Novel Entangling Gates and Scalable Trap Designs for Trapped-Ion Quantum

Computing

by

ALEXANDER D. QUINN

A dissertation accepted and approved in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

in Physics

Dissertation Committee:

Hailin Wang Chair

David Allcock Advisor & Core member

Jens Nöckel Core Member

James Prell Institutional Representative

University of Oregon
Summer 2024



© 2024 Alexander D. Quinn
All rights reserved.

2



DISSERTATION ABSTRACT

Alexander D. Quinn

Doctor of Philosophy in Physics

Title: Novel Entangling Gates and Scalable Trap Designs for Trapped-Ion Quantum
Computing

Trapped ions have received much attention as a platform for quantum

computing, a purpose they may be well-suited for on account of their natural

features: Ions of the same species are inherently identical; they can be manipulated

using electric and magnetic fields via both their net charge, which allows external

fields to couple to ions’ center-of-mass motion, and via internal electronic

transitions; and when they are isolated from the environment, their states can

remain coherent for long time spans, at least by the standards of quantum

information experiments. Despite these features, the ability to carry out useful

quantum computing in trapped ions is limited, with two major practical constraints

being 1) the difficulty of coherently controlling ions’ states and 2) the limited

physical scale of existing trapped-ion quantum computers, which typically hold

no more than tens of ions. This document present a set of projects meant to help

address these limitations through different tracks, including the development

and testing of control techniques for trapped ions and the design of new types

of traps. Firstly, we present an entangling gate carried out on quantum bits

(qubits) encoded in a set of electronic energy levels that have been relatively

unexplored until recently and whose viability for quantum information processing

may enable more efficient architectures for trapped ion quantum computing.

Specifically, we entangled a pair of qubits encoded in two Zeeman sublevels of the

D5/2 metastable excited state of a pair of trapped 40Ca+ ions using Raman laser

3



beams 10s of THz detuned from resonance to limit scattering rates (a fundamental

error source in Raman gates). We demonstrate that high-fidelity gates (98.6(1)%

subtracting state preparation and measurement error, and 99.1(1)% subtracting

both SPAM and erasure) can be performed with this encoding scheme, show

that the main source of error is technical noise, and employ a leakage detection

scheme that allows decay or deshelving from the metastable level to be heralded,

potentially making correcting this class of errors easier. After this, we shift focus

from control techniques and discuss scalable trap design, focusing on a project to

fabricate ion traps using 3D printing, a technique that could potentially enable

microfabricated traps with high harmonicity, power efficiency, and depth of

confinement relative to the 2D (planar) microfabricated traps widely used in efforts

to scale up trapped-ion quantum computing. We design, simulate, fabricate, and

carry out preliminary electrical testing on metallized trap prints, demonstrating

some minimum viability of the technique, and show computationally that traps

produced this way could have trapping characteristics similar to those of other 3D

microfabricated designs. Finally, we consider scalable trap design for continuous-

variable quantum computing (CVQC), a quantum computing scheme where, in

trapped ions, information is encoded in the states of the vibrational modes of an

ion crystal. A key requirement for universal CVQC is the ability to perform non-

Gaussian operations, which can be difficult to carry out electronically. In this

work, we present a 2D ion trap design that can perform non-Gaussian motional

operations in an all-electronic way, with the design process accounting for the

geometric limitations imposed by 2D traps. We consider some of the motional

operations possible with this trap and estimate their associated coupling rates,

finding that under a reasonable set of assumptions about operating parameters,

coupling rates for non-Gaussian operations could be achieved that are comparable

4



to those previously achieved with all-electronic Gaussian operations could likely be

achieved with this trap design. This dissertation includes co-authored, published

material.

5



ACKNOWLEDGEMENTS

None of the work presented here would have been possible without the
independent contributions (e.g. the fabrication/assembly and testing of subsystems
of the experiment apparatus as the lab was first being built up), the collaboration
(e.g. in the slow-converging feedback loop between simulation and experiment), and
the patient guidance of the many members, past and present, of the Oregon Ions
lab and those the lab has worked with.

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TABLE OF CONTENTS

Chapter Page

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 27

1.1. A conceptual overview of this document . . . . . . . . . . . . . 27

1.2. Trapped ions - a brief guide . . . . . . . . . . . . . . . . . . . 29

1.2.1. Traps . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.2.2. Ions . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.3. Quantum computing . . . . . . . . . . . . . . . . . . . . . . 32

1.3.1. Building blocks for QC . . . . . . . . . . . . . . . . . 33

1.3.2. Technical requirements . . . . . . . . . . . . . . . . . 36

1.4. TIQC architecture . . . . . . . . . . . . . . . . . . . . . . . 37

1.5. The omg architecture . . . . . . . . . . . . . . . . . . . . . 40

1.5.1. Optical, metastable, and ground qubits . . . . . . . . . . 40

1.5.2. Using omg qubits together . . . . . . . . . . . . . . . . 41

1.6. Continuous-variable quantum computing . . . . . . . . . . . . . 42

1.6.1. Building blocks for CVQC . . . . . . . . . . . . . . . . 42

1.6.2. Implementing CVQC in trapped ions . . . . . . . . . . . 43

1.7. Scalable ion trap design . . . . . . . . . . . . . . . . . . . . 44

1.7.1. The QCCD architecture and surface electrode traps. . . . . . 45

II. EXPERIMENTAL APPARATUS . . . . . . . . . . . . . . . . . 47

2.1. Ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.1.1. Trap design . . . . . . . . . . . . . . . . . . . . . . 47

2.1.2. Trapping RF drive . . . . . . . . . . . . . . . . . . . 48

2.1.3. Trapping DC potential . . . . . . . . . . . . . . . . . . 50

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Chapter Page

2.1.4. Shim potentials . . . . . . . . . . . . . . . . . . . . . 51

2.1.5. UHV system . . . . . . . . . . . . . . . . . . . . . . 51

2.2. Magnetic field control . . . . . . . . . . . . . . . . . . . . . 54

2.2.1. Magnetic field coils . . . . . . . . . . . . . . . . . . . 54

2.2.2. Line noise mitigation . . . . . . . . . . . . . . . . . . 55

2.3. Imaging and state detection . . . . . . . . . . . . . . . . . . 57

2.3.1. Imaging system overview . . . . . . . . . . . . . . . . . 58

2.3.2. Imaging system performance . . . . . . . . . . . . . . . 60

2.4. Laser system . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.4.1. Rack-mounted laser boards . . . . . . . . . . . . . . . . 62

2.4.2. Laser frequency locking . . . . . . . . . . . . . . . . . 65

2.4.3. AOM boards . . . . . . . . . . . . . . . . . . . . . . 67

2.5. Far-detuned lasers . . . . . . . . . . . . . . . . . . . . . . . 69

2.5.1. 976 nm Raman lasers . . . . . . . . . . . . . . . . . . 70

2.5.2. 854 nm AC Stark shift laser . . . . . . . . . . . . . . . 72

2.6. Beam delivery . . . . . . . . . . . . . . . . . . . . . . . . 72

2.6.1. Beam pointing . . . . . . . . . . . . . . . . . . . . . 75

2.6.2. Beam focusing . . . . . . . . . . . . . . . . . . . . . 76

2.6.3. Beam power control . . . . . . . . . . . . . . . . . . . 78

2.6.4. Polarization control . . . . . . . . . . . . . . . . . . . 80

2.7. Electronic control fields . . . . . . . . . . . . . . . . . . . . 81

2.7.1. Magnetic Rabi drive . . . . . . . . . . . . . . . . . . . 81

2.7.2. Displacement drive . . . . . . . . . . . . . . . . . . . 82

2.7.3. Squeeze/beamsplitter drives . . . . . . . . . . . . . . . 83

2.7.4. Pulse shaping . . . . . . . . . . . . . . . . . . . . . 83

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Chapter Page

2.8. Experiment control system . . . . . . . . . . . . . . . . . . . 86

2.8.1. Control hardware . . . . . . . . . . . . . . . . . . . . 86

2.8.2. Clocking . . . . . . . . . . . . . . . . . . . . . . . . 86

2.8.3. DDSes and phase-coherent RF control . . . . . . . . . . . 87

2.8.4. Experimental control codebase . . . . . . . . . . . . . . 89

2.9. Ion loading . . . . . . . . . . . . . . . . . . . . . . . . . . 91

2.9.1. Calcium oven . . . . . . . . . . . . . . . . . . . . . . 92

2.9.2. Neutral calcium detection . . . . . . . . . . . . . . . . 92

2.9.3. Photoionization . . . . . . . . . . . . . . . . . . . . . 94

III. EXPERIMENTAL METHODS . . . . . . . . . . . . . . . . . . 97

3.1. Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.2. Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.2.1. Doppler cooling . . . . . . . . . . . . . . . . . . . . . 102

