An apparatus for measuring the electron’s electric dipole
moment in trapped ThF+
byNoah Schlossberger
B.S., Indiana University Bloomington, 2018
M.S., University of Colorado Boulder, 2021

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
2023

Committee Members:
Eric A. Cornell, Chair
Jun YeJohn Bohn
Adam Kaufman
Joshua Combes

ii
Schlossberger, Noah (Ph.D., Physics)
An apparatus for measuring the electron’s electric dipole moment in trapped ThF+
Thesis directed by Prof. Eric A. Cornell

Unresolved questions in the universe, such as the matter-antimatter asymmetry, challenge
the limitations of the Standard Model. As searches for beyond-the-Standard-Model physics faceescalating energy scale requirements in particle accelerator experiments, low energy investigations
targeting symmetry-breaking properties, like the electron’s electric dipole moment (eEDM), havebecome a key tool in probing these theories. Recently, our group at JILA set a world-record limiton the eEDM, providing a new constraint on the masses of supersymmetric particles.
In order to achieve any more sensitivity in such a measurement, the next iteration of theexperiment must grow dramatically in complexity. The new generation of JILA eEDM experimentis different in three main ways. First, a new molecular ion species, ThF+ , gives inherently longer
coherence times because the experiment can be performed in the ground electronic, vibrational,
and rotational state of the molecule. In order to gain access to this long coherence time, wemust introduce cryogenics to the ion trap so as to reduce ambient black-body radiation that can
incoherently excite the relevant quantum states. Finally, in order to access long coherence times
without sacrificing count rate, the experiment is designed to eventually be multiplexed.
The resulting experiment will take of order one decade from conception to result. This
thesis will cover the design and construction of the third generation eEDM experiment, as well as
demonstrate preliminary measurements of the eEDM-sensitive states of ThF+ .

iii

Acknowledgements

I would like to express my sincere gratitude to Prof. Eric Cornell for his invaluable guidance
and patience throughout my journey. His ability to intuitively frame complex problems has played
a vital role in shaping me into the researcher I am today. I am also deeply thankful to Prof. Jun
Ye, whose technical expertise has consistently pulled me out of challenging situations.
I am indebted to my mentor, Kia Boon Ng, for his invaluable teachings and guidance. Additionally, I extend my thanks to my labmates Sun Yool Park, Anzhou Wang, Yuval Shagam, Tanya
Roussy, Trevor Wright, and Luke Caldwell, who have made the lab a friendly and enjoyable space.
I cannot give enough gratitude to the technical staff at JILA, particularly J. R. Raith, Hans
Green, and James Urich, who have not only taught me invaluable skills have also offered rewarding
friendships.
Furthermore, I am grateful to my close friends who have provided unwavering emotional support throughout this journey, including Sinéad Ryan, Drew Morrill, Anya Grafov, Hope Whitelock,and my loving family, without whom this accomplishment would not have been possible.

Contents

Chapter
1 Introduction

1

1.1

Symmetry and the electron electric dipole moment . . . . . . . . . . . . . . . . . . .

1

1.2

Thesis layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 Experiment Overview
2.1

2.2

6

Measurement concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.1

Molecular structure and the eEDM . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1.2

Laboratory fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.3

Measuring energy levels in a molecule . . . . . . . . . . . . . . . . . . . . . . 15

2.1.4

Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 State preparation

24

3.1

Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2

Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1

Continuous wave laser overview . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.2

Rb reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.3

Microwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.4

Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

v
4 Electric Fields
4.1

4.2

50

Electrode design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.2

Ponderomotive potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.3

Paul trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.4

Rotating polarizing field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.1

Amplitude and timing requirements . . . . . . . . . . . . . . . . . . . . . . . 61

4.2.2

DDS boards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2.3

Resonant transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.4

Transformer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2.5

Coupling between rods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.6

Control matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Magnetic Fields

88

5.1

Effect of magnetic fields in the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 88

5.2

f0 from anti-Helmholtz coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3

Characterization of homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3.1

Multipole expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.3.2

Sources of decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.3.3

Magnetic field measurement scheme . . . . . . . . . . . . . . . . . . . . . . . 96

5.4

Background fields from the optical table . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.5

Field-distorting effects of the optical table . . . . . . . . . . . . . . . . . . . . . . . . 99

5.6

Homogeneity of anti-Helmholtz fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.7

Shim coils for B4,0 and B4,2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6 Detection
6.1

106

Dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

vi

6.2

6.3

6.1.1

Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.1.2

Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.1.3

Hyperfine selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Ion imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2.1

Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2.2

Mesh gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.2.3

Phosphor screen gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2.4

Capacitive readout interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.2.5

Ion counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Simultaneous doublet readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7 Preliminary Results and Conclusion

130

7.1

Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.2

Ramsey oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.3

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

135

Bibliography

Appendix
A Biot-Savart Integrals
A.1 Square current loop

140
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

A.2 Circular current loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
B A modified chain rule for vector fields

143Chapter 1
Introduction

1.1

Symmetry and the electron electric dipole moment
The prevalence of matter over antimatter is one of the most important outstanding problems

in physics. Before the big bang, the universe had no matter, meaning the universe started with asymmetric amount of matter and antimatter. The asymmetry must have arisen due to the matterforming interactions in the first few moments of the universe The current most accepted description
for the formation of this asymmetry is called Baryogenesis, and it requires three elements1 [1]:• interactions that violate baryon number symmetry, meaning they can create more matterthan antimatter or vice-versa• interactions that violate charge-parity (CP) symmetry, meaning that they don’t work thesame way if you swap all charge in the universe with negative charge and flip all spatial
coordinates (⃗r → −⃗r)• the universe to be out of thermal equilibrium, meaning that energy can be unequallydistributed in the universe.
To see why we both baryon number symmetry violating and CP symmetry violating interactions
to end up with more matter than antimatter, first consider why we need both baryon number
symmetry violation and charge (C) violation. Charge symmetry implies that matter and antimattershould behave the same, meaning that for every baryon number symmetry violating interaction that
1

These requirements are known in the field of cosmology as the “Sakharov conditions.”

2
produces more matter than antimatter there is an equivalent interaction that produces equally more
antimatter than matter, so the total amount of each would still be symmetric if C symmetry ispreserved. By extension, CP symmetry would also not allow for Baryogenesis, since it would imply
equal amounts of matter of one parity and antimatter of the other. Since it would do so for bothparities, this would still overall lead to equal amounts of matter and antimatter.
The existence of the amount of CP violation that would allow for Baryogenesis is inconsistent
with our understanding of the universe. The current best understanding of the structure of matter,known as the Standard Model of particle physics, does not contain enough CP symmetry violation
to give rise to Baryogenesis [2], meaning that the Standard Model cannot adequately explain our
existence. Several extensions to the Standard Model have been proposed to fix this [3, 4]. However,these models require experimental verification. This can be done by measuring the CP-violating
properties of the universe and comparing them to the predicted values from these theories. One
such property is an electric dipole moment (EDM) of any fundamental particle with spin.
To understand why an EDM of a fundamental particle with spin is fundamentally CP violating, consider the relative orientations of the particle’s electric dipole moment and magnetic dipole
moment (arising from the particle’s spin). A diagram of the orientation of these properties undercharge and parity inversion are shown in Figure 1.1.

μ

μ

μ

Figure 1.1: Transformations of a particle’s magnetic dipole moment µ⃗ and electric dipole moment⃗d under a charge (C) and parity (P) symmetry transformation. The rotation arrow indicates the
direction of classical rotation of the negative charge that would give rise to the particle’s magnetic
moment.

In order to visualize how spin changes under various inversions, it is helpful to think of the

3particle’s spin as arising from a charge distribution rotating classically. Under charge inversion, theelectric dipole moment flips, and the sign of the classically rotating charge distribution flips meaning
the magnetic moment flips as well. Next, under parity inversion, the electric dipole moment flips,
but the spin does not (rotation direction does not flip under ⃗r → −⃗r). This means that under CP
inversion, the relative orientation of the electric dipole moment and the magnetic dipole momentof the particle is reversed. Therefore a particle having both an electric dipole moment and a spinis inherently CP violating.
Efforts to constrain particle physics include measuring the electric dipole moment of themuon, neutron, and proton. So far the measurement of these quantities is consistent with zero, so
the results are expressed as a bound on the magnitude of the dipole moments. A history of thesemeasurements is shown in Figure 1.2.
Any dipole moment measured above the Standard Model prediction would be a clear signatureof beyond-the-Standard-Model physics. Electron EDM measurements are particularly exciting
because they have been able to reach much greater sensitivity than neutron EDM measurements,
and the gap between the current experimental bounds and the Standard Model background isshrinking.

4

nEDM value (e cm)

BSM
predictions

Experimental bounds
Standard model prediction

10 14
10 17
10 20
10 23

a)

10 26

b)

10 29
10 32

1960

1970

1980

year

2000

2010

2020

BSM
predictions

Experimental bounds
Standard model prediction

10 13

eEDM value (e cm)

1990

10 17
10 21
10 25

a)

10 29

atoms

10 33
10 37

1960

1970

molecules
1980

1990

year

2000

2010

2020

Figure 1.2: Historical and theoretical context for EDM searches.
Top: Experimental upper bounds [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] on the neutron’s
electric dipole moment (nEDM) compared to the Standard Model prediction [20]. Experimental
upper bounds represent the 90 percent confidence interval for the magnitude of the EDM.
Bottom: Experimental upper bounds [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]
on the electron’s electric dipole moment (eEDM) compared to the Standard Model prediction [37].
Highlighted in red is our 2023 JILA result [38]. Vertical gray dashed lines indicate the paradigm
shifts of the best bounds being placed by eEDM searches in heavy atoms in 1967 and in polar
molecules in 2011.

b)

5

1.2

Thesis layout
This thesis was written at a similar time to that of Kia Boon Ng [39], another graduate student

working on the third generation of JILA eEDM measurements. This thesis delves more into thetechnological efforts towards the eEDM measurement while Ng’s focuses more on the physics of theexperiment. The two function well as companion references to a complete understanding of theexperiment.
The thesis is organized as follows: first Chapter 2 will describe in overview the details of
measuring the eEDM in ThF+ , providing both an abstract explanation of the measurement concept
and a complete view of the actual experiment. Subsequent chapters will describe the technicaldetails of the instrumentation developed to do this, highlighting the technical requirements, thesolution, and any data taken. Specifically, Chapter 3 will cover preparation of the ions from their
atomic sources into the relevant quantum states, Chapter 4 will cover the electric fields needed totrap and polarize the ions, and Chapter 5 the magnetic fields needed to generate the base frequency
f0 and possible sources of decoherence. Next, Chapter 6 will cover the way that the populationin the states of interest is read out, and finally Chapter 7 will present preliminary data and makeconcluding remarks.

Chapter 2
Experiment Overview

2.1

Measurement concept
An electric dipole moment has an energy associated with it when placed in an external electricfield. An intuitive explanation of this is shown in Figure 2.1.

+
-

a)

-

+
-

+

+

-

}

}

}

b)
Figure 2.1: Intuitive picture of the DC Stark effect.
a) An ideal dipole consisting of two opposite charges of charge magnitude q separated by an in⃗ = qr
finitesimal distance r, resulting in a dipole moment |d|b) The ideal dipole in an external electric field. The electric potential difference between two points
a distance r away is given by ∆V = dV
dr r = −Er. We can then calculate
P the energy of each configuration as the sum of the Coulomb potentials of each charge U = i V (⃗ri )qi . The energies areshown for several orientations, and it follows that the energy is exactly the projection of the dipole⃗
moment onto the external electric field U = −d⃗ · E.

7This energy, known as the DC Stark shift, allows us to measure the dipole moment of theelectron de . The diﬀiculty comes from the extremely small size of d⃗e . Even if one were to place
10 kV across electrodes 1 mm apart, the resulting field would produce a Stark shift of less than 1→

nHz1 (using the current bound on the magnitude of de [38]).
The solution is to measure the DC Stark shift in the valence electrons of a heavy polar
molecule, as in Figure 2.2

F

Th
e-

Figure 2.2: A valence electron in a ThF+ molecule. The internal electric field arising from the polar
nature of the molecule as well as a classical trajectory of the the eEDM-sensitive valence electron
is illustrated. While this classical trajectory is not particularly physical, it demonstrates the key
feature of an eccentric orbit, meaning that the electron experiences vastly different “velocities” atdifferent spatial locations in the molecule.

The electric fields inside the polar molecule are significantly larger than any electric field that
one can create in a lab. However, the time-averaged electric field experienced by the electron ⟨E⟩t
⃗ t canmust be zero for the electron to be bound. Even so, the time averaged Stark shift ⟨−d⃗e · E⟩
be nonzero due to the eccentric orbit of the electron.2

As the electron zips by the heavier atomin the polar molecule, the length contraction of the electron in the laboratory frame means thatd⃗e shrinks in magnitude. This gives rise to a nonzero time average Stark shift in the laboratory⃗ effframe, with the complexity of this effect absorbed into an “effective” electric field E⃗ t ≡ d⃗e · E⃗ eff .
⟨d⃗e · E⟩(2.1)

Table 2.1 lists the relativistic effective electric field experienced by a valence electron each polar
1

Here the energy is given as a frequency via implicit division by Planck’s constant h. Frequencies and energies
will be used interchangeably throughout this thesis.
2
Of the two valence electrons in ThF+ , only the σ electron exhibits this property.

8molecule in the world’s leading eEDM measurements3 .
Molecule
ThO
HfF+
ThF+

Eeff (GV/cm)
78±2 [40, 41]
21±2 [43, 44, 45]
36±1 [47, 48]

Experiment
ACME II [42]JILA Gen. I [46] and II [38]JILA Gen. III (this work)

Table 2.1: Effective electric fields of the polar molecules used in leading eEDM experiments. The
error bars presented are the standard deviation of the theory predictions over the square root of
the number of predictions.

These effective electric fields in the tens of GV/cm are much greater than one could create
with electrodes, and bring the frequency of the Stark shift at the current eEDM bounds up to theµHz scale, which can be measured in a Ramsey-type experiment.

2.1.1

Molecular structure and the eEDM

The eEDM-sensitive Stark shift described in the previous section manifests as a shift in thequantized energy levels of the polar molecule which can be measured. First, we must understandthe structure of the molecule to see where the shift shows up.
The quantum state of a molecule is described by quantization of all possible degrees of
freedom, including the configuration of electrons within the molecule, inter-atomic vibration of themolecule, and rotation of the molecule.
The electron configuration is described by the electronic state, denoted by the molecular term
symbol
2S+1

ΛΩ

(2.2)

where S is the total spin of the unpaired electrons, Λ is the projection of the orbital angularmomentum onto the internuclear axis (expressed as Greek letters where Σ = 0, Π = 1, ∆ = 2, ...),and Ω is the projection of the total angular momentum onto the internuclear axis. The groundelectronic state of ThF+ is 3 ∆1 , meaning the two valence electrons give a spin of S = 1, with orbital
3

While ThO boasts a higher Eeff than the JILA molecular ion species, the increased interrogation time offeredby using trapped ions leads to an overall increased sensitivity.

9
angular momentum along the internculear axis of Λ = 2, and with these two adding in opposite
directions to give a total angular momentum of Ω = 1 along the internuclear axis.
In addition, the vibrational mode of the two constituent atoms is quantized into the vibrational quantum number v. The bottom of the potential as a function of inter-atomic distance looks
roughly harmonic, so the vibrational levels have harmonic-oscillator-like behavior, starting at v = 0
and having roughly equal energy spacing. This is illustrated in Figure 2.3

Energy321v=0Inter-atomic distance
Figure 2.3: Illustration of the quantization of the vibrations of a molecule.

Finally, the rigid-body rotation of the molecule is quantized. Since this is an angular momentum, it can add with the angular momentum of the electronic wavefunction. The rotational
quantum number J is actually the projection of the total angular momentum (including Ω andmolecular rotation) onto the internuclear axis. For this reason, J must be at least Ω, so the groundrotational state of 3 ∆1 , v = 0 is J = 1.
Finer structure arises from the nuclear spins of the constituent atoms. The hyperfine states
are depicted in Figure 2.4. 232 Th has a nuclear spin of 0, while 19 F has a nuclear spin of 1/2 [49].
Therefore the total angular momentum quantum number F of (3 ∆1 , v = 0, J = 1) comes fromthe addition of one unit of angular momentum from the angular momentum of the electronic state(Ω = 1) and 1/2 unit of angular momentum from the nuclear spin of fluorine, with allowed valuesof either 1/2 or 3/2. Within each F , there can be a projection of the total angular momentum
mF ∈ [−F, F ]. Finally, each mF level has two nearly degenerate states corresponding to the parityof the wavefunction, with the degeneracy lifted by a higher order coupling known as Ω-doubling.

10

electronic state

rotational statevibrational state

hyperﬁne
1/2{= -1/2

1/2+{

parity= -3/2

-1/2

1/2

3/2

+

Figure 2.4: Structure of the hyperfine states within 3 ∆1 , v = 0, J = 1. Without any external fields,
nothing sets the quantization axis. In this case, parity is a good quantum number, with degeneracyof the parities lifted by Ω-doubling. The parity states are illustrated with a ± or ∓ sign because it
is not clear which parity has higher energy, and this was never experimentally determined.

We are interested in the F = 3/2 hyperfine manifold, particularly when we apply an external⃗ app . The effect of applying such a field on the quantum states is shown in Figure
electric field E
2.5.
With an external electric field, spatial symmetry is broken and the molecules become polarized. Without spatial symmetry, parity is no longer a good quantum number, and each mF level
becomes split instead into two orientation states (aligned or anti-aligned) of the molecule. Because
the molecule is polar it has its own molecular-frame electric dipole moment,
dmf = 3.37(9) Debye [50],(2.3)

the energy shift of the orientation states is then set by the DC Stark shifts from the molecule’s⃗ app . In addition, because there is a well defined orientation ofelectric dipole moment −d⃗mf · E

11= -3/2
(

-1/2
,

,

1/2

3/2

,

)

Figure 2.5: Energy level diagram of the 3 ∆1 , v = 0, J = 1, F = 3/2 manifold of ThF+ in thepresence of an external electric field. For each mF = ±3/2 state, the energy of the state (relative
to the baseline F = 3/2 center energy) are labeled and the quantum state depicting the orientationof the molecule and the pointing of the eEDM-sensitive valence electron’s spin (and therefore also
its d⃗e ) are illustrated.

the molecule and therefore its internal electric field relative to the spin4 of the eEDM-sensitive⃗ eff , with theelectron, these states become modified by the eEDM sensitive Stark splitting5 −d⃗e · E
sign of the dot product determined by the relative orientation of the molecule and the electron.
Finally, one more modification to the states is useful. In practice, energy levels of a molecule
are measured as the difference between two energy levels as a frequency. If we want to measure theenergy difference between the opposite sign mF = ±3/2 states for a given orientation, it is usefulto create an intentional energy difference between these states so that the frequency can be tuned⃗ app .
to an experimentally convenient timescale. We can do this by introducing a magnetic field B
The modification of the states in the presence of this field is shown in Figure 2.6.
⃗ app couples to the projection of the angular momentum onto the quanThe magnetic field B
tization axis mF through the Zeeman effect:⃗ app = ±mF g µB Bapp
∆UZeeman = −⃗µtotal · B(2.4)

where g is the g-factor and µB is the Bohr magneton. Note that the g-factor is slightly different
4

The electric dipole moment of the electron is defined to point the same direction of the spin, so the orientationof the spin is the same as the orientation of d⃗e .
5⃗ app coupling since Eapp ≪ Eeff by nine orders of magnitude.
Here we can neglect the −d⃗e · E

12= -3/2
(

-1/2
,

,

1/2

3/2

,

)

Figure 2.6: Energy level diagram of the 3 ∆1 , v = 0, J = 1, F = 3/2 manifold of ThF+ in thepresence of an external electric and magnetic field.

for the two orientations, so we label the g-factor as gu and gl for the upper and lower pair of states
respectively, which we measured to begu = 0.1475(30) [50]

(2.5)




gl = 0.1505(30) [50].

We call the upper pair of states the “upper doublet” and the lower pair of states the “lower doublet.”
The electric dipole moment of the electron can be measured by measuring the energy differences of each doublet fu and fl (violet in Figure 2.6), whose energies aref = 3g µ B+ 2d E
u

u B

app

e

eff




fl = 3gl µB Bapp − 2de Eeff .

(2.6)

Note that by measuring energy differences between states of the same molecular orientation, thedmf -dependent term drops out. The magnetic field component gives a background energy f0 =3gµB Bapp designed to turn the eEDM-sensitive energy from nominally zero frequency to one thatcan be easily measured (with our f0 on the order of 50Hz), turning the eEDM signal into a small
frequency modification that turns into a phase offset at late time.

13
Flipping the orientation of different fields and taking a linear combination of the measured
energies can then isolate the contribution from the electron’s electric dipole moment de Eeff .
This encapsulates the essence of the eEDM measurement: measure the energy shift due tothe Stark energy of the electron inside of a polar molecule with large internal effective electric field,
realized by measuring the energy difference between two internal quantum states of the moleculein the presence of known applied laboratory fields.

2.1.2

Laboratory fields⃗ app in order to polarize the molecules.
In the above scheme, we apply an external electric field E
However, since the molecules are ions, the resulting Coulomb force must has a nonzero time averagein order for the molecules to remain trapped, i.e.
⃗ app (t)⟩t = 0.
⟨q E

(2.7)

To get around this, we apply a rotating electric field polarize the molecules, rotating at a frequency
ωrot
E⃗rot = Erot (cos(ωrot t)x̂ + sin(ωrot t)ŷ).

(2.8)

Then the coulomb force on the ions becomes a centripetal force, and the ions move in a circular
trajectory with radius rrot . This is shown in Figure 2.7.

(+)

}
(+)
(+)

Figure 2.7: The rotating polarizing field E⃗rot giving rise to circular motion of the molecular ionswith a trajectory radius of rrot .

14
The radius of the trajectory can be calculated by setting the magnitude of the coulomb force
F = qE equal to the centripetal force in angular form F = mω 2 r and solving for the radius, yielding
rrot =

qErot
2
mωrot

(2.9)

232 + |{z}19 ) atomic mass units and q is the charge of the ion
where m is the mass of the ion = (|{z}ThF= 1 electron charge. For the typical values used in the experiment (ωrot = 2π·150 kHz, Erot = 60
V/cm), the molecules move in a circle of radius rrot = 2.6 mm.
Since the polarizing field and thus the quantization axis is now rotating, naïvely one might
think that the magnetic field we apply to generate f0 must also rotate so that there is a well definedmagnetic field along the quantization axis. However, we can instead apply a static magnetic fieldthat diverges in the plane of rotation. This is demonstrated in Figure 2.8.

⃗ · Êrot ⟩t .
Figure 2.8: The diverging (in the x − y plane) magnetic field that gives rise to a non-zero ⟨B

At each point in the trajectory, the magnetic field is exactly anti-aligned with E⃗rot andtherefore the quantization axis. We can generate such a field by applying a field with an axial
gradient (a field with a gradient in z along ẑ):⃗ = Baxgrad (xx̂ + y ŷ − 2z ẑ).
B

(2.10)

Then Brot , the value of the magnetic field along the quantization axis, is simply given by
Brot = Baxgrad rrot .

(2.11)

15
From the illustration, it may seem as though this method requires that the center of the path tracedout by the ions be exactly at the center of the divergence of the magnetic field. However, an offset(x0 , y0 ) in the position of the field just looks like a uniform background field,
Baxgrad ((x − x0 )x̂ + (y − y0 )ŷ − 2z ẑ) = Baxgrad (xx̂ + y ŷ − 2z ẑ) + Baxgrad (−x0 x̂ − y0 ŷ),
|
{z
} |
{z
}
intended field(2.12)

uniform backgroundand the contribution from this background field will cancel out as the quantization axis rotates
one cycle (this is shown analytically in Section 5.1). This means that the ions can trace a circular
trajectory of radius rrot around any point in the x − y plane and experience the same time-averaged
field into the quantization axis:⃗ · Êrot ⟩t = Brot .
⟨B

2.1.3

(2.13)

Measuring energy levels in a molecule

The method we use to measure the energy difference between states in a molecule relies on thefact for an eigenstate of the molecule’s Hamiltonian (ĤΨ = EΨ Ψ), the solution to the Schrödinger
equation has a solution of a complex exponential,iℏEΨdΨ = Ĥψ = EΨ Ψ → Ψ(t) = Ψ(t=0) e−i ℏ t ,dt(2.14)

which means the phase of the state Ψ accrues at a frequency equal to its energy divided by Planck’s
constant:
ωΨ =

EΨ
.
ℏ

(2.15)The phase difference between two states then accrues at a frequency proportional to the energy
difference between the states. We can measure this phase using a technique called Ramsey interferometry.
To understand how Ramsey interferometry works, it’s useful to parameterize a superpositionof the two states |g⟩, |e⟩ in terms of two angles|ψ⟩ = cos(θ) |g⟩ + eiϕ sin(θ) |e⟩ .

(2.16)

16
We can then represent any such state as a unit vector on a sphere with coordinates dictated by thepolar angle θ and azimuthal angle ϕ (known as the “Bloch sphere”). The space of this representation
is illustrated in Figure 2.9

population
fraction
relative
phase

Figure 2.9: The Bloch sphere representation of a superposition of two quantum states.

There are important two properties of the state in this formalism. First, if we measure a state
many times to extract the probability of it being in either eigenstate, we have essentially measured
the angle θ:P|g⟩ = | ⟨g|ψ⟩ |2 = cos2 (θ)
P|e⟩ = | ⟨e|ψ⟩ |2 = sin2 (θ).

(2.17)

Second, ϕ is the relative phase between the two states, so time evolution of the state looks likerotation about ϕ at a rate of ∆Eℏ t. where ∆E is the energy difference between the states ∆E =Ee − Eg .
With these two properties, we can understand the Ramsey-type measurement scheme, whichis as follows:• prepare a pure state, represented by θ = 0 or ẑ• rotate6 about x̂ by π/2• wait some time t for ϕ to evolve by ∆Eℏ t (a rotation about ẑ)• rotate about x̂ by π/2.
6The rotation can be in any direction (other than ẑ) as long as both π/2 rotations are along the same axis, witha rotation axis in the x-y plane giving the most contrast.

17
A representation of this scheme on the Bloch sphere is illustrated in Figure 2.10.

initial state

pulse

Ramsey evolution

pulse

Figure 2.10: Bloch sphere representation of the three operators that make up a Ramsey interferometry measurement.

This scheme works by encoding the phase evolution of ϕ (a quantum phase) onto θ (a measurable probability), which happens because rotations along different axes do not commute. Mathematically we can represent this scheme as an operator Ô operating on our initial state ẑ via 3D
rotation operators:Ô ẑ = R̂x (π/2)R̂z (∆Et)R̂x (π/2) ẑ.
ℏ(2.18)

If we write out these rotation operators as matrices and perform the operation, we can see how thisencodes the phase accrued onto θ:∆E cos( ℏ t)

0
0
1




Ô ẑ = 0 cos(π/2) − sin(π/2)  sin( ∆E t)
ℏ




0 sin(π/2) cos(π/2)
0− sin( ∆Eℏ t)cos( ∆Eℏ t)0

0  1
0
0





ẑ


0 0 cos(π/2) − sin(π/2)

1
0 sin(π/2) cos(π/2)

(2.19)∆E sin( ℏ t) 




=
.
0




− cos( ∆Et)ℏ
If we convert this back to spherical coordinates to find θ, we get
q

!
p
2 ∆E
t)
sin
(
2
2
x +y
ℏ
 = − ∆E t,
θ = tan−1= tan−1 ∆Ezℏ− cos( ℏ t)

(2.20)(2.21)

18
so the probability of measuring the either eigenstate after the Ramsey sequence becomesP = cos2 ( ∆E t)

|g⟩
ℏ



P|e⟩ = sin2 ( ∆Eℏ t).

(2.22)

We can then simply vary the evolution time t and count the number of cycles in order to measure
the energy difference ∆E between the states.
With this technique, we can measure the eEDM-sensitive frequencies fu and fl introduced in
Equation 2.6.

