ABSTRACT

Title of Dissertation: ANALOG–DIGITAL QUANTUM
SIMULATIONS WITH TRAPPED IONS

Katherine S. Collins
Doctor of Philosophy, 2023

Dissertation Directed by: Professor Christopher Monroe
Department of Physics

Since its inception in the early 1920s, the theory of quantum mechanics has provided a

framework to describe the physics of nature; or at least our interpretations about systems in

nature. However, even though quantum theory works, the unsettling question of “why?’ still

remains. The field of quantum information science and technology (QIST) has brought together a

collection of disciplines forming a united multidisciplinary collaborative effort towards realizing

a large-scale quantum processor as an attempt to understand quantum mechanics better. It has

been established in the field that the most efficient architectural design of this quantum processor

would be composed of numerous individual quantum computers, quantum simulators, quantum

networks, quantum memories, and quantum sensors that are “wired” together creating just the

hardware layer in the full stack of the machine. Realizing a module-based quantum processor on

such a macroscopic scale is an ongoing and challenging endeavor in itself.

However, existing noisy intermediate-scale quantum (NISQ) devices across all the quantum

applications above are still worth building, running, and studying. NISQ quantum computers can



still provide quantum advantages over classical computation for given algorithms, and quantum

simulators can still probe complex many-body dynamics that remain improbable to consider even

on the best supercomputer. One such system is the trapped-ion quantum simulator at the center

of this dissertation. Using 171Yb+ ions, we expand our “analog” quantum simulation toolbox by

incorporating “digital” quantum computing techniques in each of the three experiments presented

in this work.

In the first experiment, we perform a quantum approximate optimization algorithm (QAOA)

to estimate the ground-state energy of a transverse-field antiferromagnetic Ising Hamiltonian with

long-range interactions. For the second project, we develop and demonstrate dynamically decou-

pled (DD) quantum simulation sequences in which the coherence in observed dynamics evolving

under the unitary operator of the target Hamiltonian is extended while the known noise is sup-

pressed. Finally, in the third project, we implement an experimental protocol to measure the

spectral form factor (SFF) and its generalization, the partial spectral form factor (PSFF), in both

an ergodic many-body quantum system and in a many-body localized (MBL) model. As a result,

a quantum simulator can be utilized to test universal random matrix theory (RMT) predictions,

and simultaneously, probe subsystem eigenstate thermalization hypothesis (ETH) predictions of

a quantum many-body system of interest.



ANALOG–DIGITAL QUANTUM SIMULATIONS
WITH TRAPPED IONS

by

Katherine S. Collins

Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment

of the requirements for the degree of
Doctor of Philosophy

2023

Advisory Committee:
Dr. Christopher Monroe, Chair/Advisor
Dr. Zohreh Davoudi
Dr. Alexey Gorshkov
Dr. Christopher Jarzynski
Dr. Qudsia Quraishi



© Copyright by
Katherine S. Collins

2023



Dedication

For my mother, Lisa

ii



Acknowledgments

I am truly grateful to Chris Monroe for introducing me to the quantum systems of trapped

ions as an undergraduate freshman. I appreciate your continued support in allowing me to pursue

research in your group after deciding to stay at UMD for graduate school. This experience has

given me the opportunity to develop the skills needed as a researcher and to further my knowledge

of trapped ion systems in the quantum information field.

In regards to my work in the lab, QSIM, I would like to say a special thanks to our post-

doc Will Morong and fellow graduate student Arinjoy De. I appreciate your all-hands-on-deck

approach in keeping the experiment running and collecting the necessary data for the spectral

form factor project.

To Allison Carter, Wen Lin Tan, and Crystal Noel I enjoyed your friendship and mentoring

in navigating the complexities of pursuing a Ph.D. as a woman in the field of physics.

I also want to thank all of the members of the Trapped Ion Quantum Information group,

both current and past, for their help, and for taking the time to answer the countless physics

questions I pestered you with. It was a fantastic experience working with you all.

To my good and longtime friend, Emily Ackerman, thank you for your support and for

being there at a moment’s notice. Thank you to David, Sherry, and my extended family, for being

in my life and providing friendship over the years.

Most of all, thank you to my Mom. I am grateful for all of the opportunities, the endless

iii



support, and the love you have given me. I am lucky to have been adopted by you and that you

became my Mom. In the ultimate roll of the dice, you refused the Welfare Institute’s effort to

swap me out for a “healthy girl” instead choosing to stick with the “sick one.”

iv



Table of Contents

Dedication ii

Acknowledgements iii

Table of Contents v

List of Tables viii

List of Figures ix

List of Abbreviations xi

Chapter 1: Introduction 1
1.1 Trapped Ion Quantum Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Chapter 2: Ytterbium Ions 9
2.1 Continuous Wave Lasers Required for 171Yb+ . . . . . . . . . . . . . . . . . . . 14

2.1.1 Photoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Repumping Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.4 Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.5 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 The CW Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 399 nm Optical Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2 935 nm Optical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.3 369 nm Beam Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.4 739 nm Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Trapping Our Friends the Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 Difficulty in Loading Ions Troubleshooting . . . . . . . . . . . . . . . . . . . . 45

Chapter 3: Components of the Experimental System 46
3.1 Ion Trap and Vacuum Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Coherent Manipulation via Stimulated Raman Transitions . . . . . . . . . . . . . 55

3.2.1 Carrier Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.2 Sideband Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.3 Resolved Sideband Cooling . . . . . . . . . . . . . . . . . . . . . . . . 60

v



3.3 Mølmer-Sørenson Interaction to the Long-Range Ising Model . . . . . . . . . . . 61
3.3.1 Tuning Jij to Change α . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4 355 nm Pulsed-Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.1 Beam-Pointing Stabilization . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.2 Individual Addressing Beam . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5 Experimental Control Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5.1 Calibrating the Individual Addressing Beam . . . . . . . . . . . . . . . . 79

Chapter 4: A Hybrid Quantum-Classical Algorithm 86
4.1 Quantum Approximate Optimization Algorithm . . . . . . . . . . . . . . . . . . 87
4.2 The QAOA Protocol and Experimental Implementation . . . . . . . . . . . . . . 88
4.3 p=1 QAOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3.1 p=1 QAOA Exhaustive Search . . . . . . . . . . . . . . . . . . . . . . 94
4.3.2 p=1 QAOA System Size Performance . . . . . . . . . . . . . . . . . . 95

4.4 p=2 QAOA Exhaustive Search . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.5 Closed-Loop QAOA Gradient Descent Search . . . . . . . . . . . . . . . . . . . 98
4.6 QAOA Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.7 Summary of the QAOA Experiment . . . . . . . . . . . . . . . . . . . . . . . . 104

Chapter 5: Dynamically Decoupling Quantum Simulations 105
5.1 Dynamical Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2 Principles of Dynamical Decoupling . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Implementing DD Sequences in QSim . . . . . . . . . . . . . . . . . . . . . . . 110
5.4 Application to Trapped Ions and Example Sequences . . . . . . . . . . . . . . . 112

5.4.1 Example 1: A CMPG Sequence . . . . . . . . . . . . . . . . . . . . . . 114
5.4.2 Example 2: A XY Sequence . . . . . . . . . . . . . . . . . . . . . . . . 115
5.4.3 Example 3: A Decoupled Heisenberg Model . . . . . . . . . . . . . . . 116

5.5 Two-Ion Tests of Pulse-Sequence Parameters . . . . . . . . . . . . . . . . . . . 117
5.5.1 Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.5.2 Drive Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5.3 Detuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.5.4 Noise Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.6 Multi-Ion Dynamically Decoupled Realizations . . . . . . . . . . . . . . . . . . 125
5.6.1 11 Ion Dephasing Suppression . . . . . . . . . . . . . . . . . . . . . . . 126
5.6.2 Realizing an Approximate Haldane-Shastry Model . . . . . . . . . . . . 128

5.7 Summary: Adding Dynamical Decoupling to the Simulation Toolbox . . . . . . 135

Chapter 6: Spectral Form Factor 137
6.1 What is the Spectral Form Factor? . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2 Characteristic Properties of the SFF and PSFF . . . . . . . . . . . . . . . . . . . 139
6.3 Proposed Protocol and Requirements for Measuring SFF and PSFF . . . . . . . . 141

6.3.1 Estimators of the SFF and PSFF Observables . . . . . . . . . . . . . . . 143
6.3.2 General Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.4 Two Methods to Rescale the Raw Data from Decoherence Errors . . . . . . . . . 145
6.4.1 Imperfections from U and U † . . . . . . . . . . . . . . . . . . . . . . . 146

vi



6.4.2 Imperfections During T (t) . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.5 Implementing the SFF Sequence in QSim . . . . . . . . . . . . . . . . . . . . . 149

6.5.1 Implementing and Testing the Individual RUs . . . . . . . . . . . . . . . 153
6.5.2 Implementing Local Random Disorders . . . . . . . . . . . . . . . . . . 165
6.5.3 SFF Purity Sequence and Calculating the Purity . . . . . . . . . . . . . . 167

6.6 N=5 SFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.7 Variations of the Purity Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.8 N=4 SFF Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.8.1 Disorder-Averaged Ergodic SFF . . . . . . . . . . . . . . . . . . . . . . 179
6.8.2 Disorder-Averaged Localized SFF . . . . . . . . . . . . . . . . . . . . . 181
6.8.3 Single-Disorder Localized SFF . . . . . . . . . . . . . . . . . . . . . . . 182

6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Bibliography 187

vii



List of Tables

2.1 Isotopes of neutral Ytterbium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Some angular momentum quantum numbers . . . . . . . . . . . . . . . . . . . . 10
2.3 Iodine VCO control signal parameters . . . . . . . . . . . . . . . . . . . . . . . 43

3.1 Static voltage parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1 N=20 QAOA exhaustive search performances . . . . . . . . . . . . . . . . . . 98

5.1 Signs of uniform magnetic field terms for Hamiltonians in XY sequence . . . . . 116

6.1 LabView SFF sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2 Measure X, Y, Z experimental sequences . . . . . . . . . . . . . . . . . . . . . . 161
6.3 LabView state overlap sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.4 LabView purity sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

viii



List of Figures

2.1 171Yb+ energy level diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 369 nm Doppler cooling transitions . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 369 nm optical pumping transitions . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 369 nm detection transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Optical paths of 171Yb+ CW lasers . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 399 nm optical path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 935 nm optical path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.8 369 nm optical path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.9 739 nm beam path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.10 PDH current and grating locks . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.11 Iodine SAS optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.12 Feature 5 of the iodine error signal . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.13 Feature 4 of the iodine error signal . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.14 Iodine AOM RF source signal comparison . . . . . . . . . . . . . . . . . . . . . 42
2.15 CW optics right before the chamber . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1 QSim vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 The QSim ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Iseg LabView control program . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Λ system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5 355 nm optics outside of Raman enclosure . . . . . . . . . . . . . . . . . . . . . 68
3.6 Beam-pointing optical system . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.7 Raman beam paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.8 Individual addressing beam path, part 1 . . . . . . . . . . . . . . . . . . . . . . 75
3.9 Individual addressing beam path, part 2 . . . . . . . . . . . . . . . . . . . . . . 76
3.10 Pulse sequence experiment control window . . . . . . . . . . . . . . . . . . . . 78
3.11 Individual addressing beam calibrations with one ion . . . . . . . . . . . . . . . 80
3.12 Single ion Stark shift versus AWG amplitude . . . . . . . . . . . . . . . . . . . 81
3.13 Camera calibration for N=4 ions using the experimental control program . . . . 82
3.14 AOD frequency scan with N=4 ions . . . . . . . . . . . . . . . . . . . . . . . . 83
3.15 Calibrating the individual Stark shift applied to each ion . . . . . . . . . . . . . 85