3.2.2. EIT cooling . . . . . . . . . . . . . . . . . . . . . . 104

3.2.3. Pulsed sideband cooling . . . . . . . . . . . . . . . . . 105

3.2.4. Cooling testing with thermometry . . . . . . . . . . . . . 106

3.2.5. Micromotion compensation . . . . . . . . . . . . . . . . 107

3.3. m qubit preparation . . . . . . . . . . . . . . . . . . . . . . 109

3.3.1. Optical pumping . . . . . . . . . . . . . . . . . . . . 109

3.3.2. Preparation fidelity and limits . . . . . . . . . . . . . . 110

3.3.3. Isolating qubit states . . . . . . . . . . . . . . . . . . 112

3.4. State detection . . . . . . . . . . . . . . . . . . . . . . . . 114

3.4.1. State-selective deshelving . . . . . . . . . . . . . . . . 114

3.4.2. Distinguishing shelved and unshelved ions - Fluorescence . . . 115

3.5. Coherent control - Spin . . . . . . . . . . . . . . . . . . . . 116

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Chapter Page

3.5.1. RF magnetic Rabi flopping . . . . . . . . . . . . . . . . 117

3.5.2. Raman Rabi flopping . . . . . . . . . . . . . . . . . . 121

3.5.3. Pulse types and coherent errors . . . . . . . . . . . . . . 123

3.5.4. Composite pulses . . . . . . . . . . . . . . . . . . . . 124

3.5.5. AC Stark shifts . . . . . . . . . . . . . . . . . . . . . 127

3.6. Coherent control - Motion . . . . . . . . . . . . . . . . . . . 128

3.6.1. General operations . . . . . . . . . . . . . . . . . . . 130

3.6.2. Displacement . . . . . . . . . . . . . . . . . . . . . . 131

3.6.3. Single-mode squeezing and phase shifting . . . . . . . . . 131

3.6.4. Two-mode squeezing and beamsplitter . . . . . . . . . . . 132

3.7. Coherent control - Spin-motion coupling . . . . . . . . . . . . . 133

3.7.1. Raman sidebands . . . . . . . . . . . . . . . . . . . . 133

3.7.2. Spin-dependent forces . . . . . . . . . . . . . . . . . . 136

3.8. Motional state characterization . . . . . . . . . . . . . . . . . 138

3.9. Coherence time measurement . . . . . . . . . . . . . . . . . . 140

3.9.1. Ramsey experiment . . . . . . . . . . . . . . . . . . . 140

3.9.2. Ramsey with spin echoes . . . . . . . . . . . . . . . . . 141

3.9.3. Fock state Ramsey experiment . . . . . . . . . . . . . . 143

3.10. Routine calibration techniques . . . . . . . . . . . . . . . . . 143

3.10.1. Coherent drive calibrations . . . . . . . . . . . . . . . . 143

3.10.2. Light shift calibrations . . . . . . . . . . . . . . . . . . 145

3.10.3. Optical frequency calibrations . . . . . . . . . . . . . . 146

3.10.4. Two-point calibrations . . . . . . . . . . . . . . . . . . 148

IV. RAMAN GATES IN METASTABLE QUBITS . . . . . . . . . . . 150

4.1. Erasure errors in m qubits . . . . . . . . . . . . . . . . . . . 151

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Chapter Page

4.1.1. Erasure conversion with fluorescence . . . . . . . . . . . . 152

4.1.2. Leakage pathways in m qubits . . . . . . . . . . . . . . 153

4.2. Geometric phase gates . . . . . . . . . . . . . . . . . . . . . 154

4.2.1. Fidelity estimation . . . . . . . . . . . . . . . . . . . 155

4.3. Gate errors . . . . . . . . . . . . . . . . . . . . . . . . . . 158

4.3.1. Error budget . . . . . . . . . . . . . . . . . . . . . . 158

4.3.2. Addressing dominant error sources . . . . . . . . . . . . 159

4.3.3. Errors from leakage detection . . . . . . . . . . . . . . . 161

4.3.4. Outlook . . . . . . . . . . . . . . . . . . . . . . . . 164

V. 3D-PRINTED ION TRAPS . . . . . . . . . . . . . . . . . . . . 166

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.2. Quantifying Trap Performance . . . . . . . . . . . . . . . . . 169

5.3. Trench Trap Geometries . . . . . . . . . . . . . . . . . . . . 172

5.3.1. Simple Trench . . . . . . . . . . . . . . . . . . . . . 174

5.3.2. Stacked Trench . . . . . . . . . . . . . . . . . . . . . 175

5.3.3. Comparison Geometries - SET and Wafer Traps . . . . . . 176

5.3.4. Summary . . . . . . . . . . . . . . . . . . . . . . . 177

5.4. 3D Printing . . . . . . . . . . . . . . . . . . . . . . . . . 178

5.4.1. Dielectric Material . . . . . . . . . . . . . . . . . . . 179

5.4.2. Initial Fabrication . . . . . . . . . . . . . . . . . . . . 180

5.4.2.1. Print . . . . . . . . . . . . . . . . . . . . . 180

5.4.2.2. Metalization . . . . . . . . . . . . . . . . . . 182

5.4.2.3. Integrated Optics . . . . . . . . . . . . . . . . 182

5.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 182

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Chapter Page

VI. SURFACE ELECTRODE TRAPS FOR CVQC . . . . . . . . . . . 189

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 189

6.2. Single ion in a cubic potential . . . . . . . . . . . . . . . . . 190

6.2.1. Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 190

6.2.2. The ωp = 0 case (cubic phase gate) . . . . . . . . . . . . 191

6.2.3. The ωp = ωx case . . . . . . . . . . . . . . . . . . . . 192

6.2.4. The ωp = 3ωx case (trisqueezing) . . . . . . . . . . . . . 192

6.3. Two ions in a cubic potential . . . . . . . . . . . . . . . . . . 193

6.3.1. Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 195

6.3.2. The ωp = 0 case . . . . . . . . . . . . . . . . . . . . 198

6.3.3. The ωp = ωc case . . . . . . . . . . . . . . . . . . . . 199

6.3.4. The ωp = ωc + 2ωr case . . . . . . . . . . . . . . . . . 200

6.3.5. The ωp = ωc − 2ωr case (parametric oscillation) . . . . . . . 200

6.3.6. The ωp = 3ωc and ωp = 3ωr cases (trisqueezing) . . . . . . . 200

6.4. Trap design for generating cubic potentials . . . . . . . . . . . . 201

6.4.1. Multipole expansion - the hexapole potential . . . . . . . . 201

6.4.2. Surface-electrode design . . . . . . . . . . . . . . . . . 203

6.4.2.1. Equivalent cylindrical electrode systems . . . . . . 203

6.4.2.2. Designing electrode systems to produce hexapoles . . 203

6.4.2.3. Integrating hexapole electrodes into
surface trap . . . . . . . . . . . . . . . . . . 205

6.4.3. Axial cubic potentials - design limitations . . . . . . . . . 209

6.5. Practical implementation . . . . . . . . . . . . . . . . . . . . 209

6.5.1. Trisqueezing . . . . . . . . . . . . . . . . . . . . . . 209

6.5.2. Parametric oscillation . . . . . . . . . . . . . . . . . . 211

6.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 212

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Chapter Page

VII. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . 213

APPENDIX: GATE ERROR CALCULATIONS . . . . . . . . . . . . . 214

A.1. Error budget . . . . . . . . . . . . . . . . . . . . . . . . . 214

REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . 221

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LIST OF FIGURES

Figure Page

1.1. A schematic illustration of the energy levels of an ion in
a trap, showing two independent modes: A harmonic
oscillator mode (left) and the internal electronic state of
the ion (right). Line thicknesses in the right plot correspond
to the number of Zeeman/hyperfine states in a given energy
level. Solid arrows denote dipole transitions, while dashed
arrows denote quadrupoles. Splittings between energy levels
are not to scale. . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.2. Examples of commonplace one- and two-qubit gates written
in circuit notation (with each line representing an individual
qubit over time) and the operation of the gate on a specific
input state. a) The Hadamard gate, which rotates a single
qubit. b) The controlled-NOT (CNOT) gate, which carries
out a bit flip on the target qubit conditional on the state of
the control qubit, generating entanglement. . . . . . . . . . . . . . 35

1.3. An illustration of the conceptual reasoning for using mixed-
species ion chains. a) A pair of ions sharing a potential
well can be entangled by driving both ions with laser fields.
b) For carrying out auxilliary functions (e.g. sympathetic
cooling of the ions being used for gates), additional ions
need to be added to the chain. If these ions are of the same
species as those used for gates, when driven with laser fields,
they will emit photons that the other ions can absorb,
destroying the information they contain. c) If the auxilliary
functions are performed using ions of a different species
with different optical transition frequencies, then photons
scattered by one species will not be absorbed by the other. . . . . . . 38

1.4. The three qubit types considered by the omg architecture. . . . . . . . 41

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Figure Page

1.5. A cartoon illustration of a possible QCCD “unit cell,” which
can be tiled to produce arbitrarily large trap networks.
Dark blue represents DC electrodes, while pink represents
trapping RF. Ions are shown in light blue. The illustration
on the left shows the unit cell of a 2D/surface electrode
implementation of a QCCD architecture, while the one on
the right shows how the concept can be extended to three-
dimensional traps. . . . . . . . . . . . . . . . . . . . . . . . . 46

2.1. Photographs of our trap, showing (left) the trap attached
to a vacuum flange and connected to electrical feedthroughs
and (middle) a close-up shot of the unmounted trap, with
the rough location of the trapping zone marked with a blue
box. The associated inset in a picture taken with a CMOS
camera of a five-ion chain, with ion spacing marked. Finally,
a COMSOL render (right) shows trapping RF electrodes
marked in red. . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.2. Schematic illustration of the hardware setup used for
estimating the thickness of a layer of electroplated metal
by measuring the total charge passed through the system
during electroplating. . . . . . . . . . . . . . . . . . . . . . . . 49

2.3. Block diagram of the RF chain used to supply the RF
trapping voltage. The trapping RF itself is supplied by the
chain in red. Quadrupole motional drives (discussed further
in Section 2.7.3 are combined with the trapping RF at the resonator. . . 51

2.4. Circuit diagram for the electrical connections to the DC
trap electrodes (including AC drives applied to these
electrodes), with endcap needles in blue, DC trapping
electrodes in white, and compensation electrodes in light
yellow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.5. a) CAD rendering of the vacuum chamber, associated
support hardware, and components directly attached to this
hardware, including magnetic field coils (Section 2.2.1 and
b) mounts for beam delivery optics (Section 2.6). . . . . . . . . . . . 53

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Figure Page

2.6. a) A schematic of the magnetic field coil arrangement used
to set the quantization axis in the trap. Current is marked
with white arrows, with arrow thickness corresponding
the current magnitude. b) A photograph showing the five
magnetic field coils used on the trap. Worth noting are the
two smaller coils (white and blue) wrapped around the top
left coil, which are used for line noise feedforward (Section 2.2.2). . . . . 55

2.7. Schematic illustrations of our two line noise mitigation
schemes, explained in greater detail in text. . . . . . . . . . . . . . 56

2.8. Schematic illustration (and photograph below) of our
imaging system, with light emitted by the ion (a) being
captured by an objective (b), passed through a slit (c)
to reduce stray light and a narrow-band filter (d) to
prevent the system from registering light far away from the
wavelength of the cycling transition of Ca+, collimated with
a lens (f) in an adjustable barrel mount to be redirected
by a flip-mounted mirror (g), which can direct light to
a focusing lens leading to the PMT (h) and (i) or to a
focusing lens (j) and (k) heading to the camera (l). . . . . . . . . . 59

2.9. Schematic of our overall laser system (exluding far-detuned
lasers, discussed in Section 2.5) shown in terms of the
output of a single laser breadboard. Free-space laser beams
are shown with red and blue lines. Fiber optics are shown
with solid black lines. Data connections are shown with
dashed black lines. . . . . . . . . . . . . . . . . . . . . . . . . 62

2.10. Photograph of our laser rack, showing the drawers
containing laser breadboards as well as an additional rack
for housing laser controllers and a third for the wavemeter
and OSA board. Inset shows a sample laser breadboard
mounted in a drawer. . . . . . . . . . . . . . . . . . . . . . . . 64

2.11. Schematic illustrations of the three main AOM
configurations we use in the lab. These schematics omit
the focusing/recollimating lenses that focus the beams into
the AOMs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

16



Figure Page

2.12. Schematics of the two 976 nm Raman laser setups we
have used in our experiments, showing the diode laser
sources (marked in dark red) all the way through to the ion
(marked in light blue). These schematics omit details of the
beam delivery optics. . . . . . . . . . . . . . . . . . . . . . . . 70

2.13. Schematic of the laser system used to supply our high-power
854 nm beam used for generating AC Stark shifts on the
non-qubit levels of the D5/2 manifold. . . . . . . . . . . . . . . . . 72

2.14. Schematic of the optics leading from the output collimators
on our optics table to our trap. Definitions of abbreviations
are as follows - PD: photodiode, GTP: Glan-Thompson
polarizer, PM: piezo motorized mirror, PS: piezo stack
mirror, GP: Glan polarizer. . . . . . . . . . . . . . . . . . . . . 74

2.15. A photograph with labels of one of our outcoupler bases,
with the front plate of the photodiode box removed to show
internal hardware. . . . . . . . . . . . . . . . . . . . . . . . . 79

2.16. Schematic summary of the RF voltages applied to the trap
for coherent driving, marking the trap electrodes (shown in
cross-section) that they are connected to. . . . . . . . . . . . . . . 82

