phase errors by the early fringe/late fringe chop. Our best effort
at creating a nonzero fringe phase resulted in a 170 mrad phase, which is 1000 times larger than
the 1 σ limit on our measured ϕ DB , but even in that case we could resolve no frequency shift in
the corresponding frequency channel.

3.10.6

Offsets

Offsets in our Ramsey fringes can occur due to imperfections in π/2 pulses, as occurs
in O0 , O D and presumably O BR . There are scenarios where offsets in some channels can occur
due to minor spatial variations in the polarization of laser beams, such as could happen from a
dusty waveplate. This may account for the anomalous value of O DIP . Other than O0 and O D ,the magnitudes of offset channels are less than one thousandth of the average fringe amplitude.
Even we did not allow the fitting routine to relax around a nonzero offset, the spacing of ourRamsey time points is such that it would strongly suppress coupling of offsets to frequencies in
our fits.

This marks the end of the manuscript draft
3.11

Not Systematics
In this section I’m going to do a quick review of some effects we observed which are notsystematics, but which were interesting/edifying/time-wasting nonetheless.

3.11.1

Shifts vs z position

We have recorded f , f B , & f BR all varying with the z position of the ions in the trap.
Plotting script for the plots in this section are in the X drive in 211129, titled plotting_z_shim.
We were able to explain f and f B fairly quickly, but it took us some time to figure out f BR .

138

Figure 3.18: Observed frequency shifts in z

Figure 3.19: Observed frequency shifts in z

3.11.1.1

f vs zThe parabolic shift in f (z) can be explained by constructing a simple model of the axial
gradient (Brot ) coils, and expanding the resulting magnetic field out to second order. The fieldfrom a single coil is (up to a factor of current and 4πϵ0 ):k=4Rρ
( R + ρ)2 + (z − z0)2

(3.94)

where R is the radius of the coil, ρ is the radial distance from the center, and z is the distancealong the axis. We can take this expression and add a second term at the correct distance (and139
sign for the current) to model an anti-Helmholtz configuration, then expand that expression,
with our actual coil parameters substituted, to second order in ρ and z. This gives a term like(r − Krz2 ) Baxgrad , with K = 6 × 10−5 .The fit to the data gives K = 9 × 10−5 .
152
151
150
149
148
147
-20

-10

0

10

20

Figure 3.20: Fitting f (z)Note the model is Luke’s mathematica notebook titled Baxgrad model which is in 201112
as is the analysis code which generated the f (z) plot, named analyzing_frequencies_vs_zshim.

3.11.1.2

f B vs z

nr =f B (z) shows a shift of 0.007 Hz/mm or 0.07 Hz/cm. We can approximate f B ∼ (3gF µ B /h) Brotnr
(3gF µ B /h) Baxgrad
rrot , where Bnr is the non-reversing magnetic gradient. We can differentiate both∂fBsides to get ∂z = (3gF µ B /h)
∂fB

nr
nr
∂Baxgrad∂Baxgrad∂fB∂z rrot . Rearranging, ( ∂z ) / (3gF µ B rrot /h ) =∂z . Plugging in

the known values ( ∂z = 0.07 Hz/cm, rrot = 0.05 cm, etc), we getnr∂Baxgrad∂znr∂Baxgrad= 1 × 10−6 G/mm2 , or∂z

= 1 × 10−4 G/cm2 . For comparison, Brot = 11.9 mG, which for rrot = 0.5mm means we

nrhave 0.238G/cm Baxgrad . So to get our observed change in f B we need a change in Baxgrad
over 1cm which is about 1/1000 of Baxgrad , which seems quite believable.
It remains to determine the source of f BR (z). The next sections detail the our investigations.

140
3.11.1.3

f BR vs z: is it from the magnetic field shimming Z coils?

In the Gen 1 trap f B (z) turned out to be caused by the z coils - turning them off removed
the dependence (see notes in lab notebook Nov 17 2016). Simulations of the Gen 1 coils showedthat the z coils were causing a quadratic field near z = 0. We repeated this experiment in the
Gen 2 trap and found that the Z coils being either on or off had no effect on the shifts observed.

Figure 3.21: Comparing frequency shifts in z with the magnetic field shimming Z coil on or off

Figure 3.22: Comparing frequency shifts in z with the magnetic field shimming Z coil on or off

141
3.11.1.4

f BR vs z: is it from axial confinement?

Changing the axial confinement changes the spatial extent of the ion cloud in the z direction. We checked to see what kind of effect this might have on the observed frequency shifts. We
did not observe any notable changes.

Figure 3.23: Comparing observed frequency shifts in z with 2 different values for the axial confinement

Figure 3.24: Comparing observed frequency shifts in z with 2 different values for the axial confinement

142
3.11.1.5

f BR vs z: is it from anharmonicity in z?

We know that our z trapping potential is anharmonic. We found this by giving the cloudvarying magnitudes of kicks and looking at the resultant slosh frequency. See the script from Oct
22 2021 titled trap_anharmonicity_analysis (lab notebook day Oct 22 2021) (the original data
is from 2020). See also the mathematica notebook attached in the ’trap anharmonicity analysis
continued’ lab notebook, it should be titled ’modelling endcap anharmonicity 3’.
Using Mathematica, I modelled the axial confinement as quadratic with higher order even
perturbations (whose strengths are parameterized by α and β, while the perturbation is parameterized by ϵ) plus a z shift (whose strength is parameterized by k2 ):U (x) =

1
1
1
mω02 x2 + mαϵx4 + + mβϵx6 + k2 mx.
24
6(3.95)

We solve for the equilibrium position (which is a rather complicated expression), then
expand our potential about the equilibrium position for small ϵ to get an expression for the
potential that we will use which eliminates the shift. Then we take the negative gradient of thisexpression to get the force:F ( x ) = −mϵ




 !
βk25 + αk23 ω04
k2 3
k2 5
+α x−
+β x−
− mxω02 .
ω02
ω02
ω010

(3.96)We are interested in periodic functions of s ≡ ωt, so we write x (t) = z(s), which leads toẍ (t) = ω 2 z′′ (s). We define ω and z as series:ω = ω0 + ϵω1 + ...

(3.97)z = z0 + ϵz1 + ...

(3.98)

and143
then we substitute the series and x (t) = z(s) into our equation of motion:mω 2 z0′′ + ϵz1


′′

= −mϵ




 !
βk25 + αk23 ω04
k2 3
k2 5
+α z−
+β z−
− mω02 z.
ω02
ω02
ω010
(3.99)

Then we expand the result for small ϵ to get:mω02 z0′′ +ϵ 2mω0ω1z0′′ + mω02 z1′′ + O ϵ2 =5βk42 mz0 10βk32 mz02 10βk22 mz03 3αk22 mz0
+
−
−
+
ω08
ω06
ω04
ω04


5βk2 mz04 3αk2 mz02
5
3
2
2
+
+
β
(−
m
)
z0
−
αmz0
−
mω0
z1
+
O
ϵ
ω02
ω02

− mω02 z0 + ϵ −(3.100)(3.101)(3.102)

Then we group terms by order of ϵ, and solve each resulting equation using the solution
from the previous order. The zeroth order term, z0′′ (t) = z0(t) has the obvious solution:z0(t) = A sin(s).

(3.103)

We substitute this into the equation which is first order in ϵ. We set the integration constantto zero and set the “secular” terms (terms with the form s cos(s)) to zero to getω1 =5A4 βω08 + 6αA2 ω08 + 60A2 βk22 ω04 + 40βk24 + 24αk22 ω04.
16ω09(3.104)

Plugging this into the first order equation you get a pretty gnarly expression for z1(t).
Then since we have z0(t) we can write down z(t) to first order: z(t) = z0(t) + ϵz1(t). We plug
our expression for z(t) into our expression for the force, which becomes pretty gross to look at,

144then we expand the expression for the force to first order in ϵ. It’s still pretty gross but for therecord this is what we get:F (t) = − Amω02 sin(ω0 t)+


Amϵ 
3
2
3
4
4
8
128ω0
cos
(
tω0
)
8A
βk2ω0
+
20Aβk2+
6αAk2ω0−3tω0ω1
384ω08− 256Ak2ω02 cos(2tω0) ω04 3α + 5A2 β + 10βk22 + 256A3 βk2ω06 cos(4tω0)−
6 sin(tω0) A2 ω08 46α + 37A2 β + 4k22 ω04 48α + 115A2 β + 320βk24





+ 27A2 ω04 sin(3tω0) ω04 4α + 5A2 β + 40βk22 − 25A4 βω08 sin(5tω0) + O ϵ2
(3.105)
Taking the electric field Ez (t) as the force over the charge, and integrating

−1 ω
6 2π

Z 2π/ω
0

EZ3 (t)3
Erot(3.106)then expanding the result as a series in ϵ, we get an expression for the Berry’s phase we expect as a function of displacement (in terms of k2 ) and the strength of the quartic (parameterizedby α (we also let ϵ go to 1 as we finish with the perturbation theory.) This gives

αberry == −m3 ϵ 5A6 βk2ω04 + 10A4 βk23 + 3αA4 k2ω04
12Erot3 q3 ω02(3.107)

If we plug in the known values for everything in the above equation plus xeq ω02 for k2 ,where Xeq is the equilibrium position, we get that the berrys phase alpha is
αberry = xeq β −3.722534205999362`*∧ -17xeq2 − 1.861267102999681`*∧ -19 − 1.1167602617998085`*∧ -17α.
(3.108)

145
fBR is then 375000 × 3αberry so the fBR due to this quartic perturbation should scale likef BR = 375000 × 3 × xeq(−1.12 × 10−17 α + β(−1.86 × 10−19 − 3.72 × 10−17 xeq2 ))(3.109)Since we have expressions for ω0 and ω1 we can say that the frequency isf =1( ω0 + ω1 )
2π

(3.110)and for the data where we gave the cloud a kick and measured the slosh amplitude A andfrequency then the equilibrium position is zero, so k2 = 0 and our expression for the frequencyisf =5A4 β + 6αA2 + 16ω0232πω0(3.111)We then fit the slosh vs frequency data we have to determine ω0 , α and β.
Z

1635
data
fit

1630

1625

1620

1615

1610

1605
3

4

5

6

7

8

9
10-3

Figure 3.25: Fitting the measured frequency vs slosh amplitude in the non-displaced trap

Now we can plug the values for ω0 , α, and β into equation 3.109:

146

f BR = −0.000154xeq − 2.677xeq3(3.112)

We can plot this prediction on a plot with our experimental results, and we see that thiscompletely fails to account for the observed shift; the slopes differ by a factor of ∼ −10.
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-15

-10

-5

0

5

10

15

Figure 3.26: Comparing the prediction for f BR from our model of the anharmonicity to theempirical results. We see that this model does not account for the observed shifts

3.11.1.6

f BR vs z: we’re pretty sure it’s due to currents in the electrodes

Trevor Wright made a simple model of the currents in the electrodes which explained what
we observed quite well. Look forward to his thesis for the write-up :)In any case, we have no mechanism by which this could become a systematic.

3.11.2

Ion number vs f 0

At one point we saw what appeared to be a correlation between total trapped HfF+ number
and f 0 , but this has not held true through the dataset. What seems to happen reliably is that f 0
tends to drop over a day whether or not HfF+ number is dropping, and some days HfF+ number
would also drop throughout the day so we saw false correlations.

147
3.11.3fR

In general we measure a non-zero f R , the value depends on what primary fringe frequency
we are operating at (see Figure 3.27). We suspect this is just due to the DAC resolution on ourErot generating channel; it’s only a 10-bit DAC so a difference in a single DAC step would beenough to provide the f R we observe.

Figure 3.27: Averaging f R over our entire dataset, per primary fringe frequency f 0 . f R scaleswith f 0 with a slope of 0.00018, which is very close to 1/1024 = 0.00098 or a single step in a
10-bit DAC.

3.11.4

Shifts vs x-y position

We observe frequency shifts if we move the ions in the x-y direction (see Figures 3.28 and3.29). We have no clue why! Baxgrad gradients would explain f, but fR is a little harder tounderstand. This is one remains a mystery, but we have no mechanism by which this could be asystematic4 .

4 plotting script is in 220125\analyze_radial_shifts.

148

Figure 3.28: Comparing observed frequency shifts with x translation

Figure 3.29: Comparing observed frequency shifts with x translation

Chapter 4
A new limit on the electron’s electric dipole moment

This chapter is a reproduction of a manuscript in preparation co-written by myself and Luke Caldwell

4.1

Summary of the eEDM result
Physicists have been able to describe our entire universe using just two mathematical models: General Relativity, for gravity, and the Standard Model, foreverything else. While the Standard Model is generally successful it has someserious inconsistencies: it fails, for example, to explain why matter dominates
over antimatter in our universe. New physics models aim to eliminate these inconsistencies and often introduce additional symmetry violation, which in turnresults in predicted symmetry violation in fundamental particles like electrons.
In particular, many new physics models predict that the electron will have ameasurable non-zero electric dipole moment aligned with the spin axis, which
violates time-reversal symmetry. Here we present a new experimental limit on
the electron’s electric dipole moment using trapped HfF+ in rotating bias fields.
We find de = (−9.7 ± 20.1stat ± 6.2syst ) × 10−31 e·cm, which corresponds to an
upper bound of |de | < 4.4 × 10−30 e·cm (90% confidence). Our measurement is afactor of 2.5 better than the previous best limit, and confirms previous results ina radically different system. Our limit directly constrains new physics theories atenergies beyond the reach of the Large Hadron Collider [32].
The Standard Model of physics predicts that the electron has a very small displacementof its center of charge from its center of mass, giving it an electric dipole moment (EDM) which
violates time-reversal (T) symmetry. So far, nobody has detected an electron electric dipole
moment (eEDM, de ), and the Standard Model prediction lies ∼ 8 orders of magnitude below

150
current experimental sensitivity. Extensions to the Standard Model aimed at solving inconsistencies like matter-antimatter asymmetry often do so by adding new particles at energies higher
than any so far discovered, accompanied by new T-violating phases. These new particles result in a much larger prediction for the eEDM - often within reach of near-term experiments.
Measurements of the eEDM serve as a direct constraint on extensions to the Standard Model,
and current eEDM measurements probe for particles with energies above ∼ 30 TeV [32], wellbeyond the reach of any current or planned particle collider. In addition, a non-zero measurement at current experimental sensitivities would unambiguously signal new physics. Here we
present the most precise measurement of the eEDM so far. We achieved this with a uniqueexperimental setup, trapping molecular ions and polarizing them in rotating bias fields. Our
result, de = (−9.7 ± 20.1syst ± 6.2stat ) × 10−31 e·cm (’syst’, systematic uncertainty; ’stat’, statisticaluncertainty), is consistent with zero and constrains de to an upper limit of |de | < 4.4 × 10−31 e·cm
at 90% confidence.
In general, eEDM searches aim to measure the energy difference ∆U when de is aligned
vs anti-aligned with an electric field E ; ∆U = −2de · E . Assuming there are no new degrees
of freedom the eEDM must be aligned with the electron spin, and an unpaired electron spin
inside a polarized paramagnetic molecule is subject to a huge intramolecular electric field Eeff[41]. This is why our measurement uses trapped HfF+ molecules; with an applied field of ∼ 50
V/cm the science states of our molecule are oriented and fully polarised (Eeff = 23 GV/cm). By
using trapped ions we are able to realize spin coherence times 3 orders of magnitude larger thanany other molecular eEDM experiment. As shown in Figure 4.1, the intramolecular axis, which
defines the direction of Eeff , points either parallel or antiparallel to the applied electric field. A
differential measurement performed between these two orientation states is at the heart of ourexperiment.

151

4.2

Experimental Overview
We begin our experiment with the production and radiofrequency trapping of roughly

20, 000 HfF+ ions in the 1 Σ+ ground state [55] [50]. In order to both polarize and trap ourions we rotate our external bias field Erot at angular frequency ωrot = 2π × 375 kHz and do
our spectroscopy in the rotating frame. The rotating electric field polarizes the ions, whichhave a molecule-frame dipole moment, and states of good parity are mixed into states of goodorientation (either aligned or anti-aligned with Erot ), known as Stark doublets (see Figure 4.1, top).
We also apply a quadrupole magnetic field gradient B , which drives the primary component of
the spin precession (see Figure 4.2).
Once we have trapped our ions and applied our bias fields, we prepare them for spectroscopy. We perform Ramsey spectroscopy on the stretched (m F = ±3/2) states in the 3 ∆1 ,J = 1, F = 3/2 manifold of HfF+ . As described in Methods, we transfer our ensemble into 3 ∆1then prepare a pure spin state before applying a π/2 pulse to create a coherent superposition of
the m F = ±3/2 states. We allow the superposition to evolve for a variable amount of time, thenwe apply a second π/2 pulse to map the relative phase onto a population difference between them F = ±3/2 states. We clean out one of the stretched states, then we detect and count the numberof ions in the remaining stretched state by state-selectively photodissociating the molecules [19],using the opposing orientations in the trap to send the dissociation products to opposite sidesof our imaging microchannel plate and phosphor screen assembly [48, 47]. We count the ions ineach doublet then repeat the measurement reading out the other stretched state. From these two
N −N

measurements we can construct a spin asymmetry, Au/l = N↑↑ + N↓↓ , where the subscripts u/l indicate the upper or lower Stark doublet, and the subscripts ↑ / ↓ indicate the spin state / stretchedstate we read out. We repeat our measurement at different free-evolution times, which generatesa Ramsey fringe whose frequency is proportional to the energy splitting between stretched states(see Figure 4.1).

152

Figure 4.1: Ramsey spectroscopy in HfF+ . Top, level structure in the eEDM-sensitive 3 ∆1 , J = 1,
F = 3/2 manifold in external electric Erot and magnetic Brot bias fields (not to scale). The oriented
molecules are either aligned or anti-aligned with Erot , which we call the lower and upper Stark
doublets respectively. The component of the energy splitting due to the eEDM, 2de Eeff , flips sign
between doublets. Bottom, example Ramsey fringes from our dataset. The fringes from the two
doublets are collected simultaneously.

