Searching for time-reversal symmetry violation with molecular ions:
Quantum state control and photofragment imaging
byWilliam B. Cairncross
B.Sc., Queen’s University, 2013
M.S., University of Colorado Boulder, 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
2019This thesis entitled:
Searching for time-reversal symmetry violation with molecular ions:
Quantum state control and photofragment imaging
written by William B. Cairncrosshas been approved for the Department of Physics

Prof. Eric A. Cornell

Prof. Jun Ye

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Cairncross, William B. (Ph.D., Physics)
Searching for time-reversal symmetry violation with molecular ions:
Quantum state control and photofragment imaging
Thesis directed by Prof. Eric A. Cornell

The relative abundance of matter over antimatter in the observable universe cannot be accounted for by the Standard Model of particle physics (SM), and requires additional sources of timereversal symmetry violation to be explained. Theories seeking to explain this imbalance generically
predict electric dipole moments of fundamental particles such as the electron. We search for theelectron’s electric dipole moment (eEDM) using HfF+ molecular ions confined in a radiofrequency
trap, polarized by a rotating electric bias field. We have produced a first result with this systemthat is consistent with zero, setting an upper bound |de | < 1.3 × 10−28 e cm (90% confidence) –equivalent to setting a lower bound on the mass of beyond SM particles of several TeV. In thisthesis, we describe the first generation measurement and progress towards a second-generation
measurement with more than an order of magnitude improved sensitivity. In particular, we discuss
quantum state preparation of molecular ions by optical pumping, and a new technique for simultaneous differential measurement of molecules with opposite orientations, which allows us to attainthe shot-noise limit with hundreds of detected ions and multi-second coherence.

Dedication

To Mum, Dad, John, and Zoe.

v

Acknowledgements

I have been immensely lucky in my PhD to be advised by Eric Cornell and Jun Ye, twobrilliant scientists who bring a wealth of knowledge to every problem we encounter in the lab.
They have been patient teachers, enthusiastic supporters, and wise sages, guiding the experimenton our sometimes slow (but hopefully always forward) progress.
On the laboratory side, I would like to thank Matt Grau, Kevin Cossel, and Dan Gresh for
guiding my education in the ways of eEDM from Fall 2013. Since then, we have been joined in thelab by Yan Zhou, Kia Boon Ng, Tanya Roussy, Yuval Shagam, Tanner Grogan, Maddie Pettine,and Antonio Vigil. I have learned something from each and every one of these talented people,
and I am happy to be leaving the lab in their capable hands. Thanks as well to Krista Beck, Amy
Allison, the greater Cornell and Ye groups, and the JILA community as a whole, who have madethis a wonderful place to work and learn.
I have received invaluable help from JILA staff Hans Green, James Uhrich, Kim Hagen, Todd
Asnicar, Carl Sauer, Terry Brown, James Fung-A-Fat, and Felix Vietmayer. Their knowledge of allthings mechanical and electrical has been instrumental in bringing the eEDM experiment to where
it is today.
Finally, I would like to thank my friends: Gabby Subia Smith, Kurt Hill, Maggie Jacoby,
Sade Odumuye, Andrea Egan, Peter Madigan, Abhinav Prem, Pattilynn McLaughlin, Allison Duh,Will and Annie Jay, Dan Hackett, Joseph Samaniego, Mike Stefferson, Molly May, Matt Norcia,
Martha Hawkinson, Maddie Schlick, Stephen Hibbs, and others. Thank you for reminding me that
there is a world outside the lab. I hope to see each of you before too long.

vi

Contents

Chapter
1 Introduction1

1.1

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

1

1.2

EDM measurements in atoms and molecules . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Status of the field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.3.1

Diamagnetic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.3.2

Paramagnetic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4

Theoretical interpretation of EDM measurements . . . . . . . . . . . . . . . . . . . . 15

1.5

About this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5.1

Experiment overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 Theoretical calculations in HfF+
2.1

2.2

26

Molecular structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.1

Qualitative features of HfF+

2.1.2

The molecular wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1.3

Electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1.4

Rotational structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.1.5

Vibrational structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.1.6

Nuclear spin & the total wavefunction revisited . . . . . . . . . . . . . . . . . 35

. . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Energy shifts and decay rates in HfF+ . . . . . . . . . . . . . . . . . . . . . . . . . . 36

vii
2.2.1

Electric fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.2

Stark effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.3

Electronic transitions & the Franck-Condon principle . . . . . . . . . . . . . . 39

2.2.4

Vibrational decay & blackbody excitation . . . . . . . . . . . . . . . . . . . . 42

2.2.5

Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.2.6

Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.2.7

Omega doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2.8

Effect of Erot rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.2.9

eEDM shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.2.10 Differential g-factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Generation 1 eEDM Measurement

60

3.1

Summary of the eEDM result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2

Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.1

Switch state timing & control . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3

Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.4

Modeling frequency channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.5

3.4.1

Ion motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4.2

Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.4.3

Mixing of frequency channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.4.4

Non-ideal frequency shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.1

Doublet population contamination . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5.2

Non-reversing Brot and effective differential g-factor

3.5.3

Geometric phase and axial magnetic field . . . . . . . . . . . . . . . . . . . . 89

3.5.4

Harmonic distortion of Erot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.5.5

Frequency modulation due to axial secular motion . . . . . . . . . . . . . . . 91

. . . . . . . . . . . . . . 88

viii
3.5.6
3.6

Additional details on systematic effects . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.6.1

3.7

Rotation-odd Erot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Harmonic distortion of Vrot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Room for improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4 Generation 2 Ion Trap

100

4.1

Design considerations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2

Electrode shape optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.4

Trap driver electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.1

Synthesis boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.4.2

Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5 State preparation

114

5.1

Coherent methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.2

Rotational cooling & vibrational decay . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.3

Incoherent state preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.4

5.3.1

MF pumping with 3 Π0− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.3.2

Polarization measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3.3

Vibrational cleanups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Sensitivity improvement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6 Oriented photodissociation

133

6.1

Photodissociation before Generation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.2

Imaging photofragments & new ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.4

Survey spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.5

Imaging oriented photodissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

ix
6.5.1

Doublet depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7 Generation 2 eEDM signal

158

7.1

Center cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.2

Parameterizing the eEDM signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.3

Limits to precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.3.1

Noise in an ideal Ramsey experiment . . . . . . . . . . . . . . . . . . . . . . . 162

7.3.2

Varying particle number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7.3.3

State preparation and detection efficiency . . . . . . . . . . . . . . . . . . . . 167

7.3.4

Thoughts on “the QPN limit” . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.4

Methods of estimating Ramsey phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7.5

Noise on Generation 2 eEDM signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

7.6

Effect of ion cloud jitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.7

Generation 2 sensitivity estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

8 Looking forward

179

8.1

Future techniques in AMO EDM experiments . . . . . . . . . . . . . . . . . . . . . . 180

8.2

Towards eEDM Generation 2 with HfF+ . . . . . . . . . . . . . . . . . . . . . . . . . 183

Bibliography

187

Appendix
A Simulations

197

A.1 Spherical multipole fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
A.2 Space charge & thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
A.2.1 Ion cloud profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
A.2.2 Ion cloud breathing modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

x
A.2.3 Doppler thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
A.3 Trap design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
A.4 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
A.5 Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
B Analytic calculations

222

B.1 MCP Transfer Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
B.2 Relativistic & inertial effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
B.2.1 Mostly kinetic term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
B.2.2 Relativistic correction to kinetic term . . . . . . . . . . . . . . . . . . . . . . 227
B.2.3 Interaction term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
B.2.4 Total FW Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
B.3 Rotation matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
C Designs

229

C.1 Lab layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
C.2 Generation 2 Ion Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
C.3 Trap amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
C.4 Trap synthesizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
D Molecular Data

248

xi

TablesTable
2.1

Matrix elements hF 0 MF0 | HStark |F MF i for 3 ∆1 , J = 1, showing only the Ω = +1
components. Matrix elements for Ω = −1 are (−1)× the matrix elements shown,and matrix elements off-diagonal in Ω are zero. Here E⊥ = Ex − iEy , and entries left
blank are zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1

Selected frequency channels, their leading expression in terms of experimental parameters, and their interpretations. Here δg eff is half the effective magnetic g-factor
difference between Stark doublets, α is the tilt angle of E rot above the radial planeof the ion trap, and frot is the rotation frequency of Erot . . . . . . . . . . . . . . . . . 66

3.2

Systematic effects and corrections applied to the eEDM channel f BD , in units of µHz. 70

3.3

Values of selected measured values from the portion of our eEDM dataset collected
in 2017. For this data, Erot = 24 V/cm and frot = 150 kHz. For the values f B , C 0 ,and γ 0 that are subject to drift and therefore not normally distributed, we quote a
range of observed values. For various values with either nonzero measured values,
drift, or excess scatter, contributing factors are given. . . . . . . . . . . . . . . . . . 76

3.4

Constants used in the effective Hamiltonian of Eq. (3.12), for the calculation of
frequency channels and systematic corrections. The total magnetic g-factor of J =1, F = 3/2 states gF ≡ (Gk + gN µN /µB )/3 results from the combination of nuclear
and electronic Zeeman effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xii
3.5

Experimental parameters and imperfections explored as possible sources of systematic error. The letters E and M indicate that an effect was explored experimentallyor through numerical and perturbative modeling, respectively. . . . . . . . . . . . . . 84

3.6

Observed frequency shifts and their scaling with selected experimental parameters(numerical factors are omitted). Here q is a dimensionless RF trap parameter [9],R0 is the radius of the RF trap, Z1 is the amplitude of the center-of-mass secularmotion of the ion cloud along the Z axis, Celec is the capacitance of an electrode,and β ≈ 1.4 × 10−5 mm−2 is the fractional inhomogeneity of Erot along the Z axis. . 84

3.7

Experimental (in)efficiencies in Generation 1. . . . . . . . . . . . . . . . . . . . . . . 99

4.1

Optimized polynomial coefficients for the radial electrodes’ vertical profile for the
Generation 2 ion trap.

4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Results of dial indicator measurements of the radial electrode positions by Yuval
and Kim. The physical meaning of each measurement type is described in the text.
All measurements are in mil (0.001 inch). The “back” measurement for electrode4 has a large deviation, which was anticipated by Kim. However, the back of eachelectrode was not used as a mechanical reference; rather, the mounting holes are thereferences surfaces, so this measurement is not of concern.

5.1. . . . . . . . . . . . . . 109

system of HfF+ , assuming a MorseFranck-Condon factors for the 1 Σ+ ↔ 3 Σ−
0+potential. Spectroscopic constants for both electronic states are given in Ref. [28]. . 120

5.2

Franck-Condon factors for the 3 Π0− ↔ 3 ∆1 system, assuming a Morse potential.
Here I have used the known value of ∆G1/2 = 716.0 cm−1 for 3 Π0− and assumed that
ωe xe has the same value as for 3 Π0+ , approximately 3.3 cm−1 . Molecular constants
are found in Ref. [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.3

Observed line positions for the 3 Π0− ← 3 ∆1 (0,0) band. . . . . . . . . . . . . . . . . 126

xiii
6.1

Selected analytical expressions for the angular distribution of photofragments, resulting from Eq. (6.1).

In the oriented cases |Ωi | > 0, E rot is directed alongθ = 0. The laser polarizations 1,2 are represented in standard atomic notation,and σ ± = σ + + σ − , with the quantization axis set by Erot as shown in Fig. 6.5. Notethat the angular distributions are unnormalized. . . . . . . . . . . . . . . . . . . . . 142
7.1

Typical experimental parameters for the Generation 2 eEDM experiment with HfF+ .
N0 is the total number of Hf+ detected (in both Stark doublets) in one experimental
cycle near t = 0 on the side of a fringe, i.e., when Φ0 ≈ π/2.

. . . . . . . . . . . . . 177

A.1 Unnormalized real spherical harmonics used in spherical multipole fitting in thisthesis. The (lack of) normalization is chosen so that the (l = 1, m) and (l = 2, m = 0)components have the form conventionally used in past eEDM work. . . . . . . . . . . 199
A.2 Example of a table of multipole potential data for l = 2. The first two rows containl and m values, however the first three columns of the first two rows are ignored.
The remaining rows of the first three columns indicate exponents of the X, Y ,and Z coordinates. Each of the remaining columns contains coefficients of the Φlm
indicated by the first two rows. For example, reading from this table, we obtain
Φ2,−2 = 1.732 X Y , in agreement with Table A.1.

. . . . . . . . . . . . . . . . . . . 200

A.3 Example output from the VBA script that saves magnetometer coordinates and orientations. The elements R ij belong to a rotation matrix that transforms the magnetometer magnetic field measurement axes into the laboratory axes. The elements
O i are coordinates of the magnetometer’s center.
A.4 Coil unit vectors from May 2019.

. . . . . . . . . . . . . . . . . . . 217. . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

xiv
A.5 Parameters for magnet coil pairs. The number of turns is the measured value, while
the other parameters are taken from our experiment CAD model. The physicalmeanings of the parameters are shown in Fig. A.6. The parameter I rel indicatesthe relative sign of the current in the two coils – a negative sign results in a gradientrather than a uniform field. Distances are in meters, angles in radians.

. . . . . . . 221

C.1 Multipole fit coefficients up to l = 11 for electrodes of Generation 2 ion trap (Fig. 4.1),
with units of V/ml . Rows where all columns have a magnitude less than 0.02 areneglected. Note the absence of all rows (l, m = ±1) for odd values of l, which behaveas Erot ∝ Z l−1 along the line (0, 0, Z). . . . . . . . . . . . . . . . . . . . . . . . . . . 247
D.1 Summary of observed photodissociation transitions, part I. . . . . . . . . . . . . . . 249
D.2 Summary of observed photodissociation transitions, part II. . . . . . . . . . . . . . . 250

xv

FiguresFigure1.1

Basic concepts and context of EDM experiments. (a) An electric dipole moment of
a particle with spin violates time-reversal symmetry, and assuming CPT symmetry
holds, violates CP symmetry. (b) Context for EDM experiments: (i) the Baryon
asymetry of the Universe, one of several phenomena not explained within the Standard Model, requires additional sources of CP violation. (ii) BSM theories supply
sources of additional CP violation, and an effective field theory approach parameterizes the sensitivity of low-energy composite systems to BSM physics. (iii) Low
energy precision measurements of EDMs can be interpreted in a global analysis toconstrain BSM theories. Nuclear and particle EDMs contribute on equal footingwith results from atoms and molecules in a global analysis. . . . . . . . . . . . . . .

1.2

3

Energy levels of an atom or molecule in the presence of an EDM. States of opposite
parity are mixed by an applied electric field, while a magnetic bias field orients themagnetic moment. Upper panel: In atoms, only a region of partial polarization
can be accessed with typical laboratory electric fields, and excited opposite-parity
states decay on the nanosecond scale. In molecules, by contrast, opposite parity
states are closely spaced, and regions of complete polarization are accessible. Lower
panels: The signature of an EDM is detectable as an electric field-dependent shift inmagnetic sublevels |↑i and |↓i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

xvi
1.3

A standard method for precision spectroscopy in AMO systems: Ramsey’s methodof separated oscillatory fields. A pair of RF or laser pulses separated in time by tprobes the spin precession phase. The highest attainable precision is given by the√shot noise limit, which scales as 1/t N . . . . . . . . . . . . . . . . . . . . . . . . .

1.4

8

Experimental approaches leading to recent results in EDM experiments on diamagnetic and paramagnetic atoms and molecules. (a) Octupole-deformed nuclei haveclosely spaced nuclear states of opposite parity, enhancing the parity-mixing effectof T-violating matrix elements. These nuclei are orders of magnitude more sensitive
to PT-violation than undeformed nuclei. (b) Diatomic molecules for both diamagnetic and paramagnetic EDM searches. Relativistic effects and the distortion of
atomic wavefunctions by bonding in heavy polar molecules makes them maximallysensitive to T-violating effects. Molecules also have closely-spaced levels of opposite
parity, allowing them to be more easily polarized in the laboratory than atoms –this is analogous to collective Schiff enhancement in octupole-deformed nuclei. (c)Cryogenic buffer gas beams are a highly versatile source of cold, slow, bright beamsof almost any small molecule that can be introduced to the inert buffer gas environment. (d) Omega-doublet levels in paramagnetic molecules allow a differential
EDM measurement without reversal of a laboratory electric field, greatly suppressing systematics. They are also very closely spaced levels of opposite parity, allowing
complete polarization of the molecules in electric fields of tens of V/cm. . . . . . . . 12

1.5Interpretation of EDM experiments. (a) SM-EFT sources of EDMs, interpreted
from the text of Ref. [37]. Dimension-six effective operators couple Standard Modeldegrees of freedom, encapsulating the effects of BSM theories at the electroweak scale.
Engel et al. [37] trace the effects of a subset of 12 CP-violating effective operators
through the hadronic and nuclear scales to their effects on EDMs of paramagnetic
and diamagnetic atoms and molecules, nucleons, and nuclei.

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

xvii
1.6

Conceptual picture of global analysis of EDM results: Two experiments labeled 1
and 2 have different sensitivities to CPV coefficients CA and CB , parameterized by

 

dω
dω1,2
the slopes dC1,2/
dCB . A global analysis can be used to obtain simultaneousAconstraints on CA and CB from experiments 1 and 2. . . . . . . . . . . . . . . . . . . 19

1.7

Overview of the HfF+ eEDM experiment. (a) Top-down view of the main vacuum
chamber containing the RF ion trap. (b) Relevant energy levels of HfF and HfF+ ,with lasers showing main transitions. (c) Timing diagram showing the major teps ofthe experiment sequence: (1) Neutral HfF production, (2) ionization, (3) ion trapping, (4) state preparation, (5) Ramsey sequence, (6) dissociation, and (7) detection.
These steps are described in detail in Section 1.5.1 of the text. . . . . . . . . . . . . 22

2.1

Molecular orbital diagram for HfF+ . Since the molecule is ionically bonded, themolecular states correlate with atomic states of Hf2+ and F− . The ground electronic
state of Hf2+ is 3 F2 , and correlates with the excited 3 Φ2 state of HfF+ . . . . . . . . . 27

2.2

Schematic potential energy curves for HfF+ , including a few electronic states of themolecule and free atoms. The low-lying bound states of the molecule are ionicallybonded, so they correlate to electronic states of Hf2+ and F− . Shown in black at theright are the energies of low-lying electronic states of free Hf+ and F below 14000
cm−1 . Potential curves are drawn as Morse potentials. . . . . . . . . . . . . . . . . . 28

2.3

Electronic states of HfF+ up to about 15000 cm−1 , with singlet states in black andtriplet states in blue. The 3 ∆3 and 1 ∆2 states have not been observed. Red lines indicate known transitions between electronic states, with thick lines indicating dipole
allowed transitions and thin lines indicating dipole forbidden (but weakly allowed)
transitions. Equally spaced states connected by a shaded region are vibrational levels belonging to the labeled electronic state. Only the excited vibrational levels usedin the experiment are shown. Rotational levels exist on a much smaller energy scaleof . 1 cm−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

xviii
2.4

Calculated Stark shifts in 3 ∆1 , J = 1, using experimentally measured parameters forthe hyperfine splitting and the molecule-frame dipole moment. While in this figure
it appears that the stretched states (red and purple) have linear shifts all the way
to Erot = 0 indicating a permanent electric dipole moment, their shifts are in fact
quadratic at very low Erot due to Ω doubling, which is discussed in Section 2.2.7

2.5. . 40Petrov et al.’s theoretically calculated molecule-frame dipole moment of HfF+ in the3∆

1 electronic state (points), as a function of internuclear distance.

Adapted from

Ref. [99]. The red line is a quadratic fit to Petrov et al.’s calculation, while the black
dashed line is the approximation D|| = −eR, offset vertically by 2ea0 . For estimating
vibrational lifetimes, this approximation gives 20% error, which is not too bad! . . . 44
2.6

Theoretically calculated vibrational decay of 3 ∆1 , v = 3, J = 1, leading to populationof rotational levels up to J = 4 in v = 0. This phenomenon leads to an undesirable
background signal in 3 ∆1 , v = 0, J = 1 that hurts our eEDM sensitivity, as discussed
in Chapter 5. Populations Pi are expressed as a fraction of the total molecular
population. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.7

Theoretically calculated Stark effect in 3 ∆1 , J = 1, F = 3/2 for small values of theelectric bias field, showing Ω doubling – essentially a zoom in of Fig. 2.4 to the region
near Erot = 0. As judged by the linearity of the Stark shift as a function of Erot , theF = 3/2 states are essentially fully polarized at Erot ≈ 1 V/cm. Here the effects of
the rotating frame are not included. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.8

Quantum numbers and shorthand alphabetical labels for magnetic and hyperfine
sublevels of 3 ∆1 , J = 1. The quantum number MF is good for all values of Erot , but
F ceases to be a good quantum number when Dk Erot ≈ Ak . . . . . . . . . . . . . . . 53

2.9

Energy difference between states of opposite mF , but the same value of mF Ω – i.e.,
states in the same Stark Doublet, versus Erot , for Brot = 0. This is the size of the
avoided crossing |∆| in units of Hz. Here ωrot /(2π) = 375 kHz. . . . . . . . . . . . . 55

xix
2.10 Magnetic g-factor of stretched states as a function of Erot , at frot = 375 kHz. The
drop of the absolute g-factors to zero at low Erot results from mixing of MF levels
due to Hrot at order > 2 in perturbation theory. Note that we have not derived theabsolute value of gF from theory, but taken it from experiment.
3.1. . . . . . . . . . . 59(a) Apparatus schematic, (b) experimental timing, and (c) relevant energy levels (notto scale) for an eEDM measurement using trapped ions. HfF is photoionized (yellow)
to form HfF+ . A rotating electric bias field Erot (blue) polarizes the molecules, andtransfer (red) and depletion (orange) lasers perform state preparation. The π/2pulses are performed by modulating Erot . Spin state populations are detected bydepletion followed by photodissociation (purple) and counting the resulting Hf+ ionson a microchannel plate (MCP). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2

Electron spin resonance spectroscopy in HfF+ . (a) Level structure of the eEDMsensitive 3 ∆1 , F = 3/2 state in an electric bias field Erot . (b) Energies of |MF | = 3/2
states as a function of magnetic bias field Brot (not to scale), showing an avoided
crossing at Brot = 0 due to a rotation-induced coupling ∆u/l . (c) Interference fringewith interrogation time ∼ 700 ms and decoherence rate γ = 0.3(2) s−1 . . . . . . . . . 64

3.3

Frequency shifts in the f B and f BR channels due to a stray uniform magnetic field
BYnr and ion displacements Y0 Ŷ and Z0 Ẑ. (a) A shift in f B ∝ BYnr Y0 resulted froma contribution to Brot from an electric field gradient oscillating at 2frot , which we
suppressed by reducing harmonic distortion in Erot via feedforward. (b) A shift in
f BR = 3hαifrot ∝ Y02 Z0 was well modeled by the known inhomogeneity in Erot , andwas suppressed by applying feedback to the ion position between eEDM measurements. Error bars are ∼ ±0.1 Hz on all points. . . . . . . . . . . . . . . . . . . . . . 67

xx
3.4

Summary of eEDM dataset after cuts and scaling δf byp
χ2r to account for over-scatter. (a) Histogram of normalized, centered eEDM-sensitive frequency measurements (f BD − hf BD i)/δf . (b) Normal probability plot of the same dataset, showing
a linear trend suggesting that the data are consistent with a normal distribution. (c)Subsets of eEDM data taken under different values of experimental parameters, andthe overall average of f BD . Here N is the average number of trapped HfF+ ions perrun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5

Relative timing of experimental configurations during the collection of one data
block. (a) When taking data “across,” we collect in each switch state an entire
interference fringe as a function of t. (b) In the “down” configuration, we change thespin precession time t on a slower timescale than B, D, and R. (c) In the “down +scramble” configuration, we also re-randomize the order of switch states (B̃, D̃, R̃)at each value of t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.6

Schematic layout of control system for experimental switches. Blue traces indicate
digital signals, gold traces indicated DC analog signals, and red traces indicate RF
analog signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.7

Typical raw data trace from the time-of-flight microchannel plate detector, showing time-resolved Hf+ and HfF+ signals. Red markers indicate counted Hf+ ionscontributing to the asymmetry signal. . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.8

Stimulated Raman transfer from 1 Σ+ , J = 0 to 3 ∆1 , J = 1, F = 3/2, showing
Doppler-broadened resonances at the locations of Stark doublets. . . . . . . . . . . . 86

3.9

Simulated effect of doublet population contamination with a fraction w = 0.2 of the3∆

1 population in the undesired Stark doublet.

Green and blue traces in the inset

illustrate the fractional contributions from the upper and lower doublet to the totaltrue interference fringe (black line). Measuring the asymmetry at only short and longtimes (black points) leads to a misinterpretation of beating as a loss of coherence(red line), and results in an error in the fit frequency. . . . . . . . . . . . . . . . . . . 86

xxi
3.10 Systematic shift in the eEDM channel f BD due to non-reversing Brot (left hand vertical axis, solid line), and a proportional shift in the f B channel that we use to apply
a correction (right hand vertical axis, dashed line). The constant of proportionality
f BD /f B = δgeff /gF depends on Erot and frot . The data shown here were taken underconditions chosen to make it particularly large (∼ 10−2 ). . . . . . . . . . . . . . . . . 89
3.11 Phase modulation of an interference fringe at fZ = 4.01(7) kHz due to axial secularmotion of the ion cloud with an intentionally exaggerated amplitude of ∼ 3 mm,measured near the steepest portion of the fringe where 2πf t ≈ π/2. . . . . . . . . . . 92
3.12 Attainable frequency precision for a Ramsey measurement σf (in arbitrary units)
versus spin precession time for various numbers of trapped particles N , assumingthat the decoherence rate γ scales proportionally with N . In this case, higher countrate is not always advantageous for precision when it leads to a loss of coherence.
4.1

. 98

Key dimensions of Generation 2 ion trap. All dimensions in millimeters. Trap
principal axes are indicated in red, and electode numbers are in green.

. . . . . . . 104

4.2

Weight function for electrode shape optimization, given by Eq. (4.3b) . . . . . . . . 106

4.3

Percent deviation in Erot in the XY and XZ planes in the Generation 2 ion trap,
averaged over one rotation period. Plots generated using multipole fit, not direct
Comsol solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.4

Comparison of Comsol solution for Erot with multipole fit to ` = 9, for Vrot = 1 V
peak. Plots are for Erot averaged over one rotation period. . . . . . . . . . . . . . . . 108

4.5

Potential energy surfaces for trapped HfF+ ions in the Generation 2 ion trap, in unitsof Kelvin. The RF trapping ponderomotive potential (blue) closely approximates aharmonic trap (black) near the geometric center of the trap. The ponderomotive
potential imposed by Erot inhomogeneity (red) distorts the total potential (yellow)
and tends to decrease the trap depth. For this figure, Vrf = 15 V peak, Vrot = 375 V
peak, and Vdc = 0.5 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

xxii
4.6

Schematic layout of the eEDM Generation 2 ion trap synthesis board. A Spartan
6 FPGA acts as a programmer and sequencer. We communicate with the FPGA
via a USB-to-Serial interface chip from FTDI. The FPGA controls an AD9959 DDS,
each of which has an AD5543 multiplying DAC, and two DACs to generate Shimand Kick voltages. Each channel has an analog switch to turn it on and off. All six
channels are summed to produce a single analog output channel for each board. . . . 111

4.7

Labview code for writing to a Generation 2 ion trap synthesis board with identifier “handle.” Each data packet contains 7 bytes. The first byte has the format
10YX XXXX, where Y specifies read (0) or write(1), and XXXXX specifies a device numbered 0-31. The second two bytes specify the memory address on the device, andthe last four bytes carry the data to be written. The DLL function call on the righthand side is to the function FT write in the FTDI library ftd2xx.dll. . . . . . . . 112

5.1

Simplified picture of STIRAP in a three-level system. Here |g1 i represents 1 Σ+ (v =0, N = 0), while |g2 i represents 3 ∆1 (v = 0, J = 1), and |ei represents 3 Π0+ (v =1, J = 0). The pump laser Ω1 or “Stella” has λ ≈ 900 nm, and the stokes laser Ω2or “Toptica” has λ ≈ 986 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.2

Tests of the model from Eq. (5.2) for chirped Raman adiabatic rapid passage with
HfF+ . In (a) and (b), we did not account for the cloud side and laser beam profile.
In (a), we calibrated the parameters of Eq. (5.2). In (b), we reduced each laser powerby half and found that the model did not fit observations. In (c) and (d), integrating
Eq.(5.2) over a spatially varying laser intensity and ion density was able to accountfor the observed variation in transfer probability with laser power and detuning. . . 119

5.3

R branch of the REMPD spectrum of HfF+ from its 1 Σ+ ground state throughan Ω0 = 0 intermediate state at ν0 ≈ 35976 cm−1 , with chops showing rotationalcooling. Not shown: enormous background in 3 ∆1 , v = 0, J = 1. . . . . . . . . . . . 121

xxiii
5.4

Repopulation of 1 Σ+ (v = 0, N = 0) by vibrational decay after laser depletion,
detected by photodissociation. A similar repopulation will occur in 3 ∆1 (v = 0, J =1), leading to a spurious background in an eEDM measurement. . . . . . . . . . . . . 122

5.5

Incoherent state preparation scheme for HfF+ . Population is pumped out of 1 Σ+using a combination of laser light on the 3 Π0+ ← 1 Σ+ (0,0) P(1) transition (red)and microwaves connecting rotational levels in 1 Σ+ , v = 0 (green). Population in
excited vibrational states of 3 ∆1 is cleaned up by lasers on the 3 Σ−
← 3 ∆1 (1,1) and0+(2,2) transitions (blue dash and dash-dot). In the ground rovibrational state of 3 ∆1 ,quasi-cycling MF pumping is performed on the 3 Π0− ← 3 ∆1 (0,0) P(1) transition← 3 ∆1 (0,0) Q(1) transition (blue). 124(purple), and depletion is performed on the 3 Σ−
0+

5.6

REMPD spectrum of 1 Σ+ , v = 0 through an Ω = 1 intermediate state. Red shows
the initial distribution, with population distributed from N = 0 to N = 3. Adding
the 961 nm transfer laser on the P(1) line to 3 Π0+ causes depletion of N = 1,
evidenced by disappearance of the R(1) line. Further adding microwaves gives the
purple trace, where only a broad background is visible. We believe this backgroundarises from vibrational decay that occurs between the end of transfer and dissociation. 124

5.7

Timing of state preparation and readout. The 961 nm laser and microwaves transfer
population to 3 ∆1 , and remain on during MF pumping to send any population that
returns to 1 Σ+ (for example from vibrational decay) back to 3 ∆1 . After the Ramsey
sequence, the 814.5 nm laser depletes all but one MF level in 3 ∆1 . . . . . . . . . . . 127

5.8

Timescale of strobed circularly polarized state preparation and readout. (a) MF
pumping on the 3 Π0− ← 3 ∆1 (0,0) P(1) transition with about 50 mW of 1082.3 nm
laser light, with a fixed duration of MF depletion of 40 ms. Inset: Timing sequence ofσ + σ + and σ + σ − chops. (b) MF depletion on the 3 Σ−
← 3 ∆1 (0,0) P(1) transition0+
using about 400 mW of 814.5 nm laser light, with a fixed duration of MF pumpingof 60 ms. Both beam waists are roughly 5 mm.

. . . . . . . . . . . . . . . . . . . . 129

xxiv
5.9

Paths of rovibrational spontaneous decay from initially populated J = 1 levels in
the 3 ∆1 electronic state of HfF+ , showing (a) original and (b) improved vibrational
cleanup schemes for preventing background in 3 ∆1 (v = 0, J = 1). The J = 1 levels
in several vibrational states are populated by transfer from 1 Σ+ and MF pumping
in 3 ∆1 . In the original scheme, some population could still reach v = 0, J = 1. In
the improved scheme, cleanup lasers (blue) driving the 3 Σ−
← 3 ∆1 (1,1) P(1) and0+
P(2) transitions prevent any decay to 3 ∆1 (v = 0, J = 1) and reduce the decay to
3 ∆ (v = 0, J = 2).
1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131←
5.10 Depletion spectroscopy of 3 ∆1 (v = 1) using “cleanup” lasers driving the 3 Σ−0+3∆

1 (1, 1) transition at 818.4 nm. Population is transferred from1 Σ+ (v = 0, J = 0)

with 961 nm laser light driving the 3 Π0+ ← 1 Σ+ (0, 0) transition, causing somespontaneous emission into 3 ∆1 (v = 1). In the blue trace, no cleanup lasers areapplied, so three rotational lines are observed in the REMPD spectrum (b). In the
red trace, the J = 1 state is depleted, and in purple, both J = 1 and J = 2 aredepleted.
6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Top view of the Generation 2 ion trap, showing the locations of REMPD lasers andthe imaging MCP detector. Space constraints prevent us from having perfectly σ ± polarized REMPD lasers while also having Erot parallel to the MCP face for optimal
imaging.

6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134(a) Energy levels and (b) qualitative 1st and 2nd photon absorption in REMPD, asdescribed in Section 6.1. One tunable UV laser drives a bound-bound transition in
HfF+ , and a fixed UV laser drives the bound intermediate state into the continuumof dissociating states. The absorption spectrum of the tunable UV laser is typical ofbound-bound transitions in diatomics, while the absorption of the second photon isa very slowly varying function of energy, determined by the Franck-Condon overlapof the intermedate and final state wavefunctions (the reflection approximation [61]). 134

xxv
6.3

(a) Observed excess noise as a function of first dissociation laser (“Cora”) detuningfrom resonance. (b) Theoretical model of fluorescence of an atomic transition andexcess noise in photon counts as a function of laser detuning, using a phase diffusion
model of laser linewidth, from Ref. [56]. . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.4

Imaged angular distribution of photofragments resulting from REMPD of HfF+ ions
initially in the 1 Σ+ state, with the first transition tuned to the R(0), R(1), and P(1)rotational lines in the left, middle, and right columns respectively. Average of about
25 shots. In the top row, the 266 nm laser is horizontally polarized, while in thebottom row, 266 nm is vertically polarized. This shows that the bound-to-continuum
transition is perpendicular (∆Ω = ±1). . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.5

Examples of possible laser polarizations for oriented REMPD. Note that due tothe availability of optical access in our vacuum chamber and the location of theion detector, not all configurations of lasers and quantization axis (set by Erot ) arepractical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.6

Simulated photofragment angular distributions resulting from REMPD of HfF+ inthe 1 Σ+ electronic state in an incoherent mixture of all magnetic sublevels, calculatedusing Eq. (6.1). The black and red lines in the top right indicate the polarizationof the first and second dissociation photons, respectively. Three views are shown foreach distribution to give a sense of the 3D shape.

6.7. . . . . . . . . . . . . . . . . . . 144

Simulated photofragment distributions resulting from REMPD of HfF+ in the 3 ∆1electronic state, with opposite Stark doublets populated. The blue and red lines inthe top right indicate the polarization of the first and second dissociation photons,
respectively, while the thick black arrow indicates the direction of E rot . Oriented
distributions are obtained only when Ωn 6= 0. . . . . . . . . . . . . . . . . . . . . . . 145

xxvi
6.8

Comparison of theoretical oriented photofragment distributions I(θ) for REMPD
transitions Ω = 1 → 2 → 1 and Ω = 1 → 2 → 3, each with the corresponding optimal
polarizations. Initial population is only in one Stark doublet. Expressions are listed
in Table 6.1, lines 4 and 8. In the case Ωf = 1, an additional lobe of photofragments
appears on the “wrong” hemisphere, reducing the attainable contrast. . . . . . . . . 146

6.9

Schematic potential energy curves showing the effect of total dissociation energy on
photofragment distributions. Red: low energy above threshold results in a high probability of radiative decay. Green: high kinetic energy results in a lower probability ofradiative decay. Different energies above threshold may also result in multiple dissociation channels becoming available, resulting in concentric rings of photofragments.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.10 Schematic layout of the frequency doubling and pointing compensation setup for
the Cobra-Stretch dye laser, which provides the first UV photon for REMPD. The
first rotation stage optimizes the phase matching of the BBO for UV production,
while the second rotation stage compensates for changes in beam pointing due to the
BBO rotation. Abbreviations: BBO, β-barium borate crystal; RS, rotation stage;W, wedged pickoff; F, UV-passing filter; PD, photodiode; DCR, dichroic mirror.

. . 149

6.11 REMPD survey spectroscopy of HfF+ in the 26000-32500 cm−1 region for the first
photon. The second photon wavelength is 266 nm for all observed transitions. . . . . 151
6.12 Identification of the Ω value of a REMPD intermediate state by selective population of 3 ∆1 through 3 Π0+ . Part (a) shows the trajectories of population throughrotational levels of electronic states, while part (b) shows the resulting REMPD spectrum. A single strong line when populating on the P(1) line with 961 nm indicatesthat this is an Ω = 2 transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

xxvii6.13 Left-right asymmetry K of oriented photofragments as a function of dissociation
laser timing. The offset of the horizontal axis is arbitrary; the fall of trap timeoccurs around 30 µs. The flat sections indicate that our peak asymmetry is nolonger limited by the ion cloud temperature.

. . . . . . . . . . . . . . . . . . . . . . 155

6.14 Images and integrated distributions of upper and lower Stark doublets using theoriented REMPD transition at a first photon wavelength of ∼ 27137 cm−1 .

. . . . 155

6.15 Laser depletion of Stark doublets to verify orientation-selective dissociation and tomeasure the doublet contrast K. A portion of the 814.5 nm light usually used forstrobed MF depletion is picked off and applied vertically down the trap axis toattain the minimum Doppler width of about 40 MHz. While π polarization is shownin (a), the laser does not in fact have a well-defined polarization with respect tothe quantization axis set by Erot , so which MF levels are depleted is set by energy
alone. (b) Observed reduction in 3 ∆1 dissociation yield as a function of 814.5 nm
laser detuning, showing population depletion of each Stark doublet.
7.1. . . . . . . . . 157

Effect of a central signal cut on doublet contrast K, signal size N , and frequency√
sensitivity K N . The horizontal axis is the width of a rectangular region, centered√around Y = 650 pixels in Fig. 6.14. The lines K N and N are normalized to 1 atzero cut pixels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.2

Typical appearance of doublet-even and doublet-odd asymmetry signals. The peakto-peak value of A0 is C (with a typical value of ∼ 0.8) in the present parameterization, while the peak-to-peak value of AD is CK, where K has a typical value of∼ 0.75. I have reduced the value of f 0 to 15 Hz to more clearly show the Ramsey
fringe, and I have taken f D = 0.33 Hz. Here I have neglected decoherence and set
γ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

xxviii
7.3

Mean and standard deviation of the asymmetry for a single experimental trial withn = 10 particles and perfect detection efficiency. The behavior of the standard
deviation as a function of Ramsey phase φ is characteristic of quantum projection
noise.

7.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Signal-to-noise ratio (SNR) of a Ramsey phase measurement as a function of particle
number, in the case where both spin state populations cannot be measured simultaneously. In the presence of excess noise α, the SNR approaches a constant value
with increasing signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.5

Sequence of probabilities in one experimental cycle of the HfF+ eEDM experiment,from creation of NHfF+ to simultaneous detection of NA and NC Hf+ ions on theimaging MCP detector. Here pprep is the probability of a molecule being preparedin 3 ∆1 (v = 0, J = 1, F = 3/2, MF = 3/2) rather than lost to other states via
spontaneous decay, pa -pd are the probabilities of surviving the Ramsey sequence in
one of states |ai-|di, pdet is the detection probability, and pstay = (1 + K)/2 (where
K is the doublet contrast) is the probability of a detected ion being assigned to thecorrect doublet group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.6

Standard deviation of doublet-even and doublet-odd asymmetries A0 and AD asa function of ΦD , with 1000 ions detected per experiment cycle (shot noise limit√
1/ 2Nt ≈ 0.02). Points are Monte Carlo simulation results, while lines are from
Eq. (7.22). Three regimes of excess noise are shown: In (a), αeff = 0.1 and Var (Φ) =0 so that fractional noise dominates. In (b), αeff = 0 and Var (Φ) = (0.1)2 , so thatphase noise dominates. In (c), αeff = 0.05 and Var (Φ) = (0.05)2 , so that bothcontribute equally. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.7

Effect of ion cloud jitter on assignment of ions to doublet groups, here labeled A andC. A shift of the ion cloud in the positive Y direction by an amount δY relative tothe mean position Y0 causes a loss of an area κ0 δY + κ1 δY2 . . . . . . . . . . . . . . . 174

xxix
7.8

(a) Statistical sensitivity for Ttot = 1 hour of data collection with the experimental
parameters listed in Table 7.1. (b) Optimal fraction of data to be collected at t ≈ 0,
versus Ramsey time.

8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Future methods for electric dipole moment searches. (a) A “conveyor belt” of trappedions for attaining long coherence time and high repetition rates, in development
at JILA. (b) Laser cooled and trapped YbOH, a long-term goal for the PolyEDM
Collaboration between Caltech, Harvard, and University of Toronto. (c) The EDM3
method for molecules in inert gas matrices, in development at York University andthe University of Toronto.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

A.1 (a) Simulated integrated density profile of an ion cloud with (blue) and without (reddashed) space charge effects. Here N = 104 ions, T = 5 K, and the cloud is confinedin a spherically symmetric harmonic potential with ω = 2π × 2 kHz. The Y and Z
axes have been integrated out, so that this is what we would see as a voltage trace
on an MCP detector – i.e., this is not a slice through the density distribution. (b)Experimentally observed time of flight profile of the ion cloud density, with overlaysof various simulated profiles. Assuming a temperature of 5 K and 104 ions, a profilethat accounts for the space charge effects provides the best fit. . . . . . . . . . . . . 203
A.2 Examples of normal modes of a cold ion cloud. In the “axgrad” mode (a), the axial
direction is out of phase with the radial directions, and oscillates with twice theamplitude. In the “breathe” mode (b), the cloud widths along all three principal
axes oscillate in phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
A.3 (a) Damping of the “axgrad” and “breathe” collective modes of oscillation of the
HfF+ cloud, after excitation by a kick applied to both endcaps. (b) FFT of the datafrom part (a), showing the frequency of the “axgrad” mode, which is approximately
1.7ωsec . This indicates that space charge effects are significant in the ion cloud.

. . 205

xxx
A.4 Population detected by photodissociation of 3 ∆1 (v = 0, J = 1) versus frequency of
the 961 nm laser driving the 3 Π0+ ← 1 Σ+ (0,0) P (1) transition, showing the onephoton Doppler width of the HfF+ cloud. The 961 nm laser is stabilized to the
HighFinesse wavemeter. The measured width is σν = 11.8(2) MHz, corresponding
to a temperature of T = mσν2 λ2 /kB = 3.1(1) K. . . . . . . . . . . . . . . . . . . . . . 205
A.5 Rate equation prediction for state populations after optical pumping state preparation. (a) Transfer from 1 Σ+ only via 3 Π0+ , (b) with MF pumping through 3 Π0− , (c)with vibrational cleanups through 3 Σ−
. Part (b) indicates that the best route to0+
improved signal size is with a repump from 3 ∆1 (v = 1) through 3 Π0− (1178.7 nm). . 214
A.6 Parametric description of the orientation of a pair of coils in space. The vector rc
indicates the position of the center of mass of the coil pair. Each coil is located at±rrel relative to rc . The coils are rotated by a common set of Euler angles (θ, φ, χ)about the laboratory Z axis, and are then rotated by a relative set of Euler angles±(θrel , φrel , χrel )/2 relative to their common z axis. The coils have radius Rcoil , width
w, thickness t, and have numbers of turns Nw and Nt along their width and thickness. 221
B.1 Potential on one electrode for 1 V on the DAC. . . . . . . . . . . . . . . . . . . . . . 223
C.1 Geographical layout of the HfF+ eEDM lab as of June 2019. The colors of opticalfiber launchers are simply to correlate the input and output of fibers and are notrelated to the wavelength. The 818 nm, 961 nm, and 815 nm lasers are combined
and sent vertically down through the vacuum chamber from an optical breadboard
mounted above (translucent white).

. . . . . . . . . . . . . . . . . . . . . . . . . . . 230

Chapter 1

Introduction

1.1

Motivation
This section is reproduced from Ref. [17]The discovery of the Higgs boson in 2012 [23, 24] marked the completion of the Standard

Model of particle physics (SM), in the sense that all of its fundamental particles have now been
observed. Predictions of the SM have been confirmed with unparalleled precision [54], making it
the best-tested fundamental theory yet developed. Along with general relativity, the SM could in
principle model an evolving universe from times as early as nanoseconds after the Big Bang to
the present. However, such a universe would look very different from our own: it would be nearly
devoid of matter, and – even if that issue were resolved – it would lack the cold dark matter (CDM)
whose gravitation is thought to help create the large-scale structure of matter in our universe. In
addition, the mass of the SM Higgs boson is small compared to expectations – one incarnation of
an issue referred to as the hierarchy problem [38]. These discrepancies and others have for several
decades motivated searches for physics beyond the SM (BSM).
The near-absence of matter in a SM universe would result from the near-perfect annihilationof matter particles with their corresponding antiparticles, which are produced in nearly equal
quantities. In order to leave behind the matter we observe in our universe, particles and antiparticles
must have been produced in unequal amounts. A fundamental theory that can model this process
must fulfill the Sakharov conditions: charge conjugation (C) and time reversal (T) symmetries must2
be violated 1 , and the universe’s evolution must take place out of thermal equilibrium – for example,
a first-order phase transition could occur, “freezing in” a matter-antimatter imbalance [33, 106].
While T-violation within the SM has been observed in the decays of K and B mesons governed by
the Cabbibo-Kobayashi-Maskawa model of quark flavor mixing [70], it is highly suppressed in theSM, and is believed to be insufficient to explain the observed imbalance of matter over antimatterin the universe [33].
BSM theories seeking to explain the Baryon asymmetry of the universe (BAU) and otherphenomena generally include additional sources of CP-violation (Fig. 1.1b), which generate Tviolating low-energy phenomena: electric dipole moments (EDMs) of nucleons, nuclei, atoms, andmolecules (Fig. 1.1a, [37, 70, 105]). EDMs thus provide a set of observables that are complementary
to direct searches for BSM particles in collider experiments. Using only tabletop-scale apparatus,
the present generation of atomic and molecular EDM experiments provides constraints on themasses of BSM particles that are comparable to, or even exceed, the reach of the LHC [2, 31].
BSM particles at the TeV scale have long been anticipated [33], as they have the potential
to solve several outstanding problems: (i) a stable, weakly interacting massive particle (WIMP)
is a natural dark-matter candidate [31, 105]; (ii) SUSY partners at TeV energies could resolve
the hierarchy problem and stabilize the Higgs mass [33]; (iii) new CP violating particles at TeV
scale could solve the baryogenesis problem, and give rise to EDMs in existing, stable particles[37]. However, so far the LHC has discovered no BSM particles, WIMP dark matter has not been
detected, and no electric dipole moments have been discovered. These null results point to thecritical importance of enhancing EDM sensitivities, and of expanding the search for new physics inform of very weakly-coupled, lower-mass particles.
Recent progress in atomic, molecular, and optical physics (AMO) techniques such as cooling,
trapping, and quantum state control combined with advances in laser technology have led to enormous improvements in precision experiments. In the latest generation of optical atomic clocks, for1

According to the CPT Theorem, a quantum field theory conforming with the requirements of locality and Lorentz
invariance (special relativity) must be symmetric under combined C, P and T symmetry operations. We assume that
all fundamental theories obey this property, and therefore CP-violation is equivalent to T-violation.

3

T

(a)

Proposedsolutions

BSM theories(ii) Theory
Constraints

Low-E limit

SM + EFT(i) BSM Phenomena
Baryonasymmetry

Low-E limit

Nuc. & atomic
structure
theory

BSM CP
violation

Dark matter

Globalanalysis

Theoretical tools

Dark energy

Motivation

Constraints

Atomic &molecular
EDMs

Results(iii) Experiments

Hierarchy problem
AMO physics
techniques

Molecular
spectroscopy

Experimental tools...

Motivation

Nuclear & particle
EDMs

(b)

Figure 1.1: Basic concepts and context of EDM experiments. (a) An electric dipole moment of aparticle with spin violates time-reversal symmetry, and assuming CPT symmetry holds, violates
CP symmetry. (b) Context for EDM experiments: (i) the Baryon asymetry of the Universe, oneof several phenomena not explained within the Standard Model, requires additional sources of CP
violation. (ii) BSM theories supply sources of additional CP violation, and an effective field theory
approach parameterizes the sensitivity of low-energy composite systems to BSM physics. (iii) Low
energy precision measurements of EDMs can be interpreted in a global analysis to constrain BSM
theories. Nuclear and particle EDMs contribute on equal footing with results from atoms andmolecules in a global analysis.

4example, confinement of ultracold alkaline earth-like atoms a three-dimensional optical lattice andprobing with light from ultra-stable lasers has enabled fractional frequency stability on the orderof one part in 1018 and better [18]. The precision attainable in AMO experiments have led to theirapplication to a wide variety of investigations such as searches for dark matter, tests of generalrelativity, and precision measurements of atomic parity violation [105].
Amongst precision measurements, searches for fundamental symmetry violations via EDM
measurements have an exceptionally long history, and have been reviewed extensively (see Ref. [22]and references therein). Recent theoretical insights and experimental innovations have contributed
to substantial advancement in EDM experiments, but challenges and opportunities remain in bringing the full toolset of experimental AMO physics to bear on the particular atomic and molecularspecies that are sensitive to BSM T-violation through EDMs.

1.2

EDM measurements in atoms and molecules
This section is reproduced from Ref. [17]While polar molecules such as H2 O and NH3 are often described as possessing electric dipole

moments, in non-relativistic quantum mechanics and in the absence of an applied electric field, theeigenstates of all molecules – and indeed all atoms and their constituent particles – are symmetric(up to a factor ±1) under an inversion of spatial coordinates (parity, P). The application of an
external electric field mixes these opposite-parity states, causing the system to acquire a nonzero
expectation value of the electric dipole moment operator – i.e., an electric dipole moment is induced.
For polar molecules, the very small spacing of opposite parity states of ≤ 10−5 eV means that theelectric field required to fully polarize the molecule can be very small – on the order of only afew V/cm for some molecules. In atoms, by contrast, opposite-parity eigenstates are typicallyseparated by eV energy scales, meaning they can only be partially polarized by laboratory electricfields (Fig. 1.2, upper panel).
Fundamental CP-violating interactions generate a mixing of opposite-parity states even in
the absence of an applied electric field, leading to a T-violating permanent EDM of an atom or5molecule [22, 105]. This atomic or molecular EDM can arise from an EDM of a constituent electron
or nucleon, or from other T-violating interactions between constituent particles. A permanentelectric dipole moment causes a linear shift in the energies of angular momentum eigenstates thatis linear in an applied electric field, which for small applied fields can be described by an effective
interaction Hamiltonian [70]Heff = − (µ B + d E) F /F

(1.1)

where µ is the magnetic moment of the atom or molecule within a particular state of total angularmomentum F , d is the T-violating permanent electric dipole moment of the atom or molecule, B isan applied magnetic field, and E is an applied electric field. The signature of d can be detected as achange in the Zeeman splitting of total angular momentum eigenstates upon reversal of an applied
electric field with respect to a simultaneously applied magnetic field (Fig. 1.2, lower panels).
The fact that δT 6= 0 for atoms and molecules in general requires the evasion of Schiff’s
Theorem [110], which states that the total EDM of a composite system is completely shielded, even
if one of the constituent particles possesses an EDM. Schiff pointed out several possible ways to
avoid complete shielding, one of which is a difference between the charge and electric dipole moment
distributions – commonly called the Schiff moment, and is responsible for a large contribution todiamagnetic atoms’ nonzero value of δT . Intuitively, the Schiff moment is analogous to an offset
between the center of mass and the center of charge of an atomic nucleus, caused by T-violating
nuclear forces such as those generated by a nucleon EDM. A further evasion of Schiff’s theorem
was pointed out by Sandars [108], and leads to an orders of magnitude enhancement in sensitivity
to an electron EDM in paramagnetic atoms and molecules. This effect arises from the length
contraction of a molecule in the frame of a relativistically moving electron (or equivalently, thelength contraction of an electron EDM in the lab frame). A more detailed description of theevasion of Schiff’s theorem is given in Refs. [27, 62, 105].
The size of a T-violating permanent atomic or molecular EDM is approximately d ≈ 2DδT /∆E,
where D is the ordinary electric dipole matrix element between opposite-parity states, ∆E is the6EnergyOpposite parity states
mixed by E-field

Closest opposite parparity state
Atoms: ∆E/h ~ 100 - 1000 THz
Molecules: ∆E/h ~ 1 - 1000 MHz

Accessible with atoms
Accessible with molecules - Fully polarized

Figure 1.2: Energy levels of an atom or molecule in the presence of an EDM. States of opposite parity
are mixed by an applied electric field, while a magnetic bias field orients the magnetic moment.
Upper panel: In atoms, only a region of partial polarization can be accessed with typical laboratory
electric fields, and excited opposite-parity states decay on the nanosecond scale. In molecules,
by contrast, opposite parity states are closely spaced, and regions of complete polarization areaccessible. Lower panels: The signature of an EDM is detectable as an electric field-dependent
shift in magnetic sublevels |↑i and |↓i.

7
zero-field energy splitting of those states, and δT (with units of energy) is the matrix element
between opposite parity states induced by T-violating interactions, resulting from the evasion of
Schiff’s theorem [39, 121]. The large value of D and small spacing ∆E between opposite-parity
levels in polar molecules that is a manifestation of their intrinsic deformation also leads to an
enhancement of a permanent EDM for a given value of δT [121]. An analogous effect leads toenhancement of the Schiff moment in intrinsically deformed nuclei [57, 122].
Essentially all AMO EDM experiments search for a T-violating energy shift in the same way:
An atom or molecule is subjected to parallel electric and magnetic fields, and the energy difference
between states of opposite magnetic quantum number M is obtained by a measurement of the spin
precession angular frequency, ∆E = E↑ − E↓ = ~ω, where up and down arrows represent opposite
electronic or nuclear spin states. The usual procedure for such a measurement is Ramsey’s methodof separated oscillatory fields, illustrated in Fig. 1.3. A pure spin state is prepared by selective
population or depletion, and a pair of sequential radiofrequency or laser pulses spaced by a time tmaps the relative accumulated quantum phase ωt onto a final relative population between the states
being interrogated. A final measurement of populations N↑ , N↓ is used to form the asymmetryA = (N↑ − N↓ )/(N↑ + N↓ ) ≈ −C cos(ωt + φ) + O. The contrast C and offset O characterize
imperfections in state preparation and readout, while γ characterizes decoherence, for example due
to instability in the bias magnetic field B.
In one form or another, most EDM experiments are accomplished by counting the number ofparticles resulting in each spin state. The precision of such a counting experiment (excluding spinsqueezing techniques) is limited by quantum projection noise: Projecting the quantum state of a√particle in an equal superposition ∼ (|↑i + |↓i)/ 2 onto |↑i and |↓i is analogous to a coin flip, andobeys the same binomial distribution. Identical repetitions of the same projection measurement
2 = N P (1 − P ),(or coin toss) will result in scattered values of the asymmetry A with a variance σA

where N is the total number of particles measured (or coin flips) and P = 0.5 is the probability offinding |↑i (or heads). This variance results in an uncertainty in the extracted value of ω, and the8Ramsey spectroscopy sequence

precess

project

Varying spin precession time

rf amplitude

initializetR

Bloch sphere representation

Time-domain signal
1

0

tR

–1

Figure 1.3: A standard method for precision spectroscopy in AMO systems: Ramsey’s method ofseparated oscillatory fields. A pair of RF or laser pulses separated in time by t probes the spin
precession
phase. The highest attainable precision is given by the shot noise limit, which scales as√
1/t N .

9resulting standard error on a quantum projection noise limited EDM measurement is
σd =

~
p
,
2ECt t Nt frep Ttot

(1.2)

where Nt = N0 e−Γt is the number of particles counted in a single experimental cycle including aloss rate Γ, frep is the repetition rate of the experiment, Ct = Ce−γt is the measurement contrast
including decoherence rate γ, and Ttot is the total measurement time.
While quantum projection noise represents the limit of precision for EDM measurements(again neglecting spin squeezing techniques), even attaining this limit is not guaranteed. In most
instances, experimental inefficiencies lead to measurements that do not achieve the quantum projection noise limit. This fact makes several features of precision experiments invaluable: The first ishigh sensitivity to the underlying physics through enhancement factors, which reduces the requisite
measurement precision. The second is to make a differential and normalized measurement wherever
possible. Finally, sources of technical noise frequently have a “flicker” or 1/f spectral character;thus they can often be suppressed by increased measurement frequency. Each of these requirements
has played a major role in motivating the present and next generation of EDM experiments.

1.3

Status of the field
This section is reproduced from Ref. [17]1.3.1

Diamagnetic systems

Diamagnetic atoms and molecules are sensitive to up to 11 of the 13 BSM CPV effective
operators that are expected to give dominant contributions to EDMs of low-energy systems [22,37]. This wide net of possible parameter sensitivities – particularly those connected to stronglyinteracting nuclear physics – makes the theoretical interpretation of diamagnetic EDMs challenging[22, 37]. It also makes the benefits of complementary experiments and global analysis highly
applicable to diamagnetic EDM measurements [21]. Experiments are underway using a variety ofsystems, including 199 Hg [48], 225 Ra [11], 129 Xe [74], Rn [124], and TlF [94].

10
The 199 Hg EDM experiment at the University of Washington has set the most stringent
constraints on BSM parameters derived from diamagnetic systems, with their most recent limit|dHg | ≤ 7.4 × 10−30 e cm (95% confidence) in 2016 [48]. To set this limit, the group implementeda Ramsey-like laser spectroscopy scheme conducted on vapor cells of isotopically enriched 199 Hg,
where they optically pumped and detected the spin orientation of the 1S0 ground state of the atom
using near-resonant laser light at 254 nm. The 199 Hg experiment attains nuclear spin coherence
times on the order of 500 s in their room-temperature, paraffin-coated cells, owing to the shieldingof nuclear spin by the closed shells of bound electrons. Less chemically reactive species such as
129 Xe attain even longer coherence times [74].

For species such as 129 Xe and 199 Hg, which have high vapor pressures and long spin coherence
times at room temperature, vapor cell experiments are highly advantageous in attaining large signal
sizes. However, the large volume occupied by the atomic sample in vapor cell experiments makes
them susceptible to systematic errors arising from magnetic and electric field gradients. In addition,
the requirement of multiple experimental constraints and the challenges of nuclear theory for 199 Hg
and 129 Xe [37] have led experimenters to consider species such as 225 Ra, which has an enhanced
Schiff moment through its octupole-deformed nucleus (Fig. 1.4, [5]). This deformation not only
leads to an orders-of-magnitude enhancement of BSM physics sensitivity in 225 Ra, it also makes
this enhancement more readily calculable [4, 35, 41].
An ongoing experiment at Argonne National Laboratory uses AMO techniques of laser slowing, cooling and trapping to perform nuclear spin resonance spectroscopy in the ground state of
225 Ra [11, 96]. This isotope is not naturally occurring, but has a long enough half-life of 14.9 days

so that it can be produced, transported from Oak Ridge National Laboratory to Argonne National
Laboratory, and studied in the EDM experiment. An AMO style experiment allows efficient use of
the extremely small available samples by reducing their temperature to the range of 10−6 K. The
laser-cooled sample can have high optical density and excellent overlap with a probe laser beam,in contrast to the situation in vapor cells.
Despite the challenges of collecting, laser slowing and trapping 225 Ra, the Argonne group11
has produced two successive limits on dRa , with the most stringent result |dRa | < 1.4 × 10−23 e
cm (95% confidence). Since the 2016 result, the group has been working to improve the statisticalsensitivity of their experiment: A top priority is the use of electron shelving detection [89], where
the population of one nuclear spin projection will be transferred to the 3 D1 state, while the otherpopulation is measured using an optical cycling transition. This technique should allow quantum
projection noise limited detection of the atoms’ spin precession phase, representing an order ofmagnitude increase in sensitivity.
With improvements to their electric field and spin readout, the Argonne group anticipatesattaining EDM precision at the 10−25 e cm level, where 225 Ra will attain similar sensitivity to BSM
physics as 199 Hg, and the two experiments’ combined constraints on BSM parameters will be more
stringent than 199 Hg alone. The ultimate precision goal of the experiment is 10−28 e cm, which will
require improved laser slowing and cooling, as well as a larger source of 225 Ra from the Facility for
Rare Isotope Beams at Michigan State University [13].