2.1.4

Sensitivity

In our experiment, the sensitivity δde to the eEDM scales as
δde ∝

1
√ ,
Eeff τ N(2.23)

where τ is the Ramsey evolution time of the measurement (limited by the coherence time) and N is
the total number of ions measured. Several key innovations in our new experiment maximize theseparameters. By moving from HfF+ (the species used in our current best bound [38]) to ThF+ , thesensitivity increases in several ways. The coherence time in HfF+ is limited by spontaneous decayof the eEDM-sensitive states (∼2 s), while the eEDM-sensitive states in ThF+ are in the groundelectronic, vibrational, and rotational state, meaning that they do not spontaneously decay. Instead,the coherence time is limited by excitation from blackbody radiation (with cryogenic cooling of theexperiment, coherence times of 20 s are theoretically possible, with more information in Ng’s thesis[39]). Furthermore, ThF+ has a slightly higher Eeff than HfF+ , as mentioned previously.
Measuring the ions for a Ramsey time of 20 s would severely limit the rate at which themeasurements could be done. Therefore, to fully leverage the long coherence time, we design theexperiment to be multiplexed, i.e. ions will be continually loaded and prepared, then shuttled down
the length of the trap as their Ramsey phase evolves, then read out on the other side of the trap.
We can then make a measurement once per ion loading time (∼.5 s) rather than once per Ramsey
time (∼20 s).

19
The current iteration of the experiment as was developed in this thesis is not fully multiplexed,
but tests some of the key technology required by shuttling the Ramsey evolving ions down to acold region, then back to the original position for detection, demonstrating both the shuttlingcapabilities of the electrodes and the coherence time available via cryogenic cooling.
With all the planned improvements, we can make a rough estimate of the statistical sensitivity
we expect to reach with this experiment, which is done in Table 2.2.
Experiment
EeffτN/tmeastmeas
δdeGen II
21±2 GV/cm [43, 44, 45]∼2 s∼120/(2 s)∼500 hr
2.1 × 10−30 e·cm [38]Gen III
36±1 GV/cm [40, 41]∼20 s∼20/(.5 s)∼500 hr∼ 1 × 10−31 e·cm (projected)

Table 2.2: Projected sensitivity for the third generation of eEDM experiment at JILA comparedto the second generation. The projected sensitivity is calculated by scaling the Gen II sensitivity
by the factor prescribed in Equation 2.23.

This projected sensitivity hinges on the success of long coherence times and multiplexing,
and it only represents the possible statistical uncertainty of the experiment. Systematic sources of
uncertainty can limit the sensitivity of the experiment as well (see [51] for an example of a studyof systematic uncertainty).

2.2

Experimental realization
Now that we have seen how to measure the eEDM in an abstract sense and what makes sucha measurement sensitive, this section will describe how such an experiment is actually realized.
The first step of the experiment is to create ThF+ out of sulfur hexafluoride gas and solid
thorium. The technique for doing this is shown in Figure 2.11 A pulsed Nd:YAG laser ablates apellet of thorium, vaporising a small amount of the metal. At the same time, a gas valve pulses
some SF6 gas, which reacts with the Th vapor to create a beam of neutral ThF in a supersonic
expansion.
The pellet is on a rotation and translation stage that rotates after each ablation such that20

ionization
lasersThF+

skimmers
ablation laser

ThF

Th pellet

SF6 gasnozzle pulser

Figure 2.11: Setup for creation of ThF+ ions in the experiment.

each laser pulse hits a fresh polished surface of Th metal. The Th must be periodically polishedas laser ablates away the clean surface in order to maintain production eﬀiciency. The SF6 gas ispressurized to 80 pounds per square inch on the other end of the nozzle.
The ThF beam then passes through two skimmers, which are mm-scale pinholes with a conicalshape. The skimmers reduce the angle of the beam such that when it reaches the trap it is about
1cm in size. The first skimmer is held at 30V to reject any ionized species in the beam.
Between the skimmers is an intermediate vacuum, with a pressure between that of the vacuum
where the molecules are created (∼ 10−8 Torr, limited by consistent vacuum breaks to polish Th
and consistent leaking of pressurized gas) and the vacuum in the trapping region (< 10−9 Torr,
which must be low to allow for long coherence time). The small aperture of the skimmers allow forsuch a pressure gradient.7
Finally, the neutral beam of ThF is photo-ionized to create a packet of ThF+ ions. The
photoionization process is covered in Section 3.1.
7
At these pressures, intuition for vacuum behavior is governed by random scattering of individual atoms rather
than traditional properties of fluid mechanics.

21
The next steps are illustrated in Figure 2.12.

stop pulse

ramp ontrapping
potential

Figure 2.12: Illustration of the experimental procedure for stopping and trapping the ions.

Immediately after ionization, a large electric field is applied in the direction in which theions are coming from, which slows the ions to a velocity distribution that can be trapped. The
pulse is roughly 5.3 V/cm for 4.1 µs (though the actual energy delivered to the ions is obscured byinhomogeneity of the field and rise time of the electrode drivers).
At the same time, the trapping potentials are turned on so that once the ions are slowed they
are immediately trapped. The trapping potentials are discussed in Section 4.1.3
Finally, the rotating polarizing field Erot is ramped up over 1.2 ms, a time such that the ramp1
is adiabatic with respect to the rotation of the field ( ωrot
= 1501kHz ≈ 7 µs ≪ 1.2 ms). The details

on generating the polarizing field are covered in Section 4.1.4.
Once the molecules are polarized, they can be pumped into the eEDM-sensitive states withlasers and microwaves through four of the viewports of the vacuum chamber, as shown in Figure
2.13. The details of optical pumping are described in Section 3.2.
With the proper initial quantum state prepared, we can execute the Ramsey sequence. Thisis shown in Figure 2.14. First, we apply the π/2 pulse by momentarily ramping down the polarizing
field such that the hyperfine states become degenerate. Next, during the Ramsey evolution, thetrapping potentials are modulated to translate the ions down the trap into a region where the trap
is cryogenically cooled and the ions are shielded from room-temperature blackbody radiation. At

22lasersmicrowaves

Figure 2.13: Illustration of the experimental procedure for optically pumping them into the eEDMsensitive states.

cryocooler
conveyor belt
potential

translation

insulation

π/2

Figure 2.14: Illustration of the Ramsey evolution period of the experiment.

the end of the Ramsey evolution time, the ions are shuttled back and given another π/2 pulse.
More details on the π/2 pulse and blackbody radiation can be found in Ng’s thesis [39].
Finally, the state can be read out, as shown in Figure 2.15. The ions are state-selectively
photo-dissociated (details in Section 6.1), while the trapping fields are turned off and Erot is ramped
down to zero. A strong electric field in the direction of the detector (4.9 V/cm for 20 µs) isthen pulsed in order to kick the ions onto the detector. The dissociated Th+ ions as well as the23

dissociation
lasersTh+

MCP
detector

kickout pulse

ThF+

Figure 2.15: Illustration of the state detection sequence of the experiment.

undissociated ThF+ ions then arrive on the detector where they are imaged. Details of the detection
scheme are provided in Section 6.2.
The chapters that follow will describe the details of each of these steps as well as the instrumentation that was designed and built to actualize each step.

Chapter 3
State preparation

3.1

Ionization
The neutral ThF atoms are ionized in a process called resonance-enhanced multi-photon

ionization (REMPI). An energy level diagram for this process is shown in Figure 3.1.

...
autoionization...

REMPI 2(532nm)Ω = 3/2
ThF+
REMPI 1
(tunable)(304nm)
2

Δ3/2
ThF

Figure 3.1: Diagram of the two photon ionization process

The first laser, denoted REMPI 1, pumps the neutral ThF into an intermediate state, then asecond laser labeled REMPI 2 excites the atoms above the ionization threshold. Ionizing throughan intermediate state of ThF allows the ionization to be state-selective, only ionizing the ground25
state atoms of ThF and leaving the atoms with a well-defined energy, which is just barely above theground state of ThF+ . This means that when we ionize, we only populate the ground electronic(3 ∆1 ) and vibrational (v = 0) state. The energy resolution is not fine enough to select the groundrotational or hyperfine states, so we populate a mixture of mF and J levels within 3 ∆1 , v = 0.
The REMPI1 photon is provided by a Lioptec LiopStar dye laser, while the REMPI2 photon
is provided by a Quanta-Ray INDI-40-10 pulsed Nd:YAG laser (the 532 nm light from the QuantaRay also pumps the Lioptec dye laser).

3.2

Optical pumping
After the ThF+ ions are prepared into the 3 ∆1 , v = 0 manifold, we then optically pumpthem into the J = 0, mF = ±3/2 states. The lasers used to do this are shown in Figure 3.2.
Each higher vibrational state is optically pumped with lasers labeled “vibrational cleanup”
to a higher Ω = 0, v = 0 state, which then decays among the vibrational levels of 3 ∆1 . Higher
rotational sates within the these vibrational states are coupled to the J = 1 state that the respective
laser talks to using microwaves, labeled “rotational cleanup.” These frequency of these microwaves
are swept to address all the hyperfine transitions in the vibrational manifold. Together these lasers
and microwaves close all the transitions to the next two higher vibrational states of 3 ∆1 (theHonl-London factors are such that the v = 3 and above levels are not populated).
v=0 .
Similarly, the higher rotational states of v = 0 are closed with Lrc and Urc

The two remaining sets of photons are meant to move population within the v = 0, J = 1
state. In order to deplete one of the two doublets, Udepl couples the upper or lower doublet tov = 0, J = 2, where Ldepl then shelves it to a higher rotational state. After some time thismoves the population from the doublet of interest into higher rotational levels of 3 ∆1 which cannot
decay back into v = 0, J = 1. This way a single doublet within J = 1 has been moved to othermetastable states, and the population of the remaining doublet can be read out directly by readingout population in J = 1. Finally, LmF is designed to move population into either mF = 3/2 or
mF = −3/2. A diagram of how this works is shown in Figure 3.3.

26

Label
LmF
Lrc
v=0
Urc
v=1
Lvc
v=1
Urc
v=2
Lvc
v=2
Urc

Description
mF pumping
rotational cleanup
rotational cleanup
vibrational cleanup
rotational cleanup
vibrational cleanup
rotational cleanup

Udepl

depletion

Ldepl

depletion

Wavelength/Frequency
685.4436371(43) nm
685.489059(1) nm
43.627-43.667 GHz swept
717.591265(27) nm
28.81-28.90 GHz swept
752.658792(10) nm
28.94-28.98
GHz swept(
29.115642(57) lower
29.082823(29) upper
685.38160(1) nm

Polarization
circularlinear (z)linear (z)linear (z)linear (z)linear (z)linear (z)

Power
3 mW∗
80 mW
-10 dBm
15 mW-10 dBm
30 mW-10 dBm

See Fig.
3.8

linear (z)-21 dBm

3.16

linear (z)

25 mW

3.9
3.17
3.10
3.19

Figure 3.2: Lasers and microwaves for state manipulation. Red indicates photons for opticalpumping, while blue indicates photons for depletion, with solid lines representing optical transitions
and dashed lines representing microwave transitions. The powers are given as light going into thechamber for optical transitions, and power going into the microwave horn for microwave transitions(the power inside the chamber has never been empirically measured and relies on the calculationsin Section 3.2.3.2). Each laser is shaped to have a Gaussian beam width (1σ) of roughly 2.5 mm
to match the spatial extent of the ion cloud.
* LmF is strobed, so the typical power given here corresponds to a duty cycle of 0.5%, i.e. the peak power of thelaser is ∼600 mW.

27

a)

Pumping to: -3/2

+3/2

}

}
}
b)

= -3/2

opticalpumpingotherrovibrational
levels
-1/2

1/2

3/2

Figure 3.3: The mF pumping scheme using LmF .
a) LmF is circularly polarized, and is strobed so that it is only on when it is either aligned (left) or
anti-aligned (right) to the quantization axis defined by Erot (each pulse of light is only 0.5% of one
cycle of Erot ). This way each photon absorbed gives a well defined quanta of angular momentum
along the quantization axis. The pulsing of LmF is done by sending the laser tightly focused throughan acousto-optic modulator (AOM), so the laser can be turned on and off on a ∼10 ns timescalewith an RF switch on the input signal to the AOM. The pulse is then aligned to the correct phaseof Erot empirically by scanning the timing and looking at population distributions. Note that thechirality of LmF was never actually measured, so an ambiguity exists with respect to which side wepump to for a given alignment, i.e. the labels for this depiction could be reversed.
b) Energy level diagram of how pumping to mF = +3/2 works. Because each photon absorbed
increases the angular momentum by 1, absorbing a photon moves to the right. Even though eachintermediate state can decay with ∆mF = {−1, 0, +1}, the total angular momentum change fromabsorption and spontaneous emission is between 0 and +2, so they cannot move to the left and on
average they move to the right. While these states can also fall to other rotational and vibrational
levels, these are closed with the optical pumping photons discussed above. Thus if LmF and
optical pumping photons are on for long enough, population will eventually land in mF = +3/2.
Pumping to mF = −3/2 works exactly the same way, but with each photon taking away one angularmomentum.

28
3.2.1

Continuous wave laser overview

Optical pumping is done with low power (tens of mW) continuous wave lasers in the red partof the visible spectrum. Historically these lasers have been external cavity diode lasers (ECDLs)
fabricated by the JILA instrument shop, however many have been switched to fiber lasers orcommercial ECDLs in order to achieve higher power and stability. The currently used opticalpumping lasers are listed in Table 3.1.
Photon
LmF
Lrc
Ldepl
Ldepl
Lv=1
vc
Lv=2
vc

Description
Fiber laser
Fiber laser
JILA ECDL Diode
JILA TAJILA ECDL Diode
Commercial ECDL

Manufacturer
Precilaser
Precilaser
Thorlabs
Toptica Eagleyard
Toptica
Toptica

Model
PL202106532
PL202106533
HL6750MG
RLS/680NM-TA-250MW
LD-0725-0040-AR-1
DL-proTable 3.1: Model numbers for each component of the optical pumping laser system

The complete optical pumping system, highlighting the layout of the JILA ECDLs, is shownin Figure 3.4. The physical layout of one such JILA ECDL is shown in Figure 3.5. It functions
as follows: The cavity is defined between the back of the diode and the first order diffraction off
the grating. The supported wavelength of the cavity can be tuned by adjusting the angle of thegrating. The grating is mounted on a kinematic mount with a piezolectric crystal in the horizontal
axis, allowing fine control of the wavelength to be done electronically. To prevent parasitic lasing
modes, the diodes are anti-reflection coated to prevent a cavity from forming between the two
surfaces of the diode. For stability purposes, the temperature of both the diode and the cavity
are fixed using a thermoelectric cooler (TEC) and a thermistor. TEC 1 controls the temperatureof the laser diode, whereas TEC 2 stabilizes the temperature of the aluminum enclosure on which
the diode and grating are mounted, preventing cavity length drifts due to thermal expansion. The
whole system is mounted on a large block of aluminum which stabilizes the temperature and acts
as a thermal reservoir for the temperature feedback system.
The light then leaves the ECDL enclosure and passes through an optical isolator to prevent

29

Figure 3.4: Optical pumping laser system configuration. Shown is a JILA ECDL and TA pair, as
well as the frequency locking configuration. The fiber switch is a DiCon MEMS 1xN switch andthe wavemeter is a HighFinesse WS-7.

reflections from damaging the laser diode. Some light is then picked off and sent to a fiber switch,which forwards it to a wavemeter for spectral measurement. When the laser is locked, python will
extract the frequency from the wavemeter, and feed back the difference from the desired frequency
to a voltage on the piezo.
The rate at which the frequency is sampled and the feedback signal is updated is limited by

30

Thermallystabilized
block 1

Rough
align

Diode

Thermallystabilized
block 2Largethermalmass
reservoir

Grating Piezo

Figure 3.5: Photos of the JILA ECDL assembly with the lids removed.

the speed of the fiber switch. While all continuous wave lasers are being locked at once, each laser
is sampled approximately once per second, which is suﬀicient to compensate for slow thermo/mechanical fluctuations. With this system each laser is stable to within ∼5 MHz.
3.2.2

Rb reference

In order to reference the lasers to an absolute frequency standard, we lock a laser to an atomic
transition in Rubidium in a vapor cell. Then the wavemeter can be calibrated to the known energyof this transition, thereby calibrating the wavemeter’s measurements of all other lasers. A diagramof the setup is shown in Figure 3.6.
This technique, known as “saturated absorption spectroscopy” or “Doppler-free spectroscopy”
makes use of two counter-propagating laser beams such that the signal corresponds to light thatis resonant in both directions, allowing us to resolve beyond the Doppler width. This is necessary
because the one-sigma Doppler width of the transition, given by
r
∆f =

kT
f0 ,
mc2

(3.1)

is ∼200 MHz for red light on room temperature rubidium atoms, and we want to lock the lasers tothe ∼1 MHz level.
The technique works as follows: a high-power pump beam is sent one way through the Rb

31

Figure 3.6: Optical setup for locking the reference laser to the Rb line.

vapor cell, and a weak probe beam is sent collinearly the other direction and onto a photodetector.
When the pump beam is on resonance for a transition in the Rb, it saturates the transition, so theprobe beam is not absorbed, leading to a spike in the transmission of the pump beam.
When we scan the laser frequency (by scanning the voltage on the piezo controlling thegrating of the ECDL), the resulting signal looks like a broad Doppler absorption dip with local
saturation-induced transparencies at each individual transition. A copy of the probe beam with nopump is also sent through the vapor cell, which captures only the broad Doppler absorption dip.
The two signal can then be subtracted, leaving only the Doppler-free peaks. The results are shownin Figure 3.7.
The laser is locked to the (52 S1/2 → 52 P3/2 ) transition of 87 Rb, from F =2 to halfway between F ′ =2 and F ′ =3 in a so called “crossover transition”. This transition has an energy of
384.2279818773(55) THz [52]. Crossover transitions occur when the atoms are moving at a velocity
such that the Doppler shift is exactly half the spacing between two atom-frame transitions. Thisway the atom sees each beam as addressing one of the two separate transitions, which still results

Transmission (arb)

32

probe only
pump + probe

Difference (arb)(2co3)(1co3)(1co2)
(2)
(1)

(3)

Derivative (arb)

PID setpoint

-0.02

-0.01

0.00
0.01
Piezo voltage (arb)

0.02

0.03

Figure 3.7: Spectroscopy of the Rb87 F=2 line used to calibrate the wavemeter. Top: the raw
photodiode signals of the two beam paths. Middle: the difference of the two photodiode signals.
Each peak is labeled with the F’ to which the transition goes, with ‘co’ indicating a crossover
between two transitions. Bottom: the derivative of the difference signal, which is used to lock thelaser. The zero crossing corresponding to the peak of the (2co3) line is used, with the derivative
providing a signed linear feedback signal which can be used to lock the frequency of the laser.

in transparency of the atoms to the probe beam as the atoms are pumped out of F = 2. Generally,
these crossover transitions are stronger than the Doppler free transitions because of the thermal

33
distribution of atoms in the vapor cell (there are more atoms with the crossover-inducing velocity
in the axis of the beams than atoms with no velocity in the axis of the beams).
Because the crossover transition between F ′ = 2 and F ′ = 3 is the most predominant feature,
it is the most stable feature to lock to so we lock the laser to this transition. The lock is done withelectronics that produce a small dither on the piezo voltage, and takes the time derivative of the
Doppler-free signal. Because the derivative should go to zero at the peak, the derivative can beused as a signed feedback, which goes into a PID to control the central piezo voltage. In this way,
the laser can be locked to within ∼1 MHz, providing a stable absolute frequency reference to which
we can calibrate the wavemeter.

3.2.2.1

Spectroscopy of optical pumping transitions

We can measure the energy of the optical transitions by reading off the population in thestates that each laser addresses using techniques described in Chapter 6. We can then scan thefrequency of the laser, with the population of the state decreasing when the laser is on resonance.
First we measure the frequency of the LmF transition. If we were to measure the frequencyof the transition with the polarizing field Erot , the large Doppler shift caused by the rotation of theions would significantly broaden the transition (to beyond the free spectral range of the ECDL,
meaning it couldn’t be measured in a single scan), as well as shift the center frequency if the timingof the pulse is off. For this reason we measure the frequency of this transition in the absence ofErot . We can then simply look at the dip in population in v = 0, J = 1 when the laser is turned
on. This is done in Figure 3.8.
The width of the Gaussian fits to 106.8(3.4) MHz. This apparent linewidth is predominantly
due to the Doppler shift from the micromotion of the ions, rather than from a Doppler widthcorresponding to the temperature of the secular motion, a natural linewidth of the transition, or
the width corresponding to the stability of the laser (∼5 MHz).
v=2
We repeat this measurement for Lv=1
vc and Lvc in Figures 3.9 and 3.10 respectively.

34

Population in v = 0 , J = 1

50
40
30
20
10

No pumping
mF

437.36972 437.36984 437.36996 437.37008 437.37020
mF frequency (THz)

Figure 3.8: Frequency scan of the LmF laser with no Erot . Population in v = 0, J = 1 is read out
by dissociating from the (3 ∆1 , v = 0, J = 1) → (Ω = 0, J = 2) transition measured in Figure 6.3.
The dip in population is fit to a Gaussian, which yields a center frequency of 437.3699627(27) THz
or a vacuum wavelength of 685.4436371(43) nm.

Population in v = 1,J = 1

10

No optical pumping
Optical pumping
Optical pumping + vvc= 1

8
6
4
2
0
417.775700 417.775875 417.776050 417.776225 417.776400
v = 1 frequency (THz)vc

Figure 3.9: Frequency scan of the Lv=1
vc laser with no Erot . Population in v = 1, J = 1 is read out by
dissociating via the (3 ∆1 , v = 1, J = 1) → (Ω = 0, J = 2) transition measured in Figure 6.4. The
Gaussian dip fits to a frequency of 417.776069(16) THz or a vacuum wavelength of 717.591265(27)nm

35No optical pumping
Optical pumping
Optical pumping + vvc= 2

Population in v = 2,J = 1

40
30
20
10
752.6580

752.6584v=2vc

752.6588
752.6592
wavelength (nm)752.6596

Figure 3.10: Frequency scan of the Lv=2
vc laser with no Erot . Population in v = 2, J = 1 is read out by
3
dissociating from the ( ∆1 , v = 2, J = 1) → (Ω = 2, J = 2) transition measured in Figure 6.5. The
Gaussian dip fits to a frequency of 398.311241(05) THz or a vacuum wavelength of 752.658792(10)
nm.

36
3.2.3

Microwaves

3.2.3.1

TheoryTo predict the microwave transition rates, we begin with a model of the microwave-moleculeinteractions. We can use a Jaynes-Cummings model Hamiltonian, which is nice because it isphysically intuitive and time-independent, meaning that it’s mathematically simple to work withand we don’t have to do any work doing a rotating wave approximation to get out the dynamics
of the system (since this approximation is already built in to the Jaynes-Cummings model).
The Jaynes-Cummings Hamiltonian for the molecule and light field are given by
H = Hmolecule + Hlight + Hinteraction11= ℏω0 σ̂z + ℏω(â† â + ) + ℏg(σ̂+ â + â† σ̂− ).
2
2

(3.2)
(3.3)

In the first term, ℏω0 is the energy spacing between the two unperturbed molecular states. This
represents the energy of the molecule with an offset to symmetrize the matrix. In the second term,ω is the frequency of the applied light and â† and â are the photon creation and annihilation
operators. This term represents the energy of all the photons = ℏω(n + 1/2). The third term
represents the interaction whereby a photon can be annihilated and the molecule excited or aphoton can be created and the molecule de-excited. The coupling strength of the interaction ischaracterized by g, the vacuum Rabi rate given byrrd⃗ · Êℏωq ⟨g| ⃗r |e⟩ · Ê
ℏω
g=
=
,
ℏ
2ϵ0 V
ℏ
2ϵ0 V

(3.4)

where V is the volume of the cavity mode, q is the electron charge, Ê is the unit vector indicating thepolarization of the light, and d⃗ is the dipole transition matrix element of the molecular transition,with |g⟩ and |e⟩ being the lower and excited states of the molecular transition. Since the microwaves
are not actually in a cavity, it’s more instructive to write the interaction part of the Hamiltoniansemi-classically.
First, recognize that the only two relevant combined photon-molecule states are |n, e⟩ and|n + 1, g⟩, where n is the number of microwave photons which is arbitrarily large, and e and g

37
reference the excited and ground states of the molecule. In this basis, we can write the interaction
Hamiltonian as 0Hinteraction = ℏg 
√
n


n 
.
0

√

(3.5)

In this basis, the whole Hamiltonian is then







√
0  1
n 1
 1 0  n
 0
H = ℏω0 
 + ℏω  +  + ℏg  √
.
2
2
0 −1
0 n+1
n 0

(3.6)

To get from terms that depend on the cavity properties and the number of photons to terms
that depend only on the classical field strength, we can treat the microwave field in some finite
volume V as light in a cavity. We can then calculate the energy in this volume due to the classicalfield and compare this to the photon energy in order to solve for the number of photons n:
Z
nℏω =

1
1
ϵ0 |E|2 d3 r = ϵ0 |E0 |2 V.
2
V 2

(3.7)

Then we can write
√

g n=g

r

1
q ⟨g| ⃗r |e⟩ · Êϵ0 |E|2 V =
2ℏω
ℏ

r

ℏω
2ϵ0 V

r

⃗
1
1 q ⟨g| ⃗r |e⟩ · E
ϵ0 |E0 |2 V =
,
2ℏω
2
ℏ

(3.8)

and we have recovered the classical Rabi rateΩ≡⃗q ⟨g| ⃗r |e⟩ · E.
ℏ√
Knowing that g n = Ω/2, we can write the Hamiltonian as





0  1
1
 0
 1 0  n
H = ℏω0 
 + ℏω( + ) + ℏΩ 
2
2
1
0 −1
0 n+1
2

(3.9)


1
2


.

(3.10)

0

We now have a Hamiltonian in which the diagonal components describe the combined energiesof the molecule and a photon that would be absorbed or emitted to transfer molecular states in asecond quantization framework, and the off-diagonal components are semi-classical field couplings.
This representation allows both physical intuition of all parts as well as mathematical simplicity.

38
Since the dynamics of the system are invariant to overall energy shifts, we can add a constantto the Hamiltonian. If we add −ℏω(n + 12 ), the light field energy term simplifies to a symmetricenergy difference of the two states. With this the Hamiltonian becomes fully symmetrized.




1
0
−
1
0
1

 2


 0
H = ℏω0 
 + ℏω  + ℏΩ 
2
1
0 −1
0 21
2


1
Ω
2
 − 2 (ω − ω0 )

= ℏ

Ω
1
(ω
−
ω
)
0
2
2


Ω
∆
 −2 2 
= ℏ
,


Ω
2


1
2




(3.11)

0(3.12)(3.13)∆
2

where ∆ is the detuning of the light from the resonance ∆ ≡ (ω − ω0 ). If we plot the eigenenergiesof this Hamiltonian as a function of the detuning, we observe an avoided crossing, as shown in
Figure 3.11.

2

Energy10-1
Eigenvalues
Diagonal Components

-2
-4

-2

0
/

2

4

Figure 3.11: Energies of the dressed state Hamiltonian as a function of the detuning ∆.

If the microwaves are swept, then the dynamics of the dressed states can be described using theLandau-Zener formulation. The goal of the microwaves is to de-excite the molecule, i.e. population

39
starting in |n, e⟩ transfers to |n + 1, g⟩. If the microwaves are swept adiabatically, the system will
follow the avoided crossings and all population will be transferred. However, if the sweeping of the
microwave frequency is diabatic, some population will remain in the molecular excited state. In
the Landau-Zener formulation, the probability of not transferring (staying in |n, e⟩) after a single
sweep through the resonance is given by [53]Pe→e = e−2π|H12 |2 /ℏ| d (H22 −H11 )|dt(3.14).

ddℏ∆| = ℏ| dt(ω − ω0 )| = ℏ| dω
The denominator is simply the sweep rate | dt
dt |. The probability oftransferring in a single sweep is then given by
Pe→g = 1 − Pe→e = 1 − e

− π2

Ω2
| dω |
dt(3.15).

If the sweep occurs periodically with a period Ts and sweeps over a frequency range ∆ω, then we
can turn this probability into a transition rate (assuming a small portion transferred each sweep
and that transferred population is then coupled out of the new state by a laser):2Γµ = Pe→g /Ts =1−eΩ− π2 |∆ω/T|s(3.16).

