Measuring the Electron Electric Dipole Moment with
Trapped Molecular Ions
byMatt Grau
B.S., California Institute of Technology, 2009
M.S., University of Colorado Boulder, 2011

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
2016This thesis entitled:
Measuring the Electron Electric Dipole Moment with Trapped Molecular Ions
written by Matt Grau
has been approved for the Department of Physics

Prof. Eric 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.

iiiGrau, Matt (Ph.D., Physics)
Measuring the Electron Electric Dipole Moment with Trapped Molecular Ions
Thesis directed by Prof. Eric Cornell

An electron electric dipole moment (eEDM) directly violates time-reversal symmetry, a factwhich has far reaching implications for physics beyond the Standard Model. An experiment using trapped molecular ions offers high sensitivity because of the large effective electric fields andlong coherence times that are possible. We demonstrate precision spectroscopy on many trapped
HfF+ ions in a radiofrequency quadrupole trap with rotating electric and magnetic fields. The
spectroscopy performed is a Ramsey type experiment between spin states of the metastable 3 ∆1electronic state and has a coherence time exceeding 1 second. We have collected and analyzed
over 200 hours of Ramsey spectroscopy data taken under a variety of experimental conditions. We
identify several systematic errors that could potentially affect an eEDM measurement and estimate
the size of these interloping effects. By collecting data under pairs of conditions where the eEDM
has opposing signs (e.g. performing Ramsey spectroscopy on the two Λ-doublets) we are able totake frequency differences that can suppress some of these systematic effects. Although our data
set includes runs where we have intentionally varied experimental parameters to study systematiceffects, we achieve a 1σ statistical sensitivity of 2.76 × 10−28 e·cm and place a 1σ upper bound on
the size of systematic effects of 2.82 × 10−28 e·cm. Finally we present a preliminary upper boundon the eEDM of |de | < 4.6 × 10−28 e·cm with 90% confidence.

Dedication

To Mom & Dad – Thanks. Also BJ is great.

v

Acknowledgements

This thesis is only possible thanks to the help of a great many people. In particular my fellow
graduate students Dan Gresh and Will Cairncross, who not only make working in lab productive
but enjoyable as well. Proper tribute must be paid to Dan for his stoic relentlessness. Like aboulder tumbling down a hill, he will surely crush all opposition in between us and the EDM. Willis as enthusiastic as he is talented, and his zeal for simulation causes me to worry that we might
all be inside of one. Additionally, Yan Zhou, amazes me by his embodiment of the principle ofsuperposition - he seems to be everywhere and working on everything at once. Kia is new to thegroup, but I am sure that he will have the experiment control completely rewritten in Python inno time flat. Finally I must thank Yiqi Ni, our brilliant undergraduate, who did all the real workon the experiment while only complaining a little.
I’ve been very fortunate to work with exceptionally great colleagues in the past as well.
Huanqian Loh was a fantastic labmate for many years on the experiment, and a great role model
for me. I learned a great deal from her. I’ve also benefited greatly from working closely with Kevin
Cossel, ever since he lent me a drill almost a decade ago. Additionally I’d like to thank Kang-Kuen
Ni, Laura Sinclair, Russel Stutz, Tyler Yahn, and Tyler Coffey for all of their past contributions
to the experiment.
My advisers Eric Cornell and Jun Ye have been influential in more ways than I have space towrite about here. Eric taught me to question everything, and to never even think about collecting
data without a plan to analyze and interpret it first. His intuition for physics never ceases toastonish me. Jun is a constant source of energy and enthusiasm, and is never afraid to jump rightvi
into any problem that needs solving. I’d also like to thank Deborah Jin and her group for great
feedback during the Jin/Cornell meetings, and also Pam Leland and Krista Beck, who are the only
reason anything ever gets done.
I’ve had great technical support from the JILA instrument shop and electronics shop. Todd
Asnicar machined and delicately hand polished the ion trap that made all of the experiments in thisthesis possible. He did a great job, and the trap has never disappointed me. David Alchenberger
was critical in cleaning the trap electrodes for ultrahigh vacuum and gold coating them for highoptical reflectivity. Terry Brown, James Fung-A-Fat, Carl Sauer, and Chris Ho all provided expert
guidance for electronics projects large and small.
My fellow classmates at CU, Dan Weflen, Maya Fabrikant, Dan Lobser, Adam Kaufman,
and Brandon Ruzic, have provided me the occasional useful idea as well as the more occasional
entertaining diversion. In particular Josh Weiner has been a great friend for the past decade, eversince he explained to me how a rotisserie functioned in undergrad. Jon Malmaud aided me withtons of useful insight into statistics and general strategic thought. Todd Gingrich has always beena source of motivation for me. I’d also like to thank John Leichty for more than a little help withmicrocontrollers, and Dan Lo and Gabe Cohn for answering countless questions about FPGAs anddigital electronics.
I’ve had many great teachers along the way, but I’d like to especially thank Michael Cross and
Gil Refael for always having time to answer my questions. Mondania Gallagher was my wonderful
middle school science teacher who was always engaged and ever patient. She was always delighted
to share in her students’ discoveries. Karen Young was a great teacher and an awesome scoutmaster.
Without them I would be nowhere.
I’d also like to thanks my parents, my siblings Dan, Christina, and Maria for their support,
as well as all my family in Virginia, Maryland, and especially Colorado for their unwavering encouragement and cheer-leading. Finally, I want to thank my wonderful wife BJ for her love and carewhile I’ve been writing this. She also painstakingly revised the inchoate neighing and braying ofbarnyard animals into something that resembles prose. She pushes me to be all I can be.

Contents

Chapter
1 Introduction
1.1

1.2

1

The electron EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

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

6

Overview for thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

EDM searches

2 Experimental apparatus
2.1

2.2

2.3

10

Summary of techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1

Choice of molecular species . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2

Neutral HfF production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.3

Photoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.4

State transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.5

State detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Ion Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1

Refinements unique to the eEDM experiment . . . . . . . . . . . . . . . . . . 16

2.2.2

Rotating electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3

Erot amplitude pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.4

Experiment sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.5

Future trap directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Computer control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

viii
2.3.1

uBinary file format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.2

Experiment control state machine . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Statistical sensitivity
3.1

3.2

Theory of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1

Measurement principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.2

3∆

3.1.3

Effect of the rotating frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1 state Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.1

Ramsey spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.2

Experimental Ramsey sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.3

Applying π/2 pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.4

Optimizing statistical sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3

57

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1

4.2

38

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Systematic uncertainty
4.1

37

Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Understood systematic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.1

Differential g-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.2

Effective differential g-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2.3

Non reversing magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.4

Axial magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.5

Fields inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2.6

Unintentional gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Measured systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3.1

~ · ~r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
fB ∼ B

4.3.2~ × ~r) · ω
f BR ∼ (B~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.3.3

f BR ∼ zr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

ix
4.4

Potential other systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5 Conclusion
5.1

5.2

5.3

105

A precision measurement of the electron EDM . . . . . . . . . . . . . . . . . . . . . 105
5.1.1

Measurement overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.1.2

Data blinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.1.3

Blinded EDM measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.1.4

Systematic corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.1.5

Systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.1.6

Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.1.7

Total EDM measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Future direction of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2.1

Improvements to current experiment . . . . . . . . . . . . . . . . . . . . . . . 127

5.2.2

Improvements for next generation experiment . . . . . . . . . . . . . . . . . . 128

5.2.3

Potential for ThF+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Bibliography

131

Appendix
A Typical experiment parameters

136

B Matlab code to generate linear combinations channels

137

x

TablesTable
2.1

uBinary data types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2

uBinary file format specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.1

Summary of systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

xi

FiguresFigure1.1

Three-loop contribution to electron EDM from Standard Model . . . . . . . . . . . .

4

1.2

One-loop contributions to electron EDM from Beyond Standard Model theories. . .

5

2.1

Schematic of state transfer energy levels . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2

Ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3

Multiplying DAC variable attenuator schematic . . . . . . . . . . . . . . . . . . . . . 22

2.4

Multiplying DAC variable attenuator PCB . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5

Experiment timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6

uBinary LabVIEW libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1

3∆

3.2

Bloch sphere depiction of Ramsey spectroscopy . . . . . . . . . . . . . . . . . . . . . 43

3.3

Coherence time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4

Frequency sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1

Measurement of δg/gF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2

Correlation of f BD and f B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3

Removing background non-reversing magnetic field with a magnetic field gradient. . 71

4.4

Measuring non-reversing magnetic fields with a magnetometer array. . . . . . . . . . 74

4.5

Magnetometer array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.6

The effect of axial magnetic fields on f R and f DR

1 state diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

. . . . . . . . . . . . . . . . . . . 78

xii
4.7

The effect of axial magnetic field compared with prediction . . . . . . . . . . . . . . 81

4.8

The effect of axial magnetic field on Ramsey fringe contrast . . . . . . . . . . . . . . 83

4.9

Berry’s phase due to Erot inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.10 Unintentional magnetic field gradients . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.11 Magnetic field with parallel and perpendicular displacement . . . . . . . . . . . . . . 92
~ · ~r scaling with various parameters4.12 f B ∼ B. . . . . . . . . . . . . . . . . . . . . . . 94

4.13 Bounds on the effect of f B on f BD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
~ × ~r) · ω
4.14 f B ∼ (B
~ rot scaling with various parameters . . . . . . . . . . . . . . . . . . . 100
4.15 Bounds on the effect of f BR on f BD . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.16 f BR ∼ zr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1

Ramsey fringes and blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2

EDM experiment timing diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3

Expectation bias in the measurement of c . . . . . . . . . . . . . . . . . . . . . . . . 112

5.4

233 hours of blinded EDM data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.5

Systematics correction applied to the EDM channel . . . . . . . . . . . . . . . . . . . 117

Chapter 1

Introduction

1.1

The electron EDM
The electron is an elementary particle that has a well defined mass (me = 9.10938356(11) ×

10−31 kg) and a well defined charge (−e = −1.6021766208(98) × 10−19 C)[49]. It is appealing tovisualize the electron as a small clump of charged “stuff” with mass me and charge e. Additionallythe electron has intrinsic angular momentum, known as spin, that has no classical analog. The~ = 1 ~, where ~ is the reduced Planck constant. For much ofelectron spin is a vector quantity S
2
the rest of this thesis we will consider units where ~ = 1, and the electron spin is simply 12 . As
the electron has both charge and angular momentum it must have a magnetic moment, which fora non-relativistic object is
µ
~=



1 −e ~~
S = −µB S.
2 me

(1.1)

Where e/2me = µB is the Bohr magneton. However, Thompson scattering shows that the electron
radius is less than 10−15 m, which would mean that in a classical calculation of the angular momentum of the electron that the outside radius of the electron is moving at relativistic speeds. Therefore
we must use the relativistic Dirac equation to calculate the magnetic moment. The Dirac equation
finds a magnetic moment that is a factor of 2 larger than the naive non-relativistic expectation,~µ~ = −2µB S.

(1.2)

This difference from the non-relativistic expectation is typically contained in a factor g, the so~ where gs = 2 is the relativistic electron spin g-factor. In fact, thecalled g-factor, µ~ = −gs µB S,

2electron spin g-factor is not quite equal to 2, more precisely gs ≈ 2.0023. This difference from
2 is referred to as the anomalous magnetic moment of the electron, ae = (gs − 2)/2. It arises
due to higher order interactions of the electron with itself in Quantum Electrodynamics (QED). A
free electron is surrounded by a “cloud” of virtual particles: virtual photons and virtual electrons.
These virtual particles provide radiative corrections to the magnetic moment of the electron, which
can be calculated in a perturbation series,
ae =

α
π

A(2)
e +

 α 2πA(4)e + α 3πA(6)e + ...
(n)where α ≈ 1/137 is the fine structure constant, and the terms Ae(1.3)

are the integrals of the nth order

QED Feynman diagrams describing the electron magnetic dipole moment. Almost surprisingly,
this perturbative calculation agrees well with experiment out to 10th order and beyond. The
calculated value ae = 0.00115965218178(77)[4] agrees with the experimentally determined value of
ae = 0.00115965218073(28)[33] made by single electron cyclotron measurements in a Penning trap.
This extreme precision and marvelous agreement between experiment and theory are one of thecrowning successes of QED.
Seeing as the phenomenon of the electron magnetic moment is so rich we might next consider
an electric dipole moment (EDM). Unfortunately, simply referencing an elementary textbook[10]will tell us that fundamental particles (such as electrons) and nuclei (such as neutrons) must nothave any electric dipole moments because parity is a conserved quantity. The electric dipole moment
can be expressed asZde =dxj ei ri |ψ(rj )|2 ,(1.4)

where ri is the position coordinate along the ith dimension, ei is the charge distribution along that
dimension, ψ(r) is the particle wavefunction as a function of spatial coordinate, and repeated indicesrepresent a sum over that index. A parity inversion operation ri → −ri should leave the system
invariant, and we find that de = −de = 0. However, Purcell and Ramsey pointed out in 1950 that
this argument is ludicrous[55]. This sort of theoretical argument rests on an assumption that has
not been tested, namely that parity is a perfect symmetry of nature, and the proper course of action

3
is to either validate or invalidate this argument by the means of an experimental measurement.
They argued that EDMs of fundamental particles and nuclei could be used as observables forsymmetry violations. At this point in time there was not conclusive evidence against the existenceof EDMs, most experiments were not sensitive to EDMs that were smaller than 10−13 e·cm. Smith,Purcell, and Ramsey then performed an experiment to measure the EDM of the neutron. The useda polarized beam of neutrons in a 71.6 keV/cm electric field to place an upper limit on the neutron
EDM of |dn | < 5 × 10−20 e·cm[67]. This type of experiment is only suitable for experiments onneutral particles, as a charged particle such as an electron would be quickly accelerated out of thebeam path.
As it turns out, parity is not a conserved quantity, demonstrated by the angular distributionof parity violating Beta decays[74] and meson decays[29]. Lee and Yang calculated that the sizeof the neutron EDM which would be generated by the amount of parity violation measured in [74]and [29] would be well below the measured upper bound of |dn | < 5 × 10−20 e·cm[43]. Moreover,
Landau theorized that even though parity P is not a good symmetry, that CP , the product of thesymmetries of charge conjugation C and parity P , is a good symmetry. If CP is a good symmetry
then we should not expect EDMs of fundamental particles to exist.
In 1964 Cronin and Fitch found direct evidence of CP violation in the decay of the K20
meson[15]. This finding renewed interest in the measurement of fundamental particle EDMs. Sandars performed one of the first electron EDM experiments, measuring the EDM of a beam of neutral
Cesium atoms to be |dCs | < 2.2 × 10−19 e· cm[63]. While it was shown by Schiff[64] that in the
non-relativistic limit an atom such as Cesium has no EDM even if its constituent electrons do, ina relativistic treatment the EDM of the atom can be more than 100 times larger than that of theelectron[61]. Sandars was able to interpret the Cesium EDM limit as an electron EDM limit of|de | < 2 × 10−21 e·cm, a factor of 105 smaller than limits placed by scattering electrons off of 4 He
at the Stanford linear accelerator[31].
The CP violation measured so far has been in the hadronic sector of the Standard Model,
affecting primarily heavy quarks. It can be included in the Standard Model as a single CP violating

4phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix that describes the mixing of quarks[12,41]. The CP violating phase in the CKM matrix may contribute to a neutron EDM at the dn <1030 e·cm level, which is still beyond the current best limit of 2.9 × 1026 e·cm[5]. Even though thisCP violation primarily affects hadrons, it does have the potential to contribute to an electron EDM
through very high order effects. These high order effects can be represented as Feynman diagrams
where an electron couples to a quark loop containing a CP violating phase via several virtual W
bosons. These Feynman diagrams are exactly zero out to second order[9], so only third order andhigher diagrams contribute. An example of such a high order Feynman diagram is shown in fig. 1.1.

We can make a back-of-the-envelope calculation by summing over the vertices of such adiagram,de =1
eGF me α2 αs J,
(4π)5

(1.5)

where GF is the Fermi coupling constant, α is the fine structure constant, αs is the strong couplinginteraction constant, and J is an invariant combination of angles in the CKM matrix to which
any CP violating effects in the Standard Model must be proportional. If we use a value of J =2 × 10−4 we obtain an upper bound for the Standard Model prediction of the electron EDM of|de | < 10−37 e·cm.

While the amount of CP violation already included in the Standard Model is insufficient togenerate a large electron EDM, there are still compelling reasons to suspect more CP violation,

Figure 1.1: Example of a three-loop electron EDM Feynman diagram, where an electron couples
to a CP violating quark loop via W bosons.
γ
d

W

eL

d

u

W

eR

5
and thus larger electron EDMs could exist. One such reason is the asymmetry between matter andantimatter in the universe, or . The so-called baryon asymmetry can be estimated by the baryon
to photon ratio, η = (NB − NB̄ )/Nγ = 5.8 ± 0.27 × 10−10 [14]. The matter-antimatter difference isnormalized by photon number because equal portions of matter and antimatter can annihilate andproduce photons. Balloon experiments have searched for signatures of antihelium by measuring
antiproton decays with no evidence of antibaryons[2]. Sakharov proposed a potential solution tothis problem[60]. In order to produce this asymmetry we need a mechanism where matter andantimatter can be produced at different rates. In order to produce matter and antimatter at
different rates these three criteria must all be met:
(1) Baryon number violation(2) Interactions out of thermal equilibrium(3) CP violation
However, the amount of CP violation required to produce a baryon asymmetry as large as 10−10
is much more than is present in the CP violating phase in the CKM matrix. The quantity of
CP violation required to satisfy Sakharov’s criteria could generate an electron EDM as large as
10−29 e·cm[28].
Additionally, many physical theories predict new physics beyond the confines of the Standard Model. Nearly all of them introduce new CP violating phases φCP which allow for an electron EDM
that is much larger than that predicted by the Standard Model [9, 54]. These CP violating phases
can couple to the electron at the one-loop level, meaning that their effect can be greatly enhanced

Figure 1.2: Example of a one-loop contribution to the electron EDM from a Beyond Standard
Model theory where the electron couples to some source of CP violation directly via a new fermion
F or boson B.
γ
γ

B

eL

B

F

F

eR

eL

F

B

eR

6
compared to the CP violating phase found in the CKM matrix that couples to the electron only
above the three-loop level[25]. The size of an electron EDM we expect from a one-loop diagram
coupling the electron to some new physics at energy scale Λ is
 α  ~c  m c2 2ede = esin(φCP ).
2
4π
me c
Λ2

(1.6)

Without any sort of specific symmetry requirement constraining the value of φCP we can assume
that sin(φCP ) ∼ 1. If Λ ∼ 10 TeV, then de ∼ 5×10−29 . This prediction is an attractive proposition,
because it means that by constructing sensitive probes of the electron EDM we can probe for new
physics at the several TeV scale.

1.1.1

EDM searches

Following the Sandars electron EDM experiment in 1964 there have been many attempts tomeasure the electron EDM in atomic systems, including those in Cesium[72], metastable Xenon[53],and Thallium[1]. The most recent atomic beam experiment, performed by Commins, reported
de = 6.9 ± 7.4 × 10−28 e·cm, placing an upper bound on the electron EDM of |de | < 1.6 × 10−27 [57].
In this experiment Commins measured the EDM of atomic Thallium, which has an EDM with andopposite sign a factor of 585 larger than the electron EDM, by applying a 122 kV/cm electric field
to a Thallium beam with a detected count rate of almost 109 per second. The dominant systematicin this and other atomic beam experiments comes from motional magnetic fields. This systematicarises because the atoms moving in the applied lab-frame electric field experience a rest-frame
magnetic field. This rest-frame magnetic field leads to a Zeeman shift that is linear in applied
electric field and mimics an electron EDM of the size (v/c)µm ∼ 10−17 e·cm. This effect is manyorders of magnitude larger than the sensitivity that these beam experiments hope to obtain, so theexperiments require extremely precise cancellation via a co-magnetometer species. The Thallium
experiment used Sodium as a co-magnetometer species. By carefully measuring the motional fields
experienced by the co-magnetometer species this effect can be subtracted off.
A way to improve upon the electron EDM limit set by atomic beam experiments is to use a7molecular beam. The figure of merit for the sensitivity of an EDM experiment is Eeff τpṄ , where

Eeff is the size of the effective electric field that can be applied to the electron, τ is coherence time,and Ṅ is the experimental count rate. Molecules have much more closely spaced energy states,
which means that they can be fully polarized with an electric field, making the effective electricfield Eeff experienced by the electron much larger. This means that the ratio of the molecular EDM
to the electron EDM can be a factor of ∼ 1000 larger than the enhancement factor in an atomic
experiment. The first molecular experiment was performed by Imperial College, using a beam of
YbF molecules. They measured the electron EDM to be de = −2.4 ± 5.7stat ± 1.5syst × 10−28 e·cm,
placing a 90% confidence interval upper bound on the electron EDM of |de | < 10.5 × 10−28 [36].
This experiment benefited from an effective electric field that was 220 times larger than that of theThallium atomic beam experiment. With over 270 hours of data the measurement is still statistics
limited. The dominant systematics are on the 1 × 10−28 e·cm level. One systematic was due tocorrelations with their electric field switch, caused by incomplete electric field reversals. Anotherdominant systematic was a correlated signal with the detuning of the π/2 pulses, causing an initial
phase shift in their Ramsey fringe[39].
Another potential advantage of molecules is that they often possess closely spaced parity
doublets. A parity doublet can be exploited like an “internal co-magnetomoeter” to remove theeffect of any motional fields or other systematics. Additionally, these doublets are typically moreclosely spaced than rotational energy levels, allowing such molecules to be polarized by electricfields of only several V/cm. This further reduces systematic effects due to leakage currents createdby, e.g., charging large capacitor places in order to apply the very large electric fields required tomix rotational levels. This technique was first demonstrated by an experiment at Yale using a vapor
cell of PbO[23]. The species PbO has a 3 Σ+ state which possesses an Ω-doublet. This experiment
was able to achieve an electron EDM measurement of de = −4.4 ± 9.5stat ± 1.8syst × 10−27 e·cm.
Using a similar technique, the ACME collaboration at Harvard and Yale have performed
the most sensitive electron EDM experiment to date. The molecular species ThO has an effective
electric field of 75.2 GV/cm[27], and the 3 ∆1 state of ThO possesses a similar doublet structure to8
the 3 Σ+ state of PbO. ACME was able to measure the electron EDM to be de = 2.1 ± 3.7stat ±2.5syst × 10−29 e·cm, placing an upper bound of |de | < 8.7 × 10−29 e·cm with 90% confidence[7].
The largest systematic effects in this experiment are constrained to the 10−29 e·cm level:• A laser polarization gradient causes an initial phase shift of the Ramsey fringe that is at
the 10−29 e·cm level. This is caused by laser-induced thermal stress birefringence in the
optically transparent electric field plates. The ThO molecules are initially prepared withspin orientation dependent on laser polarization. This polarization gradient induces a phasebetween the dark and bright states, creating the initial phase shift that is indistinguishable
from a frequency shift at later time in the Ramsey fringe.
• Change of state preparation laser pointing of 5 µrad can have an effect that is correlatedwith the direction of the electric field that is a 10−29 e·cm effect.
• State preparation and state readout laser detuning has a 1 × 10−29 e·cm systematic effect.
In this thesis we will describe an experiment to measure the electron EDM using molecular
HfF+ ions in an rf quadrupole trap. By using a molecular species such as HfF+ we will take advantage of the large effective electric field typical of molecules, and also the systematic error rejectionthat comes from using a 3 ∆1 state with a parity-doublet internal co-magnetometer. Moreover, because we will use molecules in an ion trap rather than in a beam, we will have access to the Ramsey
fringe at all times during its evolution, compared to a beam experiment that can only measure at
the outlet of the experiment apparatus. This will make us less susceptible to systematic effects
caused by imperfect state preparation that will cause initial phase shifts of the Ramsey fringe.

1.2

Overview for thesis
This chapter provides some motivation for the importance of an electron EDM search, aswell as providing some historical perspective. In chapter 2 we will discuss the apparatus we use
to perform the eEDM measurement. We will also provide an overview of the techniques we use to9create, manipulate, trap, and detect the molecular ions used in the experiment. Additionally thischapter will cover some of the technical details involved in automating a complex experiment thatis capable of self correcting its parameters based on the results of analyzed data. We will thenreview the statistical sensitivity of the experiment in chapter 3, and describe the Ramsey sequence
we will use to measure the eEDM. We will also discuss measurement strategies we can adopt to
optimize the statistical sensitivity achievable in a fixed amount of time. Next, in chapter 4 we
explore the various systematic effects we have so far uncovered that might affect a precision eEDM
measurement. Finally, in chapter 5 we summarize the systematic effects characterized in chapter 4
and use over 200 hours of data to calculate the size of these systematic shifts and estimate theuncertainties of the corrections we must apply to cancel these shifts. With a full estimate of all
known systematic effects we calculate a preliminary upper bound on the size of the electron EDM.

Chapter 2

Experimental apparatus

In this chapter we will outline the design of the experimental apparatus, with particularattention paid to the components that make an appearance in the following chapters.

2.1

Summary of techniques
Many of the techniques we use to prepare, manipulate, and detect our molecules were pio-

neered by Huanqian Loh and Kevin Cossel. We will briefly survey these techniques for the sake ofcompleteness, but for a thorough discussion the reader is directed to the PhD theses [44] and [18].

2.1.1

Choice of molecular species

We choose to work with HfF+ as the foundation of our experiment. There are several advantages to this choice of molecule. The molecule HfF+ was pointed out to have a large internal
effective electric field, Eeff ,[48], and subsequent studies[51, 47, 26] calculated the value of Eeff to be
24–30 GV/cm. We will use the most recent determination[26] of 23.3 GV/cm in all future calculations that involve Eeff . The most recent calculation uses 34 valence electrons in its calculation.
Additionally, HfF+ possesses a 3 ∆ molecular symmetry. In particular, the 3 ∆1 electronic state
contains two closely spaced opposite parity states, allowing it to be fully polarized with electricfields as small as ∼ 1 V/cm. Once the molecule is fully polarized we can take full advantage of theinternal effective electric field in our Ramsey experiment. Finally, the 3 ∆1 state in HfF+ is very
low-lying. PFI-ZEKE studies have experimentally determined that the ground state of HfF+ is11
1 Σ+ , and the 3 ∆ state is located at 978.3 cm−1 [6]. This low energy gives the 3 ∆ a long radiative1
1
0

lifetime of 2.1(2) second[50]. This long lifetime has allowed us to achieve a long coherence time of∼ 1 second when performing Ramsey spectroscopy to measure the electron EDM. In order to takeadvantage of this long lifetime we use a radio-frequency Paul trap to confine the HfF+ ions so thatthe linewidth of our Ramsey experiment is not transit time broadened, as would be the case in abeam-type experiment.

2.1.2

Neutral HfF production

To produce the HfF+ molecules that will be used in the experiment we must first have theneutral species HfF. We produce the neutral molecules by using a small, variable repetition rate
Nd:YAG pulse laser to ablate the surface of a Hf metal rod. At the same time, a piezo-actuatedvalve releases a pulse of 99% Ar gas seeded with 1% SF6 . The ablation plume is entrained in thegas pulse and HfF molecules are produced. The HfF molecular beam is cooled through supersonic
expansion to a rotational temperature of ∼ 10 Kelvin, measured by laser-induced fluorescence
spectroscopy. Two skimmers, one 2mm in diameter and a second 3mm in diameter, collimate thebeam before it enters the ion trapping region.

2.1.3

Photoionization

Once the neutral HfF molecules enter the ion trapping region our task is to ionize them. We
employ a two-photon process to promote a valence electron of HfF to a highly excited Rydberg
state that is 54 cm−1 above the HfF ionization threshold of 59462(2) cm−1 . The first photon isthe doubled output of a 3-stage dye laser amplifier seeded by the doubled output of a 1240 nm
high power diode laser. We achieve ∼ 5 µJ per pulse at 309.388 nm. The linewidth of this laser issufficient to resolve transitions of the various HfF isotopologues, allowing us to address transitionsof only the 199 Hf19 F isotopologue. The second photon is the doubled output of a commercial dyelaser, at a wavelength of 368.351 nm with pulse energy of 100-1000 µJ depending on the number ofions we would like to produce. Once promoted to a Rydberg state above the ionization threshold,12
the HfF molecules quickly autoionize. The Rydberg state is energetically allowed to decay only to+
the 1 Σ+0 , v = 0 vibronic ground state of HfF , although this could be spread out over as many as

13 distinct rotational states. However, we observe a propensity of the Rydberg HfF molecules todecay only into a few ion rotational states with a specific parity. By choosing the correct parity of
the intermediate state we can prepare as much as 50% of the ions in the 1 Σ+
0 , v = 0, J = 0 state[46].
Once the molecules are ionized we confine them with an rf quadrupole trap. The details of
ion trapping are discussed more in section 2.2.

2.1.4

State transfer

3
After ionization we transfer population from the 1 Σ+
0 ground state to ∆1 , the state we wish

to perform Ramsey spectroscopy on. We use two cw lasers with crossed linear polarizations directedalong the axial direction of the ion trap in order to avoid sampling the Doppler shifts caused by trap
micromotion. The transfer laser wavelengths are λS = 899.670 nm and λT = 986.422 nm, chosen to3
3
couple the 1 Σ+
0 v = 0, J = 0 and ∆1 v = 0, J = 1 states via intermediate state Π0+ v = 0, J = 1(fig. 2.1). The high-lying 3 Π0+ intermediate state has strong transition dipole moments and goodFranck-Condon overlap with both lower states[45]. Both transfer lasers are external cavity diodelasers with tapered amplifiers and provide about 1.5 W of power each. The two lasers are locked
to two separate modes of a high-stability cavity. Most of the power from each laser is sent to theion trap with polarization-maintaining single-mode fiber. At the ion trap, we obtain about 200
mW of power at 899 nm and 500 mW at 986 nm. We initially operated with a total single-photon
detuning (∆) of 160 MHz. However, we found that scattered light in the trap chamber led tosingle-photon line broadening in the presence of the rotating electric-field due to the large Doppler
shifts arising from the rotational micromotion. To reduce the spontaneous emission, we increased
the single-photon detuning to ∼ 1.5 GHz.
We use an AOM to tune the frequency of the down-going transfer laser such that when weapply an electric field it addresses a single Stark level of 3 ∆1 , v = 0, J = 1. In a half period of
axial oscillation in the trap, each ion experiences a two-photon Doppler shift δ(t), causing a sweep

13

3
Figure 2.1: Schematic diagram of the experimental setup during state transfer from 1 Σ+
0 to ∆1 .
3
We use two lasers, labelled S and T to transfer population from 1 Σ+
0 , N = 0 to ∆1 , J = 1 through3
the intermediate Π0+ , J = 1 state. Both lasers are detuned ∆ ≈ 1.5 GHz from the intermediate3Π
0+ state in order to prevent populating this state. The transfer lasers also have a two-photon
detuning δ(t) from the final state within the 3 ∆1 , J = 1 state manifold. This detuning is the resultof the Doppler shift due to harmonic motion in the ion trap.