1.3.2Paramagnetic systems

Until 2011, the most precise measurement of the EDM of a paramagnetic system was set byexperiments in the Commins group at UC Berkeley [104]. Since then, a significant advancement
has resulted from several key innovations, illustrated in Figure 1.4: the use of diatomic molecules,
the cryogenic buffer gas beam, and Ω-doublet electronic states. All EDM results since 2011 inparamagnetic systems have been obtained using diatomic molecules: YbF at Imperial College
London in 2011 [64], ThO by the ACME Collaboration (Harvard/Yale) in 2013 and 2018 [2, 6],
and HfF+ at JILA in 2017 [16].
The most recent paramagnetic EDM limit |de | < 1.1 × 10−29 e cm or CS < 7.3 × 10−10 (90%confidence, using a sole-source interpretation) was set by the ACME Collaboration in 2018 [2],representing approximately an order of magnitude improvement over their 2013 limit [6]. Based on
generic expressions for how an eEDM could arise in Feynman diagrams including BSM particles,
and assuming a near-unity CP-violating phase [105], they estimate that their newest limit constrains

12

(a)

(b)

I

–
J

MHz-GHz

+
Sz

Eeff
Sintr

Eint

Closely spaced
opposite parity levels
Relativistic Eeff
enhancement

Octupole-deformed
nucleus

Diatomic molecule

Length contraction

(a)

(b)

(c)

Figure 1.4: Experimental approaches leading to recent results in EDM experiments on diamagnetic
and paramagnetic atoms and molecules. (a) Octupole-deformed nuclei have closely spaced nuclear
states of opposite parity, enhancing the parity-mixing effect of T-violating matrix elements. These
nuclei are orders of magnitude more sensitive to PT-violation than undeformed nuclei. (b) Diatomic
molecules for both diamagnetic and paramagnetic EDM searches. Relativistic effects and the
distortion of atomic wavefunctions by bonding in heavy polar molecules makes them maximallysensitive to T-violating effects. Molecules also have closely-spaced levels of opposite parity, allowing
them to be more easily polarized in the laboratory than atoms – this is analogous to collective Schiffenhancement in octupole-deformed nuclei. (c) Cryogenic buffer gas beams are a highly versatilesource of cold, slow, bright beams of almost any small molecule that can be introduced to the inert
buffer gas environment. (d) Omega-doublet levels in paramagnetic molecules allow a differential
EDM measurement without reversal of a laboratory electric field, greatly suppressing systematics.
They are also very closely spaced levels of opposite parity, allowing complete polarization of themolecules in electric fields of tens of V/cm.

13
the mass scale of new physics to lie above 30 TeV if the contributions are to arise at next-to-leadingorder in perturbation theory. This mass range is far beyond the direct reach of the LHC, where
the center of mass energy of proton-proton collisions is ∼ 14 TeV.
The unprecedented precision achieved by the ACME Collaboration’s experiments owes inlarge part to their extremely high count rate, obtained through the use of a cryogenic buffer gas
beam (CBGB, Fig. 1.4b, [67]). In this highly versatile technique, molecules are introduced in thevapor phase into a flow of inert gas inside an enclosure that is cooled to a few Kelvin. The molecules
are cooled by collisions with the buffer gas, and escape the enclosure through an aperture to form acold, bright beam. The versatility of cryogenic buffer gas beams allows molecules to be introduced
by a variety of methods: In the first two generations of ACME experiments, ThO molecules havebeen introduced by ablation of a ThO2 target. For their third generation experiment, the group
have developed a thermochemical source based on the reaction Th + ThO2 → 2ThO, which they
project will provide an order of magnitude signal increase [97].
The statistical sensitivity of the first-generation ACME experiment was limited by photon
shot noise. Sources of technical noise were highly suppressed by their ability to form a differential quantum phase measurement with high bandwidth through polarization switching. The spin
readout laser is rapidly alternated between perpendicular linear polarizations, allowing rapid sequential measurements of the projections SX and SY of the quantum state onto perpendicular axes
X and Y , respectively, to measure the spin precession angle φ. They then form the asymmetryA = (SX − SY )/(SX + SY ) ∝ cos φ for fluorescence pulses separated by only a few microseconds –a much shorter timescale than variations in the density of molecules in their cryogenic beam. The
asymmetry is thus essentially immune to technical noise arising from variations in molecule density,
which occur most severely from one pulse of the molecular beam to the next – a timescale of tensof milliseconds.
The ACME II dataset was limited by technical noise, but with a statistical sensitivity approximately 20 times higher than ACME I [6, 7]. While the exact source of excess noise was notdefinitively established during the data collection, its characteristic timescale was approximately

14that of sequential molecular beam pulses, ∼ 10 ms. The excess noise enters their EDM measurement because the measured precession phase in two Stark doublets is subtracted to isolate thecontribution due to an EDM. Elimination of the noise could be achieved by conducting simultaneous phase measurements in both Stark doublets within a single molecular beam pulse (at the costof possible systematic effects).
In 2017, our group obtained a limit |de | < 1.3 × 10−28 e cm using HfF+ molecules in an
ion trap [16]. While the measurement did not set a new limit on de and CS in a sole-source
interpretation, a joint analysis along with the 2014 ACME limit improved the global constraints
on de and CS [43]. The first generation molecular ion result was limited by statistics: in a single
experimental cycle, only of order N ≈ 10 signal molecules were detected, out of about 103 trappedmolecules. This small fraction of ions contributing to the usable signal resulted from low efficienciesof state preparation and detection. Much like the 225 Ra hadronic EDM result, systematic errors
and technical noise did not play a dominant role at the present sensitivity – a fact that will certainly
change in subsequent measurements as the statistical precision continues to improve.
Towards the second generation eEDM search with HfF+ , our group has implemented an
ion trap with a larger trapping volume, allowing the molecular cloud to be cooled by adiabatic
expansion. The strong scaling γcoll ∼ T 13/2 of the collisional decoherence rate with temperature[78] has already allowed coherence times of over 2 seconds with 104 trapped molecules. The groupis also implementing state preparation by optical pumping, which has the potential to improve theusable fraction of trapped molecules from the few-percent level to over 50%, with no change incycle time or coherence time.
Shot-to-shot variations in ion creation, state preparation and detection efficiency will present
a significant challenge for subsequent generations of molecular ion EDM experiments. It is possible
to normalize against many of these fluctuations by conducting simultaneous measurements of laser
powers and frequencies, however fluctuations in molecular beam properties such as density, velocity,and temperature are significantly more difficult to monitor in situ. A much more robust defense
against technical noise is to conduct differential phase measurements within a single experimental

15
cycle, analogous to the ACME experiment’s polarization switching technique. However, such ameasurement is difficult to conduct with our group’s present detection technique.
The present landscape of EDM searches in paramagnetic systems is one where advancements
are being made on several fronts: very large numbers of molecules are produced in a quantum statewith high efficiency by the ACME Collaboration through the use of a CBGB and STIRAP [95];and very long coherence times are being pursued at the JILA trapped molecular ion experiments
Future experiments on paramagnetic systems aim to bring together these developments into EDM
searches that will constrain BSM physics up to the PeV scale.

1.4

Theoretical interpretation of EDM measurements
This section is reproduced from Ref. [17]Historically, the possibility of particle EDMs was first pointed out in 1950 by Purcell andRamsey [103] as a possible counterexample to the common assumption that P is a symmetry ofnature – since EDMs violate both P and T symmetries. Their motivation at the time was thatlacking a fundamental theory of nuclear structure, P conservation should not be assumed. At thetime, a significant nonzero value for the neutron EDM (nEDM) would likely have guided the development of the theory of the strong and weak interactions. Lacking a theoretical interpretationof their null result, however, they did not publish until after PV was proposed and subsequently
discovered in 1957 [80, 118, 131]. In a modern context featuring the tools of quantum field theory and the renormalization group, it is possible to interpret the results of EDM experiments asconstraints on the mass of BSM particles, and to guide experimental approaches toward systems
with optimal sensitivity to BSM physics [21]. Today, EDM searches are more strongly motivatedby BSM T-violation and its connection to the BAU.
The theoretical review in Ref. [37] describes in detail the connection of BSM physics tolow-energy EDM searches. The authors begin with an effective field theory extension of the SM(SM-EFT, [3, 37, 53]), which supplements the SM with particle interactions of the form Ci Oi /Λ2 .
Operators Oi couple SM fields such as electrons and quarks, while the mass scale of BSM physics Λ16suppresses its magnitude, and a dimensionless Wilson coefficient Ci encapsulates the effects of BSM
particles that are not directly observed. They identify thirteen effective operators at the SM scalethat are expected to give dominant contributions to EDMs, and further apply effective field theory
techniques to identify these operators with T-violating observables at the atomic scale (Fig. 1.5).
At low energies, searches for EDMs can be categorized according to the physical systembeing experimented on: paramagnetic atoms or molecules, diamagnetic atoms or molecules, orparticles and nucleons. Permanent EDMs of paramagnetic atoms and molecules arise primarily
from the electron EDM de and the scalar electron-nucleon coupling CS [22]. Diamagnetic atom
or molecule EDMs arise from the nuclear Schiff moment S, which obtains dominant contributions(0)

from nucleon EDMs dn and dp and from isoscalar and isovector nucleon-pion interactions gπ and(1)gπ . Independently, diamagnetic EDMs can also arise from scalar and tensor electron-nucleon
interactions CS (at sub-leading order) and CT (at leading order) respectively [22].
No EDM of an atom, molecule, nucleus, or nucleon has yet been observed. Measurementsthat are consistent with zero are used to set constraints on CP-violating parameters: typically,
it is assumed that the Wilson coefficient C is of order unity, and an upper bound on an EDM
is used to set a lower bound on the mass scale Λ of BSM particles. Within a particular BSM
theory, limits can be set on the masses of specific particles [38]. Because different experimental
searches are sensitive to multiple BSM effects at different levels, and since multiple CPV terms
generally arise in the low-energy limit of a BSM theory, the most robust constraints are obtainedby a global analysis of experimental results [22, 43]. In the method of Ref. [22], the EDM ofP
a system is parameterized as a sum of contributions di =j αij Cj , where αij = ∂di /∂Cj is atheoretically obtained sensitivity of the EDM of system i to source Cj . The function χ2 (Cj ) =P exp− di )2 /σd2exp then expresses the deviation of the observed value from what might arisei (diifrom a superposition of multiple sources. Reasonable ranges for BSM physics parameters are thenellipsoids with boundaries of constant χ2 . This model-independent analysis provides an agnostic
interpretation of multiple EDM measurements with varying sensitivities to a large number of BSM
parameters, and allows predictions of which future measurements may most effectively improve17

Restrict to CP-violating,exclude CKM matrix

Electroweak
symmetry breaking

Experiments

QCD phasetransition
Hadronicscale> Electroweak
scale

Nuclear scale

Electron EDM
deParamagnetic atoms &molecules

Fermion EDMs (3)CfW, CfBCPV Electron-nucleon couplings
CS, CT
Semileptonic (3)Clequ(1,3), Cledq

Diamagnetic atoms &molecules

Quark EDMs
du,dd

Chromo-EDMs (2)CqG
Dimension-6
terms (59)Standard
Model

Nuclear Schiff moment
charge partSch

Four-quark (2)Cquqd(1,8)

Quark-Higgs (1)Cφud

Three-gluon (1)
CG

θQCD

Light nuclei

CPV Pion-Nucleon couplings
gπ(0),gπ(1),gπ(2)Nuclear Schiff momentnucleon EDM partSN

Induced four-quarkCφud

Nucleon EDMs
dn, dp

Nucleons

Running & matching,
non-perturbative matrix
elements

Figure 1.5: Interpretation of EDM experiments. (a) SM-EFT sources of EDMs, interpreted fromthe text of Ref. [37]. Dimension-six effective operators couple Standard Model degrees of freedom,
encapsulating the effects of BSM theories at the electroweak scale. Engel et al. [37] trace the effects
of a subset of 12 CP-violating effective operators through the hadronic and nuclear scales to theireffects on EDMs of paramagnetic and diamagnetic atoms and molecules, nucleons, and nuclei.

18
upon existing constraints (Fig. 1.6, [22, 43]).
It is also possible to extract constraints on the masses of specific BSM particles from EDM
measurements within a particular SM extension [37, 38, 105]. In this case, the contributions of
BSM particle interactions to EDMs can be evaluated explicitly, and experimental EDM limits
result in constraints on particle masses and mixing angles. In the case of supersymmetric models,the electron EDM constraints on the electron superpartner mass exceeds the reach of the LHC[2, 38].

1.5

About this thesis
This thesis is the seventh to come out of the Cornell/Ye eEDM project, where we have beenmeasuring the electron’s electric dipole moment (eEDM) with trapped molecular ions, presently
with HfF+ and in the third-generation experiment with ThF+ . At the inception of this project,
HfF+ and ThF+ were not molecules whose state assignments and spectroscopic constants could be
looked up on the NIST Webbook database 2 , nor anywhere else in the literature. As a matter offact, it was not even decided which molecular ion the experiment would use until around 2011. An
enormous amount of work carried out by the previous PhD students on the project made our firstmeasurement possible in 2017.
Survey spectroscopy of HfF+ and ThF+ was carried out by Laura Sinclair, Kevin Cossel, and
Dan Gresh in the comb velocity modulation spectroscopy lab, with survey work on ThF+ being
completed in early 2016 [28, 51, 112]. Russell Stutz, Huanqian Loh, and Matt Grau developed
techniques for producing neutral HfF in a supersonic beam [120] and conducted studies that enable
us to produce large amounts of HfF+ in their rovibronic ground state: they performed laser-induced
fluorescence (LIF) spectroscopy of HfF in the neutral beam to determine suitable intermediate states
for ionization [50], they studied the Rydberg states of HfF that autoionize to form HfF+ and studied
their propensities for producing different rotational states [83], and they used LIF of the trapped
HfF+ ions to characterize their resulting internal states [84]. Finally, after having experienced2

https://webbook.nist.gov/

19

CA
Global 2σ
confidence
region

PT 2σ

ω2

CB
ω PT
1

2σ

Figure 1.6: Conceptual picture of global analysis of EDM results: Two experiments labeled 1
and 2 havesensitivities to CPV coefficients CA and CB , parameterized by the slopes different

dω1,2
dω1,2
dCA / dCB . A global analysis can be used to obtain simultaneous constraints on CA and CB
from experiments 1 and 2.

20
the challenging (to put it mildly) signal-to-noise ratios that come with performing LIF detectionof molecular ions without optical cycling transitions, Kang-Kuen Ni, Kevin, Matt, and Huanqian
developed our rotational state-selective resonance-enhanced multiphoton dissociation (REMPD) in2013 [91].
This thesis covers the period from Fall 2013 to Summer 2019 in the HfF+ eEDM lab. In Fall
2013, Huanqian et al. had just demonstrated our first eEDM-sensitive Ramsey fringes in the 3 ∆1state of HfF+ . In 2014, we worked to understand our statistical sensitivity, in particular working
to improve the efficiency of population transfer from 1 Σ+ (v = 0, J = 0) to 3 ∆1 (v = 0, J = 1), andunderstanding the effects limiting our coherence time. This work is described in detail in Kevin
Cossel’s thesis [28].
In 2015, we studied systematic errors. The sources of systematic frequency shifts in our
experiment are in many cases not particularly comparable to those of previous eEDM experiments,
or to trapped-ion atomic clocks, and so this investigation was a significant learning experience
wherein we learned the best ways to look for and understand systematics in our unique apparatus.
A large number of effects are described in Matt Grau’s thesis [49], and others are described in
the supplemental information of Ref. [16], which I have reproduced in Section 3.5 of this thesis.
Throughout the process of our systematic error investigations, I performed a large number oftheoretical calculations regarding the structure of HfF+ in external electric and magnetic fields,
which may be useful in the future. These are described in Chapter 2.
In 2016 and early 2017, with the addition of Dan Gresh (who moved over from ThF+ spectroscopy) and Tanya Roussy to the HfF+ ion trap lab, we collected eEDM data and produced
our first measurement, which is described in Dan Gresh’s thesis [52], and in Ref. [16] which is
reproduced in Chapter 3 of this thesis.
The majority of the original content of this thesis covers our progress towards a Generation
2 experiment with at least an order of magnitude improved sensitivity, which covers the period
Spring 2017 to Summer 2019. We designed, constructed, and commissioned our Generation 2
ion trap beginning in Fall 2016 and concluding in Fall 2017, described in Chapter 4. We then21completely altered our state preparation scheme to a much lower-tech approach, the motivationand implementation of which is described in Chapter 5.
Next, inspired by a serendipitous combination of Yuval Shagam’s experience in velocity map
imaging (VMI), very nice photodissociation results from the Zelevinsky group at Columbia [85, 86],
and advice from Tanya Zelevinsky herself during a visit in 2018, we developed a scheme by which we
can use molecular orientation-resolved REMPD to conduct an eEDM-sensitive differential phasemeasurement in a single experimental cycle. I feel that this last development breathes new life
into the potential precision of this and future precision measurement experiments with trappedmolecular ions. The oriented REMPD scheme is described in Chapter 6.
Finally, in Chapter 7, I explore the implications of our improved count rate, coherence time,and new REMPD signal for a Generation 2 eEDM measurement. It is my hope that this will provide
a jumping off point for our group to produce a second generation measurement in relatively shortorder.

1.5.1

Experiment overview

The HfF+ eEDM experiment is unusually complex, even for a modern AMO physics experiment. Altogether, we have about 13 lasers (listed in Appendix C), and each step of the experimentmight have been an atomic physics experiment in its own right a few decades ago. This thesis
is primarily about improving several of these steps, rather than inventing each of them – therefore I provide a brief description of the entire experiment here. An overview of the HfF+ eEDM
experiment is shown in Fig. 1.7. One cycle of the experiment consists of the following steps:(1) Supersonic beam
We produce neutral HfF through the reaction of Hafnium with Sulfur Hexafluoride, SF6 ,in a supersonic beam of Argon. The SF6 is seeded with 1% concentration in Ar, andthe mixture is allowed into the chamber one 50 microsecond puff at a time through apiezoelectric valve in the source chamber. While the piezo valve is open, we ablate a rod of

22

(b)

CP

FM
TO

Energy(a)ionization
MF pumping
MF depletion

dissociation

266 nmfixed UV
Ω=2
3

3

–

Σ0+

Π0+

3

Π0–

tunable UV

HfF
HfF+
3

imaging
MCP

ionization

X1Σ+phosphorCCD

Δ1

Ω = 3/2

X2Δ3/2dissociation
stop ionskick to MCP(c)ttrap = 0

~ 100 ms

π/2

π/2

MF depletion

RF
trap

populationtransfer

MF pumping

ablation
ionization

rotating E fieldttrap~ 1700 ms

Figure 1.7: Overview of the HfF+ eEDM experiment. (a) Top-down view of the main vacuum
chamber containing the RF ion trap. (b) Relevant energy levels of HfF and HfF+ , with lasers
showing main transitions. (c) Timing diagram showing the major teps of the experiment sequence:(1) Neutral HfF production, (2) ionization, (3) ion trapping, (4) state preparation, (5) Ramsey
sequence, (6) dissociation, and (7) detection. These steps are described in detail in Section 1.5.1 of
the text.

23
Hafnium that is mounted next to the valve opening, causing vaporized Hf and Hf+ to be
injected into the supersonic expansion of Ar + SF6 . The HfF (along with whatever otherreaction products may be present) are rotationally and vibrationally cooled by collisions
with Argon, and accelerated to speeds of about 600 m/s. The beam passes through a pairof skimmers, which for our purposes leaves behind a cold “slug” of HfF traveling throughour vacuum chamber towards the ion trap at 600 m/s. Molecular beam production for the
HfF+ experiment was primarily developed by Russell Stutz and Aaron Leanhardt, and is
described in Russ’ thesis [120].
(2) Ionization
Just as the slug of HfF passes through the center of the ion trap, the pulses of UV light fromour two ionization lasers (“Harambe,” 309 nm and “Bertha,” 367 nm) arrive, resonantly
ionizing HfF to form HfF+ in its vibronic ground state 1 Σ+ (v = 0), with ions populating
the lowest few rotational states N = 0-3. This fairly exact timing is surprisingly easy toachieve, via knowledge of the velocity of an Argon supersonic beam from the literature,measuring the distance from the pulsed valve to the trap center (a measuring tape will
do just fine), and using fast photodiodes and an oscilloscope to compare the pulsed valve
timing with the arrival time of the laser pulses at the ion trap. The details of HfF+
resonance-enhanced multiphoton ionization (REMPI) were worked out by Huanqian Loh
and others, and are described in Huanqian’s thesis [82].
(3) Ion trapping
Following their creation, we confine the HfF+ ions in our RF trap and apply a rotating
electric bias field Erot to polarize the molecules. While the control, signal generation,
and amplification electronics have changed completely since the beginning of the second
generation HfF+ experiment, the mechanics of trapping remain very much the same, anda useful introduction is given in Ref. [82]. The Generation 2 ion trap and electronics aredescribed in Chapter 4 of this thesis.

24(4) State preparation
While the HfF+ ions are created in their 1 Σ+ ground state, we need to prepare them in asingle MF magnetic sublevel of the 3 ∆1 (v = 0, J = 1) state. In Generation 1, this was done
by a stimulated Raman process that equilibrated population between 1 Σ+ (v = 0, J = 0)
and 3 ∆1 (v = 0, J = 1), followed by selective depletion of MF levels. For Generation 2, wehave moved to an incoherent state preparation scheme, which is described in Chapter 5.
(5) Ramsey sequence
The Ramsey interferometer sequence that extracts the eEDM-induced energy shift between
MF = ±3/2 levels in 3 ∆1 (v = 0, J = 1) is carried out simply by modulating the magnitudeof Erot , which is performed by the same digital synthesis boards that control the trap
electronics as a whole. These boards were designed, constructed, and tested by Felix
Vietmayer, Dan Gresh, Carl Sauer, and Terry Brown. Following the second π/2 pulse (i.e.,
modulation of Erot ), we use a circularly polarized laser to deplete one of the MF levels, thus
projecting the population onto the MF basis. This step is described in Chapter 5.
(6) Dissociation
At the end of the Ramsey sequence and following MF depletion, the ion trap still contains
HfF+ ions in a mixture of internal states, only a small subset of which we would like toknow about – namely, those remaining in 3 ∆1 (v = 0, J = 1). We detect those ions byselectively photodissociating them into Hf+ + F, as described in Chapter 6.
(7) Ion detection
We detect both Hf+ and HfF+ ions using microchannel plate (MCP) ion detectors, which
are common tools in physical chemistry and see some use in atomic physics. While many
physical chemistry experiments use complex systems of electrostatic lenses to achieve highquality ion imaging, we are less picky, and simply use the arbitrary waveform capabilitiesof our ion trapping electronics to kick the entire bunch of both HfF+ and Hf+ ions simul-

25
taneously onto an MCP. Thanks to recent lessons from our postdoc Yuval Shagam, we now
use our imaging MCP detector to its fullest potential: we apply a gated voltage to thephosphor, allowing us to selectively image the Hf+ ions, which arrive first in time of flight
due to their lower mass than HfF+ . We simultaneously collect a time trace by AC coupling
to the middle of the pair of MCPs, giving us access to information about the number of
HfF+ ions in the same experiment cycle.

Chapter 2Theoretical calculations in HfF+

2.1

Molecular structure
For both numerical and analytic calculations of frequency shifts and population dynamics

in HfF+ , it is essential to have an effective Hamiltonian model of molecular structure. In the
Generation 1 eEDM experiment, we primarily used an effective Hamiltonian for the 3 ∆1 state only,
along with numerical simulations of ion motion, to predict the values of frequency channels f 0 , f D ,f BR and so on. Over the years, I have also found this model useful for simulations of coherent
transfer, depletion, optical pumping, and microwave spectroscopy, with varying degrees of successat reproducing experimental observations. In this section, I will describe the various components ofthe effective Hamiltonian, attempt to motivate their structure, and detail their implementation in
Matlab. Along the way, I will give an operational introduction to molecular structure and sphericaltensor operators.

2.1.1

Qualitative features of HfF+

While the details of molecular structure are complex, and our analysis is often fairly mathematical, there are certain general features of HfF+ that can be understood through freshman
undergraduate chemistry methods such as molecular orbital theory. HfF+ is an ionically bonded
molecule, meaning that in the ground and low-lying electronic states, Hf+ essentially donates anelectron to F. The fluoride anion F− has a 1 S0 configuration, so the low-lying states of HfF+ with
nonzero total angular momentum arise from the 5d and 6s orbitals of Hf 2+ , as shown in Fig. 2.1. As

27

Figure 2.1: Molecular orbital diagram for HfF+ . Since the molecule is ionically bonded, the molecular states correlate with atomic states of Hf2+ and F− . The ground electronic state of Hf2+ is
3 F , and correlates with the excited 3 Φ state of HfF+ .
2
2

discussed in Kevin Cossel’s thesis [28], the ground 1 Σ+ state arises from the sσ molecular orbital,
while the low-lying 3 ∆, 1 ∆, 3 Π, and 1 Π states (and others) arise from single excitations of the valence electrons from sσ to dδ and dπ. The fact that these molecular states primarily arise from thestates of Hf2+ allows an estimate of their spin-orbit and Λ-doubling parameters [28]. Interestingly,
the ground electronic state of Hf2+ is 3 F2 , like the isoelectronic ion La+ . While the 3 Φ2 state in
HfF+ arises from this state, it is an excited state of HfF+ by nearly 15000 cm−1 . The reasons forthe ordering of the states of HfF+ can be justified using Ligand-field theory, as discussed for theisoelectronic neutral molecule LaF in [109].
Though HfF+ is ionically bonded, a positive net energy is required to remove an electron from
Hf+ and donate it to F: In other words, the ionization energy of Hf+ is greater than the electron
affinity of F. Therefore, if the HfF+ molecule were adiabatically dissociated from its ground state,
the resulting atomic species would be Hf+ + F. As a result, the ionic and covalent states cross in
the region of internuclear distance where the ionic bond is broken, as illustrated in Figure 2.2. Thissituation results in a high probability of quasi-bound (“predissociating”) states, which presented
challenges for our photodissociation-based detection scheme, discussed in Section 6. A further
consequence of the ionic nature of the HfF+ bond is the very large dissociation energy of De ≈52000 cm−1 [91], which requires us to use multiple UV photons for photodissociation, as described

28

4
2

...

Energy (104 cm –1)

6

0
-2
-4
-6

2

3

4

5

6

Internuclear distance (Angstroms)

Figure 2.2: Schematic potential energy curves for HfF+ , including a few electronic states of themolecule and free atoms. The low-lying bound states of the molecule are ionically bonded, so theycorrelate to electronic states of Hf2+ and F− . Shown in black at the right are the energies of lowlying electronic states of free Hf+ and F below 14000 cm−1 . Potential curves are drawn as Morsepotentials.

29
in Section 6.

2.1.2

The molecular wavefunction

The complete wavefunction of a molecule in the nonrelativistic approximation is a solution
to the time-independent Schrödinger equation with the total molecular Hamiltonian
H = Hnuc + Helec + Hspin ,Hnuc =X P2ααHelec =

X p2
i

i

2me

+

2Mα

X
i<j

+

X kZα Zβα<βRαβ(2.1a)

,

X kZα
k
−
,
|ri − rj ||ri − Rα |(2.1b)(2.1c)

α,iwhere Latin indices enumerate electrons, Greek indices enumerate nuclei, me is the electron mass,
Mα is the mass and Zα is the charge of nucleus α, and k = e2 /(4π0 ). For the complete systemof HfF+ , with AHf + AF − 1 = 80 electrons and 2 nuclei, solution of the Schrödinger equation isextremely difficult. To make matters worse, the Hf nucleus with Z = 72 causes electrons to experience significant relativistic effects. Thus, strictly speaking, the already intractable Schrödinger
equation is not even the correct equation to solve to obtain the complete structure and characterof the states of HfF+ !
Despite this complexity, all is not lost for this precision measurement experiment. We seek
to measure the eEDM – a quantity that violates T symmetry – and to measure the eEDM precisely
does not rely on our knowing the energies or characters of the enormous number of highly coupledstates of HfF+ to a high degree of precision. We endeavor to know the structure of HfF+ not tofurther our understanding of molecular structure, but to develop tools for preparing HfF+ in just a
few very well-understood states, for performing a Ramsey spectroscopy sequence, and for detecting
the populations of those states with high fidelity. Thus, we primarily turn to the well-developed
tools of molecular spectroscopy to allow us to qualitatively characterize the states of our molecule.

30
To qualitatively understand the states of HfF+ and small molecules in general, we rely on theseparation of energy and time scales of electronic and nuclear motions: the electrons have typicalvelocities many thousands of times faster than the nuclei, and from their point of view the Coulomb
potential imposed by the nuclei can be thought of as classical and quasistatic. Conversely, when
considering the motion of the nuclei, we can average over the electronic motion and treat the nuclei
as moving in a potential V (R) comprising the nuclear Coulomb repulsion and the attractive force of
the average electron density. Finally, the rotational and vibrational degrees of freedom of the nuclei
can be approximately separated. Mathematically, these physical ideas are manifested in the BornOppenheimer approximation and the Born adiabatic approximation (Chapter 2 of Ref. [15]), where
we expand the total wavefunction as a product of electronic and nuclear components. Schematically,Ψ ' ψelec (ri ; R|si ) χvib (R) ψrot (ω),(2.2)

where ω is a shorthand for the Euler angles φ, θ, χ, si represents electron spin, and I have neglectednuclear spin for now.
To obtain a version of the Schrödinger equation where the electronic and nuclear degrees
of freedom are separable, we must transform the Hamiltonian from the laboratory-fixed frame[Eq. (2.1)] to the molecule-fixed frame. This requires a translation into the center of mass of thenuclei, followed by a rotation by Euler angles (θ, φ, χ) into a frame where the internuclear axis isoriented along ẑ. The derivation is lengthy, spanning Sections 2.1-2.7 of Ref. [15]! Neglecting thecenter of mass kinetic energy, the result is~2 ∂Hnuc = −2µR2 ∂R|

Helec =



~2
2 ∂
R
+ V (R) +(J − Ja )2 ,
∂R
2µR2
{z
}
{z
} |
Hvib

(2.3a)

Htum

X p2
pi · pj
k
kZ1
kZ2
i
+
+
−
−
+ Hspin ,2m
2MN|ri − rj | |ri − (µ/M1 )Rẑ| |ri + (µ/M2 )Rẑ|(2.3b)

i,j

where R is the internuclear distance, MN is the total nuclear mass, µ = M1 M2 /(M1 + M2 ) is the
reduced nuclear mass, J = L + S + R is the total angular momentum including electronic spin and31orbital (L and S) along with nuclear rotation R, and Ja = L + S is the total electronic angularmomentum. In this form, the electronic part of the Hamiltonian depends on the nuclear coordinates
only through their relative positions on the body-fixed z axis (coincident with the internuclear axisn). This form is simple enough to be attacked by numerical methods. The nuclear part is further
separable into components governing end-over-end “tumbling” Htum and vibration Hvib . We will
discuss the electronic, rotational, and vibrational parts of the wavefunction in the following sections.

2.1.3

Electronic structure

The electronic structure of a diatomic molecule in the Born-Oppenheimer and adiabatic
approximations is governed by the Hamiltonian in Eq. (2.3b). Due to the nonsymmetric natureof the Coulomb potential imposed by the two nuclei, the electronic orbital angular momentum
L is generally not conserved, in contrast to the situation in the spherically symmetric potential
of atoms. Though I have written the electronic and spin parts of the wavefunction separately in
Eq. (2.2), spin-orbit coupling couples L and S to form the total electronic angular momentum Ja ,particularly in heavy molecules such as HfF+ . Depending on the strength of the spin-orbit energy
Es-o , it may be useful to express the electronic wavefunction in Hund’s case (a) or (c):hri ; R|Λ, S, Σi case (a): Es-o  Etum ,ψelec (ri ; R|si ) =hri ; R|Ja , Ωicase (c): Es-o  Etum ,(2.4)

where Etum ' ~2 /(2µRe2 ) is the energy scale of the end-over-end tumbling (rotation) of the molecule.
While numerical calculations of electronic structure are beyond the scope of this thesis, they
have been undertaken for HfF+ by several groups, and are detailed in Refs. [28, 29, 44, 99, 100,
114, 115, 117]. In these works, the electronic Hamiltonian is augmented with relativistic corrections
which we have not discussed here. For the purposes of an experimentalist, the results of theircalculations are electronic wavefunctions that can be characterized either by case (a) or case (c)in Eq. (2.4). These wavefunctions can then be used to calculate the eEDM sensitivity parameter
Eeff , as well as expectation values of the dipole moment operator in order to estimate transition

32
strengths and radiative decay lifetimes [99]. The low-lying 1 Σ, 3 ∆, and 1 ∆ states of HfF+ are well(P )described by case (a), as indicated by our use of term symbols 2S+1 |Λ||Ω| to denote them, where
P is the overall parity. Higher lying states such as the nominal 3 Π0+ are in fact better describedin the case (c) notation |Ω|(P ) = 0+ , as evidenced by the fact that this state couples to both 1 Σ+and 3 ∆1 , which would be forbidden for a state with good quantum number S.
Figure 2.3 shows the low-lying electronic states of HfF+ , many of which can be populated
by conveniently located near-infrared wavelength lasers. Only two of the electronic states below
15000 cm−1 have not been observed; a testament to the effectiveness of the frequency comb velocity
modulation spectroscopy technique developed by the Ye & Cornell groups [28, 29, 52, 112].

2.1.4

Rotational structure

The rotational structure of HfF+ is present throughout the broadband spectroscopy performed in this thesis and earlier eEDM theses, in particular those of Kevin Cossel, Dan Gresh,
Laura Sinclair, and Huanqian Loh [82]. In this thesis in particular, we use our understanding ofrotational structure to find an Ω = 2 excited electronic state to act as an intermediate in ouroriented photodissociation in Section 6.
As discussed in Section 2.1.2, the rotational part of the molecular wavefunction is governedby the “tumbling” part of the Hamiltonian
Htum =~2(J − Ja )2 .
2µR2

(2.5)

The operator R ≡ (J − Ja )2 , which represents the purely end-over-end tumbling of the nuclei,
can be separated into parts which are diagonal and off-diagonal with respect to the electronic
wavefunction,222(J − Ja )2 = (J 2 − 2Jz Ja,z + Ja,z
) + (Ja,x+ Ja,y− 2Jx Ja,x − 2Jy Ja,y ),and we can use the fact that both Jz and Ja,z have eigenvalue Ω; the total electronic angularmomentum projection on the internuclear axis (Ref. [15], Section 2.6). Neglecting off-diagonal

33

15000

10000

5000

0
0

1

2

3

Figure 2.3: Electronic states of HfF+ up to about 15000 cm−1 , with singlet states in black andtriplet states in blue. The 3 ∆3 and 1 ∆2 states have not been observed. Red lines indicate known
transitions between electronic states, with thick lines indicating dipole allowed transitions and thinlines indicating dipole forbidden (but weakly allowed) transitions. Equally spaced states connectedby a shaded region are vibrational levels belonging to the labeled electronic state. Only the excited
vibrational levels used in the experiment are shown. Rotational levels exist on a much smaller
energy scale of . 1 cm−1 .

34electronic terms 1 , an approximate effective Hamiltonian for the end-over-end rotation is thenHtum = B(R)(J 2 − Ω2 )where the rotational constant B(R) = ~2 /(2µR2 ) depends parametrically on the internuclear distance. The constant part −B(R)Ω2 is often absorbed into the electronic energy. The eigenfunctionsof this Hamiltonian are (proportional to) the Wigner rotation matrices,r
2J + 1 (J)ψrot (ω) =DM Ω (ω)∗
8π 2

(2.6)

≡ hω|JΩM i
with eigenvalues Etum = B(R)[J(J + 1) − Ω2 ]. These eigenfunctions are discussed extensively in
Chapter 5 of Ref. [15], and have many nice identities.

2.1.5

Vibrational structure

I will not go into great detail on the vibrational structure of HfF+ , as it is addressed inessentially every textbook on molecular structure. The vibrational eigenfunctions |ψvib i ≡ |vi
are solutions to a time-independent Schrödinger equation with the nuclear Hamiltonian given in
Eq. (2.3a), but with the replacement of the rotation operator (J − Ja )2 → J(J + 1) − Ω2 due to ourseparation of the vibrational and rotational degrees of freedom. The internuclear potential V (R)is computationally expensive to calculate numerically, since this involves solution of the electronic
Schrödinger equation for each value of the internuclear distance R. A common approximation is
the Morse potential, which I have used to schematically illustrate internuclear potentials in Fig. 2.2(See e.g. Ref. [15], Section 2.9). Because the internuclear potential is approximately harmonic nearits minimum, the vibrational wavefunctions closely resemble harmonic oscillator states, and theirenergies are well approximated by
Ev = hv| Hvib |vi ≈ ~ωe (v + 1/2) − ~ωe xe (v + 1/2)2 .
1

(2.7)

Neglecting coupling between electronic states caused by nuclear motion is called the adiabatic approximation.

35
In Section 2.2.4, we will use these energies in evaluating the Einstein coefficients for spontaneous
decay and blackbody excitation of vibrational degrees of freedom in the 3 ∆1 electronic state of
HfF+ .

2.1.6

Nuclear spin & the total wavefunction revisited

To summarize the results of the previous sections, we have established that the total molecular
wavefunction in the Born-Oppenheimer and adiabatic approximations, and in a Hund’s case (a)basis, can be represented by a product state ket with the quantum numbers|ΛSΣi |vi |JΩM i ,with electronic, vibrational, and rotational components, respectively. In HfF+ , we must accountfor one final source of angular momentum: the nuclear spin I of 19 F. The coupling of I to theelectronic plus rotational angular momentum J = Ja + R can be done in the lab frame to form thetotal molecular angular momentum F = J + I with projection MF onto the laboratory-frame z
axis. The transformation from one basis to the other can be straightforwardly accomplished using
Clebsch-Gordan coefficients:|JIF MF i =X|JMJ IMI i hJMJ IMI |F MF i .

(2.8)

MJ ,MIThe only further complication is that when evaluating matrix elements of vector operators in thelab frame (as we will do in Section 2.2), there will be some spherical tensor algebra involved todecouple I from F to evaluate matrix elements only involving J . Happily, all of the expressions
we need are handily summarized in Ref. [15], Appendix 5.1. So finally, the total wavefunction of
the HfF+ molecule can be expressed as|Ψi = |ψelec i |ψvib i |ψrot i = |ΛSΣi |vi |IJΩF MF i ,and we can set about evaluating important matrix elements for the JILA eEDM experiment.

(2.9)

36

2.2

Energy shifts and decay rates in HfF+

2.2.1

Electric fields

In the HfF+ eEDM experiment, we use electric fields extensively for the manipulation of boththe internal and external degrees of freedom of HfF+ molecules. The characteristic frequencies of
our applied electric fields in our experiment range from the tens of kHz for the RF ion trap,
to tens of GHz for rotational spectroscopy with microwave fields, to over 1 PHz (1015 Hz!) for
photodissociation with 266 nm laser light. The interaction of all of these electric fields with theinternal state of the molecules can be described by the Stark Hamiltonian
HStark = −D · E = −

+1
X

(1)

(−1)p Tp(1) (D) T−p (E),

(2.10)

p=−1

where D is the electric dipole operator and E is an applied electric field. On the second line, I
have expressed the same Hamiltonian in the language of spherical tensor operators (See Ref. [15],Chapter 5.5) 2 . Spherical tensor operators are a useful way to formalize the algebra of operators
acting on angular momenta, and though the notation can be highly intimidating, essentially every
important result has been worked out in detail in Refs. [15, 136].
Unlike in atoms, the electric dipole operator acts not only on the electrons, but also on therelative positions of the nuclei:
D = DN + De =Z1 µ Z2 µ
−
M1
M2


eR n −

Xeri(2.11)i

In Eq. (2.11), I have expressed the dipole moment operator in the body-fixed frame, in the same
way that we did for the molecular Hamiltonian without external fields in Section 2.1.3 in order tosimplify the separation of rotational, vibrational, and electronic degrees of freedom. However, thecomponents of the electric field E are most easily expressed in the laboratory frame. To correctly2(1)

The spherical components of a rank-1 vector operator Tp (V ) in terms of the cartesian components of V are(1)

T±1 (V ) = ∓Vx ± iVy
√
,
2

(1)

T0 (V ) = Vz .
Substitution of these components into the second line of Eq. (2.10) straightforwardly returns the first line.

37
evaluate the matrix elements of HStark , we must transform the components of D into the laboratory
frame. This transformation is most easily done with Wigner rotation matrices, using the expression(Eq. 5.143 of Ref. [15])Tp(k) (A) =

k
X

(k)
Dpq
(ω)∗ Tq(k) (A),

(2.12)

q=−kwhere A can be any tensor operator. Equation (2.12) can be deceiving: while seemingly thisexpression represents a simple matrix multiplying a vector, the variable ω is just a shorthand forthe three Euler angles φ, θ, χ, and the rotational wavefunction of Eq. (2.6) is a function of thesevariables! To evaluate a matrix element of any molecule-frame operator transformed into the lab(k)frame in this way, these variables must be integrated over. In this sense, we must treat Dpq (ω)∗
as an operator on the rotational part of the wavefunction. Fortunately, the rotational part of the
wavefunction always takes the form of Eq. (2.6), so we need only evaluate this expression once.
The results are given in Eq. 5.146 of Ref. [15].
(1)

In the laboratory frame, the components of Tp (E) are c-numbers, not operators. Thus,
the task of evaluating the matrix elements of HStark reduces to finding the matrix elements of
(1)

(1)

(1)

Dpq (ω)∗ Tq (D). Further, the matrix elements of Dpq (ω)∗ involve only the rotational wavefunction(and therefore only Wigner D matrix identities), so we need only consider the matrix elements of
D in the electronic and vibrational parts of the wavefunction, hΛ0 S 0 Σ0 , v 0 | D |ΛSΣ, vi. In the next
few sections, we will consider the matrix elements of HStark within and between electronic states,and do some example calculations that are useful for the HfF+ eEDM experiment.

2.2.2

Stark effect

In the JILA eEDM experiment and atomic physics experiments in general, the scale of laboratory DC electric fields that can be applied is . 500 kV/cm in record-setting circumstances,and . 20 kV/cm in more typical experiments. In these situations, we can usually neglect mixing
between electronic states, so that
ΛSΣ, v 0 D |ΛSΣ, vi ≡ v 0 Dk (R) |vi n,(2.13)

38
where n is a unit vector pointing between the nuclei, which lie along the z axis in the body-fixed
frame. The nuclear contribution to the molecule-frame dipole moment is explicitly a function of
R [Eq. (2.11)], while the electronic contribution depends parametrically on R through the nuclear
Coulomb potential [Eq. (2.3b)]. The total molecule-frame dipole moment Dk (R) can thus beexpanded in a power series in the internuclear distance,00

v Dk (R) |vi = Dk v v +



dDk
dQ



d2 Dk
dQ2

0

v Q |vi +0!
v 0 Q2 |vi + . . . .

(2.14)

0where Q ≡ R−Re is the displacement of the nuclei from their equilibrium positions in the electronic
state under consideration. For feasible laboratory electric fields, we can also neglect the mixing ofvibrational states and take v 0 = v. The first term in the expansion typically dominates, so for the
DC Stark effect we can simply take the electronic and vibrational matrix element to be Dk n.
We can now put our molecule-frame dipole matrix element together with some sphericaltensor algebra from Ref. [15] to evaluate the energy shifts in HfF+ states due to an applied DC or
quasi-DC electric field E whose components are known in the lab frame – an exceedingly common
and important situation in the JILA eEDM experiment! The results we need from Ref. [15] are
Eqs. 5.172, 5.174, and 5.186. Substituting these expressions, we obtain
ΛSΣ,v, I 0 J 0 Ω0 F 0 MF0 HStark |ΛSΣ, v, IJΩF MF i
=−

X

(1)

(1)
(ω)∗ |IJΩF MF i hΛSΣ, v| Tq(1) (D) |ΛSΣ, vi(−1)p T−p (E) I 0 J 0 Ω0 F 0 MF0 Dpq

p,q

=−

+1
X



F0



F 

p MF




0

J F Ip
F +J 0 +1+I 000× δI,I (−1)(2F + 1)(2F + 1)

F 0 J 0 1 



0
+1
X
1 Jp
0
0  J
(1)
×
(−1)J −Ω  (2J 0 + 1)(2J + 1) Tq (D).
q=−1−Ω0 q Ω

00 (1)(−1)p T−p (E) (−1)F −MF 
p=−1
−MF0

1

(2.15)

While this expression may look a little bit intimidating, it simply evaluates to a sum of three(1)

numbers multiplying the components of T−p (E). This expression is applicable to the scalar product39|F, MF i
hF 0 , MF0 |

1
1
2, −2

1 1
2, 2

3
3
2, −2

3
1
2, −2

3 1
2, 2

3 3
2, 2

1
1
2, −2

− E3z

E⊥
3

E√
⊥
2 6

− 6E√⊥2

0

1 1
2, 2

Ez
3

0

E√z
3 2

− 2E√⊥6

0

− E2z

0

0

E⊥
3

0

0

3 3
2, 2

0

E√
⊥
6 2
E√z
3 2
− 2E√⊥6

E√
⊥
2 3

3 1
2, 2

E⊥
3
E√
⊥
2 6
E√z
3 2
− 6E√⊥2

E√z
3 2
E√
⊥
6 2
E√
⊥
2 3
− E6z
E⊥
3

0

0

Ez
6
E√
⊥
2 3

E√
⊥
2 3
Ez
2

3
3
2, −2
3
1
2, −2

Table 2.1: Matrix elements hF 0 MF0 | HStark |F MF i for 3 ∆1 , J = 1, showing only the Ω = +1
components. Matrix elements for Ω = −1 are (−1)× the matrix elements shown, and matrix
elements off-diagonal in Ω are zero. Here E⊥ = Ex − iEy , and entries left blank are zero.

of any pair of vector operators, when the components of one of them is best expressed in thebody-fixed frame. It will be useful in Section 2.2.5 in evaluating the Zeeman Hamiltonian, which(1)

has a very similar form. Having found just above that Tq (D) = Dk δq,0 at the present levelof approximation, and using the fact that I = I 0 = 1/2 in HfF+ , Eq. (2.15) simplifies slightly.
After choosing a basis of states with quantum numbers |IJΩF MF i, these matrix elements can betabulated using Mathematica, Matlab, or tables of Wigner 3j and 6j symbols, which can be foundonline and in many textbooks.
The most important example of the DC Stark effect for the purposes of the HfF+ eEDM
experiment is the energy shifts of the 3 ∆1 , J = 1 states under the influence of the rotating electricbias field Erot . The matrix elements of HStark in the basis of these levels are tabulated in Table 2.1.
The calculated energy shifts due to the Stark effect are shown in Fig. 2.4.

2.2.3

Electronic transitions & the Franck-Condon principle

In the electric dipole approximation, the Stark Hamiltonian also governs radiative transitions
between states of HfF+ , due to stimulated absorption or emission, or spontaneous emission. In anelectronic transition that may also change rotational and vibrational quantum numbers, we must

40

100

50

0

-50

-100
0

20

40

60

80

Figure 2.4: Calculated Stark shifts in 3 ∆1 , J = 1, using experimentally measured parameters for thehyperfine splitting and the molecule-frame dipole moment. While in this figure it appears that thestretched states (red and purple) have linear shifts all the way to Erot = 0 indicating a permanentelectric dipole moment, their shifts are in fact quadratic at very low Erot due to Ω doubling, which
is discussed in Section 2.2.7

41
consider matrix elements of the Stark Hamiltonian between any two states of the molecule:Ψ0 HStark |Ψi =

X

0
(1)∗
0
0
(−1)p+1 E −p ψrot
Dpq
|ψrot i ψelec
ψvib
Dq |ψelec ψvib i(2.16)

p,q

The electric dipole selection rules for J, Ω, F , and MF are contained in the matrix element of the
(1)∗

0 |D
rotation matrix, hψrot
pq |ψrot i, which encodes angular momentum conservation. At this point,a standard approximation to apply is the Franck-Condon principle for the dipole matrix element.
Splitting D into its electronic and nuclear parts [Eq. (2.11)] and using the fact that the nuclear
dipole moment does not act on the electronic state, we have
0

ηη
0
0
0
ψelec
(R) |vi + ψelec
Dq |ψelec ψvib i = v 0 De,q
ψvib
ψelec

v 0 DN |vi .

(2.17)The orthonormality of electronic wavefunctions makes the second term zero for electronic transitions. We have evaluated the electronic part of the dipole operator between electronic wavefunctions
to give a function of R that can be expanded as a power series, similarly to the total moleculeframe dipole moment in Eq. (2.14), but this time between electronic states and without the nuclear
contribution. The Franck-Condon principle consists of keeping only the first term in this series,
which is independent of R,
0

ηη
0
0
Dq |ψelec ψvib i ≈ De,q
η 0 v 0 ηv .
ψvib
ψelec

(2.18)

The first factor is the transition electric dipole moment between electronic states, while the second
factor squared is defined as the Franck-Condon factor (FCF), representing the relative strength oftransitions that change the degree of vibrational excitation along with the electronic state. Becauseof the different shape of the internuclear potential in different electronic states of a molecule, thevibrational wavefunctions of different electronic states are not orthonormal, and have no strict
selection rules. However, due to the normalization of individual vibrational wavefunctions, theFCFs are normalized with respect to summation over v 0 or v. After applying the Franck-Condon
principle for electronic transitions, we have for the total Stark matrix element
Ψ0 HStark |Ψi =

X
0
(1)∗
η0 η
(−1)p+1 E −p ψrot
Dpq
|ψrot i η 0 v 0 ηv De,q.
p,q(2.19)

42
Despite the large number of electric dipole transitions driven by lasers in the HfF+ eEDM experiment, we rarely have occasion to use Eq. (2.19). However, Kevin Cossel’s thesis [28] goes overthe important examples: (a) the Raman spectrum of 3 ∆1 (v = 0, J = 1) and (b) evaluating the0ηηmolecule-frame transition dipole moment De,q
for the 3 Π0+ ← 1 Σ+ and 3 Π0+ ← 3 ∆1 transitions.

2.2.4

Vibrational decay & blackbody excitation

A particularly relevant type of radiative transition for the Generation 2 JILA eEDM experiment is the spontaneous decay of vibrational levels within a particular electronic state. In thiscase, we should take v 0 6= v in Eq. (2.13), and again consider the Taylor series expansion for Dk
from Eq. (2.14),
00

v Dk (R) |vi = Dk v v +



dDk
dQ



0

v Q |vi +
0

d2 Dk
dQ2

!
v 0 Q2 |vi + . . . .
0Within a particular electronic state, the |vi are orthogonal, so the first term is zero for vibrationaltransitions. In the approximation that the vibrational levels are harmonic oscillator eigenstates,
we can see that the term proportional to the relative internuclear coordinate Q = R − Re inducestransitions between states v 0 = v ± 1, while Q2 induces transitions between states v 0 = v ± 2, andso on. The vibrational states of diatomic molecules are harmonic to a first approximation, andtransitions v 0 = v ± 1 are typically strongest.
In the JILA eEDM experiment, we are typically concerned with excitation of vibrational
degrees of freedom by blackbody radiation, and contamination of the 3 ∆1 , v = 0, J = 1 state via
decay of 3 ∆1 , v > 0 states. As an example, we can estimate the rates of these two processes in HfF+
under typical experimental conditions. To calculate these rates, we need the standard relationshipsof Einstein A and B coefficients:
3 |D |2ωij
ij,
3π0 ~c3
π 2 c3Bij =3 Aij ,
~ωij

Aij =

(2.20)

where i and j denote any two states in question, and ωij = (Ej − Ei )/~. First, let’s evaluate theEinstein A coefficient for decay of excited 3 ∆1 vibrational states. To do so, we need the modulus

43
squared of the dipole moment operator,
2
2

|D| =

X X
p

0
ψrot

0
ψelec ψvib(1)
Dpq(ω) |ψrot i

Tq(1) (D) |ψelec ψvib i.

(2.21)q

We use Eq. 5.184 of Ref. [15], along with the fact that for rovibrational transitions, q = 0. We arealso not interested in the details of magnetic sublevel populations, so we should sum over the final
state magnetic sublevels M 0 . Doing all this, we obtain


J0

2

1 J

|D|2 = (2J + 1)(2J 0 + 1) −Ω 0 Ω


v 0 Dk |vi2

J0

2

X
1 J

 .
0
0
p,M
−M p M(2.22)

We can use Eq. (12) of Ref. [55] to simplify the sum over p and M 0 , which turns out to equal simply
1/(2J + 1). The Einstein A coefficient for purely vibrational decay from a state |vJi to state |v 0 J 0 i
comes out to be
Av0 J 0 vJ ≈

ωv30 v
2J, Ω; 1, 0 J 0 , Ωv 0 Dk |vi2 ,
3π0 ~c3(2.23)

where the second factor is a Clebsch-Gordan coefficient that has absorbed the 3j symbol involvingΩ and the factor (2J 0 + 1). Happily, and not surprisingly, if we also sum over final rotational states
J 0 , all rotational and Ω dependence drops out, and we are left only with hv 0 | Dk (R) |vi2 for the
dipole matrix element. Then using the Planck formula for the spectral energy density of blackbody
radiation 3 ρ(ωij ), we can evaluate the rate of excitation Rij = Bij ρ(ωij ),ωv30 v~ ωv0 v
20
Rv0 v ≈v Dk |vi exp −.
3π0 ~c3
kB T(2.24)For HfF+ in the 3 ∆1 state, the molecule-frame dipole moment has been evaluated as a functionof R by Petrov and coworkers in Ref. [99], and their results are shown in Fig. 2.5. By performing adDpolynomial fit to their calculation, we can extract the first-order coefficient dQk ≈ −1.2 e, where0in their convention, the internuclear axis is directed from Hf+ to F. With my calculated value of
h1| Q |0i = 0.068 a0 using Morse wavefunctions and 3 ∆1 spectroscopic parameters from Ref. [28], I
obtain an estimated 3 ∆1 , v = 1 lifetime of τ ≈ 173 ms, which compares reasonably well to Petrov
et al.’s estimate of τ ≈ 184 ms. From the squared Clebsch-Gordan coefficient in Eq. 2.23, we can3with units of energy per unit volume per unit frequency

44
-1.2
-1.4
-1.6
-1.8
-2
-2.2
3

3.2

3.4

3.6

3.8

4

Figure 2.5: Petrov et al.’s theoretically calculated molecule-frame dipole moment of HfF+ in the 3 ∆1electronic state (points), as a function of internuclear distance. Adapted from Ref. [99]. The redline is a quadratic fit to Petrov et al.’s calculation, while the black dashed line is the approximation
D|| = −eR, offset vertically by 2ea0 . For estimating vibrational lifetimes, this approximation gives
20% error, which is not too bad!

also estimate the rotational branching ratios for vibrational decay. Figure 2.6 shows an exampleof the rotational branching of decay from 3 ∆1 , v = 3, J = 1. Using Eq. (2.24), and T = 300 K,
we can estimate a blackbody excitation timescale of τBB ≈ 6.4 seconds for 3 ∆1 , v = 0, accounting
only for the v = 0 → v = 1 transition.

2.2.5

Zeeman effect

The Zeeman effect describes the interaction of an applied magnetic field with a magnetic
dipole moment, and is governed by the Hamiltonian
HZeeman = −µ · B
=−

X

(1)

(−1)p Tp(1) (µ) T−p (B)(2.25)

p

In HfF+ and in molecules in general, there are multiple contributions to the total magnetic momentof the molecule, and it is easiest to treat each of them separately. These contributions are, in orderof their typical size, electronic (spin and orbital), nuclear spin, and rotational magnetic moments:µ = µe + µn + µr .

45

1

0.5

0
0.3
0.2
0.1
0
0.2

0.1

0
0.4

0.2

0
0

100

200

300

400

500

600

700

Figure 2.6: Theoretically calculated vibrational decay of 3 ∆1 , v = 3, J = 1, leading to populationof rotational levels up to J = 4 in v = 0. This phenomenon leads to an undesirable backgroundsignal in 3 ∆1 , v = 0, J = 1 that hurts our eEDM sensitivity, as discussed in Chapter 5. Populations
Pi are expressed as a fraction of the total molecular population.

46
Due to the strong coupling of the total electronic angular momentum to the internuclear axis –motivating our use of Hund’s case (a) basis functions – the total electronic magnetic moment µe is
most easily evaluated in the body-fixed frame, similarly to the electronic contribution to the electric
dipole moment in Section 2.2.1. The evaluation of the laboratory frame components of µe follows
precisely the same form as Eq. (2.15), but with the molecule-frame electronic matrix elementhΛSΣ| µe |ΛSΣi = Gk µB Ω n,

(2.26)(1)or equivalently Tq (µe ) = Gk µB Ω δq,0 . In principle, Gk might also be calculated as a functionof internuclear distance and expanded in a power series, as we did for Dk in Eq. (2.14). It seems
likely that Gk would have a much weaker dependence on internuclear distance than Dk , and to myknowledge this is not typically done. Thus the matrix elements of the electronic part of the Zeeman
Hamiltonian can be obtained from Eq. (2.15) with the replacements D → µe and E → B. For the3 ∆ (J = 1) state, in the same basis as Table 2.1, the electronic Zeeman Hamiltonian is given by1

Eq. (2.27).


− B3z



 B⊥
 3


 B√⊥
 2 6
HZ,e = −Gk µB 

 B√z
 3 2


 − B√⊥
 6 2

0

B⊥
3

B√⊥
2 6

B
√z
3 2

− 6B√⊥2

Bz
3

0

B√⊥
6 2

B
√z
3 2

0

− B2z

B√⊥
2 3

0

B√⊥
6 2

B√⊥
2 3

− B6z

B⊥
3

B
√z
3 2

0

B⊥
3

Bz
6

− 2B√⊥6

0

0

B√⊥
2 3


0



− 2B√⊥6 



0 


0 



B√⊥

2 3 


(2.27)Bz
2

The nuclear part of the magnetic moment µn ≡ gN µN I (where µN ≈ 762 Hz/G is thenuclear magneton) arises from the 19 F nuclear spin I, which we have chosen to couple to the total
electronic plus rotational angular momentum J in the laboratory frame to form F . 4 The matrix
elements of I are easy to work out by application of the Wigner-Eckart theorem (B&C Eq. 5.172),the formula for “matrix elements of a single operator in a coupled scheme” (B&C Eq. 5.175), and4

I have not considered how I might be coupled to J in the molecule frame, but I imagine it is possible. I believethat in either case, all observables should come out the same, since angular momentum coupling is just a choice ofbasis.

