ABSTRACT

Title of dissertation: EXPLORING QUANTUM MANY-BODY
SYSTEMS IN PROGRAMMABLE TRAPPED
ION QUANTUM SIMULATORS

Arinjoy De, Doctor of Philosophy, 2024

Dissertation directed by: Professor Christopher R. Monroe
Department of Physics

Quantum simulation is perhaps the most natural application of a quantum computer,

where a precisely controllable quantum system is designed to emulate a more complex

or less accessible quantum system. Significant research efforts over the last decade have

advanced quantum technology to the point where it is foreseeable to achieve ‘quantum

advantage’ over classical computers, to enable the exploration of complex phenomena in

condensed-matter physics, high-energy physics, atomic physics, quantum chemistry, and

cosmology. While the realization of a universal fault-tolerant quantum computer remains a

future goal, analog quantum simulators – featuring continuous unitary evolution of many-

body Hamiltonians – have been developed across several experimental platforms. A key

challenge in this field is balancing the control of these systems with the need to scale them

up to address more complex problems. Trapped-ion platforms, with exceptionally high

levels of control enabled by laser-cooled and electromagnetically confined ions, and all-

to-all entangling capabilities through precise control over their collective motional modes,

have emerged as a strong candidate for quantum simulation and provide a promising avenue

for scaling up the systems.



In this dissertation, I present my research work, emphasizing both the scalability and

controllability aspects of 171Yb+based trapped-ion platforms, with an underlying theme

of analog quantum simulation. The initial part of my research involves utilizing a trapped

ion apparatus operating within a cryogenic vacuum environment, suitable for scaling up to

hundreds of ions. We address various challenges associated with this approach, particularly

the impact of mechanical vibrations originating from the cryostat, which can induce phase

errors during coherent operations. Subsequently, we detail the implementation of a scheme

to generate phase-stable spin-spin interactions that are robust to vibration noise.

In the second part, we use a trapped-ion quantum simulator operating at room tem-

perature, to investigate the non-equilibrium dynamics of critical fluctuations following a

quantum quench to the critical point. Employing systems with up to 50 spins, we show that

the amplitude and timescale of post-quench fluctuations scale with system size, exhibiting

distinct universal critical exponents. While a generic quench can lead to thermal critical

behavior, a second quench from one critical state to another (i.e., double quench) results

in unique critical behavior not seen in equilibrium. Our results highlight the potential of

quantum simulators to explore universal scaling beyond the equilibrium paradigm.

In the final part of the thesis, we investigate an analog of the paradigmatic string-

breaking phenomena using a quantum spin simulator. We employ an integrated trapped-ion

apparatus with 13 spins that utilizes the individual controllability of laser beams to program

a uniform spin-spin interaction profile across the chain, alongside 3-dimensional control of

the local magnetic fields. We introduce two static probe charges, realized through local lon-

gitudinal magnetic fields, that create string tension. By implementing quantum quenches

across the string-breaking point, we monitor non-equilibrium charge evolution with spatio-

temporal resolution that elucidates the dynamical string breaking. Furthermore, by initial-

izing the charges away from the string boundary, we generate isolated charges and observe

localization effects that arise from the interplay between confinement and lattice effects.



EXPLORING QUANTUM MANY-BODY SYSTEMS IN
PROGRAMMABLE TRAPPED ION QUANTUM SIMULATORS

by

Arinjoy De

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

2024

Advisory Committee:
Professor Christopher R. Monroe, Chair/Advisor
Professor Alexey V. Gorshkov
Professor Zohreh Davoudi
Professor Norbert M. Linke
Professor Christopher Jarzynski



© Copyright by
Arinjoy De

2024



Dedication

To my family

ii



Acknowledgments

First and foremost, I would like to express my sincere gratitude to my advisor, Pro-

fessor Chris Monroe, for his unwavering support, guidance, and patience throughout my

Ph.D. journey. His expertise and encouragement have been invaluable to my growth as a

researcher. I greatly appreciate his steadfast support and mentorship during my transitions

between different labs and relocations across states.

I also extend my heartfelt thanks to the members of my dissertation committee, Pro-

fessor Alexey V. Gorshkov, Professor Zohreh Davoudi, Professor Norbert M. Linke, and

Professor Christopher Jarzynski, for their insightful feedback and constructive criticism,

which significantly enhanced the quality of my research.

Research in experimental labs is certainly a group effort, and this was especially true

for my Ph.D. work involving three different labs. Special thanks go to Will Morong, and

Lei Feng, for your scientific passion, responsible leadership, and inspiring mentorship. I

have been fortunate to work with a pool of exceptional postdocs and fellow graduate and

undergraduate students. Guido Pagano, Wen Lin Tan, and Harvey Kaplan, I am forever

indebted to you for helping me take my first steps in the trapped-ion quantum simulation

lab. I would also like to thank Abhishek Menon and Albert Chu for your contributions in

Cryo-Qsim lab. Kate Collins and Will Morong, thank you for being such amazing labmates

in the Warm-Qsim lab. I greatly benefitted from our productive scientific discussion and

collective belief in the show must go on approach, despite multiple challenges, including

a biblical flood! Or Katz, Lei Feng, Marko Cetina, and Kee Wang, thank you for your

support during my onboarding in Blue-system lab. Thank you Henry Luo for your constant

support through the long hours in the lab during the final phase of my research.

The list above is only a fraction of the people in the Monroe group who have contributed

to my development as a researcher. To name a few, I would like to express my heart-

felt gratitude to Antonis Kyprianidis, Patrick Becker, Daiwei Zhu, Laird Egan, Michael

iii



Goldman, Crystal Noel, Drew Risinger, Allison Carter, Ksenia Sosnova, Marty Lichtman,

George Toh, Jameson O’Reilly, Debopriyo Biswas, Sagnik Saha, Isabella Goetting, Alex

Kozhanov, Liudmilla Zhukas, Vivian Zhang, Bahaa Harraz, Yichao Yu, Mikhail Shalaev,

Emma Stavropoulous, Keqin Yan, Ashish Kalakuntala, and Rob Cady. I enjoyed interact-

ing with you even though I may not have directly worked with you. Additionally, I would

like to thank our excellent collaborators, Patrick Cook, Paraj Titum, Mohammad Maghrebi,

Alessio Lerose, Federica M. Surace, Elizabeth R. Bennewitz, and Alex Schuckert, for al-

ways coming up with the smartest proposals and patiently explaining the intricate concepts

of quantum many-body systems and lattice gauge theories to us.

Special thanks to the faculty and staff of the Physics Department and JQI at the Uni-

versity of Maryland and the Duke Quantum Center at Duke University for providing a

stimulating academic environment and the resources necessary for my research.

When research is not going where you want, or as fast as you would like, it’s vital to

maintain other sources of happiness, having friends and family proves to be quite valuable.

I am fortunate to be blessed with friends such as Sohitri Ghosh, Saurav Das, Spandan

Pathak, Tamoghna Barik, Rajat Roy, Kishalay Mahato, Sagnik Saha, Debopriyo Biswas–

grad school would not have been the same without our impromptu hiking trips, road trips

and venting sessions. To all the numerous friends from Maryland and Durham who I could

not mention here but have been part of my journey, a big thank you to all.

I have to especially mention Kanika for being a constant source of love, encouragement,

and support during the ups and downs of the last few years of my Ph.D. I greatly appreciate

all the sacrifices you have made for me. I am incredibly lucky to have you in my life. Thank

you!

Last but not least, I am deeply indebted to my family for their love, patience, and

unwavering belief in me. To my parents, Malay De and Mandira De, my brother Niloy De,

and sister-in-law Sahana Roy, thank you for your endless encouragement and support.

iv



Table of Contents

Dedication ii

Acknowledgements iii

1 Introduction 1
1.1 Quantum Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Quantum Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Quantum simulation of High energy physics models . . . . . . . . . 6
1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Quantum Simulation with Trapped Ions 10
2.1 Introduction to Paul Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 The 171Y b+ ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Photoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Doppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.3 State Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.4 State Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Coherent Operations with Qubits . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Raman Transition in a Λ level configuration . . . . . . . . . . . . . 28
2.3.2 Two-photon AC Stark Shift . . . . . . . . . . . . . . . . . . . . . . 32
2.3.3 Quantization of Motion . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.4 Generating the Ising interaction . . . . . . . . . . . . . . . . . . . 37
2.3.5 Transverse Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3.6 Longitudinal field . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.3.7 Undesired Phonon excitations . . . . . . . . . . . . . . . . . . . . 44

2.4 Extension to Pulsed laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.1 4-photon Stark shift . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Phase-Stable Quantum Simulation in A Cryogenic Trapped-Ion Apparatus 48
3.1 Overview of the Cryogenic Quantum Simulator . . . . . . . . . . . . . . . 48
3.2 The “Bad” Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

v



3.3 Phase Noise due to Vibration . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Beatnotes for Phase-sensitive and Phase-insensitive Schemes . . . . . . . . 53
3.5 Telecentric Optical path design . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5.1 Modified beat-note stabilization . . . . . . . . . . . . . . . . . . . 60
3.6 Characterization of the Phase-insensitive Scheme with Parity Scan . . . . . 62
3.7 Benchmarking Phase-sensitive vs -insensitive schemes . . . . . . . . . . . 65

3.7.1 Ramsey experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7.2 2-ion Mølmer-Sørensen evolution . . . . . . . . . . . . . . . . . . 68
3.7.3 Effects of Intensity Noise . . . . . . . . . . . . . . . . . . . . . . . 70

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 Ground-state Cooling with a Tripod EIT Scheme 75
4.1 EIT cooling with 171Yb+ion . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Benchmarking EIT Cooling with Single Ion . . . . . . . . . . . . . . . . . 80
4.3 EIT Cooling with Long Chains . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3.1 Motion-Sensitive Carrier Rabi Flopping . . . . . . . . . . . . . . . 82
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Non-equilibrium Critical Scaling and Universality in a Quantum Simulator 86
5.1 Overview and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Introduction of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3 The Experimental System . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3.1 Generating XX and YY Interactions . . . . . . . . . . . . . . . . . 93
5.3.2 Challenges of Trapping Large Ion Chains . . . . . . . . . . . . . . 94
5.3.3 Optimizing Ji j Profiles for Long Ion Chains . . . . . . . . . . . . . 97

5.4 Dynamics after a single quench . . . . . . . . . . . . . . . . . . . . . . . . 100
5.5 Finding Critical Exponents after Single Quench . . . . . . . . . . . . . . . 103
5.6 Critical exponents after Double Quench . . . . . . . . . . . . . . . . . . . 106

5.6.1 Experimental Error Sources . . . . . . . . . . . . . . . . . . . . . 108
5.6.2 Double Quench Switch Times . . . . . . . . . . . . . . . . . . . . 109
5.6.3 Extracting scaling expoenent . . . . . . . . . . . . . . . . . . . . . 110
5.6.4 Jackknife error estimation: . . . . . . . . . . . . . . . . . . . . . . 111