Ts

If we plug in the expression for Ω from Equation 3.9 and solve for the electric field, we can get thefield strength we need to get a desired rate

s

ℏ
⃗ =

|E|
⟨g| ⃗r |e⟩ · Ê q
1

2 ln( 1−T1s Γµ )∆ω
πTs.

(3.17)

Calculating the dipole matrix elements ⟨g| ⃗r |e⟩ · Ê is highly nontrivial and requires some advanced
angular momentum algebra, but we can say that it will be of the form1
| ⟨g| ⃗r |e⟩ | = d⃗mf Ag,e ,q(3.18)

where q is the charge of the electron, d⃗mf is the molecular frame dipole moment (coming from theelectronic configuration ⟨ψelectronic | r |ψelectronic ⟩ = dmf /q), and Ag,e is a dimensionless constant oforder unity. We can then write

s
ℏ
⃗ =
|E|
dMF Ag,e2 ln( 1−T1s Γµ )∆ω
πTs.

(3.19)

40
3.2.3.2

Power requirements

To calculate the required microwave power, we need to know the strength of the microwave
field at the ion location for a given power we apply. The microwaves are radiated with a horn
through a window in the vacuum chamber. It then passes through the trap electrodes and hits theion cloud.
A naive calculation of the field strength can be performed using the directivity DdBi of the
microwave horn measured by the manufacturer. Given that the horn is pointed directly at the ion
cloud and is located a distance rhorn away, the intensity of the electric field seen at the ions is thenI=

Phorn  DdBi /10 10.
2
4πrhorn(3.20)For our setup, the distance from the horn to the ions when placed against a window of the octagon
is rhorn = 15cm, and the directivity of the horn is 17.9 dBi. The electric field magnitude is then
s
r
r

2I
Phorn
Phorn
D
/10
dBi
· 4.05 V/cm.
(3.21)
E0 =
=
10
=
2
cϵ0
1W
2πrhorn cϵ0
Because the electrodes are spaced apart by a distance of similar magnitude to the wavelength of
the radiation, they can significantly alter the field. For a more accurate calculation the electric field
incident on the ions, we can run a finite element model of the geometry of the horn and electrodes.
The COMSOL model simulation results are shown in Figure 3.12.
The result of the simulation is that for a power of 1 W into the horn, the electric field
amplitude at the ion location is 6 V/cm. The electric field seen at the ions as a function of thepower into the horn is thus:

r
E0 =

Phorn
· 6 V/cm.
1W(3.22)The result is actually a slight enhancement over the naïve calculation (Equation 3.21) due to thefocusing of the field by the rod electrodes.
Another consideration is the loss due to reflection of the microwaves off of the UHV window.
We can treat the free space before and after the glass as well as the propagation in the glass itself

41

a)

b)

c)
10 4
1
0.5
0
-0.5
-1
0

5

10

15

20

25

d)
Figure 3.12: Finite element simulations of 30 GHz microwaves exiting the horn and passing throughthe electrodesa) 3D view of the COMSOL model.
b) The microwave field in the plane of the rod electrodes. The field is polarized into the page,
parallel to the rod electrodes. This means that the field must go to zero near the rod electrodes tosatisfy boundary conditions. The result is an interference pattern where the rods cast a ”shadow”.
c) The microwave field in the plane of the ring electrodes. Since the field is polarized up and down,
perpendicular to the ring electrodes, the boundary conditions do not require that the electric field
go to zero near these electrodes. As such, the ring electrodes have little effect on the microwave field.
However, the boundary conditions do forbid curvature of the field near them since the tangential
field must be zero, so we observe a“flattening” of the wavefront near these electrodes.
d) The electric field on the axis of the microwave horn. The origin is the injection port of the hornwhere the field originates from.

42
as propagation along a transmission line. The characteristic impedance of the air before and after
the glass is given by the free space impedance:
r
Z0 ≡

µ0
.
ϵ0

(3.23)

If the glass has a relative permeability ϵr at the microwave frequency, the glass has a characteristic
impedancer
Zwindow =µ01= √ Z0 .
ϵ0 ϵrϵr(3.24)The effective impedance of propagation through the glass of thickness l and then into free space isthen given by [54]Zprop, interface = Zwindow

Z0 + iZwindow tan(βl)
Zwindow + iZ0 tan(βl)

where β is the propagation constant in the glass β = 2π

√

ϵr f
c


,

(3.25)

where c is the speed of light. Thenthe reflection coeﬀicient of the light incident on the glass is given byΓ=

Zprop, interface − Z0.
Zprop, interface + Z0

(3.26)The total power reflected is thenPr = P0 |Γ|2 = P0

(1 − ϵr )2.
(1 + ϵr )2 + 4ϵr cot(βl)2(3.27)The vacuum windows are made from fused silica (quartz), which has a relative permittivityof about 3.8 in the Ka band [55]. Using this value we can calculate the power reflected off the glassfor varying thicknesses, as shown in Figure 3.13.
Applying this 20% reduction to our COMSOL result of Equation 3.22, we have a prediction
for the field seen by the ions as

r
E0 ≈

Phorn
· 5 V/cm.
1W(3.28)

We can now use Equation 3.19 to calculate the power we need to apply. A good approximation
for the coupling constant Ag,e is a Clebsch Gordan coeﬀicient associated with projecting the angularmomentum of the initial state and the photon absorbed into the angular momentum of the final
state. Since each hyperfine state in the higher rotational state is swept over a transition to all other43

Fraction of power reflected

0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00

30 GHz43 GHz0

1

2

3

Window thickness (mm)45Figure 3.13: The fraction of power reflected off of a glass window of a given thickness, using a
relative permittivity of 3.8. The dashed line represents the thickness of the windows used in theexperiment.

hyperfine states in the lower rotational stage, the transfer rate out of a specific hyperfine state is
at least the rate given by the transition with the highest Clebsch-Gordan coeﬀicient. The totaltransfer out of the higher rotational state is then limited by the slowest of these maximum rates.
Thus taking the minimum over the hyperfine initial states of the maximum over each state it goesto, we can establish a lower bound on the total population transfer rate.
The required microwave powers are shown in Table 3.2.
Frequency (GHz)
Directivity (dBi)
Limiting CG coeﬀicient
Required electric field (V/cm)
Required Pout (dBm)

Lower doublet
29.1038
17.92
1
.0075
-26

Upper doublet
29.0942
17.92
1
.0075
-26

J = 3→ 2
43.6450 (swept)
21.41
p
10/21
.01094-24

J = 2→ 1
28.9828 (swept)
17.90
p
1/2
.011
-23

Table 3.2: The power required to achieve a microwave coupling rate of Γµ = 200 s−1 , calculatedusing Equation 3.19 for the electric field, with powers calculated using Equation 3.28 for 30 GHz
and a 20% reduced Equation 3.21 for 43 GHz with the relevant directivity.

A microwave coupling rate of Γµ = 200 s−1 is chosen to be fast compared to the fastest
relevant spontaneous decay rate (∼20 s−1 ).

44
3.2.3.3

Microwave sources

A photograph of the setup for generating the microwaves is shown in Figure 3.14.

Figure 3.14: Photo of the microwave generating setup.

The system consists of many discrete microwave components assembled together. Each microwave component gets its own power supply that current limits to the appropriate value foreach component. Furthermore, the transistor-based microwave components will suffer damage if
the positive supply rail is applied before the negative supply rail. In this case the positive rail is
supplied through a relay which is only open if the negative rail is active. Furthermore, the negative
rail is given a much larger parallel capacitance so that if the negative rail is interrupted the voltage
slowly returns to zero giving the relay time to disengage. Each custom power supply runs on thesame ±15 V supply so that the whole system can be easily powered off or on.
A schematic of the setup is shown in Figure 3.15.
The system consists of two branches, one at 30 GHz to address J = 1 ↔ J = 2 transitions
and one at 43 GHz to address J = 2 ↔ J = 3 transitions. The signals then get combined andapplied to the ions with a custom broadband microwave horn.

45
cavityhorn (v) combiner (u)bandpass (t) quadrupler (s)to ions×4

amplifier (h)̸→←cavitybandpass (i)switch (f)̸→doubler (g)←×2isolator (j)switch (e)↗
◦variable
attenuator (k)isolator (r)cavitybandpass (q)cavitybandpass (m)×4×4quadrupler (p)quadrupler (l)switch (o)switch (d)

(c)
v=1,2
Urc

label
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v

(b)
l
Udepl

(n)

(a)
u
Udepl

description
1/8(29.082941 GHz) source
1/8(29.115175 GHz) source
1/8(28.81-28.98 GHz) swept source
switch, doublet selectswitch, pumping/depletionswitch, 30 GHz on/off
passive frequency doubler
amplifier, 20 dB
cavity bandpass 7.25 to 7.75 GHz
isolatorvariable attuenuator
active frequency quadrupler
cavity bandpass 27.5 to 31 GHz
1/16(43.63-43.67 GHz) swept source
switch, 43 GHz on/off
active frequency quadrupler
cavity bandpass 10.55 to 11.8 GHz
isolator
active frequency quadrupler
cavity bandpass 39 to 45 GHz
broadband splitter/combiner
microwave horn, broadband (28.0 to 45.0 GHz)

v=0
Urc

manufacturer
RF-Consultant
RF-Consultant
Agilent
Mini-Circuits
Mini-Circuits
Mini-Circuits
Marki
Avantek
Pasternackµ-wave Associates
narda
Marki
Pasternack
Agilent
Mini-Circuits
Marki
Pasternack
Ditom
Marki
RF-Lambda
Mini-Circuits
Dolph

Figure 3.15: The circuit for generating microwave signals

part no.
TPI-1001-B
TPI-1001-B
E8251AZYSWA-2-50DR+
ZYSWA-2-50DR+
ZYSWA-2-50DR+
D97281
APT-18646
PE8741
MA7L604
4798
AQA-2156
PE8747
N5181AZYSWA-2-50DR+
AQA-1933K
PE87FL1001
D3I8012
AQA-2156
RBPF39G45G
ZC2PD-V18443+
DH-320WDBLP122PA

46
3.2.4

Spectroscopy

Once again we can measure the exact energy for each transition by using them to deplete therelavent states and looking for a dip in the population. The two Udepl transitions are measured in
Figure 3.16.

Population in v = 0, J = 1

8070

Optical pumping
Optical pumping + Udepl

60
50
40
30
29.08129.08229.08329.08429.114
Udepl frequency (GHz)29.11529.11629.117

Figure 3.16: Frequency scan of Udepl at an Erot of 60 V/cm. The lower frequency dip corresponds
to optically pumping into the upper doublet then depleting the upper doublet, while the higher
frequency dip corresponds to optically pumping into the lower doublet, then depleting the lowerldoublet. The lower doublet transition Udepl
fits to a frequency of 29.115642(57) GHz, while theuupper doublet transition Udepl fits to a frequency of 29.082823(29) Ghz.

v=1 is measured by pumping
The swept microwaves can be measured in a similar manner. Urc

up to v = 1, J = 2 and then watching population fall back down to v = 0, J = 1 as the microwave
and laser pair couple ions out of this state. The results are shown in Figure 3.17.
An important note is that at 60 V/cm, sweeping over this whole range will also address the
mf = ± 32 transition in v = 0, thus depleting some of the states of scientific interest. This is shownv=1 to a maximum of 28.995 GHz to maintain a safein Figure 3.18. To avoid this, we only sweep Urc

distance from these transitions.
v=2 is measured in Figure 3.19.
Finally Urc

47

Population in v = 0, J = 1

70

Optical pumping +Optical pumping +60vc
v=1 v=1
vc
rc

50
40
28.85

Transition strength (arb)

v=1

28.90

28.95
fcenter (GHz)

29.00

29.05

29.00

29.05

Transitions
Theoretical pop. transfer

0.4
0.2
0.0

28.85

28.90

28.95
f (GHz)v=1 .
Figure 3.17: Microwave scan of the Urc
Top: Spectroscopy of the transition. The ions are first optically pumped into v = 1, then either
v=1
only Lv=1
vc is turned on to pump out v = 1, J = 2 which falls back down to v = 0, J = 1 or Urc
is also turned on, resulting in additional population from v = 1, J = 2. The microwaves are swept
only 5 MHz around the frequency on the axis.
Bottom: Theoretical location and relative strengths of transitions based off of a 32-level Hamiltonianas calculated by Kia Boon Ng [39]. The theoretical population transfer is found by modeling a fixed
time of sweeping the frequency over each transition using the calculated transition rates.

48

v=1 depleting the upper doublet if the frequency is swept too high.
Figure 3.18: Demonstration of Urc
Top: calculated rotational microwave transitions for each vibrational manifold at Erot = 60 V/cm.
Labeled are the lowest v = 0 transition that addresses the upper doublet as well as the transitionu
Udepl
that we use to intentionally deplete the upper doublet.
Bottom: measured population in the ro-vibrational manifold containing the states of interest. The
populations are measured after optically pumping, then either doing nothing or applying the swept
v=1 , then depleting the lower doublet. This processes leaves only population in the upper doublet,Urcthus a dip in population indicates a transfer of population from the upper doublet.

Population in v = 2, J = 2

49

80
60
No optical pumping
Optical pumping + vvc= 2
Optical pumping + vvc= 2 rcv = 2

40
20Transition Strength

28.70

28.75

28.80

28.85
28.90
fcenter (GHz)

28.95

29.00

Transitions
Theoretical pop. transfer

0.6
0.4
0.2
0.0

28.70

28.75

28.80

28.85
f (GHz)

28.90

28.95

29.00

v=2 .
Figure 3.19: Microwave scan Urc
Top: Spectroscopy of the transition. First the ions are optically pumped up to v = 2, then the
population there is depleted using the microwaves, and measured directly. The population inv = 2, J = 2 is measured by dissociating from the (3 ∆1 , v = 2, J = 2) → (Ω = 2, J = 3) transition
measured in Figure 6.5. The microwaves are swept only 5MHz around the frequency on the axis.
Bottom: Theoretical location and relative strengths of transitions based off of a 32-level Hamiltonianas calculated by Kia Boon Ng [39]. The theoretical population transfer is found by modeling a fixed
time of sweeping the frequency over each transition using the calculated transition rates.

Chapter 4
Electric Fields

4.1

Electrode design
The electrodes used to apply fields to the ions are made up of 8 long ”rod” electrodes runningthe length of the trap to apply fields in the x-y plane and many small uniformly spaced ”ring”
electrodes along the axial direction to apply fields along the z-axis. The electrodes were designed incollaboration with and fabricated by Calvin Schwadron of the JILA instrument shop. The electrodeconfiguration is displayed in Figure 4.1.

y
x
Rods

Quadrant
Rings

Special
Rings

Rings

z

Figure 4.1: CAD rendering of the finalized electrode design.

The ions enter the electrode assembly out of the page between the quadrant ring electrodes(yellow). Both of these electrodes are divided into four regions with individual control, so they canapply localized fields in the x-y plane as well as along the z axis. They can then be used to pulse
an electric field along +x right after ionization in order to stop the ions. In addition, individual

51
control of the special ring electrodes (red) allows us to pulse a high voltage electric field along −z tokick the ions to the left onto the detector. Finally, the rest of the ring electrodes (green) are wiredwith every fourth electrode held at the same potential. We can then use these rings to translate
the ions forward and backward along the z-axis.
The dimensions of the electrode assembly are given in Figure 4.2.

lDimension
drod
Drod
Lrod
dring
lring
Lring
dcollarValue
12.7 mm
11.7 cm
123.2 cm
13.5 cm
2 cm
1 mm
19.3 cm

Description
Diameter of rod electrodes
Distance between opposite pairs of rod electrodes
Length of rod electrodes
Inner diameter of the ring electrodes
Axial spacing of ring electrodes
Thickness of ring electrodes
Outer diameter of the support collars

Figure 4.2: Relevant dimensions of the electrodes

The choice to have the rod electrodes inside of the ring electrodes is made because theelectric fields that need to be generated in x̂ and ŷ are much stronger than the fields that need tobe generated along ẑ, and having another set of electrodes between the other electrodes and theions shields the ions from the electric fields generated by the outer electrodes.
The dimensions are selected with several considerations in mind. First, the whole assemblymust fit inside the vacuum chamber. The vacuum tube is a 10” outer diameter flange nipple, which52
is the largest diameter of the standard ultra-high-vacuum components and has an inner diameter of
19.7 cm, inside of which dcollar must fit. The rod electrodes are then placed as far apart as possible
in the interest of making the fields near the center of the trap as homogeneous as possible.

4.1.1

Overview

There are three main electric fields the electrodes must generate (in addition to the pulses
to stop the initial velocity of the ions and to kick them out onto the detector). These fields aredemonstrated in Figure 4.3. First, the electrodes must apply a strong electric field to polarize themolecules (Figure 4.3.a). For this field not to apply a time-averaged force on the ions, the field mustrotate. This way there is no net force on the ions over one rotation cycle, and the instantaneous
force acts as a centripetal force as the ions move in a circular trajectory. This field will be further
explained in Section 4.1.4.
Next, the electrodes must generate fields to trap the ions, and translate them along the axial
dimension of the trap. This is done with an oscillating quadrupole field (Figure 4.3.b) appliedby the rod electrodes and a sinusoidal potential along the axial direction (Figure 4.3.c) appliedwith the ring electrodes. The mechanism for which these two fields lead to trapping of the ions is
explained in Section 4.1.3

4.1.2

Ponderomotive potential

Before discussing the fields generated by the electrodes, we need to discuss the ponderomotive
effect, by which sinusoidally oscillating inhomogeneous fields impart a time-average force on theions. An argument for this force is as follows:
If there is a periodic electric field⃗ r, t) = E⃗ 0 (⃗r) cos(ωt),
E(⃗

(4.1)

then the equation of motion for a charged particle of charge q and mass m ismd2⃗r
⃗ 0 (⃗r) cos(ωt).
= qE
dt2

(4.2)

53

a)

b)

c)
Figure 4.3: Electric potentials generated with the rod and ring electrodes. Left: slice of the electric
potential for the respective type of potential applied. Potentials are generated using COMSOLMultiphysics finite element analysis software. Red represents positive potential while blue represents negative potential. Right: Electric potential surfaces for each of the fields. The two horizontal
axes represent physical dimensions while the vertical axis represents potential depth.
a) Vrot polarizing fieldb) VRF , oscillating saddle point potential that provides radial confinementc) Vaxial , sinusoidal potential that provides axial confinement at each valley of the sinusoid

54
We then assume the solution looks like an small oscillation (micromotion) at the driven frequency
ω with amplitude ⃗rµ , plus a slow spatially larger secular motion ⃗rsec (t):⃗r(t) = ⃗rµ cos(ωt) + ⃗rsec (t).

(4.3)

If the micromotion is small compared to the secular motion ⃗rµ ≪ ⃗rsec , then we can expand theelectric field about ⃗rsec :⃗⃗ ⃗⃗ 0 (⃗r) ≈ E⃗ 0 (⃗rsec ) + ∇E
E0 (⃗rsec ) · ⃗rµ cos(ωt),⃗⃗ ⃗where ∇
E0 is the gradient of the vector field, yielding a tensor
∂ ⃗ ∂x E0 


 ⃗ 
⃗⃗ ⃗
∇
E0 ≡  ∂ E.
 ∂y 0 


∂ ⃗
∂z E0

(4.4)

(4.5)

To find the amplitude of the micromotion, we can plug Equations 4.3 and 4.4 into Equation 4.2:d2⃗rsecm −ω ⃗rµ cos(ωt) +
dt2



2


⃗⃗ ⃗
⃗
= q cos(ωt) E0 (⃗rsec ) + ∇E0 (⃗rsec ) · ⃗rµ cos(ωt) .


(4.6)

Grouping terms by timescale,



2
⃗
⃗E
⃗ 0 (⃗rsec ) cos(ωt) + q ∇⃗ 0 (⃗rsec ) · ⃗rµ cos2 (ωt) − m d ⃗rsec = 0.
mω ⃗ru + q E
dt2
2





(4.7)

Note that the second two terms are small in magnitude compared to the first term, and they change
on different timescales. Therefore, we can extract the micromotion amplitude by setting the firstterm equal to zero, giving⃗ 0 (⃗rsec )
qE
.
mω 2

(4.8)

⃗ 0 (⃗r) cos(ωt)
≈ qE

(4.9)

⃗rµ ≈ −
We can then write Equation 4.2 asm⃗ 0 (⃗rsec )d2⃗rsecqEcos(ωt)
+
ω2
ω2m
dt2

!



d2⃗rsec⃗⃗ ⃗⃗⃗q E0 (⃗rsec ) cos(ωt) + m≈ q E0 (⃗rsec ) + ∇E0 (⃗rsec ) · ⃗rµ cos(ωt) cos(ωt)
dt2d2⃗rsec⃗⃗ ⃗m
E0 (⃗rsec ) · ⃗rµ cos2 (ωt).
≈ q∇
2
dt

(4.10)(4.11)

55
Finally, we can plug Equation 4.8 into this and average over a cycle of ω to getm⃗⃗ ⃗⃗ 0 (⃗rsec )d2⃗rsecq2∇
E0 (⃗rsec ) · E
=
−
.
dt2
2mω 2

(4.12)

We can use a modified version of a chain rule (proven in Appendix B),
:0


1⃗
⃗⃗ ⃗ ⃗
⃗
2
⃗
⃗×
−
(∇
E) · E = ∇|E|
E
×
(∇
E)
2

,

(4.13)

where we have ignored the cross term since electrostatic fields that obey Maxwell’s equations must
⃗ ×E
⃗ = 0. With this, we arrive at
have ∇m2
2
d2⃗rsec⃗ q |E0 (⃗rsec )| ,
= −∇
2
2
dt
4mω

(4.14)and thus the secular motion is described by an effective potentialU=

q 2 |E0 |2.
4mω 2(4.15)

This effective potential is known as the ponderomotive potential. This potential can be thought ofas the classical harmonic oscillator energy of the micromotion.
1
1
1
2
U = mω 2 ⟨rmicromotion⟩t = mω 2 ⟨(rµ cos(ωt))2 ⟩t = mω 2 rµ2 ,2
2
4(4.16)

yielding the same result if we use the expression for rµ from Equation 4.8.
4.1.3

Paul trap

The ions are trapped axially with the ring electrodes producing a sinusoid. The voltages
applied to the rings are shown in Figure 4.4.
By modulating ϕtranslate , the minima of the sinusoid are spatially translated along z. Thusby changing the phase of the ring electrodes, the ions can be translated. The relationship between
spatial translation and phase applied to the electrodes is given by
ztranslated = ϕtranslate

4lring.
2π(4.17)

While this potential is confining along z, it is anti-confining along x and y (see Figure 4.3.c). In
general, any static electric potential we can apply with the electrodes can only be confining in up

56
Ring
1
2
3
4

Potential applied
Vring,pk cos( 0π
2 + ϕtranslate (t))
1πVring,pk cos( 2 + ϕtranslate (t))
Vring,pk cos( 2π
2 + ϕtranslate (t))
3πVring,pk cos( 2 + ϕtranslate (t))

Figure 4.4: Voltages applied to the ring electrodes to trap the ions axially and to translate them.
The potentials are periodic every fourth ring, e.g. the voltage on the 5th ring is the same as thevoltage on the 1st ring.

⃗ ·E⃗ = 0). In order to overcome this anti-confining
to two dimensions since there are no sources (∇potential and confine the ions along x and y, we apply a psuedopotential that is constant alongz and confining along x and y. This is done by applying a quadrupole potential with the rod
electrodes that oscillates at ωRF = 2π × 50 kHz. The voltages applied are shown in Figure 4.5.
Rod
1
2
3
4
5
6
7
8

Potential applied
VRF,pk cos(ωRF t)
VRF,pk cos(ωRF t)
VRF,pk cos(ωRF t + π)
VRF,pk cos(ωRF t + π)
VRF,pk cos(ωRF t)
VRF,pk cos(ωRF t)
VRF,pk cos(ωRF t + π)
VRF,pk cos(ωRF t + π)

Figure 4.5: The voltages applied to the rods in order to generate the RF quadrupole potential

Even though the resulting potential is a saddle point instantaneously, the oscillations of the
field give rise to a ponderomotive potential that is proportional to |E|2 , which is radially confining.
This is displayed in Figure 4.6. The saddle point potential near the center of the trap looks like
VRF ≈ αRF VRF,pk (x2 − y 2 ),(4.18)

57

Figure 4.6: The transformation of the RF saddle point potential to its confining effective ponderomotive potential near the center of the trap.

where αRF is a scaling factor based on the geometry of the electrodes. Then the electric field is⃗ RF = −∇V⃗ RF ≈ αRF VRF,pk (−2xx̂ + 2y ŷ).
E

(4.19)

This gives rise to a ponderomotive potential of
2 V22e2 αRFe2 (αRF VRF,pk )2 (x2 + y 2 )e2 |ERF |2
RF,pk ρURF ===,(4.20)
2
2
2
4mωRF
mωRF
mωRF
p
where e is the electron charge and ρ is the cylindrical radial coordinate ρ = x2 + y 2 . The potentialfrom the axial field looks like (up to a constant which can be neglected)
Uring = e Vring ≈ e αring Vring,pk (2z 2 − ρ2 ),(4.21)

where αring is once again a scaling factor due to electrode geometry. Thus the full trapping potential
experienced by the ions is given by
1
Utrap = URF + Uring =22 V22e2 αRF
RF,pk
2
mωRF

|

!
− 2e αring Vring,pk
{z

ρ2 +
}


1
4e αring Vring,pk z 2 .
2|{z}(4.22)kzkρThe effective potential has the form of a harmonic oscillator potential. In this form it is easy to
see that the secular motion of the ions in the trap will undergo oscillations at frequencies in ρ andz respectively:
r
ωρ =
r
ωz =

s
kρ
=
m
kz
=
m

r

2 V2
2e2 αRFRF,pk2
m2 ωRF−4e αring Vring,pk.
m

2e αring Vring,pkm(4.23)(4.24)

58
We can invert these equations to find the required voltages to apply to the electrodes in order to
get the desired secular frequencies:qmωRF
VRF,pk =Vring,pk =2ωρ2 + ωz22αRF emωz2.
4αring e(4.25)(4.26)The value of the scaling constants can be found by fitting the potentials from the COMSOL
simulations, yieldingαRF = 348 m−2αring = 8.73 m−2 .

(4.27)

With this we can calculate the parameters of the trap. For example, to get a secular frequency of
2π × 2 kHz in each dimension, the voltages are VRF,pk = 25.6 V and Vring,pk = 11.8 V. The digital toanalog converters (DACs) driving the ring electrodes can only go up to 10 V, so we instead apply
a Vring,pk of 9.95 V to stay within the limit, giving us an theoretical ωz of 1.84 kHz and ωρ of 2.08
kHz.
Far enough away from the center of the trap, the harmonic approximation of the potentials
break down. Furthermore, the radial anti-confining of the ring potential grows faster than theradial confining of the RF potential, leading to a point at which the total effective potential curls
over and is no longer confining. This is shown in Figure 4.7.
The RF confining potential is stronger along the diagonal axes such as (b) than the axes ofthe saddle point such as (a), and are thus more tightly radially confined. We can assign each of
the local maxima where the potential curls over an effective trap depth ∆U , equal to the difference
between the minimum of the trap and the local maximum. These values are calculated to be∆Uaxial = 1330 K





(4.28)∆Uradial,a = 894 K







∆Uradial,b = 5070 K.

59Figure 4.7: The total effective trap potentials experienced by the ions with VRF,pk = 25.6V andVring,pk = 9.95V using |ERF |2 and Vring fields generated in COMSOL. Left: a diagram of the RF
potentials showing the two slices over which the total effective potentials are plotted. Right: Surface
plots of the total effective potentials in Kelvin (related to the potential energy by the Boltzmann
constant). Then highlighted are the potential minimum at which the ions sit and the four points
in which the potential is no longer confining, indicating the boundary of the trap.

In addition, the distance ∆r from the bottom of the trap to each of these local maxima is also a
useful criteria:∆raxial = 4.0 cm





∆rradial,a = 2.7 cm







∆rradial,b = 4.4 cm.