2.17. Block diagram showing the RF chain for three motional
drives used in our experiments. PS: pulse shaper, BS: beamsplitter. . . . 84

2.18. Schematic illustration of our two pulse-shaping schemes. a)
Pulse shaping by multiplying a fixed-amplitude RF pulse
with a nearly-trapezoidal DC envelope. b) Pulse shaping by
running the DDS in RAM mode and generating arbitrarily
shaped pulses, with the DC output on the AWG being unchanged. . . . 85

2.19. Schematic illustration of our experiment control
system. a) Block diagram showing how our experiment
control hardware (blue) interfaces with other blocks
of our experiment, including beam control (red), the
trap/imagining system (yellow), and the host PC allowing
users to interface with the system (purple). b) A schematic
of the beam power control system we use (SU-Servo). . . . . . . . . . 87

17



Figure Page

2.20. Class inheritance diagram showing the hierarchy of classes
in our in our experiment codebase, down to the level of code
for running individual experiments (e.g. an experiment for
driving Rabi flopping on a qubit, RabiFlopping). NDScan
classes are shown in yellow, in-house support classes are
shown in pink, and experiments are shown in blue. The
inset in purple shows the sequence for a generic scan
point. In both portions, the main block of code that differs
between individual experiments is shown in dark blue. . . . . . . . . 90

2.21. a) Illustration of the relevant photoionization geometry,
showing the neutral calcium beam coming from the oven
and the PI beam relative to the trap (electrodes shown
in black). The neutral calcium and PI beams are both in
the plane of the page. b) Energy level diagram of the two-
photon PI process. . . . . . . . . . . . . . . . . . . . . . . . . 91

2.22. a) Fluorescence spectrum data (black dots) on the 423 nm
line for our neutral calcium beam with a manual fit
including three calcium isotopes (individual isotopes shown
with dashed colored lines, total fluorescence shown with a
solid black line). b) Relative probability of loading 43Ca+ at
different 423 nm beam detunings and with different amounts
of Doppler broadening. . . . . . . . . . . . . . . . . . . . . . . 94

3.1. A level diagram showing a Ca+ ion trapped in a harmonic
oscillator potential (only one dimension of which is shown).
Transitions between internal electronic energy levels are
marked with the wavelengths and polarizations that we use
to drive them. Energy levels are not to scale. . . . . . . . . . . . . 98

3.2. A schematic illustration of how the RF electrodes of a linear
Paul trap generate a harmonic effective potential. . . . . . . . . . . 101

3.3. A set of diagrams showing how a) Doppler cooling, b) EIT
cooling, and c) pulsed sideband cooling are implemented in
our setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

18



Figure Page

3.4. Schematic illustrations of techniques that we use to
minimize micromotion. The offset direction each technique
allows us to optimize is marked with a black double-headed
arrow. The direction of the micromotion is marked with
a double-headed red arrow. These techniques are a)
minimization of position change with RF confinement
change, b) minimization of Doppler broadening when
driving with the 397π beam (blue arrow), and c)
minimization of the micromotion sideband associated with
our two-beam (red arrows) Raman Rabi drive. . . . . . . . . . . . . 108

3.5. a) The generic sequence for characterizing the effect of
some operation (“Expt.”) on some known state, performing
repeated experimental “shots” and calculating the
probability of different outcomes. b) The specific pulse
sequences of each experimental step (not showing cooling,
which either happens before or is interleaved with state preparation). . . 110

3.6. Sample Rabi flopping, driven with a resonant RF magnetic
field, showing two-level sinusoidal dynamics when the
854LS beam is on (blue) and precession through the D5/2

manifold before returning to the starting state when the
854LS beam is off (red). . . . . . . . . . . . . . . . . . . . . . 113

3.7. A sample count histogram for two-ion fluorescence
detection, with the “zero bright” (blue), “one bright” (red),
and “two bright” (green) Poisson distributions labelled above. . . . . . 116

3.8. Schematic illustration of our RF magnetic Rabi drive,
showing trap electrodes (pink for trap RF, blue for Rabi

drive current), the quantization axis magnetic field B⃗ (in

the xz plane), and the magnetic Rabi drive B⃗drive (in the xy plane). . . . 120

3.9. Schematic illustration of the dynamics of a Λ system,
showing a) the relationship between the three-level
dynamics of the actual system and the effective two-level
dynamics and b) the three-level system we use in our experiments. . . . 122

3.10. The results of experiments testing composite a) π/2 and
b) π pulses. Different colors denote time scans at different
ϕBB1, showing that a value of ϕBB1 can be selected such
that the output of the pulse sequence is made insensitive to
change in pulse times. . . . . . . . . . . . . . . . . . . . . . . . 126

19



Figure Page

3.11. A reference table for the motional operations performed in
our experiments, showing in the phase space picture a) the
Wigner function of the motional ground state, b the effect
of a displacement operation, c) the effect of squeezing, d)
the effect of a phase shift on a displaced state, e) the effect
of a beamsplitter operation on a pair of modes (with the
Wigner function of the second mode shown in red), and f)
the effect of two-mode squeezing on the correlations between
the quadratures of a pair of modes. . . . . . . . . . . . . . . . . . 130

3.12. Two techniques of achieving spin-motion coupling. a)
Raman sideband transitions, showing the ∆n = +1 (BSB)
and ∆n = −1 (RSB) transitions, with the spin states being
represented as a pair of harmonic oscillator energy ladders
separated by the spin state splitting. b) Spin-dependent
forces, showing how two interfering beams with the same
polarizations will generate periodic intensity gradients of
different amplitudes (and different force magnitudes) on the
two spin states. . . . . . . . . . . . . . . . . . . . . . . . . . 134

3.13. Pulse sequences for Ramsey sequences for characterizing
spin (left) and motion (right) coherence, with (bottom) and
without (top) a spin echo pulse. Carrier pulses are shown in
dark blue, blue sideband pulses in light blue. To generate
fringes at a given t, the phase ϕ of the final pulse can be scanned. . . . 142

3.14. Experiments for measuring the AC Stark shifts caused by a
given beam (red). a) Carrying out Rabi spectroscopy while
the beam is on and comparing this result to the result when
the beam is off. b) Performing a Ramsey sequence with a
spin echo, with the beam only being on during one delay so
as to cause asymmetrical accumulation of phase. . . . . . . . . . . . 146

3.15. Pulse sequences (with time running from left to right) of
our four laser line calibrations, with the state of the ion at
the end of the experiment (during a fluorescence check, FC)
shown assuming that laser associated with the line being
characterized is far off-resonance. . . . . . . . . . . . . . . . . . . 147

3.16. Illustration of the two-point calibration servo loop. . . . . . . . . . . 148

20



Figure Page

4.1. Schematic illustration of our erasure error detection scheme
in the context of a gate, showing how checks for 397
fluorescence can be used both to confirm successful state
preparation prior to running a gate and check for leakage
errors after a gate. . . . . . . . . . . . . . . . . . . . . . . . . 153

4.2. a) Gate pulse sequence showing beams relevant to the gate,
with SDF phase (0, π) denoted by color (white, blue). b) A
scan over SDF detuning from the LF OOP mode resonance,
showing the crossing point in |↓↓⟩ and |↑↑⟩ populations at
which the gate was performed. Inset shows high-shot time
series data at the gate operating point. c) A sample parity
fringe, with insets showing high-shot time-series data at the
parity fringe extrema. . . . . . . . . . . . . . . . . . . . . . . . 157

4.3. a) A flowchart illustrating the possible outcomes of a
one-ion gate sequence with an FC carried out at the end,
considering only leakage errors. Outcomes corresponding
to an erasure error are labelled in yellow, while outcomes
corresponding to a missed leakage error that is missed are
labelled in pink. b) An accounting of the error probabilities
in each category with and without leakage checks. . . . . . . . . . . 163

5.1. Cross-section of a trap fabricated using our proposed
method of 3D printing an electrodes onto a wafer containing
electrode routing and some of the potential integrated ion
control elements that have been demonstrated. . . . . . . . . . . . . 168

5.2. Cross-sectional views of of the trap designs simulated. The
ions’ location is shown as an orange dot and the trap axis is
out of the page. DC electrodes are blue, RF electrodes are
red. Numerical apertures are shown in lavender. . . . . . . . . . . . 173

21



Figure Page

5.3. Simulated parameters of surface-electrode traps vs the
symmetric and anti-symmetric ‘simple trench’ trench
geometries. (a) Trap depth at a fixed radial frequency.
(b) The quadrupole component (C2) of the trapping
potential at the ion. (c) The hexapole component (C ′

3)
of the trapping potential. (d) The octopole component (C ′

4)
of the trapping potential. (e) Electrode dimensions at a
constant ion-electrode separation for the symmetric simple
trench trap. (f) Electrode dimensions at a constant ion-
electrode separation for the anti-symmetric simple trench
trap. (g,h) NA above and below the ion. Parameters are
plotted against a generic geometric variable w, which is
related to an electrode dimension of each trap as shown in
the legend. Simulation results are shown with markers. The
representative SET, wafer, and simple trench traps whose
characteristics are summarized in Table 4 are highlighted
with larger markers. Simulation points are connected with
dashed lines only as a guide to eye. . . . . . . . . . . . . . . . . . 184

5.4. Simulated parameters of surface-electrode traps and wafer
traps vs the symmetric ‘stacked trench’ trap geometry. (a)
Trap depth at a fixed radial frequency. (b) The quadrupole
component (C2) of the trapping potential at the ion. (c)
The hexapole component (C ′

3) of the trapping potential.
(d) The octopole component (C ′

4) of the trapping potential.
(e) The hexapole component of the trapping potential,
excluding the SET data to make the other series more
clear. (f) Electrode dimensions at a constant ion-electrode
separation for the symmetric stacked trench trap. (g,h)
NA above and below the ion. Parameters are plotted
against a generic geometric variable w, which is related
to an electrode dimension of each trap as shown in the
legend. Simulation results are shown with markers. The
representative wafer and symmetric stacked trench traps
whose characteristics are summarized in Table 4 are
highlighted with larger markers. Simulation points are
connected with dashed lines only as a guide to eye. . . . . . . . . . . 185

22



Figure Page

5.5. Simulated parameters of surface-electrode traps and wafer
traps vs the anti-symmetric ‘stacked trench’ trap geometry.
(a) Trap depth at a fixed radial frequency. (b) The
quadrupole component (C2) of the trapping potential at
the ion. (c) The hexapole component (C ′

3) of the trapping
potential. (d) The octopole component (C ′

4) of the trapping
potential. (e) The hexapole component of the trapping
potential, excluding the SET data to make the other series
more clear. (f) Electrode dimensions at a constant ion-
electrode separation for the anti-symmetric stacked trench
trap. (g,h) NA above and below the ion. Parameters are
plotted against a generic geometric variable w, which is
related to an electrode dimension of each trap as shown
in the legend. Simulation results are shown with markers.
The representative wafer and anti-symmetric stacked trench
traps whose characteristics are summarized in Table 4 are
highlighted with larger markers. Simulation points are
connected with dashed lines only as a guide to eye. . . . . . . . . . . 186