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.8 Engineering a Dynamical Decoupling Sequence for Trapped-ion Quantum

Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.8.1 Principles of Dynamical Decoupling . . . . . . . . . . . . . . . . . 112
5.8.2 Example Dynamical Decoupling Sequences . . . . . . . . . . . . . 115

Example 1: CPMG sequence . . . . . . . . . . . . . . . . . . . . . 116
Example 2: XY sequence . . . . . . . . . . . . . . . . . . . . . . . 117

5.8.3 Two-ion Tests of Decoupling Sequence . . . . . . . . . . . . . . . 118
5.8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6 Real-time Dynamical String Breaking in Trapped Ions 122

vi



6.1 Experimental Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2 Experiment calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2.1 Amplitude calibration . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2.2 Motional frequency calibration . . . . . . . . . . . . . . . . . . . . 128
6.2.3 Stark Shift Compensation . . . . . . . . . . . . . . . . . . . . . . . 130
6.2.4 Infrequent Ramsey calibrations . . . . . . . . . . . . . . . . . . . . 133

6.3 Programming the spin-spin interaction Hamiltonian . . . . . . . . . . . . . 136
6.3.1 Transverse field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.3.2 Longitudinal field calibration . . . . . . . . . . . . . . . . . . . . . 142

6.4 Probing String-Breaking Dynamics . . . . . . . . . . . . . . . . . . . . . 146
6.4.1 Introduction of the model . . . . . . . . . . . . . . . . . . . . . . . 147
6.4.2 Ground state String Breaking . . . . . . . . . . . . . . . . . . . . 150
6.4.3 Effect of probe charges . . . . . . . . . . . . . . . . . . . . . . . . 151
6.4.4 Non-equilibrium dynamics in the presence of Static charges . . . . 153
6.4.5 Dynamics of Isolated Charge . . . . . . . . . . . . . . . . . . . . . 155

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7 Summary and Outlook 160

Bibliography 162

vii



Chapter 1: Introduction

1.1 Quantum Computers

Information processing technology has profoundly transformed modern society, per-

meating all facets of daily life–from accessing the internet through smartphones to the

analysis of complex global economic and climate issues, and even the exploration of outer

space via spacecraft–all critically depend on systems capable of processing information

with high accuracy and efficiency. Countries and corporations worldwide have invested

substantial resources in constructing supercomputing clusters to tackle the toughest prob-

lems and organize vast quantities of data. Nevertheless, even with these so-called “clas-

sical” computers, which can have hundreds of thousands of processing cores, billions of

transistors, and consume megawatts of energy, there are problems that would require more

energy than exists in Earth’s fossil fuels and take longer than the age of the universe to

solve. The quest for greater computational power drives the miniaturization of semicon-

ductor chips, but as Moore’s Law predicts, packing billions of transistors will soon reduce

the fundamental computational bits to atomic scales. As Richard Feynman famously an-

ticipated in 1959 [1], the microscopic world of atoms can offer much more “room” for

information processing. In this domain, the operational principles of these information

processors are governed by quantum physics, and naturally these advanced machines are

known as quantum computers.

The foundations of quantum mechanics were established about a century ago, in 1926

1



by Erwin Schrödinger [2]. The fundamental building blocks of a quantum computer are

quantum bits, or qubits. Qubits are the physical embodiment of classical bits, represented

by the states |0⟩ and |1⟩. Unlike classical bits, qubits can exist in a the supersposition of

states, expressed by [3],

|ψ⟩= α|0⟩+β |1⟩; |α|2+|β |2= 1, (1.1)

with α and β representing the probability amplitudes of the qubit being in the |0⟩ and |1⟩

states, respectively. To describe a two-qubit system, four parameters are required, and in

general, an N-qubit system necessitates 2N parameters. This exponential scaling underlies

the immense computational potential of quantum computers. Qubit states can be manipu-

lated through quantum operations that transform an initial qubit state into a different state.

These operations can generate an entangled state represented by, for example,

|Ψ⟩= 1√
2
(|00⟩+ |11⟩) , (1.2)

in a two-qubit system. Measurement of one qubit in an entangled pair immediately deter-

mines the state of the other qubit, regardless of the distance separating them. This specific

entangled pair is one of the four so-called Bell states [4] and the concept can easily be ex-

tended to multiple qubits as well. Entanglement provides the ability to efficiently hold and

process information in a quantum computer. The experimental verification of entanglement

using a Bell state was a significant milestone in the development of the field of quantum

computing, which was recognized with the award of the Nobel prize in 2022 [5, 6].

Despite their promising advantages over classical computing through superposition and

entanglement, quantum computers face significant intrinsic challenges. Quantum states are

extremely fragile, and the interaction of a qubit with its environment can cause decoherence

2



or the loss of information stored in the qubit. Maintaining coherent operations necessitates

the development of quantum error-correcting protocols, which remain an active area of

research in both theory and experiment [7–10].

Building upon Feynman’s vision of “simulating physics with computers” [11], the past

few decades have witnessed remarkable advancements in isolating and controlling quan-

tum objects. Researchers have employed a diverse range of particles and control tech-

niques, from ions trapped with electric fields [12–14], electrons in superconducting met-

als [15, 16], to atoms and molecules cooled with lasers and magnetic fields, sometimes

achieving quantum degeneracy [17, 18], and other times using trapping potentials like op-

tical lattices [19, 20]. Trapped ion platforms emerged early as a candidate for quantum

computers, demonstrating scalable qubit arrays, precise initialization, and measurement of

the qubits, long coherence times, and a universal set of logical quantum gates [21, 22], thus

satisfying the DiVincenzo criteria for a universal quantum computer [23]. The building

blocks for quantum computing have been demonstrated with few qubits in different phys-

ical platforms [15, 24–26] as well. While the realization of quantum algorithms such as

Shor’s algorithm [27] and Grover’s search algorithm [28] with many logical qubits remain

beyond the current capabilities of the experimental quantum computing platforms, these

systems hold significant promise for efficiently solving various quantum algorithms [29–

31]. There is considerable room for improving the level of control over these systems.

Inherently, a tension exists between isolation and control, as systems that hold large num-

bers of particles generally have less control over individual particles, whereas platforms

with highly developed control for each component tend to have fewer such components.

Overcoming this tension is critical for advancing the practical implementation of quantum

computing technologies.

While a fully controllable fault-tolerant quantum computer is a long-term goal is it

possible to build on the experimental advances made so far, and construct a device not

3



quite at the level of complexity of a quantum computer, but one that still could perform

some tasks that classical devices cannot? The answer lies in the quantum simulators which

is the ‘native’ and most natural application of quantum computers, where we aim to use a

quantum computer to mimic the rules that describe physical microscopic quantum systems.

Many of the short-term promising applications of quantum computers can be achieved with

systems operating with 50-100 qubits [16, 31–39] and demonstrate an advantage over their

classical counterpart [40].

1.2 Quantum Simulators

Understanding the collective behavior of interacting quantum particles has been a focus

of attention for many years, due to its relevance in a wide variety of fields ranging from

nuclear physics to the design and characterization of materials. Quantum simulators are

developed to mimic the properties of a quantum system using another more controllable

and/or accessible quantum system. As mentioned earlier, the amount of memory required

for representing quantum systems grows exponentially with the system size, and so does the

number of operations required to simulate the time evolution. The evolution of a quantum

state |Ψ⟩ acted upon by a Hamiltonian H can be reconstructed by piecewise application of

a series of Hamiltonians Hi following,

e−iHt ≈
(
e−iH1te−iH2t · · ·e−iHNt) (1.3)

where H = ∑i Hi. This decomposition is known as the Trotter formula [41] and a break-

down into finer time steps results in a more accurate description of the evolution under the

original Hamiltonian. In 1996, Seth Lloyd pointed out [42] that evolving in small time steps

would allow the complexity of the simulation to grow only polynomially with the number

4



of particles to be simulated as opposed to exponentially. Such a quantum simulator, real-

ized by unitaries implemented with a set of universal quantum operations, is referred to as

a digital quantum simulator. Although a digital simulator is applicable to study a broader

class of problems, trotterized evolution requires high-fidelity gate operations, and achieving

the latter for hundreds of qubits remains out of reach for modern quantum computers.

However, NISQ (Noisy Intermediate-Scale Quantum) devices, available with current

technology, provide another avenue for quantum simulators [31]. These so-called ana-

log quantum simulators directly implement the many-body Hamiltonian H of a model and

measure the unitary evolution, typically after a quantum quench, i.e., turning on the Hamil-

tonian faster than the relevant timescales of the system [43, 44]. While analog simulators

may not be universal, one of their major advantages is the ability to scale to large system

sizes, making them a natural frontier for achieving practical quantum advantage relative

to classical simulations. Various experimental platforms enable the realization of analog

quantum simulations, including, but not limited to, neutral atoms [20, 45, 46], supercon-

ducting systems [47, 48], trapped ions [49, 50], and photons [51, 52]. These platforms

allow the study of problems across multiple fields, including condensed-matter physics,

high-energy physics, cosmology, atomic physics, and quantum chemistry.

The work in this thesis focuses on analog quantum simulators using trapped 171Yb+ions.

The qubit is encoded in the stable hyperfine ground state of the ions, which are localized

using an oscillating electric field and further manipulated using laser or microwave fields.

Coherent operations are performed using the collective motional modes of the ions, and

by carefully choosing laser parameters, we can apply a spin-dependent Hamiltonian that

efficiently emulates an Ising Hamiltonian [53]:

H = ∑
i, j

Ji jσ
x
i σ

x
j , (1.4)

5



The interaction matrix (Ji j) represents a long-range coupling Ji j ∼ J/rα where r = |i− j|

is the separation between ions i and j and the exponent α is tunable between 0 and 3.

The ability to add a three-dimensional magnetic field (see Chapter 2) along with the Ising

interaction opens up the possibility of simulating interesting many-body models. While

trapped-ions feature excellent controllability over the simulation parameters, one of the

major challenges faced by this platform is scaling up the system sizes. In Chapter 3, we

discuss an approach to scale up the system size using a cryogenic environment. In Chapter

4, we report our work on studying critical behavior near a many-body phase transition with

up to 50 ions.