(4.29)

With cross-dimensional thermalization, the trap depth is limited by the weakest radial trap depth,
so only ions cooler than 900 K will be trapped. A final useful quantity is the maximum slope of
the ponderomotive potential before the local maximum, which tells us how much we can tilt the

60potential before it no longer has any minimum. By dividing by the charge q of the ion, this thenrepresents the maximum electric field we can apply in that direction before there is no trap:



1 ∂U



q ∂z axial = 0.049 V/cm



1 ∂U
(4.30)q ∂ρ radial,a = 0.047 V/cm







 1 ∂U= 0.228 V/cm.
q ∂ρ
radial,b

4.1.4

Rotating polarizing field

The voltages applied to the electrodes in order to generate the rotating polarizing field E⃗rot
are shown in Figure 4.8.
Rod
1
2
3
4
5
6
7
8

Potential applied
Vrot,pk cos(ωrot t + 0π
4 )1π
Vrot,pk cos(ωrot t + 4 )Vrot,pk cos(ωrot t + 2π
4 )Vrot,pk cos(ωrot t + 3π
4 )4π
Vrot,pk cos(ωrot t + 4 )Vrot,pk cos(ωrot t + 5π
4 )6π
Vrot,pk cos(ωrot t + 4 )Vrot,pk cos(ωrot t + 7π
4 )Figure 4.8: Voltages applied to the rod electrodes to apply the rotating polarizing field

The homogeneity of the field applied is very important for several reasons. Firstly, themagnitude of the field goes directly into the Ramsey frequency f0 (see Section 5.1), since
f0 ∝ Brot ∝ rrot ∝ Erot .

(4.31)−4 from Section 5.3.2 requires that in theThis means that the homogeneity requirement of ∆f
f < 10−4
rot
region in which the ions are trapped, ∆E
Erot < 10 .

Secondly, any gradients in the strength of the field applied generates a ponderomotive potential that could disrupt the ions and cause them to fall out of the trap (while the field is rotating

61
rather than oscillating meaning that it can’t be written in the form of Equation 4.1, the ponderomotive force argument can be made separately in each dimension, where the projection of the field
onto that dimension can be written in this form).
If the electric field oscillating at ωrot applied is composed of the intended uniform-magnitude⃗ dev , then the ponderomotive potential isE⃗rot and a small spatially-varying perturbation E
Upond,rot =



2
2
e2
2
⃗rot + E⃗ dev = e⃗rot · E⃗ dev +|Edev |2 .
E
|E
|
+
2
E
rot
2
2
| {z }
4mωrot4mωrot(4.32)

cross term

Since |Erot |2 is a constant (not spatially varying), it can be neglected from the potential. Further,since |Edev | ≪ |Erot |, the ponderomotive potential comes mainly from the cross term
Upond,rot ≈e2 ⃗⃗2 Erot · Edev .
2mωrot(4.33)

This places a constraint on how big the deviation can be before the ions can be lost from the trap
⃗ dev | <∆|E

22mωrotUtrap depth = 0.6 V/cm.
e2 |E⃗rot |(4.34)

given a Utrap depth of 900 K, an Erot of 60 V/cm, and an ωrot of 2π × 150 kHz.
Studies of the electrode geometry to meet these requirements is detailed in Kia Boon Ng’s
thesis [39].

4.2Electronics

4.2.1

Amplitude and timing requirements

The electric fields described in the previous section must have their amplitudes modulated
during the experiment, as well as have well defined phases with respect to laser pulses and othertrap potentials. A typical experimental sequence is shown in Figure 4.9 that demonstrates the
specific requirements of these signals.
At the beginning of stage 1, our pulsed lasers ionize the beam of neutral ThF molecules, andspecial ring electrodes apply a pulsed potential gradient to slow them. At the same time, the RF
confining potentials are diabatically turned on. The initial phase of VRF is defined such that at the62

Figure 4.9: AC trap electrode envelope profiles for a typical experiment. The experiment is broken
into different functional periods indicated with numerals, with three phase matching conditions
indicated with letters.
1) Catching the ions2) State preparation3) π/2 pulse
4) Ramsey evolution time5) π/2 pulse
6) Depletion
7) Dissociation and kickouta) and b): The trap must be turned on and off when its amplitude is at a maximum. This corresponds to the point in the ions’ micromotional trajectories where velocity is minimized, meaning
they will not be heated when caught and the cloud will not undergo excess expansion when dissociated and kicked out.
c): In order to align the axis of the molecules with the polarization of the dissociation pulse, Erotmust have a well defined phase at the time of dissociation/kickout. It must then be ramped down
so that it does not affect the trajectories of the dissociated ions as they travel to the detector.

catch time, the RF potentials are at a maximum, as shown in phase criteria a). This minimizes
possible heating of the ion cloud from the applied potentials. The ions are then held for some timeso they can thermalize and reach equilibrium mean trajectories. Erot is then adiabatically ramped
up to lift the degeneracy of the hyperfine states, thus beginning stage 2. In this stage, the ionsare optically pumped into the J = 1, mF = ±3/2 states. Then in stage 3, Erot is ramped down toallow for mixing of the mF states, then ramped back up. This results in an effective π/2 pulse for63each doublet. In stage 4, the ions are allowed to freely Ramsey evolve, before undergoing another
π/2 pulse in stage 5. Finally, we optically deplete the states in stage 6, before ramping up the trap
depth and performing dissociation in step 7. At the end of step 7, the dissociated ions are thenkicked out and more special ring electrodes accelerate them towards an imaging MCP.
Once again, the kickout must be done at a place in the micromotion where velocity is minimized to avoid additional expansion of the cloud after the trap is turned off (see phase requirementc). To achieve this, the kickout time is chosen to be an integer multiple of RF cycles from thecatch time. Finally, the polarization of the dissociation photons with respect to the quantization
axis is critical in achieving high dissociation eﬀiciency, so the phase of Erot during dissociation andkickout must also be well-defined. This is done by setting the initial phase of Erot at the start ofthe experiment such that it will match up at the dissociation time.

4.2.2

DDS boards

In order to generate these waveforms, a multiplexed system of field programmable gate arrays(FPGAs) and direct digital synthesis (DDS) signal generator chips are used. The system wasdesigned by Felix Vietmeyer of the JILA electronics. Each board, which outputs a single waveform,
contains its own FPGA and DDS. A schematic is shown in Figure 4.10.
Each DDS has four channels of output which can each be programmed with their own phaseand amplitude. These channels then each pass through their own multiplying digital to analog
converters (DACs) and switches which can ramp the amplitudes and turn the signal on and off
in real time. They then pass through a summing amplifier to combine the signals into a single
channel.
The system also has two standalone DACs which can also be controlled in real time, butthese are not used in the current configuration (the signals are AC coupled to the electrodes so DC
components cannot be applied with these boards).
The computer programs the system via a USB serial decoder which talks to the FPGA. The
FPGA logic can be divided into two devices, one labeled “A1” which forwards the serial commands

RFTransformers

1.9 MHzLow pass

Preamps
AD829J

64

Fixed-gain Switches
Mult.
JFETs
DACs
ADG5212B
AD5543 ADA4627-1
×AD5543×AD9959 DDS
Ch1

AD5433

IO update Ch2(t = 0)Ch3High Speed
Summing Amp
ADA4894×Ch4AD5433×+Out

TTL InUSB
SerialDecoder

A1 Device(DDS control)
A2 Device(Sequencer)

t=0

DACs

Amps

AD5542
DA

AD8675AD5542 ADA4627-1
DA
Spartan 6 FPGA

Figure 4.10: A schematic of the JILA-designed DDS boards. Blue wires represent synchronous,
real-time digital control while red wires represent asynchronous programming done between shotsof the experiment. Black wires represent analog signal channels.

to the DDS chip with the appropriate formatting, and another labeled “A2” which programs asequencer. The sequencer stores a list of times (as number of clock cyles) and values of multiplying
DACs, switches, and DACs. Once the system receives a TTL pulse indicating the experiment has
begun, it then executes this table synchronously.
Each channel of the DDS can be programmed asynchronously with its own amplitude andphase. Generally only relative amplitudes that remain fixed (such as the voltages prescribed in
Figure 4.22) are written to the DDS, and then their amplitudes are ramped synchronously during
the experiment using the multiplying DACs. The only synchronous component of the DDS is that
the TTL input is tied to the DDS’s ‘IO update’ pin, which resets the phase to the programmed

65
phase at the next logical clock rising edge, thus synchronizing the phases of each channel to thet = 0 of the system.
Synchronizing the t = 0 of the DDS and the sequencer to each other and also to the t = 0 of
the system supplying the TTL pulse requires some care. Generally these systems are operating atdifferent logical clock rates, which can lead to timing jitter if the timings are not synchronized to
the least common denominator of these clock periods. A diagram of the clock distribution setup is
shown in Figure 4.11.
Main 10MHz Clock
Clock Distributor
Function Generator,
125MHz clock
PLL x8,
1GHz

Other Lab
DDG,
10Hz clock Equipment

FPGA
Artiq
TTL out

DDS box

PLL x10,
100MHz‘Clock’ Distributor

Phase
Accumulator

Sequencer
FPGA

LogicalProcessor

÷4 Flip-Flop,
25MHz

Clock Distributor

Phase
Accumulator

Sequencer
FPGA

LogicalProcessor

AD9959 chip
DDS Board (Master)

Other DDS
Boards (Slave)x8

÷4 Flip-Flop,
25MHz

AD9959 chip
DDS Board (Slave)sync out

sync in

Clock Distributor

Figure 4.11: Diagram of the timing distribution system. Each solid box represents a physicallydistinct piece of hardware. Each arrow color represents a different clock frequency of the system.
Red represents 10 MHz, blue represents 100 MHz, orange represents 25 MHz, and green represents
a TTL pulse signifying the DDSs’ t = 0 of the experiment.

66
Even though the DDS’s phase accumulator runs at 100 MHz (meaning the output moves toa new discrete phase once per 10 ns), the logical processor of the DDS chip only operates at 25
MHz, meaning that the DDS only looks for the TTL rising edge once every 40 ns. The sequencer,
however, has a logic clock of 100 MHz. Since they both look for the rising edge of the TTL trigger
on each clock cycle, they can have a relative jitter of up to three cycles of 100 MHz. To avoid
this, the TTL trigger must be sent at an integer multiple of (25 MHz)−1 or 40 ns. This is doneby ensuring that Artiq sends the TTL trigger at an integer multiple of 40 ns from the rising edgeof the 10 Hz clock signal from the DDG. Because 1/10 Hz is cleanly divided by 1/25 MHz, the 10
Hz clock rising edges are separated by an integer multiple of 40 ns (one 10Hz clock cycle is exactly
25×105 cycles of the 25 MHz clock), any TTL pulse given at a 40 ns increment from a rising edgeof the 10 Hz clock will be a multiple of 40 ns from the last TTL pulse, which means no timing jitterfrom incommensurate system clocks will occur.
Furthermore, because the rising edge of the logical clock for each DDS occurs once very four
rising edges of the master clock, the logical clock for different DDS boards can be up to threemaster clock cycles apart. This is why the DDS boards have a synchronization function, whereby
the logical clocks of nine “slave” DDS boards are aligned to the logical clock of one “master” DDS
board.
Wherever timing signals are distributed to the boards, a clock distributor (Texas Instruments
LMK00301 evaluation board) is used to ensure each board receives the signal at the same time
without any relative phase delay.

4.2.3

Resonant transformers

Second harmonics of Erot on the rod electrodes can introduce systematic error into the experiment [51]. In addition, the amplitude of Erot generating voltages must be quite high (429 Vpk
in order to reach 60 V/cm). In order to deal with both of these issues, the voltages coming out of
the DDS are amplified with a PA94 high-voltage op-amp with a gain of 100, and then applied tothe electrodes through a resonant transformer circuit, which provides extra gain on resonance (re-

67
ducing the requirement on the source voltages) and also attenuation of higher harmonics generated
by nonlinearity in the voltage sources.
In addition, the rest of the voltages are introduced to the electrodes through various couplings
based around the resonant transformer design. The rod driver consists of four identical circuits
which each drive a set of two rod electrodes furthest from each other (e.g. rod 1 and rod 5). The
rod driver electronics are designed to apply the following different voltages to each of these pairsof electrodes:• An independent DC shim voltage for each rod• VRF at 50 kHz with the same phase and amplitude.
• Vrot at 150 kHz with the same phase and opposite sign. This must have suppressed higher
harmonics to prevent systematic effects
In order to apply these voltages to the same electrodes, the following scheme is used, with thecircuit diagram in Figure 4.12 and a photo in Figure 4.13:
Vrot is applied via a transformer whose inductance paired with the capacitance on the electrode forms a resonant circuit. This allows resonant enhancement of the gain and suppression ofhigher harmonics due to nonlinearity in the DDS voltage or the op-amps. In addition, groundcan be defined relative to the center tap of the transformer, allowing each rod to receive opposite
polarity of Vrot . VRF is then applied to the center tap of the secondary side of the transformer. Thisway, VRF is applied with the same polarity to both rods, and the inductance on either side of thesecondary partially cancels out, and with ωRF far away from the resulting resonance this portionof the circuit can be driven non-resonantly. Finally, DC shims are applied to each rod individuallywith a resistive coupling.
The voltages on the rods can be measured via a resistive voltage divider to ground (resistors
RQ and Rgain ). This resistance to ground both limits the quality factor of the resonance, and offers apoint to measure the voltage. The values are chosen with Rgain ≪ RQ so that the voltage is divided
from high voltage down to a voltage more suitable to read with a scope. In addition, this ratio

68

Rgain
+

RDAC

VDAC, Shim1

−

PA94

ADA4898

VDDS, RFRod 1 Readout

RRF

CDAC
RQ

Cshim,1

Csecondary

Cparasitic,1
Rod 1

VDDS, Erot

PA94

Cprimary
Cprimary
Csecondary
Rod 5

Rprimary
Cshim,5RQ

Cparasitic,5RDAC

VDAC, Shim5

−

+

Rgain

ADA4898

CDAC

Rod 5 Readout

Figure 4.12: Circuit for driving the rod electrodes.

ensures that parasitics introduced by the measurement circuit do not interfere with the resonance.
The readout is done with an isolation op-amp of unity gain, in a voltage follower configuration.
Cprimary and Rprimary are designed to protect the op-amp that drives the primary. The
capacitive coupling means that any DC signal on the input to the op-amp sees a high impedance,
and Rprimary limits the current sourced from the op-amp. In addition, the values are chosen suchthat the resonance of the primary given by1

ωprimary = q

Lprimary ( 12 Cprimary )(4.35)

69
Op-ampcontrolboardsHigh voltage
op-amps(PA94)

Water
cooling

Hand-wound
air-coretransformers

Figure 4.13: Photo of the rod-driver box. The PA94 op-amps are thermally coupled to the wallof the enclosure. On the other side of the enclousure is a thermal block containing a copper pipe
flowing cold water to regulate the temperature of the op-amps. The transformers are each placedin a shielding Faraday cage to prevent coupling between channels or pickup from external sources.

is far away from the desired resonant frequency, i.e.
ωprimary ≪ ω0 .
The value of Rprimary is discussed further in Section 4.2.4.2, and the values settled on areRprimary = 100 ΩCprimary = 33 nF.

(4.36)(4.37)

Csecondary , RDAC , and CDAC are chosen to allow the shim channels to address the DC components of the rod voltages independently from the AC components. The capacitors should look
like shorts in the 100 kHz regime but open to DC voltages. Furthermore, RDAC should be large
compared to the reactances of the circuit in this regime. CDAC protects the DAC from any RFthat has made it through RDAC by sending it to ground. The values chosen to satisfy the above

70
requirements are:
Csecondary = 100 nF





CDAC = 100 nF







RDAC = 1 MΩ.

(4.38)

Finally, Cparasitic is the parasitic capacitance both in the cabling and arising from the geometry of
the rod electrodes. The capacitance is measured to be
36 pF| {z }

Cparasitic =60 pF| {z }+

cable
transformer→monitor

+

cable
monitor→vacuum

63 pF| {z }= 158 pF.

(4.39)

inside vacuum
chamber

In order to correct for asymmetries in the system, Cshim is a variable capacitor with a range of 30
pF that can be tuned until all rods have equal capacitances to ground.

4.2.4

Transformer theory

In order to treat the center tapped transformer, we treat the center tapped transformer as
two separate transformers, as in Figure 4.14.
A
A
Lp
B

C

↙ M↘

Ls

E
D

↙ M/2↘

Lp /2

Ls /2

=
↙ M/2↘

Lp /2
B

C

E

Ls /2
D

Figure 4.14: Equivalent circuit for a center tapped transformer

We can then write out the dynamics of this split transformer in terms of the induced voltages
across each inductor. Let VA,B be the voltage between terminals A and B and iA→B be the current

71
flowing from terminal A towards terminal B. The equations governing the transformer are then:





diD→E
M diC→E

VA,B = −Lp diA→B
+
−


dt
2
dt
dt



diA→B
(4.40)
VC,E = − L2s diC→E
+M
dt
2
dt







VD,E = − Ls diD→E − M diA→B .
2
dt
2
dt

4.2.4.1

Resonance

The above equation is a fully general description of the transformer. To describe the resonanceof the circuit at ωrot , we can apply this analysis to the transformer in Figure 4.12, with a few
simplifying assumptions. First we can assume that the circuit is symmetric, i.e. the impedance oneach rod is the same. Because the voltage at the center tap is set by VRF , we can think of thisvoltage as ground in the ωrot frequency analysis. Finally, Csecondary offers a negligible reactance at
ωrot (by design), so we can ignore this component. With this assumption, the voltage at the rods
is exactly the voltage across the terminals of the secondary coils. In addition, the currents andvoltages across the output are equal and opposite due to the symmetry of the system:
iD→E = −iC→EVD,E = −VC,E ,so the system given by Equation 4.40 reduces toVA,B = −Lp diA→B+ M diC→E
dt
dt




diA→B

VC,E = 21 −Ls diC→E
+
M
.
dt
dt

(4.41)(4.42)

If the voltages and currents are periodic with frequency ω then the time derivatives in the above
equation act to simply multiply by jω, so the system of differential equations becomes a simple
system of linear equations

VA,B = jω (−Lp iA→B + M iC→E )VC,E = 12 jω (−Ls iC→E + M iA→B ) .

(4.43)

72
We can then express the currents and voltages above in terms of elements of the circuit. If Zprimaryis all impedance in series with the primary,
Zprimary = Rprimary +2
jωCprimary(4.44).

Then the voltage across the primary is equal to the voltage Vsource driven by the op-amp minus
any voltage drop due to this series impedance:
VA,B = Vsource − (−iA→B )Zprimary .

(4.45)

Next, we can write an impedance from the rod to ground
Zrod = (RQ + Rgain )||
=1
RQ + Rgain

1(4.46)jω(Cparasitic + Cshim )−1+ jω(Cparasitic + Cshim ),(4.47)

so the current across the rod is given by(4.48)

iC→E = VC,E /Zrod .
If we plug Equations 4.45 and 4.48 into Equation 4.43, we get






Vsource + iA→B Zprimary = jω −Lp iA→B + M VZC,E
rod(4.49)

VC,E = jω − L2s VZC,E + M
i
.
A→B
2
rod

We can now solve this system of equations to get the current on the source iA→B and the voltage
on a rod VC,E :
VC,E = Vsource
iA→B = VsourcejωM Zrod22
ω (Lp Ls − M ) − jω(Ls Zprimary + 2Lp Zrod ) − 2Zprimary Zrod
(2Zrod + jωLs )22
ω (Lp Ls − M ) − jω(Ls Zprimary + 2Lp Zrod ) − 2Zprimary Zrod
(4.50)
.

(4.51)

This encapsulates the resonant behavior of the circuit. The voltage is plotted in Figure 4.15. The
validity of Equation 4.50 is verified numerically in Section 4.2.4.2.

VC, E ȱȱǻǼ|VC, E ||Vsource |

73

Ś
Ř
Ŗ

ŗŖŖ

ŗŘś

ŗśŖ
fȱǻ£Ǽŗŝś

ŗśŖ
ŗŖŖśŖŘŖŖŖŗŖŖ

ŗŘś

ŗśŖ
fȱǻ£Ǽŗŝś

ŘŖŖ

Figure 4.15: Amplitude (left) and phase (right) plots of the solution to the voltage on a rod from
Equation 4.50 as a function of the driven frequency f . Standard component values are used:
Lp = Ls = 11.5 mH, M = 0.2Lp , RQ = 300 kΩ, Rgain = 1 kΩ, Cpar = 200 pF, Cprimary = 33 nF,
Rprimary = 100 Ω.

Solving analytically for the resonant frequency and quality factor is diﬀicult with the above
equation. To do this, we must first make some approximations. The capacitance in series has asmall reactance compared to other impedances in the circuit, so we can neglect its effect by taking
Cprimary → ∞. While in reality, Rprimary may not be negligible compared to other impedances,
it is instructive to take it to zero to calculate the resonant frequency, and will discuss its effectin Section 4.2.4.2. We then take Rprimary → 0. Finally, since Rgain ≪ RQ by design, we canapproximate the resistance to ground as only arising form RQ , i.e. Rgain → 0. With this, the gain
G = |VC,E /Vsource | of the transformer on the rod is
G= q

ωM RQ
2
2
2 ω 2 (C22
4
2
RQ
par ω (M − Lp Ls ) + 2Lp ) + ω (M − Lp Ls ).

(4.52)The resonant frequency can be found by setting dGdω = 0 :v
uu 1 − (Lp Ls −M 22)
u
4Lp Cpar RQ
ω0 = t Cpar
.
2)
(L
L
−
M
p
s
2Lp
With the appropriate mapping of variables (and using the definition M = K(4.53)p
Lp Ls ), this result

corroborates that of the analysis in Kia Boon Ng’s thesis [39]. If the second term in the numerator

74
L L −M 2
2 ≪ 1), then a good approximation for the resonant frequency is given by
p par RQ

is small ( 4Lp Cs


ω0 ≈

Cpar
(Lp Ls − M 2 )
2Lp

−1/2
(4.54).

0)
The quality factor can be found by solving G(ω) = G(ωfor ω to find the half-max frequencies,2and taking the difference of the two solutions. Using this method, the full width at half max ∆ω is∆ω =Cpar RQrp

12(Lp Ls − M 2 )2 − L L + M2 +4Cpar Lp RQ
p s

q

r
−

2 − L L + M2 −4Cpar Lp RQ
p s

2 − L L + M 2)
3(Lp Ls − M 2 )(8Cpar Lp RQ
p s

q

!
2 − L L + M 2 ) . (4.55)
3(Lp Ls − M 2 )(8Cpar Lp RQ
p s

Parameterizing in terms of the small variableϵ≡

Lp Ls − M 2
2
Cpar Lp RQ(which for our values is of order 10−3 ), we can expand the full width half max as
!
√
√
√
1
3 3
63 3 2∆ω =3+ϵ+ϵ + O(ϵ3 ) .
RQ Cpar16
512(4.56)(4.57)

Keeping only the first term is suﬀicient to reach 99.9% accuracy, giving us a quality factor ofQ=ω01≈ √ RQ Cpar ω0 .
∆ω
3Finally, the gain on resonance is approximately given bys√
2
Cpar Lp RQ
1 M
3M
G(ωo ) ≈ √
≈
Q.
2
L
L
L
−
M
2
Lp
2 p
p s

4.2.4.2(4.58)(4.59)

Series resistance on primary

Including effects of Cprimary and Rprimary in the analysis performed in the previous section
causes the analytic solution to become intractable. Qualitatively, Cprimary has only negligible
effect on the system in the range of nanofarads and above, while Rprimary severely affects the75
quality factor and thus the gain. To discuss this effect, we will move away from arguing via
analytic expressions and instead demonstrate the effect using numerical solutions. In addition,
we can verify our analytic expressions by comparing their numerical result with that of a circuit
simulation program (LTspice® ). This is done in Figure 4.16 to demonstrate the effect of Rprimaryon the gain.
When Rprimary becomes large enough to be the same order of magnitude as the reactances1characteristic of the resonant circuit’s capacitors and inductors ( jω(Cpar/2) ∼ |jωLp | ∼ 10 kΩ), it

starts to dampen Q and thus also reduces the gain of the circuit. The resonant frequency is alsoslightly shifted downwards.
The reason to include an Rprimary is to limit current sourced by the op-amp. The effectsof Rprimary on requirements of the op-amp are explored in Figure 4.17. We can choose its value
in order to satisfy the requirements of the PA94. The op-amp should be operated suﬀiciently far
enough away from its maximum current capability (100 mA) and from the voltage of the rails (±343 V) in order to prevent saturation and nonlinearity. In the end, a value of Rprimary =100 Ω wasused in the interest of maximizing gain, and the current draw is still far away from the limit of the
op-amp.

4.2.5

Coupling between rods

In addition to having parasitic capacitance to ground, neighboring rods also have parasitic
capacitance between them. An approximation of the coupling capacitances can be made from thegeometry of the rods. If the rods have a radius a and length L and are separated by a distance d,then for a rod with a charge per unit length q the electric field due to the rod is given byE=

q.
2πϵ0 r

(4.60)

Then the voltage between the two rods is given by
Z d−aE(r)dr =

V =
a

q
q
d−a
r=d−a
ln r r=a =
ln
.
2πϵ0
2πϵ0
a

(4.61)

76

Rprimary ( )Gain (linear), analytic

5

0
100
500
2k
5k
10k
50k

4
3
2
1
0

135

140

145

a)

150
f (kHz)

155

160

165

b)

Gain (linear), LTSpice

6

1m
100
500
2k
5k
10k
50k

4
3
2
1
0

c)

Rprimary ( )

5

135

140

145

150

f (kHz)

155

160

165

Figure 4.16: Effect of Rprimary on the gain of the resonant circuit. a) Analytic calculation using
Equation 4.50. The gain profile of the rod voltage from the source voltage is plotted for various
values of Rprimary . b) LTspice model of the resonant circuit c) LTspice calculation confirming theanalytic result. Resistances in the software cannot go to zero, so Rprimary is only probed down to
1 mΩ.

77

řŖŖŗŜŘśŖ

Vpk ȱȱǻǼ

ipk ȱȱǻǼ

ŗŞ

ŗŚ

ŘŖŖŗŘŗśŖŗŖŗŖŖ
Ş
Ŗ

ś

ŗŖ
Rprimary ȱǻ Ǽ

ŗś

ŘŖ

Figure 4.17: Effect of Rprimary on current requirements for the op-amp driving Vrot . ipk requiredis the current needed to drive 60 V/cm on the rods on resonance, while the primary is driven at
whatever voltage is needed to produce 60 V/cm. It is calculated as 429 V · iVA→B. Similarly,
C,E
ω=ω0

Vpk required is the voltage the op-amps need to put out in order to drive 60 V/cm on the rods onresonance, calculated as 429 V ·Vsource
VC,E ω=ω .

These quantities are calculated using the analytic

0expressions in Equations 4.50 and 4.51. In reality, the required voltages and currents are higher
than in this figure due to parasitics that dampen the gain, but the above plot offers a qualitativedescription of the function of Rprimary in limiting current.

The rod-rod capacitance is then given by
C=

qL
Lπϵ0
= d−a .
Vln a(4.62)

This calculation neglects edge effects, and a more accurate capacitance can be calculated using
finite element software (COMSOL) with realistic geometry. If the rods are arranged at a radiusrrod from the origin and are spaced at an angle 2π
8 radians apart, then the distance between nthnearest neighbors is given by



2π
2π
rrod cos(n ), rrod sin(n ) − (rrod , 0)
8
8
r
nπ
nπ
= rrod (cos( ) − 1)2 + sin2 ( )44rnπ= rrod 2 − 2 cos( ).
4dn =(4.63)(4.64)(4.65)

Given that the rod electrodes have a radius a = 6.35 mm, a length L = 123.24 cm, and are locatedat rrod = 5.85 cm, we can calculate the rod-rod couplings, given in Table 4.1.

Coupling
Nearest neighbor2nd nearest neighbor
3rd nearest neighbor
4th nearest neighbordn (cm)
4.48
8.27
10.81
11.70

Analytic approximation (pF)
19.0
13.8
12.3
12.0

78
Finite element calculation (pF)
18.0
13.8
12.6
12.2

Table 4.1: Capacitance between pairs of rod electrodes based on their theoretical geometric coupling. The distances dn are calculated using Equation 4.65 and the analytic approximation is
calculated using Equation 4.62. The couplings may be modified by shielding from other electrodes,
but this effect has not been modeled.