47
the reduced matrix element of a first-rank tensor between its own eigenstates (B&C Eq. 5.179).
The result is




F0

1

F 

p MF

0
0 
η 0 Tp(1) (µn ) |ηi = (−1)F −MF 
−MF0

0

0

× δJ 0 J (−1)F +J +1+I×p
(2F 0 + 1)(2F + 1)



I

F


J 0


F 0

I


1



(2.28)p
I(I + 1)(2I + 1) gN µN .

Note that here we have made no use of Wigner D matrices, but only identities related to decouplingof angular momenta in the language of spherical tensors. For the 3 ∆1 (J = 1) state, the nuclear
contribution to the Zeeman Hamiltonian is given by Eq. (2.29) below.

√
− B6z



 B⊥
 6


 B√⊥

6
HZ,n = gN µN 
 √
 2Bz
 3


 − B√⊥
 3 2

0

B⊥
6

B
√⊥
6

2Bz
3

− 3B√⊥2
√
2Bz
3

Bz
6

0

B√⊥
3 2

0

Bz
2

− 2B√⊥3

0

B√⊥
3 2
√
2Bz
3

− 2B√⊥3

Bz
6

− B3⊥

0

− B3⊥

− B6z

− B√⊥600− 2B√⊥3


0



B
⊥ 
√
− 6 


0 


0 


B√⊥ 
−2 3 
− B2z(2.29)

Within the eight sublevels of the F = 3/2 hyperfine manifold, the electronic and nuclear
contributions to the Zeeman Hamiltonian have the same form. We can therefore define an effective magnetic g-factor gF for this manifold of states only, and parameterize the total Zeeman
Hamiltonian asHZ,F = −gF µB B · F = −gF µB 




√



3B⊥
2

0

− B2z

B⊥

0

∗
B⊥

0

0

Bz
2
√ ∗
3B⊥
2

3Bz
 − 2
 √ ∗
 3B⊥
 2


0



0 

√

3B⊥ 

2


(2.30)

3Bz
2

When performing perturbation theory within F = 3/2 only, we can use this parameterization of
the Zeeman matrix elements.

48
In HfF+ , we have typically neglected the rotational contribution to the Zeeman effect, because
we expect it to be small in the ground rotational state of 3 ∆1 . In fact, I have not evaluated thematrix elements of this operator, so I will just sketch a procedure here. A definition of the rotational
magnetic moment consistent with our previous definitions, and which will work in Hund’s case (a)and (c), isµr = gr µN (J − Ja ).

(2.31)The only conserved quantum number of the total electronic angular momentum operator Ja is its
component along the internuclear axis, Ja · n = Ω. The perpendicular components of Ja act tochange Ω and therefore the electronic state, so for example they connect 3 ∆1 ↔ 3 Π0± . Thus to afirst approximation, we can replace Ja → Ω n in Eq. (2.31). Upon doing this, we can redefine theelectronic and rotational magnetic moments asµe ≡ (Gk µB − gr µN ) Ω n,
(2.32)µJ ≡ gr µN J .
We can absorb the contribution gr µN into the definition of Gk , and the matrix elements of µe areunchanged. We can then evaluate the matrix elements of our newly defined µJ ,


0
1 F 
0
0  F
η 0 Tp(1) (µJ ) |ηi = (−1)F −MF −MF0 p MF J F Ip
F +J+1+I0
× (−1)(2F + 1)(2F + 1)

F 0 J 0 1×(2.33)p
J(J + 1)(2J + 1) gr µN .

This contribution to the Zeeman Hamiltonian was not accounted for in the Generation 1 JILA
eEDM experiment. It is my impression that since we typically use very weak magnetic fields inthe HfF+ experiment, its effects can simply be absorbed into a redefinition of gF . The validity ofthis assumption has not been tested, however we can be slightly reassured by considering the sizeof the rotational Zeeman effect: From Eq. 7.226 of Ref. [15], the contribution of the nuclei to gr is
grn =m(Z1 M22 + Z2 M12 )≈ 0.00026,
M1 M2 (M1 + M2 )(2.34)

49
which is already very small. The nuclear and electronic contributions to the total g-factor should
partially cancel due to their opposite charge, so this can be taken as an upper bound on gr . Even
before any cancellations, the rotational Zeeman energy scale is grn µN ≈ 0.2 Hz/Gauss. Since therotational Zeeman effect will be indistinguishable from the electronic and nuclear spin contributions
in our measurement of the energy difference between MF = ±3/2 states, and will be divided down
by at least the Stark splitting between MF states when considering perturbative effects, I think itshould be completely negligible in considering systematic effects in the JILA eEDM experiment.
A possible systematic effect could arise from an erroneous applied doublet-odd correction for a
non-reversing magnetic field effect, where I might expect the effect to be of order (0.2 Hz/G) ×
δg/g × (0.002 G) ≈ 10−7 Hz.

2.2.6

Hyperfine structure

Due to the presence of the Fluorine nucleus in HfF+ with nuclear spin I = 1/2, we mustaccount for nuclear magnetic hyperfine structure, which gives rise to the coupling discussed in Section 2.1.6 and causes the good total angular momentum quantum numbers to be F and MF forsmall external electric and magnetic fields. Because the valence electrons in HfF+ are relativistic,
their spin and motion are highly coupled, and the positron amplitude of the electrons’ Dirac wavefunction cannot be neglected. To obtain accurate theoretical predictions, theorists performing ab
initio calculations use Dirac wavefunctions for each electron Refs. [45, 46, 116], and evaluate thematrix element
µN
Ak =
ΩI

*
ψe

X  αi × riN 
i

3
riN

+
· n ψe,(2.35)

where riN is the displacement of electron i from the Fluorine nucleus, µN is the Fluorine magnetic
moment, and α is a vector of 4 × 4 Dirac matrices [34]. For our purposes, we can simply use thiselectronic matrix element as a constant in an effective Hamiltonian, in the same way that we do
Gk and Dk . We use as an effective Hamiltonian
Hhf = Ak (I · n)(Ja · n),(2.36)

50
where Ja ·n ≡ Ω. The projection of I onto the internuclear axis can be evaluated using the projection
theorem [107], which is a corollary of the Wigner-Eckart theorem. The projection theorem is
J 0 M 0 Ω0 Vq |JM Ωi =hV · J i
J 0 M 0 Ω0 Jq |JM Ωi ,J(J + 1)(2.37)

where V is any vector operator. We can then set V → I, and use the defining property of thetotal angular momentum F that I · J = 12 (F 2 − J 2 − I 2 ). We are looking for I · n ≡ Iq=0 , but weknow that Jq=0 ≡ Ω. So we simply plug these in and find that the matrix elements of our effective
hyperfine Hamiltonian areψ 0 Hhf |ψi =F (F + 1) − I(I + 1) − J(J + 1)
Ak Ω2 δη,η0 ,2J(J + 1)

which is independent of MF and of the sign of Ω. Here η represents all quantum numbers; thehyperfine Hamiltonian is diagonal in the F , MF basis. In the 3 ∆1 state of HfF+ , we find that theF = 1/2 states have a hyperfine energy of −Ak /2, while the F = 3/2 states have energy Ak /4,
for a total separation of Ehf = −3Ak /4 ≈ 46 MHz. This energy splitting is visible in Fig. 2.4 atErot = 0. Using microwave spectroscopy, we have also confirmed this formula in 3 ∆1 , J = 2, where
the hyperfine splitting is 1/3 as large as in J = 1, or ≈ 15.5 MHz.

2.2.7

Omega doubling

In a heteronuclear diatomic molecule with nonzero total angular momentum, there are two
possible projections of the total angular momentum along the internuclear axis, given by ±Ω.
Because these states correspond to exactly the same spatial distribution of electronic charge aroundthe nuclei, one might expect the energies of these two configurations to be exactly degenerate –however, this is not the case. Ω doubling results from the coupling of the electronic and rotational
angular momentum, and arises primarily from the electronically off-diagonal part of the operator
J · Ja that we discussed in the context of rotational states in Section 2.1.4. Kevin Cossel’s thesis[28] discusses in detail the perturbations giving rise to Ω doubling, so I will not go into a detailed
derivation here.

51

-13
-14
-15
-16
-17
-18
0

0.5

1

1.5

2

Figure 2.7: Theoretically calculated Stark effect in 3 ∆1 , J = 1, F = 3/2 for small values of theelectric bias field, showing Ω doubling – essentially a zoom in of Fig. 2.4 to the region near Erot = 0.
As judged by the linearity of the Stark shift as a function of Erot , the F = 3/2 states are essentiallyfully polarized at Erot ≈ 1 V/cm. Here the effects of the rotating frame are not included.

For our purposes, we will simply use an effective Hamiltonian description within a single
electronic state (See e.g. Ref. [15], Section 7.2). The contribution from Ω doubling in the 3 ∆1 state
has the form
ψ 0 HΩ |ψi =~ωef
J(J + 1) δΩ0 ,−Ω δη0 ,η ,4(2.38)

where η represents all quantum numbers other than Ω. The result of this effective Hamiltonian isthat in the absence of an applied electric field, the eigenstates of the molecule are superpositions∼ |Ωi ± |−Ωi separated by an energy splitting ωef ≈ 2π × 700 kHz in J = 1. Figure 2.7 shows the
Stark effect in 3 ∆1 , J = 1 for small values of Erot , where the Ω-doubling causes the eigen-energies
to be quadratic near Erot = 0.

2.2.8

Effect of Erot rotation

So far, we have only discussed the energy levels of HfF+ in static electric and magnetic fields,
however in the eEDM experiment, Erot rotates with a frequency frot between 150 kHz to 400 kHz.
Our eEDM-sensitive Ramsey spectroscopy is performed between the stretched MF states referredto the rotating electric field. However, the rotating frame in which Erot lies along the ẑ axis is noninertial, and so we should expect fictitious forces – or more specifically, fictitious torques – to act on

52
the total angular momentum of a HfF+ molecule in this frame. The combination of the fictitious
torque on F and Ω-doubling in the 3 ∆1 state is a high-order coupling between MF = ±3/2 statesin the same Stark doublet, which significantly affects the way we perform an eEDM measurement,
and is even helpful for performing π/2 pulses. This effect was discussed generally in Ref. [88], but
an explicit calculation in states with nuclear spin was not given. Here I will explicitly calculate thiseffect with particular application to the HfF+ eEDM experiment.
Our applied electric and magnetic fields are most easily parameterized in the laboratory
frame, where the geometric center of the ion trap defines the origin of coordinates. In this frame,
the rotating electric field ishi
E rot = Erot X̂ cos (ωrot t) + Ŷ sin (ωrot t) ,(2.39)To transform the Cartesian vector E rot into a frame where E rot = Erot ẑ, we need to apply a rotation
operatorcos θ cos φ cos θ sin φ − sin θ
Rcart (θ, φ) = RY (−θ) RZ (−φ) =  − sin φ
,
cos
φ
0




cos φ sin θ sin θ sin φ cos θ(2.40)

where we can then take θ = π/2 and φ = ωrot t. In much the same way, we need to apply arotation operator to our Hamiltonian and our quantum states in order to express them in therotating frame. The rotation operator for an angle α about some vector v in quantum mechanics
is Rv (α) = exp(−iFv α) where Fv is the total angular momentum operator about v, so by analogy
with Eq. (2.40), the appropriate rotation operator isV = exp (iFY θ) exp (iFZ φ) .

(2.41)

If we call the Hamiltonian before the transformation H 0 (in the lab frame), then the Hamiltonianin the rotating frame is
H = V H0 V † + i

dV †
V ,
dt

(2.42)

which can be derived by requiring that the transformed state vector obey the Schrödinger equation
with a transformed Hamiltonian. We call the second term on the right hand side Hrot ; it represents

53
the fictitious torque experienced by the total angular momentum of HfF+ molecules in the rotating
frame. In terms of angular momentum operators, its value isHrot = −ωrot · F = −ωrot FZ = ωrot Fx ,(2.43)

where I have used the transformation of Cartesian coordinates in Eq. (2.40) to find that FZ = −Fx ,where uppercase coordinates are in the lab frame, and lowercase coordinates are in the rotating
frame. Combined with Ω-doubling, it is this component of the Hamiltonian that gives rise to
the coupling between MF = ±3/2 states in the rotating frame. We can derive this couplingby performing perturbation theory to fourth order, with the large and small components of the
Hamiltonian
H0 = Hhf + HStark ,(2.44)
H1 = Hrot + HΩ .
The large component H0 can be exactly diagonalized, so we can apply fourth order perturbation
theory, defining the result as the off-diagonal element of an effective Hamiltonian that couples states|ai and |bi (or |ci and |di, with labels defined in Fig. 2.8). For example, the perturbation theory

Figure 2.8: Quantum numbers and shorthand alphabetical labels for magnetic and hyperfine sublevels of 3 ∆1 , J = 1. The quantum number MF is good for all values of Erot , but F ceases to be agood quantum number when Dk Erot ≈ Ak .

54
formula applied to the upper doublet is~∆uha| Hrot |ei he| Hrot |hi hh| HΩ |f i hf | Hrot |bi ha| Hrot |ei he| HΩ |gi hg| Hrot |f i hf | Hrot |bi≡+2(Ea − Ee )(Ea − Eh )(Ea − Ef )(Ea − Ee )(Ea − Eg )(Ea − Ef )+ha| HΩ |ci hc| Hrot |gi hg| Hrot |f i hf | Hrot |bi ha| Hrot |ei he| Hrot |hi hh| Hrot |di hd| HΩ |bi+(Ea − Ec )(Ea − Eg )(Ea − Ef )(Ea − Ee )(Ea − Eh )(Ea − Ed )(2.45)

We typically express ∆ as an average part ∆ and a doublet-odd part ∆D . The result of applying
Eq. (2.45) and averaging and differencing the results for the Stark doublets is
!




2
2 2
3ωef
~ωrot 3 18Ak − 19Dk Erot~ωrot 3
∆=−
,
≈
−27ω
ef
2
2
Dk Erot
Dk Erot
A2k − Dk2 Erot
!
!
2
3ω3
9A2k − 8Dk2 Erot
Dk Erot
3ω
~
ef
rot
≈∆×
.
∆D =
2
2
2
2
2
2
2Ak
Dk Erot Ak
Ak − Dk Erot

(2.46)(2.47)

Note that these results are exact with respect to the size of Dk Erot /Ak , but not exact with respect
to ωef /Dk Erot . The approximate expressions given are for Dk Erot  Ak . In addition, the prefactor given for ∆ in Ref. [78] is incorrect by a factor of 2π. Thus we should expect deviation of
perturbation theory from exact diagonalization at small values of Erot . Also, note that ∆ here hasunits of rad/s, in contrast to Ref. [16] where it has units of Hz. Figure 2.9 shows the numericaland perturbation theory results for ∆u and ∆l as a function of Erot . At the value of Dk Erot where
the F = 3/2, MF Ω = −3/2 and F = 1/2, MF Ω = 1/2 hyperfine states cross, the value of ∆udiverges. While there is deviation between the perturbation theory and numerical results at low
Erot , it is not visible on the scale of the figure. Note that due to the logarithmic scale of the plot, theapparent divergence near Erot ≈ 31 V/cm is only a zero crossing and is not real, but the divergenceat ∼ 35 V/cm is genuine.

2.2.9

eEDM shift

The effect of a P,T-violating electron EDM in the 3 ∆1 state of HfF+ is to mix states of opposite
parity but the same MF with a constant matrix element Hd = −de (S · E int )/S. In the basis ofHund’s case (a) states and in the presence of Erot  ~ωef , we can essentially treat this Hamiltonian
at lowest order in perturbation theory by taking S · E int ≈ (S · n)(E int · n) ≡ Σ Eeff = −Ω Eeff ,

55

10 4
upper
lower
upper theory
lower theory

10 2

10 0

10 -2

0

20

40

60

80

Figure 2.9: Energy difference between states of opposite mF , but the same value of mF Ω – i.e.,
states in the same Stark Doublet, versus Erot , for Brot = 0. This is the size of the avoided crossing|∆| in units of Hz. Here ωrot /(2π) = 375 kHz.

56
where the last equality is only true in the 3 ∆1 state. We typically take the internuclear axis n topoint from the Fluorine to the Hafnium nucleus, so that Dk ≈ +1.8 MHz/(V/cm). As a result, theinternal electric field, which points from Hf to F, points against the internuclear axis. In Ref. [16],we took the eEDM energy shift to be
F 0 MF0 Ω0 Hd |F MF Ωi = −de |Eeff |Ω δF 0 F δMF0 MF δΩ0 Ω(2.48)

to make all of these signs explicit and hopefully prevent confusion.

2.2.10Differential g-factor

Conducting an eEDM measurement in a molecular state of nonzero Ω was first proposed byDave DeMille for PbO in its metastable 3 Σ state Ref. [30], motivated by the desire to reverse theelectric field experienced by a valence electron without reversing a laboratory electric field, which
suppresses major systematics. John Bohn and Ed Meyer expanded on this idea to suggest the 3 ∆1state, which not only has the Ω-doublet feature, but also has a very small magnetic g-factor due tothe cancellation of spin and orbital magnetic moments [87]. These proposals led to the use of themetastable 3 ∆1 states of ThO and HfF+ at Harvard and JILA, respectively, in what are presently
the most sensitive eEDM searches.
While the magnetic g-factors of the Ω doublets in the 3 ∆1 state of HfF+ are very similar,
they are not exactly equal, and their difference leads to some of the largest systematic effects that
we are aware of. We parameterize their difference with the differential g-factor,
δgF ≡gFu − gFl,
2

u/l

where gF are the g-factors of MF Ω < 0 and MF Ω > 0 states, respectively. We can obtain a good
approximation for δgF using perturbation theory on the effective Hamiltonian derived in the earlier
sections of this chapter, accounting for two main contributions: the first is from Stark mixing ofrotational levels J = 1 and J = 2 in 3 ∆1 , and the second is from a hyperfine-differential mixing ofneighboring MF sublevels within 3 ∆1 , J = 1, caused by the rotation of Erot . Here I will go throughthe derivations of each.

57
The contribution to δgF arising from Stark mixing of J = 1 and J = 2 can be derived using
2nd order perturbation theory, where separate our Hamiltonian into a dominant term including
tumbling and J-diagonal Stark contributions, and a perturbation including J-off-diagonal Stark
and Zeeman contributions.
H0 = Htum + Hhf + HStark δJ 0 J ,(2.49)
H1 = (HStark + HZeeman ) (1 − δJ 0 J ).
From standard 2nd order perturbation theory, the energy of a state |ii = |J = 1, Ω, F, MF i will
receive a correction
(2)

∆Ei

=

X hi| H1 |ni hn| H1 |ii.
hi| H0 |ii − hn| H0 |ni(2.50)

n6=i

Because Erot acts along the laboratory z axis (in a rotating reference frame), it has only a p = 0
spherical component, and does not couple states of different MF . In addition, Erot can only couple
states of the same Ω, because the molecule-frame dipole moment Dk has only a q = 0 sphericalcomponent. Therefore for a given |J = 1, F, MF , Ωi state, the sum in Eq. (2.50) contains only two
terms. By substituting into Eq. (2.15) and a similar expression for the Zeeman Hamiltonian as
described in Section 2.2.5, we obtain the relevant matrix elements

 √

3 3
3
3 3
2, , , ±1 H1 1, , , ±1 =Dk Erot ∓ Gk µB Brot
2 2
2 2
10
√



3
5 3
3 3
Dk Erot ∓ Gk µB Brot
2, , , ±1 H1 1, , , ±1 = −
2 2
2 2
5

(2.51)

Substituting these matrix elements into the perturbation theory expression above, we find that the
Stark contribution to the differential g-factor is(2)

δgFStark ≈(2)∆E3/2,−1 − E3/2,+1
8B gF µB Brot MF

Dk Erot Gk
3Dk Erot==
20B gF20Bg N µN1−
3gF µB(2.52)

where B = 0.29889(2) cm−1 is the 3 ∆1 rotational constant, and I have substituted for the moleculeframe electronic g-factor Gk = 3gF − gN µN /µB in terms of the experimentally measurable value
gF and the Fluorine magnetic moment gN .
The contribution to δgF arising from the rotation of Erot can be derived from second order
perturbation theory, with Hrot as the perturbing Hamiltonian. The MF = ±3/2 states gain some

58
admixture of their neighboring MF = 1/2 states due to the rotating frame, which applies a slightchange to their effective g-factor. The dominant change in the g-factor is common to both doublets,
but the presence of hyperfine structure and the subsequent quadratic Stark shifts of the F = 3/2,
MF = ±1/2 states causes the degree of mixing of the upper doublet to be less than that of thelower doublet. The energy denominator Ea − Ee from Eq. (2.50) applied to state |ai [Fig. (2.8)]
can be expanded for small Brot , and the resulting energy shift (for example in state |ai) contains aterm∆Ea(2) ⊂2 g µ B3~2 ωrot
N N rot.
Dk Erot AkThis energy shift is common to all four of the states a-d, but with a relative sign that depends onΩ. The g-factor of state |a0 i thus contains a contribution
δgFrot ≈22gN µN ~2 ωrot
gF µB Dk Erot Ak(2.53)

Note that this equation is slightly different from the result reported in Ref. [16]. That result contains
an error that has been fixed here, and the result here has been verified by direct diagonalization of
the full Hamiltonian. Thus to summarize, the total differential g-factor is2
3Dk Erot
2gN µN ~2 ωrot
gN µN
δgF
≈
1−
+
gF
gF µB Dk Erot Ak
20B
3gF µB(2.54)

Figure 2.10 shows the upper and lower magnetic g-factors, as well as their difference, as a functionof Erot . The perturbation theory estimates show good agreement with a numerical diagonalization
at Erot & 10 V/cm, however the mean value and difference do not agree well at low Erot . This isdue to higher-order coupling of MF levels by Hrot , which cannot be accounted for by second-order
perturbation theory. It is worth noting that in exactly the range of Erot where the first term in
Eq. (2.54) is supposed to dominate, namely at small values of Erot , the mean value of gF fromperturbation theory is in complete disagreement with numerical diagonalization. As a result, thefirst term above is not particularly useful, and we are generally better off just relying on numerical
diagonalization. It is also important to note that the differential g-factor given in Fig. 2.10 is not
equivalent to the frequency channel f D due to the effects of ∆ and ∆D . These effects are discussed
in Chapter 3.

59

3

-3
upper numerics
lower numerics
upper pert thy
lower pert thy

-3.02
-3.04

1

-3.06

0

-3.08

-1

-3.1
0

20

40

(a)

60

numerics
pert. thy.

2

80

-2
0

20

40

60

80

(b)

Figure 2.10: Magnetic g-factor of stretched states as a function of Erot , at frot = 375 kHz. The
drop of the absolute g-factors to zero at low Erot results from mixing of MF levels due to Hrot atorder > 2 in perturbation theory. Note that we have not derived the absolute value of gF fromtheory, but taken it from experiment.

Chapter 3

Generation 1 eEDM Measurement

3.1

Summary of the eEDM result
This section is a reproduction of the main text of Ref. [16].
A search for a nonzero permanent electric dipole moment of the electron (eEDM, de ) con-stitutes a nearly background-free test for physics beyond the Standard Model (SM), since the SM
predicts |de | . 10−38 e cm [102], while the natural scale of de in many proposed SM extensions is10−27 to 10−30 e cm [37]. Present experimental techniques now constrain these theories [6]; hence,
there have been many recent efforts to measure an eEDM [6, 36, 60, 64, 79, 104, 137].
The most precise eEDM measurements to date were performed using beams of neutral atoms
or molecules [6, 64, 104]. These experiments benefited from excellent statistical sensitivity provided
by a high flux of neutral particles, and decades of past work have produced a thorough understanding of their common sources of systematic error. Nonetheless, a crucial systematics check can beprovided by independent measurements conducted using different physical systems and experimental techniques. Moreover, techniques that allow longer interrogation times offer significant potential
for sensitivity improvements in eEDM searches and other tests of fundamental physics [117].
In this Letter, we report on a precision measurement of the eEDM using molecular ions confined in a radio frequency (RF) trap, applying the methods proposed in Ref. [78] and demonstrated
in Ref. [82]. We perform an electron spin precession experiment on 180 Hf 19 F+ molecules in theirmetastable 3 ∆1 electronic state, and extract the relativistically enhanced eEDM-induced energy
shift ∼ 2de Eeff between stretched Zeeman sublevels, where Eeff ≈ 23 GV/cm in HfF+ [44, 87, 98].

61
In addition to leveraging the high eEDM sensitivity and systematic error rejection intrinsic to a
3∆

1 state in a heavy polar molecule [36, 87], we use a unique experimental approach that is robust

against sources of systematic error common to other methods. The 2.1(1) s lifetime of the 3 ∆1 statein HfF+ [91] and our use of an RF trap allow us to attain spin precession times in excess of 700 ms– nearly three orders of magnitude longer than in contemporary neutral beam experiments. This
exceptionally long interrogation time allows us to obtain high eEDM sensitivity despite our lower
count rate. In addition, performing an experiment on trapped particles permits the measurement ofspin precession fringes at arbitrary free-evolution times, making our experiment relatively immune
to systematic errors associated with imperfectly characterized state preparation.
Our apparatus and experimental sequence, shown schematically in Fig. 3.1, have been described previously [29, 78, 82, 83, 84, 91]. We produce HfF by ablation of Hf metal into a pulsed
supersonic expansion of Ar and SF6 . The neutral beam enters the RF trap, where HfF is ionizedwith pulsed UV lasers to form HfF+ in its ground vibronic state [83, 84]. The ions are stopped bya pulsed voltage on the radial trap electrodes, then confined by DC and RF electric quadrupole
gradients (with RF frequency frf = 50 kHz). We next adiabatically turn on a spatially uniform
electric bias field Erot ≈ 24 V/cm that rotates in the radial plane of the ion trap with frequency
frot ≈ 250 kHz, causing the ions to undergo circular motion with radius rrot ≈ 0.5 mm. A pair of
0(2Z − X − Y )magnet coils aligned with the Z axis produce an axial magnetic gradient B = Baxgrad0| ≈ 40 mG/cm, which in the rotating frame of the ions creates a magnetic bias field
where |Baxgrad
00Brot ≡ |hB · E rot /Erot i| ' |Baxgrad
rrot | that is parallel (antiparallel) to Erot if Baxgrad> 0 ( < 0)[78, 82].
Our state preparation consists of population transfer to the eEDM-sensitive 3 ∆1 state andselective depletion of magnetic sublevels to produce a pure spin state [Fig. 3.1(b-c)]. Two cw lasers
co-propagating along the Ẑ axis drive a Raman transition through a 3 Π0+ intermediate state,transferring ∼ 40% of the ground state population to the 3 ∆1 , J = 1, F = 3/2 state. Figure 3.2(a)shows the structure of this state in a rotating frame defined by E rot ≡ Erot ẑ. It consists of four
Stark doublets (pairs of magnetic sublevels) separated by dmf Erot /3h ≈ 14 MHz, where dmf is the

62

(c)

HfF

fer

Π0+

HfF

tion

HfF

Ω=0

le
dep

trap
electrodes

3

–

Σ0+

trans

Hf+

3

EnergyTOF MCP

HfF+

+

3

X1Σ+
ionizationpho
CC sphorD

X2Δ3/2stop ionskick to MCP

populationtransfer~ 100 ms

π/2

π/2

depletionRF
trap

depletion

ionization

ablationrotating E field

ttrap = 0

Δ1

Ω = 3/2

imagingMCP(b)dissociationntio
cia onosti
dis eple ation
d
iz
ion

(a)

dissociation~ 850 ms

ttrap
~ 950 ms

Figure 3.1: (a) Apparatus schematic, (b) experimental timing, and (c) relevant energy levels (notto scale) for an eEDM measurement using trapped ions. HfF is photoionized (yellow) to form HfF+ .
A rotating electric bias field Erot (blue) polarizes the molecules, and transfer (red) and depletion(orange) lasers perform state preparation. The π/2 pulses are performed by modulating Erot . Spinstate populations are detected by depletion followed by photodissociation (purple) and counting
the resulting Hf+ ions on a microchannel plate (MCP).

63
3∆

1 molecule-frame dipole moment and h is Planck’s constant.

The population transfer process

produces an incoherent mixture of MF = ± 3/2 states in the upper or lower doublet, dependingon the detuning of the second transfer laser. Selective depletion is then performed by a circularly
polarized laser resonant with the Q(1) line of a 3 Σ−
← 3 ∆1 transition. The depletion laser0+
is strobed synchronously with the rotating electric field so that its wavevector is either parallelor antiparallel to E rot , thus driving a σ ± transition to an F 0 = 3/2 manifold and leaving one
MF = ± 3/2 level populated in the 3 ∆1 state.
Following strobed depletion, we perform a π/2 pulse to prepare an equal superposition of
MF = ± 3/2 states. This is accomplished by reducing Erot for a brief interval, which increases
a rotation-induced coupling ∆u/l between MF = ± 3/2 states [Fig. 3.2(b)] and causes a purespin state to evolve into an equal superposition in ∼ 1 ms [78, 82]. We return Erot to its nominal
value and allow the phase of the superposition to evolve for a variable precession time, then apply
a second π/2 pulse to map the relative phase of the superposition onto a population difference
between MF = ± 3/2 states. A second set of strobed laser pulses again depletes all but one
MF = ± 3/2 level. To selectively detect the remaining population in the 3 ∆1 , J = 1 state, we
resonantly photodissociate HfF+ using pulsed UV lasers [91]. We eject all ions from the trap, andcount both Hf+ and the temporally resolved HfF+ using a microchannel plate (MCP) detector.
We interleave experimental trials where the two sets of strobed depletion pulses have the sameor opposite phase with respect to Erot in order to alternately prepare and detect population in the
MF = ± 3/2 states. Denoting by NA (NB ) the measured population when the depletion phases are
the same (opposite), we form the asymmetry A = (NA − NB )/(NA + NB ), which normalizes drifts
in absolute 3 ∆1 population. The asymmetry forms an interference fringe that is well-approximatedby a sinusoidal function of precession time t,
A(t) ' − Ce−γt cos(2πf t + φ) + O,

(3.1)

with frequency f proportional to the energy difference between the MF = ± 3/2 states, as shownin Fig. 3.2(c). The initial contrast C, initial phase φ, offset O, and decoherence rate γ parametrize

64

3

F–

Hf2+

Ω = +1mF = –3/2

upper doublet
–1/2

+1/2

Ω = –1+3/2~14 MHz

Hf2+

F–

(b)

Δ1, J = 1, F = 3/2

Ω = –1lower doublet

Energy (arb)(a)Δu ≈ 0.5 Hz

fu ≈ 23 Hz
Δl ≈ 1 HzΩ = +1(c)
Asymmetry

0.5
0

–0.5
0

20

40
680
700
Spin precession time t (ms)

720

Figure 3.2: Electron spin resonance spectroscopy in HfF+ . (a) Level structure of the eEDMsensitive 3 ∆1 , F = 3/2 state in an electric bias field Erot . (b) Energies of |MF | = 3/2 states asa function of magnetic bias field Brot (not to scale), showing an avoided crossing at Brot = 0 due
to a rotation-induced coupling ∆u/l . (c) Interference fringe with interrogation time ∼ 700 ms anddecoherence rate γ = 0.3(2) s−1 .

65imperfect state preparation and the loss of coherence. We perform nonlinear least squares fittingof the asymmetry with Eq. (3.1), using C, γ, f , φ, and O as fit parameters. Standard errors δC,δγ, δf , δφ, and δO are estimated from the Jacobian of the fit function at the optimum parametervalues. The precession frequency contains the eEDM signal, while the other fit parameters are used
to diagnose experimental imperfections and sources of systematic error.
To isolate an eEDM-dependent frequency shift and diagnose systematic errors, we form
data “channels”: components of a measurement that have a particular parity under a set ofchosen “switches” – experimental parameters that are modulated between opposite values on ashort timescale 1 . Our switches are the sign of the magnetic bias field B̃ = sgn(hB · E rot i),
the populated Stark doublet D̃ = − sgn(MF Ω), and the sense of the electric bias field rotationR̃ = −sgn(ωrot · Ẑ). We repeat our spin precession measurement in each of the eight unique “switch
states” S̃ = (B̃, D̃, R̃) to form a “block,” and form channels X s with parities s ⊂ {B, D, R} fromlinear combinations of the eight measurements X(S̃), where X ∈ {C, γ, f, φ, O}. We estimate thestandard error δX by propagating the error estimates δX(S̃) resulting from the nonlinear leastsquares fit of Eq. (3.1).
If higher order effects are neglected, the measured spin precession frequency is dominated by
the Zeeman shift between populated magnetic sublevels, and includes a BD-odd contribution froman eEDM:
hf (S̃) ≈ −3gF µB B̃Brot + 2D̃de |Eeff |(3.2)= 3|gF |µB Brot − 2B̃ D̃ sgn(gF ) de |Eeff |.
An eEDM signal thus appears as the lowest-order contribution to the f BD frequency channel, while
any non-ideal contributions to f BD constitute sources of systematic error. The seven non-eEDM
frequency channels contain information about experimental conditions such as non-reversing magnetic fields, and we use these channels to construct and confirm models of non-ideal experimental
behavior and to correct for systematic shifts in f BD . Some examples of frequency channels, theirleading-order expressions in terms of experimental parameters, and their interpretations are shown1

Our notation for data channels closely follows that used by the authors of Refs. [6, 7].

66
Channel
f0
fB
fD
f BR
f BD

Leading term
3|gF |µB Brot /hnr /h3|gF |µB Brot3δg eff µB Brot sgn(gF )/h−3hαifrot sgn(gF )−2de |Eeff | sgn(gF )/hInterpretation
Avg. precession frequency
Non-reversing Brot
Level-dependent g-factor
Geometric phaseeEDM shift

Table 3.1: Selected frequency channels, their leading expression in terms of experimental parameters, and their interpretations. Here δg eff is half the effective magnetic g-factor difference between
Stark doublets, α is the tilt angle of E rot above the radial plane of the ion trap, and frot is therotation frequency of Erot .

in Table 3.1.
Prior to eEDM data collection, we tuned a wide variety of experimental parameters overan exaggerated dynamic range and observed the response of the data channels to study non-ideal
frequency shifts in our system that might affect an eEDM measurement. Two illustrative examplesof these effects are shown in Fig. 3.3. Through this study, we developed a unified numericalmodel of our spin precession sequence. In this model, we integrate the classical motion of ions insimulated time-varying electric and magnetic fields, then propagate the internal quantum state of
the molecules using an effective Hamiltonian that includes the two lowest rotational levels of 3 ∆1 .
Using known experimental parameters and realistic estimates of construction imperfections, ourmodel was able to reproduce all observed frequency shifts.
In total, we collected 1024 blocks (360.3 hours) of eEDM-sensitive data, with each block
resulting in one value of f BD and thus one eEDM measurement. Throughout the collection andanalysis of this eEDM data, we added to the f BD channel a hidden, computer-generated pseudorandom value drawn from a normal distribution with a standard deviation of 5 × 10−28 e cm. This“blind” allowed us to investigate systematic frequency shifts and perform statistical analysis while
mitigating the effects of operator bias. We applied cuts to the blinded data based on non-eEDM
channels indicating signal quality: blocks with C < 0.1 or Ce−γT < 0.1 were cut due to low signal
to noise (where T is the largest value of t sampled in a block). In addition, we cut data where shifts
in the “co-magnetometer” channel f B exceeded 0.4 Hz due to its contribution to systematic errors.

67

1
0
–1

(a)

2
0
–2

(b)

–4
–6

–4

–2

0

2

4

6

Figure 3.3: Frequency shifts in the f B and f BR channels due to a stray uniform magnetic field BYnr
and ion displacements Y0 Ŷ and Z0 Ẑ. (a) A shift in f B ∝ BYnr Y0 resulted from a contribution to
Brot from an electric field gradient oscillating at 2frot , which we suppressed by reducing harmonic
distortion in Erot via feedforward. (b) A shift in f BR = 3hαifrot ∝ Y02 Z0 was well modeled by the
known inhomogeneity in Erot , and was suppressed by applying feedback to the ion position between
eEDM measurements. Error bars are ∼ ±0.1 Hz on all points.

68
After these cuts, our eEDM dataset consists of 903 blocks or 313.8 hours of data. The unblinded
dataset is shown in Fig. 3.4(a-b). Normality tests and visual inspection of a normal probability plot
indicate that the distribution of normalized and centered eEDM measurements (f BD − hf BD i)/δf
is consistent with a normal distribution. The reduced chi-squared statistic for fitting a weighted
mean to the eEDM dataset is χ2r = 1.22(5). This over-scatter is present in all frequency channels,0
and is attributable to non-reversing Baxgrad
drifts on a timescale comparable to one data block. To
pcompensate for this over-scatter, we scale our final statistical error bar by χ2r ≈ 1.1.

During eEDM data collection, we suppressed sources of systematic error that appeared inour earlier model-building investigation by applying active feedback to relevant experimental parameters between the collection of data blocks. The only one of these that produced an observablenr andshift in the f BD channel was the combined effect of a non-reversing magnetic bias field Brot

the difference in effective magnetic g-factor between Stark doublets [78]. The f B and f D frequency
channels, which are acquired concurrently with f BD , provide direct measurements of these contributions. Since the value of f D ≈ 10−3 f 0 is fixed by the values of Erot , Brot , and frot , we suppress0the systematic shift in the eEDM channel by applying a compensating Baxgrad
to minimize |f B |.

We also apply a block-by-block correction to f BD based on the measured values of f B and f D , thevalidity of which was verified in our earlier model-building study.
Though they were too small to be observed at our level of sensitivity, we predicted systematic
shifts in the eEDM channel due to the frequency shifts in the f B and f BR channels shown in Fig. 3.3.
We suppressed the first of these by adding a feedforward signal to Erot to cancel the harmonic
distortion component at 2frot , reducing it from −48 dBc to −70 dBc, and by using magnet coils tonull the ambient uniform magnetic field at the RF trap center to within ∼ ±30 mG. To suppress
the shift in f BR caused by Erot inhomogeneity shown in Fig. 3.3(b), we measured the ion cloud
position once per data block on a pair of MCPs, and applied DC potentials on the trap electrodes
to position the ion cloud within ∼ 2 mm of the minimum of the quadratic shift. The residual offset
of f BR ≈ − 100 mHz and gradient of ∂f BR /∂Y0 ≈ 20 mHz/mm shown in Fig. 3.3 are consistent
with Erot inhomogeneity resulting from realistic machining, welding, and assembly imperfections in69

Occurrences

200

(a)

(c)

100

0
0.999
0.99
0.9
0.5
0.1
0.01
0.001

Normal cumulative
fraction

(b)

–4

–2

0

2

4 –10

–5

0

5

Standard deviations from meanp
Figure 3.4: Summary of eEDM dataset after cuts and scaling δf by χ2r to account for over-scatter.
(a) Histogram of normalized, centered eEDM-sensitive frequency measurements (f BD −hf BD i)/δf .
(b) Normal probability plot of the same dataset, showing a linear trend suggesting that the data
are consistent with a normal distribution. (c) Subsets of eEDM data taken under different valuesof experimental parameters, and the overall average of f BD . Here N is the average number oftrapped HfF+ ions per run.

70
EffectNon-reversing Brot
Geometric phases
Axial secular motion
Rotation-odd Erot
Doublet population backgroundTotal systematic
Statistical
Total uncertainty

Correction−1−1

Uncertainty
5
4
2
14
195
195
868
890

Table 3.2: Systematic effects and corrections applied to the eEDM channel f BD , in units of µHz.

the construction of our RF trap.
While collecting eEDM data, we also searched for new systematic errors correlated withparameters that could not be tuned over a significantly exaggerated dynamic range, including
Erot , frot , and the number of HfF+ ions trapped per experimental trial [Fig. 3.4(c)]. We did notobserve significant variation of f BD with these parameters at our current level of precision. The
variations of the non-eEDM frequency channels f 0 and f D , in which we did anticipate variation
with Erot and frot , were consistent with model predictions. Finally, we modified our data collection
by randomizing the order of switch states in each block to search for and suppress systematic errors
caused by parameter drifts correlated with our switches, and observed no significant variation ofdata channels. The final results of our systematic error searches and corrections are summarizedin Table 3.2.
We removed our blind on 31 March 2017, and obtained a final value for the eEDM-sensitivefrequency channelf BD = 0.10 ± 0.87stat ± 0.20syst mHz.

(3.3)

Dividing by −2|Eeff |sgn(gF )/h ≈ 1.13 × 1028 mHz/e cm [44, 98], we obtain a value for the eEDM
de = (0.9 ± 7.7stat ± 1.7syst ) × 10−29 e cm,(3.4)

which is consistent with zero within one standard error. The resulting upper bound is|de | < 1.3 × 10−28 e cm

(90% confidence).

(3.5)

71
Our result is consistent with the limit of |de | < 9.3 × 10−29 e cm set by the ACME Collaboration[6, 7], and we have confirmed their result using a radically different experimental approach. Our
measurement is limited by statistics, and our dominant source of systematic error can be further
suppressed to the 10−30 e cm level. Here we have assumed that parity and time-reversal violating
effects arise purely from de . An additional contribution ∼ WS CS can arise from a pseudoscalarscalar electron-nucleon coupling CS [20, 32, 69, 113].
Since the completion of this first generation eEDM measurement, we have constructed asecond generation ion trap that will confine ten times more ions over a one hundred times larger
volume, and will provide a larger, more uniform rotating electric bias field. We estimate that theseand other improvements should provide an order of magnitude higher eEDM sensitivity. In thefurther future, we plan to pursue a third generation eEDM measurement using 232 Th19 F+ , in which
the 3 ∆1 ground electronic state with Eeff ≈ 36 GV/cm may allow a coherence time up to tens ofseconds [32, 52, 116].

3.2

Data collection
This section is reproduced from the supplementary information of Ref. [16].

3.2.1

Switch state timing & control

As described in Section 3.1, a single eEDM measurement requires collecting one interference
fringe in each of the eight unique switch states (B̃, D̃, R̃). As a compromise between signal to noise
on a single fringe and reducing susceptibility to errors from drifting experimental conditions, wetypically collect twelve points per fringe, with six points spanning one fringe period at short spin
precession times 0 < t < 40 ms and six more points spanning a second fringe period at long spin
precession time, (T − 40 ms) < t < T , where during data collection we varied T between 200 and
700 ms. One measurement of the asymmetry A(t) requires two experimental trials, and we typicallyaverage eight measurements comprising sixteen trials at each value of t. As shown in Fig. 3.1, asingle experimental trial is accompanied by approximately 300 ms of “dead time” spent on state(a) “Across”(b) “Down”

72(c) “Down + scramble”Tt (ms)
0
+1
–1
+1
–1
+1
–1
0

5

10

Elapsed time (min)

15

0

5

10

15

Elapsed time (min)

0

10

15

Elapsed time (min)Figure 3.5: Relative timing of experimental configurations during the collection of one data block.
(a) When taking data “across,” we collect in each switch state an entire interference fringe as afunction of t. (b) In the “down” configuration, we change the spin precession time t on a slower
timescale than B, D, and R. (c) In the “down + scramble” configuration, we also re-randomize
the order of switch states (B̃, D̃, R̃) at each value of t.

preparation, state readout, auxiliary measurements, and saving data. Thus, a single block requires
a minimum of approximately 16 minutes to complete.
The timescale of data collection and the order of switch states and interrogation times canaffect our statistical sensitivity, susceptibility to parameter drift, and sources of systematic error,
depending primarily on the Fourier spectrum of B field drifts. We used three different ordering
schemes, shown in Fig. 3.5, to investigate and mitigate these errors. The first, “across,” makes
up the smallest fraction of our eEDM dataset. We expect the “down” timing scheme to be lesssusceptible to drifts in experimental parameters on the few-minute timescale, for example due totemperature fluctuations or the accumulation of patch potentials on ion trap surfaces. The “down +scramble” timing scheme, in which the order of switch states is randomized in each block, should be
still less susceptible to errors associated with both parameter drifts and systematic errors associatedwith the order of switch states.
As discussed in Section 3.1, the B, D, and R switches represent the sign of the axial magnetic
0
gradient Baxgrad
generating the rotating magnetic bias field, the populated Stark doublet, and thesense of electric bias field rotation, respectively. A National Instruments PCI-6733 digital-toanalog converter (DAC) supplies a control voltage to a bipolar current supply that powers the0pair of magnet coils generating Baxgrad, thus setting the value of B̃. The doublet switch D̃ is set

73

Figure 3.6: Schematic layout of control system for experimental switches. Blue traces indicate
digital signals, gold traces indicated DC analog signals, and red traces indicate RF analog signals.

by adjusting the frequency output of an Analog Devices AD9959 direct digital synthesis (DDS)ASIC between two values separated by ∼ 20 MHz. The amplified output drives an acousto-optic
modulator that controls the frequency of the second of our Raman transfer lasers via a frequency
offset lock to a stable optical cavity, thus tuning the laser to populate either the upper or lower
Stark doublet. Finally, the value of R̃ is set by adjusting the relative phase of six DDS-generated
sinusoidal signals that are amplified to produce the rotating electric bias field. These control systems
are shown schematically in Fig. 3.6. None of our switches generate or require large currents or
voltages, and each can be changed on a timescale that is short compared to collecting one 16-shotdata point. Thus we are not restricted in their order due to dead-time considerations, and canrandomize our switches without significant change in our duty cycle.

3.3

Data processing
This section is reproduced from the supplementary information of Ref. [16].
We use MATLAB to perform data analysis, which consists of counting Hf+ ions at each time

point in the fringe, calculating and fitting the asymmetry, forming data channels, blinding theeEDM channel, applying cuts, and searching for signs of systematic errors in the resulting datachannels.

74

TOF MCP signal (V)

0.5

HfF

0.4

+

0.3

Hf

0.2

+

0.1
0
18.5

19

19.5

20

20.5

21

21.5

time after kick to MCP (μs)

Figure 3.7: Typical raw data trace from the time-of-flight microchannel plate detector, showing
time-resolved Hf+ and HfF+ signals. Red markers indicate counted Hf+ ions contributing to theasymmetry signal.

A typical signal from our time-of-flight (TOF) microchannel plate (MCP) ion detector isshown in Fig. 3.7. The “signal” Hf+ ions resulting from state-selective photodissociation and the“spectator” HfF+ ions are temporally separated, and for numbers of Hf+ ions below ∼ 30, individual
Hf+ ion peaks are also well-resolved. We separately amplify the Hf+ and HfF+ signals to reducenoise on the Hf+ signal while avoiding saturation of our transimpedance amplifier by the HfF+
signal. We use a peak-finding algorithm to locate Hf+ peaks of a specified prominence above thebackground value; typically 4 times the rms voltage of an empty trace.
We compute the asymmetry A(t) from the number of counted Hf+ ions in runs of the experiment with opposite depletion phases. We fit the functional form of Eq. 3.1 to the measured
asymmetry using MATLAB’s Levenberg-Marquardt nonlinear least squares regression algorithm.
Because measurements of the asymmetry do not follow a normal distribution (see e.g. Ref. [66]),
we perform an unweighted regression, rather than binning measurements of A(t) for a given t andperforming a weighted regression. Fitting simulated data with our analysis routine does not show
any evidence of systematic shifts due to the non-gaussian distribution of A(t).
As discussed in Section 3.1 in the paragraph preceding Eq. 3.2, we form linear combinationsof frequency measurements f B̃ D̃R̃ ≡ f (B̃, D̃, R̃) to obtain frequency channels according to the75transformation


f0






+




 B 

 f 
+






 fD 
+






 BD f
 1 +
=  R  8
 f 
+






 f BR 
+






 DR 

f
+







+
f BDR

+ + + + + +− + − + − ++ − − + + −− − + + − −+ + + − − −− + − − + −+ − − − − +− − + − + +

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

(3.6)

where ± signs represent ±1. The standard error δf is equal for all frequency channels within one
block, and is given by
1
δf =
8

s

X

(δf B̃ D̃R̃ )2 ,(3.7)

B̃,D̃,R̃=±

where δf B̃ D̃R̃ are the standard error estimates obtained from nonlinear least squares regression. We
follow the same prescription to obtain the data channels for contrast, phase, decoherence rate, andoffset parameters. Within the same MATLAB script that computes this linear transformation of
measurements from the “state basis” to the “parity basis,” we apply the blinding value to the f BD
channel. The pseudo-random blind value, which was previously generated and saved to a binary
file, is read and added to the f BD channel within a single line of code. Following systematics
corrections (discussed in Section 3.5), we obtain the eEDM result by computing the weighted meanvalue of f BD across all blocks (indexed by n),
hf

BD

P
BD
n Wn fn
i= P
,
n Wn

1
hδf BD i = pP

n Wn

(3.8)

where we use weights Wn = (δfn )−2 . To correct for the over-scatter of our data, we scale ourpfinal value of hδf BD i by χ2r , where χ2r is the reduced chi squared statistic of the distribution of
normalized and centered eEDM measurements (f BD − hf BD i)/δf . Typical values of data channels
and other measured parameters are given in Table 3.3.

76

Table 3.3: Values of selected measured values from the portion of our eEDM dataset collected in2017. For this data, Erot = 24 V/cm and frot = 150 kHz. For the values f B , C 0 , and γ 0 that are
subject to drift and therefore not normally distributed, we quote a range of observed values. For
various values with either nonzero measured values, drift, or excess scatter, contributing factors are
given.

Parameter
f0
fB
fD
f BD
fR
f BR
f DR
f BDR
C0
C BD
γ0
γ BD
φ0
φBD
O0
OBD

Mean or Range
22.9985(13) Hz−0.3 - 0.3 Hz−32.0(1.0) mHz−0.1(1.0) mHz
3.1(1.1) mHz−21.8(1.3) mHz−0.8(1.0) mHz
0.1(1.1) mHz
0.55 - 0.75
0.0065(69)0.3 - 1.7 s−1
0.0127(59) s−1
0.2543(16) rad
0.0021(15) rad
0.00416(50)0.00016(47)Contributing factors0, Erot , frot
Choice of Baxgrad
0nr
Drifting ambient Baxgrad
0
Choice of Baxgrad , Erot , frot

Erot inhomogeneity + ion position drift

Photodissociation backgroundIon number, excess slosh and micromotion
Detuning of π/2 pulse due to nonzero Brot
Detuning of π/2 pulse due to nonzero Brot

77
Over-dispersion appears in our frequency channels in a way that is consistent with it arising
0nr
0nr
from a drifting ambient magnetic field gradient Baxgrad. A drifting value of Baxgrad
nominally only
0nr
affects the f B channel, however changes in Baxgrad
on a timescale comparable to the collection of one

data block causes over-scatter in all other data channels, including f BD . In future measurements,
0nr
we plan to mitigate this effect by removing sources of ambient Baxgrad
and by collecting data blocks

on a shorter timescale.
We chose our data cuts based on signal-to-noise considerations and the values of eEDMinsensitive data channels. The first data cut is performed at the ion counting level by a choice ofa time-of-flight window and minimum pulse height for Hf+ ion counting. Further cuts were basedon initial and final contrast and on the value of the f B data channel. We investigated the valuesof frequency channels as a function of the cut parameters, and found no worrisome dependencies.

3.4

Modeling frequency channels
This section is reproduced from the supplementary information of Ref. [16].
The high statistical sensitivity and the systematic error rejection features of our experimental

approach come with an associated cost in the form of an increased level of complexity in modeling frequency measurements. The Ω-doublet structure of the 3 ∆1 state, the nuclear hyperfine
structure of HfF+ , our rotating quantization axis, and the motion of ions in inhomogeneous andtime-dependent electric and magnetic fields all contribute to this complexity. As a result, we used avariety of numerical and perturbative techniques to analyze sources of non-ideal frequency shifts inour system. In this section we discuss our methods in general terms, and provide a list of observed
frequency shifts. In Section 3.5 we describe in more detail effects that systematically affect theeEDM measurement channel f BD .

3.4.1

Ion motion

The spin precession frequency of HfF+ in our experiment is set by the electric and magnetic
fields experienced by each ion. Since the electric and magnetic fields in the RF trap are inhomo-

78
geneous and (for the electric fields) time-varying, and the ions’ trajectories are modified by theelectric field (neglecting the Lorentz force), we must know the trajectories of the ions in order todetermine their spin precession frequencies.
For our perturbative analysis of frequency channels, we use the standard approximation ofharmonic motion with superimposed RF micromotion [9], and additionally superimpose rotating
micromotion due to Erot . In this case, the total electric field isE(R, t) =

i
h
Vrf
Vdc
X̂
cos
(ω
t)
−
R̃
Ŷ
sin
(ω
t)
, (3.9)
cos
(ω
t)(X
−
Y
)
+
(X
+
Y
−
2Z)
+
E
rot
rot
rot
rf
R02
Z02

where ωrot ≡ 2πfrot , ωrf ≡ 2πfrf ≈ 50 kHz, R̃ is the rotation switch sign, and R0 and Z0 are theeffective radius and height of the RF trap. However, this approximation is not able to account foreffects we observe due to electric field inhomogeneities of multipole order l > 2. We account forthese higher order effects numerically by performing a multipole fit up to l = 9 of the electric field
due to unit potential on each of the eight trap electrodes to obtain multipole coefficients cklm (wherek = 1 . . . 8), which allows us to represent the total electric field in the ion trap as a function of theelectrode voltages:
E(R, t) =8X
k=1

Vk (t)

Xh
icklm −∇ Rl Ylm (Θ, Φ) ,(3.10)

lm

where (R, Θ, Φ) are spherical polar coordinates in the laboratory frame, and Ylm are real sphericalharmonics. We then numerically integrate the equations of motion for an ion to obtain Rion (t),and substitute back into Eq. (3.10) and a corresponding expression for B(R) to obtain E ion (t) andBion (t), the electric and magnetic fields at the location of the ion. Finally, we transform these fields
into a rotating frame whose coordinate axes x̂, ŷ, ẑ are related to the laboratory frame axes X̂, Ŷ ,Ẑ byx̂ = −Ẑ,ŷ = Ŷ cos (ωrot t) + R̃X̂ sin (ωrot t),ẑ = X̂ cos (ωrot t) − R̃Ŷ sin (ωrot t).

(3.11)In this frame, Erot nominally points along the +ẑ axis. For this analysis, we neglect the effect of
ion-ion interactions.

79
3.4.2

Effective Hamiltonian

Our modeling of frequency channels relies on an effective Hamiltonian description of HfF+
molecules. We do not use optical pumping into dark states to perform π/2 pulses, so high-energy
photons are not present, and coupling to other electronic states is absent throughout our spin
precession sequence. As a result, we can build an effective Hamiltonian that includes only 3 ∆1
sublevels. The 3 ∆1 state is well-described by Hund’s case (a) basis states with coupled nuclearspin,|Λ = ±2, S = 1, Σ = ∓1, J, Ω = ±1, I = 1/2, F, MF i ,where 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 angularmomentum, Ω = 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 themolecule, and MF = F · ẑ is the projection of F on the rotating quantization axis [15]. We takethe internuclear axis n to be directed from the 19 F nucleus to the 180 Hf nucleus. We model our
spin precession experiment using an effective Hamiltonian that includes (in decreasing order of size)
molecular rotation, the nuclear spin hyperfine interaction, the Stark effect, Ω-doubling, a rotating
quantization axis, the electronic and nuclear Zeeman effects, and an eEDM:
H(E, B, ωrot ) = Htum + Hhf + HS (E) + HΩ + Hrot (ωrot ) + HZ,e (B) + HZ,N (B) + Hedm .

(3.12)We use the effective operators 2
Htum = Be J2 ,Hhf = Ak (I · n)(J · n),

HS = −dmf n · E,

HΩ = ~ωef Ωx /2,

Hrot = −~ωrot · F, HZ,e = −Gk µB (J · n)(B · n), HZ,N = −gN µN I · B, Hedm = −de |Eeff |Ω,with constants listed in Table 3.4. The effective operator Ωx has matrix elements δη0 ,η δΩ0 ,−Ω (whereη represents all other quantum numbers).
2

Matrix elements for each of these operators are discussed in Chapter 2.

80Constant
Be /h
Ak /h
dmf /hωef /(2π)ωrot /(2π)
gF
gN|Eeff |/hValue
8.983(1) GHz−62.0(2) MHz
1.79(1) MHz/(V/cm)
0.74(4) MHz
250 kHz typ.
−0.0031(1)5.25774(2)5.63 × 1024 Hz/(e cm)

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

Reference[29]
This work
This work[29]
This work
[82]3
[119]
[44]

Table 3.4: Constants used in the effective Hamiltonian of Eq. (3.12), for the calculation of frequency
channels and systematic corrections. The total magnetic g-factor of J = 1, F = 3/2 states gF ≡(Gk + gN µN /µB )/3 results from the combination of nuclear and electronic Zeeman effects.

In our perturbative model of frequency channels, we treat the pairs of levels in either Stark
doublet as a two-level system, and construct a 2×2 effective Hamiltonian for either doublet. We takethe molecular rotation, Stark and Hyperfine Hamiltonian terms as an unperturbed Hamiltonian,then include the Zeeman, frame-rotation, Ω-doubling, and eEDM terms as perturbations. While the
Zeeman effect of Brot is the dominant diagonal contribution to the two-level effective Hamiltonian,perpendicular electric and magnetic fields, rotation, Ω-doubling, and Stark mixing of rotational
levels all contribute frequency shifts that are significant at the level of an eEDM search. In thebasis of F = 3/2, MF = ±3/2 states, the two-level effective Hamiltonian has the formDh(∆ ± ∆ )1 −3(gF ± δgF )µB Brot ± 2de |Eeff | − 3αhfrot + . . .

u/lHeff = ,2Dh(∆ ± ∆ )3(gF ± δgF )µB Brot ∓ 2de |Eeff | + 3αhfrot + . . .
(3.13)
where the u/l superscript and upper or lower signs correspond to the upper or lower Stark doublet,α ≡ EZ /Erot is the tilt of the electric field away from the XY plane, and ∆ is a rotation induced
coupling with a Stark doublet-odd contribution ∆D . Here the magnetic g-factor gF results fromthe combined nuclear and electronic magnetic moments, which are coupled by the nuclear hyperfine
Hamiltonian Hhf . Notable features of this two-level system are the difference in magnetic g-factor
between Stark doublets δgF , the rotation-induced coupling ∆, and the geometric frequency shift
3αfrot . The magnetic g-factor difference arises from Stark mixing of J = 1 with J = 2 and from81
rotation-induced mixing of adjacent magnetic sublevels at second order in perturbation theory, andhas the approximate form
δgF
3dmf Erot
≈−
gF
20Be





2
gN µN
3~2 ωrot
2gN µN1−+
1−≈ −1 × 10−3 .
3gF µB
dmf Erot Ak
3gF µB(3.14)

The coupling ∆ first arises at fourth order from the combined perturbations of Hrot and HΩ , andbreaks the degeneracy of the |MF | = 3/2 states in either Stark doublet at Brot = 0 [78, 88]. The
fourth-order perturbation theory expression for ∆ and its Stark doublet-odd component ∆D aregiven by
3~ωef
h∆ =
2



~ωrot
dmf Erot3

2
18A2k − 19d2mf Erot
2
A2k − d2mf Erot

!
,

3~ωef
h∆D =
2

3
~3 ωrot
2 A
d2mf Erotk!