1.2.1 Quantum simulation of High energy physics models

In the past several years, rapid progress in the capabilities of quantum simulators has

generated strong excitement in exploring the fundamental laws of nature using these de-

vices. Interactions between the elementary constituents of matter are described by gauge

theories [54]. These theories describe the interaction between fundamental gauge and

matter fields, consistent with a set of local symmetries. The standard model of particle

physics involves local gauge symmetries of SU(3)× SU(2)×U(1) corresponding to the

strong (quantum chromodynamics), weak, and electromagnetic (quantum electrodynam-

ics) interactions respectively [55, 56]. Perturbative theoretical techniques have been applied

with great success to study quantum electrodynamics (QED) and chromodynamics (QCD)

at high energies. However, in the strong-coupling regime of quantum chromodynamics

(QCD), a non-perturbative approach is necessary [57, 58]. The lattice gauge theory (LGT)

was originally invented by Wilson as a tool for the study of the quark confinement prob-

lem [59]. These theories are formulations of gauge theories on a discretized space and/or

time [60, 61]. LGTs have been traditionally studied via Quantum Monte Carlo (QMC)

6



sampling techniques [62]. These techniques are inherently limited due to the infamous

sign problem where weights of different configurations in the QMC simulation become

negative or even complex [63, 64]. Furthermore, QMC methods fall short of shedding light

on the real-time dynamics of strongly interacting matter.

Quantum computers hold the promise of simulating real-time processes from first prin-

ciples. Currently, various proposals for quantum simulations of LGTs on different exper-

imental platforms have been put forward [65–73]. One of the first successful quantum

simulations of LGTs was achieved using trapped ions [74], where the real-time dynamics

of the Schwinger model was studied using a digital simulator. Despite this initial success,

shallow circuit depths, limited by the finite coherence time of the digital simulators have

hindered the study of long-time evolution [75]. Consequently, there has been a strong focus

on the fully analog simulation of LGT models [68, 70, 76, 77], which are now being ex-

perimentally realized [78, 79]. In the final part of this thesis, we explore the capabilities of

a trapped-ion analog quantum simulator with control over individual spin-spin interactions

to study the analog of the paradigmatic problem of quark confinement, which forbids the

existence of free quarks and rather binds them together into composite particles [59].

Evidence from heavy-ion and proton-collision experiments at the Relativistic Heavy

Ion Collider (RHIC) and the Large Hadron Collider (LHC) suggests that strongly inter-

acting matter thermalizes in a remarkably short period of time, possibly due to quantum

entanglement plays a role in the equilibration process [80–82]. The ability of quantum

simulations to perform real-time evolution can potentially reveal valuable insights into the

approach to equilibrium in QCD along with unique properties of strongly interacting quan-

tum many-body systems, that are prohibitively difficult with classical computers.

7



1.3 Thesis Overview

My PhD journey has a unique aspect: I worked in three different laboratories across two

universities (one at a time). The majority of my PhD was spent at the Joint Quantum In-

stitute (JQI) at the University of Maryland, College Park, where I worked in the Cryogenic

and room-temperature Quantum Simulation labs known as Cryo-QSim and Warm-QSim

labs, respectively. The final portion of my research work was carried out at the Duke Quan-

tum Center, Duke University, in the Blue-System lab, previously known as the EURIQA

lab. In this thesis, I will describe the research work I performed in these labs, unified by

the underlying theme of analog quantum simulations.

• Chapter 2 introduces the toolbox for quantum simulation with trapped ions, including

a review of ion traps, cooling, preparation, and measurement procedures of atomic

states. It also develops the framework to understand the principles of coherent opera-

tions with qubits, primarily assuming continuous wave lasers but extending to pulsed

lasers which are used in experiments.

• Chapter 3 details the research performed in the Cryo-QSim lab, which features a

macroscopic blade trap operating in a cryogenic vacuum chamber with global ad-

dressing optics for all the spins. Cryogenic technology provides a suitable way to

scale up quantum systems. Next, I review the advantages and challenges of operat-

ing with a cryostat, particularly the effect of mechanical vibration, which introduces

phase noise during coherent operations. The chapter then details the method of im-

plementing a phase-stable quantum simulation protocol and necessary modifications

to the experiment. Finally, I benchmark the performance of the phase-insensitive pro-

tocol against the phase-sensitive protocol in the presence of extreme vibration noise.

Additionally, analog quantum simulation experiments performed in this lab have re-

8



sulted in several publications [78, 83, 84], which are described in detail elsewhere.

• Chapter 4 describes the implementation of the Electromagnetically induced trans-

parency (EIT) cooling method for large chains of 171Yb+ions [85] in the same

Cryogenic apparatus. Unlike conventional EIT cooling, we engage a four-level tripod

structure and achieve fast sub-Doppler cooling over a large bandwidth of motional

modes. I describe the implementation and benchmarking of our scheme with a single

ion and then test the performance of the EIT cooling method with a chain of 40 ions.

• Chapter 5 covers the work conducted in the Warm-QSim lab. After providing a brief

overview of the apparatus, which features a three-layer macroscopic trap in a room-

temperature vacuum system, I highlight the ability to engineer global spin-spin in-

teraction matrices with up to 50 ions. The chapter then discusses the experimen-

tal protocol to observe nonequilibrium critical scaling after a sequence of quantum

quenches, showcasing the ability to explore quantum many-body systems beyond the

scope of classical computation [86]. At the end of the chapter, I provide a short de-

scription of the dynamical decoupling method that utilizes a repeated external drive

to improve the coherence of the many-body quantum simulators [87]. We imple-

ment and benchmark this technique and employ it in the experiment described in the

subsequent chapter.

• Chapter 6 details the research performed in the Blue-System lab, which features a

microfabricated chip trap with individual control of spin-spin interactions. I discuss

a procedure to exploit this individual control for programming uniform Ising inter-

actions, along with a magnetic field that can be arbitrarily orientated in the Bloch

sphere. These advanced capabilities facilitated the first experimental observation of

real-time dynamical string-breaking phenomena using a trapped ion quantum simu-

lator. The manuscript of these results is currently under preparation [88].

9



Chapter 2: Quantum Simulation with Trapped Ions

Trapped ions offer a promising platform for the development of practical quantum com-

puters. A fundamental step in utilizing trapped ions is to precisely localize the ions in three

dimensions, which is essential for achieving accurate control. However, Earnshaw’s theo-

rem poses a significant caveat by stating that electrostatic fields alone are insufficient for

confining a charged particle in three dimensions. This theorem is rooted in Gauss’s law,

which asserts that the divergence of any electric field in free space is zero, i.e., ∇⃗ · E⃗ = 0.

Consequently, electric field lines converging towards a point must also diverge away from

it, thereby preventing a stable equilibrium for charged particles.

To circumvent this limitation, two primary methods for confining charged particles have

been developed: the Radio Frequency (RF) Paul trap and the Penning trap. Wolfgang Paul,

who was awarded the Nobel Prize in Physics in 1989, invented the Paul trap, which employs

a combination of static and dynamic electric fields to trap ions [89, 90]. The research

presented in this thesis utilizes linear RF Paul traps, which are particularly suitable for

trapping linear ion chains. These chains can be individually manipulated using laser beams,

providing the precise control necessary for quantum computing applications.

The Penning trap, first conceptualized by Frans Penning and later constructed by Hans

Dehmelt—who shared the Nobel Prize with Wolfgang Paul—utilizes a combination of

electromagnetic fields to confine a charged particle in two dimensions via the Lorentz force,

while an inhomogeneous static electric field provides confinement along the third dimen-

sion [91]. Penning traps are highly effective for trapping large ion crystals, and numerous

10



research groups are actively exploring their potential [92].

2.1 Introduction to Paul Trap

A general three-dimensional (3D) quadrupole potential can be expressed as,

Φ(x,y,z) =
V
2
(
1+αx2 +βy2 + γz2) , (2.1)

where V is the electric potential created by a specific electrode configuration. According to

Gauss’s law, the potential must satisfy Laplace’s equation,

∇
2
Φ =

V
2
(2α +2β +2γ) = 0, (2.2)

which requires α + β + γ = 0. Disregarding the potential along x-axis, we can satisfy

Eq. 2.2 by setting γ = 0, β = −α . Under this constraint, we consider the specific case of

a linear Paul trap. In the later part we shall see that, to achieve three-dimensional confine-

ment, an electrostatic field is applied along the z-axis, referred to as the axial direction. The

potential can then be rewritten as

Φ(x,y,z) =
V
2
(
1+ x2 − y2) . (2.3)

which reveals that this static potential is confining in the y-direction but deconfining along

the z-direction.

A way around this is to make the potential amplitude V time-dependent such that the

time-averaged force experienced by an ion becomes a confining force. Consider two elec-

trodes separated by a distance d. The one-dimensional potential between them is homoge-

neous but varies with time as V cos(Ωr f t). The force on a charged particle with charge e

11



and mass m is given by:

F(t) = mẍ = e
V
d

cosΩr f t. (2.4)

The position of the particle can be determined as follows,

x(t) = x0 −
eV x0

mΩ2
r f d

cosΩr f t, (2.5)

where x0 represents the time-averaged position of the particle over many periods of the

oscillating field. Considering the inhomogeneous potential described by Eq. 2.3 along the

x-axis, the force on the particle becomes:

F(t) = eV xcosΩr f t. (2.6)

Under the assumption that the field is oscillating rapidly, such that the ion moves only a

small distance during one RF cycle, we can substitute Eq. 2.5 above to obtain:

F(t) = eV

(
x0 −

eV x0

mΩ2
r f d

cosΩr f t

)
cosΩr f t (2.7)

= eV x0 cosΩr f t −
e2V

mΩ2
r f d

(
cosΩr f t

)2
. (2.8)

Averaging over one RF period yields,

F̄ =− e2V
2mΩ2

r f d
x0. (2.9)

This represents a restoring force acting on the particle’s average position, known as the

ponderomotive force. The particle’s motion undergoes simple harmonic oscillation with a

12



secular frequency given by:

ωsec =
eV√

2mΩr f
. (2.10)

It is noteworthy that the ponderomotive force is independent of the charge’s sign, allowing

both positively and negatively charged particles to be trapped using this method.

We now extend our analysis to a two-dimensional quadrupole RF potential in the x-

y plane, accompanied by a static DC component that provides confinement along the z

direction, represented by:

Φ(x,y,z, t) =
VRF

2
cosΩRFt

(
1+

x2 − y2

R2

)
+VDC

(
z2 − x2 + y2

2

)
. (2.11)

Such a potential can be achieved using hyperbolic electrodes, where opposite electrodes—separated

by d = 2R—are grounded, and RF voltage with amplitude VRF and frequency ΩRF is ap-

plied to the remaining electrodes. Additionally, endcap electrodes at voltage VDC provide

the axial confinement.