λ/2

λ/2

λ/2

m

F

de

pl
et
io
pu
n
m
pi
ng

n
tio
ia
oc
ss
di

MCP

HfF+
HfF

B

Z

Y

Erot
X
trap
electrodes

MCP
phosphor
CCD

Figure 4.2: Top down schematic of our experiment’s main chamber. The neutral ions fly into thecenter of the trap where they are ionized and trapped. We apply the rotating electric bias field⃗rot and the quadrupole magnetic field gradient B⃗ . The ions undergo unifom circular motion,E⃗⃗rot · B⃗.
either aligned or anti-aligned with Erot , and their unpaired spins precess about E

153
In practice, our experiment is an electron spin precession measurement simultaneously
executed on two orientation states (Stark doublets, D̃ ) of trapped HfF+ . The primary componentof the spin precession frequency that we measure, f B , is due to the spin precessing about a⃗ : h f B = 3gu/l µ B Êrot · B⃗ = 3gu/l µ B Brot , where h is Planck’s constant,laboratory magnetic field BFF
gF is the magnetic g-factor, the superscripts u/l indicate the upper or lower Stark doublet, µ B
is the Bohr magneton, and Brot is the magnetic field in the ion’s rotating frame. As indicatedin Figure 4.1, the relativistically enhanced eEDM-induced energy shift, h f EDM ∼ de Eeff , adds orsubtracts from the primary frequency h f B depending on the relative orientation of E eff and B rot(aligned or anti-aligned). We take measurements in both orientation states simultaneously but
separately (on either side of the screen), which allows us to perform a differential measurement
between the D̃ states (cancelling most common-mode noise [47]).
We collect our Ramsey fringes in each one of 8 possible experimental states, where weswitch the sign of the magnetic field B̃ (in or out), the rotation direction R̃ (up or down), andthe side of the screen/image we read a given Stark doublet out on Ĩ (left or right). A set ofRamsey fringes in all possible switch states is called a block, and we fit those fringes to a set ofparameters (frequency, phase, contrast, offset and decay rate). Using all the fringes in a block weform linear combinations of the fitted fringe parameters to create what we call parity channels.
The B̃ , D̃ -odd frequency channel, f BD , contains the eEDM signal. The other channels allow us todiagnose systematics and monitor experimental performance.
We collected our blocks in “superblock” chunks where we changed an experimental parameter not listed above for a large set of blocks to search for unknown systematic effects. Our
superblocks consist of: different magnitudes of B to adjust the average fringe frequency f 0 (wetook data at f 0 = 77, 105 and 151 Hz), inverse orientations of the fixed waveplates used for state
preparation and readout (to change the handedness of the light in the lab frame), and changing
which stretched state we prepare (we routinely read out both stretched states, and we interleaved blocks preparing one or the other stretched state). During data collection and analysis, we154‘blinded’ our measurement to prevent experimenter bias from affecting our results. We generated a random number from a uniform distribution, and when frequency channels are computed
the random number is added to the f BD channel before any results are displayed. We did notremove the blind until our systematics search and analysis was complete.

4.3

Accuracy evaluation
To evaluate the accuracy of our measurement we searched extensively for systematic shifts

prior to data collection; full details are in 3 and our systematic error budget is delineated inTable 4.1. In general, we tuned a variety of experimental parameters over a large range andobserved the response in our data channels. The only shift we were able to observe directlyin the eEDM channel is that which is due to a non-reversing quadrupole magnetic field and thedifference in magnetic g-factors gF between Stark doublets δgF . The f B channel, which is acquired
concurrently with the f BD channel, provides a direct measurement of the non-reversing magnetic
BD = f B F , see Figure 4.3.
field and allows us to apply a correction to our science channel: δ f corrgF
δg

Before applying any corrections to the science channel, we suppress this systematic by actively
shimming the currents through the B coils in each direction to minimize f B . This shimming
was so effective that the mean correction we applied was well below our statistical sensitivity:
BD = 181 nHz. Unfortunately, this type of shimming and correction leaves us susceptible toδ f corrother possible effects which can cause shifts in f B but not f BD . One of the most important sourcesof such a shift was the combination of a transverse magnetic field and an Erot second harmoniccaused by harmonic distortion in our op-amps. This combination can cause shifts in both f B andf BR . We shim out the second harmonic on each electrode by feeding forward a second harmonic
signal with the opposite phase. With this technique, we are able to suppress the second harmonicby ∼ 80 dB. We also used our magnet coils to null the ambient uniform magnetic field at the RFtrap center below 10 mG. All remaining known systematic shifts are enumerated in Table 4.1.

155

Table 4.1: Systematic error budget
EffectMagnetic⃗0
Non-reversing B
Second harmonic of Erot and transverse magnetic field
Third harmonic of Erot and magnetic field gradients
Higher harmonics of Erot and higher order magnetic field gradients
Ellipticity of Erot and magnetic field gradients
Berry’s phase
Phase modulation due to axial secular motion
Axial 2nd harmonic with ellipticity of Erot
Higher harmonics of Erot
Rotation induced mixing
Leaking of f BR
Axial magnetic fields
Other frequency
Imperfect overlap of spatial distributions
Phase
Improperly characterized imaging contrast
Swatch position
Total

Correction (µHz)

Uncertainty (µHz)

0.2

< 0.1
2
2
< 0.1
1
3
2
3

< 0.2
0.4
3.5
0.7
1.2
0.2

7

156

Figure 4.3: Systematic shift in the eEDM channel f BD due to non-reversing axial magnetic field,
which shows up in the f B channel.

4.4

Results
We collected 1370 blocks over roughly two months. We applied cuts to the blinded data

based on non-EDM channels indicating signal quality: blocks with late-time contrast below 0.2
were cut due to low signal to noise, as were blocks containing fitted fringe frequencies whichwere over 3.5σ different from the mean fringe frequency for that switch state. The cuts applied
did not shift our result, see Figure 4.4. After applying cuts we were left with 1328 blocks, or∼ 620 hours of data.

Figure 4.4: Results as we change the threshold for cutting blocks from our dataset. In each plot,
only one cut is applied (the other cut is not applied). Applying cuts does not systematically shift
our result.

157
Each block results in one value of f BD and thus one eEDM measurement. In general our
fringe measurement uncertainty came in at the quantum projection noise limit [47], and the
reduced chi-squared statistic for taking a weighted mean of all the blocks is χ2r = 1.07. To correctpfor the over scatter we scale our final statistical error bar by χ2 = 1.03. The unblinded dataset
is shown in Figure 4.5. Inspection of a normal probability plot indicates that the distributionof normalized and centered eEDM measurements ( f BD − ⟨ f BD ⟩)/δ f is consistent with a normal
distribution.

p
Figure 4.5: Summary of our dataset after cuts have been applied and scaling δ f by χ2 to accountfor over-scatter. Top left, normal probability plot of f BD showing the data are consistent with anormal distribution. Bottom left, histogram of normalized and centered f BD data. Overlaid inblue is a normal distribution. Right, superblocks taken under different experimental parameterscompared to the overall average value of f BD . Here N is the average number of trapped HfF+ ions
per experimental trail.

We removed our blind on November 1, 2022 and obtained a final value for the eEDMsensitive frequency channelf BD = −10.9 ± 22.8stat ± 7syst µHz.

(4.1)

Dividing by −2|Eeff |sgn( gF )/h ≃ 1.13 × 1031 µHz/e·cm, we obtain a value for the eEDM

158

de = (−9.7 ± 20.1stat ± 6.2syst ) × 10−31 e · cm,(4.2)

which is consistent with zero within one standard error.
The resulting upper bound on de is|de | < 4.4 × 10−30 e · cm (90% confidence),(4.3)

Our result is consistent with the ACME collaboration’s 2018 findings [32], and confirms
their result with a radically different experimental approach. So far we have assumed that parity and time-reversal violation arise purely from de . Diatomic molecules are sensitive to othersymmetry violating effects [56], and we can interpret our measurement as a linear combinationof them all. Assuming that de is zero, we can instead attribute our measurement of f BD to2WS CS , which arises from a pseudoscalar-scalar electron-nucleon coupling CS [57] [58] (here themolecule-specific structure constant WS = 20.4 kHz [59]). Dividing f BD by 2WS , we find

CS = (−2.7 ± 5.6stat ± 1.7syst ) × 10−10 .

4.5

(4.4)

Conclusions
Our measurement imposes constraints on beyond standard model theories, in a regime

largely inaccessible to particle accelerators [32]. The standard model predictions for both de andCS are well below current experimental sensitivity, making this measurement a background-freeprobe for new physics. In addition, our measurement confirms previous results [32] in a radicallydifferent system, improving confidence in both experiments. Measurements like ours can also be
used to constrain possible couplings to dark matter [60].

159

4.6

Methods

4.6.1

Bias fields
We apply a rotating electric field E rot (R, t) = Erot [ X̂ cos (ωrot t) − R̃Ŷ sin (ωrot t)], whereωrot ≡ 2π f rot , f rot = 375 kHz, and R̃ indicates the rotation direction (up or down). The rotating
field orients and polarizes the ions, which are driven in uniform circular micromotion with aradius rrot ∼ 0.5mm.
Our experiment has 4 pairs of magnetic field coils and no magnetic shielding. All coils
are outside of the main vacuum chamber. Three of the coil pairs are set up along our labframe X, Y, Z axes in Helmholtz configuration for tuning the magnetic field at the position of
the molecules. The magnetic field at the center of the trap is measured by an array of fluxgate
magnetometers bolted to the main experimental chamber. The fourth coil pair generates our socalled ‘rotating’ magnetic field Brot . The coils are aligned with the lab-frame Z axis and produce
the quadrupole magnetic gradient B = B ′ axgrad (2Z − X − Y ). In the rotating frame of the ionsthis appears as a magnetic bias field Brot ≡ |⟨B ·E rot /Erot ⟩| ≃ |B ′ axgrad rrot |.

4.6.2

HfF+ 3 ∆1 details
Once we have trapped our ions and applied our bias and shimming fields, we preparethem for spectroscopy. We perform Ramsey spectroscopy on the stretched (m F = ±3/2) statesin HfF+ ’s 3 ∆1 , J = 1, F = 3/2 manifold. Here, J indicates the electronic plus rotational angularmomentum, F indicates the total angular momentum, and m F = F · ẑ is the projection of F onto
the rotating quantization axis. The 3 ∆1 state is well-described by Hund’s case a) basis states withcoupled nuclear spin from the Fluorine nucleus.

160
4.6.3

Quantum state preparation
Our state-selective two-photon ionization procedure produces HfF+ in the ground elec-

tronic and vibrational state (1 Σ+ , v = 0), primarily distributed over the lowest 4 rotational levels.
We connect J = 0 → 3 using microwaves and perform incoherent transfer to the eEDM-sensitive
3∆ , v
1= 0, J = 1 state via the 3 Π0+ , J = 0 ← 1 Σ+ transition at ∼ 961 nm, which decays

preferentially to 3 ∆1 . The decay from 3 Π0+ puts population in several vibrational levels in 3 ∆1 ,which will decay into our ‘science’ state 3 ∆1 , v = 0 if left untreated. We clean the population in 3 ∆1 , v = 1 out by illuminating the cloud with radiation at ∼ 818 nm which connects
3 Σ− , v = 1, J = 0 ← 3 ∆ , v = 1, J = 1 and preferentially decays back to 1 Σ+ .
10+This laser is on

for the duration of the experiment, and will clean out any population which leaks down fromhigher vibrational levels over the course of the measurement. The lasers used in our experimental
sequence are summarized in Figure 4.6.
After transfer to 3 ∆1 we turn on a strobed, circularly polarized, CW laser at ∼ 1082 nm
which is brought in from the side of the trap and connects 3 Π0− , v = 0, J = 0 ← 3 ∆1 , v = 0, J = 1.
The strobing is synchronous with the rotation of Erot such that the wavevector is either parallel oranti-parallel to E rot , which drives a σ+/− transition to an F ′ = 3/2 manifold in 3 Π0− , eventuallyleaving only one stretched state populated in 3 ∆1 (in both Stark doublets).

4.6.4

Ramsey spectroscopy
Once we have a pure spin state in both doublets prepared, we apply a π/2 pulse to theensemble of ions by temporarily ramping down the magnitude of Erot , which increases a rotationinduced coupling ∆u/l between m F = ±3/2 states [61]. The increased coupling causes the purespin states to evolve into a coherent superposition of the m F = +3/2 and m F = −3/2 states in∼ 1 ms [43] [50]. We ramp Erot back up and allow the superposition to evolve for a variable
amount of time. After the desired precession time has elapsed, we apply a second π/2 pulse to
map the relative phase onto a population difference between the m F = ±3/2 states. We project

161
the ions into their final state by applying the ‘depletion’ laser, which connects 3 ∆1 , v = 0, J =1 → 3 Σ0−+ , v = 0, J = 0 at ∼ 814 nm, and preferentially decays to 1 Σ+ by a 10:1 ratio having
weaker coupling to the 3 ∆1 state. This laser is circularly polarized and strobed at f rot such that it
is parallel/antiparallel to E rot , which drives a σ+/− transition to 3 Σ0−+ , leaving only one stretchedstate populated.

4.6.5

Final state readout
Finally, we detect and count the number of ions in the stretched states via state-selective

resonance-enhanced multiphoton dissociation [19] using two pulsed lasers, both of which are
circularly polarized and timed such that their k-vectors are parallel/antiparallel to E rot to drive
σ+/− transitions. The ions are oriented in the trap thanks to Erot , and we perform dissociation
when they are parallel to our imaging microchannel plate (MCP) and phosphor screen assembly.
Because our dissociation transitions obey the angular momentum requirements detailed in [13],the dissociated Hafnium atoms from the two Stark doublets will fly to opposite sides of thescreen. We gate the phosphor screen such that we only image the Hf+ atoms, the un-dissociated
HfF+ ‘spectator’ ions would otherwise ruin the image. We detect both the Hf+ and the HfF+ ionsin time of flight. The technical details of our imaging and counting system are detailed in [48]and [47].

Figure 4.6: Cartoon depicting the transitions used during our state preparation.

162
We repeat our measurement at different free-evolution times, which generates a Ramsey
fringe whose frequency is proportional to the energy splitting between stretched states. Because
we simultaneously measure both doublets, we get two Ramsey fringes. We can prepare one of
2 stretched states and read out one of 2 stretched states, and we chop between those possible
preparation and readout choices by strobing the circularly polarized pumping and depletion
lasers either in phase or out of phase with Erot to drive σ+ or σ− transitions. There are 4 possiblecombinations: in/in, in/out, out/in, and out/out.

4.6.6

de signal extraction
When we take measurements we do them in one of 8 possible experimental switch states,

where we switch the sign of the magnetic field (B̃ , in or out), the rotation direction (R̃, again in orout), and the side of the screen/image we read a given Stark doublet out on (Ĩ , left or right). Note,
we measure both doublets (D̃ ) simultaneously, but separately (on either side of the screen).
We take a set of Ramsey fringes in all possible switch states (we call this a block). We fit
each fringe in the block to extract the fringe parameters (frequency f , phase ϕ, contrast C, decayrate γ, and offset O), and we form linear combinations of those parameters to create what wecall parity channels. Taking linear combinations of the frequencies in a block gives us frequency
channels, which are of primary concern. For example, the mean of all the measured frequencies
in a block is f 0 ∼ 3gF µ B Brot /h. If instead we take the difference over the B̃ switch we getf B ∼ 3gF µ B δBnr /h ∼ f 0 ( δBBrotnr ), which measures how accurately the magnetic field is reversed.
f D is the difference in frequency between the two Stark doublets: f D ∼ −3δgF µ B Brot /h. If theg-factor was identical for each doublet and the rotation-induced coupling terms were identical,
the leading order f D term would vanish. In general, we indicate the odd parity components
with a superscript, so f BR is the the antisymmetric combination of switching both the magnetic
field and the rotation direction. Forming frequency channels allows us to diagnose experimental
issues and get a handle on many systematics.

163
The frequency channel of primary interest contains the eEDM signal: f BD ∼ 2de Eeff /h −3δgF µ B δBnr /h + . . . Here, the subscript on δB indicates a quadrupole magnetic field gradientδBnr that does not switch with our applied switching quadrupole field gradient.
Because we collect data in both doublets simultaneously, we are able to take differential
measurements which cancel out most common-mode noise. After we define the asymmetry
N −N

Au/l = N↑↑ + N↓↓ in the upper and lower doublets, we can take the difference D and sum S over the
doublets:

A D = Au − Al

(4.5)

AS = Au + Al ,which allows us to fit the difference and sum fringes to reduce uncertainty in our D̃ -odd channels.
Full details about our fitting and analysis routine are in 3, and a discussion of the advantages ofour simultaneous measurement scheme is in [47].

Chapter 5
Dark Matter Exclusion

5.1

Introduction
Sometime in 2017 Victor Flambaum approached us with a curious idea: to use our first-

generation EDM measurement as the basis for a dark matter search. Dark matter? Indeed. As
it turns out, a certain class of dark matter could leave an oscillatory echo in our spectroscopic
signal - in short, even though the DC value of the transition we measured was zero it could still
have a non-zero AC amplitude.
One might reasonably ask - why use AMO/precision measurements to try and detectdark matter at all? Isn’t new particle detection an accelerator’s job? And what about dedicatedFigure 5.1: Generation 1 EDM data vs time. Could there be an oscillatory signal buried in here?