5.6. SEM images of gold coated ormocer trap structures
printed on glass. In this structure, many trenches have
been connected up with X-junctions, as is envisaged in
a large ‘QCCD’ processor (Blakestad et al. (2009)). (a)
An overhead view of a grid of symmetric stacked trenches
showing segmentation of the top DC electrodes. (b) An
oblique view of the same grid. (c) A close up view of an
intersection of trenches. (d) A false-color version of (c)
showing DC electrodes in blue and RF electrodes in red. (e)
A close-up view of a DC electrode segment. . . . . . . . . . . . . . 187

5.7. Proposed integration of electrode metalization with printing
process shown here on a short section of the larger trap
shown in Fig 5.6. (a) Silicon wafer with electrode routing
and vias up to the surface is fabricated using standard
CMOS process (metal shown copper coloured, SiO2

dielectric shown transparent). (b) Aluminum is deposited
and patterned to define gaps between electrodes. (c) DLW
used to print ormocer trap structures (blue). (d) Gold is
deposition from multiple angles to coat all surfaces of the
print. (e) Aluminium is etched to allow liftoff of the gold
over the gaps between electrodes. Insets shown the liftoff
process in more detail. . . . . . . . . . . . . . . . . . . . . . . 188

23



Figure Page

6.1. Plot of the Fock state distribution of the state generated
by applying the ωp = ωx operation to the motional ground
state for a time t = 0.23 × 2π/g. Components n > 9 not
shown. Inset - Associated Wigner function. . . . . . . . . . . . . . 193

6.2. Plot of the Fock state distribution of of the trisqueezed
state, obtained by applying the trisqueezing Hamiltonian to
the ground state for a time t = 0.035 × 2π/g. Components
n > 9 not shown. Inset - Associated Wigner function. . . . . . . . . . 194

6.3. Left - Illustration of a two-ion chain with coordinates
labelled. Equilibrium positions are marked in light blue.
Right - Table illustrating two-ion vibrational normal modes. . . . . . . 195

6.4. Trapping potential generated by symmetrical surface-
electrode trap around ion (marked in yellow). The
decomposition of this potential into overlaid quadrupole,
hexapole, and octopole terms are represented schematically. . . . . . . 202

6.5. Cross-section of an asymmetric surface-electrode trap
through plane perpendicular to trap axis, with ion position
marked by a yellow dot. Cross-section of equivalent
cylindrical electrode system (partitioned ring of radius r0) is
overlaid on top, with the relationship between surface and
cylindrical electrode positions r⃗(θ) = −r0êx + r0 tan

(
θ
2

)
êy

illustrated geometrically. Figure adapted from Wesenberg
(Wesenberg (2008) . . . . . . . . . . . . . . . . . . . . . . . . 204

6.6. a). Quadrupole system. b). Hexapole system. c). Combined
electrode system for generating quadrupole trapping
potential and hexapole trisqueezing potential. Red -
Quadrupole electrodes. Purple - Hexapole electrodes.
Green - Inner ground. Blue - Outer ground. Notably, to
produce optimal performance under geometric constraints,
θ1 + θ2 = 60◦, and the quadrupole electrodes are adjacent to
hexapole electrodes. . . . . . . . . . . . . . . . . . . . . . . . 207

6.7. a). C2 for the quadrupole electrodes and C3 for the
hexapole electrodes plotted against θ1 when θ1 + θ2 = 60◦.
b). Widths of different electrodes in the combined electrode
system of Fig. 6.6b, normalized to ion height, under the
condition θ1 + θ2 = 60◦. . . . . . . . . . . . . . . . . . . . . . . 208

24



Figure Page

A.1. a) Motional Ramsey experiment, plotting the contrast
between the |0⟩ and |1⟩ fock states as a function of the
delay time between sideband π/2 pulses. b) Numerically
simulated Bell-state infidelity due to a white motional noise
source as a function of the coherence to gate time ratio.
Orange lines represent the infidelity expected for measured
coherence and gate times used in this work. Dashed lines
represent the 68% confidence interval. . . . . . . . . . . . . . . . 215

A.2. a) Bell-state infidelity corresponding to a normally
distributed static offset in gate mode frequency with
standard deviation σ for different orders of Walsh
modulation. b) Distribution of Bell-state infidelities
corresponding to a normal distribution of mode frequency
offsets with a standard deviation of 100 Hz for a Walsh
1 4-loop gate operating at 11.6 kHz. Dashed orange lines
represent 68% confidence interval. . . . . . . . . . . . . . . . . . 217

A.3. a) Bell-state infidelity corresponding to a normally
distributed static offset in gate mode frequency with
standard deviation σ for different orders of Walsh
modulation. b) Distribution of Bell-state infidelities
corresponding to a normal distribution of mode frequency
offsets with a standard deviation of 100 Hz for a Walsh
1 4-loop gate operating at 11.6 kHz. Dashed orange lines
represent 68% confidence interval. . . . . . . . . . . . . . . . . . 218

A.4. a) Population scattered out of the m = +5/2 and +3/2 in
the D5/2 manifold with either the 976 Rpi or Rsig beam on.
b) Population scattered out of the qubit states with the 854
beam on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

25



LIST OF TABLES

Table Page

1. A summary of projects worked on over the course of the
last five years and which areas of trapped-ion quantum
computing they are relevant to. My contribution to the
projects in the top two rows was on the experimental side
(hardware design/construction, writing/running/debugging
experiments), while my contribution to the projects on the
bottom was on the theoretical/computational side. . . . . . . . . . . 29

2. A summary of the components of our control system and
what we use them for in our experiments. . . . . . . . . . . . . . . 88

3. Dominant sources of two-qubit gate infidelity as predicted by
our error model. The lower half of the table summarizes the
expected contributions due to erasure errors. Values in bold
represent infidelity as measured in the gate . . . . . . . . . . . . . . 165

4. Summary of results for trap parameter calculations for the
SET, simple trench trap (‘Simple’), stacked trench trap
(‘Stacked’), and wafer trap. Values shown correspond to a
set of geometric parameters selected to balance trap depth
and quadrupole strength/purity. . . . . . . . . . . . . . . . . . . 177

5. The dimensions (in microns) of the traps whose parameters
are summarized in Table 4. . . . . . . . . . . . . . . . . . . . . . 177

6. Electrical and thermal properties of ormocer. . . . . . . . . . . . . . 179

A.7. Measured scattering rates out of D5/2 (in Hz) when
initialized in the m = +3/2 or m = +5/2 qubit states
for various beams used in the gate. Measurements for the
854 LS beam were taken with two trapped ions. . . . . . . . . . . . . 219

26



CHAPTER I

INTRODUCTION

1.1 A conceptual overview of this document

This thesis describes a set of projects, largely independent of each other

in terms of the work involved and the specific goals pursued, that may seem

disconnected from one another. However, an important common thread runs

between them: They are all ultimately focused on developing tools to use trapped

ions in technological applications, whether quantum sensing or trapped-ion

quantum computing (TIQC). In that light, these projects can be organized

according to how they relate to the broader project of refining, expanding the

capabilities of, and scaling up TIQC. There are two “axes” along which these

projects can be organized: The type of TIQC they work to enable and the type

of improvement they seek to make. Specifically, these projects can be in support

of discrete-variable quantum computing (i.e. quantum computing using two-level

Hilbert spaces or “qubits,” which will typically just be called “quantum computing”

in the rest of the text) or of continuous-variable quantum computing (CVQC, i.e.

quantum computing using infinite Hilbert spaces of evenly-spaced energy states).

They also broadly fall under two parallel paths of development: the development

and testing of new techniques for controlling the states of trapped ions (mostly

carried out on the experimental side), and the design of novel trap geometries

(carried out on the theoretical/computational side). These projects are listed

specifically in Table 1. Not listed in the table is a fifth project investigating non-

Hermitian quantum systems using our qubit platform. While this specific project

was less motivated by technological applications, it demonstrates features of non-

Hermitian dynamics with potential applications (e.g. the ability to implement fast

and deterministic operations) that are worth discussion.

27



Projects not covered in detail. Of these five projects, two of them (observing

of non-Hermitian dynamics in trapped-ion qubits Quinn et al. (2023) and carrying

out trapped-ion motional interferometry with phase sensitivity below the standard

quantum limit J. Metzner et al. (2024)) are not covered in detail in this thesis.

Accounts of these projects can be found in their respective papers and in the

thesis of Dr. Jeremy Metzner (see J. M. Metzner (2024)), which discusses them in

greater detail (particularly on the theory side). Much of the content laid out in this

introduction and in Chapters II and III still lays out the experimental apparatus

and methods used to carry out these projects.

Currently unpublished projects. Our project to perform entangling gates in

metastable qubits, covered in Chapter IV, is soon to be published. My role in this

project was on the experimental side, developing and debugging experiments and

setting up hardware (though two very important hardware systems enabling this

specific project, a high power injection-locked diode laser system for performing

gates 2.5.1 and a patched version of our experimental control system 2.8.3, were

built and tested by my colleagues Sean Brudney and Gabe Gregory respectively).

The work of simulating our gates and calculating an error budget was performed

by Gabe Gregory. Important context for this project is provided by a set of

photon scattering rate measurements, taken by me and analyzed and presented

in the thesis of Dr. Daniel Moore (see Moore et al. (2023)). An entirely separate

theoretical/computational project, developing a surface-electrode trap for carrying

out non-Gaussian operations on the motional modes of trapped ions for CVQC

applications, was carried out by me alone, with guidance from my advisor David

Allcock.

Projects covered in existing publications. Chapter V is based on a paper

currently available as a preprint, ”Geometries and fabrication methods for 3D

28



Discrete-variable
quantum computing

Continuous-variable
quantum computing
(CVQC)

Control
techniques

Single- and two-
qubit Raman gates in
metastable (m) qubits

Coherent control of ion
motion for motional
interferometry

Trap design Design of 3D-printable
ion traps for achieving
scalability and favorable
geometries

Design of surface-
electrode traps (SETs)
for carrying out non-
Gaussian operations on
ion motion

Table 1. A summary of projects worked on over the course of the last five years
and which areas of trapped-ion quantum computing they are relevant to. My
contribution to the projects in the top two rows was on the experimental side
(hardware design/construction, writing/running/debugging experiments), while my
contribution to the projects on the bottom was on the theoretical/computational
side.

printing ion traps” (Quinn, Brown, Gardner, and Allcock (2022)). The portions

of this paper focusing on theory and simulation were written by me, with David

Allcock writing portions of the intro and conclusion and Morgan Brown writing

the description of the 3D printing and metallization process. Morgan Brown and

David Miller carried out the fabrication and characterization of test prints, and I

simulated and analyzed the designs discussed in this chapter.

1.2 Trapped ions - a brief guide

Before discussing their technological applications in quantum information,

a short discussion is offered below of what trapped ions are, how they can be

modelled, and how, in broad strokes, they can be controlled. These topics,

particularly as they relate to our own experiments, are discussed more thoroughly

in Chapter III.

1.2.1 Traps. Because they have a net electric charge, ion experience

a Lorentz force F⃗ = q(E⃗ + v⃗ × B⃗) and this force can be used to confine ions in

29



an entirely classical way. There are two specific types of traps widely in use for

quantum information experiments: Penning traps and Paul traps.