14
through the two-photon resonance. The probability of transfer for a single ion in a half trap cycle
2π∆2

is given by the Landau-Zener formula, P = e ~δ̇ . For P ∼ 1 it is enough to pulse the transfer
lasers on for half a period of axial oscillation. However, the temperature of the ions in the axial
direction is ∼ 30 K, and P  1. In this case we pulse the transfer lasers for many axial periods
3
until the population equilibrates between the 1 Σ+
0 and ∆1 states. In this scenario we expect atransfer efficiency of ∼ 50 %, but we observe 37(2) %. This discrepancy could potentially be due
to ion-ion collisions.

2.1.5

State detection

After the HfF+ ions are transferred into the 3 ∆1 state we are ready to perform Ramsey
spectroscopy. The Ramsey spectroscopy is covered in detail in chapter 3. The final step in the
spectroscopy is to be able to detect the state of the ions. We explored two options: laser-induced
fluorescence and photodissociation.
(1) Laser-induced fluorescence
The ion trap was designed to facilitate the detection of ions using laser-induced fluorescence,
where a laser pulse would state-selectively promote the ion to an excited state from whichit could decay, emitting a photon in the process. This photon could be collected onto a
photomultiplier tube, allowing us to count the ions in a particular state. We demonstrated
this technique in an ion beam in [46] to detect the rotational states of the HfF+ ground state,
1 Σ+ . The trap was designed to optimize the signal possible with this detection technique.
0

Two ellipsoidal reflecting electrodes were placed around the trap, subtending 80% of thesolid angle, designed to focus fluorescence from the center of the ion trap to light pipes
leading to photomultiplier tubes. These reflecting surfaces needed to be metallic, because
any insulator near the trap could disrupt the uniformity of the ion trap. The reflecting
surfaces also needed to be highly polished, so we chose to work with polished aluminum.
Because aluminum typically has poor ultrahigh vacuum performance due to the porosity of

15
the oxide layer that forms on its surface, we treated the aluminum with a cleaning processthat has been shown to form a dense oxide layer, for superior vacuum performance[58].
With this aluminum treatment we achieve a base pressure of 2 × 10−9 torr. We ultimately
achieve a count rate of almost 1 photon per shot of the experiment, however this signal was
still too small to continue with LIF as a detection technique.
(2) Photodissociation
Our favored technique for molecule state detection is resonance-enhanced multiphoton dissociation. We employ two photons, the first tuned to a UV bound-bound transition of
HfF+ . Depending on which state we wish to address, this photon has a wavelength of3
277 – 286 nm, which promotes either the 1 Σ+
0 ground state or the metastable ∆1 state

to an intermediate Ω = 0 state. We are able to address this intermediate state transition
with rotational selectivity, so this is suitable as a readout technique for a single rotational
state of the molecular ion. In particular, the 3 ∆1 , v = 0, J = 1 rovibrational state can beselectively dissociated using this technique. A second photon then promotes the ion fromthe intermediate Ω = 0 state to a repulsive state which dissociates into Hf+ and F. Because
Hf+ is 10% lighter than HfF+ , it can be co-trapped in the ion trap. We may then use
an electric field pulse to eject the ions from the trap onto an ion detector. With a large
enough electric field kick the ion detector effectively subtends a 4π solid angle, allowing
us to realize good detector efficiency compared to laser-induced fluorescence, where many
optical elements were required to direct the photons onto the photon detectors. We areable to realize a single shot detection efficiency of 18%, which is about 200 times better
than we were able to achieve using LIF. For more details please see [50].

16

2.2

Ion Trap

2.2.1

Refinements unique to the eEDM experiment

The ion trap we use for our precision spectroscopy has several differences from the standard“off the shelf” rf quadrupole trap. It is a custom design, unique to our group, where we have taken
steps to optimize characteristics not commonly targeted in other ion trapping experiments. The
qualities we wanted to optimize when designing the current generation ion trap were:(1) application of additional rotating electric bias field.
(2) electric field homogeneity.
(3) fluorescence collection efficiency.
In addition to using the ion trap electrodes to apply a time-dependent electric field to confine theions in the trap, we will use the ion trap electrodes to apply an electric field to polarize the trappedmolecular ions. However, at the center of an ion trap the time averaged electric field must be zero,otherwise the ions must be accelerating. This complicates the application of a polarizing electricfield because we cannot simply apply a static, uniform electric field. Instead we apply a rotating
electric field and polarize the molecules in a rotating frame of reference. The electric potential tocreate this rotating field can be added to the electric potential used to create the rf quadrupole
trap due to the principle of superposition. The application of this additional field will have various
design consequences, discussed below.
We chose to optimize electric field homogeneity because we predicted that inhomogeneity ofthe trapping electric fields, and especially of the rotating bias field, would be a source of systematic
errors[42]. Indeed, if we skip ahead to chapter 4 we will see that we observe several systematic
shifts in the EDM measurement that we can attribute to inhomogeneity in the trap. There areseveral features of the trap that help optimize the electric field homogeneity.
• The trap is extremely large. Typical ion traps range in size from tens of µm between
electrodes, as in the case for quantum information surface traps[65], to tens of mm, as in

17the case of the multi-pole traps used to study ballistic collisions[30]. The distance between
rf electrodes in our trap is 8 cm, making it probably one of the larger rf quadrupole
traps constructed. Increasing the size of our ion trap allows us to increase the size of thehomogeneous electric field region at the center of the trap. Additionally, with such a large
trap architecture we are unlikely to experience the anomalous rf heating encountered in
microscale surface traps[35].
• The trap does not have the standard four electrodes found in a conventional linear quadrupole
trap design. We have instead opted for six electrodes, distributed radially around the trap
at 60◦ intervals. This allows us to apply a more homogeneous rotating electric field by“smoothing” out the difference in electric field magnitude during a period of rotation.
When the electric field points directly at an electrode it will be stronger than when it
points at the space between two electrodes. This effect decreased as we crowded more
electrodes in, but numerical simulations of the electric field show that operating with 6 issufficient. We do not drive the six electrodes as a hexapole, but instead electrically shortpairs of electrodes together and apply the voltage Vi to the ith electrodes (where i runs
from 1 to 6),
Vi = Vrf sin(ωrf t) cos(2π(i − 1)/3).

(2.1)• The trap has electrodes that curve in at the ends, as shown in fig. 2.2. The purpose ofthis is to compensate for the electrodes not being infinitely long in the axial direction. The
curve was calculated to cancel the second order gradient of the rotating electric field in theaxial direction using SIMION[19].
We optimized fluorescence collection efficiency because we initially planned laser-induced fluorescence to be our primary technique to read out the internal state of the molecules. As such, optics
were designed into the trap to focus fluorescence from a 1 cm diameter region at the center of thetrap to exit ports at the center of the axial endcaps. Light pipes directed the fluoresced light further out from the trap to photomultiplier tubes where the light could be collected. As fluorescence18

Figure 2.2: Depicted is the design of the current ion trap. It has six linear electrodes, in contrast
to the standard four, to provide more uniform rotating electric fields. The bottom axial endcap isshown, as well as the bottom half of the ellipsoidal reflector which focuses fluorescence from thecenter of the trap to the exit port at the center of the top endcap electrode (not shown). The top
half of the ion trap, comprising the top ellipsoidal reflector and top endcap electrode are hidden inorder to reveal the six linear electrodes.

19collection is no longer used in the experiment, these mirrors serve no purpose other than to help
define the ground plane of the trap. If they were removed this would actually ruin the uniformityof the rotating electric field, but in future iterations of the ion trap we will redesign the shape ofthe electrodes to compensate for this.

2.2.2

Rotating electric field

To apply the rotating electric field we add the sinusoidal voltage Vrot to the rf quadrupole
voltage Vrf , such that the voltage Vi on the ith electrode (where i runs from 1 to 6) is
Vi = Vrf sin(ωrf t) cos(2π(i − 1)/3) + Vrot sin(ωrot t + 2π(i − 1)/6).

(2.2)

We typically operate with ωrf = (2π) × 50 kHz and ωrot = (2π) × 250 kHz. Generating the Vrot
drive for each of the six electrodes at six different phases requires some custom control hardware.
The rf quadrupole drive is a sinusoid at frequency ωrf that is π out of phase, which can be donewith a simple inverting op amp. One potential way to generate the 60◦ phase offsets is to makelow-pass filters with corner frequencies chosen to import the 60◦ , 120◦ , . . . phase shifts necessary.
This has several disadvantages. The first is that this will only work for one particular frequency,
and there are various reasons explored in chapter 4 for why we would want to operate at a rangeof ωrot frequencies. A second disadvantage is that such a circuit would be difficult to tune, and besusceptible to drifts which would cause the relative phases to wander, consequently contributing toinhomogeneity in Erot .
Our solution to this phase problem is to generate the Vrot signal using direct digital synthesis(DDS). DDS works by keeping track of a phase in time with a digital clock, and using a sinusoid
look-up table to digitally generate a sine wave signal. We can construct six separate DDS units
with synchronized clocks and program in the exact relative phase offsets. This has the advantageof having exact correct relative phase offsets for a range of frequencies, up to the digital precision,
which is better than 0.03◦ . We use the popular Analog Devices AD9959 DDS, because it incorporates 4 separate DDS units onto one chip. This model DDS also comes on a convenient evaluation

20
board containing a microcontroller that can be used to program the frequency and phase of the
DDS with a computer.

2.2.3

Erot amplitude pulses

When applying the rotating electric field it is important that we take care not to do so tooquickly. Instantly turning Erot on or off will accelerate the trapped ions to one direction, causing
excess trap slosh and heating. To solve this problem we turn on the rotating electric field gradually
over the period of many, typically 50, rotations. Additionally, we use fast amplitude ramps of therotating electric field to apply π/2 pulses for Ramsey spectroscopy, as explained in section 3.2.3.
This requires some technology to control the amplitude of the sinusoidal voltage that we apply tothe ion trap to generate Erot .
We initially designed a board with 6 voltage controlled amplifiers (analog devices AD8337),
allowing us to control the amplitude of our applied Vrot with a single voltage generated by anarbitrary waveform generator. However, the voltage controlled amplifiers are not really variable
amplifiers, they are variable attenuators with a built-in +18dB amplifier. This has two consequences, the first being that we get extra noise from the extra built in attenuator, and the second
that we are limited in the contrast we can achieve. We would like an amplifier that operates between
0dB (unity gain), and −∞ dB (fully off). This architecture seems typical for packaged voltage controlled amplifiers, which seem to be designed for use in modulating cellphone transmission signals.
When tested in the ion trap, this design produced too much excess ion slosh due to the lack of afull off and the additional noise of the built in amplifier.
We quickly designed an alternative amplitude control board using multiplying DACs. Multiplying DACs are typically very fast digital to analog converters that can operate with a variable
voltage reference, instead of the conventional DAC design with a constant voltage reference anda variable analog output defined by the digital inputs. A multiplying DAC can essentially bethought of as voltage divider and a set of very fast switches. When all the digital inputs are high,the switches are closed and the voltage divider is bypassed, and the output voltage is equal to the21reference voltage. When all the digital inputs are low, the switches all open and the voltage output
is shorted to ground. In this sense the output of the DAC, Vout = Vref × x, where x is an unsigned
integer defined by the the digital inputs. The advantages of this method are that it is capable ofperfect contrast, by programming the value of 0 to the DAC, and it doesn’t include any additional
noise due to amplification.
The design of the amplitude control boards is shown in fig. 2.3 and fig. 2.4. The essential
component is the AD5433 10 bit multiplying DAC, which has a 10 MHz bandwidth and an update
rate of 20Mbps, which is ideal for our application where the frequency of Erot will typically be
250 kHz. By using a DAC instead of a voltage controlled amplifier we have made the control
problem an order of magnitude more difficult, because instead of one analog control voltage we
now require 10 digital controls. We used a fast ARM Cortex-M3 microcontroller running at 72
MHz seated on the same PCB as the multiplying DACs. This microcontroller should in principle
be capable of switching a digital output with a period of 28 ns, which means it could run through thesequence to either fully ramp up or ramp down the 10 bit control to the DAC in 28 µs. However,
the microcontroller has a more complicated task than simply producing square waves: it must
output a sequence of pulses on 10 separate channels that count up or count down an integer. Thisrequires storing the value of the integer in the microcontroller memory, performing arithmetic toincrement or decrement the integer, indexing a loop, and conditional statements to check if theramp has finished. All of these have overheads of several clock cycles. In order to reduce theoverhead as much as possible while still retaining flexibility we use the interrupt function of theARM microcontroller to ensure that our code is executed immediately and does not suffer overhead
from other microcontroller functions (such as communicating with the main experiment control
computer). In order to not perform the condition check of whether to increment or decrement the
integer every loop iteration, we only perform unidirectional ramps; 6 interrupts will ramp up and6 interrupts will ramp down. An example of an interrupt ramping down follows:v o i d ramp1 ( ){3i n t a = begin1 ;
1
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R12
1
4

10K
20K
6

10K
20K

R13
1
2
R14
1

1

10uF
C4 C3
+
10K
20K

R15
1
2
R16

2

2

TTL11

5

3

C18
1pF

2

1

3

2

7

C26
IN8
0.1uF

TTL12

TITLE:CODE:

COMPANY:

2

1

C25
0.1uF

2

1

8

1

1

R35
10k

+
1
-15V
4

U4

+15V
0.1uF
C17

2

R39
1
2
R38
1
6

R8
1
2

6

1K

1K
R44

R43

CW

2

2

TTL13

+15V

OP-27GP

0.1uF
C21
C24
1pF

SIZE:

2

1

1

1K

R40

3

2

2

7

8

R42
10k

+
1
-15V
4

U5

+15V

1

LTR

OUT7

CW

+15V

OP-27GP

6

0.1uF
C27

1K

R47

DRAWING NO:

1

REVISION RECORDECO NO:2OUT8APPROVED:

1K

D

7

DB0
DB1
DB2
DB3
DB4
DB5
DB6
DB7
VDD
DB8
R/W
DB9 CS
13
12
11
10
9
8
7
6
5
4
1

10K
20K

R17
1
2
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1

10K
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R19
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0.1uF
C13
10K
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1

1K
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1
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3
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4
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1K
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2

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1
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C16
0.1uF
10K
20K

R29
1
2
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1

2
R31
1
2
R32
1

10K
20K

2
R33
1
2
R34
1

2
R41
0.1uF
C23
C22
0.1uF
10K
20K

2
R48
1

1K

Figure 2.3: The multiplying DAC variable attenuator is designed to accept both the true and complementary current outputs of the DDS
6532DAC channel.
The currents are converted
into a voltage 4and buffered with a differential
amplifier (U3). The
differential output is1fed into
the VREF of a AD5433 multiplying DAC. Digital lines D0-D9 represent a 10bit unsigned integer that is effectively multiplied by VREF,
thus programming 1023 into lines D0-D9 will pass the input at VREF to the output at full scale, 512 will output it at half scale, and 0 willturn the output off. The digital lines D0-D9 come from the outputs of an ARM Cortex-M3 microcontroller buffered by three 74HC125
quad TTL buffers. The sub schematic from IN1 to OUT1 is repeated 6 times, for each of the trap DDS channels, but is driven in parallelby the same digital lines to ensure that all DDS channels have the same multiplicative attenuation. Inputs TTL1 through TTL13 areconnected to interrupt pins on the ARM microcontroller, which is programmed to initiate various ramp sequences when it detects a risingedge on one of these channels.

DATE:

REV:

22
A

B

C

D

Figure 2.4: The multiplying DAC variable attenuator has 6 parallel channels OUT1-OUT6 which attenuate the current output of a DDS
channel by a factor specified by a programmable 10 bit unsigned integer. The 10 bit unsigned integer is supplied by a ARM Cortex-M3
microcontroller on a Maple hobbyist board developed by Leaf Labs, and headers J4, J5, J6, J7 and J8 are designed to interface with the
Maple board. Channels OUT7 and OUT8 are identical to OUT1-OUT6 with the exception of the multiplying DAC hardware, they simply
provide a differential buffered output to the remaining two DDS channels. Through efficient microcontroller software design this board iscapable of ramping the DDS output amplitude smoothly through the full 1024 bit range in 100 µs.

23

24
i n t b = end1 ;int c = step1 ;f o r ( i n t i=a ; i >b ; i−=c ){GPIOC−>r e g s −>ODR = i ;GPIOC−>r e g s −>BSRR = 1<<15;}
GPIOC−>r e g s −>ODR = b ;GPIOC−>r e g s −>BSRR = 1<<15;

4
5
6
7
8
9
10
11
12
13

}

The code GPIOC->regs->ODR = i; allows us to write the bits of the integer i to the register
ODR in a single clock cycle, rather than 10. This requires careful PCB layout to ensure that theoutput pins of the register correspond to the correct digital input pins of the multiplying DACs.
The code GPIOC->regs->BSRR = 1<<15; makes one bit of the BSRR register high to latch the
DAC. This allows us to increment the values of the DAC once every 8 clock cycles, or equivalently,
perform a full ramp in 100 µs. In a typical experiment sequence we might ramp Erot on to full, andthen pulse it down to 25% of its value and back up to perform one π/2 pulse, pulse it down andup again 100 ms later for the second π/2 pulse, before finally ramping Erot down to zero at the endof the experiment. This requires triggering 4 different interrupts on the microcontroller 6 times,
e.g. in the sequence {2, 3, 4, 3, 4, 1}, and programming the thresholds begin i and end i on eachbeforehand. This is easily done using the fast digital timing card that sequences the experiment.

2.2.4

Experiment sequence

The experiment is timed by a special purpose timing card designed by Viewpoint Systems.
The card is at its core an FPGA that has a 32 × 512 bit memory bank containing timestamps anda 64 × 512 memory bank containing digital output line states. After receiving a trigger the FPGA
increments a counter every cycle of a 10 MHz external clock, comparing the counter to an entryof the timestamp memory bank indicated by a pointer. Once the counter matches a timestamp
the corresponding row of the output memory bank is latched to the outputs and the pointer isincremented such that it points at the new row of the timestamps memory bank. This allows thetiming card to specify pulses as narrow as 100 ns spaced as much as 7 minutes apart. This is25
ideal for an experiment such as ours that lasts for no more than several seconds and rarely requires
timing precision better than 100 ns. Compare this to the more typical configuration found on many
computer controlled DACs that simply playback from memory 1 sample per clock cycle, and assuch are typically limited to experiment times no greater than the product of the resolution andthe number of samples that can be stored in memory.
Additionally, the timing card can be reprogrammed by transferring several kB of data at atime. This is because of its efficient nature at representing sparse pulse sequence. To reprogram thesame pulse sequence on a traditional DAC we would have to transfer several GB of data, because wemust specify the dead time between pulses. A potential disadvantage of this specific timing card isthat Viewpoint Systems is all but no longer supporting it. While they are still available for sale, it isincreasingly difficult to make them run on newer computer systems. A potential solution is internal
development of a new timing board. We currently have prototype designs of a timing sequencer
that can run on a Xilinx FPGA and should replicate the features described above, although due
to improvements in FPGA technology since the design of the Viewpoint card it is possible to store
more than 512 pulses in memory. This new timing board design can communicate with computers
using the standard RS232 protocol over USB2.0 at several MB/s, and could potentially be extended
to even higher speeds using PCIe.
The timing sequence for a typical Ramsey experiment is shown in fig. 2.5. There are primarily
three sets of things sequenced in the experiment: lasers, trap electronics, and the rotating electricfield. Timing pulses fire the Q-switch of the ablation laser to produce neutral molecules, open
shutters to allow the two transfer lasers to irradiate the ions, pulse on the amplifier to the AOMs that
shutter the depletion pulse train, and open the shutter for the 266 nm laser that dissociates the ions.
In synchrony with this, various pieces of electronics attached to the ion trap driver receive pulses
to increase confinement of the ion trap in the radial direction to overlap the ions with the transfer
lasers, then to relax the trap to suppress collisions during the Ramsey experiment, and finally tocompress the ions so that the 266 nm laser can be focused on them during dissociation. Pulses arealso sent to the rotating electric field amplitude modulator described in section 2.2.3 to ensure the26rotating electric field is at one value for transfer, is at another value for the Ramsey experiment,
and is pulsed to a lower value to apply π/2 pulses before and after the Ramsey experiment.

2.2.5

Future trap directions

The current ion trap has been very useful for the study of EDM systematics and in demonstrating the rotating electric field technique. It does have some minor shortcomings, which we hope
to correct in a 3rd generation version ion trap. Perhaps the most immediately obvious when looking
at the trap is that it was designed to be optimized for fluorescence collection, which is no longer
used in the experiment. This design constraint dictated much of the shape of the current trap. As
discussed in chapter 4 it has also become clear that trap is not as uniform as we would like, and thiscan potentially be improved by revisiting the shape of the trap electrodes, the number of electrodes,
and the size of the trap itself. Finally, the trap vacuum chamber and electrodes were fabricatedfrom stainless steel, which while nominally nonmagnetic, might perhaps be magnetic enough to bea source of potential systematic errors, see section 4.2.3. Indeed, the table upon which the experiment rests is also made of stainless steel. This could be replaced by a skeleton structure made of
extruded aluminum, which in addition to being nonmagnetic might provide easier access to partsof the experiment.

2.3

Computer control
Sophisticated computer control of experiments is very commonplace in the field of physics

in this day and age. While every experiment has its own peculiarities, there are a few interesting
features of our experiment control that are worth mentioning, because their complexity requires
some documentation, and also because they might be useful in general.

2.3.1

uBinary file format

In the EDM experiment we record photo-dissociated ion counts using a digital oscilloscope.
A single shot of the experiment corresponds to a single trace of the oscilloscope which is 2500

27

V rf (V)

LasersFigure 2.5: Experiment timing pulses for a typical run of a Ramsey fringe experiment.

ablationtransferon

dissociation

off

depletion pulses

100

Tightened Trap For Transfer

Tightened Trap For Dissociation

Normal Trap Operation50

E rot (V/cm)

030

Erot max

20

transfer

10
0

0

:/2
100

200

300

Time (ms)

:/2
400

500

28
double precision floating point numbers, which if taken at a rate of 2 Hz is a modest data rateof about 40 kB/s. A precision measurement of the electron EDM will require about 100 hours ofdata, or more than 14 GB. Additionally, this 100 hours of data will be taken under a large numberof deliberately varied experimental conditions, and we will want to track these along with the raw
oscilloscope data. We need to store the data in some manner that is consistent, unambiguous, andreasonably compact. For example, a large text file with lists of human readable numbers and atext header describing each section is reasonably unambiguous, but it is not very compact. A large
text file is about a factor of 10 times larger than the minimal machine binary representation of thedata (140 GB), and the time to read or write data to disk will be similarly increased. Moreover,there is the additional overhead of parsing the text to and from machine data whenever the datais stored or retrieved. Conversely, a pure machine binary representation of the data will be very
efficient in terms of storage space and read/write times, but will be completely unintelligible to ahuman reader. Additionally, both data formats are inflexible to changes in the sets of parameters
we wish to store in the data file. If, for example, we increase the resolution of the oscilloscope to
10000 points per trace, or if we wish to also begin tracking the relative humidity in the lab, thiswill require changes made to the computer code that reads and writes the data to disk.
Our solution to these problems is to design a new data file format unique to our experiment.
The file format is self descriptive so that it can remain flexible to changes in the quantity of typeof data contained, and it is in machine binary so it can be efficiently stored and retrieved to disk.
Even though it is in a binary format it is still suitable for archival purposes due to its self descriptive
header. There are similar existing file formats, such as the Hierarchical Data Format (HDF)[71]
developed by the NCSA, or the Common Data Format (CDF)[68] developed by NASA. These wereconsidered, but ultimately rejected due to their complexity and lack of suitable libraries for thesoftware environments we were currently using at the time.
We refer to this file format as “uBinary”, which is short for Universal Binary, because it
allows us to store, retrieve, and manipulate our data in a platform independent manner while stilltaking advantage of the speed and compactness of a binary format. In particular it allows us to29
share data between the MATLAB and LabView platforms, our choice platforms for data analysis
and data acquisition. A description of the file format is outlined in table 2.2, with the relevant datatypes enumerated in table 2.1. The file has gone through several iterations, but the most current,
used in all the data presented in chapter 5, is the 4th iteration. All versions at their core share thesame structure, but the newer versions divide data in ways to make it more easily searched.
Table 2.1: uBinary data types
Type
1
2
3
4
5
6
7
8
9
10
22
23
33
48
50
51
55
64
80
84
112

MATLAB
int8
int16
int32
int64
uint8
uint16
uint32
uint64
singledoubleuint16
uint32logicalstringstringstringstringarray
structwaveform
stringLabVIEW (G)
int8
int16
int32
int64
uint8
uint16
uint32
uint64
singledoubleenum
tab controlboolstringpath
text picture
DAQmx resource
arraycluster
waveform
VISA resource

The first 18 bytes of the uBinary format is the string “@@@@@uBinary4}}}}}”, identifying
it as a file with a format described in table 2.2. Next is a 4 byte integer that describes how many“chunks” the file contains. Each chunk contains 4 parts:(1) the name of the chunk.
(2) a list of strings called names, naming the data in the chunk.

30

Table 2.2: uBinary file format specification
Field

Description
Type
Value
file header
filetype
string to identify file as ubinary4 char[18]ubinary4
N chunks
number of chunks of data
uint32
list of N chunks data chunks
name lenlength of chunk nameuint32
chunk namename of the chunkchar[name len]
N namelength of list of chunk element uint32
names
names
names of the chunk elements
string[N name]
N typelength of list of chunk element uint32
types
types
types of the chunk elements
uint8[N type]
N datasize of chunk data
uint32data
chunk data
uint8[N data]
format of data is defined by parsing through types[i]. Each parsed element has a name innames[j]. The number of elements in types[] and in names[] may be different, so they mustbe parsed through in order. For fields and values that reference types[] and names[], these areparsed from those arrays, and not from data.
types[i]string48
str namename of the string
stringnames[j]str lenlength of the string
uint32
str data
string datachar[str len]types[i]arraydimsnumber of array dimensions
uint8
e typetype of array element
uint8
array namename of the arraystringN array[dims] size of each array dimension
uint32[dims]
Q
array
data
array
data
e type[ n N array[n]]Qn N array[n] array elements must be parsed from data[]

64
types[i+1]types[i+2]names[j]types[i]cluster80
N clusternumber of elements in cluster
uint8
types[i+1]cluster namename of the cluster
stringnames[j]N cluster cluster contents types and names must be parsed from types[i+2:...] andfrom names[j+1:...]
types[i]type in table 2.1
1 – 33
namename of the data element
stringnames[j]element data
data element
types[i]N cluster cluster contents types and names must be parsed from types[i+2:...] andfrom names[j+1:...]

31(3) a list of types describing the data in the chunk.
(4) the data in the chunk.
Each of these 3 parts is preceded by a 4 byte integer describing the length of each list in bytes.
This allows for chunks to be easily skipped over if we wish to only read or write to a single chunk.
As an interesting historical note, with the chunk name and the integer describing the data length
removed, a chunk is actually the first iteration of uBinary. Once the data from a chunk is read in
it must be parsed, according to the prescription in the bottom part of table 2.2. This is done byiterating through the list types, an array of 8 bit integers. These numbers enumerate the type ofthe data to be parsed, and are listed in table 2.1. For example, if the next integer parsed off thelist types is 10, this means that the next parcel of data is an 8 byte double. We then remove 8
bytes from data, typecast it as a double, and label it with the next string parsed off of the list
names. The matter is more complicated for the composite types of string, array, and cluster.
These require recursive parsing. For example, a cluster might contain 3 elements, a double, a 32
bit integer, and another cluster containing two 8 bit integers. The list of types describing this
scenario would be [80 3 10 3 80 2 1 1]: 80 for the cluster, 3 for the three elements, [10 3 80]
for double, int32, and cluster, and then finally 2 for the two int8s each specified by 1.
Currently we have software libraries to read and write uBinarys in LabVIEW, our dataacquisition platform, and to read and write uBinarys in MATLAB, our data analysis platform. In
principle this can be extended to other languages as well. For example, preliminary code exists
to read uBinarys in Python, and libraries for other software environments can be implemented asneeded. The peculiarities of the format are due to it being based around the format LabVIEW
uses for storing its own data types internally, which we pieced together from some small amountsof documentation and trial and error. Because it is based on the LabVIEW internal data format,
all data in the uBinary file is stored in the big-endian[69] byte order, with the most-significant
byte stored in the first indexed memory address. Because of this relationship with the internal
LabVIEW data format we can very easily generate uBinary files by using the LabVIEW built in32function “Flatten To String” to generate the binary data. By iterating through all controls in
a LabVIEW program by reference we can automatically save every parameter in our LabVIEW
control program using the uBinary data, as shown in fig. 2.6. The top level LabVIEW experimentcontrol has 368 independent controls that can be written along with data to a uBinary file in under
80 ms. This time is primarily limited by how long it takes to parse the data to be saved andconstruct the appropriately descriptive header. Additionally, a uBinary file can be loaded backinto the data acquisition program, effectively allowing us to reset the experiment parameters tomatch that of any recorded set of data.