165
dark matter detection experiments? To answer the first question, I think it’s fair to predict that
accelerator energies are likely to remain on a plateau for quite a while to come. So if the particles
we are searching for are higher energy than the reach of the LHC, the dark matter detective isout of luck. So we really need to look outside the accelerator paradigm to solve the issue of dark
matter. Which leads us to the next question: what about dedicated direct detection searches?
Well, of course there are plenty of those, and they’re all interesting and worth doing. But one
big advantage that AMO/precision measurements have over their direct detection counterpartsis their ability to exclude huge portions of the parameter space (in a relative sense) at once.
For example, our exclusion covered seven orders of magnitude: from 10−22 − 10−15 eV in mass.
Competing direct detection searches are often limited to tiny fractions of a single decade (see, forexample, this paper [62] from the ADMX collaboration, which excluded axion dark matter in the
3.3-4.3 eV mass range). There are, of course, some direct detection experiments that have large
mass ranges, but while we still have no idea what dark matter is and we are still searching over
40 mass decades, it seems prudent to expand our toolbox to include a few tools that allow forbroadband searches.
Separate from the question of the width of the search in mass range, there is also thedepth of the search in coupling strength, which we can call our “sensitivity”. Because AMO
experiments are not designed specifically for dark matter detection they can be (and in our case,was) much less sensitive than one might hope. But again, I think pursuing these searches is still
worth it - simply because of how quickly precision measurements are becoming more sensitive.
The electron EDM limit has moved a factor of 100 in less than 10 years, and one can see similarly
impressive progress with things like atomic clocks. There are tons of precision measurements
like this out there, and a lot of them are so sensitive now that they are really well-positioned tomake a big dent in the search for dark matter. This is exciting, and something that I think we asa field should take advantage of. Assuming we do, this leads us to the most important questionof all: at the threshold of this era, we should not be asking ‘how is the analysis of hypothesis

166
exclusion typically done?’ but rather ’how can we get the analysis of hypothesis exclusion right?’
You see, after taking on this project there was about a year of confusion and suffering trying
to understand dark matter (in general) and statistical exclusion methods for dark matter searches(in particular). Part of the difficulty was that it became clear that nobody really agreed on how to
do dark matter exclusions, and based on my preliminary and fairly shaky analysis it seemed likesome groups were doing stuff that was patently wrong. Most of the stuff I read about was fairly
standard threshold-based frequentist methods. One issue I noticed right away was that noneof those methods had a particularly appealing way of dealing with the look-elsewhere effect,
and after more reading and struggling it seemed like none of the methods I looked into had areasonable way of dealing with the stochastic nature of the dark matter field itself. There weremore issues: the scale of the stochasticity changing as your measurement time approaches thedark matter coherence time, the differential sensitivity of different measurement quadratures,and even the finite sample time for fast hypothesized oscillations. All of these had us (Eric andI) spinning our wheels somewhat, trying to make it all work using threshold-based frequentist
methods.
Ultimately, and thankfully, we teamed up with Dan Palken and Ben Brubaker from the
Lehnert group for their much-needed expertise on statistical methods (Dan) and dark matter(Ben). Dan ultimately confirmed my suspicion about the general confusion and disagreement
and was able to demonstrate exactly what some groups were doing wrong. I’d recommend
reading his thesis for all the gory details [63]. Dan introduced us to a Bayesian exclusion method
which could neatly deal with the issues we had identified, and we set to work extending theanalysis framework he had devised for to suit our needs. The details of that are, obviously,
below.
Before we get into the details I’d like to make two last points. The first is that the issues
listed above, that we were the first group to properly deal with, really are the difference between

167
getting it wrong and getting it at least closer to right. In fact, Gary Centers and co [64] wrote
a lovely paper about how failing to address the stochasticity issue (the field amplitude itself is
temporally stochastic) can means that seven currently published exclusions are too stringent,some by about a factor of 10! So one really must try to address these things if they want toclaim they gave it the ol’ college try. The second point is that some folks can be hesistant about
Bayesian analysis methods, because they feel that things like the ‘prior belief’ are too subjective. I completely understand this sentiment, I actually began my dark matter experimentalist
journey as a staunch frequentist for this exact reason. It wasn’t until I was doing this analysis
that it became clear how superior the Bayesian approach is. But to ease your mind before you
dig into the details, dear reader, I will point out three things. The first is that our analysis islargely independent of the priors. The prior update, introduced to you soon, is actually the key
parameter in this search and this is really just the ratio of two probabilities: that of getting yourmeasurement if dark matter really exists vs that of getting your measurement if it’s just noise. We
also made our matrix of prior updates freely available to anyone who wants it so that they can
explore the results with whatever priors they want. Our exclusion does not even depend on theabsolute scale of the priors, but only on their profile. But the profile really matters, which brings
me to the second point. We chose a log-uniform prior profile and we believe that it is a natural
choice given that there isn’t really any reason for the new dark matter particle to appear in any
specific mass decade. The masses of known standard model particles are distributed over many
decades, and there is no particular reason to expect new physics to be any different. In addition,
the body of literature surrounding this type of dark matter has only constrained it to exist within
25 (!) mass decades, which means that we have already implicitly agreed, as a community, on
logarithmically uniform priors. The third point is that the choice of priors is isomorphic to the
choice of power threshold set for detection used in a frequentist analysis. You’ll often read about
a 95% confidence interval for each frequency examined. All of the published frequentist analysesof this type that I’ve seen use a constant threshold for detection over a large frequency range
which is, frankly, silly - because it is equivalent to choosing uniform priors over several mass

168
decades, which amounts to only actually doing your search in the highest decade. So if you’regoing to do that, you might as well throw out the lower decades entirely. A frequentist analysis
would solve this problem by choosing a frequency-dependent threshold, but then you have done
something which is isomorphic to choosing a profile for the priors, and then you are being justas “subjective" as Bayesians. I really think that it is a meaningful practical advantage, both as
the scientist doing the analysis, and for the critical readers examining the results, to have clearlystates priors and justifications for them - precisely because it makes your assumptions explicit.
One can compare this situation to a frequentist analysis where they can be hidden, unexamined,
and sometimes, indefensible.
Well I’m sure you’ve heard just about enough of me by now. Let’s get on to the main event:a copy-paste of the paper and supplement :)5.2

Experimental Constraint on Axionlike Particles over Seven Orders of Magnitude in Mass
This section is a reproduction of the main text of ref. [60]A measurement of the permanent electric dipole moment of the electron (eEDM, de ) well

above the standard model prediction of |de | ≤ 10−38 e cm would be evidence of new physics[65, 66, 67, 68, 69]. The search for permanent EDMs of fundamental particles was launchedby Purcell and Ramsey in the 1950s [21, 70], but it was not until recently that physicists began
to consider the possibility of time-varying EDMs inspired in large part by hypothetical axionlike particles (ALPs) [71, 72, 6, 73, 74, 75]. Coupling to ALPs can cause EDMs of paramagnetic
atoms and molecules to vary, due to variations in de or the scalar-pseudoscalar nucleon-electron
coupling CS . In this paper, we extend our analysis of the EDM data we collected in 2016 and 2017[2] to place a limit on the possibility that CS oscillates in time. This allows us to put a constraint
on the hypothetical ALP-gluon coupling over seven orders of magnitude in mass.

169
Dark matter is one of the largest unsolved mysteries in modern physics: the microscopic
nature of 84% of our universe’s cosmic matter density remains inscrutable [10]. A family ofcandidate dark matter particles called axion-like particles are pseudoscalar fields favored for theirpotential role in solving problems such as the strong CP problem, baryogenesis, the cosmologicalconstant, and small-scale cosmic structure formation [76]. ALPs can be anything from the QCD
axion [77, 78, 79, 80] to ultralight axions from string theory [81], with different models favouring
different mass ranges [82, 83, 84, 76, 85, 5]. In this paper, we will consider ultralight spin-0
bosonic fields whose mass m a can range from 10−24 − 101 eV (or 10−10 − 1015 Hz) [5]. ALPs in
the low end of this mass range would solve some issues pertaining to small scale structure raisedby astrophysical results [85, 86, 76, 87].
Ultralight dark matter particles such as these must be bosonic: the observed matter density
can’t be reproduced with light fermions because their Fermi velocity would exceed the galactic
escape velocity. The resulting field has high mode occupation, is well described classically as awave, and oscillates primarily at the Compton frequency ν = m a c2 /h. Astrophysical observations tell us that a quasi-Maxwellian dark matter velocity distribution would have, near earth,a dispersion of δv ∼ 10−3 c, which in turn leads to a coherence for the dark matter (DM) wave
of about 106 cycles [88]. This finite linewidth ultimately limits the sensitivity of any experiment
whose total observation time approaches or exceeds the coherence time.
Several distinct couplings to ALP fields can result in an oscillating CP-odd quantity suchas de or CS , which amounts to an oscillating molecular EDM signal in our data. The moststraightforward effect is the derivative coupling of the ALP field to the pseudovector electron
current operator — in other words, a direct coupling between ALP and electron. Unfortunately,in an atomic or molecular system the observable effect scales linearly with m a , leading to a large
suppression for the mass range we explore [75]. A second potential coupling is from the ALPinduced modification of the electron-nucleon Coulomb interaction [74]. Regrettably, the natureof this effect has not been elucidated in detail, but it is also expected to scale linearly with m a

170[89]. Finally, there are two distinct effects due to the coupling of ALPs to gluons. The first leads
to a partially screened nuclear EDM which would be enhanced in polar molecules but scales
linearly with m a , which again is problematic in the mass range we probe [90]. The second effectwill result in an oscillating CS and this is independent of m a [91, 92, 93]. In this paper, we use thehypothesized oscillation in CS to constrain the ALP-gluon coupling.
To understand the analysis discussed herein, the reader should be familiar with a few key
details from the original measurement. Complete details can be found in [2] and references
therein. The heart of our experiment is an electron spin precession measurement of 180 Hf19 F+
molecular ions confined in a radio-frequency Paul trap and polarized in a rotating bias field to
extract the relativistically-enhanced CP-violating energy shift 2hWS CS between the stretched andoriented m F Ω Stark sublevels [94]. The molecule-specific structure constant WS = 20.4 kHz [59],h is Planck’s constant, and we assume for the remainder of this paper that de is zero.
In an individual run (or shot) of the experimental procedure, many HfF+ ions are loaded
into an ion trap, and the internal quantum population is concentrated into one of four internal
sublevels by means of pulses of polarized light. The energy difference between two sublevels isinterrogated with Ramsey spectroscopy [2]. The population is put into a superposition of two levels by means of a coherent mixing pulse, and then allowed to coherently evolve for a variable timet R , before a second coherent pulse remixes the two levels. The resulting population in a sublevelis destructively read out by means of final series of laser pulses, which induce photodissociationof HfF+ ions into Hf+ ions with a probability proportional to the population in the interrogatedsublevel. The procedure requires about one second and the ion dissociation signal has a component which oscillates with t R as cos(2π f R t R ), where f R is proportional to the energy difference(averaged over the ∼0.5 second coherent evolution time t R ) between the m F = ±3/2 magnetic
sublevels of a given Stark level. There are multiple contributions to f R , including Zeeman shifts
and Berry’s phase shifts. Over the course of a block of data, comprising 1536 experimental shots,
we chop the direction of the ambient magnetic field, the rotation direction of the bias field, and171
the internal Stark level probed, and for each of these eight different experimental configurations
we sample multiple Ramsey times, in order to extract f R under eight different laboratory conditions. A certain linear combination of all the various f R measurements isolates, to first order,
the CP-violating contribution to the energy difference, f CS = 2WS CS , where in this case we aresensitive to the average value of CS during the ∼22 minutes required to collect a block of data.
To probe for a possible time-variation in CS , our analysis is broken into two parts: ‘lowfrequency’ (27 nHz – 126 µHz), corresponding to hypothesized variations in WS with periods
shorter than the overall 14 month span of data collection but still long compared to the durationof a block, and ‘high frequency’ (126 µHz – 400 mHz), corresponding to periods shorter than our
most sensitive period of data collection (∼11 days) but still long compared to the duration of ashot (∼1 second)[95, 96].
The low-frequency Least Squares Spectral Analysis (LSSA) is straightforward: we takethe bottom-line results for each of our 842 blocks of data ( f CSi , σi , Ti ), where Ti is the time inseconds at which the ith block was collected, relative to the center of our data set (at 00:17 UTC,
Aug 16, 2016), and σi is the standard error on f CSi . Then for each hypothesized ALP frequency
νi = ω/(2π ) in a set of equally spaced trial frequencies in the low-frequency range, we assume
f CS ( T ) = A cos(ωT ) + B sin(ωT ),(5.1)

and do a weighted least-squares fit [96] to extract our best values A0 (ν) and B0 (ν).
For the high-frequency LSSA analysis we want to resolve hypothesized changes in f CS that
occur over the time scales ≤ 22 minutes (the time required to collect a block), so we disaggregate
each block into its 1536 component measurements: the number of dissociated ions N measured
for each of the shots that make up a block. The expected value for N for each shot is an oscillatory
function of the Ramsey time t R . The frequency of the oscillation in t R is modulated because acomponent of f R is proportional to CS , hypothesized to be an oscillatory function of Ti , and thesign of the modulation amplitude depends on the sign of the bias magnetic field applied for

172

Figure 5.2: Least-squares spectral analysis of our EDM data.

that shot, and on the Stark doublet selected for probing for that shot. The least-squares fitting
routine is correspondingly more complicated [96] but at the end of the day we again extract bestfit values A0 (ν) and B0 (ν) for each hypothesized value of the ALP oscillation frequency ν in theequally spaced grid of trial frequencies across the high-frequency range.
Combining the two analyses, we obtain a measure of the best fit oscillation amplitude
over seven orders of magnitude (Figure 5.2). Now we can ask: to what extent do our fit values
convince us that CS is actually oscillating, given experimental noise and the look-elsewhere effect[97]? To answer this question we used the Bayesian Power Measured (BPM) analysis framework
developed by Palken et al. [3].
The essence of the BPM framework is a comparison between the probability of measuring agiven oscillation amplitude assuming the existence of an ALP with a specified mass and coupling
and the probability of measuring the same amplitude assuming it doesn’t exist. This means weneed probability distributions for both cases at each frequency ν, which we can evaluate at theactual measured values A0 , B0 . For the case where ALPs do not exist, we get the distributions bygenerating simulated data which by construction has no coherent CS oscillation in it. For the low

173
frequency data we generate a single point of ‘noise’ at each acquisition time in the real dataset.
Each point of noise is drawn from a Gaussian distribution with mean of zero and a variancewhich matches that of the real dataset. For the high frequency analysis we randomly shuffle
the timestamps of the individual Ramsey measurements in such a manner as to effectively erase
any coherent oscillation which may be present in the CS channel while preserving both the basic
structure of the data (including the general coherent oscillation present in each Ramsey dataset)and the technical noise (see [96] for more details).
Once we have this simulated data we perform LSSA on it to get best fit values for A, B
at each frequency ν of interest. We repeat this process 1000 times. The resultant distribution of
A0 , B0 at each frequency ν represents what we would expect to find in a universe where ALPs
do not exist. Each distribution is bivariate normal with some rotation angle. We rotate each
distribution into the primary axes A′ , B′ where the variance of the distribution is maximized
along A′ and minimized along B′ , to define the no-ALP distribution:
1

N =

A ′2

B ′2

−2( 2 + 2 )
1
σ ′
σ ′
A
B .
e
2πσA′ σB′

(5.2)

We still need to determine the probability distributions in the case that ALPs do exist.
Note that A′ and B′ are still random variables in this case due to the stochastic nature of the ALP
field: different spatio-temporal modes of the field interfere with each other (the ALP field at any
point in space is a sum over many contributions with random phases), so the field amplitude isstochastic over timescales longer than the coherence time [98, 99]. Except at the very high end of
our analysis range, the ALP coherence time is much larger than the duration of our measurement,
so we resolve only a single mode of the ALP field. In this limit the local ALP field may be written
in the form√a( T ) = α ρDM /m a cos(ωT + ϕ),(5.3)

where we take the mean local dark matter density ρDM to be 0.4 GeV/cm3 [4], ω includes a small
contribution from the ALP kinetic energy, ϕ ∈ [0, 2π ) is a uniform-distributed random variable,174
2and α ≥ 0 is a Rayleigh-distributed random variable: P(α) = αe−α /2 . The modifications to ouranalysis due to the decoherence of the ALP field are discussed in our Supplemental Material [96],otherwise we assume α and ϕ are constant over the entire data collection time. The oscillating CS
term in the Ramsey fringe frequency is then written as
f CS ( T ) = 2WS ηCG / f a a( T ),(5.4)

where CG / f a is the ALP-gluon coupling (see [96] for alternative parameterizations) and η ≈ 0.03
is a coefficient relating CS induced in a generic heavy nucleus to the QCD theta angle [91, 100].
The quadrature amplitudes of the ALP signal are then√A′ = 2WS ηCG / f a α ρDM /m a cos(ϕ)
√

′

(5.5)

B = −2WS ηCG / f a α ρDM /m a sin(ϕ).
As mentioned earlier, A′ and B′ must still be treated as random variables even in the absence ofnoise; the fact that α is Rayleigh-distributed implies that A′ and B′ are each Gaussian distributedwith mean zero and variance√σX2 = (2WS ηCG / f a ρDM /m a )2 .

(5.6)

The equal quadrature variances reflect our ignorance of the local phase of the ALP field oscillations. The fact that A′ and B′ can simultaneously be small reflects the possibility that the earth
may have been near a null in ALP density during our data collection time. We can now construct
the expected ‘ALP distribution’ by adding the variance σX2 to our no-ALP variances (the no-ALP
variances characterize the experimental noise). This defines a new bivariate normal distribution

X =1
2πσAX σBX

e

′2

′2

σ
AX

σ
BX

− 12 ( A2 + B2 )

,

(5.7)

2 = σ2 + ζσ2 and σ2 = σ2 + ζσ2 . We encapsuwhich we call our ‘ALP distribution’. Here, σAX
X
BX
X
A′
B′

late sources of attenuation, including decoherence of the ALP field due to the finite linewidth, inthe frequency-dependent factor ζ which we discuss in detail in the Supplemental Material [96].

175
Now we can compare probabilities for the ALP vs no-ALP case. For each hypothesized
coupling strength CG /( f a m a ) and frequency ν of interest we take the ratio of the probability
distributions N and X evaluated at our actual measured amplitudes A0′ and B0′ :Ui (νi , CG /( f a m a )) =X ( A0′ (νi ), B0′ (νi ), νi , CG /( f a m a )),N ( A0′ (νi ), B0′ (νi ), νi )(5.8)

which we call the prior update: the number Ui is a multiplicative factor which, when multipliedwith our logarithmically-uniform priors, updates our prior belief to our post-experimental (orposterior) belief (see Figure 5.3). Note the prior update is equal to the Bayes factor in the limit ofvery low priors, which is the the limit we are operating in. A 95% exclusion corresponds to theprior update dropping to 0.05 [3].
So far, our analysis has not taken into account the look-elsewhere effect even though weare looking at more than 106 frequencies. The colormap of U in Figure 5.3 simply shows how ourbelief in the existence of ALPs has changed as a function of frequency and coupling. To accountfor the look-elsewhere effect one must move from a notion of how locally unlikely an event is
to how globally unlikely it is. For example, what might seem unlikely in a single frequency bin(say, you only expect it to happen 1/1000 times) becomes rather unsurprising when you lookin 1000 bins - in this case you should expect to see that event approximately once. Our local
measure, the prior update, compares the probability distributions in each bin for the ALP vs theno-ALP case. To move to a global measure we generate the aggregate prior update:

U (CG /( f a m a )) =aggregate posterior∑ U ϵ/νi= i i,aggregate prior∑i ϵ/νi(5.9)

where ϵ/νi is our prior belief (for an ALP of frequency νi ), using logarithmically-uniform priors(ϵ being constant). The aggregate prior update correctly accounts for our logarithmic priors,
which is necessary in broad searches like ours where the number of hypotheses tested varies
strongly as a function of frequency but we expect the ALP is equally likely to be found in any
frequency decade. Using equation 5.9 we can sub-aggregate over any subset of the analysis range
and at any frequency we like, and any interested reader can do the same - the full un-aggregated176

Figure 5.3: Bayesian power measured analysis of the EDM data. The grayscale matrix corresponds to a sub-aggregation of the prior updates U over 100 logarithmically-spaced frequency
bins. Darker shading on the logarithmic color scale corresponds to increased prior update, which
can be interpreted as an increased belief in an ALP of given frequency ν at ALP-gluon couplingCG /( f a m a ). We use a blue line to indicate the 95% exclusion (which corresponds to the subaggregated update dropping to 0.05 [3]) for sub-aggregation over each decade. The corresponding greyscale matrix for this sub-aggregation is not shown. Finally, a dotted line corresponds tothe aggregated exclusion over the full analysis range. The full un-aggregated matrix of U values
is available upon request.