Penning traps. Penning traps confine ions using a combination of static

electric and magnetic fields. A pair of endcaps held at a static voltage offer

confinement in one direction while a magnetic field parallel to the endcap axis

forces ions to undergo cyclotron motion, providing confinement in the other two

directions. The ions’ continuous cyclotron motion (and the large magnetic fields

required to confine them) makes them more difficult to work with in certain ways,

and Penning traps are less widely used in research on scalable trapped-ion quantum

computing than Paul traps (though there has been work done on microfabricated,

scalable Penning traps. See Jain et al. (2024) among others).

Paul traps. Paul traps confine ions using only electric fields. By Earnshaw’s

theorem, it is impossible to achieve confinement in more than two directions

at once using static electric fields (an immediate consequence of the fact that

∇·E⃗ = 0 in the absence of charge). Paul traps work around this constraint by using

dynamic electric fields instead of static ones. A deeper treatment of this is given in

Section 3.1. In short though, if an AC multipole potential is applied to a charged

particle, in certain parameter regimes, the time-averaged motion of the particle

shows the particle experiencing an effective potential (or “pseudopotential”) ϕeff

proportional to the energy density of the AC electric field (i.e. ϕeff ∝ |E⃗|2). For a

“linear” Paul trap (shown schematically in Figure 3.2), the trap electrodes are very

long relative to the spacing between them, and they generate a 2D potential in the

trapping zone. This potential only confines in two directions, with confinement in

the third direction being supplied by a DC electric field, as in Penning traps. Most

Paul traps used in trapped-ion work generate, to lowest-order, an AC quadrupole

potential, which in turn corresponds to a harmonic pseudopotential. An ion

30



trapped in this 2D harmonic pseudopotential and the 1D harmonic potential

supplied by the DC electric field (which tends to be anti-trapping in the other two

directions) is, effectively, a 3D harmonic oscillator. This feature is important for

much of the work done in trapped ions.

1.2.2 Ions. A neutral atom (or molecule) that has had one (or more)

electrons removed (or added) is an ion. For all discussions below, “ion” will refer to

a more narrow subset of this category: alkaline earth atoms that are singly-ionized,

such that they have a single valence electron in a hydrogen-like potential. This

accounts for most of the species used in trapped-ion experiments (though other

elements like ytterbium have been used in experiments requiring the control and

detection of individual ions (Balzer et al. (2006)), as have molecules (Drewsen,

Mortensen, Martinussen, Staanum, and Sørensen (2004))).

Electronic level structure. The generic energy level structure of an alkaline

earth ion is shown in Figure 1.1. The relevant energy levels are the electronic

ground state S1/2, the short-lived excited states P1/2 and P3/2, and the D3/2 and

D5/2 excited states (which are long-lived in most ion species). The S and D levels

can both couple to the P levels (where other selection rules are not violated)

through electric dipole interactions. On the other hand, transitions between the

D levels (also called metastable levels) and the S level are dipole-forbidden and can

only take place through electric quadrupole interactions.

Zeeman/hyperfine splitting. Within an energy level (e.g. within S1/2),

the valence electron of the ion can occupy different angular momentum states

based on the orbital angular momentum and its own inherent spin. Uncoupled to

other systems, these states would be degenerate in energy. However, applying an

external magnetic field will lift this degeneracy via the Zeeman effect, defining a

quantization axis (i.e. an axis along which the angular momentum state is more

31



Vibrational 
energy levels

      Electronic
energy levels

Figure 1.1. A schematic illustration of the energy levels of an ion in a trap,
showing two independent modes: A harmonic oscillator mode (left) and the internal
electronic state of the ion (right). Line thicknesses in the right plot correspond to
the number of Zeeman/hyperfine states in a given energy level. Solid arrows denote
dipole transitions, while dashed arrows denote quadrupoles. Splittings between
energy levels are not to scale.

or less aligned with the magnetic field) and a set of non-degenerate states. In

addition, if the ion nucleus has an inherent magnetic moment, this will also couple

with the electron spin and generate hyperfine splitting, lifting the degeneracy of

states based on the (anti-)alignment of the electron and nuclear spins. Both of

these processes yield a set of energetically distinct magnetic sublevels in each

energy level. In all of the work presented here, we use an ion species with no

nuclear spin, meaning degeneracy in our system is lifted only by Zeeman splitting.

1.3 Quantum computing

A quantum computer is a device that processes information that is encoded

in quantum states. Typically when people refer to quantum computing, they

implicitly mean computing done using information encoded in discrete two-level

systems (a quantum bit or “qubit”). Qubits can be encoded in the states of many

quantum systems: the current through a superconducting circuit, the spin state

of a nitrogen-vacancy center, or the polarization of an individual photon, among

32



other systems (see Cheng et al. (2023) for a broad overview of the state of field

around the time of writing). While as of now, no quantum computing platform

has seen the implementation of large-scale “fault-tolerant” quantum computing

(described in greater detail later in this section), trapped ions are currently among

the leading platforms, with some of the highest reported single-(Löschnauer et

al. (2024)) and two-qubit (Clark et al. (2021); Löschnauer et al. (2024)) gate

performances reported in the literature, as well as recent reports of better-than-

break-even error correction in small-scale experiments (Hong, Durso-Sabina, Hayes,

and Lucas (2024)). As such, there is currently much interest in scaling up trapped-

ion quantum computing.

These next three sections will offer a broad overview of quantum

computing. (More detailed discussions can be found in review papers or in

books, e.g. Nielsen and Chuang (2010), with this source informing much of the

discussion below), Quantum computing’s basic building blocks and the associated

technical requirements that must be met for it to be practically useful, its typical

implementation in trapped ions, and a recently proposed alternative trapped-ion

architecture (the omg qubit scheme Allcock et al. (2021)) that motivates much of

the work we do in our lab will be discussed.

1.3.1 Building blocks for QC. A quantum algorithm (a set of

calculations run on a quantum computer) can be carried out by initializing a

quantum register (a set of qubits) to some known state and carrying out a set of

state-changing operations and measurements on it. The output of this sequence

when the final state of the register is measured will be some tensor product of

measurement basis eigenstates (usually written as |0⟩ and |1⟩ or |↓⟩ and |↑⟩). To

determine the output state of the algorithm in this basis, the algorithm must (in

33



general) be repeated many times to build up statistics, though computations can

also have deterministic outputs.

The state manipulations performed on qubits during a quantum algorithm

are often called “gates,” in analogy to classical computing. It can be shown that a

set of single-qubit gates together with a single two-qubit gate are “complete” (i.e.

that any quantum algorithm can be carried out using a combination of these gates).

The meanings of these gate types are clarified here.

Single-qubit gates. In classical computing, where an individual bit can only

be in a 0 or 1 state, the only meaningful “single-bit gate” is a NOT operation,

flipping the state of a bit. In quantum computing however, where information in

encoded in a coherent superposition of states |0⟩ and |1⟩ (|Ψ⟩ = C0 |0⟩ + C1e
iϕ |0⟩,

where |C0|2 + |C1|2 = 1), each qubit has two continuous degrees of freedom to

control. In fact, the state of a qubit can be represented as a vector on a unit sphere

(the Bloch sphere, discussed in greater detail in Section 3.5), and the general set of

single-qubit gates can be regarded as “rotations” of this vector. An example of a

single-qubit gate is the Hadamard (or H) gate, illustrated in Figure 1.2a.

Multi-qubit (entangling) gates. Entanglement is a key computational

resource available to quantum systems. Multi-qubit gates (or entangling gates)

generate non-classical correlations between two or more qubits by carrying out

an operation that affects the state of one qubit differently depending on the state

of the other. A prototypical example of this type of gate, the controlled-NOT (or

CNOT) gate, is illustrated in Figure 1.2b.

Gate errors. Like in classical computing, errors can occur during algorithms,

leading to incorrect output states. In the classical case, the only single-bit error is

an unexpected bit flip. In the quantum case, this idea can be extended to a general

set of undesired rotations of the state vector, which can be coherent (i.e. consistent

34



}a b

Figure 1.2. Examples of commonplace one- and two-qubit gates written in circuit
notation (with each line representing an individual qubit over time) and the
operation of the gate on a specific input state. a) The Hadamard gate, which
rotates a single qubit. b) The controlled-NOT (CNOT) gate, which carries out a
bit flip on the target qubit conditional on the state of the control qubit, generating
entanglement.

between runs of the algorithm or over time scales allowing coherent cancellation)

or incoherent (i.e. random over the same time scales). Other classes of error exist

as well, with one notable class (discussed in greater detail in Chapter IV) being

leakage errors, i.e. the system entering a state not in the two-level qubit manifold.

These different error types can be handled in different ways with different levels of

ease. For a gate, the error rate can be quantified in terms of a gate fidelity, which

is the degree of overlap between the expected and true outputs of the gate, defined

as F = ⟨ΨT | ρ |ΨT ⟩, where |ΨT ⟩ is the target output state and ρ is the density

operator of the true output.

Error correction. Error correcting codes exist for the most common types

of errors in quantum computing. Typically, these codes require that each qubit

encoding data must be entangled with a set of auxilliary qubits whose states can

be measured to check whether an error occurred on the data qubit. Together,

these qubits are said to make up one “logical qubit” in an error-corrected quantum

algorithm. At a given rate of gate errors, in order for a quantum computer to be

fault-tolerant (i.e. able to push rates of logical errors to arbitrarily low levels), each

logical qubit must consist of some minimum number of physical qubits. In practice,

fault-tolerance is difficult to achieve because the number of physical qubits required

35



for each logical qubit grows quickly with error rate. This problem is discussed

further below.

1.3.2 Technical requirements. For quantum computing to be

practically useful, two distinct but related technical requirements must be

considered: scalability and gate fidelity.

Scalability. A useful quantum computer is one that can perform certain

calculations too difficult for classical computers (e.g. the factoring of large

prime numbers or the simulation of large molecules). By this benchmark, a

quantum computer must be large enough that it cannot be simulated classically.

A quantum computer of two logical qubits can be “simulated” with pencil and

paper. One made up of 10s can be simulated in a reasonable amount of time on

a supercomputer. Beyond this, a quantum computer can perform calculations

that are classically infeasible. Quantum computers with 10s to 100s of physical

qubits have already been implemented (Cheng et al. (2023)). However, as discussed

above, error correction requires multiple physical qubits per logical qubit, and

in fact, there may need to be orders of magnitude more of the former than the

latter, as discussed below. In practice this means that the design of a practical

quantum computer must be scalable, i.e. able to generate, store, control, and read

out arbitrarily large numbers of qubits with high accuracy. Scalability is the main

motivation for the work presented in Chapter V.

Gate fidelity. Related to scalability (through the physical overhead required

by error correction) is gate fidelity. Given an error rate of 10−4 per gate (and

making some assumptions about the distributions of these errors among the

different types), fault-tolerance can be acheived using a Steane code for error

correction, encoding each logical qubit in seven physical qubits (Nielsen and

Chuang (2010)). For higher error rates and different error correcting codes, this

36



ratio of physical qubits needed for one fault-tolerant logical qubit grows rapidly.