The capacitance leads to two overall effects. Firstly, the capacitances act to reduce the main
resonant frequency of the circuit, since it adds additional capacitance pathways to ground effectively
increasing Cpar .
Secondly, when only one channel is driven and the other voltage supplies are held at zero
volts a second resonance arises. To understand this, first let’s look at what happens when a channel
is not driven. In this case, each other voltage supply looks like a short, so each transformer justlooks like an inductor (argued in Figure 4.18).
Ls − M Ls − M

Ls

Lp

0 V = Ls

Lp

=

M

=

Lp Ls −M 2
Lp

Figure 4.18: Treatment of a transformer driven at 0 V, equivalent to an inductor. The transformation of the third equivalence is given by the equivalent circuit prescribed in [56]. While thistransformation makes assumptions about relative grounds, the same result arises from writing out
the governing equations of the transformer (allowing for different relative grounds) and requiring
the voltage across the primary to be 0 V.

We can then think of each coupling as an additional impedance network between the rods.
The conceptual reduction is shown in Figure 4.19.
So each parasitic coupling to an undriven transformer can be modeled as an additional
impedance between rods of
Lp Ls − M 21
11
Zcoup,i =++ RQ |||| jω
.
jωCcoup,ijωCpar
LpjωCcoup,i(4.66)

79
Ccoup,i
Rod1

RQ

Lp

Cpar Ls

0V

Rod5
Ccoup,iCcoup,i
Rod1

Rod1

=

RQ

Lp Ls −M 2
Lp

Cpar

=

Rod5

Zcoup,iRod5Ccoup,iFigure 4.19: Schematic reduction of coupling of the rod electrodes to another (undriven) rod-pair
into a single complex impedance Zcoup,iWe can then include the effect of the couplings by modifying Equation 4.50 with additional
impedances to ground
Zrod →

1
Zrod

+

X

1

i

Zcoup,i /2!−1.

(4.67)

Each coupling adds an additional resonance, since each coupling has its own set of inductance andcapacitance which can behave resonantly.
An expanded LTspice model, shown in Figure 4.20, contains four separately driven transformers with all the relevant rod-rod couplings. The values of the coupling capacitors are taken to
be the COMSOL values from Table 4.1.
The effect of the capacitances is shown in Figure 4.21. The qualitative effect of the coupling
is to create a dual resonance when only one channel is driven. However, if all channels are driven attheir respective phase, the effect of the coupling cancels out and the system exhibits a symmetricK1 L1 L2 {kappa}
K2 L3 L4 {kappa}
K3 L7 L9 {kappa}
K4 L5 L6 {kappa}.param Lprimary=11.5m Lsecondary = 11.5m kappa = 0.2.ac lin 1000 110000 150000.param Ccoup=18p Ccoupsec = 13.8p Ccoupthird =12.6p Ccoupfourth = 12.2p Cpar=160p.step param symdrive list 0 1.param Rcoil=25 RQ=301k Rprimary=100 Cprimary=33n Csec=100n

C7

Rod_6

{Cpar}{Cpar}

rod5{Rprimary}{Cprimary}rod6{Cpar}

R6

Rod_8

Rod_7

{Cpar}{Cpar}{Lsecondary}

rod8

{RQ}

rod2

rod6

{Ccoupsec}{Ccoupsec}{Ccoupsec}{Ccoupsec}

rod3{Ccoupthird}{Ccoupthird}{Ccoupthird}{Ccoupthird}

rod3
rod1

rod5
{Ccoupfourth}

rod6

rod8

rod2

rod4

rod7

{Ccoupsec}{Ccoupsec}

rod8{Ccoupsec}{Ccoupsec}

Second Nearest Neighbor Coupling

rod5

rod7

rod1

{Rprimary}{Cprimary}{Csec} {Rcoil}rod6{Ccoupthird}{Ccoupthird}

C9

rod7

Nearest Neighbor Coupling{Ccoupthird}{Ccoupthird}

L7

rod5

rod7
{Ccoup}{Ccoup}

L9

R8Resonant Circuits

rod6

rod5
rod4

rod3

{Ccoup}{Ccoup}

rod2

rod1

{Rcoil} {Csec}{RQ}

AC {symdrive} 90{Cpar}{Csec} {Rcoil}

R7{Lsecondary}

{RQ}

{RQ}

{Rprimary}{Cprimary}{Cprimary}
Rod_3

R5

L6

rod3

Rod_4

rod1

{Lprimary}AC {symdrive} 135

rod4{Rcoil} {Csec}{Ccoup}{Ccoup}

L4{Csec} {Rcoil}{Cprimary}{Ccoup}{Ccoup}

L3{RQ}{Rcoil} {Csec}

L5

AC {symdrive} 45

Rod_5{Csec} {Rcoil}
{RQ}

rod8

R20

{RQ}

{Cpar}{Lprimary}{Rprimary}{Cprimary}

Rod_2{Cpar}

rod4

L1

Rod_1

{Lprimary}{RQ}{Lsecondary}

L2

80{Cprimary}rod2{Lsecondary}rod1{Rcoil} {Csec}{Lprimary}AC 1 0{Cprimary}rod2{Ccoupfourth}

rod7

rod3
{Ccoupfourth}

rod8

rod4
{Ccoupfourth}

Third Nearest Neighbor Coupling

Fourth Nearest Neighbor Coupling

Figure 4.20: LTspice model for the four parasitically coupled resonant circuits, including all capacitive rod-rod couplings

single resonance. The figure also shows the measured response of the system, which exhibits thesame qualitative behavior.
The model differs from the measured behavior in two ways. First, the resonant frequencyof the actual transformer system is higher than that predicted by the model. In addition, thequality factor of the actual transformer system is lower than that of the model. This is likely due

81
to additional parasitic impedances in the system that are not modeled.
In addition, the difference between the resonant response of each channel is due to a combination of different inductances and couplings of each transformer, as well as asymmetry in thecapacitive coupling network.
5

This rod-pair driven only
All rods driven in desired phase

4

Gain

3
2
1
110

115

120

125

a)

135

140

145

150

ŗŚŖ

ŗŚś

ŗśŖ

ŗśś

ŗŜŖ

¡Ȭȱȱ
ŗ
Ř
ř
Ś

ŖǯŚ
	ȱǻǼ130

f (kHz)Ŗǯř
ŖǯŘ
Ŗǯŗ
ŗŘŖb)ŗŘś

ŗřŖ

ŗřś

fȱȱǻ£Ǽ

Figure 4.21: Demonstration of the dual resonance behavior arising from parasitic capacitive coupling between rods.
a) LTspice simulation of the coupled resonance.
b) Measured gain profile of the resonant circuit using the monitor box readout. The gain of eachchannel represents writing a fixed amplitude to a single transformer and taking the difference of thetwo relevant rod electrodes. For example, Ch1 represents the difference of rod 5 and rod 1, while
Ch2 represents the difference between rod 6 and rod 2. The black line represents using the method
described in Section 4.2.6 to produce a symmetric set of voltages on the rods at each frequency,
and the value reported is the maximum symmetric amplitude that can be made.

82
4.2.6

Control matrix

Due to asymmetry the coupling between adjacent rods described in the previous section, thevoltages on the rods can have significant difference in both phase and relative amplitude to thevoltages applied to the sources. In order to generate the voltages we desire on the rods, some
control theory must be applied in order to choose the voltages applied via the sources.
Because we are driving the system with four sources, it is convenient to only have four output
voltages to keep track of. We can separate the fields produced by the rods into their common modeand differential voltages (Vdiff,1,5 = Vrod1 − Vrod5 , Vcomm,1,5 = Vrod1 + Vrod5 . If each transformer
circuit is suﬀiciently symmetric (Vrod1 ≈ −Vrod5 ), then we can ignore the common mode term,
as its effect looks like a small oscillating quadrupole field. While this term can actually generateenough of a ponderomotive potential to lose ions if it is as large as 0.5 V/cm within the ion cloud(given by Equation 4.34), it can be compensated by applying a small amount of ωrot oscillating
potential onto the RF-generating DDS until the ions are no longer kicked out of the trap. The
differential term then becomes the one that contributes to Erot . Thus, rather than measuring all 8
rod voltages, we need only to keep track of the difference of the voltage on opposite rod pairs. We
can then characterize the system with a mapping from each voltage source to the difference of thetwo rods connected to that source’s transformer.
While applying a resonant voltage to each rod does drive all other rods with varying amplitudes and phases, the response of the system is linear in the in phase (I) and quadrature (Q) basis.
Therefore, the response of the system can be represented with a matrix.
The basis for describing the Vrot on each rod-pair is defined to be

83



I
 1
Q 
 1
 
 I2 
 
 
Q 
⃗
V =  2 I3 
 
 
Q3 
 
I 
 4
Q4

(4.68)

where each element is the in-phase and quadrature component for each channel, i.e. the voltage onchannel 1 is given byv1 (t) ≡ I1 cos(2πωrot t) + Q1 sin(2πωrot t).

(4.69)

We can then write a matrix M such that⃗rods = M V⃗dds
V(4.70)

which fully characterizes the gain and phase shift of each channel, as well as all parasitic couplings
between channels. We can then produce any arbitrary set of voltages on the rods by inverting thematrix:⃗dds = M −1 V⃗rods .
V

(4.71)

A symmetry requirement of the matrix is that a time shift on the input will shift the output by thesame amount without changing anything else. This is equivalent to saying that global phase does
not change the response of the system. This can be represented mathematically with a global phaseshift operator R(θ). Phase shifting the input and looking at the output (represented as R(θ)M )should be the same as just phase shifting the output and phase shifting (represented as M R(θ)).
In other words, our matrix must commute with the global phase operator.
Global phase is achieved by applying a phase shift to each channel, meaning applying a

84rotation matrix to each pair of I,Q. This is represented in our space as a block diagonal matrix:


0
0
0
0
0
0

 cos(θ) − sin(θ)





 sin(θ) cos(θ)
0
0
0
0
0
0






0
0
cos(θ) − sin(θ)
0
0
0
0








0
0
sin(θ)
cos(θ)
0000(4.72)
R(θ) = 
.




0
0
0
0
cos(θ) − sin(θ)
0
0






0
0
0
0
sin(θ) cos(θ)
0
0








0
0
0
0
0
0
cos(θ)−sin(θ)




0
0
0
0
0
0
sin(θ) cos(θ)
For a general matrix M with elements mi,j , our symmetry requirement is that the commutator
R(θ)M − M R(θ) = −m1,2 − m2,1 m1,1 − m2,2 −m1,4 − m2,3 m1,3 − m2,4 −m1,6 − m2,5 m1,5 − m2,6 −m1,8 − m2,7 m1,7 − m2,8  m1,1 − m2,2 m1,2 + m2,1 m1,3 − m2,4 m1,4 + m2,3 m1,5 − m2,6 m1,6 + m2,5 m1,7 − m2,8 m1,8 + m2,7  −m3,2 − m4,1 m3,1 − m4,2 −m3,4 − m4,3 m3,3 − m4,4 −m3,6 − m4,5 m3,5 − m4,6 −m3,8 − m4,7 m3,7 − m4,8  m −m
m3,2 + m4,1 m3,3 − m4,4 m3,4 + m4,3 m3,5 − m4,6 m3,6 + m4,5 m3,7 − m4,8 m3,8 + m4,7 
4,2
 3,1
sin(θ)  −m − m m − m −m − m m − m −m − m m − m −m − m m − m 
6,1
5,1
6,2
5,4
6,3
5,3
6,4
5,6
6,5
5,5
6,6
5,8
6,7
5,7
6,8 
 5,2




 m5,1 − m6,2 m5,2 + m6,1 m5,3 − m6,4 m5,4 + m6,3 m5,5 − m6,6 m5,6 + m6,5 m5,7 − m6,8 m5,8 + m6,7  −m7,2 − m8,1 m7,1 − m8,2 −m7,4 − m8,3 m7,3 − m8,4 −m7,6 − m8,5 m7,5 − m8,6 −m7,8 − m8,7 m7,7 − m8,8 m7,1 − m8,2

m7,2 + m8,1

m7,3 − m8,4

m7,4 + m8,3

m7,5 − m8,6

m7,6 + m8,5

m7,7 − m8,8

m7,8 + m8,7(4.73)
must be identically zero for any phase θ. This means that each element of the above array mustbe zero, which establishes a relationship between the elements of M. Put succinctly, if you divide
up M into 2x2 blocks, the diagonal components of each block must be equal and the off-diagonalcomponents must be opposite in sign but equal in magnitude.
One result of this symmetry is that the matrix can be measured in 4 steps by applying onlyan I-type signal to each input and then reading out the odd rows of the matrix. The even rows arethen defined by the symmetry relationships.

85
Another result of the symmetry is that any feedback algorithms to fine-tune the matrix must
be carefully crafted to ensure the symmetry is preserved. One such algorithm is as follows: once
we have measured M, pick the desired set of outputs V⃗d and use the measured matrix M to findthe DDS voltages⃗d .
⃗vDDS = M −1 V

(4.74)

When ⃗vDDS is actually applied, a slightly different output is measured V⃗m ̸= V⃗d . Then a feedback
matrix can chosenN=

Vd1 Vm1 +Vd2 Vm2Vm 21 +Vm 22

Vd1 Vm2 −Vd2 Vm1Vm 21 +Vm 22

0

0

0

0

0

0

d2 m1 −Vd1 Vm2
2
2
m 1 +Vm 2

Vd1 Vm1 +Vd2 Vm2Vm 21 +Vm 22

0

0

0

0

0

0

0

0

Vd3 Vm3 +Vd4 Vm4Vm 23 +Vm 24

Vd3 Vm4 −Vd4 Vm3Vm 23 +Vm 24

0

0

0

0

0

0

Vd4 Vm3 −Vd3 Vm4Vm 23 +Vm 24

Vd3 Vm3 +Vd4 Vm4Vm 23 +Vm 24

0

0

0

0

0

0

0

0

Vd5 Vm5 +Vd6 Vm6Vm 25 +Vm 26

Vd5 Vm6 −Vd6 Vm5Vm 25 +Vm 26

0

0

0

0

0

0

Vd6 Vm5 −Vd5 Vm6Vm 25 +Vm 26

Vd5 Vm5 +Vd6 Vm6Vm 25 +Vm 26

0

0

0

0

0

0

0

0

Vd7 Vm7 +Vd8 Vm8Vm 27 +Vm 28

Vd7 Vm8 −Vd8 Vm7Vm 27 +Vm 28

0

0

0

0

0

0

Vd8 Vm7 −Vd7 Vm8Vm 27 +Vm 28

Vd7 Vm7 +Vd8 Vm8Vm 27 +Vm 28



V V
 V






































(4.75)
such that⃗d = N V⃗m .
V(4.76)⃗m ,⃗vDDS = M −1 N V(4.77)

Substituting this into Equation 4.74,

and M −1 N serves as a more accurate inverse transform matrix. We then update our characteristic
matrix:M ′ = (M −1 N )−1 .

(4.78)

Note that the matrix N was formulated specifically to commute with R(θ) so that M ′ preservesglobal phase invariance. If a diagonal matrix is used instead, measurement noise can break thesymmetry, causing instability when scanning trap times.

86
The measured matrix M of the system is displayed in Figure 4.22. Each 2x2 block represents
a coupling from an input channel to an output channel. If there were no crosstalk between channels,
the matrix would be block diagonal, and deviation from a block-diagonal matrix indicates strong
couplings between channels. Within each 2x2 block, the diagonal components represent coupling
without phase shift, while the off-diagonal components represent coupling with π/2 phase shift.
After several iterations of feedback as in Equation 4.78, we are able to reach an Erot where
each pair of rods has the same amplitude to within 0.1% and each phase is correct to within one
tenth of a degree.

87Ŗ ȱŖǯśŝ ȬŖǯŗŜ ȱŖǯřŜ ȱŖǯřŞ ȬŖǯŖŝ ȱŖǯŗş ȬŖǯśŖ ȬŖǯŘŚŖǯŝśŗ ȱŖǯŗŜ ȱŖǯśŝ ȬŖǯřŞ ȱŖǯřŜ ȬŖǯŗş ȬŖǯŖŝ ȱŖǯŘŚ ȬŖǯśŖŖǯśŖŘ ȱŖǯřş ȱŖǯřŚ ȱŖǯŜŖ ȬŖǯŗş ȱŖǯśŗ ȱŖǯŗś ȱŖǯŖŖ ȬŖǯŘŜŖǯŘśř ȬŖǯřŚ ȱŖǯřş ȱŖǯŗş ȱŖǯŜŖ ȬŖǯŗś ȱŖǯśŗ ȱŖǯŘŜ ȱŖǯŖŖŖǯŖŖŚ ȱŖǯŖŝ ȬŖǯŘŖ ȬŖǯśŗ ȬŖǯŘŖ ȬŖǯŞŝ ȱŖǯśŗ ȬŖǯśŝ ȱŖǯŗŚŖǯŘśś ȱŖǯŘŖ ȱŖǯŖŝ ȱŖǯŘŖ ȬŖǯśŗ ȬŖǯśŗ ȬŖǯŞŝ ȬŖǯŗŚ ȬŖǯśŝŖǯśŖŜ ȱŖǯśŚ ȱŖǯŗŞ ȱŖǯŖŘ ȱŖǯŘŜ ȬŖǯśś ȱŖǯŘŘ ȬŖǯŝś ȱŖǯśŝŖǯŝśŝ ȬŖǯŗŞ ȱŖǯśŚ ȬŖǯŘŜ ȱŖǯŖŘ ȬŖǯŘŘ ȬŖǯśś ȬŖǯśŝ ȬŖǯŝś
Ŗ

ŗ

Ř

ř

Ś

ś

Ŝ

ŝ

Ch1
Ch2
Ch3
Ch4

VDDS

a)

0

5

b)

10
t ( s)

15

Vrod

Ch1
Ch2
Ch3
Ch4
0

c)

20

5

10
t ( s)

15

20

Figure 4.22: The control parameters for making a uniform Erot with the rod electrodes.
a) The measured (normalized) control matrix M relating VDDS to the voltage on the rods Vrod .
b) The voltages applied to the DDSs to generate the desired Vrot given by inverting the matrix.
c) The resulting (desired) voltages applied to the rods.

Chapter 5
Magnetic Fields

5.1

Effect of magnetic fields in the Hamiltonian
Any magnetic field that is aligned or anti-aligned with the quantization axis (defined by

Erot ) will cause phase accumulation due to the spin of the electron. Because one rotation of thequantization axis is much faster than all other motions of the ions, phase will be accumulated if⃗ · ⃗rrot over one cycle of Erot is nonzero, and the corresponding frequency f 0 is
the time-average of B
given by
f0 =

3gµB
Brot
h

with

⃗ · r̂rot ⟩t ,Brot ≡ ⟨B(5.1)

where g is the g-factor, µB is the Bohr magneton, h is Planck’s constant, and Brot is the magnetic
field introduced in Section 2.1.2. The simplest field that will do this if of the form
⃗ = Baxgrad (x x̂ + y ŷ − 2z ẑ) .
B

(5.2)

Note that as an ion traces out a circle of radius rrot in the x-y plane, the time-average over once
cycle is exactly
Brot = Baxgrad rrot .

(5.3)

89
This is in fact the case for an ion anywhere in the x-y plane. To see this, consider an ion at location(x0 , y0 , z0 ), rotating around this point with a radius rrot and period T . We can then calculate

 

2π
2π
2π
2π
⃗
B x0 + rrot cos( t), y0 + rrot sin( t), z0 · cos( t), sin( t), 0 dt
T
T
T
T
0







x0 + rrot cos( 2π

T t)



Z T
 

2π
1
2π
Baxgrad y0 + rrot sin( 2π t) · cos( t), sin( t), 0 dt
=
T


T 0
T
T










−2z0
(5.4)


Z T
Baxgrad
2π
2π
2π
2π
=
x0 cos( t) + y0 sin( t) + rrot cos( t) + sin( t)
dt
T
T
T
T
T
0



BaxgradTT=+
x0 (0) + y0 (0) + rrot
T2
2

⃗ · r̂rot ⟩t = 1
⟨B
T

Z T

= Baxgrad rrot .
The result is independent of (x0 , y0 , z0 ). This means that all ions at all locations in the trap
experience the same phase accumulation.
The ions only start to see different phase accumulations if Baxgrad starts to vary over thelength scale of the size of the ion cloud. A first order derivative would lead to temporary phasedecoherence, but would average to zero over a longer timescale as the ions slosh back and forthharmonically in the trap. A second order derivative, however, would not cancel, and would lead todecoherence.

5.2

f0 from anti-Helmholtz coils
We can use a Baxgrad field configuration as the main source of frequency in our Ramsey

measurement. Such a field can be created with a set of anti-Helmholtz coils. For practical reasons,a set of square coils are used. The analytical result for the field produced by a square coil is derived
in Appendix A.1.
If we place two coils centered around the same axis with opposite a distance d apart andflow current in the opposite directions (as we would do for an anti-Helmholtz configuration), the90resulting fields are given by⃗ anti-Helmholtz = B⃗ coil (x, y, z + d ) − B⃗ coil (x, y, z − d ).
B
2
2

(5.5)

⃗ coil is given by Equation A.5. Physically, the coils must rest outside of the cryogenic
where B
insulation foam block, so the length of each side of the square coils is set to L = 54 cm. To findthe correct spacing d between the coils for an anti-Helmholtz configuration, we look at the on-axis
equation and Taylor expand about the origin:⃗ anti-Helmholtz (0, 0, z) =z
B

−

+z

3

16µ0 I L2 d 3d2 + 5L2

 !

π (d2 + L2 )2 (d2 + 2L2 )3/2
64µ0 IL2 −10d9 − 40d7 L2 − 39d5 L4 + 26d3 L6 + 43dL8!

π (d2 + L2 )4 (d2 + 2L2 )7/2

+ O(z 5 ).

(5.6)

The first term leads to the linear gradient we want for the anti-Helmholtz configuration, but thesecond term represents an unwanted third derivative. We can then select d so that the second term
goes to zero. This occurs when d = 51 cm.
The expected Baxgrad produced by the coils is then given by the derivative⃗ anti-Helmholtz (0, 0, z)∂ẑ
·B
8µ0 IdL2 3d2 + 5L2
1
Baxgrad ==.
2∂z
π (d2 + L2 )2 (d2 + 2L2 )3/2

(5.7)

The coils are wound on aluminum u-channels to set the dimensions. Additional coil sets in all
dimensions are added in Helmholtz configurations to shim away the background field, forming an
aluminum cage structure that also acts to set the spacing of each coil set. Details of the setup arein Figure 5.1

5.3

Characterization of homogeneity
In Equation 5.2, Baxgrad is treated as a constant. However, one can imagine that fields ofthis form can have a spatially varying Baxgrad (or other phase-contributing forms with an effective
Baxgrad that is spatially varying). Since the phase seen by the ions is no longer independent of

91
c

Name

a

a

L (cm)d (cm)

54.6

50.8

46.2

52.7

46.2
43.3

52.7

Anti-Helmholtz,
North-South HelmholtzbEast-West HelmholtzcUp-Down Helmholtz(bFigure 5.1: The field-generating coils for both shimming away background fields and creating
Baxgrad . The coils that lie on the North-South axis (a) serve both purposes. The dimensions ofeach coil’s side length (L) and spacing (d) are listed in the table on the right, with allowances for a
1/2 cm offset between the current and the u-channel to account for the finite thickness of the wire
assembly. Note that the Up-Down coils (c) were reduced in size along the East-West axis so thatthey could fit between the two pieces of aluminum used to clamp the foam assembly.

their position, this can cause the ions to decohere. Such field can be caused by inhomogeneity
in the applied fields as well as any background magnetic fields. To place a bound on this effect,
we aim to measure the magnetic field around the ions’ resting place. Since we are interested indeviations on the order of a part in 104 of the fields over the spatial extent of the ion cloud, of order
1 cm, directly measuring the fields is impractical. Instead, we can measure the fields on a muchlarger surface enclosing the ion trap, with Maxwell’s equations prescribing the fields inside. The
proper machinery for this technique is to perform a multipole expansion of the fields about the ion
location. Once we measure enough points to determine the multipole coeﬀicients, the fields and its
derivatives can be sensitively found from the analytic form of the multipole fields.

5.3.1

Multipole expansion

In the region where the ions are there are no current sources. In the magnetostatic regime,


⃗ ·B
⃗ =0
 ∇
(5.8)

 ∇
⃗ ×B
⃗ = 0.

92
We can then write the magnetic field as the gradient of a scalar potential ΦB :⃗ = −∇Φ
⃗ B.
B

(5.9)

This scalar potential can be decomposed into a multipole expansion [57]
ΦB =

∞ X
l
X
l=0 m=−l12l + 11
alm r′l + blm ′l+1rYlm (θ, ϕ),(5.10)

where alm and blm are the multipole moments and Ylm are the spherical harmonics. (For convenience, we define r′ as a dimensionless quantity r′ ≡ rr0 to absorb the spatial dimensions of thecoeﬀicients alm and blm . This is so that each coeﬀicient has the same units. For further convenience we can define r0 as 1 meter.) Since there are no currents near the origin, it is unphysicalto have terms that go to infinity at the origin. Therefore all blm terms must be zero, and we canleave them out of our expansion. Furthermore, since we are working with real magnetic fields, it
is convenient to work with the “real spherical harmonics” Ylm , which give rise to physical fields.
These are defined as [58]
√1 Y m + (−1)m Y −m

m>0

l
l
2


Ylm ≡
Ylm
m=0






 √1 Y m − (−1)m Y −m m < 0.
ll
i 2

(5.11)

We can then define our expansion as
ΦB =

∞ X
l
X


alm ϕlm

where

ϕlm ≡

l=0 m=−l1′lr Ylm (θ, ϕ) .
2l + 1(5.12)In this formulation, the fields are fully described by the real scalar multipole coeﬀicients alm .
⃗ lm (1 T · m) are tabulated
The basis scalar fields ϕlm and the resulting basis vector fields Blm ≡ −∇ϕin Table 5.1. Note that the l = 0 term is not included since it is a constant, so it goes to zero when
the gradient is taken and therefore it does not produce a magnetic field.
Note that the l = 2, m = 0 term gives rise to a Baxgrad :⃗ 2,0 ∝ √1 (x x̂ + y ŷ − 2z ẑ) .
−∇ϕ
2 5π

(5.13)

93
l

1

2

m

ϕlm in Cartesian Coordinates

−1

√y
2 3π

0

√z
2 3π

1

√x
2 3π

−2

1
2

−1

1
2

q
2

1
4

2
−3

q

− 41

−2
−1

q

− 41

q

− 41

2

1
4

−4

1
12

2

q

3
14π y

y 2 − 3x215
7π xyzx2 + y 2 − 4z 2



2z 3 −3z (x2 +y 2 )
√
4 7π
3
14π x



15
7π z(x − y)(x + y)qqx2 + y 2 − 4z 2

5
14π x

x2 − 3y 235
π xy(x − y)(x + y)

−1

q

1
35
2
2
− 12
2π yz y − 3x
q

1
5
2
2
2
− 12
π xy x + y − 6z
q

1
5
2
2
2
12
2π yz 4z − 3 x + y

0−24z 2 (x2 +y 2 )+3(x2 +y 2 ) +8z 4
√
48 π

−3
−2

2

4

1
12

1
2

1
− 24

q

q

1
48

q

4z 2 − 3 x2 + y 2

5
π (x − y)(x + y)
1
12

3
4

5
2π xz

35
π

q

35
2π xz

x2 + y 2

2

x2 + y 2 − 6z 2x2 − 3y 2



cos 4 tan−1 (x, y)

0, − 2√13π , 0
0, 0, − 2√13πooo− 2√13π , 0, 0
q
o
n q
3
3
y, − 21 5π
x, 0− 21 5πqqon
33z, − 21 5π
y0, − 21 5πon√x , √y , − √z
2 5π 2 5π
5π
q
o
n q
3
3
1
1
x
− 2 5π z, 0, − 2 5π
q
n q
o
3
3
x, 21 5πy, 0− 21 5π
q
n q
 o
5
5
xy, 43 14π
y 2 − x2 , 0− 23 14π
q
q
o
n q
15
15
15
yz, − 21 7π
xz, − 21 7π
xy− 21 7πqn q
 q6 o
3
3
1
1
2
2
2 ,−
2
14π xy, 4
14π x + 3y − 4z
7π yz


2
2
2
3(x +y −2z )3xz√ , 3yz√ ,√2 7π 2 7π
4 7π
q
q
o
n q

3
3
6
1
2 + y 2 − 4z 2 , 1
3x
xy,
xz
−
4
14π
2
14π
7π
q
q
o
n q
15
15
15
1
1
1
− 2 7π xz, 2 7π yz, − 4 7π (x − y)(x + y)
q
n q
o
5
5
(x − y)(x + y), 23 14π
xy, 0− 43 14π
q
n q

 o
35
35
1
2 − 3x2 , − 1
2
2
y
y
12
π
12
π x x − 3y , 0q
n q
 1 q 35
o
35
35
2
2
xyz, 41 2π
z y 2 − x2 , 12
− 21 2π
2π y y − 3x
q
n q
o

 q5
5
5
1
2 + y 2 − 6z 2 , 1
2 + 3y 2 − 6z 2 , −
3x
x
y
x
xyz
12
π
12
π
π
q
n q
o
 1q 5
5
5
1
1
2
2
2
2
2
, 4 2π y x + y − 4z 222π xyz, 12
2π z 3x + 9y − 4zy (x2 +y 2 −4z 2 ) 3z (x2 +y 2 )−2z 3x(x2 +y 2 −4z 2 )
√
√
√
,−
,
−
4 π
4 π
3 π
q
q
n q
o

5
5
5
1
1
2
2
2
2
2
2 ,1
12
2π z 9x + 3y − 4z22π xyz, 4
2π x x + y − 4z
q
q
o
n q


5
5
5
1
2 − 3z 2 , − 1
2 − 3z 2 , − 1
x
x
y
y
z(x
−
y)(x
+
y)
6
π
6
π
2
π
q
n q
 1 q 35
o
35
35
1
1
2
2
2
2
4
2π z y − x , 22π xyz, − 12
2π x x − 3y
q
q
n

 o
1
35
35
1
2
2
2
2
− 12
π x x − 3y , − 12
π y y − 3x , 0
n

3
5π yz
2

5
14π y
1
2

q
1
4

3

n

3
5π (x − y)(x + y)

q

0
1

n

3
5π xy

√ −2z
− x +y
4 5π
q
1
3
2
5π xz

0
1

3

q

Blm

Table 5.1: Mathematical formulations of the ϕlm multipole expansion functions up to l = 4 andthe resulting magnetic fields. Dimensional constants are dropped.