4.1 QAOA experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2 QAOA p=1 example energy landscape . . . . . . . . . . . . . . . . . . . . . . 93
4.3 N=20 QAOA p=1 exhaustive search performance landscape . . . . . . . . . . 94
4.4 QAOA p=1 performance as a function of system size . . . . . . . . . . . . . . . 95
4.5 N=20 QAOA p=2 exhaustive search . . . . . . . . . . . . . . . . . . . . . . . 97

ix



4.6 N=12 QAOA p=1 gradient search . . . . . . . . . . . . . . . . . . . . . . . . 99
4.7 N=20 QAOA p=1 gradient search . . . . . . . . . . . . . . . . . . . . . . . . 100
4.8 Sampling from p=1 QAOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.1 Generalized decoupling sequence for quantum simulation . . . . . . . . . . . . . 108
5.2 An example of dynamical decoupling for a two-ion interaction . . . . . . . . . . 119
5.3 Dependence of dynamical decoupling pulse sequence parameters . . . . . . . . . 122
5.4 Relative contribution of the error terms in noise model . . . . . . . . . . . . . . 125
5.5 Performance of many-body Hamiltonian evolution without and with dynamical

decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.6 Generalized imbalance comparison . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.7 Quantum simulation of the approximate Haldane-Shastry model . . . . . . . . . 131
5.8 Haldane-Shastry numerics with decoherence . . . . . . . . . . . . . . . . . . . . 132
5.9 Additional Haldane-Shastry data . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.10 Modified Haldane-Shastry sequence results . . . . . . . . . . . . . . . . . . . . 134

6.1 Uniform sampling of θ and ϕ but nonuniform distribution of points on a sphere . 156
6.2 Defined random angle functions . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3 Uniform sampling of points on the Bloch sphere . . . . . . . . . . . . . . . . . . 157
6.4 Converting random Stark shifts to AWG amplitudes . . . . . . . . . . . . . . . . 159
6.5 Converting Stark shifts to corresponding AWG amplitudes . . . . . . . . . . . . 160
6.6 Individual ion average measure x, y, z results . . . . . . . . . . . . . . . . . . . 163
6.7 RU ensemble average connected correlations . . . . . . . . . . . . . . . . . . . 163
6.8 Individual ion average state overlap scan . . . . . . . . . . . . . . . . . . . . . . 166
6.9 Converting random disorder strengths to AWG amplitudes . . . . . . . . . . . . 166
6.10 N=5 ergodic SFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.11 N=5 ergodic average subsystem size PSFF . . . . . . . . . . . . . . . . . . . . 171
6.12 N=5 individual ion average fidelity . . . . . . . . . . . . . . . . . . . . . . . . 172
6.13 N=5 ergodic SFF purity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.14 N=5 ergodic SFF purity sequence z-magnetizations . . . . . . . . . . . . . . . 174
6.15 Average z-magnetizations of dark product state . . . . . . . . . . . . . . . . . . 175
6.16 Average z-magnetizations of bright product state . . . . . . . . . . . . . . . . . 175
6.17 Average z-magnetizations of arbitrary product state . . . . . . . . . . . . . . . . 176
6.18 Average z-magnetizations of SFF purity sequence with only random disorders . . 177
6.19 Average z-magnetizations of SFF purity sequence with only global interactions . 177
6.20 Average z-magnetizations of SFF purity sequence with only global interactions

take two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.21 N=4 dynamically decoupled results . . . . . . . . . . . . . . . . . . . . . . . . 179
6.22 N=4 ergodic disorder-averaged average subsystem size PSFF . . . . . . . . . . 180
6.23 N=4 ergodic SFF purity and individual ion average fidelity. . . . . . . . . . . . 181
6.24 N=4 localized disorder-averaged average subsystem size PSFF . . . . . . . . . 182
6.25 N=4 localized disorder-averaged SFF purity . . . . . . . . . . . . . . . . . . . 183
6.26 N=4 localized single-disorder unitary-averaged average subsystem size PSFF . 184
6.27 N=4 localized single-disorder SFF purity . . . . . . . . . . . . . . . . . . . . . 185

x



List of Abbreviations

AC Alternating Current
AHT Average Hamiltonian Theory
AMO Atomic Molecular and Optical
AOD Acousto-Optic Deflector
AOM Acousto-Optic Modulator
AWG Arbitrary Waveform Generator

BSB Blue Sideband
BS Beam Splitter
BW Bandwidth

CA Central Average
CD Central Difference
CPMG Carr-Purcell-Meiboom-Gill
COM Center-of-Mass
CW Continuous Wave

DAC Digital-to-Analog Converter
DC Direct Current
DD Dynamical Decoupling

EA End Average
ECDL External Cavity Diode Laser
EMCCD Electron Multiplying Charge-Coupled Device
ENVD End Near Vertical Difference
EOM Electro-Optic Modulator
ETH Eigenstate Thermalization Hypothesis

FM Frequency Modulation
FP Fabry-Perot
FPGA Field-Programmable Gate Array
FWHM Full Width Half Maximum

GR(s) Global Rotation(s)

HV High Voltage

xi



HWP Half-Wave Plate

MBL Many-Body Localization
MS Mølmer-Sørensen

NA Numerical Aperture
NISQ Noisy Intermediate-Scale Quantum
NMR Nuclear Magnetic Resonance

PBS Polarizing Beam Splitter
PDH Pound Drever Hall
PID Proportional Integral Differential
PM Phase Modulation
PMT Photo-Multiplier Tube
PSFF Partial Spectral Form Factor

QAOA Quantum Approximate Optimization Algorithm
QIST Quantum Information Science and Technology
QWP Quarter-Wave Plate

RD(s) Random Disorder
RF Radio Frequency
RMT Random Matrix Theory
RSB Red Sideband
RU(s) Random Unitary
RWA Rotating-Wave Approximation

SAS Saturated Absorption Spectroscopy
SB(s) Sideband(s)
SBC Sideband Cooling
SD Standard Deviation
SEM Standard Error of the Mean
SFF Spectral Form Factor
SHG Second Harmonic Generation
SIR(s) Single Ion Rotation(s)
SM Supplementary Material
SS(s) Stark Shift(s)

TA Tapered Amplifier
TTL(s) Transistor-Transistor Logic
TVD Total Variation Distance

xii



UHV Ultra High Vacuum
UV Ultra-violet

VQE Variational Quantum Eigensolver
VCA Voltage-Controlled Amplifier
VCO Voltage-Controlled Oscillator

XHV Extreme High Vacuum

Yb Ytterbium

xiii



Chapter 1: Introduction

The field of quantum information science brings together the disciplines of physics, en-

gineering, computer science, mathematics, and information science all in the pursuit of under-

standing quantum mechanics.Since multiple areas of study are involved in this quest, there are,

of course, many different questions that exist and numerous research avenues to explore. The

common “object” that is shared amongst all fields involved is a large-scale quantum computer.

The physics that describes a quantum computer is the theory of quantum mechanics which

describes the physics of particles (molecules, atoms, electrons, protons, neutrons, etc.) and pho-

tons (light) at a microscopic scale as opposed to the theory of classical mechanics that governs

the physics of macroscopic objects. As my research advisor (Chris Monroe) says, “There are two

rules in quantum physics. (1) Everything is a wave and can exist in superposition, (2) until you

look and then, the system randomly assumes a definite state.” The first rule refers to the fact that

a particle (assumed to be in a quantum system) at some time t is described by its wavefunction

|ψ(t)⟩ that corresponds to single quantum state or occupying several quantum states simultane-

ously, also known as a quantum superposition. However, when the particle’s state is measured,

the quantum superposition is destroyed, and the particle will be measured in only one of the pos-

sible states. Rule 1 highlights the “quantumness” of the particle existing in more than one state at

time t which under classical physics is physically impossible. Rule 2 summarizes two concepts

1



in quantum physics. The first is that the act of measuring the quantum system (the particle) col-

lapses the wavefunction such that the particle is detected in only a single state which reassuringly

agrees with classical expectations. Second, measurements in quantum systems are probabilistic

in nature. There exists uncertainty in the measurement, an associated probability in measuring

the particle in a given state. Because of the uncertainty in the measurement, the inability to place

the result in context, multiple measurements must be made (assuming the quantum particle can be

prepared in the same state |ψ(t)⟩, which we will assume that it can) of the particle’s wavefunction

|ψ(t)⟩ at time t, forming a probability distribution of the measured outcomes.

Keeping these two rules of quantum physics in mind, let us return to the discussion of a

quantum computer. A quantum computer is the quantum version of the “classical” computer.

Computers encode information in computer bits: 0s and 1s. In a similar fashion, a quantum

processor encodes its quantum information in quantum bits, or qubits. Unlike a classical bit,

a qubit can be in a superposition of |0⟩ and |1⟩, in addition to occupying only |0⟩ or only |1⟩.

A qubit can be any two-state quantum system in which information between the qubit states

can be accessed. The strength of a quantum processor is demonstrated when the size of the

quantum system consists of N qubits. For N qubits, there are 2N quantum states in which

quantum information can be stored due to the property of quantum superposition and something

called quantum entanglement can be generated which is still a process to be understood. In

comparison, a classical computer would need 2N bits to store just the information whereas the

storage capabilities already exist in a system of N qubits. However, having an isolated system

of N qubits is different from performing a set of coherent operations on the N qubits for a long

enough time while minimizing any unwanted disturbances or decoherence to the system.

The quantum computer and quantum simulator are very similar in foundation; however, a

2



quantum computer has universal control, access to any quantum state, while a quantum simulator

is restricted in the types of operations that can be performed. Both represent quantum systems

with realizable qubits that can be well-controlled, coherently manipulated, and measured. On a

quantum computer, a series of quantum gates similar to classical computer gates are executed in

order to realize a specific quantum algorithm or procedure. The quantum circuit picture often

comes to mind in which each qubit is placed on its own row connected to a wire. The wires are

connected to different gates or operations allowing the quantum information to flow through the

circuit before a measurement is made. This description is similar to boolean operations that might

be executed on a computer processor. However, in quantum simulators, interactions described by

a Hamiltonian are applied to the qubits in which the dynamics are allowed to evolve for varying

amounts of time before measuring the system.

The quest of developing a large-scale quantum computer is still an ongoing process. Many

technical challenges exist when designing a quantum processor large enough that can run the

already identified algorithms in which a quantum system outperforms the current classical su-

percomputer. An early question proposed in the quantum information science community was

whether it would be better to create one large quantum device or connect together many smaller

devices. I believe it has been answered. The modular approach seems to be the way to go for

many reasons that are not discussed here. Another question still remains regarding whether the

quantum architecture should be based only on a specific quantum platform that has already been

demonstrated at a smaller scale or on a combination of different quantum systems. The jury is

still out on that verdict. Regardless, there is still much of work to be done to realize fully a useful

large-scale quantum processor. Not only is there the needed improvement of the performances of

existing quantum platforms in what has been coined noisy intermediate-scale quantum (NISQ)

3



devices [1], there is also the question of the connection to the different modular systems. Many

experiments exist that examine quantum communication and networking in which the qubit is

entangled with an information bus or carrier of the information such as a photon, a phonon or

some other degree of freedom. Experiments have already demonstrated the quantum teleporta-

tion of information rather than matter and have shown that quantum information can be sent and

received between different modules that are not physically connected; assuming that there was

prior interaction and classical communication between the modules.