The Coulomb force exerted on ions with charge q is given by:

F =−q∇Φ(x,y,z, t) (2.12)

=−qVRF cosΩRFt
(

xx̂− yŷ
R2

)
+qVDC (xx̂+ yŷ−2zẑ) . (2.13)

The equation of motion for the ion can thus be derived as:

ẍ+
[

qVRF

mR2 cosΩRFt − qVDC

m

]
x = 0 (2.14)

ÿ+
[
−qVRF

mR2 cosΩRFt − qVDC

m

]
y = 0 (2.15)

z̈+
2qVDC

m
z = 0. (2.16)

13



From these equations, we observe that the ion undergoes harmonic motion along z, indicat-

ing axial confinement. The equations of motion along the x-y plane are uncoupled and take

the general form of the Mathieu equation:

d2u
dζ 2 +[au +2qu cos2ζ ]u = 0. (2.17)

We can identify the constants of Eq. 2.16 with the generalized Mathieu equation as:

ζ =
ΩRFt

2
; qx =

2eVRF

mR2Ω2
RF

; ax =
4eVDC

mR2Ω2
RF

, (2.18)

A detailed solution to the Mathieu equations is discussed in [93, 94]. Here, we present the

final result:

x(t) = Acos(ωxt)
[
1+

qx

2
cos(ΩRFt)

]
, (2.19)

where ωx is the secular frequency from Eq. 2.10. Motion along y-axis follows a similar so-

lution. The equation of motion exhibits a component oscillating at the RF driving frequency

ΩRF , referred to as micromotion. Micromotion introduces additional motional sidebands,

reducing the efficiency of Doppler cooling and state detection, and impacting the effective

simple harmonic model of trapped ion systems. Notably, when VRF = 0, qx = 0 and micro-

motion is eliminated in Eq. 2.19, the ion to undergoes simple harmonic motion in the x-y

plane with secular frequency. The region where VRF = 0 is termed as the RF null. Depend-

ing on the geometry of the trap, the RF null may be a point, a collection of discrete points,

or a line. Stray charges can displace ions from the RF null, causing excessive micromo-

tion. Various techniques, including applying additional DC voltages, are implemented to

maintain ions in the RF null region [95]. Intuitively, micromotion effects can be mitigated

by ensuring qx ≪ 1, dependent on trap voltage (VRF ), electrode spacing (R), and ion mass

(m).

14



So far, we have examined hyperbolic trap geometry for mathematical simplicity. How-

ever, manufacturing hyperbolic traps proves to be quite inconvenient in practice. Over the

past decade, trapped ion technology has witnessed rapid advancements in performance,

robustness, and versatility. Various ion trap designs have been developed based on dif-

ferent applications, as detailed in [96], broadly categorized into two classes: macroscopic

traps and micro-fabricated traps. Macroscopic traps typically comprise four rods or blades

providing RF and DC connectivity. They are often hand-assembled, leading to potential

non-repeatability. On the other hand, micro-fabricated traps utilize precise machining tech-

niques for integrated circuit designs on silicon chips, specifically tailored for ion trapping

applications. During my tenure in the Monroe Group, I have worked in labs employing

both types of traps. In the following, I will provide a brief overview of each type.

Blade trap The Cryogenic Quantum Simulation (Cryo Qsim) laboratory in the Monroe

group employs a blade trap capable of trapping more than 120 ions [98]. True to its name,

this trap comprises four electrodes shaped like blades, affixed to a sapphire holder in a

60◦/30◦ angle configuration. This arrangement ensures excellent optical access both within

the x− y plane and along the vertical z-direction (refer to Fig.2.1). The blades are made of

alumina and coated with 1 µm gold layer. They were manufactured at the Sandia National

Laboratories and hand-assembled with ∼ 5µm resolution in the laboratory. The distances

between electrode tips measure 340 µm/140µm, with an ion-electrode distance of 180 µm.

Of the four blades, two diagonally opposite ones are grounded, while the remaining blades

are driven by an RF voltage source. We empirically found that, applying an RF voltage

of amplitude about VRF = 480 V and frequency ΩRF = 2π × 24 MHz, yields a secular

frequency of ωsec = 2π × 4 MHz. The blades can be further segmented (five segments in

this example) to allow precise control over the static electric fields.

15



a) Blade Trap

b) Drawing of a Blade Trap

Figure 2.1: (a.) Image of a segmented blade trap used in the Cryo Qsim lab. (b) Computer-aided
Design of a blade trap. Image courtesy of [97]

a b

Figure 2.2: (a.) Image of the 3-layer macroscopic trap used in the Warm Qsim lab. (b). Schematic
diagram of the electrodes along with physical axis reference. Most of the laser beams are incident
at an angle from above or below the page.

16



3-layer Paul trap A 3-layer rf Paul trap was used in the Warm QSim laboratory of the

Monroe Group. The trap features three layers of electrodes the central RF layer of the trap

provides pseudopotential confinement in the x− y-plane, while the outer two layers, each

containing six electrodes, facilitate static axial (z) confinement and electric field compen-

sation (refer to Fig. 5.2). One significant advantage of this geometry is its capability to

independently rotate the trap’s principal axes and nullify micromotion effects [99, 100].

Due to restricted optical access, most lasers used in this experiment traverse the trapping

region at 45◦ angle between the x and z axes (refer to Fig. 5.2). The RF electrodes of

this three-layer trap operate at ΩRF = 2π × 38.8 MHz. The drive frequency is generated

by an HP 8640B, amplified to +25 dBm (approximately 300 mW), and transmitted into

a helical quarter-wave resonator. Typically, this setup yields a secular frequency of about

ωsec = 2π ×4.7 MHz. For a comprehensive overview of trap parameters and control mech-

anisms, please refer to Rajibul Islam’s thesis [101].

Microfabricated trap Among the array of trap designs available, the microfabricated

trap, often referred to as the “chip trap”, stands out as one of the most technologically ad-

vanced and versatile options. By capitalizing on the well-established CMOS manufacturing

processes from the silicon chip industry, two-dimensional chips with multiple segmented

electrodes can be precisely fabricated. For a comprehensive understanding of the opera-

tional principles of a chip trap, I recommend consulting the following articles [103, 104].

The Blue-system, previously known as the EURIQA laboratory, employs the High Optical

Access Trap 2.0 (HOA2.0), a popular trap developed and distributed by Sandia National

Laboratory [105] (See Fig.2.3a). This trap features a ≈ 3 mm linear slot in the middle of

the trap, designated as the “quantum region”, where quantum computing experiments are

performed and all laser beams for these operations are precisely aligned to the “quantum

region”. As shown in Fig. 2.3b, on each end of the quantum region, there is a Y-junction

17



a)

b)

Figure 2.3: (a.) Image of the HOA 2.1.1 microfabricated chip trap used in the Blue-systems formerly
known as the EURIQA laboratory. (b). Schematic diagram of the loading slot, Y- junction, and the
quantum region. Image courtesy of [102]

that splits into two arms each. Each of the four arms has a central slot known as the “load

slot”. The trap comprises 94 independent DC control electrodes enabling ion shuttling and

transportation from the load slot to the quantum region.We apply VRF = 220 V to the RF

electrodes at a frequency of ΩRF = 2π ×36.06 MHz, resulting in a radial secular frequency

of ωsec = 2π ×3.155 MHz. The RF null is positioned 68 µm above the surface of the near-

est metal electrode. However, one drawback of the chip trap is its small structure, which

imposes limitations on the application of high RF voltage to the trap electrodes (< 300

V). Lower RF voltages render the trapping potential shallow, increasing susceptibility to

background gas collisions. Furthermore, ions are typically trapped in close proximity to

the electrode surface (approximately 68 µm above the surface in this trap), increasing the

sensitivity to stray electric fields and imperfections in electrode geometry. For detailed

information on the trap used in this experiment, please refer to [102].

18



Other Designs Various other trap geometries are currently under active exploration in

both industry and academia for the development of a scalable trapped ion quantum com-

puting platform. Recent advancements in ion trap architectures include race-track type

geometries [106], integrated photonics [107], and monolithic glass traps.

2.2 The 171Y b+ ion

Perhaps one of the biggest advantages of atomic platforms in developing a quantum

computer is that all the atoms are identical by nature. Trapped ions with a hydrogen-like

atomic-level structure (e.g. alkali earth metals) provide additional advantage of simpler

control of atomic levels. The list of practical ion choices are typically limited to Ba+,

Be+, Ca+, Cd+, Hg+, Mg+, Sr+, and Yb+. Additional factors such as the availability of

commercial lasers to address the energy levels further dictate certain elements than oth-

ers. 171Yb+ has a number of characteristic properties that make it a suitable candidate for

quantum computing and simulation applications considered in this thesis.

The 171Yb isotope has a nuclear spin of I = 1/2, which splits the ground state hyperfine

|S1/2⟩ manifold into two first-order magnetically insensitive atomic energy levels. Each ion

encodes an effective spin-1/2 system into these two levels as |↓⟩z = |0⟩= |F = 0,mF = 0⟩

and |↑⟩z = |1⟩ = |F = 1,mF = 1⟩. Here, F is the quantum number for the total angular

momentum of the ion and mF is the projection along the quantization axis, set by a mag-

netic field of magnitude ≈ 5 Gauss. The energy difference between the two ground states

is ωHF = 2π × 12.64 GHz. These levels can be addressed using a Radio Frequency (RF)

signal or an optical Raman process, which will be discussed later in the thesis. Since the

energy levels of the ion are mapped to a single spin, I will use “ion” and “spin” inter-

changeably. In the following sections, I will describe the processes of trapping, cooling,

preparing, and detecting the qubit states of 171Yb+ and mention relevant lasers. Although

19



Figure 2.4: Relevant energy levels of 171Yb+ion

this thesis will not cover the details of the optical paths for realizing each of the processes,

I would point to the previous thesis from relevant experiments, as a reference for optical

path designs. All levels and wavelengths in this section are based on the NIST Atomic

Spectra Database [108]. See Fig. 2.4 for relevant energy levels, frequencies, linewidths,

and lifetimes.

2.2.1 Photoionization

To trap an ion, the first step involves ionizing a neutral atom close to the RF null point.

Neutral atoms are obtained from an isotopically enriched (∼ 90%) 171Yb source contained

within a stainless steel tube. The tube is heated resistively by passing current ≈ 2.06-2.5A

with a tungsten filament to evaporate atoms. In the ultrahigh vacuum (UHV) environment

of the oven, evaporated atoms form a directed flux beam.

20



continuum
38

9.9
 nm

369.5nm 
or

355 nm
393.14 nm

1P1

1S0

171Yb

Figure 2.5: Two stage ionization process of neutral 171Yb

21



Neutral atoms are then photo-ionized in a two-stage process (Fig. 2.5). Initially, a 398.9

nm laser excites electrons from the 1S0 →1 P1 . Subsequently, ionization occurs, with a

continuum limit being around ≈ 394 nm. One should keep in mind that orienting the 399

nm laser perpendicular to the atomic flux allows the Doppler-free linewidth which provides

isotope selectivity. The 369.53 nm Doppler cooling diode laser or 355 nm Raman laser are

suitable for the second stage ionization. Usually, there is limited laser power available at

369.53 nm (∼ 1 mW), however, there is usually plenty of 355 nm laser power, either from

the 0th order or from the 1st order diffraction from an acousto-optic modulator (AOM). In

addition, one can also use a 393 nm laser to achieve a loading rate of about 0.5 s/load[102].