177
matrix of U values is available upon request. The aggregate prior update taken over the entire
set of frequencies analyzed represents the fractional change in our belief that an ALP of a given
coupling strength exists anywhere in the full analysis range, which appropriately accounts forthe look-elsewhere efect. The aggregate prior update as a function of coupling strength is plottedin the Supplemental Material, where we also discuss our selection of the priors [96].
Our results yield no strong indication of a dark matter signal over the range 10−22 − 10−15
eV, so we use the above procedure to exclude CG / f a m a > 1.72 × 1016 GeV−2 at 95% confidence.
If we assume the density of the ALP field is held constant at its mean value, we can convert
our exclusion back to frequency modulation amplitude in Hz via eq (5.3) and (5.4) which gives
us an exclusion of 37 mHz. This is considerably less impressive-seeming than the limit of 1.2
mHz which we extracted from our original DC analysis [2]. This relaxation of the exclusion isan unavoidable consequence of correctly accounting for the stochastic nature of the ALP field
and the look-elsewhere effect associated with setting a limit across seven orders of magnitude in
frequency.
Our constraint on the ALP-gluon coupling CG / f a is plotted in Figure 5.4 along with existing
direct and indirect limits. Our results are the first laboratory constraints on the ALP-gluon
coupling in the mass range 10−17 − 10−15 eV.
The analysis techniques applied herein are general enough to apply to most time-series
datastreams, and in particular will be applied to our next-generation EDM measurement which
we expect to be about an order of magnitude more sensitive. More generally, the field of precision
measurement is in a remarkable period of progress reminiscent of Moore’s Law, and we expectthat analyses of this type will become more and more common. At the threshold of this era, we
believe that it is critical to ask: how we can get the analysis of hypothesis exclusion right?
This is especially pertinent given that the existing frameworks and analyses already published have so far failed to account for important effects, such as the stochastic nature of the ALP

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Figure 5.4: Astrophysical and laboratory limits on the ALP-gluon interaction. All limits assume
that ALPs saturate the local dark matter content and we have taken the mean local dark matter
density to be 0.4 GeV/cm3 [4]. All constraints correspond to a 95% confidence level. The purple
region embodies the constraint that the virial deBroglie wavelength is smaller than the size ofa dwarf galaxy [5]. The blue region is the constraint derived from looking at neutron EDM
oscillation [6]. The green region represents constraints from big bang nucleosynthesis [7], andthe yellow region represents constraints from supernova energy loss bounds [8, 9]. The pinkregion is the constraint described in this work.

179
field or the look-elsewhere effect. At least seven other dark matter exclusion publications haveoverestimated their exclusion by factors ranging from 3 to 10 in their failure to account for thestochasticity of the field [98]. To the best of our knowledge, the Bayesian analysis framework
we have chosen here is optimal for this kind exclusion for several reasons: first, it easily incorporates the stochastic nature of the dark matter field, as well as the fact that the scale of thestochastic fluctuations changes as the measurement time approaches the coherence time of thedark matter field. In addition, this framework seamlessly accounts for the differential sensitivityof the the two quadratures of our measurement to the dark matter field, and can be easily scaled
up to even higher dimensional parameter spaces. Finally, this framework correctly accounts forthe the look-elsewhere effect with respect to exclusion [63]. Note that [6] in Figure 5.4 fails toaccount for the stochastic fluctuations in the ALP field, which could mean that their exclusion
is overestimated by nearly an order of magnitude [98]. More generally, we appreciate how thisframework preserves all the information in the signal compared to a threshold-based framework,
which reduces the information in the signal to only ‘above’ or ‘below’ threshold.

5.3

Experimental Constraint on Axionlike Particles over Seven Orders of Magnitude in Mass - Supplemental Material
this section is a reproduction of the supplemental material of ref. [60]5.3.1Frequency Ranges
Our analysis was broken into two regimes: ‘low frequency’ (27 nHz – 126 µHz) and ‘highfrequency’ (126 µHz – 400 mHz). The fundamental limits of the low frequency analysis were set
by the inverse timespan of the entire dataset (27 nHz, see Figure 5.5a) and the ‘Nyquist’ limit forthe ‘block’ collection (each block took 22 minutes to collect, so this is 378 µHz). The fundamental
lower limit of the high frequency analysis was set by the inverse timespan of the most sensitive
data, which was taken over an 11-day period in 2017 (1.12 µHz, see Figure 5.5b). The upper limit
is slightly below the ‘Nyquist’ limit – each point took about 1 second to collect and we chose180
400 mHz. Our data is not evenly spaced in time, so we are using the term ‘Nyquist limit’ in aloose sense: with unevenly spaced data, the Nyquist frequency is often higher than one might
expect [101]. The frequency ranges above have considerable overlap, and we used this overlap

Figure 5.5: (Top) Individual CS measurements plotted as a function of their acquisition timesince August 16 2016 (the temporal ‘center’ of our dataset). (Bottom) Our most sensitive CS
measurements were collected over 11 days in 2017. For the low frequency analysis we used theentire dataset (top), and for the high frequency analysis we used only the most sensitive 11 days(bottom).

to validate our high frequency analysis. We analyzed the overlapping portion of the dataset
using both the ‘low frequency’ and ‘high frequency’ methods and found virtually the same LSSA
result in both cases. Taken together, the entire dataset was enormous, so we chose to truncate
both datasets to eliminate overlap and reduce the size of the dataset, which made computing
slightly less resource-intensive.
In each regime, the frequencies we interrogated were evenly spaced between maxima andminima; the spacing was chosen as 1/6 of our spectral resolution – intentionally oversamplingso that we wouldn’t miss any potential peaks. Our spectral resolution is given by the inverse of
the timespan to collect each dataset, in the low frequency case this is ∼450 days and in the high181
frequency case this is ∼11 days.

5.3.2

Generating Least-Squares Spectra
Our experimental signal is a count of Hf+ ions generated by state-selective dissociation of+

Hf ⟩the HfF+ ions. From this signal we generate an asymmetry Ai′ = ± Ni⟨−⟨. Here, Ni is the
Hf+ ⟩Hf+ count in a single shot (run) of the experiment and ⟨Hf+ ⟩ is the mean Hf+ count for thedata point (set of shots, averaged) to which the shot belongs. The sign of Ai′ is given by thespin prepration/readout conditions [102, 103]. A set of asymmetries, taken by sandwiching avariable free-evolution time t R between 2 coherent m F mixing pulses, generates a Ramsey fringe.
The frequency of the Ramsey fringe is directly proportional to the energy difference between them F = ±3/2 states in a given Stark doublet.
The Ramsey fringe frequency has multiple contributions:f R = 3gF µB Brot /h + 3α′ f rot R̃ B̃ − 2WS CS D̃ B̃

(5.10)+ ...,
where gF is the F = 3/2 state g-factor, µB is the Bohr magneton, Brot is the magnitude of the rotating bias magnetic field, f rot is the frequency of rotation, B̃ is the sign of the magnetic field, D̃ is the
populated Stark doublet, R̃ is the sense of the electric bias field rotation, and α′ is a parameter describing Berry’s phase [45]. For any given measurement, the experiment will be in a given switch
state, characterized by the values of { B̃, D̃, R̃}={±1, ±1 ± 1}. To isolate the CP-violating energy
2hWS CS , we repeat our spin precession measurement in each of the 23 unique states at varying
free-evolution times to form eight different Ramsey fringes and respective switch frequency measurements f { B̃ D̃ R̃} . From each set of eight fringes (called a block) we form linear combinations ofour measured switch frequencies to form parity frequencies: fparity = M−1 f{B̃D̃R̃} , where M−1
is a transformation matrix and f indicates the set of parity or switch frequencies. One of theselinear combinations isolates, to first order, the CP-violating CS term f CS = 2WS CS .
As mentioned above, our analysis is broken into two parts, low and high frequency. In

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general terms, the low frequency analysis consists of taking the f CS measurement from each block
as a function of block acquisition time (Figure 1) and reanalyzing the data to check for oscillations
in our signal. In contrast, for the the high frequency analysis we break apart each block and fringe
to extract a (potentially) time-varying CS signal from each shot of the experiment, allowing us toextend the bandwidth of our search over an additional three orders of magnitude.
In detail, the low frequency analysis consists of least-squares spectral analysis (LSSA) on
our set of f CS measurements as a function of block acquisition time. We minimizeN

1[ A cos(ωTi ) + B sin(ωTi ) − f CS ( Ti )]22
σi =1 iχ2 = ∑(5.11)

over A and B for every potential ALP oscillation frequency ν = ω/(2π ) of interest, where σi isthe standard error on a given f CS measurement and Ti is the block acquisition time in seconds
with respect to the temporal center of the dataset. This equation is analytically solvable, so we
directly compute the minima of the resulting quadratic equation to obtain the “best fit” values of
A and B: A0 (ν) and B0 (ν).
For the high frequency analysis, in contrast, we have to perform LSSA directly on the set ofasymmetry measurements as a function of shot acquisition time T. It may be helpful at this point
to emphasize that we have two different time streams: the free-evolution time t R (which ranges
between 0 − 0.7 seconds) which is an entirely distinct parameter from the shot/data acquisition
time T (which ranges between ±2 · 107 seconds). The asymmetry measurements already oscillate
as a function of free-evolution time (t R ), and for a static (DC) EDM measurement it is assumed
this oscillation frequency is constant (for a given switch state, up to experimental noise). In thepresence of an oscillating f CS ( T ), however, we expect the already-present oscillation in t R to be
frequency modulated in T, and we need to determine the form of this frequency modulation.
We begin with the fits generated from our DC analysis in [2]: the asymmetry points A′ (t R ) fromeach fringe in a given switch state were fit to Afit (t R ) = −Ce−γtR cos(2π f R t R + ϕ) + O, where
C is the fringe contrast, f R is the fringe frequency, ϕ is the initial phase, O is the offset, and γ

183
is the coherence decay rate. As mentioned earlier, the frequency f R measured in a given fringe
does not provide a direct measurement of CS ; it contains multiple contributions. The frequency
due to non-CS terms is f 0 = f R − M′ f CS , where M′ are the matrix elements used to transformbetween switch and parity bases and f CS is the frequency component due to CS . Since we are
only interested in the potentially time-varying CS component (and we assume it has a mean ofzero [2]) we writef R ( T ) = f 0 + M′ f CS ( T ),f

CS

(5.12)( T ) = A cos(ωT ) + B sin(ωT ).

If we include this frequency modulation (which we assume is very small), we can expand our fit
asymmetry Afit to second order in f CS ( T ) to obtain

Aosc ( T ) = −Ce−γtR cos(2π ( f 0 + M′ f CS ( T ))t R + ϕ) + O∼ −Ce−γtR cos(2π f 0 t R + ϕ) + O
+ Ce

−γt R

′ CS

(2πt R M f(5.13)( T )) sin(2π f 0 t R + ϕ)

+ Ce−γtR 2(πt R M′ f CS ( T ))2 cos(2π f 0 t R + ϕ),which is our desired expression for the f CS ( T )-induced frequency modulation. With this, we
perform LSSA on the difference between Aosc ( T ) and A′ ( T ),
N

1
′ 2
[Aosc
i − Ai ] ,2
σ
i =1 iχ2 = ∑(5.14)

where the subscript indicates the ith acquisition time point in our dataset.

5.3.3

Timestamp Shuffling
As described in the main text, our statistical analysis requires simulated data which, byconstruction, contains no coherent oscillations in the f CS channel. For the low frequency analysis
this is a rather simple procedure which relies on our knowledge that each block-based f CS measurement is drawn from a Gaussian distribution with mean zero and well-characterized variance[2]. For the high frequency analysis our simulation technique warrants further discussion.

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The challenge is to simulate shot-by-shot data which preserves the basic structure of thefringes and blocks as well as the inherent noise of the measurements. The fringes themselves
have a coherent oscillation in them, and each block contained eight fringes – each of which may
oscillate at a slightly different frequency depending on the switch state. The fringes also had
a limited coherence time, which varied throughout the dataset as we changed experimental parameters. While generating simulated data that faithfully preserves these features is in principle
possible, the most difficult part for our purposes was preserving the noise. There were a numberof technical noise sources which were not easily characterized, i.e. they didn’t conform to someknown distribution, and were transient, i.e. they were present or absent to different degrees
through the dataset.
We found that the most effective way to preserve the qualitative features of the noise andthe specific details of the fringes and blocks while erasing any real-time coherent oscillation in
the f CS channel was to randomly shuffle the timestamps of the data by block. As described in
the previous section, a block of data formed one single f CS measurement and was composed ofeight Ramsey fringes, which together comprise 96 data points or 1536 shots. When we collecteda block of data we would not collect each Ramsey fringe one at a time, slowly increasing thefree-evolution time until completion then changing the switch state to collect the next fringe. Instead, we would increase the Ramsey time while scrambling the various switch states, effectively
collecting points from each fringe in random order. In other words, we randomized the order inwhich we collected the 96 data points corresponding to each block, such that there is no fixed
relationship between either the switch state or Ramsey time of adjacent data points. When we
shuffle the timestamps by block, that means we take all timestamps of the data points belonging
to a given block and swap them with timestamps for data points belonging to some other block.
One can imagine this as swapping whole blocks around in time. Since a block corresponds to
a f CS measurement, this will eliminate any coherent oscillation in the f CS channel which may
have been present, so that it is broken instead into many lower-amplitude features that neither

185
appear as candidates for discovery, nor, when combined with many reshuffles, as significant
perturbation to the noise distributions we use for determining exclusions.
We checked to make sure the reshuffled data quantitatively reproduced the coarse-grained
frequency structure of the noise in the actual data. We were also able to compare the spectra generated by shuffling the f Cs data to spectra of the non- f Cs parity frequencies, which byconstruction should be free of coherent oscillation. Each such non- f Cs channel yields a single
representation of a noise spectrum which we should expect to be similar to that of f Cs in theabsence of an ALP signal – indeed, these shuffled spectra qualitatively reproduced our non- f Cs
noise spectra quite well. As an additional check, we can take an individual data set from ourcollection of reshuffled data sets and treat it to the full analysis as if it were our actual data set,
including creating a subaggregated exclusion heat-map plot as in Figure 3 of our main text, andan aggregate prior update ‘discovery plot’ as in Figure 4 below. The fake exclusion heat maps
reproduce the qualitative features of our actual exclusion heat maps. For the twenty fake ‘discovery plots’ we generated, the largest peak aggregate update value was < 20, and all but thethree largest were < 10; i.e., none were indicative of discovery. The peak aggregate update value
occasionally being slightly larger than in our real data set is to be expected as a consequence ofa sort of higher-order look-elsewhere effect.
To confirm that our simulation method really did sufficiently attenuate coherent oscillations
while preserving the spectral structure of the noise in our dataset, we performed least-squares
spectral analysis (LSSA) on simpler versions of simulated datasets which by construction contained a coherent oscillation as well as noise. The simulated data was sampled at the same times
as our real dataset, and we performed simulations at frequencies throughout the bandwidth analyzed. In each case we found that the detected amplitude of the oscillation was significantly
attenuated while the noise at other frequencies was not affected. Conversely, when we shuffledby data point or shot we found that while any coherent oscillation would be erased, the noise atother frequencies would also be attenuated.

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It is clear why shuffling the timestamps by block should erase any oscillations on thetimescale of the block itself or longer, but it may not be clear why the technique works foroscillations with a higher frequency. The reason this works is due to the manner in which
the data was collected – the random scrambling of when we collected the Ramsey times andswitch states is unique to every block, so the shuffle introduces phase jumps into the data on thetimescale of single data points – in other words, it effectively shuffles the data on the data point(16 second) timescale.
One might ask: why not just shuffle the timestamps by data point? The reason is thatshuffling by data point, while very effective at erasing any coherent oscillation present, also
appears to drastically change the spectral structure of the noise in our dataset. One might thenask why we don’t simply simulate the data directly, and again we find that assuming a specificnoise model (such as shot noise in the HfF+ count) is insufficient for simulating the real noise
we had present. It should go without saying that arbitrarily increasing the noise in the simulated
data led to the risk of hiding any real oscillation which would otherwise be detected, has no realscientific basis, and should be avoided.

5.3.4

Estimating the no-ALP distribution
Our analysis hinges on having a good estimate of the no-ALP distribution, which itself

relies on knowing the variances σA2 ′ and σB2 ′ . Because we are searching over such a large frequency
range, and examining so many different frequencies, there are many upward fluctuations in
the oscillation amplitudes obtained from LSSA, in both the real and simulated datasets. Small
fluctuations in the estimated variances can result in large changes in the apparent significance of
these amplitudes. If our estimates of σA2 ′ and σB2 ′ are poor, our exclusion will reflect that.
One might think that the best way to ensure we have good estimates of the variances
would be to do many more simulations. Unfortunately, especially for the high frequency data,
the simulations were very computationally expensive so running more simulations was not an

187
option. This high computational cost is not a feature specific to our choice of Bayesian analysis
framework, but rather a generic consequence of the large number of independent frequencies atwhich we must construct the no-ALP distribution, taken together with the size of the full set ofasymmetry measurements on which the high frequency LSSA is performed. Because we onlyran 1000 simulations our error in estimating the variances is dominated by statistical noise. To
account for this, we chose to adjust the raw variances obtained from the simulated data to better
match what we know to be true.
In the low frequency range, we made no adjustments to the variances. We know that there
will be some phases and frequencies which are favoured due to the temporal structure of thedata (see Figure 5.5, top), so we wanted to preserve any structure in the variances which we
found via simulation (including significant differences between the two quadratures A′ , B′ ), even
if it was a bit more noisy than desired due to the limited number of simulations.
In contrast, for the high frequency range we have no reason to believe there would be
any meaningful difference between the two quadratures thanks to the near-continuous temporal
spacing of the data we analyze (see Figure 5.5, bottom). In addition, we know that statisticalfluctuations are independent, bin-to-bin. So to attenuate the effect of statistical fluctuations wesmoothed the data using a 36-bin running average (this is 6 times our spectral resolution) andthen averaged the the two quadratures A′ , B′ .