Despite the ongoing quest for the creation a large-scale quantum processor and determin-

ing useful applications for the device to run, there still remain the current NISQ-era quantum

computers and simulators. Several experimental platforms exist in which qubits can be realized:

trapped ions, ultra-cold neutral atoms, Rydberg atoms, superconducting transmons, solid-state

nitrogen-vacancy centers in diamond, photon-coupled cavity systems, as well as possibly-to-

be-determined topological qubits or further experimental development of vibrational modes of

molecules. There are pros and cons (or challenges) for each of these systems. However, the

quantum platform of interest in this dissertation focuses on trapped-ion qubits.

1.1 Trapped Ion Quantum Simulators

Experiments using trapped ions for quantum information processing have been in existence

for over three decades and counting. There are many ion species that can be chosen: Beryl-

lium (Be+), Magnesium (Mg+), Calcium (Ca+), Strontium (Sr+), Barium (Ba+), Zinc (Zn+),

Cadmium (Cd+), Mercury (Hg+), Boron (B+), Aluminum (Al+), Gallium (Ga+), Indium (In+),

Thallium (Tl+), Ytterbium (Yb+), and Lutetium (Lu+). The research presented in this disser-

4



tation focuses on using trapped 171Yb+ ions as the qubits in a quantum simulator. The atomic

structure of 171Yb+ makes it a good qubit candidate. The large fine-structure splitting of the

P-orbital excited states and the large hyperfine splitting between different F levels in the atom’s

ground state level are well suited for both coherent and incoherent manipulation of its internal

states.

1.2 Outline

The experimental apparatus at the heart of this dissertation is the trapped-ion quantum sim-

ulator system in Chris Monroe’s Trapped Ion Quantum Information group originally located in

the Atlantic Building on the University of Maryland, College Park campus. The current exper-

imental system has been well documented in recent dissertations, [2] and [3], which includes

the presentation of the redesign of the Raman optical setup. This dissertation will talk a little bit

more about the quadrant photodetector and the output voltage amplifying gain circuits that were

designed for the active stabilization of the lateral beam-pointing of the Raman beams at the plane

of the ions.

In addition, this dissertation presents three different quantum simulation experiments real-

ized in QSim. The brief topics of the three projects are in order: (1) implementing a quantum

approximate optimization algorithm (QAOA) in order to estimate the ground-state energy of a

transverse field antiferromagnetic Ising Hamiltonian with long-range interactions; (2) develop-

ing and demonstrating dynamically decoupled (DD) quantum simulation sequences to extend

the coherence of the intended simulation Hamiltonian dynamics; and (3) implementing the ex-

perimental sequence measuring the spectral form factor (SFF) as well as its generalization, the

5



partial spectral form factor (PSFF), in different types of many-body systems and demonstrating

the feasibility of incorporating both of these observables into the quantum simulation toolbox

which can be used directly to probe predictions of random matrix theory (RMT) and predictions

from the eigenstate thermalization hypothesis (ETH). Although all three of these experiments are

still evolving the quantum spin system under a Hamiltonian, aspects from quantum computations

arise in both. “Digital” operations on individual ions are interwoven with the “analog” time evo-

lution of the Ising-like Hamiltonians in the last project while the quantum circuit representation

of the experimental sequence is present in all three projects. By this statement, I mean that the

Hamiltonian interactions are segmented into chunks of operations as opposed to a continuous

quench. Also, the third project utilizes the individual addressing beam to incorporate methods

from the random operation toolbox instead only being used for state preparation. The SFF project

has tested the QSim apparatus like no other in both experimental complexity of the implementa-

tion as well as the computational demands on the compiling of the waveforms. The SFF project is

an example of some of the challenges that must be faced in order to create a large-scale quantum

module-based processor.

Chapter 2 begins by introducing the Ytterbium atom and its 171-isotope ion, 171Yb+. The

qubit states and the energy levels are presented leading to an overview of how the internal energy

levels of 171Yb+ ions can be controlled and manipulated. The different continuous-wave lasers

are introduced which are a crucial component to address the trapped-ion system. In context to

171Yb+ ions, laser cooling, state preparation, and state-dependent detection of the qubit states

are discussed as well as their corresponding experimental implementation for the trapped-ion

quantum simulator.

Chapter 3 presents the other components of the experimental system which combined make

6



up a tunable quantum simulator of trapped 171Yb+ ions. The vacuum chamber and its history are

briefly introduced before discussing the important ingredients housed inside the chamber: the

linear RF Paul trap and the atomic sources of neutral Yb. Following the ion trap design and fea-

tures, we break take a break from the physical implementation of the experiment to discuss the

physics of the atom-laser interactions describing coherent manipulation of a single ion’s qubit

states. Generalizing the coherent control from one ion to N ions, spin-spin interactions are de-

rived in which the collective transverse modes of motion are coupled to the internal degrees of

freedom to create what is called the long-range antiferromagnetic Ising Hamiltonian. After in-

troducing the atom-laser interactions, the quantum simulation toolbox of Hamiltonian terms is

presented since it provides background for the experiments presented in Ch. 4, Ch. 5, and Ch. 6.

In the remaining sections of Ch. 3, the experimental setups for realizing the coherent interactions

are presented as well as the experimental control software interface connecting all the optical and

electrical components together.

Chapter 4 demonstrates the first implementation of using QAOA to approximate the ground-

state energy specifically of a transverse-field antiferromagnetic long-range Ising Hamiltonian.

Focusing on this quantum many-body Hamiltonian, the scaling of the optimal QAOA perfor-

mance is investigated in both the number of layers of the routine and the number of ions in the

system. In addition, both a brute-force approach and a closed-loop gradient descent was used in

experimentally determining the optimal QAOA variational parameters. Lastly, QAOA was used

to sample the optimized solution to a classical combinatorial problem in which it has already

been proved to be inefficient to realize by a computer.

Chapter 5 develops a general strategy for dynamically decoupling (DD) a quantum sim-

ulation experiment from a known error while preserving the intended coherent interactions of

7



the quantum many-body system that are described by the unitary operator of the desired target

Hamiltonian. We demonstrate this technique on our trapped-ion quantum simulator in which

AC Stark shift noise is identified as the primary cause of decoherence observed in the measured

time-evolved dynamics. With a multi-ion system evolving under the simple long-range Ising

Hamiltonian, we experimentally demonstrate that our method can suppress the noise and, in fact,

extend the coherence of the time-evolved dynamics by more than a factor of 5 in comparison to

coherence observed in the non-decoupled sequence. Lastly, combining of dynamical decoupling

technique with Floquet Hamiltonian engineering, we realize an approximate Haldane-Shastry

model that would be difficult to simulate without the decoupling.

Chapter 6 lastly implements a proposal for measuring the SFF and PSFF of the time-

evolution operator for two known types of quantum many-body spin models: an ergodic system

and a many-body localized (MBL) one. Both the SFF and PSFF observables will be explained

and how they are related to RMT and ETH. This experiment is a prime example of the “analog-

digital” quantum simulation in which single-ion operations are applied not only for state prepara-

tion but also as part of the Hamiltonian interactions that we wish to simulate. Combining methods

from the random measurement toolbox, we demonstrate that universal predictions of RMT and

properties of ETH properties can be probed experimentally. These demonstrations will expand

the quantum simulation toolbox of today’s current quantum simulators.

8



Chapter 2: Ytterbium Ions

The atom of choice for the trapped-ion system discussed in this dissertation is Ytterbium

(Yb). Yb is a lanthanide rare-earth metal whose atomic number is Z = 70 and whose ground-

state electron configuration is [Xe] 4f 14 6s2. As seen in Table 2.1, neutral Yb has seven stable

isotopes [4]. Five of these isotopes have a spinless nucleus while the other two isotopes have

a nonzero nuclear spin. We primarily trap Ytterbium-171 ions (171Yb+); however, Ytterbium-

174 ions (174Yb+) can also be trapped as well under given circumstances that are discussed in

Sec. 2.4. The significance in writing out the electron configuration for the ground state of the

Isotope Abundance (%) Nuclear Spin (I)
168 0.13 0
170 3.05 0
171 14.30 1/2
172 21.90 0
173 16.12 5/2
174 31.80 0
176 12.70 0

Table 2.1. Isotopes of neutral Ytterbium. The isotopes of Yb are listed in ascending order.
The natural abundance and the spin of the nucleus, represented by the quantum number I , are
specified for each of the seven isotopes. The natural abundance values listed in the table are
from [4].

neutral atom, Yb, is to highlight the similarities in the atomic structure of a 171Yb+ ion to that of

a Hydrogen (H) atom. In fact, if one disregards the principal number quantum number n, then

9



the low-energy electronic states in 171Yb+ ions are labeled by the exact same quantum numbers

as that of Hydrogen, obviously, with different hyperfine splittings.

The electronic states (or internal energy levels) of an atom are usually denoted by “good”

quantum numbers which provide a complete description of the atom’s angular momentum. Ta-

ble 2.2 defines some of these different quantum numbers and the corresponding total angular mo-

mentum vector operator. In addition to the quantum numbers listed in Table 2.2, there are “mag-

Quantum Number Angular Momentum Quantity

F magnitude of the total atomic angular momentum F⃗ of the atom

I magnitude of the nuclear spin I⃗

J magnitude of the total electronic angular momentum J⃗

L magnitude of the total electronic orbital momentum L⃗

S magnitude of the total electronic spin S⃗

Table 2.2. Some angular momentum quantum numbers. Often, the internal energy states of an
atom are specified by a set of “good” quantum numbers that are simultaneous eigenstates of the
corresponding angular momentum operators.

netic” or Zeeman quantum numbers corresponding to the spatial component of the total angular

momentum vector that is parallel to the direction of an externally applied magnetic field B⃗ext.

Typically, these magnetic quantum numbers are defined as the z-component of the relavent total

angular momentum vector. For example, quantum number mF corresponds to the z-component

of the atom’s total angular momentum, F⃗z which is better defined in the presence of an magnetic

field. In general, the internal states of an atom with hyperfine structure are denoted by |F, mF ⟩

and their associated energy level (or orbital shell) that is, usually, denoted by the Russel-Saunders

term symbol without the principal quantum number n, i.e. 2S+1LJ . This is the notation that is

adopted for this dissertation when discussing the atomic structure of 171Yb+ ions unless explicitly

10



stated otherwise.

