Towards Measuring the Electron Electric Dipole Moment
Using Trapped Molecular Ions
by
R.P. Stutz
B.S., University of Kansas, 1998

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
2010This thesis entitled:
Towards Measuring the Electron Electric Dipole Moment Using Trapped Molecular Ions
written by R.P. Stutz
has been approved for the Department of Physics

Eric Cornell

Jun YeDate

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

iii
Stutz, R.P. (Ph.D., Physics)
Towards Measuring the Electron Electric Dipole Moment Using Trapped Molecular Ions
Thesis directed by Prof. Eric Cornell

Permanent electric dipole moments have been the subject of experimental investigation forthe past sixty years, as they entail the breaking of fundamental symmetries and provide a sensitive
probe for physics beyond the Standard Model. This thesis describes an experiment aimed atmeasuring the electron electric dipole moment (eEDM) using trapped molecular ions. The 3 ∆1level of certain diatomic ions are desirable in eEDM searches due to their high polarizability, large
eEDM enhancement factor, and relative insensitivity to magnetic fields. Ions allow for simple
trapping and long interrogation times, but require a time-varying electric bias field in order toprobe the eEDM.
I will discuss the criteria for molecular ions in our experiment and our current candidates. A
laser-ablation supersonic-expansion beam source has been developed to create and cool molecularions. These ions have been loaded into a linear rf Paul trap and alternative photoionization methods
for state-selective ion creation have been tested. Various experimental methods for performing thenecessary spin resonance measurement are discussed. Sources of both decoherence and systematic
errors have been identified and estimated. The experiment described in this thesis should be capableof a factor of 30 improvement on the current limit of the eEDM.

Dedication

To my brothers, Christopher and Curtis.

v

Acknowledgements

I would first like to thank my advisor, Eric Cornell. His enthusiasm for physics was contagious
and he always seemed to have the most powerful way to view any physics problem in front of him.
Unfortunately, Eric was dealt a large dose of adversity during my studies here. Abraham Lincoln
once said,“Nearly all men can stand adversity, but if you want to test a man’s character, give
him power.” Well, I’ve also observed Eric with power. In fact, I can think of few better ways toobserve someone having power than to be their graduate student. In dealing with both adversity
and power, Eric’s true character was revealed. I found him to not only be an amazing scientist,
but also a thoroughly decent human being. I couldn’t have asked for a better advisor.
Carl Wiemen and Debbie Jin have both been exceedingly helpful during the course of mystudies. Carl stepped into the role of advisor during those difficult days of 2004. His advice on
topics from experimental techniques to improving the way I presented work was always useful.
Debbie had a tendency to always ask the right question that kept us thinking about the important
issues in our experiment.
John Bohn and his graduate student Ed Meyer provided a great deal of theory support.
Their insights were crucial in finding credible molecular ion candidates for our experiment. They
performed many calculations and a lot of the ideas of how the experiment will ultimately be
performed came from discussions with them. The group from St. Petersburg, Anatoly Titov,
Mikhail Kozlov, Alexander Petrov and Timur Iseav, also did many calculations that helped guideour way. Also, many thanks to them for showing me St. Petersburg.
The community of scientists within JILA is truly impressive, and no matter what issue camevi
up an expert was always down the hall. Jun Ye has been involved with the experiment from the
beginning, and has given much helpful advice on everything from lasers to molecular beams. I would
also like to thank Jun and his group for allowing us to win so many of those JILA Cup challenges,
although it is with great shame that I will likely be leaving JILA without the cup being possessedby its rightful owners, the Cornell and Jin groups. I would like to thank Carl Lineberger, David
Nesbitt and their students for teaching us so much about molecular spectroscopy and molecularbeams. Heather Lewandowski, a.k.a. Coach Lew, and her group have been a great resource on all
things molecules as well.
Thanks to Robert Field for teaching us everything we know about autoionization of Rydberg
states. He has been very gracious in providing us advice and his trips to Boulder were always
greatly anticipated.
I’ve had the great pleasure of working with so many talented graduate students and postdocs
during my time at JILA. Thanks to Aaron Leanhardt who helped with a great deal of the workin this thesis. He has an amazing drive to do physics and his tireless work rate and great gifts asa scientist were inspiring to observe. I’d also like to thank him for road trips to Kansas to watch
opening day baseball and other such distractions. Laura Sinclair was here from almost the very
beginning of the experiment and has been a great lab mate and friend. Patrick Maletinsky visited
JILA for only a short time, but he made a big impact and was crucial in the first ion trapping
experiments. Herbert Loser visited us on a sabbatical and was a great help in making our LabView
code much more professional and also with photoionization experiments. Huanqian Loh has been an
absolute joy to work with, a great young scientist and a really nice person to be around. Matthew
Grau has made an impact right from the start and has a great future ahead of him. Tyler Yahn
has been an undergraduate researcher on the experiment and a great help.
I’d like to thank all the members of the “tri-group” for the helpful suggestions throughout
the years and for some great friendships. In particular, I would like to thank Josh Zirbel and Scott
Papp for being my surrogate lab mates during my first year. It was invaluable having them there
to bounce ideas off of or to ask the truly stupid questions you’d rather not take to your advisor as

vii
a first-year student. Thanks to all of the students and post-docs who I shared lab space with inB224/228, especially my roommate Giacomo Lamporesi and my good friend Shih Kuang “Zeke”
Tung. Thanks to Juan Pino, Jason Stewart, and John Gaebler for useful discussions, most of whichwere happily not about physics.
The amount of support we receive as graduate students in JILA is exceptional. Thanks to
all the folks in the machine shops, electronic shops, computing staff, supply office, and the office
staff. Special thanks to Hans Green, Todd Asnicar, Blaine Horner, Terry Brown, and Carl Sauer
for producing so much of the experimental apparatus described here. Also, Pam Leland was always
amazing in helping make things run smoothly.
I would have never made it to JILA without the help of many people along the way. Kenneth
Dinndorf and Harold Miller were my supervisors at Kirtland AFB and taught me all about lasers
and optics. They were also great mentors and friends and always encouraged me to pursue graduate school. David Besson, my undergraduate advisor at KU, taught me a great deal about how
experimental science is done. He was a great mentor and friend. Thanks to the “fab five physics
majors” at KU who I learned so much from during my undergraduate days while doing homework,
performing research, and enjoying life with them.
I couldn’t have done this without the love and support from all of my family. My mother andfather have always supported me, even when I decided to do crazy things like join the Air Force oreven crazier things like go to physics graduate school. I dedicate this thesis to my brothers, whowere always there for me and who taught me so much about life.
My greatest accomplishment during my graduate school career was convincing my loving andlovely wife Oraya to marry me. She has been a constant source of support throughout the writingof this thesis. I can’t wait for life’s next adventure knowing my best friend will be by my side.

viii

Contents

Chapter
1 Introduction [1]1

1.1

Motivation for Electric Dipole Moment Searches . . . . . . . . . . . . . . . . . . . .

1

1.2

Schiff’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3

Direction of de . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.4

Overview of Ongoing Experimental Work . . . . . . . . . . . . . . . . . . . . . . . .

7

1.5

Overview of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92 Molecular Structure and the Basic Spectroscopic Idea [1]

12

2.1

Molecular Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2

Choosing a Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3|Ω| = 1 vs. |Ω| = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4|Ω| = 1, J = 1 Λ-doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5

Electronic Level Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Ion Trapping

21

3.1

Ions in Two-Dimensional Time-Varying Quadrupole Fields . . . . . . . . . . . . . . . 21

3.2

3-D Ion Trapping, the Linear Paul Trap . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3

Pseudo-Potential Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4

Trapping Multiple Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.5

Our Ion Trap, v0.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

ix
3.6

Trap Drive Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7

Ion Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8

Early Ion Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.9

Second-Generation Ion Trap and Fields in eEDM Measurement . . . . . . . . . . . . 40

4 Electron Spin Read-Out

46

4.1

LIF Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2

Photodissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.3

4.2.1

Predissociation and CH+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.2

Photodissociation to a Repulsive Curve . . . . . . . . . . . . . . . . . . . . . 52

Light-Induced Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Creation, Cooling, and Trap Loading of Ions

65

5.1

Ion Cooling Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2

Buffer-Gas Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3

Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.4

Supersonic Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.5

Laser Ablation and HV Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.6

Measuring Beam Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.7

5.6.1

Translational Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.6.2

Laser-Induced Fluorescence in the Beam . . . . . . . . . . . . . . . . . . . . . 78

Loading Ions Into the rf Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.7.1

Ion Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.7.2

State-Selective Photoionization . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.7.3

Photoionization of HfF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6 Spectroscopy in Rotating and Trapping Fields [1]6.1

99

Basic Molecular Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

x6.2

Effect of Non-rotating Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . 102

6.3

Rotating Fields, Small-Angle Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4

Rotating Fields, Large-Angle Limit (Dressed States) . . . . . . . . . . . . . . . . . . 107

6.5

Frequency- or Phase-Modulation of Axial Oscillation . . . . . . . . . . . . . . . . . . 115

6.6

Creating Coherent Mixtures of mF = +3/2 and mF = -3/2 . . . . . . . . . . . . . . 119

6.7

Structure of the Measurements. What Quantities Matter . . . . . . . . . . . . . . . . 121

6.8

An Estimate of δgF =3/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.9

Dependencies on Erot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.10 Perpendicular B-Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.11 Stray Contributions to B|| : Uniform or Time-Varying B-Fields . . . . . . . . . . . . . 126
6.12 Stray Contributions to B|| : Static B-Field Gradients . . . . . . . . . . . . . . . . . . 131
6.13 Alternative Application of Brot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.14 Relativistic (Ion-Motion-Induced) Fields . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.15 Effect of RF Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.16 Systematic Errors Associated with Trap Asymmetries . . . . . . . . . . . . . . . . . 136
7 Other Sources of Decoherence, Collisions and Black-Body Radiation [1]
7.1

138

Ion-Ion Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.1.1

Mean-Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.1.2

Geometric Phases Accumulated During an Ion-Ion Collision . . . . . . . . . . 140

7.1.3

m-Level Changing Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.2

Ion-Neutral Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.3

Rotational and Vibrational Excitations from Black-Body Radiation . . . . . . . . . . 147

8 Conclusions and Sensitivity Estimate [1]
8.1

150

Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.1.1

Coherence Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

8.1.2

Eeff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

xi
8.1.3

Count Rate and Summary of Expected Precision . . . . . . . . . . . . . . . . 151

8.2

Accuracy, Systematic Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.3

The Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Bibliography

154

Appendix
A Typical Experimental Parameter Values

160

xii

TablesTable
1.1

Figure-of-merit comparison between several recently completed and ongoing eEDM
experiments. For ongoing experiments these numbers are subject to change and areoften order-of-magnitude estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1

Electrical inputs to ion trap driver box. . . . . . . . . . . . . . . . . . . . . . . . . . 36

xiii

FiguresFigure1.1

One-loop diagram that leads to a correction of the electron g-factor. The electroninteracts with a photon (γ) from an external field as well as a “virtual” photon. . . .

1.2

2

If an electron EDM exists, the orientation between the electron’s electric (de ) andmagnetic (µ) dipole moments will change under a parity (P) or time-reversal (T)transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

3

Feynman diagram from Ref. [2] for a three-loop contribution to the eEDM. The
sum of three-loop diagrams cancels to zero [3] and four-loop diagrams are needed to
produce a non-zero eEDM in the Standard Model. . . . . . . . . . . . . . . . . . . .

2.1

4

Energy levels of HfF+ in the 3 ∆1 , J = 1 state including hyperfine structure associated with the fluorine I = 1/2 nucleus. Λ and Σ are defined as the projection along
the molecular axis of the electronic orbital angular momentum, and spin, respectively.
Ω = Λ + Σ. (a) In zero electric field, the eigenstates of the system are states of good√√parity, |ei = (|Ω = +1i − |Ω = −1i)/ 2 and |f i = (|Ω = +1i + |Ω = −1i)/ 2,
separated by a small Λ-doublet splitting. (b) An electric field, Erot , mixes the parity eigenstates yielding states with well defined Ω. (c) A small magnetic field lifts
the degeneracy between states with the same value of mF Ω. A permanent electron
electric dipole moment further breaks this degeneracy, but with opposite sign for theupper (solid arrow) and lower (dotted arrow) transition. Energy splittings not to scale. 17

xiv
2.2

Potential energy curves for select states of HfF+ [4]. The b(1) and c(1) states are
well-mixed combinations of 1 Π1 ,3 Π1 , and 3 Σ−
1 states. . . . . . . . . . . . . . . . . . . 19

2.3

Molecular levels in HfF and HfF+ discussed throughout this thesis. (a) A schematicof levels in HfF+ useful in measuring the eEDM are shown. Ion population in
the 1 Σ ground state might be transferred, using STIRAP (blue), to the 3 ∆1 level
where precision Ramsey spectroscopy is performed to search for the eEDM. Afterthe Ramsey sequence, a spin-dependent transition (green) using σ-polarized light
can be used to measure spin populations, either through fluorescence (red) decay orby making a photo-dissociation transition (orange) to Hf+ + F. (b) A schematic oflevels useful in creating HfF+ via photoionization of HfF are shown. A transition(Orange) to an intermediate state of HfF is followed by a transition to a Rydberg
state with a vibrationally excited HfF+ core in the 1 Σ (red) or 3 ∆1 state, which canthen autoionize to the v=0 level of the respective states. This process is discussed
in Chapter 5.7. Energy splittings not to scale. . . . . . . . . . . . . . . . . . . . . . . 20

3.1

Electrodes used to produce a two-dimensional quadrupole field. For a quadrupole
mass spectrometer, Φ0 = (U − Vrf cos ωrf t)/2. Adapted from [5]. . . . . . . . . . . . . 23

3.2

Ion stability diagram in terms of the dimensionless trapping parameters a = M4eU
r2 ω2
0

rf

rf
and q = M2eV
2 . The bounds of stability are the βx,y = 0 and βx,y = 1 contours. . . .
r02 ωrf

3.3

25

Ions of different masses will have different stability regions when graphed in terms of
the DC (U ) and rf (Vrf ) trapping voltages. This graph is for a constant rf frequency
ωrf . By scanning along a line of constant U/Vrf and recording ion signal, a mass
spectrum is produced. Greater resolution is achieved, at the expense of sensitivity,
by increasing the slope of the scan line. . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4

The ax and ay trapping parameters are shifted in the case of the linear Paul trap
due to an axial trapping potential, Uz , and also from space charge effects due to atrapped ion density of ρ. aeff = 2M4e
r2 ω2
0

rf



ρr 2



U ∓ kU2 z ∓ ǫ00 . . . . . . . . . . . . . . . . 28

xv
3.5

Our ion trap design. Potentials are shown that create the necessary fields for trapping, mass spectrometry, and applying Erot . . . . . . . . . . . . . . . . . . . . . . . . 29

3.6

Image of our first ion trap. The electrodes are obscured by the grounded wire meshat R = 3.54 r0 . Note the channeltron ion detector mounted below the trap.

3.7

Axial trapping potentials for different central rod lengths, lc , holding the overall trap
length constant. Our first trap was built with lc = 88 mm.

3.8. . . . 34

. . . . . . . . . . . . . . 35

Schematic of an integrator used to sum and amplify our ion signal when dumping
our ion trap into a channeltron detector. Gains of either 1010 or 108 were typicallyused. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.9a) Ion number in trap after trap potential U has been ramped to various values as afunction of rf voltage. b)Plotting the onset of loss in the a, q stability diagram shows
the ion loss was associated with a β = 2/3 resonance, implying imperfections in our
trapping fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.10 Mass spectrometry performed in our ion trap showing a) the five dominant isotopesof xenon, data taken at an rf frequency of 100 kHz, with U/Vrf = 0.16 and b) the fivenaturally occurring isotopes of hafnium, data taken at an rf frequency of 250 kHz,and U/Vrf = 0.1695. Mass resolutions of better than 1 amu allowed us to distinguish
Hf+ from HfH+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1

Potential energy curves of a molecule containing predissociating states as a functionof the internuclear spacing R. The bound energy curve A dissociates to atoms, withat least one atom in an excited state. The repulsive curve B dissociates to atoms at
a lower energy. Couplings between states A and B lead to an avoided crossing. An
excited vibrational level of state A can tunnel through the resulting potential into
the continuum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

xvi
4.2

CH+ held in our ion trap appeared to react with CH4 gas. After trap loading, theions are initially CH+ , but after a 100 ms hold time there is no evidence of CH+ in
the trap. Instead a mass peak around 24 amu is found. This required trap loading
at lower CH4 pressures and made detection of CH+ predissociation more difficult.

4.3

. 51

Data showing photodissociation of CH+ to C+ and H through a predissociation
resonance. This data corresponds to a transition to the J = 3, v = 12, A1 Π stateof CH+ . This data was taken with a 5 second ion hold time, during which time 50
pulses, with 10mJ/pulse, from a frequency doubled dye laser were incident on the
ions.

4.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Lattice potential depth as a function of the laser detuning δωL divided by the onresonance single beam Rabi frequency ΩR , in the limit ΩR ≫ γn , with γn the naturallinewidth of the transition.

4.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Data showing ion heating in a Yb+ beam as the lattice laser was tuned over a Yb+
resonance at 27,061.8 cm−1 (369.4 nm). . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.6

Experimental set-up for light-induced heating measurements. Ions are created using
laser ablation of a metal target and are swept up in a supersonic expansion (morebeam details are found in Chapter 5). The ions are collimated using two skimmers totransverse temperatures < 500 mK. An ion lens is used to minimize the beam width
on the imaging MCP detector. An intense 1-D optical standing wave heats ions if
the laser frequency is near an ion resonance. A 45◦ electrode is used to mass-selections. The heated clouds are detected by an imaging MCP. . . . . . . . . . . . . . . . 61

4.7

Mass spectrometry data using the 45◦ deflection plate showing the ability to seperate
HfF+ ions from Hf+ and HfF+
2 . This was achieved while maintaining < 500 mKtransverse beam temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.8

Data from the imaging MCP showing a Yb+ cloud deflected by the mass-selecting 45◦
plate, with (a) the one-dimensional standing wave off, and (b) the one-dimensional
standing wave applied on resonance, causing heating of the ion beam. . . . . . . . . 64

xvii
5.1

Occupation fraction for rotational states with quantum number J, for 300 K (red)and 10 K (blue) rotational temperatures given a typical fluoride rotational constantof Be = 0.3 cm−1 in an Ω = 0 state. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2

Mass spectrum of ablation products from a nickel target in a xenon expansion. Clustering occurs when a Xe atom attaches itself to a Ni+ or Ni+
2 ion. . . . . . . . . . . . 71

5.3

Diagram showing the pulsed valve set-up. An ablation target rod was held justoutside the pulsed valve opening and was followed by an expansion cone with a 40◦
opening angle.

5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Number of a) Hf+ and b) HfF+ ions created, as a function of 1064nm laser ablationenergy. The channeltron detector was operated at a lower than normal voltage (1,400
V here vs 2,000 V normally) in order to keep the detector response linear.

5.5. . . . . 75

The density of HfF neutral molecules created in our molecular beam as a functionof 1064 nm laser ablation energy. This data was obtained using photoionization
techniques discussed in Chapter 5.7.3 to measure HfF density.

5.6. . . . . . . . . . . . 76

FIG data taken with the filament turned off, making the device only sensitive toions. This data was taken with Hf+ ions in a xenon expansion and corresponds toan ion beam temperature of approximately 1 K. . . . . . . . . . . . . . . . . . . . . . 79

xviii
5.7

Experimental setup. Laser ablation of a metal Hf target creates neutral Hf atoms and
Hf+ ions that react with SF6 to produce neutral HfF molecules and HfF+ molecularions, respectively (Eqns. 5.5 and 5.6). The molecules (both neutral and ionic) arecooled in a supersonic expansion with an Ar buffer gas. The molecular beam isilluminated with a pulsed dye laser beam and the resulting fluorescence is collectedwith a photomultiplier tube (PMT) yielding laser induced fluorescence (LIF) spectra(Fig. 5.9). At the end of the beamline, the ions can be loaded into an rf Paul trap
where the electron spin resonance experiment will be performed. The Paul trap
also acts as a quadrupole mass filter and ions of a particular mass/charge ratio aredetected with a microchannel plate (MCP) (Fig. 3.10). Additionally, the spatial
resolution of the MCP allows for the temperature of the ion cloud to be determinedfrom the detected cloud size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.8

LIF data taken with a cw Ti:Saph laser showing hyperfine structure in the 3 F2 → 3 D1transition of hafnium. Similar data was taken of the 3 F4 → 3 D3 transition as thefine structure was not completely cooled in the supersonic expansion. . . . . . . . . . 83

5.9

Data from an Ω = 3/2 → 3/2 transition in HfF. The smooth curve is the fit obtained
while ignoring the unresolved Q-branch lines in the 12-14 cm−1 region of this graph.
Small differences in the rotational constant Be allowed us to ascribe vibrational statesto the levels involved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.10 Schematic of HfF transitions seen from LIF in our molecular beam showing approximate relative intensities. Lines connected by blue(orange) arrows are believed tobe different vibrational transitions between the same Ω = 3/2(1/2) electronic levels, as suggested by slight differences in Be′ and Be′′ , the rotational constants in theupper and lower states respectively. Preliminary vibrational assignments are givenas (v ′ ,v ′′ ), again with the ′ (′′ ) denoting the upper(lower) state. The ground-statevibrational spacing ωe is measured to be 670 cm−1 , consistent with a previous measurement [6], while an excited electronic state has a measured ωe of 595 cm−1 .

. . . 86

xix
5.11 Mass spectrum of ions created in laser-ablation supersonic-expansion with a hafnium
ablation target and an expansion gas of 1% SF6 in 99% He. The ion number is a
non-linear function of the y-axis value, but a level of 0.4 corresponds to ∼ 100,000
ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.12 a) Spectrum of ion motions for values of trapping parameters a = 0 and q = 0.785
that can be used for operating an rf quadrupole as a mass-to-charge ratio sensitive
ion lens. b) Dots denote values of q, with a = 0, that can be used in operating ourion lens. c)Numerical simulation of ion motion for different ion starting positions
Ri , and two different singly-charged ions. The simulation was done with a = 0 and
q = 0.785 for the mass = 181 ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.13 (a) Schematic of the time-of-flight mass spectrometer built for our experiment. The
first plate is used to further collimate the neutral molecular beam and has a 6 mmdiameter opening. Photoionization occurred at the approximate location marked by*. Region (I) could be used in delayed-pulse field ionization experiments to separate
prompt ions from pulsed-field ions. Regions (II) and (III) were used to focus initial
spatial and velocity spread of the created ions. Dashed lines correspond to hightransmission (90%) wire mesh used to improve electric field uniformity. (b) An
image of the TOFMS, before a grounded cylinder of wire mesh was placed aroundthe device to shield it from external electric fields. . . . . . . . . . . . . . . . . . . . 94
5.14 (a) Data from a two-photon one-color survey scan, with many UV transitions foundin HfF. (b) False-color image showing our ability to separate different isotopes of
HfF using our TOFMS, giving us the ability to measure isotope shifts and helping
to identify the change in vibrational quantum number in these transitions. (c) Twocolor photoionization showing a spectral resolution of approximately 100 MHz. The
two traces correspond to the two dominant (m = 178 and m = 180) isotopes of HfF.

96

xx
5.15 Autoionization spectra showing (a) the same ionization threshold for HfF using two
different intermediate states and (b) large isotope shifts of the resonant features
between Hf178 F and Hf180 F. A change of one vibrational quanta between the intermediate HfF state and the HfF Rydberg state that undergoes autoionization corresponds to an isotope shift between Hf178 F and Hf180 F of 0.32 cm−1 . This data wastaken using two-color photoionization, starting in the HfF ground electronic andro-vibrational state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.1a) Small-angle limit. When the quantization axis F follows a slow periodic perturbation characterized by tilt angle θ, angular frequency ω and enclosed solid angle
A, two states whose instantaneous projection of angular momentum along F differs by δm will have their effective relative energy displaced by a Berry’s energy
ωAδm/2π. b) Large-angle limit. When instead the quantization axis sweeps out afull 2π steradians per cycle (α=0), the differential phase shift between the two levels
is indistinguishable from zero, and in the most natural conceptual framework, theBerry’s energy vanishes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2

The apparent energy shifts between mF = +3/2 and mF = −3/2 states in upper(a,b) and lower (c,d) Λ-doublet levels versus α, the angle of the electric field to theplane orthogonal the rotation axis of Erot (α is shown in Fig. 6.1(b)). (a) At α = 0,
there is an avoided crossing that mixes mF = ±3/2 states, with an energy splittingat the crossing of ∆u/ℓ . (b) Since α = 0 at the axial trap center, and since weneed mF to be a signed quantity in order to measure de , we will bias away fromthe avoided crossing using a magnetic field Brot . δmF gF µB Brot > ∆u/ℓ is required
for mF to be a quantity of definite sign. This picture is intuitively correct in thelimit that ∆u/ℓ > ωmax . The experiment will be performed in the opposite limit.
However, solving the time dependent Schrödinger equation (Eq. 6.42) gives the samerequirement of δmF gF µB Brot > ∆u/ℓ in both limits. . . . . . . . . . . . . . . . . . . 111

xxi
6.3

The apparent energy shifts between mF = +3/2 and mF = −3/2 states in upper(a,b) and lower (c,d) Λ-doublet levels versus Brot ,“dressed” by both the electric fieldrotation (ωrot ) and by the ion’s axial trap oscillation (ωz ). At Brot = 0, there isan avoided crossing that mixes mF = ±3/2 states, with an energy splitting at theu/ℓcrossing of ∆eff . In the limit δmF gF µB Brot ≫ ∆eff , the dressed states are of goodmF with an energy splitting slightly modified by ∆eff . . . . . . . . . . . . . . . . . . 118
6.4

Population of states |+i (red) and |−i (blue) with parameters ωz = 2 π ×1000 Hz,
u/ℓωmax = 2 π ×500 Hz, ∆ = 2 π ×2 Hz, and δmF gF µB B rot = π ×10 Hz. The population is initially prepared in |+i and results were found by numerically integratingEq. 6.42. (a) Measured at integer periods of ωz the |ai and |bi states appear stationary, leading to smooth Rabi oscillations if the state is prepared in state |+i or|−i. (b) If the measurement cannot be made at integer periods of ωz , frequency
modulation at ωmax is observed, leading to reduced accuracy when measuring theprecession frequency. Note here that ωmax /ωz = 1/2. As discussed in Chapter 6.11,
similar modulation can arise from static, uniform magnetic fields that are averaged
over during a period of ωrot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5

Over one rotation of Erot , both Erf and Ez are quasistatic. The total electric field isthe sum of all three and its trajectory over one cycle of Erot is plotted as the dottedline projected onto (a) the x-y and (b) the x-z planes. The electric field trajectory
is a circle of radius Erot , parallel to and displaced from x-y plane, a circle whose
center is offset from the z-axis by Erf . In the limit | Erf | ≪ | Erot |, the solid angle
subtended from the origin by this circle differs only slighlty from that subtended bya circle with vanishing Erf . The magnitudes of both Erf and Ez relative to Erot arevery much exaggerated for clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

xxii
7.1

Geometric phases accumulated during an ion-ion collision. (a) A typical ion-ion
collision trajectory (red), resultant Rutherford scattering angle, θ, and ion-ion interaction electric field, Eion , are shown in the collision plane (blue). For clarity, thecollision plane has been taken perpendicular to the instantaneous direction of Erot .
(b) During an ion-ion collision the molecular axis adiabatically follows the net electric field vector, E~rot + E~ion , and traces out the contour (black) on the unit sphere(yellow). The solid angle, ∆A(θ), subtended by this contour gives rise to a geometricphase accumulated by the eigenstates during the collision. This leads to decoherenceof the spectroscopic transition, see text. . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.2

Inverse coherence times, τ −1 , due to geometric phases accumulated during ion-ion
collisions as a function of (a) collision energy in temperature units and (b) Erot .
Dotted lines are approximations given in Eqns. 7.4 and 7.6. Solid lines are more
involved estimates based on integrals over collision parameters, but are still basedon approximations so as to be conservative. The ion density was taken to be n =1000 cm−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.3

The effective helium-ion scattering potential for b = 0 and b ¿ 0. . . . . . . . . . . . 146

7.4

Relative black-body excitation rates assuming a constant transition dipole moment. The peak lies near our vibrational energy
versus the dimensionless parameter khν
BTspacing for a 300 K black body. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Chapter 1
Introduction [1]1.1Motivation for Electric Dipole Moment Searches
At first glance, the electron appears to be a trivial physical system to study. An elementary

particle in modern physics, it has only a handful of measureable properties, including mass, charge,
and spin. Perhaps the most interesting property, due to its historical impact on modern physics,
is the magnetic g-factor, ge . The prediction of ge = 2 was a major triumph of the Dirac equation,and thus aided in the development of relativistic quantum mechanics. In order to account for slightdeviations of ge from 2, QED is needed. Fig. 1.1 shows a one-loop correction in QED to ge , wherea virtual photon interacts with the electron, adding a factor of α/π to the electrons g-factor. The
current measurement of the electron’s magnetic moment is accurate to 14 digits [7] and agrees withtheory out to eighth-order corrections [8]. At this level of precision, hadronic and weak corrections
must be included. In fact the value of ge will be influenced by the properties of all particles thatinteract with the electron. This is true of not only ge , but also of other properties of the electron,
including its mass and charge. This illustrates how a precision measurement of the electron can inprinciple give information about less well understood particles, for instance the Higgs Boson. In
practice, measurements of ge are not very sensitive to exotic massive particles, due to the small
electron mass. For this reason, measurements of the muon g-factor can offer better constraints
on exotic particle physics [9], even though the accuracy of these measurements are far worse thansimilar measurements on the electron. Extracting information from these measurements is difficult
since the Standard Model hadronic corrections are uncertain [10].

2

g

e

-

g

e

-

..
Figure 1.1: One-loop diagram that leads to a correction of the electron g-factor. The electroninteracts with a photon (γ) from an external field as well as a “virtual” photon.

3

m

de

m

de

P

+
_

m

de

T

+
_

+
_

Figure 1.2: If an electron EDM exists, the orientation between the electron’s electric (de ) andmagnetic (µ) dipole moments will change under a parity (P) or time-reversal (T) transformation.

Measurements of the electron electric dipole moment (eEDM) are a sensitive probe to new
physics as it violates time-reversal (T) and parity (P) symmetries. This is seen by first assuming
the eEDM lies along the same axis as the electron’s magnetic dipole moment, an assumption thatwill be discussed below. The magnetic moment is T-odd and P-even, while an electric moment
would be T-even and P-odd. This means that parity or time-reversal transformation would change
the relative orientation between the electron’s magnetic and electric moments and an inversion of
space or reversal of time would be observable by measuring the orientation of electron moments.
This is illustrated schematically in Fig. 1.2. The P and T violation of an electron EDM implies CP
violation if the widely held CPT conservation theorem is assumed correct.
CP violation has been experimentally observed in both Kaons [11] and B mesons [12, 13]. The
observed CP violation is accounted for in the Standard Model through a CP violating phase in the
Kobayashi-Maskawa (KM) quark-mixing matrix. This leads to an electron EDM through Feynman
diagrams that include electroweak couplings to quarks, but at higher orders of perturbation thanfor quark systems. Hoogeveen [14] attempted to estimate de from three-loop diagrams, an example
of which is shown in Fig. 1.3 and obtained a value of ∼ 10−38 e cm. It was later shown that
diagrams even at the three-loop level will cancel [3] and Khriplovich and Lamoreaux estimate theStandard Model electron EDM as < 10−40 e cm [15].
Another possible source of CP violation is in the strong nuclear interaction where another4

Figure 1.3: Feynman diagram from Ref. [2] for a three-loop contribution to the eEDM. The sum of
three-loop diagrams cancels to zero [3] and four-loop diagrams are needed to produce a non-zeroeEDM in the Standard Model.

5
CP violating phase ΘQCD is found. This phase would give rise to a very large neutron EDM given
its “natural” size O(1). The current experimental limit on the neutron EDM requires ΘQCD to beO(10−10 ) [16], giving rise to the strong CP problem in particle physics. This source of CP violationis unimportant when determining the size of the electron EDM.
Observations of neutrino oscillations and their implication of neutrino masses gives rise toa third possible mechanism for an electron EDM in the Standard Model. Massive neutrinos giverise to a lepton mixing matrix analogous to the KM matrix, with the possibility of CP violatingphases. Contributions to an electron EDM from this source tend to be smaller than for the KM< 10−33 e cm is found with two “extra” massive neutrinos
mechanism, although an estimates of de ∼and fine-tuning of neutrino masses [17].
Many extensions to the Standard Model predict far greater values for the eEDM [2]. This isperhaps not surprising, since a major failing of the Standard Model, that these extensions wish tocorrect, is the fact that it does not contain enough CP violation to explain the matter-antimatterasymmetry in the Universe. Extensions to the Standard Model predict a value for the eEDM as highas the current experimental limit of 1.6 × 10−27 e cm, many orders of magnitude above the Standard
Model background. With the range of theoretical predictions starting at the experimental limit,
and Standard Model backgrounds orders of magnitude below this limit, an improved measurementof the eEDM is a tempting goal. This thesis describes an experiment that hopes to achieve a factorof thirty improvement on the experimental upper limit for an eEDM.