5.3.5

Sources of Attenuation
To reduce the risk of artificially over-constraining the axion-like particle (ALP) coupling it

was important to account for all the known sources of attenuation present in our experimental
signal. To date we are aware of three possible sources of attenuation:(1) When the total observation time T is smaller than the coherence time τc , i.e. T < τc ,the ALP field is built from a sum over many phases of a freely oscillating bosonic field

188
in three dimensions, resulting in a random distribution of the field amplitude. We mustaccount for the probability that the local ALP field amplitude is significantly smaller thanits RMS value throughout the galaxy. For example it is possible that when we took thedata we could have been, by sheer bad luck, be sitting near a null of the field [98].
(2) When the total observation time T is comparable to the coherence time τc , i.e. T ∼ τc ,decoherence due to the finite linewidth of the ALP field will decrease the measured signal
amplitude [99].
(3) When the observation time for a given sample (i.e. a block in the low frequency analysis
or a shot in the high frequency analysis) is not small compared to the period of the ALPinduced oscillation, we are effectively averaging over some portion of the wave, leading
to attenuation of the signal.

5.3.5.1

Effect 1: Random distribution of the ALP field amplitude

This effect is dealt with automatically in the Bayesian Power Measured analysis framework.
When we generate the ALP distribution X from the no-ALP distribution N we increase thevariance of both quadratures without increasing the mean. See the main text between equations(3)-(7) for more details.

5.3.5.2

Effect 2: Decoherence

To quantify the effect of decoherence we compared a realistic (finite linewidth) ALP field
to an infinitely narrow linewidth field at several possible mass values. We built the realistic fieldfrom its constituent parts as described in Ref. [99], implementing eqs (3) – (10) numerically. The
result is an expression for the local ALP field, which we can ‘fit’ using LSSA as described in themain text. To generate the ‘infinitely narrow’ ALP distribution, we follow the same steps but
instead sample the velocity distribution only once before building up the full field. We ‘fit’ thisfield with LSSA as well, then repeat the process hundreds of times to get distributions for the189
quadrature amplitudes A0 , B0 in both cases. We call the amplitudes A, B and Anarrow , Bnarrow .
We rotate these distributions into the primary axes where the variance is maximized along one
axis, then define the attenuation due to decoherence in each quadrature ζ d,i as the ratio of the22. In practice we find ζ d,A ∼ ζ d,B , so we define
variances: ζ d,A = σA2 /σA,narrow, ζ d,B = σB2 /σB,narrowζ d = (ζ d,A + ζ d,B )/2. Unsurprisingly we see no decoherence effect in our low frequency band,
but we do at the extreme high end of the high frequency band. We fit the results using a secondorder polynomial to obtain a functional form for the attenuation due to decoherence over theentire analysis band: ζ d (ν) (Figure 5.6).

Figure 5.6: Attenuation factor due to ALP field decoherence, calculated over the high frequency
analysis range. Grey-filled circles are the calculated values, and the blue curve is the fit.

5.3.5.3

Effect 3: Finite sample time

At the high end of each analysis regime, we are analyzing frequencies fast enough that
there is a non-negligible change in the signal during the sample time. This results in an ‘averaging’ of the oscillation over the sample time and an effective attenuation in perceived signal
amplitude. For instance, at the upper end of our high frequency range we analyze 0.4 Hz but
each sample takes about one second to acquire. We have a similar issue at the high frequency

190
end of the low frequency analysis. To quantify this attenuation, we generated artificial sinusoidal
data sampled more finely than the timestamps in our real dataset, then averaged the signal over
the actual intervals of data collection. We did this for a number of frequencies spaced throughour analysis band, then averaged these results over phases spanning [0,2π). We then perform
LSSA on both the original simulated data and the averaged simulated data and define the attenuation due to finite sample time ζ s,i in each quadrature as ζ s,A = σA2 /σA2 ′ , ζ s,B = σB2 /σB2 ′ , where
A, B are from the averaged signal and A′ , B′ are from the original signal. We average the twoquadrature attenuation factors to determine the attenuation due to finite sample time ζ s . We did
this for both the high frequency analysis and the low frequency analysis, and used the fits to thedata to generate a piecewise function describing the attenuation due to this effect as a functionof frequency (Figure 5.7).
Finally, our overall attenuation factor over the entire analysis band is given by the productof the attenuation factors for the two effects: ζ (ν) = ζ d (ν)ζ s (ν). This is the quantity introduced
below Eq. (7) in the main text.

5.3.6

Parameterizing the ALP-gluon coupling
Several different conventions have been used in the literature to parameterize the ALP-

gluon coupling we constrain in this letter. Here, we use the parameterization of Ref. [6]. The
most fundamental parameter in any ALP model is the energy scale f a at which a new U (1) symmetry is spontaneously broken. The ALP field then arises as the pseudo-Goldstone boson of thisbroken symmetry; its mass and all its couplings will generically scale as f a−1 , with different dimensionless coefficients CG , CN , Cγ , Ce for couplings to gluons, nucleons, photons, and electrons,
respectively. For the canonical QCD axion, the gluon coupling coefficient is fixed at CG = 1 by
the requirement that the axion solve the Strong CP problem, but for more general ALPs CG canbe regarded as a free parameter, which we would typically expect to be O(1). Because of this,191

Figure 5.7: Attenuation due to finite sample time over the entire analysis range. The blue curveshows the piecewise function generated by the fit to the grey circles.

many papers (such as Ref. [9]) express bounds on the gluon coupling CG / f a as bounds on f a .
Other papers (e.g., Refs. [7, 8]) parameterize the ALP-gluon coupling as the coefficient gd of theoscillating nucleon EDM that the ALP field would generate. Because an EDM has dimensionsof ecm ∼ eV−1 in natural units and the ALP field itself has dimensions of eV, this implies gd
has dimensions of eV−2 , and is related to our parameterization through Eq. (7) of Ref. [8], withf a → f a /CG .

5.3.7

Aggregate Prior Update
As described in the main text, the aggegate prior update (Figure 5.8) describes the fractionalchange in our belief that an ALP of a given coupling strength exists anywhere in the full analysis
range, which accounts for the look-elsewhere effect. Computing the aggregate prior update requires making a choice about the frequency-dependence of the priors (though not their absolute
scale). While this may seem to introduce an element of subjectivity into the analysis, it in fact

192
merely makes explicit a choice of assumptions which is also present, though implicit, in otheranalysis frameworks [3]. In equation (9) of the main text and Fig 4 below, we adopt logarithmically uniform priors, which are simple, well-suited to broadband exclusion, and well-motivatedby how particle masses seem to distribute themselves in nature.
If our analysis were composed of evenly-spaced frequencies between 27 nHz — 400 mHz,
computing the aggregate update would have been a direct implementation of equation (9) of themain text, where the logarithmically uniform priors account for the fact that there are far fewer
frequencies in the lowest bin vs the highest bin. But since our analysis is split into two regimes,
so-called ‘low frequency’ and ‘high frequency’, each with different frequency spacing, we needed
to ensure our computation of the aggregate prior update accounted for the fact that the numberof frequency bins in each decade wasn’t a smooth function of the frequency (or, in other words,
that it is a smooth function of the frequency, except for the discontinuity at the point where thetwo analysis regimes are stitched together). To account for this we computed the aggregate priorupdate in each regime separately, then took a weighted average of the two results, where theweights were the logarithm of the number of frequency bins in each analysis regime.

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Figure 5.8: Aggregate prior update U (equivalently, exclusion) as a function of ALP-gluon coupling CG /( f a m a ). The horizontal dotted line indicates 95% exclusion (where the aggregate priorupdate drops to 0.05), while the vertical dotted line indicates the lower limit of CG /( f a m a ) whichis excluded at 95% confidence over the entire analysis band. Had our data presented us withstrong affirmative evidence for the existence of an ALP in our frequency range, the curve of U vs
CG /( f a m a ) might have peaked not at U = 2.5 but for instance at U = 1, 000 or higher.

Chapter 6
Conclusion

Conclusion chapters are often corny, pedantic, and/or unnecessary. “Tell me what you
told me” is the general approach to chapters with this title. Why bother with that? Even I am
tired of listening to myself talk at this point, and you, dear reader, have already slogged through5 chapters with me. But I made the executive decision (it is my thesis after all), to include thischapter for one specific reason: this thesis marks an end to the original conception of the JILA
trapped ion eEDM experiment. Moving forwards, JILA’s eEDM experiment will take on a totallynew, and rather revolutionary1 , form.
Let’s try to re-adjust our perspective for a moment: this thesis was about performing aprecision measurement of the electron’s electric dipole moment using trapped molecular ions.
The measurement detailed in this thesis was the second, and last, of its kind: no other group in
the world is using trapped ions to perform any measurement that requires electrically polarizing
the species - the idea of a rotating electric field is still so new no-one else has picked it up yet.
Since 2014 I have been working on a physics experiment which, in form at least, had no peers.
Honestly, I found that delightful.
With such a unique experimental setup it almost seems a shame to put the whole thing out
1 In my opinion.

195
to pasture2 , but we are left with little choice (plus, and it’s hard to understate how important thisis to the whole operation, Eric was obviously already kind of bored with it). Why is our handforced? Well, let’s imagine what we need to do to squeeze another result out: if we aim for a 10x
improvement in sensitivity with each subsequent measurement (which is admittedly ambitious
but also in keeping with our current pace) then we would need to either read out 100x more ionswith each shot or find a way to get our fringe contrast up to 1, then read out 25x more ions each
shot, or use a new molecule to get higher | Eeff |, then get more contrast, then, etc etc. Choose any
way you like to manipulate our old friend, equation 2.1 (reproduced below as Eq. 6.1 for yourconvenience), to get a factor of 10 out, and then think about what’s actually possible.

δde ∼

1

√
| Eeff |τ N

(6.1)

Let’s consider τ first: we could get a straight factor of 10 in sensitivity if we can make τ
10× larger. Imagine we have a molecule which has the science state as a ground state, so that weare not limited by excited state lifetimes. Then what happens? Our experimental rep rate goesway down. We would prepare our ions, start the Ramsey sequence, then wait for something like
15 seconds before reading out the result and starting all over again. We would still need lots ofdata to get the statistical sensitivity we need, and now what previously took about 4 weeks ofdedicated data-taking time now takes a year! Not to mention the 1/ f noise issues that would
inevitably crop up with such a low rep rate. So in general, if we are looking to extend τ we arelooking immediately at having a rep rate, and a pink noise, problem. Not acceptable.
What about√

N? Could we leave τ alone and just read out 100× more ions per shot? Well I

don’t want to declare anything like that impossible3 , but I can confidently say (quite sarcastically):
good luck. First of all, we can’t really trap more ions: space charge effects will always put a
2 Presumably, some kind of UHV safe pasture with no shortage of copper gaskets and pure quantum states
3 I am far too cowardly for that

196
fairly hard limit on the total number of ions one can trap in one place. But! We were routinely
trapping something on the order of 20, 000 ions while we only read out ∼ 500 of them. Thismassive discrepancy is due to state preparation issues: each step in our state prep was much lessefficient than 1 and so the ultimately useful fraction of ions we could read out is the result of abunch of values < 1 being multiplied together. Ok fine, then why can’t we just ‘fix’ each step inthe state preparation to massively improve our efficiency? I suppose we could, in theory. But inpractice? Nah. Our species was selected for its sensitivity to the eEDM, not its amenability toquantum state control. It is possible that there is a magic species out there, with highly diagonal
frank-condon factors as well as a large enough Eeff etc, but if it exists nobody has found it yet.
Even if we did know of this magical species, and we could, through sheer force of will, prepare
all 20, 000 ions in the trap in the science state and read them out with perfect efficiency, thispwould still only offer us (40) ∼ 6× improvement in sensitivity, so we would also need to fightsome other battles to get the factor of 10 we require. But more realistically: this magic species is
unknown and the time it would take to find it, do all the required spectroscopy, etc etc, would
put us so far behind our competitors temporally that we would suddenly be needing a factor of
100 in improvement to stay competitive. So again, we are stuck.
That leave us with Eeff . This is (obviously) fixed for our current species, but of course one isfree to change species! Could we get a factor of 10 that way? Maybe. But changing species is nosmall task: since we work with bespoke molecules (they simply must be sensitive to the eEDM,
darling) we have to do all the spectroscopy and figure out all the state preparation steps ourselves
before we can even think about performing an EDM measurement. And given a quick survey ofthe molecules currently in use for eEDM experiments, it’s unlikely that one would enjoy a full
10× improvement in Eeff . It’s more likely to be something like 3×. So again, as expected, we arestuck.
Now you may be thinking: ‘well, just figure out the right combination of those three things,
dummy’! And sure, that is a possibility. But how far will that get us? Another 10×, then after

197
that? At some point the potential of this approach is limited: you can’t, even if you reeeeally wantto, trap arbitrarily large amounts of charged species in one location, so you will always end up
severely statistics limited. And so we end back up where we started: trapped ion experiments
have a rep rate problem.
What one needs is a way to combine the ‘unlimited’ rep rate of a beam experiment with thelong coherence times available with trapped ions. Eric realized that instead of trapping the ionsin one place one could instead ’trap’ the ions in a clumpy sort of beam : one can shuffle the ion
trap itself along in space, then load another one behind it and another one behind that and so
on, until what one has is a beam (more like a train) of N ion clumps, with state preparation being
done at one end, free evolution happening along the travel distance, and state readout happening
at the other end. With this method, it’s like running N traps in parallel, without multiplying thesurrounding infrastructure by the same factor. We have been calling this approach the ‘bucket
brigade’. Thankfully4 , the details of the bucket brigade are well beyond the scope of this thesis.
So while our once-revolutionary trapped ion EDM experiment is being burnt to the groundout behind the institute by a group of disgruntled graduate students (proverbially, of course), a
new phoenix rises from the ashes, crying out ’good luck, suckers’.
Oh! And before I go, one last thing: unionize your workplace. United we bargain, divided
we beg.

4 At this point we are all getting pretty tired of my cutesy use of italics and breaking the 4th wall

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[103] In our previous constraint of the static eEDM, we used a slightly different asymmetry,
N −N
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Appendix A
LasersOur experiment used a lot of lasers. Here is a master list, much too late to be of any use foranyone actually doing the experiment.

Neutral HfF production / ablation: “Abby" 532 nm

A.1

Continuum ‘Minilite’ pulsed Nd:YAG Laser
Abby is a Class 4 Pulsed Q-switched Nd:YAG laser, with a 2nd harmonic crystal for 532
nm output. The output pulses are 10 ns long and deliver ∼ 10 mJ per pulse.
We use a dichroic mirror on the output to dump 1064, while 532 goes to the chamber. Abby
is a variable rep rate laser, and is triggered by the LabView program’s DIO.
The table below is from the manual, which is stored in the bottom drawer of the filingcabinet.

207
Wavelength
EnergyPeak Power
Average Power
Pulsewidth
Stability
Polarization
JitterBeam Size
Divergence
Rep Rate

1064 nm
50 mJ
8.3 MW
750 mW
5-7 nsec
2%
Horz±0.5 nsec<3mm<3 mrad1-15 Hz532 nm25 mJ
6.3 MW
375 mW
3-4 nsec3%
Vert±0.5 nsec<3mm<3 mrad1-15 Hz

Table A.1: Minilite II specs from manual

Photionization 1+2 pump: “Mia" 532 nm

A.2

Spectra Physics ‘Quanta Ray Pro Series’ pulsed Nd:YAG laser
Part specification: Quanta-Ray-Pro-250-10. This 10 Hz YAG pumps Harambe and Bertha.
The undoubled 1064 nm fundamental gets dumped internally. 532 nm is the only output. We
use about 25 mJ to pump Harambe and 250 mJ to pump Bertha. The serial number is 2344PR.
The table below is from the manual, which is stored in the bottom drawer of the filingcabinet.
Wavelength
EnergyPulsewidth
Short Term Energy Stability

1064 nm1250 mJ/p
8-12 ns±2%532 nm
650 mJ/p
1-2 ns±3%

266 nm
130 mJ/p
3-4 ns±8%

Table A.2: Quanta-Ray-Pro-250-10 specs from manual

208

A.3

Photoionization 1 seed: “Tuna" 1237.55598 nm

Toptica ‘TA Pro’ external cavity diode laser
Tuna is a commercial ECDL with a TA, whose fundamental wavelength is set to 1237.55598
nm. We send about 350 mW into a fibre, which goes to a waveguide doubler (NTT Electronics
Wavelength Conversion Module WH-0619-000-A-B-C) with a maximum power input of 200 mW.
Out of the waveguide doubler we get about 70 mW of doubled output (618.77799) which is used
to seed Harambe, our pulsed dye amplifier. The wavelength of the doubled light is controlled by
the temperature of waveguide doubler. The serial number is 025112, The TA Article number is
TA-1250-1500-1.
Tuna’s datasheets are stored in the tall black cabinet as well as digitally in the ’Lasers’
lab notebook. The table below summarizes a few key parameters. The ’actual’ parameters drift
around a bit, but they give a good starting point if you’re looking to get it locked to the nominal
wavelength.
Wavelength (nominal)
Output Power (Actual)
Output Power (Max)
Diode Current (Actual)
Diode Current (Max)
Piezo Voltage (Actual)
Lasing Threshold
TA Current (Actual)
TA Current (Max)Linewidth (from OSA spectra I took)

1237.5 nm
370 mW into fiber
2000 mW
267.9 mA
272 mA
94.643 V
43 mA
3840 mA
7000 mA
0.1 nm FWHM

Table A.3: A few Tuna (Toptica TA Pro) specs from the data sheet

209

A.4

Photoionization 1 (first photon): “Harambe" 309.388 nm

Radiant ‘NarrowAmp’ pulsed dye amplifier
Harambe is a 3 stage pulsed dye amplifier1 . Note there isn’t much of a manual for this(apparently it was shipped without one and when Yan requested one they sent him a hastily
written 10 page Word document).
Harambe is pumped by Mia (532 nm) and seeded by doubled Tuna (618.778 nm). The
photon used for ionization is the doubled output of Harambe (309.388 nm). We use around 30µJ
per pulse, and when the amplifier is well-aligned with 50 mJ of pump power and 60 mW of seed
power we are able to get ∼ 150 uJ out.
Harambe’s dyes are Rhodamine 610 chloride and Rhodamine 640 perchlorate. We premake 1 g/L concentrate of both dyes which we then dilute when we fill the circulators. Harambe’s
dye volumes are (all assuming 1g/L concentrate):• first stage: 165 mL Rhodamine 610, 185 mL Rhodamine 640, 725 mL ethanol• second stage: 90 mL Rhodamine 610, 100 mL Rhodamine 640, 390 mL ethanol• third stage: 90 mL Rhodamine 610, 100 mL Rhodamine 640, 880 mL ethanol
Photoionization is a two-photon process, which promotes a valence electron to a Rydberg
state 54 cm−1 above the ionization potential. Once in this state, the molecule quickly autoionizes.
Here are some specs I found at the following website: https://www.radiant-dyes.com/
PDF/NarrowAmp.pdf, plus specs for our specific usage - generating 618nm light then doubling it
to 309 nm.
1 Technically not a laser, just an amplifier, but nonetheless relevant to put here. Sorry to be pedantic, but this is a

PHD thesis after all.