Depending on the type of error, this number can potentially be much larger

or smaller, but in general, 10−4 error rates are regarded as the “threshold” for

technically feasible fault-tolerant quantum computing, and much of the current

work being done focuses on reducing these rates, whether by improving coherence

times (the length of time for which the phase between qubit states remains well

defined), improving qubit lifetimes, developing coherent control techniques more

robust to decoherence and/or calibration errors, etc. Gate fidelity is a motivating

factor and important consideration for the work presented in Chapter IV.

Though these requirements may seem to set very stringent standards on the

size and performance of quantum computers, there has been much work on how

NISQ (noisy intermediate-scale quantum) trapped-ion devices could be optimally

designed for scales/performances expected in the near term (Murali, Debroy,

Brown, and Martonosi (2020)) and how NISQ devices in general could be used for

practical applications (Bharti et al. (2022)).

1.4 TIQC architecture

In trapped-ion quantum computing (TIQC), quantum information is

encoded in the Hilbert space available to an ion (or set of ions) in a trap. As

discussed above, a single ion has a large set of long-lived electronic energy levels

as well as three harmonic oscillator modes and their associated Fock spaces. In

principle, any pair of electronic states can be selected as the |0⟩ and |1⟩ states of

a qubit (which we will from here on out write as |↓⟩ and |↑⟩ to distinguish them

from Fock states in discussions where these are relevant). Typically however, qubits

are encoded in Zeeman/hyperfine sublevels of the S1/2 state (ground-state qubits,

or just “qubits” for the remainder of this section), which has an effectively infinite

lifetime.

37



To explain the standard TIQC architecture and some of its core techniques

(techniques for generating entanglement, handling error correction, and reducing

unwanted cross-talk between qubits with different functions), we can consider the

process of building up a chain of ions in a shared potential well and the operations

that need to be performed on them for carrying out a quantum algorithm. This

building-up process is represented schematically in Figure 1.3.

a

b

c

Figure 1.3. An illustration of the conceptual reasoning for using mixed-species ion
chains. a) A pair of ions sharing a potential well can be entangled by driving both
ions with laser fields. b) For carrying out auxilliary functions (e.g. sympathetic
cooling of the ions being used for gates), additional ions need to be added to the
chain. If these ions are of the same species as those used for gates, when driven
with laser fields, they will emit photons that the other ions can absorb, destroying
the information they contain. c) If the auxilliary functions are performed using
ions of a different species with different optical transition frequencies, then photons
scattered by one species will not be absorbed by the other.

Qubit coupling through shared motion. Two ions sharing a harmonic

potential well will be driven together by the well and pushed apart by their

electrostatic repulsion. The upshot of this is that the ions effectively make up a

two-atom solid. (In general, this solid can be called an ion crystal, or an ion chain

if the crystal is one-dimensional). This effective solid has a set of shared vibrational

38



modes. Because both ions couple to the same modes, these modes can serve to

mediate entanglement between between the ions if there is a way of driving the

modes that depends on the state of each qubit. In other words, if there is a way

of entangling the qubit state with the state of the motional mode, then there is a

way of entangling two qubit states sharing a motional mode. This entanglement is

usually carried out with lasers but can also be performed with strong magnetic field

gradients (e.g. Löschnauer et al. (2024)).

Ancilla ions. As discussed in Section 1.4 above, fault-tolerant quantum

computing requires extra error-correcting qubits. We can imagine adding additional

ions to the two-ion chain discussed above to serve this purpose. An issue arises

with the fact that, to carry out certain error correction schemes, the ancilla ions

need to be read out “mid-circuit” (i.e. before the data qubits are to be measured).

This readout process involves driving an optical transition and scattering photons.

While this optical drive can be targeted to a single ion (i.e. ions can be individually

addressed), some scattered photons will inevitably be absorbed by data qubits if

the ancilla and data ions couple to the same optical transition (e.g. if they are both

ground-state qubits in the same species of ion). This absorption and spontaneous

decay destroys any information in the qubit and induces an error. Similar issues

arise when trying to use ancilla ions for other tasks (e.g. sympathetically cooling

the chain, as discussed in later chapters).

Multi-species ion chains. To work around the problem of cross-coupling

between ancilla and data ions, common practice is to use different ion species

(e.g. different alkaline earth metals) for different purposes. These elements have

ground state transitions at very different optical wavelengths (e.g. the S1/2 →

P1/2 transition being 397 nm in Ca+ and 422 nm in Sr+), meaning that photons

39



scattered during ancilla readout or sympathetic cooling will not be absorbed by the

data ions.

1.5 The omg architecture

While dual-species ion chains resolve the issue of separating the functions

of data and ancilla qubits, they have their own set of drawbacks. Dual-species

chains require sources for two types of ion species, two separate laser systems for

addressing a different set of optical transitions, and the ability to detect different

wavelengths of emitted light. In addition to this increased technical overhead,

mixed-species chains see less coupling through common motional modes (due to

mass differences between the species), making both mixed-species entangling gates

and sympathetic cooling less efficient, and the mixed masses makes the chain more

vulnerable to distortion by stray electric fields. To work around these issues and

achieve separation of ancilla/data functions with a single ion species, the omg

qubit scheme was proposed (Allcock et al. (2021)), building off a body of previous

work done using optical (Monz et al. (2016); Roos et al. (2004)) and metastable

(Sherman et al. (2013)) qubits for multi- and single-qubit operations respectively

and proposed in parallel with efforts to implement a similar scheme in ytterbium

ions (Yang et al. (2022)). This scheme suggests using the fuller range of electronic

states available for encoding for elimination of cross-talk between qubit types.

1.5.1 Optical, metastable, and ground qubits. The omg scheme

draws its name from the three qubit types it identifies: optical (o), metastable

(m), and ground (g) qubits. These qubit encodings are shown in Figure 1.4. As

discussed above, g qubits, encoded in Zeeman/hyperfine splitting in the electronic

ground manifold, are the typical choice for encoding information in trapped ions. In

contrast, o qubits, encoded in the narrow-linewidth quadrupole transition between

ground and metastable states, are sometimes used for information processing, but

40



typically see more use as tools for state detection in g qubits (since this transition

can be used to selectively “shelve” one of the g qubit states where it will not

couple to the transition used for readout). Finally, m qubits, encoded in the

Zeeman/hyperfine splitting of the metastable manifold, were not widely explored

at the time that the work described in this document was done, but can be useful

in conjunction with these other qubit types as described below.

Ground state
(g) qubit

Metastable state
(m) qubit

Op�cal
(o) qubit

Figure 1.4. The three qubit types considered by the omg architecture.

1.5.2 Using omg qubits together. To illustrate how the omg

scheme might be used, an example will be given which specifically motivates the

need to investigate the prospect of doing gates in m qubits. One can consider two

ions (one in g, the other in m) sharing a potential well. The g qubit could be used

for some sort of dissipative operation (e.g. sympathetic cooling of the chain or

mid-circuit readout of some ancilla qubit previously encoded in the g ion) while

information is stored (or is potentially even being actively manipulated) in the m

qubit. In a setup like this, the separation of functions discussed in Section 1.4 is

achieved in a single ion species, where g qubits are used for dissipative operations

and m qubits are used for encoding and processing information.

41



1.6 Continuous-variable quantum computing

In typical quantum computing schemes, as discussed above, information

is encoded in two-level Hilbert spaces. Often, as in the electronic state of an ion,

this Hilbert space is a subspace of some larger but still functionally finite space

with varied splitting between levels. In continuous-variable quantum computing

(CVQC), information is instead encoded in an infinite Hilbert space of states evenly

spaced in energy. Much of the literature on CVQC focuses on its implementation

in photons, where photon number states (i.e. the number of excitations of some

bosonic mode) are the suggested encoding space (see Fukui and Takeda (2022) for

a recent review of the state of photonic CVQC). The harmonic oscillator modes

of a trapped ion can be used for much the same purpose, with phonon number

state used as a computational basis (Chen, Gan, Zhang, Matuskevich, and Kim

(2021); Ortiz-Gutiérrez et al. (2017)), and approaches employing hybrid discrete-

variable/continuous-variable approaches using spin and motional modes together

have also been proposed (Sutherland and Srinivas (2021)).

1.6.1 Building blocks for CVQC. In the way that a discrete-

variable quantum computing scheme requires both single-qubit gates and multi-

qubit entangling gates to be computationally complete, CVQC requires a set of

Gaussian operations and at least one non-Gaussian operation that can be carried

out on a bosonic mode (Braunstein and Van Loock (2005); Lloyd and Braunstein

(1999)), as well as operations for entangling modes. The theory behind these

operations are discussed in greater detail in Chapter III, but a brief overview of

these techniques is given below.

Gaussian and non-Gaussian operations. The terms “Gaussian” and “non-

Gaussian” refer to the shape of the Wigner function associated with a continuous-

variable mode’s state. The Wigner function (described more fully in Section 3.6)

42



is a quasiprobability distribution in the two quadratures of a mode. In a trapped

ion, these quadratures may be the position and momentum associated with a

motional mode, potentially in some rotating frame. A Gaussian state is one for

which the Wigner function is a two-dimensional Gaussian function, and operations

which preserve this feature are said to be Gaussian. Notably, the bosonic ground

state (the zero-photon/phonon state) and the familiar coherent state (the quantum

analog to classical mechanical oscillators / coherent light fields) are both Gaussian,

as is the operation for promoting one to the other. Operations which are first- or

second-order in the raising and lowering operators of a mode will be Gaussian

(since, in the case of mechanical oscillators, they preserve the parabolic shape of

the harmonic potential well, just changing its curvature or position). Higher-order

operations (corresponding in light to interactions of two fields in e.g. nonlinear

optical media) will distort the state’s Gaussian shape and are therefore called non-

Gaussian. Non-Gaussian operations can also be implemented by coupling a mode

to some non-bosonic system (e.g. through spin-motion entanglement in ions).

Entangling operations. Like two-level systems, continuous-variable modes

can be entangled with each other in ways useful for quantum information

processing. In photonic systems, two modes can be entangled through interference

(e.g. combining two modes on a beamsplitter), or entangled modes can be

generated through correlated generation of photons in different spatial modes

(as in e.g. spontaneous parametric downconversion). Analogues for both types of

operations exist for motional modes in trapped ions, as discussed briefly below and

in greater detail in Section 3.6.

1.6.2 Implementing CVQC in trapped ions. A trapped ion chain

has an associated set of harmonic oscillator modes. Because ions are charged, their

motion will couple to electric fields, and this presents a direct way of controlling

43



these modes. This means that the operations described above can be achieved in

ions in a deterministic, all-electronic fashion. Gaussian operations are routinely

performed on ion motion for mode/trap characterization and are sometimes

used for sensing applications (Burd et al. (2019)), and electronic operations that

generate entanglement between modes by exchanging energy between them have

also been demonstrated (Gorman, Schindler, Selvarajan, Daniilidis, and Häffner

(2014); Hou et al. (2024); Toyoda, Hiji, Noguchi, and Urabe (2015)). (The ways

that we implement these operations in our setup are described in Section 2.7). The

final piece required for all-electronic trapped-ion CVQC, the ability to perform

non-Gaussian motional operations efficiently, is relatively difficult to achieve,

as it requires strong electric potentials that are third-order or higher in ion

position, which are generally not desirable in a trap. Overcoming this difficulty

and implementing non-Gaussian operations motivates the work presented in

Chapter VI.