1.2

Schiff ’s Theorem
In general, an electron EDM measurement is made by applying an electric field to an electron

and looking for a response. Applying large electric fields to bare electrons results in a rapid
acceleration of the electrons out of the experimental apparatus. Using an unpaired electron in aneutral atom or molecule seems problematic. Naively, one would assume that the electrons inside
the atom would just arrange themselves such that the average electric field on any charged particle
in the atom would time average to zero, as the neutral atom is not accelerated by the external

6field. This principle is known as Schiff’s theorem [18, 19] and is largely the reason why the firstexperiments specifically designed to measure a T-violating EDM were made on neutrons [20]. A
similar thing occurs when an external electric field is applied to an ion in an rf Paul trap, moving
the ion to a new position where the time-averaged electric field is zero.
Loopholes in Schiff’s theorem include finite-size effects of the nucleus and relativistic corrections. The latter tends to dominate for electron EDM measurements and is discussed in Ref. [21].
The time-averaged electric field hEi on an atomic electron, even when relativistic effects are taken
into account, is zero. However, due to length contraction, the time-averaged value of de · E will in
general not equal zero. The interaction energy with an electric dipole moving at a velocity cβ, as
observed in the laboratory frame, will beγ
We = −de · E −β · Eβ ,
1+γ




(1.1)

with γ = (1-β 2 )1/2 . Ref. [21] shows it is essentially the second term in Eq. 1.1 that gives rise to aninteraction energy between an atomic or molecular valence electron EDM and an external electricfield. Despite the average electric field seen by an atomic electron being zero, it is customary todefine an effective electric field, Eeff , such that Eedm = −de · Eeff .

1.3

Direction of de
Earlier, the assumption was made that the eEDM lies along the same axis as the electron’smagnetic dipole moment. To test this, let us assume that the eEDM is not at a fixed angle relative
to the electron’s spin axis. If that were the case, an extra quantum number would be required todescribe the state of an electron and more than the experimentally observed two electrons would
be allowed in a 1S atomic orbital. Clearly, an eEDM must have a fixed orientation with respect tothe electron’s spin axis.
It is less clear if de⊥ , a component of de perpendicular to the spin axis, is allowed. Certainly,
a fundamental spin-1/2 particle has only one vector, the spin axis, available to orient the particle.
The situation is less clear if the electron is allowed internal structure. An experiment sensitive to7a de⊥ is beyond the imagination of this author, and certainly our experiment will only be sensitive
to a de component parallel or anti-parallel to the spin axis. If our experiment yields a non-zero dethe symmetry breaking arguments made above will therefore be valid.

1.4

Overview of Ongoing Experimental WorkFor 20 years the most stringent limits on the eEDM have been the atomic-beam experimentsof Commins’ group [22, 23, 24]. That work set a standard against which one can compare ongoingand proposed experiments to improve the limit. Here is a brief survey of ongoing experiments ofwhich we are aware.
√
For evaluating the sensitivity of an eEDM experiment the key figure-of-merit is Eeff τ N ,where Eeff is the effective electric field on any unpaired electrons, τ is the coherence time of theresonance, and N is the number of spin-flips that can be counted in some reasonable experimental
integration time, for instance one week. We will discuss these three terms in order.
In practice all eEDM experiments involve heavy atoms with unpaired electron spins, andmost involve measuring the atomic spin-flip frequency ωd . An applied laboratory electric fielddistorts the atomic wavefunction, and the eEDM contribution to the atomic spin-flip frequency ωdis enhanced by relativistic effects occurring near the high-Z nucleus [25, 26], so that ωd = de Eeff [27],where the effective electric field Eeff can be many times larger than the laboratory electric field Elab .
The enhancement factor is roughly proportional to Z 3 although details of the atomic structure
come into play such that the enhancement factors for thallium (Z = 81) and cesium (Z = 55)
are −585 [28] and +114 [29], respectively. Practical DC electric fields in a laboratory vacuum arelimited by electric breakdown to about 105 V/cm. The Commins experiment used a very high-Z
atom, thallium, and achieved an Eeff of about 7 × 107 V/cm [22]. There have been proposed anumber of experiments in cesium [30, 31, 32] that expect to achieve Eeff of about 107 V/cm. A
completed experiment by Hunter [33] achieved Eeff = 4.6×105 V/cm in Cs by using Elab = 4 kV/cm.
It was pointed out by Sandars [34] that much larger Eeff can be achieved in polar diatomic
molecules. In these experiments, the atomic wavefunctions of the high-Z atom are distorted by the8effects of a molecular bond, typically to a much lighter partner atom, rather than by a laboratory
electric field. One still applies a laboratory electric field, but it need be only large enough to alignthe polar molecule in the lab frame. The Hinds group [35] is working with YbF, for which theasymptotic value of Eeff is 26 GV/cm [35, 36, 37, 38, 39, 40, 41]. DeMille’s group [42, 43, 44]
uses PbO, with an asymptotic value of Eeff ≃ 25 GV/cm [45, 46, 47]. Shafer-Ray’s group [48] hasproposed to work with PbF, which has a limiting value of Eeff ≃ 29 GV/cm [49, 50]. The ACME
collaboration [51] will use ThO, with Eeff ≃ 100 GV/cm [52]. The Leanhardt group is working
with WC, with Eeff ≃ 54 GV/cm [53]. We will discuss candidate molecules for our experiment in
Chapter 2.2; we anticipate having an Eeff of around 25 to 90 GV/cm [52, 54].
After Eeff , the next most important quantity for comparison is the coherence time τ , which
determines the linewidth in the spectroscopic measurement of ωd . In Commins’ beams experiment,τ was limited by transit time to 2.4 ms. Future beams experiments may do better with a longer
beam line [48], or with a decelerated beam [55]. Groups working in laser-cooled cesium anticipate
coherence times of around 1 s, using either a fountain [30] or an optical trap [31, 32]. The PbO
experiment has τ limited to 80 µs by spontaneous decay of the metastable electronic level in which
they perform their ESR. Coherence in ThO experiment will be limited by the excited-state lifetime
to 2 ms [51]. A now discontinued experiment of Hunter [33] achieved τ = 15 ms in a vapor cell withcoated walls and a buffer gas. The JILA experiment will work with trapped ions. The mechanismsthat will limit the coherence time in our trapped ions are discussed in Chapters 6, 7, and 8. We
anticipate a value in the vicinity of 300 ms.
The quantity Eeff converts a hypothetical value of de into a frequency ωd , and τ sets the√experimental linewidth of ωd . The final component of the overall figure-of-merit is N , which
determines the precision with which the resonance line can be determined. Vapor-cell experiments
such as those of Hunter or DeMille can achieve very high values of effective N , atomic beams
machines are usually somewhat lower, and molecular beams usually lower yet (due to greater
multiplicity of thermally occupied states.) Atomic fountains and atomic traps have still lower
count rates, but the worst performers in this category are ion traps. The JILA experiment may

9trap as few as 100 ions at a time, and observe only 10 transitions in a second.
The discussion above is summarized in Table 1.1. To improve on the experiment of Commins,
it is necessary to do significantly better in at least one of the three main components of the
figure-of-merit. The various ongoing or proposed eEDM experiments can be sorted into categories
according to the component or components in which they represent a potential improvement over
the Commins’ benchmark. The prospects of large improvements in both τ and Eeff put JILA’s
experiment in its own category. This combination means that our resonance linewidth, expressedin units of a potential eEDM shift, will be 105 times narrower than was Commins’. Splitting ourresonance line by even a factor of 100 could lead to an improved limit on the eEDM. This is anadvantage we absolutely must have, because by choosing to work with trapped, charged molecules,
we have guaranteed that our count rate, Ṅ , will be far smaller than those of any of the competing
experiments.
We note that there are in addition ongoing experiments attempting to measure the eEDM in
solid-state systems [56, 57, 58, 59]. These experiments may also realize very high sensitivity, but
because they are not strictly speaking spectroscopic measurements, it is not easy to compare themto the other proposals by means of the same figure-of-merit.

1.5

Overview of Thesis
Chapter 2 of this thesis will be a brief walk through of the general spectroscopic idea behind

our experiment. The next four chapters are dedicated to experimental progress and the manypieces of technology that needs to come together to make an eEDM measurement with trappedmolecular ions possible. Along the way we were faced with many alternative visions of how tomake this experiment work, and I will in general discuss some of the alternatives we investigated.
Chapter 3 focuses on ion trapping basics. Also included in this chapter is discussion of our first
generation ion trap, some experiments done in this first trap, and a brief description of our next
generation trap. One challenge of our experiment is efficient read-out of our spin populations after
the Ramsey experiment is completed. This will be covered in Chapter 4. The process of creating,10

Table 1.1: Figure-of-merit comparison between several recently completed and ongoing eEDM
experiments. For ongoing experiments these numbers are subject to change and are often order-ofmagnitude estimates.
Group
Commins
Hunter
Gould
Heinzen
Weiss
DeMilleHinds
Shafer-Ray
ACME
Leanhardt
JILA

Species
Tl
Cs
Cs
Cs
Cs
PbO
YbF
PbF
ThO
WC
+
HfF , ThF+ , etc.

Elab [V/cm]
1.23 × 105
4 × 103
105
105
105
10
8.3 × 103
7 × 104102
5

Eeff [V/cm]
7 × 107
4.6 × 105
107
107
107
2.5 × 1010
1.3 × 1010
2.9 × 1010
1011
5.4 × 1010
2 − 8 × 1010

τ [s]2.4 × 10−3
1.5 × 10−2
1
1
1
8 × 10−5
10−3Ṅ [s−1 ]109

2 × 10−3
10−3
0.3

105

109

10

Refs.
[22]
[33]
[30]
[32]
[31]
[42, 43, 44, 45, 47][35, 60][48, 49, 50][51, 52][53]

11
cooling and loading into a trap our molecular ions is covered in Chapter 5. This will include ourlaser-ablation supersonic-expansion source, a brief discussion of alternative cooling methods, andphotoionization techniques for state-selective trap loading. Chapter 6 will discuss some of the issuesof performing precision spectroscopy in time-varying, inhomegenous fields. Ion-ion and ion-neutral
collisions will be discussed in Chapter 7. Things will get wrapped up in Chapter 8 with conclusions
and an estimate of sensitivity to the eEDM.

Chapter 2
Molecular Structure and the Basic Spectroscopic Idea [1]2.1

Molecular Notation
A final decision as to which molecule we will use has not been made. For reasons discussed

below, the main candidates are diatomic molecular ions Mx+ , where M = Hf, Pt, or Th and x = H
or F. In the case of molecules such as HfF+ , ab initio methods [4, 54] enable us to determine thatthe 3 ∆ state is well described by a set of Hund’s case (a) quantum numbers: J, S, Σ, Λ, Ω, MJ , e/f .
Here J is the sum of electronic plus rotational angular momentum, S the total electronic spin
angular momentum, Σ the projection of S onto the molecular axis, Λ the projection of L, theelectronic orbital angular momentum, onto the molecular axis, and Ω the projection of J onto themolecular axis. In a case (a) 3 ∆ molecule |Ω| can take the values one, two or three. MJ is the
projection of J along the quantization axis and the labels e/f specify the parity of the molecularstate.
In addition to these quantum numbers, the experiment will be concerned with the nuclear
spin quantum number I, the total angular momentum quantum number F , given by the vector
sum of J and I, and mF the projection of F along the quantization axis. Throughout this thesisa total nuclear spin of I = 1/2 is assumed, the nuclear spin of fluorine or hydrogen.

2.2

Choosing a Molecule
In selecting a molecular ion for this experiment we have several criteria. First, we want a

simple spectrum. Ideally, we would like the supersonic expansion to be able to cool the molecules

13
into a single internal quantum state so that every trapped molecule could contribute to the contrastof the spectroscopic transition. Failing that, we want to minimize the partition function by using
a molecule with a large rotational constant, most likely a diatomic molecule with one of its atoms
being relatively light. Small or vanishing nuclear spin is to be preferred, as are atoms with only
one abundant isotope. Second, we need to be able to make the molecule. This requirement favors
more deeply bound molecules and is the main reason we anticipate working with fluorides rather
than hydrides. Third, the molecule should be polarizable with a small applied electric field, i.e. it
should have a relatively small Λ-doublet splitting, ωef , a quantity that is described in Chapter 6.1.
Fourth, and most important, the molecule should have unpaired electron spin that experiences alarge value of Eeff .
These latter two requirements would appear to be mutually exclusive: a small Λ-doublet
splitting requires a large electronic orbital angular momentum, which prohibits good overlap with
the nucleus required for a large Eeff . Fortunately, working with two valence electrons in a triplet state
allows us to satisfy our needs. One valence electron can carry a large orbital angular momentum
making the molecule easily polarizable, while the other can carry zero orbital angular momentum
giving it good overlap with the nucleus and generating a large Eeff . This concept was detailed by
Bohn and Meyer in Ref. [4] and for the 3 ∆1 state of interest here, the two valence electrons occupy
molecular σ and δ orbitals. Calculations [54, 52] indicate that in the 3 ∆1 state of HfH+ and HfF++
< 2π × 1 MHz and Eωef ∼eff ≈ 30 GV/cm should be expected. ThF has Eeff ≈ 90 GV/cm [52].

2.3|Ω| = 1 vs. |Ω| = 3
One final valuable feature to look for in a candidate molecule is a small magnetic g-factor, so

as to reduce the vulnerability to decoherence and systematic errors arising from magnetic fields. To
the extent that spin-orbit mixing does not mix other |Ω| = 1 states into a nominally 3 ∆1 molecularlevel, it will have a very small magnetic moment, a feature shared by PbF in the 2 Π1/2 state [48].
This is because Σ = −Λ/2, and because the spin g-factor is ∼ 2 times the orbital g-factor. Under
these conditions, the contributions of the electronic spin and orbital angular momentum to the net

14molecular magnetic dipole moment nominally cancel. In HfF+ , the magnetic moment of a stretchedmagnetic sublevel level of the 3 ∆1 , J = 1 rotational ground state is about 0.05 µB [61]. This is afactor of 20 less than the magnetic moment of ground state atomic cesium. In the 3 ∆3 level, on
the other hand, the magnetic moment in the stretched Zeeman level is 4.0 µB . The |Ω| = 3 state
may nonetheless be of scientific interest. The 3 ∆1 and 3 ∆3 levels have Eeff equal in magnitude but
opposite in sign. If one could accurately measure the science signal, ωd , in the 3 ∆3 level despite its
larger sensitivity to magnetic field background (and despite its shorter spontaneous-decay lifetime),
the comparison with the 3 ∆1 result would allow one to reject many systematic errors.

2.4|Ω| = 1, J = 1 Λ-doublet
Since we have not made a final decision as to which molecule we will use, and also because

we have yet to measure the hyperfine constants of our candidate molecules, the discussion of level
schemes in this section will be qualitative in nature, usually emphasizing general properties sharedby all the molecules we are investigating. To simplify the discussion, it will be specialized to
discussing spectroscopy within the J = 1 rotational manifold of a molecular 3 ∆1 level.
For Hunds’ case (a) molecular levels with |Λ| ≥ 1, each rotational level is a Λ-doublet, thatis, it consists of two closely spaced levels of opposite parity. We can think of the even (odd) parity
level as the symmetric (antisymmetric) superposition of the electronic angular momentum lying
predominantly parallel and antiparallel to the molecular axis [Fig. 2.1(a)]. The parity doublet
is split by the Λ-doubling energy ωef . A polar diatomic molecule will have a permanent electric
dipole moment, d~mf , aligned along the internuclear axis n̂, but in parity eigenstates, there will bevanishing expectation value hn̂i in the lab frame. An applied laboratory electric field, Erot , will
act on dmf to mix the states of good parity. In the limit of dmf Erot ≫ ωef , energy eigenstates will
have nonvanishing hn̂i in the lab frame. More to the point, Ω, a signed quantity given by the~ + S)~ · n̂, can also haveprojection of the electron angular momentum on the molecular axis, (La nonzero expectation value [Fig. 2.1(b)]. Heuristically, it is the large electric fields developed
internal to the molecule, along n̂, that gives rise to the large value of Eeff that the electron spin

15
can experience in polar molecules. In the absence of the Λ-doublet mechanism for polarizing themolecule, a much larger field would be necessary, dmf Elab ≫ 2Be , to mix rotational states withsplitting typically twice the rotational constant Be . For HfF+ , it is estimated ωef will be something
under 2π × 50 kHz, whereas Be will be about 2π × 10 GHz. For a dipole moment dmf = 4.3 D,
mixing the Λ-doublet levels will take a field well under 1 V/cm, whereas “brute force” mixing ofrotational levels would require around 10 kV/cm. For an experiment on trapped ions, the smaller
electric fields are essential.
In the context of their eEDM experiment on the a3 Σ1 level in PbO, DeMille and his colleagues
have explored in some detail [42, 43, 44] the convenient features of an |Ω| = 1, J = 1 state, especiallywith respect to the suppression of systematic error. Our experiment liberally borrows from thoseideas. In a molecule with at least one high-Z atom, 3 ∆1 states will be very similar to the a3 Σ1state of PbO, but with typically smaller values of ωef and much smaller values of magnetic g-factor.
Singly charged molecules with spin triplet states will necessarily have an odd-Z atom, and thus theunavoidable complication of hyperfine structure, not present in PbO.
In Fig. 2.1 the 3 ∆1 , J = 1 state with hyperfine splitting due to the fluorine I = 1/2 nucleus
is presented. A key feature is the existence of two near-identical pairs of mF -levels with opposite
parity due to Λ-doubling. As seen in Fig. 2.1(b), an external electric field, Erot , mixes these opposite
parity states to yield pairs of mF -levels with opposite sign of Eeff [44] relative to the external field.
Fig. 2.1(c) shows the effect of a rotating magnetic bias field, parallel with the electric field, applied
to break a degeneracy as described in Chapter 6.4. Note that any two levels connected by arrows
in Fig. 2.1(c) transform into each other under time reversal. Time reversal takes mF → −mF ,Ω → −Ω, and B → −B, where B is the magnetic field. If we measure the resonant frequency for thetransition indicated by the solid (or dashed) line once before and once after inverting the directionof the magnetic field, time reversal invariance tells us the difference between the two measurements
should be zero. In the presence of an eEDM, which violates time-reversal invariance, this energy
difference W u (B)−W u (−B) will give 2de Eeff . As well, under the same magnetic field the transitions
indicated by the solid and dashed lines should be degenerate, if the magnetic g-factors are identical

16
for the states involved [62]. With non-zero eEDM the energy difference W u − W l also gives 2de Eeff .
Potential additional shifts, due predominantly to Berry’s phase [63], are discussed in Chapter 6 but for now note only that in the absence of new physics (such as a nonzero eEDM) the energy
levels of a molecule in time-varying electromagnetic fields obey time-reversal symmetry. Reversing
the direction of the electric field rotation while chopping the sign of the magnetic field amounts
to cleanly reversing the direction of time, and will leave certain transition energies rigorously unchanged if de = 0. These are our “science transitions”, which we will measure with our highest
precision.

2.5

Electronic Level Structure
There are ∼ 104 possible diatomic molecules and one might hope that humanity has knowledgeof all their spectra. Alas, that is not the case. It appears that too many people have been at home
watching American Idol instead of being in the laboratory late at night taking molecular spectra. In
general, diatomic molecules containing atoms as heavy as hafnium, thorium, or platinum have very
little known spectra. Diatomic molecular ions are less known still and no previous experimental
studies existed for our most promising eEDM molecular ion candidates.
We identified a handful of candidates that might posses low-lying 3 ∆1 states, mostly by looking at known or theoretically predicted spectra of isoelectronic neutral molecules. These molecules
typically have low-lying 1 Σ and 3 ∆1 states with the energy splitting between them predicted tobe small enough that either might be the ground state. ab initio calculations have been performed [4, 54, 64, 61] that give us a picture of what to expect for states from 0 ∼ 14,000 cm−1
above the ground state for many of our candidate molecules. Theoretical curves are shown in
Fig. 2.2 for the case of HfF+ . There are several states in this energy range of potential interest forstate preparation and read-out purposes in our experiment. Spin-orbit mixing between states ofidentical |Ω| are enhanced by relativistic effects in the high-Z Hf atom. The b(1) and c(1) states
are well-mixed combinations of 1 Π1 ,3 Π1 , and 3 Σ−
1 states, allowing for electric dipole transitions toand from these states that do not respect spin selection rules. The 1 Σ0 state, on the other hand,

(a)

(b)

(c)
|Ω = +1〉|Ω = -1〉
mF
-3/2
-1/2
+1/2
+3/2
FF
|b〉

Σ = -1F = 3/2|f〉-3/2 -1/2 +1/2 +3/2 mF

|f〉
|e〉

-1/2 +1/2 mF

3gFµBB+deEeff

Wu

Erot

Erot B E
eff

Hf++
3γFErotdmf

Hf++

Hf++

n^

n^

Σ = +1
EHF

^

n

n^
Hf++

ωef

|e〉

|a〉 Σ = +1

mF -3/2 -1/2 +1/2 +3/2

|d〉

|c〉

F|Ω = -1〉

Erot

3gFµBB−deEeff

Erot B E
l

W

eff

Σ = -1

F|Ω = +1〉

-1/2 +1/2 mF

mF-1/2 +1/2

F = 1/2
Figure 2.1: Energy levels of HfF+ in the 3 ∆1 , J = 1 state including hyperfine structure associated with the fluorine I = 1/2 nucleus.
Λ and Σ are defined as the projection along the molecular axis of the electronic orbital angular momentum, and spin, respectively.
√Ω = Λ + Σ. (a) In zero electricfield,
theeigenstatesofthesystemarestatesofgoodparity,
|ei
=
(|Ω
=
+1i
−
|Ω
=
−1i)/
2 and
√
|f i = (|Ω = +1i + |Ω = −1i)/ 2, separated by a small Λ-doublet splitting. (b) An electric field, Erot , mixes the parity eigenstates yielding
states with well defined Ω. (c) A small magnetic field lifts the degeneracy between states with the same value of mF Ω. A permanentelectron electric dipole moment further breaks this degeneracy, but with opposite sign for the upper (solid arrow) and lower (dotted arrow)transition. Energy splittings not to scale.
17

18
has no nearby |Ω| = 0 state with which to mix, and thus Σ and Λ are good quantum numbers.
Similarly, the 3 ∆1 state has so little contamination of 1 Π1 in it that a rough calculation indicatesthat it is metastable against spontaneous decay, with a lifetime of order 300 ms [4, 54]. Levels in
HfF and HfF+ that are relevant to material throughout this thesis are shown in Fig. 2.3.
We have spent considerable effort attempting to measure the energy of these excited statesin HfF+ . This work was complicated by the one difference between our group and molecular
spectroscopists; we were not molecular spectroscopists. Laura Sinclair [65] has since measured one
electronic transition in HfF+ , an (Ω = 1) ← 1 Σ transition, likely to either the b(1) or c(1) state,
at 13,000 cm−1 . This transition could be used to link the 1 Σ ground state with the 3 ∆1 state wewish to use for eEDM spectroscopy. Laura and her co-workers are currently working on measuring
more spectra in HfF+ and potentially other candidate molecules. Rumor is they don’t even like
American Idol, so progress should be swift.

19

3

P2
c(1)
b(1)
3
P0
3
D3
3
D2
3
D1
1
S0

Figure 2.2: Potential energy curves for select states of HfF+ [4]. The b(1) and c(1) states are
well-mixed combinations of 1 Π1 ,3 Π1 , and 3 Σ−
1 states.

20+

Hf + F

+

HfF
(b)

(a)
c(1)

D1

v=0

1

S

v=0

3

v=1

v=1

b(1)
+

HfF

int. level3-3/2 -1/2 +1/2 +3/2
1

..

mF

S

D1
2

D3/2

HfF

Figure 2.3: Molecular levels in HfF and HfF+ discussed throughout this thesis. (a) A schematic oflevels in HfF+ useful in measuring the eEDM are shown. Ion population in the 1 Σ ground statemight be transferred, using STIRAP (blue), to the 3 ∆1 level where precision Ramsey spectroscopy
is performed to search for the eEDM. After the Ramsey sequence, a spin-dependent transition(green) using σ-polarized light can be used to measure spin populations, either through fluorescence(red) decay or by making a photo-dissociation transition (orange) to Hf+ + F. (b) A schematic oflevels useful in creating HfF+ via photoionization of HfF are shown. A transition (Orange) to anintermediate state of HfF is followed by a transition to a Rydberg state with a vibrationally excited
HfF+ core in the 1 Σ (red) or 3 ∆1 (blue) state, which can then autoionize to the v=0 continuum(grey boxes) of the respective states. This process is discussed in Chapter 5.7. Energy splittings
not to scale.

Chapter 3
Ion Trapping

In a charge free region, electric potentials obey Laplace’s equation,∇2 V = 0(3.1)

Solutions to this equation allow no local extremum, i.e. all maximums and minimums must occur
at the boundaries. Therefore, it is impossible to produce a three-dimensional trapping potential in
free space from static electric fields alone. Two solutions to this problem are found in the Penning
trap and the rf Paul trap. In a Penning trap [66], the combination of static electric and magnetic
fields allows for 3-D confinement. In an rf Paul trap [67, 68], time-varying electric fields are used
to create a time-averaged potential with a minimum at the trap center. In our eEDM experiment,
the energy difference we wish to precisely measure is, to a very good approximation as discussed
in Chapter 6, independent of the electric field, while being quite sensitive to magnetic fields. For
this reason, an rf Paul trap seems the better choice here.

3.1

Ions in Two-Dimensional Time-Varying Quadrupole Fields
A general three dimensional quadrupole field has a potential,
Φ(x, y, z) = Φ0 (λx2 + σy 2 + γz 2 )(3.2)

where Φ0 is an externally applied, position independent electric potential and λ, σ, and γ areconstants depending on the nature of the field. With this potential, the three dimensions are not22coupled, and motion along these three directions can be treated independently. Laplace’s equation
requiresλ+σ+γ =0

(3.3)

When λ = −σ and γ = 0, we obtain a two-dimensional quadrupole field. This field isutilized in quadrupole mass spectrometers, and is convenient to discuss first before tackling threedimensional ion trapping fields. Using four rods with a hyperbolic structure, as seen in Fig. 3.1,
and applying a potential of (U − Vrf cos ωrf t)/2 to one pair of oppositely placed rods and (−U +Vrf cos ωrf t)/2 to the other pair, the following potential is achieved [5, 69],Φ(x, y, t) = (U − Vrf cos ωrf t)

x2 − y 2
2r02

(3.4)

with 2r0 the smallest distance between opposite hyperbolic rods. Using
M

−e∂Φ
d2 xi
=
2
dt
∂xi

(3.5)

we obtain the following equations of motion,ed2 x+(U − 2Vrf cos ωrf t)x = 0
2
dt
M r02
e
d2 y
(U − 2Vrf cos ωrf t)y = 0
−
2
dt
M r02
d2 z
=0
dt2

(3.6)
(3.7)
(3.8)

Making substitutions to dimensionless variables, we can rewrite the above equations for x- andy-motions as
d2 x+ (a − 2q cos 2ζ)x = 0
dζ 2
d2 y
− (a − 2q cos 2ζ)y = 0
dζ 2

(3.9)
(3.10)

with
a=

2eVrf
ωrf t
4eU
2 ; q = M r2 ω2 ; ζ = 2
M r02 ωrf
0 rf

(3.11)

23

Figure 3.1: Electrodes used to produce a two-dimensional quadrupole field. For a quadrupole mass
spectrometer, Φ0 = (U − Vrf cos ωrf t)/2. Adapted from [5].

24
Eqn. 3.9 is the canonical form of the Mathieu equation, a well studied equation used in many
areas of physics. Floquet’s Theorem guarantees solutions of the form,
′ iβζ

x=αe

∞
X

2inζ

C2n e

′′ −iβζ

+α e

n=−∞

∞
X

C2n e−2inζ(3.12)

n=−∞Here, α′ and α′′ are constants whose values depend on the initial conditions of the ions. The C2nconstants and the characteristic exponent, β, depend on the parameters a and q. From this we seethat periodic solutions exist if β is purely real, otherwise x increases without bound as ζ → ∞. If
β is a real integer, the solution is periodic, but unbounded. The β = n contours in the a-q plane,
with n a real integer, serve as the boundaries between stable and unstable regions. Eqn. 3.10 has
identical solutions with a replaced by −a. The four contours of β = 0 and β = 1 for both the x andy directions bounds the first stability zone as seen in Fig. 3.2. This is the region in which almost
all ion quadrupole traps and guides are operated.
From Eqn. 3.12, when β is real, the ion motion frequencies are,ωn =(2n ± β)ωrf
2(3.13)

This shows the frequency spectrum is the same along contours of constant β in the stability diagram,′ s, being determined only by thewith the weighting factors of the various frequencies, the C2nparameters a and q. However, the exact trajectory of the ions depends on their initial position andvelocity, and the initial phase of the rf. Along the β = 0 line, the lowest order frequencies are 0
and ωrf . Along the β = 1 line, the fundamental frequency of ion motion is ωrf /2. As discussed
in Chapter 3.3 the rf voltage Vrf can be viewed as giving a time-averaged psuedo-potential thatis focusing in both the x- and y-directions. The DC potential U breaks the symmetry between
the x- and y-directions, with a positive U creating a focusing potential in the x-direction and a
defocusing potential in the y-direction. The β = 0 contours in the stability diagram (Fig. 3.2)
then correspond to values of U and Vrf such that the focusing potential from the rf fields is exactly
cancelled with the defocusing caused from the DC fields in either the x- or y-directions, leading tounstable ion motion. The β = 1 contours have trapping frequencies due to the combination of rf

25

a
0.2

y stablex stable0.1

x and ystable

0.0
0.2

0.4

0.6

0.8

q

- 0.1

y stable-- 0.2x stable

Figure 3.2: Ion stability diagram in terms of the dimensionless trapping parameters a = M4eU
r2 ω2
0

rf

rf
and q = M2eV
. The bounds of stability are the βx,y = 0 and βx,y = 1 contours.
r2 ω2
0

rf

and dc voltages that are exactly ωrf /2, here the rf drive resonantly excites the trapping motion andleads to unbounded ion motion.
Since the parameters a and q depend on the mass-to-charge ratio, M/e, of the ions being
trapped, the rf quadrupole can be used to measure M/e of an ion sample. This is most readily
accomplished by utilizing the narrow tip of stability at q ≈ 0.706 and scanning along a line ofconstant U/Vrf as seen in Fig. 3.3. Here, we plot the stability diagram as a function of U and Vrf at
a fixed rf drive frequency ωrf for ions with different values of M/e. As seen in the figure, as U/Vrf
is scanned only a small range of mass-to-charge ratios are stable. This method of mass filtering has
the advantage of simple tuning of the resolution, at the expense of sensitivity, by picking a slightly
different constant for the U/Vrf scan. Sensitivity is decreased at higher values of U/Vrf due to thelower trapping potential as you approach the unstable region and the lower phase space of ions thatwill make it through the quadrupole. This utilization of the rf quadrupole as a mass filter inspires
the choice of applying a potential U to one set of opposing rods, thus breaking the x-y symmetryof the ion guide.

26

Figure 3.3: Ions of different masses will have different stability regions when graphed in terms of the
DC (U ) and rf (Vrf ) trapping voltages. This graph is for a constant rf frequency ωrf . By scanningalong a line of constant U/Vrf and recording ion signal, a mass spectrum is produced. Greater
resolution is achieved, at the expense of sensitivity, by increasing the slope of the scan line.