2.2.2 Doppler Cooling

Once an 171Yb+is trapped, it must be laser-cooled to perform coherent operation with

it. Doppler cooling is one of the primary methods of cooling used in atomic systems.

Doppler cooling relies on the frequency-dependent scattering of photons [109, 110]. For

171Yb+we use the cycling transition between the 2S1/2 ↔2 P1/2 levels with a 369.53 nm

laser beam. The scattering rate of photons from a laser detuned by ∆ is given by,

Γs =
sΓ/2

1+ s+4∆2/Γ2 , (2.20)

where s = I/Isat ; I is the intensity of the incident laser, and Isat = πhcΓ/3λ 3Rbr is defined

as the intensity necessary to create an equal population in the two levels. For 171Yb+;

λ = 369.53 nm, Γ = 2π ×19.6 MHz (the natural linewidth of the transition), and Rbr ∼ 1

is the branching ratio [111].

Ions moving towards (opposite to) the laser beam experience blue (red) shifted detun-

ings. If the laser is initially red-detuned, i.e., ∆ < 0, ions moving towards (opposite to) the

laser scatter more (fewer) photons. Each absorption imparts a momentum kick along the

22



F=1

F=1

F=0

F=0

F=1

F=2

F=1

F=0
2.1 GHz 2.2 GHz

0.86 GHz

2P1/2

2S1/2

2D3/2

3[3/2]1/2

σ+ σ-�

93
5.

1 
nm

369.5 nm

Stimulated 
absorption

Spontaneous 
emission0

1

ωhf=12.6 GHz

Figure 2.6: Relevant energy levels and transitions involved in Doppler cooling of 171Yb+ion (not
to scale).

photon direction, but the emission of the photons is isotropic, resulting in no net momen-

tum transfer. As a result, ions moving towards the laser beam slow down and get cooled. To

efficiently cool the ion in all spatial directions, it is essential that the k⃗ of the Doppler cool-

ing beam has overlap in all three dimensions. Optimal Doppler cooling can be achieved

at a detuning of ∆ = Γ/2. For details on cooling path configurations in the Cryo-Qsim,

Warm-Qsim, and Blue-system labs, please refer to the theses of Wen Lin Tan [97], Kate

Collins [112], and Laird Egan [102], respectively.

The level structure of 171Yb+causes the above description to deviate from the ideal

two-level systems (See Fig. 2.6).

• The |2S1/2,F = 1⟩ Zeeman levels: The excited 2P1/2 level can decay to any of the

three Zeeman sublevels. Therefore, a mixture of π and σ± polarization is required to

ensure continuous cooling transitions. A 5 Gauss magnetic field removes the degen-

eracy of the Zeeman levels, preventing coherent population trapping [113].

23



• Off resonant coupling to the |2P1/2,F = 1⟩ level: The hyperfine splitting between

the |2P1/2,F = 0⟩ ↔ |2P1/2,F = 1⟩ states is 2.105 GHz. As a result, there is a finite

probability exciting the |2P1/2,F = 1⟩ state, which can decay to the |2S1/2,F = 0⟩

state, stopping the cooling transition as shown in Fig. 2.6. To address this, we apply

an additional 12.64+ 2.10 = 14.74 GHz frequency to the cooling laser to cover the

|2S1/2,F = 0⟩ ↔ |2P1/2,F = 1⟩ transition. This is accomplished by driving an EOM

at 7.37 GHz and using its 2nd order sidebands.

• Leakage to D states: Population from the 2P1/2 manifold can decay to the |2D3/2⟩

manifold, which includes both the |F = 1⟩ and |F = 2⟩ states. The |2D3/2,F = 1⟩

state is metastable with a lifetime of τ ≈ 52.7 ms, which stops the cooling transition.

A 935 nm laser depopulated this level by exciting the ion to the 2[3/2]1/2 bracket

state. This state predominantly decays back to the S-manifold, resuming the cooling

transition. To effectively depopulate the two hyperfine levels of the |2D3/2⟩- mani-

fold, we add a 3.07 GHz sideband to the 935 nm laser using an EOM. Additionally,

the 935 lase should also have π,σ± polarization to address all the Zeman levels.

Furthermore, the ion can decay to the 2F7/2 state from the 2P1/2 state, which has a

lifetime of ≈ 10 yrs. Experimentally, we find that applying a high-power 355 laser

can pump the population back to the S manifold.

2.2.3 State Initialization

At the beginning of an experimental sequence, the ion is initialized in the |↓⟩z = |0⟩=

|F = 0,mF = 0⟩ state. For state initialization, also known as optical pumping, we apply

the same 369.53 nm laser beam but add a 2.10 GHz sideband to it and turn off the 14.74

GHz sideband. After Doppler cooling, the ion is in one of the states in the S-manifold. If it

is in the 2S1/2|F = 0,mF = 0⟩ state, there is no resonant transition available and the ion’s

24



F=1

F=1

F=0

F=0

F=1

F=2

F=1

F=02.1 GHz
2.2 GHz

0.86 GHz

2P1/2

2S1/2

2D3/2

3[3/2]1/2

93
5.

1 
nm

Stimulated 
absorption

Spontaneous 
emission0

1

ωhf=12.6 GHz

Figure 2.7: Relevant energy levels and transitions involved in state initialization (optical pumping)
of 171Yb+ion (not to scale).

state remains unchanged. If it is in the 2S1/2|F = 1⟩ manifold, the ion gets excited to the

2P1/2|F = 1⟩ manifold, which has a 1/3 probability of decaying to the 2S1/2|F = 0,mF = 0⟩

state see Fig. 2.7. Similar to the Doppler cooling transitions, the optical pumping process

also needs to depopulate the 2D3/2 states using a 935 nm laser and sidebands. Usually,

experiments can achieve > 99% optical pumping fidelity within a few microseconds of

pumping time.

2.2.4 State Detection

After an experiment, we detect the ions’s state using a state-dependent fluorescence

technique. During the detection process, we apply a 369.53 nm laser beam resonant with

the 2S1/2|F = 1⟩ ↔ 2P1/2|F = 0⟩ transition. This beam includes all the π and σ± polar-

izations. If the ion is in |↓⟩z =
2S1/2|F = 0⟩ state, it is about 14.7 GHz detuned from the

2P1/2|F = 1⟩ state; as a result it scatters very few photons and appears “dark”. However, if

25



F=1

F=1

F=0

F=0

F=1

F=2

F=1

F=0
2.1 GHz 2.2 GHz

0.86 GHz

2P1/2

2S1/2

2D3/2

3[3/2]1/2

93
5.

1 
nm

369.5 nm

Stimulated 
absorption

Spontaneous 
emission0

1

ωhf=12.6 GHz

Figure 2.8: Relevant energy levels and transitions involved in state detection of 171Yb+ion (not to
scale).

the ion is in the |↑⟩z =
2 S1/2|F = 1⟩ state, it is nearly resonant with the detection light and

scatters many photons, appearing “bright” (See Fig. 2.8).

A typical detection window ranges from 100-500 µs, depending on the available laser

power, in which about 1000 scattering events can occur. The detection light is typically

collected using a high numerical aperture (NA) objective (NA ∼ 0.4−0.6) and then passes

through an imaging subsystem that provides a magnification of about 30-70×. This light

is then captured either using an EMCCD camera [97, 114], or a multi-channel PMT ar-

ray [102]. The detection efficiency resulting from these imaging systems is about 1-5%, so

if the ion is in the bright state, we would get a few tens of counts. We apply a threshold of

1 photon to distinguish between bright and dark ions. A theoretical maximum detection ef-

ficiency has been reported to be 99.86% (assuming 1% photon collection efficiency) [115].

Several error sources can limit the observed detection efficiency.

• The primary source of error is coupling to the 2P1/2|F = 1⟩ state, which can decay

26



to the 2S1/2|F = 0⟩ state, making a bright ion appear dark and vice-versa. Although

the possibility of the dark state appearing bright is usually much lower as it is far off-

resonant. The 2P1/2|F = 1⟩ state can further decay to the 2D3/2 state, which would

slow down the photon scattering. Applying 935 nm laser and sidebands brings the

ion back to the cycling transition. see Steven Olmshecnk’s thesis for details [111].

• The detection window should be optimized such that the bright and dark state his-

tograms are separated by an ‘appropriate’ amount. If the detection time is too low,

the bright histograms would overlap with dark ones, causing the thresholding to fail.

However, if the detection time is too high, there would be an excess separation of

bright and dark counts, this might cause (a) the detector to saturate and cause back-

ground dark counts, (b) photons from neighboring ions can overlap causing detec-

tion crosstalk and also (c) dark to bright off-resonant excitation (2S1/2|F = 0⟩ ↔2

P1/2|F = 1⟩).

In a typical experiment we calibrate the bright and dark counts by running two separate

experiments. First, we prepare the ion in the |↓⟩z state using Doppler cooling and optical

pumping and then detect the state. This produces a reference for dark counts. Then in the

second experiment, we apply a coherent π (using Raman process or microwaves) rotation

after optical pumping to prepare the ion in the |↑⟩ state and then detect the ion. This pro-

vides a reference for the bright state. Investigating the histograms from these calibrations

can help debug lower detection fidelity.

2.3 Coherent Operations with Qubits

Reliable entangling operations require the ability to coherently change the quantum

states of the qubits, encompassing both single-qubit and multi-qubit operations. For 171Yb+,

virtually all coherent operations are performed using a mode-locked 355 nm laser. In

27



the following sections, I will describe the basics of various single-qubit and multi-qubit

operations. Much of the physics described here has been discussed at length elsewhere

[101, 116], but I will give a brief account again to provide the background needed for later

parts of this thesis.

2.3.1 Raman Transition in a Λ level configuration

As described earlier, the qubit states are encoded in the |↓⟩z = |0⟩ = |F = 0,mF = 0⟩

and |↑⟩z = |1⟩ = |F = 1,mF = 1⟩ levels. The aim of coherent operations is to coherently

manipulate the occupancy in these two levels. Although real experiments typically use

two frequency combs of the pulsed laser to address the qubit levels via a virtual meta-

stable state [117], for the sake of developing a simplified intuition, I will assume a pair of

continuous wave (CW) lasers addressing a Λ-level structure (see Fig. 2.9. Subsequently, I

would connect the frequency comb picture with this current description 2.4.