1.7 Scalable ion trap design

As discussed in Section 1.3, if a quantum computer carried out gates

perfectly, even a small one (e.g. 100 qubits) could be used to solve useful problems.

However, error rates are likely to remain high (e.g. > 10−4 for two-qubit gates)

for the near future, meaning that to carry out useful computations, much larger

qubit numbers will be needed for performing error correction. Because of this,

a practical hardware architecture is one that is scalable, e.g. one that can be

expanded from controlling hundreds of physical qubits to hundreds of thousands of

them with minimal increase in technical overhead and minimal reduction in quality

of control. One key consideration in scalable quantum computer design is all-to-all

connectivity, or the ability to perform entangling gates between arbitrary pairs of

qubits. In trapped ions, there are two widely-investigated systems for achieving this

44



all-to-all connectivity. In architectures relying on quantum networking (e.g. the

modular universal scalable ion trap quantum-computer or MUSIQ system, Monroe

et al. (2014)), ions in entirely separate traps are entangled via single photons.

Most proposed architectures however rely on the shuttling of ions through a trap

network using electric fields. The discussions in the rest of this document will be

focused on this second type of trap architecture, though many of the same design

considerations apply for both.

Ideally, a scalable trap architecture is one that can be microfabricated (e.g.

with CMOS or MEMS techniques) and integrated easily with components for ion

control and state detection (which themselves could be microfabricated, potentially

in ways such that integration is inherent to the fabrication process) and that

can be driven with off-trap resources (e.g. laser power, trapping power, etc.) for

which demand scales in a manageable way with the number of ions trapped. These

considerations are discussed more thoroughly in Chapter V, but a brief account is

given below of the trap architecture most widely in use and the trap designs most

commonly used in research seeking to implement scalable TIQC.

1.7.1 The QCCD architecture and surface electrode traps..

A quantum charge-coupled device (QCCD, Kielpinski, Monroe, and Wineland

(2002), first described conceptually in D. Wineland et al. (1998)) is a set of traps,

individually controllable segments (generally sharing a trapping RF voltage),

making up interconnected “channels” in which ions can be confined and through

which they can be transported by adjusting the DC voltage on each segment. Using

these DC voltages, pairs of ions can be brought together in short crystals, made to

interact, and pulled apart for further operations. A schematic view of this type

of trap is shown in Figure 1.5. The scalability of this architecture comes from

the fact that a block like the one shown in Figure 1.5 represents a sort of “unit

45



3D implementation

QCCD 
“unit cell”
concept

Figure 1.5. A cartoon illustration of a possible QCCD “unit cell,” which can
be tiled to produce arbitrarily large trap networks. Dark blue represents DC
electrodes, while pink represents trapping RF. Ions are shown in light blue. The
illustration on the left shows the unit cell of a 2D/surface electrode implementation
of a QCCD architecture, while the one on the right shows how the concept can be
extended to three-dimensional traps.

cell” which can be repeated endlessly. Each of these unit cells introduces some

fixed overhead in terms of power requirements and number of control lines, but

this overhead can be reduced by engineering e.g. new approaches for ion shutting

and swapping (Malinowski, Allcock, and Ballance (2023)). Implementing the

QCCD architecture, or something like it, motivates much of the work carried out

in Chapter V, where we also discuss the typical hardware approach for seeking to

implement this architecture, the surface electrode trap (or SET Chiaverini et al.

(2005)), which adapts the three-dimensional Paul trap to a set of electrodes in a

two-dimensional plane.

46



CHAPTER II

EXPERIMENTAL APPARATUS

2.1 Ion trap

To trap ions, we need to subject them to electric fields, and to generate

these electric fields, we use an arrangement of electrodes to which appropriate

voltages can be applied. This arrangement, shown in photographs before and after

mounting on a vacuum flange (Figure 2.1), is our trap. The details of this trap’s

design, geometric features, electrical drives, and vacuum environment are presented

in greater detail below.

2.1.1 Trap design. We run experiments using a macroscopic Paul

trap. Our trap electrodes are 0.5mm dia. rods, with the end cap electrodes

(American Probe 72BE-E3/1000x1.25”) having conical tips with ≈ 11◦ internal

angle and 200µm dia. spherical ends. These ten electrodes are arranged in an

X-configuration, with four RF trapping electrodes surrounding a pair end cap

electrodes, with a set of DC compensation electrodes ringing the outside, with

uniform spaces to maximize numerical aperture (NA) for both imaging and laser

access. This trap has an 0.75mm ion-electrode separation, and separation between

end cap needle tips is ≈3mm.

Simulation. The design process for this trap was informed by simulations

performed using the AC/DC module of COMSOL Multiphysics to generate

electrostatic potentials associated with voltages applied to individual sets of

rods. These electrostatic potentials/fields were further analyzed in the Electrode

simulation suite from NIST (NIST (2017)), allowing calculation of expected trap

parameters. From simulations, the trap design used should have a geometric

efficiency of 0.90 (defined as the ratio of the mode frequency of a trapped ion at a

47



10mm

7μm

Figure 2.1. Photographs of our trap, showing (left) the trap attached to a vacuum
flange and connected to electrical feedthroughs and (middle) a close-up shot of the
unmounted trap, with the rough location of the trapping zone marked with a blue
box. The associated inset in a picture taken with a CMOS camera of a five-ion
chain, with ion spacing marked. Finally, a COMSOL render (right) shows trapping
RF electrodes marked in red.

given set of parameters to the mode frequency at the same parameters given ideal,

i.e. hyperbolic, electrodes).

Fabrication. The electrode rods and needles are made of beryllium-copper

(98% Cu 2% Be with a hard temper) electroplated (to prevent oxidation of the

electrode surface) with a ∼100 nm thick layer of gold plated on top of a palladium

diffusion barrier (which is non-magnetic). To ensure that the electroplated gold was

sufficiently thick, we used the scheme shown in Figure 2.2, measuring the number

of gold atoms deposited onto the rod by measuring the total charge passed through

the electroplating circuit. To electrically insulate the trap, the rods are held in a

pair of end plates machined from Shapal Hi-M Soft, a machinable ceramic.

2.1.2 Trapping RF drive. To generate an effective potential (see

Section 3.1) to confine the ions along the trap axis we need an RF voltage that

can be applied to the two RF trap electrodes. This voltage is supplied by four-

stage RF chain shown schematically in Figure 2.3 which can be summarized as

follows: 1) an AWG supplies a sinusoidal voltage with high amplitude noise to 2) a

Squareatron, which generates a stable-amplitude signal by turning the AWG input

48



DC power
   supply

Multimeter

+

_

I

Ethernet

PC

Figure 2.2. Schematic illustration of the hardware setup used for estimating the
thickness of a layer of electroplated metal by measuring the total charge passed
through the system during electroplating.

into an effectively digital signal and stripping off all higher harmonics, with this

signal in turn being fed into 3) an amplifier (Minicircuits ZX60-100VH+) which

drives 4) our resonator cavity. These components are discussed in greater detail

below, but for a full accounting, see J. M. Metzner (2024).

AWG. A Rigol DG1022Z arbitrary waveform generator (AWG) generates

an RF sinusoid voltage. In the context of the rest of the chain, this signal acts as a

“frequency reference,” and can be controlled over ethernet to change the trap RF

frequency when needed (e.g. for automatically reducing radial confinement for ion

chain recrystallization). The output of this AWG has a large amount of amplitude

noise, so we do not use it to drive our resonator directly, since fluctuations in the

trap RF voltage translate to fluctuations in the mode frequencies of the ions.

Instead, we take a 0 dBm AWG output and feed it into a second stage of RF

conditioning.

Squareatron. The Squareatron (the design for which originates at the ion

trapping group at Oxford with modifications developed in our own lab) is a device

that takes in a frequency reference with large amounts of amplitude noise and

outputs a sinusoid with a stable amplitude. The basic operating principle is that

49



the input signal saturates an amplifier, generating an effectively digital signal, from

which the higher harmonics are stripped with filters. More complete accounts of

how this device works are offered in the theses of Tom Harty (Harty (2013)) and

Jeremy Metzner (J. M. Metzner (2024)). The Squareatron allows ≈1 dB of digital

tuning of its output amplitude, which is fed into an amplifier before being sent to

our resonator. The Squareatron and amplifier are both attached to a temperature-

stabilized aluminum plate in a box intended to reduce airflow.

Helical resonator. To get a factor of ≈80 step up in the RF voltage going to

the trap, we use a helical resonator with a resonant frequency of ≈14.5MHz with

Q-factor of up to 150. Coils are mechanically stabilized with low-loss polyethylene

foam supports to prevent pickup of acoustic-frequency vibrational noise. The step-

up coil includes a connection for applying other signals to the trap electrodes,

discussed in greater detail in Section 2.7. The output of the resonator is connected

to the trap electrodes through a single wire in a vacuum feedthrough to which the

RF trap electrodes are connected.

Under typical operating conditions, we see radial center-of-mass (COM)

mode frequencies of ∼1.8MHz, corresponding to RF voltages at the trap of

≈0.8 kV.

2.1.3 Trapping DC potential. To generate confinement along the

trap axis, an equal DC voltage is supplied to the two endcap needles using a DC

source (Stanford Research Systems DC205), which can output up to ±100V. The

output of this voltage source is connected to the trap through a pair of wires on a

vacuum feedthrough, with these connections being grounded at RF through 0.68 nF

capacitors, as shown in Figure 2.4. Under typical operating conditions (e.g. ≈60V

DC), we see ≈0.55MHz axial mode frequencies.

50



Trapping RF

Quadrupole
mo�onal drives

AWG Squareatron Amplifier

Resonator

DDS AWGFilter box

Trap

Figure 2.3. Block diagram of the RF chain used to supply the RF trapping
voltage. The trapping RF itself is supplied by the chain in red. Quadrupole
motional drives (discussed further in Section 2.7.3 are combined with the trapping
RF at the resonator.

2.1.4 Shim potentials. Unwanted electric fields in the trap (typically

called “stray fields” and coming from e.g. charging of nearby dielectric materials

by stray UV laser light) will tend to move the ion off of the trap center (the

“RF null” discussed further in Section 3.2.5). To counteract these fields, we use

our compensation electrodes to provide “shim potentials,” small DC potentials

that shift the trap equilibrium point. We drive these electrodes using digital-

to-analog converters (DACs) incorporated into our ARTIQ control system,

described in Section 2.8 below. These DACs can supply up to ±10V DC, and each

compensation electrode can generate an electric field of magnitude 19.9 (V/m)/V

at the ion. Connections are through four wires on a vacuum feedthrough, each

connection being grounded at RF through a 2.2 nF capacitor, as shown in

Figure 2.4.