27

3.2

3-D Ion Trapping, the Linear Paul Trap
One way of achieving a three dimensional trapping potential is the linear rf Paul trap. The

previously discussed four-rod geometry is used with the addition of a DC potential that is axially
focusing and radially defocusing,Φax =kUz 2 x2 y 2
(z −
− )
2
2
2r02

(3.14)

with Uz an applied potential and k a dimensionless geometric factor. This potential is achieved byapplying a DC voltage to end-caps. The geometry of these end-caps depends on trap design, but
can entail needle electrodes or segmenting each quadrupole rod into two end-caps and a central
electrode. The radial focusing from the rf induced pseudo-potential must then overcome the radial
defocusing from the DC potentials. Since the fields in the three dimensions are de-coupled, we
can treat each direction separately. The z-direction is trivial, with a DC harmonic potential wellleading to simple harmonic motion. In the x- and y-directions, the defocusing from this new DC
potential can be quantified by modifying the a parameter,aeff =

kUz
4e
(U ∓
)
2
2
2
M r0 ωrf(3.15)

with the -(+) corresponding to the x(y) direction. This leads to a shift in the first stability region
to higher q values as seen in Fig. 3.4.
Another often used geometry is commonly referred to as the 3-D Paul trap. Here, a ring
electrode with hyperbolic cross-section is used in conjunction with two end-cap electrodes which are
hyperboloids of revolution around the z-axis and complementary to the intermediate ring electrode.
The potential is then,Φ = Φ0 (x2 + y 2 − 2z 2)
4z02

(3.16)

with r02 = 2z02 . This leads to a similar stability diagram as the two dimensional case, except there
is an asymmetry between the r and z directions.
Historically, this has been the more common geometry. However, the linear Paul trap has
been adopted for many AMO experiments due to several advantages. Its initial interest was due to28

Figure 3.4: The ax and ay trapping parameters are shifted in the case of the linear Paul trap due
to an axial trappingpotential, Uz , and also from space charge effects due to a trapped ion density
ρr02
4e
kU z
of ρ. aeff = 2M r2 ω2 U ∓ 2 ∓ ǫ0 .
0

rf

+VDC

+VDC

29

(U+V cos(Ω t - π))/2 + V cos(ω t)
rf
rf
rot
rot
(-U+V cos(Ω t ))/2+ V cos(ω t - π/2)
rf
rf
rot
rot

2r

0

(-U+V cos(ω t))/2+ V cos(ω t - 3π/2)2r
rf
rf
rot
rot
(U+V cos(Ω t - π))/2 + V cos(ω t - π)
rf
rf
rot
rot

+VDC

+VDC

Figure 3.5: Our ion trap design. Potentials are shown that create the necessary fields for trapping,mass spectrometry, and applying Erot .

its smaller second order doppler shift in frequency standards applications [70]. The linear geometry
gives a line down the axial center of the trap where the rf voltage is zero as opposed to a single
point in the earlier geometry. This allows for a greater number of ions to be stored at low rf voltage
when space charge effects are taken into account. Heating in the trap occurs due to collisions thatdephase the ion motion from the applied rf fields. Therefore, the lower the rf fields the ions are
stored in, the lower the heating rates. Linear traps also offer greater optical and ion access forloading the trap and manipulating and detecting the stored ions. The linear trap is especiallysuited to our experiment due to the rotating bias field that we require. The four rods in our ion
trap can be driven 90◦ out of phase, as seen in Fig. 3.5 in order to produce the rotating field.
This also means that all of the time-varying fields are in the radial plane, which greatly simplifies
analysis of our experimental systematics.

3.3

Pseudo-Potential Well
Although the stability diagram of Fig. 3.2 is quite useful, it does not tell the whole story.

Whether an ion will remain within the physical confines of the trap depend not only on the a andq trapping parameters, but also on the ion’s initial position, velocity, and the initial rf phase. In
general, the ion motion will be a complicated sum of several motional frequencies, all with similar

30
weighting. A complete description requires numerical methods, and methods based on phasespace dynamics have been developed [69]. However, if the trap is operated in a region of small q
and smaller still a, analytic methods [5] are useful. We will operate in this regime, with typicalexperimental values as seen in Appendix A, giving a = 0.01 and q = 0.2.
In this limit of small q and smaller a, and thus small β, we see from Eqn. 3.13 that the lowest
order frequencies are well separated, with ω0 = βωrf /2 and ω1 ∼ ωrf . We then expect a slowly
varying, large amplitude motion X (the secular motion) and a rapidly varying, small amplitude
motion, δ (the micromotion),x=δ+X(3.17)d2 δ+ (a − 2q cos 2ζ)X = 0dζ 2(3.18)

Then, Eqn. 3.9 can be re-written as,

Since a is assumed to be smaller than q, and X is assumed to be slowly varying compared to the rf
drive frequency, we integrate Eqn. 3.18 to obtain,δ=−qX
cos 2ζ
2(3.19)

Substituting in Eqn. 3.17 and Eqn. 3.19 to Eqn. 3.9 we obtain,d2 x
aqX= −aX +
cos 2ζ + 2qX cos 2ζ − q 2 X cos2 2ζ
dζ 2
2(3.20)Taking the average acceleration of X to be the acceleration of x over one rf cycle we obtain,
*

d2 X
dζ 2

+

1
=
π

Z π
0

q2
d2 x
dζ
=
−
a
+
dζ 2
2

!

X

(3.21)

This shows, averaged over an rf cycle, that we have simple harmonic motion with a frequency of(a+q 2 /2)1/2 ωrf /2. Knowing that this is equal to βωrf /2, we see that β ≈ (a+q 2 /2)1/2 in the limitof small a and q. This approximation is good at the 5% level for q < 0.4. We can associate a trap
depth to this simple harmonic potential, and for a = 0 we obtain,2 r2
q 2 ωrfeVrf2
1
0=
Dx = ωx2 r02 =2
2
8
4M r02 ωrf(3.22)

31
The separation of the ion motion into a slow secular motion and a fast micromotion is strong
motivation to work in the limit of small a and q during our eEDM measurement.

3.4

Trapping Multiple Ions
To this point, we have discussed the trapping of a single ion with an idealized quadrupoleelectric field. The situation becomes considerably more complex when multiple ions are trapped
simultaneously. When considering neutral-neutral collisions, satisfactory results are usually foundby assuming hard sphere collisions described by a single scattering length. For 1/r interactions,
hard collisions that entail momentum transfer on order of the average thermal momentum are but asmall part of the story. The distance over which long range collisions are important is approximately
the Debye length, or Debye shielding distance,
λD =



ǫ0 kT
ni (Ze)2

1/2(3.23)

where ǫ0 is the permittivity of free space, k is the Boltzmann constant, T is the ion temperature,ni is the ion density, and Ze is the ion charge.
In the case of a 10 K cloud of 1000 trapped ions, and an ion cloud diameter of 1 cm, λD
will be ∼ 7 mm, or about the ion cloud size. The ion-ion interaction energy at a distance of 1 cm
is ∼ 1 mK. This means that a single ion will be in constant collision with ∼ 1000 ions, all withcollision energies of 1 mK or above. An individual ion motion, even in the absence of collisions
with ions or neutrals, can be quite complicated with at least the two frequency components ofrf micromotion and trapping secular motion. It is easy to imagine that the combination of thesecomplicated motions, with a thousand ion-ion interactions at an appreciable energy compared tothe ion temperature, will lead to chaotic motion in the trap and a very broad frequency spectrum.
This broad frequency spectrum will likely include components that are resonantly excited by therf drive, thus extracting energy from the rf field and heating the ions. This is one component ofwhat is broadly referred to as rf heating. Other components can include the effect of ion-neutral
collisions and hard ion-ion collisions that dephase the ion motion from the rf drive. The latter

32
effect is sometimes referred to as instability heating and can be viewed as a momentary increase in
the q parameter such that the ion temporarily falls into an unstable region where the ion motion
increases with time.
If the initial ion velocity distribution is anisotropic or non-Maxwellian, ion-ion collisions willredistribute the ions. The rate at which this is done can be estimated using the self-collisiontime [71],tc =11.4M 1/2 T 3/2
ni (Ze)4 lnΛ

(3.24)

where M is the ion mass in amu, T the ion temperature in Kelvin, ni the ion density in cm−3 , andΛ is a shielding parameter related to the Debye length. This has been confirmed for the case of
protons trapped in a 3-D quadrupole ion trap [72]. For our case of a 10 K ion cloud with 1000 ionsin a 1 cm ion cloud, and assuming singly charged ions of mass 200 amu, tc = 640 ms.
Outside of collisions, ion-ion interactions will also add another term to the electric potential.
For simplicity, take the case of a spherical symmetric trapping potential such that ωx = ωy = ωz .
In the limit of space charge energy less than the thermal energy of the ions, the limit we will likely
perform our experiment in, the ion cloud will have a Gaussian shape. The situation is easier toevaluate in the opposite limit, where the space charge energy is much greater than the thermalenergy of the ions and, assuming a harmonic pseudo-potential well, where the ion cloud will be asphere of uniform density. The potential due to space charge will then be,
−ρ
Φsc =
ǫ0

x2 y 2 z 2
+
+
2
2
2

!

(3.25)

where ρ is the charge density of the ions. This will modify the a trapping parameter which will
now read,
4eaeff =
2
M r02 ωrf

kUz
ρr2
U∓
∓ 0
2
ǫ0

!

(3.26)

with the -(+) corresponding to the x(y) direction. The situation is more complex at values of a andq such that the motion cannot be approximated as a fast micromotion and a slow secular motion.
In this case, the ion cloud can be significantly distorted from a uniform, spherical object during a
trapping period.

33

3.5

Our Ion Trap, v0.0
Often quadrupole construction is done with cylindrical rods in place of hyperbolic rods due tothe relative ease of construction of the former. Denison [73] has investigated the optimum electrodesize and spacing for cylindrical electrodes. The symmetry of the problem requires that the lowest
order terms are a quadrupole and then a 12-pole term. Denison looks for a minimum in the 12-pole
coefficient as r/r0 , the ratio of the electrode radius to half the spacing between opposite rods (seeFig. 3.5), is varied. This was done with the ground potential at infinity and at 3.54 r0 (the smallest
practical radius for a grounded cylinder placed around the quadrupole), yielding optimum ratios
for r/r0 of 1.146 and 1.1468 respectively.
An image of our trap is shown in Fig. 3.6. We utilized a grounded wire mesh at R = 3.54
r0 , non-magnetic 316 stainless steel electrodes of 25 mm diameter, with a spacing 2r0 = 21.8 mm,yielding the above stated optimum ratios. Each rod was segmented into three pieces, with theaxially confining DC voltages applied to the eight end-pieces. The length of the center electrodes,
lc , was 88 mm, and the length of the end electrodes, le , was 25 mm. Vespel spacers were used
to electrically isolate the three segments of the electrodes, chosen for vespel’s ease of machining,as well as its vacuum, electrical, and mechanical characteristics. Two stainless steel end-plates
were used to define the electrode spacing, with threaded rods running down the center of thehollow electrodes held by washers at the end-plates providing rigidity to the whole structure. The
threaded rods extended into a 6” CF vacuum flange that contained all of the electrical feedthroughsfor the trapping voltages as well as supporting a channeltron ion detector. The openings at theaxial ends of the the trap were covered by a 75% visibility wire mesh, making the entire trapping
volume well isolated from external electric fields.
The axial trapping potential was simulated using the electrostatic program SIMION. Fig. 3.7
shows the axial potential at the radial trap center for various ratios of the center electrode to
end-cap electrode length. The choice of an 88 mm center rod length gives rise to an approximate
square well potential. If a harmonic axial potential is desired, a fit to the potential over the range

34

electrodechannetlron

Figure 3.6: Image of our first ion trap. The electrodes are obscured by the grounded wire mesh at
R = 3.54 r0 . Note the channeltron ion detector mounted below the trap.

35

Figure 3.7: Axial trapping potentials for different central rod lengths, lc , holding the overall trap
length constant. Our first trap was built with lc = 88 mm.

of its full width half maximum yields an optimized center rod length of ≈ 25mm for our value of
21.8 mm for r0 . However, for a 10 K ion cloud, only 2 V must be applied to the end-caps in thecase of lc = 88mm in order to confine the ions to a 1cm length along z. In this case, the trapping
potential seen by the ions is extremely harmonic.

3.6

Trap Drive Electronics
We wish to apply rotating electric fields with the trap electrodes at frequencies ωrot ∼ 2π×100

kHz, and superimpose rf trapping fields that oscillate at frequencies ωrf ∼ 2π×15 kHz. This gives
wavelengths on the order of 1 km, much larger than the distance between the source electronics
and the trap. Therefore, impedance matching was of no concern in designing our drive electronics.
Flexibility was essential, as we desired to operate the trap with different ion species and at very
different trapping frequencies and trapping parameters a and q, depending on the nature of theexperiment being performed. We chose to use a system of op-amps as opposed to more traditional
methods of ion trap drivers such as resonant LRC circuits or quarter-wave resonators [74].
We used buffers/pre-amplifiers with standard ± 15 Volt op amps (OPA627) and their outputs
were used as inputs to simple summing amplifiers. In order to achieve relatively high trapping

36
Table 3.1: Electrical inputs to ion trap driver box.
Name of input
RF
DC A rod
DC D rod
DC bottom
DC A,B top
DC C,D top
DC center rods

Purpose of applied voltage
Oscillating trapping voltage
for radial confinement
Radial defocussing voltage, seperate
to move trap center radiallyaxial confinement, ability to load trap
axial confinement, ability
to steer ions into detector
alternative axial confinement

Electrode segments
phase: A and D rods
antiphase: B and C rods
A rod
D rod
A,B,C,D bottom end-caps
A, B top
C, D top
A,B,C,D center

potentials, high voltage OPA98 op-amps were used for the summing amplifiers. These have theability to drive up to ± 225 Volts, with maximum slew rates of 1000 V/µs. Radial trapping
potentials, for Hf+ ions, as high as 16 eV were achieved with rf trapping voltages of 190 Volts anda trap frequency of 250 kHz. At these high trapping voltages and frequencies, forced air cooling
was found to be insufficient, and the OPA98 op-amps were mounted on water-cooled copper heatsinks. The summing amplifiers included a trim input used to reduce DC offsets to ∼ 5 mV, withdrifts over the course of 10 hours of similar magnitude.
Each electrode segment was driven by a separate OPA98, and several inputs were available,
listed in Table 3.1, for applying rf and DC voltages. These inputs allowed us to manipulate the ionsin many ways, including ejecting ions from the trap, moving the axial and radial trapping center,and performing mass spectrometry. The DC voltages were supplied by an NI6733 DAQ board, witheight sixteen bit analog outputs and with maximum update rates of 1000 kS/s. We buffered theoutput of this board in order to source more current and to separate the computer ground from theexperiment ground. The rf voltages were obtained from an Agilent 33220A waveform generator,
capable of being AM or FM modulated. The rf was AC coupled to inverting (A,D rods) and noninverting (B,C rods) pre-amplifiers to produce the π phase delay between adjacent electrodes. Boththe DAQ and the waveform generator were computer controlled via LabView code.

37

Figure 3.8: Schematic of an integrator used to sum and amplify our ion signal when dumping ourion trap into a channeltron detector. Gains of either 1010 or 108 were typically used.

3.7

Ion Detection
Detection of ions can be highly efficient with the use of a channeltron, a form of electron

multiplier. Here an ion is accelerated to 1000’s of eV before striking the detector, with close to unit
probability of dislodging an electron from the surface. This primary electron is then accelerated
down the detector where it again collides with the detector walls and is amplified in an electron
avalanche, in the same way a primary electron is amplified in a photomultiplier tube. This canresult in ion detection efficiencies of ∼ 1, and gains of 106 or more.
Our initial ion trapping experiments utilized a channeltron, placed after the axial opening of
the trap, with the input of the detector biased at -2000 Volts and the output at ground potential.
Ions were ejected from the trap by dropping the end-caps nearest the detector to 0 V or to a slightnegative voltage, while the end-caps farthest from the detector remained at a positive bias. A
simple electronic integrator was used (Fig. 3.8) in order to further amplify the signal, and to reduce
saturation effects due to the channeltrons inability to source much voltage. Two gain settings
were typically used of 1010 and 108 V/A with corresponding time constants of 30 µs and 10 ms
respectively. Dark count rates on the order of 1/s were achieved when operating in pulse counting
mode.
In order to attempt an absolute calibration of ion number, we employed a Faraday cup. Thisis simply a metal can that collects the positive ions. A commercial CR-110 charge sensitive pre-

38
amplifier was placed directly after the vacuum electrical feedthrough that connected the Faradaycup. This configuration produced a minimum noise equivalent to 300 ions per trap “dump”.
A device similar to the channeltron is the microchannel plate detector, or MCP. An MCP
can be viewed as a 2-D array of micro-channeltrons. Typical dimensions are a plate thickness of
500 µm, with hole diameters of 10 µm, and an open area of 60%. Each hole acts as a channeltron,as incoming ions are accelerated and collide into the hole walls they create primary electrons thatare themselves accelerated into the walls creating an electron avalanche. Gains of a single plate
tend to be 104 , with higher gains achieved by placing two or more plates in series. Two main typesof MCPs were used during this experiment, an imaging MCP and a “fast” MCP.
In the imaging MCP a phosphor screen is placed behind the output of the MCP, and is biased
at a large positive voltage. The electrons output from the MCP are accelerated onto the phosphor
screen and a CCD camera records the resulting fluorescence. This detector is obviously helpful indetermining the spatial profile of the ion cloud, but is poor at recording arrival times. In orderto get some information on arrival times, we typically pulsed on the negative bias voltage to the
MCP. This was limited to a time resolution of ∼ 1 µs by the 5 mm distance between the MCP anda grounded wire mesh placed in front of the MCP.
For the “fast” MCP, a single electrode, at ground potential, is placed at the output. Thiscan lead to much faster detection times than with a channeltron, since the electron transit times
are much smaller in the extremely thin MCP. This device gives no ion position information, otherthan the fact the ion fell somewhere on the device, but the time resolution can be as fast as 100
ps, although typically the read-out electronics limits this to slower values. These devices are often
used in time-of-flight mass spectrometers due to their time resolution and we utilized them in thiscapacity as discussed in Chapter 5. A fast transimpedance amplifier was used in conjuction withthese MCPs. This reduced the time resolution to ∼ 20 ns, but this was still sufficient for our needs.

39

3.8

Early Ion Trapping
We loaded ions into our trap using electron impact ionization in our earliest ion trapping

experiments. A solenoid driven pulsed valve was opened typically with a 200 Volt, 300µs drive
pulse and 1 Torr of Xe backing pressure. The gas pulse was intersected by a 200 ms electron gun
pulse, both being directed through the center of the ion trap. This was done while rf trapping
voltages of ∼ 20 Volts were applied at 100kHz, and the end-caps were biased at 15 Volts DC.
Trapping times on the order of minutes were achieved after an initial ion decay that was due toincreased background pressure from the xenon gas pulse. Xenon was used since it had a similar
mass to our leading eEDM molecules at the time, HI+ . Xenon has seven naturally occurringisotopes, five of them have greater than 5% abundance, and they are spaced by one or two amu.
This isotope structure allowed us to test our ability to perform mass spectrometry in our trap. As
discussed in Chapter 4, mass spectrometry could possibly be used to read-out the electron spin in
our eEDM measurement. Therefore, demonstrating mass spectrometry in our ion trap with 1 amu
mass resolution and high sensitivity was one of our first ion trapping objectives.
Mass spectrometry was done by loading the ion trap with the voltage U = 0, and thenramping U to obtain a set value of U/Vrf , the voltage U was ramped back to zero and then the ionswere ejected from the trap using the end-cap voltages and sent into a channeltron for detection.
The rf Voltage Vrf was then stepped to a higher value and the experiment was repeated. In thisway we probed the typical U/Vrf operating line of a quadrupole mass spectrometer. One initialdifficulty in achieving sensitive mass spectrometry at good resolution was higher order resonancesin our ion trap. Due to details of the electronics and DAQ boards used, the ramp of U was done indiscrete steps with 1 ms between steps. As we stepped across the β = 2/3 contour, ions would be
lost from the trap. The data showing at what rf voltage ion loss began, for different U ramp values,
is shown in Fig. 3.9. Plotting these points out on the a, q stability graph, we see that the loss wasindeed associated with the β = 2/3 line. This problem was circumvented simply by increasing our
U voltage ramp speed, but it is worth pointing out that the next generation ion trap, discussed

40
below, will likely be driven in such a way as to increase these higher order resonances that arise
due to imperfections of the trapping fields. It will likely be important to operate the trap at valuesof a and q that are far removed from these resonances. Discussion of these trap resonances can befound in the literature [75].
Resolution of the mass spectrometer tended to improve at higher trapping frequencies. Data
for Xe+ and Hf+ showing ∼ 1 amu mass resolution can be seen in Fig. 3.10. The trapping frequencies
used were 100kHz for the Xe+ data and 250 kHz for the Hf+ data. Typical rf frequencies forcommercial quadrupole mass spectrometers are 1-3 MHz. In order to reach higher resolution, thescan line should be increased to higher values of U/Vrf . This pushes you closer and closer to theedge of the stability region, where the trap depth is close to zero for the x direction and close to
the β = 1 contour in the y direction, were the trapping frequency is one-half the rf drive frequency
and resonant excitation occurs. Many factors come into play in this region that are not containedin the analytical approach of this chapter that lead to the ion stability diagram. These include thefinite size of the trap, patch charge potentials on the electrodes, ion-ion and ion-neutral collisions,and field imperfections due to the cylindrical rods.

3.9

Second-Generation Ion Trap and Fields in eEDM Measurement
The first-generation trap was designed to minimize errors in our quadrupole trapping field,

largely driven by our desire for ∼ 1 amu mass resolution. Since building our first trap, the requirements for mass spectrometry in our ion trap have been greatly reduced, as we now will likely use
a fluoride instead of a hydride for the eEDM measurement. Mass spectrometry in our next generation trap will be performed using time-of-flight, and the necessary resolution of 19 amu shouldbe trivial to achieve. Time-of-flight mass spectrometry has the advantage of measuring multiple
charge-to-mass ratio ions for a single trap loading at efficiencies close to unity. This also meansthat the uniformities in the trapping fields can be sacrificed for uniformity in the rotating electricfield. The first trap suffered from non-uniformities in the rotating electric field of ≈ 20% over a
1 cm diameter ion cloud. The next generation trap will have six electrodes instead of four, and41a)Channeltron signal [a.u.]

Mass scan line

Applied RF voltage V [V]b)Figure 3.9: a) Ion number in trap after trap potential U has been ramped to various values as afunction of rf voltage. b)Plotting the onset of loss in the a, q stability diagram shows the ion loss
was associated with a β = 2/3 resonance, implying imperfections in our trapping fields.

42

0.40

132Xe+

ion signal (arb units)

0.35

131Xe+

0.30
0.25

134Xe+

129Xe+

136Xe+

0.20
0.15
0.10
0.05
0.00

126

128

130

132

134

136

138

mass (amu)
+

+
+
+

+

Figure 3.10: Mass spectrometry performed in our ion trap showing a) the five dominant isotopesof xenon, data taken at an rf frequency of 100 kHz, with U/Vrf = 0.16 and b) the five naturally
occurring isotopes of hafnium, data taken at an rf frequency of 250 kHz, and U/Vrf = 0.1695. Mass
resolutions of better than 1 amu allowed us to distinguish Hf+ from HfH+ .

43will be spaced farther apart (r0 = 4.0 cm). Numerical modelling shows the rotating electric field
uniformity should be better than 0.5% over a 1 cm diameter ion cloud.
Another motivation in designing a new trap is the desire to make ion measurements throughlaser induced fluorescence. The maximum solid angle for optical detection achievable in our first
ion trap design was limited by the large area occupied by the rod electrodes. The new trap design
replaces the cylindrical rods with electrodes of thin cross-sections when viewed from the trap center.
Two large ellipsoidal mirrors surround the ion trap covering 80% of the solid angle. These mirrors
have a bare gold coating, as conducting surfaces are preferred to minimize patch charge effects on
the trapping fields. The center sections of the mirrors are electrically isolated from the bulk of themirrors and act as the end-cap electrodes for z-axis ion confinement.
The perfect ion trap would have idealized electric and magnetic fields as follows
E~ = Erot r̂′ + Erf′ (xx̂ − y ŷ) cos(ωrf t) + Ez′ (−z ẑ + y ŷ/2 + xx̂/2)(3.27)~ = Brot r̂′B(3.28)

where r̂′ = cos(ωrot t)x̂ + sin(ωrot t)ŷ and Erf′ ≈ −2Vrf /r02 .
2 ≫ eE ′ /M ,If we assume ωrot ≫ ωrf , that ωrot /ωrf is not a rational fraction, and that ωrf
rf

then we can cleanly separate out the ion motion into three components: rf micromotion, circular
micromotion, and secular motion.
As discussed above, rf micromotion involves a rapid oscillation at ωrf whose amplitude grows
as the ion’s secular trajectory takes it away from trap center. The kinetic energy of this motion,
averaged over an rf cycle, is given by
Erf = (x2 + y 2 )e2 Erf′ 2
2
4M ωrf(3.29)

where x and y in this case refer to the displacement of the ion’s secular motion.
The displacement of the ion’s circular micromotion is given by~rrot = −eE~rot2 .
M ωrot(3.30)

44
The kinetic energy of the circular motion, averaged over a rotation cycle, is given by
Erot =

2e2 Erot
2 .
2M ωrot(3.31)The time-averaged kinetic energies of the two micromotions act as ponderomotive potentials that
contribute to the potential that determines the relatively slowly varying secular motion:
Usec = Erot (x, y, z) + Erf (x, y, z) + eEz′ (2z 2 − y 2 − x2 )/4.

(3.32)In the idealized case, the secular motion corresponds to 3-d harmonic confinement with secular or“confining” frequencies
ωi =1 ∂ 2 Usec
M ∂i2

!1/2

,

(3.33)

for i = x, y, z. In the idealized case, confinement is cylindrically symmetric, ωx = ωy , and Erot is
spatially uniform, so the circular micromotion does not contribute to the confining frequencies.
If the density of ions is low enough such that few momentum-changing collisions occur during
a single measurement, any given ion’s trajectory will be well approximated by the simple sum of
three contributions:(i) a 3-d sinusoidal secular motion, specified by a magnitude and initial phase for each of
the x̂, ŷ, and ẑ directions. In a thermal ensemble of ions, the distribution of initial phases will
be random and the magnitudes, Maxwell-Boltzmannian. For typical experimental parameters (seeAppendix A) the secular frequencies ωi will each be about 2π × 1 kHz and the typical magnitudeof motions, r, will be about 0.5 cm.
(ii) the more rapid, smaller amplitude rf micromotion, of characteristic frequency about
2π × 15 kHz and radius perhaps 0.05 cm. This rf micromotion, purely in the x-y plane, is strongly
modulated by the instantaneous displacement of the secular motion in the x-y plane, and vanishes
at secular displacement x=y=0.
(iii) The still more rapid rotational micromotion, purely circular motion in the x-y plane, at
frequency ωrot about 2π × 100 kHz and of radius comparable to the rf motion, around 0.05 cm. In
the idealized case, the rotational micromotion (in contrast to the rf micromotion) is not modulatedby the secular motion.

45
As described in Chapters 6.5 and 7.1, for spectroscopic reasons we must operate with trapping> 30k T . Under that condition, relatively small imperfections in E ,parameters such that Erot ∼rotB

say a spatial variation of 1.5%, can give rise to contributions to Usec of the same scale as the ions’
thermal energy, and thus significantly distort the shape of the trapped ion cloud or even deconfine
the ions.

Chapter 4
Electron Spin Read-Out

As mentioned previously, spectroscopic eEDM measurements perform an electron spin resonance in the presence of electric and magnetic fields. Of course, a crucial component of a spinresonance experiment is measuring the final spin populations. Typically, this is done by making
a spin dependent electronic transition and then collecting fluorescence from the subsequent decay(LIF). Our trapped molecular ion experiment is limited to low ion numbers due to strong ion-ion
interactions (see Chapter 7). Also, since molecules are being used, there are no cycling transitions
to increase LIF detection efficiency. With multiple vibrational and rotational levels to decay to,approximately one photon will be scattered from any given molecular ion. With a limited solid
angle of collection and photon detection efficiencies of ∼ 0.10, LIF could have poor efficiency.
As mentioned in Chapter 3.7, ion detection using channeltron and MCP detectors can approach unit efficiency. It is tempting to leverage this high efficiency of ion detection to read-out
the electron spin populations in our experiment. One approach would be to make a spin-dependent
electronic transition to an excited state and then photodissociate this excited state. This would
map the electron spin information onto the mass-to-charge ratio of the trapped ions. As discussed
in Chapter 3, the ion trap can act as a mass filter, either by operating in the a, q stability plane ata point where a narrow range of M/e ions are trapped, or by using the trap as a time-of-flight mass
spectrometer. We have also experimented with using a one-dimensional standing wave of intenselaser light in order to create a spin-dependent heating of the ion cloud. This technique has also
been explored as a method for performing broad survey spectroscopy of our molecular ions.

47

4.1

LIF Detection
ab initio calculations [4, 54, 64, 61] suggest there are Ω = 1 states ∼ 10,000-15,000 cm−1

above the 1 Σ ground state and ∼ 7,000-14,000 cm−1 above the 3 ∆1 level where we will perform
our spin resonance measurement. Using σ ± polarized light, we can excite only the population in
the mF = ∓3/2 level to the J = 1 level of one of these higher lying Ω = 1 states. A measurementof the fluorescent decay from this higher lying state will then yield the final spin population of ourRamsey experiment. The efficiency for LIF can be approximated, in the strongly saturated limit,
ηLIF =

1× A × RNb × ηdet × T
2

(4.1)

with A the solid-angle of fluorescence collected onto the detector, R the mirror reflectivity, Nb the
average number of reflections off the mirrors, ηdet the detector efficiency, and T the transmissionof any filters, windows, or lenses in the imaging system.
Pulsed laser excitation is preferred for several reasons. The quantization axis will be rotating
at ωrot , and we wish to make transitions only of δmF = +1 or -1. Therefore the laser k̂-vector
should point along the quantization axis with left- or right-circularly polarized light being used.
The excitation pulse must then be short compared to 1/ωrot so the quantization axis can be treatedas stationary during the pulse. Also, reaching saturation intensity on the molecular transition could
be difficult with a cw laser. We might wish to drive a weaker transition that then decays to thelower 1 Σ level as this could allow us to frequency filter the signal light from the scattered lightof the excitation laser. Also, the decay will then be at higher frequencies and therefore a higher
detection efficiency for our PMT can be realized. This, combined with the relatively large (r ∼5 mm) ion clouds, requires more laser power than could be conveniently achieved with cw lasers.
Pulsed excitation also allows time gating of the detector in order to decrease background signal
due to light scatter of the excitation beam into our detector.
If the Ω = 1 excited levels around 13,000 cm−1 are used, we will require a detector efficientat the corresponding wavelength of 770 nm. Photodiodes have high quantum efficiencies at suchwavelengths, but photodiodes are not well suited to measuring the expected low light intensities

48
due to the inherent electronic noise, e.g. it will be difficult to reach shot-noise limited detection.
PMTs have extremely low dark count rates, and can reach shot-noise limited detection at very low
count rates. The detection efficiencies of PMTs tend to fall off dramatically at near-IR wavelengths
since cathodes with low enough work functions are difficult to produce. The highest efficiency PMT
in these wavelength regions that we have been able to find are the Hammamatsu R3896. We can
expect quantum efficiencies, at best, of 10% in our region of interest, and more likely 1%. Perhaps
transitions can be found at higher frequencies, but our current form of survey spectroscopy requiresa Ti:Saph laser with minimum wavelength ≈ 700 nm.
With the new ion trap allowing for 80% of the solid angle to be imaged onto two photon
detectors, the maximum spin detection efficiency we expect is 4% using LIF.

4.2

Photodissociation

4.2.1

Predissociation and CH+

A process known as predissociation can occur in molecules if the molecular potential undergoes an avoided crossing, giving rise to excited vibrational levels that are allowed to tunnel out
to the dissociation continuum as seen in Fig. 4.1. This situation is known to occur in CH+ andhas been subject to several experimental and theoretical treatments [76, 77, 78, 79, 80, 81]. The
predissociating states in CH+ occur due to coupling between the A1 Π state and the c3 Σ+ repulsive
curve. The A1 Π state dissociates to the slightly higher energy C+ (2 P3/2 ) + H(2 S) configurationcompared to the C+ (2 P1/2 ) + H(2 S) state that the c3 Σ+ level dissociates to. A combination of
spin-orbit coupling and nuclei rotation lead to couplings of the two molecular potentials. The potential advantage of using a predissociating state is that the bound-to-bound transition involved
will often have a larger cross-section than a bound-to-continuum transition. After building an ion
trap capable of 1 amu mass resolution, we decided to perform a proof-of-principle photodissociation
experiment using the known CH+ transitions.
CH+ was produced and trapped by pulsing in CH4 gas from a pulsed valve and using electron

49

B
A
vexcited

E
R
Figure 4.1: Potential energy curves of a molecule containing predissociating states as a functionof the internuclear spacing R. The bound energy curve A dissociates to atoms, with at least one
atom in an excited state. The repulsive curve B dissociates to atoms at a lower energy. Couplings
between states A and B lead to an avoided crossing. An excited vibrational level of state A cantunnel through the resulting potential into the continuum.