2
9A2k − 8d2mf Erot
2
A2k − d2mf Erot
(3.15)

These expressions are valid as long as dmf Erot  ~ωef and dmf Erot  ~ωrot . The strong scaling of∆ with Erot allows us to perform off-resonant π/2 pulses by modulating the magnitude of Erot , asdiscussed in Section 3.1. Finally, the geometric phase shift arises from the fact that if EZ is nonzero,
the solid angle swept out by the rotating electric field differs from its nominal value of 2π. Thisu/leffect is discussed extensively in Ref. [87]. Higher order contributions to Heff , denoted by ellipses
in Eq. (3.13), are due to higher-order combinations of perturbing Hamiltonian components, andprimarily result in small corrections to the terms already discussed.
For our numerical studies, we use E ion (t) and Bion (t) resulting from the simulated ion motion (Section 3.4.1) to construct Hion (t) = H(E ion (t), Bion (t), ωrot ), and numerically integrate the
Schrödinger equation for a single ion by exponentiating Hion (t) at each value of t to obtain thetime-evolution operator Uion (t), which we apply to an initial state vector |ψ0 i. For various sets ofsimulations we have either taken |ψ0 i to be one of states |ai, |bi, |ci, or |di [Fig. 3.2] and included
realistic π/2 pulses by modulating the simulated value of Erot , or we have assumed an ideal π/2
pulse by initializing |ψ0 i as an equal superposition of states |ai and |bi or |ci and |di. Finally, weconstruct the (simulated) asymmetry A(t) by projecting |ψ(t)i onto the upper or lower doublet
states, and fit A(t) using the functional form of Eq. 3.1 in Section 3.1.

!
.

82
3.4.3

Mixing of frequency channels

The non-negligible size of ∆ and ∆D throughout our spin precession experiment produces aunique structure of frequency channels that affects our systematics analysis, and warrants describing
in more detail. To do so, we parametrize the two-state effective Hamiltonian in terms of “diagonal”
and “off-diagonal” parity components f0s and ∆s ,0BDh  f0 + B̃f0 + D̃f0 + . . .
Heff (S̃) = 
2∆ + B̃∆B + D̃∆D + . . .

∆ + B̃∆B + D̃∆D . . .
f00 + B̃f0B + D̃f0D + . . .

.

(3.16)

Expanding the frequency channels obtained from this Hamiltonian about f00 , we obtain
B
D BD + . . .
D
B BD + . . .
∆2 + (∆B )2 + (∆D )2 + . . .
B ∆∆ + ∆ ∆D ∆∆ + ∆ ∆
−
f
−
f
+ ...
f 0 = |f00 | 1 +00
00
2|f0 |2|f0 |2|f00 |2(3.17)
BD + ∆B ∆D + . . .
∆2 + (∆B )2 + (∆D )2 + . . .
∆∆B + ∆D ∆BD + . . .
D ∆∆f B = f0B 1 −
+
−
f
+ ...
0
2|f00 |2
|f00 ||f00 |2(3.18)
f D = f0D 1 −∆2 + (∆B )2 + (∆D )2 + . . .
2|f00 |2

+BD + ∆B ∆D + . . .
∆∆D + ∆B ∆BD + . . .
B ∆∆
−
f
+ ...
0
|f00 ||f00 |2(3.19)

D
B BD + . . .
∆2 + (∆B )2 + (∆D )2 + . . .
∆∆BD + ∆B ∆D + . . .
B ∆∆ + ∆ ∆f BD = f0BD 1 −
−
f
+
+ ...
0
2|f00 |2
|f00 ||f00 |2(3.20)
with similar results for f R , f BR , f DR and f BDR . Thus the nonzero value of ∆, as well as any
component of it that is odd under B, D, or R, causes mixing of “diagonal” parity components
between measured frequency channels. This can cause systematic effects, the largest of which
are described in Section 3.5. However, all B-odd components of ∆ (∆B , ∆BD etc.) are negligible,
greatly reducing the number of terms that must be considered. The regular form of these frequency
channels also allows a straightforward correction that removes mixing terms up to third order in
perturbation theory of Heff ,f0BD = f BD + f B

f D − f0D
f0


+f

BR



f DR − f0DR
f0


+f

BDR



f R − f0R
f0

+ h. o.

(3.21)

83
The frequency channels f 0 , f B , f D , f BR , f DR , f BDR , and f R are measured simultaneously withf BD , while the “diagonal” components f0D , f0DR and f0R must be estimated from theoretical models
and auxiliary measurements. Note that in general the corrected value of f0BD includes terms in
addition to 2de |Eeff |. The corrections in Eq. (3.21) account only for those systematics arising fromthe nonzero value of ∆, and does not include “diagonal” systematics such as those arising froma difference in magnetic g-factor between Stark doublets and a non-reversing magnetic bias field.
Under typical experimental conditions, the above corrections cancel mixing to the level of ∼ 10 µHz(∼ 10−30 e cm). Correction terms are discussed in Section 3.5 and are included in our uncertainty
budget.

3.4.4

Non-ideal frequency shifts

A detailed description of every non-ideal frequency shift observed during our model-building
phase is beyond the scope of this supplement, and will be given in a future publication. Here we list
the parameters and experimental imperfections we explored in Table 3.5, and list observed effects
and their observation channels in Table 3.6. More detailed descriptions of effects included in our
uncertainty budget are given in Section 3.5.

3.5

Systematics
This section is reproduced from the supplementary information of Ref. [16].
In perturbation theory of our effective Hamiltonian [Eq. (3.12)], an eEDM signal appears

as the lowest-order contribution to the f BD frequency channel, and its value is independent of
all experimental parameters. Any other contributions to this channel constitute systematic errors
and must be corrected if they are large enough to cause a significant shift. Systematic shifts cangenerally be grouped into one of two categories: real frequency shifts arising from higher-orderterms in the effective Hamiltonian (introduced in Section 3.4.3), and apparent shifts arising frommeasurement errors. We have identified several possible sources of both types, and observed afrequency shift in f BD due to one effect.

84
Description
Rotating E field
Axial magnetic bias gradientNon-reversing uniform B fields
Non-reversing magnetic gradients
Ion cloud position
Ion cloud secular motion amplitude
Trap RF amplitude
Trap RF frequency
Number of trapped HfF+ ions
Vrot harmonic distortion
Currents in electrodes
Depletion laser polarization
Thermal drifts in trap amplifiers
Accumulating patch potentials

Parameters
Erot , ωrot0
Baxgrad, (B2,m6=0 , BX , BY , BZ )4
nr , B nr , B nr
BX
Y
Z
nr
B2,m6
=0
X0 , Y0 , Z0
X1 , Y1 , Z1
Vrf
ωrf
N
Vnf , φnf , n = 2 . . . 6

Study method
E,M
E,M
E,M
E,M
E,M
E,M
E,M
M
E
E,M
E,M
E
E,M
E,M

Table 3.5: Experimental parameters and imperfections explored as possible sources of systematic
error. The letters E and M indicate that an effect was explored experimentally or through numericaland perturbative modeling, respectively.

Description
0
RF micromotion in Baxgrad
Non-reversing Brotnr
nr
nr due to E
Brot
rot 2nd harmonic, BX , BY

Stark-induced D-odd g-factor
Rotation-induced D-odd g-factor∆-induced effective D-odd g-factor
Non-reversing axial B-field
Erot inhomogeneity-induced geometric phase
Brot due to electrode currents
Axial secular motion frequency modulation

Channel
f0
fB
f BD
fB
f BR
fD
fD
fD
fR
f DR
f BR
f BR
f BR

Scaling
0(X02 + Y02 )q/R0
gF µB Baxgradnr
gF µB Brotnrδgeff µB Brot
nr X + B nr Y )/E R2
gF µB V2f (BX
rot 0
0
Y 0
nr
gF µB V2f (BX Y0 − BYnr X0 )/Erot R02
gF µB Brot dmf Erot /Be
gF µB Brot (~ωrot )2 /dmf Erot Ak∆D ∆/f 0(∆2 /f 0 )(gF µB BZ /~ωrot )(∆D ∆/f 0 )(gF µB BZ /~ωrot )frot (X02 + Y02 )Z0 /R03
gF µB µ0 ωrot Celec Erot
βZ0 eErot /(mωrot ωZ T )Table 3.6: Observed frequency shifts and their scaling with selected experimental parameters (numerical factors are omitted). Here q is a dimensionless RF trap parameter [9], R0 is the radiusof the RF trap, Z1 is the amplitude of the center-of-mass secular motion of the ion cloud along
the Z axis, Celec is the capacitance of an electrode, and β ≈ 1.4 × 10−5 mm−2 is the fractional
inhomogeneity of Erot along the Z axis.

85
We calculate systematic corrections according to a procedure similar to that of Ref. [6].
BD = P S where P is a parameter andParametrizing a systematic shift in the eEDM channel as fsyst,i
i i
i
P BD
Si ≡ ∂f BD /∂Pi , the corrected eEDM measurement in the nth block is fneEDM = fnBD + i fcorr,i,n=P
fnBD − i Pi,n Si,n . We compute systematic corrections on a block-by-block basis, and we obtain the
BD i and its uncertainty hδf BD i by propagating the standard errors δPtotal ith correction hfcorr,ii,ncorr,iand δSi,n through the weighted averaging of blocks. For systematics that produced an observed
shift in the eEDM channel (of which there was only one), we apply the correction and include
BD i in our uncertainty budget (Table II). For systematics that we did not observe directly inhδfcorr,iq
BD i ≡
BD i2 + hδf BD i2 in our
the eEDM channel, we include a systematic uncertainty hδftot,ihfcorr,icorr,i

uncertainty budget.

3.5.1

Doublet population contamination

When population is transferred from 1 Σ+ to 3 ∆1 , the detuning of the second transfer laser is
set by an acousto-optic modulator to select either the upper or lower Stark doublet to be populated.
The upper and lower Stark doublets are resolved by approximately nine times the 1σ Doppler
width of each transfer resonance (Fig. 3.8), so population of the undesired Stark doublet is highly
suppressed. During both population transfer and strobed depletion, however, spontaneous decay
excited states to all hyperfine levels in 3 ∆1 , J = 1 can occur, albeit with afrom the 3 Π0+ and 3 Σ−0+very small probability. Because our depletion and dissociation state readout processes are not Stark
doublet-selective, population in the undesired Stark doublet will lead to uncharacterized beatingin our interference fringes that will be misidentified as a loss of coherence and a frequency shift, asshown in Fig. 3.9. The effect of the apparent frequency shift is to suppress the measured value off D , leading to two sources of systematic error that we have identified.
The first systematic error arising from population in the wrong Stark doublet is that a
suppressed value of f D will suppress the correction of Eq. (3.26). Including this effect, we predicta value for f D of
D
D
fmeas
= ftrue−w
Dsin (4πftrue
T ),
2πT

(3.22)

60

mFΩ = +3/2

–3/2
–1/2

40

+1/2

20

1

Σ+ to 3Δ1 transfer (arb)

86

0

–30

–20

–10
0
10
Energy/h (MHz)20

30

Figure 3.8: Stimulated Raman transfer from 1 Σ+ , J = 0 to 3 ∆1 , J = 1, F = 3/2, showing
Doppler-broadened resonances at the locations of Stark doublets.

upper

8x
0.

measuredtrue
fit

Asymmetry

lower

0.2x

Spin precession time

Figure 3.9: Simulated effect of doublet population contamination with a fraction w = 0.2 of the
3 ∆ population in the undesired Stark doublet. Green and blue traces in the inset illustrate the1
fractional contributions from the upper and lower doublet to the total true interference fringe(black line). Measuring the asymmetry at only short and long times (black points) leads to amisinterpretation of beating as a loss of coherence (red line), and results in an error in the fit
frequency.

87
where T is the temporal separation between early and late time fringe measurements, and w is
the fraction of the 3 ∆1 state population in the lower (upper) Stark doublet when only the upper(lower) doublet should be populated. The resulting systematic error in the correction (3.26) isBD
fsyst,1=wf B
Dsin (4πftrue
T ).
2πf 0 T

(3.23)

Using Eq. (3.22) and the difference between our measured and predicted values of f D , we estimate
w = 0.02(1). However, w was not directly monitored during our data collection and systematic
errors in the measured parameters contributing to the predicted value of f D may dominate overany true population contamination. We estimate the total systematic error due to this effect to be
BD i = 8 µHz.
hδftot,1

While in neutral beam experiments, an uncharacterized φBD shift leads directly to a systematic frequency shift φBD /2πT , our practice of measuring both early and late time phase nominallydistinguishes φBD from f BD . In the presence of population contamination, this systematic shift
makes a small reappearance proportional to w. To derive the shift, we consider the sum of cosineswith amplitudes C+ = −(1 − w) and C− = −w, and overall phases
Φ± (B, D) = (ω 0 + B ω B ± D ω D ± BD ω BD )t + (φ0 + B φB ± D φD ± BD φBD ),where here we are neglecting the R switch. Assuming w  1, the amplitude and phase of the sum
are
C ' C+ + w cos [2(Φ+ − Φ− )],Φ ' Φ+ − w sin [2(Φ+ − Φ− )].

(3.24)

We approximate our measured fringe frequency as f ' (ΦT − Φ0 )/2πT , and upon transforming a
block of phase measurements of the form of Eq. (3.24) to the parity basis according to Eq. (3.6),
we obtain a systematic contribution to the f BD channelBDfsyst,2'w
sin (2φBD ) sin2 (2πf D T ).
πT

(3.25)We also obtain a similar term proportional to sin (2φD ) sin2 (2πf BD T ), however we neglect it since
it results in a multiplicative, rather than additive error on f BD and because φD and f BD are

88
uncorrelated in our eEDM dataset. Again, since w was not directly monitored during data collection, and since inconsistencies in our measured and predicted values of f D could be due to other
BD i = 195 µHz in our uncertainty budget.
sources, we include a total systematic uncertainty hftot,2
BD , and is the largest contributor to systematic error in ourThis contribution dominates over ftot,1

experiment. Improved monitoring of w in the next generation of this experiment (via, e.g., microwave spectroscopy of the 3 ∆1 J = 1 → J = 2 transition), as well as operating in a regime ofErot and frot where f D is suppressed, will reduce this systematic to the order of ∼ 1 µHz.

3.5.2

Non-reversing Brot and effective differential g-factor

nr is an imperfect
The simplest contributor to a non-reversing rotating magnetic bias field Brot0
reversal of the applied axial gradient Baxgrad. Contributions can also arise from other sources, in-cluding for example higher order magnetic gradients or time-dependent magnetic fields. Figure 3.10
0nr.
shows the only observed shift in the f BD channel, caused by a non-reversing axial gradient Baxgrad

Non-reversing B fields appear in the f BD channel due to the difference in the magnetic g-factors of
the upper and lower doublet states δgF , and due to the non-negligible size of ∆ and ∆D comparedto the Zeeman shift. Fortunately, an amplified shift appears in the simultaneously collected f Bchannel, allowing us to apply a proportional correction. In terms of experimental parameters, theshift in the eEDM channel due to δgF and ∆D isBD
nrfsyst,3= 3gF µB Brot



δgF
∆∆D
−
gF
|3gF µB Brot |2+ h. o.

(3.26)

BD is of order ∼ 10−3 f B and the sum of higher order terms is ∼ 1 µHz.
Under typical conditions, fsyst,7

By comparison of the terms in Eq. (3.26) to the leading order expressions for frequency channels
in Table I in Section 3.1, we find that we can apply a block-by-block correction (suppressing thesubscript n)
δgeff
BD
= fB
fcorr,3
= fB
gF



fD
δgF
−2
0
f
gF


.

(3.27)The typical value of f B for a given block was . 100 mHz, and due to our applied feedback to reducethe value of |f B |, its average value over many consecutive blocks was much smaller. The small value

89
0.4

30

0.2

20

0

10

–0.2

0

–0.4

–10

–0.6
–1.5

–1

–0.5

0

0.5

1

1.5

Figure 3.10: Systematic shift in the eEDM channel f BD due to non-reversing Brot (left hand verticalaxis, solid line), and a proportional shift in the f B channel that we use to apply a correction (righthand vertical axis, dashed line). The constant of proportionality f BD /f B = δgeff /gF depends on
Erot and frot . The data shown here were taken under conditions chosen to make it particularly
large (∼ 10−2 ).

BD i = −1(5) µHz
of δgeff /gF makes the correction to f BD still smaller, with an average value of hfcorr,3over the entire eEDM dataset.

3.5.3

Geometric phase and axial magnetic field

While the six radial electrodes of our RF trap are designed to optimize uniformity of therotating electric bias field, there nonetheless exist inhomogeneities in Erot that are well-describedby spherical multipoles, as discussed in Section 3.4.1. The (l, m) = (3, ±1) and (5, ±1) spherical multipole components of Erot cause a nonzero time-average tilt of the rotating electric field
hαi = hEZ /Erot i and a corresponding geometric frequency shift f BR ≈ 3hαifrot , shown in Fig. 3.3.
Independently, a non-reversing axial magnetic field BZnr produces a nonzero value of ∆DR , whichforms the dominant contribution to the f DR channel. The mixing mechanism described in Section 3.4.3 leads to a shift in f BD equal to −f0BR ∆DR ∆/|f00 |2 . Since f BR ≈ f0BR = 3hαifrot andf DR ≈ ∆DR ∆/|f00 |, we can in principle apply a corresponding correctionBDfcorr,4=f BR f DR,
f0(3.28)

BD i = −3(2) µHz over the eEDM dataset. However, since thiswhich has an average value of hfcorr,4

shift was too small for us to observe in the eEDM channel, we include a total systematic uncertainty
BD i = 4 µHz in our uncertainty budget.
hδftot,4

90Comparing Eqs. (3.21) and (3.28), we have neglected a contribution −f BR f0DR /|f00 | thatcannot be obtained from our measured data channels. While f0DR is typically negligible, a contribution to this “diagonal” channel can arise from an R-odd contribution to Erot , and is discussed in
Section 3.5.6.

3.5.4

Harmonic distortion of Erot

The systematic effects discussed in Sections 3.5.2 and 3.5.3 each arise from a single physicalmechanism generating a frequency shift in f B and f BR respectively, which are subsequently “mixed”
into f BD by the nonzero values of δgF , ∆D , and ∆DR . Other physical effects that generate“diagonal” frequency shifts in f B and f BR enter into f BD in precisely the same way, and are thus
contained in the corrections already applied. An illustrative example is the harmonic distortionnr and B nr produces a B nr
of Erot , which together with non-reversing uniform magnetic fields BXrotYB ) and a B BR (an R-odd contribution to B nr ).
(≡ Brotrot
rot

The rotating electric bias field Erot is generated by sinusoidal voltages of equal amplitude oneach of the six radial electrodes, oscillating with a frequency of frot and with a relative phase ofπ/3 between adjacent electrodes. These voltages are generated by power operational amplifiers,
which inevitably suffer from harmonic distortion. While exploring sources of systematic error, weobserved frequency shifts up to several Hz in the f B and f BR channels. The observed shifts had
a linear dependence on both transverse uniform magnetic fields and the equilibrium position of
the ion cloud during the spin precession experiment. These shifts, shown in Fig. 3.2, were caused
nr from the combined effect of transverse magnetic fields and an oscillatingby a contribution to Brot

electric field gradient generated by the second harmonic of Erot . From a simple model of electricfields in the ion trap and using 2nd order perturbation theory of our effective Hamiltonian 5 , weobtained model expressions that matched the observed frequency shifts,
fB = −
5

3gF µB V2f(BX X0 + BY Y0 ) cos φ2f ,4Erot R2

See Section 3.6.1.

f BR = −3gF µB V2f(BX Y0 − BY X0 ) sin φ2f , (3.29)
4Erot R2

91
where V2f and φ2f are the amplitude and phase of the 2nd harmonic. During eEDM data collection,
we suppressed V2f to −70 dBc by adding a feedforward signal to the voltages generating Erot , andcanceled BX and BY to within 30 mG of zero at the RF trap center using magnet coils. Bothof these frequency shifts can cause a false eEDM; f B through the effective differential g-factor,
BD andand f BR through a shift in the f DR channel. Both, however, are already corrected by fcorr,3
BD . Higher harmonics of f
fcorr,4rot combined with magnetic gradients can contribute higher-orderterms to Eq. (3.29), however the resulting shifts in the f BD channel are similarly accounted for by
the corrections already applied.

3.5.5

Frequency modulation due to axial secular motion

The geometric frequency shift 3αfrot in Eq. (3.13) generates a BR-odd frequency contribution
proportional to an axial electric field EZ , as was already discussed in the context of geometric phases.
While coherent axial secular motion of the ion cloud at frequency fZ  f 0 will not produce atime-average nonzero value of α, it does cause a BR-odd modulation of the instantaneous spin
precession frequency, as shown for a deliberately large secular motion amplitude in Fig. 3.11.
The ponderomotive potential associated with Erot inhomogeneity provides a source for axial secularmotion, since our π/2 pulses involve modulating Erot and thus applying position-dependent impulses
to the ion cloud. Further, we apply π/2 pulses of different lengths in the upper or lower Stark
doublet, which can lead to a D-odd slosh amplitude and therefore a BDR-odd modulation.
We perform a frequency measurement by collecting a set of (typically) six equally spaced
points at short interrogation time t . 50 ms and a second set at late time t . 700 ms, each set
spanning approximately one period of spin precession. The frequency resulting from a nonlinear
least squares fit to the functional form of Eq. 3.1 is approximately equal to the difference in phasebetween early and late time, f ' (φT − φ0 )/2πT . If aliasing this frequency modulation causes a
systematic phase shift, a systematic frequency shift f BR = ∆φBR /2πT will result (with a similarfrequency shift due to φBDR ). To the extent that the spacing of our asymmetry points in interrogation time is incommensurate with 1/fZ , the shift in f BR and f BDR will be suppressed. However,

92

0.6
(simulated fringe)

Asymmetry

0.4
0.2
0.3
0
0.2
–0.2
0.1
–0.4
–0.6

0
–0.1
10.6

10.8

11.4
11
11.2
Interrogation time t (ms)11.6

Figure 3.11: Phase modulation of an interference fringe at fZ = 4.01(7) kHz due to axial secularmotion of the ion cloud with an intentionally exaggerated amplitude of ∼ 3 mm, measured near
the steepest portion of the fringe where 2πf t ≈ π/2.

93
we did not vary either fZ or the spacing of our interrogation times during eEDM data collectionin order to enhance this suppression. Thus a contribution to offsets in f BR and f BDR from thiseffect may be present.
While any BR-odd contribution from frequency modulation is a “diagonal” frequency shift
BD , the BDR-odd frequency shift has not yet been accounted for. Thisand is accounted for by fcorr,6BD and f BD , as outlined in Section 3.4.3. The corresponding
is done so in the same manner as fcorr,7corr,6

correction isBD
fcorr,5=f BDR f R.
f0

(3.30)

Since we did not observe a shift in the eEDM channel due to this systematic, we include an overall
BD i = 2 µHz. As in the case of f BD where we neglected f DR , here we have
uncertainty hδftot,5corr,60neglected f0R . The main known source of f0R is discussed in the next section.
3.5.6

Rotation-odd Erot

As discussed in Section 3.2, the R switch is controlled digitally by adjusting the relative
phases of six DDS channels, each of which provides the Erot signal for one electrode. As a result,
R to be very small, possibly dominated by RF pickup
we expect an R-odd rotating electric field ErotR
between neighboring amplifier circuits in our ion trap driver electronics. To the extent that Erotdoes exist, it could potentially cause a systematic error through accidental cancellation of the BZnr R produces R- andinduced f R and f DR described in Section 3.5.3. This could occur because Erot

DR-odd “diagonal” frequency components,
R
f0R = 3gF µB Brot,

R
f0DR = 3δgeff µB Brot,

(3.31)

while the shifts in the same channels from BZnr arise from ∆R and ∆DR . The two sources of f R andf DR cannot be distinguished, and lead to systematic shifts of opposite sign in the eEDM channel.
However, at our present level of sensitivity, all of these shifts were small compared to our statistical
R areuncertainty. The resulting systematic frequency shifts in f BD due to ErotBD
fsyst,6= f BRR
δgeff Erot,gF ErotERBD
fsyst,7= f BDR rot .
Erot(3.32)

94
R throughout data collection, we use a very conservative estimate
Since we did not monitor Erot
R /E
Rof Erotrot = 0.01 with an uncertainty of δErot /Erot = 0.01, and calculate systematic uncertainBD i and hδf BD i using the measured values of hf BR i = −36.4(1.8) mHz and hf BDR i =ties hδftot,6
tot,7
BD to ∼ 1 µHz, so we include
0.55(93) mHz. The very small value of δgeff /gF ≈ 10−3 suppresses ftot,6
BD i = 14 µHz in our uncertainty budget.
only hδftot,7

3.6

Additional details on systematic effects

3.6.1

Harmonic distortion of Vrot

Because we amplify our ion trap waveforms with op-amps rather than resonant circuits,
harmonic distortion is an inevitability. We can generally anticipate spurious signals arising at
frequencies satisfying
fspur = n frf + m frot ,where n and m are integers. It so happens that certain ones of these are more prone to generating
systematics than others, with (n, m) = (0, 2) being a particularly interesting one! We can use themultipole fit coefficients of the electrodes in Table C.1 to predict the potential generated at the ion
trap center by a particular configuration of voltages on the trap electrodes. We simply decide what
configuration of voltages we are going to apply on each of the electrodes, and sum the multipole
coefficients of each electrode (which have units of m−l ) multiplied by the voltage on that electrode,

clm =

X

Vk cklm .

(3.33)k

For the rotating electric field, the voltage applied on each of the electrodes is Vrot cos (ωrot t + φrot,k ),with phase φrot,k = (9 − 2k)π/8 + ωrot t for E rot that initially points along X̂ and rotates “up” in aright hand sense with respect to Ẑ. (For the opposite rotation sense, just flip the sign of the phaseoffsets ∼ 9 − 2k). While we do not know in detail the specific mechanisms of harmonic distortion
at work in our trap amplifiers (though Terry Brown certainly does), we are reasonably certain thatthey include some degree of slew rate limiting and voltage clipping. In general, we can parameterize

95
the voltage output of our amplifiers as a power series with a time delay,
V actual (t) = V nominal (t) + α [V nominal (t − τ2 )]2 + β [V nominal (t − τ3 )]3 + . . . .

(3.34)

Based on this parameterization and assuming the degree of harmonic distortion in all electrodes is
the same, we can expect a term of the form
V2h,k (t) = V2h,k cos [(9 − 2k)π/4 + 2ωrot t + φ2h ]Summing over electrodes, we find that to lowest order, the resulting multipole electric field in theion trap is
E(R, t) = Erot X̂ cos (ωrot t) + Ŷ sin (ωrot t)+ E2h!
X Ŷ + Y X̂X X̂ − Y Ŷcos (2ωrot t + φ2h ) +sin (2ωrot t + φ2h ) ,R0R0(3.35)

where R0 = 4.8 cm is the radius of the ion trap. From the coefficients in Table C.1, I findE2h ≈ 0.8 V/m per volt of second harmonic (amplitude). We can also parameterize the laboratoryframe magnetic field as B(R, t) = Brot Êrot (t) + BX X̂ + BY Ŷ . We can use the transformation in
Eq. (2.40) to transform the electric and magnetic fields into the rotating frame of the molecule.
The results areE(r, t) = Erot ẑ + E2h
X ẑ − Y ŷ
X ŷ + Y ẑcos (ωrot t + φ2h ) +sin (ωrot t + φ2h ) ,R0R0(3.36)

B(r, t) = Brot ẑ + (BX ẑ + BY ŷ) cos (ωrot t) + (BY ẑ − BX ŷ) sin (ωrot t),
where lowercase coordinate axes are in the in the rotating frame. Crucially, the ẑ componentsof the 2nd harmonic electric field are completely dwarfed by Erot , and furthermore lead only tocommon-mode energy shifts for the MF stretched states. Also crucially, moving to the rotating
frame has reduced the frequency dependence of the 2nd harmonic electric field from 2ωrot to ωrot .
We can now perform perturbation theory to determine the frequency shift we will observe in Ramsey
spectroscopy between states |ai and |bi, for example. To perform perturbation theory, we need toknow the contribution to the 3 ∆1 Hamiltonian due to the electric and magnetic fields in Eq. (3.36).
Fortunately, due to the work we did in deriving the matrix elements of various operators in earlier

96chapters, this is quite straightforward. We can simply pull the relevant matrix elements from
Eq. (2.30) and Table 2.1, finding√

3
gF µB B⊥ ,2√
3
hb| HZeeman |f i = hd| HZeeman |hi = −gF µB B⊥ ,2
D|| E⊥
ha| HStark |ei = hc| HStark |gi = √ ,2 3
D|| E⊥
hb| HStark |f i = hd| HStark |hi = − √ ,2 3
ha| HZeeman |ei = hc| HZeeman |gi = −(3.37)

where here E⊥ = Ex − iEy and B⊥ = Bx − iBy . Using 2nd order perturbation theory with theperpendicular components of the Zeeman and Stark effects as our perturbing Hamiltonian H1 , wefind the lowest-order contribution to the relative energy shift in the upper doublet,(2)∆Eab =

ha| H1 |ei he| H1 |ai hb| H1 |f i hf | H1 |bi−
Ea − EeEb − Ef

3gF µB E2h=[(BX X + BY Y ) cos (φ2h ) + (BX Y − BY X) sin (φ2h )] .
2Erot R0

(3.38)

Both the cos φ2h term and the sin φ2h term are odd with respect to B̃ and even with respect to D̃.
With respect to R̃, however, the sin φ2h term is odd while the cos φ2h term is even. This is because
fundamentally, φ2h arises from something akin to a time-delay mechanism in the HV amplifiers
and other synthesis electronics that generate Erot . Thus, we get contributions to the f B and f BR
channels
3gF µB E2h(BX X + BY Y ) cos (φ2h ) + . . . ,2Erot R0
3gF µB E2h
f BR = · · · +(BX Y − BY X) sin (φ2h ) + . . . .
2Erot R0
fB = · · · +3.7(3.39)

Room for improvement
While the Generation 1 eEDM measurement with HfF+ was in many ways a success, setting

an upper bound on |de | that was comparable to the ACME I experiment, it was clear that we would
have to make significant gains in order to remain competitive in the next generation. That reality
became even clearer all too soon, when the ACME II result of |de | < 1.1 × 10−29 e cm was releasedin Fall 2018 [2] – a full 12× more stringent than our Generation 1 result, and about 9× more

97
stringent than ACME I. Thus, we developed plans for improvements in the only two categories
available: coherence time and count rate.
In the Generation 1 eEDM data, our spin coherence time was typically γ −1 ≈ 800 ms. The
methods for improving this were already well known to us, from early theory work by Eric andAaron Leanhardt [78] and from experimental results in 2014, summarized in Kevin Cossel’s thesis[28]: we should attempt to have lower ion cloud temperature for weaker collisions, lower density
for a lower collision rate, and a larger value of Erot to suppress the geometric phase accumulated ineach collision. To achieve each of these goals, we needed to design and build a new ion trap. This
step towards Generation 2 is described in Chapter 4.
How much room for sensitivity improvement was available from gains in coherence time?
It is often approximated that precision scales inversely with interrogation time, however this is
only true when the spin coherence time is short compared to dead time – time spent on state
preparation and readout. In addition, in the HfF+ experiment we suffer from both collisional
dephasing parameterized by γ as well as loss of 3 ∆1 population at a rate Γ due to spontaneousemission, which combine to give an effective coherence time of τeff = (γ +Γ/2)−1 . Using an estimatethat takes all of these effects into account 6 , I estimate that an improvement in γ −1 from 800 ms
to 10 s would result in a factor of about 2.5× sensitivity gain.
The HfF+ experiment had significantly more potential for improvement in the category ofcount rate. Increases in count rate can come from simply trapping a larger number of molecules
per experiment cycle, which offers obvious gains but comes with drawbacks: Larger ion number
in a fixed volume leads to a higher collision rate and thus faster collisional decoherence. We thus
anticipated that we would like to “cash in” some of our gains that we expected to make in coherencetime for an increase in count rate; in particular because the gains of improved spin coherence time
are reduced once γ  Γ. Thus, the question becomes: What factor increase in ion number can we
6

See Section 7.7.

98

5

4

3

2
N = 320N = 1000
N = 3200
N = 10,000

1

0

1

2

3

4

Figure 3.12: Attainable frequency precision for a Ramsey measurement σf (in arbitrary units) versus spin precession time for various numbers of trapped particles N , assuming that the decoherence
rate γ scales proportionally with N . In this case, higher count rate is not always advantageous for
precision when it leads to a loss of coherence.

attain and still maintain the γ  Γ? For example, if we assume the scalingγnew = γold

Nnew
Nold


,

(3.40)

we might find a behavior as shown in Fig. 3.12. In the end, this question will have to be answered
empirically, because while the relation of γ to N at a fixed temperature is known approximately, thenumber of trapped ions affects the heating rate [19], and the temperature has a much stronger effecton the coherence time [78]. Fortunately, much like the result shown in Fig. 3.12, our frequency
precision is generally a weak function of both the number and the interrogation time near the
optimum value, so a crude optimization will be sufficient for our purposes.
A more fruitful but also more challenging avenue for improved precision is to improve theefficiency of our state preparation and readout processes, so that we could have larger signal sizes
with the same number of trapped molecules, and thus the same decoherence rate. In this category,there was much to be gained at the conclusion of Generation 1, as shown in Table 3.7. In particular,
if we are able to essentially lump together the ionization, transfer, and depletion processes inTable 3.7, we can imagine gaining up to a factor of ∼ 10 in count rate, and thus about 3× inprecision, with no sacrifices made in coherence time. Our efforts in this direction are described in99
Experiment step
Ionization
Transfer
Depletion 1
Depletion 2
Dissociation

Step efficiency
0.2
0.5
0.5
0.5
0.1

Overall efficiency
0.2
0.1
0.05
0.025
0.0025

Table 3.7: Experimental (in)efficiencies in Generation 1.

Chapter 5.

Chapter 4

Generation 2 Ion Trap

Even before the main thrust of Generation 1 eEDM data collection in 2017, it was clear that
there was room for improvement upon that experiment in terms of both signal size and coherence
time. To that end, we began design of a second generation ion trap and accompanying electronics,
with a few goals in mind:(1) Trap approximately 10× more ions for improved signal size. As it turns out, attaining
this change was more a question of ionization laser beam shaping and pulse fluence thanany property of the ion trap itself, but subsequently attaining long coherence times withthose many ions depends on the trap parameters. To further increase our ion number
substantially beyond the present value of . 5 × 104 , we would likely need to modify ourpulsed molecular beam source to provide a higher density of neutral HfF.
(2) Eliminate the ellipsoidal mirrors from the Generation 1 ion trap, and redesign the electrodeshape in order to have a more uniform Erot over a large volume. This should allow us toadiabatically expand the ion cloud (now with about 104 ions) and attain longer coherence
times through colder temperatures and lower ion density [28].
(3) With new electronics, generate a much larger value of Erot to suppress collisional decoherence [28]. Also with the new electronics, suppress ground loops, which caused significant
heating of the ion cloud in the Generation 1 trap.
This chapter centers around the latter two goals, which were carried out between Spring 2016 and101
Fall 2017. In the next few sections, I discuss the design considerations and numerical optimizationof the Generation 2 ion trap, as well as our benchmarking of the final product. This is followed bya brief discussion of the Generation 2 trap electronics, the design and construction of which was
spearheaded by Matt Grau, Dan Gresh, Terry Brown, Felix Vietmayer, and Carl Sauer.

4.1

Design considerations
The main goal of our design for the Generation 2 ion trap was to obtain the trap parametersthat would provide the most uniform E rot over the largest possible volume. Not only do certain
systematic errors depend on the uniformity of E rot (see Chapter 3), but the ponderomotive potential

Upond (r) =e2
2
2 |E rot (r)|4mωrot(4.1)

tends to reduce the overall trap depth in a way that is very sensitive to the spatial uniformityof E rot . This tendency of the ponderomotive potential to reduce the trap depth was one of the
primary motivations to move to eight radial electrodes over six: In the Generation 1 trap, therelative asymmetry between Upond along the X̂ and Ŷ directions set a limit on the degree to which
the trap could be “relaxed.” A second motivation to move to eight electrodes was the increased
optical access, in particular allowing us to use separate windows for the ionization and dissociation
lasers, greatly simplifying our beam-overlapping optics [82] (and reducing the cost of replacement
dichroics when the 266 nm laser inevitably burns everything in its path).
In addition to the chosen design constraint of using eight radial electrodes, we also assessedthat to avoid making the distance from the pulsed valve to the trap center much longer (which
would reduce the neutral HfF density at the point of ionization), we should attempt to make theGeneration 2 trap fit in approximately the same size of vacuum chamber as the Generation 1 trap
– i.e., one with 10CF flanges on top and bottom. Thus, the design choices to be made for the
Generation 2 trap were relatively few:
(1) Material: The electrodes in the Generation 1 trap were fabricated from Stainless Steel102
sheet metal, and were welded onto their supports. From the point of view of avoiding
poorly characterized magnetic fields, this was not ideal. For Generation 2, we opted forTitanium, a very non-magnetic material which was favored by Kim Hagen for machining.
We had some concerns about the UHV properties and electrical conductivity of Ti, butfrom various online databases these both appeared to be similar to SS, and we have had
no issues.
(2) Trap height: For maximum trapping volume, this should be made as large as possible. As
it turned out, a constraint was set on this parameter by the availability of 10CF-flanged,
non-magnetic vacuum chambers. To reduce the presence of ferromagnetic material in thetrap vicinity, we opted to purchase a custom Kimball Physics spherical octagon, which ismachined directly from 316LN Stainless Steel and thus should avoid the phase changes andmagnetism that unavoidably are present on welded chambers. The height of the chamberis 6.824” (see drawings in Appendix C), which we extended a further ∼ 2” by adding 10CF
double-sided flanges on top and bottom, sandwiched between the sealing 10CF flanges withfeedthroughs.
(3) Endcap electrode radius: The endcap electrodes create the DC potential that confines
the ions in the Z direction. The only hard constraint on their radius is that there remains
adequate room outside for mounting of the radial electrodes. However, their potential isin general significantly shielded by the radial electrodes, and their radius was simply choseto be a convenient Rec = 25 mm (Fig. 4.1). Upon later consideration, it became evident
that the finiteness of the radius of the endcaps, along with their shielding by the radial
electrodes, contributes significantly to trap anharmonicity in the axial direction.
(4) Inner and outer trap radius: These parameters are set by the constraints that thetrap must fit through the bore of a 10CF flange, that the inner radius should be as large
as possible for a high degree of E rot uniformity, and that the inside of the trap should bewell-shielded from outside sources – e.g., patch charges on the chamber windows.

103(5) Radial electrode thickness: The radial electrodes act as vertical supports for the trap,
so they need to have some degree of rigidity. In addition, the flat surfaces of the machinedtabs on the top and bottom of the electrodes serve to locate them. As a compromise
between rigidity, ease of locating, and allowing optical access, we chose 4 mm thickness.
(6) Radial electrode shape: Having decided on the inner and outer trap radius, the numberof electrodes, and the trap height, the shape of the radial electrodes’ profile along the Z
axis was the only remaining parameter that has an appreciable effect on the uniformity of
E rot . This shape is the subject of our numerical optimization.
Key dimensions of the Generation 2 ion trap are shown in Figure 4.1. The dimensions shown
were chosen to fit the design constraints described in the items above, and allow a starting point
for the numerical optimization of Erot uniformity. The shape of the radial electrodes shown in the
figure is as constructed.

4.2Electrode shape optimization
We opted to parameterize the shape of the radial electrodes along the Z axis by a polynomial

Xel (Z) = Rin

1−

nX
max
n=1


C2n

Z
Zel

2n !
,

(4.2)

where Zel = 75 mm is the maximum vertical extent of the radial electrodes, and Rin = 48 mm is theinside radius of the trap at the Z = 0 plane (Fig. 4.1). To the extent that the trap is infinitely tallcompared to its radius, optimal uniformity will be obtained with all C2n = 0. We expect C2n < 1P
for all coefficients, and alson C2n < 1 to prevent intersection of the radial electrodes in thecomputer model. In the design of the Generation 1 ion trap by Matt Grau in 2009, the electrodes
had a parabolic shape – i.e., only the C2 coefficient in Eq. (4.2) was nonzero. This allowed forthe cancellation of Erot deviations up to 2nd order along the Z axis. In the Generation 2 trap, weallowed for coefficients up to C10 , which allows for cancellation of Erot inhomogeneity up to 10thorder.

104

75.00
25.00

Ø 5.00Ø 190.00
5.00

2.00

5.00

T

2

3
15.00

8.0
0

50.00
52.63

1

4
R2.
00

150.00160.00

72.
00
86.
00

Ø5
4.0
0
Ø5
0.0
0

8

R1.11750

6
B

36.
00

57.
68

5

4.0
0

7
N

E

48.00
84.00

Figure 4.1: Key dimensions of Generation 2 ion trap. All dimensions in millimeters. Trap principal
axes are indicated in red, and electode numbers are in green.

105
We typically characterize the degree of inhomogeneity in Erot with a spherical multipole
expansion about the ion trap center, which has the form of interior solutions to Laplace’s equationΦlm ≡ rl Ylm (θ, φ). Through numerical electrostatics simulations in Comsol, we found that thechanging the polynomial shape of the radial electrodes along the Z direction primarily affected theamplitudes of the multipole components having (l, m) = (2n + 1, ±1). Along the line (0, 0, Z), thenorm of the electric field arising from these components behaves as E ∼ Z l−1 .
When optimizing the uniformity of Erot , we would like to prioritize a very high degree offlatness about the trap center; equivalent to weighting the coefficients Cn inversely with increasingn. We achieved this using a cost function
Z Zmax
Cost =[Erot (0, 0, Z) − Erot (0, 0, 0)]2 W (Z) dZ,(4.3a)−Zmax

W (Z) =







1 − sin10 2ZπZ ,max

Z < Zmax




0,

Z ≥ Zmax(4.3b)

where we took Zmax = 3 cm, and W (Z) is a weight function shown in Fig. 4.2. A drawback of thiscomputationally inexpensive scheme is that in principle, this optimization attempts to minimize allterms Erot ∼ Z 2n rather than only up to Z 10 . If we made Zmax too large, the values of C2 . . . C10
would converge such that Erot was on average flat over a larger region, but had high-order variations
that would have created a significant ponderomotive potential. In the end, this choice of Zmax andweight function worked reasonably well, and succeeded at canceling terms up to Erot ∼ Z 10 to
the few percent level. The values of the optimized electrode shape coefficients are listed in Table
4.1, and the corresponding amplitudes of the multipole components for each electrode are given in
Appendix C, Table C.1.
The optimization itself was carried out using the Livelink for Matlab add-on for the Comsol
finite element analysis software suite. Livelink provides an application programming interface (API)– essentially a collection of functions callable from Matlab – that allows Comsol to be controlled
programmatically from the Matlab command line, or better yet, from callable Matlab scripts and

106

1
0.8
0.6
0.4
0.2
0
-3

-2

-1

0

1

2

3

Figure 4.2: Weight function for electrode shape optimization, given by Eq. (4.3b)

C2
C4
C6
C8
C10

0.067403−0.119970−0.167233
0.696342
0.092692

Table 4.1: Optimized polynomial coefficients for the radial electrodes’ vertical profile for the Generation 2 ion trap.

107
functions. For iterative processes such as optimization, this is highly preferable to a GUI such as
is available in standard Comsol. In Appendix A, I have included some example Matlab code for
optimization, multipole fitting, and plotting of electrostatics results with Comsol via Livelink.

4.3

Results
We verified that our optimization of electrode shape had been successful through a variety

of simulations, mechanical measurements, and experiments. The most straightforward verificationthat the optimization procedure has converged is to investigate the uniformity of Erot , averaged
over a single rotation period. Figure 4.3 shows the percent deviation of Erot in the XY and XZ
planes, where deviations appear to be less than 0.5% over a cylinder of radius 20 mm and height 60
mm. Figure 4.4 compares the numerical results for Erot imported from Comsol with the multipole
fit up to l = 11 along key trap directions: The X axis makes an angle π/8 with pairs of neighboring
electrodes, while the line X cos(π/8) + Y sin(π/8) runs along a line between opposing electrodes.
The comparison confirms that the multipole fit to this order provides a reasonable approximation
to the numerical results.
After assembling the ion trap but before installing in the vacuum chamber, Yuval and Kim
Hagen conducted some measurements of the assembled trap using a dial indicator to verify thelocations of the radial electrodes. First, they inserted the indicator along the Z axis of the trapwith the top endcap removed, and after running the indicator along the inner edge of the top
and bottom ground planes to verify that the trap was mounted flat, they then measured thedifferential radial displacement of each electrode near the midplane of the trap (“radial”). Second,
they measured the relative vertical displacement of each of the electrodes’ top surfaces, just inside
the ground plane (“top”). Third, they ran the indicator along the back face of each electrode tocheck that they were mounted vertically (“back”). Finally, they ran the indicator vertically from
top to bottom along either side of each electrode to measure their tilt (“side”). Kim recalled fromthe fabrication process that the back face was not particularly well registered to the mounting
holes of the electrodes, so the “back” measurement is to be trusted least. With each measurement,

108

Figure 4.3: Percent deviation in Erot in the XY and XZ planes in the Generation 2 ion trap,
averaged over one rotation period. Plots generated using multipole fit, not direct Comsol solution.

17.8
COMSOL
multipole

17.6

17.4

17.2
-40

-20

0

20

40

-40

-20

0

20

40

-60 -40 -20

020 40 60

Figure 4.4: Comparison of Comsol solution for Erot with multipole fit to ` = 9, for Vrot = 1 V peak.
Plots are for Erot averaged over one rotation period.

109
electrode
1
2
3
4
5
6
7
8

radial
1.5
0
0
0
0
0
0
1.5

top
-1.5
-1.5
-2.5
-3
-3
-0.5
-0.5
0

back
3.5
-3
-2
11
-0.25
2
2
4

side
4.25
4
2.25
1.5
-0.75
-0.5
2
2

Table 4.2: Results of dial indicator measurements of the radial electrode positions by Yuval andKim. The physical meaning of each measurement type is described in the text. All measurements
are in mil (0.001 inch). The “back” measurement for electrode 4 has a large deviation, which was
anticipated by Kim. However, the back of each electrode was not used as a mechanical reference;
rather, the mounting holes are the references surfaces, so this measurement is not of concern.

they performed minor adjustments by slightly loosening and re-tightening the mounting bolts, inorder to try to minimize the runout of each electrode relative to the average. The results of thesemeasurements are listed in Table 4.2. Overall, the measurements indicate a cos2 φ -like deviationof the height of the electrodes, and a slight overall twisting of the trap from top to bottom. Whilethese deviations are likely too small to notice their effects on the trapping and ponderomotive
potentials, these measurements may be important when considering geometric phase effects for the
Generation 2 eEDM measurement.
Experimentally, we investigated the uniformity of Erot following the Generation 2 trap installation in Fall 2017 via its effects on ion motion through the ponderomotive potential [Eq. (4.1)], andthrough ion loss at the extremes of ion cloud position. Figure 4.5 shows potential energy surfaces
along various trap axes, illustrating the anti-trapping behavior of Erot . Evidently, a uniform applied
electric field will result in a shift of the potential minimum away from (0, 0, 0) and a reduction in
the trap depth. A large enough applied field will result in loss of ions from the trap.

110

1000
harm
trap
pond
tot

500
0
-500
-40

-20

0

20

40

-40

-20

0

20

40

-40

-20

0

20

40

-60 -40 -20 0

20 40 60

Figure 4.5: Potential energy surfaces for trapped HfF+ ions in the Generation 2 ion trap, inunits of Kelvin. The RF trapping ponderomotive potential (blue) closely approximates a harmonic
trap (black) near the geometric center of the trap. The ponderomotive potential imposed by Erotinhomogeneity (red) distorts the total potential (yellow) and tends to decrease the trap depth. For
this figure, Vrf = 15 V peak, Vrot = 375 V peak, and Vdc = 0.5 V.

4.4

Trap driver electronics
In our studies of the effects of the effects of Erot amplitude and ion number on coherencetime, we found that coherence time scales approximately as [28]
τ∼

2.4
Erot
,
NHfF+

(4.4)

for fixed values of the trap secular frequencies and ion cloud temperature. To improve our statisticalsensitivity in Generation 2, we planned to increase NHfF+ and decrease the trap frequencies, as
discussed in Section 4.1. However, we also opted to improve our chances for long coherence times
with high ion numbers by developing new ion trap driver electronics that could provide a higher
value of Erot . Additionally, we aimed to consolidate our trap controls and reduce RF pickup on theion trap by developing new low-voltage trap waveform synthesis boards. We undertook these tasks
with the help of Terry Brown, Felix Vietmayer, and Carl Sauer from the JILA electronics shop.

4.4.1

Synthesis boards

The design of our Generation 2 trap synthesis boards was initiated by Matt Grau before
his graduation, and taken over and completed by Felix Vietmayer. The synthesis boards are built
around the AD9959 direct digital synthesis (DDS) integrated circuit (IC) from Analog Devices,which is able to simultaneously generate four sinusoidal waveforms with digitally tunable amplitude,

111
AD5433
MultDAC

AD5543
MultDAC2x2xAD5542
ShimDAC

AD5542
KickDAC

Figure 4.6: Schematic layout of the eEDM Generation 2 ion trap synthesis board. A Spartan 6
FPGA acts as a programmer and sequencer. We communicate with the FPGA via a USB-to-Serialinterface chip from FTDI. The FPGA controls an AD9959 DDS, each of which has an AD5543
multiplying DAC, and two DACs to generate Shim and Kick voltages. Each channel has an analog
switch to turn it on and off. All six channels are summed to produce a single analog output channel
for each board.

phase, and frequency. Alongside the DDS IC, we have a pair of digital to analog converters (DACs)
that generate DC signals for fine adjustments of ion cloud position (“shims”) and sudden voltage
jumps that suddenly impart a large velocity to the ion cloud (“kicks”). The DDS and DACs foreach electrode are controlled by a field-programmable gate array (FPGA, one per electrode), whichorchestrates the sequence of voltages to be applied. Our control software, which is based in Labview,communicates with the FPGA through a USB-to-serial interface chip from FTDI. The layout of
the synthesis boards is shown schematically in Fig. 4.6.
Programming and testing of the synthesis boards was carried out by Felix Vietmayer and Dan
Gresh in Summer 2017. To communicate with the synthesis boards, we use the functions FT Write
and FT Read from the FTDI library ftd2xx.dll, which can be called from any Labview script (orfrom scripts written in C, C++ and a wide variety of other languages). Felix has programmed
the FPGAs to expect data in a particular 7-byte format, implemented in Labview as shown in
Fig. 4.7. We specify a “device” such as the AD9959 DDS, ShimDAC, or KickDAC; a memory
address containing a parameter such as the start value of a voltage ramp or a DDS frequency;
and finally the value to be assigned to that address. Using this tool, we write all of the sequenceparameters to each board, then trigger all boards to output their sequence simultaneously using
our Viewpoint DIO64. Documentation for the memory addresses of all sequencer parameters have

112

Figure 4.7: Labview code for writing to a Generation 2 ion trap synthesis board with identifier“handle.” Each data packet contains 7 bytes. The first byte has the format 10YX XXXX, where Y
specifies read (0) or write(1), and XXXXX specifies a device numbered 0-31. The second two bytes
specify the memory address on the device, and the last four bytes carry the data to be written.
The DLL function call on the right hand side is to the function FT write in the FTDI library
ftd2xx.dll.

been provided by Felix in a pair of documentation files, “wb slave AD9959.html” for the AD9959(device 1) and “edm seq register level documentation.html” for the sequencer and DACs (device
0).

4.4.2

Amplifiers

Terry Brown designed our Generation 2 trap driver amplifiers, which are built around theApex PA94 op-amp. These op-amps can accommodate ±450 V rails, as compared to the ±200 V
rails that were used in Generation 1 with PA98 op-amps [82]. Assuming that we operate withthe rails set to 425 V for a margin of safety, and dedicate 25 V worth of amplitude to the RFtrapping fields and shim voltages, the resulting Vrot = 400 V will generate Erot = 69.9 V/cm in
the Generation 2 ion trap – a factor of 2.9 increase from Generation 1. The drawback of the PA94
is that it has a lower specified slew rate limit of dV /dt = 700 V/µs, as compared to 1000 V/µs
specified for the PA98. In a basic amplifier design operating at Vrot = 400 V, this would result in a
maximum value of ωrot ≈ 2π × 280 kHz. In Generation 1, our typical values of ωrot were between
2π × 150 kHz and 2π × 250 kHz, however for systematic checks we wanted as much simultaneous
range on both Erot and ωrot as possible. To help in this area, Terry added two advanced design

113
features to the feedback loop built around the PA94. The following two paragraphs describing
these features are essentially paraphrasing a discussion about these features.
The first feature Terry included in the Generation 2 trap amplifiers is a composite amplifier,which uses a pair of op-amps with different frequency response characteristics to improve the overall
response of the amplifier as a whole [130]. Our amplifiers use the AD4898, which is a high-speed,
low-noise op-amp that should improve the noise characteristics of the amplifier at low frequencies,
and may also reduce harmonic distortion.
Secondly, Terry created a circuit design where current from the feedback loop is injected into
the compensation pin (pin 6) of the PA94, whereas in their suggested design this pin is simply connected by a compensation capacitor to part of the input stage (pin 4). The compensation capacitor
is typically chosen to maintain stability depending on the choice of gain, but also detrimentallyaffects the slew rate limit when the amplifier is driving a capacitive load. Terry’s design essentiallybypasses the input stage of the amplifier to drive the output stage directly, reducing crossover distortion and ringing in the step response when operating close to the slew rate limit. A limitationof this design is that the compensation pin can only access the output stage asymmetrically, so theslew rate limits for dV /dt > 0 and dV /dt < 0 are asymmetric in the final design. This asymmetryis visible on an oscilloscope trace of the amplifier output when attempting to drive the amplifier
beyond the slew rate limit (or under normal conditions when the amplifier is not properly heat-sunkand begins to heat up).

Chapter 5

State preparation

One of the major inefficiencies of the Generation 1 eEDM experiment with HfF+ was in state
preparation, starting at the very beginning of the experiment sequence with the initial rotational
distribution in the 1 Σ+ state. As shown in Table 3.7, the combined efficiency of ionization and
Raman transfer was about 10%. Unfortunately, state preparation was also one of the areas whereat the conclusion of the Generation 1 data collection, we had limited promising ideas for dramatic
improvement. It took several unsuccessful attempts at coherent state preparation via stimulated
Raman adiabatic passage (STIRAP) and adiabatic rapid passage (ARP) for us to realize thatincoherent methods were not only easier to apply in our system, but dramatically more effective atconcentrating population in the 3 ∆1 state.

5.1

Coherent methods
Adiabatic techniques are widely applied for coherent internal state manipulation of atoms

and molecules. Among the most commonly applied are ARP, which involves a frequency sweep ofan electromagnetic driving field, and STIRAP, which involves amplitude modulation of a pair oflasers [126]. STIRAP was notably applied by the ACME Collaboration to improve their signal sizeby a factor of 14 in their second-generation measurement [2, 95]. As the name suggests, STIRAP
involves adiabatically inducing a Raman transition between two low-lying states |g1 i, |g2 i (Fig. 5.1)
using a pair of phase-coherent lasers of different frequencies that are detuned from resonance withan excited state |ei that is subject to spontaneous decay with rate Γ. The intensities of the lasers

Rabi frequency (MHz)

115
6
1

4

2

2
0

Population

1

0.5

0
-60

-40

-20

0

20

40

60

t (μs)

Figure 5.1: Simplified picture of STIRAP in a three-level system. Here |g1 i represents 1 Σ+ (v =0, N = 0), while |g2 i represents 3 ∆1 (v = 0, J = 1), and |ei represents 3 Π0+ (v = 1, J = 0). The
pump laser Ω1 or “Stella” has λ ≈ 900 nm, and the stokes laser Ω2 or “Toptica” has λ ≈ 986 nm.

are modulated, creating an intensity-dependent dark state that adiabatically changes characterfrom |g1 i to |g2 i, maintaining zero amplitude for populating |ei and thus avoiding spontaneousemission.
In HfF+ , the 3 Π0+ , v = 1, J = 1 state was chosen as an intermediate state for coherent
transfer because it has a moderately strong coupling to both 1 Σ+ , v = 0, J = 0 and 3 ∆1 , v = 0,
J = 1, and because both of these transitions had been observed in near-IR survey spectroscopy of
HfF+ by Laura Sinclair, Kevin Cossel, and Dan Gresh [29]. These transitions have wavelengths ofapproximately 899.6665 nm and 986.4175 nm, respectively, both of which can be addressed withcommercially available laser diodes. The details of our transfer laser setup and stabilization are
described in Ref. [28].
Successful application of STIRAP can lead to population transfer efficiencies between lowlying states in excess of 90% [126], however it has stringent requirements that make it difficult
to implement in general, and in our system in particular. Specifically, adiabaticity and phasecoherence are the two main competing requirements for successful population transfer. Adiabaticity
requires high Rabi frequencies and low rates of change of the Rabi frequencies, and can be roughly
characterized by two requirements, one “local” and one “global.” The local requirement, which

116
ensures the Rabi frequencies do not change too quickly, ishi1/3Ωeff (t)  Ω1 (t)Ω̇2 (t) + Ω̇1 (t)Ω2 (t)
,

(5.1)

p
where Ωeff (t) = Ω1 (t)2 + Ω2 (t)2 . The global requirement, which ensures large enough pulse areas,R∞is −∞ Ωeff (t) dt  π/2 [127]. In the HfF+ eEDM experiment, a major challenge for STIRAP is thehigh temperature of the ions, which creates Doppler widths of ≈ 40 MHz for near-infrared (NIR)
one-photon transitions and ≈ 5 MHz for the 1 Σ+ → 3 Π0+ → 3 ∆1 Raman transition. The ion
velocities contributes to the two-photon detuning, increasing the intensity requirement for efficient
STIRAP [127]. In our most recent STIRAP attempts led by Yan and Yuval in Summer 2017
following the completion of the Generation 1 eEDM measurement, we were not able to observe any
coherent transfer, which is likely attributable to high ion temperatures and low Rabi frequencies.
We also attempted a Raman ARP, which potentially avoids the issues of temperature bysweeping the second transfer laser’s frequency across the entire Doppler width, thereby addressing
the entire molecular population with the full two-photon Rabi frequency. To sweep Toptica’s
frequency, we simply applied one period of a triangle wave to its external modulation port, which
has a spec bandwidth of 5 MHz. Both Raman lasers were locked to the stable cavity, and Toptica’s
lock does not have the bandwidth to respond to the modulation on this timescale. We explored theparameter space of transfer fraction versus laser intensity and detuning as shown in Fig. 5.2. These
tests were performed in the absence of Erot , so the maximum transfer fraction we could obtain with
the incoherent method described in Ref. [28] is 2/3.
We fit the results of our ARP explorations with a semi-analytical model that combines theLandau-Zener transfer efficiency of ARP with the effects of decoherence due to laser phase noiseor spontaneous scattering from the intermediate state, as described in Refs. [75, 93]. Following
Ref. [93], we take the probability of transfer to be the sum of a contribution from incoherent
transfer with a maximum probability of 2/3 given by the state degeneracy, and a coherent partproportional to the Landau-Zener transfer probability. Because in Fig. 5.2 we looked only for loss
from 1 Σ+ (v = 0, J = 0), we also must multiply by a probability of spontaneous scattering. We

117
obtain a model equation for the population lost from 1 Σ+ (v = 0, J = 0) as a function of sweep
rate,

 


2
2
− πΩ
−πγΩ/α−πγΩ/αe−Rsc t ,Ploss =
1 − e 2α
1−e
+e
3

(5.2)

where α is the energy sweep rate of the second transfer laser, Ω = Ω1 Ω2 /(2∆) is the 2-photon
Rabi frequency, ∆ is the 1-photon detuning, and Ω1,2 are the 1-photon Rabi frequencies. The
spontaneous scattering rate is
 
Γ
I/I0
Rsc =
,
2 1 + (2∆/Γ)2 + I/I0

(5.3)

with excited state linewidth Γ, first photon (Stella or Ω1 ) laser intensity I, and saturation intensity
I0 .
Throughout Fig. 5.2, we plot the fraction of molecules lost from 1 Σ+ (v = 0, J = 0) as afunction of the sweep duration Tsweep ∝ 1/α. In Fig. 5.2a, we assumed a flat intensity profile
across the ion cloud, and fit the transferred fraction with γ, Ω1 , and Ω2 as free parameters. Thenin Fig. 5.2b, we reduced both laser powers by half, and found that the experimental results fortransfer fraction as a function of sweep time did not qualitatively match the model expectation.
Dashed lines in parts (a) and (b) of Fig. 5.2 also show the factors contributing to the overall
probability, including the probability of surviving spontaneous emission and the Landau-Zener
probability. Fig. 5.2c-5.2d, we found significantly better agreement between our model and theexperiment by including the effects of finite laser beam size and finite ion cloud size. We expressed
the overall transfer probability as an integral over the X-Y plane, where Ω1 (X, Y ) and Ω2 (X, Y )are 2D Gaussian functions and the transfer probability is weighted by the ion density ρ(X, Y ).
In all of our model fits to attempted ARPs, the parameter γ characterizing decoherence would
essentially diverge. We interpreted this to mean that we were far from the region of parameter
space where we could attain high efficiency transfer. This can also be anticipated from the factthat we attained our highest transfer fractions in the vicinity of 1 ms sweep duration, which islong enough that the ions’ velocities would undergo a full cycle of oscillation in the Z axis. In thisregime, we expect the model of Eq. (5.2) to break down, and we cannot reasonably expect high118transfer efficiency. However, the good agreement between experiment and our model in the shorttransfer duration region near 0.1 ms, where transfer is well approximated by the Landau-Zener
formula alone, can help us to conclude that to achieve high transfer efficiency we would need ordersof magnitude more laser intensity or a stronger transition. Unfortunately, higher intensity is notavailable, and we soon found that incoherent transfer was a better option.