As shown in the Fig. 2.9 the energy levels |0⟩ and |1⟩ are the hyperfine ground states

with an energy splitting ωh f = 12.6GHz (h̄ = 1). Two CW laser beams, with frequencies

ωL
0 and ωL

1 , off-resonantly couple these ground states to an excited state |e⟩. We define

ωi j = ωi−ω j for i, j ∈ {0,1,e} and the relationships between the frequencies are given by:

ωe0 −ω
L
0 = ∆, (2.21)

ω
L
0 −ω

L
1 = δω. (2.22)

The bare atomic Hamiltonian H0 is given by,

H0 = ω0|0⟩⟨0|+ω1|1⟩⟨1|+ωe|e⟩⟨e|. (2.23)

We now adopt a semi-classical description in which the atomic levels are quantized

28



0

1

e

δω

∆

ω0

ωe

ω1

ωe0

ω0
L

ω1
L

ωhf

g0
g1

Figure 2.9: Schematic of the Λ system involving the three levels |0⟩, |1⟩, |e⟩ states which are ad-
dressed by two lasers (glowing lines).

29



while the electromagnetic fields are treated classically. This approximation is generally

valid for the high-intensity laser beams considered here. The electric field of the two lasers

can be expressed as:

E⃗(r, t) = E⃗0(r, t)+ E⃗1(r, t)

= Re[E0(r, t)ε̂0ei(k0.r−ωL
0 t)+E1(r, t)ε̂1ei(k1.r−ωL

1 t)], (2.24)

where ε̂L is the polarization vector of laser field L.

The interaction between the atomic dipole and the laser fields is described by the Hamil-

tonian:

HI =−d⃗ · E⃗(r, t), (2.25)

where the dipole operator is given by d⃗ = de0|e⟩⟨0|+de1|e⟩⟨1|+h.c, with de0,de1 being the

dipole matrix elements. If the wavefunction at time t is:

|ψ(t)⟩=C0(t)|0⟩+C1(t)|1⟩+Ce(t)|e⟩, (2.26)

we apply the Schrödinger equation ι
∂ |ψ(t)⟩

∂ t = H|ψ(t)⟩ and equate the coefficients of each

state to obtain a set of coupled differential equations,

iĊ0 = ω0C0 −gCe[cos(k0.r−ω
L
0 t)+ cos(k1.r−ω

L
1 t)], (2.27)

iĊ1 = ω1C1 −gCe[cos(k0.r−ω
L
0 t)+ cos(k1.r−ω

L
1 t)], (2.28)

iĊe = ωeCe −g∗[C0 +C1][cos(k0.r−ω
L
0 t)+ cos(k1.r−ω

L
1 t)], (2.29)

where g = de0.E0 = de1.E1 s the single-photon Rabi rate between the two levels. Next,

we move to a rotating frame of reference for the atom by substituting Ci(t) = C̃i(t)e−iωit ,

i, j ∈ {0,1,e}. Applying a rotating wave approximation (RWA), which neglects the terms

30



involving the sums of large frequencies under the assumption ∆ ≪ ωi. We further assume

that the laser fields are weak, i.e. |g|≪ ∆ and the detuning from the excited levels are

large compared to energy level separations ∆ ≫ ω10,δω . These approximations allow us

to assume that the populations in the states |0⟩ ↔ |1⟩ are changing slowly and the excited

|e⟩ state can be effectively eliminated to obtain a two-level system evolving as:

i ˙̃C0 =−|g|2

2∆
C̃0 −

|g|2

4∆
C̃1[e−i[∆k·r−(δω−ω10)t]], (2.30)

i ˙̃C1 =−|g|2

2∆
C̃1 −

|g|2

4∆
C̃0[ei[∆k·r−(δω−ω10)t]], (2.31)

where ∆k = k0 − k1. We can also introduce a global phase difference between the two

laser beams using AOMs, i.e. φ = φ0 − φ1 and identify the effective 2-photon detuning

µ = δω −ω10. The two-level Hamiltonian in the interaction frame can be written as:

HI =
Ω

2

(
ei[∆k·r−µt−φ ]

σ
++ e−i[∆k·r−µt−φ ]

σ
−
)
+

g2

2∆
I, (2.32)

where σ+ = |1⟩⟨0| and σ− = |0⟩⟨1| are the spin raising and lowering operators, and Ω =

|g|2/2∆ = g0g1/2∆ is the two-photon Rabi frequency. The second identity term can be

absorbed by moving to a rotating frame. However, this term represents the two-photon dif-

ferential AC stark shift (SS) that introduces a |0⟩⟨0|− |1⟩⟨1| ≡ σ z term in the Hamiltonian

(see the subsection 2.3.2). This simplification resulted from the RWA, where we assumed

∆ ≈ ∆−ω10.

We identify the Λ system with 171Yb+energy levels as |0⟩=2 S1/2|F = 0,mF = 0⟩ and

|1⟩=2 S1/2|F = 1,mF = 0⟩ with ω10 = 2π ×12.64 GHz. The excited state |e⟩ lies between

the 2P1/2 and 2P3/2 levels which have a energy splitting of about 2π ×100 THz. The two

laser beams would eventually be mapped to the frequency combs of the Raman laser (see

subsec 2.4. We choose the laser parameters such as polarization and detunings to minimize

31



the AC stark shift as described in the following subsection. Therefore, in the subsequent

analysis, we will ignore the two-photon SS. The modified interaction Hamiltonian under

this assumption is,

HI =
Ω

2

(
ei[∆k·r−µt−φ ]

σ
++ e−i[∆k·r−µt−φ ]

σ
−
)
. (2.33)

2.3.2 Two-photon AC Stark Shift

In the previous derivation, we assumed a simplified three-level system. However, gen-

eralizing this to a more complex energy-level structure (such as the real atomic levels of

171Yb+) is not overly difficult. As a first step, we consider stimulated Raman transitions

involving the energy levels of both 2P1/2 and 2P3/2 manifolds. These levels are separated

by the fine-structure splitting of ωFS ≈ 2π ×100 THz. For the following analysis, we will

ignore the hyperfine structure of the 2P3/2 manifold.

To find the modified Hamiltonian terms, let’s recall from Eq. 2.24 that the electric field

vector of a laser beam L can be expressed in terms of the polarization vector as E⃗(r, t) =

|E(r, t)|ε̂L. The generalized polarization vector can be written in terms of left circular

(σ̂−), linear (π̂), and right circular (σ̂+) polarizations as ε̂L = ε
−
L σ̂−+ επ

L π̂ + ε
+
L σ̂+ with

|ε−L |2+|ε0
L|2+|ε+L |2= 1. The dipole Hamiltonian from Eq. 2.25 can be expressed as HI =

−|E(r, t)|d⃗ · ε̂ . The two-photon rabi frequency between the levels |α⟩ ↔ |β ⟩ mediated by

laser field E⃗i is given by gαβ = Ei|d⃗αβ · ε̂i|. Now, if we consider that the laser fields E⃗0

and E⃗1 are driving a stimulated Raman transition between states |0⟩ and |1⟩, as shown in

fig. 2.9, but instead of only one excited state (|e⟩), they couple to multiple excited states,

the stimulated Rabi frequency between |0⟩ and |1⟩ is given by [118],

Ω =
1
4 ∑

e
E0E1

⟨1|d⃗.ε̂0|e⟩⟨e|d⃗.ε̂1|0⟩
∆e

, (2.34)

32



with ∆e as the detuning of the laser from the state |e⟩. Note that, here we have assumed

∆e ≫ ωh f∀ e. The two-photon stark shift of the energy level |α⟩ by the laser field E⃗i is

given by,

δ
(2)
α =

1
4 ∑

e
Ei
|⟨α|d⃗.ε̂i|e⟩|2

∆e
. (2.35)

If we now consider the specific example of 171Yb+ions, the relevant excited states are

in the 2P1/2 and 2P3/2 manifolds, with all other states having negligible contributions.

The dipole matrix element of Eq. 2.342.35 can be expressed using Clebsch-Gordan co-

efficients [119], and the states with non-zero coefficients will be coupled by laser fields

with appropriate polarization. The qubit states |0⟩ and |1⟩ are coupled through excited 2P

states via σ̂± polarized light, as transitions from π polarized light are forbidden by selection

rules. The Raman Rabi frequency for this transition is given by[118, 120]:

Ω =
g0g1

6

(
1
∆
+

1
ωF −∆

)
, (2.36)

where g0,g1 are the resonant two-level Rabi frequencies. Note that the two-photon Rabi

frequency Ω ∼ E0E1 ∼
√

I1I2, meaning the Rabi frequency is proportional to the total laser

intensity for identical laser fields.

For 171Yb+however, all the hyperfine ground states |F = 0,MF = 0⟩ and |F = 1,MF =

0,±1⟩ can be coupled by 2-photon Raman process. Writing the states as |0,0⟩, |1,0⟩,

|1,1⟩, |1,−1⟩, respectively. Using the generalized polarization vector and Clebsch-Gordan

33



coefficients, the Rabi frequency between the states can be written as[121]:

Ω00,10 =
(
ε
−
0 ε

−
1 − ε

+
0 ε

+
1
)

Ω, (2.37)

Ω00,1−1 =−
(
ε
−
0 ε

π
1 + ε

π
0 ε

+
1
)

Ω, (2.38)

Ω00,11 =
(
ε
+
0 ε

π
1 + ε

π
0 ε

−
1
)

Ω, (2.39)

Ω10,1−1 =
(
ε
−
0 ε

π
1 + ε

π
0 ε

+
1
)

Ω, (2.40)

Ω10,11 =
(
ε
+
0 ε

π
1 − ε

π
0 ε

−
1
)

Ω. (2.41)

From these equations, we see that the Rabi frequency between all the |1,−1⟩ Zeeman levels

can be minimized if there is no π polarization. In addition, if both laser beams are linearly

polarized but perpendicular to each other, i.e. ε
−
0 = ε

+
0 = ε

−
1 =−ε

+
1 = 1/

√
2 the Rabi rate

between the qubit states Ω00,11 is maximized and equals to Ω. Using a similar selection

rule, we can also calculate the two-photon Stark-shift of either qubit levels as[122],

δ
(2)
0 =

g0g1

12

(
1
∆
− 2

ωF −∆

)
, (2.42)

δ
(2)
1 =

g0g1

12

(
1

∆+ωh f
− 2

ωF −
(
∆+ωh f

)) , (2.43)

δ
(2)
1 −δ

(2)
0 ≈−

g0g1ωh f

12

(
1

∆2 +
2

(ωF −∆)2

)
. (2.44)

Stark shifts of each qubit state cross near a detuning ∆opt ≈ ωFS/3 ≈ 33 THz, correspond-

ing to a laser wavelength of 355 nm for driving the Raman transitions. However, at this

detuning, the differential Stark shift (Eq. 2.44) is not nulled. By carefully optimizing the

polarizations, the differential Stark shift can be minimized to ≈ 1.1×10−4Ω [122]. From

Eq. 2.33, we can see that a differential Stark shift between the states |0⟩ and |1⟩ can be

expressed as an effective σ z term in the Hamiltonian.