A 171Yb+ ion has a nuclear spin of I=1/2 and, together with its unpaired valence electron

that has a spin angular momentum of S=1/2, gives rise to the hyperfine structure of its internal

energy levels. As a result, the fine structure 6 2S1/2 ground state for even-isotope Yb+ ions is

split into two 6 2S1/2 hyperfine ground states, |F =0⟩ and |F =1⟩, for 171Yb+ ions. The quantum

number F =I+J denotes the total angular momentum of the atom due to the hyperfine interaction

between the nucleus and the electron (F⃗ ∝ I⃗ ·J⃗) where J=L+S is the total angular momentum

of the electron and L is its orbital angular momentum. From principles governing the addition

(or coupling) of angular momenta [5], each hyperfine level with quantum number F has (2F+1)

sublevels or Zeeman states denoted by “magnetic” quantum number mF whose value, typically,

corresponds to the magnitude of F⃗z, the z-component of the atom’s total angular momentum

vector operator. In the absence of an externally applied magnetic field, Zeeman sublevels with

the same F are degenerate in energy. Thus, the 2S1/2 |F =1⟩ hyperfine ground state has three

Zeeman (sub)levels, mF =−{1, 0, 1}, whereas the 2S1/2 |F =0⟩ hyperfine ground state remains a

singlet with mF =0. The degeneracy within each hyperfine manifold is easily lifted by applying

an external magnetic field that occurs due to the Zeeman effect.

In the presence of an external magnetic field, B⃗ext=Bextẑ, Zeeman levels within the same

F -manifold experience an energy shift, or a Zeeman shift away from the energy eigenstates at

zero-field, also known as the Zeeman effect. When the interaction between the atom’s magnetic

dipole moment and the external magnetic field is much weaker than the hyperfine interaction

between the nucleus and the electron, − ˆ⃗µm ·B⃗ext ≪ , the degeneracy of the Zeeman sublevels

within the same F manifold is lifted and F , mF , I , and J are still good quantum numbers that

describe the electronic states of the atom. In this weak-field limit, sublevels with nonzero mF

11



numbers experience a first-order correction to their zero-field energies:

∆EZeeman = µBgFmF |B⃗ext|, (2.1a)

gF = gJ
F (F + 1) + J(J + 1)− I(I + 1)

2F (F + 1)
, (2.1b)

gJ = 1 +
J(J + 1) + S(S + 1)− L(L+ 1)

2J(J + 1)
. (2.1c)

In Eq. (??), µB ≡ e0ℏ/(2me) is the Bohr magneton [6] with elementary charge e0 and electron

mass me, gF is the Landé g-factor [7] for hyperfine structure defined in Eq. (2.1b), and mF

is the total atom angular momentum component parallel to the direction of external field with

magnitude Bext. For example, the ∆mF =±1 Zeeman splittings of the 2S1/2 |F =1⟩ sublevels

corresponds to µB/ℏ = ±1.4 MHz/Gauss. In comparison, the two mF = 0 hyperfine ground

states are first-order magnetic-field insensitive and do not experience a first-order Zeeman shift.

Instead, the lowest-order Zeeman correction to the qubit hyperfine splitting is a second-order

term,

δ2Z = 310.8×B2
ext, (2.2)

where Bext is the magnitude of the applied magnetic field in Gauss (1 Gauss=1×10−4 Tesla) cor-

responding to a total hyperfine qubit splitting of ωhf/(2π) = 12.642812118466+δ2Z GHz [8].

This large separation in the 2S1/2 mF =0 states as well as other desired properties of the atomic

structure of 171Yb+ (see Fig. 2.1) which will be discussed shortly, allow us to define an effective

two-level spin-1/2 system in each trapped ion. The 171Yb+ hyperfine qubit states are defined in

the 2S1/2 ground-state manifold as:

|0⟩ ≡ |↓⟩ ≡ |↓⟩z ≡
∣∣2S1/2 F =0,mF =0

〉
, (2.3a)

|1⟩ ≡ |↑⟩ ≡ |↑⟩z ≡
∣∣2S1/2 F =1,mF =0

〉
. (2.3b)

12



Resonant transitions between these qubit states can conveniently be addressed by a ∼12.6 GHz

Figure 2.1. 171Yb+ energy level diagram. This diagram displays an extensive energy levels
in 171Yb+ The atomic transitions between different hyperfine states are given with respect to
wavelength of the optical light while the hyperfine splitting between different F quantum num-
bers (within each energy level) is specified by a (linear) frequency. The magnitude of the Zee-
man shift with respect to the externally applied magnetic field strength (MHz/G) for the 2S1/2

|F =1, mF =±1⟩ levels are also displayed. Lastly, the blue and red arrows denote the primary
electric dipole transitions that we drive with the 369 nm and 935 nm lasers, respectively.

microwave source or by way of two-photon processes called stimulated Raman absorption and

emission. In QSim, we use two 355 nm pulsed-laser beams to transfer the population coherently

between the two qubit states. Further discussion about these stimulated Raman transitions and

the way they are experimentally implemented are discussed in the next chapter, Ch. 3.

However, before we discuss the atom-laser interactions describing the coherent manipula-

13



tion of the qubit, we first present the required fundamental optical processes that enable basic

control and manipulation of 171Yb+ ions: laser cooling, state initialization, and state detection.

In addition, photoionization of neutral Yb is also presented along with the other three processes

in the next section.

2.1 Continuous Wave Lasers Required for 171Yb+

As observed in Fig. 2.1, the atomic transitions between different energy levels of 171Yb+

can be addressed by shining the appropriate optical wavelength of light on the ion. Fortunately,

continuous-wave (CW) lasers at the wavelengths—λ=369 nm, λ=399 nm, λ=935 nm—are all

now commercially available. The processes and atomic transitions that these three lasers drive

are summarized next.

The 399 nm and 369 nm lasers are used to generate Yb+ ions from neutral Yb atoms

through a two-step photoionization method [9]. Conveniently, the 369 nm laser also happens

to be the wavelength of the light to laser cool the thermal average motion of these “hot” ions.

The 369 nm laser addresses the main electric dipole transition in 171Yb+ that connects the 2S1/2

ground states to the 2P1/2 first-excited states. Utilizing this nearly-closed cycling transition, we

can Doppler cool the ions in their electronic ground states, optically pump the ions to prepare an

initial product state of all spins down, and detect each qubit’s state using a state-dependent proto-

col. Collected photons are imaged on an electron multiplying charge-coupled device (EMCCD),

also known as a camera, or counted on a photomultiplier tube (PMT). The phrase “nearly-closed

cycling transition” refers to the ∼0.005 branching ratio in which there exists a 0.5% probabil-

ity that population in the 2P1/2 state will decay to the metastable 2D3/2 level instead of to the

14



2S1/2 level [10]. The lifetime of the 2D3/2 level is τD3/2
=52.7 ms [11] which is about ∼ 6×106

times longer than the lifetime of the 2P1/2 state which is τP1/2
=8.12 ns [12]. As a result, we use

the 935 nm laser to repopulate (or repump) any population that has spontaneously decayed into

the 2D3/2 level back into to the 2S1/2 ground state by driving it to the 3[3/2]1/2 level [13] which

has a natural lifetime of τ[3/2]1/2 =37.7 ns [14]. From the 3[3/2]1/2 energy level, ions will pri-

marily decay back to 2S1/2 manifold allowing them to reenter the nearly-closed 369 nm cooling

cycle [15].

Each of the atomic transitions using 399 nm, 935 nm, and 369 nm for 171Yb+ are presented

in more detail next. The optical paths of how these lasers were configured in the lab space at the

University of Maryland will also be discussed.

2.1.1 Photoionization

The photoionization of neutral Yb atoms to obtain 171Yb+ ions will now be discussed. We

will assume that there exists an atomic oven containing some Yb metal that can be resistively

heated to create a beam of neutral atoms. A discussion on the way we actually trap ions in the

ion trap will be discussed later on after describing the relevant transitions in the ion and the

experimental implementation. So assuming that a beam of Yb atoms is being ejected from the

oven, these atoms can be ionized, using a two-step, or two-photon transition. First, the 399 nm

wavelength laser is used to excite resonantly the 1S0 ↔ 1P1 transition in neutral Yb. Second,

any laser beam with a wavelength λ<394.1 nm can be used to drive the electron from 1P1 to the

continuum resulting in a Yb+ ion. Thus satisfying this wavelength requirement, the 369 nm laser

cooling beam is used to promote the atoms in the first-excited state to become ions. The 355 nm

15



laser also satisfies the condition, and we actually quickly flash on and off one of the Raman beams

to load ions faster into the trap.

This dichroic two-photon transition enables isotope selective trapping of Yb+ ions [9] by

tuning the frequency of the 399 nm to the resonance of the corresponding isotope that we wish

to trap. We take care to have the alignment of the 399 nm beam roughly perpendicular to the

direction of the Yb oven in order to minimize the Doppler shift in frequency observed by the

high-velocity atoms. We have often observed in QSim that when the ion loading rate is very slow

that it is due to a slight misalignment of the 399 nm beam.

2.1.2 Repumping Light

The 935 nm laser is used to “retrieve” ions that spontaneously decay from 2P1/2 to 2D3/2

which can occur one out of two hundred scattering events when the 369 nm laser is used to

drive the 2S1/2 ↔ 2P1/2 transition. The 369 nm laser does not does not interact with the 2D3/2

hyperfine states, and as a result, the fidelity of a given 2S1/2 to 2P1/2 transition would be reduced

without the presence of the 935 nm laser. Fortunately, the electric dipole transition between the

2D3/2 and 3[3/2]1/2 can be addressed using 935 nm light in which the natural lifetime in the

3[3/2]1/2 energy level is τ[3/2]1/2=37.7 ns [14], and thus, spontaneous emission happens quickly.

The spontaneous emission from the bracket energy level has a strong branching ratio of 0.982

(98.2%) to the 2S1/2 manifold while the remaining branching ratio of 0.018 (1.8%) goes back to

the 2D3/2 states [10, 13, 15]. Thus, the 935 nm laser acts to repump ions that have decayed into

the 2D3/2 states back to the 2S1/2 manifold by driving the 2D3/2 ↔ 3[3/2]1/2 transition.

This repumping laser addresses both ∆F = 1 2D3/2 ↔ 3[3/2]1/2 transitions. The 935 nm

16



laser’s (carrier) frequency, ω935, resonantly addresses the 2D3/2 |F =1⟩ ↔ 3[3/2]1/2 |F =0⟩ tran-

sition and and an electro-optic modulator (EOM) is used to apply a negative first-order sideband at

3.0695 GHz to the 935 nm laser such that this second frequency tone at (ω935−2π×3.0695GHz)

is resonant with the 2D3/2 |F =2⟩ ↔ 3[3/2]1/2 |F =1⟩ transition. The magnitude of the 935 nm

EOM RF sideband is equal to the sum of the 2D3/2 hyperfine splitting and 3[3/2]1/2 hyperfine

splitting, whose values are 0.86 GHz and 2.2095 GHz, respectively. Ions in the 3[3/2]1/2 |F =0⟩

level will spontaneously decay back down into one of the states in the 2S1/2 |F =1⟩ manifold

while ions in any of the 3[3/2]1/2 |F =1⟩ states can spontaneously decay to any of the 2S1/2 hy-

perfine levels, due to the selections rules of allowed transitions. The 935 nm laser remains on at

all times, i.e. during the processes of loading, Doppler cooling, optical pumping, and detecting

the ions.

2.1.3 Doppler Cooling

Conveniently, the 369 nm laser that is used in the second step of the ionization process

of neutral Yb, described in Sec. 2.1.1, also happens to be the laser cooling beam. Thus, laser

cooling of any trapped ions can occur simultaneously if more ions need to be trapped before the

current to the Yb oven is turned off. We laser cool the ions using a well-known technique called

Doppler cooling [16,17]. This cooling method utilizes the Doppler effect to maximize the photon

scattering rate of the 2S1/2 ↔ 2P1/2 transition in order to reduce the average kinetic energy of the

ion [16, 17].