2.1.5 UHV system. The trap is kept in an ultrahigh vacuum (UHV)

chamber held at 10−11 Torr. Low pressure is required in order to minimize rates of

collision between trapped ions and background gasses (mainly hydrogen) that can

cause decrystallization, dephasing, etc. The system is designed to be held under

UHV continuously and never brought up to atmosphere, and the hardware used to

51



DAC

DAC

DAC

D
AC

Bias
Tee

DC
RF

Tickle

71 dB att

RF Rabi 

3 dB att

RF

RF

2.2 nF

0.68 nF

2.2 nF 2.2 nF

2.2 nF

Needle
DC

0.68 nF

Trap

Figure 2.4. Circuit diagram for the electrical connections to the DC trap
electrodes (including AC drives applied to these electrodes), with endcap needles in
blue, DC trapping electrodes in white, and compensation electrodes in light yellow.

maintain vacuum is not connected to air. A render of this system, with components

labelled, is presented in Figure 2.5.

The chamber. The ion trap sits in an octagonal chamber with six viewports

on 2-3/4” flanges for laser entry, one viewport (used for imaging/state detection)

on a 6” flange with electrical feedthroughs for the trap, and one custom made

high-NA viewport for future ion-photon entanglement experiments. The top of

the chamber is connected to a custom-fabricated spherical manifold attaching to

hardware (described below) for maintaining/measuring UHV as well as a valve

for pumping the system down with mechanical pumps. The bottom of chamber

is covered with a 2-3/4” blank flange. The chamber is mounted in stainless steel

bracketing designed in-house. Chamber components are held together with silver-

52



a

b

Figure 2.5. a) CAD rendering of the vacuum chamber, associated support
hardware, and components directly attached to this hardware, including magnetic
field coils (Section 2.2.1 and b) mounts for beam delivery optics (Section 2.6).

coated screws to prevent parts from seizing during bakeout. Conflat (CF) gaskets

used at the interfaces of flanges are silver-coated for the same reason.

Maintaining and measuring vacuum. Vacuum is maintained using a passive

non-evaporable getter pump (a NEG, SAES CapaciTorr Z 100) and an active ion

pump (Agilent StarCell, 20). These parts hold the system at UHV without being

connected to any mechanical pumps. Low pressures can be measured using an ion

gauge (Agilent UHV-24P), which measures the current generated by ionized gasses.

53



Preparing chamber hardware. Before assembly and pumpdown of the

vacuum system, most of the stainless steel components were baked at 450◦C under

atmosphere for several days. This baking process causes the formation of a bronze-

colored oxide layer that reduces the amount of hydrogen outgassing in the interior

of the system (Sefa, Fedchak, and Scherschligt (2017)). Hardware was washed in

isopropanol and acetone before and after pre-assembly baking.

2.2 Magnetic field control

Qubits are encoded in the magnetic moments of our trapped ions. In

40Ca+, which has no nuclear spin, the quantization axis and energies associated

with these moments are set by an external magnetic field, which induces Zeeman

splitting and breaks the degeneracy of spin states in each energy level. In our

system, this magnetic field is at a 45◦ angle to the trap axis and is a combination

of the magnetic field generated by a set of wire coils and the local magnetic

field of the earth. All in all, the magnitude of our magnetic field is ≈1.56G,

producing a Zeeman splitting (between spin states, e.g. m=+3/2 and m=+5/2)

of 2.63MHz in the D5/2 metastable level of 40Ca+. Noise in this magnetic field is

the limiting factor in the coherence of spins encoded in these Zeeman sublevels

(since their splitting is first-order sensitive to magnetic field strength). Techniques

for characterizing the strength and (to some extent) spectrum of this noise will be

discussed in Section 3.9, but an important noise type to discuss here is line noise

(noise coming from the 60Hz mains), which we cancel using a feed-forward system.

This system is described in greater detail below in Section 2.2.2.

2.2.1 Magnetic field coils. To control the magnitude and direction

of the magnetic field at the ion, we use a set of five wire coils, wound with 20 gauge

copper wire. Opposite pairs (spaced about 8.6” apart) of 3.6”x3.0” rectangular coils

with ∼300 windings per coil provide magnetic field control in the xz-plane (in the

54



trap coordinate system defined above), while a single circular 6.1” dia. coil (about

1.4” from the ion) is used for magnetic field cancellation in the y-direction. These

coils are driven with a set of three DC currents (with opposing coil pairs wired in

series) supplied by a regulated DC power supply (QL355TP), which can provide up

to 500mA in its low-noise mode. This maximum current is used to drive the coils

along the quantization axis. An image of the coil setup is shown in Figure 2.6.

a b
Figure 2.6. a) A schematic of the magnetic field coil arrangement used to set the
quantization axis in the trap. Current is marked with white arrows, with arrow
thickness corresponding the current magnitude. b) A photograph showing the
five magnetic field coils used on the trap. Worth noting are the two smaller coils
(white and blue) wrapped around the top left coil, which are used for line noise
feedforward (Section 2.2.2).

2.2.2 Line noise mitigation. Line noise refers to the magnetic

fields produced by the 60Hz (in America) alternating current used to supply

electric power to the lab. The noise itself consists of a stable, narrow-band

60Hz component and its harmonics (with the leading harmonic ordinarily being

the first odd harmonic, 180Hz, produced by e.g. clipping of a 60Hz sinusoid).

Essentially, line noise causes the frequency of our Zeeman qubits to oscillate on

a 60Hz cycle. If we are running an experiment that is not itself in-phase with

55



this line noise (i.e. shots of the experiment are being performed at randomly

sampled points over a cycle), then the line noise will look like a broadening of

our qubit frequency / a reduction in spin coherence time. There are two ways

of addressing this issue, discussed below: 1) running experiments in sync with

the 60Hz line, and 2) cancelling out 60Hz noise using feed-forward. These

techniques are represented schematically in Figure 2.7a and b respectively. (As an

important note, both techniques involve the use of a Line Trigger (Sinara (2020)),

a component developed as part of the Sinara open source hardware project. This

device generates a 5V digital output in-phase with an input from a 60Hz power

supply connected to mains).

LT

Experiment
control sys

60 Hz
AWG

180 Hz
AWG

TTL
trigger

Ion

Control
pulses out

60 Hz in

Digital out

LT
60 Hz in

Digital out

Experiment
control sys

Ion

Control
pulses out

B-�eld
at ion

TTL
trigger

TTL
trigger

180 Hz coil
60 Hz coil

B-�eld
at ion

V180, φ180 V60, φ60

Triggering experiment B-field feed-forward

Figure 2.7. Schematic illustrations of our two line noise mitigation schemes,
explained in greater detail in text.

Line triggering. A simple solution to the problem of line noise is to run

experiments in sync with the 60Hz line. This can be done by taking a Line Trigger

output, reading it in with a TTL channel on the ARTIQ crate (described in more

56



detail below), and using that input to time experiments to run at consistent points

on the line cycle, ideally selecting a time where the line noise is at a crest/trough,

so that the qubit frequency will be first-order insensitive to experiment run time,

for experiments ≪16.7ms. This technique introduces little technical overhead,

but it can make experiments substantially slower, since it forces the time between

experimental shots to be 16.7ms (or multiples of this). In addition, for experiments

lasting an appreciable portion of the line cycle, the qubit frequency will still be

shifted over the duration of the experiment.

Line noise feed-forward. A technique that requires more technical overhead,

but which we generally use in our experiments, is to cancel out line noise with a

feed-forward loop. In this setup, we drive small coils (∼5 windings each) with

60Hz and 180Hz voltages generated by a pair of AWG channels triggered using

the Line Trigger. (As shown in Figure 2.6b, we only wind feed-forward coils

around one of the quantization axis coils. In our case, our line noise magnetic

field is small relative to our quantization field, so our qubit frequency will only be

first-order sensitive to line noise along the quantization axis). We calibrate the

amplitudes and phases of the 60 and 180Hz voltages by selecting values which

maximize the contrast of our spin Ramsey fringes at long delay times (the details

of this technique are explained further in Section 3.9.1). While this technique

requires substantially more calibration than line triggering the experiment, it allows

experiments to be run much more quickly and ensures there will no systematic

effects from coherent changes in qubit frequency over the course of an experiment.

2.3 Imaging and state detection

If we have the ability to generate and trap ions, before we can use them

for experiments, we need to be able to 1) confirm when ions are in the trap and

how many of them we have and 2) detect the states of these ions during/after

57



experiments. For both of these purposes, we use fluorescence on the 397 nm

S1/2 ↔ P1/2 “cycling transition” in 40Ca+. Illuminating an ion with 397 nm laser

light, it can scatter photons at a rate proportional to 1/τP1/2, where τP1/2 is the

lifetime of the P1/2 excited state. This means that an ion can scatter ∼ 108 photons

per second. These photons can be detected and used to infer that ions are in the

trap and that they are in the electronic ground state (since if they were e.g. shelved

in a metastable state, the cycling transition could not be driven). This being the

case, we built a high-NA (numerical aperture) microscope for detecting 397 nm

photons, both allowing ions in a chain to be imaged on a camera in a spatially

resolved way and for photons to be counted using a photomultiplier tube (PMT).

The hardware details and performance specifications of this imaging system are

discussed below.

2.3.1 Imaging system overview. This discussion of the imaging

system will start with a hardware overview and an accounting of the design process.

A schematic of the imaging system, along with a photograph of the system itself

in our experimental setup, is presented in Figure 2.8. The design of the objective

was based on one used at NIST, with parameters optimized for 397 nm light

using Zemax OpticStudio. OpticStudio was used for optimizing the placement

and selection of optical elements and for predicting performance factors like

magnification and spot size.

Objective. The objective consists of, in order of nearest- to farthest-from-ion

components, a 2” dia. aspheric convex lens (Asphericon AFL50-60-S-U), a 2” dia.

aspheric convex lens (Asphericon AFL50-60-S-U), and a 1” dia. planoconvex lens

(Edmund 48-321). At the end of the objective, an iris (not shown in schematic)

allows the NA to be restricted. By itself, this objective should provide -3.8x

magnification and a 2-3 micron spot size.

58



Figure 2.8. Schematic illustration (and photograph below) of our imaging system,
with light emitted by the ion (a) being captured by an objective (b), passed
through a slit (c) to reduce stray light and a narrow-band filter (d) to prevent the
system from registering light far away from the wavelength of the cycling transition
of Ca+, collimated with a lens (f) in an adjustable barrel mount to be redirected
by a flip-mounted mirror (g), which can direct light to a focusing lens leading to
the PMT (h) and (i) or to a focusing lens (j) and (k) heading to the camera (l).

Stray/unwanted light reduction. After the objective, a slit (Owis SP-40) is

used to reduce stray light (i.e. light entering the system from sources other than

the ions at the objective focus). After this, a narrow-band optical filter (Semrock

FF01-390/18-25) is used to screen out light at wavelengths other than 397 nm.

Collimation and reimaging. Focused light is collimated using a convex lens

(Thorlabs AC127-030-A) and sent along a path with a mirror (Edmund Optics

85-311) mounted in a motorized flippable mount (Owis KSHM 40) which can

59



direct light down two paths: To a PMT (Hamamatsu H10682-210) with just a

1” dia. lens (Thorlabs LA1540-A) used to focus the collimated light onto the

PMT’s active area, or to a CMOS camera (Kiralux CS505MU) with a combination

con