50
impact ionization in an identical set-up to that mentioned in Chapter 3.8. The trap was loadedwith trapping parameters only allowing stable trajectories for CH+ ions. The trap parameters
were then ramped to a more stable configuration where both C+ and CH+ were stably trapped asa 10 ns laser pulse was sent through the ion cloud. This pulse was frequency doubled to the UV(the relevant transitions were in the range of 33,000 cm−1 ) and approximately 10 mJ/pulse wasavailable. The trap was then ramped to parameters compatible with only C+ and then dumped
into a channeltron ion detector. Ions would be detected only if CH+ had been dissociated to C+
by the laser pulse.
Despite the known transitions of CH+ , we experienced two problems that frustrated ourattempts to observe autoionization in our set-up. The first was charge-exchange collisions between
CH+ ions and neutral CH4 background gas that severely limited our CH+ trapping lifetime. Fig. 4.2
shows the results of holding CH+ in our trap for 1 ms and 100 ms. The 100 ms data shows almost
no evidence of CH+ and instead it appears reactions produced C+
2 ions. C2 H2 and CHF3 were also
used as sources of CH+ ions, all with similar effects limiting the ion storage time. The only way
we were able to mitigate this problem was by decreasing the backing pressure, but this limited thenumber of CH+ ions we were able to load into our trap. The optimum backing pressure of CH4
was found to be ∼ 1 Torr.
The other problem was a misunderstanding of our needed laser pulse intensity. We believed
the cross-section for the transition from the ground state to the A1 Π predissociating state was afactor of 104 higher than what was ultimately measured. Looking at results from [80], we expectedcross-sections on the order of 10−16 cm2 for the transitions of interest. This meant for our experiment we should have easily saturated our ∼ 1 cm diameter ion cloud with a 10 mJ laser pulse.
When a signal was found, we observed that the transition was far from saturated. In the end, CH+
was trapped for 5-10 seconds as laser pulses at 10 Hz were fired into the cloud, allowing for saturation of the transition. Fig. 4.3 shows data taken in this way. Our results were consistent with otherestimates for the cross-section [81, 76, 77] of ∼ 10−20 cm−1 . It appears there was an error in [80]and we chose the wrong reference to help us estimate our needed saturation intensity. Ultimately,

51

12

+

CH

peak

C

2

10

+

peak?

ion signal (arb units)

1 msec hold
100 msec hold

8
6
4
2
0
10

15

20

25

30

35

mass (amu)

Figure 4.2: CH+ held in our ion trap appeared to react with CH4 gas. After trap loading, the ions
are initially CH+ , but after a 100 ms hold time there is no evidence of CH+ in the trap. Instead amass peak around 24 amu is found. This required trap loading at lower CH4 pressures and madedetection of CH+ predissociation more difficult.

52
we were able to dissociate through multiple rotational lines as many ground state rotational levels
were populated from the electron-impact ionization process.

4.2.2

Photodissociation to a Repulsive Curve

The experience with CH+ was not very encouraging in our efforts to readout electron spin
information using dissociation. However, the first difficulty in observing the CH+ dissociation,
the charge-exchange collisions limiting our CH+ lifetimes, should cause no concern for our eEDM
measurement using ions such as HfF+ . We have not observed such fast charge-exchange processes tooccur in HfF+ . Background pressures when trapping ions such as HfF+ from our pulsed molecularsource discussed in Chapter 5 are made small by differential pumping. As we will see in Chapter 7.2,
other considerations require strong enough differential pumping such that ion-neutral collisions arerare during the coherence time of our measurement.
The small photodissociation cross sections are an area of concern. It will not be possible towait several seconds while multiple laser pulses saturate the transition, as the lifetime of the 3 ∆1state is not expected to be that long. The CH+ cross sections are small due to the very poor overlap
between the vibrational wave function of the ground state and that of the predissociating state.
According to the Franck-Condon principle, the transition moment between two electronic states will
be proportional to the overlap integral between the two vibrational levels, Ψ′v Ψ′′v dr, with Ψ′v (′′ )R

the vibrational wave function of the excited (ground) state involved in the transition [82]. The
transition strength is proportional to the square of this overlap integral or Franck-Condon factor.
The Franck-Condon principle is based on the idea that during an electronic transition, nuclei haveno time to respond and therefore will have the same position and momentum immediately before
and after the transition. A Franck-Condon matrix will have off-diagonal terms in v ′ − v ′′ if the twoelectronic levels have different vibrational energy spacings or bond lengths. However, the FranckCondon factor between a v ′′ = 1 state and a v ′ = 12 state, the case for our CH+ predisocciatingtransition, is typically quite small and might be small enough to negate the advantage of a boundbound transition. An estimate of the cross section to a repulsive potential curve will reveal to what

53

uv vacuum wavevector [cm^-1]
32810

32809

32808

32807

32806

32805

32804

140

120

C+ counts

100

80

60

40

20

0
609.40609.42609.44609.46609.48609.50laser fundamental wavelength [nm in air]Figure 4.3: Data showing photodissociation of CH+ to C+ and H through a predissociation resonance. This data corresponds to a transition to the J = 3, v = 12, A1 Π state of CH+ . This data
was taken with a 5 second ion hold time, during which time 50 pulses, with 10mJ/pulse, from afrequency doubled dye laser were incident on the ions.

54
extent this is true.
The width of a photodissociation resonance in a diatomic molecule involving a repulsive curve
will be dominated simply by the Frank-Condon overlap between the initial state vibrational wave
function and the repulsive curve. A good approximation is to treat the unbound molecular wave
functions as delta functions at the classical turning points [82]. Then, one needs only to know theslope of the repulsive curve at the equilibrium internuclear distance of the initial state and the widthof the initial state vibrational wave function. These two numbers will then give the approximate
width of the photodissociation resonance.
In order to approximate the slope of a repulsive curve in HfF+ , we look at a repulsive curvein the isoelectronic PtH+ . ab initio calculations [61] of PtH+ electronic levels show a 3 Σ repulsive
state. One could imagine a similar state being easily accessed from the 3 ∆1 level of HfF+ through anintermediate 3 Π state. We then simply scale the energy of dissociation for PtH+ to that calculatedfor HfF+ and use that same scaling factor for the energy of the repulsive curve. Doing this givesa 3 Σ repulsive curve in HfF+ that is at 71,000 cm−1 with a slope of about 80,000 cm−1 /Å at themolecular bond length.
A quantum harmonic oscillator wave function is given by:1Ψ = (2n n!)− 2 (mω 1 −ξ2 /2
Hn (ξ))4 eπh̄where Hn is a Hermite polynomial of order n, ξ =

q

(4.2)

mω
h̄ x, and m is the reduced mass of thetwo atom system. The ground state vibrational wave function is then just a Gaussian with width
wgs =

q

h̄
mω .

Using h̄ω ≈ 700 cm−1 in HfF+ , we get wgs = 5.3 × 10−12 m or about 0.1ao . This

gives a width for our photodissociation resonance of wpd =4,200 cm−1 .
The cross-section at a given frequency is given by:σ(ω) =

1 g2 2λ g(ω)A21
4 g1 21(4.3)

where g2 and g1 are degeneracies of the upper and lower levels respectively and g(ω) is a lineshapefunction normalized such thatR∞−∞ g(ω)dω = 1.

55
Using
3ǫ0 hc3
3
2ω21

(4.4)

4π 3 g2 g(ω)µ221
3 g1 ǫ0 hλ21

(4.5)µ221 = A21
this becomesσ(ω) =

if guessing a transition dipole moment is preferred to guessing a spontaneous decay lifetime. Now
the assumptions come in. Assume λ21 =600 nm (more on this later). Looking at transition dipole
moments from Petrov et al. [64], µ21 = 0.3 a.u. seems like a conservative estimate. This is about thesmallest µ21 for an allowed transition in HfF+ that Petrov calculated. Note that this corresponds
to a spontaneous lifetime of ≈ 1.2µs. We assume that the full dipole moment is realized since all
rotational and vibrational transitions will fall within the broad photodissociation resonance. We
assume the degeneracy of the intermediate state and the repulsive curve are equal and set gg21 = 1.
Using
Z ∞

x2

e− 2w2 dx =

√

2πw2

(4.6)

−∞

we obtaing(ω0 ) = q

1
2
2πwpd

(4.7)

at the center of the resonance. This gives σ(ω0 ) = 3.8 × 10−19 cm2 . Note that this is an orderof magnitude larger than the cross section for the predissociating CH+ state used in our earlier
experiment (σ = 2.4 × 10−20 cm2 was the calculated cross section for the strongest line [81, 76, 77]).
Define a saturation pulse energy
E=

hcA
λσ

(4.8)

where A is the cross-sectional area of both the laser beam and ion cloud. Assuming a 100 mJ
pulse at 600 nm, this gives 0.12 cm2 as the area that can be saturated. This corresponds to a 1.9mm beam radius. If the radius of the ion cloud needs to be decreased, it would only have to bedone for a very short time, since the laser pulse is only 10 ns long, and heating of the ion cloudshould not be a concern once the ions are dissociated. The focusing of the ions would have to be56
done while still maintaining a well defined quantization axis. The 0.3 ea0 transition dipole moment
seems very conservative, as long as repulsive states exist. One could imagine that there would bemany repulsive states with allowed transitions to the intermediate state within this broad 4000
cm−1 window, yielding a larger cross section.
If the dissociating pulse is at 600 nm, we would be able to use the higher pulse energy of thefundamental. However, this would require us to use an intermediate state that is very close to thedissociating limit, which might not be feasible. If you assume the pulse must be at a shorter λ,the saturation energy remains unchanged, but the available pulse energy would decrease by abouta factor of four. If 532 nm (doubled Nd:YAG) light could be used for the dissociating photon, anorder of magnitude more pulse energy would be available.
It seems the most likely scheme would involve a narrow cw laser operating on a transition
that selects the spin using polarization. The second laser would probably have a wavelength longer
than the first (decreasing the chance that we photodissociate without the first laser) and might beresonant with another intermediate state that then dissociates. Since the dissociation step is so
broad the requirement on the second laser wavelength is mainly determined by the bound-boundresonance.
It seems possible that photodissociation to a repulsive curve could be used to read out the
spin information in an efficient way in our experiment.

4.3

Light-Induced Heating
Another possible way of detecting our spin populations is to create a spin-dependent heatingof our molecular ions. This was inspired by work on slowing molecular neutrals using a deep (≈ 20K) optical lattice potential. In the work from Ref. [83] molecules from a pulsed supersonic beamsource interact with a travelling lattice potential given by,
2αa q
1U (z, t) = −I1 (t)I2 (t) cos2 (kL z − δωrel t)
ǫ0 c
2




(4.9)

57
where αa is the effective polarizability, I1 (t) and I2 (t) are the intensities of two crossed laser beams,
δωrel is the frequency difference between the two laser beams, and kL is the lattice wavenumber
given by kL = (4π sin φ)/λ with λ the mean laser wavelength on φ the half-angle between the two
fields. Typical laser pulsewidths for high pulse energy Q-switched lasers used in these experimentsare ∼ 10 ns. Molecular neutrals were created in a supersonic expansion beam and their initial
velocity was slowed from 400 m/s to as low as 270 m/s, with the goal of producing slow cold
molecules in the laboratory frame.
Our goals and approach differed from the above work. As will be discussed in Chapter 5, webuilt a laser ablation supersonic molecular beam source containing both ion and neutral molecules.
We wished to use a deep one-dimensional optical lattice to read-out spin populations as well as
to perform survey spectroscopy on molecular ions, with HfF+ as our first target. As molecularion densities in the beam were quite low compared to neutral densities of the above experiment,
we used a much larger diameter laser beam (3 mm diameter vs 60 µm). The scheme was toshine the one-dimensional lattice perpendicular to the beam direction, and attempt to heat the
low transverse temperatures of our molecular beam. This heating could be detected simply byusing an imaging MCP detector (described in Chapter 3) and measuring the resulting increasein the ion beam width. The AC polarizability of the molecules is strongly frequency dependentnear the region of a resonant transition. Scanning the laser wavelength across a transition will
therefore show a frequency dependent ion heating and, if σ ± -polarized light is used, a potentiallyspin-state-dependent heating.
Eq. 4.9 is most useful far from resonance, were the polarizability αa can be well approximatedas constant. Close to a two-level resonance, the potential due to the dipole interaction of a standing
wave is [84]


1 +

1
Udip = δωL ln 2

8Ω2R cos2 (kL z)
+
2
γn

1+






2δωL 2γn
2δωL 2γn,(4.10)

with ΩR the on-resonance Rabi frequency of each of the standing wave laser beams, δωL the laser
frequency detuning, and γn the natural linewidth of the excited state. The depth of the lattice

58
potential wells is then Ul d = |Udip (kL z = 0)|. Fig. 4.4 shows the lattice depth in the limit ofΩR ≫ γn as a function of the dimensionless ratio δωL /ΩR . The maximum lattice depth of ≈ 0.569
× ΩR is at δωL /ΩR ≈ 0.714. As seen in the figure, the lattice depth peak is quite broad, with thehalf-maximum point occurring at a detuning of ≈ 3.2 × ΩR . The ion heating will depend not only
on the lattice depth, but also on the initial ion beam temperature, the velocity of the ions in the
direction of the lattice, and the ion’s motional period in the trapping potential compared to thelaser pulse time. If the trap period is ≈ 4 times the laser pulse length then maximum heating will
occur. However, if the ions have an initial velocity such that they travel an appreciable fraction ofa lattice wavenumber during the time of the ∼ 10 ns pulse, the ion heating will be greatly reduced.
The fundamental advantage for spectroscopy is the large linewidths associated with the technique. Well depths on the order of a few Kelvin are reasonably achieved, making possible linewidthsof the heating signal greater than 500 GHz. This is a nice feature when attempting to do survey
spectroscopy over several thousand cm−1 , which is the position we found ourselves in with ab initio
calculations having uncertainties in this range for molecular ions containing heavy transition metals
such as hafnium. The potential advantage for spin read-out is again the leveraging of high detection
efficiencies for ions.
We chose to test this technique on Yb+ ions as they posses a known transition we could
easily access with the same frequency-doubled pulsed dye laser that was used for CH+ dissociation
studies and also since Yb+ was easily produced in our laser-ablation supersonic-expansion source.
A collimated Yb+ ion beam was produced using two beam skimmers, with transverse temperatures< 500 mK observed. The ion beam was ≈ 17 cm long, while the laser beam diameter was 3 mm.
In order to maximize the signal-to-noise ratio we gated the MCP voltage on for 1 µs and therefore
needed precise timing in order to image the ions that had been subjected to the optical latticepotential. An ion lens was placed in the beam, just before the location of the lattice, in order tominimize the ion beam width on the MCP. This lens could also produce timing information bypulsing the lens on and noting at what time the minimum ion width was achieved. The timing
resolution of this method was insufficient and instead we used two small “timing rods” with voltages

59pulsed on to deflect the ion beam. Patch charges on these rods caused distortions in the transverse
beam profile and the rods were therefore lowered out of the beam when heating data was taken.
The dye laser beam was split into two beams that entered our vacuum chamber from opposite
directions. This was done because of the short (∼ 10 cm) coherence length of the laser. Placing a
retro-reflecting mirror inside the vacuum chamber proved problematic as the laser pulse lead to aslight ionization of the mirror that distorted the ion beam. Fig. 4.5 shows results of Yb+ heatingwhen the optical lattice laser was tuned across the 2 S1/2 →2 P1/2 resonance at 369.4 nm.
Moving to molecular ions provided several challenges. The production of molecular ions suchas HfF+ in our beam source was accompanied by the creation of many other molecular and atomic
species. In order to maximize our signal-to-noise ratio, we wished to separate out the HfF+ ions ina way that left the transverse temperatures < 500 mK. This was accomplished by placing a large(9 cm diameter) plate electrode at a 45◦ angle to the ion beam and pulsing a positive voltage on
this plate for ≈ 1 µs in order to deflect the ion beam 90◦ in a mass-dependent way into our imaging
MCP. The experimental set up is shown in Fig. 4.6 and mass resolution using the 45◦ plate is shownin Fig. 4.7. This technique maintained the spatial profile of the ion beam along the direction of the
optical lattice and beam temperatures of < 500 mK were maintained. Fig. 4.8 shows MCP imagesof a Yb+ ion beam taken with this configuration.
Transition strengths are expected to be significantly lower in the case of HfF+ compared toYb+ , leading to smaller lattice well depths. The transition dipole moment for the 2 S1/2 →2 P1/2
transition in Yb+ is 1.75 a.u. compared to the estimates for transition strengths in HfF+ of 0.3-0.6
a.u. for transitions of interest from the ground 1 Σ state [64]. Transition strength is also spread acrossrotational and vibrational levels in molecules and many rotational levels are thermally populated
in our molecular ion beam. Transitions from the thermally populated states that don’t change therotational quantum number, “Q-branch” transitions, will all fall comfortably within the powerbroadened linewidth and therefore all ions should be exposed to the lattice potential as the laser
frequency is scanned over Q-branch transitions. However, the transition strength will still be
spread across all allowed rotational transitions, leading to an approximate factor of three decrease

60

0.5

ULD/WR

0.4
0.3
0.2
0.1

-5

0

5

10

dwL/WR
Figure 4.4: Lattice potential depth as a function of the laser detuning δωL divided by the onresonance single beam Rabi frequency ΩR , in the limit ΩR ≫ γn , with γn the natural linewidth of
the transition.

Figure 4.5: Data showing ion heating in a Yb+ beam as the lattice laser was tuned over a Yb+
resonance at 27,061.8 cm−1 (369.4 nm).

MCP detector

Ion lens
Ablation Laser pulse
Pulse
valve

Skimmer

Skimmerv ~ 1000 - 2000 m/s

10 nsec dye laser pulse

Hf ablation target
He (99%) +SF6 (1%)
Seed Gas

45° electrode
Intense 1-D standing wave

Figure 4.6: Experimental set-up for light-induced heating measurements. Ions are created using laser ablation of a metal target and areswept up in a supersonic expansion (more beam details are found in Chapter 5). The ions are collimated using two skimmers to transverse
temperatures < 500 mK. An ion lens is used to minimize the beam width on the imaging MCP detector. An intense 1-D optical standing
wave heats ions if the laser frequency is near an ion resonance. A 45◦ electrode is used to mass-select ions. The heated clouds are detectedby an imaging MCP.

61

62
in transition strength, as well as a non-diagonal Franck-Condon matrix leading to at least a factorof two decrease in transition strength. It is possible all transitions for the low rotational states will
fall under the FWHM of the lattice potential, retrieving some of this lost transition strength. The
range we wished to scan in search of HfF+ transitions was also at a factor of two longer wavelengths,
making it more difficult to reach the regime were the lattice trap period is ≈ 4 times the laser pulse
duration. The one advantage relative to the Yb+ case is the laser pulse energy available in thevisible wavelengths of interest for HfF+ was ∼ 10 times that available at the UV wavelengths of
Yb+ , but this corresponds to only a factor of about 3 in the lattice depth. Certainly the molecularsignal was expected to be weaker.
Obtaining a reliable signal, even in the much favored case of Yb+ , proved to be difficult. An
ion beam temperature of 500 mK corresponds to an electric potential of only 40 µV. Small patch
potentials on components inside the vacuum chamber were capable of wreaking havoc on our ion
beams transverse profile. Also, the angle between the lattice laser and the ion beam was critical.
If the angle between the two was greater than 5 mrad, the ions would traverse a full lattice spacing
during the laser pulse, greatly reducing the signal. Aligning the laser beams to this accuracy is not
a problem, but knowing the exact ion beam direction is difficult as small potential gradients willsteer the beam. Given these issues we decided to abandon this approach, at least as a means ofbroad survey spectroscopy. However, performing similar experiments inside the ion trap would havecertain advantages. Ion-laser overlap and timing issues would be mitigated. If used as a methodof electron spin read-out the ions should have fewer rotational levels populated and a transition
with a diagonal Franck-Condon matrix might be used. Another possible technique to salvage thismethod is to sympathetically cool the molecular ions with laser-cooled atomic ions. This should
lead to large improvements in signal-to-noise.

63

Hf

+

HfF

+
+

HfF2 HfF+3

Figure 4.7: Mass spectrometry data using the 45◦ deflection plate showing the ability to seperate
HfF+ ions from Hf+ and HfF+
2 . This was achieved while maintaining < 500 mK transverse beam
temperatures.

(a)

(b)

Figure 4.8: Data from the imaging MCP showing a Yb+ cloud deflected by the mass-selecting 45◦ plate, with (a) the one-dimensional
standing wave off, and (b) the one-dimensional standing wave applied on resonance, causing heating of the ion beam.

64

Chapter 5
Creation, Cooling, and Trap Loading of Ions

5.1

Ion Cooling Introduction
Our eEDM experiment requires cooling of both the internal (rotational, vibrational) andexternal degrees of freedom. Internal cooling is needed as we wish to perform the eEDM measurement on only a single ro-vibrational state and we must prepare the ions in only a single mF -level
before performing our measurement. Typical rotational constants of heavy metal fluorides are ≈0.3 cm−1 . This means 0.14% of ions will be in the ground rotational level at 300 K for an Ω =0 state and 0.43% for an Ω = 1 state. Those percentages increase to 4% and 12% if the ions arerotationally cooled to 10 K (see Fig. 5.1). Working with the ground rotational level is preferable,
as it provides the lowest number of mF levels. If the molecular ions are created at 300 K, withinternal and external degrees of freedom in equilibrium, we would expect the ions to occupy almost
exclusively the vibrational ground state. Below we will see that is not always the case. Any ion
not in the vibrational ground state will not be useful in our EDM measurement.
Low translational temperatures are important in order to maximize our electron spin coherence time. As discussed in Chapters 6 and 7, high ion temperatures can lead to decoherence
through ion-ion collisions above a threshold momentum transfer, as well as from inhomogeneous
Berry’s phase contributions. Below we discuss several cooling schemes we explored; buffer-gas
cooling, laser cooling, and supersonic expansion.

Occupation Fraction

66

0.15

0.10

0.05

10

20

30

40

50

60

J
Figure 5.1: Occupation fraction for rotational states with quantum number J, for 300 K (red) and
10 K (blue) rotational temperatures given a typical fluoride rotational constant of Be = 0.3 cm−1
in an Ω = 0 state.

67

5.2

Buffer-Gas Cooling
Although buffer-gas cooling of neutral species is a relatively new field, with pioneering workfrom Doyle et al. [85], it has long been used as a means to cool trapped ions [86]. Trapping depthsof thousands of Kelvin are relatively trivial to achieve in ion traps, therefore even room temperaturebuffer gas can provide dramatic cooling effects for trapped ions. Our initial attempts to demonstrate
improved trap lifetimes and mass spectrometry resolution in our ion trap using room temperature
helium buffer gas were unsuccessful due to contaminants in the helium. These contaminants werelikely coming from the stainless steel tubing of our gas handling system. Once a liquid nitrogen cold
trap was used to eliminate these contaminants, we were able to achieve improved trap lifetimes.
As discussed above, ion temperatures of 300 K still leave only a small portion of the molecular ions in the rotational ground state. Therefore, if buffer gas were to be utilized in our EDM
measurement, helium cooled in an ≈ 4 K liquid helium dewar, or a dilution refrigerator, would
be employed. While this scheme would cool both the internal and external degrees of freedom tosufficiently cold temperatures, there are potential issues. Neutral-ion collisions during the Ramsey
resonance portion of the experiment would destroy our spin coherence, due to inelastic collisionschanging magnetic and rotational levels. Even in the absence of inelastic collision, elastic collisions occurring while Erot is applied would have a characteristic energy of Erot , 1800 K for typicalexperimental parameters listed in Appendix A.
For the above reasons, the ions would have to be cooled in a relatively high pressure of helium
and then either moved into a low pressure region, or the helium must be removed, before Erot could
be applied and the spin resonance performed. The ions are easily moved from one trapping region
to another through the use of electric fields. This has been demonstrated [87] on trapped ions forquantum information applications with heating as low as a few quanta (∼ 1 mK). However, in ourcase differential pumping would require the ions to travel through a small aperture. Patch charges
on this aperture, combined with the rf trapping fields, would likely lead to significant heating and ion
loss. Quickly removing buffer gas at cryogenic temperatures has been demonstrated [88]. However,68
the minimum helium pressure, due to desorption of helium from the cell walls, was measured to be
4.5× 108 cm−3 , a value that will likely be problematic (see Chapter 7.2).

5.3

Laser Cooling
Direct laser cooling of molecules is complicated by the added degrees of freedom involved,

specifically rotation and vibration.

There are several proposals for laser cooling of diatomic

molecules [89, 90, 91, 92, 93] but they typically require very diagonal Franck-Condon matrices
or complicated cooling schemes. As seen in Chapter 2.2, we already have many constraints inchoosing a molecule which discourages us from also adding an extremely diagonal Frank-Condon
matrix to that list. Even if possible, direct laser cooling of our molecules would be a significant
technical challenge.
Sympathetic cooling with laser-cooled atomic ions has been used for cooling both atomic [94]and molecular [95] ions. The long range nature of ion-ion collisions cool the translational degrees of
freedom very well, but the internal states will not be cooled. This has been experimentally verified
in Ref. [96], with differences between translational and rotational temperatures of > 2 orders ofmagnitude, consistent with a zero coupling between the two.
Sympathetic cooling by overlapping laser cooled neutral atoms with trapped molecular ionsshould cool both the internal and external degrees of freedom. However, charge exchange collisions
between the atomic neutrals and molecular ions would destroy ion trapping times. Hudson [97]has proposed to circumvent this problem by selecting molecular ions whose neutral species possess
lower ionization potential than the laser-cooled atomic neutrals. This should mitigate the problemof charge exchange collisions. We have measured the ionization potential of HfF to be 59,458 cm−1 ,higher than most atomic species that are available for laser cooling. Potentially Mg (IP = 75,000
cm−1 ) or Be (IP = 61,000 cm−1 ) could be used to cool HfF+ but these are difficult atoms to lasercool. Also, other reactions beyond simple charge exchange might occur. Again, the large numberof requirements on our molecule selection (Chapter 2.2) precludes us from choosing a molecule thatwould be more amenable to this technique.

69

5.4

Supersonic Expansion
Molecular-beam experiments are often carried out through the use of supersonic expansions.

A reservoir of gas at high pressure, P0 , is separated from a region of lower base pressure Pb , witha small nozzle connecting the two. If the pressure ratio exceeds a critical value, G ≡ ((γSH +1)/2)γSH /(γSH −1) , the mean velocity of the gas exiting the high pressure region will reach the local
speed of sound. Here γSH is the ratio of the gas’s specific heats of constant pressure and volume.
For the noble gases γSH = 5/3 and G ≈ 2.05. In the case of P0 /Pb > G the exit pressure at thenozzle will be P0 /G ≈ 1/2P0 and the gas will continue to expand. As it expands, the beam velocity
will exceed the local speed of sound and downstream of the nozzle shockwaves can occur since theexpanding gas is moving faster than information about it can be propagated to the surroundinggas. The distance from the nozzle to the first shockwave is approximately 0.67(P0 /Pb )1 /2d, withd the nozzle diameter. In modern beams the base pressure is often quite low and the transition
from continuum to free-molecular flow is often smooth, without shockwaves present. In our casethe distance from the nozzle to the shockwave would be on order 50 meters, much larger than thedimensions of our experiment.
The final velocity can be approximated, for the case of an ideal gas, from energy conservationarguments [98]. For a flow driven by a pressure gradient, the first law of thermodynamics takes theform, h + v2 /2 = h0 , with h and h0 being the enthalpy per unit mass of the expanded and initial
gas respectively and v the velocity of the expanded gas. Using the relation dh/dt = Cp the velocity
can be related to the temperature of the gas,
v2 = 2(h − h0 ) = 2

Z T0
T

Cp dT

(5.1)

If Cp can be approximated as constant over the integration, and using the relationship for an ideal
gas Cp = (γSH /(γSH − 1)(R/M ), with R the universal gas constant, we obtain,
vf (Tf ) =

s

2R
M



γSH
(T0 − Tf )γSH − 1


(5.2)

If there is significant cooling of the gas, the final velocity approaches a limiting value, and higher

70
initial pressures have very little effect on the final beam velocity. When dealing with a mixture ofgases the molar average molecular mass and molar average heat capacity can be used.
The final beam velocity scales asp

T0 /M , meaning lower beam velocities can be achieved withheavier expansion gases and by cooling the gas reservoir. This is often done in stark decelleration
experiments, since much effort is required in order to slow the beam to rest in the lab frame. In
comparison, the slowing of molecular ions is trivial as one needs only to apply a modest uniform
electric field to the ions in order to stop them. Our choice of expansion gas was therefore dominatedby other considerations. In early experiments with trapped atomic ions such as Hf+ , the largest
numbers of ions that could be loaded into our trap was achieved using xenon. However, heavier
noble gases have the disadvantage of higher polarizabilities that can lead to clusters. Evidence ofclusters is seen in Fig. 5.2 where mass spectroscopy reveals strong signals at mass-to-charge ratios
corresponding to NiXe+ and Ni2 Xe+ when nickel ions were introduced into a Xe expansion via
ablation. Also, we tended to achieve better rotational cooling using neon and argon as they better
matched the reduced mass of our diatomic fluorides.

5.5

Laser Ablation and HV Discharge
An eEDM search generally requires atoms or molecules with unpaired electron spins. The

presence of unpaired electron spins make these free-radicals highly reactive. Due to this highreactivity, free radicals must typically first be produced in an experiment before studying them.
Two typical approaches are laser ablation and high-voltage discharge. During the course of thiswork we briefly explored both techniques. A solenoid pulsed valve (General Valve Series 99) wasused with an 800 µm opening. Connected to the face plate of the valve was a holder for the “target”
metal rod as well as an opening for either a stainless steel electrode for discharge studies or for
an ablation laser beam (Fig. 5.3). An expansion cone followed with a 40◦ full opening angle. Asan alternative to the solenoid valve, a home-built valve utilizing a piezo-electric transducer was
also used. A schematic of the design can be found in Ref. [99]. These valves seem to be preferred
in JILA Stark deceleration experiments [100], with claims of greater pulse-to-pulse and long-term

71

calculated mass [amu]
40

60

80

100 120 140 160 180 200 220 240 260 280 300 320 340

5.0
4.5
4.0

Xe

ion signal (arb. units)

3.5

+

3.0
2.5
2.0
1.5

NiXe
1.0

Ni

+

Ni Xe

+

+

2

0.5
0.0
-0.5
40

80

120

160

200

240

280

rf amplitude [Vppk]

Figure 5.2: Mass spectrum of ablation products from a nickel target in a xenon expansion. Clustering occurs when a Xe atom attaches itself to a Ni+ or Ni+
2 ion.

72
stability. We achieved similar results for both valves and preferred the solenoid valve due to its
robustness, as a careless graduate student can easily break the PZT with excess tension. Unlike
Stark deceleration experiments, our early experiments were not very sensitive to variations in beam
velocities. This might explain the apparent discrepancy in valve preference. In future experiments,
beam velocity fluctuations will likely be more important and another investigation of the PZT valve
might be appropriate.
The pulsed valve was operated with several noble gases, He, Ne, Ar, and Xe, at backing
pressures of 20-100 psig. A 200 µs, +200V pulse was typically sent to the solenoid valve. Thiswas followed by either a laser ablation pulse focused on the metal target, or an electrical discharge
between the metal target and stainless steel electrode. The discharge voltage was typically run
at DC, with the increase in pressure from the pulsed valve firing acting as the discharge gate,
since the higher pressure corresponded to a lower breakdown voltage. An Nd:YAG laser served as
the ablation laser. This laser was capable of producing 50 mJ at 1064 nm in a 10 ns pulse, andwas focused with a 50 cm focal length lens onto the ablation target to a 230 µm beam diametercorresponding to a 3.8 cm Rayleigh range.
The discharge or laser ablation pulse created both neutral and ionic atoms from the metaltarget. In order to produce diatomic molecular ions, gas containing hydrogen or fluorine was mixed
with the noble gas backing the pulsed valve. We attempted to produce HfH using a mixture of 1%
CH4 with the balance being a noble gas, with the hope of seeing the following reactions,
Hf + CH4 −→ HfH + CH3 ,(5.3)

Hf+ + CH4 −→ HfH+ + CH3 .

(5.4)

Unfortunately, the bond energy of a hydrogen atom in CH4 is 36,686 cm−1 compared to ab initio
calculations that predict a bond energy of 22,000 cm−1 for HfH+ [4]. Therefore, in order to efficiently
produce the intended HfH+ molecules, large amounts of hydrogen would have to be dissociatedfrom CH4 during the ablation or discharge process. The failure to produce these molecular species
suggests large numbers of atomic hydrogen were not produced.