The atomic transitions that are driven by the cooling beam in 171Yb+ are shown in Fig. 2.2.

The blue arrows indicate the dipole allowed stimulated absorption and emission processes while

17



the gray arrows display only the spontaneous emission paths back to the 2S1/2 states. A single

σ-σ+ π

π σ+σ-

2S1/2

2P1/2 2.105 GHz

~12.6428 GHz

F = 1

F = 0

F = 1

F = 0

mF = −1 mF = 0 mF = +1

Figure 2.2. 369 nm Doppler cooling transitions. The 369 nm Doppler cooling beam off-
resonantly drives both allowed ∆F = 1 2S1/2 ↔ 2P1/2 transitions in order to cool efficiently
both the |↓⟩ and |↑⟩ states in 171Yb+. The blue arrows depict the stimulated absorption and emis-
sion processes from the 2S1/2 while the gray arrows indicate spontaneous emission transitions.
The cooling beam also contains all polarizations components in order to cool population that de-
cays in the the nonzero 2S1/2 Zeeman sublevels.

369 nm (cooling) beam is used to Doppler cool the ions by orienting it through the chamber and

ion trap slot such that the beam’s k⃗ has components along all three axes. The cooling beam

frequency is red-detuned from the 2S1/2 |F =1⟩ ↔ 2P1/2 |F =0⟩ resonant frequency by slightly

more than half the linewidth, ∼15 MHz. In addition, we apply an external static magnetic field of

about ∼5.22 G along the y principal axis of the ion trap (z axis in the lab frame). The externally

applied B-field not only defines the spin quantization axis, but it also helps to prevent population

from getting stuck in coherent dark states, superpositions of the 2S1/2 |F =1⟩ Zeeman levels,

that exist in the absence of one [18]. The |B⃗ext| =5.22 G magnetic field shifts the Zeeman

|F =1, mF =±1⟩ states, to first-order, by ∼ ±7 MHz. In order to efficiently cool on both allowed

18



electric dipole transitions between the 2S1/2 and 2P1/2 states, the 369 nm cooling beam is sent

through an EOM that applies a sideband frequency at 14.748 GHz that is red-detuned from the

2S1/2 |F =0⟩ ↔ 2P1/2 |F =1⟩ transition. In addition, to the 369 cooling beam, the 935 nm beam

is also critical in order to return population that spontaneously decays into the 2D3/2 manifold

from the 2P1/2 states back to the main cooling transitions.

2.1.4 Optical Pumping

State preparation or initialization of a single trapped ion (or a linear chain of ions) is ac-

complished through the process of optical pumping. By applying a separate 369 nm laser beam

(not the cooling beam) that addresses the 2S1/2 |F =1⟩ ↔ 2P1/2 |F =1⟩ transition, the ion is effi-

ciently prepared in the |↓⟩ state. The optical pumping beam frequency is 2.105 GHz shifted above

2S1/2

2P1/2 2.105 GHz
F = 1

F = 0

F = 1

F = 0

mF = −1 mF = 0 mF = +1

σ-σ+π π

σ+σ-

Figure 2.3. 369 nm optical pumping transitions. The 369 nm optical pumping beam frequency is
resonant with the transition, 2S1/2 |F =1⟩ ↔ 2P1/2 |F =1⟩. The stimulated absorption and emis-
sion transitions of the optical pumping process are represented by blue arrows while spontaneous
emission from the 2P1/2 |F =1⟩ excited states are represented by gray arrows.

19



the 2S1/2 |F =1⟩ ↔ 2P1/2 |F =0⟩ resonant frequency and contains both π- and σ- polarizations

as shown in Fig. 2.3. When the ion is pumped to any of the 2P1/2 |F =1⟩ hyperfine states, se-

lection rules dictate that the ion can spontaneously decay to a state in either 2S1/2 F manifold or

to a state in the 2D3/2 |F =2⟩ level. If the ion decays back into any of the 2S1/2 |F =1⟩ states, it

will be optically pumped back to the 2P1/2 |F =1⟩ manifold. However, if the ion spontaneously

emits into the 2S1/2 |F =0⟩ (|↓⟩) state, it will be unaffected by the optical pumping beam since

the frequency of the beam is ∼12.6 GHz off-resonant from the |↓⟩ ↔ 2P1/2 |F =1⟩ transition. If

the ion spontaneously emits into the 2D3/2 |F =2⟩ level, the 935 nm beam will return the ion to

the |↓⟩ state. The optical pumping beam only needs to be turned on for 15–25 µs before the ion

is initialized in the |↓⟩ state.

2.1.5 Detection

We can determine the spin state of a single ion through a spin-dependent fluorescence

method illustrated in Fig. 2.4. The 369 nm detection light frequency is exactly on resonance with

the |↑⟩ ↔ 2P1/2 |F =0⟩ transition. If an ion is in the |↑⟩ state, then it will absorb a photon and

transition to the 2P1/2 |F =0⟩ state. As mentioned previously, an ion in the 2P1/2 |F =0⟩ state will

preferentially spontaneously decay back to one of the 2S1/2 |F =1⟩ hyperfine states, scattering

a photon in the process. The detection beam is composed of all three polarizations in order to

address all three of the 2S1/2 Zeeman states. In comparison, an ion that is in the |↓⟩ state will

not be excited by the detection beam since the frequency of the light is ∼12.6 GHz off-resonant

from the 2P1/2 |F =0⟩ state, which is also a transition that is forbidden. In summary, when the

detection beam is applied, an ion in the |↑⟩ state will fluoresce and appear “bright” while an ion

20



σ+ π σ-

2S1/2

2P1/2

F = 1

F = 0

F = 1

F = 0

mF = −1 mF = 0 mF = +1

Figure 2.4. 369 nm detection transitions. The 369 nm detection beam frequency is resonant with
the transition, 2S1/2 |F =1, mF =0⟩ ↔ 2P1/2 |F =0, mF =0⟩. However, the detection beam po-
larization contains π and both σ polarizations in order to drive population from the 2S1/2 |F =1⟩
Zeeman levels as well. Stimulated absorption and emission transitions are represented by blue
arrows while spontaneous emission from the 2P1/2 |F =0⟩ excited state are represented by gray
arrows.

in the |↓⟩ state will not fluoresce and appear “dark.” Further discussion regarding the collection

of the photons and discriminating the measured counts for a single ion versus a linear chain of

ions will be discussed in the following chapter in Sec. ??. Off-resonant excitation to the 2P1/2

|F =1⟩ states that spontaneously decay to either 2D3/2 F -manifold (due to selection rules) can

cause detection errors since an ion decaying into 2D3/2 |F =2⟩ could be repumped back the |↓⟩

state. However, the 935 nm light is still needed to retrieve ions that decay from the 2P1/2 |F =0⟩

to 2D3/2 |F =1⟩. This type of spontaneous emission transition would not cause a detection error

since the 3[3/2]1/2 |F =0⟩ state can only decay back into the 2S1/2 |F =1⟩ manifold.

21



2.2 The CW Lasers

In this section, we will present how the atomic transitions in 171Yb+ are physically imple-

mented. Before presenting the beam paths of the 369 nm, 399 nm, and 935 nm lasers; I should

mention that there is a fourth CW laser that is also utilized. This laser has a wavelength at

λ=739 nm and it was the one that was alluded to back in Sec. 2.1. The reasons for why QSim uses

this 739 nm laser system instead of a direct diode laser at 369 nm will be explained in Sec. 2.2.4.

Figure 2.5 displays the entire optical layout for the 369 nm, 399 nm, 739 nm, and 935 nm

lasers that are before a dividing wall. This dividing wall is a composite aluminum panel, sold by

the 8020 company, that separates the area surrounding the vacuum chamber from these optical

elements that make up the beam paths for the CW lasers. As observed in Fig. 2.5, the optical

93
5 

nm
L

as
er

to 935nm Fiber EOM Input

fib
er

wavem
ete

r
935nm

pick off

500mm

λ/2λ/4 λ/4

λ/2

λ/2

200mm

Isolator Isolator 200mm 200mm

achromatic
prism pair

to monitor
cavity

935nm

fiber
to wavemeter

935nm

1000mm

PD
935nm

50mm

50mm
PD

399nm

piezo HV BNC

monitor cavity
diffraction

grating

39
9n

m
L

as
er

75mm

200mm

pick off

75mm

75mm

λ/2

λ/2

λ/2

iris Isolator

399nm to
wavemeter

fiber

QSim's
399nm flag

399nm path 
for QSim

to Newport
flipper mirror

TTL switch box

switch state controlled
by LabJack TTL

from Iseg LabView

399nm
to monitor

cavity

399nm path
for Gates'

flipper mirror
399nm

Gates' 

flag
399nm399nm 

wavemeter
fiber

Gates'
399nm fiber

λ/2

λ/4

PDH cavity175mm

50mm

739nm
Laser

λ/2

λ/2λ/2

λ/2

NE10A

PDH
EOM

@20MHz
RF

to PDH
optics

BS10:90
(R:T)

BS10:90
(R:T)

transmission
PDH

signal

to Iodine
SAS setup

to Doubler

to wavemeter FC

739nm
wavemeter

fiber
reflection

PDH

signal

PDH cavity 
piezo HV BNC

fiber EOM input
to Iodine

FET BNC
diode current

grating piezo
voltage BNC

Frequency
Doubler

739nm 369nm

λ/2 λ/2

λ/2

λ/2

λ/2

λ/2

λ/2

λ/2

λ/2

Detection/

Fiber Input
Optical Pumping

Detection/

Fiber Output
Optical Pumping

Fiber Input
Cooling

Cooling Fiber Output

Optical Pumping
& Detection

Combined Here

Cooling &
Optical Pumping

& Detection
Combined Here

Optical
Pumping

Back Table
Optical

Pumping
Fiber Input

Cooling

C
oo

li
ng

AOM AOM

AOM

EOM
2.105
GHz

EOM

GHz
7.343

Vertical
Cylindrical

369 nm
EOM SB
Cavity

Detection

iris

iris

iris

Lens

iris

iris

Optical Pumping

Detectiondoubler measured here

this is iris effect
bright to dark

ratio and difference

50/50 BS
369nm & 399nm

combing

hole in 8020 panel
going to chamber

Iodine
Saturated Absorption

Spectroscopy
Optics

Figure 2.5. Optical paths of 171Yb+ CW lasers. The beam paths for the 369 nm, 399 nm, 739 nm,
and 935 nm laser light are displayed roughly to scale in the configuration that they were in at the
University of Maryland.

paths are quite spread out. In the following subsections, each laser path will be discussed and the

corresponding optical setup will be displayed again by itself so that it can be seen better.

22



2.2.1 399 nm Optical Path

The 399 nm laser is an off-the-shelf commercial continuous wave (CW) tunable diode laser

from Toptica Photonics. A diagram of the 399 nm optical path is shown in Fig. 2.6. After the

50mm
PD

399nm

piezo HV BNC

monitor cavity
diffraction

grating

39
9n

m
L

as
er

75mm

200mm

pick off

75mm

75mm

λ/2

λ/2

λ/2

iris Isolator

399nm to
wavemeter

fiber

QSim's
399nm flag

399nm path 
for QSim

to Newport
flipper mirror

TTL switch box

switch state controlled
by LabJack TTL

from Iseg LabView

399nm
to monitor

cavity

399nm path
for Gates'

flipper mirror
399nm

Gates' 

flag
399nm399nm 

wavemeter
fiber

Gates'
399nm fiber

50/50 BS
369nm & 399nm

combing

hole in 8020 panel
going to chamber

Figure 2.6. 399 nm optical path. The 399 nm beam is shared with another lab. There is a
motorized flipper mirror that allows us to switch which the position that determines which way
the main 399 nm light is directed.