5.2

Rotational cooling & vibrational decay
Upon ionization, the HfF+ population is spread approximately equally over rotational levels

J = 0 through J = 2, plus a little bit in J = 3 – comprising 32 total quantum states, when nuclear
spin is accounted for. About 20% of the total HfF+ population is in 1 Σ+ , v = 0, J = 0, which
has two nuclear spin states. If we use only coherent methods of state preparation, we can expect
to prepare at best 10% of the total ion population in a single |F, MF , Ωi state of 3 ∆1 , v = 0,
J = 1. With a single quantum projection step and unit detection efficiency, we could then in
principle detect 5% of the total ion population as eEDM signal. On the other hand, if we use aform of dissipative state preparation that removes entropy from the initial ion ensemble throughspontaneous emission, we might hope to significantly improve our overall efficiency. Vibrational
branching ratios are not in our favor in HfF+ , so tasks requiring thousands of photon cycles such as
laser cooling are certainly out of the question. However, in the eEDM spirit of trying new things,
in Summer 2017 after the completion of the Generation 1 dataset, we decided to try some basicrotational cooling in 1 Σ+ , v = 0, using the 3 Σ−
, v = 0, J = 1 excited state. The motivations for0+
attempting rotational cooling through 3 Σ−
were0+
• It has a strongly allowed electric dipole transition with 1 Σ+ (D = 0.23 e a0 theoretically[28]) and a weak electric dipole transition with 3 ∆1 (D = 0.08 e a0 theoretically). Thus,
contamination of 3 ∆1 excited vibrational states should in theory be kept to a minimum.
• It has relatively favorable Franck-Condon overlap with 1 Σ+ , as shown in Table 5.1.
• It is an Ω = 0 state of definite parity, so only P and R branches exist in the transition

119

(a)

(c)

(b)

(d)

Figure 5.2: Tests of the model from Eq. (5.2) for chirped Raman adiabatic rapid passage with
HfF+ . In (a) and (b), we did not account for the cloud side and laser beam profile. In (a), we
calibrated the parameters of Eq. (5.2). In (b), we reduced each laser power by half and found thatthe model did not fit observations. In (c) and (d), integrating Eq.(5.2) over a spatially varying
laser intensity and ion density was able to account for the observed variation in transfer probability
with laser power and detuning.

120

v0
v 00
0
1
2
3

0
0.66480.2771
0.05190.00571
0.26410.22980.36190.1222

20.05950.32450.04300.3364

Table 5.1: Franck-Condon factors for the 1 Σ+ ↔ 3 Σ−
system of HfF+ , assuming a Morse potential.
0+Spectroscopic constants for both electronic states are given in Ref. [28].

with 1 Σ+ . This removes one possible decay path, allowing for higher-efficiency rotationalcooling.
(0, 0) transition has wavelength λ ≈ 754 nm, which can be reached by• The 1 Σ+ ← 3 Σ−0+our cw Ti:Sapph laser, as well as a diode available from Toptica.
The 1 Σ+ ← 3 Σ−(0, 0) transition was observed and fitted in Comb-vms survey spectroscopy0+of HfF+ conducted by Laura Sinclair, Kevin, and Dan. The fit constants, found in Ref. [28], Table
5.2 line 12, give that the desired P (2) line will be found at ν = 13253.053(7) cm−1 . By scanning
the Ti:Sapph by hand while monitoring the dissociated population from 1 Σ+ , v = 0, J = 2, wefound the P (2) transition at ν = 13253.059(1) cm−1 , or λ = 754.54280(6) nm, consistent withthe fit within the reported 1σ statistical error bar. In addition to a laser resonant with the P (2)transition, we also added microwaves (with frequencies in multiples of 18277.55 GHz) to couple
the J = 1, 2, 3 states of 1 Σ+ , v = 0, causing the entire population of these three rotational states
to be electronically excited and (potentially) decay to J = 0. While in molecular laser coolingexperiments, this method of microwave remixing can be detrimental to MOT performance because
it reduces the scattering rate [26, 134], this slowdown is relatively small compared to our coherence
times of ∼ 2 s, and should not significantly impact our statistical sensitivity.
The results of rotation cooling with and without microwave remixing are shown in Fig. 5.3.
With microwave remixing, we attained about a factor of 2 increase in the J = 0 population. This
represented a modest win, and was our first foray into incoherent state preparation. At this earlypoint in the Generation 2 experiment, we had not yet returned to dissociation of the 3 ∆1 state,

121

2500
DissociationOP
OP + Microwaves

2000

1500

1000

500

0
7.6

7.8

8.0

8.2

8.4

8.6

8.8

9.0

Figure 5.3: R branch of the REMPD spectrum of HfF+ from its 1 Σ+ ground state through anΩ0 = 0 intermediate state at ν0 ≈ 35976 cm−1 , with chops showing rotational cooling. Not shown:enormous background in 3 ∆1 , v = 0, J = 1.

and so we did not yet know of the unfortunate news waiting for us there – vibrational decay!
As discussed in Section 2.2.4 and in early theoretical work by Petrov et al. [99], the spontaneous decay lifetimes of excited vibrational states of 3 ∆1 have lifetimes on the order of ∼ 100 ms(Fig. 5.4). By contrast, the lifetime for decay of 3 ∆1 to 1 Σ+ manifold is on the order of 3 s, fromour latest measurements. Any population reaching the excited vibrational states of 3 ∆1 early inthe experiment sequence will tend to decay to 3 ∆1 , v = 0 by the end of an experiment cycle ∼ 1.5 slater, and some of this population will end up in J = 1. That population will be dissociated, anddetected as a background, reducing the contrast of our eEDM signal and hurting our signal-to-noise
ratio. This was precisely the situation we found upon performing optical pumping through 3 Σ−.
0+Even with the 814.5 nm depletion laser depopulating 3 ∆1 , v = 0, J = 1 for the first ∼ 100 ms
of trap time, hundreds of ions would again be detected in 3 ∆1 , v = 0, J = 1 several hundred mslater – a sure sign that population was reaching excited vibrational states. This was a substantial
disappointment, and essentially put a stop to 754 nm rotational cooling. Fortunately, this initialinvestigation into incoherent state preparation inspired bigger and better ideas in first the ThF+
experiment and subsequently in HfF+ .

122

80

60

40

20

0
0

200

400

600

800

1000

Figure 5.4: Repopulation of 1 Σ+ (v = 0, N = 0) by vibrational decay after laser depletion, detectedby photodissociation. A similar repopulation will occur in 3 ∆1 (v = 0, J = 1), leading to a spurious
background in an eEDM measurement.

123

5.3

Incoherent state preparation
While our rotational cooling on the 3 Σ−
← 1 Σ+ (0,0) P(2) transition was a limited success,0+

it provided the inspiration for a more extensive optical pumping-based state preparation schemein ThF+ by Yan and Kia Boon, which we readapted for use in HfF+ (Fig. 5.5). The general ideaof this scheme in HfF+ is to drive the 3 Π0+ ← 1 Σ+ (0,0) P(1) transition at 961.434 nm withsimultaneous application of microwaves that link N = 0 through N = 3 in 1 Σ+ , thereby removing
all 1 Σ+ , v = 0 population and nominally transferring it to 3 ∆1 , v = 0. In reality, of course, somedecay from 3 Π0+ , v = 0 will end up in excited vibrational levels of both 1 Σ+ and 3 ∆1 . Populationin excited vibrational levels of 1 Σ+ will simply be lost, but the excited vibrational levels of 3 ∆1
must be depleted to prevent background due to their decay. We do this through the P(1) rotational
state, which primarily sends the population back to 1 Σ+lines of ∆v = 0 transitions to the 3 Σ−0+and eliminates the risk of background. To develop this scheme, we had to search for several newtransitions that had not previously been observed in HfF+ , and build or acquire cw lasers at several
new NIR wavelengths.
For the first step of this state preparation scheme, no new lasers were required. We simply
tuned our Toptica laser that had previously been driving the 3 Π0+ ← 3 ∆1 (1,0) Q(1) transition at
986.4 nm over to the 3 Π0+ ← 1 Σ+ (0,0) band at 961.4 nm, which had been observed in comb-vms(Ref. [28], Table 5.2). We also already had microwaves set up to address the N = 0 ↔ 1, 1 ↔ 2,
and 2 ↔ 3 transitions in the ground vibrational level of 1 Σ+ at multiples of 18277.55 MHz. We
verified that this optical pumping was successful by REMPD of 1 Σ+ , v = 0 at a first dissociation
photon wavelength of 27154 cm−1 while chopping the 961 nm laser and the microwaves on and off,
as shown in Fig. 5.6.
Between Generations 1 and 2, the experiment underwent simultaneous changes to both ourstate preparation method and our REMPD transition, and thus we cannot exactly distinguish the
improvement in state preparation and the slight decrease in our readout efficiency (discussed in
Section 6). Nonetheless, we can compare the number of ions we detected in 3 ∆1 (v = 0, J = 1)

124

Coriolis MF coupling

Figure 5.5: Incoherent state preparation scheme for HfF+ . Population is pumped out of 1 Σ+using a combination of laser light on the 3 Π0+ ← 1 Σ+ (0,0) P(1) transition (red) and microwaves
connecting rotational levels in 1 Σ+ , v = 0 (green). Population in excited vibrational states of 3 ∆1
← 3 ∆1 (1,1) and (2,2) transitions (blue dash and dash-dot). In
is cleaned up by lasers on the 3 Σ−0+
the ground rovibrational state of 3 ∆1 , quasi-cycling MF pumping is performed on the 3 Π0− ← 3 ∆1
← 3 ∆1 (0,0) Q(1) transition(0,0) P(1) transition (purple), and depletion is performed on the 3 Σ−0+(blue).

1200
1000
800
600
400
200
0
27152

27152.827153.627154.427155.2

Figure 5.6: REMPD spectrum of 1 Σ+ , v = 0 through an Ω = 1 intermediate state. Red shows theinitial distribution, with population distributed from N = 0 to N = 3. Adding the 961 nm transfer
laser on the P(1) line to 3 Π0+ causes depletion of N = 1, evidenced by disappearance of the R(1)line. Further adding microwaves gives the purple trace, where only a broad background is visible.
We believe this background arises from vibrational decay that occurs between the end of transfer
and dissociation.

125
before MF pumping. In Generation 1, this number was on the order of 50 detected ions spread
between two MF levels, while in Generation 2 we detect about 2500 ions spread amongst 12 MF
sublevels.

5.3.1

MF pumping with 3 Π0−

To prepare a pure MF state in 3 ∆1 , we need to do further optical pumping with strobed,
circularly polarized light applied from the side of the ion trap. In Generation 1, this was done
only on the 3 Σ−
← 3 ∆1 (0,0) P(1) transition at 814.5463 nm. However, a greater population in0+
3 ∆ , v = 0, J = 1 can be maintained if we instead use the 3 Π , v = 0 state, which due to parity
10−

selection rules, forms an electronically closed system with 3 ∆1 . We therefore decided to locate
the 3 Π0− ← 3 ∆1 (0,0) transition, which had not been previously observed in velocity modulation
spectroscopy as at about 1083 nm, it was too far into the infrared.
While the ground vibrational state of 3 Π0− had not been observed, the comb-vms resultsin Kevin’s thesis [28] do list the 3 Π0− ← 3 ∆1 (1,0) transition with ν0 = 9948.624(6) cm−1 . The∆G1/2 value (the energy difference between the first two vibration levels) for 3 Π0− is not known,
however assuming it is approximately the same as 3 Π0+ , we would expect to find the 3 Π0− ← 3 ∆1
transition at ν0 = 9235.694(16) cm−1 or λ0 = 1082.756(2) nm. This is a very fortunate wavelength,as DBR lasers designed for metastable Helium spectroscopy are readily available in the 1083 nm
region from Eagleyard and Photodigm. Also fortunately, 3 Π0− ← 3 ∆1 is a strongly allowed electric
dipole transition with favorable Franck-Condon factors (Table 5.2), so a few tens of mW from a
DBR laser is acceptable for depletion, in contrast to the hundreds of mW needed for depletion at
814.5 nm.
We first attempted to find the 3 Π0− ← 3 ∆1 (0,0) transition with a DBR laser borrowed
from Yuval’s former group, that of Edvardas Narevicius at the Weizmann Institute. We populated
3∆

1 using 961 nm laser light, then looked for depletion of the

3 ∆ , v = 0 population detected via
1

dissociation as a function of the DBR laser wavelength. Unfortunately, the DBR laser from the
Narevicius group was centered too far towards the red, and we were unable to find the transition

126

v0
v 00
0
1
2
3

0
0.69160.26580.03950.0001

1
0.24220.27260.37450.00062
0.05400.30450.06990.1635

Table 5.2: Franck-Condon factors for the 3 Π0− ↔ 3 ∆1 system, assuming a Morse potential. Here I
have used the known value of ∆G1/2 = 716.0 cm−1 for 3 Π0− and assumed that ωe xe has the samevalue as for 3 Π0+ , approximately 3.3 cm−1 . Molecular constants are found in Ref. [28].

without pulling the laser temperature dangerously close to the dew point, around 10 Celsius. We
then purchased a DBR centered around a bluer wavelength, and we were able to locate the bandin the vicinity of 1082.3 nm. The precise values of the P(1), Q(1) and R(1) lines are recorded inTable 5.3. From these values, we find that the band origin is ν0 = 9239.2090(7) for this transition,and ∆G1/2 = 709.415(6) cm−1 for the 3 Π0− state. Having only observed a pair of vibrationaltransitions, we cannot estimate ωe and ωe xe for the 3 Π0− state, but if ωe xe ≈ 3.3 cm−1 (the samevalue as 3 Π0+ ), then ωe = ∆G1/2 + 2ωe xe ≈ 716.0 cm−1 . This result is in agreement with thetheory from Skripnikov et al. in Ref. [29].
Having found the 3 Π0− transition, we can then use the P(1) line to perform quasi-cycling
optical pumping of MF in order to accumulate population in a single stretched MF level of 3 ∆1 ,v = 0, J = 1, as illustrated in Fig. 5.5. Using the P(1) transition keeps the transition closed withrespect to the rotational degree of freedom, and prevents population from decaying to J = 2. Dueto the presence of Ω doubling, both Stark doublets in 3 ∆1 (which have completely mixed parity inthe presence of Erot ) are populated equally by decay from 3 Π0− . While this would have presented anissue in the Generation 1 experiment, it works well in combination with our Stark doublet-selective
Line
P(1)
Q(1)
R(1)

λ (nm)
1082.4137(1)1082.3472(1)1082.2139(1)ν (cm−1 )9238.6118(9)9239.1794(9)9240.3174(9)Table 5.3: Observed line positions for the 3 Π0− ← 3 ∆1 (0,0) band.

127

Figure 5.7: Timing of state preparation and readout. The 961 nm laser and microwaves transfer
population to 3 ∆1 , and remain on during MF pumping to send any population that returns to 1 Σ+(for example from vibrational decay) back to 3 ∆1 . After the Ramsey sequence, the 814.5 nm laser
depletes all but one MF level in 3 ∆1 .

readout, which is decribed in Section 6.

5.3.2

Polarization measurement

While the 3 Π0− state is ideal for preparation of a large population in a single MF level, its
quasi-cycling nature is not optimal for readout of the MF level populations. For state readout,
we would like to be able to selectively remove population from 3 ∆1 , J = 1 via an electronic statethat has a very small probability of returning via spontaneous emission. The best accessible state, which is predicted theoretically to decay with 88% probability
for this purpose in HfF+ is 3 Σ−0+
to 1 Σ+ rather than 3 ∆1 [28] (in the approximation that these are the only states it decays to).
, v = 0 with considerable success for several years [52],We have performed depletion through 3 Σ−
0+using the Coherent cw Ti:sapphire previously used by Dan, Kevin, and Laura in the discharge lab
tuned to the transition at 814.5463(1) nm. Due to the usefulness of the Ti:sapph as a tunable laser
for spectroscopy work, we purchased a new Toptica TA Pro system at 814.5 nm for the Generation
2 experiment. While previously this transition was used both for state preparation (before theRamsey sequence) and readout, we now use it only for readout after the 2nd π/2 pulse.
Figure 5.7 shows the timing sequence of our state preparation and readout. First, 961 nm
and microwaves are applied to transfer population to 3 ∆1 , as described in Section 5.3. We thenperform MF pumping with 1082 nm, which pumps all population in 3 ∆1 , v = 0, J = 1 into a single

128
MF level in each stretched Stark doublet. Figure 5.8 shows the dissociation yield from 3 ∆1 J = 1 asa function of MF pumping and depletion durations. The cycling nature of the 1082 nm transitionis visible in the increase of dissociation yield in the σ + σ + chop as a function of pumping duration.
The comparable timescales of each process despite the very different laser powers highlight the very
different transition strengths of 3 Π0− ↔ 3 ∆1 and 3 Σ−
↔ 3 ∆1 .
0+
5.3.3

Vibrational cleanups

The excited vibrational states of 3 ∆1 are populated by decay of 3 Π0+ in proportion to the(0, v 00 ) Franck-Condon factors given in Table 5.7 of Ref. [28]. Because we drive only the P(1) linewith our 961 nm laser in transferring population from 1 Σ+ to 3 ∆1 , only the J = 1 levels of each
vibrational state in 3 ∆1 should initially be populated. As discussed in Section 2.2.4, these levels
decay with lifetimes on the order of ∼ 100 ms, which presents a source of background for our secondstate all
scale eEDM measurement. Happily, the progression of ∆v = 0 transitions to the 3 Σ−
0+have similarly favorable wavelengths to our projection laser at 814.5 nm, with the (1,1) and (2,2)
transitions being found at approximately 818.4 nm and 822.4 nm, respectively. The AR-coatedlaser diodes from Eagleyard can easily be tuned to these wavelengths in a homebuilt ECDL, andbecause we are simply performing incoherent optical pumping on a timescale of many milliseconds,
a few mW of each to the chamber is enough for our purposes. We implemented two vibrational
cleanup lasers for these purposes, which enabled the high depletion contrast shown in Fig. 5.8b.
Our initial vibrational repump scheme as shown in Fig. 5.9a has us driving the P(1) lines
to 3 Σ−(v = 1, 2) out of 3 ∆1 (v = 1, 2), respectively. We chose these transitions based on the idea0+that maintaining rotational closure is very important to attaining high overall efficiency. While
this is generally true, we have not yet implemented any repumping lasers to recover the population
initially sent to excited vibrational levels of 3 ∆1 , and this population is essentially just considered
to be lost. As a result, we are not particularly concerned with driving only P(1) transitions, andwe realized that we can in theory do better at preventing background by driving the P(1) and P(2)transitions from 3 ∆1 (v = 1), as shown in Fig. 5.9b. Empirically, we have not observed a significant

129

1500
1500

1000

500

1082 nm

815 nm1082 nm

815 nm

1000

500

0

0
0

20

40

(a)

60

80

100

0

20

10

30

40

(b)

Figure 5.8: Timescale of strobed circularly polarized state preparation and readout. (a) MF pumping on the 3 Π0− ← 3 ∆1 (0,0) P(1) transition with about 50 mW of 1082.3 nm laser light, with afixed duration of MF depletion of 40 ms. Inset: Timing sequence of σ + σ + and σ + σ − chops. (b)← 3 ∆1 (0,0) P(1) transition using about 400 mW of 814.5 nm laser
MF depletion on the 3 Σ−
0+light, with a fixed duration of MF pumping of 60 ms. Both beam waists are roughly 5 mm.

130
difference between these two methods, in particular because the dissociation background is a muchstronger function of the intensity of the two REMPD lasers.
There remains one background phenomenon related to our incoherent state preparation thatis not well understood. This is a REMPD background that is independent of the first REMPD
photon wavelength, leading us to think that it is off-resonant 2-photon dissociation of excitedmolecules by 266 nm. This background appears only if the 961 nm laser drives the 3 Π0+ ← 1 Σ+transition, and it is larger if we drive the R branch of this transition than if we drive the P branch.
All of this is consistent with the idea that vibrationally excited 3 ∆1 molecules can be off-resonantly
dissociated by two photons of 266 nm, however the more mysterious part is illustrated in Fig. 5.10:
The 818 nm vibrational cleanup lasers tend to reduce this background substantially. To my eyes,
the most probable explanation for this is that these vibrationally excited 3 ∆1 molecules can be
dissociated by Cora+266 or by 266+266, so that both the sharp transitions shown in Fig. 5.10 andthe broad background arise from the same population.

5.4

Sensitivity improvement
In the HfF+ eEDM experiment, absolutely efficiencies are often slightly squirrely quantities.

Even when pulsed laser fluences are carefully measured, the beam profiles tend to change dramatically upon dye laser realignment, and beam pointing is only controlled on the millimeter scale; thesame scale as the ion cloud size. The nonlinear nature of frequency doubling of pulsed lasers also
tends to exacerbate intensity inhomogeneities across beam profiles, which are further convolved
with their overlap with the ion density. As a result of all these factors, the overall efficiency of ourstate preparation and detection is most easily characterized by the number of Hf+ ions detected
as eEDM signal, and is least subject to errors when averaged over days or even weeks. Thus, oursensitivity improvement in going from Generation 1 to Generation 2 can be summed up in the comparison of two numbers: Averaged over the Generation 1 eEDM dataset, our Hf+ signal was about
7 ions per experiment cycle at late Ramsey times (where we get most of our phase sensitivity).
In Generation 2 as of May 2019, we typically count about 400 ions per experiment cycle at late131(b) Improved scheme(a) Original scheme

Figure 5.9: Paths of rovibrational spontaneous decay from initially populated J = 1 levels in the+
1 electronic state of HfF , showing (a) original and (b) improved vibrational cleanup schemes
for preventing background in 3 ∆1 (v = 0, J = 1). The J = 1 levels in several vibrational states are
populated by transfer from 1 Σ+ and MF pumping in 3 ∆1 . In the original scheme, some population← 3 ∆1
could still reach v = 0, J = 1. In the improved scheme, cleanup lasers (blue) driving the 3 Σ−0+3(1,1) P(1) and P(2) transitions prevent any decay to ∆1 (v = 0, J = 1) and reduce the decay to
3 ∆ (v = 0, J = 2).
1
3∆

Detect

266 nm

up

an

cle
REMPD

transfer
961 nm

–6.5

(a)

–6

–5.5

–5

–4.5

–4

–3.5

(b)

Figure 5.10: Depletion spectroscopy of 3 ∆1 (v = 1) using “cleanup” lasers driving the 3 Σ−←0+
3 ∆ (1, 1) transition at 818.4 nm. Population is transferred from 1 Σ+ (v = 0, J = 0) with 961 nm
1
laser light driving the 3 Π0+ ← 1 Σ+ (0, 0) transition, causing some spontaneous emission into
3 ∆ (v = 1). In the blue trace, no cleanup lasers are applied, so three rotational lines are ob1
served in the REMPD spectrum (b). In the red trace, the J = 1 state is depleted, and in purple,
both J = 1 and J = 2 are depleted.

132
Ramsey time. We have thus attained a factor of about 50 in count rate, or about a factor of 7 inoverall sensitivity based on count rate alone.

Chapter 6

Oriented photodissociation

6.1

Photodissociation before Generation 2
Until Fall 2018, we performed resonance-enhanced multiphoton dissociation (REMPD) inthe same way as described in Refs. [82, 91]: We use a pair of UV lasers, one a tunable gratingstabilized dye laser that is frequency doubled with a nonlinear crystal (See Appendix C) and one
a fixed-frequency 266 nm quadrupled YAG, to photodissociate HfF+ into Hf+ and F. The Hf+ arethen detected on an MCP (Fig. 6.1). The “resonance-enhanced” aspect of REMPD allows us toresolve rotational states of HfF+ with the first photon. The second photon dissociates the moleculefrom the intermediate state, and its resonance width is essentially determined by the width of thevibrational wavefuction of the intermediate state (Fig. 6.2, [61]).
Very little is known qualitatively about the states of HfF+ in the vicinity of 30000 cm−1 ,where our REMPD intermediate state is located. In the group’s initial studies, performed by
Huanqian, Kang-Kuen, Matt, and Kevin, essentially every available laser was sent through thechamber simultaneously, and the frequency of one laser was tuned while monitoring for photodissociation [82]. The survey scan found several transitions, all of which had intermediate states withquantum number Ω = 0. Due to the large number of electronic states in HfF+ and the high degreeof mixing, particularly amongst excited states, J and Ω are expected to be the only good quantum
numbers for REMPD intermediate states. Some of the observed transitions appeared to follow avibrational progression with ωe ≈ 600 cm−1 , some variation in relative coupling strength to 1 Σ+vs. 3 ∆1 was observed.

134
TOF MCP
Ionization lasers

REMPD lasers

Molecular beam

Pumping

HfF+
Hf+ photofragments
Imaging
detector

Figure 6.1: Top view of the Generation 2 ion trap, showing the locations of REMPD lasers and theimaging MCP detector. Space constraints prevent us from having perfectly σ ± -polarized REMPD
lasers while also having Erot parallel to the MCP face for optimal imaging.

Classical turning pointContinuum state

Fixed
UV

Intermediate state
TunableUV
Initial state

Figure 6.2: (a) Energy levels and (b) qualitative 1st and 2nd photon absorption in REMPD, asdescribed in Section 6.1. One tunable UV laser drives a bound-bound transition in HfF+ , and afixed UV laser drives the bound intermediate state into the continuum of dissociating states. The
absorption spectrum of the tunable UV laser is typical of bound-bound transitions in diatomics,
while the absorption of the second photon is a very slowly varying function of energy, determinedby the Franck-Condon overlap of the intermedate and final state wavefunctions (the reflection
approximation [61]).

135
The Generation 1 REMPD scheme has the advantage of very high quantum efficiency, but
its drawback is excess noise arising from our grating-stabilized pulsed dye laser. The particularvariety of dye laser we use for REMPD lases on several longitudinal modes simultaneously, leading
to a spectral width on the order of 0.06 cm−1 or 1.8 GHz. Worse still is that the relative strengthof these longitudinal modes varies stochastically from one laser pulse to the next. We believe thisbroad spectral width is the cause of excess noise we observe in our photodissociation yield. A
comparable situation was considered in a 1988 theory [56], where the authors considered the excess
noise that would be observed in the intensity of resonance fluorescence of an atomic vapor in thepresence of laser phase noise. Their model, shown in Fig. 6.3, qualitatively agrees with the behaviorof our observed excess noise as a function of the first REMPD laser detuning.
While in the Generation 1 eEDM measurement, this excess noise was undoubtedly present,
it was simply drowned out by the shot noise on our . 10 ions per experiment cycle. Now withhundreds to thousand of detected Hf+ per cycle, excess noise has become impossible to ignore.
After struggling against this and other sources of noise for several months, we decided to pursue
a scheme for a differential measurement that had been floating around the lab for a few months:
oriented photodissociation. With a single-shot differential measurement of relative ion populations,
all noise arising from pulsed lasers (include ablation, ionization, and dissociation) becomes common
mode, and is greatly suppressed. In this chapter, I describe the implementation of this scheme,
which has been a resounding success.

6.2

Imaging photofragments & new ideas
In Fall 2017, led by Yuval, we implemented imaging of Hf+ ions via rapidly (. 500 ns)

switching off the high voltage bias on the phosphor screen in between the arrival times of Hf+ and
HfF+ . Upon our first imaging of dissociation from 1 Σ+ , v = 0, J = 0, we observed a distinctive
dipolar pattern of the photofragments, shown in Fig. 6.4. Our immediate thoughts were along
the lines of “that’s cool” and “is that useful?” Some thought led us to the conclusion that it
might be possible for the orientation of the molecules set by E rot to be preserved through the

136

(b)
(a)

Figure 6.3: (a) Observed excess noise as a function of first dissociation laser (“Cora”) detuningfrom resonance. (b) Theoretical model of fluorescence of an atomic transition and excess noise inphoton counts as a function of laser detuning, using a phase diffusion model of laser linewidth, from
Ref. [56].

137
photodissociation process, resulting in a distribution of photofragments that depended on the Stark
doublet, i.e., on MF Ω.
The initial outlook for the usefulness of this sort of oriented photodissociation seemed discouraging, as we were under the impression that a simultaneous detection was of limited usefulness,
unless it was a simultaneous detection of both MF = ±3/2 states in the same Stark doublet. However, as discussed in Chapter 7, this is not the case. In fact, a simultaneous measurement of a single
MF state in both Stark doublets is extremely useful for removing excess noise due to ion creation,state preparation, and photodissociation. Until fully appreciating this fact somewhat later, mypersonal motivation for pursuing the theoretical development described in Section 6.3 was simply
that it was interesting.
The primary outcome of the theoretical and numerical investigation detailed in Section 6.3
was the conclusion that in order to attain this doublet-resolved photodissociation, we needed anintermediate state for REMPD that had total angular momentum projection quantum number|Ω| > 0. At that time, all observed intermediate states for photodissociation had |Ω| = 0; they were
the results of the investigation described in Refs. [82, 91]. To find at state with |Ω| > 0, we would
need to perform more survey spectroscopy. Embarking on several weeks of survey spectroscopy
can seem like a daunting task at the outset, and much of the delay between having this idea
and completing our survey scan was due to the discouraging prospect that we might not find any
states with the desired property. A further concern was the fact that the only practical direction
available for us to scan was towards lower total dissociation energy (a redder first photon), resulting
inevitably in lower-energy products and degraded radial resolution set by the finite temperature of
our ion cloud. The reason for this limitation is that the scans in Ref. [82] cover the spectral region
from frequency doubled 548 nm (∼ 36500 cm−1 ) to about 618 nm (∼ 32360 cm−1 ). Generating
shorter wavelengths than 548 nm with a dye laser necessitates pumping with 355 nm light, anddealing with shorter dye degradation times – something we were not willing to do, as the eEDM
experiment is sufficiently bogged down by equipment maintenance as it is.
Fortunately, we received a push in the right direction from Prof. Tanya Zelevinsky from

138

Figure 6.4: Imaged angular distribution of photofragments resulting from REMPD of HfF+ ions
initially in the 1 Σ+ state, with the first transition tuned to the R(0), R(1), and P(1) rotationallines in the left, middle, and right columns respectively. Average of about 25 shots. In the top row,the 266 nm laser is horizontally polarized, while in the bottom row, 266 nm is vertically polarized.
This shows that the bound-to-continuum transition is perpendicular (∆Ω = ±1).

139
Columbia University, who visited Jun’s group in Summer 2018. Her group has had considerable
success studying photodissociation of ultracold 88 Sr2 [86], and her familiarity with the theory
provided us with confidence in an idea that we did not have much background knowledge on. She
felt strongly that the doublet-resolved dissociation should work, as our theory (and the semiclassicalapproximation) predicted. Motivated by her expertise, we collected several thousand cm−1 worthof spectra in Fall 2018, as described in Section 6.4.

6.3

TheoryAfter our initial observation of the dumbbell-like angular distribution of photofragments

resulting from dissociation of 1 Σ+ molecules, we applied the well-established theory of photofragmentation to address one central question: How does the angular distribution of photofragmentsin the lab frame I(θ, φ) depend on the quantum numbers of the initial, intermediate, and final
states, and on the laser polarizations? As discussed in Section 2.1, the high Z of the Hf nucleus,
the presence of f-shell electrons in Hf, and the ionic nature of the Hf2+ − F− bond led us to expectthat the high-lying and dissociating states of HfF+ are many in number and complex in structure.
We therefore do not expect to be able to make any predictions of the radial distribution of thephotofragments beyond the maximum value of their kinetic energy, which is set by the UV lasers’
wavelengths compared to the dissociation threshold.
Photofragment distributions are particularly pretty to look at, and they have been studied insome detail since initial work by Zare and others in the 1960s, summarized in the review in Ref. [135].
Due to the very large multiplicity of possible interfering channels in a photofragmentation process,the application of a fully quantum mechanical theory is quite complicated. This has motivatedquasiclassical expressions, which were explored in some detail for homonuclear molecules in recent
work from Prof. Tanya Zelevinsky’s group [85].
Because the theory side of photofragmentation I describe here is primarily motivational rather
than quantitative, I will make essentially every available approximation, retaining only what isnecessary to determine the requirements for orienting the photofragments. We begin with Eq. (7)140
of Ref. [10], and adapt it to describe the angular distribution of photofragments from a two-photon
dissociation process. We take our initial state to be a mixture of magnetic sublevels in a single
rotational level represented by a density matrix ρ, our intermediate states are denoted |Jn Ωn Mn iwhere we fix Jn and Ωn , and our final states are denoted by a subscript f . We assume the final
state has a well-defined value of |Ωf | (the axial recoil approximation), but multiple J and M levels
are accessible. The resulting formula for the photofragment distribution isI(θ, φ) ∝

X X

q
J0
J ∗
DMff Ωf (φ, θ, 0) DMf 0 Ω (φ, θ, 0) (2Jf0 + 1)(2Jf + 1)

Xf±|Ωf | Jf0 Jf MF0 MF

X

X

X

X

f

hJf Ωf Mf | T2 |Jn Ωn Mn i hJn Ωn Mn | T1 |Ji Ωi Mi i

(6.1)(Ωn Mn ) (Ω0n Mn0 ) (Ωi Mi ) (Ω0i Mi0 )× Ji Ω0i Mi0 T1† Jn Ω0n Mn0

Jn Ω0n Mn0 T2† Jf0 Ωf Mf0

Ji Ω0i Mi0 ρ |Ji Ωi Mi i ,where the parentheses under the summation signs indicate that this sum simply runs over magnetic
sublevels and parity doublet states within a single rotational level. Note that because REMPD isable to resolve rotational levels of the initial and intermediate states, there is no summation overJi or Jn . The spherical tensor operators T1 and T2 represent electric dipole transition operators,
and their matrix elements are given by
J 0 Ω0 M 0 (−D · E) |JΩM i = −X00 (−1)p E−p (−1)M −Ω 
p


×

X

q

J0
−Ω0

J0
−M 0


1

J

p M



(6.2)

1 Jp
 (2J + 1)(2J 0 + 1) Dq ,q Ωwhere E−p should be replaced with the appropriate components for the corresponding dissociation
laser polarization.
In this treatment, I have not included nuclear spin. However, as long as we are primarily
interested in the angular distribution created by a single |Ji Ωi Mi i or an incoherent mixture of thesame Mi and opposite Ωi (and thus opposite Stark doublet), the angular distribution will be thesame as if we were considering a single |F ΩMF i level including nuclear spin. Alternatively, if thehyperfine structure is not resolved by the first dissociation laser (which is the situation in the HfF+

141
eEDM experiment), then we can simply trace over the nuclear spin degree of freedom to obtain thematrix elements of the density matrix hJi Ω0i Mi0 | ρ |Ji Ωi Mi i, as discussed in Ref. [10].
In late 2017, I implemented Eq. (6.1) in Matlab in order to generate simulated angular
distribution images 1 . The adjustable parameters when generating an angular distribution are theinitial, intermediate, and final state quantum numbers |Ωi |, Ji , |Ωn |, Jn , and |Ωf |; the REMPD laser
electric fields E 1,2 ; and the rotating electric field E rot , which sets the quantization axis with respect
to which we define the density matrix elements. In the rotating wave approximation, and since theintensity distribution is unnormalized (so the magnitude of E does not matter), the electric fieldsof the lasers can simply be represented as complex unit vectors similar to Jones vectors [58]. For√
example E = (X̂ + iŶ )/ 2 represents a circularly polarized laser propagating along Ẑ. For thetransition dipole matrix elements, the choice of |Ω| for each of the initial, intermediate, and final
states sets ahead of time whether a given transition is parallel or perpendicular, and we can simply
set q = Ω0 − Ω and Dq = 1 in Eq. (6.2). In order to reduce the enormous multiplicity of possible
channels [suggested by the number of summation symbols in Eq. (6.1)], I first eliminated possiblecombinations of quantum numbers that would be zero by standard selection rules such as angularmomentum conservation in 3j symbols. I then evaluated Eq. (6.1), with the summations essentiallyrestricted to the nonzero channels.
Figure 6.6 shows simulated photofragment angular distributions arising from REMPD of
1 Σ+ , generated using Eq. (6.1).

I show photofragment distributions for a selection of REMPD

rotational lines with Ωn = 0 and |Ωf | = 1, which is the choice of intermediate and final Ω thatshows good agreement with our experimental observations. In all of the images in Fig. 6.6, I
calculated the distribution for an incoherent mixture of initial magnetic sublevels Mi in order tomatch our experimental conditions immediately after REMPI of HfF to form HfF+ in the ion trap.
1

Strictly speaking, I implemented a simulation that was equivalent to Eq. (6.1) but not exactly the same. In orderto account for nuclear spin, I simulated the first transition using a master equation approach for the bound-boundsystem. I then traced over nuclear spin degrees of freedom and applied Eq. (7) of Ref. [10] for the bound-continuum
transition. The problem with this implementation is that if the first transition is completely forbidden (for example
due to poor choice of laser polarization), the intermediate state will remain empty. Then since I renormalized theintermediate state wavefunction for the bound-continuum transition, the results would essentially be junk. In a more
recent implementation, I used Eq. (6.1) directly, which gives the same results in all situations where the first transitionis allowed.

142

|Ωi |

|Ωn |

|Ωf |

Line

|Ωi Mi i

12

I(θ, φ)

0

0

1

R(0)

Mixed

σ±

π

sin4 θ cos2 φsin2 θ

P(1)
1

2

3

R(1)

|−1, 1iσ+σ+cos12 (θ/2)|1, 1i
1

2

1

R(1)

|−1, 1i

σ+

σ−

1

1

Q(1)

|−1, 1i|1, 1isin12 (θ/2)
cos12 (θ/2)|1, 1i1

sin12 (θ/2)ππ

cos2 θ sin4 (θ/2)
cos2 θ cos4 (θ/2)

Table 6.1: Selected analytical expressions for the angular distribution of photofragments, resulting
from Eq. (6.1). In the oriented cases |Ωi | > 0, E rot is directed along θ = 0. The laser polarizations
1,2 are represented in standard atomic notation, and σ ± = σ + + σ − , with the quantization axis
set by Erot as shown in Fig. 6.5. Note that the angular distributions are unnormalized.

Figure 6.5: Examples of possible laser polarizations for oriented REMPD. Note that due to theavailability of optical access in our vacuum chamber and the location of the ion detector, not all
configurations of lasers and quantization axis (set by Erot ) are practical.

143
Apart from the aspect ratio, which is set by the relative trap secular frequencies and the time
between dissociation and detection, the results match our experimental observations in Fig. 6.4
qualitatively very well.
Figure 6.7 shows simulated photofragment distributions arising from REMPD of oriented populations 2 in the stretched magnetic sublevels of 3 ∆1 , J = 1. The selection of transitions shown
illustrates the main finding of this theoretical investigation, which is that an oriented photofragment distribution can only be obtained when |Ωn | > 0. By eye, the theoretically “most oriented”
distribution is obtained when |Ωn | = 1, |Ωf | = 1, and REMPD is performed on the Q(1) line.
However, as shown in Fig. 6.8, this distribution in fact has a small amplitude for photofragments
to be ejected in the “wrong” hemisphere. If we define the doublet contrast as
K=

N1 − N2
N1 + N2(6.3)

where N1,2 are
N
N1 =
4π

Z 2π Z π/2I(θ, φ) sin θ dθ dφ,
0

0

N
N2 =
4π

Z 2π Z πI(θ, φ) sin θ dθ dφ,0

π/2

then we find a theoretical maximum value of 15/16 ≈ 0.94 for the |Ωf | = 1 case. Experimentally,
Q branch transitions are problematic because they are likely to be contaminated with backgroundfrom population in higher J levels. The recipe for the overall best contrast in dissociation of 3 ∆1(albeit by a slim margin) is to use the R(1) line of a transition where |Ωn | = 2 and |Ωf | = 1 or 3,
attaining a theoretical value K = 63/64 ≈ 0.98. In reality, of course, there are other factors that
go into the choice of an intermediate state for oriented photodissociation, which are discussed in
the next section.

6.4

Survey spectroscopy
In Fall 2018, we decided to pursue oriented photodissociation in HfF+ , and therefore we hadto perform survey spectroscopy in order to find an appropriate transition. Our search was subject
2

By “oriented populations,” I mean a pure quantum state with a well-defined orientation of the Hf and F nuclei
in the laboratory frame.

144

Figure 6.6: Simulated photofragment angular distributions resulting from REMPD of HfF+ in the
1 Σ+ electronic state in an incoherent mixture of all magnetic sublevels, calculated using Eq. (6.1).
The black and red lines in the top right indicate the polarization of the first and second dissociation
photons, respectively. Three views are shown for each distribution to give a sense of the 3D shape.

145

Figure 6.7: Simulated photofragment distributions resulting from REMPD of HfF+ in the 3 ∆1electronic state, with opposite Stark doublets populated. The blue and red lines in the top rightindicate the polarization of the first and second dissociation photons, respectively, while the thickblack arrow indicates the direction of E rot . Oriented distributions are obtained only when Ωn 6= 0.

146

10 0

10 -2

10 -4
0

0.5

1

Figure 6.8: Comparison of theoretical oriented photofragment distributions I(θ) for REMPD transitions Ω = 1 → 2 → 1 and Ω = 1 → 2 → 3, each with the corresponding optimal polarizations.
Initial population is only in one Stark doublet. Expressions are listed in Table 6.1, lines 4 and
8. In the case Ωf = 1, an additional lobe of photofragments appears on the “wrong” hemisphere,
reducing the attainable contrast.

147
to several constraints, some of which were self-imposed: Firstly, the second photon must be either
266 nm, or have the same frequency as the first. This is because 266 nm is the highest energy photon
we can straightforwardly produce with our existing lasers, and we would like the photofragments tohave as much kinetic energy as possible, both for the purposes of spatially separating the oriented
photofragments in time of flight, and to decrease the probability of predissociation. Secondly, thefirst photon, which drives a bound-bound transition in HfF+ , must be a wavelength that we canproduce with our Sirah Cobra-Stretch dye laser, pumped with 532 nm light from our SpectraPhysics LAB-170 pulsed YAG. Thus, it must lie between 274 nm (∼ 36500 cm−1 ) and about
380 nm (∼ 26300 cm−1 ). The region from 274 nm to 309 nm (∼ 32400 cm−1 ) was surveyed by
Matt Grau, Huanqian Loh, Kevin Cossel, and Kang-Kuen Ni in Refs. [82, 91] with no |Ωn | > 0transitions found, so this left us only with the region 26300-32400 cm−1 to search.
Aside from the technical requirements, there are other qualitative features that are desirable
in an oriented REMPD transition. The most obvious is strength: We would like the largest signal possible, and we are restricted to a single dissociation laser pulse in order to image oriented
photofragments. Thus, the strongest transition is desirable. Unfortunately, this desirable property
runs counter to the necessity of scanning at a lower first photon energy than had been done in
Ref. [91], because a lower energy bound-free transition will result in a higher probability of predissociation. The effects of total dissociation energy on the strength and angular distribution of
a REMPD transition are shown in Figure 6.9. In red, a low energy above threshold results in ahigher probability of predissociation and radiative decay back to bound states and thus a weakertransition, as well as lower kinetic energy of the photofragments. A higher energy above threshold(green) results in a stronger dissociation transition, as more kinetic energy is gained by the nuclei
before they encounter an avoided crossing and have the potential to undergo radiative decay. An
additional qualitative feature of REMPD transitions shown in Fig. 6.9 is the possibility of multiple
dissociation channels resulting in different kinetic energies of the products, and potentially evendifferent angular distributions.
Before beginning survey scans, we had to implement a frequency doubling and pointing

148

Imaged distribution

Figure 6.9: Schematic potential energy curves showing the effect of total dissociation energy on
photofragment distributions. Red: low energy above threshold results in a high probability ofradiative decay. Green: high kinetic energy results in a lower probability of radiative decay. Different energies above threshold may also result in multiple dissociation channels becoming available,
resulting in concentric rings of photofragments.

149
PD
RS

F

BBO
NIR from Cora

RS
W

PB
PD

UV to expt.
DCR
W

Figure 6.10: Schematic layout of the frequency doubling and pointing compensation setup for
the Cobra-Stretch dye laser, which provides the first UV photon for REMPD. The first rotation
stage optimizes the phase matching of the BBO for UV production, while the second rotation
stage compensates for changes in beam pointing due to the BBO rotation. Abbreviations: BBO, βbarium borate crystal; RS, rotation stage; W, wedged pickoff; F, UV-passing filter; PD, photodiode;
DCR, dichroic mirror.

compensation setup for the Cobra-Stretch dye laser (Cora), which is approximately equivalent
to the commercial autotracker we have for the PrecisionScan dye laser (Bertha). This setup is
shown schematically in Fig. 6.10. Visible to NIR light produced by the dye laser is sent through a
BBO crystal mounted on a rotation stage (RS) to produce UV light. A wedged glass pickoff (W)and photodiode (PD) with a UV-passing filter positioned close to the BBO detects the UV light,
providing information on the optimal angle of the BBO. A second pickoff and photodiode positioned
close to the experiment provides information on the UV pointing, which is fed back to the angleof a Pellin-Broca prism (PB) mounted on a second rotation stage. All feedback is performed ina Labview script of the “state machine” type described in Matt Grau’s thesis [49] running on acontrol PC (Pikachu). While in principle the photodiodes and rotation stages could be used in aclosed-loop control system to servo Cora’s UV fluence being provided to the experiment, we foundit more useful and straightforward to create a lookup table of BBO and PB angles versus Cora
wavelength, then use that lookup table during the scan. The procedure for calibrating the angles
is as follows:(1) Go to the next Cora resonator position in the scan.

150(2) Perform a 1D scan of the BBO angle and find the maximum UV energy measured by PD1.
Go to this angle with the BBO.
(3) Perform a 1D scan of the PB angle and find the maximum UV energy measured by PD2.
Go to this angle with the PB.
(4) Record the wavelength as measured by the Bristol wavemeter, as well as the two rotation
stage angles.
We found that the positions of the rotation stages were sufficiently repeatable and the pointing of
the vis-NIR light from the dye laser was sufficiently stable that a single coarse scan over the lasing
range of a dye would remain valid essentially indefinitely (months and counting), requiring only
occasional small adjustments to the pointing of the fundamental light into the BBO.
We performed REMPD scans over the 26300-32400 cm−1 range using Rhodamine 640, Pyridine 1, Pyridine 2, and Styryl 8 dyes3 . An approximately 800 cm−1 region from 30000-30800 cm−1
was inaccessible with these dyes, and would only have been accessible with DCM dye in DMSO
solvent, which we were hesitant to use due to the dye’s probable carcinogenic nature [132] and thepropensity of DMSO solvent to transport dye molecules through the skin [63]. Figure 6.11 shows
the raw survey spectroscopy data from the coarse scans, which were performed at a resolution of
about 0.2 cm−1 . We performed three main “chops” to distinguish transitions originating from different low-lying states of HfF+ : dissociation products in the “ionization only” chop can arise only
from the 1 Σ+ , v = 0 state, as our REMPI process is vibronically selective by energy conservation.
In the “754 nm” chop, we applied cw laser light to drive the 3 Σ−
← 1 Σ+ (0, 0) transition for a0+
few tens of milliseconds, causing population to decay into excited vibrational levels of 1 Σ+ and 3 ∆1with a slight preference for 1 Σ+ , which can subsequently be photodissociated. In the “961 nm”
chop, we applied cw light to drive the 3 Π0+ ← 1 Σ+ (0, 0) transition, which leads to decay primarily
to the 3 ∆1 state, albeit to multiple vibrational levels as dictated by Franck-Condon factors (seeRef. [28], Table 5.7).
3

These are Sirah Lasertechnik brand names for dyes. Their proper chemical names are much less concise.

151

3

ionization only
754 nm961 nm

2.5

2

1.5

1

0.5

0
-4000

-3000

-2000

-1000

0

1000

2000

3000

Figure 6.11: REMPD survey spectroscopy of HfF+ in the 26000-32500 cm−1 region for the first
photon. The second photon wavelength is 266 nm for all observed transitions.

152
For all of the strong transitions we observed in the coarse REMPD survey scans, we performed
fine scans in order to identify the Ω value of the intermediate state. For transitions originatingfrom 1 Σ+ , we know that the only possible values of |Ωn | were 0 and 1, so we needed only to look
for a Q branch in the rotational band to identify the character of the intermediate state. For
transitions originating from 3 ∆1 , we identified |Ωn | by optical pumping from 1 Σ+ using the 961 nm
laser (Toptica) driving the 3 Π0+ ← 1 Σ+ (0, 0) transition to populate the lowest rotatioal levelsof 3 ∆1 . The lowest allowed value of J is Ω, so if Ωn = 0, 1, or 2, there should be 3, 2, or 1line(s), respectively, visible in the dissociation spectrum of 3 ∆1 , J = 1. Successively more lines
become visible if 3 ∆1 is populated via the P(1), R(1), or R(2) lines of the 3 Π0+ ← 1 Σ+ (0, 0) band.
Figure 6.12 shows this process for a transition to an Ω = 2 state observed at a first dissociation
photon energy ≈ 27752 cm−1 .
A table of observed dissociation transitions and their Ω values (where identified) is given
in Table D.1 in Appendix D. The most important results of our search are an Ω = 2 transition
at 27137 cm−1 and an Ω = 1 transition at 26516 cm−1 both originating from 3 ∆1 , v = 0. Also
important is an Ω = 1 transition at 27154 cm−1 originating from 1 Σ+ , v = 0, which is useful for
diagnostic purposes, for example for checking the efficiency and timescale of our microwave andoptical pumping-based transfer scheme.

6.5

Imaging oriented photodissociation
Having performed survey spectroscopy and found a few candidate REMPD transitions fororiented detection, we then had to attempt to observe oriented photofragments, and find the setof experimental parameters that provides the best separation of the Stark doublets for eEDM
spectroscopy. In contrast to the conditions in the experiments described in Ref. [85] and otherworks from the Zelevinsky lab, our molecules are not particularly cold or tightly confined. As aresult, the effects of finite temperature may degrade our ability to resolve the Stark doublets.
Photodissociation imparts the Hf+ molecules with a momentum kick with an angular probability distribution given by I(θ, φ) of the form discussed in Section 6.3. Thus at zero elapsed

153

961 nm on R(1) line

Time

266 nm961 nm on R(0) line

Cora

Spontaneousemission

961 nm on P(1) line

961 nm

749

750

751

752

753

754

755

(a)
(b)

Figure 6.12: Identification of the Ω value of a REMPD intermediate state by selective populationof 3 ∆1 through 3 Π0+ . Part (a) shows the trajectories of population through rotational levels ofelectronic states, while part (b) shows the resulting REMPD spectrum. A single strong line whenpopulating on the P(1) line with 961 nm indicates that this is an Ω = 2 transition.

154
time after REMPD, the ion cloud (now composed of HfF+ + Hf+ ) would be indistinguishable fromthe distribution before dissociation. In theory, to observe I(θ, φ), we should wait a time π/(2ωsec )where ωsec is the trap secular frequency (assumed isotropic). In reality, however, the trap secularfrequencies are not typically isotropic, and the travel time of the ion cloud to the imaging MCP
detector is not infinitesimal compared to a trap secular period. Furthermore, we need to temporally
separate Hf+ from HfF+ by ≈ 1 µs in order to image only Hf+ and prevent contamination of ourimages by a HfF+ background. Thus, the optimal imaging of Stark doublets could not be “dialed
in,” and involved some searching in the parameter space spanned by the time of dissociation, thephase of the trap rf, and the duration and strength of the kick to the imaging MCP.
The Imaging MCP kick settings were set roughly by hand in order to temporally resolve Hf+and HfF+ , and the timing of dissociation was scanned (via serial communication with the DDG
controlling Olga’s flashlamps and Q-switch) to find the maximum contrast, as shown in Fig. 6.13.
The fact that the asymmetry plateaus for a range of Cora delay while the number of counted Hf+ions is not depleted indicates that we have overcome the limitation set by the ion cloud’s initialsize and temperature, and are instead limited by some other factor setting our doublet contrast. A
reduction of the number of counted Hf+ accompanied by a plateau of the contrast would indicatethat we are losing Hf+ ions off the side of the detector.
Figure 6.14 shows images of photofragment distributions resulting from REMPD of the upperand lower Stark doublets using the transition at 27137 cm−1 . Conditions in these images in termsof energy resolution are about as good as we could achieve. Two distinct radii of photofragments
are visible, showing that (at least) two channels are populated by the 266 nm dissociation transition– a possibility described in Fig. 6.9. The inner radius (lower energy) photofragments have a muchless shell-like and more spherical appearance, which is likely a result of the convolution of the finite
initial ion cloud size and temperature with the kick imparted by photodissociation.

155

Hf+

400
200
0

asymmetry

1
0.5
0
-0.5
-1
-40

-30

-20

-10

0

10

20

30

cora delay ( s)Figure 6.13: Left-right asymmetry K of oriented photofragments as a function of dissociation laser
timing. The offset of the horizontal axis is arbitrary; the fall of trap time occurs around 30 µs. The
flat sections indicate that our peak asymmetry is no longer limited by the ion cloud temperature.

Figure 6.14: Images and integrated distributions of upper and lower Stark doublets using theoriented REMPD transition at a first photon wavelength of ∼ 27137 cm−1 .

156
6.5.1

Doublet depletion

Our first studies of oriented dissociation were performed using our old “semi-coherent” Raman transfer with Stella (899.7 nm) and Toptica (tuned to 986.4 nm), where we can selectively
populate a single Stark doublet. While this technique is good for populating only one doublet, it
is not particularly good for attaining high signal-to-noise. Subsequently, we were able to produce
population in a single doublet by first populating all of 3 ∆1 by optical pumping, then performing
our 1083 nm MF pumping to push all of the 3 ∆1 population to the same MF level in both Stark
doublets, and finally depleting one doublet using an 814.5 nm beam tuned to the 3 Σ−
← 3 ∆1 (0, 0),0+Q(1) transition (Fig. 6.15a). This doublet depletion beam is picked off from the 814.5 nm depletion
laser and sent along the −Ẑ direction via the breadboard above the chamber in order to avoid
the Doppler shifts caused by the RF trap and Erot . At early trap time, our Doppler FWHM is on
the order of 40 MHz, so with Erot ≈ 60 V/cm, the Stark separation of ∼ 110 MHz allows us toselectively deplete either the upper or lower doublet.

157

400

3

1

dissociation

600

200

0
814.5072

814.5076

814.508

Depletion wavelength (nm)

(a)

(b)

Figure 6.15: Laser depletion of Stark doublets to verify orientation-selective dissociation and tomeasure the doublet contrast K. A portion of the 814.5 nm light usually used for strobed MF
depletion is picked off and applied vertically down the trap axis to attain the minimum Doppler
width of about 40 MHz. While π polarization is shown in (a), the laser does not in fact have awell-defined polarization with respect to the quantization axis set by Erot , so which MF levels are
depleted is set by energy alone. (b) Observed reduction in 3 ∆1 dissociation yield as a function of
814.5 nm laser detuning, showing population depletion of each Stark doublet.

Chapter 7

Generation 2 eEDM signal

In the Generation 2 eEDM measurement with HfF+ , we have made significant improvements
to our signal size, at the cost of increased complexity of analysis. In addition, with our increased
signal size, sources of excess noise are no longer drowned out by the shot noise on Hf+ . As a result,
we will need to be much more quantitative in our understanding of sources of excess noise andsystematic effects arising from incorrect signal parameterization. In this chapter, I provide somepreliminary thoughts in the direction of signal parameterization for Generation 2, consider thefundamental limits to our measurement precision, and estimate our statistical sensitivity based onpresent performance and known sources of excess noise. These ideas are by no means complete, andI hope they will simply contribute as a jumping off point for what will certainly be a challenging
analysis.

7.1

Center cut
In our Generation 1 eEDM experiment with HfF+ , we would intentionally populate only one

Stark doublet at a time, and our analysis in fact relied on this assumption to the extent that we had
to include a “doublet contamination” systematic to account for spurious population in the “wrong”
Stark doublet (See supplemental material of Ref. [16]). Now in our Generation 2 measurement, weare populating both Stark doublets by design, and doublet contrast K is never perfect. As a result,
our analysis must take into account the doublet contamination that will always be present, in orderto prevent a host of dangerous systematic effects.

159

1.2
1
0.8
0.6
0.4
0.2
0
0

200

400

600

800

Figure 7.1: √Effect of a central signal cut on doublet contrast K, signal size N , and frequency
sensitivity K N . The horizontal√ axis is the width of a rectangular region, centered around Y = 650
pixels in Fig. 6.14. The lines K N and N are normalized to 1 at zero cut pixels.

One step we can take in the early stages of data processing to suppress the effects of finite
doublet contrast is to make a signal cut in the region of our Hf+ images where an ion has approximately equal probability of belonging to either Stark doublet, and so cannot meaningfully
contribute to a doublet-odd signal such as the eEDM. Such a cut must be made after many images
are collected at different Ramsey times, so that averaged over all images, the doublet populations
will be very nearly equal. We can then average together all images and take the center of massin order to find the central Y value, Y0 . We then reprocess each image, throwing away ion counts
that occurred within a band |Y − Y0 | ≤ Ycut /2. The remaining ion counts N , doublet contrast√
K, and sensitivity K N resulting from this “center cut” procedure is shown in Fig. 7.1. A centercut of about 60 pixels results in a slight increase in eEDM sensitivity, but the doublet contrast K
increases significantly. To suppress systematic effects, we will likely want to cut even more thanthe value of Ycut that gives the highest sensitivity.