73

Expansion
Cone

Ablation target

Pulsed Valve

Ablation Laser
Pulse

Figure 5.3: Diagram showing the pulsed valve set-up. An ablation target rod was held just outside
the pulsed valve opening and was followed by an expansion cone with a 40◦ opening angle.

74
The bond energies of fluorine-containing diatomic molecules tends to be higher than thosecontaining hydrogen, while the electronic structure is analogous. HfF has a measured bond energyof 53,966 cm−1 [101], and ab initio calculations predict a bond energy of 51,107cm−1 for HfF+ [64],compared to a fluorine bond energy of 35,098 cm−1 in SF6 . Seeding our noble gas with 1% SF6 ,we observed the following exothermic reactions, among others,
Hf + SF6 −→ HfF + SF5 ,(5.5)

Hf+ + SF6 −→ HfF+ + SF5 .

(5.6)

Laser ablation yielded better results for hafnium ion creation than the HV discharge and thedischarge was quickly abandoned. The number of Hf+ and HfF+ ions created increased withincreasing ablation laser energy up to the 50 mJ pulse energy maximum of our ablation laser. We
also frequency doubled the ablation laser to 532 nm and obtained a maximum pulse energy of 25
mJ. The ion number achieved with 25 mJ of green light was equivalent to that obtained with 50
mJ of IR light so we chose to use IR light for the rest of our experiments. A larger ablation laser
was used in order to look for a maximum in ion creation with ablation energy. The data in Fig 5.4
shows a maximum in Hf+ ion creation at 135 mJ, and a maximum in HfF+ ion creation at 200
mJ. In later experiments we were more concerned with maximizing the number of neutral HfF
molecules, and we found a much lower optimum ablation energy (Fig. 5.5).
After many ablation pulses, large pits would form on the metal target and ion production
would decrease. This required a vacuum break and the pitted end of the metal target would be
removed. Highest ion numbers were achieved with a slightly rough metal target surface, and we
preferred to use a hacksaw to produce a fresh, fairly rough surface. Often times researchers will use
metal ablation targets that rotate and translate, leaving a fresh target surface. This was tried, but
it was found that shot-to-shot number variations did not improve, at least in the case of ablating
hafnium. Many different metals were used as ablation targets in our experiment, with hafnium
yielding the best results for shot-to-shot stability and number of ablation pulses incident on thetarget before needing to saw the pitted end off. At 50 mJ ablation energies, hafnium targets would

75

a)

Hf+

b)
+

HfF

Figure 5.4: Number of a) Hf+ and b) HfF+ ions created, as a function of 1064nm laser ablationenergy. The channeltron detector was operated at a lower than normal voltage (1,400 V here vs
2,000 V normally) in order to keep the detector response linear.

76

HfF neutral density (arb. units)

2000

1500

1000

500

0
20

25

30

35

40

45

50

Ablation pulse energy (mJ)

Figure 5.5: The density of HfF neutral molecules created in our molecular beam as a function of1064 nm laser ablation energy. This data was obtained using photoionization techniques discussed
in Chapter 5.7.3 to measure HfF density.

77
receive millions of ablation pulses before needing to be re-cut. Since our most recent experiments aredone at lower pulse energies, even more ablation pulses are possible. Perhaps if thorium or anothermetal is used in the final experiment a rotating and translating target might become necessary.

5.6

Measuring Beam Temperatures
Collisions with the expansion gas will cool the molecular ions and neutrals created from theablation process. As mentioned above, we care about cooling of not only the translational degrees
of freedom, but also the rotational, vibrational, and electronic degrees of freedom. Below we discuss
measurements made of our molecular beam using various methods.

5.6.1

Translational Temperatures

Some of the first measurements of our molecular beam were made using a commercial fast ion
gauge (FIG). The FIG was mounted on a linear motion vacuum feedthrough such that it could bescanned across the beam, although it could only scan slightly past the centerline of the beam dueto the finite travel of the feedthrough. The FIG is very similar to a traditional vacuum ion gauge,
with the main differences being small filament-grid and grid-collector distances and fast electronics
mounted to the device inside the vacuum chamber. The response time of the device is < 5 µs, witha sensitivity, 1 × 105 V/(Torr mA), that depends on the adjustable filament emission current.
Early experiments used a 2 mm diameter skimmer to separate the source chamber (a sixway cross with 6” CF flanges) from lower pressure chambers (also 6” six-way crosses) downstream.
Differential pumping was employed via turbomolecular pumps attached to each chamber and backed
by mechanical scroll pumps. These were used instead of oil diffusion pumps to reduce backstreamingof oil and other contaminants from the pumps into the vacuum chambers. The pulsed valve toskimmer distance was varied while using a channeltron to measure the Hf+ ion number seen after
the skimmer. The ion signal when the valve-to-skimmer distance was scanned showed a broad
maximum after the separation was at least 5 cm. We operated most experiments with a pulsed
valve to skimmer distance of ≈ 12 cm. The distance between the skimmer and the FIG was ≈ 18

78
cm. Since the beam is much larger than 2 mm in diameter as it reaches the skimmer, the skimmer
position and diameter define an initial beam position and diameter. By measuring the width of
the beam at a known distance from the skimmer, the transverse velocity spread can be calculated.
Normally the FIG is used to measure neutral signals by ionizing gas from the filament and thencollecting and counting those ions. However, if you turn on the detection electronics for the FIG
while leaving the filament off, the instrument will be sensitive to ion signals only. Fig. 5.6 shows
data taken in this way, for a beam of ions created through ablation of hafnium in the presence of axenon expansion with 52 psig backing pressure. The data corresponds to a transverse temperatureof the ion beam that is ∼ 1 K. Similar temperatures were found for the neutral signal when the
FIG filament was turned on.
Axial temperatures are more difficult to define and to measure in our beam. The gas pulse
width was ∼ 400 µs when measured at the FIG position 30 cm downstream from the pulsed valve.
The axial velocity of a gas molecule should become strongly correlated to its axial position as thebeam travels through the vacuum chamber. Axial temperature is perhaps less important than anaxial velocity spread within a given axial slice of the beam. This was measured indirectly whentime-of-flight mass spectroscopy was performed as described in Chapter 5.7.3. Simulations of our
TOFMS put an upper limit of 20 m/s velocity spread in our beam over a 3 mm axial slice of thecloud, “measured” at a distance of 60 cm after the pulsed valve.

5.6.2

Laser-Induced Fluorescence in the BeamIn hopes of measuring HfF+ spectrum, laser-induced fluorescence was performed on themolecular beam using the set-up shown in Fig. 5.7. The vacuum chamber after the first skimmer
was painted with a vacuum compatible black paint and light baffles with imaging optics integrated
into them were installed between the chamber center and a photomultiplier tube (PMT) detector
that sat underneath the chamber. Lasers could pass through the center via two long arms mounted
to the vacuum chamber with Brewster windows attached and containing baffles to reduce scattered
light. Data could be recorded in pulse-counting mode or, for larger signals, in analog mode. For

79

ion number [arb. units]

0.125
0.100
0.075
0.050
0.025

T~1K
+
Hf beam

0.000
10 12 14 16 18 20 22 24 26 28 30

Detector position [mm]Figure 5.6: FIG data taken with the filament turned off, making the device only sensitive toions. This data was taken with Hf+ ions in a xenon expansion and corresponds to an ion beam
temperature of approximately 1 K.

80large amplitude signals, a transimpedance amplifier was used to reduce PMT saturation effects.
Two different laser systems were generally used. One was a pulsed dye laser, capable ofoperating throughout the visible and near-IR wavelength ranges. Frequency doubled light could beobtained through simple single-pass frequency doubling with BBO or KTP crystals, depending on
the specific operating wavelength. The laser produced 10 ns pulsewidths with typical linewidths
less than 0.1 cm−1 . Maximum pulse energy was dependent on wavelength but was generally 50-300
mJ in the fundamental and 5-50 mJ of frequency-doubled light. A cw Ti:Saphire ring laser, witha specified linewidth of 100 kHz, was also used. This laser was capable of > 4 W at 800 nm,
the center of the gain curve. The laser’s tunable range, through the use of three different output
couplers, extended from 700 - 1030 nm.
When the pulsed laser was used, we gated the PMT detector on using JILA built electronics
in order to limit scattered light backgrounds and PMT “after-pulsing”. After-pulsing occurs when
positive ions are created due to electron impact ionization of residual gas inside the PMT. These ionsare accelerated upstream in the PMT and eventually collide with an electrode, creating electrons
that imitate photoelectrons. Delay times between firing the laser and gating the detector were
between 150 and 650 ns and the gate was left on for up to 20 µs, though usually useful signal was
only obtained for shorter times due to the molecules travelling outside the imaging region of our
optics or electronic decay of the molecules. After-pulsing was suppressed by a factor of ∼ 40 using
the gating circuit. The ultimate limit to light background was dominated by an unknown sourceof scattered light from the molecular beam. This scattered light background was approximately
constant over entire laser dye curves (∼ 30 nm), and only appeared when the LIF laser, ablation
laser, and pulsed valve were all operated. Typical background count rates were 4 photons/shot ina 5 µs counting window with a 3 mJ laser pulse and a 150 ns delay between firing the laser andgating on the PMT. Background rates using the cw Ti:Saphire laser were never an issue, which
had the advantage that the fluence needed to saturate the transition is lowered by (3 GHz)/(0.05
GHz), the ratio of the pulsed laser linewidth to the typical Doppler-broadened linewidth in ourbeam. However, the cw laser was not very useful in looking for unknown HfF+ transitions as step

dye laser
700 nm"click"+Nd:YAG
1064 nm++"click"

+
+

+

+

+

pulse valve

+

+

+

+

skimmer++~100 pisg
Ar + 1% SF6

skimmermicrochannel plate

Hftargetphotomultiplier tube

rf (Paul) trap & quadrupole mass filter

Figure 5.7: Experimental setup. Laser ablation of a metal Hf target creates neutral Hf atoms and Hf+ ions that react with SF6 to produce
neutral HfF molecules and HfF+ molecular ions, respectively (Eqns. 5.5 and 5.6). The molecules (both neutral and ionic) are cooled in asupersonic expansion with an Ar buffer gas. The molecular beam is illuminated with a pulsed dye laser beam and the resulting fluorescenceis collected with a photomultiplier tube (PMT) yielding laser induced fluorescence (LIF) spectra (Fig. 5.9). At the end of the beamline,
the ions can be loaded into an rf Paul trap where the electron spin resonance experiment will be performed. The Paul trap also acts as aquadrupole mass filter and ions of a particular mass/charge ratio are detected with a microchannel plate (MCP) (Fig. 3.10). Additionally,
the spatial resolution of the MCP allows for the temperature of the ion cloud to be determined from the detected cloud size.

81

82
sizes of 50 MHz would be necessary and the uncertainty in the position of the transitions was onorder of 1,000 cm−1 .
The first transitions seen in the supersonic beam were neutral atomic hafnium transitions.
The ground state of neutral hafnium has 3 F2 symmetry with J = 3 and J = 4 excited states at
2,357 and 4,568 cm−1 respectively. Transitions were seen not only from the J = 2 ground state, butalso from excited fine-structure states. Fig. 5.8 shows LIF data taken for the 3 F2 → 3 D1 transition
using the cw Ti:Saph laser for excitation. Similar data on the 3 F4 → 3 D3 transition showed apopulation ratio of ∼ 500. If the fine-structure distribution was in thermal equilibrium with thetranslational degree of freedom we would expect a ratio of 10286 , given the <10K translationaltemperatures observed in the supersonic beam.
Previously unmeasured transitions in neutral HfF were also detected in the beam, and these
allowed us to measure the rotational and vibrational temperatures, although the rotational andvibrational temperatures of HfF+ could be significantly different due to the different collisionalcross sections involved. HfF is known to have a 2 ∆3/2 ground electronic level, and transitions to Ω
= 1/2 and Ω = 3/2 states were found. The transitions were fit using the same effective Hamiltonians
as used in Ref. [6],H = T + Be [J(J + 1) − Ω2 ] − D[J(J + 1) − Ω2 ]2(5.7)

was used for states of Ω = 3/2 character, with Be the rotational constant, T the band origin, andD a constant characterizing the effect of centrifugal distortion on rotation. Excited states showingΣ character were fit using,γs[N + 1]2γs
Hb = T + Be N (N + 1) − D[N (N + 1)]2 + N,
2

Ha = T + Be N (N + 1) − D[N (N + 1)]2 −

(5.8)
(5.9)

with N the rotational quantum number, γs a doubling constant, and where a and b denote statesof different parity. States that demonstrated 2 Π1/2 character were fit to,
1
1
Ha/b = T + Be (J + )2 − D J +
2
2


4

1
1
,
∓ (p + 2q) J +
2
2




(5.10)

Fluorescence signal (arb. units)

83

Figure 5.8: LIF data taken with a cw Ti:Saph laser showing hyperfine structure in the 3 F2 → 3 D1transition of hafnium. Similar data was taken of the 3 F4 → 3 D3 transition as the fine structure
was not completely cooled in the supersonic expansion.

84
with 1/2(p + 2q) a constant describing the Λ-doublet splitting. We typically set D to zero as mostof our data involved only low rotational states where the energy term from non-zero D was muchsmaller than our experimental resolution. The resolution was determined by the laser linewidthand was approximately 0.1 cm−1 . Nonlinear least-squares fitting routines were used.
Through the examination of “missing” lines around the band origin, assignments of Ω could
be made. In order to find rotational temperatures, the rotational line intensities were fit to thepartition function and Hönl-London factors. The typical value of the rotational temperature was 10
K. Several states were found that appeared to be different vibrational transitions between the same
electronic states. To test this we carefully fit the rotational lines as follows. First, rotational lines inthe “P-branch” (δJ = -1 transitions) and “R-branch” (δJ = +1 transitions) were fit to Lorentzianline shapes, with the peak intensities allowed to float. The “Q-branch” (δJ = 0 transitions) lines
were excluded as the resolution was insufficient to resolve them. The line positions were then fit
to the model Hamiltonian to determine a value for the ground and excited state Be . Preliminary
assignments of some of these lines are shown in Fig. 5.10, where two electronic states appear to
be responsible for eight rotational bands. It appears that vibrational levels are not in thermal
equilibrium with the translational and rotational temperatures, as transitions from v ′′ = 2 areobserved.

5.7

Loading Ions Into the rf Trap
Our experimental count rate will be limited by space charge effects of the trapped ions.

Therefore, any ions trapped that are not used in measuring the eEDM limit the statistical sensitivityof our measurement. In order to maximize our count rate, we would ideally create and trap only
HfF+ ions of a single Hf isotope and in a single internal quantum state. Fig. 5.11 shows a mass
spectrum of ions loaded from the laser-ablation supersonic-expansion source into the ion trap. As
seen in the figure, many ion species are created, from SF+ to HfF+
3 and perhaps beyond as therange of mass-to-charge ratios we were able to trap was limited.
The ions created during the ablation process populate many rotational and magnetic sub-

Fluorescence Signal (arb. units)

6

x 10

6

5

4
Data
Fit to Hamiltonian

3

2

1

0

4

6

8

10

12

14

Laser Energy - 14,209.83 cm

16

18

-1

Figure 5.9: Data from an Ω = 3/2 → 3/2 transition in HfF. The smooth curve is the fit obtained while ignoring the unresolved Q-branchlines in the 12-14 cm−1 region of this graph. Small differences in the rotational constant Be allowed us to ascribe vibrational states to thelevels involved.
85

we > 670 cm-1

we > 595 cm

-1

(0,0)

(0,0)
(1,0)

(1,1)
(0,1)
(1,2)

D1

(2,1)

(1,1)

D2

D1

D2

Figure 5.10: Schematic of HfF transitions seen from LIF in our molecular beam showing approximate relative intensities. Lines connectedby blue(orange) arrows are believed to be different vibrational transitions between the same Ω = 3/2(1/2) electronic levels, as suggested
by slight differences in Be′ and Be′′ , the rotational constants in the upper and lower states respectively. Preliminary vibrational assignments
are given as (v ′ ,v ′′ ), again with the ′ (′′ ) denoting the upper(lower) state. The ground-state vibrational spacing ωe is measured to be 670
cm−1 , consistent with a previous measurement [6], while an excited electronic state has a measured ωe of 595 cm−1 .

86

87levels. Above thermal population of excited electronic levels, such as the 3 ∆1 state in which oureEDM experiment will be performed, might also be present as the supersonic expansion cooling ofthese levels might be less than complete. However, the likelihood of significant population of the3∆

1 level surviving the expansion seems unlikely. It is therefore necessary to find another method

to populate our 3 ∆1 “science” level. One potential method is a two-photon stimulated Raman
adiabatic passage or STIRAP, utilizing as the intermediate state a spin-orbit mixed Ω = 1 statesuch as the b(1) and c(1) states of Fig. 2.2.
Theoretical predictions [64] indicate the transition dipole moments should be on the order of
0.3 a.u. for transitions to the 1 Σ (3 ∆1 ) level while the transition dipole moment to the 3 ∆1 (1 Σ)should be 0.03 a.u., with which transition is stronger depending on which intermediate Ω = 1 stateis chosen. The transition strength will be decreased when vibrational and rotational wavefunction
overlap is taken into account. Since the ion cloud diameter will be ∼ 1 cm, the large laser power
necessary for efficient STIRAP with cw lasers will be impractical for these transition strengths.
When performing STIRAP using pulsed laser sources, laser bandwidths that are transform-limitedyield the minimum required pulse energies. The minimum pulse energy required scales as the squareof the ratio of the laser bandwidth to the transform-limited bandwidth [102]. With our HfF+ states,in the ideal case of a transform-limited pulse, a 10 ns pulse of at least 100 µJ would be required forthe weaker of the two transitions involved in the SITRAP. If a pulsed dye laser with a 0.1 cm−1
bandwidth is used, the minimum requirement would be ∼ 1 J laser pulses.
5.7.1

Ion Lens

As shown in Chapter 3, it is possible to mass-selectively remove ions from our trap by
operating at certain a and q trapping parameters. The issue with this approach is that heating of
the desired ions tends to occur due to the proximity to the β = 1 contour of the stability diagram.
A potential solution to this is inserting a quadrupole ion lens along the beam path, between thesource chamber and the ion trap. As shown in Eqn. 3.13, the lowest order frequencies of ion motion
in an rf quadrupole at rf drive frequency Ωrf will be ωsec ≡ βΩrf /2, Ωrf − ωsec , and Ωrf + ωsec . If

ionionnumbersignal[arb.
[arb.units]
units]

88

0.8

SF+3

0.7

SF+5
Hf+

0.6

HfF+HfF+
2
HfF+3

0.5
0.4

SF+4

0.3

+
SF
2
0.2
+
SF
0.1

SF+6

0.0

50

75 100 125 150 175 200 225 250

Mass [amu]Figure 5.11: Mass spectrum of ions created in laser-ablation supersonic-expansion with a hafnium
ablation target and an expansion gas of 1% SF6 in 99% He. The ion number is a non-linear functionof the y-axis value, but a level of 0.4 corresponds to ∼ 100,000 ions.

89
these three frequencies are made to be commensurate for ions of a desired mass-to-charge ratio, andthe rf is applied for an integer number of secular periods, the desired ions will be imaged from one
end of the lens to the other, while ions of different mass-to-charge ratios will undergo defocusing.
If a small aperture is placed between the lens and the ion trap, the desired ions might be separated
from the ion ablation soup. Fig. 5.12 shows the frequency spectrum of ion motion at certain valuesof a and q where our lens conditions are met, as well as results of a numerical simulation of ion
motion under these conditions for two different ion masses. The radial velocity change the ions will
experience is proportional to qΩrf Ri | δM
M | with Ri the initial radial displacement from the center ofthe lens and δM is the mass difference from the ions that are imaged by the lens.
The ion lens was constructed with solid stainless-steel electrodes of 25 mm radius and spacing
2r0 = 21.8 mm, i.e. the identical radial dimensions of our first ion trap. Two of the electrodes weregrounded by mounting them directly to a vacuum flange with the first skimmer being mounted on
the opposite face of this flange. The other two electrodes were attached to similar drive electronics
as those described in Chapter 3.6.
We operated the lens over a range of parameters, but satisfactory separation of different
massed ions was not achieved. It was discovered that the ion cloud after the first skimmer was
best described not as free ions, but rather as being one component of an overall neutral plasma.
Experimental evidence of this came from placing electrode plates, parallel to and straddling themolecular beam, after the first skimmer and measuring the ion deflection caused by a potential
difference between the plates. Greater voltages were necessary for a given ion deflection thanwould be required if the ion cloud was strictly composed of positive ions. The voltage required
was also a function of ablation pulse energy, and therefore ion density. The fact that negative ions
and/or electrons formed a neutral plasma with the ablation ions is likely responsible for us being
able to obtain large numbers of translationally cold ions. Had this not been the case, patch charge
effects would have likely lead to heating and deflections of our ion cloud at the skimmer. This
neutral plasma character of our ion beam would likely also cause complications in any scheme togently slow ions into our ion trap using the ions from our ablation source.

90

b)

a)

c)
50
40

radial position (mm)30

mass 170

20
10mass 181

0
-10
-20
-30
-40
-50
0.0

1.0x10-5

2.0x10-5

3.0x10-5

4.0x10-5

time (sec)Figure 5.12: a) Spectrum of ion motions for values of trapping parameters a = 0 and q = 0.785 thatcan be used for operating an rf quadrupole as a mass-to-charge ratio sensitive ion lens. b) Dotsdenote values of q, with a = 0, that can be used in operating our ion lens. c)Numerical simulationof ion motion for different ion starting positions Ri , and two different singly-charged ions. The
simulation was done with a = 0 and q = 0.785 for the mass = 181 ions.

91
5.7.2

State-Selective Photoionization

One alternative solution to our selective ion loading problem is to get rid of all of the ionscreated from laser ablation and use photoionization techniques to ionize neutral HfF in as stateselective a way as possible. Since the density of neutral HfF is much greater than the densityof HfF+ in our molecular beam, even inefficient photoionization could lead to higher densities oftrapped HfF+ than what can be achieved by loading the HfF+ ions created from laser ablation.
We have attempted two techniques, with both involving a transition to a Rydberg state of neutral
HfF. Every ro-vibrational state in the molecular ion is the termination of a Rydberg series from theneutral molecule. These Rydberg states can be populated in our molecular beam experiment by
first driving a transition from the electronic ground state to some intermediate bound state. Thistransition can be narrow and allow the excitation of only a single ro-vibrational state, of a single
isotope. Another transition can then be made to a high-lying Rydberg level. If that Rydberg stateis close enough to the ionization threshold, the application of a pulsed electric field can ionize themolecule while leaving the ro-vibrational level of the ion-core undisturbed. After the excitation
laser pulses, a small electric field can be applied to give any molecules that were directly ionizedby the lasers an acceleration. After waiting some time, a larger pulsed electric field can then be
applied to ionize Rydberg-state molecules. The ions can then be accelerated in a time-of-flight mass
spectrometer (TOFMS) and the field-ionized molecules will arrive at a different time compared tothe direct ions. This process is referred to as mass analyzed threshold ionization spectroscopy, or
MATI. Near the threshold of a molecular ion there will be a near continuum of Rydberg levels
with different principle quantum number n and rotational quantum number. In order to populate
only one rotational level in the ion, two field-ionizing pulses can be used, separated by some time.
The first pulse ionizes all Rydberg states some energy from the continuum given roughly by E =4(Ep )1/2 with Ep the magnitude of the pulsed electric field and with the energy in units of cm−1 andthe electric field in units of V/cm. Those ions then experience an acceleration from a small applied“spoiling” electric field. The second field pulse is slightly larger in magnitude, thus ionizing a small

92
number of Rydberg states lying in a region δE = (4F −δEp )1/2 −4F , with F the value of the spoilingfield [103]. There is a limit to the resolution that can be achieved by this method that depends
on the value of the spoiling electric field and can be estimated as δE = 0.8F 1/2 . The resolution
is therefore limited by the need of a sizable spoiling electric field in order to separate ions createdat different times. For this reason, pulsed-field ionization techniques that measure the electronsas opposed to the ions created tend to have higher resolutions. However, similar resolutions havebeen demonstrated when measuring ions compared to electrons, when the proper pulse sequences
are utilized [104]. The Rydberg states involved in this process can be largely immune from decay
processes such as dissociation of the ion core, and electronic decay [103]. They are often referred
to as ZEKE states, an acronym used to describe the technique when the free electron is measured
instead of the molecular ion.
The second method we explored was a process known as autoionization. Here a neutral
molecule is excited to a Rydberg state, in the same two-step process as before, but now the Rydberg
molecule has a molecular-ion core that is in an excited state. This can be an electronic, vibrational,
rotational or spin-orbit excitation. We will focus on vibrational autoionization, where the excitedmolecular core can interact with the orbiting Rydberg electron in an inelastic scattering process thatleaves the Rydberg electron in the continuum and the molecular ion core in a relaxed vibrational
state. This process has been known to leave the molecular ion core’s rotational state largely
unperturbed [105]. In this way we hope to excite a Rydberg level that corresponds to an excited
3∆

1 ion core with v = 1, J = 1.

The Rydberg state might then vibrationally autoionize to thev = 0, J = 1 3 ∆1 level that will be used to measure the eEDM. Perhaps the Rydberg states
involved have a faster decay rate via electronic autoionization to the ground 1 Σ level than to
the 3 ∆1 level via vibrational autoionization. In this case autoionization should still allow thecreation of only the molecular ion species we desire in our ion trap, without the worry of imperfectelectrostatics anywhere along our entire beam line distorting and heating the ion beam, and with
the ion population in far fewer states than possible with molecular ions created from our ablation
source.

93
5.7.3

Photoionization of HfF

A TOFMS was built and placed after two beam skimmers that were used to collimate themolecular beam, and after an electrode that could be biased at high positive voltages in orderto reject ions from the ablation process. The TOFMS design involved the competition between
simplicity, uniformity of electric fields, and a long drift chamber to allow the separation of ionscreated at slightly different times in delayed pulse measurements. A compromise was found using
90 mm diameter disk electrodes with 10 mm diameter holes for the ions to travel through, spacedby 52 mm for the MATI drift chamber, with a third electrode placed equidistant from them having
a 38 mm diamter hole to create a more uniform field. These plates were followed by two moreplates with spacings of 21 mm that allowed for focusing at the MCP detector of the initial spatial
and velocity spread in the ion beam (see Fig. 5.13). This TOFMS was designed to be used withboth ionization techniques discussed above, and was initially operated in a way consistent with the
MATI techniques. For some of the later autoionization experiments the TOFMS was operated ina simpler Wiley-McLaren mode [106]. We were able to achieve better than 1 amu mass resolution
when detecting HfF+ with the isotope structure of Hf well resolved. This allowed us to see isotope
shifts of transitions even though our laser resolution would normally have not allowed this. The
isotope shifts were very useful in determining the change in vibrational state of the transition, δv.
The isotope shift due to a change of vibrational quantum number is very easily estimated, and for
HfF it should be ∼ 0.16 cm−1 /(δv amu), assuming a 600 cm−1 vibrational spacing.
Once the ionization threshold was found, we performed single-color experiments in searchof intermediate HfF levels that could be useful in our ionization schemes. Fig. 5.14(a) shows theresults of such a scan over an ≈ 3,000 cm−1 scan range that corresponded to a dye tuning curve.
As seen in the figure, a high density of states were found, and many were scanned with single-color
photoionization with enough resolution to roughly determine Ω and Be . A few states were scanned
at higher resolution in two-color ionization measurements. The first transition was driven by a cw
ring-cavity dye laser that was pulsed amplified using a two-stage home-built dye cell amplifier, and

94

(a)

*

(I)

(II)

(III)

(b)

Figure 5.13: (a) Schematic of the time-of-flight mass spectrometer built for our experiment. The
first plate is used to further collimate the neutral molecular beam and has a 6 mm diameter opening.
Photoionization occurred at the approximate location marked by *. Region (I) could be used indelayed-pulse field ionization experiments to separate prompt ions from pulsed-field ions. Regions(II) and (III) were used to focus initial spatial and velocity spread of the created ions. Dashed lines
correspond to high transmission (90%) wire mesh used to improve electric field uniformity. (b) An
image of the TOFMS, before a grounded cylinder of wire mesh was placed around the device toshield it from external electric fields.

95
was then single-pass frequency doubled using a BBO crystal. This laser had a measured linewidthof ∼ 100 MHz and frequency-doubled pulse energies of 20-30 µJ in the UV. The second transition
was driven by the large-linewidth (0.1 cm−1 ) pulsed dye laser, with the laser tuned well above thethreshold for ionization. Typical data is shown in Fig. 5.14(c).
A better measurement of the ionization threshold was made, taking into account the reductionin threshold energy due to an electric field. The electric field during the ionizing laser pulses wasvaried through zero while recording the onset of ionization. A value of 59,458 cm−1 was obtained
for the ionization potential of HfF. Combining this with the published value of the HfF bondenergy [101] of 53,966 cm−1 and the known value of Hf’s ionization potential of 55,048 cm−1 gives
a value of 49,556 cm−1 for the HfF+ bond energy. ab initio calculations have estimated the HfF+
bond energy at 51,107 cm−1 [64]. The measured HfF bond energy and the theoretical HfF+ bondenergy are both likely to have large (∼ 1,000 cm−1 ) errors, given the nature of their techniques.
Attempts at delayed pulse field ionization were unsuccessful. Although the literature suggeststhat ZEKE states can be long-lived, even in the presence of otherwise strong relaxation processes,a likely reason for our lack of signal was predissociation of our ZEKE states. It appears in somemolecules decay channels simply win out, as many experiments find some molecules posses a delayed
ionization signal and others do not (see Ref. [107] for an example).
We focused our efforts on our second scheme, vibrational autoionization. Fig. 5.15 shows
data taken with two-color excitation, with the first transition again being driven by our narrowlaser, and the second by the broad laser. As seen in Fig. 5.15(a), the spectrum is quite complex.
The density of lines is increased by the fact that many large δv autoionization peaks are found,as determined by the isotope shifts associated with them (Fig 5.15(b)). This is surprising asvibrational autoionization is expected to have a strong propensity rule for δv = 1 transitions [108].
As understanding the autoionization is not necessary for our goals, our current work involves
directly measuring the rotational state populations of the molecular ions produced in this process.
This is done by performing LIF using the only known electronic transition in HfF+ at ∼ 13,000
cm−1 . This work is ongoing and in the future we hope to detect ions created directly in the 3 ∆1

(a)Mass/Charge Ratio

HfF intermediate states (1+1’)(b)

Laser Frequency(c)Figure 5.14: (a) Data from a two-photon one-color survey scan, with many UV transitions found in HfF. (b) False-color image showing ourability to separate different isotopes of HfF using our TOFMS, giving us the ability to measure isotope shifts and helping to identify thechange in vibrational quantum number in these transitions. (c) Two-color photoionization showing a spectral resolution of approximately
100 MHz. The two traces correspond to the two dominant (m = 178 and m = 180) isotopes of HfF.
96

97
level, although that work is awaiting the discovery of a known electronic transition from the 3 ∆1level that might be used in LIF.

(a) Measuring Ionization Potential (1+1’) (b) Auto-ionization resonances (1+1’)
1.5

1.3
0.7

2

P1/2 [32.5]
int. state

Isotope shift
-1
(cm )
2.0

1.0

2

P1/2 [32.3]
int. state

Figure 5.15: Autoionization spectra showing (a) the same ionization threshold for HfF using two different intermediate states and (b) large
isotope shifts of the resonant features between Hf178 F and Hf180 F. A change of one vibrational quanta between the intermediate HfF state
and the HfF Rydberg state that undergoes autoionization corresponds to an isotope shift between Hf178 F and Hf180 F of 0.32 cm−1 . Thisdata was taken using two-color photoionization, starting in the HfF ground electronic and ro-vibrational state.

98Chapter 6
Spectroscopy in Rotating and Trapping Fields [1]On the face of it, an ion trap, with its inhomogeneous and rapidly time-varying electricfields, is not necessarily a promising environment in which to perform sub-Hertz spectroscopic
measurements on a polar molecule. In this section the effects of the various components of theelectric and magnetic fields on the transition energies relevant to our science goals will be exploredin more detail. The theoretical determination of the energy levels of heavy diatomic moleculesin the presence of time-varying electric and magnetic fields is a tremendously involved problem in
relativistic few-body quantum mechanics. State-of-the-art ab initio molecular structure calculations
are limited to an energy accuracy of perhaps 1013 Hz, a quantity which could be compared withthe size of a hypothetical “science signal”, which could be on the order of 10−3 Hz or smaller.
Fortunately, we can take advantage of the fact that at the energy scales of molecular physics,
time-reversal invariance is an exact symmetry except to the extent that there is a time-violating
moment associated with the electron (or nuclear) spin. In this section, except in those terms
explicitly involving de , it is assumed that time-reversal invariance is a perfect symmetry in orderto analyze how various laboratory effects can cause decoherence or systematic shifts in the relevant
resonance measurements. The results can be compared to the size of the line shift that would arise
from a given value of the electron EDM, which is treated theoretically as a very small first-order
perturbation on the otherwise T-symmetric system.
In the subsections below, sequentially more realistic features of the trapping fields are brought
in.