399 nm light passes through an isolator and a couple of lenses, a small portion of the light is

picked off from the main beam path. The beam that is reflected from the front surface of the

optical element (labeled as “pick off” in Fig. 2.6 and referred to as such) goes to a fiber that is

sent to a wavemeter that measures the frequency of the 399 nm laser. The beam that is reflected

from the back surface of the pick off is aligned through a home-built scanning confocal “monitor”

cavity in which a photodetector is used to measure the 399 nm light transmitted through the cavity.

By scanning the length of the cavity via applying a triangle-ramp voltage to the cavity piezo, we

can monitor whether the 399 nm frequency is single-mode or multi-mode.

Returning to the main 399 nm beam path, the next element in the beam path is a mirror

23



on a motorized flipper-mount allowing the beam to be shared with another lab that also traps

171Yb+ ions. When the 399 nm flipper-mirror is flipped up into the beam path, the 399 nm light is

directed to the rest of the free-space optical path. A 399 nm flag (or mechanical shutter) by default

prevents (blocks) the beam from continuing to the rest of the beam path and whose positioned can

be triggered by our experimental control software that raises the flag out of the 399 nm beam path

when we wish to load ions into the trap. With the 399 nm flag raised out of the beam path, the

beam passes through a couple of beam-shaping lenses and is eventually combined with the three

369 nm beam paths (cooling, optical pumping, and detection) at a non-polarizing beam splitter.

The reflected 399 nm beam and the transmitted 369 nm beams pass through a small hole made

in the 8020 aluminum composite panel that separates optical paths of the CW lasers from the

remaining optics leading to the vacuum chamber and the vacuum chamber itself.

2.2.2 935 nm Optical Setup

The 935 nm laser is a CW tunable diode laser also from Toptica Photonics. A diagram of

the 935 nm optical path is shown in Fig. 2.7. Similar to the 399 nm optical path, a small portion

of the light is picked off soon after the isolators. The front surface of the pick off sends the light

to the monitor cavity so that the 935 nm transmission peaks can be observed on an oscilloscope.

Assuming that the 935 nm beam is well-coupled to the monitor cavity, we can look at the 935 nm

oscilloscope signal to guarantee that the laser frequency is single mode. The back surface of the

935 nm pick off is coupled to a fiber that goes to the wavemeter in order to measure the value of

the laser’s frequency.

We use the wavemeter frequency reading of the 935 nm laser to stabilize the laser’s fre-

24



93
5 

nm
L

as
er

to 935nm Fiber EOM Input

fib
er

wavem
ete

r
935nm

pick off

500mm

λ/2λ/4 λ/4

λ/2

λ/2

200mm

Isolator Isolator 200mm 200mm

achromatic
prism pair

to monitor
cavity

935nm

fiber
to wavemeter

935nm

1000mm

diffraction
grating

PD
935nm

50mm
piezo HV BNC

monitor cavity

Figure 2.7. 935 nm optical path. The 935 nm optical path is very simple. The main beam path
goes to a fiber coupler sending the light to the input of the fiber EOM. The other two beams paths
are one that goes to the wavemeter to read and lock the laser frequency while the other path goes
to a “monitor” cavity. The monitor cavity is used to determine ensure that the 935 nm laser is
single mode.

quency with a software (or “wavemeter”) lock. The 935 nm wavemeter lock compares the

wavemeter measured frequency value of the laser to the set frequency that is typed into the web

browser software interface. The wavemeter lock sends a feedback signal to the voltage con-

trol module that supplies the laser’s grating piezo voltage. The wavemeter itself is calibrated

with a laser whose frequency is stabilized to an absolute reference (the locked 739 nm laser, see

Sec. 2.2.4.2). The 935 nm wavemeter lock set frequency is typically at 320.569251 THz which

resonantly addresses the 2D3/2 |F =1⟩ ↔ 3[3/2]1/2 |F =0⟩ transition in 171Yb+.

Returning to the main 935 nm beam path, the light is coupled into the input port of a fiber

EOM in which the negative first-order sideband at 3.0695 GHz is applied to the carrier frequency

of the 935 nm light. The sideband frequency is the sum of the hyperfine splittings of the 2D3/2

25



and 3[3/2]1/2 levels which enables the single 935 nm beam to re-pump ions from any of the 2D3/2

hyperfine states. The RF amplitude of the 935 nm sideband is about 2% of the initial carrier

signal. The output port of the 935 nm fiber EOM is connected to a fiber coupler that is placed

closer to the vacuum chamber, on the other side of the dividing 8020 panel, where the light is

directed towards the entrance of the back chamber window.

2.2.3 369 nm Beam Paths

The 739 nm laser will be discussed in the next section, Sec. 2.2.4. The majority of the

739 nm laser light is sent to the input of a WaveTrain frequency doubler which outputs blue

369 nm light. Figure 2.8 shows the optical setup exiting from the doubler. The frequency of the

light exiting the doubler is ∼ 430MHz red-detuned from the resonant frequency that addresses

the 2S1/2 |F =1⟩ ↔2P1/2 |F =0⟩ transition. By splitting the 369 nm light into three different

paths and using optical modulators to shift the frequency of (using an AOM) and/or apply addi-

tional frequency tones (using an EOM) to the original 369 nm light, we can use the 369 nm light

to address transitions between the 2S1/2 and 2P1/2 hyperfine levels. Each of the 369 nm beam

paths are separated into two parts in which a fiber is used to create the separation. The optics that

lie before the corresponding fiber input, all have an EOM (except for detection) and an AOM in

their path while the path after the fiber output provides the degrees of freedom to optimize the

alignment of the beam interacting with the ions. We often find that the AOMs need to be recou-

pled or at least optimized slightly every few months, so having the defined separation ensures

that the beams still remain aligned to the ions once the fiber coupling has been optimized.

We typically measure the 369 nm power coming out of the doubler each day and usually

26



Frequency
Doubler

739nm 369nm

λ/2 λ/2

λ/2

λ/2

λ/2

λ/2

λ/2

λ/2

λ/2

Detection/

Fiber Input
Optical Pumping

Detection/

Fiber Output
Optical Pumping

Fiber Input
Cooling

Cooling Fiber Output

Optical Pumping
& Detection

Combined Here

Cooling &
Optical Pumping

& Detection
Combined Here

Optical
Pumping

Back Table
Optical

Pumping
Fiber Input

Cooling

C
oo

li
ng

AOM AOM

AOM

EOM
2.105
GHz

EOM

GHz
7.343

Vertical
Cylindrical

369 nm
EOM SB
Cavity

Detection

iris

iris

iris

Lens

iris

iris

Optical Pumping

Detectiondoubler measured here

this is iris effect
bright to dark

ratio and difference

50/50 BS
369nm & 399nm

combing

hole in 8020 panel
going to chamber

Figure 2.8. 369 nm optical path. The 369 nm beam paths is a lot more complicated in com-
parison to the 399 nm and 935 nm beam paths. The 369 nm light exiting the doubler system is
immediately split into three beam paths: cooling, optical pumping, and detection.

the power coming out of the cooling fiber output if we do not see any of the usual background

scattered cooling light on the camera. The detection beam transmits through the first PBS after

the doubler while cooling and optical pumping are reflected by the PBS. The orange rectangles

in Fig. 2.8 (and in the other beam path sketches) are half-wave plates (HWPs) and are used to

control the power in each beam path such as the ones that are placed before PBSs or to set the

polarization of the light.

The detection beam path only has an AOM in its path which shifts the frequency of the

light to be resonant with the 2S1/2 |F =1⟩ ↔2P1/2 |F =0⟩ transition, and it’s AOM’s RF power

is kept lower than the max in order to not saturate the transition. In addition, less power in the

detection beam helps to reduce off-resonantly scattering to the 2P1/2 |F =1, mF =±1⟩ sublevels

which can off-resonantly decay into either F hyperfine level in the 2D3/2 spectrum. Unlike the

off-resonant process that can occur during cooling and optical pumping, if an ion is off-resonantly

27



pumped into either of the decays into the 2P1/2 |F =1, mF =±1⟩ states and ends up decaying

into one of the 2D3/2 |2, mF ⟩ sublevels, it will transition to the 3[3/2]1/2 |F =1⟩ in which the

only decay path back to the ground state manifold is to |↓⟩ due to selection rules. This particular

process with correspond to a detection error in which an ion that is detected “bright” becomes

stuck in the “dark” state.

In the 369 nm beam path, shown in Fig. 2.8, there are two removable mirror mounts that

can be placed in the cooling and optical pumping paths which are depicted as grey dashed circles.

These are convenient to have in place. The removable mirror in the cooling beam path that is after

the AOM but before the cooling fiber input allows us to direct the beam into a scanning confocal

cavity to check the the second-order EOM sideband (2×7.374 GHz=14.748 GHz) coupling while

the other removable mirror mount enables us to redirect the optical pumping beam. The optical

pumping light goes to a fiber coupler that is connected to a fiber nicknamed, the “back table

optical pumping fiber” (see Fig. 2.8) which delivers the optical pumping beam across the back of

the optics table, naturally, to be combined with the individual addressing beam. It has been very

handy to have the optical pumping beam to be able to realign that 355ṅm beam since it is a tightly

focused beam going down the objective lens stack to the trap. The procedure for overlapping the

back-table optical pumping beam with the individual addressing beam is discussed in Ref. [3].

2.2.4 739 nm Laser

As previously mentioned, the electric dipole transitions between the hyperfine levels, 2S1/2

and 2P1/2, are addressable with a laser at 369 nm. However, we use Toptica Photonics TA739, a

tapered amplifier (TA) external cavity diode laser (ECDL), that is at a wavelength of λ=739 nm.

28



The 739 nm laser is sent through a frequency doubler in which 369 nm light is generated at exactly

twice the frequency of the fundamental 739 nm frequency. Frequency doublers, generally, use

a nonlinear optical crystal to provide the second-harmonic generalization (SHG) or frequency

doubling when a high intensity laser traverses through the crystal and utilize optical resonators to

amplify SHG converted power.

The 739 nm laser can also be used to probe absorption features in molecular iodine that

are within ±20 GHz of the 739 nm light. Molecular iodine has an “atlas” of narrow absorption

lines that can be used as a frequency reference. Accordingly, the 739 nm laser frequency can be

stabilized by referencing its frequency to an absorption line in molecular iodine which will be

inherited by the SHG 369 nm light coming out of the doubler system.