210
Tuning Range
Linewidth(dependent on pump laser)
Maximum Output Power, doubled (309 nm)(50 mJ pump, 60 mW seed)Maximum Output Power, doubled (309 nm)(21 mJ pump, 60 mW seed)
Output pulse energy (from url)
(at 730 nm, pumped 45 mJ with 532 nm)
Power Stability
Pointing Stability
Divergence

540-800 nm
approx. < MHz 100 mS
150 µJ
70 µJ

> 9mJ
< 6%
< 50µrad< 0.5mrad

Table A.4: Select Harambe (Radiant Narrow Amp) specs

A.5

Photoionization 2 (second photon): “Bertha" 735.464 nm

Sirah ‘Precision Scan’ dye laser
Bertha is a tunable pulsed dye laser pumped by Mia (we typically use about 250 mJ pump
power). Currently, we don’t use the tuneability. The output beam is S polarized. The fundamental is doubled, and it is the doubled output which is used for ionization. We use around 2 mJ of
doubled power for ionization.
Bertha’s dye is Pyridine 2 (LDS 722). We pre-make 0.5 g/L concentrate we then dilute
when we fill the circulators. Bertha’s dye volumes are (all assuming 0.5 g/L concentrate):• Resonator: 252 mL dye concentrate, 448 mL ethanol• Pre-Amp: 173 mL dye concentrate, 1027 mL ethanol• Cap Amp: 90 mL dye concentrate, 1910 mL ethanol
Here are some specs I found at the following website: https://www.sirah.com/wp-content/
uploads/documents/PrecisionScan.pdf, plus specs for our specific usage - generating 735 nm
light then doubling it to 367.7 nm.

211
Linewidth(dependent on grating, our specific are unknown)
Absolute wavelength accuracy
Wavelength Stability
Divergence (typical)
Polarization
Pump Power (532 nm, our use)
Maximum Output Power (doubled, with 250 mJ pump)

1 − 4 pm
< 15 pm
1.5pm/◦ C
0.4mrad> 98% (vertical)
250 mJ
5 mJ

Table A.5: Select Harambe (Radiant Narrow Amp) specs

A.6

State transfer: “Toptica" 961.43495 nm

Toptica ‘TA Pro’ external cavity diode laser
Toptica is a commercial ECDL with a TA, it’s named ‘Toptica’ because it was the very first
Toptica laser we got in the lab and apparently it never occurred to anyone that we would get
another one. It’s probably easier to call this laser ‘Transfer’ but it’s too late for that! We couple
about 1W into a PM fiber which is sent over to the chamber then directed through the top of thetrap (with linear polarization). We send about 600 mW into the trap.
Toptica’s datasheets are stored in the tall black cabinet as well as digitally in the ’Lasers’
lab notebook. The table below summarizes a few key parameters.
Wavelength (nominal)
Output Power (Actual)
Output Power (Max)
Diode Current (Actual)
Diode Current (Max)
Lasing Threshold
TA Current (Actual)
TA Current (Max)986.45 nm
1.3 W before fiber, 620 mW after fiber
2060 mW
94 mA
110 mA
27 mA
2877 mA
2930 mA

Table A.6: A few Toptica/Transfer (Toptica TA Pro) specs

212

A.7

Cleanup: “Vibational Cleanup" 818.372 nm

Homebuilt external cavity diode laser
Vibrational Cleanup is a homebuilt ECDL, using a ThorLabs diode centered at 818 nm. We
get about 68 mW out which we send into a fiber then into the top of the trap (linearly polarized).
We get about 17.5 mW into the chamber.
There are no datasheets or manuals for this laser. Here’s our current operating parameters:
Wavelength (nominal)
Output Power (Actual)
Diode Current (Actual)
Piezo Voltage (Actual)

818 nm68.5 mW before the fiber, 17.5 mW into the chamber
291.4 mA
73.18 mV

Table A.7: A few Vibrational Cleanup (Homebuilt ECDL) specs

213

A.8

mF pumping: “DBR/1083" 1082.4137 nm
I couldn’t find much documentation for this one
Operating Temperature (Actual)
Diode Current (Actual)
Power into isolator (Actual)
Power into AOM (Actual)
Power into fiber at 49% duty cycle (Actual)
Power into trap at 49% duty cycle (Actual)

7.5 ◦ C
318.36 mA
170 mW
120 mW
46 mW
21 mW

Table A.8: A few Vibrational Cleanup (Homebuilt ECDL) specs

A.9

Depletion: “814/Depletion" 814.508 nm

Toptica ‘TA Pro’ external cavity diode laser
Depletion is a commercial ECDL with TA. The laser puts out about 1W. We couple thelight into an AOM which is modulated at a 50% duty cycle at f rot . The AOM’s zeroth order
output is coupled into a fiber and sent into the top of the trap for single doublet depletion. The
AOM’s first order diffracted light is sent into a polarization setup (known as a Berek Polarization
Compensator) to obtain circularly polarized light, which is then sent into the side of the trap. We
send about 500 mW into the side of the trap for mF level depletion and about 50 mW into the
top of the trap for single doublet depletion.
Depletion’s datasheets are stored in the tall black cabinet as well as digitally in the ’Lasers’
lab notebook. The table below summarizes a few key parameters.

214
Wavelength (nominal)
Output Power (Actual)
Output Power (Max)TA Current (Set)
TA Current (Max)
Diode Current (Set)Diode Current (Max)
Piezo Voltage (Actual)
Lasing Threshold

815.0 nm
540 mW into trap at 50% duty cycle
2100 mW
2610 mA
2800 mA
111.665 mA
128 mA
90.83 V
56 mA

Table A.9: A few Depletion (Toptica TA Pro) specs

A.10

Photodissociation 1 (first photon): “Cora" 736.989/2 nm or 736.986/2 nm

Sirah ’Cobra Stretch’ dye laser
Cora is a tunable pulsed dye laser pumped by Olga. We use the tunability to dissociate
out of different states. All values here are for dissociating out of 3 ∆1 , v = 0, J = 1. The output
beam is S polarized. The fundamental is doubled, and it is the doubled output which is used for
dissociation. We use around 1.5 mJ of doubled power for dissociation.
Cora’s dye is Pyridine 2 (LDS 722). We pre-make 0.5 g/L concentrate we then dilute when
we fill the circulators. Cora’s dye volumes are (all assuming 0.5 g/L concentrate):• Resonator: 280 mL dye concentrate, 420 mL ethanol• Amplifier: 144 mL dye concentrate, 936 mL ethanol
Here are some specs I found at the following website: https://www.sirah.com/wp-content/
uploads/documents/CobraStretch.pdf, plus specs for our specific usage - generating 736 nm
light then doubling it to 368.5 nm.

215
Linewidth(dependent on grating, our specific are unknown)
Absolute wavelength accuracy
Wavelength Stability
Divergence (typical)
Polarization
Pump Power (532 nm, our use)
Fundamental measured before doubling (at 736.989 nm)
Power measured right after doubling (at 736.989/2 nm)
Power measured after the chamber, (at 736.989/2 nm)

1 − 4 pm
< 15 pm
1.5pm/◦ C
1.0mrad> 98% (vertical)
170 mJ
20 mJ
3 mJ
1.6 mJ

Table A.10: Select Cora (Syrah Cobra Stretch) specs

Photodissociation 2 (second photon) + Cora pump: “Olga" 266 nm

A.11

Spectra Physics ‘Quanta Ray Lab Series’ pulsed Nd:YAG laser
Part specification: Quanta-Ray-Lab-170-30H - note we bought this as a 30 Hz laser so thelast part of the part number is 30 H. But we had it converted to 10 Hz, so the last part of thepart number should be changed to 10H. This 10 Hz YAG pumps Cora and provides 266 nm
for dissociation. The undoubled 1064 nm fundamental gets dumped internally. 532 nm output
pumps Cora (170 mJ, remainder is dumped). 266 nm output is sent straight to the trap for
dissociation. The serial number is 4364L.
The table below is from the manual, which is stored in the bottom drawer of the filingcabinet.
Wavelength
EnergyPulsewidth
Short Term Energy Stability

1064 nm
850 mJ/p8-12 ns±2%532 nm
450 mJ/p
1-2 ns±3%

266 nm
90 mJ/p
3-4 ns±8%

Table A.11: Quanta-Ray-Lab-170-10H specs from manual. Note the manual specified the pulse
width in nm but I’m pretty sure that’s a typo, especially given footnote 1 on table 3-2.

Appendix B
One shot asymmetry

We normally define our asymmetry as A = ( N↓ − N↑ )/( N↓ + N↑ ), which takes two ’shots’of the experiment, but one can instead define it with only one shot. We did this for our dark
matter analysis [60]. We define the single-shot asymmetry A as

A=±

Ni − ⟨ N ⟩
,
⟨N⟩

(B.1)

where we choose + when the preparation and the depletion have the same phase (known colloquially as ‘in-in’ or ‘anti-anti’) and − when preparation and depletion have the opposite phase.
Ni is the Hf+ count in a single shot, and ⟨ N ⟩ is the mean Hf+ count for the set of shots (datapoint)
to which the single shot belongs.
Below is a little thing I worked out on how background affects the measured fringe contrast,
in case it’s useful to someone one day.

B.0.1

One-shot asymmetry and measured fringe contrast
As mentioned above, we define the single-shot asymmetry A as

A=±

Ni − ⟨ N ⟩
⟨N⟩

(B.2)

where we choose + when the preparation and the depletion have the same phase (known colloquially as ‘in-in’ or ‘anti-anti’) and − when preparation and depletion have the opposite phase.

217
Ni is the Hf+ count in a single shot, and ⟨ N ⟩ is the mean Hf+ count for the set of shots (datapoint)
to which the single shot belongs. Let’s see how this definition works. Assume, in some perfect
experiment, that we prepare the same number of ions each time, and that our state preparationand dissociation are perfect. If you consider zero free-evolution time, then the in-in and anti-anti
shots will result in N Hf+ ions being read out, while the in-anti and anti-in shots will result in 0
Hf+ ions being read out. For a given set of shots, called a data point we will alternate between
in-in and anti-in or anti-anti and in-anti. For any set of these chops, the mean number of ions weread out will be N/2. For the in-in and anti-anti shots we have the asymmetryA=+

N − N/2
=1
N/2

(B.3)

while for the in-anti or anti-in shots we will have the asymmetryA=−

0 − N/2
=1
N/2

(B.4)

so you see that we can combine the shots with opposite depletion appropriately. In this perfect
experiment, as we increase the free-evolution time, the asymmetry A traces out a sinusoid which
oscillates between ± 1 at a frequency proportional to the energy splitting.
Now what happens if you add a constant background? This is an important question
because we know that there is a background which is independent of the free-evolution time. It
seems to be due to imperfect ionization and dissociation. For simplicity let’s continue to assume
the ion number is constant and so on, and that the background is constant. Call the backgroundHf number b. Now the mean Hf (for a data point) is N/2 + b, the minimum number is b, and218
the maximum number is N + b. Our in-in or anti-anti shots will have the asymmetry( N + b) − ( N/2 + b)
( N/2 + b)
N+b=
−1
N/2 + b2N + 2b=−1
N + 2b2b2N+−1=
N + 2b
N + 2bA=+

while the anti-in or in-anti shots will giveb
N/2 + b2b
= 1−
N + 2b

A = −1 −Averaging these two will give
 2N
2b
2b 
/2
+
−1+1−
N + 2b
N + 2b
N + 2bN=
N + 2bA=<1
Which we see leads to a loss in contrast.
Our measured fringe contrast varies between 0.6-0.85. It is possible that the imperfectionis due to more than just a constant background, such as imperfect depletion, pumping, or π/2
pulses, but it is at least in part due to the background that we always observe.

Appendix C
A list of useful transitions

There’s a lot to keep track of in our experiment, and at one point I made the diagram below
and put it as the desktop background on the main experiment computer. Until then, there wasn’t
any central place where all the useful transitions we use were catalogued. Turns out that kind ofthing is quite useful so I am saving that figure here, for posterity1 . Enjoy.

1 I don’t actually know what the word ’posterity’ means and I will not look it up

220

Figure C.1: The infamous ’laser diagram’Appendix D
An approximation to our shot noise limit

Here’s my derivation of the shot noise limit if we knew the total number of ions we werestarting with.
Ultimately, the core of our experiment is spin precession experiment (Ramsey spectroscopy)
between stretched states. We prepare an ensemble of ions (for now, let’s say that we prepare thesame number each time Ntot ) whose valence electrons are in a pure spin state |↑⟩ or |↓⟩, we take√that pure spin state and rotate it by π/2 into an equal superposition (|↑⟩ + |↓⟩)/ 2, then we waita varying amount of time before we project the superposition back onto one of the basis vectors(|↑⟩ or |↓⟩), and we count N↑ or N↓ (for now let’s assume that N↑ + N↑ = Ntot ) . For any quantum
two-level system prepared in a superposition |ψ⟩ = c↑ |↑⟩ + c↓ |↓⟩, the probability of measuring(projecting the state into) a given basis vector is given by the modulus squared of the coefficient1:p↑ = |c↑ |2 , p↓ = |c↓ |2 . This probabilistic/non-deterministic essence is inescapable (it’s one of

the core features of quantum mechanics, after all2 ), so unless either c↓ or c↑ is 0 or 1, there will
always be some uncertainty in the outcome. When an experiment has a binary outcome withsome probability (eg p↓ , like we have here), and it is repeated over and over, one will find thatthe measured outcomes obey a binomial distribution: if we have Ntot ions, and N↑ is the numberof ions in state |↑⟩ (and N↓ is the number of ions in state |↓⟩, N↑ + N↓ = Ntot ), then the probability
1 Assuming that the state vector is properly normalized, ie | c |2 + | c |2 = 1.
↑↓
2 As far as we know

222of measuring a given value of N↑ , ie P( N↑ ), is3P( N↑ , Ntot , p↑ ) =Ntot !
Np↑ ↑ (1 − p↑ ) Ntot − N↑ .
N↑ !( Ntot − N↑ )!

(D.1)

The variance of the binomial distribution isσ2 = Ntot p↑ (1 − p↑ ),(D.2)

which goes to zero when p↑ = 0 or p↑ = 1, and has its maximum value of Ntot /4 when p↑ = 1/2.
It is this variance in the measured outcome that is the ultimate limit to our errorbars 4 .
Now there’s one final detail: during our Ramsey experiment the spin precesses, so the initial√state (|↑⟩ + |↓⟩)/ 2 evolves such that at a later time t R after a second π/2 pulse the state vector
can be written as ∼ cos(ωt R ) |↑⟩ + sin(ωt R ) |↓⟩. So now the probabilities of measuring a givenbasis vector become sinusoidally time-varying: p↑ = |c↑ |2 = | cos(ωt R )|2 , which goes right into
the binomial distribution of equation D.1 and the corresponding variance (eq D.2):
σ2 (t R ) = Ntot | cos(ωt R )|2 (1 − | cos(ωt R )|2 ).

(D.3)

This is the quantum projection noise limited uncertainty for any measurement we will make ofan ensemble of independent ions. In other words, this is the minimum uncertainty we can expect
in any measurement of the number of ions in a given stretched state. How will this uncertainty
affect our total uncertainty?
To compute the quantum projection noise limited total uncertainty, we need to go froman uncertainty in a single spin asymmetry measurement to an uncertainty in the precession
frequency, then from that to a measurement of de . I’ll sketch out a heuristic argument using only
the sides of the fringe, I expect the rigorous argument is more complex than we need. Let’s step
3 If you haven’t studied probability distributions, the way to read P ( N , N , p ) = is “the probability of measuring
tot ↑↑N↑ given Ntot and p↑ is...
4 If I’ve explained this poorly, of if you’re just looking for more details, I would strongly recommend you read [104]223
through a perfect, quantum-projection-noise-limited version of our experiment and keep track of
the uncertainties:√
First, we prepare an ensemble of ions Ntot in a superposition state (|↑⟩ + |↓⟩)/ 2, let it
evolve, apply a second π/2 pulse, then measure the number of ions in the up state, N↑ . The
probability of measuring a given value of N↑ (on the side of a fringe, we expect N↑ = Ntot /2) isgiven by the binomial distribution with variance σ↑2/↓ = Ntot p↑ (1 − p↑ ) = Ntot /4, since p↑ = 1/2
on the side of a fringe.
Then we repeat the same process to measure N↓ and again σ↑2/↓ = Ntot /2 since we are on
the side of a fringe (p↑ = p↓ = 1/2).
Now we take these two measurements and form the fringe asymmetry: A = ( N↓ −N↑ )/( N↓ + N↑ ). Combining the errors we have a new uncertainty: the numerator and denominator have the same variance δA2n/d = Ntot /2, and the quotient of that has varianceδ A2 =N↓ − N↑ 2
Ntot /2 
Ntot /2+
2
N↓ + N↑( N↓ − N↑ )( N↓ + N↑ )2=

Ntot /2( N↓ − N↑ )2
Ntot /2+( N↓ + N↑ )2( N↓ + N↑ )4

=

1
2Ntot

(D.4)

where in the last line we used N↓ + N↑ = Ntot
Now we repeat this procedure r times, and we get r/2 asymmetry measurements. We
average these measurements to get a better estimate of the spin asymmetry, and the sampling
distribution of the mean5 is usually approximated by a normal distribution, so the quantumprojection-noise-limited uncertainty on an asymmetry measurement is δA2mean = δA2 /r:
δA2QPN =5 For N
tot >> 1

1
Ntot f rep Ttot

,

(D.5)

224
f rep is the rep rate of the experiment, and Ttot is the total time the experiment was running, andr = f rep Ttot /2.
On the side (zero-crossing) of a sinusoisally-varying asymmetry measurement, the slope(in units 1/s) is equal to ω 6 , which we approximate as a line with zero offset: y = ωt. If we are
fitting a line to a point with errorbar σ′ (and σ′ << 1), then the line can be moved up or down σ′ ,so the fitted line can range from y = ωt to y = ωt ± σ′ . The horizontal distance (in this case, thetime shift) between the "true line" y = ωt and the "line inferred due to the errorbar" y = ωt ± σ′
is given by finding the point on the x-axis where y = ωt = σ′ , so t = σ′ /ω. In a cycle, the time
for a single degree is tdeg = 1/360 f = T/360, where f is the frequency and T is the period. The
time uncertainty due to our errorbar divided by the time for a single degree is the uncertainty
in degrees: σ′ /ω/1/360 f = σ′ /2π f /1/360 f = σ′ (remembering that the 2π is radians and 360
is degrees). So our phase error, in radians on the side of a fringe, is the same as the errorbar:
σ′ . Now if we say that our frequency measurement is given by (ϕL − ϕE )/2πt R , where ϕL is thelate-time phase, ϕE is the early-time phase and t R is the ramsey free evolution time, then the√ ′2 ′2√ ′(σ ) +(σ )2σfrequency uncertainty is δ f ==2πt R
2πt R . This is all for a fringe with perfect contrast, for√′2σ. Finally, to get from a frequency uncertainty toa non-unity contrast C we can scale it δ f = C2πtR

a de uncertainty, we divide by −2|Eeff |sgn(gF )/h [2], where |Eeff | is the effective field experienced
by the valence electron and sgn(gF ) is the sign on the g-factor. Thus,

√

δde =

2σ′ h
2|Eeff |C2πt R(D.6)

is the uncertainty in de for a single frequency measurement.