34



2.3.3 Quantization of Motion

The term ei∆k·r in Eq. 2.33 contains the coupling of the ion’s motion with lasers the dif-

ference of the laser beams’ momentum vector (∆(k)). In a simplified model, we can assume

that the momentum vector is parallel to one direction of the ion’s motion, i.e. ∆k||r = x

taken to be along the x-direction. This alignment can be achieved by rotating the principal

axis of the trap using DC electrodes. If the ion’s motion is also cooled to the motional

ground state, it can be expressed as a quantized simple harmonic oscillator (SHO) with

motional quanta represented by phonons with frequency ω . The state of the ion in atomic

level i and phonon level n would now be represented as |i,n⟩. The position operator of the

ion can be represented as:

x̂(t) = x0[ae−iωt +a†eiωt ], (2.45)

where x0 =
√

h̄/(2mω) is the zero-point spread of the ground state wave function of the

SHO with mass m. Now we introduce the Lamb-Dicke (LD) parameter η = ∆kx0 to write

the Hamiltonian as:

HI =
Ω

2
exp[iη(ae−iωt +a†eiωt)]e−i[µt+φ ]

σ
++h.c. (2.46)

From Eq. 2.46, we note that depending on the laser detuning µ , certain motional and atomic

states will be coupled to satisfy RWA. Let’s consider three different cases:

Carrier Transition, µ=0 In this case, only the ω = 0 state is allowed by RWA, so no

motional excitation is possible. Eq. 2.46 then reduces to:

HI =
Ω

2
exp[iη(a+a†)]e−iφ

σ
++h.c. (2.47)

35



C
ar

ri
erRed sideband Bl

ue
 s
id

eb
an

d

Figure 2.10: Schematic of the Carrier (black), Red sideband (red), and Blue sideband (blue) transi-
tions.

Assuming the ion is confined to the LD regime, where the extension of the ion’s wave func-

tion is much smaller than 1/∆k, the inequality η2(2n+1)≪ 1 must hold at all times [123].

In this regime, we can expand the exponential and keep only the first order terms in η ,

resulting in eiηx ≈ 1+ iηx+O(η2). Thus, the carrier Hamiltonian becomes:

Hcarr =
Ωn,n

2
[σ+e−iφ +σ

−eiφ ], (2.48)

=
Ωn,n

2
[σ x cosφ +σ

y sinφ ], (2.49)

=
Ωn,n

2
σ

φ , (2.50)

where σ± = σ x ± iσ y, σφ = σ x cosφ +σ y sinφ , and Ωmn = Ω|⟨m|eiη(a+a†)|n⟩| is the gen-

eralized Rabi frequency where Dmn is the Debye-Waller factor given by [123],

Dm,n = e−η2/2
η
|m−n|

√
m!
n!

L|m−n|
m (η2), n ≥ m, (2.51)

and Lα
n (X) is the generalized Laguerre polynomial. During the carrier transition, only

the spin state changes while the phonon state remains unchanged. Therefore, if the ion is

initially in the motional ground state it would only undergo a Rabi oscillation corresponding

to the transition |0,n⟩ ↔ |1,n⟩ (see Fig. 2.10). However, if the initial n ̸= 0, there will

36



be multiple Rabi frequencies corresponding to different motional levels (n), and the Rabi

oscillation would dephase [124]. Under the LD regime, the effective carrier Rabi rate is

given by Ωcarr ∼ Ω[1− (n+ 1/2)/η2]. The phase φ in Eq. 2.50 can be rotated along any

direction in the x− y plane by controlling the laser phase via AOMs.

Red Sideband Transition (RSB), µ =−ωm By choosing the laser frequency red-detuned

by one motional quantum, we can address the energy levels between |0,n⟩↔ |1,n−1⟩ (see

Fig. 2.10). The effective Hamiltonian under the same LD regime is,

Hrsb =
iΩn,n−1

2
[aσ

+e−iφr −a†
σ
−eiφr ]. (2.52)

Note that Ωn,n−1 = ηΩn,n ≈ ηΩ
√

n.

Blue Sideband Transition (BSB), µ =ωm By choosing the laser frequency blue detuned

by one motional quantum, we can address the energy levels between |0,n⟩↔ |1,n+1⟩ (see

Fig. 2.10). The effective Hamiltonian under the same LD regime is,

Hbsb =
iΩn,n+1

2
[a†

σ
+e−iφb −aσ

−eiφb ]. (2.53)

Note that Ωn,n+1 = ηΩn+1,n+1 ≈ ηΩ
√

n+1.

2.3.4 Generating the Ising interaction

Trapped ion quantum computers primarily utilize the Mølmer-Sørensen (MS) scheme [125]

to perform entangling operations. In this section, I will describe how we control the

spin-motion entanglement to create an Ising-type Hamiltonian for analog quantum sim-

ulations. The basic requirement of the MS scheme is to simultaneously apply a pair of

laser beams near resonance with the RSB and BSB transitions. We define the MS detun-

37



ing δm = |µ −ωm| as the shift of the laser frequency from the motional frequency ωm.

The RSB/BSB tones have frequencies µrsb/µbsb =−ωm −δm/ωm +δm and phases φr/φb.

These frequency and phase controls can be obtained by applying the appropriate microwave

signal to AOMs in each beam path.

Next, I will consider a more complex scenario involving a system of N ions, which, in

principle, possess 3N collective motional mode frequencies along all spatial dimensions.

However, we can control the principal axis of the motion and the laser beam geometry to

primarily address one set of transverse motional modes [97]. Generalizing Eqs. 2.53, and

2.52 for all the motional modes, we can write the Hamiltonian for a system on N ions as

follows:

HMS = Hbsb +Hrsb, (2.54)

HMS = i
N

∑
i=1

N

∑
m=1

ηi,mΩi

2

[
amσ

+
i ei(δmt+φr)−a†

mσ
−
i e−i(δmt+φr)

]
+
[
a†

mσ
+
i e−i(δmt+φb)−amσ

−
i ei(δmt+φb)

]
, (2.55)

with the generalized LD factor given by ηi,m = bi,m∆k
√

h̄/2mωm. Here, bi,m are the i-th

component of the m-th motional mode, normalized such that ∑i,m|bi,m|2= 1. Rearranging

the terms we get,

HMS = i
N

∑
i=1

N

∑
m=1

ηi,mΩi

2

[
ameiδmt

(
σ
+
i e−iφr −σ

−
i eiφb

)
+a†

me−iδmt
(

σ
+
i e−iφb −σ

−
i eiφr

)]
,

=
N

∑
i=1

N

∑
m=1

ηi,mΩi

2

[
amei(δmt+φM)+a†

me−i(δmt+φM)
][

σ
+
i e−iφS +σ

−
i eiφS

]
, (2.56)

=
N

∑
i=1

N

∑
m=1

ηi,mΩi

2

[
amei(δmt+φM)+a†

me−i(δmt+φM)
]

σ
φS
i . (2.57)

Here we have substituted φS = φb+φr+π

2 and φM = φb−φr
2 . Note that we have absorbed

38



imaginary unit i into the exponent, i = eiπ/2. If you are reading this, I agree with you

that this is some algebra trickery, but substitute the phases and make sure you get back

Eq.2.55. φS is defined as the spin phase as it determines the phase of the spin operator in

the Bloch sphere, while φM is defined as the motional phase as it governs the evolution of

the motional state. All the sideband phases, therefore the spin and motional phases can

be controlled using AOMs. If we choose φb = π,φr = 0, we get φS = π , and φM = π/2.

Consequently, the Hamiltonian of Eq.2.57 becomes,

HMS =
N

∑
i=1

N

∑
m=1

iηi,mΩi

2

[
ameiδmt −a†

me−iδmt
]

σ
x
i . (2.58)

In deriving the equation above, I have already considered the RWA to consider motional

modes close to the laser frequency (µ −ωm ≪ µ,ωm). However, in the presence of many

motional modes, a more comprehensive treatment should account for all the rotating and

counter-rotating terms. Eq. 2.58 can be generalized starting from Eq. 2.46. If we consider

both RSB and BSB tones at frequencies µrsb = −µbsb = µ and phases φb = π,φr = 0, the

generalized MS hamiltonian of Eq. 2.55 can be written as:

HMS =
N

∑
i=1

N

∑
m=1

ηi,mΩi

2
(ae−iωt +a†eiωt)

[
e−i(µt)

σ
+
i + ei(µt)

σ
−
i

]
+

N

∑
i=1

N

∑
m=1

ηi,mΩi

2
(ae−iωt +a†eiωt)

[
ei(µt−π)

σ
+
i + e−i(µt−π)

σ
−
i

]
=

N

∑
i=1

N

∑
m=1

ηi,mΩi

2
(ae−iωt +a†eiωt)

[
σ
+
i
(
e−iµt − eiµt)+σ

−
i
(
eiµt − e−iµt)]

HMS =
N

∑
i=1

N

∑
m=1

ηi,mΩi(ae−iωt +a†eiωt)(−sin µt)σ x
i . (2.59)

In the next step, we proceed to calculate the time evolution operator of the time-

dependent Hamiltonian H(t) using U(t) = T
[
e−i

∫ t
0 dtH(t)

]
, where T is the time ordering

operator. We will utilize the Magnus expansion to calculate the time-ordered unitary oper-

39



ator [53, 126]:

U(t) = T [e−i
∫ 1

0 dtH(t)] = eΩ̄1+Ω̄1+Ω̄1+.... (2.60)

Thankfully, we do not need to calculate an infinite series of exponentials. Examining the

first few terms helps build intuition for the time evolution operator. The first three terms

are:

Ω̄1 =−i
∫ t

0
dtH(t), (2.61)

Ω̄2 =−1
2

∫ t

0
dt1
∫ t1

0
dt2[H(t1),H(t2)], (2.62)

Ω̄3 =
i
6

∫ t

0
dt1
∫ t1

0
dt2
∫ t2

0
dt3 ([H(t1), [H(t2),H(t3)]]+ [H(t3), [H(t2),H(t1)]]) . (2.63)

Looks daunting! I will not calculate all the integrals here, which are quite straightforward,

but I would develop some intuition for it. First, let’s recall that the phonon operators follow

the commutation relation
[
am,a

†
m′

]
= δm,m′ . Considering the Hamiltonian of Eq. 2.59, the

first term is a relatively simple integration. Taking a closer look at the second integral, we

find that it effectively reduces to commutators of [am,a†
m] = 1. So, the phonon operators

drop off from the second term, and as a result, the higher order commutators in the third

term and so on become zero! Therefore, we need to primarily consider the two terms

Ω̄1, and Ω̄2. However, as I will describe later (see subsection 2.3.7, the commutators are

non-zero if the Hamiltonian contains non-commuting σ y,σ z terms. Further, note that the

first term contains only a single spin operator σ x
i , and the second term contains spin-spin

operators σ x
i σ x

j . The coefficients of these spin operators are derived in detail in references

[101, 116]. The final time evolution operator then takes the form:

40



U = exp

(
∑
i,m

σ
x
i (α

∗
i,m(t)am +αi,m(t)a†

m)+
N

∑
i, j

χi, j(t)σ x
i σ

x
j

)
, (2.64)

αi,m(t) =
iηi,mΩi

µ2 −ω2
m

(
µ − eiωmt[µ cos µt−iωmsinµt]

)
, (2.65)

χi, j(t) = ΩiΩ j ∑
m

iηi,mη j,m

2(µ2 −ω2
m)

(
ωm sin2µt

2µ
+

µ sin(µ −ωm)t
µ −ωm

− µ sin(µ +ωm)t
µ +ωm

−ωmt
)
.