The 739 nm optical setup is shown in Fig. 2.9. The light from the 739 nm laser diode is

coupled into a tapered amplifier (TA) chip which is an active gain-medium that produces a high-

power output typically between 450 mW to 520 mW (depending on the laser mode). The first

element in the 739 nm beam path is a 10:90 beam splitter (BS) that reflects about 10% of the

light exiting from the laser system, to a single-mode fiber that goes to the wavemeter, allowing

us to observe the laser’s frequency. The remaining 90% of the laser light is transmitted through

the BS. We measure and record the output power of the 739 nm laser after the first mirror which

has a typical value of ∼470 mW. The main beam path is split in two by a PBS in which the

reflected beam path leads to the optical setups for the PDH and iodine locks while the transmitted

beam path directs the light to the input of the doubler. Before the PBS, we have a 739 nm half-

wave plate (HWP) in an adjustable rotation mount. By rotating the HWP, we can change the

polarization of the light, resulting in how much power is reflected by and/or transmitted through

the PBS. We typically send the majority of the 739 nm power to the doubler input in which typical

29



λ/2

λ/4

PDH Cavity175mm

50mm

Iodine Fiber739nm
Laser

λ/2

λ/2λ/2

λ/2

NE10A

BS10:90
(R:T)

BS10:90
(R:T)

739nm
Wavemeter

Fiber

20MHz SBs
PDH
EOM

Grating Piezo
Voltage BNC

FET BNC
Diode Current

PDH Cavity Piezo Voltage BNC

Reflection
PDH

Signal

Transmission
PDH

Signal

Frequency
Doubler

739nm 369nm

Figure 2.9. 739 nm beam path. 739 nm beam path is the only one of the four CW lasers that has a
nice compact beam path excluding the molecular iodine saturated absorption spectroscopy setup.
The majority of the 739 nm light is sent to the input of the doubler system while the remaining
light is reflected by a PBS for the 739 nm laser locks.

values of the power measured after the second HWP are between 365–420 mW depending on the

total output power from the laser and the amount of power is reflected by the PBS for the locks.

The power measured in the PBS reflected path is typically set to a value between 45–110 mW.

The PBS reflected beam path is further split in two by another 10:90 BS and distinguishes the

PDH optical setup from the iodine setup. Thus, the majority of the light designated for the 739 nm

stabilization locks is sent to the input of a fiber that delivers the light to the iodine SAS system.

The discussion of locking the 739 nm laser current, piezo grating voltage, and laser frequency

will be presented next.

Unlike the 935 nm laser whose frequency is locked using the wavemeter, the stabilization

of the 739 nm laser is slightly more complicated. We use the Pound-Drever-Hall (PDH) tech-

nique [19, 20] to lock the 739 nm laser frequency to a scanning confocal cavity. In turn, the

30



length of this PDH cavity is stabilized using a Doppler-free saturated absorption spectroscopy

(SAS) signal from a heated cell of molecular iodine. The following sub-subsections detail the

PDH and iodine SAS optical setups in which the signals obtained are used to stabilize the 739 nm

laser.

2.2.4.1 739 nm PDH Locks

The frequency of the 739 nm laser is mainly controlled by the amount of current supplied

to the laser diode and the angular position of the diffraction grating. The temperature of the laser

diode is another parameter that can be adjusted to tune the frequency but generally is not the

first “knob” of choice. From day-to-day operation, the laser frequency is usually tuned by finely

adjusting the angular position of the diffraction grating with respect to the incident light emitted

from the laser diode. The grating itself is epoxied onto to the front surface of a flexure mount

in which the flexure angle can be opened or closed by tightening or loosening a piezo actuated

screw, respectively. The grating flexure angle can be adjusted to a finer degree by changing the

voltage applied to the piezoelectric material of the actuated screw. By scanning the voltage of the

grating piezo, the laser’s frequency can be tuned at the MHz resolution and over a range of a few

GHz. Frequency tuning the laser via the piezo voltage is slow in comparison to scanning the laser

diode’s current which also changes the intensity of the laser. Thus, in order to guarantee a stable

laser frequency, both the current of the laser and the grating piezo voltage must be controlled.

The PDH technique [19, 20] provides a straightforward solution for stabilizing a laser’s

frequency using a Fabry-Perot (FP) cavity. The PDH method provides a protocol to measure

the derivative of the cavity-reflected intensity with respect to changes in the laser’s frequency

31



which indicates how the frequency of the laser should be corrected to remain on-resonance with

the cavity. By using the reflected intensity to measure the frequency of the laser, the signal is

decoupled from fluctuations in the laser’s intensity which is not the case when measuring the

transmitted cavity signal. The implementation of the PDH optical setup is shown in Fig. 2.9 in

which the beam path is defined by the reflected 739 nm beam from the half-inch diameter 10:90

BS. The 739 nm beam in the PDH path is phase-modulated using a resonant EOM that adds

weakly-coupled frequency sidebands at ±20 MHz about the carrier frequency of the beam. Our

setup differs from previous implementations of the 739 nm PDH lock, discussed in dissertations

of former group members [21, 22], in which previous schemes frequency-modulated (FM) the

current of the laser diode.

The phase modulated (PM) 739 nm light is aligned to the PDH scanning confocal cavity in

which photodetectors before and after the cavity have been positioned to measure the reflected

and transmitted intensity signals, respectively. The HWP before the PBS ensures that the modu-

lated beam incident from the left is transmitted through the PBS while the QWP just before the

PDH cavity ensures that the reflected signal from the cavity is reflected by the PBS and directed

towards the PDH reflection photodetector. The PDH cavity itself has one of its mirrors epoxied

on the outside of the hollow Invar-36 rod which has a low coefficient of thermal expansion. The

piezoelectric tube and the other cavity mirror are mounted inside the Invar-36 rod. A homebuilt

electronics board supplies the high-voltage (HV) to the piezo enabling the length of the PDH

cavity to be scanned using the electronic board’s tunable internal ramp signal or with an external

signal that can be fed as an input connection to HV board. Which ramp setting is determined by

the internal/external toggle switch on the HV board. This manual control allows us to optimize

the PDH signal when the switch is set to the internal setting, and then, have the length of the

32



cavity itself be locked to a stable reference frequency when the switch is flipped to the external

state.

Before we are able to reference the length of the PDH cavity using a Doppler-free saturated

absorption signal from molecular iodine, the 739 nm laser’s current and grating piezo voltage

must first be locked to the PDH cavity. Figure 2.10 shows the electronic diagram of the PDH

lock. The PDH cavity reflection signal is compared to the 20 MHz local oscillator signal using a

frequency mixer. The output of the mixer is sent through a low-pass filter resulting in the PDH

error signal. This error signal is split and sent to the inputs of two different proportional integral

differential (PID) servo controllers that are used to stabilize the laser’s frequency. The PDH error

signal that is sent to the PID controller used for the grating piezo voltage passes through another

low-pass filter before entering the controller’s input. The grating PID stabilizes the 739 nm laser’s

frequency against slow drifts while the current PID counteracts faster noise. The 739 nm laser

frequency is locked to the length of the PDH cavity after engaging the grating and current PID

controllers. However, the PDH cavity length is still susceptible to slow thermal drifts and as a

result, stabilization of its length is required in order to maintain frequency stability of the 739 nm

laser which is imparted in the resulting 369 nm light exiting the doubler.

2.2.4.2 739 nm Iodine Reference Frequency Lock

The PDH cavity length is referenced to a narrow absorption line in molecular iodine. More

specifically, our lab has an optical setup to perform Doppler-free SAS of a heated glass cell

of molecular iodine. Conveniently, molecular iodine has three absorption lines that are within

± 15 GHz of the 739 nm laser frequency value required to address 171Yb+ ions. However, the

33



ZMDC-10-1+
Directional CouplerHP 8656B RF Output

20 MHz
Frequency

+10.0 dB
Amplitude

Mixer
ZAD-1-1+

LO
RF

IF

Low Pass Filter
≤1.9 MHz

MiniCircuits BLP-1.0

Low Pass Filter
≤10 kHz

Thorlabs EF120

Low Pass Filter
≤10 kHz

Thorlabs EF120

HV Ramp
Input

Output

Error
Signal

Iodine PID
Input

Output

Error
Signal

Current PID
Input

Output

Error
Signal

Grating PID
Input

Output

Error
Signal

to PDH EOM RF Input

Iodine Lock-In Amplifier Output

PDH PD Reflection Signal

Thorlabs Piezo Voltage
Control Box

Vmod Vout

to Grating Piezo
to Current FET Board Input

to PDH Cavity Piezo
to Oscilloscope Channels

to Oscilloscope
Channels

Figure 2.10. PDH current and grating locks. These are the electronic components of the PDH
stabilization scheme used to lock the 739 nm laser diode’s current and grating piezo voltage. The
four PIDs are a type of rack mount card that can be plugged into a Toptica controller main-frame
slot.

strengths of these absorption lines are quite weak with the iodine cell at room-temperature. As a

result, our iodine cell, shown in Fig. 2.11, is heated to a temperature of about 560 °C. This section

will highlight improvements that have been made to both the optical and electrical components of

the iodine lock during my time in the lab. For more information about the design and theoretical

principles of SAS, please refer to [21–23].

I will first summarize the iodine SAS setup. The 739 nm light is sent through a fiber EOM

shifting the 739 nm frequency on the order of GHz in order to address the desired iodine line. The

EOM fiber output is connected to a fiber coupler located at the top right corner of the elevated

breadboard platform, the input of the iodine SAS setup. Immediately, the 739 nm beam is split

into two beams in which the higher-intensity beam is transmitted through the pick off and is

denoted as the pump beam. The beam reflected from the pick off is further split in two by a BS

in which the reflected beam is the reference and the transmitted beam is the probe. The reference

beam is aligned to the reference input of a balanced photodetector. Meanwhile, the probe beam

34



Figure 2.11. Iodine SAS optical setup. The large cylinder blob that is wrapped in aluminum,
a good thermal conductor, is the molecular iodine cell. In order to be able to see an absorption
feature of molecular iodine at room temperature it must be heated to over 500 °C. The 739 nm
light that has sidebands that are on the order of 13 GHz enters the iodine SAS setup in the top
right corner of the elevated bread board. The probe beam goes through the cell from right to left
with respect to this image and the pump goes from left to right.

travels through the iodine cell from right to left and is aligned to the signal input of the balanced

photodetector. Returning to the pump beam path, a small portion of the light is picked off and

aligned to a FP cavity. By observing the FP transmission signal, we can check the 739 nm laser

mode and measure the GHz sidebands that are applied to the light from the fiber EOM. The

carrier peak should be suppressed and only the first-order sidebands peaks should be observed

in the transmission signal. The majority of the pump light is sent to the input of an AOM that

has a center frequency at 80 MHz and the optimized first-order diffracted beam is used as the

pump. The RF signal applied to the AOM is frequency modulated by a 11 kHz sine wave. The

35



modulated pump beam is aligned through the iodine cell traveling left to right. In our setup, the

ratio between the intensities of pump and probe beams is about 3.5:1. The power in each beam

is measured right before it enters the iodine cell which corresponds to the position just before the

spherical lens.

The signal from the balanced photodetector and the 11 kHz modulation signal are sent to

the input and reference ports of a lock-in amplifier electronics box, respectively. The output

from the lock-in amplifier is the derivative of the Doppler-free signal measured by the balanced

photodetector. This iodine reference frequency error signal is sent to the input of another PID

servo controller that is used to lock the length of the PDH cavity. More specifically, the iodine

error signal as a function of the 739 nm laser frequency can be observed on an oscilloscope.