2.3.2

Experiment control state machine

We choose to work with two separate software environments, LabVIEW for data acquisition,and MATLAB for data analysis, to exploit the separate strengths of each system. The LabVIEW
development environment, based on the graphical G data-flow based programming language, is well
suited for quick and easy production of experiment control graphical user interfaces. Additionally,
LabVIEW is produced by National Instruments (NI), an industry leader in the production of dataacquisition and control hardware, and programmatically interfacing with these devices is madeextremely simple using NI’s libraries. Conversely, MATLAB was designed with scientific computing
in mind, and is well suited for performing arbitrarily complex analysis on data. While either system
could be used for both tasks, it typically is much more tedious, e.g. designing a GUI in MATLAB
to control a complex device might take roughly ten times longer, and that time could be betterused serving the needs of science.
During the course of an EDM measurement we will want to interweave various different typesof data, such as periodically taking trap slosh measurements to check the position and oscillation
amplitude of the ion cloud in between Ramsey fringe measurements. To achieve this goal we need aflexible computer control scheme that allows us to perform many different types of experiments inarbitrary sequence without having to retool large portions of the experiment control code. We use astate machine with about 140 different states corresponding to sending various types of commands

33

Figure 2.6(a) ubinary save.vi – This subvi works by assembling a list of references to every control in the program(via All Refs subvi). It parses out those controls which cannot be saved, either because we do not wish tosave them, or because they cannot be accessed via reference due to race conditions (e.g. latching boolean
controls). It iterates through these control references, first writing the Label.Text as the chunk name, thenwriting the list of control names and types returned from the uBin Single subvi, and finally writing the valueof the control, accessed by reference.

(b) ubinary load.vi – This subvi reverses the hard work done by ubinary save.vi. It reads in a uBinary
file, checking first that the header string matches the expected “@@@@@ubinary4}}}}}”, and then iterates
through the chunks of the file. For each chunk it checks to see if the chunk name matches an entry in thelist of control references returned by All Refs. If it does discover a match, then it will try to typecast the
chunk data to the type of the matching control. If this typecast succeeds then it inserts the chunk data into
the control, otherwise it quietly returns an error and continues iterating. This subvi also has the optional
input of “control”, allowing us to specify a single control to match to. This is useful if we wish to use auBinary file to roll back changes to a single control, such as the DAC voltages, while leaving the whole ofthe experimental parameters unchanged.

34
to pieces of acquisition or control hardware, e.g. “Scope: Read” or “TrapDDS: Upload”. By
specifying a short sequence of states we can efficiently describe an experiment sequence, as in thefollowing case describing a sequence to measure ion trap slosh on two different ion detectors:
Frames >> 1 , 1
Scope : S e t t i n g s >> TDS5034B
3 Scope : T r i g g e r S e t t i n g s >> TDS5034B
4 Trap DDS: Upload
12

5
6 DAC:

Upload2 >> 1 , Fast
Scope : S t a r t >> TDS5034B
8 DIO64 : P r e p a r e >> 5
9 DIO64 : Run
10 Scope : Read >> TDS5034B
11 Scope : Stop >> TDS5034B
12 Scope : D i s p l a y T r a c e s >> TDS5034B
13 Count I o n s
7

14

IMG MCP: C l e a r
Upload2 >> 1 ,IMG
17 DIO64 : P r e p a r e >> 6
18 MV500 : T r i g g e r
19 Wait >> 50
20 DIO64 : Run
21 MV500 : Read
22 Count I o n s
15

16 DAC:In principle such a state machine is Turing complete and capable of arbitrary computation,
but this would be tedious, especially given the number of already existing scripting languages
with much more well thought out control flow. We chose the MATLAB language as the master
arbiter of the control flow of the state machine, as we were already using it for data analysis.
Given an appropriate interface layer between the LabVIEW state machine and MATLAB, we canscript experiments in the MATLAB language, taking full advantage of that language’s control flow
features, function stack, and numerical analysis techniques.
Because of the flexible nature of the uBinary file format it is well suited for use as a serialized
format for real time communication between LabVIEW and MATLAB. By creating a parallel threadof execution within the LabVIEW state machine we can implement a TCP server that listens forcommunications requests from clients. MATLAB can talk to this TCP server and manipulate thestatus of the state machine queue by adding states to the front or back. In other words, MATLAB
commands LabVIEW via TCP to execute a particular set of commands to communicate with

35
hardware and collect data before turning over control back to MATLAB. Additionally, the uBinary
data format can be used to pass experiment control parameters and measurement data between
LabVIEW and MATLAB. This allows MATLAB to instruct the LabVIEW to make a sequence ofmeasurements, request the data and analyze it, and then in real time make some decision about
what nature of experiment to run next. This blurring of the boundaries of data acquisition and data
analysis allows us to create very sophisticated experimental control sequences, where for example
we record EDM data until the contrast of our Ramsey fringe begins to fall, at which point thedata analysis can identify the decrease in contrast and add some diagnostic sequences to the queue,
make a decision, and correct.
The experiment control TCP server runs on port 8888, and it listens until a TCP request
is made. Once communication is initiated it expects one uint32 describing how many bytes will
follow. It then parses these bytes for text strings, recognizing the following commands:(1) “commands” - will return an array of strings containing all of the available states of thestate machine.
(2) “documentation” - will return an array of strings containing the documentation of theavailable states of the state machine.
(3) “controls” - will return an array of strings containing the names of all of the available
controls in the data acquisition program.
(4) “type” - will attempt to read a string, match that string to a control, and return an array ofnames and types as specified by table 2.2 describing how to parse the uBinary data chunkassociated with that control.
(5) “read” - will attempt to read a string, match that string to a control, and return theuBinary data of that control.
(6) “write” - will attempt to read a string, match that string to a control, and then readuBinary data and load that into that control.

36(7) “sync” - will wait until the state machine reaches the state “Sync”, and then return a 1.
Additionally, if a string is not recognized as a command it will try to load this onto the backof the state machine queue. The state machine is robust to unrecognized states, removing themfrom the queue when they reach the front, and passing a descriptive error to a text control facing
the user.

Chapter 3

Statistical sensitivity

3.1

Theory of measurement

3.1.1

Measurement principles

The electron EDM manifests as a torque on an electron spin due to the interaction with
an electric field. Much like the interaction of the magnetic moment with the magnetic field, the~ · E,~ where S~ is the spin ofelectron EDM de , appears in the Hamiltonian as the energy shift −2de S
the electron. The Hamiltonian for an electron in and electric and a magnetic field is~ + 2de E~ · S,~
H = − gS µB B

(3.1)

where µB is the Bohr magneton and gS is the electron spin g-factor. To measure the 2de E term
we can perform spectroscopy between the two spin states. This alone is not sufficient to accurately
measure 2de E, because under typical conditions 2de E  gs µB B, and any small deviation fromthe expected value for the measured energy could be attributed to uncertainty in magnetic field.
We can improve this by making one measurement for the energy between the two spin states,~ and then making a second measurement. In this differential
reversing the direction of electric field E,
measurement the uncertainty in B drops out, and half the difference of the two measurements is
2de E.

38
3.1.2

3∆

1 state Hamiltonian

To achieve the best upper bound on the EDM it is imperative to amplify the effect of de E in
the experiment as much as possible. To provide the largest possible energy shift for a finite EDM
we strive to apply the largest electric field possible. The largest electric field that is feasible toapply with parallel plates in a vacuum is < 1 MV/cm[73], typically somewhere before this point
microscopic surface roughness causes breakdown to occur in vacuum. Classes of diatomic molecules
have internal electric fields[62, 16] potentially as large as ∼ 100 GV/cm[47]. Our molecule of choiceis HfF+ . In the 3 ∆1 electronic state of HfF+ one of the valence electrons experiences an effective
electric field of Eeff = 23.3 GV/cm[48, 51, 52, 26]. An electric field of this magnitude will cause a1.13 mHz energy shift between the mF = +3/2 and mF = −3/2 stretch states of the J = 1, F = 3/2
hyperfine level. As we will see in later sections, this size energy shift is challenging to measure, but
is technically feasible.
We can write down the approximate Hamiltonian for the 3 ∆1 state of the HfF+ diatomic
molecule using Brown and Carrington[11] as a template,3

H ∆1 = He + Hvib + Hrot + HHF + HΛD + HStark + HZeeman + HEDM . . . ,(3.2)

where He is the electronic contribution, Hvib describes the molecule vibration, Hrot describes themolecule rotation, HHF is the hyperfine interaction with the fluorine nucleus, HΛD is the Λ-doubling
interaction, HEDM is the possible contribution from the electron EDM, and the other terms describeinteractions with external fields. We will restrict ourselves to eigenstates of He + Hvib + Hrot , inparticular 3 ∆1 , v = 0, J = 1. In the 3 ∆1 state |Σ| = 1, |Λ| = 2, and |Ω| = 1, where Σ, Λ, and~ L,~ and J~ on to the internuclear axis. Within this manifold of statesΩ are the projections of S,
there is both hyperfine and Λ-doubling structure. The fluorine nucleus has a spin I = 12 , which
enters into the HHF term and provides an energy splitting of EHF = −46.6(1) MHz between the~ A Coriolis type effect mixes states of opposite sign Λ.
F = 1/2 and F = 3/2 states (F~ = I~ + J).
Perturbations with nearby electronic states creates a splitting between Ω = +1 and Ω = −1 statesthat is EΛD = 740(40) kHz. A diagram of the states in the 3 ∆1 , J = 1 manifold is shown in fig. 3.1.

39
~ where d~mf is the molecule
In the presence of an applied external electric field a HStark = d~mf · E,
frame dipole moment. For HfF+ this is dmf = 1.411(6) e a0 . At energies where dmf Elab  EΛD themolecule is completely polarized, and there is a 2EStark energy splitting between states of different|F, (mF Ω)i,
EStark = −mF ΩγF dmf Elab ,(3.3)

where γF = 1/3 for F = 3/2 and γF = 2/3 for F = 1/2. In the F = 3/2 manifold of states thisyields four states with a positive EStark ,|F = 3/2, Ω = −1, mF = +3/2i ,|F = 3/2, Ω = −1, mF = +1/2i ,|F = 3/2, Ω = +1, mF = −3/2i ,|F = 3/2, Ω = +1, mF = −1/2i ,and four complementary states where mF Ω > 0 and EStark < 0. We refer to the states with
EStark > 0 as the upper manifold of states, denoted with u superscripts, and those states with
EStark < 0 as the lower manifold, denoted with l superscripts.
~ term in the
The application of an external magnetic field gives rise to a HZeeman = −~µ·B
Hamiltonian. The Zeeman effect can be very complicated, so we will only discuss the leading terms
for now. They are:HZeeman = ((gL + gr )γF ΛΩ + (gS + gr )γF ΣΩ − gr γF J(J + 1) − gI kF )mF µB B.

(3.4)

The contributions to the Zeeman Hamiltonian from nuclear spin (gI ) and molecule rotation (gr )are small, suppressed by the ratio of the mass of the electron to the mass of the molecule (< 10−3 ).
This leaves the terms
HZeeman = (gL Λ + gS Σ)γF mF ΩµB B = gF mF |Ω|µB B.

(3.5)

For an ideal 3 ∆1 molecular state Λ = ±2, Σ = ∓1, and Ω = Λ+Σ = 1. Considering that the orbital
angular momentum g-factor is gL = 1 and the spin angular momentum g-factor is gS ≈ 2 + α/π,40Figure 3.1: The state diagram of the 12 states contained in the 3 ∆1 , J = 1 manifold. In the absenceof any external electric or magnetic fields the states are denoted by dashed lines. The F = 1/2
and F = 3/2 states are split by the hyperfine energy EHF , and within each hyperfine level is aΛ-doublet split by EΛD . When an electric field is applied the states pick up energy EStark , andwhen a magnetic field is applied the states pick up energy EZeeman . In the presence of external fields
the states are drawn as solid lines. For the precision EDM experiment we will perform spectroscopyon the transitions between the two pairs of mF = ±3/2 states in the upper and lower doublet.

F=1/2

3guF μB Brot

EHF
EZeeman
EStarkEΛD

F=3/2

3glF μB BrotmF =-3/2

mF =-1/2

mF =1/2

mF =3/2

41
where α ≈ 1/137 is the fine structure constant, we would expect a cancellation to order α. We
estimate the g-factor of the F = 3/2 state to be as small as gF ≈ 0.00077. Measurements ofthis value show that it is gF = 0.00305(10). This discrepancy can be due to 3 ∆1 actually being
an electronic state of predominantly 3 ∆1 character but with slight admixtures of other electronic
states, where this near cancellation of (gL Λ + gS Σ) no longer holds. Moreover, while the g-factor
for the upper and lower Λ doublet states are nominally the same, effects such as differential mixing
with the J = 2 levels can create fractional differences on the 10−3 level.
Finally we must consider the effect of the electron EDM. The 3 ∆1 state has an Eeff = 23.3
GV/cm electric field that points along the internuclear axis. Choosing the externally applied
electric field Elab as the direction of our quantization axis n̂ we find that for the upper doublet
E~eff · n̂ = +23.3 GV/cm, and for the lower doublet E~eff · n̂ = −23.3 GV/cm. The EDM shift is~ which is−de E~eff · S,u/l

HEDM = ±Σde Eeff ,(3.6)

where we note that for mF = ±3/2 the projection of the electron spin on the internuclear axis isΣ = ∓1/2.
We will proceed by separately considering the two pairs of states in the F = 3/2 manifold,the |mF = ±3/2, Ω = ∓1i upper doublet states, and the |mF = ±3/2, Ω = ±1i lower doublet states.
The Hamiltonians describing these two subspaces (with superscripts u and l denoting doublet) are


3(g
±
δg)µ
B
±
d
E
0
e eff
B
1 F

H u/l = 
(3.7)
,
2
0
−3(gF ± δg)µB B ∓ de Eeff
where δgF = (gFu − gFl )/2 describes the difference between gF in the upper and lower doublets. In
this reduced space Hamiltonian the sign of the electric field in the EDM term, Eeff , depends onwhether we are in the upper or lower Λ-doublet. This is because between the two doublets, themolecular axis, and hence the 23.3 GV/cm electric field, points in opposite directions with respectto Σ, the electron spin.

42
3.1.3

Effect of the rotating frame

The application of the bias electric field Elab is not trivial in an ion trap, where the trapped
D E
region must by definition have E~ = 0, otherwise the ions would be accelerated out of thettrap. In order to work around this limitation of ion trapping technology we must apply a field
D
E
Elab where the time average E~lab = 0. We apply a rotating electric field in the x − y plane,t

E~lab = Erot (cos(ωrot t)x̂ + sin(ωrot t)ŷ)[45]. There are primarily two effects of this:(1) The ions are driven in uniform circular motion by the electric field. They make small circulareErot
orbits with radius r = mω2 , where r ≈ 0.47 mm for typical operating conditions. In therotpresence of a quadrupole gradient ∇B these circular orbits have the effect in the molecule
frame of a static magnetic field that is aligned with the quantization axis, Brot = ∇Brrot .
(2) The ions are accelerated into a rotating frame relative to the lab. In the lab frame thisinduces a coupling between neighboring mF states. In the |mF = ±3/2i reduced state
space Hamiltonian the coupling is third order, and can be written as follows,∆ ± δ∆
1 3(gF ± δg)µB Brot − 3αωrot ± de Eeff
H u/l = ,2
∆ ± δ∆−3(gF ± δg)µB Brot + 3αωrot ∓ de Eeff
(3.8)
where the off-diagonal coupling terms are [42, 44]




 !
1 dmf Erot 2
9 dmf Erot 4~ωrot 3
1−−
+ ...,∆ = 27ωef
dmf Erot
32
EHF
512
EHF





 !
57 dmf Erot 43
dmf Erot
3 dmf Erot 2δ∆ = ∆
1+++ ....
8EHF
32 EHF1024 EHF


3.2

Measurement

3.2.1

Ramsey spectroscopy(3.9)(3.10)To measure the energy shift provided by de Eef f we use Ramsey spectroscopy[56]. The generalprinciple of Ramsey spectroscopy is to measure the energy difference, ∆E, of a two level quantum

43
system. The system can be described by the Hamiltonian,


∆E
0
1

H= 
.
2
0 −∆E

(3.11)

We identify the two eigenstates of the system as |↑i and |↓i, motivated by a spin- 21 system. First
Figure 3.2: Depicted is a representation of Ramsey spectroscopy on the Bloch sphere. In (a) thesystem is initially prepared in |↑i before√a π/2 pulse rotates the system about the |xi axis, suchthat the system is in |yi = (|↑i + i |↓i)/ 2. In (b) the system precesses about the equator as |↑iacquires phase et(∆E/2)/i~ and |↓i acquires phase e−t(∆E/2)/i~ . Finally in (c) the second π/2 pulse
is applied,

(a)

(b)

j"i

(c)

j"i

j"i

:
2

jyi

jxi

:
2

t"E=7h

jyi

jxi

j#i

jyi

jxi

j#i

j#i

we prepare the system in one of two eigenstates, say |ψi = |↑i. As shown in fig. 3.2, we then apply
a π/2 pulse operation, so called because the state is rotated by π/2 on the Bloch sphere to theequator. The state is now in the superposition |ψi = √12 (|↑i − i |↓i). If we allow the state to evolve
forward in time under the Hamiltonian H, the phase of each component of the superposition will
advance at a different rate,

t
t
1 
|ψ(t)i = √ |↑i e i~ ∆E/2 − i |↓i e− i~ ∆E/2 .
2(3.12)We then apply a second π/2 identical to the first and measure the probability that the state is in|↑i or |↓i,∆E
P (↑) = |h↑|ψi| = sin
t ,
2~


2
2 ∆E
P (↓) = |h↓|ψi| = cos
t .
2~
2

2



(3.13)(3.14)

44
We construct the population asymmetry by taking the probability difference over the probabilitysum,P (↑) − P (↓)∆E
A(t) == − cost .
P (↑) + P (↓)~(3.15)

By carefully measuring the frequency of oscillation of the asymmetry we can determine the energy
difference of the two states ∆E.

3.2.1.1

Time Dependence

If the Hamiltonian in eq. (3.11) has some time dependence we require a more careful treatment,

01 ∆E + α(t)H(t) = .
20−∆E − α(t)(3.16)The propagator that we use to express how the phase between |↑i and |↓i must be explicitly
integrated with respect to time,

Rt
Rt
1
1
1 
∆E+α(t0 )dt0
− 2i~
∆E+α(t0 )dt0
0
0
2i~
√
− i |↓i e.
|↑i e|ψ(t)i =2(3.17)

With the addition of this time dependent term the asymmetry gain new time dependence,
 

Z t
1
0
0
A(t) = − cost∆E +α(t )dt.
~
0(3.18)

If we assume that α(t) has some sinusoidal dependence, e.g. it is due to some oscillatory motion in
the ion trap, we can write α(t) = α0 cos(ωt). We can now integrate α(t) in the asymmetry,1(t∆E + α0 sin(ωt)/ω) ,
~



 
α


∆E
∆E
α0
0
t cos
sin(ωt) ) + sin
t sin
sin(ωt) ).
= − cos~~ω~~ωA(t) = − cos(3.19)(3.20)

45
We may use 9.1.42 and 9.1.43 of Abramowitz and Stegun[3] to expand this as a series of Bessel
functions,!
∞α  Xα ∆E
00A(t) = − costJ0+cos(2kωt)
2J2k
~
~ω
~ω
k=1
 ∞

α 
∆E X
0
t
sin((2k + 1)ωt),+ sin
2J2k+1
~
~ω
k=0

  
∆E
α0
≈ − cos
t J0
~
~ω


(3.21)(α0  ~ω).

(3.22)

This time dependence has the effect of placing sidebands on the asymmetry. As long as themodulation index α0 /~ω  1 then the carrier frequency ∆E/~ should dominate and the spectralline will not shift.

3.2.2

Experimental Ramsey sequence

Using the coherent transfer process described in section 2.1.4 we prepare the HfF+ ions ineither the upper or lower doublet of the 3 ∆1 , v = 0, J = 1, F = 3/2 state, and an incoherent equal
mixture of the mF = +3/2 and mF = −3/2 states. To achieve a pure initial state, e.g. |↑i =|mF = 3/2i, we must deplete the population in the undesired fraction of the incoherent mixture.
We apply a series of laser pulses resonant with the 3 ∆1 , J = 1, F = 3/2 →3 Π0 , J = 1, F = 3/2
transition for state purification. We use the same laser that was used earlier in the experimental
sequence to transfer the ions into the F = 3/2 state. The coherent state transfer laser is locked
to a cavity to keep its wavelength stable throughout the transfer process. During the transfer
procedure the laser was detuned from the 3 ∆1 →3 Π0 transition by one cavity FSR to prevent theintermediate state from becoming populated. For depletion the laser must be made resonant withthe transition. We use a microcontroller to ramp the laser frequency one FSR of the cavity suchthat it is resonant with the depletion transition. We use a microcontroller because it allows us toramp the laser frequency faster than the bandwidth of the electronics used to lock it to the cavity,
such that when the ramp is finished a new lock is acquired now 1 FSR away.
The laser pulses are strobed at a frequency synchronous with the rotating electric field such

46
that the laser is only on when the k-vector is nearly parallel or antiparallel to the direction of theelectric field. The laser light is circularly polarized, and by choosing the phase of the strobe suchthat k-vector of the laser beam points along or against the electric field direction, we can apply
either σ + or σ − circular polarization in the frame of our chosen quantization axis. Because weaddress a ∆J = 0 transition, for the case of of σ + (σ − ) polarized light, only ions in the mF = −3/2
(mF = +3/2) have an allowed transition to the 3 Π0 state. Once promoted, the ions quickly
spontaneously decay to various long lived states, with only a fraction returning to the 3 ∆1 , J = 1,
leaving us with a pure state of ions in a single mF = 3/2 or mF = −3/2 in the 3 ∆1 , J = 1
rotational manifold. For clarity we will now only consider without loss of generality the state
preparation that yields ions in upper doublet and the mF = +3/2 state. We will use the shorthand|↑i = |mF = +3/2i , |↓i = |mF = −3/2i, ∆u = ∆ + δ∆, and g u = gF + δg.

3.2.3

Applying π/2 pulses

To apply the π/2 pulse we take advantage of the rotation induced coupling between |mF = +3/2i
and |mF = −3/2i in eq. (3.9),


u
u
∆
1 3g µB B + de Eeff
Hu = 
,
2uu∆−3g µB B − de Eeff

(3.23)

By quickly decreasing the magnitude of the rotating electric bias field Erot , we access a regime
where ∆u  (3g u µB B + de Eeff ). In this regime the eigenstates are no longer pure |↑i and |↓i, but
the symmetric and antisymmetric superpositions
1|ψS i = √ (|↑i + |↓i),
2
1|ψA i = √ (|↑i − |↓i).
2(3.24)(3.25)The state |↑i = √12 (|ψS i + |ψA i) will rotate into √12 (|↑i − i |↓i) in time tπ/2 = π2 ∆~u . After waitingtπ/2 we quickly ramp the rotating electric bias field back up to return to the regime where ∆u
is small. After applying the π/2 pulse, the ions sit undisturbed in the ion trap for a time T .
The phase between |↑i and |↓i will precess at the frequency 3g u µB B + 2de Eeff . We then apply

47
a second π/2 pulse and a second series of depletion laser pulses synchronized with the rotating
electric field. By performing successive experiments where the second depletion strobe is in phasewith the rotating electric field and where the strobe is π out of phase with the rotating electric field
we are able to alternatively measure the final population of ions in mF = + 23 and in mF = − 32 .
We perform several sets of measurements of the asymmetry at different times T to create a Ramsey
fringe. We fit the frequency f of the fringe with a nonlinear least squares algorithm to extract
3g u µB B + 2de Eeff = 2π~f .

3.2.4

Optimizing statistical sensitivity

The statistical uncertainty on our measurement of the Ramsey frequency f is directly proportional to the statistical uncertainty we can ascribe to the energy shift due to de Eeff . In the previous
section we considered an ideal Ramsey fringe with perfect initial state contrast, precise π/2 pulses,
and infinite lifetime. Unfortunately, nature conspires against us, and none of these conditions are
fully realized. A more realistic Ramsey fringe might look like this:A(t) = −C(t/τ ) cos(2π(f t + φ)) + O.

(3.26)

Here we have allowed for a contrast C < 1 that can decay as some function of t in a characteristic
time τ . Also included are an intial phase φ in units of fractional periods (i.e., radians ×2π) and~a dimensionless fringe offset O, which can result from π/2 pulses that are not precisely π2 ∆induration, of if the approximation that ∆  3gF µB B does not strongly hold. To maximize thestatistical sensitivity of our experiment we should device a scheme where we can minimize σf , thestatistical uncertainty of frequency. If we propagate uncertainty through this expression, we see
that

2
σA
= (2πtC(t/τ ) sin(2π(f t + φ)))2 σf2 + σφ2 /t2 +(cos(2π(f t + φ)))2t 2 C(t/τ ) 2
2
σC0 +σO.
C (t/τ ) 2 στ +τC00

(3.27)
This expression is fairly tedious, and what we are looking for is to minimize σf . There are a few
reasonable strategies we can employ that will both simplify eq. (3.27) and minimize σf .:48• Take data at fringe zero crossings if we take data at the peaks of the Ramsey fringe,2(f t + φ) = n, where n is an integer, then the coefficient of σf diverges, and we have nosensitivity to frequency. If instead we choose 2(f t + φ) = n + 21 , then we maximize oursensitivity to f , and the terms for the uncertainty in the contrast, στ and σC0 , drop out.
This can be a bit tricky, because choosing the correct time t requires prior knowledge ofthe frequency, which is the quantity we are trying to measure. We must first make a roughmeasurement of frequency that is precise enough that we can choose t to ensure that nreally is an integer. The uncertainty in n is σn = nσf /f , and we would like σn  1/2. Thisrequires that σf /f  1/2n, so if we are trying to make a measurement 10 periods into a
Ramsey fringe (n = 20), then we must first measure the frequency to much better than 1
part in 40. If we can successfully do this then eq. (3.27) simplifies to
σf2 + σφ2 /t2 =2 − σ2σAO(2πtC(t/τ ) sin(2π(f t + φ)))2.

(3.28)• Fix φ and O.
The parameters φ and O are mostly affected by the preparation of the Ramsey fringe,
the initial state contrast, the purity of the π/2 pulse, etc. We can make a rough initial
measurement of φ and O and hold these values fixed in our nonlinear fitting. This will havethe effect of potentially shifting the value of f that we determine from out fitting routine,
but because we are interested in differential shifts in f , these potential shifts drop out.
Also, we note that at t = 0 we can make an independent measure of φ and O and theiruncertainty where there is no dependence on the frequency of the Ramsey fringe. If weallow φ to float in our nonlinear fitting routine, then σf2 ≈ σφ2 /t2 , which will have the effect√of increasing the uncertainty in f by a factor of 2. With these simplications eq. (3.28)
becomesσf =• Minimize σAσA.
2πtC(t/τ ) sin(2π(f t + φ))(3.29)

49
The uncertainty in frequency is proportional to the uncertainty in the measurement of the√asymmetry. For a measurement that is Poisson distributed, σA ∼ 1/ N , where N is thenumber of ion counts. Generally, to minimize σA we will want to operate with the highest
possible number of experimental counts. Nicholas Hutzler points out in his thesis [38] thatthe asymmetry is not actually a Poisson distributed random variable, and we will consider
that below. The asymmetry A is a dimensionless quantity that is the ratio of two Poisson
statistics and is not itself Poisson distributed, which is why σA is not the typically expected√N.
The asymmetry is the ratio of the difference and the sum of two Poisson distributed variables, N↑ and N↓ . Assuming no other sources of noise, the uncertainty in these will be the
2 = N + bg, where bg is the background counts. If we
Poisson distribution uncertainty, σN↑↑

propagate uncertainties through the asymmetry, we find4N↑2 σ↓2 + N↓2 σ↑2 ,4(N↑ + N↓ )(3.30)

=


4
2
2
(N
+
bg)
,
(N
+
bg)
+
N
N
↑
↓
↓
↑
(N↑ + N↓ )4(3.31)=

4(N↑ + N↓ )bg
4N↑ N↓+,
3(N↑ + N↓ )(N↑ + N↓ )4

2
σA
=

2

2

2(1 + A2 )bg
1 − A2+,(N↑ + N↓ )(N↑ + N↓ )2
r
1p
σA =
1 − A2 + 2(1 + A2 )bg/N ,N=(3.32)(3.33)(3.34)

where N = N↑ + N↓ is the total number of potential signal ion counts. When we are mea√suring at the zero crossing of a fringe where A = 0 the asymmetry uncertainty σA ∼ 1/ N ,pand that this is increased in the presence of background noise by a factor of 1 + 2bg/N .
• Maximize t
The frequency uncertainty σf ∼ 1/t, so we achieve the best precision when we can measure
our fringes out at long times. This is the primary motivation for performing the experimentin an ion trap, where long coherence times are possible.

50• Choose t < τThe experimental reality in the ion trap is that decohering processes do exist, and cause C(t)to trend towards 0 as t → ∞. It’s unclear what the exact functional form of C should be, as
this depends on the exact mechanisms causing the decoherence, but we can experimentally
determine a characteristic time in which the coherence decays, which is referred to as τ .
This aim is directly at odds with the previous point, where taking data at long fringe times
yields the minimal frequency uncertainty. We can determine the optimal time to focus ourdata collection efforts by finding the time that minimizes σf ,p
1 + 2bg/N (t)pσf =.
2πtC(t/τ ) N (t)(3.35)

Additionally, a time dependence has been included in the total number of experimentally
accessible ions, N (t). We perform Ramsey spectroscopy in the 3 ∆1 electronic state of HfF+ ,which we have experimentally determined to have a τ3 = 2.1(1) seconds [50]. Including
this exponential decay in signal further complicates the expression,p
σf =1 + 2bg/(N0 e−t/τ3 )p.
2πtC(t/τ ) N0 e−t/τ3(3.36)

Indeed, it does seem difficult to find a fully optimal time to collect data without knowing
the exact functional form with which the coherence decays. A good starting point is tosimply assume any functional form that monotonically decreases to 0 as t → ∞. One suchfunction is the exponential function, and if we optimize σf , assuming bg  N , then we findthe frequency uncertainty is minimized at
t=

2τ
2τ3
2τ3 τ
=
=
,
2τ3 + τ2 + τ /τ3
1 + 2τ3 /τ

(3.37)

which suggest that if the coherence time τ is much less than the 3 ∆1 lifetime of 2 seconds,
then we should take data when t = τ . On the other hand, if decoherence is not the main

51
limitation, and τ  τ3 , then we can achieve the most frequency sensitivity by choosingt = 2τ3 ≈ 4.2 seconds.