100

6.1

Basic Molecular Structure
We begin by considering in detail the relevant molecular structure in zero electric and mag-netic fields, thus quantifying the qualitative discussion of the experiment given in Chapter 2. Although the molecular structure cannot be calculated in detail from ab initio structure calculations,nevertheless its analytic structure is well known. Because the measurements will take place in nominally a single electronic, vibrational, and rotational state, we will employ an effective Hamiltonian
within this state, as elaborated by Brown and Carrington [109]. This approach will specify a few
undetermined numerical coefficients, whose values can be approximated from perturbation theory,
but which will ultimately be measured.
Brown [110, 111, 112] and co-workers have done thorough work on deriving an effective
Hamiltonian for 3 ∆ molecules. The complete Hamiltonian in the absence of de is given by
Hstruct = Helec + Hvib + HSO + Htum + HSS + HSR + HHFS + HLD ,(6.1)

listed in rough order of decreasing magnitude. Since we are concerned only with terms acting
within the subspace of the 3 ∆ manifold, other electronic and vibrational states will enter only as
perturbations that help to determine the effective Hamiltonian. Thus we consider eigenstates ofHelec and Hvib .
The remaining terms in Eq. (6.1) are corrections to the Born-Oppenheimer curves. They
describe couplings between various angular momenta (HSR , HHFS ), parity splittings (HLD , HHFS ),and spin-dipolar interactions (HSS , HHFS ). In typical Hund’s case (a) molecules these interactions
are small compared to the rotational energy governed by Htum . The relevant interactions that act
within the |Ω| = 1 manifold of states take the explicit form
HSO = AΛΣ

(6.2)

Htum = Be (J − S)2 − D(J − S)4

(6.3)

2
λ(3Σ2 − S2 )
3

(6.4)

HSR = γSR (J − S) · S

(6.5)

HSS =

101
c
1
2
2
HHFS = aIz Lz + bF I · S + (3Iz Sz − I · S) + e∆ (J+ I+ S++ J− I− S−
)
3
2
1
2 2
2 2
(o∆ + 3p∆ + 6q∆ )(S+
J+ + S−
J− ).
HLD =
2

(6.6)
(6.7)

The constants in the first four terms are as follows: A is the molecular spin-orbit constant, Be
the rotational constant for the electronic level of interest, D the effect of centrifugal distortion on
rotation (typically D ∼ Be (me /mmol )2 , with me the electron mass and mmol the reduced mass ofthe molecule), λ governs the strength of the spin-spin dipolar interaction, and γSR determines thestrength of the interaction of the spin with the end-over-end rotation of the molecule. These fourterms primarily describe an overall shift of the 3 ∆1 J-level, and can be ignored in evaluating energy
differences in the states we care about. They can, however, contribute small perturbations to thesebasic levels, as described below.
Within the 3 ∆1 , J = 1 manifold of interest, the energy levels are distinguished by thehyperfine and Λ-doubling terms. The hyperfine Hamiltonian HHFS includes the familiar contact(bF ), nuclear-spin-orbit (a) and spin-nuclear spin terms (c). By estimating the parameters in
perturbation theory, it is expected that the resulting hyperfine splitting is on the order of 50
MHz [54]. The hyperfine interaction also contains a previously unreported term, with constantdenoted e∆ , that is connected to the Λ-doubling. This term is expected to be even smaller than
the already small Λ-doublet splitting itself, however, [113] and will be ignored.
The Λ-doubling Hamiltonian arises from Coriolis-type mixing of states with differing signsof Λ due to end-over-end rotation of the molecule. For a 3 ∆ state this interaction is characterized
by three constants, of which the parameter o∆ is the dominant one. These terms describe how
the 3 ∆ state is perturbed by electronic states with 2S+1 Π and 2S+1 Σ symmetry. Since we are
primarily concerned with terms in the Hamiltonian that affect the ground rotational state of the3∆

1 electronic level, we only need to keep the term which connects Ω = 1 to Ω = −1.

This term

has the general form, with numerical prefactors CΠ,Σ,Π′ that depend on Clebsch-Gordon coefficients
and wavefunction overlap, [110]|o∆ + 3p∆ + 6q∆ | = õ∆ ≈XΠ,Σ,Π′

CΠ,Σ,Π′A2 Be2,(E∆ − EΠ )(E∆ − EΣ )(E∆ − EΠ′ )

(6.8)

102
where the sum is over all intermediate Σ and Π states of singlet and triplet spin symmetries. For
HfF+ this perturbation leads to a Λ-doublet splitting on the order of 10 kHz. From here on, weshall express the energy difference in parity levels for the J = 1 as ωef = 4õ∆ , rather than õ∆ itself.
Thus the basic molecular structure of interest to the 3 ∆1 , J = 1 state is governed by two
constants: the hyperfine splitting Ehf (given by 3A|| /4 for J = 1, I = 1/2) and the Λ-doublet
splitting ωef . These constants give the structure depicted in Fig. 2.1(a). These basic levels may be
perturbed by couplings to other levels, especially rotational or electronic excited states. However,for the J = 1 state of interest, some simplifications are possible, namely: (1) Off-diagonal couplingsin Ω are zero since J · S preserves the value of J (there is no level with J = 1 and Ω = 2); (2)Off-diagonal contributions that mix J = 2 into the J = 1 manifold thus depend solely on theapplied fields and the hyperfine interactions. Since the value of the spin-orbit constant is expected
to be far larger than the rotational constant and we are concerned with a J = 1 state, the operatorsthat connect Ω to Ω ± 1 will be ignored. The contributions to the ground state characteristics byterms off diagonal in Ω are smaller by a factor of the hyperfine interaction energy to the spin-orbit
separation energy, hence a factor of 10−6 . This is the value which appears in front of any term
connecting Ω to Ω ± 1 in the ground J = 1 state.

6.2

Effect of Non-rotating Electric and Magnetic Fields
The influence of external fields presents new terms in the Hamiltonian of the form
HStark = −~dmf · E~

(6.9)

~
HZeeman = −~µ · B.

(6.10)~ are the electric and magnetic fields, assumed for the moment to be collinear so that
Here E~ and B
they define the axis along which mF is a good quantum number; while ~dmf and µ~ are the electricand magnetic dipole moments of the molecule.
The electric dipole moment arises from the body-fixed molecular dipole moment, at fields
sufficiently small not to disturb the electronic structure. We assume that the field is sufficiently

103
large to completely polarize this dipole moment, i.e., dmf E ≫ ωef , in which case the Stark energies
are given by
EStark = −mF ΩγF dmf E,(6.11)

where γF is a geometric factor, analogous to a Landé g-factor, which accounts for the Stark effectin the total angular momentum basis F . In the limit where the electric field is weak compared torotational splittings, it is given byγF =J(J + 1) + F (F + 1) − I(I + 1).
2F (F + 1)J(J + 1)

(6.12)

Its numerical values in the J = 1 state are therefore γF =3/2 = 1/3 and γF =1/2 = 2/3. The
electric field therefore raises the energy of the states with mF Ω < 0 (denoted “upper” states withsuperscript u), and lowers the energy of states with mF Ω > 0 (“lower” states with superscript ℓ).
This shift in energy levels is shown in Fig. 2.1(b), where |ai and |bi are upper and |ci and |di arelower states.
The form of the Zeeman interaction is somewhat more elaborate, as the magnetic moment of
the molecule can arise from any of the angular momenta L, S, J, and I. Quite generally, however, inu/ℓ

the weak-field limit where µB B ≪ Ehf , the Zeeman energies are given by mF gF µB B, where µB isu/ℓ

the Bohr magneton and gF

are g-factors for the upper and lower states. In general, gFu 6= gFℓ , andthis difference can depend on electric field, a possible source of systematic error. We will discuss
this in Chapter 6.8.
The leading order terms in the Zeeman energy are those that preserve the signed value of Ω.
They are given by
HZeeman = (γF [((gL + gr )Λ + (gS + gr )Σ) Ω − gr J(J + 1)] − gI κF ) mF µB B,(6.13)

where κF = (F (F + 1) + I(I + 1) − J(J + 1))/2F (F + 1) is another Landé-type g-factor, but for
nuclear spin. The orbital and spin g-factors are gL and gS , while the rotation and nuclear sping-factors are gr and gI . Both gr and gI are small, being on the order of the electron-to-molecularmass ratio ∼ me /mmol ∼ 10−3 . Thus for an idealized 3 ∆1 molecule where gL = 1, Λ = ±2, gS = 2,

104
Σ = ∓1, we would expect molecular g-factors on the order of 10−3 . In a slightly less ideal situation,
gs differs from 2 by a number on the order of α, the fine structure constant, and a g-factor ∼ 10−2
might be expected. In heavy-atom molecules such as ours for which spin-orbit effects mix Λ, we
may expect instead the difference 2gL − gS to be as large as ∼ 0.1 in magnitude. If we assume thedominant contribution comes from these spin-orbit type effects, we can define the g-factor for theJ = 1 state asgF =3/2 = γF =3/2 (gL Λ + gS Σ)Ω <≈ 0.03,(6.14)

gF =1/2 = 2gF =3/2 .

(6.15)

while

Finally, the effect of the EDM itself introduces a small energy shift
HEDM = −d~e · E~eff = de Eeff ~σ1 · n̂,(6.16)

where ~σ1 is the spin of the s-electron contributing to the EDM signal; and n̂ denotes the intermolecular axis, with n̂ pointing from the more negative atom to the more positive one; in our casefrom the fluorine or hydrogen to thorium, platinum, or hafnium. Also in this convention we take
Eeff as positive if it is anti-parallel to n̂. The energy shift arising from this Hamiltonian depends
only on the relative direction of the electron spin and the internuclear axis, and is given by
EEDM = −de EeffΩ.
2|Ω|(6.17)

Polarizing the molecule in the external field selects a definite value of Ω, hence a definite energy
shift, positive or negative, due to the EDM. This additional shift is illustrated in Fig. 2.1(c).
For a range of field strengths and parameters, the energies of the sublevels within the J = 1
manifold are well approximated by a linear expansion in the electric and magnetic fields. We define~~ = B|| E + B
~⊥
B
|E|

(6.18)Taking ωef ≪ dmf E ≪ Ehf and dmf E ≫ gF µB B|| , and setting B⊥ = 0, we get for the non-rotatingenergies,
11
1u/ℓ)Ehf − mF ΩγF dmf E + mF gF µB B − (de Eeff /2|Ω|)Ω, (6.19)
Eu/ℓ
nr (F, mF , Ω; E, B) ≈ (F (F + 1) −
3
4

105
where Ω is either 1 or -1, and the prefactor in front of Ehf is such that for the J = 1 level, E(F = 3/2)
- E(F = 1/2) = 3A|| /4 = Ehf . F and Ω are good quantum numbers only to the extent that theelectric field is neither too large nor too small, but we will use F and Ω as labels for levels even as
these approximations begin to break down.
For notational compactness, we introduce special labels for particular states as follows (seeFig. 2.1(b)):|ai = |F = 3/2, m = 3/2, Ω = −1i(6.20)|bi = |F = 3/2, m = −3/2, Ω = 1i|ci = |F = 3/2, m = 3/2, Ω = 1i|di = |F = 3/2, m = −3/2, Ω = −1iwith corresponding energies, Ea , Eb , Ec , and Ed , and identify the energies of two particularlyinteresting transitions, W u = Ea − Eb , and W ℓ = Ec − Ed such thatW u = 3gFu µB B + de Eeff
W ℓ = 3gFℓ µB B − de Eeff .

(6.21)The equations of this section have so far been to one degree or another approximate results.
But in the absence of exotic particle physics we can invoke time-reversal symmetry and write exact
relations:
Enr (F, mF , Ω; E, B) − Enr (F, −mF , −Ω; E, B) = Enr (F, −mF , −Ω; E, −B) − Enr (F, mF , Ω; E, −B)(6.22)
which, for B = 0, becomes
Enr (F, mF , Ω; E) = Enr (F, −mF , −Ω; E).

(6.23)

This exact degeneracy is, in fact, an example of the Kramers degeneracy that follows from timereversal invariance [114]. For our purposes, the key result here is that, in the limit of non-rotating
fields, zero applied magnetic field, and an electron EDM, the energy of the science transitions

106|mF , Ωi ↔ | − mF , −Ωi (and in particular, W u and W ℓ ) are independent of the magnitude of theelectric field. This is an important property because we are using spatially inhomogeneous electricfields to confine the ions in the trap, and we want to minimize the resulting decoherence.
This degeneracy in turn means that the energy differences W u and W l depend only on themagnetic field and, of course, the EDM term as shown in Eq. (6.21). The magnetic contribution
reverses sign upon reversing the direction of B with respect to the electric field direction (which
also sets the quantization axis, since dmf E ≫ µB B). Therefore the science measurement is given by
the combinations
W u (E, B) + W u (E, −B) = 2de Eeff
W l (E, B) + W l (E, −B) = −2de Eeff ,(6.24)

where a + sign on B denotes that it points in the same direction as E.

6.3

Rotating Fields, Small-Angle Limit
Many EDM experiments over the years have been complicated by the problem of “Berry’s

phase”, the term in this context used as a catch-all to describe a variety of effects related to the
motion of the particles in inhomogeneous fields.
The sketch in Fig. 6.1(a) illustrates the classic Berry’s phase result: if the field that defines
the quantization axis, as experienced locally by a particle (or atom, or molecule), precesses about
the laboratory axis at some angle, θ, then, in the limit of slow precession, with each cycle of theprecession the wave-function Ψ picks up a phase given by mF A, where mF is the instantaneous
projection of the particle’s total angular momentum on the quantization axis, and A is the solid
angle subtended by the cone. If the precession is periodic with period τ , one can (with provisos,
as we will discuss) think of this phase-shift as being associated with a frequency, or indeed energy,
mF A/τ . In a spectroscopic measurement of the energy difference between two states whose mF
values differ by δmF , there will be a contribution to the transition angular frequency AδmF /τ .
In neutron EDM experiments, motional magnetic fields, in combination with uncharacterized

107
fixed gradients from magnetic impurities, Berry’s phase can be a dangerous systematic whose
dependence on applied fields can mimic an EDM signal [115]. In Chapter 6.14 we will see that theeffects of motional fields in our experiment are negligible.
Neutral atoms or molecules may be confined in traps consisting of static configurations ofelectric or magnetic fields. These traps are based on the interaction between the trapped species’
magnetic or electric dipoles and the inhomogeneous magnetic or electric fields, respectively, of the
trap. Especially in cases where the traps are axially symmetric, so that the single-particle trajectoryof an atom can orbit many times one way or the other about the axis of the trap, the coherence timeof an ensemble of atoms with a thermal distribution of trajectories can be severely restricted [116].
Our system is quite different, because in an ion trap the forces arise from the interaction between
the trapping fields and the monopole moment of our trapped ion. Assuming the temperature, sizeof bias field, and radius of confinement are the same, the trapping fields for an ion are spatiallymuch more homogeneous than would be those for a neutral molecule or atom.
That said, the fact that we can speak of a “bias” electric field at all in an ion trap comes atthe cost of having the applied electric field constantly rotating.

6.4

Rotating Fields, Large-Angle Limit (Dressed States)
The basic dressed-state idea is an extension of the more common idea of an energy eigenstate:a system governed by a time-invariant Hamiltonian H will have solutions Ψ such that Ψ(t + T ) =e−iωT Ψ(t) for all T and t; such a solution Ψ is called an energy eigenstate, with ω being then thecorresponding energy. Similarly, a system governed by a periodic Hamiltonian with period τ suchthat H(t + τ ) = H(t) for all values of t, will have so-called “dressed-state” solutions Ψ such thatΨ(t + nτ ) = e−inφ Ψ(t) for all t and all integer values of n. The quantity φ/τ is sometimes referred
to as the dressed state energy.
Operationally, the dressed state energies are derived from the eigenvalues of a formally timeindependent Hamiltonian. If H0 denotes the Hamiltonian in the absence of the field, then the108

A ~ pq2

F

w
q

(a)

A ~ 2p(1-a)

Fw

a = p/2-q(b)

Figure 6.1: a) Small-angle limit. When the quantization axis F follows a slow periodic perturbation
characterized by tilt angle θ, angular frequency ω and enclosed solid angle A, two states whose
instantaneous projection of angular momentum along F differs by δm will have their effective
relative energy displaced by a Berry’s energy ωAδm/2π. b) Large-angle limit. When instead thequantization axis sweeps out a full 2π steradians per cycle (α=0), the differential phase shift between
the two levels is indistinguishable from zero, and in the most natural conceptual framework, theBerry’s energy vanishes.

109
appropriate rotation-dressed Hamiltonian is given by
Hdressed = H0 − d~mf · E~rot + Hrot ,(6.25)

Hrot = −ωrot (cos(θ)Fz − sin(θ)Fx )(6.26)

Hrot is defined as [117]where Fz and Fx are the projections of the total angular momentum F~ into a set of axes where z
coincides with the instantaneous direction of the electric field. We now make explicit the rotating
electric field with E~rot . The cos(θ) term thus provides an energy which, when multiplied by therotational period τ = 2π/ωrot , gives the ordinary Berry phase,−2π cos(θ)mF → 2π(1 − cos(θ))mF(6.27)

where we have taken the liberty of adding an arbitrary phase 2πmF to reveal explicitly the solid
angle 2π(1 − cos θ).
In the experiment, the applied electric field should lie very nearly in the plane orthogonal to
the rotation axis, i.e., θ ≈ π/2. It is therefore useful to consider the small angular deviation fromthis plane, α = π/2 − θ (Fig. 6.1). Then the apparent energy shift arising from the geometric phaseisEgeo = −mF ωrot sin(α) ≈ −mF ωrot α.

(6.28)

Now consider two states which are, in the absence of rotation, degenerate, say the states |ai, with
m = 3/2, Ω = −1, and the state |bi, with m = −3/2, Ω = 1, indicated in Fig 2.1(b). Rotation
breaks this degeneracy, by adding the energies ∼ ±(3/2)ωrot α, as shown by the dashed lines inFig. 6.2. These levels cross at α = 0, leading to their apparent degeneracy when the electric fieldlies in the horizontal plane.
In addition, the rotation of the field also incurs coupling between states with different mF
values, arising from the sin(θ) term in Eq. (6.26). This perturbation, treated at third-order in
perturbation theory, connects the two levels and turns the crossing into an avoided one, as shown

110
by the solid lines in Fig. 6.2. This has been worked out by Meyer in Ref. [113]. The result, for the
mF = ±3/2 states in Fig. 2.1(b), is∆u/ℓ ≈ 170ωefωrot
dmf Erot3,(6.29)

where ∆ is the energy splitting at the level crossing between otherwise degenerate states withmF > 0 and mF < 0. The superscript u refers to mixing between the |ai and |bi states, and thesuperscript ℓ to mixing between |ci and |di states. In the absence of the hyperfine interaction, theaverage value of the numerical prefactor is 170 and the upper and lower states have the same avoided
crossing. However, small fractional differences between ∆u and ∆ℓ turn out to be significant, andare discussed further below.
The presence of the electric field causes the states with |F = 1/2, mF = ±1/2i and |F =3/2, mF = ±1/2i to mix. Including the hyperfine interaction into the numerical diagonalization
yields31ωrot∆ = (∆u + ∆ℓ ) ≈ 170 ωef,
2dmf Erot

3 

ωrot
dmf Erot 2
1.
δ∆ = (∆u − ∆ℓ ) ≈ 127 ωef
2dmf ErotEhf(6.30)(6.31)

It is evident that the average shift is the same, but now the upper and lower levels acquire a different
splitting due to the rotation-induced mixing within the sublevels. The difference is suppressed
relative to the average value of the splitting by a factor of (dmf E/Ehf )2 , reflecting the fact thathigher orders of perturbation theory are needed to include the effects of the hyperfine interaction.
For ωef = 2π×10 kHz, ωrot = 2π×100 kHz, dmf Erot = 2π×10 MHz, Ehf = 2π×40 MHz, then ∆ =2π×1.7 Hz and δ∆ = 2π×0.08 Hz.
The magnitude of the rotation-induced mixing within any of the four pairs of otherwise
degenerate m = ±1/2 states is much larger than the mixing within either pair of m = ±3/2 states,∆u or ∆ℓ . For this reason, the m = ±1/2 levels are probably not great candidates for precision
metrology in rotating fields.
An ion in a trap will feel an axial force pushing it towards the axial position where the axial
electric field vanishes, that is, the location at which α is identically zero. This poses a problem,

E/D

E/D

111

dmFwrota/D
Figure 6.2: The apparent energy shifts between mF = +3/2 and mF = −3/2 states in upper (a,b)
and lower (c,d) Λ-doublet levels versus α, the angle of the electric field to the plane orthogonal
the rotation axis of Erot (α is shown in Fig. 6.1(b)). (a) At α = 0, there is an avoided crossingthat mixes mF = ±3/2 states, with an energy splitting at the crossing of ∆u/ℓ . (b) Since α = 0
at the axial trap center, and since we need mF to be a signed quantity in order to measure de ,we will bias away from the avoided crossing using a magnetic field Brot . δmF gF µB Brot > ∆u/ℓ isrequired for mF to be a quantity of definite sign. This picture is intuitively correct in the limit that∆u/ℓ > ωmax . The experiment will be performed in the opposite limit. However, solving the timedependent Schrödinger equation (Eq. 6.42) gives the same requirement of δmF gF µB Brot > ∆u/ℓ inboth limits.

112
because at α = 0, each dressed state is an equal mixture of states with Ω = 1 and with Ω = −1. Inother words, the dressed states right at the avoided crossing will have vanishing eEDM signal. The
solution is to bias the avoided crossing away from α = 0 by adding to the trapping fields a uniform,
rotating magnetic field which is instantaneously always parallel or anti-parallel to vector Erot .
~rot = Brot E~rot
B
Erot(6.32)

In our convention, E~rot defines the quantization axis, so that the number Erot will always be taken
to be positive. The sign of Brot then determines whether the co-rotating magnetic field is parallel(Brot > 0) or anti-parallel (Brot < 0).
The energy levels are now as shown in Fig. 6.2(b). As derived in Chapter 6.5, in the limit
Brot gF µB ≫ ∆u/ℓ , the dressed states near α = 0 are once again states of good mF and Ω. The energy splitting between the two states, as altered by the rotation of the field, are given approximatelyby(∆u/ℓ )2u/ℓ

W u/ℓ (Erot , Brot ) = Ea/c − Eb/d = −3αωrot + 3gF µB Brot +u/ℓ

6gF µB Brot − 6αωrot± de Eeff , (6.33)

where the + sign corresponds to u states, and the − sign to ℓ states.
Over the course of one axial oscillation of the ion in the trap, α which is approximately
proportional to the axial electric field, will average to zero. Unfortunately, the contributions toδW from Erot and from Brot are larger than that from the scale of the physics we most care about,de Eeff , and the spatial and temporal variation in Erot and in Brot will reduce the coherence timeof the spectroscopy, as discussed in Chapter 6.9 - 6.11. But to the extent that one is able quite
precisely to chop Brot to −Brot on alternate measurements, the science signal still arises from thesame combination as in Eq. 6.24:
hW u/ℓ (Erot , Brot )i + hW u/ℓ (Erot , −Brot )i ≈ ±2de Eeff ,(6.34)

where the +/- corresponds to the u/ℓ superscripts respectively, and the brackets denote averaging
over the excursions of α, which is assumed to vary symmetrically about zero.

113
The equation above relies on several approximations. One needs in particular that dmf Erot ≫ωef , 3gF µB Brot ≫ 170ωef (ωrot /dmf Erot )3 , and dmf Erot ≫ ωrot , α ≪ 1, and dmf Erot < Ehf . These
are all good approximations, but they are not perfect. For example, using values from Appendix
A, ωrot /(dmf Erot ) ≈ 0.01, a small number, but not zero. To what extent will imperfections in theseapproximations mimic an eEDM signal?
The driving principle of our experimental design is to measure de with as close to a nullbackground as possible. We are not especially concerned if the right hand side of Eq. (6.34) is
1.9 de Eeff rather than 2.0 de Eeff . More important to us is that, if de = 0, the right-hand side ofEq. (6.34) be as close to zero as possible. As we shall see, as long as we preserve certain symmetriesof the system we are guaranteed a very high quality null. A preliminary remark is that the “energy”
of a dressed state, or more precisely the phase shift per period τ , is unaffected by an offset in how
the zero of time is defined. A second observation is that, in the absence of exotic particle physics(such as nonzero eEDM), the energy levels of a diatomic molecule in external electromagnetic fields
are not affected by a global parity inversion.
Under the action of this inversion, all the fields and interactions in the Hamiltonian transform
according to their classical prescriptions, whereas quantum states are transformed into their parityrelated partners. In a parity-invariant system, parity thus changes quantum numbers, but leaves
energies of the eigenstates unchanged. This is true for the dressed states as well, since their eigenenergies emerge formally from a time-independent Hamiltonian.
To formulate the effect of inversion symmetry we write the electric and magnetic fields as
E~ = Erot r̂′ + Ez ẑ

(6.35)~ = Brot r̂′B(6.36)

where r̂′ = cos(ωrot t)x̂ + sin(ωrot t)ŷ and α = tan−1 (Ez /Erot ). The dressed states defined by therotating field are characterized by the projection mF of total angular momentum on the axis definedby the rotating electric field, E~rot /Erot . Because the magnetic field is not strictly collinear with theelectric field, and because of the field rotation, mF is only approximately a good quantum number.

114
Nevertheless, considering the effect of parity on all the mF ’s simultaneously, we can still map eachdressed eigenstate into its parity-reversed partner.
Assuming the ions are “nailed down” in their axial oscillation, at a particular value ofEz and thus α, our various spectroscopic measurements would give dressed energy differences
E(Erot , Brot , α, mF , Ω) − E(Erot , Brot , α, −mF , −Ω). Now we invoke the following symmetry argument: if we take the entire system, electric fields, magnetic field, and molecule, and apply aparity inversion, that will leave the energy of the corresponding levels unchanged. If further wethen shift the zero of time by π/ωrot , in effect letting the system advance through half a cycle of the
field rotation, that also will not change the corresponding energy levels of the dressed state, which
are after all defined over an entire period of the rotation. This transformation effectively connects
measurements made for α > 0, above the mid-plane, to those with α < 0, below the mid-plane.
The combined transform acts as follows:
F→ F(6.37)~rot → −B~rot
B
E~rot → E~rot
Ez → −Ez
α → −αω~ rot → ω
~ rot
mF

→ −mF

~ ·B
~ → S
~ ·B
~
S
dˆ · E~ → dˆ · E~
Ω → −Ω
The last of these is equivalent to n̂ · ~σ1 , i.e., our symmetry operation would change the sign of the
EDM energy shift. However, in the absence of this shift we can expect the following exact relations

115
between the dressed state energies:
E(Erot , Brot , α, mF , Ω) − E(Erot , −Brot , −α, −mF , −Ω) = 0(6.38)

E(Erot , −Brot , α, mF , Ω) − E(Erot , Brot , −α, −mF , −Ω) = 0
E(Erot , Brot , −α, mF , Ω) − E(Erot , −Brot , α, −mF , −Ω) = 0
E(Erot , −Brot , −α, mF , Ω) − E(Erot , Brot , α, −mF , −Ω) = 0.
Summing four equations and rearranging terms, we get that
W u/ℓ (Erot , Brot , α)+W u/ℓ (Erot , −Brot , α)+W u/ℓ (Erot , −Brot , −α)+W u/ℓ (Erot , Brot , −α) = 0. (6.39)
If we assume that the axial confinement is symmetric (not necessarily harmonic), and that
our spectroscopy averages over an ensemble of ions oscillating in the axial motion with no preferredinitial phase of the axial motion (we will later explore the consequences of relaxing this assumption)then the ions will spend the same amount of time on average at any given positive value of α asthey do at the corresponding negative value of α, and thus the averaged results yield:< W u/ℓ (Erot , Brot ) > + < W u/ℓ (Erot , −Brot ) >= 0.

(6.40)

The combined result, in the absence of exotic particle physics, is zero by symmetry. We did
not need to invoke the various approximations that went into Eq. 6.34. In particular, this null
result is, unlike the traditional Berry’s phase result, not based on the assumption of very small(ωrot /dmf Erot ). Also, for conceptual simplicity we have discussed the result as being based on anaverage over quasi-static values of α, but the symmetry argument does not hinge on the axial
frequency being infinitely slow compared to ωrot .

6.5

Frequency- or Phase-Modulation of Axial Oscillation
The trapped ions will oscillate in the axial direction at a frequency ωz , confined by anapproximately harmonic axial trapping potential Uz = (1/2)M ωz2 z 2 . Upon moving away from the
mid-plane z = 0, the ions will experience an oscillating axial electric field Ez (t) = −M ωz2 z(t)/e. The

116
geometric phase correction to the energy is then −mF ωrot αmax cos(ωz t), where αmax = Ez,max /Erotis the maximum excursion of the tilt angle. Because the product ωrot αmax is again an energy, it isconvenient to redefine the geometric energy contribution in terms of a frequency ωmax ,Egeo = ωmax cos(ωz t),(6.41)

with ωmax = −δmF ωrot αmax .
For ωz = 2π×1 kHz, an ion cloud temperature of 15 K, an ion whose axial energy Ez is twice
the thermal value, for ωrot as shown in Appendix A, and Erot = 5 V/cm, then a transition such as
W u/ℓ with δm = 3 will have a maximum frequency modulation ωmax = 2π×400 Hz.
Thus the electric field undergoes two motions, the comparatively fast radial rotation, andthe comparatively slow axial wobble. We exploit the different time scales to create, for each
instantaneous value of α, the rotation-dressed states worked out in the previous section. The effectof the axial wobble is then described by the time variation of the amplitudes in these dressed states.
The time-dependent Schrödinger equation of motion for this is

i







a  
∂ 
=

 

∂t
b

ωmax
3
2 gF µB Brot + 2 cos(ωz t)

∆
2

∆
2

− 23 gF µB Brot − ωmax
2 cos(ωz t) a 
,

(6.42)bwhere a and b are the probability amplitudes for being in the |ai and |bi states, respectively. For
typical experimental values, ωz is about 2 π ×1 kHz, ωmax will range as high as 2 π ×1000 Hz, and∆ (given by Eq. 6.29) is perhaps 2 π ×2 Hz, and 3µB gF Brot is about 2 π ×10 Hz.
Eqn. 6.42 describes a system again governed by a periodic Hamiltonian, and we will therefore
follow a similar course to Chapter 6.4 and search for dressed-state solutions Ψ such that Ψ(t+nτ ) =e−inφ Ψ(t). Of course, this will only be valid in the limit that ωrot ≫ ωz , a necessary condition towrite the time-dependent Hamiltonian in Eqn. 6.42. First, we get rid of fast time-dependence byguessing solutions:a(t) = A(t)

∞
X

n=−∞

Jn



ωmax −iωz nte2ωz(6.43)

117
b(t) = B(t)

∞
X

Jn

n=−∞



−ωmax −iωz nt
e
,
2ωz


where Jn are Bessel’s functions of the first kind and A(t) and B(t) are slowly varying
functions. We then substitute our trial solutions into Eqn. 6.42 and use the recurrence relation (2n/x)Jn (x) = Jn−1 (x) + Jn+1 (x). We multiply through by

∞
X

Jn′

n′ =−∞∞XJn′n′ =−∞



ωmax iωz n′ t
e
and
2ωz


−ωmax iωz n′ te
as appropriate. We then integrate over an axial time period, 2π/ωz ,2ωz

and make the approximation that A(t) and B(t) are unchanged over this small time interval. Thisapproximation should be good as long as ωz ≫ ωmax and ωz ≫ gF µB Brot . The integration thenyields,

i
with







A  
∂ 
=

 
∂t 
B

∆eff =

∞
X

n=−∞

Jn



3
2 gF µB Brot

∆eff
2

∆eff
2

− 32 gF µB Brot A 





ωmax
−ωmax
ωmax
Jn
∆ = J0 (
)∆.
2ωz
2ωz
ωz




(6.44)

B

(6.45)

This results in dressed-state energies, now as a function of Brot , and not α, as seen in Fig. 6.3.
This clearly shows the requirement of 3gF µB Brot > ∆eff in order to keep |ai and |bi as the dressedstates. This is true despite the fact that an ion will sample the avoided crossing in Fig. 6.2 during
its axial oscillation in the trap, as ωmax ≫ 3gF µB Brot in our experiment. ∆eff will have a maximum
value of ∆ at ωmax /ωz = 0 and will oscillate about zero according to J0 (ωmax /ωz ).
For finite ωmax /ωz , the dressed states from Eqn. 6.44 only appear stationary if measured at
integer multiples of the axial trapping period, 2π/ωz . Consider states |+i and |−i, symmetric andantisymmetric combinations of states |ai and |bi, respectively. In the limit that δmF gF µB Brot ≫∆eff , an ion initially in state |+i will oscillate between |+i and |−i at the precession frequency ω0 =u/ℓ

1((3gF µB B rot +de Eeff )2 +∆2eff ) 2 , when measured at integer multiples of the axial trapping frequency.
However, if our EDM measurement is made after a non-integer number of axial oscillations, or if
the ions have different axial frequencies in the trap, the |+i to |−i oscillation will be frequency
modulated at ωmax (see Fig. 6.4). For the example parameters, the frequency-modulation index

118

E/Deffeff

dmFgFmBBrot/Deff

Figure 6.3: The apparent energy shifts between mF = +3/2 and mF = −3/2 states in upper (a,b)
and lower (c,d) Λ-doublet levels versus Brot ,“dressed” by both the electric field rotation (ωrot ) and by
the ion’s axial trap oscillation (ωz ). At Brot = 0, there is an avoided crossing that mixes mF = ±3/2
u/ℓ
states, with an energy splitting at the crossing of ∆eff . In the limit δmF gF µB Brot ≫ ∆eff , thedressed states are of good mF with an energy splitting slightly modified by ∆eff .