Once the 739 nm current and grating piezo are locked to the PDH cavity, scanning the PDH

cavity length results in scanning the frequency of the 739 nm laser. Thus, by scanning the 739 nm

frequency, the scanned iodine error signal will show the derivative of the Doppler-free iodine

signal as a function of the laser frequency. The PDH cavity length is locked to the zero-crossing

of the Doppler-free derivative iodine signal using the iodine PID controller. The output of the

iodine PID is sent to the control port of the PDH HV ramp card that sends the voltage signal to

the cavity as shown in Fig. 2.10.

The frequency detuning with respect to the 739 nm frequency of the three closest io-

dine absorption lines are about -5 GHz, +10 GHz, and +13 GHz and correspond to transitions

at 13530.6745 cm−1, 13531.1823 cm−1, and 13531.2773 cm−1 respectively [24]. Further spec-

troscopy at each absorption line was performed in Ref. [21] to determine which hyperfine-

structure feature would be the best to use as an absolute reference. The lock-in amplifier output

signal which is equivalent to the derivative of the Doppler-free hyperfine-structure of the +13 GHz

36



line and the +10 GHz line is plotted in Ref. [23] and Ref. [10], respectively. Reference [22] has

the signal of the largest hyperfine structure feature of the -5 GHz line.

Figure 2.12 displays the trace of the observed unlocked iodine error signal as the 739 nm

laser frequency is changed by scanning the PDH cavity length and it corresponds to “Feature 5”

of the lock error signal plotted in Ref. [23]. Feature 5 is one of six features that are observed in

the +13 GHz absorption line. This iodine absorption feature has the largest amplitude, and as a

result, the zero-crossing of the furthest right positive slope of Fig. 2.12 was chosen as the absolute

frequency reference to stabilize the 739 nm laser. Originally in QSim, we used the second zero-

Figure 2.12. Feature 5 of the iodine error signal. Sweeping the unlocked iodine error signal over
the doublet feature in the +13 GHz absorption line. The PDH cavity length is referenced to the
zero-crossing that’s farthest to the right (or the second zero-crossing). When the iodine PID is
locked, the frequency of the 739 nm laser is stabilized at 405.644325 THz, the value measured on
the wavemeter.

crossing in the doublet feature, shown in Fig. 2.12, in the +13 GHz detuned iodine absorption line

as the lock-point of the iodine PID. To observe this feature, 13.315 GHz was the RF frequency

applied to the iodine fiber EOM in which the first-order sideband was optimized and the carrier

amplitude suppressed.

37



In December 2019, we took some time to optimize the Doppler-free iodine signal measured

by the balanced photodetector since we had noticed that the unlocked iodine error signal had be-

come smaller and noisier leading to the iodine PID unlocking shortly after being locked. Several

improvements were made to the iodine optical setup:

1. Improved the polarization stability of the 739 nm light entering the iodine setup. The po-

larization of the 739 nm light entering the iodine setup is set such that the light coupled

through the fiber EOM maximizes the power in the first-order sideband while minimizing

the power in the carrier frequency. Fluctuations in the polarization of the 739 nm light en-

tering the iodine setup will cause the optimized first-order sideband signal to decrease and

the carrier peak to emerge. We noticed this polarization instability when we were looking

at the transmitted FP signal. The amplitude of the sideband became noisy as we wiggled

iodine EOM’s fibers. We improved the polarization stability of the entering light by taping

down the EOM’s input and output fibers making sure the fibers secure but not under strain.

In addition, we used some optical clamps from Thorlabs to “hold” the EOM body itself to

the elevated optical breadboard. Lastly, we switched out the single-mode fiber delivering

the 739 nm light to the fiber EOM input with a polarization maintaining fiber and we made

sure that this fiber was secured well.

2. Removed the nonzero offset observed in the error signal. We discovered that a small

amount of scattered light from the pump beam was being measured by the balanced pho-

todetector. As a result, the iodine error signal had an overall offset from zero. We added

lens tubes about the reference and probe paths leading to the balanced photodetector. In

addition, black plates were placed about these paths in locations where space was limited.

Lastly, we made sure that the zeroth and other diffracted orders of the modulated 80 MHz

AOM were picked off from the main pump path and truly centered in the copper beam

dump.

3. Optimized the alignment of the pump and probe beams through the cell. We added a x2

beam expander which defocused the pump and probe beams inside the cell and increased

their beam sizes. Previously, the alignment of the pump beam inside the iodine cell crossed

38



the probe beam at a small angle. The beam expander increased the waists of the pump

and probe beams enabling their alignment to be fully overlapped inside the cell. Increasing

the spatial overlap between the two beams inside the cell helped to enhance the size of the

Doppler-free signal.

4. Increased the temperature of the iodine cell. Originally, the iodine cell was heated to a tem-

perature of 400 °C. After reviewing the iodine setups previously implemented in other labs

within the group, we realized that temperature of the cell could be heated to a higher tem-

perature as long as the value was below 600 °C. Increasing the cell’s temperature beyond

600 °C, one risked cracking the iodine cell. As a result, we slowly increased the temper-

ature of the iodine cell, and at each temperature increment, we observed the strength of

the error signal once the cell temperature reached equilibrium. We decided to keep the

iodine cell temperature heated to 560 °C since the largest error signal was observed at this

location, and we didn’t want to risk heating the cell temperature any higher.

After incorporating all of the changes made to the iodine setup, we obtained an improved

error signal that had a larger peak-to-peak amplitude, no offset from zero, and less noise. It was

easier to identify the correct zero-crossing, and once the iodine PID was engaged, the lock re-

mained locked for a longer period of time. However, despite having the PDH and iodine PIDs all

locked, thus locking the frequency of the 739 nm and 369 nm light, we still had the issue of iodine

frequency reference jumping from the second to the first positive slope zero-crossing observed

in Fig. 2.12. The doublet structure of this fifth feature had two positive slope zero-crossings

that were separated by about 15 MHz. When the iodine PID locked to the lower frequency zero-

crossing, the frequencies of 369 nm beams for cooling, optical pumping, and detection were all

red-detuned from their set values and the efficiency of these processes decreased. To solve this

problem, we decided to change the iodine feature that we used as the absolute reference. We

chose the fourth feature (“Feature 4”) appearing in the +13 GHz absorption line Doppler-free

39



signal as a function of detuning frequency [23]. Figure 2.13 shows the oscilloscope trace of the

unlocked iodine error signal in which the length of the PDH cavity has been scanned to show

Feature 4. Locking the PDH cavity length to the absolute frequency reference now corresponds

Figure 2.13. Feature 4 of the iodine error signal. The unlocked iodine error signal is the derivative
of the swept over the Doppler-free doublet feature in the +13 GHz absorption line. The PDH
cavity length is referenced to the zero-crossing of the positive slope in which only one exists.
When the Iodine PID is locked, the frequency of the 739 nm laser is stabilized at 405.644318 THz,
the value measured on the wavemeter.

to locking the iodine PID at the single zero-crossing that the feature possess. In switching from

the fifth to fourth feature in the spectroscopy of the +13 GHz absorption line, the iodine fiber

EOM RF frequency was changed from 13.315 GHz to 13.161 GHz. This change in the iodine

lock has made it more robust and corresponds to having only to lock once or twice a day the three

PIDs required to stabilize the 739 nm laser frequency.

The last major change to the iodine reference frequency lock that has occurred during my

time in QSim is the RF source used for the 80 MHz AOM. Originally, a HP8660C synthesized

40



signal generator along with its modules (HP86635A modulation selection and HP86602B RF

selection) was used to generate the frequency modulated RF signal for the iodine AOM. Unfor-

tunately, the modulation section of the RF unit stopped working in May 2021. We were able

to use a spare HP8640B, a standard RF source, to supply the 80 MHz frequency and apply the

11 kHz frequency modulation to the iodine AOM. However, the maximum peak deviation that the

HP8640B unit could supply was ±640 kHz (based on the electronic control box specifications).

In comparison, the maximum peak deviation of the HP8660C system was set at ±1.2 MHz. It

is important to note here that both the modulation bandwidth and intensity of the pump beam

influence the size of the error signal [23]. The bandwidth, fBW , of a FM signal is related to

the frequency of the modulation, fFM , and the maximum peak deviation, ∆f , from the center

frequency and is given by the following equation:

fBW = 2(∆f + fFM). (2.4)

Based on Eq. (2.4), the modulation bandwidth of the HP8640B RF source was slightly more than

half of the modulation bandwidth that was achieved with the now broken HP8660C RF source.

In order to compensate for the the reduced modulation bandwidth, we increased the amount of

power going to the 739 nm iodine fiber so that the power measured on the iodine breadboard

platform was twice the original value.

The solution of using a HP8640B to supply the RF signal to the iodine AOM with having

more power sent to the iodine setup worked temporarily. Diverting the increased amount of power

to the iodine system consequently reduced the amount of power being sent to the frequency dou-

bler corresponding to a reduced doubler output power (less 369 nm power). So instead of using a

HP8640B to provide the iodine AOM RF frequency signal, I specifically figured out the param-

41



eters required to use a voltage controlled oscillator (VCO) that could be used instead. Figure ??

shows the RF output at 80 MHz with and without the 11 kHz frequency modulation generated by

the HP8640B (left) versus a VCO (right) using a spectrum analyzer (Agilent E4440A) to mea-

sure the average trace “max-hold” signal. In Fig. 2.14a, the yellow trace shows the unmodulated

(a) RF signal from the HP8640B (b) RF signal from the ZOS-150+ VCO

Figure 2.14. Iodine AOM RF source signal comparison. (a)

80 MHz output while the blue trace displays the average maximum FM signal generated from the

HP8640B source. From the x-axis value of markers ‘4’ (80.60 MHz), the measured maximum

peak deviation is about 600 kHz, corresponding to a modulation bandwidth of ∼1.2 MHz.

In comparison, the yellow trace of the FM signal in Fig. 2.14b has a measured maximum

peak deviation of 3.6 MHz, corresponding to a modulation bandwidth of ∼7.2 MHz. An even

larger maximum peak deviation of the FM signal centered at 80 MHz could be achieved, indicated

by the pink trace in Fig. 2.14b, by tuning the control voltage signal that’s sent to the input of the

VCO (Mini Circuits part number: ZOS-150+). However, we chose to use the control voltage

settings that produced the VCO RF output with a maximum peak deviation of 3.6 MHz since this

peak deviation value was three times the original peak deviation produced by the HP8660 system.

The FM signal parameters that were sent to the control port of the VCO are displayed in Table 2.3.

42



Since implementing the VCO to send the RF signal to the iodine AOM, we have been able to

Type Frequency (kHz) Amplitude (Vpk−pk) Offset (Vpk−pk) Phase (◦)

sinusoidal 11.0 0.60 1.268 0.0

Table 2.3. Iodine VCO control signal parameters. The parameters of the FM signal sent into the
control port of the VCO are displayed. A Stanford Research System function generator was used
to generate this input signal. Vpk−pk represents the peak-to-peak voltage.

reduce the power measured on the iodine platform while still maintaining an adequate iodine

error signal in which the measured 739 nm locked laser frequency on the wavemeter fluctuates

less than 2 MHz.

2.3 Trapping Our Friends the Ions

The 935 nm beam size is much larger than the beam sizes of the 399 nm and all the 369 nm

laser beams. All three beams are combined on the other side of the 8020 wall. In a similar

manner, even though they have not been discussed yet, both Raman beams, named “Raman 1”

and “Raman 2” also enter and exit the chamber. Raman 1 is usually the beam 355 nm that is

used to load ions faster. In QSim, the vacuum chamb