7.2

Parameterizing the eEDM signal
In order to fit our Ramsey fringes and extract an eEDM signal, we need to develop a param-

eterized model of the fringes’ functional form. Just as in Generation 1, the population in each of

160
the MF levels after the Ramsey sequence can be parameterized asN0 e−Γt1 ∓ C u e−γt cos Φu ± Ou ,2N0 e−Γt 
1 ∓ C l e−γt cos Φl ± Ol ,Nc/d =
2

Na/b =(7.1)

where t is the Ramsey time, Γ ≈ (2 s)−1 is the 3 ∆1 decay rate, γ is the spin coherence decay rate,
C u/l is the fringe contrast, N0 is the initial Ramsey-phase-averaged ion number, and Φu/l is theRamsey phase. I have allowed some parameters to have a doublet dependence (for example C u/l ),but kept others such as γ and N0 with only an average value. The ion numbers in each state from
Equation (7.1) can also be expressed in the form
Na = Nt pu ,Nb = Nt (1 − pu ),Nc = Nt pl ,Nd = Nt (1 − pl )(7.2)

where Nt = N0 e−Γt , which more clearly expresses the fact that the Ramsey experiment has abinomially distributed outcome.
At this stage, our finite doublet contrast presents a complication: we cannot directly detect
the populations Nabcd , but only mixtures Na/c and Nb/d . I will refer to the detected ion number
mostly consisting of state a with an uppercase subscript NA , and similarly for the other states.
Using our existing definition of the doublet contrast from Eq. (6.3), the detected number in eachof these populations is





1+K
1−K
NA/B =
Na/b +
Nc/d
2
2




1+K
1−K
NC/D =
Na/b +
Nc/d
2
2

(7.3)

From each of these populations, we form an asymmetry A = ±(2N/Nt − 1), where Nt = N0 e−Γt ,and the upper and lower signs depend on the sign of the nominal MF of the state detected. The
asymmetries resulting from states in the same Stark doublet are nominally equal, so we are left
with two asymmetries that are each a sum of interference fringes of the form familiar from ourGeneration 1 eEDM measurement, ∼ −Ce−γt cos Φ + O, with weights (1 ± K)/2. We then separate
the contrast, offset, and phase into their doublet-even and doublet-odd parts, e.g. C u/l = C 0 ± C D ,

1.0

1.0

0.5

0.5

–0.5

–0.5

–1.0

161

–1.0
0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Figure 7.2: Typical appearance of doublet-even and doublet-odd asymmetry signals. The peak-topeak value of A0 is C (with a typical value of ∼ 0.8) in the present parameterization, while thepeak-to-peak value of AD is CK, where K has a typical value of ∼ 0.75. I have reduced the value off 0 to 15 Hz to more clearly show the Ramsey fringe, and I have taken f D = 0.33 Hz. Here I haveneglected decoherence and set γ = 0.

and rearrange the asymmetries to the doublet-even and doublet odd forms,
A0 = C 0 e−γt cos Φ0 cos ΦD − C D e−γt sin Φ0 sin ΦD + O0 ,
(7.4)
D

D

A = C Ke

−γt

0

D

0

−γt

cos Φ cos Φ − C Ke

0

D

D

sin Φ sin Φ + KO .

With a parameterized model for our Ramsey signal, we can investigate the effects of fundamental
and technical noise sources, and develop a data collection scheme to extract a quiet measurementof ΦD , which contains the eEDM signal.

7.3

Limits to precision
The form of the doublet-even and -odd asymmetries given in Eq. (7.4) is somewhat hairy, anderror propagation on these expressions will quickly become a mess. In this section, I will step backfrom a full error propagation calculation to think about the limits to the precision of a basic Ramsey
measurement 1 . I then investigate the effects of some realistic experimental imperfections suchas noisy particle number, and finite and noisy state preparation and detection efficiency. These
simpler situations will introduce standard mathematical tools for propagation of uncertainties,
1

Strictly I will be discussing the standard quantum limit (SQL), which is less fundamental than the Heisenberg
limit. Experiments that do not utilize spin-spin correlations cannot beat the SQL.

162
identify some reasonable approximations we should make in subsequent calculations, and highlightthe ways in which experimental imperfections limit our precision.

7.3.1

Noise in an ideal Ramsey experiment

For experiments on single trapped atomic ions, where essentially an infinite number of photons
can be scattered from a trapped ion and its presence in the trap can be determined with extremely
high fidelity, the fundamental limit to the measurement of a quantum phase is quantum projection
noise. Quantum projection noise for a single trapped ion is the variance in the number of “successes”(say, finding the particle in the state |ai) for a given number of trials N ,E (Na ) = N p,

Var (Na ) = N p(1 − p),(7.5)

where E () and Var () denote the expectation and variance, p = cos2 (φ/2) is the probability offinding the state |ai after a Ramsey sequence that starts in |bi, and where φ = ωt is the quantum
phase evolved between states |ai and |bi.
Of course, N repeated Ramsey experiments on a single particle is equivalent to one Ramsey
experiment on an ensemble of a fixed number of particles N , where both Na and Nb can bemeasured with perfect fidelity. In this situation, if we to form the asymmetry (sometimes also
called the polarization or the visibility) A = (Na − Nb )/(Na + Nb ), we would findA = 2p − 1 = cos φ,

Var (A) =4p(1 − p)sin2 φ
=
.
N
N

(7.6)

In an experiment where we would like to measure an energy splitting ~ω = ∆E with the highest
precision possible, we should repeat trials with the Ramsey time t chosen so that E (φ) = π/2. (Thisfact is fairly intuitive, but it is also discussed in the context of maximum likelihood estimation in
Ref. [133]). At φ ≈ π/2, the variance of the asymmetry is Var (A) ≈ 1/N , leading to Var (φ) = N1 .
In the case that the Ramsey fringe has imperfect contrast characterized by 0 < C < 1, we simply
obtain Var (φ) = (C 2 N )−1 . The variance on φ leads to an uncertainty on ω and thus and uncertainty
on the measured energy ∆E,
Std (∆E) =

~
√ .
Ct N

(7.7)

163

Figure 7.3: Mean and standard deviation of the asymmetry for a single experimental trial withn = 10 particles and perfect detection efficiency. The behavior of the standard deviation as afunction of Ramsey phase φ is characteristic of quantum projection noise.

This result is known as the standard quantum limit or the quantum projection noise limit, andcan be beaten by techniques such as spin squeezing, which take advantage of correlations between
particles [14].

7.3.2

Varying particle number

In contrast to the situation in experiments with one or a few trapped atomic ions, our state
detection on the HfF+ eEDM experiment is destructive, and we must produce a new sample withevery experimental cycle. The number of ions we create and trap is probabilistic, and has an
unknown distribution. In the most optimistic scenario, where we produce molecules from a muchlarger sample with a small probability, we should expect N ∼ Poisson N . However, due toa variety of factors including fluctuations in ionization laser intensity and frequency, we insteadobserve a distribution that can be approximated as a Gaussian with a mean E (N ) = N andvariance Var (N ) = N + (αN )2 . We refer to the parameter α as “fractional noise,” and it typicallytakes a value between 0.2 - 0.4 for our HfF+ creation.
We might want to ask how a varying particle number affects our ability to reach the quantum
projection noise limit Var (φ) = 1/(C 2 N ). In particular, investigating this question will introduce
several useful concepts, which we can then apply to questions of how more complex and realistic
effects affect our measurement noise on the HfF+ eEDM experiment. When N is a random vari-164
able, the number of particles ending up in state |ai can be thought of as a sum over N independent
Bernoulli distributed variables di , where i runs from 1 to N . For a Bernoulli variable with probability p, we have E (di ) = p and Var (di ) = p(1 − p). The total number of particles ending up inPstate |ai and |bi is Na/b = N
i=1 di , but now N is also a random variable. We can evaluate themean and variance of Na/b using Wald’s lemma and the law of total variance [12],!
NX
E (Na ) = E
di = E (N ) E (d) = N pi=1Var (Na ) = E (Var (Na |N )) + Var (E (Na |N )) ,(7.8)

= N p(1 − p) + p2 Var (N )where the vertical bar notation in E (Na |N ) means “the expectation of Na conditional on a fixedvalue of N .” The law of total variance is used when random variables depend on the values of otherrandom variables: the total variance of Na depends on the fact that it is a Binomially distributed
variable in its own right, and would have a variance N p(1 − p) for a fixed value of N – this accounts
for the first term in Eq. (7.8). The second term arises from the random variation in N . Similarly,
the total variance on Nb is
Var (Nb ) = N p(1 − p) + (1 − p)2 Var (N ) ,(7.9)

where 1 − p has replaced p since particles that are not projected onto state |ai end up in state |bi.
When calculating the asymmetry, it’s therefore also important to consider the covariance between
the populations in state |ai and |bi, which we can do with the law of total covariance,
Cov (Na , Nb ) = E (Cov (Na , Nb |N )) + Cov (E (Na |N ) , E (Nb |N )) .
(7.10)= −N p(1 − p) + p(1 − p)Var (N ) .
The first term in Eq. (7.10) is the covariance of successes and failures in a binomial, and is negative
because of the inherent anticorrelation between Na and Nb .
Now, armed with expressions for the means, variances, and covariance of Na and Nb in termsof the ion number and projection probability, we can express the variance of A similarly. Strictly
speaking, A is a ratio of random variables and follows a ratio distribution, which has fairly nasty

165
properties [66, 76]. Provided N is large enough, we can approximate Var (A) with a Taylor series
expansion, obtaining
2

2

4N a Var (Nb ) + 4N b Var (Na ) − 8N a N b Cov (Na , Nb )Var (A) ≈.
(N a + N b )4

(7.11)

Substituting in our expressions from Eqs. (7.8) to (7.10), we again find the same result as Eq. (7.6),
Var (A) = 4p(1 − p)/N . Thus we see that measuring the populations of states |ai and |bi simultaneously in an idealized experiment allows for normalization of fluctuations in particle number,even in the presence of excess noise on ion production.
What if we cannot simultaneously detect states |ai and |bi in a single experimental cycle? In
this case, the ion number is essentially uncorrelated from one experimental cycle to the next, andwe can take Cov (Na , Nb ) = 0. We will no longer have the favorable cancellation of excess noisethat we had due to the covariance in Eq. (7.11), and instead we get
Var (A) =4p(1 − p)sin2 φ α2 sin4 φ+ 8α2 p2 (1 − p)2 =
+
.
2
N
N

(7.12)

While the first term decreases with increasing N , the second term is independent of N and thus
represents a limit to measurement precision. We see that if we cannot simultaneously measure bothspin states in order to normalize away ion number noise, the fractional noise will become dominant
when (αN )2  2N , which for our typical values of α ≈ 0.3 occurs by about N ≈ 50. While in
the Generation 1 eEDM experiment with HfF+ we had N . 10 so that quantum projection noise
was dominant, in the Generation 2 experiment with N & 500, excess noise will easily dominate.
Figure 7.4 shows the behavior of the signal-to-noise ratio (SNR) 1/σA as a function of N (assuming
perfect contrast).
One aspect of Eq. (7.12) worth mentioning is that even in the limit α → 0, we have required
two separate experimental cycles to obtain a measurement of A. We can no longer think of N
as the number of particles detected in a single experiment cycle; rather it is the average number
detected in two experimental cycles. To obtain the same number of measurements of A, we mustrun the experiment for twice as long.

166

10
8
6
4
2

20

40

60

80

100

Figure 7.4: Signal-to-noise ratio (SNR) of a Ramsey phase measurement as a function of particle
number, in the case where both spin state populations cannot be measured simultaneously. In thepresence of excess noise α, the SNR approaches a constant value with increasing signal.

167
7.3.3

State preparation and detection efficiency

In Sections 7.3.1 and 7.3.2, we considered an experiment with perfect state preparation anddetection efficiency, but in Section 7.3.2 allowed the particle number to be a random variable. What
are the effects of finite state preparation and detection efficiency? We should expect that finite
state preparation efficiency behaves in the same way as ion creation, since our only mathematicaldescription of ion production so far has been to assign a mean N and variance N + (αN )2 to thenumber of particles taking part in the Ramsey sequence. Indeed, this is the case: if we begin
with our excessively noisy ion creation and follow this with a state preparation step with efficiency
pT (with subscript T alluding to transfer to the 3 ∆1 state), we can use the law of total varianceto obtain the mean and variance of the number Nu of molecules that participate in the Ramsey
sequence in the upper Stark doublet,
E (Nu ) = N pT ≡ N T
2

Var (Nu ) = N pT (1 − pT ) + p2T Var (N ) + N Var (pT )Var (pT )2= N T + N T α2 +p2T

(7.13)= N T + (αT N T )2 ,where in contrast to the previous two sections, I have now allowed the probability of transfer pT tohave a variance, which could arise from laser intensity or frequency noise. In the last line, I havedefined a new effective ion number N T and a new fractional noise parameter αT2 = α2 +Var (pT ) /p2T .
Evidently, adding a binomial state preparation step has not changed the general characteristics of
our ion number distribution, but has only reduced the mean ion number by its efficiency and addedin quadrature to the fractional noise parameter. Thus, from now on we can lump together all ion
creation and state preparation processes, and just take the number of ions participating in the
Ramsey sequence to have a mean N and variance N + (αN )2 , as in the previous two sections.
What about a finite (and potentially varying) detection efficiency pdet ? We can start herewith Eqs. (7.8)-(7.10), and investigate the variance on the asymmetry calculated from Nadet andNbdet , the detected ion numbers in states |ai and |bi. Again using the law of total variance, and168
continuing under the assumption that Na and Nb can only be measured on separate experimental
trials, we find
Var (pdet )4p(1 − p)222
Var (A) =+ 8p (1 − p) α +,p2det
N pdet(7.14)

where as before, p = cos2 (φ/2) is the probability of projection onto state |ai. Thus in the case that
Na and Nb can only be detected on separate trials, finite and noisy detection efficiency has essentiallythe same effects as state preparation efficiency: it reduces the effective number of particles thatparticipate in the Ramsey experiment through the denominator N pdet , and it contributes to excess
noise that is independent of N .

7.3.4

Thoughts on “the QPN limit”

In experiments with optical cycling, the effects of finite detection efficiency can be overcome,√and it is possible to achieve σφ = (C N )−1 even without collecting every photon emitted by theatoms or molecules, where N is the number of particles participating in the Ramsey sequence.
However, the requirements on how many photons must be scattered by each particle are quite
onerous, and there are some surprising requirements – for example, scattering photons until there
is a high probability that each particle in the experiment will reach a dark state is in fact detrimental
to the final SNR [76, 77].
On the HfF+ eEDM experiment, we cannot presently detect via optical cycling, and the limit
to our variance on the asymmetry is given by the first term in Eq. (7.14). In light of the progress wehave made on differential detection as discussed in Chapter 6, the quantitative effects of which are
discussed in the next few sections, we have been motivated to ask ourselves questions like “What is√
the standard quantum limit for our experiment?”, and “If we measure σφ = (C Nd )−1 where Nd isthe number of particles detected [in two experiment cycles, see discussion below Eq. (7.12)], is thisthe quantum projection noise limit?” Having seen the essentially interchangeable nature of finite
state preparation and detection efficiency in the preceding two sections, my answer to this question√is that yes, we can consider σφ = (C Nd )−1 to be limited by quantum projection noise. This isbecause our finite detection efficiency is interchangeable with a finite state preparation efficiency,

169
and due to this fact, the total number of particles participating in the Ramsey sequence is not aparticularly important concept. Of course, our precision is also very much limited by our finite
state preparation and detection efficiency! This is a semantic point, but the real takeaway is that√when we find σφ = (C Nd )−1 , we should be happy that technical noise sources are not contributing
significantly to our measurement noise.

7.4

Methods of estimating Ramsey phase
In the preceding section, I used a form of the asymmetryA=Na − Nb,
Na + Nb(7.15)

assuming that Na and Nb could result from any pair of experimental cycles. However, other methodsof assembling the asymmetry are possible. In particular, since the number of HfF+ ions created(and thus Hf+ detected) in neighboring experimental cycles are essentially uncorrelated, there is
no reason to choose any particular pair of Na and Nb . For this reason, we considered various
possibilities for calculating A using various combinations of averaged values of Na and Nb . For
example, we can define two versions of the asymmetry,
Aa = 2Na− 1,
N

Ab = 1 − 2Nb,N(7.16)

where N is the total average number of particles taking part in the experiment. Both of thesefunctions take the same value on average as A. Of course, in reality, we cannot know the true mean
N exactly, and must estimate it with a sample mean. We can then aggregate the datasets Aa and
Ab together to form A.
In the past, we have considered other forms of the asymmetry. Take, for example, the form
A0a =Na − N b,
Na + N b

A0b =N a − Nb.
N a + Nb

(7.17)The variance on A0a/b is smaller than on the form in Eq. (7.16) by a factor of two, and for largenumbers of trials so that Var N a/b is small, the mean is nearly identical. However, through170
Monte-Carlo simulation, we have found that this form of the asymmetry is pathological: it isalways over-scattered, i.e., the reduced χ2 statistic is much greater than 1 for typical parameters.
While I have not been able to show this analytically, for the rest of this thesis I will stick to theform of Eq. (7.16).

7.5

Noise on Generation 2 eEDM signal
Returning at last to the full form of the asymmetry from Section 7.2, we have that the

doublet-even and doublet-odd asymmetries can be expressed in terms of the counted ion numbers
in groups A through D as
A0 =NA − NB + NC − ND
,
2Nt

AD =

NA − NB − NC + ND,
2Nt

(7.18)

where Nt = E (NA + NB + NC + ND ) /2 is the average number of ions in a shot at a given Ramsey
time t, where there are two groups per shot as in Fig. 6.14. Fortunately, error propagation on these
expressions is very straightforward, and we obtain
"
#
X

1
Var (Ns ) + 2Cov (NA , NC ) + 2Cov (NB , ND )Var A0 =(2Nt )2 s#"
X
1
Var AD =Var (Ns ) − 2Cov (NA , NC ) − 2Cov (NB , ND ) ,(2Nt )2 s(7.19)

where s runs over A-D. At this stage, we need to account for finite doublet contrast, which causesgroup A to consist of ions derived from both states |ai and |ci. In Var A0 above, we simply
replace NA → Na , NB → Nb and so on, where Na is the number of detected ions from state |ai.
For the variance on the doublet-odd asymmetry, we obtain
Var AD =
1 (Na + Nc )(1 − K2 ) + K2 (Var (Na ) + Var (Nc ) − 2Cov (Na , Nc ))2(2Nt )(7.20)+ identical term with a → b and c → d,where K is the doublet contrast, as defined in Eq. (6.3).
The variances and covariances on the number of detected ions from each MF state can bederived by considering a series of cascaded binomial processes shown in Fig. 7.5. As discussed in

171
Sections 7.3.2 and 7.3.3, all of this apparent complexity boils down to the result that each stateremains approximately Poisson distributed, with an effective fractional noise constant αeff thatreceives contributions from the fractional variance of each binomial process. We can therefore takethe number variances to have a simple form:
Var (Na ) = Nt pu (1 − pu ) + p2u Var (Nt ) + Nt2 Var (pu ) ,Var (Nc ) = Nt pl (1 − pl ) + p2l Var (Nt ) + Nt2 Var (pl ) ,(7.21)Cov (Na , Nc ) = Cov (pu , pl ) Var (Nt ) + pu pl Var (Nt ) + Nt Cov (pu , pl ) ,2 N 2 to account for fractional noise. The variances
and we can then substitute in Var (Nt ) = Nt + αefftof Nb and Nd can be obtained from Eq. (7.21) with the replacements a → b, c → d, pu → 1 − pu ,and pl → 1 − pl . Here pu and pl are the binomial probabilities of quantum projection onto state|ai or |ci in the upper or lower-doublet Ramsey sequence, as defined in Eqs. (7.1), (7.2).
For simplicity, I will assume that the upper and lower Stark doublets have the same contrast
and coherence time, and call the contrast as a function of time Ct ≡ C 0 e−γt . Substituting Eqs. (7.21)
into Eqs. (7.19) and neglecting higher order terms, we obtain estimates for the variances of the
doublet-even and doublet-odd asymmetries,
 C2
α2 
1
+ eff 1 + Ct2 cos2 (Φ0 ) cos2 (ΦD ) + t Var (Φ) sin2 (Φ0 ) cos2 (ΦD ),
2Nt
2
2
2
1
(Ct Kαeff )(Ct K)2
Var (A)D ≈+sin2 (Φ0 ) sin2 (ΦD ) +Var (Φ) cos2 (Φ0 ) sin2 (ΦD ).
2Nt
22
Var (A)0 ≈(7.22)

The first term in both equations arises from quantum projection noise. The second term in each
expression arises from fractional noise αeff , which may arise from variations in ablation, ionization,transfer, or photodissociation yield. The last term in each expression arises from phase noise,parameterized by Var (Φ) – it may arise from magnetic field noise, or jitter of the ion cloud positionin the trap. We see that when ΦD is a multiple of π, meaning the upper and lower doublet Ramseyfringes are in phase, sin2 ΦD vanishes and we are left with Var AD ≈ 1/(2Nt ), which is the
shot-noise limit.
Figure 7.6 shows the standard deviation of A0 and AD as a function of ΦD , with Φ0 = 10ΦD .
The analytical expressions in Eq. (7.22) agree well with Monte Carlo simulation results, and show

172

Figure 7.5: Sequence of probabilities in one experimental cycle of the HfF+ eEDM experiment, fromcreation of NHfF+ to simultaneous detection of NA and NC Hf+ ions on the imaging MCP detector.
Here pprep is the probability of a molecule being prepared in 3 ∆1 (v = 0, J = 1, F = 3/2, MF = 3/2)
rather than lost to other states via spontaneous decay, pa -pd are the probabilities of surviving theRamsey sequence in one of states |ai-|di, pdet is the detection probability, and pstay = (1 + K)/2
(where K is the doublet contrast) is the probability of a detected ion being assigned to the correct
doublet group.

0.1

173

0.08

0.08

0.06

0.1
0.06

0.04
0.04

0.05

0.02

0.02
0

0
0

0.5

1

0
0

(a)

0.5

(b)

1

0

0.5

1

(c)

Figure 7.6: Standard deviation of doublet-even and doublet-odd asymmetries A0√and AD as afunction of ΦD , with 1000 ions detected per experiment cycle (shot noise limit 1/ 2Nt ≈ 0.02).
Points are Monte Carlo simulation results, while lines are from Eq. (7.22). Three regimes ofexcess noise are shown: In (a), αeff = 0.1 and Var (Φ) = 0 so that fractional noise dominates.
In (b), αeff = 0 and Var (Φ) = (0.1)2 , so that phase noise dominates. In (c), αeff = 0.05 andVar (Φ) = (0.05)2 , so that both contribute equally.

that when ΦD = 0 or π, the differential asymmetry is able to achieve the shot noise limit evenin the presence of both phase noise and ion number noise. Thus, as long as we measure ΦD in
the vicinity of these values, we should feel reassured that we can take Var AD ≈ 1/(2Nt ) whenestimating the Generation 2 eEDM sensitivity.

7.6

Effect of ion cloud jitter
While we have shown in the preceding section that our differential measurement is robust

to technical noise in ion number and photodissociation efficiency, it is not robust to jitter in theion cloud position along the axis separating the upper and lower doublets, which is typically the
Y axis in the lab frame (horizontal in Fig. 6.14). Technical noise of this type produces negative
correlation between the populations detected in each Stark doublet, which leads to noise on the
difference NA − NC and thus on the eEDM signal.
After imaging photodissociation products, we assign each of the imaged ions to the left or
right group depending on their position in the image. As in earlier sections, I will label thesegroups A and C, corresponding to the phase of the depletion lasers where only states |ai and
R Y0|ci are detected (Fig. 2.8). Ions coming from state |ai have a probability ps = −∞dY P (Y ) of

174

Figure 7.7: Effect of ion cloud jitter on assignment of ions to doublet groups, here labeled A and C.
A shift of the ion cloud in the positive Y direction by an amount δY relative to the mean position
Y0 causes a loss of an area κ0 δY + κ1 δY2 .

being assigned to group A, and a probability pg = 1 − ps of being assigned to group C. We canparameterize jitter of the ion cloud along Y by the random variable δY , such that E (δY ) ≡ 0 andE δY2 ≡ Var (Y ), as shown in Fig. 7.7. A jitter of the ion cloud in the +Y direction effectively
corresponds to a shift of the integration limit in the −Y direction, away from Y0 . As a function ofδY , the numbers of ions counted in groups A and C in a single shot are thus
NA = Nad (ps − κ0 δY − κ1 δY2 ) + Ncd (1 − ps − κ0 δY + κ1 δY2 ),(7.23)

NC = Ncd (ps + κ0 δY − κ1 δY2 ) + Nad (1 − ps + κ0 δY + κ1 δY2 ),where I have defined the parameters κ0 and κ1 as the value and half the derivative of the Hf+probability density function at Y0 , as shown in Fig. 7.7. Here Nad and Ncd are the numbers of ionsdetected from states |ai and |ci, respectively.
Using the definitions of the variance and covariance 2 along with our chosen properties ofδY , we can derive the variances and covariances of NA and NC . The results are quite messy andnot extremely illuminating. More interesting is the variance of the population difference, since near
φD = nπ, the doublet-odd asymmetry (for one choice of depletion laser phase) is approximately
2

For any two random variables X and Y ,Var (X) ≡ E (X − E (X))2

Cov (X, Y ) ≡ E ((X − E (X))(Y − E (Y ))) .

175
equal toAD (ΦD ≈ nπ) ≈NA − NC.
Nt(7.24)

I find an excess noise contribution due to jitter of Var (NA − NC )2κ1 Var (Y )(K − κ1 Var (Y ))Var AD ≈= κ20 Var (Y ) +2
Nt
Nt+ 4κ21(Nad − Ncd )2(Kurt (Y ) − Var (Y )2 )Nt2

(7.25)+ terms not dependent on Y ,where Kurt (X) = E (X − E (X))4 is the kurtosis. We find that nonzero jitter in the ion cloud
position contributes excess noise directly to our eEDM measurement, and it can be suppressedby having better resolved Stark doublets, characterized by the parameters κ0 and κ1 . The mostconcerning of these effects is the first term, which can limit our signal to noise independent of N if
Var (Y ) is large.
Using Fig. 6.14, we can make an estimate of the contribution of κ20 Var (Y ) to our excess
noise: Very roughly, the ions in one doublet are spread over about 300 pixels on the detector, so we
can take κ0 ≈ 0.003 pix−1 . The ion trap is approximately a harmonic oscillator, but by very long
confinement times the ion cloud position has essentially dephased. The probability density for the
cloud center position in a single shot Y is thus
s
1
1
ρ=
,
2
π Ymax − Y 2(7.26)

where Ymax is the maximum excursion in Y , which depends on the ion cloud’s energy. This gives
2 /2. A typical value for Ya mean square value Var (Y ) = Ymax
max is perhaps 20 pixels (≈ 1 mm,corresponding to an energy of about kB × (4 Kelvin) – about the HfF+ temperature), so that we
obtain an excess noise contribution Std AD excess ≈ 0.05. Given that we typically have about 1000
detected ions per shot leading to shot noise of ≈ 0.02, this contribution may well dominate overshot noise.

176

7.7

Generation 2 sensitivity estimate
Assuming that we are able to attain the shot noise limit, we would like to know (a) what is

the optimal Ramsey time given our counts per shot, decoherence and loss rates, and dead time,and (b) what is our expected precision after a given data collection time? We can safely assume
that much like in the Generation 1 experiment described in Chapter 3, we will measure the Ramsey
phase Φ at short and long Ramsey times ≈ 0 and t, and extract the spin precession frequencyf ≈ ∆Φ/(2πt). However, in contrast to Generation 1, we can now measure both the Stark doubleteven and doublet-odd components of the phase Φ0 and ΦD simultaneously. So from each measured
sum and difference Ramsey fringe A0 and AD , we will extractΦ0 (t) − Φ0 (0),2πt(7.27)
ΦD (t) − ΦD (0)
D
f =
.
2πt
The phases Φ0 and ΦD will likely be extracted from a nonlinear least squares fit of A0 and AD ,f0 =as in the Generation 1 measurement. We will then repeat the Ramsey measurement for opposite0
to extract the eEDM channel frequency f BD , as discussed in Section 3.3.
signs of Baxgrad

In order to determine the optimal parameters for data collection, we can estimate Var f BD ≈Var f D by straightforward error propagation on Eq. (7.27). Assuming the variance of AD is dom
inated by shot noise, and by propagating uncertainty on Eq. (7.4), we can relate Var ΦD (t) to
Var AD :


1
0
−γt 2
D
Var AD =
≈
(C
Ke
)
Var
Φ
.
(7.28)
2N0 e−ΓtPropagating errors on Eq. (7.27) and using f BD ≈ 2de Eeff /h, we find a shot noise limited uncertainty
on de ofqD
Var ΦD
0 + Var Φt
s


~
1
e(2γ+Γ)tt(1 − se ) + td
√
=
+
,
se
1 − seTtot
2Eeff tKC 0 N0~σde =2Eeff t(7.29)

where N0 is the average number 3 of detected Hf+ per shot at t = 0 (before any loss), Γ and γ are
the Hf+ loss and decoherence rates, Ttot is the total amount of data collection time, td is the “dead3

I have skipped several steps of algebra in writing down Eq.(7.29). More explicitly, starting with the first line of

177
Parameter
N0
Γ
γ
td
Eeff
C0
K

Value
103
1/2
1/5
0.4
23 × 109
0.85
0.8

Units

Description
Initial detected Hf+
Hf+ loss rate
Spin decoherence rate
Dead time
Effective electric field
Initial spin contrast
Doublet contrast

s−1
s−1
s
V/cm

Table 7.1: Typical experimental parameters for the Generation 2 eEDM experiment with HfF+ .
N0 is the total number of Hf+ detected (in both Stark doublets) in one experimental cycle neart = 0 on the side of a fringe, i.e., when Φ0 ≈ π/2.

time” spent on state preparation and readout rather than phase evolution, and se is the fraction ofexperiment cycles where we choose t ≈ 0 in order to measure ΦD (0). We can optimize Eq. (7.29)
with respect to se , finding
se,best =1
1 + e(γ+Γ/2)tp(1 − D),(7.31)

where D ≡ t/(t + td ) is the duty cycle; the fraction of total data collection time spent on spin
precession.
Optimization of Eq. (7.29) with respect to t is easily done graphically. It so happens that σdeis quite insensitive to the choice of t, as shown in Fig. 7.8, for the typical experimental parameterslisted in Table 7.1. I find that for an approximately optimal choice of t, the shot noise limit forour statistical sensitivity is about 3 × 10−29 e cm for 1 hour of data collection. Comparing to the
ACME II statistical error bar of 3.6 × 10−30 e cm for 350 hours of data collection, we should be able
to do about as well in 100 hours. Of course, this strongly depends on whether we can acheive shotnoise limited sensitivity! As of this writing, we have observed shot noise limited measurements of
AD at t ≈ 1.5 s, but challenges continue with extracting shot noise limited phase measurements.
Eq. (7.29), we have~
2Eeff tqsD
Var (ΦD0 ) + Var (Φt ) =~√2Eeff tKC 0 2N0 S

1
1
+
,
se
(1 − se )e−2γt e−Γt(7.30)

where S is the total number of phase measurements performed. The total time taken for S is Ttot =2S [se td + (1 − se )(t + td )], where the factor of 2 accounts for the fact that we need two experiment cycles to measure
the populations of all four states |ai - |di. Solving this expression for S and substituting the result into Eq. (7.30)
gives Eq.(7.29).

178

1.0

8

0.8
6

0.6
4

0.4
2

0.2

0

1

2

(a)

3

4

0

1

2

3

4

(b)

Figure 7.8: (a) Statistical sensitivity for Ttot = 1 hour of data collection with the experimental
parameters listed in Table 7.1. (b) Optimal fraction of data to be collected at t ≈ 0, versus Ramsey
time.

Chapter 8

Looking forward

This section is reproduced from Ref. [17].
Searches for electric dipole moments and other T-violating interactions in atoms and molecules
have great potential to host some of the first signs of microscopic BSM physics. Their high sensitivity, relatively low cost and short time between successive measurements (when compared tocollider experiments) has brought them to the forefront of new physics searches, particularly inview of the non-detection of new TeV-scale particles at the LHC.
The opposing requirements of high absolute frequency precision and strong relativistic effects
for EDM-sensitive systems have pushed experimenters to use specialized approaches that are often
distinct from what is applicable in systems chosen for other precision applications, such as atomic
clocks, gyroscopes, and accelerometers. However, these challenging systems inspire innovation,
which can contribute to advancement in other fields – laser cooling of polyatomic molecules forultracold chemistry, for example [72].
While the careful approach that is required of experimenters in EDMs and other precision
measurements does not generally result in overnight advances, the past decade has seen dramatic
improvements to EDM bounds both in paramagnetic and diamagnetic systems. For diamagnetic
systems in particular, the abundance of BSM parameter sensitivities encourages a wide variety of
approaches: the most stringent constraints will likely require combined results from Hg, Xe, Rn,
TlF, and the neutron. For paramagnetic systems, attaining orders of magnitude further improvement will necessitate systems that achieve both high count rate and long coherence times. The

180
confluence of techniques required to improve both paramagnetic and diamagnetic EDM bounds will
require drawing from, and contributing to, advances in techniques in the broader world of AMO
physics and emerging quantum technologies.

8.1

Future techniques in AMO EDM experiments
This section is reproduced from Ref. [17].
The near future holds great potential for improvements to EDM experiments. Further de-velopment of the 225 Ra apparatus with established techniques can be expected to lead straightforwardly to orders of magnitude higher statistical sensitivity, and improvements in quantum state
control of molecules will allow the application of laser cooling and trapping techniques to both diamagnetic and paramagnetic EDM-sensitive molecular species. Also being developed are additional
experiments on paramagnetic atoms Cesium and Francium [68, 137], a paramagnetic molecule experiment using a Stark-decelerated beam of BaF [25], and a diamagnetic molecule experiment using
TlF in collaboration between Yale, Columbia, and the University of Massachusetts Amherst [31]. At
present, molecules appear to provide the highest raw sensitivity to paramagnetic and diamagnetic
EDMs, so essentially all near-future techniques focus on improving statistical sensitivity throughhigher count rates and longer coherence times. Long coherence times provide another important
advantage in significantly suppressing systematic effects.
One possibility for improving global constraints on BSM physics parameters without dramatically changing the absolute precision of an individual EDM experiment is through the nuclear
magnetic quadrupole moment (MQM, M ) in isotopologues with spin I > 1/2 for the heavy nucleus[40]. The operators giving rise to a MQM are complementary to those constrained by diamagnetic
systems: the dimension-four CP-violating QCD parameter θ and the proton EDM dp [40]. Species
such as 177 HfF+ and 181 Ta16 O+ have accessible 3 ∆1 states and comparable sensitivity to de , CS
and M [42, 117]. The neutral species 173 YbF and the isovalent triatomic molecule 173 YbOH are
expected to have comparable sensitivity to a MQM [40, 73].
In HfF+ , the eEDM search already underway at JILA has the potential to search for a MQM

181
without significant changes to the experimental approach. However, the experimental sensitivity
required for a nuclear MQM measurement to improve upon constraints to the underlying BSM
physics parameters is on the order of 30 µHz; approximately 20× smaller than the 1σ uncertainty
in the most recent eEDM measurement using HfF+ . Particularly in view of multiple parameter
sensitivities and the necessity of a global analysis, ion trap measurements must improve by at least
an order of magnitude in order to constrain BSM physics. This goal could be within reach in thesecond generation JILA eEDM experiment.
Apart from direct BSM physics sensitivity resulting from a choice of molecule, there areseveral possible routes to improved precision in molecular ion EDM experiments. The JILA group
has improved the spin coherence time in HfF+ to ∼ 2.3 s through adiabatic expansion cooling,
reducing decoherence due to ion-ion collisions. Significant further improvements will require acryogenic environment to reduce the vibrational excitation rate from blackbody radiation, and theuse of a 3 ∆1 ground state molecule, which the JILA group is also pursuing in a separate eEDM
search using ThF+ . For a MQM search, the 3 ∆1 ground state molecule TaO+ , possibly in acryogenic environment, may attain coherence times on the scale of many seconds [42].
EDM experiments with coherence times on the scale of seconds operate in a regime where
the experimental repetition rate is limited by spin interrogation time, rather than dead time. In√
this regime, experimental sensitivity scales as 1/ τ for a fixed total data collection time (which isoften constrained by practicality). Both the HfF+ eEDM search and the 225 Ra EDM search areapproaching this limit. Substantial improvement could be attained by multiplexing experiments
in a conveyor-belt style apparatus (Fig. 8.1a): Simultaneous interrogation of 100 bunches of 103(detected) particles with τ = 10 s and a repetition rate of 10 Hz would result in a sensitivity of
approximately σf ∼ 10 µHz in one hour of data collection, or σd ∼ 10−31 e cm for Eeff = 35 GV/cm(Fig. 8.1a). Such techniques are likely more straightforward to apply in an ion-trapping experiment,
where the technical challenges will primarily be centered around development of electronics. The
JILA group is investigating this possibility for the ThF+ eEDM search.
Ideas are also being pursued for attaining long coherence times with neutral molecules

182
through laser cooling and trapping. Magneto-optical traps (MOTs) have been an essential toolin AMO experiments for several decades, and show promise for application to EDM experimentswith molecules. These non-conservative traps allow population to be accumulated from a slowed
atomic beam or a vapor, and in atomic systems reach ∼ 109 atoms at temperatures on the orderof ∼ 100 µK. The first MOTs for diatomic molecules have been demonstrated since 2011, for the
species SrF, CaF and YO [1, 8, 26, 125]. These molecules were chosen for their decoupling ofvibrational and electronic degrees of freedom, and their low mass, which allows the molecules tobe slowed to MOT capture velocities with as few scattered photons as possible while maintaining
optical cycling. The molecules used in EDM experiments are typically very heavy (∼ 200 u), however YbF in particular is isovalent with SrF and CaF, and is thus reasonably well-suited to laser
cooling. The Imperial College group has demonstrated one-dimensional laser cooling for YbF [81],and has proposed a molecular fountain design for measuring the eEDM [123]. They estimate over
100 ms coherence time will be achievable with packets of 105 molecules, leading to an estimatedsensitivity in the 10−30 e cm range.
In a recent proposal [73], the triatomic molecule YbOH is proposed as a species that is
amenable to both laser cooling and high eEDM sensitivity, while maintaining the systematicsuppressing properties of 3 ∆1 states that are being applied in the ThO, HfF+ and ThF+ eEDM
searches (Fig. 8.1b). YbOH has a similar electronic structure to YbF, SrF and CaF, leading toa low probability of vibrational excitation in electronic decay, and the possibility of laser slowing
and trapping. Experiments sensitive to BSM physics would take place in the first excited state of
the bending vibrational mode, using its parity doublet structure to allow full polarization in smalllaboratory electric fields and suppress systematic effects. As it is isovalent with YbF, YbOH is also
anticipated to be sensitive to a nuclear MQM at a similar level to YbF and HfF+ [40]. A MQM
search is under development in the Hutzler group at Caltech [90], along with a parallel effort toimplement laser cooling and trapping for a future eEDM measurement (Fig. 8.1b).
In upcoming experiments using molecules and atoms with ground or low-lying EDM-sensitive
states (e.g. 225 Ra, ThF+ , YbF, and YbOH), the spin precession time will likely not be limited by

183
spontaneous decay. For YbF and YbOH in CBGBs, for example, the spin interrogation time will
be limited by the length of the apparatus and the beam velocity. In this regime, spin-squeezing
techniques that use quantum correlations between particles to overcome the quantum projection
noise limit become applicable [14, 65, 101]. To our knowledge, these techniques have not yet been
demonstrated with cold molecules, but they have the potential to further improve the precision of
future EDM experiments.
Several past EDM searches have used solid-state systems [36, 60], which have exceptionally high statistical sensitivity due to their sample sizes of order 1020 particles. However, theseexperiments have faced challenges with diagnosing and suppressing systematic effects. Other proposals have suggested performing an EDM measurement on molecules confined in inert gas matrices[71, 128], which has the potential to combine very high count rate with coherence times on the orderof 100 ms, and robustness against systematic effects through highly variable experimental parameters (Fig. 8.1c). Such an experiment would yield a statistical sensitivity on the order of 10−36 e cm
in 100 hours – a value beyond any other proposed technique. The more recent proposal suggests
using fluorescence detection and an orientation-dependent hyperfine structure [129] to both prepare
and read out individual molecules in an orientation-specific way. The proposal [128] is currently
under development at York University and the University of Toronto.

8.2

Towards eEDM Generation 2 with HfF+
I recall saying to Matt Grau and Kevin Cossel in 2014, just as I was learning to operate the

HfF+ apparatus, that I thought our experiment was “designed into a corner,” and that I couldn’t
think of a single way we could do things differently and still have the experiment work at all. At
that time, I had evidently not yet meditated enough on the ways of eEDM. Much of this thesis has
been about revisiting aspects of the experiment that seemed to me as a first year graduate student
to be written in stone, and thinking hard about whether they could be done differently.
Chapters 2 and 3 of this thesis covered basic aspects of HfF+ and progress in the lab until
2017, which to me feels like just the beginning. The Generation 1 result, while competitive at the

184

(i)

(ii)

(iii)
MCP

~104 ThF+/bunchIo
st niz
at a
e tio
pr n
ep &
ar
at
io
Pulsed supersonic beam

n

e t
at ou
St ad
re

T = 77K Cu shield
Magnetic shield
Vrot cos(ωrott)Vrf cos(ωrft)
Vax cos(ωaxt)(a)Orientation-dependent
hyperfine shift
EnergyMolecules oriented in FCC matrix
Optical trapping

DetectFluorescence

Fluorescence detection
with optical cycling

Microwave and opticalstate preparation

Probe laser

(b)

(c)

Figure 8.1: Future methods for electric dipole moment searches. (a) A “conveyor belt” of trappedions for attaining long coherence time and high repetition rates, in development at JILA. (b) Lasercooled and trapped YbOH, a long-term goal for the PolyEDM Collaboration between Caltech,
Harvard, and University of Toronto. (c) The EDM3 method for molecules in inert gas matrices, indevelopment at York University and the University of Toronto.

185
time, has now been thoroughly left in the dust by ACME II, and we once again need to catch up.
Our 1σ statistical error bar for Generation 1 with HfF+ was 8.7 × 10−29 e cm, while ACME II
had σstat = 3.6 × 10−30 e cm; a factor of 24 better. Fortunately, thanks to recent developments,
catching up remains a possibility.
In Chapter 4, we discussed the design and implementation of the Generation 2 RF trap. The
trap, built by the JILA machine shop and powered by drivers from the JILA electronics shop, hasbeen happily confining ions for nearly 2 years. The ideas going into the design of the Generation
2 trap were essentially the same as for the Generation 1 trap designed by Matt Grau, minus theellipsoidal mirrors for laser-induced fluorescence detection of HfF+ . The main goal was that Erotshould be uniform over a large region so that the ion cloud can have very low density for longcoherence times. I think it has met that goal.
In Chapter 5, we discussed our old and new state preparation in HfF+ , with a few central
lessons that are all too obvious in retrospect:(1) a low-tech solution is often the best one,(2) a molecule does not need to be laser-coolable for optical pumping to work well, and(3) in an ion trap, we have time to spare.
For years, we worked under the impression that the electronic and vibrational branching ratios
for spontaneous emission in HfF+ were too poor, so that resonantly driving electronic transitions
would immediately lead to our molecules being “lost in Hilbert space,” never to be seen again,
except perhaps as incoherent background. Then in Summer 2017, thanks to our privilege of owninga widely tunable cw Ti:sapphire laser, we tried on a whim to rotationally cool 1 Σ+ molecules via
the 3 Σ−state as discussed in Section 5.3, and had a modest success. From there, Yan and Kia
0+
Boon decided to perform all-incoherent state preparation in ThF+ , and we on the HfF+ side –not being so proud as to not adopt a good idea when we see it – followed suit. We begged andborrowed laser diodes from all around JILA, hacked together lasers, and in the course of a few
days in 2018 were able to observe as many as 4000 Hf+ ions as dissociation products of 3 ∆1 – an

186
unprecedented yield for our experiment. Better still, our incoherent state preparation has very low
requirements for laser stability and power: we presently have 7 lasers stabilized to about 20 MHz
with our HighFinesse wavemeter, and some send as little as 10 mW to the experiment.
The developments described in Chapter 6 and 7, all of which transpired in 2018-2019, havebeen some of the most exciting of my PhD. I hope that this work may be valuable for future
experiments with molecules beyond eEDM, such as quantum information studies [18, 92]. As
discussed in Section 6.1, the signal gains we reaped from the Generation 2 ion trap and incoherent
state preparation would be essentially useless if we continued to be hampered by excess noise fromour multimode pulsed dye lasers. Inspired by classic work in photofragmentation from the chemical
physics community [135] and new work with ultracold molecules [85], we developed a new state
detection technique that provides us with a differential quantum phase measurement in a single
experiment cycle. This means that noise in ion production, state preparation, and dissociation are
all common-mode and can be normalized away, leaving only the fundamental quantum projection
noise. I believe that without this development, which came about thanks to our collaboration with
Tanya Zelevinsky, we would have had a long and arduous battle against virtually endless sourcesof excess noise.
I am sometimes asked by others in the AMO field if it is dull working on a precision measurement experiment, where we spend years at a time studying excess noise and systematic effects,
with the likely outcome of these years of effort being a measurement of a quantity that is consistent
with zero. My answer is emphatically “no.” Why? In precision measurement, life is always better
than it used to be. In 2013, a typical Ramsey fringe had about 5 signal Hf+ ions per shot, and had
a coherence time on the order of 150 ms. Now in 2019, our Ramsey fringes have about 1000 signal
Hf+ ions per shot, and we have coherence times of order 5 s. Not only do these gains keep us in
the running for record sensitivity, they allow us to perform diagnostic studies with unprecedented
ease. In 2013, I could not have imagined where we would be today.

Bibliography[1] L. Anderegg, B. L. Augenbraun, E. Chae, B. Hemmerling, N. R. Hutzler, A. Ravi, A. Collopy,
J. Ye, W. Ketterle, and J. M. Doyle. Radio Frequency Magneto-Optical Trapping of CaF
with High Density. Physical Review Letters, 119(10):103201, 9 2017.
[2] V. Andreev, D. G. Ang, D. DeMille, J. M. Doyle, G. Gabrielse, J. Haefner, N. R. Hutzler,
Z. Lasner, C. Meisenhelder, B. R. OLeary, C. D. Panda, A. D. West, E. P. West, and X. Wu.
Improved limit on the electric dipole moment of the electron. Nature, 562(7727):355–360,
2018.
[3] T. Appelquist and J. Carazzone. Infrared singularities and massive fields. Physical Review
D, 11(10):2856–2861, 1975.
[4] N. Auerbach, V. V. Flambaum, and V. Spevak. Collective T- and P-odd electromagnetic
moments in nuclei with octupole deformations. Physical Review Letters, 76(23):4316–4319,
6 1996.
[5] S. Ban, J. Dobaczewski, J. Engel, and A. Shukla. Fully self-consistent calculations of nuclear
Schiff moments. Physical Review C, 82(1):15501–15507, 7 2010.
[6] J. Baron, W. C. Campbell, D. DeMille, J. M. Doyle, G. Gabrielse, Y. V. Gurevich, P. W.
Hess, N. R. Hutzler, E. Kirilov, I. Kozyryev, B. R. O’Leary, C. D. Panda, M. F. Parsons,
E. S. Petrik, B. Spaun, A. C. Vutha, and A. D. West. Order of magnitude smaller limit on
the electric dipole moment of the electron. Science, 343(6168):269–272, 1 2014.
[7] J. Baron, W. C. Campbell, D. DeMille, J. M. Doyle, G. Gabrielse, Y. V. Gurevich, P. W.
Hess, N. R. Hutzler, E. Kirilov, I. Kozyryev, B. R. O’Leary, C. D. Panda, M. F. Parsons,
B. Spaun, A. C. Vutha, A. D. West, and E. P. West. Methods, analysis, and the treatment ofsystematic errors for the electron electric dipole moment search in thorium monoxide. New
Journal of Physics, 19(7):73029–73068, 5 2017.
[8] J. F. Barry, D. J. McCarron, E. B. Norrgard, M. H. Steinecker, and D. DeMille. Magnetooptical trapping of a diatomic molecule. Nature, 512(7514):286–289, 8 2014.
[9] D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano, and D. J. Wineland. Minimizationof ion micromotion in a Paul trap. Journal of Applied Physics, 83(10):5025–5033, 1 1998.
[10] J. A. Beswick and R. N. Zare. On the quantum and quasiclassical angular distributions ofphotofragments. Journal of Chemical Physics, 129(16):164310–164315, 10 2008.

188[11] M. Bishof, R. H. Parker, K. G. Bailey, J. P. Greene, R. J. Holt, M. R. Kalita, W. Korsch,N. D. Lemke, Z. T. Lu, P. Mueller, T. P. O’Connor, J. T. Singh, and M. R. Dietrich. Improved
limit on the 225 Ra electric dipole moment. Physical Review C, 94(2):25501, 8 2016.
[12] J. K. Blitzstein and J. Hwang. Introduction to Probability. Chapman & Hall/CRC Texts in
Statistical Science. CRC Press/Taylor & Francis Group, 2014.
[13] G. Bollen. FRIB - Facility for Rare Isotope Beams. In AIP Conference Proceedings, volume1224, pages 432–441. American Institute of Physics, 5 2010.
[14] B. Braverman, A. Kawasaki, and V. Vuletić. Impact of non-unitary spin squeezing on atomic
clock performance. New Journal of Physics, 20(10):103019, 10 2018.
[15] J. M. Brown and A. Carrington. Rotational Spectroscopy of Diatomic Molecules. Cambridge
University Press, 2003.
[16] W. B. Cairncross, D. N. Gresh, M. Grau, K. C. Cossel, T. S. Roussy, Y. Ni, Y. Zhou, J. Ye,and E. A. Cornell. Precision Measurement of the Electron’s Electric Dipole Moment Using
Trapped Molecular Ions. Physical Review Letters, 119(15), 2017.
[17] W. B. Cairncross and J. Ye. Atoms and molecules in the search for time-reversal symmetry
violation. Nature Reviews Physics, submitted, 2019.
[18] S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. Darkwah Oppong, R. L. McNally,
L. Sonderhouse, J. M. Robinson, W. Zhang, B. J. Bloom, and J. Ye. A Fermi-degenerate
three-dimensional optical lattice clock. Science, 358(6359):90–94, 2017.
[19] K. Chen, S. T. Sullivan, W. G. Rellergert, and E. R. Hudson. Measurement of the Coulomb
logarithm in a radio-frequency Paul trap. Physical Review Letters, 110(17), 2013.
[20] T. Chupp, P. Fierlinger, M. Ramsey-Musolf, and J. Singh. Electric Dipole Moments of theAtoms, Molecules, Nuclei and Particles. Reviews of Modern Physics, 10 2017.
[21] T. Chupp and M. Ramsey-Musolf. Electric Dipole Moments: A Global Analysis. PhysicalReview C, 91(3):035502, 3 2014.
[22] T. E. Chupp, P. Fierlinger, M. J. Ramsey-Musolf, and J. T. Singh. Electric dipole moments
of atoms, molecules, nuclei, and particles. Reviews of Modern Physics, 91(1):015001, 1 2019.
[23] A. Collaboration. Observation of a new particle in the search for the Standard Model Higgs
boson with the ATLAS detector at the LHC. Physics Letters B, 716(1):1–29, 9 2012.
[24] C. Collaboration. Observation of a new boson at a mass of 125 GeV with the CMS experimentat the LHC. Physics Letters B, 716(1):30–61, 9 2012.
[25] T. N.-e. collaboration. Measuring the electric dipole moment of the electron in BaF. 4 2018.
[26] A. L. Collopy, S. Ding, Y. Wu, I. A. Finneran, L. Anderegg, B. L. Augenbraun, J. M.
Doyle, and J. Ye. 3D Magneto-Optical Trap of Yttrium Monoxide. Physical Review Letters,
121(21):213201, 11 2018.

189[27] E. D. Commins, J. D. Jackson, and D. P. DeMille. The electric dipole moment of the electron:
An intuitive explanation for the evasion of Schiff’s theorem. American Journal of Physics,
75(6):532–536, 6 2007.
[28] K. C. Cossel. Techniques in molecular spectroscopy: from broad bandwidth to high resolution.
PhD thesis, University of Colorado Boulder, 10 2014.
[29] K. C. Cossel, D. N. Gresh, L. C. Sinclair, T. Coffey, L. V. Skripnikov, A. N. Petrov, N. S.
Mosyagin, A. V. Titov, R. W. Field, E. R. Meyer, E. A. Cornell, and J. Ye. Broadband
velocity modulation spectroscopy of HfF+ : Towards a measurement of the electron electric
dipole moment. Chemical Physics Letters, 546(C):1–11, 9 2012.
[30] D. DeMille. Search for the electric dipole moment of the electron using metastable PbO. AIP
Conference Proceedings, 596:72–83, 1 2001.
[31] D. DeMille. Towards a new generation of EDM experiments using molecules. Bulletin of theAmerican Physical Society, 62, 2017.
[32] M. Denis, M. S. Norby, H. J. A. Jensen, A. S. P. Gomes, M. K. Nayak, S. Knecht, and T. Fleig.
Theoretical study on ThF+ , a prospective system in search of time-reversal violation. New
Journal of Physics, 17(4):43005–43014, 4 2015.
[33] M. Dine and A. Kusenko. Origin of the matter-antimatter asymmetry. Reviews of Modern
Physics, 76(January):1–30, 2004.
[34] P. A. M. Dirac. The principles of quantum mechanics. Clarendon Press, Oxford, 1958.
[35] J. Dobaczewski and J. Engel. Nuclear time-reversal violation and the schiff moment of 225Ra.
Physical Review Letters, 94(23):232502, 2005.
[36] S. Eckel, A. O. Sushkov, and S. K. Lamoreaux. Limit on the electron electric dipole momentusing paramagnetic ferroelectric Eu 0.5Ba 0.5TiO 3. Physical Review Letters, 109(19):193003,
11 2012.
[37] J. Engel, M. J. Ramsey-Musolf, and U. Van Kolck. Electric dipole moments of nucleons,
nuclei, and atoms: The Standard Model and beyond. Progress in Particle and Nuclear
Physics, 71:21–74, 7 2013.
[38] J. L. Feng. Naturalness and the Status of Supersymmetry. Annual Review of Nuclear andParticle Science, 63(1):351–382, 10 2013.
[39] V. V. Flambaum. Enhanced nuclear Schiff moment and time reversal violation in 229 Thcontaining molecules. arXiv.org, page 1808.03629, 2018.
[40] V. V. Flambaum, D. Demille, and M. G. Kozlov. Time-reversal symmetry violation inmolecules induced by nuclear magnetic quadrupole moments. Physical Review Letters,
113(10):103003, 2014.
[41] V. V. Flambaum and V. G. Zelevinsky. Enhancement of nuclear Schiff moments and timereversal violation in atoms due to soft nuclear octupole vibrations. Physical Review C, 68(3):7,
2003.

190
[42] T. Fleig. TaO+ as a candidate molecular ion for searches of physics beyond the standard
model. Physical Review A, 95(2):022504, 2017.
[43] T. Fleig and M. Jung. Model-independent determinations of the electron EDM and the roleof diamagnetic atoms. Journal of High Energy Physics, 2018(7):718–795, 7 2018.
[44] T. Fleig and M. K. Nayak. Electron electric-dipole-moment interaction constant for HfF+
from relativistic correlated all-electron theory. Physical Review A, 88(3):32514–32516, 9 2013.
[45] T. Fleig and M. K. Nayak. Electron electric dipole moment and hyperfine interaction constants for ThO. Journal of Molecular Spectroscopy, 300(C):16–21, 6 2014.
[46] T. Fleig, M. K. Nayak, and M. G. Kozlov. TaN, a molecular system for probing P, T -violating
hadron physics. Physical Review A, 93(1):810–873, 1 2016.
[47] L. L. Foldy and S. A. Wouthuysen. On the Dirac theory of spin 1/2 particles and its nonrelativistic limit. Physical Review, 78(1):29–36, 4 1950.
[48] B. Graner, Y. Chen, E. G. Lindahl, and B. R. Heckel. Reduced Limit on the Permanent
Electric Dipole Moment of 199 Hg. Physical Review Letters, 116(16):161601, 4 2016.
[49] M. Grau.
Measuring the Electron Electric Dipole Moment with Trapped Molecular Ions.
PhD thesis, University of Colorado Boulder, 1 2016.
[50] M. Grau, A. E. Leanhardt, H. Loh, L. C. Sinclair, R. P. Stutz, T. S. Yahn, and E. A. Cornell.
Near-infrared LIF spectroscopy of HfF. Journal of Molecular Spectroscopy, 272(1):32–35, 2
2012.
[51] D. Gresh. The first precision measurement of the electron electric dipole moment in trappedmolecular ions. PhD thesis, University of Colorado Boulder, 2018.
[52] D. N. Gresh, K. C. Cossel, Y. Zhou, J. Ye, and E. A. Cornell. Broadband velocity modulation
spectroscopy of ThF+ for use in a measurement of the electron electric dipole moment. Journalof Molecular Spectroscopy, 319(C):1–9, 1 2016.
[53] B. Grzadkowski, M. Iskrzyński, M. Misiaka, and J. Rosieka. Dimension-six terms in theStandard Model Lagrangian. Journal of High Energy Physics, 2010(10):75018–75021, 10
2010.
[54] D. Hanneke, S. Fogwell, and G. Gabrielse. New Measurement of the Electron Magnetic
Moment and the Fine Structure Constant. Physical Review Letters, 100(12):120801, 2008.
[55] A. Hansson and J. K. Watson. A comment on Hönl-London factors. Journal of Molecular
Spectroscopy, 233(2):169–173, 2005.
[56] T. Haslwanter, H. Ritsch, J. Cooper, and P. Zoller. Laser-noise-induced population fluctuations in two- and three-level systems. Physical Review A, 38(11):5652–5659, 1988.
[57] W. C. Haxton and E. M. Henley. Enhanced T-nonconserving nuclear moments. PhysicalReview Letters, 51(21):1937–1940, 11 1983.
[58] E. Hecht. Optics. Pearson education. Addison-Wesley, 2002.