119ωmax /ωz is less than 1, and thus the spectral power of transition is overwhelmingly at ω0 , thequantity which symmetry arguments above show is unaffected by Berry’s phase. In an ensembleof ions which have a random distribution of initial axial motions, the sidebands on the transition
average to zero, and won’t pull the frequency of the measured central transition. If instead theprocess of loading ions into the trap has left the ions with an initial nonzero axial velocity or axial
offset from trap center, the measured frequency can be systematically pulled from ω0 .
Note that increasing Erot or decreasing ωrot reduces the value of ωmax and thus the frequency
modulation index. On the other hand, these changes also would have the effect of increasing theenergy Emicro of the micromotion of the ions in the rotating fields. For harmonic axial confinement,
we find the frequency modulation for a δm = 3 transition obeys the following relation ωmax /ωz =3(Ez /Erot )1/2 . Thus to keep the modulation safely under unity for a comfortable majority of an
ensemble of ions with an average Ez given by Tz , one needs to choose operating parameters suchthat Erot > 30kB Tz . This inequality in turn places stringent requirements on the spatial uniformityof Erot . On a time-scale slow compared to 1/ωrot , Erot acts like a sort of ponderomotive potential
analogous to the effective confining potential in a Paul trap. If Erot = 30kB Tz , then a spatial
inhomogeneity in Erot of only 1.5% already gives rise to structure in the ponderomotive potential
comparable to Tz .
To summarize the effect of axial motion: in the limit 3gF µB Brot > ∆, ions prepared, forinstance by optical pumping, in state |ai (or |bi) will remain in |ai (or |bi). The energy difference
between dressed states which are predominantly either |ai or |bi will be slightly modified by the
avoided crossing. But the important combined measurement described by Eq. (6.34) will continue
to yield zero in the absence of EDM signal, and the sensitivity of that combined measurement to
the EDM signal will not be much affected as long as ωmax /ωz <≈ 1.

6.6

Creating Coherent Mixtures of mF = +3/2 and mF = -3/2
The above discussion reveals a possible way to deal with an experimental issue. The Ramsey-

type experiment we wish to perform in order to measure de requires a coherent superposition of

120

1.0

(a)

Population

0.8

0.6

0.4

0.2

0.25
50

0.75
150

100
0.5

1.0
200

1.25
250

t/dmFgFmBBrot

(b)

1.0

Population

0.8

0.6

0.4

0.2

0.5
0.05

1.0
0.10

t/dmFgFmBBrot

1.5
0.15

2.0
0.20

Figure 6.4: Population of states |+i (red) and |−i (blue) with parameters ωz = 2 π ×1000 Hz,
u/ℓωmax = 2 π ×500 Hz, ∆ = 2 π ×2 Hz, and δmF gF µB B rot = π ×10 Hz. The population is initiallyprepared in |+i and results were found by numerically integrating Eq. 6.42. (a) Measured at integer
periods of ωz the |ai and |bi states appear stationary, leading to smooth Rabi oscillations if thestate is prepared in state |+i or |−i. (b) If the measurement cannot be made at integer periodsof ωz , frequency modulation at ωmax is observed, leading to reduced accuracy when measuring
the precession frequency. Note here that ωmax /ωz = 1/2. As discussed in Chapter 6.11, similar
modulation can arise from static, uniform magnetic fields that are averaged over during a periodof ωrot .

121electron spin up and spin down states. One possible way of achieving this is to start with ions in
the 1 Σ, F = ± 1/2 states and then populate the 3 ∆1 , J = 1, mF = ± 3/2 states using STIRAP,
similar to the discussion in Chapter 5.7. The quantum beat signal from the radiative decay could
be measured, or another STIRAP could be performed and the transfer percentage measured. As
discussed in Chapter 5.7, STIRAP could prove challenging with our weak transitions and large ion
clouds.
Typically, when performing a Ramsey measurement between magnetic sub-levels, a π/2 pulse
is performed with rf photons. If the ions can be created directly in the 3 ∆1 , J = 1 state throughautoionization as described in Chapter 5.7.3, a statistical mixture of mF levels would exist. Opticaldepletion of all but the mF = +3/2 level could be done with σ + polarized light driving a transition
with a small decay branching fraction back to the 3 ∆1 level. For our case of F = 3/2 states, ourπ/2 pulse would have to come from a three-photon transition.
A typical JILAn laughs in the face of a challenge such as three-photon rf transitions. However,the preceding section points to another possible solution. The coupling need not be performed by an
rf photon field, instead the rotational electric field can create the coupling, as seen by the quantity∆ above. After the other spin levels have been optically depleted, ∆eff can be ramped up to somevalue much larger than 3gF µB Brot , by decreasing the value of Erot . After waiting a time 1/(2∆eff )and ramping back to ∆eff < 3gF µB Brot , the mF = +3/2 population can be coherently transferred toan equal superposition of mF = ± 3/2 states. The spin is then allowed to precess at the frequency
u/ℓ1ω0 = ((3Brot gF µB + de Eeff )2 + ∆2eff ) 2 for some time τ , ∆eff is again ramped to perform the second
π/2 pulse of the Ramsey sequence, and the final spin population can be read out as discussed in
Chapter 4.

6.7

Structure of the Measurements. What Quantities MatterIn the remainder of this section, the possible effects of various experimental imperfections on

our measurement are explored.
The symmetry argument in Chapter 6.2 presupposes the ability to impose a perfect “B-chop”,

122i.e., to collect data with alternating measurements changing quite precisely only the sign of Brot . If
not only the sign but the magnitude of the rotating magnetic field alternates, the situation is more
complicated. There will likely be contributions to the rotating magnetic field that are not perfectly
reversed in our B-chop, including displacement currents associated with sinusoidally charging thestrayelectrodes that create the rotating electric field. These effects can be quantified with a value Brot,

and to lowest order they would appear as a frequency offset in the chopped measurement:~ B~rot + B~ stray ) + W u/ℓ (E,~ −B~rot + B~ stray ) = 6g u/ℓ µB B stray ± 2de Eeff
W u/ℓ (E,
rot
rot
rot
F

(6.46)

This offset is very nearly the same for the upper and lower states, to the extent that gFu ≈ gFℓ ,i.e., to the extent that δgF ≡ 1/2(gFu − gFℓ ) ≪ gF . The effect of the stray field is reduced by
combining measurements from the upper and lower states, in the form of a “four-way chop”:
hi~rot + B~ stray ) + W u (E~rot , −B~rot + B~ stray )W u (E~rot , B
rot
rot
h

i

~rot + B~ stray ) + W ℓ (E~rot , −B~rot + B~ stray ) = 12δgF µB B stray + 4de Eeff (6.47)− W ℓ (E~rot , B
rot
rot
rot
It may prove to be advantageous to shim the B-chop by deliberately adding a non-chopped
shim , and adjusting its value until experimentally we measurerotating magnetic field, Brot
stray
stray
shim
shim
W u (Brot + Brot
+ Brot) + W u (−Brot + Brot
+ Brot) = 0.

(6.48)

Then, a measurement in the lower Λ-doublet state gives
stray
stray
shim
shim
W ℓ (Brot + Brot
+ Brot) + W ℓ (−Brot + Brot
+ Brot) = −2(1 +gFℓ)de Eeff ,gFu(6.49)

yielding a still more accurate value for 4de Eeff .
What we care about most then are: (1) Things that perturb W u and W ℓ differently, in particular the quantity δgF , but also the analogous electrical quantity, δη, to be defined and estimatedin Chapter 6.9, and (2) to a lesser extent, we care about effects which affect W u (Brot ) + W u (−Brot )the same way as they affect W ℓ (Brot ) + W ℓ (−Brot ), because, to the extent that they lead to ameasurement
W u (Brot ) + W u (−Brot ) = +2de Eeff + δsyst ,(6.50)

123
stray
we can mistake a nonzero value for δsyst as an indicator for a nonzero value of Brot
. In that case,stray
the shimming procedure discussed above to remove Brot
would lead to a combined result from thefour-way chop of 4de Eeff + (4δgF /gF )δsyst . This is down by a relative factor of (δgF /gF ) comparedto the effects that differentially perturb W u versus W ℓ , but they could still be troublesome. And(3) to a still lesser extent, we care about imperfections that perturb individual measurements suchas W u (Brot ), even if they do not perturb the B-chop measurement, W u (Brot )−W u (−Brot ), because,to the extent that they vary over time, or depend on the trajectory of an individual ion in the trap,
they can reduce coherence times. This leads not to systematic errors, but to a reduction in theoverall precision.
In addition to the B chop, state chop, and four-way chop discussed above, we can perform arotation chop, by changing the sign of ωrot . Our hope is to keep experimental imperfections to alevel where the four-way chop is by itself already good enough to suppress systematic error below
the desired level. Then repeating the entire series of measurements with the opposite sign of ωrot
(rotating the field CW instead of CCW) will to the extent it yields the same final value of 4Eeff deprovide a useful redundant check.

6.8

An Estimate of δgF =3/2
There are three leading contributions to δgF =3/2 in our molecule: a zero field difference, adifference due to static electric fields, and a difference due to rotating electric fields. The zero field
difference is due to terms omitted in the Zeeman Hamiltonian in Eq. 6.13. For our molecule, theseterms give rise to a difference in gF =3/2 in terms of the Λ-doubling J = 1 energy splitting ωef [113],
δgF =3/2
ωef≈.
gF =3/2
Be gF =3/2

(6.51)

This is on the order of 10−5 for HfF+ .
The electric field dependent g-factor arises due to the mixing of rotational levels J in themolecule. The levels with J = 2, while far away in energy compared to the Stark energy dmf Erot ,are perturbers. In the signed Ω basis, the mF sub-levels in the J = 2 level have a smaller γF value

124
than do the mF sub-levels in the J = 1 level. Therefore, the states which go up (down) in energy
in the J = 1 level “gain” (“run”) on (from) the J = 2 level. When one includes the effects ofHyperfine interactions, there are multiple connections to each sub-level. In the J = 1, mF = ±3/2
levels that we are interested in this contribution is [113],
δgF =3/2
9 dmf Erot
(Erot ) =,gF =3/2
40 Be(6.52)

which gives a fractional shift δgF =3/2 /gF =3/2 of a few 10−4 . The same approach gives that theelectric field “g” factor, γF , will shift in the same manner such that δγF /γF ≈ 10−4 .
For rotating fields, another contribution to δgF arises from non-vanishing value of ωrot /(dmf Erot ).
The states with Ω = 1 and Ω = −1 are equally affected by the rotating field since they havean equal Stark shift in the absence of hyperfine interactions. However, because the levels with
|F = 3/2, mF = ±1/2i are repelled by the lower |F = 1/2, mF = ±1/2i states, the effective Stark
difference between mF levels with Ω = −1 (upper levels) is smaller than the same mF levels with
Ω = +1 (lower levels). The scale at which this difference will appear is then determined by how
much the lower hyperfine state pushes on the upper due to the coupling induced by the electricfield.

√
2
δgF =3/2
6
ωrot
= 2
.
gF =3/2
γF =3/2 dmf Erot Ehf(6.53)

This fractional shift is of the order a few 10−4 and is therefore about the same magnitude as theelectric field induced mixing of higher rotational levels.

6.9

Dependencies on Erot
Proximity to the avoided crossing shown in Fig. 6.2(b) means that the transitions W u andW ℓ will have residual dependencies on Erot , which in turn may lead to decoherence or systematic
errors. We characterize the sensitivity of W u/ℓ to small changes in Erot with the following expansion00
W u/ℓ (Erot+ δErot , Brot ) = W u/ℓ (Erot, Brot ) + η u/ℓ δErot(6.54)

125with
η u/ℓ ≡−(∆u/ℓ )2∂W u/ℓ
,
=
|Erot
0 ,B
rot
u/ℓ
∂Erot
gF µB Brot Erot

(6.55)

using the expressions in Eqs. (6.29) and (6.33). Any spatial inhomogeneity in Erot that does notaverage away with ion motion will lead to a decoherence rate given approximately by ηδErot .
In terms of systematic errors, if chopping the sign of Brot gives rise to an unintended systematic change in the magnitude of Erot (call it δEchop ), for instance due to motional fields discussed
later, or due to ohmic voltages generated by the eddy currents, then there will be a frequency shift
in a B-chop combination, 2η u/ℓ δEchop . To the extent that δη = 12 (η u − η ℓ ) is nonzero, some of thisshift will survive a four-way chop as well. The dominant contribution to δη is likely from δ∆, rather
than from δgF . Assuming this limit, the systematic error surviving is
8δ∆
dmf ErotηδEchop ≈ 6
∆
Ehf


2

ηδEchop .

(6.56)For a large but not inconceivable value for δEchop of 100 µV/cm, and for other values as inAppendix A, this works out to comfortably less than 100 µHz, and is therefore not a problem. But−5, and thus could cause trouble if for other reasons we chose to decrease
this error would scale as Erot

Erot . The science signal is roughly independent of Erot , which should allow for the source of error
to be readily identified.

6.10

Perpendicular B-Fields
The quantization axis is essentially defined by Erot . The shift of the various levels |ai, |bi,|ci, |di due to a component of the magnetic field perpendicular to Erot is on the order of±

3 (gF µB B⊥ )24 (γF dmf Erot )(6.57)

for the upper/lower states. In the absence of rotation, the lowest-order correction to W u/ℓ (Brot )goes as−2B
3 gF3 µ3B B⊥
rot.
2 (γF dmf Erot )2(6.58)

126
For reasonable experimental parameters, this will be a negligible number. The lowest-order correction to the state-chop combination, W u (Brot ) − W ℓ (Brot ) is smaller still and goes as
2B
3 gF3 µ3B B⊥
rot.
2 γF dmf Erot Ehf(6.59)

It is similar in form to the difference in g-factors caused by the rotation of the field.
When we turn on rotation, there is an additional larger contribution to W u/ℓ (Brot ). If weassume (as a worst case) that B⊥ is purely axial, not azimuthal, then the lowest-order effect of B⊥
is to tilt the quantization axis by angle given by±gF µB B⊥,(γF dmf Erot ± µB gF Brot )(6.60)

with the +(-) in the numerator corresponding to the upper(lower) states and the +(-) in the
denominator corresponding to the Ω = -1(+1) states. This has the leading order effect on W u/ℓ of3ωrot gF2 µ2B B⊥ Brot
(γF dmf Erot )2(6.61)

even a rudimentary nulling of the Earth’s magnetic field, say to below 25 mG, will leave this term
negligible, for parameters in Appendix A. Its contribution to the state chop, W u (Brot ) − W ℓ (Brot ),is still smaller by dmf Erot /Ehf

6.11

ωrot gF2 µ2B B⊥ Brot
γF dmf Erot Ehf(6.62)

Stray Contributions to B|| : Uniform or Time-Varying B-Fields
In the previous section we’ve seen that the effects of B⊥ are small. Spatial or shot-to-shot

variation in B|| , on the other hand, can limit coherence time through its contribution to W u/ℓ . The
biggest contribution to B|| is of course the intentionally applied rotating field Brot . Let us examine
the various other contributions to B|| .
Static, uniform fields: B fields of this nature are relatively harmless. B|| is defined relative~ · Êrot i. Since Êrot sweeps out ato the quantization axis Êrot . The time-average of B|| is hB|| i = hB
circle with angular velocity ωrot , the contribution to the time-averaged B|| from a uniform, static

127
magnetic field averages nearly to zero in a single rotation of the bias electric field, and still moreaccurately after a few cycles of axial and radial motion in the trap. The average electric field in
the ion trap must be very close to zero, or the ions would not remain trapped. In the case ofcertain anharmonicities in the trapping potential, however, one can find that the average value ofÊrot is nonzero, even if the average value of E~rot is zero. For instance, an electrostatic potential
term proportional to z 3 , along with a uniform axial magnetic field Bz , will for an ion with nonzero
axial secular motion, yield a nonzero hB|| i. In addition, nonzero Bz will interact with the tilt ofE~rot oscillating with an ion’s axial motion at ωz to cause a frequency modulation similar to theone discussed in Chapter 6.5. A uniform magnetic field in the x-y plane will cause a frequency
modulation at ωrot . If the modulation index for either of these modulations approaches one, themodulation will begin to suppress the contrast of spectroscopy performed at the carrier frequency.
For uniform magnetic fields with amplitude less than 10 mG (achievable for instance by roughly
nulling the earth’s field with Helmholtz coils), frequency modulation indices will be small, and,barring pathologically large z 3 electrostatic terms, the mean shifts from uniform, static B-fields
shim as discussed inwill be less than 1 Hz and can be can be dealt with by means of an applied Brot

Chapter 6.7.
Time-varying magnetic fields with frequency near ωrot can cause more trouble. If the time
between the two Ramsey pulses used to interrogate the frequency is tRamsey , then the dangerous
bandwidth is 1/tRamsey , centered on ωrot . We discuss in order (i) thermally generated fields fromthe electrodes, (ii) ambient magnetic field noise in laboratory, (iii) magnetic fields associated withthe application of Erot , oscillating coherently with Erot , (iv) shot to-shot variation in magnitude ofapplied Brot , and (v) spatial inhomogeneities in Brot .
(i) Proposed EDM experiments on trapped atomic species such as Cesium are vulnerable
to magnetic field noise generated by thermally excited currents in conductors located close to thetrapped species [118]. In our case, the effect is less worrisome because, vis-a-vis the trapped atom
experiments, our bandwidth of vulnerability is centered at much higher frequency fields, because ourmolecules are trapped considerably further from the nearest conductors, and because the sensitivity

128
of our measurement of de to magnetic field noise, which goes as gF µB /Eeff is down by a factor of
104 . The spectral density of thermal magnetic field noise (which is calculated in reference [119] inthe simplified geometry of a semi-infinite planar conductor) will surely be less than 1 pG/Hz1/2 inour bandwidth of vulnerability. This effect is negligible.
(ii) Like thermal magnetic noise, technical magnetic noise in our lab arising for instancefrom various nearby equipment will not so much decohere an individual measurement as generate
shot-to-shot irreproducibility between measurements. What level of noise are we sensitive to? As
we discuss in Sec 8, the precision of a single trap load is unlikely to be better than 300 mHz,
meaning magnetic field noise less than 0.2 µG/Hz1/2 won’t hurt us, for a 1 s interrogation time.
Measurements made in our lab show that there are a number of magnetic field “tones” of very
narrow bandwidth, associated with harmonics of 60 Hz power and various power supplies. As longas we choose ωrot to not coincide with one of these frequencies, in the range of 50 kHz to 300 kHz
ambient magnetic frequency noise in our lab has spectral density typically less than 0.02 µG/Hz1/2 .
For this reason, at least for the first generation experiment, there will be no explicit effort to shieldambient magnetic field other than to use Helmholtz coils to roughly null the earth’s dc field. The
steel vacuum chamber will in addition provide some shielding at 100 kHz.
(iii) In traditional eEDM experiments, one of the most difficult unwanted effects to characterize and bring under control is magnetic fields generated by leakage currents associated with thehigh voltages on the electrodes that generate the principal electric field. In our case the bulk of
the electric field Eeff is generated inside the molecule. The laboratory electric fields are measured
in V/cm, not kV/cm, and leakage currents as traditionally conceived will not be a problem for
us. On the other hand, the electric field does rotate at hundreds of kHz, and thus the electrodepotentials must constantly oscillate. Displacement currents in the trapping volume between theelectrodes, and real currents in the electrodes themselves and in the wire leads leading to them,will generate magnetic fields with spatial gradients and strengths that oscillate coherently with Erotat the frequency ωrot .
The spatial structure of the oscillating magnetic fields will depend on the geometry of the129electrodes and in particular on the layout of the wire leads that provide the current to charge them.
In principle, shim coils can be constructed just outside the trap electrodes and driven with various
phases and amplitude of current oscillating at ωrot , all in order to further control the shape of the~ = c−2 ∂ E/∂t.
~magnetic field. The one immutable fact is the Maxwell equation, ∇ × B
The dominant time dependence of the electric field is from the spatially uniform rotating field,
and thus for a circular field trajectory, the dominant contribution to the magnetic field structuregoes as~ = kŷ ′∇×B(6.63)withk = c−2 Erot ωrot ,(6.64)= 350 nG/cm ×ωrot
2π × 100 kHz



Erot
,
5 V/cm

(6.65)

where ŷ ′ is the direction in the x-y plane orthogonal to the instantaneous electric field.
The curl determines only the spatial derivatives of B; B itself only depends on the boundary
conditions. An idealized arrangement of current carrying leads and shim coils could in principle
force the B field to be~ ideal = kx′ ẑ.
B(6.66)

where k is given by Eq. 6.64 and x′ is displacement in the x-y plane along the direction of theinstantaneous rotating electric field. These fields would be perpendicular to the quantization axis
provided by the electric field, and would have negligible effect on the transitions of interest.
While realizing such an idealized displacement field would be very difficult, there are relatively
simple steps to take to minimize the displacement fields. For instance, each rod-like electrode canbe charged up by two leads, one connected to each end of the rod, with the leads running along
respective paths symmetric in reflection in the z=0 plane to a common oscillating voltage source
outside of the vacuum can, at z=0. It is worth considering a maximally bad electrode layout, to put
a limit on worst-case performance. Our electrodes will be spaced by about 10 cm and mounted in130such a way that their capacitance to each other or to ground will be at worst 5 pF. If the charging
current is provided entirely by a single lead connected to one end of the rod, the peak current
running down the rod near its center will be 80 µA, leading to a worst-case field magnitude at thetrap center of about 20 µG, and a contribution to W u/ℓ of perhaps 2.5 Hz. Spatial gradients of thiseffect, and shot-to-shot irreproducibility of this effect will not contribute to decoherence at the 0.1
Hz level. As for its contribution to systematic error, this shift will survive the B-chop, but will be
suppressed in the four-way chop by the factor (δgF /gF ), perhaps a factor of a thousand. For still
shim , as discussed
better accuracy the shift should be nulled out of the B-field chop by adjusting Brot

in Chapter 6.7.
(iv)Given that the main effect of Brot is to apply an offset frequency, 3gF µB Brot of perhaps
8 Hz, and given that (see Sec 8) the single-shot precision is unlikely to be any better than 300
mHz, the shot-to-shot reproducibility of Brot need be no better than a part in 30, a very modest
requirement on stability. Decoherence then is not a problem, but a potential source of systematic
error arises if the the B-chop is not “clean” that is if Brot before the chop is not exactly equal to−Brot after the chop. This sort of error could arise for instance from certain offset errors in op-amps
shim to cancel these offsets, but
generating the oscillating current. Experimentally, one adjusts Brot

even in the absence of that procedure, the four-way chop cleans up these sorts of errors. For a rather
egregious fractional deviation from B-chop cleanliness of, for instance, 1%, and for (δgF /gF ) < .001,
the systematic error remaining after the four-way chop is 10−5 of the offset frequency of perhaps 8
Hz. In HfF+ this is a systematic error on de of 10−29 e cm. For ThF+ the error as referred to deshim so as to null the post B-chop signal tois smaller still, and of course if we avail ourselves of Brot

<100 mHz, the systematic error on de will be considerably less than 10−29 e cm for either species.
(v) The largest single contribution to decoherence (with the exception of spontaneous decayof the 3 ∆1 line to a lower electronic state) will likely be due to spatial inhomogeneity in the applied~ that rotate in the x-y plane atrotating bias field Brot . That is to say, spatial inhomogeneities in B
frequency ωrot . First-order spatial gradients in Brot are not important, because ion secular motionin the trap will average away the effects of these gradients leaving only the value of Brot at the

131center of the trap. Second-order spatial gradients on the other hand will lead to nonzero averagefrequency shifts whose value will vary from ion to ion in a thermal sample of ions, depending
on conserved quantities of individual ion motion like the axial secular energy Ez or radial secularenergy Er , quantities with thermally averaged values of kTz and kTr , respectively, and with ion-toion variation comparable to their mean values. The Brot might be generated by current-carrying
rods which are of necessity within the vacuum chamber because of the screening effects of a metal
vacuum chamber. Unless particular care is taken in the design of these rods, the second-order
spatial gradients in Brot will scale as 1/X 2 , where X is the characteristic size (and spacing) of
the current carrying rods. The contribution to the inhomogeneity of the time-averaged value ofBrot experienced by a thermal sample of ions orbiting in a cloud with r.m.s size r is then of order(r2 /X 2 )Brot , leading to an ion-to-ion frequency variability of order (r2 /X 2 )3gF µB Brot For planned
parameters of the experiment, (r2 /X 2 ) is of order 0.01. We’ve seen from Chapter 6.5 above thatthe quantity 3gF µB Brot must be at least about five times larger than ∆ in order to make theeigenstates in the rotating fields be states of good mF . Thus in the absence of explicit apparatus
design to null the second-order spatial gradient in Brot (The rod-like electrodes that bear the chargethat generates Erot are in the second-generation trap the same objects that carry the current thatgenerates Brot and thus their shape is already subject to multiple design constraints) we may haveto live with a decoherence rate from this effect on the order of 0.05∆, perhaps 0.5 s−1 , for theexperimental values given in Appendix A.
The inhomogeneity in Brot should reverse quite cleanly with the B-chop, and residual imperfections there will be cleaned up with the four-way-chop, and thus the effects of the second-order
gradients in Brot are expected to be predominantly a source of decoherence, rather than systematic
error on measured de .

6.12

Stray Contributions to B|| : Static B-Field Gradients
We now return to discussing static magnetic fields, now including the effects of spatial gradi-

ents. With the characteristic size of the ion cloud r being smaller than the characteristic distance X

132from cloud center to source of magnetic field by a ratio of 0.1 or smaller, it makes sense to expand
the field about the uniform value at the trap center. The most general first-order correction to astatic magnetic field in the absence of local sources can be characterized by five linearly independent
components as follows:yx′~ = BaxgradB(z ẑ − x̂ − ŷ)22′+Btrans(xx̂ − y ŷ)+B1′ (yx̂ + xŷ)+B2′ (z x̂ + xẑ)+B3′ (yẑ + z ŷ)(6.67)

By far the most important effect of these terms is the “micromotion-axial gradient interaction.”
As discussed in Chapter 3.9, the displacement of an ion’s circular micromotion ~rrot is exactly outof phase with the rotation of its quantization axis Ê, see Eq. 3.30 . Averaged over a cycle of ωrot ,this will give rise to a nonzero average contribution to B|| and cause a shift in W u/ℓ given by′′
2 ). A guess for a possible value of stray B ′3gF µB Baxgrad
rrot = 3gF µB Baxgrad
eErot /(M ωrot
axgrad is

2 mG/cm, which for anticipated experimental parameters would lead to a shift in W u/ℓ of order
4 Hz, and this shift would survive the B-chop. As with the effect of displacement currents, one
expects the systematic effect of the shift to be reduced after the four-way chop by (δgF /gF ), but
for maximum accuracy the effect should be shimmed out of the B-field chop, either by adjusting
shim , or by applying (say with anti-Helmholtz coils external to the vacuum chamber)the value of Brot′
a compensating value of Baxgrad.

A smaller effect arises from the interaction of the magnetic field gradient with the componentof the electric fields responsible for providing ion confinement, which after averaging over cycles ofωrot and ωrf , always point inward, giving rise to a net inward-pointing time average of Ê. If we lookat only the component of the first-order magnetic field gradient that points towards or away fromthe trap center

133

′
′
′
′
′
~ central = (BtransB− Baxgrad/2)xx̂ + (−Btrans− Baxgrad/2)y ŷ + Baxgrad
z ẑ(6.68)The net contribution to B|| comes from integrating, along the rf and rotation micromotion
trajectories, over first a rotational cycle, and then an rf cycle, and then a secular cycle in a given
direction. We assume that the trap is sufficiently harmonic that there is no cross-dimensional
mixing of secular energy, and that hard-momentum-changing collisions are rare enough so that,during the duration of a spectroscopic measurement, there is no change in Ei , the sum of the
kinetic and potential energy associated with an individual ion’s secular motion in the ith direction.
The contribution to B|| is then,
′
′
′
′
′
δB|| = ((Btrans/2 − Baxgrad/4)Ex + (−Btrans/2 − Baxgrad/4)Ey + Baxgrad
Ez )/(e Erot ).

(6.69)

The contribution to B|| averaged over a thermal sample of ions is given by the above expression with
Ei replaced by Ti . Note that for Tx = Ty = Tz , several terms cancel and the thermally averaged′
contribution to B|| is just Baxgrad
Ez /(2eErot ). The decohering effect is comparable because within athermal sample, Ex , Ey , and Ez will in general differ from one another for a given ion, and between′′
different ions. For Baxgrad
and Btranseach about 2 mG/cm, ion temperatures about 15 K, the mean

shift in Wu/ℓ for typical experimental parameters given in Appendix A might be 30 mHz, with acomparable contribution to dephasing.
The three remaining terms in the first-order gradient, B1′ , B2′ and B3′ , will contribute to a shift
in B|| only when combined with other (usually small) trap imperfections, for instance the plane ofrotation of Erot being tilted with respect to principal axes of the confining potential. The net effects′.
will be correspondingly smaller than those from Btrans′, coupled
Just as with the second spatial derivative of Brot , the spatial derivative of Baxgrad

to a thermal spread in the size of ion orbits, can give rise to decoherence. Of course, Baxgrad isdefined already as a first spatial derivative of a magnetic field, thus the dephasing arises from athird derivative of the field, and its rate should be down from the mean size of the shift (roughly
estimated above at 4 Hz) by a factor of order (r/X)2 , or a factor of one hundred. Even spatially134′
uniform Baxgrad
could give rise to decoherence if there is a spatial dependence in rrot . The fractionalchange in rrot is the same as the fractional change in Erot . As discussed in Chapter 3.9, this should
be smaller than 0.5% over the typical size of the ion sample.
To sum up Chapter 6.11 and 6.12, we have looked at a range of ways in which various
contributions to B|| can shift Wu and Wℓ . Decoherence due to shot-to-shot fluctuations or spatial
inhomogeneity should not be a problem out to beyond 1 s coherence times. Various effects canshift Wu and Wℓ by as much as a few Hz, and this shift can survive a B-chop. With δgF /gF
on order of 10−4 , and Eeff estimated at 90 GV/cm in ThF+ , after a four-way chop the remaining
systematic error will be a few 10−29 e cm, but this can be dramatically reduced by tuning away
the post-B-chop signal with B shim . The most dangerous systematic error would be if Brot weresystematically different between measurements on the upper and on the lower states. Chopping
between upper and lower states will be determined by variations in optical pumping, which should
be well decoupled from the mechanisms that generate Brot .