3.2.4.1

Coherence TimeThe factors limiting our coherence time are explored in detail by Kevin Cossel in his PhD
thesis[17], but we will review them here. There are two dominant decoherence mechanisms that
we have observed, decoherence due to long range ion-ion collision via the Coulomb interaction anddecoherence due to spatial inhomogeneity of magnetic fields.
• Ion-ion collisions can occur when ions in the trap get close enough such the electric fieldof the colliding ion is larger than that from that rotating bias field Erot . This causes thepolarization of the molecule to drift off the quantization axis, causing the phase between
mF states to become scrambled. Intuitively this will happen more often when the ionsare at higher temperatures, as the ions will have sufficient kinetic energy to overcome the
Coulomb barrier and come close enough to collide. Additionally it should occur less often
for larger Erot , as more kinetic energy will be required to tilt the quantization axis. The
decoherence rate due to this high energy, hard sphere type of collision is√TΓhot ∼ n,
Erot(3.38)

where n is the ion density. Additionally there is another collision mechanism in the low
energy limit. Glancing collisions of ions can tip the molecular axis off of the quantization
axis slightly. Over the course of a transit of a colliding ion, the molecule axis will tilt away
and return, tracing out a geometric phase. Over the course of many collisions the phasewill randomly walk off, leading to decoherence. Intuitively this will also be suppressed atlower temperatures and higher Erot , but with very different power scaling,√
T T6
Γcold ∼ n
3 .
Erot Erot(3.39)−2.3
Empirically we observe a scaling of decoherence with Erotin fig. 3.3, which implies thatthe temperature is such that we are at an intermediate situation between these two effects.

52
Figure 3.3: Measurements of coherence time when changing ion density n and Erot . The coherencetime in (a) varies linearly with ion density, as in eqs. (3.38) and (3.39) until it reaches a ceiling due
2 .3,
to the decoherence due to field inhomogeneities. The coherence time in (b) scales roughly as Erotpossibly indicating the combination of the effects in eqs. (3.38) and (3.39).

(a)

Coherence time (ms)

103

102

101
102

103
Ion number

Coherence time (ms)

(b)

2

10

1

10

0

10

10

Erot (V/cm)

104

53
• An inhomogeneous magnetic field Brot will result in some ions experiencing one fringe frequency and other ions experiencing a different fringe frequency. Like an ensemble of initiallysynchronized clocks ticking with spread of frequencies ∆f , in some time τ ∼ 1/(2π∆f )the phase between the ions in the ensemble will decohere. The Ramsey fringe frequency
f = 3gF µB Brot , where Brot is applied via the combination of the circular micromotion in
the ion trap and a quadrupole gradient ∇B, Brot = ∇Brrot . Any fractional inhomogeneity
in Brot due to e.g. trapping field inhomogeneities can be expressed as ∆f /f and includedin τ ∼ 1/(2π(∆f /f )f ), and we see that
Γinhomogeneity ∼1.
f(3.40)
2

2

In general this type of mechanism should cause the coherence to decay as e−t /τ [59], but we2

2

have been unable to differentiate this effect by fitting coherence decays to e−t/τ c.f. e−t /τ :both functional forms appear reasonable descriptions. Measurements of coherence time fordifference fringe frequencies show that there is a fractional inhomogeneity of ∆f /f ∼ 0.5%
over the size of the ion cloud.

3.2.4.2

Data rate and duty cycle

So far we have only discussed the frequency uncertainty of a single run , or “shot” of theexperiment. To achieve a sufficiently high number of counts, N , it is important to run many
experimental shots in series, potentially hundreds of hours of data comprising millions of ion counts.
As such, it is important to not only consider the single shot frequency sensitivity but also thefrequency sensitivity as a function of experimenter’s time.
We will use eq. (3.36) and assume without loss of generality the limiting case of no backgroundcounts. If we perform several shots of the experiment in series, we will acquire both backgroundcounts and signal counts, but the ratio bg/N will remain unchanged, and the numerator in eq. (3.36)
can be normalized to 1 and safely included later. Considering the case of running the experiment
for a sufficiently long integration time, Tint , then the total number of ion counts we expect to see is

54
N = Ṅ Tint , where Ṅ is the count rate of the experiment and Ṅ = Nshot /Tshot . The quantity Nshot
is the number of ion counts in a single shot of the experiment, or N0 e−t/τ3 , and Tshot is the amountof time between the start of one experimental shot and the next. The time of an experiment shot
should contain t, the amount of time we spend performing the Ramsey spectroscopy, but should
also include some measure of dead time. This experiment dead time has several sources, but they
are primarily the time spent in initial state preparation (≈ 150 ms), and the time spent in state
readout (≈ 250 ms). Additionally, because the ionization lasers used to create HfF+ operate at
10 Hz, the experiment is triggered off this 10 Hz signal. This means there is the potential to be
waiting up to 100 ms between experimental sequences if the time it takes to perform an experimentslightly exceeds an integer multiple of 100 ms. On average this contributes to a dead time of ≈ 50
ms. Adding in the contribution of running the experiment for many runs into eq. (3.36) we find√t + tdeadp.
(3.41)√
2πtC(−t/τ ) N0 e−t/τ3 Tint√
The first point to note is the inclusion of a new factor of t in the numerator of the expression.
σf =There are two limiting cases to consider. The first is when tdead  t, when the experiment isrelatively unoptimized and there is much dead time between Ramsey sequences. This case removes√
the extra t dependence from σf and is now the same as above in eq. (3.36), where the optimal
t = τ . The second limiting case is when tdead  t, and in this case we find that the new optimal
time to take data at is
t=

τ3 τ
,
2τ3 + τ(3.42)

which is a full factor of 2 shorter than eq. (3.37). In practice the experiment operates under thisregime, where τ3 = 2.1(1)s, τ ≈ 1s, and tdead ≈ 0.45s, and the optimal t ≈ 0.58s, assuming exponential decay in coherence. It is worth pointing out that a precise value for t does not really matter,as σf is very flat - 10% variations in t about the maximum only contribute to a 0.3% variationin σf . Moreover, the minimum value of σf does not have a strong dependence on the functional
form of the decoherence - trying a variety of monotonically decreasing functions yield about thesame minimum, as seen in fig. 3.4. If we insert some realistic experimental parameters to estimate

55p<f = Ti(mHz/hour1=2 )Figure 3.4: In this figure we have plotted examples of the frequency uncertaintyeq. (3.36) for arange of parameters. The default parameters are dead time tdead = 0.45 s, coherence time τ = 1.0
s, 3 ∆1 radiative lifetime τ3 = 2.1 s, initial contrast C0 = 1, and N0 = 1 intial counts per shot. The
defeault functional form used to describe the decay of coherence it C(t) = exp(−t/τ ).

50= = 0.5 s= = 1.0 s= = 2.0 s

40
30
20
10
0

0

0.5

1

1.5

2

2.5

3

p
<f = Ti(mHz/hour1=2 )

t (s)
50
40
30
20

t dead = 0.5 s
t dead = 1.0 s
t dead = 2.0 s

10
0

0

0.5

1

1.5

2

2.5

3

p
<f = Ti(mHz/hour1=2 )

t (s)
50
C(t) = exp(-t/=)C(t) = exp(-t 2/=2)
C(t) = (t/=)-1
C(t) = (t/=)-2

40
30
20
10
0

0

0.5

1

1.5

t (s)

2

2.5

3

56
the frequency uncertainty, we find that for N = 30 counts per shot, with 3 background counts,√
an initial contrast C(0) = 0.3, and a coherence time of τ = 1 second, that σf ≈ 6.3 mHz/ hour.
√With the effective electric field Eeff = 23.3 × 109 V/cm this is about 5.6 × 10−28 e·cm/ hour. A√
typical run of experiments with these parameters achieves a sensitivity of ∼ 22mHz/ hour. Thisis a factor of 3.5 worse than our expectation. The primary reason is that we spent a portion of ourduty cycle acquiring Ramsey fringe data at short times to measure the initial phase φ. This both
impacts out duty cycle and also the statistical sensitivity by introducing competing uncertainty
quadratures σφ and σO . Together these effects decrease our frequency sensitivty by a factor of 2.
The additional discrepancy is due to the counts on our ion detector are a factor of 1.5 above theshot noise limit.

Chapter 4

Systematic uncertainty

4.1

Overview
What we have primarily discussed up to this point is the statistical sensitivity of the ex-periment. All of this sensitivity is pointless unless we can isolate the effect of the EDM. As wediscussed in the previous chapter, when we apply rotating electric and magnetic fields to the HfF+
molecule it is polarized along the electric field direction, and in the rotating frame we can consider
the following effective two state Hamiltonian∆ ± δ∆
1 3(gF ± δg)µB (Brot + δB) + 3αωrot ± de Eeff
H u/l = ,2
∆ ± δ∆−3(gF ± δg)µB (Brot + δB) − 3αωrot ∓ de Eeff
(4.1)
which describes the |mF = ±3/2i states in the upper and lower doublets. The terms δg and δ∆
are included to describe the differences in g-factor and rotation induced coupling between the two
doublets, δB describes a residual magnetic field, and 3αωrot describes the perturbation due to theelectric field making a small angle α with the plane of rotation. In the limit that 3gF µB Brot  ∆then we observe a Ramsey fringe frequency on the 3 ∆1 , J = 1, F = 3/2, mF = ±3/2 transition thatis approximately equal tof = 3gF µB (Brot + δB) + de Eeff .

(4.2)

We will develop the various other terms that affect this frequency and gradually add more realism.
In principle we could operate with Brot = 0, so that the entire measured effect is due to de Eeff .
There are several problems with this. The first is that the frequency shift due to even a very

58
small magnetic field B will overwhelm de Eeff . An EDM that is 10−28 e cm contributes a frequency
shift of 1.1 mHz, which will be overwhelmed by a rotating magnetic field as small as 100 nGauss.
The second is that unless 3gF µB Brot  ∆, then the |mF = ±3/2i states are highly mixed and aRamsey experiment is impossible. To try to cancel out the effect due to Brot , we can make two
measurements, one at Brot + δB, and one at −Brot + δB, where the sign change in Brot corresponds
to having Brot point in the opposite direction with respect to the electric field. We only measure theabsolute value of frequency, so switching the sign of Brot has the effect of changing the relative signof δB in the absolute value frequency measurement. We can consider the sum and the difference of
these two measurements,
1
f 0 = (f (+B) + f (−B)) = 3gF µB Brot ,2
1
f B = (f (+B) − f (−B)) = 3gF µB δB + 2de Eeff .
2

(4.3)
(4.4)

Here we introduce the notation of the frequency components f 0 and f B , which are the linearly
independent contributions to the frequency that are even and odd under the reversal of the rotating
magnetic field. We can see that by calculating these two different linear combinations of the twodata with a switched magnetic field, we divide the signal into two channels, one that is the Zeeman
contribution to the Ramsey frequency, and one that is the electron EDM and the Zeeman effect, but
now suppressed by a factor of δB/Brot . It is important to note that these are sums and differencesof frequency measurements, and the propagation of uncertainty works out such that the uncertaintyof f 0 and f B is no more than if we had taken measurements of only f (+B) or f (−B) during thesame duration of time. Partitioning the data in this fashion yields clearer information. We havecleanly divided our data into a portion that is purely due to Brot and a portion that is due to the
EDM and some residual part of Brot . By performing the same sort of switch between positive andnegative values of other parameters in the experiment we can continue to peel off other important
quantities that are useful for analysis and diagnostics, e.g. ∆, leaving a core that is the electron
EDM contribution and suppressed systematic effects. Generally, the more switches are added, themore suppression we can achieve on some of these systematics. Additionally, by measuring the

59diagnostics in the other channels, the residual systematic contributions in the EDM channel can becorrected for, see section 5.1.4. For the single switch of Brot that we have considered, the remaining
systematic in the EDM channel is 3gF µB Brot (δB/Brot ). Ideally (δB/Brot ) will be zero, but reality
is not so kind. We will not be able to perfectly reverse the rotating magnetic field, and this willaccount for some (δB/Brot ), perhaps ∼ 10−3 . Additionally there will be ambient nonreversing
magnetic fields present in the experiment due to the circular micromotion of the ions in the trap
coupled to ambient magnetic field gradients in the lab due to ferromagnetic materials. Finally,
rotating electric field is created by sinusoidal voltages on the trap electrodes. Any capacitance inthe electrodes and cables leading to them can create currents running in the electrodes which will
have the effect at the center of the trap of a magnetic field that rotates in synchrony with Brot ,but does not change sign when Brot is switched. These contributions are all summarized in δB. In
order to further suppress (δB/Brot ) we can take advantage of the two Λ-doublets. Because they
have very nearly the same g-factor they can be considered like a co-magnetometer that is internal tothe molecule[20]. This comagnetometer can be used to measure the offending δB and compensate
for it. We can exploit this opportunity by taking sequential measurements where the preparationis switched between transferring populating into the upper and into the lower Λ-doublets. We
refer to this switch with superscript D, and the switch Brot with superscript B, and by taking four
frequency measurements f (±B, ±D) we construct the four linearly independent combinations,1
f 0 = (f (+B, +D) + f (−B, +D) + f (+B, −D) + f (−B, −D)) ≈ 3gF µB Brot ,
(4.5)
4


1
δB
f B = (f (+B, +D) − f (−B, +D) + f (+B, −D) − f (−B, −D)) ≈ 3gF µB Brot
,
(4.6)
4
Brot
1
∆δ∆
f D = (f (+B, +D) + f (−B, +D) − f (+B, −D) − f (−B, −D)) ≈ 3gF µB Brot,
4
(3gF µB Brot )2(4.7)
1
f BD = (f (+B, +D) − f (−B, +D) − f (+B, −D) + f (−B, −D))
4





δB
δg
∆δ∆
fD
B δg
≈ 3gF µB Brot−
+ 2dE Eeff ≈ f− 0 + 2dE Eeff .
Brot
gF(3gF µB Brot )2
gF
f

(4.8)

60
Combining the frequency measurements has two advantages. The first is that we may take anappropriate combination of the four terms f 0 , f B , f D , f BD , and in conjunction with a careful characterization of δg/g, extract de Eeff while removing all first order effects. The second advantage isthat these frequency linear combinations are a good tool for discovering systematic errors. Broadly
speaking, a systematic error is any effect that could be confused for the EDM, which could be anyother term that appears in f BD . For every additional switched parameter we see that systematiceffects are suppressed, and in some other linear combination the same systematic effect appears in
an unsuppressed way. Consider the linear combinations f B and f BD . The systematic effect of non


δg
∆δ∆
reversing magnetic fields appears as a systematic in f BD as 3gF µB Brot BδB−,
2gF(3gF µB Brot )rotbut a nearly identical term appears in f B as 3gF µB Brot (δB/Brot ). Because gδgF ≈ 10−3 we can see
the effect due to any unintentionally applied non-reversing magnetic fields much more easily in f B
than in f BD . By deliberately tuning various parameters of the experiment to extreme operating
conditions (e.g. large stray magnetic fields, detuned π/2 pulses, etc.) we can monitor f B and theother linear combinations for effects that might appear in the f BD , or “EDM channel”, but in aless detectable way.
We introduce a general notation of keeping track of these various systematic terms, following
the example of Sandars[34], Hinds[37], and ACME[13]. For a set of different possible switchedparameters S = {B, D, . . . } with length n(S), there exist 2n possible different states of the experiment, with 2n possible frequency measurements, f (±B, ±D, ± . . . ). This yields 2n possible
different linear combinations of frequency measurements,
fR =

1 X
(−1)n(R∪s) f (−s, S \ s),
2n (S)(4.9)

s⊆S

where R is the set of switched parameters we are taking the antisymmetric difference over, andwe sum over all 2n possible subsets s of S, and f (−s, S \ s) is a frequency measurement withthe switched parameters in subset s in the negative configuration, and the switched parameters in
subset complement S \ s in the positive configuration. For example, in the above discussion we
considered S = {B, D}, where magnetic field and doublet were switched parameters. The linear

61
combination corresponding to the non-reversing magnetic field channel
1f B = (f (+B, +D) − f (−B, +D) + f (+B, −D) − f (−B, −D)),4(4.10)

where f (+B, +D) means the Ramsey frequency measurement made with Brot pointing in the same
direction as Erot and in the upper doublet.
In addition to switching the magnetic field direction (B), and the Λ-doublet (D), Anotherimportant parameter to switch is the direction of rotation of the rotating electric field (R). There
are potential systematic effects due to Berry’s phase that might arise because we are deliberately
rotating our quantization axis. By performing a rotation switch we can characterize and suppress
these systematic effects. The de Eeff term does not change sign under rotation direction, and so the
EDM appears in the f BD channel and the f BDR channel. Adding in this third switch to the set ofswitches we can write down eight linearly independent combinations of the frequency, and performthe series expansions of the small parameters δB, δg, δ∆, ∆, and α,∆2δ∆2++ . . . .,2(3µB gF Brot ) 2(3µB gF Brot
2 !!
2 + δ∆2∆3αωrot
f B = 3gF µB δB 1 −
1+3+ ....
2(3gF µB Brot )2
3gF µB Brot
f 0 = 3µB gF Brot +

∆2
δg
∆δ∆
+ 3δgµB Brot −+ . . . .,3gF µB Brot
2(3gF µB Brot ) gF
27∆2 δ∆2 δB
3(∆2 + δ∆2 ) δB
− αωrot+ . . . .,f R = αωrot
2(3gF µB Brot ) Brot
(3gF µB Brot )4 Brot
∆2 + δ∆29∆2 δ∆2
f BR = 3αωrot − 3αωrot+
3αω+ ....
rot
2(3gF µB Brot )24(3gF µB Brot )4
3gF µB δBδ∆
2de Eeff ∆2
f DR = 3αωrot+
3αω+ ....
rot(3gF µB Brot )3(3gF µB Brot )3∆δ∆ − (∆2 + δ∆2 )(δg/gF )f BDR = − 3αωrot+ ....
(3gF µB Brot )2

2 !!
δg
∆δ∆
3αω
rot
f BD = 3gF µB δB−
1+3+ 2de Eeff + . . . .
gF(3gF µB Brot )2
3gF µB Brot
fD =(4.11)(4.12)(4.13)(4.14)(4.15)(4.16)(4.17)(4.18)We have expanded the expressions for the frequency combinations out to second order in all of
the parameters included in eq. (4.1) and truncated the series at a level where the contribution for
the remaining terms, for a realistic choice of parameters, falls below 10−3 relative to the leading62
term, or 1 µHz, whichever is smaller. The expansions in eq. (4.11) through eq. (4.18) will form
the basis of our understanding of systematic effects in the following sections. They were derived
from eq. (4.1), and any terms not included in the original effective two-state Hamiltonian (e.g.
oscillating magnetic fields transverse to the internuclear axis), will not appear in these equations.
However, eq. (4.1) represents a reasonable basis of effects to consider, and any additional effects
can be considered as corrections to the original Hamiltonian, and included in the expansions in
eq. (4.11) through eq. (4.18) as similar corrections.

4.1.1

Strategy

At this point we should discuss strategy. Our objective is to measure f BD with as muchsensitivity as possible, and also to be able to place good upper bounds on terms that might accompany de Eeff in f BD . The expression eq. (4.18) includes terms that, under standard operating
parameters of the experiment, extend down to 10−31 e · cm. There are terms included that arelarger than de Eeff , and they must be accounted for. The larger worry is that some small anomaly
will also enter into f BD in a hitherto unanticipated way at the < 10−28 e·cm level. Systematics
that enter at the same level as the statistical sensitivity of the experiment after a 100 hour run will
be incredibly difficult to differentiate from an EDM. The way to smoke them out and head themoff is to operate the experiment in a deliberate way to increase the size of the systematic effects.
For example, typically we will operate the experiment between pairs of Helmholtz coils designed
to null out Earth’s magnetic field. We can use these same coils to apply a 5 Gauss field, which isan order of magnitude larger than Earth’s field. Additionally, we have the other frequency linear
combinations besides f BD to aid us. If applying a large magnetic field in the lab frame has someeffect that looks like a non-reversing magnetic field in the rotating frame, then it will show up in f B
anywhere from 100 to 1000 times larger, depending on the exact set of experimental parameters.
We have a good understanding of the leading order effects in eq. (4.11) through eq. (4.18) due tothe terms that are included in the effective two state Hamiltonian, eq. (4.1). Any effect that is
not already taken into account in the effective two state Hamiltonian will be much more visible in63
the other frequency linear combinations. Moreover, there are data channels at our disposal otherthan frequency. Recalling eq. (3.26), there are up to four other parameters that can be extracted
from a Ramsey fringe, the initial phase φ, the initial contrast C0 , the offset o, and the coherencetime τ . The most statistically sensitive times to take Ramsey fringe data are typically at t < τ ,and as such we don’t typically get good measurements of τ , but for every Ramsey fringe frequency
measurement there is an accompanying φ, C, and o measurement. These data channels can also be
separated into different linearly independent components as was done with frequency above, which
can be helpful in diagnosing systematics that affect the initial state preparation or π/2 pulses.
These quantities are difficult to predict analytically, so we treat them numerically. Generally allof the linear combinations of φ, C, and o will be zero, with the exception of the total sum linear
combination. When examining systematics we will take special note of any time these channels
deviate from zero.

4.2

Understood systematic effects
There are several effects that we can immediately predict from eq. (4.18), in particular theeffects due to differential g-factor δg, and the effect of rotational coupling ∆, and δ∆. Using thestrategies outlined above we can characterise and measure these effects.

4.2.1

Differential g-factor

The first systematic effect we will consider is the term in f BD , 3gF µB δB(δg/gF ). We cancorrect for this term if the dominant contribution to f B is 3gF µB δB. This is a good approximation
if the following conditions are met:• ∆2 + δ∆2  2(f 0 )2 ,• 3αωrot  f 0 ,where we have substituted the leading order contribution to f 0 = 3gF µB Brot into eq. (4.12). In
order to achieve good contrast π/2 pulses we already operate with f 0 > ∆ > δ∆. The Berry’s phase64
α due to things like the axial electric field in the trap will oscillate at the axial trap frequency ωz , andso the contribution to eq. (4.1) will be a phase modulation at ωz , as discussed in section 3.2.1.1, andnot a frequency shift. As long as α oscillates around a central value of zero, then the contribution
to the frequency linear combinations eq. (4.11) through eq. (4.18) will be zero as well. Therefore
measurements of f B can be used to provide a good estimate of 3gF µB δB to use in correcting
f BD . The other piece of information we need for this correction of 3gF µB δB(δg/gF ) is δg/gF ,the fractional differential g-factor between the upper and lower doublets. This differential g-factor
arises from two different mechanisms. The first is mixing from 3 ∆1 , J = 2 states into the J = 1
states. Because the upper Λ-doublet is closer to the J = 2 than the lower doublet by 2dmf Erot ,we expect a differential effect on the order of the ratio of the rotational splitting 4Be to the J = 1
Stark splitting dmf Erot . The exact analytical expression can be found in [42],dmf Erot 1/γF F
[F, F 0 , J, J 0 ]2 
Be (J + 1) ΩJ =2 F 0 ={3/2,5/2}
−mF

X
δg
=
gF
0

X

1

F0

2 

  J 1
 
0 mF
−Ω 0

2
2 


0
0


 F J I,Ω  J F 1J0

(4.19)=

9dmf Erot.
40Be(4.20)

This is about 10−3 for a 24 V/cm electric field. The other contribution to δg/gF comes from Stark
mixing within the J = 1 states. At large electric fields the mF = ±1/2 states in the F = 3/2
hyperfine level mix with the mF = ±1/2 states in the F = 1/2 level. Because the upper doublet
states are closer to the F = 1/2 states than the lower doublet the perturbation is more pronounced,
and via rotation induced coupling to the mF = ±3/2 states there is a differential g-factor. We
2 /(d f E ))2 (d f E /E
expect this to scale like (ωrot
m
rot
m
rot
HF ). Calculating this difference out to threeorders in perturbation theory we find that2
dmf Erot
δg
9
ωrot
=
,
gF
2 dmf Erot
EHF

(4.21)

which for the parameters in table A.1 is about 1.4 × 10−4 . We expect the combined effect to be on
the order of 10−3 for our typical electric field magnitude and frequency. As we will discuss below, we

65
can experimentally measure this difference by looking at the component of Ramsey fringe frequency
that changes sign with the doublet switch. We measure δg /gF = 0.0008(1). Additionally, numerical
diagonalization of a 32 state Hamiltonian comprising the 3 ∆1 J = 1 and J = 2 predicts a combined
δg/gF that is 0.000993 for the parameters in table A.1, see fig. 4.1.

4.2.2

Effective differential g-factor

There is another systematic term that looks similar to a differential g-factor, but arises froma different mechanism. This is the term f BD = (∆δ∆/3gF µB Brot )(δB/Brot ). This arises from thedifference in the rotation induced coupling between mF = ±3/2 states due to the hyperfine mixingof the mF = ±1/2 states, quantified in the term δ∆. This term is distinct from the δg/gF term
above in that it will also produce a systematic dependence of f BD on f B for effects other thanthe Zeeman interaction, such as effects arising from geometric phase. For example, a geometricphase term that didn’t change sign under switching rotation direction, mF α|ωrot |, could yield asystematic in f B and f BD that would look likef B =3α|ωrot |,f BD =3α|ωrot |∆δ∆.
(3gF µB Brot )2(4.22)(4.23)

Effects that might look like this could be imperfect reversals in rotation, such as components fo Erotthat are different for the two rotation directions. This can also potentially be used as a method of
distinguishing the effect that arises from the Zeeman interaction (e.g. non-reversing magnetic fields)from something else. Fortunately, this so-called “effective” δg has a strong inverse dependence on
the Ramsey fringe frequency, and at a frequency of 70 Hz it is comparable to the Zeeman differential
g-factor. This presents an interesting trade-off between statistical sensitivity and systematic error
susceptibility. Spatial inhomogeneity of Erot and Brot in the trap causes a decoherence rate thatscales with frequency, effectively causing decoherence after a fixed number of periods. For maximum
coherence time we are compelled to operate at low Ramsey fringe frequencies. However, systematiceffects due to higher order terms such as eq. (4.23) have inverse dependence on frequency. The

66

Figure 4.1: Measurement of δg/gF and ∆δ∆ by varying the applied ∇B to vary the average Ramsey
fringe frequency across a range of about 20 Hz to 300 Hz and comparing the component of thefrequency that is odd under change of Λ-doublet (f D ) with the average frequency (f 0 ). This wasdone under the conditions table A.1. The effect in the f D channel is the combination of several
effects, from eq. (4.19), eq. (4.21), and eq. (4.24). Numerical theory on a 32 state Hamiltonian
comprising the 3 ∆1 J = 1 and J = 2 states is in good agreement with the data.

0
theory
fit
data

-0.05

f D (Hz)

-0.1
-0.15
-0.2
fD = (/g/gF) f0 + " /"/f 0

-0.25
/g/gF
"/"/(Hz)

-0.3
-0.35

0

50

2

fit

theory-0.0008(1)-0.000993-5.3(3)

-5.27

100

150

f 0 (Hz)

200

250

300

67
ramifications of this trade-off will be explored in chapter 5.
In total, for effects arising from the Zeeman interaction, the “effective” g-factor is
δgeff
δg
∆δ∆
=
− 0 2.
gF
gF(f )(4.24)To characterize this term for a particular ωrot and Erot , we can choose parameters from table A.1
and take several blocks of Ramsey fringes with varying fringe frequencies, and compare f D to f 0 ,as in fig. 4.1, noting that to leading order, f D /f 0 equals the expression in eq. (4.24). This allows
us to extract measurements of δg/gF = −0.0008(1) and the product ∆δ∆ = −5.3(3) (Hz)2 . Thisagrees quite well with a numerical calculation of δg/gF = −0.000993 and ∆δ∆ = −5.27 (Hz)2 , azero free parameter calculation that only relies on measured values of gF , EHF , dmf , and ωrot .

4.2.3

Non reversing magnetic field

From the previous section it should be clear that a non reversing magnetic field, δB, cancause a systematic error in the EDM channel f BD . Because the frequency linear combination f 0
averages over both directions of the applied rotating magnetic field no residue of δB will appear
there, but instead will appear in f B , the component of the frequency that is odd under magnetic
field reversal. We will refer to f B as the “non-reversing magnetic field” channel, as non-reversing
magnetic fields are the dominant term in f B , see eq. (4.12), and belowfB
= 3gF µB Brot

δB
Brot


+ ....

(4.25)The non-reversing magnetic field appears in f BD suppressed by a factor of δgeff /gF ,fBD= 2de Eeff + 3gF µB Brot

δB
Brot



δg
∆δ∆
−
gF
(3gF µB Brot )2≈ 2de Eeff + f B

δgef f,gF(4.26)

and we see that f BD should have a linear relationship with f B . We can test the validity of thisrelationship by intentionally creating a large δB by deliberately making our magnetic field reversals
bad. We can quantify this effect with f B , and we can observe the systematic shift in f BD , the EDM
channel. Because we can easily vary f B by tens of Hz, we can expect to see a shift in the EDM
channel on the order of hundreds of mHz, as in fig. 4.2. Several other experiments have confirmed

68that de < 10−27 e cm, which places an upper bound on de Eeff being less than about 10 mHz, so thisterm can be neglected in comparison to the > 100 mHz shifts observed.
There are several sources of non-reversing magnetic field. The most obvious is any incomplete
magnetic field reversal on our part. We apply Brot by applying current to a pair of coils in an anti~ = ∇B(2z ẑ−xx̂+−y ŷ) centered
Helmholtz configuration to generate a quadrupole gradient field ∇B
at the trap center. Due to the circular motion of the ions in the trap this produces the effect in
the lab frame of reference of a rotating magnetic field Brot that is equal to rrot ∇B, where rrot
is the radius of the circular micromotion due to Erot . We choose our quantization axis to point
along the electric field, which points from the ion to the center of the ion’s circular orbit, and we
define +∇B and +Brot as pointing along the electric field, and −∇B and −Brot as pointing against
the electric field. We switch the direction of Brot by changing the sign of ∇B, which is done byreversing the direction of the current through the quadrupole coils. The current is applied using
a programmable bipolar current source controlled by an external voltage from a digital to analog
converter. In addition to the current source internal monitoring we use an external Hall probe tomeasure the applied current. Offsets and errors in any of the monitoring devices may contributeto a δB, as well as any offsets in current or the position of the coils.
An additional source of δB is due to static gradients. Because Brot is created by applied∇B, any static quadrupole gradient ∇Bstatic can contribute to a δB = ∇Bstatic rrot . Higher ordermultipoles can contribute to spatial inhomogeneity of δB. A quick glance at the experiment reveals
several potential sources of ∇Bstatic . The ion trap is inside a stainless steel vacuum chamber
constructed of several nominally nonmagnetic stainless steel alloy tubes that have been welded
together. The ion trap itself is made of two large aluminum end cap electrodes and 6 stainless steel
electrodes that have been welded to steel rods. Stainless steel has the potential to be magnetizedand produce a static gradient, and in particular stainless steel welds have the potential to become
ferromagnetic because of the introduction of filler material into the weld that is of a different crystal
structure. Finally, the entire experiment sits on top of a stainless steel optical table that has become
permanently magnetized from years of abuse.