94
5.3.2

Sources of decoherence

We can repeat the integral from Equation 5.4 to find which of the basis fields contribute tophase of the ions. The fields with non-zero phase contributions are listed in Table 5.2.
l
2

3

m

∂z Bzlm{∂x2 ∂z , ∂y2 ∂z , ∂z2 ∂z }Bzlm

0− √15π{0, 0, 0}

−1

−

6
7π y

{0, 0, 0}0− √3z7π
q
6
− 7π
x
q
− π5 xy
q
− 10
π yz

{0, 0, 0}

1
−2
−1
4

q

0

2
2
x2 +y
√ −2z
π

1

−

2

− 21

q

10
π xz

q

5
π (x − y)(x + y){0, 0, 0}{0, 0, 0}{0, 0, 0}no√2 , √2 , − √4ππ
π{0, 0, 0}n q qo− π5 , π5 , 0D⃗ lm (⃗r0 + ⃗rrot ) · r̂rot
B

E
t

√1 r
2 5π rotq

3
14π rrot y0√3 rz
2 7π rot 0

q
1
2

3
14π rrot x0qq5π rrot x0 y05
2π rrot y0 z02 + 2 x2 + y 2 − 2z 2
− 4√1 π rrot rrot
0
0
0
q
5
2π rrot x0 z0
q

1
5
2
2
4
π rrot x0 − y0and decoherTable 5.2: The multipole expansion fields up to l = 4 that lead to phase accumulation
DE⃗ lm (⃗r0 + ⃗rrot ) · r̂rot ,ence. Dimensional constants are dropped. To recover dimensions for quoted Btuse rrot and spatial initial positions in m and the result is given in T.

From this we can infer the sources of decoherence. While the l=3, m={0,±1} terms contributea frequency that depends linearly on an ion’s position in a single dimension during the cycle ofErot , the harmonic motion of the ions in the trap means that this phase contribution will average
to zero over the timescale of the Ramsey fringe.
Next, consider the l=4,m={-2,-1,1} terms, which have spatial dependence on products ofdifferent dimensions. Consider the sign of such a frequency in each octant. The term x0 y0 is
positive in half the octants and negative in the other half. If the ions move in a harmonic trap withthe same trapping frequency in each dimension, then assuming the ions traveled in a trajectorythrough the origin, the each ion will only occupy two opposite octants which contribute frequencies

95
of the same sign, so they will contribute an overall frequency. However, if the trapping frequencies
are different in each dimension, the ions trajectories will occupy all eight octants for equal times,
so the overall frequency contribution cancels out.
This means that the leading cause of decoherence in the ion cloud terms from the l=4,m={0,2} terms, which have quadratic dependence on ion location. These terms are synonymouswith a ∂x2i ∂z Bz since they are the only terms that have such nonzero third derivatives.
To quantify how much decoherence these terms cause, consider these derivatives as derivativesof the effective Baxgrad , as in ∂x2i ∂z Bz = 2∂x2i Baxgrad . Consider a hard shell of ions with radius rcloud .
The difference between the highest and lowest phase-contributing Baxgrad that the ions see due toeach derivative is then
1
2.
∆Baxgrad = − ∂x2i ∂z Bz rcloud2(5.14)

We want the spread in frequencies to be such that after 20s of our Ramsey fringe we want thehighest deviating frequencies to have only lost or gained one tenth of a period.
The number of cycles of Ramsey fringe is given by
N = f t,where t is the evolution time and f is the frequency. For a decohering ion, we require that the
number N of cycles it undergoes is within 1/10 of the mean number of cycles N0|N − N0 | <1,
10

or in terms of frequencies|(f0 + ∆f )t − f0 t| <1.
10

From this we can derive our constraint on the frequency deviation ∆f :|∆f | <

1 1
.
10 t

From Equations 5.1 and 5.3 we can rewrite ∆f in terms of ∆Baxgrad :3gµB
1 1(∆Baxgrad ) rrot <
.
h
10 t

96
Then using Equation 5.14 we have
3gµB
h




1 2
1 1
2
∂xi ∂z Bz rcloud rrot <
,
2
10 t

which finally gives us a limit on the third derivatives of the magnetic field:∂x2i ∂z Bz <h.
2rrot t15gµB rcloud(5.15)

Given an rcloud of 0.5 cm, an rrot of 2.6 mm, and a Ramsey evolution time of 20 s, this gives∂x2i ∂z Bz < 2.5 × 10−3 mG/cm3 .

(5.16)For a Baxgrad of 3.07 mG/cm (corresponding to an f0 of 50 Hz), a fractional representation of thisbound is given by∆Baxgrad< 10−4 .
Baxgrad

5.3.3(5.17)

Magnetic field measurement scheme

To determine the coeﬀicients that describe the background magnetic field near the ions, wemeasure the field at grid points on the surface of a cuboid centered at the ion location with sides oflength 20 cm (details in Figure 5.2). There are a few key points of this measurement grid. Firstly,
each Ylm has a spatial dependence of order (cos(θ))l eimϕ , meaning some component of the fields
have l/2 periods over the domain of θ (since θ only goes from 0 to π) and m periods over the domainof ϕ. To accurately sample the fields, the measurement grid must contain at least l points about
θ and 2m points about ϕ to reach the spatial Nyquist frequency. This is demonstrated visually in
Figure 5.2c.

a)

b)

c)
25

22
19
10cm
uniform
grid
10

21

16

17

North

East

18
15

14
11

12

7

8
5

2

Up

24

20

4
1

27

26

23

13

97

z

9

y

6

x

3

Figure 5.2: The setup for measuring the magnetic field around the ion location.
a) an LSM303 triple-axis magnetometer, read by a microcontroller, is used to measure the fields.
The magnetometer is mounted on non-magnetic plastic and aluminum components that hold it
above the table. A grid of nine points spaced 10 cm apart are measured at three different heights.
b) the resulting grid of points. The green point at the center of the grid represents the location of
the ions at their resting point in the prototype experiment.
c) The radial component of B4,±4 in the context of the measurement grid. This view is looking
down ẑ. Blue represents negative values and red represents positive values. Each lobe has its own
measurement point, just meeting the spatial Nyquist sampling criteria.

We can then use the measured fields to fit the multipole coeﬀicients, and then multiply eachcoeﬀicient by its corresponding field’s theoretical contribution to Baxgrad and its derivatives.
Let B meas be a matrix where each row corresponds to a different spatial location on themeasurement grid, and the columns are the Bx , By , and Bz measured at each location. One canthen evaluate the value of Blm at each measurement point and construct the corresponding B lm
basis fields. Then the alm that fully describe the fields inside the region can be found via
B meas =X

al,m B lm .

(5.18)

l,m

We truncate the expansion at l = 4 since higher order terms do not occur far from field sources. If
we flatten the matrices into one-dimensional row vectors (B meas → B meas , B lm → B lm ), then this98
relation can be written asa (1,−1) 





 a(1,0) B meas =  B (−1,1) , B (−1,0) , ..., B (4,4) .
 ... 
a(4,4)(5.19)

This can now be solved with traditional multiple linear regression techniques, with appropriate
error estimation.

5.4

Background fields from the optical table
Standard optical tables are made from ferromagnetic steel. Since our experiment is sensitive

to magnetic fields, we need to characterize the effect of the tables on the relevant magnetic fieldparameters at the location of the ions. The magnetic nature of the table is demonstrated in Figure
5.3, which maps out the magnetic domains by measuring the fields on the surface of the table. The
Position (in)Position (in)Baxial (Gauss)

Ion location

Melles Griot 4’x8’x12” optical table

𝐵! !"#!$

Figure 5.3: Axial magnetic field measured at the surface of the table. The fields are measured
at points on a 2 inch by 2 inch grid, and the measurement is then smoothed out with a gridded
interpolation.

99
spatial size of the domains is of order a few inches, which looks like high order multipole terms
when near to the ions. We characterize the relevant fields using the technique described in Section
5.3.3 at various points in relation to the optical table. First, we measure 35 cm above the table,
which represents the ideal location of the ion location for practical engineering-related concerns.
Next, we measure 45 cm above the table, which represents lifting the ion trap up by (or equivalently
lowering the optical table) by 10 cm to reduce the effects of the optical table. Finally, we measure
35 cm above the table and 90 cm off the end of the table, which represents the effect of the second
nearest optical table if we were to remove the optical table from underneath the trap and replace
it with a non-magnetic 80-20 structure. The fields at each location are listed in Table 5.3

35 cm above
45 cm above
90 cm off edge

Baxgrad(mG/cm)∂x2 Baxgrad(mG/cm3 )∂y2 Baxgrad(mG/cm3 )∂z2 Baxgrad(mG/cm3 )0.20 ± 0.15
0.01 ± 0.06
−0.46 ± 0.58

0.0034 ± 0.0043−0.0004 ± 0.0020−0.0003 ± 0.0004

0.0039 ± 0.0043
0.0015 ± 0.0019
0.0004 ± 0.0004−0.0073 ± 0.0055−0.0011 ± 0.0024−0.0004 ± 0.0005

Table 5.3: Measurements of the relevant derivatives of the background magnetic fields with 95%
confidence intervals

To meet the requirement prescribed by Equation 5.16 with some margin for error, we decideto lower the table by 15 cm such that the ions sit 50 cm above the table.

5.5

Field-distorting effects of the optical table
The ferromagnetic properties of the table not only produce spatially inhomogeneous back-ground magnetic fields, but can also have their spins aligned by experimental field sources, thus
distorting the field lines near the surface of the table. To characterize this effect, we take a single
coil from the up-down coil pair of the apparatus described in Figure 5.1 and place it so that it’saxis is pointing into the table. We then measure the on-axis field due to the coil both with andwithout the presence of the table. Each measurement is performed once with the coil turned off
and once with the coil turned on, so the difference represents the background-free fields from thecoils. The results are shown in Figure 5.4.

100
-10
-15
-20
-25
-30
-35
-40
-45
-50
-55
-10

-8

-6

-4

-2

0

2

4

6

8

Figure 5.4: The effect of the optical table on the fields produced by a current-carrying coil with its
axis perpendicular to the surface of the table

We see significant enhancement of the field produced by the coil near the surface of the table,consistent with the field from the coil magnetizing the table and aligning its spins. The experimentis performed again with the current running in the opposite direction, and the magnitude of the
fields was found to be the same within errorbar.

5.6

Homogeneity of anti-Helmholtz fields
As you move away from the center of a pair of anti-Helmholtz coils, the higher order deriva-

tives start to become nonzero. This could look like deviations of Baxgrad that lead to decoherence.
To verify our coil design, we can use the full analytic fields of our anti-Helmholtz coils given by
Equations 5.5, A.4, and A.3. If we evaluate take their derivative along z, we can find Baxgrad andthen we can look for deviations of Baxgrad in the size of the ion cloud. The results are shown in
Figure 5.5.
The saddle point behavior of Baxgrad is characteristic of the l=4, m=0 multipole term from
Table 5.2. Comparing the inhomogeneity to the benchmark prescribed by Equation 5.17, we can

101
10

10 -5

5

0

-5
-1

-0.5

0

0.5

1

Figure 5.5: Homogeneity of the Baxgrad generated by the analytic fields from the designed squareanti-Helmholtz coils over the spatial extent of the ion cloud. Each line represents a path along z
at a different displacement along x, representing the radial distance from the trap’s z-axis.

see that over the spatial extent of a 0.5cm radius cloud, the inhomogeneities are two orders ofmagnitude smaller than what is required.
While the theoretical fields are suﬀiciently homogeneous, the actual dimensions of the constructed coils will be slightly different than their theoretical counterparts. In addition, we areconcerned that the ferromagnetic table underneath the coils might distort the fields in a way thatcauses decoherence. For these reasons, we perform the measurement described in Section 5.3.3 atthe center of the anti-Helmholtz configuration both with the coils suspended on a nonmagnetic
frame and above the optical table.
We pass 3.2 A of current through the coils, which are each three turns. Using Equation
5.7, the theoretical Baxgrad produced is 4.3 mG/cm. The measurements are listed in Figure 5.6.
From these measurements we can determine that the optical table does not significantly affect thefields from the coils near the ions, and in both cases the homogeneity is suﬀicient to satisfy therequirement of Equation 5.16.

102

Baxgrad∂x2 Baxgrad∂y2 Baxgrad∂z2 Baxgrad(mG/cm)(mG/cm3 )(mG/cm3 )(mG/cm3 )4.67 ± .05.0004 ± .0014.0014 ± .0014-.0017 ± .0018

4.8 ± .05

.0011 ± .0016.0007 ± .0016-.0018 ± .0021

Figure 5.6: Measurements of the fields from the anti-Helmholtz coils. Top: measured away fromthe optical table on a non-magnetic aluminum surface. Bottom: measured on the surface of the
optical table. Measurements at all 27 points are taken with the coils turned off, then measured
again with the coils turned on. We then subtract the two to isolate the fields due to the coils, andperform the multipole expansion fits on this differential set of measurements.

103

5.7

Shim coils for B4,0 and B4,2
If we are unable to keep the area of interest suﬀiciently nonmagnetic to avoid decoherence

due to magnetic field effects, we wish to be able to shim away the responsible magnetic sourcesof decoherence. As demonstrated in Section 5.3.2, the primary source of magnetic-field based
decoherence arises from the l = 4, m = 0 and l = 4, m = 2 terms of the multiple expansion of themagnetic field. If we can re-create these fields using coils, we should be able to shim away thesefields.
A coil array is designed to be low-profile and fit within the existing configuration, while still
providing the necessary degrees of freedom required for generating the higher-order magnetic fields.
The design is shown in Figure 5.7.

2

1
3

4
a

b

a

b

Figure 5.7: Layout of the B4,0 and B4,2 shim coils.

Each coil is placed on the surface of the foam, 25.4 cm away from the axis of the trap. Eachcoil has a radius of 5 cm, and adjacent coils are spaced 26 cm apart.
We will use the following notation, as labeled in Figure 5.7. coil set 1 is the existing Baxgrad
coil set, coil set 2 are the four coils closest to the source chamber, coil set 4 is the four coils closest
to the cryocooler, and coil set 3 is the last four coils. Each coil set has a subset a of two horizontal

104
coils and a subset b of two vertical coils.
Observing symmetries of the basis fields reduces the degrees of freedom of the relative currents
through each coil. The following symmetries can be observed. For both fields coil sets 2 and 4
have the same current magnitude. For B4,0 , each coil set has its a subset with equal current to its
b subset, and in B4,2 each coil has its a subset with opposite current to its b subset.
Not counting the degree of freedom of overall amp-turns, there are then two degrees of freedom
for the amp-turns of the m = 0 coil pair and one for the m = 2 coil pair. The fields are then given
by




⃗
B

4,0coils = Bcoils1 − α4,0 (Bcoils2 + Bcoils4 ) + β4,0 Bcoils3

⃗

⃗

⃗

⃗




⃗ 4,2coils = (B⃗ coils2a − Bcoils2b + B⃗ coils4a − B⃗ coils4b ) + α4,2 (B⃗ coils3b − B⃗ coils3a ).
B(5.20)The relevant amp-turn ratio for each coil subset can be found using the analytic form of the fields,
which are given in Appendix A.2. We can then rotate and translate these fields to model thefield from each coil subset and decompose these fields into the spherical harmonic basis field set
described in Section 5.3.
For the B4,0 coils, the two degrees of freedom are constrained to set the B2,0 and B4,2
components to zero, while for the B4,2 coils, the single degree of freedom is constrained such thatthe B2,2 component is zero. The resulting amp-turn ratios can be found in Table 5.4.
α4,0
β4,0
α4,2

0.93
7.56
6.50

Table 5.4: Solved relative amp-turn ratios for the shim coils

The fields resulting in these amp-turn ratios for this coil configuration are shown in Figure5.8. The discrete nature of the coils results in an artifact on the generated B4,0 fields, which shows
up as an l = 4, m = 4 contribution. Since this term does not contribute to decoherence, and infact does not generate a Brot at all, this artifact is not a problem.

105⃗ l,m · r̂B⃗ generated · r̂BSpherical Harmonic Decomposition
2.5

10-11

X l=4, m=0
Y 2.185e-11

2

1.5

1
X l=4, m=4
Y 4.893e-12
0.5

0

l=

1,
l= m=
1
l= , m -1
l= 1, m=02
l= , m =12, =l= m=-12
l= , m -2
2 =l= , m 0
l= 2, m=13, =
l= m 23 =l= , m -1
3, =l= m=-23
l= , m -3
3 =l= , m 0
3 =l= , m 1
l= 3, m=24, =
l= m 34 =l= , m -1
4 =l= , m -24, =l= m=-34
l= , m -4
4 =l= , m 0
4 =l= , m 1
4 =l= , m 2
4, =
m 3
=4

-0.5

l = 4,m = 0
0.5

10-11

0
-0.5
-1
-1.5
-2
-2.5

X l=4, m=2
Y -3.011e-11

-3

l=

1,
l= m=
1
l= , m -1
l= 1, m=02
l= , m =12, =l= m=-12
l= , m -2
2 =l= , m 0
l= 2, m=13
l= , m =2
3 =
l= , m -1
3, =l= m=-23
l= , m -3
3 =l= , m 0
3 =l= , m 1
l= 3, m=24, =
l= m 34 =l= , m -1
4 =l= , m -24, =l= m=-34
l= , m -4
4 =l= , m 0
4 =l= , m 1
4 =l= , m 2
4, =
m 3
=4

-3.5

l = 4,m = 2
Figure 5.8: Analytic fields generated by the B4,0 and B4,2 shim coil configurations. Left: the⃗ l,m of the two decoherence-causing fields. Middle: The radialradial component of the theoretical B
component generated by the coil configuration. Right: A decomposition of the resulting field into
the basis fields.

Chapter 6
Detection

The population in the state of interest is read out by dissociating the ions state-selectively.
As the molecular ions dissociate, the thorium keeps the ion’s charge, leaving the fluorine atoms
neutral. A high-voltage electric field is then pulsed to accelerate the thorium ions towards a
detector. Because the dissociated Th+ ions are lighter than the un-dissociated ThF+ ions, theyacquire more speed while traversing the same potential gradient, so they arrive at the detector
sooner. This allows the two species to be differentiated, and the dissociated Th+ cloud representing
the population in the state of interest (times the dissociation eﬀiciency) can be read off separatelyfrom the un-dissociated ThF+ cloud which represents the remaining total ion number.

6.1

Dissociation
Just as the neutral atoms are state-selectively ionized using a two photon process, the ionsare dissociated state-selectively using a two photon process with pulsed lasers. This method is
illustrated in Figure 6.1.

6.1.1

OpticsThe REMPD1 photon is generated using a Sirah CBST-LG-18 dye laser pumped by a QuantaRay PRO pulsed Nd:YAG laser. The REMPD2 photon is generated using doubled light from thesame Nd:YAG laser. A diagram for generating the tunable REMPD1 photon is shown in Figure 6.2.
The cavity of the laser is formed between the diffraction grating and the reflective surface of the107

Dissociated Continuum

REMPD2 (312nm)...

Intermediate StateREMPD1 (tunable)

Figure 6.1: Energy level diagram for state detection using resonance-enhanced multi-photon ionization (REMPD). The first photon, denoted REMPD1, excites the ions from the state of interest
to an intermediate state. The wavelength of this photon can be tuned to excite from a different
initial state to the same intermediate state, thus providing state selectivity for the dissociation.
Then the second photon, denoted REMPD2, excites the ions from the intermediate state to above
the dissociation threshold, allowing the molecule to fly apart.

output coupler. Similarly to the ECDL configuration, the wavelength can be tuned by changing theangle of the diffraction grating with a piezoelectric crystal. The gain medium is a dye cell pumped
with 532nm light from the same Nd:YAG laser that generates the REMPD2 photon. The pumplight is formed into a sheet with a diverging cylindrical lens which allows the pump to spatiallyoverlap with the profile of the lasing beam inside of the dye.
Once the light leaves the cavity, it passes through the cell again in a ‘‘preamplification’’ stage.
It then is expanded with a set of telescoping lenses to prevent saturation of the gain medium, and is
amplified in the main amplifier dye cell. Once the light leaves the laser, a small amount of power ispicked off to a wavemeter and Fabry-Perot cavity to read the wavelength and image the spectrum

108
python control

quadrant
photodetector

to experimentphotodetector

downstream

towavemeter
to
Fabry-Perotpump
532nm from Nd:YAG

osc./preamp
dye celltelescope

Pellin-Broca
prism
partialbeamsplitter

outputcouplerpiezo

aperature

PBS

ampliﬁer
dye cellpartialbeamsplitter

diﬀraction
gratings

BBO
crystalelectronicrotation stages

Sirah Commercial Dye laser

Figure 6.2: Optics layout for generating the tunable REMPD1 photon.

of the light. The rest of the power goes into a beta barium borate (BBO) crystal in which second
harmonic generation (SHG) produces frequency doubled light. The light then passes through a
Pellin-Broca prism in which dispersion causes the two frequencies of light (fundamental and second
harmonic) to become angularly resolved. The doubled light is then selected with an aperture. The
doubled power is measured with a photodetector. Finally, the beam travels of order one meter tothe experiment. Some of the light is sent into a quadrant photodetector, so that the downstream
pointing of the beam can be measured.
The eﬀiciency of the process of SHG in the BBO crystal is sensitive to the angle of the
BBO, and the optimal angle changes with the wavelength of the light. For each wavelength, theangle of the BBO can be optimized using the intensity on the photodetector as a feedback. After
performing this calibration at many different wavelengths, an optimal angle for all wavelengths canbe calculated with a piecewise cubic interpolation of the optimal angle as a function of wavelength.

109
The downstream beam pointing can also change with the wavelength, as well as with thermal
fluctuations. Since the horizontal beam pointing can be adjusted with the angle of the Pellin-Broca
prism, the angle of the Pellin-Broca can be actively closed-loop controlled such that the beam points
to the center of the quadrant photodetector, providing a spatially stable beam to the experiment.
The dye used in the cell is 4-Dicyanomethylene-2-methyl-6-p-dimethylaminostyryl-4H-pyran(DCM) dye in a 0.3 g/L solution in ethanol (with a diluted solution of 0.09 g/L in the amplifiercell). This dye is chosen because it fluoresces over a broad range, offering a flat response to within
15% over the range of 600-650 nm [59], allowing the doubled photon to be tunable over a range ofroughly 30800-33300 cm−1 .

6.1.2

Spectroscopy

Reading the population in the different ro-vibrational states of 3 ∆1 is done by changing the
wavelength of REMPD1. Spectroscopy of the different transitions gives us a roadmap for which
REMPD1 energies we can use to measure the population in each ro-vibrational state. Spectroscopyof the 3 ∆1 , v = 0 manifold is shown in Figure 6.3.
The population in the state of interest to the eEDM measurement (3 ∆1 , v = 0, J = 1) ismeasured by choosing the (3 ∆1 , v = 0, J = 1) → (Ω = 0, J = 2) transition.
Similarly, the populations the states of interest in the higher vibrational manifolds can beread off by tuning REMPD1 to a transition from these manifold. Transitions from 3 ∆1 , v = 1 areshown in Figure 6.4, while transitions from v = 2 are shown in Figure 6.5.

110

a)
b)

Transition(3 ∆1 , v = 0, J = 1) → (Ω = 0, J = 2)
(3 ∆1 , v = 0, J = 2) → (Ω = 0, J = 3)

Energy (cm−1 )32098.16(05)32098.53(05)Figure 6.3: Dissociation spectroscopy for v = 0. The electronic configuration of intermediate stateis unknown. However Ω can be inferred by the spacing and relative amplitudes of the transitions.
a) Transitions to an Ω = 2 intermediate state. Each peak represents a single transition, except
for the central peak in which all of the transitions which preserve J overlap. b) Energies of theprominent and well-isolated transitions.

a)b)Transition(3 ∆1 , v = 1, J = 1) → (Ω = 0, J = 2)
(3 ∆1 , v = 1, J = 2) → (Ω = 0, J = 3)

Energy (cm−1 )31973.94(05)31974.31(05)Figure 6.4: Dissociation spectroscopy for v = 1. a) Transitions to an Ω = 2 intermediate state. b)
Energies of the prominent and well-isolated transitions.

111

a)

b)

c)

Transition(3 ∆1 , v = 2, J = 2) → (Ω = 0, J = 1)
(3 ∆1 , v = 2, J = 2) → (Ω = 0, J = 3)
(3 ∆1 , v = 2, J = 1) → (Ω = 2, J = 2)
(3 ∆1 , v = 2, J = 2) → (Ω = 2, J = 3)

Energy (cm−1 )32040.18(05)32042.40(05)31202.31(05)31202.74(05)Figure 6.5: Dissociation spectroscopy for v = 2. a) Transitions to an Ω = 0 intermediate state.
b) Transitions to an Ω = 2 intermediate state. c) Energies of the prominent and well-isolatedtransitions.

112
6.1.3

Hyperfine selectivity

Because the Ramsey spectroscopy involves the hyperfine states within 3 ∆1 , v = 0, J = 1, it
is not enough to read out the whole of the population in this manifold, as we can do by dissociatingfrom the (3 ∆1 , v = 0, J = 1) → (Ω = 0, J = 2) transition from Figure 6.3. We additionally need to
be able to distinguish population in mF = 3/2 and mF = −3/2. We can do this by turning backon LmF without closing any of the other vibrational and rotational levels. The effect of doing thisis shown in Figure 6.6}}= -3/2otherrovibrational
levels
-1/2

1/2

3/2

Figure 6.6: Shelving of the unwanted hyperfine states into higher ro-vibrational states with LmF .

Because the excited Ω = 0− state does not have an mF = 5/2 state, LmF does not talk to
the mF = 3/2 state. Meanwhile, all other mF levels will be excited to the Ω = 0− state and fall
down predominantly into other ro-vibrational manifolds. If we then dissociate from J = 1, thepopulation we dissociate will represent the population that was in mF = 3/2.
Alternatively, we can do the exact procedure for mF = −3/2 by swapping the orientation of
LmF with respect to the quantization axis.