191[59] F. Hehl and W. Ni. Inertial effect of a dirac particle. Physical Review D, 42(6):2045, 1 1990.
[60] B. J. Heidenreich, O. T. Elliott, N. D. Charney, K. A. Virgien, A. W. Bridges, M. A. McKeon,
S. K. Peck, D. Krause, J. E. Gordon, L. R. Hunter, and S. K. Lamoreaux. Limit on the electron
electric dipole moment in gadolinium-iron garnet. Physical Review Letters, 95(25):253004,
12 2005.
[61] G. Herzberg. Molecular Spectra and Molecular Structure. Prentice-Hall, Inc., New York,1939.
[62] E. a. Hinds. Testing time reversal symmetry using molecules. Physica Scripta, T70:34–41, 1
1997.
[63] A. Horita and L. J. Weber. Skin penetrating property of drugs dissolved in dimethylsulfoxide(DMSO) and other vehicles. Life Sciences, 3(12):1389–1395, 12 1964.
[64] J. J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer, M. R. Tarbutt, and E. A. Hinds.
Improved measurement of the shape of the electron. Nature, 473(7348):493–496, 4 2011.
[65] S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac.
Improvement of frequency standards with quantum entanglement. Physical Review Letters,
79(20):3865–3868, 1997.
[66] N. R. Hutzler. A New Limit on the Electron Electric Dipole Moment. PhD thesis, Harvard
University, 3 2014.
[67] N. R. Hutzler, H.-i. Lu, and J. M. Doyle. The Bu ff er Gas Beam : An Intense , Cold , andSlow Source for Atoms and Molecules. Chemical Reviews, 112(9):4803–4827, 4 2012.
[68] T. Inoue, S. Ando, T. Aoki, H. Arikawa, S. Ezure, K. Harada, T. Hayamizu, T. Ishikawa,
M. Itoh, K. Kato, H. Kawamura, A. Uchiyama, T. Aoki, K. Asahi, T. Furukawa,
A. Hatakeyama, K. Hatanaka, K. Imai, T. Murakami, H. S. Nataraj, T. Sato, Y. Shimizu,
T. Wakasa, H. P. Yoshida, A. Yoshimi, and Y. Sakemi. Experimental search for the electron
electric dipole moment with laser cooled francium atoms. Hyperfine Interactions, 231(13):157–162, 12 2015.
[69] M. Jung. A robust limit for the electric dipole moment of the electron. Journal of HighEnergy Physics, 2013(5):71805–71815, 5 2013.
[70] I. B. Khriplovich and S. K. Lamoreaux. CP Violation Without Strangeness. Springer, Berlin,
1997.
[71] M. G. Kozlov and A. Derevianko. Proposal for a sensitive search for the electric dipole
moment of the electron with matrix-isolated radicals. Physical Review Letters, 97(6):63001,
8 2006.
[72] I. Kozyryev, L. Baum, K. Matsuda, and J. M. Doyle. Proposal for Laser Cooling of Complex
Polyatomic Molecules. ChemPhysChem, 17(22):3641–3648, 11 2016.
[73] I. Kozyryev and N. R. Hutzler. Precision Measurement of Time-Reversal Symmetry Violation
with Laser-Cooled Polyatomic Molecules. Physical Review Letters, 119(13):133002–133006,
9 2017.

192[74] F. Kuchler, E. Babcock, M. Burghoff, T. Chupp, S. Degenkolb, I. Fan, P. Fierlinger, F. Gong,
E. Kraegeloh, W. Kilian, S. Knappe-Grüneberg, T. Lins, M. Marino, J. Meinel, B. Niessen,
N. Sachdeva, Z. Salhi, A. Schnabel, F. Seifert, J. Singh, S. Stuiber, L. Trahms, and J. Voigt.
A new search for the atomic EDM of 129Xe at FRM-II. Hyperfine Interactions, 237(1):1–5,
12 2016.
[75] X. Lacour, S. Guérin, L. P. Yatsenko, N. V. Vitanov, and H. R. Jauslin. Uniform analytic
description of dephasing effects in two-state transitions. Physical Review A, 75(3), 2007.
[76] Z. Lasner. Order-of-magnitude-tighter bound on the electron electric dipole moment. PhD
thesis, Yale University, 2019.
[77] Z. Lasner and D. Demille. Statistical sensitivity of phase measurements via laser-induced
fluorescence with optical cycling detection. Physical Review A, 98(5):53823, 2018.
[78] A. E. Leanhardt, J. L. Bohn, H. Loh, P. Maletinsky, E. R. Meyer, L. C. Sinclair, R. P.
Stutz, and E. A. Cornell. High-resolution spectroscopy on trapped molecular ions in rotating
electric fields: A new approach for measuring the electron electric dipole moment. Journal of
Molecular Spectroscopy, 270(1):1–25, 11 2011.
[79] J. Lee, J. Chen, L. V. Skripnikov, A. N. Petrov, A. V. Titov, N. S. Mosyagin, and A. E.
Leanhardt. Optical spectroscopy of tungsten carbide for uncertainty analysis in electron
electric-dipole-moment search. Physical Review A, 87(2):22512–22516, 2 2013.
[80] T. D. Lee and C. N. Yang. Question of parity conservation in weak interactions. PhysicalReview, 104(1):254–258, 1 1956.
[81] J. Lim, J. R. Almond, M. A. Trigatzis, J. A. Devlin, N. J. Fitch, B. E. Sauer, M. R. Tarbutt,
and E. A. Hinds. Laser Cooled YbF Molecules for Measuring the Electron’s Electric Dipole
Moment. Physical Review Letters, 120(12):123201, 3 2018.
[82] H. Loh. Search for an electron electric dipole moment with trapped molecular ions.
thesis, University of Colorado Boulder, University of Colorado Boulder, 7 2013.

PhD[83] H. Loh, R. P. Stutz, T. S. Yahn, H. Looser, R. W. Field, and E. A. Cornell. REMPI
spectroscopy of HfF. Journal of Molecular Spectroscopy, 276-277(1):49–56, 7 2012.
[84] H. Loh, J. Wang, M. Grau, T. S. Yahn, R. W. Field, C. H. Greene, and E. A. Cornell.
Laser-induced fluorescence studies of HfF produced by autoionization. Journal of Chemical
Physics, 135(15):154308, 1 2011.
[85] M. McDonald.
High Precision Optical Spectroscopy and Quantum State Selected
Photodissociation of Ultracold 88 Sr2 Molecules in an Optical Lattice. PhD thesis, Columbia
University, 2017.
[86] M. McDonald, B. H. McGuyer, F. Apfelbeck, C. H. Lee, I. Majewska, R. Moszynski, and
T. Zelevinsky. Photodissociation of ultracold diatomic strontium molecules with quantum
state control. Nature, 534(7610):122–126, 7 2016.
[87] E. R. Meyer, J. L. Bohn, and M. P. Deskevich. Candidate molecular ions for an electron
electric dipole moment experiment. Physical Review A, 73(6):62108–62110, 6 2006.

193[88] E. R. Meyer, A. E. Leanhardt, E. A. Cornell, and J. L. Bohn. Berry-like phases in structured
atoms and molecules. Physical Review A, 80(6):62110–62113, 12 2009.
[89] W. Nagourney, J. Sandberg, and H. Dehmelt. Shelved optical electron amplifier: Observationof quantum jumps. Physical Review Letters, 56(26):2797–2799, 1986.
[90] S. Nakhate, T. C. Steimle, N. H. Pilgram, and N. R. Hutzler. The pure rotational spectrumof YbOH. Chemical Physics Letters, 715:105–108, 1 2019.
[91] K. K. Ni, H. Loh, M. Grau, K. C. Cossel, J. Ye, and E. A. Cornell. State-specific detection oftrapped HfF+ by photodissociation. Journal of Molecular Spectroscopy, 300:12–15, 3 2014.
[92] K. K. Ni, T. Rosenband, and D. D. Grimes. Dipolar exchange quantum logic gate with polar
molecules. Chemical Science, 9(33):6830–6838, 8 2018.
[93] T. Noel, M. R. Dietrich, N. Kurz, G. Shu, J. Wright, and B. B. Blinov. Adiabatic passage inthe presence of noise. Physical Review A, 85(2):23401, 2012.
[94] E. B. Norrgard, E. R. Edwards, D. J. McCarron, M. H. Steinecker, D. DeMille, S. S. Alam,S. K. Peck, N. S. Wadia, and L. R. Hunter. Hyperfine structure of the B 3 Π1 state andpredictions of optical cycling behavior in the X → B transition of TlF. Physical Review A,
95(6):062506, 2017.
[95] C. D. Panda, B. R. O’Leary, A. D. West, J. Baron, P. W. Hess, C. Hoffman, E. Kirilov, C. B.
Overstreet, E. P. West, D. DeMille, J. M. Doyle, and G. Gabrielse. Stimulated Raman adiabatic passage preparation of a coherent superposition of ThO H 3 ∆1 states for an improved
electron electric-dipole-moment measurement. Physical Review A, 93(5):052110, 5 2016.
[96] R. H. Parker, M. R. Dietrich, M. R. Kalita, N. D. Lemke, K. G. Bailey, M. Bishof, J. P.
Greene, R. J. Holt, W. Korsch, Z. T. Lu, P. Mueller, T. P. O’Connor, and J. T. Singh.
First measurement of the atomic electric dipole moment of 225 Ra. Physical Review Letters,
114(23):233002, 6 2015.
[97] E. Petrik West. A Thermochemical Cryogenic Buffer Gas Beam Source of ThO for Measuring
the Electric Dipole Moment of the Electron. PhD thesis, Harvard University, 2017.
[98] A. N. Petrov, N. S. Mosyagin, T. A. Isaev, and A. V. Titov. Theoretical study of HfF+ insearch of the electron electric dipole moment. Physical Review A, 76(3):30501–30504, 9 2007.
[99] A. N. Petrov, N. S. Mosyagin, and A. V. Titov. Theoretical study of low-lying electronic terms
and transition moments for HfF+ for the electron electric-dipole-moment search. PhysicalReview A, 79(1):607–608, 1 2009.
[100] A. N. Petrov, L. V. Skripnikov, and A. V. Titov. Zeeman interaction in the 3 ∆1 state of
HfF+ to search for the electron electric dipole moment. Physical Review A, 96(2):22508, 4
2017.
[101] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein. Quantum metrology
with nonclassical states of atomic ensembles. Reviews of Modern Physics, 90:035005, 2016.
[102] M. E. Pospelov and I. B. Khriplovich. Electric dipole moment of the W boson and the electronin the Kobayashi-Maskawa model. Sov. J. Nucl. Phys., 53:638–640, 1991.

194[103] E. M. Purcell and N. F. Ramsey. On the Possibility of Electric Dipole Moments for Elementary
Particles and Nuclei.pdf. Physical Review, 78(6):807, 1 1950.
[104] B. C. Regan, E. D. Commins, C. J. Schmidt, and D. DeMille. New limit on the electron
electric dipole moment. Physical Review Letters, 88(7):718051–718054, 2 2002.
[105] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark.
Search for New Physics with Atoms and Molecules. Reviews of Modern Physics, 90(2):025008,
6 2017.
[106] A. D. Sakharov. Violation of CP-Invariance, C-Asymmetry, and Baryon Asymmetry of the
Universe. ZhETF Pis’ma, 5:24, 1 1967.
[107] J. J. Sakurai and S. F. Tuan. Modern quantum mechanics. Addison-Wesley Pub. Co, 1994.
[108] P. G. Sandars. The electric dipole moment of an atom. Physics Letters, 14(3):194–196, 2
1965.
[109] H. Schall, M. Dulick, and R. W. Field. The electronic structure of LaF: A multiconfiguration
ligand field calculation. The Journal of Chemical Physics, 87(5):2898–2912, 1987.
[110] L. I. Schiff. Measurability of nuclear electric dipole moments. Physical Review, 132(5):2194–
2200, 1 1963.
[111] E. Schmutzer and J. Plebanski. Quantum mechanics in non inertial frames of reference.
Fortschritte der Physik, 25(1):37–82, 1 1977.
[112] L. C. Sinclair.
Development of Frequency Comb Velocity-Modulation Spectroscopy,
Spectroscopy of HfF+ and the JILA eEDM Experiment. PhD thesis, University of Colorado
Boulder, 2012.
[113] L. V. Skripnikov. Combined 4-component and relativistic pseudopotential study of ThO forthe electron electric dipole moment search. Journal of Chemical Physics, 145(21):214301–
214311, 12 2016.
[114] L. V. Skripnikov. Communication: Theoretical study of HfF+ cation to search for the T,P-odd
interactions. Journal of Chemical Physics, 147(2):1–5, 7 2017.
[115] L. V. Skripnikov, A. N. Petrov, and A. V. Titov. Communication: Theoretical study of ThO
for the electron electric dipole moment search. Journal of Chemical Physics, 139(22):0–4, 1
2013.
[116] L. V. Skripnikov and A. V. Titov. Theoretical study of ThF+ in the search for T,P-violation
effects: Effective state of a Th atom in ThF+ and ThO compounds. Physical Review A,
91(4), 1 2015.
[117] L. V. Skripnikov, A. V. Titov, and V. V. Flambaum. Enhanced effect of CP -violating nuclear
magnetic quadrupole moment in a HfF+ molecule. Physical Review A, 95(2):1220, 2017.
[118] J. Smith, E. Purcell, and N. Ramsey. Experimental measurement to the electric dipole
moment of the neutron. Physical Review, 108(1):120–122, 1 1957.

195[119] N. J. Stone. Table of nuclear magnetic dipole and electric quadrupole moments. Atomic Data
and Nuclear Data Tables, 90(1):75–176, 1 2005.
[120] R. P. Stutz. Towards Measuring the Electron Electric Dipole Moment Using Trapped
Molecular Ions. PhD thesis, University of Colorado Boulder, University of Colorado Boulder,
8 2010.
[121] O. P. Sushkov and V. V. Flambaum. Parity breaking effects in diatomic molecules. Sov.
Phys. JETP, 48(4):608, 1978.
[122] P. Sushkov, V. V. Flambaum, and I. B. Khriplovich. Possibility of investigating P-and Todd nuclear forces in atomic and molecular experiments. Zh. Eksp. Teor. Fiz., 87(November
1984):1521–1540, 1984.
[123] M. R. Tarbutt, B. E. Sauer, J. J. Hudson, and E. A. Hinds. Design for a fountain of
YbF molecules to measure the electron’s electric dipole moment. New Journal of Physics,
15(5):53018–53034, 5 2013.
[124] E. R. Tardiff, E. T. Rand, G. C. Ball, T. E. Chupp, A. B. Garnsworthy, P. Garrett, M. E.
Hayden, C. A. Kierans, W. Lorenzon, M. R. Pearson, C. Schaub, and C. E. Svensson. The
radon EDM apparatus. In ISAC and ARIEL: The TRIUMF Radioactive Beam Facilities andthe Scientific Program, volume 225, pages 197–206. Springer Netherlands, 1 2014.
[125] S. Truppe, H. J. Williams, M. Hambach, L. Caldwell, N. J. Fitch, E. A. Hinds, B. E. Sauer,
and M. R. Tarbutt. Molecules cooled below the Doppler limit. Nature Physics, 13(12):1173–
1176, 8 2017.
[126] N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann. Laser-induced populationtransfer by adiabatic passage techniques. Annual Review of Physical Chemistry, 52(1):763–
809, 3 2001.
[127] N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann. Stimulated Raman adiabatic
passage in physics, chemistry, and beyond. Reviews of Modern Physics, 89(1):015006, 3 2017.
[128] A. C. Vutha, M. Horbatsch, and E. A. Hessels. Oriented polar molecules in a solid inert-gas
matrix: a proposed method for measuring the electric dipole moment of the electron. Atoms,
6(1):3–10, 3 2017.
[129] A. C. Vutha, M. Horbatsch, and E. A. Hessels. Orientation-dependent hyperfine structureof polar molecules in a rare-gas matrix: a scheme for measuring the electron electric dipole
moment. Physical Review A, 98:032513, 2018.
[130] J. Williams. Application Note 21 - Composite Amplifiers. Technical Report July, Linear
Technology, 1986.
[131] C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson. Experimental testof parity conservation in beta decay. Physical Review, 105(4):1413–1415, 1 1957.
[132] B. J. Y. Wuebbles and J. S. Felton. Evaluation of laser dye mutagenicity using theames/salmonella microsome test. Environmental Mutagenesis, 7(4):511–522, 1985.

196[133] Y. X. Yao and S. M. Pandit. Cramér-Rao Lower Bounds for a Damped Sinusoidal Process.
IEEE Transactions on Signal Processing, 43(4):878–885, 4 1995.
[134] M. Yeo, M. T. Hummon, A. L. Collopy, B. Yan, B. Hemmerling, E. Chae, J. M. Doyle, andJ. Ye. Rotational State Microwave Mixing for Laser Cooling of Complex Diatomic Molecules.
Physical Review Letters, 114(22):223003–223004, 2015.
[135] R. N. Zare. Photoejection Dynamics. Molecular Photochemistry, 4(1):1–37, 1 1972.
[136] R. N. Zare. Angular momentum: understanding spatial aspects in chemistry and physics.
Wiley, New York, 1988.
[137] K. Zhu, N. Solmeyer, C. Tang, and D. S. Weiss. Absolute polarization measurement using avector light shift. Physical Review Letters, 111(24):243005–243006, 12 2013.

Appendix A

Simulations

The goal of this section is to make a selection of past simulations and numerical calculations
accessible to future generations of students on the EDM experiment.

A.1

Spherical multipole fitting
Since 2015, we have used a simple suite of tools in Matlab for linear least squares fitting ofinterior spherical multipole solutions to the Laplace equation,−∇2 Φ(r) = 0.

(A.1)

Here “interior” refers to the fact that the potential Φ(r) remains finite as r → 0. The potential Φ
can be decomposed into multipole components,
Φ(r) =∞ XlX

clm Φlm (r)(A.2)

Φlm (r, θ, φ) = rl Ylm (θ, φ),(A.3)

l=0 m=−l

The multipole components of the potential areeach of which is itself a solution to the Laplace equation. For the spherical harmonics Ylm , we use
a semi-normalized real form √i Y m − (−1)m Y −m

l

l
2

r


4π
Ylm =
× Y0
l
2l + 1 







 √1 Yl−m + (−1)m Ylm
2

if m < 0,
if m = 0,
if m > 0,

(A.4)

198
which are convenient in that the first few multipole components are easy to memorize. Some
examples of these are shown in Table A.1.
As can be seen from Table A.1, when expressed in Cartesian coordinates, the potentials
Φlm (r) take the form of polynomials where the exponents in each term sums to l. This makes aparticularly convenient form for creating a lookup table, rather than evaluating them in spherical coordinates (which is computationally expensive). I used Mathematica to generate tables ofcoefficients as shown in Table A.2, and imported these into Matlab for fitting and further use.
To fit with multipoles, we perform linear regression using QR decomposition. The total
potential is to be expressed as a sum over components as in Eq. (A.2). Accounting for errors andnoise, we can write this as a matrix equation,
Φ = M C + ,

(A.5)

where the observed potential isΦ = [Φ(r0 ) Φ(r1 ) . . . ]T ,the coefficient vector is
C = [c0,0 c1,−1 c1,0 . . . ]T ,and the design matrix is


Φ
(r
)
Φ
(r
)
Φ
(r
)
.
.
.
1,−1 01,0 0
 0,0 0Φ0,0 (r1 ) Φ1,−1 (r1 ) Φ1,0 (r1 ) . . .


M =

Φ0,0 (r2 ) Φ1,−1 (r2 ) Φ1,0 (r2 ) . . .




..
.
We evaluate M and use QR factorization in Matlab to solve the least squares problem, with thesolution for the coefficients
C = R−1 QT Φ.

(A.6)

We have also implemented regression using observations of the fields E~ = −∇Φ, by simply concatenating the observations of the X, Y , and Z components of the fields, and evaluating a 3×

2

1

0

−1

−2

1

1

2

2

2

2

2

3

4

2Z 2 − X 2 − Y
√
3Y Z
√
3XY

3XZ

3
2
2
2 (X − Y )

X


2

3Y
q

3
2
− 8 3X + Y 2 − 4Z 2
q

5
2
2
2
8 Z −9X − 3Y + 4Z

√

−X

3X
√
3Z

√

1

0

E lm,X

−Y
√
3Z
√
3X
q
− 32 XYq−3 52 XY Z

1
√
− 3Y

0

E lm,Y

−

q

45
8 X

3X

X 2 + Y 2 − 4Z 2

√
2 6XZ

2Z
√
3Y

√

1

0

E lm,Z

Table A.1: Unnormalized real spherical harmonics used in spherical multipole fitting in this thesis. The (lack of) normalization is chosen
so that the (l = 1, m) and (l = 2, m = 0) components have the form conventionally used in past eEDM work.

q− 38 X X 2 + Y 2 − 4Z 2q
− 58 XZ 3X 2 + 3Y 2 − 4Z 2

1
2

−1

1
√

Y

0

1
√

Z

1

1

1

0

0

Φlm

m

l

199

200
0
0
0
0
1
0
1
2

0
0
0
1
0
2
1
0

0
0
2
1
1
0
0
0

2
-2
0
0
0
0
1.732
0

2
-1
0
1.732
0
0
0
0

2
0
1
0
0
-0.5
0
-0.5

2
1
0
0
1.732
0
0
0

2
2
0
0
0
-0.8660
0.866

Table A.2: Example of a table of multipole potential data for l = 2. The first two rows containl and m values, however the first three columns of the first two rows are ignored. The remaining
rows of the first three columns indicate exponents of the X, Y , and Z coordinates. Each of theremaining columns contains coefficients of the Φlm indicated by the first two rows. For example,
reading from this table, we obtain Φ2,−2 = 1.732 X Y , in agreement with Table A.1.

larger design matrix using the analytical form of the gradient of the potentials, which can be easily
obtained by matrix operations on the Mathematica output shown in Table A.2.
As of July 2019, these functions are available on the JILA network drive in the directory
smb:\\jilafile\scratch\cornell\common\edm\MATLAB\multipole , which includes a Mathematica script to generate text files, and Matlab scripts to process the text files into .mat files that
contain the same information. It also takes an analytical derivative of the tabular data to obtainlookup tables for the multipole fields.

A.2

Space charge & thermometry
In Fall 2017, just after installing the Generation 2 ion trap, we were creating about 104 HfF+

ions per shot, and we wanted to know their temperature. We calibrated the MCP transfer functions
as described in Section B.1 (as well as in Huanqian’s thesis), but found that our fitted ion cloud
temperatures were quite high, on the order of 30 to 50 Kelvin. This caused us to suspect significant
space charge effects, which tends to make the peak density lower and the width higher than fora non-interacting gas. We subsequently investigated the temperature and the presence of spacecharge effects through three main avenues:• The profile of the ion cloud in time of flight201• Frequency and damping of breathing modes of the ion cloud• Depletion spectroscopy of 1 Σ+ [0, 0]

A.2.1

Ion cloud profile

The mutual Coulomb repulsion between the HfF+ ions in our RF trap affects the density distribution of the ion cloud n(r). In the extreme limit of low temperatures, the ion cloud crystallizes
into a Coulomb crystal, but even at much higher temperatures, the effects of interactions should
be visible in the density profile. However there are several confounding factors: we only observe
the ion cloud in time of flight, the ions may be flying through a distorting potential landscape, andthere are other HfF+ isotopes present in the trap. Altogether, the ion cloud profile is likely theleast reliable way of characterizing the temperature and the extent of space charge effects.
To try to back out the temperature, I wrote a simulation in Matlab that uses successive
relaxation to obtain the equilibrium density distribution in a given trapping potential. In fact,
this simulation was initially motivated by my brief forays into molecular dynamics simulations,where I have found that one of the most time-consuming steps in a Coulomb problem is allowing
the ensemble to reach an equilibrium spatial distribution. In future simulations of Ramsey fringes,
when the effects of space-charge on fringe frequencies may become noticeable, it will likely bedesirable to simulate the electric fields experienced by a small fraction of some ∼ 104 ions whose
trajectories are simulated. In this case, we probably will want to start from approximately anequilibrium configuration.
To solve for the equilibrium distribution n(r), I begin with a guess n0 (r), which is usuallyeither a Gaussian corresponding to a non-interacting cloud, or a flat-top distribution (which ofthese converges faster depends on the Coulomb interaction energy compared to kinetic energy).
From our initial guess, we can solve the discrete Poisson equation with a simple matrix inversion
to obtain the electric potential,
Φ(r) = −

e 1
n(r),
0 ∇2

(A.7)

202
where 1/∇2 denotes the inverse of the discrete Laplacian operator. I then update the density
distribution according ton(r) → (1 − α)n(r) + α nnew (r),(A.8)

where 0 < α < 1 is a constant chosen to decide how quickly the solution attempts to converge, andthe “new” density distribution is
V (r) + eΦ(r)nnew (r) = n0 exp −
,
kB T

(A.9)

where n0 is a normalization factor chosen so that the new density distribution integrates to N , thetotal ion number. If α is too large, the solution will become unstable; typically α ≈ 0.1 is a good
choice. The updated density distribution is then used to solve for Φ(r), and the process is repeateduntil a stationary solution is reached. An example result is shown in Fig. A.1, where even at atemperature of 5 K, a cloud of 104 ions shows significant effects of reduced density due to spacecharge.

A.2.2

Ion cloud breathing modes

A useful way to inquire about the relative energy scales of Coulomb interactions versus
thermal motion is via the collective normal modes of “breathing” or “jiggling” of the ion cloud.
In a hot and therefore weakly interacting cloud of ions, collisions have a relatively small effect on
the secular (sloshing) motion of each ion in the trap. In this limit, the apparent breathe frequencyof the ion cloud along trap axis i is ωbreathe = 2ωsec,i , and energy is very slow to transfer between
principal axes. Here, the breathe modes of the ion cloud are not collective modes in the true sense,
but just single particle motion of many independent particles in the trap with some degree of phasecoherence.
In a cold ion cloud, the Coulomb potential energy of the cloud is not small compared to
its thermal kinetic energy. In this limit, collective breathing modes of the ions clouds exist, withfrequencies ωbreathe < 2ωsec,i . Experimentally, we investigated the “axgrad” mode, which is shown
schematically in Fig. A.2(a). We adjusted the trap parameters so that the secular frequencies were

203

4000

3000

2000

1000

0
-5

0

(a)

5

(b)

Figure A.1: (a) Simulated integrated density profile of an ion cloud with (blue) and without (reddashed) space charge effects. Here N = 104 ions, T = 5 K, and the cloud is confined in a sphericallysymmetric harmonic potential with ω = 2π × 2 kHz. The Y and Z axes have been integrated out,
so that this is what we would see as a voltage trace on an MCP detector – i.e., this is not a slice
through the density distribution. (b) Experimentally observed time of flight profile of the ion clouddensity, with overlays of various simulated profiles. Assuming a temperature of 5 K and 104 ions,a profile that accounts for the space charge effects provides the best fit.

204

Figure A.2: Examples of normal modes of a cold ion cloud. In the “axgrad” mode (a), the axial
direction is out of phase with the radial directions, and oscillates with twice the amplitude. In the“breathe” mode (b), the cloud widths along all three principal axes oscillate in phase.

nearly equal at ωsec = 2.05(1) Hz, and excited the “axgrad” mode by applying a simultaneousvoltage pulse of equal magnitude on the endcap electrodes. The observed oscillation is shownin Fig. A.3. The 1/e time of the oscillation is of order 2 ms, and the oscillation frequency isabout 3 kHz – significantly less than 2ωsec . Qualitatively, this shows that space charge effects
are significant in our HfF+ cloud. Comparison with a numerical simulation of the cloud breathe
frequencies could extract the temperature, however we have not yet performed such a simulation,
as we found Doppler thermometry to be more straightforward.

A.2.3

Doppler thermometry

Having found that the spatial width of the cloud in time of flight was not representativeof the temperature (unless space charge is accounted for), we found that the simplest way to
perform thermometry is by measuring the Doppler width of the cloud via laser spectroscopy. For
this purpose, we primarily used the 961 nm laser (“Toptica”) driving the 3 Π0+ ← 1 Σ+ (0,0) P (1)transition, and tuned its wavelength via the lock to our HighFinesse wavemeter. The intensity of
the 961 nm laser had to be reduced significantly to prevent saturation of the transition. With a few
millisecond probe time, we observed widths between σn u ≈ 10 MHz and 30 MHz, corresponding totemperatures in the 3 K to 30 K range.

205

(a)

(b)

Figure A.3: (a) Damping of the “axgrad” and “breathe” collective modes of oscillation of the HfF+
cloud, after excitation by a kick applied to both endcaps. (b) FFT of the data from part (a),showing the frequency of the “axgrad” mode, which is approximately 1.7ωsec . This indicates thatspace charge effects are significant in the ion cloud.

Figure A.4: Population detected by photodissociation of 3 ∆1 (v = 0, J = 1) versus frequency of the
961 nm laser driving the 3 Π0+ ← 1 Σ+ (0,0) P (1) transition, showing the one-photon Doppler widthof the HfF+ cloud. The 961 nm laser is stabilized to the HighFinesse wavemeter. The measured
width is σν = 11.8(2) MHz, corresponding to a temperature of T = mσν2 λ2 /kB = 3.1(1) K.

206

A.3

Trap design
Here I include a my Matlab code for optimization of the Generation 2 ion trap geometry.

This may serve as a useful reference for future use of the Livelink for Matlab toolbox for Comsol,as the documentation for this Comsol add-on is somewhat limited. The Matlab script below
programmatically generates a basic version of the Generation 2 ion trap geometry with certain
predefined parameters (such as the ion trap height and radius) and numerically optimizes theelectrode shapes as described in Chapter 4.
First, one needs to start the Comsol Server and Livelink for Matlab. On a PC with Comsol
installed, the Comsol Server application can be found in the directory
C:\Program Files\COMSOL\COMSOL53a\Multiphysics\COMSOL Launchers
where 5.3a is replaced by the Comsol version installed. Then in Matlab we add paths to the Livelink
API functions and start the Livelink interface using the following commands:

1
2
3
4

import com.comsol.model.*;import com.comsol.model.util.*;
addpath('C:\Program Files\COMSOL\COMSOL53a\Multiphysics\mli\');
mphstart();I then initialize the Comsol model with the following:1

function model = model init(mesh size)

2
3
4
5
6

% dependencies
import com.comsol.model.*;import com.comsol.model.util.*;% addpath('C:\Program Files\COMSOL\COMSOL52\Multiphysics\mli\');

7
8
9
10
11

% input arguments
if nargin<1
mesh size = 7;
end

12
13
14
15
16
17

params = trap geom param defaults();
filename = ['trap ' datestr(today,'YYmmDD')];model = mphopen('optimization models/chamber simple 160901.mph');% model = ModelUtil.create(filename);model.name([filename '.mph']);18
19% set model parameters

207
20

model = set model params(model,params);

21
22
23

% build geometry[model,rad elec domains] = build trap geom(model);

24
25
26

% assign materials
model = set trap materials(model,rad elec domains);

27
28
29

% add electrostatics & define potentials
model = set electrostatics(model);

30
31
32

% build meshmodel = build mesh(model,mesh size);

33
34
35

% don't show progress bar
ModelUtil.showProgress(false);

36
37
38
39

figure(1); clf;
mphviewselection(model,'geom1',[rad elec domains],'entity','domain',...
'facemode','off');

40
41

end

The function mphopen is part of the Livelink API, and takes as an argument an .mph file
path as a string. I then pass the variable model into and out of various functions that initialize
different aspects of the geometry, materials, and mesh. I set certain default parameters with thefollowing code:

1
2

params.pi = pi;params.N elec = 8;

3
4
5
6
7
8
9

params.Vrot = 1;
params.phi rot = 0;params.Vtrap = 0;params.phi trap = 0;params.Vtop = 0;params.Vbot = 0;

10
11
12
13
14
15

params.c2 = 0.053811014;
params.c4 = −0.114742513;
params.c6 = −0.149448808;
params.c8 = 0.693544237;
params.c10 = 0.099128725;

16
17
18
19
20
21
22

params.r trap inner = 0.048;
params.r trap outer = 0.084;
params.w elec tab = 0.010;
params.h elec tab = 0.020;
params.h rad = 0.150;
params.t rad = 0.004;

23
24

params.fillet radius = 0.001;

25
26
27

f = fields(params);
for i = 1:numel(f)208model.param.set(f{i},params.(f{i}));

28
29

end

The function build trap geom defines the geometry of the ion trap in terms of the existing
parameters.

1function [model,rad elec domains] = build trap geom(model)

2
3
4
5

import com.comsol.model.*;import com.comsol.model.util.*;% addpath('C:\Program Files\COMSOL\COMSOL52\Multiphysics\mli\');

6
7
8
9
10

if nargin<1 | | isempty(model)
filename = ['trap ' datestr(today,'YYmmDD')];model = ModelUtil.create(filename);
end

11
12
13

% set model params just in caseparams = get model params(model);

14
15
16
17
18
19
20

% create geometry
trygeom = model.geom('geom1');
catch
geom = model.geom.create('geom1',3);
end

21
22
23

% parametric definitions of radial electrode geometry
for i = 1:params.N elec

24
25
26
27
28
29
30

% define a workplane
workplane tag = char(geom.feature.uniquetag('workplane'));workplane = geom.feature.create(workplane tag,'WorkPlane');workplane.set('planetype', 'quick');
workplane.set('quickplane', 'xz');
workplane.set('quicky','t rad/2');

31
32
33

34

35
36
37
38
39
40
41
42
43
44
45

% define a bezier polygon that outlines the electrodes
rad elec body tag = ...
char(workplane.geom.feature.uniquetag('rad elec body'));rad elec body = ...
workplane.geom.feature.create(rad elec body tag,'BezierPolygon');rad elec body.set('type','open');rad elec body.set('degree',1);rad elec body.set('p',...
{'r trap inner*(1−c2−c4−c6−c8−c10)','r trap outer−w elec tab',...
'r trap outer−w elec tab','r trap outer','r trap outer',...
'r trap outer−w elec tab','r trap outer−w elec tab',...
'r trap inner*(1−c2−c4−c6−c8−c10)';'−h rad/2','−h rad/2','−(h rad/2+h elec tab)','−(h rad/2+h elec tab)',...
'h rad/2+h elec tab','h rad/2+h elec tab','h rad/2','h rad/2'
});

209
rad elec body.set('createselection','on');46
47% front face of the electrode −− a polynomial
rad elec front tag = ...
char(workplane.geom.feature.uniquetag('rad elec front'));rad elec front = ...
workplane.geom.feature.create(rad elec front tag,'ParametricCurve');rad elec front.set('rtol',1e−9);
rad elec front.set('parname','s');rad elec front.set('parmin',−1);
rad elec front.set('parmax',1);rad elec front.set('coord',{'r trap inner *(1 − c2*sˆ2 − c4*sˆ4 − c6*sˆ6 ...
− c8*sˆ8 − c10*sˆ10)','h rad/2*s'});rad elec front.set('createselection','on');

48
49

50

51
52
53
54
55

56
57

% turn the line defining the electrode into a surface
rad elec surf tag = ...
char(workplane.geom.feature.uniquetag('rad elec surf'));rad elec surf = workplane.geom.feature.create(rad elec surf tag, ...
'ConvertToSolid');
rad elec surf.selection('input').set({rad elec front tag,...
rad elec body tag});rad elec surf.set('createselection','on');

58
59

60

61
62
63
64

% fillet some of the corners
fillet tag = workplane.geom.feature.uniquetag('fillet');fillet = workplane.geom.feature.create(fillet tag,'Fillet');fillet.selection('point').set([rad elec surf tag '(1)'],[1 2 4 5]);fillet.set('radius','fillet radius');

65
66
67
68
69
70

% define an extrusion to make the electrode 3D
rad elec ext tag = char(geom.feature.uniquetag('rad elec ext'));rad elec ext = geom.feature.create(rad elec ext tag, 'Extrude');rad elec ext.set('distance','t rad');rad elec ext.selection('input').set({workplane tag});rad elec ext.set('createselection','on');

71
72
73
74
75
76
77

% rotate the 3D shape into its final position
rad elec tag = geom.feature.uniquetag('rad elec');
rad elec = geom.feature.create(rad elec tag,'Rotate');rad elec.selection('input').set({rad elec ext tag});rad elec.set('keep','off');
rad elec.set('createselection','on');rad elec.set('pos',{'0','0','0'});rad elec.set('rot',['(360/N elec)*' num2str(i−1) '+(360/(2* N elec))']);

78
79
80
81
82
83
84
85
86

end

87
88
89

% build geometry
model.geom.run();

90
91
92
93

% get domain numbers
rad elec domains = double(arrayfun(@(i) model.selection(['geom1 rad elec',...
num2str(i) ' dom']).inputEntities,1:params.N elec));

94
95

end

Then we define the materials of the electrodes:

210

1
2
3
4

mat = model.material.create('steel');mat.materialModel('def').set('relpermittivity',1);
mat.materialModel('def').set('electricconductivity','4.032e6[S/m]');mat.selection.set(rad elec domains);

We then introduce a stationary study with electrostatics physics, and assign potentials to theelectrodes:

1function model = set electrostatics(model)

23params = get model params(model);

4
5
6
7

% create a study, as in the Comsol GUI
study = model.study.create('study');
study.feature.create('stationary', 'Stationary');

8
9
10
11
12
13
14

% if electrostatics physics has not been defined, do thattry
es = model.physics('es');catch
es = model.physics.create('es','Electrostatics','geom1');
end

15
16
17

% numbers of the electrodesn el = 1:params.N elec;

18
19
20
21

% loop through electrodes and assign potentials
for i = 1:params.N elec
pot tag = ['Vrad' num2str(i)];22% create the potential object
pot = es.feature.create(pot tag, 'ElectricPotential', 2);

23
24
25

% set the potential value
pot.set('V0',['Vrot*cos(phi rot−' num2str(i−1+1/2) '*2*pi/N elec) + ...
Vtrap*cos(phi trap−4*' num2str(i−1+1/2) '*pi/N elec)']);

26
27

28

% create a named selection for this potential
pot.selection.named({['geom1 rad elec' num2str(i) ' bnd']});

29
30
31

end

32
33

end

We then build the mesh:

1

try

2

mesh = model.mesh('mesh1');catchmesh = model.mesh.create('mesh1','geom1');
end

3
4
5

211
6
7
8

mesh.run();mesh.feature('size').set('hauto',mesh size);mesh.run();Then finally we execute the code below for optimization. Here the only new commands used
are model.study('study').run and mphinterp, which runs the computation of the electric
potential and imports the results to Matlab, interpolated on a grid.

1
2
3
4
5

% regression setupz = linspace(−zmax,zmax,Nz)';
powers = 0:2:fit poly order;
VanderM = bsxfun(@power,z/zmax,powers);[VanderQ,VanderR] = qr(VanderM,0);

6
7
8
9

% optimization
optim table = table();weight fun = 1e6*(1−cos(linspace(0,pi,numel(z))').ˆ10);

10
11
12
13
14
15
16
17
18
19
20
21
22
23

old data = dirfun(@(x) readtable(x),'optimization results','pattern','.csv');
old data = vertcat(old data{:});
if isempty(C guess)if ¬isempty(old data)
C init = old data{old data.resid==min(old data.resid),...
{'c2','c4','c6','c8','c10'}};C init = C init(1:round(elec poly order/2));
else
C init = zeros(1,round(elec poly order/2));
end
else
C init(1:numel(C guess)) = C guess;
end

24
25
26
27

iter = 1;
C final = fminsearchbnd(@optimfun,C init,C init−0.02,C init+0.02,...
optimset('Disp','iter','maxiter',1));

28
29
30
31
32

function resid = optimfun(C)for j = 1:numel(C)model.param.set(['c' num2str(2*j)],C(j));
end

33
34
35
36
37

model.study('study').run;
Enormz = mphinterp(model,'es.normE','coord',[0*z 0*z z]')';
pfit = VanderR\(VanderQ'*Enormz);
Enormz fit = VanderM*pfit;

38
39

resid = simps(z,(Enormz fit−Enormz fit(z==0)).ˆ2.*weight fun);

40
41
42
43
44

figure(2); clf;
subplot(2,1,1);plot(z,Enormz,z,Enormz fit,'linewidth',2);title(['Iteration ' num2str(iter)]);212
ylabel(' | E {rot } | (V/m)');xlabel('z (m)');

45
46
47

subplot(2,1,2);
plot(z,(Enormz fit−Enormz fit(z==0)).ˆ2.*weight fun,'linewidth',2);title(['Iteration ' num2str(iter)]);ylabel('cost function');xlabel('z (m)');drawnow();

48
49
50
51
52
53
54

C = padarray(C,[0 5−numel(C)],0,'post');optim table = [optim table; array2table([iter C ...
resid],'variablenames',{'iter','c2','c4','c6','c8','c10','resid'})];
writetable(optim table,results filename);

55
56

57
58

iter = iter+1;

59
60

if mod(iter−1,10)==0
mphsave(model,[pwd filesep filename '.mph']);
end

61
62
63
64

end

65
66
67
68
69
70
71
72

% plot results
for k = 1:numel(C final)
model.param.set(['c' num2str(2*k)],C final(k));
endmodel.geom('geom1').run;model.mesh('mesh1').run;model.study('study').run;

73
74
75
76
77

pg = model.result.create('pg', 'PlotGroup3D');
slice = pg.feature.create('slice', 'Multislice');slice.set('expr', 'V');mphplot(model,'pg');

78
79

80

set(gcf,'units','normalized','position',[0.3536
0.8259]);view([0 90])0.0787

0.2917

...

81
82
83

A.4

% save model
mphsave(model,[pwd filesep folder filesep filename '.mph']);Optical pumping
Prior to implementing optical pumping state preparation, I performed a simple rate equation

simulation to estimate our overall efficiency for preparing population in 3 ∆1 (J = 1). I used the setof first order differential equations for state populations ni ,dni X=
Ωi←j nj + Ai←j nj − Γi ni
dt
j6=i(A.10)

213
Typically, the matrix A of Einstein A coefficients will look something like
0 A1←2 A1←3 A = 0 −Γ2 A2←3  ,




0
0
−Γ3
where Γj =(A.11)P

i Aij is the total decay rate of state j. Here I have imagined that the energies of thestates are ordered E1 < E2 < E3 , and that E1 is the ground state and does not decay. The matrixΩ represents light absorption and thus depends on which lasers are present. For example, it mightlook like





0
0
0





Ω = B2←1 ρ(ω21 )
,
0
0




B3←1 ρ(ω31 ) B3←2 ρ(ω32 ) 0(A.12)

if we have laser fields driving every transition, where Bi←j are Einstein B coefficients and ρ(ωij )is the spectral energy density of radiation at angular frequency ωij = (Ei − Ej )/~. In terms ofmolecular parameters and constants, the Einstein coefficients are
3 |D |2ωij
ij
Ai←j =,
3π0 ~c3
π 2 c3Bi←j =
3 Aj←i ,~ωij(A.13)

where D is the electric dipole matrix element between states i and j.
This model neglects all coherences and thus of course cannot produce phenomena such as
Rabi flopping. However (due to Doppler broadening) we are also unable to observe any Rabi
flopping in the experiment, so this does not seem like a significant limitation. I estimated theelectric dipole matrix elements between various electronic and vibrational states according to themethods described in Chapter 2, and solved the rate equations in Matlab. For molecule-frame
dipole matrix elements, I used the data from Kevin’s thesis and Ref. [99] in order to include as
many electronic states as possible. Rate equation simulation results are shown in Fig. A.5.

214

1

Population fraction

0.8

0.6

M 1

0.4

0.2
M=1

0
(0,0)

(0,1)

(1,1)

(2,1)

(3,1)

(0,1)

(0,1)

(1,1)

(2,1)

(3,1)

(0,0)

(0,0)

(1,0)

(2,0)

(2,1)

(3,1)

(0,0)

(0,0)

(1,0)

(2,0)

(2,1)

(3,1)

(0,0)

(0,0)

(1,0)

(2,0)

State (v,J)(a)
1Population fraction

0.8

0.6

0.4
M=1

0.2
M 1

0
(0,0)

(0,1)

(1,1)

(2,1)

(3,1)

(0,1)

(0,1)

(1,1)

State (v,J)(b)
1Population fraction

0.8

0.6

0.4

M=1

0.2
M 1

0
(0,0)

(0,1)

(1,1)

(2,1)

(3,1)

(0,1)

(0,1)

(1,1)

State (v,J)(c)

Figure A.5: Rate equation prediction for state populations after optical pumping state preparation.
(a) Transfer from 1 Σ+ only via 3 Π0+ , (b) with MF pumping through 3 Π0− , (c) with vibrational
cleanups through 3 Σ−
. Part (b) indicates that the best route to improved signal size is with a0+repump from 3 ∆1 (v = 1) through 3 Π0− (1178.7 nm).

215

A.5

Magnetic fields
In the Generation 2 experiment with HfF+ , we have implemented magnetometry using a net-

work of eight Bartington Instruments fluxgate sensors mounted to the vacuum chamber. Much moredetails about the hardware aspects of the magnetometry will be included in a future work (likely
Tanya Roussy’s thesis), but here I provide a brief overview of the process of fitting magnetometerdata.
The positions of the magnetometers are registered to the vacuum chamber via the mounting
holes machined by Kimball physics. We have a CAD model of the Generation 2 chamber alongwith the magnetometers located in
X:\Magnetometers Mark_2\Inventor\gen2_chamber_with_mags.iam
for which I wrote a VBA macro (quoted below) to export the magnetometers’ coordinates andorientations [in the form of a rotation matrix a la Eq. (2.40)]. An example of the output of thisscript is shown in Table A.3.

1

Public Sub PrintMagCoordsGen2()

2
3
4
5

' Get the active assembly.
Dim oAsmDoc As AssemblyDocument
Set oAsmDoc = ThisApplication.ActiveDocument

6
7
8
9
10

'Set up the CSV output file
Dim OutputFile As String
OutputFile = "X:\Magnetometers Mark 2\Inventor\sensor coordinates.csv"
Open OutputFile For Output As #1

11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26

' Generate header text
Dim FileHeader As String
FileHeader = "sensor" & ", "Dim i As Integer
Dim j As Integer' Rotation matrix elements (9 columns)
For i = 1 To 3
For j = 1 To 3
FileHeader = FileHeader & "R " & i & j & ", "
Next
Next
' Origin coordinates (3 columns)
j = 4
For i = 1 To 3
FileHeader = FileHeader & "O " & i & ", "

216
27
28
29

Next
' Trim final comma and space
FileHeader = Left(FileHeader, Len(FileHeader) − 2)

30
31
32
33

' Print header to file
Debug.Print FileHeader
Print #1, FileHeader

34
35
36
37
38

Dim oOcc As ComponentOccurrence
For Each oOcc In oAsmDoc.ComponentDefinition.Occurrences' Loop through sensor assemblies
If InStr(oOcc.Name, "Mag612") = 1 Then

39

' Get sensor number 1−8
Dim sensorNum As Integer
sensorNum = CInt(Right(oOcc.Name, Len(oOcc.Name) − InStr(oOcc.Name, ...
":")))

40
41
42

43

' Get origin transformation matrix
Dim oMatrix As Matrix
Set oMatrix = oOcc.Transformation

44
45
46
47

' Start this row of coordinate data
Dim RowStr As String
RowStr = sensorNum & ", "

48
49
50
51

' Rotation matrix elements
For i = 1 To 3
For j = 1 To 3
RowStr = RowStr & oMatrix.Cell(i, j) & ", "
Next
Next

52
53
54
55
56
57
58

' Origin coordinatesj = 4
For i = 1 To 3
RowStr = RowStr & oMatrix.Cell(i, j) & ", "
Next

59
60
61
62
63
64

' Trim final comma and space
RowStr = Left(RowStr, Len(RowStr) − 2)

65
66
67

' Print row for this chip
Debug.Print RowStr
Print #1, RowStr

68
69
70
71
72

End If

73
74

Next

75
76

Close #1

77
78

End Sub

In general, we would like to use our X, Y , Z, and axgrad coils to apply magnetic fields to0null out Earth’s field at the location of the ions, and apply as pure and constant a Baxgrad
field

-0.89101
0.45399-0.19186

0.191865

0.376556

0.376556

0.191865-0.19186-0.37656

2

3

4

5

6

7

8

-0.80753-0.41146

0.411455

0.807526

0.807526

0.411455-0.41146-0.80753

R 13

0.191865

0.376556

0.376556

0.191865-0.19186-0.37656-0.37656-0.19186

R 21-0.89101

0.45399

0.45399-0.89101

0.891007-0.45399-0.45399

0.891007

R 22

0.411455

0.807526

0.807526

0.411455-0.41146-0.80753-0.80753-0.41146

R 23-0.90631

0.906308-0.90631

0.906308-0.90631

0.906308-0.90631

0.906308

R 31

0

0

0

0

0

0

0

0

R 32

0.422618-0.42262

0.422618-0.42262

0.422618-0.42262

0.422618-0.42262

R 33

13.52751

6.892613-6.89261-13.5275-13.5275-6.89261

6.892613

13.52751

O1-6.89261-13.5275-13.5275-6.89261

6.892613

13.52751

13.52751

6.892613

O2-7.07962

7.079616-7.07962

7.079616-7.07962

7.079616-7.07962

7.079616

O3

Table A.3: Example output from the VBA script that saves magnetometer coordinates and orientations. The elements R ij belong toa rotation matrix that transforms the magnetometer magnetic field measurement axes into the laboratory axes. The elements O i arecoordinates of the magnetometer’s center.

-0.45399

0.891007-0.89101

0.45399

0.891007-0.45399-0.37656

1

R 12

R 11

sensor217

218
gradient as possible. We would like to simultaneously monitor the magnetic field in the vicinity of
the ion trap using our fluxgate magnetometers, and potentially apply real-time feedback. However,a challenge is presented by the fact that our magnetometers are situated very close to our magnetic
field coils, and thus cannot be used to accurately parameterize the magnetic field in terms of the“interior” multipoles described in Section A.1. We therefore have to adopt a technique of what wecall “coil basis vectors” in order to obtain magnetic field measurements that we can accurately fit
with interior multipoles. The general idea is:(1) At some earlier time (prior to a Ramsey experiment), we collect “coil basis vectors:” valuesof the magnetic field components Bx , By , Bz experienced by each magnetometer due to aunit current in a particular coil pair. These are shown in Table A.4.
(2) When measuring the magnetic field at later times, subtract the coil basis vectors, scaledby the known current in each coil pair, from each magnetometer’s measurement. As longas the magnetometers are operating in a linear regime, this will result in a measurement of
the background magnetic field only.
It is important to re-measure the coil unit vectors any time the coils or magnetometers may have
moved, or if the current supply for any set of coils is changed. Otherwise, the unit vectors should
be fixed by the geometry and the current supply.
Having subtracted the coil unit vectors to obtain the background magnetic fields, we can thenperform spherical multipole fitting to obtain estimates of the (L, M ) components of the magnetic
field about the trap center. Because we only have eight magnetometers with three axes each, we
can fit at most up to L = 4 (and preferably lower order than this). Typically we extract only themultipole components up to L = 2. This fitting is performed by the process magnetometers.mfunction, which is propagated forward into each day’s matlab directory on the HfF+ jilafile
directory.
As a final step towards predicting the magnetic fields experienced by the ions at the trap
center, we must predict the magnetic fields generated by the X, Y , Z, and axgrad coils at that

0.558

3.880
3.313
3.315
4.020
3.949
3.362

2

3

4

5

6

7

8

0.984

-0.4050.561-1.1771.098-0.639

3.881

1

BX IY
-1.004

BX IX
3.338

Sensor

BX IZ

2.120

-0.454-0.882

2.124-2.486

1.288

1.394

-2.457

BY IX-0.2930.383-0.5720.275-0.3520.736-0.676

0.234

BY IY

4.993

3.682

3.745

4.927

5.162

3.633

3.710

5.128

BY IZ

-0.9642.490-2.196

0.927

1.240

-1.8942.163-1.115-1.981-3.865-3.902-1.961

2.031

3.783

3.880

1.852

BY Iaxgrad

BZ IX

0.499

0.184

0.138

0.412

-0.439-0.596-0.324-0.331

Table A.4: Coil unit vectors from May 2019.

3.929

1.945

-2.123-3.910-3.798-1.990

1.932

3.871

BX Iaxgrad

BZ IY-0.524-0.654

0.578

0.643

0.416

0.557

-0.557-0.580

BZ IZ

5.175

5.439

5.207

5.629

4.794

5.658

5.103

5.269

-1.6141.797-1.5651.784-1.7411.859-1.6411.859

BZ Iaxgrad

219

220
location. We do so using a simple parametric model for each pair of coils, with parameters listed
in Table A.5, and whose geometric interpretations are shown in Fig. A.6. Initially, I had aimed
to tweak these parameters using magnetometer measurements of the fields generated by each coilpair, however it subsequently appeared that the magnetometers are simply too close to the coils
to generate sensible results – the model has too many free parameters, and the resulting geometry
was clearly inconsistent with the known coil positions. One important lesson from this process wasthat the number of turns in each coil should remain fixed to the known value, listed in Table A.5.
It may be worthwhile to attempt this fitting again in the future, with certain parameters (such as
the number of turns, width, thickness, and radius of the coils) held fixed. In this instance, it may
be possible to fit the center position of the coil pair rc and their azimuthal rotation phi, which
may be important for systematics studies.

221
Parameter
rc
Rcoil
theta
phi
r rel
theta relphi rel
turns
w
t
Nw
Nt
I rel

X coils[0,0,0]0.21721.57
0[0,0,0.2159]
00800.00760.010288
1

Y coils[0,0,0]0.19811.57
1.57[0,0,0.1968]0085
0.00760.01028
9
1

Z coils[0,0,0]0.17130
0[0,0,0.1716]
00570.00890.00498
91axgrad coils[0,0,0]0.13460
0[0,0,0.1922]
0085
0.00760.01028
9-1

Table A.5: Parameters for magnet coil pairs. The number of turns is the measured value, while
the other parameters are taken from our experiment CAD model. The physical meanings of theparameters are shown in Fig. A.6. The parameter I rel indicates the relative sign of the current
in the two coils – a negative sign results in a gradient rather than a uniform field. Distances are inmeters, angles in radians.

Figure A.6: Parametric description of the orientation of a pair of coils in space. The vector rc
indicates the position of the center of mass of the coil pair. Each coil is located at ±rrel relative
to rc . The coils are rotated by a common set of Euler angles (θ, φ, χ) about the laboratory Z axis,
and are then rotated by a relative set of Euler angles ±(θrel , φrel , χrel )/2 relative to their common
z axis. The coils have radius Rcoil , width w, thickness t, and have numbers of turns Nw and Nt
along their width and thickness.

Appendix B

Analytic calculations

The goal of this section is to reproduce various analytic calculations done over the years, for
the use of future students.

B.1

MCP Transfer Matrices
I assume that the position xd,j of the cloud on the detector along the j th coordinate axis islinearly related to the position of the cloud in the trap:
xd,j = m11,j xt,j + m12,j Td ẋt,j(B.1)

where Td is the time of flight to the detector.
The cloud is given a kick along one of the principal axes of the trap by applying a voltage
pulse to two or more of the electrodes. Figure B.1 shows a triangular pulse with a 2 µs rise time andan amplitude of 1 V programmed on the experiment control DAC and measured on one electrode.
Integrating the measured potential on one fin for a programmed 1 V, 2 µs pulse (amplified 25×in the trap driver electronics) gives 50.5(3) V µs. The electric field at the center of the cloud is
obtained by COMSOL simulation, with the following results for a 1 V potential applied to one fin
Ey
57.21(1) mV/cm
=
Vy
1V
Ez
8.75(1) mV/cm
=
Vz
1V

(B.2)

where the uncertainty is assumed to be simply the magnitude of the smallest digit calculated by
COMSOL. To account for trap geometry and amplification, I define the scale factors χj which

Electrode potential V [V]223
0 V dt = 50.46 V µs

30
20
10
0
0

2

4

6

8

10

12

Time [µs]Figure B.1: Potential on one electrode for 1 V on the DAC.

convert the amplitude of a 2 µs pulse to the time-integrated electric field component in direction j
at the center of the trap, which I call Kj . This assumes that voltage pulses of various amplitudes
applied to different electrodes will have the same overall shape. The χj carry units of µs/cm.
Numerically, these factors are:√
10.00(5) V/cm µs
χx = 2 3 5.721 × 10−2 cm−1 (50.5 µs) =1V
5.774(5) V/cm µs
χy = 2 5.721 × 10−2 cm−1 (50.5 µs) =1V
0.8831(5) V/cm µs
χz = 2 0.875 × 10−2 cm−1 (50.5 µs) =
1V

(B.3)

Assuming that the cloud is stationary before receiving a pulse along direction j, the in-trap
position and velocity after the pulse are
xt,j (t) =eKjsin(ωj t)
mωj

(B.4)

ẋt,j (t) =eKjcos(ωj t)
m

(B.5)

where
Z ∞
Kj ≡

Ej (t)dt.
−∞

(B.6)

224
The measured position on a detector measuring ion motion along direction j isxt,j (t) = Ad,j sin(ωj t + ϕj )(B.7)

= Ad,j cos(ϕj ) sin(ωj t) + Ad,j sin(ϕj ) cos(ωj t)

(B.8)

eKj
eKj
sin(ωj t) + m12,j Tdcos(ωj t).
mωj
m

(B.9)

= m11,jComparing coefficients, I findm11,j =

Ad,j cos(ϕj )mωj
eKj

(B.10)

m12,j =

Ad,j sin(ϕj )meKj Td(B.11)

Then incorporating the fact that Kj = χj Vj where Vj is the amplitude of the voltage pulse applied
to opposing fins,
m11,j =

Ad,j cos(ϕj )mωjeχj Vj(B.12)

m12,j =

Ad,j sin(ϕj )meχj Vj Td(B.13)For the “fast” multi-channel plate (MCP), which can resolve only cloud motion in the x direction,
the transfer matrix elements have dimensions of µs/mm.

B.2

Relativistic & inertial effects
At one point during the challenging times of systematics investigations, I became curious if

there might be additional effects of the rotating, accelerating frame of our measurement, beyond
the usual term Hrot = −ω · F . Not surprisingly, this was not a particularly fruitful avenue ofinvestigation, but I reproduce my notes here in case some offshoot becomes useful in the future.
The goal of these notes is to determine in a moderately rigorous way any corrections to theelectronic spectrum of an accelerating, rotating atom or molecule in its rest frame, in the presenceof external electric and magnetic fields. The approach is to obtain an approximate non-relativistic
Hamiltonian for a single electron in (nuclear, other-electronic, or externally applied) electric andmagnetic fields by taking the non-relativistic limit of the generally covariant Dirac equation. This

225
problem is addressed in the case of no external fields by Hehl & Ni in Ref. [59], while the caseof rotation but no linear acceleration was addressed in Ref. [111]. The style of Ref. [111] is fairly
old-fashioned, confusing, and probably less approximate than we need to be, so we refer only toRef. [59] here.
Following Hehl & Ni, the Dirac equation for a spin-1/2 particle of charge q viewed from aframe with acceleration a and rotation ω is(i~γ µ̂ Dµ̂ − mc)ψ = 0(B.14)

where the covariant derivatives Dµ̂ are
1
iqφ a · α
i
1∂
D0̂ =
+
+
−
ω·J ,
1 + a·x
c ∂t
~c
2c2
2~c
c2

(B.15)
∂
iqA
Dî =
−
,
∂xi
~
and J = L + S is the particle’s total angular momentum. Here we have simply used the expression

from Hehl & Ni and added the electromagnetic four-potential, assuming it transforms into the
non-inertial frame in the same way as the partial derivative ∂µ .
The Dirac equation is then


i~γ 0
1∂
iqφ a · αi++−
ω · J ψ = [mc + γ · (p − qA)] ψ.
1 + a·x
c ∂t
~c
2c2
2~c
c2

(B.16)

We obtain the effective Dirac Hamiltonian by isolating i~ ∂ψ∂t , finding
a · x
i~a · x
HDirac = βmc2 1 + 2 + qφ − ω · J − α · a + c 1 + 2 α · π,c2cc(B.17)

where π ≡ p − qA, β ≡ γ 0 , and α ≡ γ 0 γ. We would then like to take the low-energy limit of
the Dirac Hamiltonian, using the Foldy-Wouthuysen transformation [15, 47]. First, we write the
Hamiltonian as
HDirac = βmc2 + E + O(B.18)

where E and O are the even and odd parts of the Hamiltonian with respect to exchanging particles
and antiparticles (odd terms contain α):
E = eφ + m(a · x) − ω · J,1
O = cα · π +[(α · π)(a · x) + (a · x)(α · π)] .
2c(B.19)

226
The Foldy-Wouthuysen transformation results in the approximate Hamiltonian
HFW = mc2 + E +

i
O2
O4
1 h
−
−
O,
[O,
E]
+
i~
Ȯ
,
2mc2 8m3 c2 8m2 c4(B.20)

consisting of even operators only. We will evaluate HFW still more approximately since we areconsidering non-inertial effects to be small. To be precise, we will drop all terms proportional toa · x/c2 where the other factor is of lower order than the kinetic energy p2 /(2m). That is, we would
keep p2 (a·x)/(2mc2 ), but drop −2µB (S·B)(a·x)/c2 . In addition, we will keep terms only up to firstorder in ω (this turns out to be all ω-containing terms to this order of the F-W transformation).