69

Figure 4.2: A non-reversing magnetic field is an EDM mimicking systematic effect. From eq. (4.12)
and eq. (4.18) we see that a non-reversing magnetic field δB will appear in the f BD frequency linear
combination with a δgeff coefficient. We also observe that δB appears in the non-reversing magnetic
field linear combination, f B , with the much larger gF coefficient. This correlation is shown below,
where the application of a large second order magnetic field gradient in the axial trap direction
coupled with displacement along that direction resulted in a large effective non-reversing magnetic
field, appearing in the the f B channel as a ±15 Hz shift. In the EDM channel, f BD , this effect issuppressed to the ±400 mHz level. For this particular δgeff , experimentally shimming f B to ±30mHz will result in an uncorrected systematic shift of 800µHz in the EDM channel. As discussed in
section 5.1.4, this contribution may be characterized and subtracted away. Additionally, operatingwith smaller ∆δ∆, and larger Ramsey fringe frequency, 3gF µB Brot , can reduce δgeff and suppress
this effect even more.

0.5

f BD (Hz) [EDM channel]

0.4
fBD = f B (/g/gF + " /"/(3 g F 7 B B rot ))

0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-20

B

-10

0

10

f (Hz) [non-reversing B field channel]

20

70
For the standard values of experiment parameters in table A.1, a static gradient at the centerof the ion trap of as little as 2 mGauss/cm can contribute a 630 mHz shift in f B , which translates toa 0.5 mHz shift in f BD taking only δg/gF into account, which is a 5×10−29 e·cm effect. Welds on thevacuum chamber typically have very high order gradients that are 10 − 20 mGauss magnetic field 1
mm away, which quickly drops to the µGauss level 1 cm away. The optical table magnetic field varies
considerably over the length of the table, with local “hot spots”, corresponding to welds on the edge
and at various support points on the interior. Several cm away from the table the gradient becomesmoother, and the field component in the z direction is about (18(z/cm) − 0.6(z/cm)2 )mGauss.
The center of the ion trap is about 28 cm from the table, which corresponds to a 1.2 mGauss/cm
gradient. We also measured the gradient at the center of the trapping region by scanning theposition of a magneto-resistive magnetometer device while the chamber was vented to atmosphere.
These measurements show the background gradient to be on the order of ∼ 2 mGauss/cm. At thesame time we discovered the presence of a large static gradient due to a cold cathode type vacuum
gauge. The cold cathode works by circulating an electron plasma in a magnetic field on the orderof 1000 Gauss. Even though the gauge was more than a a meter away from the center of the trap,
the gradient experienced there was > 5 mGauss/cm. This was an easy problem to solve, we simply
had to remove the gauge and replace it with a hot filament gauge that did not produce a large
magnetic field.
These static gradients pose two problems. The first is that they have the effect of a nonreversing magnetic field, which is an EDM mimicking systematic. The 1.2 mGauss/cm gradient of
the stainless steel table produces a 3 × 10−29 e · cm systematic. Fortunately we can use f B as arealtime diagnostic of the non-reversing magnetic field, and deliberately apply an oppositely signed
quadrupole gradient to cancel this effect out, as in fig. 4.3. We can easily apply a gradient to shimf B to less than 50 mHz, which is 40µHz in f BD , a systematic ∼ 3 × 10−30 e·cm. After removing
the cold cathode gauge we saw a change in the required magnetic field gradient required to yield anominally zero f B of 4.9 mGauss/cm, or 2.9 Hz.
We must be a bit cautious with this technique, as it relies on us having exact knowledge of the

71

Figure 4.3: A non-reversing magnetic field can be measured by f B , the non-reversing magnetic
frequency combination. The rotating frame magnetic field that the ions experience is the result of
the combination effect of an applied magnetic field gradient and the circular ion micromotion. The
rotating frame magnetic field direction is reversed by applying a lab frame magnetic field gradientof the opposite sign, effectively done by reversing the direction of current through a pair of coils
in an anti-Helmholtz configuration. By deliberately performing incomplete reversals of the current
we are able to apply a non-reversing gradient, ∇B. With the freedom to apply a ∇B of our choice,
we are able to shim the measured f B to within 30 mHz of zero.

2

1.5

fB = ,3g F7 B rBrrot + offset
offset = 1.22(3)f B (Hz), = 1.03(3)

1

0.5

0

-0.5
-2.5

-2

-1.5

-1

-0.5

0

0.5

applied non-reversing rB (mgauss/cm)

1

72
relationship between f B and f BD . This works because we know that a non-reversing magnetic field
can be described by δB in eq. (4.1), and the correlation between the effects in f B and f BD is welldescribed by the effective difference in g-factor, in eq. (4.24). However, an effect due to somethingother than the Zeeman interaction, if it is odd under reversal of magnetic field, will appear in f B
and f BD , but with a different correlation between the two channels than a non-reversing magnetic
field, as in eq. (4.22). If we add in a new non-reversing magnetic field to null out the signal in f B ,we could inadvertently introduce a new systematic on the order of ∼ (δg/gF )f B .
The second problem with the static gradients is that perhaps they aren’t so static. Thismight be caused by changing parameters in the lab. Temperature changes could potentially cause
the magnetization of the optical table and the welds to drift. If they fluctuate over fast time scales
this will decrease the coherence time of the Ramsey fringes by adding noise onto the frequency.
What is more probable is that they fluctuate over longer timescales. We observed fluctuationsof f B with a standard deviation of σ = 2.8 Hz over the span of several days. After removing
the cold cathode gauge causing the large stray gradient we saw a reduction in these fluctuations
to a standard deviation of σ = 0.74 Hz. The problem arises when the gradients fluctuate overa timescale comparable to the time it takes the experiment to acquire a Ramsey fringe. In the
proposed scheme in section 3.2.4, maximal sensitivity was assumed by the ability to be able toknow the frequency well enough to only take data points on the zero crossings of the Ramsey
fringes. This required two measurements: one rough and fast measurement to get an accurateenough idea of the frequency to allow us to dial in the times between the π/2 pulses for a second,
longer, high sensitivity measurement. If on a similar timescale the background gradients drift
enough to cause the frequency to change by a quarter of a period, then instead of collecting dataon the zero crossing of a fringe, with maximal sensitivity, we will collect data on the peak of afringe, with minimal sensitivity. A one quarter fringe slip at 500 ms corresponds to a 0.5 Hz, or
0.8 mGauss/cm drift. Unfortunately this is about at the same level of our measured day to day
variations of f B . In order to reliably park the experiment on the side of a fringe we require ameasurement of f B that is much faster than a Ramsey fringe measurement time.

73
Our proposed scheme to measure the background gradients is a cluster of small, inexpensive,
magnetoresistive chip scale magnetometers. The HMC5883l is a 3 axis magnetometer made by
Honeywell that can measure fields up to ±2 Gauss with a 4 mGauss resolution. It has an integrated
12 bit ADC, and a small 3mmx3mm package. Two of these sensors placed 20 cm apart should be
able to measure a gradient with a sensitivity of 0.2 mGauss/cm. Measurements can be made very
quickly, in about 1.7s, and should be able to provide information of the background gradients thatis sensitive enough for us to apply a corrective gradient in real time. Doing so will allow us tofeed forward our measurements to stay locked to the side of a Ramsey fringe. To this end we haveconstructed an array of 8 magnetometers positioned around the experiment.
A benchmark test of this magnetometer array is presented in fig. 4.4. In this test the six
magnetometers placed in the region around the equatorial plane of the ion trap chamber wereused. The two other magnetometers, placed above and below the ion trap, had measurements thatskewed far from the expected gradients. We determined that this was due to high order magnetic
field gradients emanating from the magnetized stainless steel optical table the experiment chamber
was mounted to, so we removed these two magnetometers from our data set. The remaining
magnetometers can predict 80% of the variation in the measured non-reversing magnetic fieldchannel, but still leave much to be desired in terms of performance. Some of the problems withthis system that might account for the unexplained variation are:(1) The magnetometers were small evaluation boards purchased from Adafruit Industries for$10 a piece. The actual magnetometer chip is designed to be used in portable electronic
applications as as digital compass with ∼ 2◦ heading accuracy. The evaluation boards weremeant to be used by hobbyists in applications that typically only required roughly identifying magnetic north, and are provided without any calibration. While we did individuallycalibrate each magnetometer against a reference Helmholtz coil, we found variations in theirreadings of earth’s magnetic field and we found variations in their readings when placed
inside a zero-Gauss chamber. Closer scrutiny of the evaluation boards led to the discovery

74

Figure 4.4: We make measurements of the lab frame residual magnetic field using an array of 6
chip scale magnetometers placed roughly around the equatorial plane of the ion trap chamber.
Each device measures the magnetic field along three axes, providing in total 24 magnetic fieldmeasurements at 6 distinct locations. This is globally fit to a static magnetic field and 5 linearly
independent first order gradients. From the quadrupole gradient we can infer an effective nonreversing magnetic field, and its effect on the non-reversing magnetic field channel, f B , plottedB
below on the horizontal axis as fmagnetometer. This is compared to the actual f B channel measured
B
via Ramsey spectroscopy. The fmagnetometer
inferred from the magnetometer array can explain 0.8
Bof the variation measured in f . This is good as a proof of principle of the ability to measure andpredict the effect of a non-reversing magnetic field, but to be able to actively cancel out fluctuating
non-reversing magnetic fields in real time before a Ramsey measurement is made requires moreaccuracy. Specific deficiencies of the magnetometer array and their solutions are discussed in the
text.

-0.2
-0.4
-0.6

f B (Hz)-0.8

fB = , + - f B
magnetometer, = -0.25(6)- = 0.87(5)R 2 = 0.804

-1
-1.2
-1.4
-1.6
-1.8
-1.4

-1.2

-1
-0.8
-0.6
B
f magnetometer (Hz)

-0.4

75

Figure 4.5: The magneto-resistive sensors described in the text. On the upper left is the previous
sensor purchased from Adafruit. The components are placed close to the sensor and the PCB
contacts are coated in a gold film, contributing to the magnetization of the board. we replacedthese with a PCB, shown on the right, with the componets removed from the board and placed on
the host side. The connector is now four brass screws, which in addition to providing an electricalconnection also locate and position the magnetometer. Eight magnetometers are attached to amotherboard, shown below, which provides power and ground connections and routes the signalsof the eight magnetometers to the host microcontroller.

76that they were constructed of magnetic materials. The printed circuit board contacts weregold coated, but underneath the layer of gold a layer of nickel was used to adhere the gold
plating. The evaluation board also contained a voltage regulator, bypass capacitors, resistors, and transistors to facilitate the logic level conversion from 5V (TTL) to 3.3V (voltage
level of magnetometer). All of these components exhibited some amount of magnetism.
With the exception of two bypassing capacitors, these components could be removed fromthe board, or from the circuit entirely. We designed a new PCB for the magnetometer thatis not magnetic, and used nonmagnetic surface mount capacitors placed further from thesensor. This new sensor board is already performing with less variability than the Adafruit
evaluation board. Additionally, we purchased a single high accuracy three axis fluxgate
magnetometer with a NIST traceable calibration and constructed a larger and more uniform calibration coil jig. By placing the magnetoresistive magnetometers next to the highaccuracy fluxgate magnetometer in the calibration setup, we can potentially achieve muchbetter calibrations of the magnetometers.
(2) The position and heading information of each magnetometer as mounted around the experiment is not good enough. The magnetometers are mounted through two holes in theevaluation board using brass or nylon screws to aluminum arms mounted to 1” indexedholes in the optical table breadboard. We machined the height of the aluminum arms toplace the magnetometer at the desired height. The new PCB design of the magnetometer
mounts directly to a larger PCB that is designed to attach directly to the vaccuum chamberthat contains the ion trap, providing a much more reliable position reference.
(3) When fitting the 24 magnetic field measurements to the 8 linearly independent 0th and1st order magnetic field terms we found large variations not explained by the fits, which
we interpret as the existence of higher order gradient terms. These higher order gradientsalong with a position uncertainty can add error to the 0th and 1st order terms we hope tomeasure. To combat this effect, the second generation magnetometer sensor array design

77
has increased the number of magnetometers by an order of magnitude. A “motherboard”
printed circuit board contains mounting points for 8 magnetometers. The magnetometersattach to the motherboard with 4 brass screws, giving them a well defined position andheading relative to the motherboard, as well as providing the electrical connection for
power, ground, signal, and clock. The motherboard also contains a mux to route thecommunication from the host to each of the 8 magnetometers. Multiple motherboards canbe queried for measurements by a host, in this case an Arduino Due microprocessor. We
plan to incorporate as many as 6 of these motherboard around the experiment, for a totalof 144 independent single axis magnetic field measurements.
This new sensor array is currently being assembled and calibrated. Two motherboard are
installed on the experiment currently, and are in the process of being characterized. For moredetailed technical information see the Yiqi Ni’s Honors Thesis. A discussion of the future benefitsof the new sensor array will be continued in section 5.2.

4.2.4

Axial magnetic fields

Because the experiment takes place in a rotating frame, it is primarily sensitive to AC magnetic fields at the rotating frequency, and the effect of temporally constant and spatially uniform
magnetic fields are suppressed. We deliberately choose to operate the experiment without any
magnetic shielding[42] for two reasons, the first simply being that it is easier not to, and the second
being that it would affect the uniformity of our deliberately applied magnetic field gradient. In thissection we examine the effect of DC magnetic fields in the rotating frame.
The application of static magnetic fields will shift the various Zeeman sublevels, affecting thestrength of the 3rd and 4th order couplings between pairs of mF = ±3/2 states in perturbation~lab as Bx x̂ + By ŷ + Bz ẑ, where ẑ is thetheory. We will express static lab frame magnetic fields B
direction opposite of gravity and x̂ and ŷ are chosen to align with the radial trap axes. The effect78
Figure 4.6: A magnetic field Bz , applied along the axial direction of the trap, parallel to thedirection of the rotation vector, results in a shift in f R and f DR . As described in eq. (4.30), theaxial magnetic field shifts the Zeeman sublevels which results in an effective ∆ rotational couplingthat is increased by a factor of (1 ± 3gF µB Bz /ωrot ). The effective rotation of the axial magnetic
field removes the degeneracy of the two rotation directions, which is why the effect is odd under
the rotation switch R and appears in f R , and because it modulates ∆, as well as δ∆, it also has
an effect that is odd under the simultaneous switch of rotation direction and doublet and appears
in f DR .

3
E

rot

= 12 V/cm, Brot= 9.5 mgauss

E = 18 V/cm, B = 3.0 mgaussrot
rot
E = 24 V/cm, B = 1.7 mgauss

2

rot

rot

f R (Hz)

1
0
R

-1

f (12V/cm) = 0.50(4)B + -0.3(1)zR
f (18V/cm) = 0.132(7)B + -0.09(1)zRf (24V/cm) = 0.044(2)B + -0.035(5)

-2

z

-3
1
0.8

f DR (Hz)

0.6
0.4
0.2
0
-0.2
-0.4

f

DR

(12V/cm) = -0.17(4)B + 0.1(1)fDR(18V/cm) = -0.050(9)B + 0.02(2)fDR(24V/cm) = -0.029(4)B + 0.00(1)

z

z
z

-0.6
-0.8
-5

-4

-3

-2

-1

0

1

B (gauss)
z

2

3

4

5

79
of this on the effective two state Hamiltonian eq. (4.1) isωrot + gF µB (Bz − iBx sin(ωrot t) + iBy cos(ωrot t)) 3∆ →27ωef,
dmf Erot2 
dmf Erot
δg
ωrot
9
,
=
gF J=1 2 dmf Erot
EHF



2
Bx cos(ωrot t) + By sin(ωrot t)
2gF µB Bz
9ωrot
1+
1+
+
→
2dmf Erot EHF
ωrot
Brot



2
gF µB By sin(ωrot t)
9ωrot
gF µB Bx cos(ωrot t),
2dmf Erot EHF
ωrot
ωrot


Brot →Brot + Bx cos(ωrot t) + By sin(ωrot t)(4.27)(4.28)(4.29)

The component of δg/gF due to mixing in the J = 1 level is perturbed by terms that go asgF µB Blab /ωrot . This term only approaches 1 for Blab > 10 Gauss, which is already too large withthe current shim coils installed on the experiment, and much larger than the ambient field due
to Earth’s magnetic field. Moreover, the contribution to δg/gF from J = 1 mixing is alreadydominated by the contribution from mixing with J = 2 states, as in eq. (4.19), so the effect ofstatic lab frame fields on δg/gF can be safely ignored.
We see that the effective mixing between mF = ±3/2 states, ∆, is affected by axial magnetic
fields in much the same way as it is affected by rotation. This is because the axial magnetic field Bz
is always perpendicular to the quantization axis, chosen to point in the n̂ = x̂ cos(ωrot ) + ŷ sin(ωrot )direction, and so it will exert a torque on the spin, mixing adjacent mF states in much the same
way that an actual rotation does. The radial magnetic field x̂Bx + ŷBy will torque the molecule
whenever it does not point along the quantization axis, and will shift the mF levels whenever it
does, which is why the Bx and By terms appearing in ∆ and Brot are phase offset by π/2. Because
these terms have a time dependence we can’t simply treat them as a frequency, as in section 3.2.1,
but we have to explicitly treat the time dependence as in section 3.2.1.1. They will contribute phasemodulation to the Zeeman shift and to ∆, which we don’t have to worry about for modulations ofamplitude smaller than gF µB Bx,y /ωrot . For now we will focus our attention on the time independent
contribution of Bz . The effect of Bx and By will be treated in section 4.3. Expanding eq. (4.27) to80
first order we see
3gF µB Bz∆ ± δ∆ → (∆ ± δ∆) 1 ++ ....
ωrot(4.30)

This term is odd under switches of the electric field rotation direction, R. We expect to see large
effects in all of the frequency linear combinations that are odd under the R switch. Also because
dmf Erot3δ∆ = 8 ∆ EHF , there will be an effect that is odd under reversal of the doublet, because of
how Bz can modulate δ∆, so we expect effects in the channels that are odd under doublet switch.
Incorporating this effect into the expansions of frequency linear combinations eq. (4.11)-eq. (4.18)
we find new terms proportional to Bz , the largest of which are∆δ∆
gF µB Bz 2
f =f +
15+ . . . .,3gF µB Brot
ωrot


∆δ∆
gF µB Bz 2
0BD
BD
f
=f
+ 3gF µB δB
15+ . . . .,(3gF µB Brot )2ωrotgF µB Bz
∆2 + δ∆2
0RR
3
f =f ++ . . . .,3gF µB Brotωrot
gF µB Bz
∆δ∆
0DR
DR
6
+ ....
f=f+
3gF µB Brot
ωrot
0D

D

(4.31)(4.32)(4.33)(4.34)

There is also an effect in the f BDR channel, but it is further attenuated by a factor of δB/Brot , so wewill neglect it. For typical experimental values, and a 1 Gauss Bz , the term (gF µB Bz /ωrot ) ≈ 0.016.
The magnitude of the effect in f R and f DR are 30 mHz and -20 mHz respectively per mGauss ofmagnetic field pointing in the same direction as the rotation vector. To validate the existenceof these terms we can plot them against the measured effects in the frequency combinations, seefig. 4.7. If the predicted effects in eq. (4.33) and eq. (4.34) are correct we expect to see an exact
liner relationship with a coefficient of 1. The effect agrees quite well with what we expect fromeq. (4.11)-eq. (4.18).
There will also be an effect that will appear in the f BD channel that is quadratic in theaxial magnetic field, eq. (4.32). This term arises because of how the axial magnetic field increases
the differential coupling of the two doublets δg, effectively increasing the effective differential gfactor between the two levels, spoiling the suppression of any non-reversing magnetic fields. The

81
Figure 4.7: The effect of axial magnetic field on f R and f DR from fig. 4.6 plotted against the axial
magnetic field Bz scaled by the the predicted terms in eq. (4.33) and eq. (4.34). For this dataωrot = (2π)250 kHz, and ∆, δ∆, and Brot were inferred from the f D and f 0 channels.

3
E

f R (Hz)2rot= 12 V/cm, Brot= 9.5 mgauss

E = 18 V/cm, B = 3.0 mgaussrot
rot
E = 24 V/cm, B = 1.7 mgauss
rot

1

rot

0
-1
R

-2

f = 1.15(3)"(B /B

-3
-2

-1

z

"(B /B
z

rot

)("/!

rot

) -0.09(2))

0

rot

1

2

)("/! ) (Hz)
rot

1
f

DR

= 1.11(7)/"(B /B
z

rot

)("/!

rot

) 0.02(1))

f DR (Hz)

0.5

0

-0.5

-1

-0.5

-0.25

2/"(B /B
z

0

rot

0.25

)("/! ) (Hz)
rot

0.5

82
correction to the effective g-factor is

 !
δgeff
δg
∆δ∆
gF µB Bz 2=−
1 + 15.
gF
gF(3gF µB Brot )2
ωrot(4.35)

The correction due to Earth’s magnetic field will be smaller than 0.001, and if any attempt is madeto cancel Earth’s field to even the 10% level, this correction drops to 10−4 . For the full Earth’s
field this effect is a systematic shift on f BD of less than 3µHz, or 5 × 10−31 e·cm. In any case,the size of this effect is smaller than the measurement uncertainty on δg/gF , so the effect of axial
magnetic fields is completely irrelevant from the perspective of systematics for our current level ofstatistical sensitivity.
Another potential effect of axial magnetic fields is on the statistical sensitivity of the experiment. Because the axial magnetic field affects the coupling between adjacent mF levels the same
way that the rotation of the molecules does, it effectively increases or decreases ∆, the effective coupling between mF = ±3/2 levels. Recalling section 3.2.3, this coupling is what is used to perform
π/2 pulses at the beginning and end of the Ramsey sequence. This means that Bz will detune ourπ/2 pulses, which can lead to a decrease in contrast, which will decrease the statistical sensitivityof the experiment. The asymmetry of a Ramsey fringe, taking into account π/2 pulse timings, is2

A = − sin




∆
2 ∆
t
t
cos(2π(f t + φ)) + cos.
2~ π/22~ π/2(4.36)

Substituting the modification of ∆ due to Bz , (∆ → ∆(1 + (3gF µB Bz /ωrot )2 ) into the above
assymetry, and then expanding to first order in Bz , we find
 !



π 3gF µB Bz 2
π 3gF µB Bz 2
cos(2π(f t + φ)) +.
A=− 1−
2 ωrot2 ωrot(4.37)

While we do see a quadratic dependence of the Ramsey fringe offset and contrast on Bz , it isroughly a factor of 2 too small, see fig. 4.8. The data agrees better if we remove the factor of (π/2)2
from the prediction, although there is no justification for this. As a final note, this will only affectthe statistical sensitivity if there is a secular drift in the axial magnetic field between experiments.
We typically determine the optimal π/2 pulse time tπ/2 empirically, scanning tπ/2 until we achieve
optimal fringe contrast and offset.

83
Figure 4.8: An axial magnetic field Bz mimics a faster or slower electric field rotation, which willchange the effective coupling between mF = ±3/2 states ∆ by a factor of (1 + (3gF µB Bz /ωrot ).
Because the π/2 pulses of the Ramsey sequence are executed using ∆ to rotate into a superpositionof mF = ±3/2, if the π/2 pulse timing has previous been tuned to tπ/2 = π~/2∆, then in thepresence of an axial magnetic field shifting ∆ the π/2 pulses will now be off of resonance anda contrast and offset will be introduced. The offset will be increased and the contrast will bedecreased by a factor of (3gF µB Bz /ωrot )2 .

0.38
0.36

Contrast

0.34
0.32
0.3
0.28
0.26

C = -1.8(5)(3 g 7 B /!
F

B

z

2

rot

) + 0.34(1)

0.24
0.22
0.07
0.06

O = 1.04(7)(3 g 7 B /!
F

B

z

2

rot

) + 0.007(2)Offset

0.05
0.04
0.03
0.02
0.01
0
-5

-4

-3

-2

-1

0

1

B (gauss)
z

2

3

4

5

84
4.2.5

Fields inhomogeneities

Another potential source of systematic errors is spatial inhomogeneities in the electric andmagnetic fields that the ions experience. As the ions traverse the trap, either by secular motion,
radio-frequency micromotion, or rotational micromotion, the ions sample the inhomogeneities ofthe electric and magnetic fields and experience energy shifts. The magnetic field inhomogeneities
can cause Zeeman shifts, and the electric field inhomogeneities can tip the molecule axis out of theplane of rotation and cause a berry’s phase shift.
First we will consider inhomogeneities in the rotating electric field. The trap was designedwith the application of a homogeneous rotating electric field in mind, so ideally the electric field
will be Erot (x̂ cos(ωrot t) + ŷ sin(ωrot t)), but there will always be some degree of inhomogeneity. The
trap is symmetric about the xy plane, and also the zx and zy plane, so first order gradients in themagnitude of Erot should be heavily suppressed. However, we do anticiapte second order gradients
due to the cylindical symmetry of the trap.
Electrostatic calculations done using COMSOL allow us to predict the inhomogeneity in Erot
for a perfectly constructed trap. If we write Erot out with second order inhomogeneities,

Erot (x, y, z) = Erot 1 + βx x2 + βy y 2 + βz z 2 ,(4.38)

then we predict roughly βx = −βy = −0.0027cm−2 , and βz = 0.001cm−2 . Of course, the construction of the trap will never be exactly perfect, nor will the amplitude and phase of the voltage
drive on each of the electrodes comprising the trap. We can characterize the inhomogeneity bymeasuring the shift in the ponderomotive potential when the rotating electric field is applied. In
the absence of a rotating electric field, the secular motion in an ion trap is to lowest order harmonic.
A rotating electric field with no inhomogeneity will not have any confining or deconfining influenceon the ions because it has no gradient. A rotating electric field with a finite inhomogeneity will
induce an additional ponderomotive potential, which can be observed by measuring the change intrap frequency. The change in trap frequency can be used to extract the inhomogeneity β. The

85
ponderomotive potential is21e2 Erot
2
Vpon = mωpon
x2i =
2 .
2
4mωrot(4.39)

We can match terms with x2 , y 2 , and z 2 to solve for β,βi = 22 ω2
m2 ωroti,pon2e2 Erot.

(4.40)The trap frequencies can be measured by giving a small kick to the ions and measuring the oscillationof the center of mass of the ion cloud in the trap. These frequencies can be measured with andwithout the rotating electric field. The difference of the squares of the frequencies of these two
measurements yields the squared ponderomotive frequency. We applied a rotating electric field of2
2
24 V/cm at ωrot = (2π)250 kHz and measured ωx,pon= −(2π)2 1.75 kHz2 , ωy,pon= (2π)2 0.795
2
kHz2 , and ωx,pon= (2π)2 0.43 kHz2 , which yields βx = −0.025cm−2 , βy = 0.0114cm−2 , and βz =0.0061cm−2 . This inhomogeneity has the potential to be a systematic error when coupled withdisplacement from the center of the trap. One potential way this can lead to a systematic frequency
shift is by contributing to an oscillating Berry’s phase that is not centered about zero. The angle
α that the molecule makes with the xy plane is the same as the angle with the total electric field.
Assuming the radial trapping fields are much smaller than the rotating electric field, then the angle
α as a function of axial position z isα=Ez (z)mωz2 zmωz2 z=≈(1 − βz z 2 ),Erot (x, y, z)eErot (1 + βz z 2 )eErot(4.41)

If we write z as having a constant offset z0 and a sinusoidal oscillation at the trap frequency ωz ,then we find that the time average of α has a finite value,

hαi =


3mωz2 z12
3kb Tzmωz2 (z0 + z1 sin(ωz t))
2(1 − βz (z0 + z1 sin(ωz t)) ) =βz z0 =
βz z0 . (4.42)
eErot
2eErot2eErotAt the center of the trap the ions will oscillate in the axial direction, with the restoring force either
pointing up or down, and similarly the axial electric field tipping the molecules up or down as well.
Over the time of a period of axial oscillation the angle the molecules makes with the xy plane will
average to zero. When an ion is offset from the center of the trap in the axial direction, this is

86
Figure 4.9: Spatial inhomogeneity in Erot and displacement from the center of the ion trap causes
a Berry’s frequency shift to appear in the f BR frequency linear combination equal to 3ωrot α. The
angle the molecules makes with the plane of rotation, α = Ez /Erot , oscillates about a non-zero value.
The frequency shift in f BR should increase linearly with z displacement, and also be proportionalto ωrot /Erot , see eq. (4.43). We plot f BR against the scaled axis 3frot (kB /Erot )z to collapse thefour data sets onto a single line. This also allows us to extract a single fit value to compare to theinhomogeneity. This agrees well with the previously measured inhomogeneity, βz = 0.0061, and anaxial temperature of 15 Kelvin.