6.2

Ion imaging
After being state-selectively dissociated, the ions are imaged using a micro-channel plate(MCP) detector. The setup is shown in Figure 6.7. The mesh is a grid-like electrode designed
to prevent electric fields from leaking out and causing lensing of the ions as they approach thedetector. The MCP itself is held at a negative high voltage so that the ions accelerate towards the

113

MCP
Mesh
0V
ThF+

Front
-2611V

Back-712V

Phosphor Screen+3000VCCD Camera

Th+

ThF+
Th+

x-y integrated
Time of ﬂight trace

Time-integrated
x,y Image

Figure 6.7: Ion imaging scheme using a micro-channel plate (MCP) detector.

detector and strike the detector with around 2700 eV of kinetic energy. This collision induces anelectron avalanche along the potential gradient between the front and back plate of the MCP. The
electron avalanche produces a current which can be capacitively read out from the MCP, resulting
in a time-of-flight (TOF) signal of ions arriving.
The detector consists of an array many small channels. Electron avalanches happen individually in each of these small channels, keeping the detection spatially resolved. The electrons thenfly out the back of the MCP and are then accelerated towards a phosphor screen a short distanceaway held at a high voltage. Once again, the potential gradient caused by the high voltage means
the electrons hit the phosphor screen with a kinetic energy of over 3 keV, which is enough energy
to cause the phosphor screen to fluoresce. Because the lifetime of the fluorescing states of thephosphor is long compared to the timescales of the ion detection (tens of µs), the phosphor screen
effectively integrates the 2-D projection of ions incident on the detector. A charge coupled device(CCD) camera is then used to image the fluorescing phosphor plate.

114
6.2.1

Electronics

Mesh

Front Plate
550 Ω

Phosphor Plate

Cp,f

Cp,b

118 pF

174 pF

150 kΩ

SWm

Back Plate

3.2 MΩ

Ccoup

SWp

1.2 MΩ

Ccoup

820 Ω

820 Ω
Rgain

Rgain

Cp,s

Cp,s

10 nF

10 nF
110 pF-2611V550 Ω-2700V

Front Readout

110 pF
Back Readout

3000V

Figure 6.8: Schematic for the MCP readout circuit. Cp,f and Cp,b are the parasitic capacitance toground of the front and back plate of the MCP respectively, while Cp,s is the parasitic capacitanceof the oscilloscope and the cabling to the scope. The MCP is a Beam Imaging Solutions BOS-40OPT01-MS dual matched MCP with a BOS-40-92% transmission grid mesh and a P42 phosphor
screen. The switches SWm and SWp are Behlke HTS 31-03-HB-C fast HV transistor switches,
which are controlled with an opto-isolator to protect the TTL generator from high voltage shorts.
Additional protection to the power supplies on the switching circuits is provided by a set of diodes
which allow high-voltage transients induced by the switching to be dumped from the output of theswitch to ground.

The electronics used to drive the detector are shown in Figure 6.8. If the time domain signal
is a Gaussian of width σt then the corresponding Fourier domain signal is also a Gaussian, of widthσω = σ1t To faithfully capture the signal, we want to capture frequencies up to 2σω , which includes
97% of all frequency components in the signal. The characteristic frequency ω that we need to besensitive to is then ω = σ2t .

115This frequency informs our choice of values for the components in the readout circuit. If
we want the current induced on the MCP to flow primarily through the coupling capacitor andthrough the gain resistor, we want the impedance of this path to be small compared to the direct
path to ground through the parasitic capacitance.
ZCcoup + (Rgain ||ZCp,s ) ≪ ZCp,f

−1
1
1
1
+ iωCp,s+≪iωCcoup
RgainiωCp,f
1
≪
− iω
Ccoup
Cp,f
1

Ccoup ≫

(6.1)
(6.2)


1

−1

+ iωCp,sRgain−1 !−1
1
2
1
2
,
−i
+ i Cp,sCp,fσt Rgain
σt

(6.3)
(6.4)

where || represents impedances in parallel (A||B) ≡ (A−1 + B −1 )−1 . We also want the RC timeof the coupling capacitor and the gain resistor to be small compared to the timescale of the signal
so that the voltage at the readout port can rise and fall quickly enough to faithfully reproduce thesignal, i.e.
Rgain Ccoup ≪ σt .

(6.5)

The final requirement is one of signal to noise. The voltage readout on the port will be equal to
the gain resistance times the current induced on the MCP plate. If we require a signal of at least
Vmin for adequate signal to noise, this means that
Iinduced Rgain > Vmin .

(6.6)

The parameters can then be chosen to satisfy Equations 6.4, 6.5, and 6.6. Given the measured valuesof the relevant parameters, as given in Table 6.1, a set of parameters that satisfy the requirements
are:




R

gain = 200 ΩCcoup = 235 pF.

(6.7)

σt
Cp,s
Cp,f
Cp,b
IinducedVmin

116

0.5 µs110 pF118 pF
174 pF
0.1 mA
20 mV

Table 6.1: Measured values relevant to the MCP readout electronics

6.2.2

Mesh gating

While holding the mesh at ground does minimize ion lensing effect from electric fields leaking
out from the detector attracting the ions, it also leaves a ”shadow” artifact on the ion cloud. Thiseffect is shown in Figure 6.9. We leave the mesh at the same voltage as the front plate of the MCP,
because the lensing can be accounted for and the grid resulting from the alternative interferes withindividual ion counting.
tmesh = 162.86 s

167.14 s

171.43 s

173.57 s

175.71 s

180.00 s

Figure 6.9: Effect of the mesh of the ion image. The mesh is held at ground until the time tmesh
listed above each image, at which point the voltage is switched to be the same as the front plate of
the MCP. In the time scale of the ion arrivals, the left image is equivalent to leaving the mesh at
the same voltage as the front plate the whole time, while the right image is equivalent to leaving
the mesh at ground the whole time. The images in between represent switching the voltage during
the ion arrival, resulting in extreme lensing. When the mesh is grounded, lensing is suppressed,
but the mesh leaves an artifact on the image. When the mesh is held at the same voltage as thefront plate of the MCP, lensing focuses the image of the cloud and the grid artifact is removed.
The images are taken using ThF+ without dissociation.

The image of the mesh projected on the ion cloud can be used to calibrate the distance scaleon the image to the distance scale on the plane of arrival of the ions onto the MCP. This calibration
is done in Figure 6.10.

117
25.7(2) px

1.04(1) mm

4(5

3.6
)c
m

a)

b)

Figure 6.10: Comparison of the MCP mesh geometry and image artifact. a) An image of the MCP
detector. b) An image of a ThF+ cloud with the mesh grounded. From these the pixel size can becalibrated to a size of the ion cloud when it hits the detector. 1 pixel represents a size of 40.5(5)
µm.

6.2.3

Phosphor screen gating

The dissociated Th+ ion number constitutes the signal of interest. In general in our system,
there are no more than 200 dissociated ions, meaning we can image them in 2-D on our detector
and can count individual ions based on their location without significant overlap. However, thenon-dissociated ThF+ ions will also arrive at the detector. The slight difference in mass between
the two species means that the Th+ ions will arrive at the detector slightly before the ThF+ ions.
We can then selectively image the Th+ ions by turning off the phosphor screen in the time between
the arrival of the two species. This is demonstrated in Figure 6.11.
High temperature ions on the tail of the ThF+ cloud can still be captured by the phosphor
screen and show up as background on the signal. To prevent this, the time at which the phosphor
screen is turned off is tuned to maximally capture the Th+ cloud without any pollution of ThF+
ions. Because this timing changes with arrival time, it must be scanned every time the trap isshimmed. Figure 6.12 shows an example of such a scan.

118

ThF+

Th+

Captured by phosphor screen

Figure 6.11: MCP capacitive readout signal with phosphor gating. Top: the set voltage on thephosphor plate as a function of time. The potential is turned off between the Th+ and ThF+ arrival
times so that only the Th+ is imaged. Middle: the resulting scope trace of the capacitive readoutof the front. Switching the phosphor voltage to ground creates a pickup on the signal via parasitic
capacitance between the mesh and the MCP (orange). However, this pickup is repeatable, so thedifference between the scope trace and a background trace with no ions gives the ion signal, shownin the bottom. The data shown is a single shot, with a single shot background. This data wastaken with high REMPD power and optically pumping all ions into the dissociated state, so theTh+ count in this trace is higher than in normal operation.

119

No REMPD
REMPD

1000

n, ions counted

800
600
400
200

150

160

165

170

No REMPD
REMPD
TOF trace

400
300
dn ( s 1 )
dt

155

0.04
0.03

200

0.02

100

0.01

0

0.00

100

150

155
160
165
t, phosphor screen turn off time ( s)170

TOF trace (V)

0

0.01

Figure 6.12: Scan of phosphor gate turn off time. Top: ions counted on the phosphor image. The
data without REMPD is fitted to a single erf function representing the integral of the ThF+ cloud,while the data with REMPD is fitted to a sum of two erf functions representing the integral ofboth the Th+ and ThF+ clouds. Bottom: the derivative of the above curves, corresponding to theactual ion intensity profile. Overlayed in green is a time of flight trace from another experiment.
The dashed vertical line represents the time in which no ThF+ ions pollute the phosphor signal,
i.e. the blue signal on the top plot is negligible.

120
6.2.4

Capacitive readout interpretation

6.2.4.1

TheoryThe MCP plate is held at −2700 V, so the kinetic energy at the time of impact is 2700 eV. For
ThF ions, this equates to 45 km/s, meaning the ions spend only 0.2 µs within 1 cm of the detector.
The electrons knocked loose then undergo an acceleration in an electric field of order kV/cm, so
the timescale of the electron avalanche effect is similarly short. Since this is small compared to RC
times of the detector, the shape of a single ion’s response is dominated by the RC response of thesystem rather than the shape of the individual ion’s interaction. If the detector has a rise time ofτrise and a fall time of τfall , then the voltage response of an individual ion on the detector is givenby
RV (t) =e

1− τ trise

τrise −tet/τrise − 1 θ(τrise − t, t)+ θ(t − τrise )e τfall ,
e−1

(6.8)

where θ is the Heaviside step function. Since we are reading out the current via a capacitive
coupling, the current response is given bydRVe
RI (t) = C∝
dtτrise −t1− τ tθ(τrise − t, t) e τfall θ(t − τrise )−.
(e − 1)τrise
τfall

rise

(6.9)

If Nions fall on the detector in the form of a Gaussian with width σion , then the signal read on thedetector look like a convolution of this with the current response function:2

Ireadout (t) ∝ σ2 −2τe=Nions Nions − 2σt2
√ e ionσion 2π!
∗ RI (t)rise t +1 (
)
( 2
))
( 2
ion
σ
−τ
t
σ
+τ
(τ
−t)
2τ 2
rise
−erf √ion rise
erf ion√ rise rise
2σion τrise
2σion τrise2(e−1)τrise

−

2 +2τ
σion
fall τrise −2τfall t
)
( 2
σ
+τ
(τ
−t)
2τ 2
fall
e
erfc ion√ fall rise
2σion τfall

2τfall.


(6.10)
This current is converted to a voltage ready by the scope by the gain-setting resistor Rgain . The
impulse response functions (Equation 6.8 and 6.9) and the resulting readout signal (Equation 6.10)

121
are shown in Figure 6.13, wherein a fit to the TOF signal yields time constants:τ = 2.50 ± .04 µs
rise(6.11)

Data
Theory fit

0.05
0.00

0

10

20

30
40
time ( s)50

60

RI (arb)

RV (arb)

TOF trace (V)τfall = 44 ± 3 µs.

050time ( s)

100

0

50
time ( s)100

Figure 6.13: The impulse response functions of the MPC capacitive readout. Top: The TOF traceof a ThF+ cloud, and a fit of the theoretical form given by Equation 6.10. Bottom: The voltage
and current impulse response given by Equations 6.8 and 6.9, where the time constants are givenby the above fit.

The fall time can be understood to as the RC time for the power supply to recharge thefront plate through the 150 kΩ current-limiting resistor, with the relevant capacitance being all
capacitances from the front plate to ground:

−1
−1 −1
τf all ∼ (150 kΩ)(Cp,f + Ccoup+ Cp,s)= 30 µs.

(6.12)

One thing to note is that integrating the whole waveform is not a measure of Ion number since itevaluates identically to zero:Z ∞−∞Ireadout (t)dt = 0.

(6.13)

122
Instead, one can first take the indefinite integral of the measured waveform and then integrate
this signal over all time. Using first the relation of convolution to indefinite integrals and then therelation of convolution to integrals over all space, we can see how this is proportional to the numberof ions:
Z ∞ Z t′−∞−∞Ireadout (t)dtdt′ ∝Z ∞ Z t′
−∞

−∞

−∞

Nions − 2σt 2√ e ionσion 2π

Z ∞
=
∝

!
2
Nions − 2σt2
√ e ion ∗ RI (t) dtdt′
σion 2π
!
!
Z ′
′2

Z ∞

−∞

= Nions∗t−∞

RI (t) dt(6.14)
dt′!
2
Nions − 2σt2
√ e ion ∗ RV (t) dt
σion 2π! ! Z

Z ∞
2
∞
− t2
1
2σ
√ e ion dt
RV (t) dt
σion 2π−∞−∞∝ Nions .

(6.15)(6.16)(6.17)(6.18)

since the first integral evaluates to 1 and RV does not depend on any ion-dependent features. In
practice, this method for ion counting is impractical because a) any low-pass filtering from parasitic
capacitance interferes with this measure, b) this requires that multiple ion clouds arriving on thedetector be much farther apart than the length of the MCP recharge, and c) this method is more
susceptible to noise and offsets on the signal.
Alternatively, we can recover the signal representing the time density of ions hitting thedetector. If we know the impulse response function RI using the time constants fit in Figure 6.17,
we can simply deconvolve the measured signal with this impulse response to recover the time-domain
ion density.
Generally, deconvolution is sensitive to noise, particularly high frequency noise. This isbecause deconvolution of A and B is mathematically equivalent to dividing A by B in Fourier
space. The regions of frequency space where the Fourier transform of B, F{B}, goes to zero are
divided by a small number, so any noise in F{A} in these regions is greatly enhanced.
To treat this, we can apply a low-pass bidirectional Butterworth filter with cutoff frequency
2at ωc = 2π τrise. Because frequencies above this are naturally filtered away by the RC response123of the MCP, we do not lose any signal information by applying this filter, and we only remove
detection noise. With this treatment, we are able to successfully deconvolve the impulse response
RI from the detector to remove the negative tail corresponding to MCP recharging. The result isshown in Figure 6.14.

0.10

Raw Data
Deconvolution
Filter & Deconvolution

TOF trace (V)

0.08
0.06
0.04
0.02
0.00
0

5

10

15
time ( s)

20

25

30

Figure 6.14: Recovery of the ion signal of a ThF+ cloud via deconvolution with the current impulse
response function, given by Equation 6.9 with values from Equation 6.11. The deconvolution
removes the negative tail of the signal, allowing the signal to be directly integrated to recover theion number corresponding to the peak.

The deconvolution algorithm still requires suﬀicient signal to noise ratio, since some frequencies below ωc are still enhanced. For lower ion numbers with insuﬀicient signal to noise, directly
integrating the positive part of the signal is a good approximation of the area, and a much morerobust algorithm.

6.2.4.2

Area algorithm and uncertainty

To count the number of ThF+ ions in the TOF trace, we must integrate the area under the
TOF signal. There is some nuance in the way this is done in order to allow for noise and offsets,
as well as to characterize the uncertainty in the measurement. The algorithm used to extract thearea is as follows:(1) Use timings to predict rough arrival time of ion cloud124(2) Fit the ion signal to a Gaussian using the prediction of arrival time as the initial fit condition. From the fit extract the arrival time and the Gaussian width(3) Take the mean of the portion of the signal that is 3.5σ or more away from the fit locationin order to find the background on the signal. Subtract this background away
(4) Numerically integrate the background-subtracted signal to 3.5σ on either side of the arrival
time to extract the area under the curve.
In order to place an uncertainty prediction on this measurement, we must consider the following:(a) Statistical uncertainty in the background of the signal. This uncertainty is given by the
product σbackground · (width of area integrated), where σbackground is standard deviation of
the points used to determine the background divided by the square root of the number ofpoints.
(b) Algorithmic uncertainty from the arbitrary assignment of 3.5σ as the region of interest. To
probe this, we can vary the width integrated over and look at the standard deviation of
the results.
(c) Statistical uncertainty from the noise on the signal. Since this cannot be directly measured,
we can approximate it by assuming that the noise profile in the background is the same as
the noise profile on the signal. This is mathematically equivalent to the value calculated in
the uncertainty of the background.
The total uncertainty can be estimated by adding these three uncertainties in quadrature. To
verify the validity of this error characterization method, a Monte Carlo simulation is done with aGaussian signal in the presence of white noise. The results are displayed in Figure 6.15.

Single simulation

1.4

y (arb)

1.2

4

1.0
0.8

2
0

Percent contained

6

0.6
0.0

0.5

1.0

x (arb)

1.5

2.0

0

5

10

15

20

Iteration Number

25

30

125

500 Simulations

100

1.6

Measured Area

Background
Region of Interest
Gaussian Fit
Area Integrated

830 Simulations

80
60
40
20
0

Simulated
Ideal Gaussian
0

2

4

6

Number of

8

10

Figure 6.15: Monte Carlo simulation of the area counting algorithm for a small Gaussian signal
in the presence of significant noise. Left: A single simulated signal. Middle: The area counted by
the algorithm for 30 simulations, with error bars representing the uncertainty calculated according
the the method proposed. The actual area of the simulated Gaussian is 1. Right: statistics on
500 simulations. The percent of simulations in which the area measured by the algorithm is Nσ
away from the true value is shown for varying N. The black line represents the ideal behavior of a
Gaussian distribution. Agreement with this line indicates that the σ calculated by the algorithm
is correct.

6.2.5

Ion counting

Ions imaged on the phosphor screen are counted individually using the phython package
skimage. The code is listed in Figure 6.16. First, a smoothing filter is applied with a width of two
pixels, so that pixel-to-pixel CCD noise is removed. Then, an erosion filter is applied such thatnearby, partially overlapping ions have the bridging region between them carved away. This helps
in uniquely identifying the two neighboring ions. Next, the peaks are found using a threshold of
three pixel brightness units and a minimum ion-ion distance of five pixels.
import skimage as sk
import scipy . ndimage as sn
def get_img_info_count (img , IMG_threshold = 3, IMG_min_distance = 5):# Smooth image
img = sn. median_filter (img , size=2)# Erode between ionsimg = sk. morphology . erosion (img)# Find peaks
peaks = sk. feature . peak_local_max (img , threshold_abs = IMG_threshold , min_distance =IMG_min_distance )return peaks

Figure 6.16: Python code to count individual ions on the phosphor screen image

126
Using ions identified with this algorithm, a point spread function for an ion incident on thedetector can be measured. This is done in Figure 6.17.
If the ion density is high enough that the distance between adjacent ions is smaller than
the width of the point spread function, the ions become indiscernible in the image, and cannot becounted separately. This means that the algorithm that counts the number of ions in the image
will begin to saturate. This is demonstrated in Figure 6.18 using a Monte Carlo method.

6.3

Simultaneous doublet readout
Reading the dissociated Th+ ions in a spatially resolved way allows additional information

to be extracted from the readout. In a single image, populations in the two different doublets canbe differentiated by making use of the fact that the two doublet states have opposite orientation
relative to Erot . During photodissociation, the excess energy of the absorbed photons relative to thedissociation threshold translates into kinetic energy of the dissociated atoms, which then fly apartfrom each other along their inter-nuclear axis. As the neutral flourine atom flies away from theheavier thorium ion, the direction of the kinetic energy imparted to the thorium is determined by
the molecule orientation. The result of this process is that we see two spatially separated clouds ofThorium ions on our detector, each cloud corresponding to different orientation and thus a different
doublet state. This is demonstrated in Figure 6.19.
We can then count individual ions in each cloud, thereby directly reading out the number ofions in each state. This dissociation-based state readout technique is further detailed in [60], as
well as in Ng’s thesis [39].

127Mean ion point spread function

100
0

0

y (px)

0

10

20

100
0

30

0

10

20

30

0

0

5

5

5

5

10

10

10

10

15

15

15

15

20

20

20

20

25

25

25

25

0

10

a)

0.0006x (px)

20

30

Intensity

010x (px)

20

Width

30

0

100

0.5

0.00040.4

0.00030.3

0.00020.2

0.00010.000025507500.6

0.0005b)

Example ion

0.1
0

2000
4000
Brightness (counts px2 )

0.0

4

6
8
Lorentzian (px)10

Figure 6.17: Phosphor image of Th+ ions. 2153 captured images of individual ions are analyzed.
a) The point spread function of each ion on the detector. On the left is the mean image of all ionscounted, and on the right is an image of a single ion. Each dimension is integrated along both axes.
The spatial distribution fits nicely to a Lorentzian as a function of radius.
b) Statistics on the ion pictures. On the left is the integrated brightness on the detector due tothe ion. The greyed histogram bar indicated where the algorithm used to count the ions is undercounting due to low signal to noise ratio. This bin is excluded from the fit of the distribution. On
the right is the distribution of widths of a Lorentzian fit to each ion image. Both distributions fit
nicely to normal distributions.

= 20 px

0

0
200

200

400

400

400

600
0

200

400

600

600
0

200

400

600

0

0

0

200

200

200

400

400

400

600

600

600

0

200

300

600

0

200

400

600

0

200

400

600

0

200

400

600

= 5 px= 10 px= 20 px= 50 px= 100 px

250
Ions Counted

400

128

= 100 px

0

200

600

a)

= 50 px

200
150
100
50
0
0

b)

50

100

150
Ions Generated

200

250

300

Figure 6.18: Demonstration of ion counting algorithm and its saturation via a Monte Carlo simulation. a) Top: simulated images of ion clouds using the point spread function from Figure 6.17
for each of 300 ions in a normal spatial distribution of varying widths. Bottom: The same images,
with the ions detected by the counting algorithm overlaid in red. Significant under-counting of theions can be observed in the image with distribution width 20 px. b) Resulting saturation of thecounting algorithm at different cloud widths. Each point represents five simulations.

129

0
100

y (px)

200
300
400
500

Lower doublet
Upper doublet

600
0

200

x (px)

400

600

0
2
x-integral (arb)

Figure 6.19: Spatially resolved images of the two doublets. For each image, the one double isdepleted using the appropriate microwave transition, leaving only population in the remaining
doublet. Each image represents the ion locations cumulatively counted from 3 images. Plotted to
the right is the integral of each cloud across the horizontal axis. The shaded gray region indicatesthe region where the doublets overlap, beyond which the signal predominantly represents a single
doublet.

Chapter 7
Preliminary Results and Conclusion

7.1

Rabi oscillations
With the trapping, state preparation, and readout techniques previously described we are able

to perform quantum operations in the eEDM sensitive states. A demonstration of Rabi flopping isshown in Figure 7.1.
The Rabi flopping is done by reducing the magnitude of the polarizing field until the hyperfine
states are near-degenerate leading to population mixing. An Erot of 13.7 V/cm is chosen such that
the Rabi rate is equal for both doublets. More information on this can be found in Ng’s thesis [39].
Each of the ion number data points corresponds to either preparing one hyperfine level1
mF = ±3/2 and reading out the population2 in the same state mF = ±3/2 (labeled as “In”)
or preparing in the other hyperfine level mF = ∓3/2 and reading out from mF = ±3/2 (labeled
as “Anti”). We vary the prepared state instead of the final state because the polarization of
the dissociation laser changes the eﬀiciency of dissociating from different final states. From the
population in each of the two states we can extract the “asymmetry.” The asymmetry between theion numbers NIn and NAnti measured in this way is defined as
Asymmetry ≡NIn − NAnti.
NIn + NAnti(7.1)

This quantity is normalized and is therefore immune certain types of common mode noise, such as
slow drift in ion creation or dissociation eﬀiciency.
12Here we denote a the mF level with ± because of the ambiguity as to which state is prepared.
The method for reading out population in a specific hyperfine state is described in Section 6.1.3.

131

1.2 ms60 V/cm

1.2 ms13.7 V/cm
timea)

Ion number

120

NIn
NAnti

100
80
60
40
20
0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.025

0.030

Asymmetry

0.50
0.25
0.00
−0.25

Fit
Data

−0.50
0.000

0.005

0.010

0.015
0.020
TRabi (s)b)

Figure 7.1: Rabi flopping of the eEDM-sensitive states.
a) Modulation of the magnitude of Erot that gives rise to the Rabi oscillations. The magnitude isramped in 1.2 ms, the maximum speed at which the resonant circuit can ramp linearly based on
its response time (which is different for each channel).
b) Oscillations in the population in each state. First the ions are optically pumped to one sideof the hyperfine manifold, then the Rabi sequence is performed. Finally, the population on one
side of the hyperfine manifold is read out by turning on LmF and dissociating. The asymmetry of
the two curves (defined in Equation 7.1) is calculated and the results fit to a sinusoid. The Rabi
frequency ΩRabi is fit to 60.30(49) Hz. From this we can determine the time at which the phase of
the sinusoid is π/2 (marked with a dashed black line). The time for an effective π/2 pulse can thenbe extracted from this phase, yielding Tπ/2 =3.95(6) ms.
.

132
We can then fit this quantity to extract the Rabi frequency and the phase. The phase of theasymmetry does not quite go to zero for zero TRabi since the ramps have finite duration, so a small
amount of phase accrues during each ramp. An effective π/2 pulse can then be chosen by finding
where the asymmetry first crosses zero.

7.2

Ramsey oscillations
With a π/2 pulse demonstrated in the Rabi oscillation, we then have the tools to perform theRamsey scheme discussed in Section 2.1.3. Since our state preparation sequence populates bothorientation states of the mF = ±3/2 hyperfine state and the π/2 pulse is designed to be effective
for both doublets, the Ramsey experiment will work on both doublets simultaneously. If we thenuse the spatially selective readout technique discussed in Section 6.3 to measure the population ineach doublet independently, we can measure the frequencies of both doublets simultaneously in asingle Ramsey experiment. This is done in Figure 7.2
We apply a magnetic field with coils described in Section 5.2 to generate an f0 of ∼110 Hz.
At early times the doublets are in phase, but after some time the doublets go out of phase dueto their different g-factors (as in Equation 2.5). At around 1 s, the doublets are maximally out ofphase.
With this, we have demonstrated simultaneous measurement of the eEDM sensitive frequency
channels fu and fl from Equation 2.6.

7.3

Concluding remarks
This thesis has documented the building of an experiment to measure the eEDM from con-

ceptual design of the experiment to the sensitive measurement of eEDM sensitive states. Along theway, we have provided many engineering and technological solutions to the unique problems of anexperiment as complicated as this one.
Now that Ramsey fringes have been demonstrated, the next steps towards realizing an eEDM
measurement in this system are:

Upper Doublet

Upper Doublet

133

Asymmetry

NIn
NAnti

Lower Doublet

Early Time

Late Time

Asymmetry

Lower Doublet

Figure 7.2: Ramsey fringes for the two doublets measured simultaneously via angular resolved
photo-dissociation.

• Demonstrate blackbody-limited coherence time (∼ 20 s).
• Multiplex the experiment to allow for continuous loading and readout using the demonstrated ion shuttling electrode design.
• Characterize sources of systematic error in the experiment (as was done in [51]).
• Collect blinded eEDM-sensitive measurements until the statistical uncertainty is similar inmagnitude to the characterized systematic uncertainty.

134
Just as several graduate student years had been spent on production techniques and spectroscopy of ThF+ before I joined, many graduate student years will be spent after I leave completingthese four steps, and I am confident that they are left in capable hands.