6.13

Alternative Application of Brot
It is worth considering that applying a very spatially uniform Brot may be very challenging

because of difficult-to-model eddy currents induced in electrodes and light-gathering mirrors. On′
the other hand a purposely applied static Baxgrad
would be perturbed only by the magnetic perme-ability of trap construction materials, which can be minimized and modeled. One way or another
we will need to bias away from the avoided crossing discussed in Chapter 6.4, but it may turn outthat this can be accomplished with a lower total decoherence rate by omitting the applied Brot′
altogether, and providing the bias with a deliberately applied Baxgrad
field. The B-field chop could′
be accomplished by chopping the sign of Baxgrad. The parity invariance argument of Chapter 6.4′
rather than a chop in Brot .
can readily be modified to describe a chop of Baxgrad

ey
a)

erf

135

ez
erot
b)

ex

ex

Figure 6.5: Over one rotation of Erot , both Erf and Ez are quasistatic. The total electric field is thesum of all three and its trajectory over one cycle of Erot is plotted as the dotted line projected onto(a) the x-y and (b) the x-z planes. The electric field trajectory is a circle of radius Erot , parallelto and displaced from x-y plane, a circle whose center is offset from the z-axis by Erf . In the limit| Erf | ≪ | Erot |, the solid angle subtended from the origin by this circle differs only slighlty fromthat subtended by a circle with vanishing Erf . The magnitudes of both Erf and Ez relative to Erot
are very much exaggerated for clarity.

6.14

Relativistic (Ion-Motion-Induced) Fields
The largest component of the velocity on the ions is that of the micromotion induced by Erot ;for reasonable experimental parameters it will be less than 1000 m/s. In typical lab-frame magnetic
fields of a few mG, the motion will give rise, through relativistic transformation, to electric fields oforder of a few µV/cm, which are irrelevant to our measurement. Conversely, motion at 1000 m/s
in typical lab-frame electric fields of 10 V/cm generates a magnetic field of 0.1 µG. This field will
be rigorously perpendicular to the electric field, the quantization axis, and thus represents only anegligible modification to the generally unimportant B⊥ .

6.15

Effect of RF Fields
The effects of the rf electric fields providing Paul trap confinement are best understood byputting them in the context of a three-tier hierarchy of electric field magnitudes and frequencies.
(i) Erot , the nominally uniform, rotating electric field, with field magnitude of perhaps 10
V/cm and frequency ωrot = 2π×100 kHz.
(ii) Erf , the Paul-trap fields, are highly inhomogeneous, but at a typical displacement in thex-y plane of perhaps 0.5 cm, the field strength might be 75 mV/cm, or two orders of magnitude lessthan that of Erot , oscillating at a frequency, ωrf = 2π×15 kHz which is one order of magnitude less

136
than ωrot . At a fixed point in space, the rf fields average rigorously to zero over time, but averaged
instead along an ion’s rf micromotion trajectory, the rf fields contribute to(iii) the inward-pointing trapping electric field, again very inhomogeneous but with typicalstrength down from peak rf-field values by factor of (ωrf /ωi ), another order of magnitude, to perhaps
5 mV/cm. From the ion’s perspective, the direction of the trapping fields oscillate with the ion’s
secular motions in the trap, at frequencies ωi of perhaps 2π×1 kHz, the slowest time scale by anorder of magnitude.
The effects of the strong, fast Erot have been discussed extensively throughout Chapter 6, andthose of the weak, slow trapping fields were covered in Chapter 6.12. In this subsection we argue
that the rf electric fields, intermediate in both frequency and strength, are the least significant of
the three categories.
The effects of the rf fields averaged over the rf micromotion trajectory are discussed in
Chapter 6.12. The remaining part averages to zero in one rf cycle, but is roughly frozen at a single
value over the duration of one cycle of ωrot . The dominant source of the rf fields’ time-averaged
contribution to transitions Wu/l is in very small corrections to Berry’s phase energy associatedwith the rotation of Erot . See Fig. 6.5. The correction to the solid angle arising from Erf goes as(Ez /Erot )(Erf /Erot )2 . If we include a factor of ωrot to get a Berry’s energy shift and evaluate fortypical experimental parameters, the magnitude of the resulting frequency shift will be about 20
mHz, and will oscillate in sign with the axial secular motion. The magnitude of radial rf fields scales
linearly with the radial secular displacement. If secular freqencies were commensurate, in particularif ωz = 2ωr , then this 20 mHz shift could contribute to a decoherence rate at the negligible levelof a few tens of mHz. For incommensurate ratios of ωz /ωx or ωz /ωy , the rf fields will be still lessimportant.

6.16

Systematic Errors Associated with Trap Asymmetries
The symmetry argument of Chapter 6.4 was based on parity invariance. This argument isonly as good as reflection symmetry of the electric and magnetic fields in the region of the trapped

137
ions. In this section we look, as an example, at the consequences of a symmetry imperfection.
The electrodes used to generate Erot have been numerically designed to make Erot as spatially
uniform as possible, but imperfections in design and construction of the trap and imperfect drive
electronics will lead to some residual field nonuniformity. Suppose that the magnitude of the Erot
was consistently larger in the region of the trap for which z>0, so that the value of Erot over thez>0 half of an axial secular oscillation is about 0.3% larger than that experienced over the z<0
half. Thus the frequency modulation of perhaps ± 500 Hz, discussed in Chapter 6.5 will no longer
average to precisely zero over an axial cycle but instead a net contribution of about 1.5 Hz to Wu .
Such a frequency shift would survive a B-field chop, and, following the protocol discussed in section
IV.F, we could very likely incorrectly identify this shift as arising from the presence of a Bstrayrot , and
apply a value of Bshim to largely null the 1.5 Hz shift. After a complete four-way chop, we would
be left with a systematic error on the order of (δgF /gF )×1.5 Hz, or about 0.4 mHz.
For the value of Eeff estimated for HfF+ , a 0.4 mHz error corresponds to a systematic error
on de of the order of a few 10−29 e cm. For ThF+ , the error on de would be about three times
smaller. We continue a more general discussion on systematic errors in Chapter 8.2.

Chapter 7
Other Sources of Decoherence, Collisions and Black-Body Radiation [1]

7.1

Ion-Ion Collisions
The overarching strategy of the trapped-ion approach to precision spectroscopy is to accept

low count rates in exchange for very long coherence times. In some previous precision measurement
experiments with trapped ions, the very best results have come from taking this to the extreme
limit of working with only one ion [120, 121, 122, 123, 124, 125, 126, 127], or in some cases a pairof ions [128], in the trap at any given time. More often however, optimal precision is achieved
working with a small cloud of trapped ions. In this section various detrimental effects of ion-ion
interactions are evaluated.

7.1.1

Mean-Field

With no electrons present to neutralize overall charge, even a relatively low density cloudof ions can have a significant mean-field potential. A spherically symmetric sample of Nion ionsconfined within a sphere of radius r will give rise to a mean-field potential
Umean−field
≈3 K×
kB



Nion
1000



r
0.5 cm

−1

.

(7.1)

At values of the mean-field interaction energy comparable to or larger than kB T , there is a riskof instabilities, viscous heating, and other undesirable effects; even in their absence, systematic
errors are more difficult to analyze in the strong mean-field limit. Ion-trap experiments have been
performed at much higher mean-field strengths, and indeed there have been precision spectroscopy

(a)

(b)

εrot

139

εrot

θ

εion
Figure 7.1: Geometric phases accumulated during an ion-ion collision. (a) A typical ion-ion collision
trajectory (red), resultant Rutherford scattering angle, θ, and ion-ion interaction electric field, Eion ,are shown in the collision plane (blue). For clarity, the collision plane has been taken perpendicular
to the instantaneous direction of Erot . (b) During an ion-ion collision the molecular axis adiabaticallyfollows the net electric field vector, E~rot + E~ion , and traces out the contour (black) on the unit
sphere (yellow). The solid angle, ∆A(θ), subtended by this contour gives rise to a geometric phaseaccumulated by the eigenstates during the collision. This leads to decoherence of the spectroscopictransition, see text.

experiments done in systems for which the interaction potential even between an individual pairof nearest-neighbor ions is much larger than kB T . However, these systems exhibit a high degree of
spontaneous symmetry breaking including crystallization [129].
Throughout this thesis it is assumed the experiments will be done in the low mean-field limit,
say<
Umean−field ∼

1
kB T.
3

(7.2)

In this limit, mean-field effects are relatively benign, and can be modeled as a modest decrease in
the trap confining frequencies, ωi , plus the addition of some anharmonic terms to the potential.
Crucially for the arguments presented in Chapter 6.4, these additional modifications do not break
any of the reflection- or rotation-based symmetries of the trapping fields. Eqns. 7.1 and 7.2 combine
to set limits on various combinations of the ion number, Nion , ion temperature, T , cloud radii,
ri ∝

r

kB T
, and mean ion density, n ∝ rxNrion
.
M ωi2
y rz

This necessitates making various compromises inselecting operating parameters.
In Chapter 6.5, the axial component of the electric field at the ion’s location, Ez , was shownto tilt the rotating electric field and give rise to an apparent shift of the energy of our spectroscopictransition, linear in Ez . This energy shift integrated over time in turn gives rise to an oscillatory

140
rot
phase shift, ∆φ = ωErotR

Ez dt. In a one-component ion cloud, the effects of long-range, grazing-angle

ion-ion collisions may be thought of as simply a fluctuating component to the local electric field,
and the integrated effect of those fluctuations will make a random contribution to the phase shift.
We present a simple argument to show that the resulting rms spread in phase does not continue
to increase with time but reaches a steady-state asymptote. This is because Ez not only shifts thetransition energy, it also causes an axial force and corresponding acceleration, which, like the shift
Rin transition energy, is linear in Ez . Integrated over time, ∆pz = e Ez dt, this fluctuating force
results in a fluctuating momentum. But we know that the combined effect of a trapping field anda large number of random collisions will not cause the rms momentum to randomly walk withoutp√bound but rather to be loosely bounded by a characteristic thermal value, hp2z i ≈ M kB Tz .
This is the nature of the thermal equilibration process – once an ion has developed a super-thermal

momentum, further collisions are biased to reduce the momentum. Since both the phase excursion
and the momentum excursion are linear in the time-integrated axial electric field, we can estimateroth∆φ2 i ≈ ωErotpp

hp2z i ≈q

kB Tz
2Erot . Again, as discussed in Chapter 6.5, if Erot >∼ 30kB Tz , the phasefluctuations for each ion’s spectroscopic transition will be bounded by a value less than one radian,so that there will be no loss in spectroscopic contrast in a Ramsey-type experiment.
The argument in the paragraph above hinges on the assumption that the electric field arisingfrom the ion cloud’s mean-field distribution and from grazing-angle collisions is small in magnitude
compared to Erot , so that the shift in Berry energy is linear in the axial component of the electricfield. For higher values of the ion temperature or lower values of Erot , a pair of colliding ions canget so close to each other that the electric field is, transiently, comparable to or larger than Erot .
The consequences are discussed in the next subsection.

7.1.2

Geometric Phases Accumulated During an Ion-Ion Collision

As discussed in Chapter 6.4, when a spin adiabatically follows a time-varying quantization
axis it acquires a geometric (Berry’s) phase. For the eigenstates in Fig. 2.1(b), the geometric phasefactor can be written as exp (±imF A), where A is the solid angle subtended by the contour on the141unit sphere traced out by the time-varying quantization axis. Thus, the relative phase generated
between the |F = 3/2, mF = ±3/2i states used for spectroscopy is φ = 3A. The concern of thissubsection is how ion-ion collisions cause uncontrolled excursions of the quantization axis leading
to random geometric phase shifts and decoherence between spin states. These uncontrolled phaseshifts will be written as ∆φ = 3∆A to distinguish them from the calibrated geometric phases inthe experiment.
The instantaneous quantization axis for the molecular ion eigensates is defined by the net
electric field vector at the location of the ion. During a collision, this axis is defined by the vector
sum of the rotating electric field, E~rot , and the ion-ion interaction electric field, E~ion . Both of these
are time-varying vectors, however typical ion-ion collisions have a duration short compared to therotation period of Erot so for the purpose of this discussion Erot will be taken as stationary. Thus, theproblem is reduced to calculating the excursion of the quantization axis under the time variationof Eion . A typical ion-ion collision is shown in Fig. 7.1(a) and the effect of this collision on thequantization axis is shown in Fig. 7.1(b).
At the temperatures of our trapped ion samples, no two ions are ever close enough for thedetails of the intermolecular potential to matter. Only monopole-monopole and monopole-dipole
interactions matter. Further, the translational degree of freedom may be treated as purely classicalmotion in a 1/r ion-ion potential, with the initial condition of a given collisional event characterized
by an impact parameter and relative velocity. The outcome of the collision depends not only on themagnitudes of the impact parameter and of the velocity, but also on their angles with respect tothe ambient electric bias field, Erot . Each initial condition contributes a particular amount to thevariance in the phase between the relevant internal states. These contributions can be converted
to partial contributions to a decoherence rate, and a numerical integral over a thermal distributionof collisional initial conditions can yield the total decoherence rate. We’ve pursued this program
to a greater or lesser extent with the decoherence mechanisms discussed in this subsection andthe one immediately following, but the results are not especially illuminating and we’ve used them
primarily to confirm that the power-law expressions discussed below represent only overestimates142of the decoherence rate, and that for experimental parameters of interest, the decoherence rate will
be conservatively less than 1 s−1 .
The main question is whether T is high enough to include significant phase space for collision
trajectories for which the peak value of Eion > Erot (which is to say, large enough to transiently
tip the direction of the total field by more than a radian). If so, then a single collision can cause
decoherence and one can get a simple estimate of the cross-section for decoherence simply fromthe size of the impact parameter that leads to those events. There is a significant probability forcollisions with Eion >∼ Erot whenT ≫ 18 KErot
5 V/cm

1/2

,

(7.3)

which leads to a decoherence rate
τ −1 ≈ 0.47 ×n1000 cm−3



T
15 K

1/2 

Erot
5 V/cm

−1

s−1 .

(7.4)

If T is instead so low that the Coulomb barrier suppresses collisions that could lead to a
sufficiently large value of A and cause decoherenc with a single collision, then decoherence will arise
only from the combined effects of many collisions each causing small phase shifts that eventuallyrandom walk the science transition into decoherence. In this regime, the decoherence rate falls offvery fast at low temperatures. For
T ≪ 18 KErot
5 V/cm

1/2

,

(7.5)

typical collisions have Eion <∼ Erot and the decoherence rate isτ −1 ≈ 0.13 ×n1000 cm−3



T
15 K

13/2 Erot
5 V/cm

−4

s−1 .

(7.6)

Both Eqns. 7.4 and 7.6 represent conservative estimates of the decoherence rate, and for anintermediate range of temperature, the decoherence rate will be less than whichever estimate gives
the smaller value (Fig. 7.2).

143

(a)
102

τ-1 [s-1]

100
10-2
10-4

ε rot

=

1

cm
V/

ε rot

=

10

cm
V/

ε rot

=

0
10

cm
V/

10-6
10-8
1

5

2

20

10
T [K]

50

100

(b)
102

τ-1 [s-1]

100
10-2
10

T
T

-4

=

T
2K

=

10

K

10

20

=

50

K

10-6
10-8
0.2

0.5

1

2

5

εrot [V/cm]

50 100 200

Figure 7.2: Inverse coherence times, τ −1 , due to geometric phases accumulated during ion-ion
collisions as a function of (a) collision energy in temperature units and (b) Erot . Dotted lines
are approximations given in Eqns. 7.4 and 7.6. Solid lines are more involved estimates based on
integrals over collision parameters, but are still based on approximations so as to be conservative.
The ion density was taken to be n = 1000 cm−3 .

144
7.1.3

m-Level Changing Collisions

A second source of decoherence can arise from ion-ion collisions that induce transitions between internal levels of a molecule. The dominant inelastic channel will be transitions between mF
levels induced by a sufficiently sudden tilt in the quantization axis defined by the instantaneous
local electric field. There are two conditions for such a transition to occur: (i) the direction of thetotal field must change by nearly a radian or more, so that there is significant amplitude for, e.g.,
an mF = +3/2 level in the unperturbed electric field to suddenly have non-negligible projection on
an mF = +1/2 level in the collision-perturbed field, and (ii) the time rate of change of the electricfield direction must be comparable to or larger than the energy splitting between an mF = 3/2
level and its nearest mF = 1/2 level in the field Erot .
Note that the first requirement is the same as the requirement for picking up an appreciable
single collision Berry’s phase. However, not all collisions that satisfy the first requirement will
satisfy the second requirement. In particular, if the relative velocity in a collision is too low,then the time rate of change of the electric field direction will be too slow to satisfy the second
requirement. Thus, given that the first requirement is satisfied, then the second requirement will
not be satisfied when
T <5 K×Erot,
5 V/cm

(7.7)

In this limit, the second requirement is more stringent than the first requirement, which means thatthe rate of m-level changing collisions will be smaller than the rate of single-collision Berry’s phaseinduced decoherence. In the opposite limit, we expect the second requirement will be met whenever
the first requirement is met, and thus we would expect that the two channels of decoherence, m-levelchanging and single-collision Berry’s phase, will be comparable in magnitude.
Looking at particular collision trajectories in more detail, we see that there are trajectories
that can cause an m-level change but for which there is no contribution to Berry’s phase because
the electric field traces out a trajectory with no solid angle (for instance, if the classical impact
parameter ~b is parallel to E~rot ). We also note that our formulation of the requirement of sweep

145
rate for m-level changing collisions neglects the fact that Eion will not only change the direction of
the total electric field (E~ion + E~rot ) but also in general will change its magnitude. For most impact
parameters, the magnitude of the total electric field will increase, thus suppressing nonadiabatic
effects. However, a narrow range of impact parameters exists where the magnitude of the total electric field decreases, thus enhancing nonadiabatic effects. However, the above scaling laws accountfor the vast majority of collisions.
In the end, we are less interested in the actual rates than we are in putting conservative limits
on decoherence rates. For instance, in calculating the curves in Fig. 7.2, we pessimistically took
a worst-case geometry, Erot ⊥ Eion , which gives an upper limit on the size of the effect. Thus weestimate that:rot
the total collisional decoherence, including both m-level-changing• For T < 5 K × 5 EV/cm

and Berry’s-inducing effects, will be less than or equal to the value given by solid curves inFig. 7.2, whilerot• For T > 5 K × 5 EV/cm, the total collisional decoherence will be no greater than twice aslarge as the rate indicated by those curves.

7.2

Ion-Neutral Collisions
Ion-neutral collisions are troubling in our experiment due to their potential to change theinternal states of our ions as well as their potential to increase our ion temperatures. The electricfield of the ion will polarize a neutral atom or molecule giving rise to the attractive helium-ion
interaction potential,
1 2
1e
Vint (r) = − αEion(r) = − α
2
2
4πǫ0


2

1
,
r4

(7.8)

where α is the DC electric polarizability of the neutral particle and Eion is the electric field of
the ion. An effective scattering potential can be defined as the sum of the attractive helium-ion
interaction potential and the repulsive “angular momentum barrier” associated with the collision146

Veff(r)
Vmax
b>0
r

b=0

Figure 7.3: The effective helium-ion scattering potential for b = 0 and b ¿ 0.

(Fig. 7.3),
Veff (r) = Vint (r) +

L2
e
1
=− α
2
2mr
2
4πǫ0


2

b2
1
+
U
kin 2 .
r4
r

(7.9)

with m the reduced mass of the coliding particles, Ukin the kinetic energy of the collision, and L
the angular momentum about the center of mass of the colliding particles. The effective neutralU 2 b4

kin
ion scattering potential has a maximum value Vmax (b) = 21 α(e/4πǫ
2 . If the kinetic energy of the0)collision exceeds Vmax the particles will approach r = 0 until a sharp repulsive potential, due toFermi repulsion of the electron clouds, is reached. The cross section for these close-range collisions
is referred to as the Langevin capture cross section,
σLC = πb20 = π

e
4πǫ0

s

2α
.
Ukin

(7.10)√
The collision rate, ΓLC = nσLC vrel , will be independent of temperature since σLC ∝ 1/ Ukin while√vrel ∝ Ukin .
◦ 3

For argon-ion collisions (using αargon = 1.64 A ) we obtain,
σLC vrel ≈ 4.6 × 10−10 cm3 /s.

(7.11)

Noble gases have increasing values of α as you move to heavier atoms, with the effects of higher
mass and higher α mostly cancelling in the collision rate, therefore our choice of carrier gas is notstrongly influenced by this issue. The 1% of SF6 mixed into our expansion gas will have a similarvalue of σLC vrel , making it unimportant in this analysis. The cross section for rotation changing

147
collisions will likely be approximately equal to σLC [130], giving a limit on our background gas
pressure of ∼ 5×10−8 Torr for a 1 s−1 decoherence rate.
Our ultimate limit on background pressure will likely come from another ion-neutral collision
issue. The polarized helium atom produces a dipole electric field at the molecular ion given by
E~He (r) = EHe r̂ =

2αe 1
1 2α
Eion r̂ =r̂.
34πǫ0 r(4πǫ0 )2 r5(7.12)

Large values of E~He will lead to Landau-Zener non-adiabatic transitions of our magnetic sub-levels,as the quantization axes is defined by the electric field. A naive estimate involves finding theion-neutral separation such that E~He = Erot and assuming this is the critical impact parameter for
non-adiabatic transitions. This gives,
σLZ ≈ π2αe
12(4πǫ0 ) Erot

2/5

.

(7.13)

Increasing Erot would seem to mitigate this issue. However, the rotational micro-motion of ourions will begin to dominate vrel as Erot is increased. For typical values found in the appendix,
the rotational micromotion will have a velocity of ≈ 350 m/s, while the thermal velocity of argon
atoms is ≈ 250 m/s at 300 K. The non-adiabatic collision cross section using an argon buffer gas< 1 × 10−9 Torr. This is approachingand Erot = 5 V/cm is ≈ 120 nm2 , limiting our pressure to ∼the limit of our turbomolecular pumps and requires significant differential pumping between thesource chamber (which will have pressure spikes in the 10−4 Torr range when the pulse valve fires)and the ion trapping chamber. The 1/e pumping rate of our vacuum system is on order 10 ms,
and some time could be allowed between ion creation and state preparation in order to reduce thedifferential pumping requirements. If the ions are created in the 1 Σ state and transferred to the3∆

1 state, more wait time could be afforded than if the ions are created directly in the

7.3

3∆

1 level.

Rotational and Vibrational Excitations from Black-Body Radiation
Our spin coherence time can also be limited by excitations to higher rotational and vibrational

states from black-body radiation. The number of photons per mode, n̄, in a black-body radiation

148
field is:
n̄ =

1
e

hν
kT

−1

(7.14)

which, in the limit of hν >> kT reduces to kThν . The ratio of stimulated to spontaneous emission
for a transition is simply n̄. The rotational constant for a typical fluoride containing candidate
molecule in this experiment is ∼ 0.3 cm−1 , which gives an ∼ 1.2 cm−1 splitting between the J =1 and J = 2 rotational levels. This means that 300 K black-body radiation will excite the J =1 state to the J = 2 state at a rate ≈ 174 times faster than the spontaneous decay rate from J= 2 to J = 1. Fortunately, radiative decay rates for rotational transitions tend to be quite slow.
To estimate this, we take the transition dipole moment,µ21 , to equal the molecular frame dipole
moment of ≈ 1.2 atomic units [64]. The spontaneous decay time is then given byτspon =

3ǫo hc3
3 µ2 ,
2ω21
21(7.15)

with ω23 being the transition frequency. This gives a spontaneous lifetime of ≈ 1.2×105 s, andblack-body radiation at 300 K should take ≈ 670 s to excite a 3 ∆1 molecule from the J = 1 state
to J = 2.
Given a transition dipole moment, it is interesting to ask at what resonance frequency will
black-body radiation have the highest excitation rate. Fig. 7.4 shows a plot of black-body excitation
rate versus the dimensionless quantity hν/kT . We see that the peak rate lies at ≈ 2.82. At 300 K
this corresponds to 590 cm−1 , which happens to lie near the vibrational transitions of our molecules.
For HfF+ the vibrational constant is ≈ 750 cm−1 , with an n̄ ≈ 0.028 for a 300 K black-body. The
vibrational decay rate for v = 2 to v = 1 in the 3 ∆1 state is estimated [64] to be 184 ms. Black-body< 6 s.
radiation will therefore limit our coherence time to ∼

149

1.4

G (arb. units)

1.2
1.0
0.8
0.6
0.4
0.2

2

4

6

8

10

hn/kBT
Figure 7.4: Relative black-body excitation rates assuming a constant transition dipole moment. The peak lies near our vibrational energy spacing for a
versus the dimensionless parameter khν
BT
300 K black body.

Chapter 8
Conclusions and Sensitivity Estimate [1]8.1

Precision
Recall from Chapter 1.4 the three components to the sensitivity figure-of-merit: coherencetime, Eeff , and count rate. We estimate these three quantities below.
8.1.1

Coherence Time

Conclusions of Chapters 6 and 7: Taking into account only collisional decoherence, and all
the questions associated with being in rotating fields and in trapping fields, we would anticipate
a coherence time longer than one second. Black-body thermal excitation of the J = 1 rotational
level will also be well over one second. Vibrational black-body excitation for the v = 0 state isestimated at 6 s for HfF+ in a 300 K environment. Thus the dominant limitation to coherence will
likely be the radiative lifetime of the 3 ∆1 state, estimated [64] at 390 ms for HfF+ , and still longer
for ThF+ , for which the 3 ∆1 state is predicted to be still lower in energy. Lifetime estimates of the3∆

1 level are made difficult by the small transition energy, calculated to be 1600 cm−1 in HfF+

with large relative error, and the uncertain transition dipole moment, due to a large uncertainty indetermining the mixture of 1 Π character into the nominally 3 ∆1 state.

8.1.2

Eeff

Eeff in HfF+ is calculated by Meyer and coworkers to be 30 GV/cm [52], and by Titov etal. to be 24 GV/cm [54]. For ThF+ , Meyer calculates 90 GV/cm [52]. The uncertainties in these151
numbers are hard to assess, but they are very likely accurate to better than a factor of two and,
if ongoing spectroscopic studies provide experimental values of hyperfine and fine structure that
confirm the ab initio values predicted by the St. Petersburg group, our confidence in the precisionof calculated Eeff may be much higher.
8.1.3

Count Rate and Summary of Expected Precision

We are producing HfF+ ions by photoionization in a relatively narrow range of quantum
states, and can estimate yield per quantum level within the desired trapping volume at perhaps
100 ions per shot, but we have just begun to characterize the efficiency of the process and very little
optimization has been done. Our efficiency for reading out spin states of trapped ions via laserinduced fluorescence (LIF) is at best 4%, and has not yet been tested. With a large uncertainty,
then, we may detect about one ion per shot with four shots per second. Even if we detect as manyas four ions in a shot, single shot precision will be no better than 300 mHz, but overall, precisionin one hour could be about 10 mHz. For ten hours of data, we anticipate (very roughly) a raw
precision at 5 × 10−28 e cm in HfF+ , and 1.5 × 10−28 e cm in ThF+ . We are investigating several
alternatives to LIF for spin readout as discussed in Chapter 4.

8.2

Accuracy, Systematic Error
We have not completed a systematic study of the consequences of all possible violations ofreflection symmetry in the trapping fields, but work in this direction is ongoing.
For now, we make the following three observations:i) For the field asymmetries we have analyzed to date, realistic estimates for the magnitude
in as-constructed field imperfections lead to systematic errors on the order of a few 10−29 e cm
or less. While this is not yet as accurate as our ultimate ambitions, it would represent roughly afactor of thirty improvement on the existing best experimental limit.
ii) Asymmetries analyzed to date lead to systematic errors whose signs reverse when the
direction of rotation ωrot reverses. If we combine measurements made with clockwise and coun-152
terclockwise field rotation, the errors vanish. Ideally, we’d like to design sufficient accuracy into
the experiment so that the chop in field rotation is not needed to achieve desired accuracy, but
as a practical matter we will of course run the experiment both ways, averaging the results to getultimate accuracy, and differencing them to diagnose experimental flaws.
iii) Auxiliary measurements are envisioned to characterize and shim out flaws in the asconstructed trap. For instance, we plan to be able to shim the equilibrium position of the ion
cloud up and down along the trap axis, and at each location measure the energy difference Eb -Ed .
Unlike Ea -Eb , Eb -Ed is highly electric-field sensitive. The result will be a precise measurement of
any spatial gradient in Erot .
iv) All systematic errors we have analyzed to date have strong dependencies on quantities
such as ωrot , Brot , Erot , and on the ion-cloud temperature and trap confining frequencies. A truesignal from a nonzero value of de will be largely independent of all those quantities. We anticipate
making a number of auxiliary measurements with the experimental parameters tuned far away fromtheir optimal values to deliberately exaggerate the size of systematic errors and allow us thus tocharacterize their dependencies in less integration time than that required for ultimate sensitivity.
Even so, and as is often the case in precision measurement experiments, sensitivity and accuracy arecoupled. To the extent we can measure de to high precision at many combinations of experimental
parameters, we will better be able to detect and reject false signals.
As an overarching conclusion, the experiment as described here should have the capability to
improve the limit on the electron’s electric dipole moment to 10−29 e cm.

8.3

The Future
Experimentally there are two major ongoing efforts. Sinclair et al. are working on measuring

excited electronic states of HfF+ and ThF+ using novel femto-second comb, cavity enhanced spectroscopy. On the ion trapping side of things, Huanquian is leading the efforts to state-selectively
load HfF+ ions by photoionization and recent progress has been made in measuring the rotational
populations of ions produced via photoionization. The construction of the next generation ion trap

153
is also ongoing. Obviously, I have graciously left much work for my labmates and future generationsof students to perform before a measurement of the electron EDM can be made. As Eric mightsay, it could take weeks of effort after I’m gone to publish a new limit (or value!) on the eEDM.
To the younger generations, I say good luck!
Our scientific goal has always been clear, measure the eEDM, while the means to that goalhas been the focus of my graduate school efforts. We have been building a highly specialized hammer for a very particular nail. Of course, it has been said that once you have a hammer, every
problem looks like a nail. It is therefore tempting to think of possible nails lying around thatmight suit our particular hammer once we have completed the eEDM work. There has been a
proposal [131] that molecules such as HfF+ could be useful in measuring time variations of fundamental constants. Here near degenerate levels whose energies have very different dependencies on α
and me /mp might be found using excited vibrational levels of the 1 Σ and 3 ∆1 levels. The potential
for long interrogation times of our trapped ions could yield sensitive measurements. Molecularions sympathetically cooled by laser cooled atomic ions have many potential uses as discussed
in [132], including studies of cold ion-neutral chemistry. There have also been proposals [133, 134]
to use molecules for quantum information processing with the quantum information encoded in
the rovibrational eigenstates of the molecule. Such a scheme with trapped molecular ions could
have practical advantages over alternative molecular quantum information proposals that involve
trapping ultracold neutral molecules and using dipole-dipole interactions for entanglement [135].
Trapped molecular ions, in combination with a superconducting microwave resonator, could also
be used in quantum information systems [136].

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Appendix A
Typical Experimental Parameter Valuesde Eeff = 2π × 0.36 mHz, transition energy between mF = +3/2 and mF = −3/2 states in ThF+ if
de = 1.7 x 10−29 e cm.
dmf = +1.50 a.u. ≈ 2π × 2 MHz/(V/cm), electric dipole moment of HfF+ in the molecular rest
frame.
Erot = 5 V/cm, rotating electric field.
ωrot = 2π × 100 kHz, frequency of rotating electric field.
Erot = 1800 K, typical kinetic energy in rotational micromotion.
rrot = 0.6 mm, radius of circular micromotion.
dmf Erot = 2π × 10 MHz.
(3/2)γF =3/2 dmf Erot = 2π ×5 MHz, Stark shift of mF = ±3/2 states of 3 ∆1 levels in rotating electricfield.
ωef = 2π × 10 kHz, Λ-doublet splitting between opposite parity 3 ∆1 J=1 states.
gF =3/2 = 0.03, magnetic g-factor in 3 ∆1 mF = ±3/2 states.
Brot = 70 µG, rotating magnetic field.
3gF µB Brot = 2π × 8 Hz, Zeeman splitting between mF = +3/2 and mF = −3/2 states due to Brot .
δgF =3/2 /gF =3/2 ≈ 3 × 10−4 , fractional difference of magnetic g-factor for upper and lower levels,.
∆ ≈ 2π × 2 Hz, splitting at the avoided crossing between mF = +3/2 and mF = -3/2 levels.
B⊥ = 25 mG, anticipated scale of transverse magnetic field.
r = 0.5 cm, characteristic rms radius of trapped ion cloud.

161
T = 15 K, characteristic temperature of trapped cloud.
ωi = 2π × 1 kHz, typical trap confining frequency.
Ez = 5 mV/cm, typical axial electric field applied for confinement.
Erf = 75 mV/cm, typical Paul trap electric field strength, at typical cloud radius.
< Erf > = 5 mV/cm, typical radial confining electric field, averaged over one Paul cycle.
ωrf = 2π × 15 kHz, typical “rf freq” for Paul trap.
Erf = 15 K, typical kinetic energy in Paul micromotion.
Ehf = 2π × 45 MHz, hyperfine splitting between F = 1/2 and F = 3/2 states of 3 ∆1 J=1 level.