1.5
1f BR (Hz)

0.5

f

BR

(12V/cm) = 0.72(6) z + 0.03(7)BR

f (18V/cm) = 0.52(3) z + 0.11(3)BRf (19V/cm) = 0.44(3) z + 0.11(3)BR
f (24V/cm) = 0.34(4) z + 0.21(5)

0
-0.5
E

-1

rot

= 12 V/cm, frot= 250 kHzE = 18 V/cm, f = 250 kHz
rot
rot
E = 19 V/cm, f = 200 kHz

-1.5

rot

-2

E

-2.5
-3

-2

rot

-1

rot

= 24 V/cm, frot= 250 kHz

0

1

2

0

5
-2

10

z (cm)
2
1.5

f

BR

=3f

rot

(k /E
B

rot

) 0.136(6) z + 0.35(3)f BR (Hz)

1
0.5
0
-0.5
-1
-1.5
-2
-15

-10

3f

rot

-5

(k /E
B

rot

) z (Hz/Kelvin cm )87
no longer the case. The rotating electric field is stronger the further the ion is from the center, so
for the portion of the oscillation that the ion is furthest from the center, the angle α the moleculemakes with the xy plane is smaller, and for the portion of the oscillation when the ion is closer tothe center, the angle α is larger. Breaking the symmetry of Erot in the axial direction causes α tohave a non-zero mean value. This leads to a Ramsey fringe frequency shift 3αωrot , which is odd
under the switch of rotation direction because of the inclusion of ωrot , and is also odd under theswitch of rotating magnetic field direction. This means we expect to see this type of effect in thef BR frequency linear combination,f BR = 3ωrot

3kb Tzβz z0 .
2eErot(4.43)We expect to see this effect in f BR as we scan the axial position z0 in the trap. This effectshould scale as ωrot /Erot . This is exactly what we observe in fig. 4.9. In this set of experiments
we moved the ions over a range of 3 cm in the axial direction (the spacing between axial endcap
electrodes is about 10cm). We see several Hz shifts in f BR , and the slope of the effect scales−1like Erot. Plotting the observed effect in f BR against an axis scaled by ωrot and Erot we can fit

the slope 0.136(6)Kelvin/cm2 = (3/2)Tz βz . This agrees well with the measured inhomogeneityβz = 0.0061 cm−2 and an axial temperature of Tz = 15 Kelvin.
Next we examine how such a Berry’s frequency shift might affect the EDM channel, f BD .
We can combine higher order effects of terms with DR parity and BR parity to create potential
terms with the correct BD parity to appear in the EDM channel, f BD . An axial magnetic field has
an understood effect on f DR , described in eq. (4.34), which can couple to a non-reversing magnetic
field that appears in f BR . Additionally, the effect of αωrot can directly perturb the typical f BD
terms at a higher order in the series expansion, where an even power of ωrot doesn’t change signunder rotation switch. Thus there are two leading order terms for Berry’s phase appearing in the
EDM channel,
f

0BD

=f

BD

∆δ∆
−
3gF µB Brot

3αωrot3gF µB Brot


3gF µB Bz
3αωrot δB
2
−3
.
ωrot
3gF µB Brot Brot

(4.44)Taking no care to control the z position of the ions in the trap, with the modest requirement of

88
nulling the axial components of Earth’s field to 10%, and shimming δB to 10%, these effects are 50µHz and 10 µHz for the axial magnetic field and α2 correction respectively, which is a systematicof 10−29 e·cm. Additionally, there will be a very obvious 2 Hz shift in the f BR channel. We caneither choose to correct for the effect in f BR after the measurement, or servo the axial position of
the ions to z = 0 in the trap. This can be done to better than 0.5 cm, in which case the systematic
shift of f BD is suppressed well below the 10−29 e·cm level.

4.2.6

Unintentional gradients

We observed large frequency shifts in the non-reversing magnetic field channel f B when weapplied a magnetic field along the axial direction of the trap and then displaced the ion cloudalong the same axis. This was troublesome, because there were various predicted effects of axial
magnetic fields explored in section 4.2.4 and this was not one of them. Additionally, the nonreversing magnetic field channel is the one that appears most strongly in the EDM channel f BD ,where δg/gF is typically larger than 0.001, meaning that a 10 Hz shift in f B , as observed in fig. 4.10
will be an EDM mimicking systematic to the tune of 10 mHz, or 1.7 × 10−27 e cm. The resolutionof this problem is a cautionary tale. We performed these experiments by quickly building a pair ofquadrupole coils in a Helmholtz configuration around the ion trap vacuum chamber and applying
current to them. What is plotted in fig. 4.10 is the effect we observed in f B against the inferred Bz .
We are implicitly assuming that the magnetic field the coils are applying is uniform, an assumptionthat turns out to be completely invalid. If we consider a magnetic field that has a second order
gradient Bz (z) = Bz (1 + βz 2 ), then in addition to applying a zero order magnetic field Bz weare also adding a strong magnetic field gradient with a position dependence, δz Bz (z) = Bz 2βz.
Satisfying Maxwell’s equations requires there also be a radial gradient ∇B = Bz βz. This is exactly
a non-reversing magnetic field gradient. We therefore expect to see an effect in f B that is
f 0B = f B + 3gF µB Bz βz

eErot
2 .
mωrot

(4.45)

To substantiate this theory we moved one of the coils we used to apply Bz , reasoning that

89
Figure 4.10: We observed a large systematic shift in f B when we applied a magnetic field in the axial
direction and also displaced the ion cloud axially in the trap. This is due to an accidentally applied
gradient that is proportional to the intentionally applied Bz . This is confirmed by moving themagnetic field coils such that a different unintentional gradient would be applied, and seeing thatthe effect in f B changes. Additionally we can see the effect of poor reversals of rotation direction,
where Erot has a different magnitude and inhomogeneity for the different rotation directions. ThisR shim changed.
can be observed in f BR when the Erot20
15

f B (Hz)

10
5
0
-5
-10
-15
-20
-5

0

5

0

5

B (gauss)z2

z = 0 cmz = 1 cmz = -2z = -2 cm, ER
shimmedrot

1.5

z = -0.5 cm, B coils moved1

z = 0 cm, B coils movedzz = 0.5 cm, B coils moved

f BR (Hz)

z

z

0.5
0
-0.5
-1
-5

B (gauss)z90
we could decrease the uniformity of the applied field and increased the size of the second order
gradient β. The three sets of data in fig. 4.10 after the coils had been moved bear proof to thistheory. After the coils were shifted the second order gradient increased, and the effect in f B was
greatly enhanced. The fix for this is relatively simple, to remove the coils. They were not used in
the experiment except to apply axial magnetic fields and then to see what would happen. The coils
were never designed to be particularly uniform, in fact they were taken from a decommissioned
MOT and essentially draped over the vacuum chamber in an attempt to quickly begin the study oflab frame magnetic fields. We have replaced these coils with a new pair of coils specifically designed
to be uniform, and when we installed these new coils we inserted a magnetic field probe into thecenter of the ion trap to characterize the applied field and any incidental gradients. The new coils
have an incidental gradient of < 1 mGauss/cm when used to apply a uniform field of 2.6 Gauss.

4.3

Measured systematics
In addition to the effects that we can measure and characterize, there are additional system-

atic shifts that we have observed that still elude our understanding. In this section we describe
those effects whose mechanisms are still unknown to us. We have attempted to study these effects
by measuring how they scale with various experimental parameters, and we will catalogue theseexperimental relations and summarize our understanding.
Both effects arise from the application of a static lab frame magnetic field, Blab , in concert
with radial displacement of the ion cloud from the nominal center of the trap. When we first
applied a magnetic field in the xy plane of the experiment, we expected no effect, or perhaps asmall effect from a previously neglected geometric phase term. Instead we observed a several Hz
shift in f B , the non-reversing magnetic field frequency combination, and paradoxically, an evenlarger shift in f BR , the “Berry’s phase” channel. The larger shift in f BR was surprising because
we expected that frequency shifts in f B would be suppressed by the change in rotation direction.
We determined that these shifts were linearly correlated with both the magnitude of the magnetic
field but also the displacement from the center of the ion trap. Additionally we observe f B only

91
for displacement parallel to the applied Blab , and f BR only for displacement perpendicular to theto applied Blab , as depicted in fig. 4.11. For this reason we colloquially label the effect that appears~ × ~r · ω~ · ~r” and the effect that appears in f BR as “ B~ rot ”. Both of these effects arein f B as “Blinear with displacement from the center of the trap. This is already an important clue, because
there are a limited number of ways that the ions can even know where the center of the trap is.
They are:• ∇B
The rotating magnetic field is applied via the combined effect of a static lab gradiant,∇B, and the circular micromotion of the ions driven by the rotating electric field. The
quadrupole gradient is designed such that the magnetic field minimum should roughly
coincide with the center of the ion trap. The biggest argument against this being the causeof the observed effects is that we change the size of ∇B we do not see any change in thesize of the effect in f B or f BR , as in fig. 4.12 and fig. 4.14.
• Trap rf micromotion
The ion trap is a radio-frequency Paul trap that uses a time varying quadrupole electric field
to trap the ions. As the ions move away from the center of the ion trap they experience more
rf micromotion, creating a ponderomotive restoring force pushing them back to the centerof the trap. We displace the ion cloud by applying additional dipolar patch charges to thetrap electrodes, and as they are displaced their rf micromotion increases. The argument
against this rf micromotion related to the observed effect is that for an increase in theamplitude of the rf drive, as in fig. 4.12 and fig. 4.14, we expect an increase in micromotion
amplitude, but do not observe a change in the effects in f B and f BR when we change theamplitude of the rf drive, Vrot .
• Accidental gradients in Blab
We apply Blab by using two pairs of coils each in Helmholtz configuration to null Earth’s

92

Figure 4.11: Here we have nulled Earth’s ambient magnetic field and applied an additional magnetic
field aligned with the ŷ trap axis with magnitude 2.5 Gauss. We then performed two experiments,
one experiment where the ion cloud’s displacement in the ŷ direction of the trap was nominallyzero and we scanned the x̂ position of the ion cloud, the second experiment where the x̂ directionof the trap was nominally zero and we scanned the ŷ position of the trap several mm. The two ion
detectors used are microchannel plates on opposite sides of the trap, parallel to the x̂ axis. One
detector is a time of flight detector, and the arrival time of the ions corresponds to their position
along the x̂ vector of the trap. The other detector is a imaging detector, which gives the position of
the ion cloud along the ŷ vector of the trap. We see that the f B is only affected by displacementsof the ions parallel to the magnetic field, and f BR is only affected by displacements of the ionsperpendicular to the magnetic field.

2

2
0

fB

fB

0
-2

-2

-4
-6

-4
20.8

21

-6

21.2

-2

0

2

4

Y detector (mm)

10

10

5

5
BR

0

0

f

f BR

X detector (7s)-5-10B y = 2.5 gauss, scan y
B y = 2.5 gauss, scan x

-5
20.8

21

X detector (7s)

21.2

-10

-2

0

2

Y detector (mm)4

93
field and then apply fields that align with either the trap x̂ or ŷ axis. Neither coil axis isaligned with a trap axis. If in addition to a static magnetic field we are also applying astatic gradient, as we displace the ions we could be sampling the effect of this gradient.
The problem with this theory is that the effects can be observed in the ambient magnetic
field, and the effects go away when the ambient magnetic field is actively cancelled with thetwo pairs of coils. It is highly unlikely that the ambient lab gradients are also identicallycancelled by any accidental gradients applied by the magnetic field coils.
• Erot inhomogeneity
As we have seen in previous sections, inhomogeneities in Erot can be the source of othersystematic effects, in particular those arising due to a Berry’s phase in the f BR frequency
linear combination. The rotating electric field Erot was designed to be most homogeneousat the center of the trap, so it is not unreasonable that as the ion cloud is displaced fromthe center that some effect due to the inhomogeneity of Erot will enter play. Guided by thiswe will examine how both of these effects scale with Erot in the following sections.

4.3.1~ · ~rfB ∼ B

Informed by our suspicions established in the previous section, we examined the dependenceof this effect on f B as a function of ion trap rf quadrupole drive, see fig. 4.12. The drive is varied
by 50% and there is no strong change in the relation between f B and displacement. This could
indicate that rf micromotion has nothing to do with this effect. There are a couple of problems
with this sort of experiment. The first is that we have done measurements like this twice and founddifferent results. Earlier tests where we varied the strength of the rf drive found that there wasa dependence of f B on the strength of the drive. The difference between the older experiments
and the experiments presented in fig. 4.12 is how the ion position was shifted. In the previous
experiments to displace the ion cloud a dipolar patch potential was applied to the trap electrodes
calculated to displace the cloud by a certain amount for a given secular trap frequency. For the more

94

~ r effect scales with various experimental parameters.
Figure 4.12: Here we examine how the f B ∼ B·~
We apply various magnetic fields aligned with the ŷ axis of the ion trap and then displace the ionsalong that direction. We see confirmation of the linear scaling in both By and y, as well as a lackof dependence on displacements in z. Additionally we see that the effect increases with increased
ωrot and decreases with increased Erot (more data will be needed to establish the exact scaling).
We also see a lack of dependence on ∇B or f 0 . We have changed the magnetic field gradient andthe Ramsey fringe frequency by a factor of 4 with no effect. Additionally there is no large effectwith a change in the ion trap rf quadrupole drive, Vrf .

fB (Hz)10
50B

f = -0.09(1)y + 0.11(5)f B = 0.85(3)y + -1.8(1)f B = 1.77(2)y + -3.51(8)f B = 2.66(3)y + -6.4(1)

-5
-10
-15
-4

15
10
fB (Hz)15

B y=0 gauss, z=+3mm
B y=0 gauss, z=0mm
B y=0 gauss, z=-3mm
B y=2.5 gauss, z=+3mm
B y=5 gauss, z=+3mm
B y=5 gauss, z=-1.5mm
B y=7.5 gauss, z=+3mm

B-5-2 0 2 4 6
Y Detector (mm)f = 23 Hz
f = 93 Hz

5
0
-5

0
-5
V rf = 76 Vpp
V rf = 101 Vpp

-10
-15
-4

f = 2.10(9)y + -0.4(2)f B = 3.7(1)y + -2.5(5)f B = 3.47y + -3.36
f B = 3.322(4)y + -4.47(1)f B = 3.0(2)y + -6.1(6)f B = 2.70(1)y + -7.86(3)5fB (Hz)fB (Hz)

0

-15
-4

15
10

5

-10

-2 0 2 4 6
Y Detector (mm)E rot = 24 V/cm, f rot = 150 kHz
E rot = 16 V/cm, f rot = 250 kHz
E rot = 18 V/cm, f rot = 250 kHz
E rot = 20 V/cm, f rot = 250 kHz
E rot = 22 V/cm, f rot = 250 kHz
E rot = 24 V/cm, f rot = 250 kHz-2 0 2 4 6
Y Detector (mm)-10-4-2 0 2 4 6
Y Detector (mm)95
recent batch of experiments a dipolar patch correction is iteratively applied to the trap electrodes
until the ion cloud center of mass converges to a desired position on the ion detector. We findthis method to be more robust to changes in quadrupole drive, which will change the secular trap
frequency, although there is still a secular frequency dependent transfer function that translates
the detector ion position to trap ion position, which leaves open the potential to misinterpret theion cloud position.
There is no dependence of f B on either the Ramsey fringe frequency f 0 or lab frame magnetic
field gradient. There are several important consequences of this measurement. The first is that is
rules out the intentionally applied magnetic field gradient ∇B from being the “ruler” by which theion cloud determines its displacement from the center of the trap. Secondly, because we were able
to vary the frequency by a factor of 4, it shows that the effect on f B must have a very direct, up
front mechanism. In contrast to the effect of axial magnetic field on f R , or the effect of electricfield on differential rotational coupling δ∆ on f BD , where terms appear in a perturbation series
expanded in terms of (f 0 )−n , these effects have no f 0 dependence. The inference we draw from thisis that whatever the mechanism, it must affect the energy levels of the mF = ±3/2 levels, and not
the perturbative coupling between them. Potential types of mechanisms that directly affect theselevels are things like the Zeeman interaction or a Berry’s phase that doesn’t average to zero.
We also note the scaling of f B with ωrot and Erot . More data is currently being taken on these
scalings, but the trend established so far is that f B increases with increasing ωrot and decreasing
Erot . This is evocative of a Berry’s phase type of effect, where the frequency shift is proportionalωrot α, where α is the angle the molecule makes with the xy plane. This angle is defined by theelectric field pointing direction, which in the small angle approximation is Ez /Erot . What is unclear
is how to ensure that the effect is even under switching rotation directions if there is a dependence
on an odd power of ωrot . Presumably the dependence on ωrot must be even, or some other factor
must be included that is also odd under switching rotation direction.
The size of this effect is about 3.5 Hz/(Gauss cm) at Erot = 24 V/cm and ωrot = (2π)250
kHz. Using dimensional analysis we can write possible terms that have the right dependence on B

96
and r and see what size frequency they predict. A term that has approximately the correct scaling
could be something like2?
~ · ~r mωrot ,f B ∼ 3gF µB B
eErot

(4.46)

however this effect is about a factor of 105 too large. We would need another dimensionless factor
that doesn’t depend on Ramsey fringe frequency, rf drive amplitude, or have the wrong dependence
on Erot or ωrot . Another possible guess for an effect inspired by previously discussed Berry’s phase
effects could be?

f B ∼ 3ωrot
~ · ~r ωrot
B
Erot

.

(4.47)

This term alone is a factor of about 20 larger than the observed effect in f B . It looks like a Berry’s~ · ~r)ωrot . This hasphase where the angle α of the molecule is defined by Ez /Erot , where Ez ≈ (Bthe appearance of a Lorenz force, where rωrot is a velocity. Beyond that we can say no more about
this effect, other than that complex numerical simulations that include both the external degrees
of freedom of the ions in realistic trapping electric fields and electric and magnetic bias fields, andalso including internal states comprising the entire 3 ∆1 , J = 1 rovibronic manifold of states, does
not find any sort of significant motional effect of the ions that contributes to f B in this way.
Failing to find an understanding of the mechanism behind this effect, the next best thing we
can do is place an upper limit on how much this effect will appear as a systematic in the EDM
channel, f BD . We know that effects in f B that are due to a Zeeman type interaction appearin f BD suppresed by δgeff /gF , and that other effects in f B , such as Berry’s phase, appear inf BD suppressed by a slightly different δgeff /gF that includes the contribution from the differential
rotational coupling δ∆ but not the differential g-factor δg/gF . We can place an upper bound on
the effect of f B in the EDM channel f BD by operating at large Blab and large displacement toenchance f B as much as possible. We can shift f B by as much as 30 Hz peak to peak. This large ofan effect is sure to be seen in f BD , especially for smaller Erot where the effective δgeff /gF is larger.
There is a correction developed in section 5.2 that we can use to remove the expected contribution
from f BD due to f B in a self consistent way. We can apply this correction to f BD and look for

97

Figure 4.13: Here we take data with large trap displacements of ±5mm, and parallel lab frame
magnetic fields of up to 7.5 Gauss to observe a frequency shift in the non-reversing magnetic fieldchannel of up to 30 Hz. Using the self consistent correction developed in section 5.1.4, we subtractthe expected contribution of f B from the EDM channel f BD and look for any residual trend. The
fit to this residual trend can be used as an upper bound on the systematic effect of the anomalous~ · ~r”.
f B due to “B

0.5
0.4

fBD = -0.0047(28) f B

fBD corrected (Hz)

0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-15

-10

-5

0

fB (Hz)

5

10

15

98
any remaining residual correlation with f B . We find that f BD = −0.0047(28)f B , which has a 1.6σ
significance in its difference from zero. If we can cancel out the ambient lab frame magnetic field
to the 10% level to ∼ 0.05 Gauss, and control the ion cloud center of mass position to 0.05 cm,
then we should be able to keep f B below 10 mHz. The systematic effect on the EDM channel f BD
will be −47(28)µHz, or 8.3 ± 5 × 10−30 e cm.

4.3.2~ × ~r) · ω
f BR ∼ (B~

Similar to the anomalous f B effect there is no dependence of f BR on rf quadrupole drive,
as seen in fig. 4.14. This is strong evidence that this effect cannot be due to inhomogeneity in thetrapping fields or be related to rf micromotion. Likewise there is no dependence on ∇B or theRamsey fringe frequency f 0 . This indicates the effect isn’t due to the displacement of the ions from~ ×~r)· ω~rot is a direct effect,
the center of the magnetic quadrupole gradient. This also implies that (Bsimilar to the anomalous effect in f B . We do see that the f BR effect has a dependence on Erot andωrot , which points to some interaction with the spatial inhomogeneity of Erot . The dependence isroughly linear in each, f BR ∝ ωrot Erot . In total the unknown effect in f BR can be summarized as~ × ~r · ω
f BR ∝ B~ rot Erot .

(4.48)


~ × ~r · ωThis term B~ rot Erot has units of electric field squared. There has to be another field orfields to compare this to in order to get something with units of frequency. These comparison fields
cannot be Erot or Etrap because this would change the nature of the dependence on Erot and because
it’s already been established that there is no dependence on the trapping electric fields. It can’t be
the field from ∇B because this has also been ruled out of the f BR dependence. Other possibilities
are that the dependence of f BR on ωrot and Erot is not truly linear, in which case this should be
reexamined more carefully. Possible types of effects that increase with increasing ωrot are effects
arising from the Berry’s phase. Quantities that increase with Erot are the circular micromotion
2 and the micromotion velocity v
radius rrot ∼ Erot /ωrot
rot ∼ Erot /ωrot . Both of these have the wrong

dependence on ωrot , which could possibly be explained by an additional dependence on higher

99
powers of ωrot . As in the previous section, without any concrete understanding of the mechanism
behind this effect the best we can achieve is to place bounds on the size of any systematic that might
appear in the EDM channel f BD . For typical experimental parameters this effect is f BR ∼ 8.9(1)Hz/(cm Gauss). If we can shim Earth’s magnetic field to 0.05 Gauss and reproducibly place theions to within 0.05 cm of the trap center then f BR can be kept below 20 mHz. We can exaggerate~ × ~r) · ω
the anomalous f BR ∼ (B~ effect by displacing the ion cloud several mm from the centerof the trap and applying a large perpendicular magnetic field. Inevitably when we do this we will
also apply an incidental parallel magnetic field component. We can subtract out the f B effect due
to this incidental parallel magnetic field component as best we can by applying the correction insection 5.1.4. Fitting the residual effect we find f BD = −0.0018(11)f BR . Dependent on successfully
controlling the anomalous f BR to 20 mHz we can place an upper bound on the residual effect on
the EDM channel to be −36(22) µHz, or 3 ± 2 × 10−30 e·cm. Even making no attempt to shim
Earth’s field or place the ions in the center of the trap yields a residual systematic in f BD of
1.6 ± 9 × 10−28 e·cm.

4.3.3

f BR ∼ zr2

We also observe a quadratic effect in f BR with respect to displacements in the trap radial
direction, as shown in fig. 4.16. Such an effect could potentially be attributed to Erot inhomogeneity.
The effect on the time independent portion of Berry’s phase due to Erot inhomogeneity is
α=
hαi =

Ez (z)mωz2 zmωz2 z=≈(1 − βy y 2 − βz z 2 − 2βy βz y 2 z 2 ),Erot (x, y, z)eErot (1 + βy y 2 + βz z 2 )eErot(4.49)

3mωz2 z12
3kb Tzβz z0 (1 − 2βy y 2 ) =βz z0 (1 − 2βy y 2 ).
2eErot2eErot(4.50)

Here we have chosen to consider displacements y along the ŷ direction instead of the more generaldisplacements along a ray in the xy plane. We see that similar to the eq. (4.42) there is an α that
doesn’t time average to zero due to Erot inhomogeneity that scales likes zy 2 . This effect will appear
in the channel f BR asf BR = 3ωrot

3kb Tzβz z0 (1 − 2βy y 2 ).
2eErot(4.51)

100~ × ~r) · ω
Figure 4.14: This figure summarizes various experiments done to establish how f B ∼ (B
~ rot
scales with various experimental parameters. We apply magnetic fields along the x̂ direction of
the trap axis, and apply dipolar potentials to the trap electrodes to displace the ion cloud in the ŷ
direction. We see that the effect is linear in box Bx and in y. Additionally we see that the effect on
f BR increases with increasing Erot and ωrot . Finally, similar to the anomalous effect in f B , there isno observed dependence of f BR on magnetic field gradient ∇B, Ramsey fringe frequency f 0 , or rf
quadrupole drive Vrf .

30

B x= 0 gauss
B x= -4 gauss
B x=+4 gauss

5

20
fBR (Hz)fBR (Hz)

10

0
-5
-2

10
0
-10

f BR = -0.3(2)y + -0.1(3)f BR = 2.52(3)y + -1.57(6)f BR = -3.1(1)y + 1.4(2)

-20
-4

0
2
Y Detector (mm)

40

-2
0
2
4
Y Detector (mm)

10

0
-20
-40
-10

f BR = 3.55(7)y + 0.8(2)f BR = 2.54(6)y + 1.5(2)f BR = 2.97(2)y + 0.81(8)f BR = 2.03(6)y + 0.3(2)f0 = 285 Hz
f0 = 40 Hz

fBR (Hz)fBR (Hz)20

E = 24 V/cm, f = 250 kHzrotrot
E rot = 16 V/cm, frot = 250 kHz
E rot = 24 V/cm, frot = 222 kHz
E rot = 24 V/cm, frot = 150 kHz

50
V rf = 76 Vpp
V rf = 101 Vpp0
B y (Gauss)

10

-5
-4

-2 0 2 4 6
Y Detector (mm)

6

101

Figure 4.15: Here we take data with trap displacements of several mm, and perpendicular lab
frame magnetic fields of ±7.5 Gauss to observe a frequency shift in the f BR channel of up to 60
Hz. We can use the self consistent correction developed in section 5.1.4 to subtract the expected
contribution of any f B that might arise due to accidentally applied parallel lab frame magnetic~ · ~r” from the EDM channel f BD . We then take the remaining effect and fit any
fields and the “Bresidual trend. This can be used as an upper bound on the systematic effect of the anomalous f BR
~ × ~r) · ω~rot ”.
due to “(B

0.3
0.2

f BD corrected (Hz)

0.1
0
-0.1
-0.2
-0.3
-0.4
fBD = -0.0018(11) f BR

-0.5
-0.6
-40

-20

f

BR

0

(Hz)

20

40

102
We have already determined (3/2)Tz βz = 0.136(4) Kelvin/cm2 . Plugging this into eq. (4.51) andequating this with the measured constant of proportionality from fig. 4.10,
f BR ∝ −0.022(1)(z/cm)(y/mm)2 Hz,(4.52)

we find that βy ≈ 30cm−2 . This is extremely large compared to the measured βx = −0.025cm−2
and βy = 0.0114cm−2 .

4.4

Potential other systematics
In addition to the systematics discussed in the previous sections there are a number of otherpotential sources for systematic errors. We haven’t yet thoroughly investigated many of theseeffects, but we will list them below with a brief description and estimation their size.
• Detuning the π/2 pulses could potentially lead to a systematic effect in the doublet switch.
The differential rotation-induced mixing δ∆ requires that the timing of the π/2 pulses be
different for each doublet. The π/2 pulse timings will inevitably be imperfect, and they
will undoubtedly not be equally imperfect between the two doublets, which will lead to a
φD term. We empirically see that we do as well as a φD = 0.0082(8), measured in fractionsof a period. If we did not measure Ramsey fringes at short time to determine initial phaseshifts, this could contribute to a 16 mHz frequency shift to the Ramsey fringe at 500 ms.
With any frequency shift that is odd in B this can lead to a systematic in the EDM channel.
However, we already see that δ∆ leads to a f D of several hundred mHz, so this effect will
be an order of magnitude smaller.
• The ion trap electrodes have finite capacitance, which means that as we apply time dependent voltages to generate the rotating electric field we will incidentally also be applying
time dependent currents. These currents can have the effect at the center of the ion trap of
a rotating magnetic field that has the same rotation frequency and phase as Erot . This rotating magnetic field created by electrode currents will be odd under the rotation direction

103
Figure 4.16: We observe a quadratic shift in the f BR channel as we displace the ions radially fromthe center of the ion trap. The amplitude of the quadratic term appears to be linearly proportional
to displacement from the center of the ion trap in the axial direction, even changing sign as we movethrough the trap center. We anticipate an effect of this nature due to the trap electric field and Erotinhomogeneity, but this effect is larger than that of our predictions based on other inhomogeneity
measurements.

2
1

BR

f
= -0.514(7) + 0.224(2)y -0.0460(5)yBR2f= -0.67(10) + 0.27(4)y -0.113(8)yBR
2f= -0.03(7) + 0.08(2)y + 0.018(5)y2

f BR (Hz)

0
-1
-2
-3
-4
-5
-4

z = 0mmz = +3mm
z = -3mm

-2

0

2

4

6

y (mm)
0.05
2 BR= -0.022(1)z + -0.047(3)y(1/2)/2y f BR (Hz)(1/2)/ f

0

-0.05

-0.1

-0.15
-4

-2

0

z (mm)

24

104
switch R, and will appear primarily in f R . Indeed, we initially observed f R shifts of several
Hz when we generated the rotating electric field with cables attached to only one end of
the trap electrodes. Currently we drive the electrodes symmetrically from both ends, andthe typical f R we measure is ∼ 10 mHz.
• Motional fields are a systematic effect that have long plagued atomic beam electron EDM
experiments. Motional magnetic fields due to the rotational micromotion in the trap will
~ rot × ~vrot /c2 , and which for typical experiment parameters will be 0.2
be on the order of E
µGauss, and point in the axial direction. This is much less than Earth’s ambient magnetic
field in the axial direction of several hundred mGauss. The motional electric field due to~ × ~vrot . This can create a Berry’s phase α by tipping therotational micromotion will be B
molecular axis away from the horizontal plane. The Berry’s phase α due to Brot will be on
the order of 5 × 10−8 for typical experiment parameters. The Berry’s phase due to static
lab-frame fields will oscillate sinusoidally at frot with amplitude 10−5 .
• Earth rotates once on it’s axis every 24 hours, which is a frequency of 11.7 µHz. Thishas the potential to break the rotation symmetry of the electric field rotating with angularmomentum up and down in our experiment at the 11.7 µHz level in f R .
• Time varying magnetic fields or magnetic field noise at frequencies near frot could potentially contribute to a non-reversing magnetic field observed in the f B channel. We don’t
measure any significant noise features in the lab above 100 kHz. Additionally, the stainlesssteel ion trap chamber should provide some amount of shielding to rf noise such as this.