B.2.1

Mostly kinetic term
2O
By “mostly kinetic,” I’m referring to the term 2mc
2 . As a reminder,O = cα · π +1[(cα · π)(a · x) + (a · x)(cα · π)]2c2(B.21)≡ cα · Π
So that using the commutation properties of α, O2 can be written [15]O2 /c2 = Π2 + iσ · (Π × Π)

(B.22)

Keeping terms only up to first order in a, we get


1  2
1 
· π (a · x) + π · (a · x)π + 2 π(a · x) · π + (a · x)π 222c2ca · x= π 2 + 2π ·π
c2(B.23)

π × [π(a · x) + (a · x)π] [π(a · x) + (a · x)π] × π
+
2c2
2c2
2i~q(a · x)i~
B − 2 (a × π)
= i~qB +2cc(B.24)

Π2 ≈ π 2 +andΠ×Π≈π×π+

So then2~q(a · x)~
σ · B + 2 σ · (a × π),2cc(B.25)O2π2π(a · x) · π
~q
~
≈
+
−
σ·B+
σ · (a × π)
2mc2
2m
mc2
2m
2mc2

(B.26)

iσ · (Π × Π) ≈ −~qσ · B −and finallywhere we have neglected the Zeeman effect reduced by a · x/c2 .

227
B.2.2

Relativistic correction to kinetic term

Here we will keep only the usual relativistic correction to the kinetic term, since everything
proportional to a · x/c2 will be on the order of 10−20 times smaller than that.
−

B.2.3

O4
π4
≈
−
2m3 c28m3 c2(B.27)

Interaction term

Let’s now think about the interaction term, − 8m12 c4 [O, [O, E] + i~Ȯ]. We will neglect completely the terms of O proportional to a · x/c2 . Start with the commutator [O, E]:[O, E] ≈ [cα · π, qφ + ma · x − ω · J]

(B.28)For each of these terms, we find[cα · π, qφ] = i~qcα · (−∇φ),(B.29)[cα · π, ma · x] = −i~mcα · a,

(B.30)[cα · π, −ω · J] = i~cα · (π × ω).

(B.31)

Combined with
i~Ȯ ≈ −i~qcα ·∂A∂t(B.32)

we get[cα · π, qφ + i~Ȯ] ≈ i~qcα · E.

(B.33)The outer commutators are[cα · π, i~qcα · E] = ~2 c2 q∇ · E − ~c2 qσ · (π × E − E × π),[cα · π, −i~cmα · a] = −2~mc2 σ · (a × π),[cα · π, i~cα · (π × ω)] = 2~c2 [π 2 (σ · ω) − (ω · π)(σ · ω)].
So this contribution to the Hamiltonian is
i
1 h
q~2
q~
− 2 4 O, [O, E] + i~Ȯ ≈ − 2 2 ∇ · E −
σ · (E × π)
8m c
8m c
4m2 c2~~+σ · (a × π) −[π 2 (σ · ω) − (ω · π)(σ · π)]24mc
4m2 c2(B.34)(B.35)(B.36)(B.37)

228
B.2.4

Total FW Hamiltonian

Adding together the terms from the previous sections, the single particle Hamiltonian tolowest(-ish) order in a and ω is
π2
π4
q~
q~2
q~
−
+
qφ
−
σ
·
B
−
∇·E −
σ · (E × π)
2m 8m3 c22m
8m2 c24m2 c23~~π(a · x) · π+
σ · (a × π) −[π 2 (σ · ω) − (ω · π)(σ · π)].
+ ma · x − ω · J +mc2
4mc2
4m2 c2

HFW ≈ mc2 +(B.38)
Here the top line contains “usual” terms, while the bottom line contains inertial terms. In thethird and fourth inertial terms, this expression differs from Ref. [59] by factors of order 1. On theother hand, the potential, kinetic, Darwin, spin-orbit, and anomalous Zeeman terms appear withthe correct sign and numerical factors.

B.3

Rotation matrix elements
First, apply the Wigner-Eckart theorem:


F0
(k)
∗
F 0 −MF0 
0 0 0 00
I J Ω F MF Dpq (ω) |IJΩF MF i = (−1)

−MF0


E
F  D 0 0 0 0 (k) ∗ I J Ω F D·q (ω) IJΩF
p MF

k

(B.39)
Now, apply Eq. 5.174 of Ref. [15] for decoupling J and I. We only want the Wigner D matrix toact on the J part of the wavefunction:D(k)

I 0 J 0 Ω0 F 0 D·q (ω)∗ IJΩF
F +J 0 +k+I 0

δI,I 0 (−1)

E=p(2F 0 + 1)(2F + 1)



J



F


I 0 DF 0 J 0


k





0

0

JΩ

(k)
D·q (ω)∗E(B.40)JΩ .

Finally, apply Eq. 5.186 of Ref. [15]:D

J0k J p00 (k)
J 0 Ω0 D·q (ω)∗ JΩ = (−1)J −Ω  (2J 0 + 1)(2J + 1).
0−Ω q Ω
E

Multiplying everything together gives Eq. (2.15).

(B.41)

Appendix C

Designs

C.1

Lab layout
As is evident from the lengthy experiment sequence, the experiment uses a large numberof lasers, as shown schematically in Fig. C.1. Several of our lasers are water-cooled pulsed YAGs
and several more are dye lasers; each of these varieties is quite prone to catastrophic failure, which
consumes substantial time and resources. The following is a complete list of lasers used in the HfF+
experiment:(1) Continuum Minilite (“Abby”)
A Q-switched Nd:YAG 1 . Used for laser ablation of Hafnium metal in our supersonic beamsource. The Minilite has a variable repetition rate from 0 Hz to 30 Hz, allowing us totrigger it irregularly for ablation and conserve the lifetime of its flashlamps. The priceto be paid for the triggerable Q-switch is that the lamps do not “simmer” 2 indefinitely,
and the laser shuts off if it is not triggered every few minutes. Huanqian Loh wired an
external switch into the simmer restart button, and made the external switch accessible
from the main experiment control PC. Around 2015, I wired up a relay to a cheap DAC
controlled by the main experiment control Labview program, allowing Abby to be “primed”
1

Nd:YAG, or usually just abbreviated YAG, meaning a Neodymium-doped Yttrium Aluminium Garnet crystal,
Y3 Al5 O12
2“Simmering” is a technique used to improve repeatability and preserve the lifetime of flashlamps by continuously
running a small discharge through the lamp. The small discharge “seeds” the much larger flash that occurs when thelamp is fired, causing the large flash to develop in the center of the lamp and preventing damage to the surroundingarea.

Spectra-Physics
Quanta-Ray Pro“Mia”
1064 / 532 nm

Spectra-Physics
Quanta-Ray Lab 170“Olga”
1064 / 532 / 266 nm

230

Toptica TA Pro
815 nmRadiant Dyes NarrowAmp“Harambe”
619 nmSirah
Autotracker

Sirah PrecisionScan“Mia”
735 nmVacuum chamber
N

Sirah Cobra-Stretch“Cora”
736 nmMinilite
532 nm532 nm
266 nm
367 nm
368 nm
Fiber
BBO

PBS

309 nm
618 nm
800-1300 nm
735-736 nm
cw lasers
pulsed lasers
NEL doubler

ECDL
815 nm

ECDL
818 nm

DBR
1083 nmToptica TA Pro
1238 nmToptica TA Pro
916 nm

ECDL
818 nmFigure C.1: Geographical layout of the HfF+ eEDM lab as of June 2019. The colors of opticalfiber launchers are simply to correlate the input and output of fibers and are not related to thewavelength. The 818 nm, 961 nm, and 815 nm lasers are combined and sent vertically down throughthe vacuum chamber from an optical breadboard mounted above (translucent white).

231
programmatically.
(2) Spectra-Physics Quanta-Ray Pro (“Mia”)
The Quanta-Ray Pro is a 10 ns Q-switched YAG operating at 1064 nm with a 10 Hz
repetition rate. The Pro is equipped with a second harmonic generator module, producing
about 1 Joule of light at 532 nm. The 532 nm light is used to pump the PrecisionScan and
the NarrowAmp.
(3) Spectra-Physics Quanta-Ray Lab 170 (“Olga”)
The Quanta-Ray Lab is another 10 ns Q-switched YAG with its fundamental at 1064 nm,
and with a repetition rate of 30 Hz. It is equipped with 2nd harmonic and 4th harmonic
generating crystals. The 532 nm light pumps the Cobra-Stretch, while the 266 nm fourthharmonic is used as the second photon of photodissociation.
(4) Radiant Dyes NarrowAmp (“Harambe”)
This pulsed dye amplifier replaced a homebuilt setup in Summer 2016, and has madean enormous difference to the stability of the HfF+ eEDM experiment as a whole. It
doesn’t achieve this by doing anything particularly clever; it simply has a solid aluminium
baseplate with three dye cells and various optics mounted in a simple configuration. Wepump Harambe with approximately 50 mJ of 532 nm light from Mia, which amplifies a
619 nm seed produced by doubling 1238 nm light from Tuna. On a good day, we get about
2 mJ of 618 nm pulsed light from Harambe, which we then double using a BBO to get about
200 µJ of 309 nm UV light. This UV light drives the first transition in our two-photon
photoionization of HfF.
(5) Sirah PrecisionScan (“Bertha”)
The PrecisionScan is a YAG-pumped, grating stabilized pulsed dye laser from Sirah Lasertechnik, equipped with a 2nd harmonic generation module and “autotracker,” which compensates for changes in beam pointing when the frequency is changed. The PrecisionScan is232
equipped with three dye cells, the third of which is a “capillary amplifier,” a special tubular dye cell meant to improve the output beam shape, which it does to a limited extent.
We typically operate this laser at 735.4650(5) nm or 13596.84(1) cm−1 , and use the 2nd
harmonic generated by a nonlinear crystal housed in the autotracker (either BBO or KDP,
dpeending on the wavelength range) for the second photon of photoionization of HfF.
(6) Sirah Cobra-Stretch (“Cora”)
The Sirah Cobra-Stretch is another YAG-pumped, grating-stabilized pulsed dye laser. Thislaser has only two dye cells, but the beam is double-passed through the first cell to form
an oscillator plus an amplifier stage. With this laser, we produce the first photon of our
two-photon resonance-enhanced photodissociation detection. In the course of this thesis,
we have used this laser to its full tunable potential, scanning it from about 570 nm to760 nm to find suitable intermediate states for photodissociation (Chapter 6).
(7) Toptica TA Pro (961 nm) (“Toptica”)“Toptica” was our first laser purchased from Toptica Photonics, and it has been the most reliable laser in the lab. Until Fall 2018, we operated this laser on the 3 Π0+ ← 1 Σ+ (1, 0) R(0)transition for stimulated Raman transfer, as described in Refs. [28, 82]. In Fall 2018,
we used this laser’s impressive tunability to move over to 961 nm and start driving the
3Π

0+ ←

1 Σ+ (0, 0) P(1) line for incoherent transfer (Chapter 5).

(8) Toptica TA Pro (1238 nm) (“Tuna”)
We purchased this TA Pro in Fall 2016 to replace a homebuilt ECDL that featured a highpower 1238 nm gain chip from Innolume. While the details are proprietary, we believe that
this new laser uses a very similar gain chip, but also includes a tapered amplifier (TA).
This laser produces several hundred mW at 1238 nm, which we double to 619 nm usinga ridge waveguide doubler from NEL Photonics. The 619 nm light acts as a seed for the
Radiant Dyes NarrowAmp pulsed dye amplifier, Harambe.

233(9) Toptica TA Pro (815 nm)
The 815 nm TA Pro is our most recent purchase from Toptica, and replaced a cw Ti:sapphirelaser that was adopted by Yan and Kia Boon on the ThF+ eEDM experiment, due totheir need for a widely tunable NIR cw laser. The 815 nm laser has been slightly morechallenging than our earlier TA Pros, as it has a much worse beam profile, suffers fromsignificant thermal lensing in the TA chip, and has had issues with drifting alignment
into the TA. Still, when optimized, the laser produces several Watts of light at 815 nm,
which is important for driving the weak 3 Σ−
← 3 ∆1 (0, 0) transition for strobed depletion
0+(Chapter 5).
(10) Photodigm TO-8 DBR (1083 nm)
It is always nice when there is not much to say about a laser. This DBR laser from
Photodigm does what a DBR laser should do: It is always single-mode, it only mode-hops
when you expect it to, and it doesn’t complain. This laser is used for optical pumping ofMF levels on the 3 Π0− ← 3 ∆1 (0, 0) P(1) transition.
(11) JILA ECDL (818 nm) (“Stella”)(12) JILA ECDL (818 nm)(13) JILA ECDL (815 nm)
These grating-stabilized ECDLs are of the JILA design originated by Scott Papp someyears ago. We have swapped out their diodes and gratings a number of times, and theirreliability seems to be limited by the quality of diode installed in them, and the degree ofcare taken in optimizing the optical feedback to achieve lasing on a single external cavity
mode. At this point, only Stella has an anti-reflection (AR) coated diode installed, and so
it is the most tunable and reliable. For improved future performance, we could purchaseAR-coated diodes for these ECDLs. These lasers are all used as cleanups, as described in
Chapter 5.

234(14) New-Focus Vortex
This laser is stabilized to the 87 Rb 5S1/2 F = 2 → 5P3/2 F 0 = 3 crossover peak at
780.246291 nm. We use this reference to calibrate our HighFinesse WS7 wavemeter, towhich we lock all of our other cw lasers. The high temperature of our HfF+ ions of about
5 Kelvin leads to Doppler widths on the order of 40 MHz, so that the 5 MHz relative
precision of the WS7 is sufficient to lock on resonance.

C.2

Generation 2 Ion Trap
Table C.1 contains multipole fit coefficients up to l = 11 for the Generation 2 ion trap

electrodes, numbered 1-8, T (top), and B (bottom).
The following page shows the mechanical drawings for the Generation 2 custom vacuum
chamber, provided by Kimball Physics. These may be useful for future mechanical designs involving this chamber, in particular for mounting electronics or optics to the machined mounting
surfaces.

C

SECTION C-C
SCALE 1 : 2

1.25 REF(16) PLCS

8-32 UNC X 1/2"
EQ. SPA. ON
1.00 B.C.
(80) PLCS

.625
.500

C

7

8

7

NOTES:
1. MAKE FROM 53-183112
2. ESTIMATED FINISHED WEIGHT: 37 lbs./16.8 kg.

6.009
REF

1.250
REF

B

'B' SIZE PART TEMPLATE; REV. 6

A

B

C

D

8

B

18° REF
TYP.

45° REFTYP.

MCF1000 CF PORT
(2) PLCS

6

INITIAL RELEASEREVISION DESCRIPTION

A
REV

5

2 1/8" CF PORT
(8) PLCS

4

BY

CHK

IMK

07/12/16 IMK
DATE

IMK

07/13/16 IMKECOSECTION B-B

5.947 REF
(8) PLCS

23.65°
TYP.

5.885 REF
(4) PLCSMCF275 CF PORT
(4) PLCS

2 1/8" CF WAS 1.33 CF. 1.25 FLAT WAS 1.33. (80) 8-32 WAS (64)12.082 REF
SPHERICAL

8.300 REF
THRU

4

MCF600 CF PORT
(4) PLCS

B

10.200 REF
SPHERICAL

A

A

6.824 REF

5

4.300 REF
THRU

USED ON

0.01
0.003
0.0010ANGULAR:
X
0.5
X.X
0.1

DATE:___________PRELIMINARYFOR CUSTOMERAPPROVAL

1B SIZE

1:4

SCALE

2

0.005/0.010 SHEET 1 OF 1

BREAK EDGESNOTE 1

MODEL No. QQ-000369-P3

1

QQ-000369-P3

DWG. No.

CHMBR, MCF1000-Cust-H2F4C4B8

B

REV.

KIMBALL HILL RD., WILTON, NH 03086-9742 U.S.A. (603) 878-1616
THIS DOCUMENTATION IS THE PROPERTY OF
KIMBALL PHYSICS, INC. AND IS PROVIDED TO
THE RECEIVING PARTY (AND ITS EMPLOYEESMATERIAL
HAVING A NEED TO KNOW) SOLELY FOREVALUATION. IT MAY NOT BE USED BY THE
RECEIVING PARTY FOR ANY OTHER PURPOSEOR DISCLOSED TO ANY THIRD PARTY WITHOUT FINISH 16
THE WRITTEN CONSENT OF KIMBALL PHYSICS, INC.

INTERNAL CORNERS R.015 MAX
COAXIAL 's CONCENTRIC WITHIN .003 TIR

310.504REF

2SIGNED___________________UNLESS OTHERWISE SPECIFIED DIMENSIONSARE IN INCHES AND TOLERANCES ARE:

X.XX
X.XXX
X.XXXXNOTE 11.000 REF
THRU (8) PLCS

1.500 REF
THRU (4) PLCSSECTION A-A

3

A

B

C

D

236

C.3

Trap amplifier
The following pages contain a reproduction of Terry Brown’s circuit designs for the Generation

2 ion trap amplifiers. These are also retained in the electronics shop’s record-keeping system.

A

B

C

D

0805

C1
0.1uF

0805
0.1%

1%
0805

R2
100R

R37
1.000M

J5
INPUT

Out/200
J3
OUT_MON

8

0.1%
0603

100R
R4

6

8

-

4

-V1

U1

+

7

0.1uF
C35

0603

2

3

0603

0603
0.1%

R12
DNI

1%
0603

1%
0603

C37
1.0uF

R76
1.00M

R75
150k

D1
BAV99T

CW

0R
R79

0.1%
0603

MON_OS
R77
10k

0603
1%

R78
0R

-V1

+V1

TP8
Out_MON

0805
0.1%

1

ADA4898-2YRDZ

U3-A
3
+

2

R39
100R

-V1

4

0.1uF
C23

0603

ADA4898-2YRDZ
U3-C

8

0603

+V1 C25
0.1uF

1

8

5

7

HV_RTN

0.1uF
C15

10pF
C12

0.1%
0805

AD829JR

6

1.00k
R25

0603

U5

-V1

4

+

-

7

C26
5.2-30pF

C27
22pF

1.00k
R34

3

2

C11
10pF

0805
0.1%

0.1%
0805

0805
0.1%

Slew Rate limitor circuit to lesson large signal saturation
issues. Adjust C26/C27 such that slew rate is ~18 V/us.
This rate yields the ~maximum PA94 slew rate.

D3
BAV99T

0.1%
0805

R45
100R

0.1%
0805

R1
100R

R27
1.00k

0.1%
0805

100R
R44

7

ADA4898-2YRDZ

+
U3-B
6
-

5

R33
1.00k

0603

+V1 C22
0.1uF

TP3
SR_MON

0805
1%

R87
51.1R

0603
1%

R68
511R

0603
1%

R32
49.9R

0805
0.1%

R29
249R

0603

2

0603

U4

8

R46
1k
PI Frequency

0603
1%

CW

6

0603

C16
470pF

0603
1%
1/8W

R15
150k

6

5

JP2

JP1
2

2

Closed Loop
Jumper 1&2 for Closed Loop

1

1

1%
0603

R71
1.00M

511R
R48

1%
0603

0805
1%

C43
5pF

4

47V

4

C44
5pF

-V1

+V1

-V1

1206
0.1%
1/4W

R14
10.0k

1210
+-0.5pF
500V

C45
5pF

1206
0.1%
1/4W

R18
10.0k

1210
+-0.5pF
500V

C46
5pF

1206
0.1%
1/4W

R19
10.0k

1210
+-0.5pF
500V

C47
5pF

3

1206
0.1%
1/4W

R20
10.0k

1210
+-0.5pF
500V

C48
5pF

1206
0.1%
1/4W

R21
10.0k

1210
+-0.5pF
500V

C49
5pF

316V

271V

226V

181V

137V

92V

1206
0.1%
1/4W

R22
10.0k

1210
+-0.5pF
500V

C50
5pF

5

4

3

2

1

Vc RTN

Vc-15V IN+-RTN+15V INJ9TO HV BOARD

3-15V IN+-RTN+15V IN

3

2

1

2

0805

C6
0.1uF

0805

1.0R
R6

1%
0805

0805
1%

R16
1.0R

1206
0.1%
1/4W

R23
10.0k

1210
+-0.5pF
500V

C51
5pF

C8
0.1uF

360V

47uH
L2

L3
47uH

1206
0.1%
1/4W

R24
10.0k

1210
+-0.5pF
500V

C52
5pF

+

+

C7
33uF

C10
33uF

-V1

+V1

0805

C38
DNI

0603
1%

R70
511R

0805
1%

M1
4-40

HV RTN

R80
DNI

NC

HV OUT
2

M2
4-40

VERSION:

DATE: 06/06/2017

DN: CE085A3.1

V3.1

SHEET 1

T. Brown

Cornell / Ye

1

OF 2

DRAWN BY:BUILT FOR:COMPOSITE_AMPLIFIERUPDATE: 7/31/2017

2

M3
4-40

V3.1

High Voltage Composite Amplifier
T. Brown

CU-Boulder & NISTDESIGN BY:

ELECTRONICS

JILA

1

J6
HV_IN from HV AMP

3

1

TP12
HV_OUT

DNI part values = Do Not Install.

J4
POWER INPUTHV Output offset null. Terminate Sig Input with 50 ohms.
While measuring HV Out, adjust R72 until HV Out is ~0V.
Typically HV Out zero is within ~1mV.

1206
0.1%
1/4W

R9
10.0k

1210
+-0.5pF
500V

+V1

0603
1%

R74
0R

OUT_OS
R72
10k

0603
1%

R73
0R

1206
0.1%
1/4W

R10
10.0k

1210
+-0.5pF
500V

R69
51.1R

CW

HV_RTN

1206
0.1%

R11
499

2.23V

TP11
DIV_MON

C36
1.0uF

1210
+-5%
500V

C5
22pF

Jumper 1&2 for Open Loop
Open Loop

0603
0.1%

R8
DNI

D2
BAV99T

0603
1%

R52
511R

Jump JP1, or JP2, not both.

Vc
TP10

TP2
ERROR

5

ADA4898-1YRDZ

0603

0.1uF
-V1 C13

4

-

+

7

R47
49.9R

0603
1%

R35
0

2

3

4

-

+

0.1uF
-V1 C2

U7

3

0603

+V1 C24
0.1uF

0603
1%

R64
511R

0R
R7

0.1%
0603

6

8

7

+V1 C40.1uF

ADA4898-1YRDZ

1k
R31
Gain

CW

0603
1%

R81
100R

D4
BAV99T

0805
0.1%

R28
1.00k

TP4
GND

6

It is important to use thin- film resistor (not thick-film) for the noise critcal paths
due to thin-film superior noise properties.

0603
1%

R65
511R

Output Monitor offset null. Adjust HV Output Offset Adjustment first.
Terminate Input with 50 ohms. While measuring Output Monitor,
adjust R77 until Out Monitor is ~0V (10's of microvolts).

D6
BAV99T

0805
1%

R3
51.1

ADA4898-1YRDZ

Output range of U1 is scaled to +/- 2.5V maximum
to reduce 2nd harmonic distrotion in U1.
+V1 C34
0.1uF

7

405V

8

450V

A

B

C

D

A

B

C

D

8

1

2

3

4

5

+15V IN+-RTN-15V IN

Vc

Vc RTN

5

-V3

+V3

6
LF356M

J8
From Control Board

TP14
OUT

4

+

-

-V3

1

U8

7

+V3

0603
C20
0.1uF

0603

0.1uF
C33

0603
10%
50V

C40
1.5nF

0603
1%
1/10W

R85
1.0k

R49
316

0603
0.1%

316
R53

0.1%
0603

0603
0.1%

R59
10.0k

0603
0.1%

R66
10.0k

5

6

0603
0.1%

0.1uF
C30

DNI
C29

1.00k
R55

6

100
R51

CW

1

JP3
2

AC Channel

1%
0805

R67
121R

TP7
ILCOM
Q1
MMBT3904

1%
0805

1%
0805

R61
100

Remove Pins 2 & 4

-450V

NC

HV_RTN

NC

+450V

J1
High Voltage Input
1

5

3

3.3R
R13

1%
0805

0805
1%

R5
3.3R

5

47uH
L4

L1
47uH

-V2

+V2

1%
0805

R58
121R

TP6
-IL

Q2
MMBT3906

Differentiator V to I converter.
Iout = (jw*R50*C17/Rs)*Vc, (Rs = R54+R51)
Gc = Vout/Vc = -Iout/(jw*C14) = (R55/Rs)*(C21/C14).
with R50 = 1k, Rs = 143R, C17, C21 = 33pF, and C14 = 4.7pF
Gc = ~-50. Gc should match PA94 gain from DC control input.

D_Out
TP9

Rs, Resistor Sum 143 ohms Nominal

1%
0805

R54
75

TP5
+IL

R60
100

JP5

1

2

2

4

M5
4-40

DC-OFF
JUMP ONE NOT BOTH

JP4
1

DC-ON

BAV99T
D7

M6
4-40

RTN

FFWD

2

1

1%
0805

R62
DNI

2

0805
0.1%

D5
BAV99T

Error
TP1

0.1%
1206
1/4W

R40
49.9k

1206
10%
1kV

C32
10nF

2

1

0.1%
1206
1/4W

R36
49.9k

0.27uF
C9

8

6

3.9
R26

PA94

7

1111
+-.25pF
1.5 kV

C14
4.7pF

Q3

C3
0.27uF

U2

4

-V2

+
10

-

12

+V2

0.1%
1206
1/4W

R38
49.9k

0.1%
1206
1/4W

R30
49.9k

2W

R17
50

3

DATE: 06/06/2017

DN: CE085A3.1

V3.1

VERSION:

2

REMOVE PIN 2

HV RTNNC

HV OUT

J2
High Voltage Output

V3.1

SHEET 2

T. Brown

Cornell / Ye

1

OF 2

DRAWN BY:BUILT FOR:

HIGH_VOLTAGE_AMPLIFIERUPDATE: 7/31/2017

T. Brown

CU-Boulder & NISTDESIGN BY:

ELECTRONICS

JILA

3

1

1

High Voltage Composite Amplifier

R43/C28 are used to reduce the bandwidth of PA94 to around 24 kHz.
This shuts off the input stage of the PA94, so the PA94 feedback circuit does notcancel the higer frequency bandwidth of the compensation input (pin 4). The
Feedback network effectively forms a low pass (DC input to R42) / high pass
network via Pin 4 w/crossover frequency ~ 24 kHz. The bandwidth through pin 4 is ~ 4 MHz.

1206

C28
4.7uF

0805
0.1%

R43
49.9R

0603
1/10W

R63
511R

0.1%
1206
1/4W

R41
49.9k

Under normal operation, Pin 4 of PA94, sits 5.4V below the plus power supply (Pin 12). The maximum
voltage swings on Pin 4 is +2.0V, -1.5V, or Pin 4 voltage is between 3.4V and 6.9V below the positive
supply voltage. Q3 forms a low capacitance clamp to protect pin 4. Q3_Vbe zeners at around -9V,
then forward bias at +0.7V. This clamps pin 4 of the PA94 to (+V2 + 0.7), to (+V2 - 9V). Pin 6
tracks the output voltage, pin 7.

R86
DNI, or 4.99k

0805
0.1%

R42
4.99k

0603
1%

0805
1%

R56
100k

R84
681R

J7
Alternate FFWD Input

R51 Adjustment: While in Open Loop Mode (JP1 Shorted, JP2 open, page 1).
With 1Vpp signal input at 100 kHz, adjust HV Out to be 100 Vpp. Comfirm
HV Out is 100 Vpp at 100 Hz.

IN
TP13

AD829JR

0603

U6

8

0.1%
0603

R50
1.00k

C18
DNI

10uF
-V3 C42

4

-

2

3.9pF
C19

+

3

0603

C31
0.1uF

71

C41 +V3
10uF

DNI: R66, and R59 fix issue

1%
0603

7

0603
1%
1/10W

R82
10.0k

116 kHz LP

33pF
C21

1%
0603

R57
150k

0603
1%

C17
33pF

C39
1.5nF

0603

R83
1.00M

0603

3

2

AC Control Current Input GAIN:

U8 Restores U6's DC offset due to asymmetry in PA94
HV Amp's positive and negative slew rate response. At highsquare-wave frequencies this aysmmetry can put U6 on the rails.

0V

+

3

90V

4

180V

5

270V

6

360V

7

+

8

450V

A

B

C

D

239

C.4

Trap synthesizer
The following pages contain a reproduction of Felix Vietmayer’s circuit designs for the Gen-eration 2 ion trap synthesizer. These are also retained in the electronics shop’s record-keeping
system.

A

B

C

D

SCLK

DIN

DGND

8

10

12

FPGA1-V6

FPGA1-P7

SCLK

DIN

8

10

FPGA1-T5

FPGA1-T6

8

LDAC

11

FPGA1-R7

0.1uF
C112

+15V

RFB

AGNDF

AGNDS

V_OUT

RFB

1

GND
2

IN OUT

U20
LM7806CT

3

3

7

0.1uF
C113

AGNDF

AGNDS

4

2

1

0
R32

0 R34

INF
R31

A

A

C

C

K

K

VO_S

VO_F
6

7

GND
5 +5V
0.025%

SHDN

1
3
G1*
4
G2*
8
G3*

VIN

2

U19
LTC6655B-5V

0
R37

0 R39

INF
R36

R35
0
C79
0.1uF

+VREF

13

6

REFF

INV

5

3

4

2

1

REFS

V_OUT

U16

AD5542BRZ

DGND

CS

7

FPGA1-V4

12

REFF

0.1%

R30
0
C3
0.1uF

+VREF

13

6

INV

5

7

REFS

"shim DAC"

VDD

L27
1kohm

14

C107
0.1uF

LDAC

11

FPGA1-U5

+5VA

CS

7

FPGA1-T7

U1

AD5542BRZ

VDD

L17
1kohm

"kick DAC"

14

C4
0.1uF

+5VA

8

3

2

6

8

0.1uF
C108

1

U17

-15V

4

+

-

C111
10uF

+VREF

BAV99TD7

32AD8675ARZ

6

10pF
C140

C106
0.1uF

+15V

7

ADA4627-1ARZ

5

6

C139
10pF

0.1uF
C78

1

U6

-15V

4

+

-

7

C77
0.1uF

+15V

BAV99T
D8

6

5

5

2.49k
R65

0.1%

332
R64

9

D 10

8

D 7

1

D 2

D 15

1

D 2

D 15

-15V

-15V

4

V- G
4 5
C81
10nF

0R

0R

C40

ASHIM

-15V

V- G
4 5
C10
10nF

U31-E
13 ADG5212
V+3

ec t ro ni cs

ASHIM

FPGA1-C5

FPGA1-A7

FPGA1-M18

A3

FPGA1-V9

A2

FPGA1-A15

FPGA1-R8

A0

AKICK

A1

6
ADA4898-1YRDZ

8

0.1%

R5
499

C7
33pF

0.1uF
C129

U7

-15V

4

+

-

7

+15V

J2

A3

9

D 10

8

D 7

1

D 2

D 15

9

D 10

6 S
8

D 7

U30-D
ADG5212

11 S

U30-C
ADG5212

16

14 S

U31-B
ADG5212

3 S

U31-A
ADG5212

6 S

U31-D
ADG5212

11 S

U31-C
ADG5212

0.1%

R8
50

1

DATE:

DN: <DN>

2

VERSION:<DESIGNER><DRAWNBY><BUILTFOR>

1<DATE> SHEET 1 OF 7

A3 DRAWN BY:

BUILT FOR:ANALOG_1

JILA ELECTRONICS LABFPGA / DDS SYNTHESIZER

R6
0

3

2

C11
0.1uF

2

UPDATE: 9/14/2017

DESIGN BY:

el

0.1%

R41
75.0

0.1%
0R

AKICK

R40
75.0

0.1%

R4
75.0

0.1%

R3
75.0

0.1%

R2
75.0

0.1%

R1
75.0

C43

0R

C42

0R

C41

A1

A3

3

C12

A0

A2

C9
10nF

+15V

FPGA1-T9

FPGA1-T8

FPGA1-T4

FPGA1-R5

FPGA1-T3

FPGA1-R3

U30-E
13 ADG5212
V+

C80
10nF

+15V

16

14 S

U30-B
ADG5212

3 S

U30-A
ADG5212

16

14 S

U8-B
ADG5212

3 S

U8-A
ADG5212

6 S

U8-D
ADG5212

11 S

V- G
4 5
C110
10nF

U8-E
13 ADG5212
V+

C109
10nF

+15V

SHIM

KICK

MULT3

MULT2

MULT1

MULT0

U8-C
ADG5212

4

A

B

C

D

A

B

C

D

8

FPGA1-V11

FPGA1-N17

DDS3_AC6
0.1uF

FPGA1-N10
FPGA1-P11
FPGA1-M14
FPGA1-N14
FPGA1-T12
FPGA1-V12
FPGA1-N11
FPGA1-M11
FPGA1-N15
FPGA1-N16

8

+5VA

GND

3

CLK

1

U9

GND

I_OUT

7

6

5

8

RFB

CS

3

AD5543BRZ

SDI

VREF

4

2

VDD

7

L21
1kohm

2

20

1

19

18

16
CS
17
R/W

IOUT_2RFB

IOUT_1

VREF

VDD

AD5433YRUZ

DB9
DB8
DB7
DB6
DB5
DB4
DB3
DB2
DB1
DB0

4
5
6
7
8
9
10
11
12
13

U3

0

R45 0.1%

FPGA1-T14
FPGA1-P15

0

R43 0.1%

DDS1_A

L18
1kohm
C5
0.1uF

+5VA

7

D4

C125
10 pF

K

C

6

K

C

A

A

BAV99T

D6
BAV99TFPGA1-U18

C119

6

3

2

3

2

7

47.5k

R51
0.1%

5

6

5

MULT3

C38
6.8nF

ADA4627-1ARZ750
R62
0.1uF
C124

1

U24

-15V

4

+

-

C123
0.1uF

+15V

6

GND

3

VREF

VDD

2

20

DDS2_AFPGA1-N18

2

U4
CS

GND

I_OUT

RFB

C116

AD5543BRZ

SDI

CLK

VREF

4

1

VDD

7

L20
1kohm

FPGA1-R11
FPGA1-T110

R42 0.1%

DDS0_A

L19
1kohm
C2
0.1uF

+5VA

4

C1
0.1uF

MULT1

FPGA1-V14

C37
68pF

19

18

1

4
+5VA

16
CS
17
R/W

IOUT_2RFB

IOUT_1

U2

AD5433YRUZ

DB9
DB8
DB7
DB6
DB5
DB4
DB3
DB2
DB1
DB0

4
5
6
7
8
9
10
11
12
13

ADA4627-1ARZ

5

0.1uF
C118

1

U22

-15V

4

+

-

7

C117
0.1uF

+15V

FPGA1-V5
FPGA1-V7
FPGA1-U7
FPGA1-V8
FPGA1-U8
FPGA1-N6
FPGA1-N7
FPGA1-P8
FPGA1-N8
FPGA1-M8

5

3

6

5

3

8

K

C

ec t ro ni cs

A

D5

C

750

R7 0.1%

A

3

2

7

5

0.1uF
C121

1

U23

-15V

4

+

-

MULT0

A3

MULT2

C39
2.2nF

ADA4627-1ARZ
3.01k
R63

6

C36
6.8nF

C120
0.1uF

+15V

ADA4627-1ARZ

6

1

DATE:

DN: <DN>

2

VERSION:<DESIGNER><DRAWNBY><BUILTFOR>

1<DATE> SHEET 2 OF 7

A3 DRAWN BY:

BUILT FOR:MDACSJILA ELECTRONICS LABFPGA / DDS SYNTHESIZER

BAV99T K

C122
10 pF

5

0.1uF
C115

1

U21

-15V

4

+

-

FPGA1-P16

3

2

7

C114
0.1uF

+15V

2

UPDATE: 9/14/2017

DESIGN BY:

el0

R44 0.1%

D3
BAV99T

3

A

B

C

D

A

B

C

D

8

DDS1_F

DDS0_F

8

R67
49.9

R66
49.9

C

K

C

K

7

BAV99T
D11

A

BAV99T
D9

A

7

R58
100

49R9

R54
R55

4

-

6

-15V

2

3

4

-

+

7

8

2.74k

0.1uF

6

C49

C30

C32

1

U28

5

DNI

DNI

R75
49.9

5

0603
0.1%
1/10W

R78
49.9

TP6

TP5

5

0603
0.1%
1/10W

C34 0.1uF

2.74k

6

C48

C28 0.1uF

0.1uF

C8

8

5

C138

1

U26

R59

AD829JR

+15V

-15V

2

AD829JR 7
3
+

+15V

6

DDS1_A

DDS0_A

DDS3_F

4

DDS2_F

4

R69
49.9

R68
49.9

C

K

C

K

3

BAV99T
D10

A

BAV99TD12

A

3

ec t ro ni cs

R57

4

-

+

7

8

5

DNI

2.74k

0.1uF

AD829JR6

C51

C31 0.1uF

2.74k

C27

C29

6

DNI

AD829JR

0.1uF

U27
1

8
C33

C35

1

U29

5

C50

R79
49.9

TP8

0603
0.1%
1/10W

R80
49.9

0603
0.1%
1/10W

C137 0.1uF

DDS3_A

DDS2_A

A3

TP7

1

DATE:

DN: <DN>

2

VERSION:<DESIGNER><DRAWNBY><BUILTFOR>

1<DATE> SHEET 3 OF 7

A3 DRAWN BY:

BUILT FOR:

PREAMPJILA ELECTRONICS LABFPGA / DDS SYNTHESIZER

-15V

2

3

4

-

R61

+15V

-15V

7
+

UPDATE: 9/14/2017

DESIGN BY:

el

R56
49R9

R60
49R9

2

3

+15V

2

A

B

C

D

A

B

C

D

8

CLK_IN_N

CLK_IN_P

8

10

+3.3V

100
R74

9

7

REF_IN-REF_IN+

VDD1

1

0.1uF

VDD2

7

8

C44
0.1uF

U5
PI6C4911506LIE

VDD3

13

C45
0.1uF

VDD4
20
C46
0.1uF

VEE
12
C47

VBB
11

6

CLK0
CLK0#
CLK1
CLK1#
CLK2
CLK2#
CLK5
CLK5#
CLK4
CLK4#
CLK3
CLK3#

6

3
2
5
4
7
6
19
18
17
16
15
14

4

0.1uF

R19
130

R22
130

R71
100

R73
100

R77
130

5

Place Rs close to FPGA
100 Ohm diff pair

56R2

R72

56R2

R70

R76
130

4

Place R76 & R77 close to U5
100 Ohm diff pair

R21
130

CLK_OUTB_N

CLK_OUTB_P

CLK_OUTA_N

CLK_OUTA_P

REFCLKBREFCLKFPGA1-T10

FPGA1-R10

0.1uF
C83

R20
100

Place R21 & R22 close to U5
100 Ohm diff pair

R18
130

Place R18 & R19 close to U5
Place R20 close to DDS
C84
100 Ohm diff pair

5

3

3

ec t ro ni cs

5
4

6

10

8

C133
10nF

SYNC_CLK

0603

R50 SYNC_OUT
49.9

0603

R49
49.9

+3.3V

1

A3

DATE:

DN: <DN>

2

VERSION:

UPDATE: 9/14/2017<DESIGNER><DRAWNBY><BUILTFOR>

1<DATE> SHEET 4 OF 7

A3 DRAWN BY:

BUILT FOR:

CLOCKING

JILA ELECTRONICS LABFPGA / DDS SYNTHESIZER

U25-B

SYNC_OUTDESIGN BY:

el

9

U25-C

SYNC_CLKFPGA1-V13

2

A

B

C

D

A

B

C

8

10nF
C76

0.1uF
C74

L22
1kohm
C71
4.7uF

3 MASTER_RESET

2 SYNC_OUT

1 SYNC_IN

7

R52
500

R53
500

SYNC_CLK54

FPGA1-U16 53
SYNC_CLKSDIO_3

FPGA1-V16 52
SDIO_2

SDIO_150 SDIO_0

FPGA1-U15 51

FPGA1-V15

CS

FPGA1-M16 47

FPGA1-T17

I/O_UPDATEFPGA1-L12 46

P2

FPGA1-U17 42
P3

P1

FPGA1-U11 41

FPGA1-L13 43

P0

48 SCLK

R47
1.2k

R46
1.0k

REF_CLK

624 CLK_MODE_SEL

23

FPGA1-P18 40

FPGA1-T18

REFCLK

4 PWR_DWN_CTL
R481.91k 17
DAC_RSET
0603
22
REFCLKB
REF_CLKFPGA1-P17

FPGA1-U13

SYNC_OUTAVDD

FPGA1-C13

TRIGGERSYNC_IN

J5

6

30
35
36
8
9

13
14

CH0_IOUT
CH1_IOUT
CH1_IOUT
CH2_IOUT
CH2_IOUT

CH3_IOUT
CH3_IOUT

AD9959BCPZ

LOOP_FILTER 27

29

CH0_IOUT

U11

0.1uF
C73

L23
1kohm
C72
4.7uF

+1.8V +3.3V+

AVDD 5+7

DVDD 45
DVDD1 55
6 AGNDD8

DVDD_I/O 49

44 DGND
56 DGND1

AVDD

680pF
C82

R17
0

5

5

C53

C52

10nF

C55

10nF

C54

10nF

10nF

4

3

4

3

3

2

3

2

AVDD

4

R15ADTT1-6
T4
50

AVDD

R16
50
1

AVDD
AVDD

R13ADTT1-6
T3
50

AVDD

R14
50
1

AVDD
AVDD

4

5

6

4

5

6

5

2

R11ADTT1-6
T2
50

AVDD

R12
50
1

AVDD
AVDD

6

5

6

2

1

R9 ADTT1-6
T1
50

AVDD

R10
50

AVDD

4

DDS3

DDS2

DDS1

DDS0

TP4

TP3

TP2

TP1

1

1

8

8

3

AGND:2,3
4,5,6,7
2.3 MHzF4
PLP-1.9+

AGND:2,3
4,5,6,7
2.3 MHzF3
PLP-1.9+

AGND:2,3
4,5,6,7
2.3 MHz8

8

F2
PLP-1.9+

AGND:2,3
4,5,6,7
2.3 MHz1

1

F1
PLP-1.9+3

ec t ro ni cs

1

A3

DATE:

DN: <DN>

2

VERSION:<DESIGNER><DRAWNBY><BUILTFOR>

1<DATE> SHEET 5 OF 7

A3 DRAWN BY:

BUILT FOR:DDSJILA ELECTRONICS LABFPGA / DDS SYNTHESIZERUPDATE: 9/14/2017

DESIGN BY:

el

DDS3_F

DDS2_F

DDS1_F

DDS0_F

2

A

B

C

D

A

B

C

D

D1
TVS
24V

D2
TVS
24V

8

93LC56

DI

1

3

VSS DO

4

CLK

2

5

CS

U13

VCC

1

C102
30pF

2

7

Y2
12 MHz

R28
2k

C103
30pF

0603

62 EECLK

0603

OSCO

OSCI

13 TEST

3

2

61 EEDATA

63 EECS

14 RESET#

6 REF

8 DP

7 DM

R29
10k

R23
10k

0603

10R 0603
R25
12k

4

2

6

6

49 VREGOUT

50 VREGIN

0.1uF

C95

R27

4.7uF

+ C92

3

R24
15k

0.1uF

C94

USB_3V3

10R 0603
R26

C87
10nF

L26
600ohmUSB_1V8

USB_VPLL

USB_VPHY

1

6

USB_VBUS

USB1

USB_VBUS

7

VPLL 98

FPGA1-F17
FPGA1-F18
FPGA1-L18
FPGA1-L17
FPGA1-C18
FPGA1-C17
FPGA1-K13
FPGA1-K12
FPGA1-G18
FPGA1-G16
FPGA1-H14

ADBUS1 17
ADBUS2 18
ADBUS3 19
ADBUS4 21
ADBUS5 22
ADBUS6 23
ADBUS7 24
ACBUS0 26
ACBUS1 27
ACBUS2 28
ACBUS3 29
ACBUS4 30

FT2232HLFPGA1-L14

ADBUS0 16

FPGA1-D16
FPGA1-B18

BDBUS2
BDBUS3

BDBUS2 40
BDBUS3 41

BCBUS1BCBUS2
BCBUS3
BCBUS4
BCBUS5
BCBUS6
BCBUS7

BCBUS1 52
BCBUS2 53
BCBUS3 54
BCBUS4 55
BCBUS5 57
BCBUS6 58
BCBUS7 59

5

SUSPEND# 36

4

0.1uF

C105
10uF

0.1uF

C98

4.7uF

+ C88

3

A3

4.7uF

+ C89

0.1uF

C99

0.1uF

C100

USB_1V8

0.1uF

C91

DATE:

DN: <DN>

2

VERSION:<DESIGNER><DRAWNBY><BUILTFOR>

1<DATE> SHEET 6 OF 7

A3 DRAWN BY:

BUILT FOR:

USB/FPGA

0.1uF

C101

USB_VPHY

USB_VPLL

USB_VBUS

1

JILA ELECTRONICS LABFPGA / DDS SYNTHESIZER

0.1uF

I

C97

0.1uF

USB_3V3

C96

G

C90

1

O

UPDATE: 9/14/2017

DESIGN BY:

BCBUS0BCBUS0 48

ec t ro ni cs

0.1uF

BDBUS7

BDBUS7 46

FPGA1-L16

C93

BDBUS6

BDBUS6 45

2+

BCBUS7

BCBUS6

BCBUS5

U14
LM1117-3.3V

C104
10uF

L25
600ohm

BDBUS5

el

+

15
16

14

BCBUS4

BCBUS313

BCBUS2

12

BCBUS1

11

10

BCBUS0BDBUS7

9BDBUS6

8

BDBUS5

7

6

BDBUS4

BDBUS3

5

BDBUS2

BDBUS1

BDBUS04

23

BDBUS5 44

PWREN# 60

1
2

L24
600ohmUSB_3V3

J15

BDBUS4

3

3

BDBUS4 43

TMS P2 - 85

TDO P2 - 88

P2 - 86

FPGA1-D15

BDBUS1

BDBUS1 39

TDI

FPGA1-A17 TCK P2 - 87

BDBUS0

SIWUA

WR#

RD#

TXE#

RXF#

D7

D6

D5

D4

D3

D2

D1

D0

4

BDBUS0 38

ACBUS7 34

ACBUS6 33

ACBUS5 32

FPGA1-M13

U10

5
USB_3V3

USB_1V8

VCORE0 12
VCORE1 37
VCORE2 64
GND0
GND1
GND2
GND3
GND4
GND5
GND6
GND7
1
5
11
15
25
35
47
51

VPHY 410 AGND

20
31
42
56
VCCIO0VCCIO1VCCIO2VCCIO3

A

B

C

D

A

B

C

D

+5VD

M2
4-40

J3

J81

8

2

1

7

M6
4-40

M7
4-40

+

6

C128
33uF

GND
2

6

IN OUT

U12
LM7805CT

C24
33uF

+1.8V

+

1

C21
33uF

+3.3V

+

+5VD

M8
4-40

C127
1.0uF

C23 C22
0.1uF1.0uF

L4
1kohm

C20 C19
0.1uF1.0uF

L3
1kohm

-15V

1kohm
L1

C14
33uF

C13
33uF

C18 C16
0.1uF1.0uF

+

+15V

+

C17 C15
0.1uF1.0uF

L2
1kohm

7

U15
LT1963AET-1.8
2
4
IN
OUT
1
5
SEN
SHDN
RTN
3

3

2

1

M1 M3 M4 M54-40 4-40 4-40 4-40

+3.3V

U18
LT1963AET-3.3
2
4
IN
OUT
1
5
SHDN SEN
RTN
3

8

3
+

C26
4.7uF

FPGA1-GND0
FPGA1-GND1
FPGA1-GND2
FPGA1-GND3
FPGA1-GND4
FPGA1-GND5
FPGA1-GND6
FPGA1-GND7
FPGA1-GND8
FPGA1-GND9
FPGA1-GND10
FPGA1-GND11
FPGA1-GND12FPGA1-GND13
FPGA1-GND14
FPGA1-GND15
FPGA1-GND16
FPGA1-GND17
FPGA1-GND18
FPGA1-GND19

FPGA1-VCCIN0
FPGA1-VCCIN1
FPGA1-VCCIN2
FPGA1-VCCIN3
FPGA1-VCCIN4
FPGA1-VCCIN5
FPGA1-VCCIN6
FPGA1-VCCIN7
FPGA1-VCCIN8
FPGA1-VCCIN9
FPGA1-VCCIN10
FPGA1-VCCIN11
FPGA1-VCCIN12
FPGA1-VCCIN13

C25
1.0uF

+5VA

5

+5VD

5

C132
0.1uF

C130
0.1uF

C136
0.1uF

C135
0.1uF

C134
0.1uF

C126
0.1uF

C75
0.1uF

C131
0.1uF

4

4

3

3

ec t ro ni cs

J1-1
J1-2
J1-3
J1-4
J1-5
J1-6
J1-7
J1-8
J1-9
J1-10
J1-11
J1-12
J1-13
J1-14
J1-15
J1-16
J1-17
J1-18
J1-19
J1-20
J1-21
J1-22
J1-23
J1-24
J1-25
J1-26
J1-27
J1-28
J1-29
J1-30
J1-31
J1-32
J1-33
J1-34
J1-35
J1-36
J1-37
J1-38
J1-39
J1-40
J1-41
J1-42
J1-43
J1-44
J1-45
J1-46
J1-47
J1-48
J1-49
J1-50

+5VD
GND

GND
GND

GND
GND

GND
GND

GND
GND

+5VD
GND

1

A3

DATE:

DN: <DN>

2

VERSION:<DESIGNER><DRAWNBY><BUILTFOR>

1<DATE> SHEET 7 OF 7

A3 DRAWN BY:

BUILT FOR:POWER

JILA ELECTRONICS LABFPGA / DDS SYNTHESIZERUPDATE: 9/14/2017

DESIGN BY:

el

FPGA1-B11
FPGA1-E16
FPGA1-A11
FPGA1-E18
FPGA1-B12
FPGA1-B16

FPGA1-A13
FPGA1-D14
FPGA1-F9
FPGA1-F13
FPGA1-G9
FPGA1-E13
FPGA1-C15
FPGA1-F15

FPGA1-D9
FPGA1-H16
FPGA1-C9
FPGA1-H15
FPGA1-B9
FPGA1-C10
FPGA1-A9
FPGA1-A10

FPGA1-A5
FPGA1-A6
FPGA1-B4
FPGA1-D11
FPGA1-A4
FPGA1-C11
FPGA1-C7
FPGA1-B8

FPGA1-B2
FPGA1-D8
FPGA1-A2
FPGA1-C8
FPGA1-B3
FPGA1-D6
FPGA1-A3
FPGA1-C6

2

A

B

C

D

247

l
1

2

3

4

5

6

7

8

9

10

11

m
−1
0
1
−2
0
2
−3
0
3
−4
−2
0
2
−5
−3
0
3
5
−6
−4
−2
0
6
−7
−5
−3
0
3
5
7
−8
0
2
8
−9
0
8
9
−10
8
10
−11
11

electrode
5
6
1.67
4.04

1
−1.67

2
−4.04

3
−4.04

4
−1.67

−4.04
−2.96
0.33
−2.96
−3.30

−1.67
−2.96
0.33
2.96
1.36

1.67
2.96
0.33
2.96
1.36

4.04
2.96
0.33
−2.96
−3.30

4.04
−2.96
0.33
−2.96
3.30

−1.36
−2.82
−0.03
0.11
−0.06
−1.86
−0.04

3.30
2.82
−0.03
0.11
0.05
0.76
0.03

−3.30
−2.82
0.03
0.11
0.06
0.75
0.03

1.36
2.82
0.03
0.11
−0.06
−1.86
−0.04

−0.03
0.75
−0.91
−0.06
−0.03
0.02
0.91
−0.26
−0.04
−0.03

0.04
−1.86
−0.92
0.06
−0.03
0.02
−0.91
−0.59
0.02

−0.04
1.86
0.91
−0.06
0.03
0.02
−0.91
−0.59
0.02

0.03
−0.75
0.92
0.06
0.03
0.02
0.91
−0.26
−0.04
−0.03

0.02
0.59
−0.36

0.03
−0.04
0.26
0.36

−0.03
0.04
−0.25
−0.36

−0.02
−0.59
0.36

−0.03
0.10
−0.21

0.03
0.10
0.38

0.03
0.10
0.37

−0.03
0.10
−0.21

0.37
0.12
0.02
0.55
0.44
0.38

−0.21
0.12
0.02
−0.55
−0.38
−0.44

0.21
−0.12
0.02
−0.55
−0.38
0.44

−0.38
−0.12
0.02
0.55
0.44
−0.38

7
4.04

8
1.67

1.67
−2.96
0.33
2.96
−1.36

−1.67
2.96
0.33
2.96
−1.36

−4.04
2.96
0.33
−2.96
3.30

1.36
−2.82
−0.03
0.10
−0.06
1.86
0.04

−3.30
2.82
−0.03
0.11
0.06
−0.76
−0.03

3.30
−2.82
0.03
0.11
0.06
−0.75
−0.03

−1.36
2.82
0.03
0.11
−0.06
1.86
0.04

0.03
−0.75
−0.92
−0.06
−0.03
0.02
0.91
0.26
0.04
0.03

−0.04
1.86
−0.91
0.06
−0.03
0.02
−0.91
0.59
−0.02

0.04
−1.86
0.91
−0.06
0.03
0.02
−0.91
0.59
−0.02

−0.03
0.75
0.91
0.06
0.03
0.02
0.91
0.25
0.04
0.03

−0.02
−0.59
−0.36

−0.03
0.04
−0.25
0.36

0.03
−0.04
0.26
−0.36

0.02
0.58
0.36

−0.03
0.10
0.21

0.03
0.10
−0.38

0.03
0.10
−0.38

−0.03
0.10
0.21

−0.38
0.12
0.02
0.55
−0.44
−0.38

0.21
0.12
0.02
−0.55
0.38
0.44

−0.21
−0.12
0.02
−0.55
0.38
−0.44

0.38
−0.12
0.02
0.55
−0.44
0.38

T

B

−0.73

0.73

−0.82

−0.82

−0.65

0.65

−0.42

−0.42

−0.25

0.25

−0.15

−0.15

−0.08

0.08

−0.04

−0.04

−0.05

−0.05

−0.02
−0.02

0.02
0.02

−0.01

−0.01

Table C.1: Multipole fit coefficients up to l = 11 for electrodes of Generation 2 ion trap (Fig. 4.1),
with units of V/ml . Rows where all columns have a magnitude less than 0.02 are neglected. Notethe absence of all rows (l, m = ±1) for odd values of l, which behave as Erot ∝ Z l−1 along the line
(0, 0, Z).

Appendix D

Molecular Data

In Tables D.1 and D.2, Maddie Pettine and I have tabulated the observed transitions in our
REMPD survey spectrum collected in Fall 2018. For transitions where we performed fine scans to
identify Ω of the intermediate state, those are listed. Transition center frequencies ν and strengths
are very approximate – they are only eyeballed, and are not from any fit.

249

Dye
Styryl 8

Pyridine 2

ν (cm−1 )
26096
26177
26202
26369
26381
26393
26518
26546
26686
26707
26759
26777
26796
26864
26960
26988
27137
27154
27178
27264
27345
27394
27399
27615
27659
27683
27735
27752
27775
27861
27942
28001*
28001*
28111
28135
28194
28322
28376

Chop
754
754
754
754
754
754
754
I
I
I
754
I
961
I
I
754
961
I
I
I
961
I
961
I
I
I
I
961
I
754
961
I
I
961
754
754
I
I

Ω00
3∆
1
3∆
1
3∆
1
X 1 Σ+ , v = 1
3∆ , v = 1
1
X 1 Σ+ , v = 1
3∆

1

X 1 Σ+ , v = 0
3∆
1
3∆
1
3∆
1
X 1 Σ+ , v = 0
3∆
1
1
X Σ+ , v = 1
X 1 Σ+ , v = 1
X 1 Σ+ , v = 1
3∆ , v = 0
1
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0
X 1 Σ+ , v = 1
3∆ , v = 2
1
X 1 Σ+ , v = 0
3∆
1
1
X Σ+ , v = 1
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0
3∆
1
1
X Σ+ , v = 0
X 1 Σ+ , v = 1
X 1 Σ+ , v = 1
X 1 Σ+ , v = 0
3∆
1
3∆
1
X 1 Σ+ , v = 1
X 1 Σ+ , v = 1
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0

Ω0
2
1
1
1
2
1
2
1
2
1
1
0
1
2
1
1
2
1
1
1
1
0
1
1
2
1
1
2
1
1
2
0
1
0
1
1
1
1

Strength
0.17
0.20
0.18
0.24
0.14
0.15
0.28
1.07
0.09
0.09
0.25
0.09
0.40
0.10
0.17
0.30
0.84
0.51
0.05
0.08
0.67
0.05
0.64
0.06
0.18
0.11
0.26
0.67
0.25
0.22
0.62
0.30
0.30
0.69
0.18
0.31
0.28
0.14

Table D.1: Summary of observed photodissociation transitions, part I.

250

Dye
Pyridine 1

Rhodamine 640

ν (cm−1 )
28602
28635
28650
28738
28795
28920
28969
29088
29146
29203
29269
29328
29346
29505
29524
29530
29579
29671
29700
29831
29879
30922
30946
31111
31290
31511
31595
31703
31761
31895
32094
32255
32289
32489

Chop
754
754
754
754
754
I
754
I
I
754
754
I
754
I
I
754
I
754
I
I
754
754
I
754
I
754
I
754
I
I
754
I
754
I

Ω00
3∆
1
3∆
1
X 1 Σ+ , v = 0
X 1 Σ+ , v = 1
X 1 Σ+ , v = 1
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0
3∆
1
3∆
1
X 1 Σ+ , v = 0
X 1 Σ+ , v = 1
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0
3∆
1
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0
X 1 Σ+ , v = 1
3∆
1
X 1 Σ+ , v = 0
X 1 Σ+ , v = 1
X 1 Σ+ , v = 0
3∆
1
1
X Σ+ , v = 0
X 1 Σ+ , v = 1
X 1 Σ+ , v = 0
X 1 Σ+ , v = 0
3∆
1
1
X Σ+ , v = 0
X 1 Σ+ , v = 1
X 1 Σ+ , v = 0

Ω0
1
2
1
1
1
1
1
0
0
1
2
1
0
1
1
1
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0

Strength
0.20
0.15
0.13
0.29
0.39
0.08
0.12
0.38
0.26
0.16
0.12
0.57
0.16
0.24
0.29
0.25
0.12
0.16
0.45
0.16
0.27
0.09
0.09
0.25
1.11
0.10
0.09
0.25
0.12
0.96
0.25
0.09
0.37
1.46

Table D.2: Summary of observed photodissociation transitions, part II.