Finally, we will take the mean of all our frequency measurements, and the distributionof frequency measurements is again normal (we go to some effort to quantify this, see Chapter
4.) So our new uncertainty is just equation D.6 divided by the square root of the number of
6 The side of the fringe cos( ωt ) is at ωt = π/2, the derivative is d/dt cos( ωt ) = − ω sin( ωt ), plug in ωt = π/2

225
measurements:

√

δde =

2σ′ h
p
.
2|Eeff |C2πt R f rep Ttot /2(D.7)

Let’s clean this up by taking h̄ = h/2π and cleaning up the factors of 2:
δde =

σ′ h̄
p

|Eeff |Ct R

f rep Ttot

.

(D.8)

Now all that remains is plug in the quantum-projection noise limited errorbar on a singlepfrequency measurement, σ′ = 1/ Ntot f rep Ttot . Finally, we have the quantum-projection-noiselimited expression for our uncertainty in de :
δde =

h̄

√
.
|Eeff |Ct R f rep Ttot Ntot

(D.9)

The expression above assumes that in each experimental cycle we read out an identicalnumber of ions, Ntot . In reality, N varies in every cycle due to fluctuations in the spatial modeand power of the lasers that ablate, ionize, and dissociate, as well as other experimental noise
sources. We avoid most of that noise by performing a differential measurement between theupper and lower doublets: in a single shot of the experiment we measure both the lower andupper Stark doublets, and we perform our measurements when the fringes for both are in phase,so the proportional experimental noise is effectively cancelled out. Details of this scheme are in[47] and [48].

Appendix E
All parity channel results

Below are all parity channel results for our 2022 dataset. Because we took data at three distinct values of f 0 , the results are disaggregated over those values then re-aggregated afterwards
to give an overall average and error. All errorbars are adjusted for χ2 . Before taking weighted
means and adjusting for χ2 , the value and error for each parity channel for each block was multiplied by 1000, so the units for contrast are milli “contrast units”, for decay rate the units are per
millisecond, for frequency the units milli hertz, and for offset the units are milli offset units.

14.23
14.23
1.18
1.18
1.33
1.33
1.31
1.31
1.15
1.15
1.34
1.34
1.30
1.30
1.25
1.25
4.86
4.86
1.24
1.24
1.19
1.19
1.50
1.50
1.32
1.32
1.09
1.09
1.32
1.32
1.46
1.46

χ2
mean/σ296.29
1.08
1.24
0.62
3.63
0.96
2.02
1.63
1.27
0.22
0.94
0.23
6.72
1.71
0.27
1.75
27.51
0.63
0.13
2.11
5.65
1.64
26.19
0.00
0.53
0.07
0.05
0.43
1.49
0.21
0.70
0.37

80 < f 0 < 140meanσ541.93 1.49
-0.98
1.49
-0.22
0.41
-0.02
0.41
-1.62
0.41
-0.07
0.41
1.19
0.42
0.05
0.42
1.01
0.42
0.11
0.42
0.85
0.39
0.08
0.39
5.26
0.44
0.79
0.44
-0.69
0.42
-0.81
0.42
16.70 0.24
-0.25
0.24
-0.14
0.14
-0.03
0.14
-0.31
0.14
0.08
0.14
6.93
0.14
0.15
0.14
-0.32
0.13
-0.01
0.13
-0.03
0.14
-0.09
0.14
0.43
0.15
0.15
0.15
0.00
0.14
-0.01
0.14
16.08
16.08
1.24
1.24
1.24
1.24
1.31
1.31
1.26
1.26
1.13
1.13
1.40
1.40
1.29
1.29
3.46
3.46
1.14
1.14
1.16
1.16
1.28
1.28
1.09
1.09
1.12
1.12
1.32
1.32
1.18
1.18

χ2
mean/σ364.09
0.66
0.54
0.06
3.92
0.16
2.80
0.12
2.43
0.26
2.15
0.21
11.99
1.79
1.63
1.92
70.35
1.07
1.06
0.19
2.28
0.55
48.01
1.06
2.36
0.07
0.21
0.64
2.93
1.01
0.02
0.09

f 0 > 140mean
σ545.97 1.05
-1.74
1.05
-0.25
0.48
0.07
0.48
-3.82
0.49
0.35
0.49
-1.57
0.53
-0.56
0.53
0.56
0.53
-0.88
0.53
0.29
0.50
-0.32
0.50
6.42
0.51
-0.58
0.51
-1.11
0.54
0.56
0.54
24.34 0.39
0.18
0.39
-0.08
0.19
0.16
0.19
-0.91
0.18
-0.02
0.18
8.71
0.18
0.16
0.18
-0.61
0.17
-0.28
0.17
-0.20
0.19
-0.20
0.19
0.55
0.19
0.25
0.19
-0.13
0.19
0.16
0.19
5.53
5.53
1.17
1.17
1.18
1.18
1.39
1.39
1.39
1.39
1.26
1.26
1.29
1.29
1.43
1.43
5.48
5.48
1.25
1.25
1.23
1.23
1.18
1.18
1.04
1.04
1.27
1.27
1.30
1.30
1.28
1.28

χ2
mean/σ519.81
1.66
0.52
0.15
7.85
0.71
2.98
1.05
1.06
1.68
0.58
0.63
12.67
1.14
2.07
1.05
62.62
0.45
0.41
0.87
4.94
0.10
48.14
0.91
3.63
1.63
1.08
1.05
2.93
1.33
0.69
0.83

aggregate over f 0
mean
σ546.00
0.78
-1.58
0.78
-0.34
0.27
-0.07
0.27
-2.42
0.28
-0.05
0.28
-0.19
0.29
-0.37
0.29
0.42
0.28
-0.23
0.28
0.37
0.27
-0.03
0.27
5.24
0.29
0.40
0.29
-0.58
0.28
-0.47
0.28
17.52
0.18
-0.06
0.18
-0.09
0.10
-0.07
0.10
-0.68
0.10
-0.04
0.10
7.37
0.10
0.13
0.10
-0.33
0.09
-0.09
0.09
-0.07
0.10
-0.08
0.10
0.44
0.10
0.14
0.10
-0.07
0.10
0.02
0.10

Table E.1: All contrast parity channels, disaggregated over distinct f 0 values and aggregated over all 3 f 0 values.

parameter
C0
C0P
C0B
C0BP
C0R
C0RP
C0I
C0IP
C0BR
C0BRP
C0BI
C0BIP
C0RI
C0RIP
C0BRI
C0BRIP
C0D
C0DP
C0DB
C0DBP
C0DR
C0DRP
C0DI
C0DIP
C0DBR
C0DBRP
C0DBI
C0DBIP
C0DRI
C0DRIP
C0DBRIC0DBRIP

f 0 < 80meanσ552.43 1.86
-2.01
1.86
-0.67
0.54
-0.33
0.54
-2.07
0.57
-0.55
0.57
-1.14
0.57
-0.92
0.57
-0.67
0.53
-0.12
0.53
-0.54
0.57
0.13
0.57
3.79
0.56
0.96
0.56
0.15
0.55
-0.97
0.55
11.85 0.43
0.27
0.43
0.03
0.22
-0.46
0.22
-1.21
0.21
-0.35
0.21
6.28
0.24
0.00
0.24
0.12
0.22
-0.02
0.22
-0.01
0.20
0.09
0.20
0.33
0.22
-0.05
0.22
-0.17
0.24
-0.09
0.24

mean/σ700.61
-2.03
-1.27
-0.26
-8.79
-0.18
-0.67
-1.29
1.49
-0.82
1.38
-0.10
18.31
1.41
-2.05
-1.67
94.89
-0.31
-0.88
-0.66
-6.98
-0.44
73.42
1.26
-3.51
-1.00
-0.72
-0.77
4.27
1.30
-0.67
0.24

227

9.64
9.64
1.16
1.16
1.22
1.22
1.12
1.12
1.02
1.02
1.30
1.30
1.11
1.11
1.10
1.10
1.78
1.78
1.15
1.15
1.31
1.31
1.34
1.34
1.29
1.29
1.20
1.20
1.22
1.22
1.45
1.45

χ2
mean/σ
44.68
1.35
2.03
1.36
4.42
0.08
0.49
2.07
2.58
0.86
0.70
0.39
9.33
1.04
1.20
1.84
4.95
0.71
0.12
0.21
3.07
0.54
6.99
0.05
0.12
0.62
1.06
0.79
0.89
0.81
1.62
1.25

80 < f 0 < 140
mean
σ
χ2
87.81 1.57 8.30
-0.97 1.57 8.30
0.57
0.59 1.18
0.15
0.59 1.18
0.93
0.59 1.18-0.59 0.59 1.18
1.74
0.61 1.24-0.12 0.61 1.24
-0.33 0.61 1.25
0.44
0.61 1.25
0.36
0.58 1.14-0.05 0.58 1.14
6.41
0.64 1.37
0.12
0.64 1.37-0.60 0.59 1.15
-1.41 0.59 1.15
-6.16 0.38 2.12
0.05
0.38 2.12-0.02 0.29 1.18
-0.30 0.29 1.18
-0.44 0.30 1.32
-0.11 0.30 1.32
4.14
0.30 1.32-0.42 0.30 1.32
-0.09 0.29 1.20
-0.08 0.29 1.20
0.06
0.29 1.19
0.28
0.29 1.19-0.58 0.30 1.33
0.72
0.30 1.33-0.43 0.28 1.15
0.58
0.28 1.15
mean/σ
56.01
0.62
0.96
0.25
1.58
0.99
2.87
0.19
0.54
0.72
0.61
0.09
10.07
0.19
1.02
2.41
16.00
0.13
0.06
1.04
1.45
0.35
13.64
1.40
0.31
0.28
0.20
0.98
1.90
2.35
1.52
2.04

f 0 > 140mean
σ121.69 2.41
0.26
2.41
0.37
1.00
-0.38
1.00
-11.29 1.00
0.13
1.00
2.83
0.99
-0.61
0.99
0.46
1.00
-0.71
1.00
-0.95
1.00
-0.46
1.00
8.36
0.96
-1.38
0.96
-0.72
0.98
0.91
0.98
-6.25
0.59
0.20
0.59
0.03
0.48
0.11
0.48
-0.39
0.48
-0.05
0.48
4.97
0.49
0.12
0.49
-0.36
0.47
-0.53
0.47
-0.41
0.50
0.19
0.50
-0.48
0.50
0.57
0.50
-0.60
0.48
-0.05
0.48
7.52
7.52
1.29
1.29
1.27
1.27
1.27
1.27
1.28
1.28
1.30
1.30
1.19
1.19
1.25
1.25
1.71
1.71
1.15
1.15
1.15
1.15
1.21
1.21
1.10
1.10
1.23
1.23
1.22
1.22
1.16
1.16

χ2
mean/σ
50.50
0.11
0.37
0.38
11.34
0.13
2.85
0.62
0.46
0.71
0.94
0.45
8.72
1.44
0.73
0.93
10.68
0.35
0.07
0.24
0.81
0.09
10.06
0.24
0.77
1.14
0.83
0.37
0.96
1.15
1.23
0.11

aggregate over f 0
mean
σ
89.65
1.04
-1.24
1.04
-0.21
0.39
-0.33
0.39
-0.18
0.39
-0.26
0.39
1.00
0.39
-0.68
0.39
-0.73
0.38
0.29
0.38
-0.16
0.39
-0.19
0.39
6.26
0.39
0.09
0.39
-0.05
0.38
-0.91
0.38
-4.74
0.26
-0.05
0.26
0.01
0.20
-0.15
0.20
-0.65
0.21
-0.13
0.21
3.79
0.21
-0.18
0.21
-0.12
0.20
-0.21
0.20
0.09
0.20
0.27
0.20
-0.47
0.21
0.34
0.21
-0.52
0.21
0.17
0.21
mean/σ
86.17
-1.19
-0.55
-0.87
-0.47
-0.66
2.59
-1.75
-1.94
0.78
-0.40
-0.48
15.97
0.23
-0.13
-2.40
-18.48
-0.18
0.05
-0.74
-3.11
-0.61
18.02
-0.85
-0.61
-1.04
0.43
1.30
-2.28
1.64
-2.51
0.83

Table E.2: All decay rate parity channels, disaggregated over distinct f 0 values and aggregated over all 3 f 0 values.

parameter
g
gP
gB
gBP
gR
gRP
gI
gIP
gBR
gBRP
gBI
gBIP
gRI
gRIP
gBRI
gBRIP
gD
gDP
gDB
gDBP
gDR
gDRP
gDI
gDIP
gDBR
gDBRP
gDBI
gDBIP
gDRI
gDRIP
gDBRI
gDBRIPf 0 < 80mean
σ75.87 1.70
-2.29 1.70
-1.20 0.59
-0.80 0.59
2.67
0.60
-0.05 0.60
-0.29 0.58
-1.20 0.58
-1.42 0.55
0.48
0.55
-0.44 0.62
-0.24 0.62
5.38
0.58
0.60
0.58
0.69
0.57
-1.05 0.57
-2.12 0.43
-0.30 0.43
0.04
0.34
-0.07 0.34
-1.13 0.37
-0.20 0.37
2.59
0.37
0.02
0.37
-0.04 0.36
-0.23 0.36
0.37
0.35
0.28
0.35
-0.32 0.35
-0.29 0.35
-0.63 0.39
-0.48 0.39

228

σ
1.06
1.06
0.14
0.14
0.13
0.13
0.10
0.10
0.09
0.09
0.09
0.09
0.09
0.09
0.09
0.09
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.04
167.84167.84
3.05
3.05
2.44
2.44
1.50
1.50
1.30
1.30
1.14
1.14
1.23
1.23
1.31
1.31
1.09
1.09
1.03
1.03
1.18
1.18
1.03
1.03
1.16
1.16
1.04
1.04
1.11
1.11
1.05
1.05

χ2
mean/σ72780.41
0.16
0.79
0.34
12.68
1.71
6.61
0.66
2294.25
1.14
3.09
0.71
19.05
0.77
1.10
1.10
4394.161.66
423.35
3.39
1.15
0.34
0.18
0.57
9.13
1.12
0.19
1.22
0.93
0.77
0.16
1.08

80 < f 0 < 140meanσ105043.72 1.35
-0.04
1.35
0.00
0.15
0.06
0.15
-4.75
0.14
0.02
0.14
-0.95
0.12
0.00
0.12
210.86
0.10
-0.15
0.10
-0.54
0.09
0.04
0.09
-2.51
0.10
0.00
0.10
-0.05
0.09
-0.03
0.09
-230.12
0.03
-0.04
0.03
16.12
0.03
0.01
0.03
0.08
0.03
0.00
0.03
-0.09
0.03
-0.02
0.03
-0.45
0.03
0.05
0.03
0.05
0.03
-0.03
0.03
0.00
0.03
0.03
0.03
0.02
0.03
0.01
0.03
263.99263.99
3.07
3.07
2.73
2.73
2.02
2.02
1.56
1.56
1.28
1.28
1.45
1.45
1.25
1.25
1.06
1.06
1.08
1.08
1.03
1.03
1.08
1.08
0.99
0.99
1.06
1.06
0.99
0.99
1.11
1.11

χ2
mean/σ77654.72
0.03
0.01
0.39
34.50
0.17
8.04
0.01
2029.86
1.48
5.68
0.47
25.08
0.02
0.51
0.31
7089.881.11
491.54
0.19
2.49
0.03
2.72
0.63
14.41
1.47
1.58
0.98
0.15
0.94
0.52
0.25

f 0 > 140mean
151225.70
-3.18
0.04
0.30
-15.01
0.06
-1.17
-0.24
209.14
-0.41
-0.62
-0.05
-3.35
-0.09
-0.10
-0.19
-327.98
-0.07
16.11
0.01
0.12
-0.11
0.01
-0.04
-0.35
0.06
0.05
0.05
0.06
-0.01
-0.01
-0.03
σ
4.66
4.66
0.25
0.25
0.28
0.28
0.23
0.23
0.18
0.18
0.17
0.17
0.18
0.18
0.16
0.16
0.06
0.06
0.06
0.06
0.05
0.05
0.06
0.06
0.06
0.06
0.05
0.05
0.06
0.06
0.06
0.06
944.34944.34
2.76
2.76
3.51
3.51
2.21
2.21
1.46
1.46
1.25
1.25
1.39
1.39
1.12
1.12
1.18
1.18
1.06
1.06
0.87
0.87
1.07
1.07
1.03
1.03
0.96
0.96
1.11
1.11
1.02
1.02