Bibliography[1] A. D. Sakharov, “Violation of CP invariance, C asymmetry, and baryon asymmetry of theuniverse,” Soviet Physics Uspekhi, vol. 34, p. 392, may 1991.
[2] M. Dine and A. Kusenko, “Origin of the matter-antimatter asymmetry,” Rev. Mod. Phys.,vol. 76, pp. 1–30, Dec 2003.
[3] S. Sarkar, “Big bang nucleosynthesis and physics beyond the standard model,” Reports on
Progress in Physics, vol. 59, no. 12, p. 1493, 1996.
[4] J. Ellis, “Beyond the standard model with the LHC,” Nature, vol. 448, no. 7151, p. 297, 2007.
[5] J. H. Smith, E. M. Purcell, and N. F. Ramsey, “Experimental limit to the electric dipole
moment of the neutron,” Phys. Rev., vol. 108, pp. 120–122, Oct 1957.
[6] C. G. Shull and R. Nathans, “Search for a neutron electric dipole moment by a scatteringexperiment,” Phys. Rev. Lett., vol. 19, pp. 384–386, Aug 1967.
[7] W. B. Dress, J. K. Baird, P. D. Miller, and N. F. Ramsey, “Upper limit for the electric dipole
moment of the neutron,” Phys. Rev., vol. 170, pp. 1200–1206, Jun 1968.
[8] J. K. Baird, P. D. Miller, W. B. Dress, and N. F. Ramsey, “Improved upper limit to the electric
dipole moment of the neutron,” Phys. Rev., vol. 179, pp. 1285–1291, Mar 1969.
[9] W. B. Dress, P. D. Miller, and N. F. Ramsey, “Improved upper limit for the electric dipole
moment of the neutron,” Phys. Rev. D, vol. 7, pp. 3147–3149, Jun 1973.
[10] W. B. Dress, P. D. Miller, J. M. Pendlebury, P. Perrin, and N. F. Ramsey, “Search for anelectric dipole moment of the neutron,” Phys. Rev. D, vol. 15, pp. 9–21, Jan 1977.
[11] I. Altarev, Y. Borisov, A. Brandin, A. Egorov, V. Ezhov, S. Ivanov, V. Lobashov, V. Nazarenko,
G. Porsev, V. Ryabov, A. Serebrov, and R. Taldaev, “A search for the electric dipole momentof the neutron using ultracold neutrons,” Nuclear Physics A, vol. 341, no. 2, pp. 269–283, 1980.
[12] J. Pendlebury, K. Smith, R. Golub, J. Byrne, T. McComb, T. Sumner, S. Burnett, A. Taylor,
B. Heckel, N. Ramsey, K. Green, J. Morse, A. Kilvington, C. Baker, S. Clark, W. Mampe,
P. Ageron, and P. Miranda, “Search for a neutron electric dipole moment,” Physics Letters B,
vol. 136, no. 5, pp. 327–330, 1984.

136[13] K. Smith, N. Crampin, J. Pendlebury, D. Richardson, D. Shiers, K. Green, A. Kilvington, J. Moir, H. Prosper, D. Thompson, N. Ramsey, B. Heckel, S. Lamoreaux, P. Ageron,W. Mampe, and A. Steyerl, “A search for the electric dipole moment of the neutron,” Physics
Letters B, vol. 234, no. 1, pp. 191–196, 1990.
[14] I. Altarev, Y. Borisov, N. Borovikova, S. Ivanov, E. Kolomensky, M. Lasakov, V. Lobashev,
V. Nazarenko, A. Pirozhkov, A. Serebrov, Y. Sobolev, E. Shulgina, and A. Yegorov, “Newmeasurement of the electric dipole moment of the neutron,” Physics Letters B, vol. 276, no. 1,
pp. 242–246, 1992.
[15] E. Kolomensky, M. Lasakov, V. Lobashevº, and V. Nazarenko, “Search for the neutron electric
dipole moment,” Physics of Atomic Nuclei, vol. 59, no. 7, pp. 1152–1170, 1996.
[16] P. G. Harris, C. A. Baker, K. Green, P. Iaydjiev, S. Ivanov, D. J. R. May, J. M. Pendlebury,
D. Shiers, K. F. Smith, M. van der Grinten, and P. Geltenbort, “New experimental limit on
the electric dipole moment of the neutron,” Phys. Rev. Lett., vol. 82, pp. 904–907, Feb 1999.
[17] C. A. Baker, D. D. Doyle, P. Geltenbort, K. Green, M. G. D. van der Grinten, P. G. Harris,
P. Iaydjiev, S. N. Ivanov, D. J. R. May, J. M. Pendlebury, J. D. Richardson, D. Shiers, andK. F. Smith, “Improved experimental limit on the electric dipole moment of the neutron,”Phys. Rev. Lett., vol. 97, p. 131801, Sep 2006.
[18] A. P. Serebrov, E. A. Kolomenskiy, A. N. Pirozhkov, I. A. Krasnoschekova, A. V. Vassiljev,
A. O. Polyushkin, M. S. Lasakov, A. N. Murashkin, V. A. Solovey, A. K. Fomin, I. V. Shoka,
O. M. Zherebtsov, P. Geltenbort, S. N. Ivanov, O. Zimmer, E. B. Alexandrov, S. P. Dmitriev,
and N. A. Dovator, “New search for the neutron electric dipole moment with ultracold neutrons
at ILL,” Phys. Rev. C, vol. 92, p. 055501, Nov 2015.
[19] C. Abel, S. Afach, N. J. Ayres, C. A. Baker, G. Ban, G. Bison, K. Bodek, V. Bondar,
M. Burghoff, E. Chanel, Z. Chowdhuri, P.-J. Chiu, B. Clement, C. B. Crawford, M. Daum,
S. Emmenegger, L. Ferraris-Bouchez, M. Fertl, P. Flaux, B. Franke, A. Fratangelo, P. Geltenbort, K. Green, W. C. Griﬀith, M. van der Grinten, Z. D. Grujić, P. G. Harris, L. Hayen,
W. Heil, R. Henneck, V. Hélaine, N. Hild, Z. Hodge, M. Horras, P. Iaydjiev, S. N. Ivanov,
M. Kasprzak, Y. Kermaidic, K. Kirch, A. Knecht, P. Knowles, H.-C. Koch, P. A. Koss,
S. Komposch, A. Kozela, A. Kraft, J. Krempel, M. Kuźniak, B. Lauss, T. Lefort, Y. Lemière,
A. Leredde, P. Mohanmurthy, A. Mtchedlishvili, M. Musgrave, O. Naviliat-Cuncic, D. Pais,
F. M. Piegsa, E. Pierre, G. Pignol, C. Plonka-Spehr, P. N. Prashanth, G. Quéméner, M. Rawlik, D. Rebreyend, I. Rienäcker, D. Ries, S. Roccia, G. Rogel, D. Rozpedzik, A. Schnabel,
P. Schmidt-Wellenburg, N. Severijns, D. Shiers, R. Tavakoli Dinani, J. A. Thorne, R. Virot,
J. Voigt, A. Weis, E. Wursten, G. Wyszynski, J. Zejma, J. Zenner, and G. Zsigmond, “Measurement of the permanent electric dipole moment of the neutron,” Phys. Rev. Lett., vol. 124,
p. 081803, Feb 2020.
[20] I. Khriplovich and A. Zhitnitsky, “What is the value of the neutron electric dipole moment in
the kobayashi-maskawa model?,” Physics Letters B, vol. 109, no. 6, pp. 490–492, 1982.
[21] E. E. Salpeter, “Some atomic effects of an electronic electric dipole moment,” Phys. Rev.,
vol. 112, pp. 1642–1648, Dec 1958.
[22] D. F. Nelson, A. A. Schupp, R. W. Pidd, and H. R. Crane, “Search for an electric dipole
moment of the electron,” Phys. Rev. Lett., vol. 2, pp. 492–495, Jun 1959.

137[23] J. Goldemberg and Y. Torizuka, “Upper limit of the electric dipole moment of the electron,”Phys. Rev., vol. 129, pp. 2580–2581, Mar 1963.
[24] E. S. Ensberg, “Experimental upper limit for the permanent electric dipole moment of Rb85
by optical-pumping techniques,” Phys. Rev., vol. 153, pp. 36–43, Jan 1967.
[25] J. Angel, P. Sandars, and M. Tinker, “Observation of a v × E effect in an electric dipole
moment experiment using a reversible atomic beam machine,” Physics Letters A, vol. 25,
no. 2, pp. 160–161, 1967.
[26] M. C. Weisskopf, J. P. Carrico, H. Gould, E. Lipworth, and T. S. Stein, “Electric dipole
moment of the cesium atom. a new upper limit to the electric dipole moment of the electron,”Phys. Rev. Lett., vol. 21, pp. 1645–1648, Dec 1968.
[27] M. A. Player and P. G. H. Sandars, “An experiment to search for an electric dipole momentin the 3 P2 metastable state of xenon,” Journal of Physics B: Atomic and Molecular Physics,
vol. 3, p. 1620, dec 1970.
[28] T. G. Vold, F. J. Raab, B. Heckel, and E. N. Fortson, “Search for a permanent electric dipole
moment on the 129 Xe atom,” Phys. Rev. Lett., vol. 52, pp. 2229–2232, Jun 1984.
[29] D. Cho, K. Sangster, and E. A. Hinds, “Tenfold improvement of limits on t violation in thallium
fluoride,” Phys. Rev. Lett., vol. 63, pp. 2559–2562, Dec 1989.
[30] K. Abdullah, C. Carlberg, E. D. Commins, H. Gould, and S. B. Ross, “New experimental limiton the electron electric dipole moment,” Phys. Rev. Lett., vol. 65, pp. 2347–2350, Nov 1990.
[31] E. D. Commins, S. B. Ross, D. DeMille, and B. C. Regan, “Improved experimental limit on
the electric dipole moment of the electron,” Phys. Rev. A, vol. 50, pp. 2960–2977, Oct 1994.
[32] B. C. Regan, E. D. Commins, C. J. Schmidt, and D. DeMille, “New limit on the electron
electric dipole moment,” Phys. Rev. Lett., vol. 88, p. 071805, Feb 2002.
[33] J. J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer, M. R. Tarbutt, and E. A. Hinds,“Improved measurement of the shape of the electron,” Nature, vol. 473, pp. 493–496, May
2011.
[34] T. A. Collaboration, J. Baron, W. C. Campbell, D. DeMille, J. M. Doyle, G. Gabrielse, Y. V.
Gurevich, P. W. Hess, N. R. Hutzler, E. Kirilov, I. Kozyryev, B. R. O’Leary, C. D. Panda, M. F.
Parsons, E. S. Petrik, B. Spaun, A. C. Vutha, and A. D. West, “Order of magnitude smaller
limit on the electric dipole moment of the electron,” Science, vol. 343, no. 6168, pp. 269–272,
2014.
[35] W. B. Cairncross, D. N. Gresh, M. Grau, K. C. Cossel, T. S. Roussy, Y. Ni, Y. Zhou, J. Ye, andE. A. Cornell, “Precision measurement of the electron’s electric dipole moment using trappedmolecular ions,” Phys. Rev. Lett., vol. 119, p. 153001, Oct 2017.
[36] V. Andreev, D. G. Ang, D. DeMille, J. M. Doyle, G. Gabrielse, J. Haefner, N. R. Hutzler,
Z. Lasner, C. Meisenhelder, B. R. O’Leary, C. D. Panda, A. D. West, E. P. West, X. Wu, and
A. C. M. E. Collaboration, “Improved limit on the electric dipole moment of the electron,”
Nature, vol. 562, pp. 355–360, Oct 2018.

138[37] Y. Ema, T. Gao, and M. Pospelov, “Standard model prediction for paramagnetic electric dipole
moments,” Phys. Rev. Lett., vol. 129, p. 231801, Nov 2022.
[38] T. S. Roussy, L. Caldwell, T. Wright, W. B. Cairncross, Y. Shagam, K. B. Ng, N. Schlossberger,
S. Y. Park, A. Wang, J. Ye, and E. A. Cornell, “A new bound on the electron’s electric dipole
moment,” 2022.
[39] K. B. Ng, The ThF+ eEDM experiment: concept, design, and characterization. PhD thesis,
University of Colorado Boulder, 2023.
[40] M. Denis and T. Fleig, “In search of discrete symmetry violations beyond the standard model:
Thorium monoxide reloaded,” The Journal of Chemical Physics, vol. 145, Dec. 2016.
[41] L. V. Skripnikov, “Combined 4-component and relativistic pseudopotential study of ThO forthe electron electric dipole moment search,” The Journal of Chemical Physics, vol. 145, Dec.
2016.
[42] V. Andreev, D. G. Ang, D. DeMille, J. M. Doyle, G. Gabrielse, J. Haefner, N. R. Hutzler,
Z. Lasner, C. Meisenhelder, B. R. O’Leary, C. D. Panda, A. D. West, E. P. West, X. Wu, and
A. C. M. E. Collaboration, “Improved limit on the electric dipole moment of the electron,”
Nature, vol. 562, pp. 355–360, Oct 2018.
[43] E. R. Meyer, J. L. Bohn, and M. P. Deskevich, “Candidate molecular ions for an electron
electric dipole moment experiment,” Phys. Rev. A, vol. 73, p. 062108, Jun 2006.
[44] A. N. Petrov, N. S. Mosyagin, T. A. Isaev, and A. V. Titov, “Theoretical study of Hff + insearch of the electron electric dipole moment,” Phys. Rev. A, vol. 76, p. 030501, Sep 2007.
[45] T. Fleig, “P, T -odd and magnetic hyperfine-interaction constants and excited-state lifetime
for hﬀ + ,” Phys. Rev. A, vol. 96, p. 040502, Oct 2017.
[46] W. B. Cairncross, D. N. Gresh, M. Grau, K. C. Cossel, T. S. Roussy, Y. Ni, Y. Zhou, J. Ye, andE. A. Cornell, “Precision measurement of the electron’s electric dipole moment using trappedmolecular ions,” Phys. Rev. Lett., vol. 119, p. 153001, Oct 2017.
[47] M. Denis, M. S. Nørby, H. J. A. Jensen, A. S. P. Gomes, M. K. Nayak, S. Knecht, and T. Fleig,“Theoretical study on ThF+, a prospective system in search of time-reversal violation,” New
Journal of Physics, vol. 17, p. 043005, apr 2015.
[48] L. V. Skripnikov and A. V. Titov, “Theoretical study of ThF+ in the search for T, P -violation
effects: Effective state of a Th atom in ThF+ and ThO compounds,” Phys. Rev. A, vol. 91,
p. 042504, Apr 2015.
[49] R. D. Freeman, “Quantities, units, and symbols in physical chemistry (mills, ian; cvitas,
tomislav; homann, klaus; kallay, nikola; kuchitsu, kozo),” Journal of Chemical Education,
vol. 66, p. A188, Jul 1989.
[50] K. B. Ng, Y. Zhou, L. Cheng, N. Schlossberger, S. Y. Park, T. S. Roussy, L. Caldwell,Y. Shagam, A. J. Vigil, E. A. Cornell, and J. Ye, “Spectroscopy on the electron-electricdipole-moment–sensitive states of thf + ,” Phys. Rev. A, vol. 105, p. 022823, Feb 2022.

139[51] L. Caldwell, T. S. Roussy, T. Wright, W. B. Cairncross, Y. Shagam, K. B. Ng, N. Schlossberger, S. Y. Park, A. Wang, J. Ye, and E. A. Cornell, “Systematic and statistical uncertainty
evaluation of the HfF+ electron electric dipole moment experiment,” Phys. Rev. A, vol. 108,
p. 012804, Jul 2023.
[52] J. Ye, S. Swartz, P. Jungner, and J. L. Hall, “Hyperfine structure and absolute frequency of
the 87 Rb 5P3/2 state,” Opt. Lett., vol. 21, pp. 1280–1282, Aug 1996.
[53] J. R. Rubbmark, M. M. Kash, M. G. Littman, and D. Kleppner, “Dynamical effects at avoided
level crossings: A study of the landau-zener effect using Rydberg atoms,” Phys. Rev. A, vol. 23,
pp. 3107–3117, Jun 1981.
[54] D. M. Pozar, Microwave engineering; 3rd ed. Hoboken, NJ: Wiley, 2005.
[55] P. Sarafis and A. G. Nassiopoulou, “Dielectric properties of porous silicon for use as a substrate
for the on-chip integration of millimeter-wave devices in the frequency range 140 to 210 GHz,”
Nanoscale Research Letters, vol. 9, p. 418, Aug 2014.
[56] J. W. Nilsson, Electric circuits / James W. Nilsson, Susan A. Riedel. Upper Saddle River,
N.J: Prentice Hall, 6th ed. [rev. print.] ed., 2001.
[57] J. D. Jackson, Classical electrodynamics; 2nd ed. New York, NY: Wiley, 1975.
[58] C. D. H. Chrisholm, Group theoretical techniques in quantum chemistry. 1976.
[59] P. Hammond, “Laser dye DCM, its spectral properties, synthesis and comparison with otherdyes in the red,” Optics Communications, vol. 29, no. 3, pp. 331–333, 1979.
[60] Y. Zhou, Y. Shagam, W. B. Cairncross, K. B. Ng, T. S. Roussy, T. Grogan, K. Boyce, A. Vigil,
M. Pettine, T. Zelevinsky, J. Ye, and E. A. Cornell, “Second-scale coherence measured at thequantum projection noise limit with hundreds of molecular ions,” Phys. Rev. Lett., vol. 124,
p. 053201, Feb 2020.

Appendix A
Biot-Savart Integrals

In order to design coil geometries for generating desired magnetic fields, it is useful to haveanalytic expressions for the general building blocks for these coils. While the resulting expressions
are complicated, they can be evaluated numerically in order to simulate the fields of a superpositionof these coils without the need for approximation. The square coil expression is used in Sections
5.2 and 5.6, while the circular coil expression is used in Section 5.7.

A.1

Square current loop
Here we derive the analytic expression for the fields from a square loop of wire. Begin with asingle square current loop with sides of length L. The relevant geometry is depicted in Figure A.1.
The Biot-Savart law [57] tells us that the magnetic field from a length of current is given by⃗ r ) = µ0 I
B(⃗
4π

Z

⃗′
dr⃗′ × R.
⃗ ′ |3|R(A.1)

Considering each segment separately we can write this as⃗ coil (⃗r) = B⃗ 12 + B⃗ 23 + B⃗ 34 + B
⃗ 41
B


Z L
Z L
L
L
′
′
′
′
2 {0, −dy , 0} × x +2 {dx , 0, 0} × x − x , y +µ0 I
2,y − y ,z
2,z


=
+
L
L
′
3
′
3
L
L
4π
| x + 2,y − y ,z || x − x ,y + 2,z |
−2
−2
!


L
Z L
Z
L
L
′
′
′, z
′
2 {0, dy , 0} × x −2 {−dx , 0, 0} × x − x , y −
,
,
y
−
y
z
2
2


+
+
.
L
′ , y − L , z |3
′ , z |3
L
−
|
x
−
,
|
x
−
x
y
y
−L
−
2
2
2
2
(A.2)

141
(𝑥, 𝑦, 𝑧)

𝑟Ԧ
4

𝑅’

3
L/2
𝑧

𝑟′
Ԧ

𝑥

𝑦

1

2

Figure A.1: The geometry of a square loop of current. The origin is taken to be the center of theloop. The current is taken to run anticlockwise in the order of nodes 1, 2, 3, and 4. ⃗r is defined as
the vector to an arbitrary point in space (x, y, z) at which we wish to know the field, ⃗r′ is defined⃗ ′ isas the vector to the differential part of the current loop which we are integrating over, and R′′⃗
defined as the distance from this differential wire length to (x, y, z). Note that R = ⃗r − ⃗r .

If we define a function that looks like the solution to an integral of this form,
H(a, b, i, j, k) ≡

µ0 I
4π
−



− b + L2 (−1)i− b + L2 (−1)i

2

2

− z2

− z2

q

q

a − L2L(k+1)δj,0 +i + x 2 +2 (−1)

a + L2Lkδj,0 +i + y 2 + z 2
2 (−1)!

L(k+1)δj,1 +i + x 2 +2 (−1),

Lkδj,1 +i + y 2 + z 2
2 (−1)(A.3)
where δi,j is the Kronecker delta, then the solution to the integral from Equation A.2 can be writtenas⃗ coil (⃗r)=
B

















zH(y, x, 1, 1, 1) − zH(y, x, 0, 0, 1)
zH(x, y, 1, 1, 0) − zH(x, y, 0, 0, 0)LL
2 + y H(x, y, 0, 0, 0) + 2 − y H(x, y, 1, 1, 0) + 2 + x H(y, x, 0, 0, 1) +






 L



L
2 −x



H(y, x, 1, 1, 1)








. (A.4)

If we evaluate this expression at x = y = 0, this turns into the familiar expression for the on-axis
field from a square loop of wire:⃗ coil (0, 0, z) =
B









0, 0,
2π

µ0 IL2
L2
2
4 +z

q




L2
2

2 +z

.

(A.5)

142

A.2

Circular current loop
Direct integration of the Biot-Savart equation (Equation A.1) tells us the magnetic field froma circular loop of current centered at the origin with its axis aligned in ẑ with a radius or R passing
current I:

⃗ =
B

µ0 I
4π

ϱ2

( (
(
(
)
)
(
)))
)
) (
) (
(
ζ r 2 + R2 E − 4Rρ− ζ 2 ϱK 4Rρ
+ ζK − 4Rρ+ ζ 2 ϱ r 2 + R2 E 4Rρ
2
2
2
2
ϱ

ρ2

((

ζ

ϱ

ζ

)3/2
)
r 2 + R2 2 − 4ρ2 R2

·




xz






yz
 (
)
√(

,
(
(
)
(
))
(
(
)
(
))
(
)
(
)
)2
(
)
(
)
 2 2 2

4Rρ
4Rρ
4Rρ
4Rρ
4Rρ
4Rρ
2
2
2
2
2
2
2
+ζK − 2+2R
r 2 +R2−4ρ2 R2 ζE+ϱE − 2−ζ ϱ r +R E
−ζϱ r +R E − 2 ρ ζ ϱ ϱK



ζ2
ϱ
ζ2
ϱ
ζ2
ϱ
(
(
(
)
(
))
)
))


(
) (
(
) (
ζϱ −ζϱ ϱK

4Rρ
ζ2

+ζK −

4Rρ
ϱ2

+ζ r 2 +R2 E

4Rρ
ζ2+ϱ r 2 +R2 E −

4Rρ
ϱ2

(A.6)

where





ρ = x2 + y 2








 r = x2 + y 2 + z 2

p



ϱ = (R − ρ)2 + z 2






p

ζ = (ρ + R)2 + z 2

and E and K are the elliptic E and K integrals, respectively.

Appendix B
A modified chain rule for vector fields

In the derivation for the ponderomotive potential in Section 4.1.2, in order to expand theelectric field about the secular position, we had to introduce an operator for the gradient of a vectorfield:∂ ⃗ ∂x E ⃗⃗ ⃗  ∂ ⃗ ∇
E ≡  E .
 ∂y 


∂ ⃗
E

(B.1)

∂z

We then later used a modified version of the chain rule in this higher-dimensional space to arriveat our result:1⃗⃗⃗ ⃗ ⃗
2⃗⃗ × (∇⃗ × E).
(∇
E) · E = ∇|E|
−E
2

(B.2)

We will prove this relation here. Begin with the left hand side:

  
∂ ⃗
 ∂x E  Ex 



  
∂ ⃗
 ∂ ⃗   X⃗⃗ ⃗ ⃗(∇E) · E =  E  · Ey  =E ,Ei ∂y   
∂ri
i

  
∂ ⃗
Ez
∂z E

(B.3)

where the sum over i is a sum over the dimensions ri = {x, y, z} and Ei = {Ex , Ey , Ez }. We canwrite this in vector form and distribute the factor of Ei , giving⃗⃗ ⃗ ⃗
E) · E =
(∇

X
i







∂
∂
 ∂ri Ex 
Ei ( ∂ri Ex )

 X





∂
Ei  ∂ Ey  =.
 ∂ri Ei ( ∂ri Ey )
i




∂
∂
E
E
(
E
)
z
i
z
∂ri
∂ri

(B.4)

144
We can add a clever zero to this equation of the form∂
Ei ( ∂ri Ex )

∂
Ei ( ∂x Ei )

∂
Ei ( ∂x Ei )

 X
 X

X






⃗⃗ ⃗ ⃗∂∂∂(∇E) · E =Ei ( ∂r Ey ) −
Ei ( ∂y Ei ) +
Ei ( ∂y Ei ),
i






i 
i 
i 



∂
∂
Ei ( ∂r∂ i Ez )Ei ( ∂z
Ei )Ei ( ∂z
Ei )
|
{z
}

(B.5)

=0

then combine the first two sums to get




∂
∂
Ei ( ∂ri Ex − ∂x Ei )

∂
Ei ( ∂x Ei )

 X

X




⃗⃗ ⃗ ⃗∂∂(∇
E) · E =+Ei ( ∂r∂ Ey − ∂y


.
E
)
E
(
E
)
i
i
i
∂y
i




i 
i 


∂
∂
∂
Ei ( ∂ri Ez − ∂z Ei )Ei ( ∂z Ei )(B.6)

We can then move the sum inside the vector:




P
∂
∂
∂
 i Ei ( ∂ri Ex − ∂x Ei )
Ei ( ∂x Ei )

 X

P



⃗⃗ ⃗ ⃗∂∂∂(∇E) · E =  i Ei ( Ey − Ei ) +
Ei ( ∂y Ei ) .
∂ri
∂y




i 
P


∂
∂
∂
E
(
E
−
E
)
E
(
E
)
i ∂z ii i ∂ri z
∂z i

(B.7)

Note that the term in parenthesis goes to zero if i = x in the first row, i = y in the second row, or
i = z in the third row, so we can exclude his from the sum, giving




P
∂
∂
∂
 i̸=x Ei ( ∂ri Ex − ∂x Ei )
Ei ( ∂x Ei )

 X

P



⃗⃗ ⃗ ⃗∂
E) · E =  i̸=y Ei ( ∂ Ey − ∂ Ei )  +
(∇
,

∂ri
∂y


Ei ( ∂y Ei )
i
P



∂
∂
∂
Ei ( ∂zEi )i̸=z Ei ( ∂ri Ez − ∂z Ei )(B.8)

or explicitly




∂
∂
∂
∂
Ez ( ∂z Ex − ∂x Ez ) + Ey ( ∂y Ex − ∂x Ey )

∂
Ei ( ∂x Ei )


 X





⃗⃗ ⃗ ⃗∂(∇
E) · E = Ex ( ∂ Ey − ∂ Ex ) + Ez ( ∂ Ey − ∂ Ez ) +
Ei ( ∂y Ei ) .
∂x
∂y
∂z
∂y




i 



∂
∂
∂
∂
∂
Ey ( ∂yEz − ∂zEy ) + Ex ( ∂xEz − ∂zEx )Ei ( ∂zEi )⃗We can then recognize each term in parentheses as a component of the curl of E:




∂
⃗
⃗
⃗
⃗
 Ez (∇ × E)y − Ey (∇ × E)z Ei ( ∂x Ei )

 X





⃗⃗ ⃗ ⃗∂⃗⃗⃗⃗(∇E) · E = Ex (∇ × E)z − Ez (∇ × E)x  +Ei ( ∂y Ei ) ,




i 



∂
⃗ × E)⃗ x − Ex ( ∇⃗ × E)⃗ y
Ey ( ∇Ei ( ∂z
Ei )

(B.9)

(B.10)

145⃗ with its curl:
and then recognize this vector as (negative) the cross product of E∂Ei ( ∂x Ei )


X


⃗⃗ ⃗ ⃗⃗ × (∇⃗ × E)⃗ +∂(∇E) · E = −E.
Ei ( ∂yEi ) 
i 
∂Ei ( ∂zEi )(B.11)∂
With the remaining vector, we can then use the one dimensional chain rule ∂x(f (x)2 ) = 2f ∂f
∂x to



arrive at1 ∂2 2 ( ∂x Ei )

⃗⃗ ⃗ ⃗⃗ × (∇⃗ × E)⃗ +(∇E) · E = −EX1 ∂ 2  2 ( ∂y Ei ) .


i 

1 ∂
2)
(
E
i
2 ∂z

(B.12)

We can pull out the factor of 1/2 from the vector and move the sum inside the derivatives:


∂ P
2
 ∂x ( i Ei )
⃗⃗ ⃗ ⃗⃗ × (∇⃗ × E)⃗ +1
∂ P2 ) ,
E) · E = −E(∇
(B.13)
 ∂y
E
(
i
i


2
 P

∂
2
∂z ( i Ei )and finally, recognizing thatP

2
2
i Ei is exactly the magnitude |E| , we arrive at the result:⃗⃗ ⃗ ⃗
2⃗ × (∇⃗ × E)⃗ + 1 ∇|E|⃗E) · E = −E.
(∇2

(B.14)

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