Chapter 5

Conclusion

5.1

A precision measurement of the electron EDM

5.1.1

Measurement overview

In this chapter we will discuss the logistics of compiling various pieces of data into a measurement of the electron EDM. We need to take lots of data in order to achieve good statisticalsensitivity, and we need to account for all of the various potential systematic errors that werediscussed in chapter 4. We will begin with a description of how the data is compiled.
The fundamental unit of data is the Ramsey fringe. To collect the data for the Ramsey fringe
we step the time t between the π/2 pulses through several different values. We typically choose six
to eight time steps per period of Ramsey oscillation and two to three periods, at least one at shorttime and one at long time. The points at short time are chosen from the first period, with the firstof these points very close to time t = 0, see fig. 5.1. This allows us to get a good measurementof the initial phase φ of the Ramsey fringe. The second period is chosen at a time t ≈ τ /2, thetime during the Ramsey fringe that has the most sensitivity to the Ramsey fringe frequency, as
explained in chapter 3. We also have the option to not take the early time period and only take thelong time period, assuming we have previously characterized the phase. This can gain us not quite√√a factor of 2 increase in sensitivity per time because we are not wasting half of our time taking√
short time data that contributes no frequency sensitivity. The increase is less than 2 because fora very long fringe the data rate for short time data is faster than the data rate of long time data.

106
For a typical fringe where the dead time tdead is 300 ms and the long time data is at t = 500 ms wep√can expect an increase in sensitivity/ time of 1100/800, or 17%. Additionally, when not taking
the short time data we can fix the values φ and fringe offset as long as φBD = 0 and OBD = 0,√
which also increased the frequency sensitivity by a factor of 3.
The data points are dispersed evenly throughout each period which means that no particularcare needs to be taken to ensure that at least a couple of the points are on the sides of a fringe,
where the maximum sensitivity can be attained. This is done at the cost of a factor of roughly√
2 frequency sensitivity. We choose to pay this cost initially when we are exploring the space ofexperimental parameters because as we vary certain parameters the Ramsey fringe frequency canshift in sometimes difficult-to-predict ways, causing us to drift off the side of a Ramsey fringe andlose statistical sensitivity. The data that will be presented in this chapter was taken in the manner
described above, but a set of data collected with the aim of truly attaining the highest sensitivity
would have the times of the data points dialed in to the side of the Ramsey fringe.
At each time step in the Ramsey fringe we take several data points and average them together
to achieve a better signal to noise ratio on the ion number measurement. We typically take fewer
data points for early time periods (∼ 12) and more data points for long time periods (∼ 24) because
the our signal is smaller and our duty cycle is longer at longer Ramsey fringe times. Additionally,
we alternate data points between measuring the population in the mF = +3/2 state and themF = −3/2 state. We refer to this as “chopping” between measuring the two different spin states.
As described in chapter 2, we do this by changing the phase of the rotating electric field relative tothe depletion pulse sequence. In a truly symmetric experiment we would chop between measuring
the two different spin states and also preparing the initial population before the first π/2 pulse of
the Ramsey experiment in the two different spin states. We currently do not do this, but there is notechnical limitation preventing us, and we will implement this feature in our experimental sequenceshortly. This fast chop between measuring the two different spin states allows us to measure P (↑)
and P (↓) and construct the fringe asymmetry, eq. (3.15). After a single Ramsey fringe is acquired

107
Figure 5.1: Plotted in each sub-figure of (a) is the fringe asymmetry A against the time between
π/2 pulses for a typical set of fringes in a block. The fringes oscillate at about 20 Hz, but thefrequency is slightly different for each configuration of the switches B, D, and R. These 8 fringes
can be combined according to the prescription in eq. (4.9) to make the frequency linear combinationf BD . In sub-figure (b) the EDM channel f BD is shown for 6 different blocks, each comprising 8
Ramsey fringes, with the first block corresponding to the set of fringes shown in (a). This particular
data were taken when we were investigating a large systematic effect (the anomalous effect that~ · ~r), which is why the f BD channel in (b) varies over such a large range.
looks like f B ∼ B(a)Ramsey fringe asymetry

B=BoutD=lower
R=CW
B=BoutD=lower
R=CCW
B=BoutD=upper
R=CW
B=BoutD=upper
R=CCW
B=Bin
D=lower
R=CW
B=Bin
D=lower
R=CCW
B=Bin
D=upper
R=CW
B=Bin
D=upper
R=CCW

0.5
0
-0.5
0.5
0
-0.5
0.5
0
-0.5
0.5
0
-0.5
0.5
0
-0.5
0.5
0
-0.5
0.5
0
-0.5
0.5
0
-0.5
0

100

200

300

400

500

Time between :/2 pulses (ms)

(b)
1
0.8

f BD (Hz)

0.6
0.4
0.2
0
-0.2
-0.4
-0.6
09:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00

Time (hours)

108
we fit it to the functional form eq. (3.26)
A(t) = −C(t/τ ) cos(2π(f t + φ)) + O,

(5.1)

using nonlinear least squares analysis to extract the fringe frequency and other parameters. This
Ramsey fringe is taken with a particular configuration of switches, i.e. rotating magnetic field
Brot , doublet, and rotation direction. Several Ramsey fringes with parameters spanning the setof all possible parameter switch configurations are compiled into a “block”. For the set of thethree switches of B (rotating magnetic field direction), D (upper/lower doublet), and R (rotation
direction), eight different Ramsey fringes will comprise a block. For a single block we are able toconstruct all of the various linear combinations of frequency, e.g. f B , f BD , as well as the linear
combinations of the other data channels, e.g. φ, C, φB , OBD . A set of 8 Ramsey fringes comprisinga block and the value of f BD for this block and 5 others taken in an afternoon are shown in fig. 5.1.
With the current signal level of the experiment it typically takes about 32 minutes to acquire a
block of 8 Ramsey fringes.
As discussed in chapter 4, the addition of each new switched parameter to the set of switches
tends to suppress systematics in the EDM channel f BD and isolate the effect of those systematics
to other channels. For example, f B has an exaggerated effect due to non-reversing magnetic fields
compared to the EDM channel. While the addition of each new switched parameter does not√decrease the frequency sensitivity per time, it does have the practical disadvantage of increasing
the time it takes to acquire a single block of data by a factor of 2n , where n is the number of
additional switched parameters. We therefore choose to operate with a reduced set of switches,{B, D, R}, in contrast to the larger set of all possible switches. We choose this set because it isa minimal set required to track the systematics that we have so far observed, i.e. f B is a goodmeasure of non-reversing magnetic fields, f D is a good measure of rotation mixing effects ∆ andδ∆, etc. The timing of the various switches that make up a block are shown in fig. 5.2. We areaware of the danger of unknown, perhaps smaller systematic errors that we have not observed,
possibly because we have chosen to focus so exclusively on a minimal set of switches. To defend

109
Figure 5.2: This the an example of a typical timing sequence of a series of switches in the EDM
experiment.

(a) The fastest time scale of the experiment sequence. Every shot of the experiment, which in this case lasts
0.5 seconds, we switch between detecting ions in the mF = 3/2 state and mF = −3/2 state.

mF=3/2mF = -3/2
00

01

02

03
04
Time (seconds)

05

06

Ramsey fringetime (ms)(b) Shown below is a single Ramsey fringe of a single block of data. The time between π/2 pulses is steppedthrough values ranging from 0 ms to 500 ms.

400
200
0
00:00

00:40

01:20

02:00
02:40
Time (minutes)

03:20

04:00

(c) Shown below are 4 blocks of data, with B (rotating magetic field direction switched at the fastest
rate, followed by D (doublet), followed by R, (rotation direction). At the “super-block” level an additionalparameter S is varied between 4 different values.

R=CCW
R=CW
D=upper
D=lower
B=Bout
B=B

inS
00:00 00:10 00:20 00:30 00:40 00:50 01:00 01:10
Time (hours)

110
against this we can perform additional switches above the block level, for example in a sequenceof 100 blocks, we will alternate one of these switches every 10 blocks. We then have the ability toaverage over these switches or to form new linear combinations of data channels that are odd underthese switches. Examples of switches we might consider performing at the “super-block” level are:• Depletion laser polarization σ + /σ −• Rotating electric field magnitude (between two values, e.g. 24 V/cm and 16 V/cm)• Rotating magnetic field value (between two values, e.g. 2 mGauss and 4 mGauss)• Ion cloud center of mass trap position (+offset/-offset) in x̂, ŷ, ẑ directions• Ramsey fringe time t (between two values e.g. 400 ms and 600 ms)• Number of ions loaded into the trap
This prescription can be used to assemble a large amount of data suitable for a robust EDM
measurement. For a measurement of the electron EDM with the current level of demonstrated
statistical sensitivity we expect to take about 100 hours of data, comprising several hundred blocks
and achieve a total statistical uncertainty of several 10−28 e·cm .

5.1.2

Data blinding

In the conclusion of his paper on the measurement of e/m of the electron, Frank Dunnington
had the following to say about the role of an expectation bias in experiment[22],It is also desirable to emphasize the importance of the human equation in accurate
measurements such as these. It is easier than is generally realized to unconsciously
work toward a certain value. One cannot, of course, alter or change natural phenomena ... but one can, for instance, seek for those corrections and refinements
which shift the results in the desired direction. Every effort has been made to avoid
such tendencies in the present work.
Indeed, Dunnington blinded his own measurement of e/m by having the angle between his electron
source and his electron detector kept secret from him, only measuring it once he finished his

111
final analysis of all the other factors that could affect e/m. By blinding himself from his data inthis way he prevented himself from unintentionally biasing his final result by the results of otherexperiments that came before him. Expectation bias in an electron EDM precision measurement isa real concern, because the difference between a zero and nonzero result has profound implications
for physical theory. Many other groups have made very precise measurements that are consistent
with zero, and a worry is that we will unintentionally bias our result in some way such that it
is consistent with zero as well. This concern that expectation bias can affect the outcome of a
precision measurement is well founded, if we look at historical measurements of the speed of lightin fig. 5.3, we see a possible example of expectation bias influencing the outcome of an experiment.
The series of measurements made in the 1930s are consistent with each other, but deviate from thetrue value by several standard deviations. A possible scenario where this can occur in a precision
measurement experiment is if systematic errors are searched for and eliminated until the outcomeof the experiment matches the experimenter’s expectation, and then the search is ceased.
It is important that we don’t bias ourselves on the outstanding electron EDM measurements
made in the past, and blinding our data as Dunnington did is a potential way to ensure that we
don’t accidentally do so. At the same time, the experiment we have described is rather complicated,
and we fear that blinding might obscure some problematic irregularity in the experiment, or makediagnostic analysis more difficult. There is also the concern that the blind might be difficult oreven impossible to remove, ruining large amounts of data. We therefore adopt the philosophy that
we should be “blind, but not dumb”. We will choose to blind our data by adding an unknown
offset to the EDM channel, and the offset will be dynamically added in at the last possible step inprocessing. Additionally, we choose to apply a small blind offset. We choose a random offset thatis on the order of ±1 × 10−28 e·cm, as this will be undetectably small for any but the largest sets ofour data, but it is still larger than the statistical uncertainty of the most recent EDM experiment(0.41 × 10−28 )[7]. We will also add to the blind a known offset of 3 × 10−28 e·cm, which will serve
to reassure us that the blind is in place and operating in the analysis like we expect. Dunnington
did something similar in his e/m measurement, when he ordered the angle between his detector

112

Figure 5.3: Historical measurements of the speed of light, taken from [40]. The four measurements
made in the 1930s are each consistent with the previous measurement but are several σ away fromthe true value.

1.0014

measured c/299792458 m/s

1.00121.001
1.00081.00061.00041.0002
1
0.9998

1880

1900

1920
Year

1940

1960

113
and source to be constructed at an angle of approximately 140◦ , but that he not be told the exact
angle chosen.
Whenever we look at the frequency linear combinations of a block or blocks, a function iscalled which loads the raw data containing the frequencies of the Ramsey fringes that comprise theblocks. This function then calculates the frequency linear combinations according to the prescription eq. (4.9). Then, in what is quite literally the last step before the function returns the linear
combinations to the user, we load the value of the blind from the network disk and add it to f BD ,1 l c o m t a b l e . fBD = l c o m t a b l e . fBD + f r e a d ( f o p e n ( 'X: \ BLIND\BLIND ' ) , 1 , ' d o u b l e ' ) ;2 fclose ( ' all ') ;The blind is added to the EDM channel at the last step before it is passed back to the user for
further analysis. The full analysis procedure is detailed in appendix B. The blind that we apply isstored in a file on the network file system accessible to every computer in the lab. The file contains
an 8 byte binary representation of one double precision floating point number. This number was
the result of choosing a random number from a Gaussian distribution centered at 3.38 mHz with
1σ width of 1.13 mHz. This is a distribution equivalent to 3 ± 1 × 10−28 e·cm in EDM units for
HfF+ . This random number was chosen once on January 9th, 2015, and it has been used in all
analysis since then. This is important because it allows us to compare various blinded data sets,
as they all have the same blind offset added. We chose a blind that intentionally has a bias of
3 × 10−28 e·cm. The reason is to be sure that we are actually applying the blind, and that we areapplying it correctly, with the correct sign. Removing the blind is trivial, we simply comment out
the relevant line from our analysis file, and look for a ∼ 3.38 mHz shift in the f BD channel. We
can then take the binary number from the BLIND file and convert it to a human readable double
precision floating point and confirm that it matches the observed shift.
It is important to remember that this blind only affects the mean value of the f BD , or EDM,
channel. The other channels are not blinded, that is to say the values of f B and f R are thetrue values the experiment measures. This is important because these channels will be used as
diagnostics to make sure the experiment is working correctly, and if a blind offset were applied to114them it would cause us to think that there existed a fictitious non-reversing magnetic field, in thecase of f B , or that there was some problem with the reversibility of rotation, in the case of f R .
Another feature of the blind is that it does not affect the variance of the data in the f BD channel
at all, so we can analyze the statistics of blinded f BD data.

5.1.3

Blinded EDM measurement

We have collected almost 500 blocks comprising over 380 hours of EDM data since January
2015. Of this data, almost 150 hours was taken at deliberately offset axial positions in the trap
and with deliberately applied static axial magnetic fields, often with unintentionally applied axial
magnetic field gradients. We won’t consider any of this data taken to study systematics in thefollowing sections. Of the remaining 233 hours of data we will veto 2 hours worth of data blocks
with fitting error bars for frequency greater than 150 mHz. The rationale behind this is that thisdata is the most likely to have Ramsey fringes that misfit, or was taken under conditions with notenough statistical sensitivity (i.e. short Ramsey fringe times or poor Ramsey fringe contrast). Thiscut is not biased by any of the actual measured values so it seems fair to do. We will also veto 8hours of data with f B > 2 Hz. This is necessary, as much of this data was taken under extreme
conditions as we studied systematic effects, and vetoing this data allows us to remove the mostextreme of those experiments. Note that 2 Hz is a much wider range than the ultimately possible
30 mHz we hope to achieve in the future by actively shimming non-reversing gradients and ion
cloud trap position. For the majority of the remaining data no attempt whatsoever was made tostabilize f B or the ion cloud trap position on even a day-to-day basis. A summary of this data isshow in fig. 5.4.
After applying the above vetoes we are left with a data set of 223 hours. For measurements
between the mF = +3/2 and mF = −3/2 state of 3 ∆1 of HfF+ the 23.3 GV/cm internal electricfield means a 1.13 mHz shift corresponds to a de of 10−28 e·cm. The f BD EDM channel, after the
3×10−28 e·cm offset has been removed, has a mean of 0.015±3.13 mHz, or 0.013±2.77×10−28 e·cm,
which is consistent with 0 and the 1 × 10−28 e·cm blind. The non-reversing magnetic field channel

115
Figure 5.4: This is 500 blocks of EDM data, roughly 380 hours, all the data taken to date. From
this we removed almost 150 hours of data: 140 hours that was part of studies of systematic effects
where we varied the axial position of the ion cloud in the trap and where we varied the axial
magnetic field, and 10 hours where f B , the non-reversing magnetic field, was larger than 2 Hz, or
where the frequency fit error bars were large than 150 mHz. These vetos can be seen as gaps in thedata in (a), in particular around measurements 250 and 420. Additionally the normal probability
plot (b) shows that the data is well Gaussian distributed.

(a) Time series of f BD taken to date. The gaps are due to the vetos described above.

0.4

f BD (Hz)

0.2
0
-0.2
-0.4

0

100

200

300

400

500

Measurement Number

Probability(b) Normal probability plot of (f BD − f BD )/σf . The plot is constructed by ordering the data and thenplotting the sample data on the x-axis against the theoretical quantiles of the data on the y-axis. Gaussian
distributed data will lie along the straight dashed line. Distributions with thin tails will curve down on theleft and up on the right (like an ’N’), and distributions with fat tails will do curve up on the left and downon the right (like an ’S’).

0.997
0.98
0.90
0.50
0.10
0.02
0.003
-4

-3

-2

-1

(f

BD

0

! f BD )=<

1
f

2

116
mean value is f B = −21.3 ± 3.1 mHz. The uncertainty stated for the mean value is the rms value√of the individual f B measurement uncertainties divided by N . The variation in f B in this data
set can be quantified by the block rms value of 700 mHz. The mean, uncertainty, and variance off B will be of particular importance in the upcoming discussion of the systematic effects present inthis data.

5.1.4

Systematic corrections

Many of the systematic terms appearing in f BD discussed in section 5.1.5 can be efficiently
diagnosed with the remaining frequency linear combinations, allowing us to calculate corrections
and their uncertainties from the data itself. For example, a non-reversing magnetic field can beeffectively measured with f B because the leading term is 3gF µB δB. We may use these diagnostic
terms to subtract off understood systematic effects from the f BD channel. For example, the leading
order terms of f 0 , f B , f D , and f BD due to non-reversing magnetic field δB and rotation terms ∆and δ∆ are:
∆2 + δ∆2
f = 3µB gF Brot 1 ++ h.o.,
2(3µB gF Brot )2



δB
∆2 + δ∆2
B
f = 3gF µB Brot 1 −+ h.o.
2(3gF µB Brot )2
Brot


δg
∆δ∆
D
f = 3gF µB Brot+
+ h.o.,
gF(3gF µB Brot )2



∆δ∆
δB
δg
BD
f
= 3gF µB Brot−
+ 2de Eeff + h.o.
gF
(3gF µB Brot )2
Brot
0

(5.2)
(5.3)
(5.4)
(5.5)

We see there is a high degree of overlap between these channels. This allows us to use this linear
vector space to construct a nonlinear subspace that is more closely aligned with the 2de Eeff termthan f BD ,
f

BD

+f

Df

B

f0


= 2de Eeff + 3gF µB Brot

δg
gF



δB
Brot


2−

∆2 + δ∆2(3gF µB Brot )2+ h.o.

(5.6)

The three highest order terms remaining after this correction are roughly 100µHz, 1µHz, and
0.1µHz respectively, or roughly 1.1 × 10−29 e·cm, 1.1 × 10−31 e·cm, and 1.1 × 10−32 e·cm. With

117

Figure 5.5: The systematic correction f D f B /f 0 − 2(δg/gF )f B is in good agreement with themeasured shift in f BD . The dashed line has a slope of 1 and intersects the origin. This data wastaken with a non-reversing magnetic field contribution spanning 30 Hz, and under an assortment ofaxial magnetic fields Bz , contributing to a range of f D measurements spanning ranging from −0.1
to −0.6 Hz. Even so, when synthesized in this way we can essentially entirely remove the effect of
the non-reversing mangetic field and Bz from f BD .

0.5

f BD (Hz) [EDM channel]

0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.5

-0.4

D 0

-0.3

-0.2

-0.1

B

0

0.1

0.2

0.3

(f /f - 2 /g/g F) f (Hz) [EDM correction]

0.4

118
prior knowledge of the quantity (δg/gF ) we can go even further,f EDM ≡ f BD + f D

fB2∆2 δ∆2
B
−
2f
(δg/g
)
=
2d
E
−
e
F
eff
f0
(3gF µB Brot )3



δg
gF



δB
Brot


+ h.o,(5.7)

where now the final term in eq. (5.7) is 10 nHz.
B

Constructing f BD + f D ff 0 − 2f B (δg/gF ) can very efficiently isolate the EDM contribution
to the Ramsey frequency linear combination f BD , and there are advantages and disadvantages tothis treatment. The advantage to this type of correction is that it is internally self consistent. It
will work for a variety of Brot , Erot , ωrot , ∆, and δ∆, as demonstrated in fig. 5.5. But does thisconstruction add statistical or systematic error to our measurement of the EDM? When we makea measurement and label it f B , it is to leading order 3gF µB Brot (δB/Brot ) plus some error, f B .
When we make measurements of f B with some uncertainty σf B , that is a statement of the size ofthe upper bound of f B , i.e. f B is a random number drawn from a Gaussian distribution withstandard deviation σf B . We can estimate the uncertainty of the systematic corrections to the EDM
measurement by propagating errors on eq. (5.7).
2
 D 2
 B D 2

fB
δg 2 2
f
f f
2
2
2
B 2 2
σ
σ
σ
2
σ
σδg/gF ,
0 +
D +
B +
B + 2f
f
f
f
f
f0
f0
f0 f0
gF
!
 B 2  D 2  B D 2 

2 2
f
f
f f
δg 2
1+
+
+
+ 2
+ 2f B σδg/g
,
(5.8)
0
0
0
0
F
f
f
f f
gF

2
σEDM
=σf2BD +=σf2



Where σδg/gF is the uncertainty in the determination of δg/gF . Above we have used the fact thateach of the channels f 0 , f B , etc. all have the same uncertainty σf , because they were formed bysums or differences of the same eight frequency measurements. We find that in the case f B  f 0and f D  f 0 the correction f B f D /f 0 introduces systematic error at the level of a fraction of thestatistical uncertainty of our measurement. For typical parameters f 0 ≈ 20 Hz, f D ≈ −0.2Hz,
and f B  0.2 Hz, so these conditions are satisfied and the systematic error is at the level of
1% of the statistical uncertainty. For the correction −2f B (δg/gF ), the systematic error amounts
to replacing (δg/gF ) with σδg/gF in our expansions, so as long as we measure (δg/gF ) to enoughprecision such that σδg/gF  (δg/gF ) then we are also better off with this correction term. For
typical values of the experimental parameters we have experimentally verified that δg/gF agrees

119well with the theoretical calculations from eqs. (4.19) and (4.21). We will therefore use calculated
value δg/gF = 0.000993 and use the measurement verification uncertainty for σδg/gF = 0.0001.
This will be our general strategy for calculating systematic error corrections and estimating
the uncertainty of these corrections in section 5.1.5. For any of the systematic errors will discuss
we can use a combination of the diagnostic frequency channels f B , f D , etc. and other measurably quantities to calculate the mean value of the correction and then perform propagation of
uncertainties to estimate the total uncertainty of the correciton.
A potential danger of this technique is systematic effects that appear in f B and other channels~ · ~r. It
that we do not understand. Recall from section 4.3 the systematic that appears as f B ∼ B
is still unclear whether this effect appears in the effective two state Hamiltonian as a coefficientof (gF ± δg) or not. However, due to the lack of any dependence on Ramsey fringe frequency it
must at the very least appear as a B odd term in the diagonal of the Hamiltonian, therefore thistechnique should still be applicable. Indeed, this is established in fig. 4.13, where the correction~ · ~r is suppressed in the EDM channel. Therefore, we
f D f B /f 0 is applied and the effect due to B
expect applying this correction to have a systematic residual of f B (δg/gF ), due to the uncertainty~ · ~r effect interacts with (gF ± δg). We could settle this question byof whether or not the f B ∼ B~ · ~r at very high frequency f 0 and looking formaking a very precise measurement of the f B ∼ B
any correlation in f BD . At f 0 = 200 Hz the component of δgef f /gF due to ∆δ∆ is about 0.0001,
which is much less than the contribution due to δg/gF ≈ 0.001. We can apply a 30 Hz shift to f B
and try to observe the 30 mHz shift in f BD . This will probably require about 10 hours of data to
be convinced of the presence of absence of any effect.
Another possible avenue of exposure to systematic errors is if the assumption that terms thatenter into the corrections are independent. If we consider the correction to the systematic error
due to only non-reversing magnetic field and differential g-factor,f δg/gF = −2

δg B
f ,
gF

(5.9)

120
then the uncertainty of this correction, calculated by propagation of uncertainty is,
σf2δg/gF =






δg 2 2
δg B
B 2 2
−2
σf B + fσδg/gF ,f B .
σδg/gF + 2 −2 f
gF
gF

(5.10)In the above discussion we have tacitly assumed that all covariance terms such as σδg/gF ,f B are
zero, i.e. that all of the quantities that enter into the corrective terms are independent. Thisis true for the various frequency combinations f B , f D , etc., which by construction are linearly
independent. However it is possible that the differential g-factor could have some dependence on
non-reversing magnetic field, which would yield a σδg/gF ,f B that is nonzero. This is perhaps a safe
assumption because eqs. (4.19) and (4.21) show that δg/gF has a dependence on Erot and ωrot , but
not non-reversing magnetic field. However these calculations were only made to first order, and it
is possible that the quadratic Zeeman effect could contribute to a δg/gF that depends on magnetic
field, which would leave us vulnerable to additional systematics not discussed here.

5.1.5

Systematics

The data presented in fig. 5.4 are influenced by several potential systematic errors. We
reviewed these possible effects in chapter 4, and we will summarize them here and quantify theireffect for this data. Essentially all of these systematic effects arise because of some interaction with
non-reversing magnetic field δB. We will separate the discussion into several parts based on thevarious exact mechanisms through which non-reversing magnetic field can present as a systematic,
but the treatment overall will be very similar.

5.1.5.1

Non-reversing magnetic field

Non-reversing magnetic fields are odd under the switch B, and they can combine with any
effect that is odd under doublet switch D to provide a systematic in f BD . The two primary effects
that are odd under doublet switch are the differential g-factor between the two doublets, (δg/gF ),and the difference in rotation-induced mixing ∆, the δ∆ term. The predicted contribution to f BD

121
due to these two effects is given in eq. (4.26),
f BD ∼ 3gF µB Brot



δB
Brot




δg
∆δ∆
−
.
gF
(3gF µB Brot )2(5.11)

We will separately consider the contributions due to (δg/gF ) and δ∆ and develop systematic corrections for each. We can use the correction developed in section 5.1.4 to estimate the size of thisterm in a self consistent way using the diagnostic channels f B , f D , and f 0 . First we consider thesystematic due to δ∆. The correction for this term isf ∆δ∆ =

f Bf D,f0(5.12)and the uncertainty of this term is
σf2∆δ∆ = σf

s

fB
f0

2


+

fD
f0

2


+

fB fD
f0 f0

2
.

(5.13)Taking the mean of f ∆δ∆ across our data set yields as correction of 0.238 mHz. The uncertainty ofthis correction can be found by calculating σ ∆δ∆ for every point in the data set and then taking the√block rms value divided by N , where N is the length of the data set. This yields an uncertaintyof 0.114 mHz. This uncertainty is dominated by the size of f B , the block rms value of which is
700 mHz. If instead f B was actively servoed to an rms value of 30 mHz, and the Ramsey fringe
frequency was changed to 50 Hz, then the total systematic error contribution due to f B would drop
to 2 µHz. The next largest contribution is from f D , which has an average value of −320 mHz due
to ∆δ∆. With a ramsey fringe frequency of 50 Hz the total systematic uncertainty contribution
from this correction is 20 µHz.
The correction to the systematic term due to (δg/gF ) isf δg/gF = −2

δg B
f ,
gF

(5.14)

which using the calculated value of δg/gF = −0.000993 and the mean value of f B = −21.3 mHz isa correction of f δg/gF = −0.043 mHz. The uncertainty of this term is
σf2δg/gF =






δg 2 2
δg B
B 2 2
−2
σf B + fσδg/gF + 2 −2 fσδg/gF ,f B .
gF
gF(5.15)

122
If we proceed with the assumption that δg/gF is independent of f B , then this is σ δg/gF = 10 µHz.
This is about evenly split between the contribution due to the uncertainty of δg/gF , σδg/gF = 0.0001
and the single frequency measurement uncertainty σf B . If we were to decrease the uncertainty of√
δg/gF or decrease the rms value of f B this will decrease this uncertainty by about a factor of 2.
Moreover, it will limit our exposure due to underestimating the systematic error uncertainty due
to correlated effects, i.e. σδg/gF ,f B 6= 0.
5.1.5.2

Axial magnetic field

Axial magnetic fields have the effect of shifting ∆ and δ∆, as described by eq. (4.30). This
combines with any non-reversing magnetic field to give a systematic effect in f BD , described ineq. (4.35),
f BD ∼ 3gF µB Brot (−15)

gF µB Bz
ωrot

2 

δB
Brot




∆δ∆
.
(3gF µB Brot )2(5.16)For this set of data no effort was made to null out the axial magnetic field. We measured thatthe axial magnetic field at the center of the trap due to the ambient Earth’s magnetic field is
Bz = −0.35(1) Gauss, where negative is in the same direction as gravity. We can use the ∆δ∆component of the non-reversing magnetic field and multiply by −15(gµB Bz /ωrot )2 = −0.0005 to
get the systematic error contribution of axial magnetic fields, which is 0.122 ± 0.103 µHz. This ison the order of 10−31 e·cm, and we note it will only get smaller when the non-reversing magnetic
field is actively shimmed and Ramsey fringe frequency increased.

5.1.5.3

Field inhomogeneities

There are two systematic errors that can affect f BD that are due to field inhomogeneities.
As described in chapter 4 field inhomogeneities can give rise to an average tipping angle α of themolecule axis with respect to the plane of rotation. A non-zero tipping angle results in a Berry’s
phase effect fBerry = 3ωrot α. which is odd under magnetic field reversal B, which can couple withaxial magnetic fields to be even under rotation direction R, as well as couple to the differential

123
rotation-induced coupling δ∆ to give an effect that is odd in D, as explained in eq. (4.44),
f

BD



3αωrot∼ 3gF µB Brot (6)3gF µB BrotgF µB Bz
ωrot




∆δ∆
.
(3gF µB Brot )2(5.17)

Additionally there is a second order Berry’s phase effect that goes as α2 that is also even under R,
which can couple to any non-reversing magnetic field and δ∆ to be odd under BD reversal,
f

BD


2 


3αωrotδB∆δ∆∼ 3gF µB Brot (−3).
3gF µB Brot
Brot
(3gF µB Brot )2(5.18)The tipping angle α is zero when the ions are at the center of the trap, but any displacement awayfrom center in the axial direction and the ions can sample the field inhomogeneities. Because on
average the ions are at the center of the trap, α = 0 and there is no systematic correction to be
made. The uncertainty on the axial displacement is 2.5 mm, which we can determine from the data
blocks where axial displacement was recorded. Using the measured value of Tz βz = (2/3)0.136(6)Kelvin/cm2 from eq. (4.43) we can get an uncertainty on α, σα ≈ 10−7 . The relevant parameter,(3σα ωrot /3gB µB Brot ) = 0.00375 for ωrot = 250 kHz, Erot = 24 V/cm, and Ramsey fringe frequencyof 20Hz. We can of course calculate the uncertainty on this parameter for every single block inthe data set and take the rms value. When we do this the systematic error contribution from the
Berry’s phase/axial magnetic field term from eq. (5.17) is 3.46 µHz, and the contribution due tothe second order effect from eq. (5.18) is 63.9 nHz. In total this is about a 3 × 10−31 e·cm effect.