χ2
mean/σ32485.90
0.68
0.14
1.21
52.97
0.21
5.21
1.07
1141.35
2.23
3.66
0.27
18.77
0.49
0.63
1.16
5507.941.25
286.34
0.20
2.25
2.08
0.23
0.70
6.21
1.14
0.85
0.91
1.05
0.18
0.27
0.47

aggregate over f 0
mean
σ
89661.08
0.82
-0.22
0.82
-0.04
0.09
0.09
0.09
-4.31
0.09
0.12
0.09
-0.82
0.07
-0.06
0.07
211.83
0.06
-0.06
0.06
-0.43
0.06
-0.01
0.06
-2.23
0.06
0.02
0.06
0.01
0.06
-0.08
0.06
-225.35
0.02
-0.01
0.02
16.10
0.02
-0.03
0.02
0.08
0.02
-0.02
0.02
-0.05
0.02
-0.01
0.02
-0.41
0.02
0.02
0.02
0.03
0.02
-0.02
0.02
0.02
0.02
0.01
0.02
0.01
0.02
0.01
0.02

Table E.3: All frequency parity channels, disaggregated over distinct f 0 values and aggregated over all 3 f 0 values.

parameter
f
fP
fB
fBP
fR
fRP
fI
fIP
fBR
fBRP
fBI
fBIP
fRI
fRIP
fBRI
fBRIP
fD
fDP
fDB
fDBP
fDR
fDRP
fDI
fDIP
fDBR
fDBRP
fDBI
fDBIP
fDRI
fDRIP
fDBRI
fDBRIPf 0 < 80mean
76991.99
-0.17
-0.11
0.05
-1.62
0.22
-0.66
-0.07
213.28
0.11
-0.27
-0.06
-1.72
0.07
0.10
-0.10
-171.27
0.06
16.07
-0.13
0.05
-0.01
-0.01
0.02
-0.37
-0.05
-0.01
-0.05
0.04
-0.03
-0.01
0.04

229

mean/σ109252.06
-0.26
-0.47
0.95
-47.771.34-11.23-0.84
3378.08
-0.88
-7.24
-0.25
-35.54
0.37
0.13
-1.39
-10113.08-0.59
722.65
-1.50
3.64
-1.09
-2.31
-0.46
-18.46
0.91
1.50
-0.96
0.93
0.26
0.29
0.62

parameter
p
pP
pB
pBP
pR
pRP
pI
pIP
pBR
pBRP
pBI
pBIP
pRI
pRIP
pBRI
pBRIP
pD
pDP
pDB
pDBP
pDR
pDRP
pDI
pDIP
pDBR
pDBRP
pDBI
pDBIP
pDRI
pDRIP
pDBRI
pDBRIP
39670.19
39670.19
1.09
1.09
1.22
1.22
0.92
0.92
1.07
1.07
1.17
1.17
1.09
1.09
1.08
1.08
1.06
1.06
1.14
1.14
1.15
1.15
1.03
1.03
1.14
1.14
0.96
0.96
1.12
1.12
1.09
1.09

χ2
mean/σ
4.81
1.17
0.97
0.67
1.16
1.39
0.32
1.39
6.02
1.09
2.20
0.22
0.87
1.44
0.21
1.21
33.85
1.02
0.97
0.88
0.03
0.15
0.74
0.63
2.17
1.62
0.47
4.21
0.77
0.10
0.12
1.12

80 < f 0 < 140meanσ-3045.98 0.64
-0.85
0.64
-0.95
0.66
-0.39
0.66
0.36
0.61
0.25
0.61
-0.22
0.61
-0.52
0.61
5.97
0.67
1.15
0.67
-1.50
0.65
0.77
0.65
-0.85
0.61
-0.12
0.61
2.08
0.62
0.74
0.62
-15.66
0.20
-0.07
0.20
0.18
0.20
0.27
0.20
-0.28
0.19
0.05
0.19
0.88
0.20
-0.24
0.20
-0.57
0.19
-0.34
0.19
0.16
0.21
0.13
0.21
0.12
0.20
-0.03
0.20
-0.75
0.20
0.03
0.20
1.13
1.13
1.22
1.22
1.05
1.05
1.04
1.04
1.26
1.26
1.18
1.18
1.03
1.03
1.08
1.08
1.12
1.12
1.14
1.14
1.02
1.02
1.18
1.18
1.02
1.02
1.21
1.21
1.19
1.19
1.14
1.14

χ2
mean/σ4790.86
1.33
1.45
0.59
0.58
0.41
0.37
0.85
8.90
1.72
2.30
1.19
1.40
0.20
3.35
1.20
79.00
0.37
0.90
1.38
1.51
0.27
4.31
1.17
3.04
1.77
0.80
0.62
0.58
0.16
3.75
0.16

f 0 > 140mean
-3017.21
0.93
-3.11
0.31
0.04
-0.15
-1.00
-0.27
7.27
1.74
-4.55
-0.39
0.21
0.54
3.24
1.48
-11.45
-0.28
-0.20
-0.40
0.10
0.55
0.17
0.51
-1.23
-0.16
0.48
1.07
-0.41
0.24
-0.61
0.55
σ
0.80
0.80
0.83
0.83
0.77
0.77
0.78
0.78
0.80
0.80
0.78
0.78
0.74
0.74
0.77
0.77
0.26
0.26
0.27
0.27
0.27
0.27
0.28
0.28
0.27
0.27
0.26
0.26
0.27
0.27
0.26
0.26
1.18
1.18
1.25
1.25
1.09
1.09
1.12
1.12
1.15
1.15
1.12
1.12
1.00
1.00
1.08
1.08
1.04
1.04
1.11
1.11
1.14
1.14
1.19
1.19
1.11
1.11
1.03
1.03
1.09
1.09
1.04
1.04

χ2
mean/σ3749.48
1.15
3.75
0.38
0.05
0.19
1.27
0.34
9.13
2.19
5.80
0.50
0.28
0.73
4.21
1.92
43.70
1.06
0.73
1.48
0.36
2.02
0.61
1.81
4.52
0.59
1.83
4.11
1.52
0.90
2.31
2.10

aggregate over f 0
meanσ-3034.73
0.50
-0.16
0.50
-1.11
0.45
0.06
0.45
-0.04
0.43
-0.19
0.43
-0.31
0.41
-0.02
0.41
6.18
0.44
1.28
0.44
-2.58
0.44
0.18
0.44
-0.50
0.42
-0.17
0.42
1.99
0.42
1.04
0.42
-12.66
0.14
-0.18
0.14
-0.03
0.14
0.09
0.14
-0.12
0.13
0.17
0.13
0.41
0.14
0.05
0.14
-0.74
0.13
-0.10
0.13
0.24
0.14
0.04
0.14
-0.10
0.14
0.06
0.14
-0.50
0.14
0.08
0.14

Table E.4: All phase parity channels, disaggregated over distinct f 0 values and aggregated over all 3 f 0 values.

f 0 < 80mean
σ-804.15 167.09-195.53 167.09
0.85
0.88
0.59
0.88
-1.07
0.93
-1.29
0.93
0.26
0.81
1.12
0.81
5.23
0.87
0.94
0.87
-2.00
0.91
-0.20
0.91
-0.76
0.88
-1.26
0.88
0.18
0.88
1.06
0.88
-8.80
0.26
-0.27
0.26
-0.26
0.27
0.24
0.27
-0.01
0.27
0.04
0.27
-0.19
0.26
0.16
0.26
-0.59
0.27
0.44
0.27
0.12
0.25
-1.05
0.25
-0.21
0.27
0.03
0.27
0.03
0.26
-0.30
0.26

mean/σ-6072.46
-0.31
-2.49
0.14
-0.09
-0.46
-0.76
-0.06
13.96
2.88
-5.89
0.41
-1.20
-0.40
4.70
2.45
-93.53
-1.34
-0.25
0.63
-0.88
1.28
2.95
0.40
-5.49
-0.75
1.73
0.27
-0.76
0.41
-3.70
0.60

230

1.86
1.86
1.54
1.54
1.55
1.55
1.36
1.36
1.22
1.22
1.25
1.25
1.67
1.67
1.27
1.27
2.34
2.34
1.12
1.12
1.13
1.13
1.20
1.20
1.41
1.41
1.13
1.13
1.19
1.19
1.26
1.26

χ2
mean/σ
76.65
2.00
0.75
0.45
0.73
0.51
0.52
1.86
0.66
0.99
0.60
1.66
0.61
0.03
0.09
0.95
67.88
0.82
1.27
0.42
0.26
0.13
0.45
6.08
2.35
0.48
0.36
0.43
1.65
1.63
0.68
0.63

80 < f 0 < 140
mean
σ
χ2
26.11 0.23 1.61
-0.33 0.23 1.61
-0.10 0.23 1.52
-0.27 0.23 1.52
0.04
0.22 1.39-0.19 0.22 1.39
-0.11 0.21 1.28
0.05
0.21 1.28
0.19
0.22 1.50
0.20
0.22 1.50
0.30
0.22 1.48-0.12 0.22 1.48
-0.22 0.23 1.53
0.05
0.23 1.53-0.61 0.22 1.45
-0.50 0.22 1.45
10.86 0.11 2.69
-0.09 0.11 2.69
-0.10 0.07 1.17
0.03
0.07 1.17-0.03 0.08 1.34
0.04
0.08 1.34-0.01 0.07 1.14
0.50
0.07 1.14
0.28
0.07 1.17-0.01 0.07 1.17
-0.07 0.07 1.15
0.10
0.07 1.15-0.58 0.08 1.26
-0.13 0.08 1.26
-0.01 0.07 1.21
0.10
0.07 1.21
mean/σ112.75
1.45
0.45
1.19
0.19
0.90
0.52
0.25
0.85
0.91
1.36
0.52
0.96
0.22
2.78
2.28
98.81
0.80
1.37
0.39
0.40
0.55
0.09
7.01
3.90
0.19
0.94
1.39
7.72
1.76
0.15
1.36

f 0 > 140
mean
σ24.39 0.30
0.00
0.30
-0.10 0.26
-0.35 0.26
-0.03 0.24
0.22
0.24
-0.23 0.28
0.17
0.28
0.67
0.25
-0.05 0.25
-0.50 0.27
-0.12 0.27
-0.28 0.27
-0.06 0.27
0.03
0.26
-0.34 0.26
1.39
0.34
-0.17 0.34
-0.15 0.10
-0.05 0.10
0.00
0.09
-0.04 0.09
0.15
0.10
0.56
0.10
0.36
0.10
0.03
0.10
-0.02 0.09
-0.03 0.09
-0.46 0.10
0.08
0.10
0.05
0.10
0.07
0.10
1.76
1.76
1.39
1.39
1.15
1.15
1.52
1.52
1.24
1.24
1.42
1.42
1.43
1.43
1.36
1.36
15.01
15.01
1.28
1.28
1.17
1.17
1.24
1.24
1.20
1.20
1.16
1.16
1.24
1.24
1.20
1.20

χ2
mean/σ
81.85
0.00
0.38
1.32
0.14
0.92
0.83
0.62
2.67
0.21
1.87
0.44
1.06
0.22
0.10
1.28
4.10
0.51
1.56
0.51
0.03
0.43
1.50
5.75
3.76
0.36
0.21
0.30
4.70
0.79
0.51
0.72

aggregate over f 0
mean
σ
25.61
0.16
-0.01
0.16
-0.13
0.15
-0.27
0.15
0.06
0.14
-0.04
0.14
-0.08
0.15
-0.07
0.15
0.34
0.14
-0.00
0.14
-0.06
0.15
0.04
0.15
-0.23
0.15
0.00
0.15
-0.25
0.14
-0.39
0.14
10.10
0.09
-0.03
0.09
-0.06
0.05
0.01
0.05
-0.01
0.05
0.01
0.05
0.04
0.05
0.55
0.05
0.30
0.05
0.01
0.05
-0.05
0.05
0.03
0.05
-0.45
0.05
-0.08
0.05
-0.01
0.05
0.06
0.05

Table E.5: All offset parity channels, disaggregated over distinct f 0 values and aggregated over all 3 f 0 values.

parameter
O
OP
OB
OBP
OR
ORP
OI
OIP
OBR
OBRP
OBI
OBIP
ORI
ORIP
OBRI
OBRIP
OD
ODP
ODB
ODBP
ODR
ODRP
ODI
ODIP
ODBR
ODBRP
ODBI
ODBIP
ODRI
ODRIP
ODBRI
ODBRIPf 0 < 80mean
σ26.07 0.34
0.68
0.34
-0.23 0.31
-0.14 0.31
0.23
0.31
-0.16 0.31
0.15
0.29
-0.54 0.29
0.18
0.28
-0.27 0.28
-0.17 0.28
0.46
0.28
-0.20 0.32
-0.01 0.32
0.02
0.28
-0.27 0.28
10.38 0.15
0.12
0.15
0.13
0.11
0.04
0.11
0.03
0.11
0.01
0.11
0.05
0.11
0.66
0.11
0.28
0.12
0.06
0.12
-0.04 0.11
-0.05 0.11
-0.18 0.11
-0.18 0.11
-0.08 0.11
-0.07 0.11
mean/σ159.23
-0.06
-0.87
-1.77
0.39
-0.25
-0.53
-0.45
2.41
-0.02
-0.41
0.25
-1.53
0.00
-1.71
-2.71
117.66
-0.31
-1.25
0.23
-0.12
0.11
0.87
10.81
5.83
0.27
-0.99
0.62
-8.23
-1.48
-0.19
1.10

231

76991.994-0.167-0.1130.048-1.6170.218-0.660-0.066213.275
0.106-0.269-0.062-1.7230.070
0.103-0.102-171.268
0.065
16.072-0.1290.047-0.014-0.0070.021-0.369-0.045-0.007-0.0470.037-0.030-0.006
0.041

f
fP
fB
fBP
fR
fRP
fI
fIP
fBR
fBRP
fBI
fBIP
fRI
fRIP
fBRI
fBRIP
fD
fDP
fDB
fDBP
fDR
fDRP
fDI
fDIP
fDBR
fDBRP
fDBI
fDBIP
fDRI
fDRIP
fDBRI
fDBRIP

1.058
1.058
0.143
0.143
0.127
0.127
0.100
0.100
0.093
0.093
0.087
0.087
0.090
0.090
0.093
0.093
0.039
0.039
0.038
0.038
0.041
0.041
0.038
0.038
0.040
0.040
0.038
0.038
0.039
0.039
0.038
0.038

σ
167.844167.844
3.054
3.054
2.436
2.436
1.496
1.496
1.296
1.296
1.142
1.142
1.226
1.226
1.305
1.305
1.089
1.089
1.029
1.029
1.179
1.179
1.029
1.029
1.163
1.163
1.041
1.041
1.114
1.114
1.052
1.052

χ2
72780.413
0.158
0.795
0.339
12.685
1.707
6.606
0.663
2294.245
1.142
3.086
0.708
19.054
0.769
1.099
1.098
4394.159
1.656
423.351
3.387
1.153
0.335
0.175
0.565
9.129
1.119
0.189
1.221
0.930
0.774
0.164
1.084

mean/σ105043.721-0.0420.001
0.056-4.751
0.024-0.951-0.001210.862-0.154-0.5350.045-2.510-0.002-0.047-0.029-230.122-0.036
16.1220.006
0.080-0.001-0.089-0.021-0.4530.046
0.052-0.032-0.005
0.029
0.017
0.008

1.353
1.353
0.146
0.146
0.138
0.138
0.118
0.118
0.104
0.104
0.094
0.094
0.100
0.100
0.093
0.093
0.032
0.032
0.033
0.033
0.032
0.032
0.033
0.033
0.031
0.031
0.033
0.033
0.031
0.031
0.033
0.033

80 < f 0 < 140meanσ263.993263.993
3.067
3.067
2.734
2.734
2.022
2.022
1.555
1.555
1.281
1.281
1.446
1.446
1.249
1.249
1.057
1.057
1.079
1.079
1.033
1.033
1.083
1.083
0.990
0.990
1.062
1.062
0.985
0.985
1.111
1.111

χ2
77654.719
0.031
0.008
0.387
34.497
0.174
8.035
0.012
2029.864
1.479
5.678
0.475
25.081
0.016
0.505
0.307
7089.878
1.110
491.535
0.191
2.487
0.033
2.719
0.627
14.405
1.468
1.585
0.984
0.153
0.940
0.518
0.251

mean/σ151225.696-3.1830.036
0.303-15.005
0.059-1.173-0.241209.142-0.409-0.618-0.046-3.346-0.088-0.101-0.186-327.977-0.07416.1110.011
0.115-0.1070.013-0.039-0.346
0.063
0.046
0.049
0.061
-0.011-0.015-0.026

f 0 > 140
mean
4.655
4.655
0.252
0.252
0.283
0.283
0.225
0.225
0.183
0.183
0.169
0.169
0.178
0.178
0.161
0.161
0.060
0.060
0.056
0.056
0.051
0.051
0.057
0.057
0.056
0.056
0.054
0.054
0.058
0.058
0.055
0.055

σ
944.337
944.337
2.759
2.759
3.505
3.505
2.206
2.206
1.463
1.463
1.249
1.249
1.386
1.386
1.122
1.122
1.182
1.182
1.056
1.056
0.872
0.872
1.067
1.067
1.033
1.033
0.963
0.963
1.111
1.111
1.021
1.021

χ2
32485.897
0.684
0.143
1.205
52.970
0.210
5.210
1.073
1141.354
2.230
3.661
0.270
18.774
0.495
0.628
1.160
5507.936
1.250
286.337
0.201
2.252
2.083
0.229
0.696
6.214
1.138
0.854
0.909
1.050
0.184
0.267
0.472

mean/σ89611.303-0.215-0.0440.087-4.2290.122-0.822-0.060211.819-0.060-0.420-0.017-2.2320.022
0.010-0.083-224.100-0.007
16.102-0.0420.077-0.026-0.043-0.009-0.4090.020
0.030-0.023
0.019
0.004
0.003
0.014

0.820
0.820
0.095
0.095
0.089
0.089
0.072
0.072
0.065
0.065
0.060
0.060
0.063
0.063
0.061
0.061
0.023
0.023
0.023
0.023
0.023
0.023
0.023
0.023
0.022
0.022
0.023
0.023
0.022
0.022
0.023
0.023

aggregate over f 0mean
σmean/σ109229.509-0.262-0.4650.923-47.6591.377-11.377-0.8263267.343-0.920-7.032-0.281-35.651
0.355
0.157-1.359-9798.597-0.297707.340-1.8283.398-1.141-1.885-0.382-18.2350.894
1.328-1.021
0.834
0.161
0.147
0.616

Table E.6: All frequency parity channels (but with 3 digits after the decimal), disaggregated over distinct f 0 values and aggregated overall 3 f 0 values.

f 0 < 80mean

parameter

232