5.1.5.4~ · ~r
The effect that looks like B~ r has worst case limits placed on the contribution
The unknown effect that is proportional to B·~
to f BD using the data in fig. 4.13. In this worst case limit we have removed the contribution to f B
due to ∆δ∆/(f 0 )2 , which we empirically observe is also true for this effect. The worse case limit
we place on any residual effect that remains after the correction due to the partially understoodeffect. This limit is f BD = −0.0047(28)f B . The correction and uncertainty of this correction for124
any residual effects are~f B·~r = 0.0047f B ,(5.19)

22 B
σf2B·~~ r = (0.0047) σf B + (−0.0028) f ,(5.20)

which yields a correction and uncertainty of −0.102 ± 0.112 mHz, which is consistent with zero.
The uncertainty of this systematic is dominated by the block rms value of f B for the data set. If wereduce the rms value of f B to 30 mHz via an active servo of the non-reversing magnetic field thenthe uncertainty of this systematic term drops to the 15 µHz level, or about 10−30 e·cm. We havealready established the Ramsey fringe frequency has no effect on this systematic. Additionally,
more careful measurements of the residual effect on f BD due to large f B yield better uncertainties
on this characterization and reduce the systematic error. A potential error in this analysis cancome from correlations of the characterized worse cast upper bound, f BD = 0.0047(28)f B , with
the non-reversing magnetic field. This could potentially be limited by taking data to place upperbounds in the presence of deliberately applied non-reversing magnetic fields.

5.1.5.5~ × ~r
The effect that looks like B~ × ~r has limits placed in its contribution to the EDM
The unknown effect that looks like B
channel by the measurement made in fig. 4.15. These limits are f BD = −0.0018(11)f BR . We canuse the mean value of f BR to calculate the appropriate systematic correction and its uncertainty,~

f B×~r = 0.0018f B ,(5.21)

22 BR,
σf2B×~~ r = (0.0018) σf BR + (−0.0011) f(5.22)

which yields a correction and correction uncertainty of 66.4 ± 64.8 µHz. This is dominated by theblock rms value of f BR , which is about 1 Hz for this set of data. In the past several months we have
implemented a more rigorous ion cloud trap displacement shim routine in the experiment control.
Under this new control the block rms value of f BR for this data is 0.4 Hz. With f BR reduced to
0.4 Hz the systematic effect in f BD will be about 25 µHz, and it can be suppressed even more, as
discussed in section 4.3.2, to well below the 10−30 e·cm level.

125
5.1.6

Statistics

We note that several of the systematic errors summarized in section 5.1.5 decrease as theRamsey fringe frequency increases. On the other hand the statistical sensitivity decreases as theRamsey fringe frequency increases, as explained in section 3.2.4.1. For coherence times τ  τ3 ∆1
the statistical sensitivity of each data point in a Ramsey fringe σf ∼ 1/τ ∼ f . For the set ofdata considered in this section, the ∼ 20 Hz Ramsey fringe frequency used happens to provide agood balance between statistical uncertainty and systematic uncertainty. Optimizing this trade-off
depends on how much data we intend to take in a run and also how well we think we can control√
systematic errors. For the rate of 30 mHz/ hour quoted in chapter 3, and assuming 100 hours ofdata, this is a 3 mHz statistical uncertainty. The dominant systematic is currently non-reversing
magnetic field, and is about 0.17 mHz. If we were faced with the problem of systematic errorsthat were drastically larger than the statistical uncertainty we could trade statistical uncertainty
by decreasing Ramsey fringe frequency to decrease the systematic uncertainty.

5.1.7

Total EDM measurement

The statistical uncertainty in the EDM measurement is 3.11 mHz after the correction in theprevious section has been applied. The systematic errors are cataloged in table 5.1. The total
systematic uncertainty is 172 µHz. We can compile these into an EDM measurement,
2de Eeff = 0.532 ± 3.12stat ± 0.173syst ± 1.13blind mHz,

(5.23)or translated to units of e·cm using Eeff = 23.3 GV/cm,
de = 0.472 ± 2.77stat ± 0.153syst ± 1.00blind × 10−28 e·cm.

(5.24)

It is also customary to quote measurements consistent with zero as an upper bound. Analyticallyderiving the single sided upper bound requires making various assumptions about the shape of the
distribution of the uncertainties. The best measure of the distribution of these terms we possess
is actually the data itself. We will use a numerical technique to calculate the single sided upper126
bound, the bootstrap method[24]. This is a Monte Carlo method, where we generate fictitious
data sets by sampling from f BD with replacement. By sampling our own data set we are able
to generate random numbers that are guaranteed to match the sample distribution. With thisfictitious data set we can calculate the value of de with systematic corrections. We can then repeat
this procedure 105 times and calculate what value is greater than de for 90% of the fictitious data
sets. This places an upper bound on the EDM of |de | < 4.6 × 10−28 e · cm with 90% confidence(where we are now quoting a 90% confidence interval, rather than 1σ errorbars). This improveson the 2011 limit set by Imperial[36], and is less than a factor of 5 away from the current best
limit set by ACME in 2014[7, 27]. This has the potential to be improved upon in the near future.
This limit was created using a set of data taken with the intent to study systematic errors. It
wasn’t optimized for statistical sensitivity, and in contrast to taking care to decrease the impact ofsystematic errors, we specifically tried to increase the effect of systematics. In the next section wewill discuss improvements planned for future experiments.

Table 5.1: Summary of systematic errors. The blind offset is included in this table and treated as asystematic effect. Entries also included at the bottom of the table are the statistical measurementof f BD and the total corrected EDM measurement.
Systematic Effects
blind
non-reversing magnetic field (∆δ∆)
non-reversing magnetic field (δg/gF )axial magnetic field
field inhomogeneity ∼ α1
field inhomogeneity ∼ α2
~ · ~rfB ∼ B~ × ~rf BR ∼ B
Total Systematics

Correction (mHz)
-3.38
0.238
-0.04300.000122-0.102
0.06640.160

Uncertainty (mHz)
1.13
0.114
0.01000.000103
0.003460.0000639
0.112
0.06480.173

Measurementf BD
Total EDM Measurement

Mean (mHz)
3.72
0.496

Uncertainty (mHz)
3.12
3.32

127

5.2

Future direction of the experiment
In this previous section we present a preliminary measurement, but work is not finished. We

arrived at this preliminary measurement with data that was almost entirely meant for studying
systematics. There are a number of improvements that are possible in the very short term, near
future, and distant future that should make it possible to meet and then exceed the current best
limit placed on the electron EDM.

5.2.1

Improvements to current experiment

There are several things that can be done to improve the statistical sensitivity of the current
experiment, and also to limit the susceptibility to systematic errors. In the discussed data set we√managed to achieve a statistical sensitivity of almost 40 mHz/ hour. It should be possible to do
significantly better than this without any material changes to the physical experiment. Blocks ofdata taken under so-called baseline conditions, which are block taken with parameters meant toinvestigate the statistical sensitivity of the experiment, i.e. a data point that was not investigating√
systematic effects, have an average sensitivity that is 22 mHz/ hour, which is already a factor of
two better than this measurement. Additionally, we can optimize our data acquisition for statisticalsensitivity by only taking data on the zero-crossings of Ramsey fringes, fixing the initial phase of
the Ramsey fringe fit, and only taking data at long times of Ramsey fringes. Succeeding in theseobjectives, we can hope to do even better than our current measurement sensitivity by a factor of
two, as discussed in chapter 3. This should allow us to potentially achieve statistical uncertaintyof ∼ 1 mHz for 100 hours of data.
There are also easy gains possible on the front of systematic errors. Many of the limiting
systematics in table 5.1 are essentially a result of large f B readings, in particular the non-reversing~ · ~r effect. It should be possible tomagnetic field systematic and the not yet understood f B ∼ B
actively shim the value of f B to within 30 mHz of zero by applying a non-reversing magnetic field
gradient. We currently perform this shim sporadically whenever the value of f B for a baseline

128
set of data becomes too large, but it could easily be added into the experimental sequence after
every block. This should reduce both of these systematics to below the 1 mHz level. Additionally
we can null out the ambient DC magnetic field in the lab with 3 pairs of coils to reduce the~ · ~r effect. We are also in the process of improving the experiment’s magnetometry, via thefB ∼ B
improvements to the cluster of magnetometers described in section 4.2.3. This will allow us to getbetter measurements of the ambient magnetic fields and magnetic field gradients, which will give
us greater confidence in our abilities to null out Earth’s field and shim the non-reversing magnetic
field gradient.
We are currently in the process of preparing the experiment to run for 100 continuous hours
in a mode that is optimized for statistical sensitivity and will be more resilient to systematic errors.
We hope to learn a great deal about how far we can push the statistical sensitivity of this iterationof the experiment, and we look forward to uncovering new potential sources of systematic errorsthat are only visible below the 1 mHz threshold.

5.2.2

Improvements for next generation experiment

Ultimately we would like the statistical sensitivity of this experiment to persist well belowp
the 1 mHz level. The figure of merit for sensitivity is Eeff τ Ṅ , the product of effective electric field,
coherence time, and count rate. The effective electric field simply is what it is for our species, andthe coherence time τ is already approaching the lifetime of the 3 ∆1 electronic state. The remaining
sensitivity gains will come from improvements to our count rate. The first area we will focus on is3
the efficiency of transfer of ions from the 1 Σ+
0 state where they are created to the ∆1 state where we

perform our spectroscopy. As discussed in section 2.1.4, this is currently 40%. By employing pulsed
tapered amplifiers[70] we hope to achieve enough power to be able to perform stimulated Raman
adiabatic passage (STIRAP)[8] on the ions, where we hope to realize ∼ 90% transfer efficiency.
This will yield a factor of 2 increase in count rate, as well as potentially increasing Ramsey fringe
contrast.
A second avenue where we hope to improve count rate is by designing and building a 3rd

129
generation ion trap. The current 2nd generation trap was designed to optimize the efficiency
of collection of fluorescence photons, which necessitated most of the surface in the trap to behighly reflective and metallic. We have since switched to using photo-dissociation as our detection
technique, obviating many of the current trap features. By painting the trap black with a conductive
paint we hope to minimize the influence of patch charges accumulating on trap electrodes, allowing
us to achieve better Erot field uniformity and to ramp down the confinement of the ion trap. Better
Erot uniformity will improve systematics, and the ability to ramp down the trap confinement willallow us to load more ions into the trap without increasing ion density, an effect that limits theRamsey fringe coherence time. By adiabatically ramping down the trap frequencies a factor of 3 in√each direction we can increase the trap volume by a factor of 27, increasing possible ion numberin the trap by a factor of 5.
An additional planned improvement includes changing how we detect the ions of dissociation.
Currently we dissociate HfF+ ions and count the Hf+ products. Because of the inefficiencies in thevarious processes of creating ions, state transfer, detection, etc. we might end up counting only
tens of Hf+ ions in the presence of ∼1000 HfF ions. The dynamic range of the detector is notgood enough to count both Hf+ and HfF+ , so we count the Hf+ and allow the HfF+ mass peakto saturate. This is discarding valuable information that we could use to normalize out technicalnoise that occurs during the neutral molecule creation, ionization, and state transfer processes.
By adding a reflectron in our time of flight (TOF) mass spectrometer we can try to direct the
Hf+ and HfF+ TOF packets onto separate detectors with different gains, allowing us to accurately
count both the number of un-dissociated HfF+ ions and the number of dissociation Hf+ products.
Additionally, we can also try to use this reflectron to stretch the Hf+ ion packet out in time, allowing
us to operate the ion detector in digital mode, rather than analog mode. The advantage of this isthat the digital counting mode has better counting statistics, because the signal is not convolved
with an ion peak distribution function.

130
5.2.3

Potential for ThF+

Earlier we mentioned that the value of Eeff is fixed, but this can be tuned slightly with the
choice of a different molecular species. The diatomic ion ThF+ is isoelectronic to HfF+ , but withthe advantage of many more protons in the thorium nuclear, enhancing the effective electric field
to 36(2) GV/cm[47, 66, 21], roughly 1.5 times larger than HfF+ . This factor goes directly into thepotential statistical sensitivity. Moreover, the 3 ∆1 electronic state is the ground state for ThF+ [32],which means that the state lifetime will be limited by excitation from blackbody radiation to excitedrotational and vibrational states. With the right ion trap design this could potentially allow us
to achieve Ramsey fringes with > 2s coherence times. The downside is that very little is known
about ThF+ that is useful for an EDM measurement. We currently do not have techniques forstate-selective ionization, state-selective readout, or state transfer, but experiments are currently
underway to investigate these possibilities.

5.3

Conclusion
In conclusion, we present a preliminary upper bound of the electron EDM of |de | < 4.6 ×

10−28 e·cm with 90% confidence. This limit is derived from data taken with the intent of characterizing systematic shifts of the EDM, and moreover is still blinded. As the measurement appears to
be statistics limited, there is a clear short term path to decreasing the upper bound to the 1 × 10−28
level by simply taking data in a more efficient manner. Furthermore, signal enhancement provided
by improving state transfer and by constructing a new ion trap capable of storing significantly
larger numbers of ion should be able to perform a measurement of the electron EDM at the several
10−29 e·cm level.

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Appendix A

Typical experiment parameters

Table A.1: Typical parameter values for the experiment. We have also collected data at othervalues of Erot , ωrot , ∇B, etc., which change many of the parameters below.

N
m
Erot
ωrot
ωrf
ωsec
T
Trelaxed
rrot

Typical Value2000
199 amu
24 V/cm
2π × 250 kHz
2π × 50 kHz
2π × (5, 4, 1.5) kHz
30 Kelvin
10 Kelvin
0.47 mm

vrot
∇B
Brot
gF

740.8 m/s
40 mGauss/cm
1.88 mGauss+0.00305(10)δg/gF-0.00099(1)

f0
τ
ωef
EHF
dmf

23.7 Hz1s
2π × 740(40) kHz-46.6(1) MHz
1.805(8) MHz/(V/cm)∆

3.84 Hzδ∆-1.34 HzfD
Eeff
2de Eeff-0.217 Hz
23.3 GV/cm
1.13 mHz/(10−28 e·cm)

Source

eErot
2
mωrot
eErot
mωrot

∇Brrot[45]
u −g l )(gFFu +g l )(gFF

3gF µB Brot
[18]
1.411(6) e a0

3
ωrot
27ωef dmfErot
dmf Erot
3
8∆
EHF
∆δ∆
3gF µB Brot

[26]

Description
number of trapped ionsmass of 180 Hf19 F+
rotating electric fieldrotational micromotion frequency
ion trap rf frequency(x, y, z) trap secular frequencies
initial ion cloud temperaturerelaxed trap temperature
rotational micromotion radiusrotational micromotion velocity
magnetic quadrupole gradientrotating magnetic field
3 ∆ , J = 1, F = 3/2 g-factor
1
differential g-factor
mean Ramsey fringe frequency
Ramsey fringe coherence time3 ∆ Λ-doubling1
3 ∆ hyperfine splitting
1
molecule frame dipole moment
rotation-induced mixing
differential rotation-induced mixing
doublet odd frequency combination
effective electric field
electron EDM splittingAppendix B

Matlab code to generate linear combinations channels

1 function lcom table = generate lcoms ( varargin )2 %GENERATE LCOMS
g e n e r a t e l i n e a r c o m b i n a t i o n s o f EDM data .
3%l c o m t a b l e = GENERATE LCOMS( data ) p r o d u c e s t h e l i n e a r c o m b i n a t i o n s o f
4%t h e v a r i o u s c h a n n e l s i n t h e t a b l e data . The f u n c t i o n has t h e f o l l o w i n g
5%three input f l a g s :6%7%l c o m t a b l e = GENERATE LCOMS( data , ' s w i t c h e s ' , s w i t c h e s )8%
w i l l c a l c u l a t e the l i n e a r combination over the subset o f s w i t c h e s
9%p r o v i d e d i n ' s w i t c h e s ' . GENERATE LCOMS w i l l produce 2ˆN l i n e a r
10 %c o m b i n a t i o n s , where N i s t h e l e n g t h o f ' s w i t c h e s ' . The i n p u t
11 %' s w i t c h e s ' must be a c e l l a r r a y o f column l a b e l s f o r t h e t a b l e
12 %data . The d e f a u l t v a l u e o f ' s w i t c h e s ' i s { ' Brot ' , ' Doublet ' , ' Rotation ' }
13 %
14 %
l c o m t a b l e = GENERATE LCOMS( data , ' a v g i n d e x ' , i n d )15 %w i l l c a l c u l a t e the l i n e a r combinations taking averages over16 %p a r a m e t e r s whose v a l u e s change more q u i c k l y than we might wish t o
17 %d i f f e r e n t i a t e i n t o i n d i v i d u a l b l o c k s . The v a r i a b l e ' ind ' must be a18 %c e l l a r r a y o f column l a b e l s t o a v e r a g e o v e r . The d e f a u l t v a l u e o f
19 %' ind ' i s { ' h f f ' } , where we a v e r a g e o v e r t h e number o f HfF c o u n t s .
20 %
21 %
l c o m t a b l e = GENERATE LCOMS( data , ' f i l t e r ' , i n d )2223 %% Parse the inputs
24 p = i n p u t P a r s e r ;25 addRequired ( p , ' data ' ,@ i s t a b l e ) ;26 addParameter ( p , ' s w i t c h e s ' , { ' Brot ' , ' Doublet ' , ' R o t a t i o n ' },@ i s c e l l ) ;27 addParameter ( p , ' a v g i n d e x ' , { ' h f f ' },@ i s c e l l ) ;28 p a r s e ( p , v a r a r g i n { : } ) ;29
30 data = p . R e s u l t s . data ;31 a v g i n d e x = p . R e s u l t s . a v g i n d e x ;32 s w i t c h e s = p . R e s u l t s . s w i t c h e s ;3334 %% Descriptors and Symbols
35 % d e f i n e symbols f o r each s w i t c h
36 k n o w n s w i t c h l a b e l s = { ' Brot ' , ' Doublet ' , ' R o t a t i o n ' } ;37 k n o w n s w i t c h s y m b o l s = { 'B ' , 'D ' , 'R ' } ;138
38 % d e f i n e symbols f o r each c h a n n e l
39 c h a n n e l s = { ' f r e q ' , ' phase ' , ' o f f s e t ' , ' c o n t r a s t ' , ' tau ' } ;40 c h a n n e l s y m b o l s = { ' f ' , ' p ' , 'O ' , 'C ' , ' t ' } ;41 % check f o r unknown s w i t c h e s . I f we f i n d any t h i s f u n c t i o n s h o u l d be
42 % updated !
43 i f ˜ a l l ( a r r a y f u n (@(x ) ismember ( x , k n o w n s w i t c h l a b e l s ) , s w i t c h e s ) )44
e r r o r ( ' s o r r y , g e n e r a t e l c o m s c a n t r e c o g n i z e on o f your s w i t c h l a b e l s ' ) ;45 end
46 % c o n c a t e n a t e a l l o f t h e s w i t c h symbols i n t o a s t r i n g .
47 s w i t c h s y m b o l s = s t r j o i n ( known switch symbols , ' ' ) ;48
49 %% Generate the parity sums
50 % t u r n t h e r e c o r d e d s w i t c h s t a t e o f 0 o r 1 i n t o −1 o r +1.
51 d a t a p a r i t y = ( −1) .ˆ(1+ data { : , s w i t c h e s } ) ;52 % p i c k out a matrix o f t h e measured v a l u e o f each data c h a n n e l ( f o r s p e e d ) .
53 d a t a c h a n n e l s = data { : , c h a n n e l s } ;54 % f i n d t h e columns t h a t have t e x t data
55 t e x t c o l u m n s = data . P r o p e r t i e s . VariableNames ( a r r a y f u n (@( i ) i s c e l l ( data { 1 , i } ), 1 : s i z e ( data , 2 ) ) ) ;56 % c o n s t r u c t t h e columns t h a t we w i l l u s e t o i n d e x i n d i v i d u a l b l o c k s .
57 b l o c k i n d e x = s e t x o r ( data . P r o p e r t i e s . VariableNames , [ s w i t c h e s c h a n n e l s c e l l f u n(@(x ) [ ' s ' x ] , c h a n n e l s , ' u n i ' , f a l s e ) a v g i n d e x ] ) ;58 b l o c k i n d e x = b l o c k i n d e x ( ˜ ismember ( b l o c k i n d e x , t e x t c o l u m n s ) ) ;59 % put t h e t e x t columns a t t h e b e g i n n i n g o f t h e output t a b l e
60 b l o c k i n d e x = [ t e x t c o l u m n s b l o c k i n d e x ( ˜ any ( i s n a n ( data { : , b l o c k i n d e x } ) ) ) ] ;61 % a s s i g n a b l o c k i n d e x t o each row o f i n p u t data
62 [ ˜ , ˜ , i n d e x ] = unique ( data ( : , b l o c k i n d e x ) ) ;63 % Choose o t h e r columns t o s a v e . These w i l l a l s o be a v e r a g e d o v e r
64 columns = s e t x o r ( data . P r o p e r t i e s . VariableNames , [ t e x t c o l u m n s s w i t c h e s c h a n n e l s
c e l l f u n (@(x ) [ ' s ' x ] , c h a n n e l s , ' u n i ' , f a l s e ) ] ) ;65 % Generate a p t a b l e o f p a r i t i e s . This i s a t a b l e o f 1 s and 0 s t h a t
66 % d e s c r i b e s a l l t h e v a r i o u s ways t o combine t h e s w i t c h e s f o r a b l o c k . I t
67 % has 2ˆN rows and N columns , where N i s t h e number o f s w i t c h e s .
68 p a r i t y = l o g i c a l ( d e 2 b i ( ( 1 : 2 ˆ l e n g t h ( s w i t c h e s ) ) −1 ') ) ;69 % Generate symbols f o r a l l t h e v a r i o u s c o m b i n a t i o n s f o r each c h a n n e l . This
70 % s h o u l d have l e n g t h M∗2ˆN, where M i s t h e number o f c h a n n e l s ( f , phi , . . . )71 % and N i s t h e number o f s w i t c h e s . This w i l l i n c l u d e e n t r i e s l i k e ' f ' ,72 % ' fBD ' , ' tD ' , e t c .
73 l i n e a r c o m b i n a t i o n s = c e l l ( 1 , l e n g t h ( c h a n n e l s ) ∗ s i z e ( p a r i t y , 1 ) ) ;74 f o r c =1: l e n g t h ( c h a n n e l s )75
f o r p=1: s i z e ( p a r i t y , 1 )76
l i n e a r c o m b i n a t i o n s { ( c −1)∗ s i z e ( p a r i t y , 1 ) + p} = [ c h a n n e l s y m b o l s { c }switch symbols ( parity (p , : ) ) ] ;77
end
78 end
79 % f i n d which column i n t h e data each l i n e a r c o m b i n a t i o n r e f e r s t o .
80 l i n e a r c o m b i n a t i o n c o l u m n s = a r r a y f u n (@(x ) f i n d ( ismember ( c h a n n e l s y m b o l s , x ) , 1 ,' f i r s t ' ) , c e l l f u n (@(x ) x ( 1 ) , l i n e a r c o m b i n a t i o n s ) ) ;81 % l c o m t a b l e i s t h e v a r i a b l e we w i l l s t o r e a l l o f t h e v a r i o u s l i n e a r
82 % c o m b i n a t i o n s .
83 lcom columns = [ c e l l f u n (@(x ) s t r c a t ( ' s ' , c h a n n e l s y m b o l s { ismember ( c h a n n e l s , x ) } ), channels , ' uni ' , f a l s e ) l i n e a r c o m b i n a t i o n s ] ;84 n rows = l e n g t h ( min ( i n d e x ) : max( i n d e x ) ) ;139
85 l c o m t a b l e = a r r a y 2 t a b l e ( z e r o s ( n rows , l e n g t h ( columns ) ) , ' VariableNames ' , columns);
86 l c o m t a b l e ( : , t e x t c o l u m n s ) = c e l l 2 t a b l e ( repmat ( { ' ' } , n rows , l e n g t h ( t e x t c o l u m n s)));
87 l c o m t a b l e ( : , lcom columns ) = a r r a y 2 t a b l e ( z e r o s ( n rows , l e n g t h ( lcom columns ) ) , 'VariableNames ' , lcom columns ) ;88 % I t e r a t e through each b l o c k o f data by b l o c k i n d e x .
89 f o r i = min ( i n d e x ) : max( i n d e x )90% e x t r a c t t h e s u b t a b l e o f data t h a t d e s c r i b e s b l o c k with i n d e x i91
b l o c k = data ( i==index , : ) ;92% f i n d t h e p a r i t y t h a t d e s c r i b e s how t o c o b i n a t e t h e data c h a n n e l s i n
93% t h i s block i n t o the l i n e a r combinations
94
b l o c k p a r i t y = d a t a p a r i t y ( i==index , : ) ;95% f i n d t h e matrix o f n u m e r i c a l data t h a t i s t h e data c h a n n e l s f o r t h i s
96% subblock97
b l o c k c h a n n e l s = d a t a c h a n n e l s ( i==index , : ) ;98
n = s i z e ( blo ck , 1 ) ;99% c r e a t e a t a b l e , ' row ' , t h a t i s t h e a v e r a g e o f t h e columns we want100
% ( t h i n g s l i k e i o n number and d a t e ) . This i s a temporary t a b l e f o r t h i s
101% block
102
row = c e l l 2 s t r u c t ( n u m 2 c e l l ( mean ( b l o c k { : , columns } , 1 ) ) ' , columns ) ;103
f o r j =1: l e n g t h ( t e x t c o l u m n s )104
row . ( t e x t c o l u m n s { j } ) = b l o c k { 1 , t e x t c o l u m n s { j } } ;
105
end
106
% append t o t h i s t a b l e t h e e r r o r b a r s combined i n q u a d r a t u r e f o r each107
% channel , t o g e t t h i n g s l i k e ' s f ' , ' sO ' , e t c . .
108
for c = 1: length ( channels )109
row . ( [ ' s ' , c h a n n e l s y m b o l s { c } ] ) = s q r t ( sum ( b l o c k . ( [ ' s ' , c h a n n e l s { c } ] ). ˆ 2 ) ) /n ;
110
end
111
% Now we l o o p through a l l t h e l i n e a r c o m b i n a t i o n s we would l i k e t o make112
for p = 1: length ( linear combinations )113% lcom w i l l be s h o r t h a nd f o r t h e s p e c i f i c l i n e a r c o m b i n a t i o n we a r e114% c o n s i d e r i n g . I t i s a t e x t s t r i n g t h a t d e s c r i b e s t h e c h a n n e l and115% a l s o t h e p a r i t y o f t h e c o m b i n a t i o n s , e . g . fBD
116
lcom = l i n e a r c o m b i n a t i o n s {p } ;117% Channel data i s a v e c t o r o f t h e data f o r t h a t p a r t i c u l a r channel ,118% f o r t h e p a r t i c u l a r b l o c k we a r e c o n s i d e r i n g . lcom ( 1 ) h o l d s t h e
119% c h a r a c t e r t h a t i n d e n t i f i e s t h i s channel , e . g . ' f '120channel data = block channels ( : , linear combination columns (p) ) ;121% Now l o o p through t h e r e s t o f t h e c o m b i n a t i o n t e x t s t r i n g t o
122% f i g u r e out t h e p a r i t y o f t h e sums . c h a n n e l d a t a i s m u l t i p l i e d by123
% ( −1) anytime i t c o i n c i d e s with t h e ' 0 ' s t a t e o f any s w i t c h .
124
f o r s = 2 : l e n g t h ( lcom )125l c o m p a r i t y b i t s = b l o c k p a r i t y ( : , s w i t c h s y m b o l s==lcom ( s ) ) ;126
channel data = channel data .∗ lcom parity bits ;
127
end
128
% append t h i s onto temporary t a b l e f o r t h i s b l o c k
129row . ( lcom ) = sum ( c h a n n e l d a t a ) /n ;
130
end
131
% Append t h e t a b l e f o r t h i s b l o c k onto t h e o v e r a l l t a b l e
132
l c o m t a b l e ( i , : ) = s t r u c t 2 t a b l e ( row ) ;133 end140
134 l c o m t a b l e = [ t a b l e ( ( min ( i n d e x ) : max( i n d e x ) ) ' , ' VariableNames ' , { ' i n d e x ' } )lcom table ] ;135
136 %% Find the EDM channel and apply the blind .
137 % i f t h e b l i n d e x i s t s , l o a d i t i n t o memory and apply i t t o t h e fBD c h a n n e l .
138 i f e x i s t ( 'X: \ BLIND\BLIND ' , ' f i l e ' )139
l c o m t a b l e . fBD = l c o m t a b l e . fBD + f r e a d ( f o p e n ( 'X: \ BLIND\BLIND ' ) , 1 , ' d o u b l e ') ; fclose ( ' all ') ;140 % i f t h e b l i n d doesn ' t e x i s t , don ' t l o o k a t t h e data u n b l i n d e d ! Apply a
141 % v e r y l a r g e b l i n d i n s t e a d (4+/−1 10ˆ27 e−cm) .
142 e l s e
143% 2∗(10ˆ −27 cm) ∗ ( 2 3 . 3 e9 eV/cm) /h = 0 . 0 1 1 2 6 7 8 3 0 4 h e r t z
144
edm27 = 0 . 0 1 1 2 6 7 8 3 0 4 ;145
l c o m t a b l e . fBD = l c o m t a b l e . fBD + normrnd ( 4 ∗ edm27 , edm27 ) ;146
f p r i n t f ( 'BLIND not found . Using new random b l i n d 4+/−1 10ˆ27 e−cm\n ' ) ;147 end
148 end