(2.66)

As expected, Eq. 2.64 has both single and two-qubit terms. The phonon part of the

single-qubit term is exactly in the form of a displacement operator D(α) =αa†+α∗a. This

term essentially represents a spin-motion coupling where the trajectory of the ion motion

can be represented by displacement of a coherent state characterized by αi,m(t). However,

during the detection of the ion’s spin state, the motional state gets traced over. Therefore, it

is desirable to have αi,m(τ) = 0 for all i,m after an entangling operation for duration τ . For

more details of this, please refer to [127]. In the Quantum simulation regime discussed in

this thesis, we detune the laser frequencies far away from the motional mode frequencies,

i.e. |µ −ωm|≫ ηi,mΩi. In this regime αi,m(t)≪ 1, hence the phase space trajectories are

very small and we can ignore them up to first order. In addition, we also assume a slow gate

regime where we evolve for longer duration and for large t we note that only the ωmt term

dominates in χi, j(t). After making these approximations, the unitary operation reduces to:

U ≈ exp

(
−i

N

∑
i, j

∑
m

(
ηi,mη j,mΩiΩ j

2(µ2 −ω2
m)

ωmt
)

σ
x
i σ

x
j

)
. (2.67)

Identifying the terms with U(t) = e−iHt we obtain:

H =
N

∑
i, j

Ji jσ
x
i σ

x
j , (2.68)

41



with

Ji j = ∑
m

ηi,mη j,mΩiΩ j

2(µ2 −ω2
m)

ωm. (2.69)

Variation of the interaction matrix elements with between ion i and j is often approximated

by a power-law decay,

Ji j ≈
J̄

|i− j|α
, 0 < α < 3. (2.70)

However this decay profile can be tuned to an exponential or a combination of power-law

and exponential decay by adjusting the detuning parameter (δm = |µ −ωm|)), as we will

explore Chapters 4 and 5. Eq. 2.68 represents the native long-range Ising Hamiltonian of

trapped-ion quantum spin simulators. In principle, the interaction matrix offers tunability

through detuning (δm = |µ −ωm|), Rabi rates (Ωi and participation matrix (ηi,m), which we

will leverage while applying the Ising interaction for various models.

2.3.5 Transverse Field

Many compelling applications of the long-range Ising Hamiltonian involve introducing

a transverse field σ y/σ z to the σ xσ x term. The most common way to apply a transverse

field along the z-axis with strength +|Bz| is by adding an additional ∓2|Bz| detuning to

the RSB/BSB tones. Comparing with Section 1.3.3, the RSB/BSB frequencies become

µrsb/µbsb =−ωm−δm−2Bz/+ωm+δm−2Bz. Note that in deriving Eq.2.55, we assumed

a rotating frame of the qubit (rotating at the qubit frequency ω0 = 12.64 GHz). After the

additional shift, the RSB and BSB tones are however rotating at ω0∓2Bz, which effectively

manifests itself as a qubit shift term Bzσ z. Although the frequency shift described here

appears to add a global transverse field term, using individual control we can independently

shift the frequency of the MS tones of each ion (see Chapter 5). This would allow us to

create a local site-dependent transverse field. Nevertheless, achieving local site-dependent

42



field control can be realized by applying an additional tightly focused beam to stark shift

the qubit levels of each ion. For a detailed description of this process, refer to Aaron Lee’s

thesis [128]. The Hamiltonian in the presence of a transverse field is now expressed as:

HT FIM =
N

∑
i, j

Ji jσ
x
i σ

x
j +

N

∑
i

Bz
i σ

z
i . (2.71)

Applying these transverse fields intorduces two major caveats. Firstly, in the original MS

scheme, the Magnus expansion (Eq. 2.63) terminated after the second term due to commu-

tators. However, if the transverse fields are applied (individually or together) the expansion

does not terminate, resulting in cross-terms in the Hamiltonian (see subsection 2.3.7. he

condition to neglect terms beyond the second term is Bz ≪ ηΩ, where ηΩ is the reso-

nant sideband Rabi frequency. Secondly, due to the asymmetric frequency shift of the two

tones, the qubit frame is now rotating at ω0 +Bz, and all the subsequent operations must

incorporate a phase advance by Bz in order to remain phase coherent with the interaction

Hamiltonian [128].

2.3.6 Longitudinal field

In addition to the transverse field, we can apply a laser beam resonant with the carrier

frequency to realize a Bφ σφ term where φ is a phase in the x− y plane. If we choose

the phase to be along the interaction direction,x, we realize a longitudinal magnetic field.

Using individual control of the beams, we can arbitrarily change the phase and amplitude

of the carrier tone to generate local longitudinal fields. The final Hamiltonian then takes

the form:

H =
N

∑
i, j

Ji jσ
x
i σ

x
j +

N

∑
i

Bx
i σ

x
i +

N

∑
i

Bz
i σ

z
i . (2.72)

43



2.3.7 Undesired Phonon excitations

In the preceding subsections, I have introduced additional magnetic field terms into

the Ising Hamiltonian. The Ising Hamiltonian derived from Eq. 2.68 originated from the

initial MS Hamiltonian of Eq. 2.59 or 2.58, which contained only one σ x term. However,

in the presence of additional magnetic field terms, the Magnus expansions of Eq. 2.63 do

not simply terminate, but result in new terms in the Hamiltonian which effectively causes

phonon errors in Ising simulation. In this subsection, I will briefly describe the generation

of these terms.

Consider the MS Hamiltonian of Eq. 2.58 along with the magnetic field terms HB. The

total Hamiltonian is now given by,

Htot = HMS +HB, (2.73)

HMS = ∑
i

∑
m

iηi,mΩi

2

[
ameiδmt −a†

me−iδmt
]

σ
x
i , (2.74)

HB = ∑
i

Bζ

i σ
ζ

i , (2.75)

where the magnetic field can be along any direction along the Bloch sphere, i.e. ζ = x,y,z.

Now if we consider the Magnus expansion terms from Eq. 2.63 with Htot , the Ω̄1 term

would have an additional term ∑i Bζ

i σ
ζ

i t which is the desired evolution under the magnetic

field term. The second term Ω̄2 would however have four different commutator given by,

Ω̄2 =−1
2

∫ t

0
dt1
∫ t1

0
dt2[HMS(t1),HMS(t2)]+ [HMS(t1),HB(t2)]

+ [HB(t1),HMS(t2)]+ [HB(t1),HB(t2)]. (2.76)

The first commutator yields a similar evolution operator of Eq. 2.64. The last commutator

vanishes as HB is independent of time. The other two commutators effectively involve

44



time

E
le

ct
ri

c 
fie

ld

frequency

po
w

er
 s

pe
ct

ra
l

 d
en

si
ty

1/frep

frep

comb 1
comb 2

∆f

0

0

nA

nB

fA

fA

fB

fB

AOM AAOM B

0 
m

od
ul

at
io

n 
lin

e

a) b)

Figure 2.11: (a) Time-dependent electric fields of a pulsed laser. In the frequency domain, this
produces a beatnote with frequencies separated by the repetition rate of the laser frep. AOMs shift
the frequencies of the beatnote. (b) Interference of two frequency combs shifted by fA and fB by two
AOMs A and B, respectively in each path. A transition would be driven if the resonant condition is
satisfied by the frequency difference between the nA-th and the nB-th comb teeth and the effective
shifts by the AOMs (∆ f ).

[
σ x

i ,σ
ζ

i

]
. If ζ = x then there is no additional term, but for ζ = y,z there would be an

additional term with σ
z
i ,σ

y
i respectively. If we consider the case ζ = z, which is most

common for quantum simulation experiments, the additional term from Eq.2.76 is given

by,

Hcross = ∑
i

∑
m

ηi,mΩiBz
i

2δm

[
2

δm
(a−a†)+ it(a+a†)

]
σ

y
i . (2.77)

In the far-detuned limit, δm ≫ Ωi,Bz
i , only the second term becomes significant. Note that

the cross term depends on the phonon operator (a+a†) which introduces a spin-flip error

in the Hamiltonian [129]. Various pulse-shaping techniques are implemented to minimize

the effects of phonons. The reference [114] has a detailed account of mitigating phonon

error in the context of quantum simulations. Furthermore, due to this additional term, the

Magnus expansion of Eq. 2.63 does not terminate and higher-order spin-dependent phonon

terms contribute to the time evolution operator [130].

45



2.4 Extension to Pulsed laser

The description of stimulated Raman transition described in the previous section con-

sidered a 3-level Λ system addressed by a pair of CW laser beams. However, due to the

hyperfine qubit splitting of 171Yb+energy levels, it is challenging to use CW lasers. There-

fore, we utilize a mode-locked 355 nm laser to address the Raman transitions. This laser

beam consists of an optical frequency comb with equally spaced teeth separated by 2π frep,

where frep is the repetition rate of the pulsed laser (See Fig. 2.11). The laser beam is split

between the two arms (A and B in Fig. 2.11 of an interferometer, which is then spatio-

temporally overlapped at the ion to drive qubit transitions. We can shift the frequency of

the combs using AOMs in either arm of the beam path, denoted as fA and fB, respectively.

The interference of the two frequency combs generates a beatnote comb at frequencies

2πn frep ± 2π∆ f , where ∆ f = fA − fB (See Fig. 2.11). The up-shifted (+∆ f ) and down-

shifted (−∆ f ) beatnote combs are identified as co-rotating and counter-rotating terms, cor-

responding to a photon being absorbed from beam A and emitted in B, and vice versa. This

terminology is useful for comparing the “phase-sensitive” and “phase-insensitive” schemes

discussed in Chapter 3. Most of the stimulated Raman transition calculations in the pre-

vious sections can be extended to the two frequency combs with an additional caveat that

the electric-field amplitudes are now varying with time characterized by the pulse shape

of the frequency combs[117]. If the nA-th beatnote of beam A is resonant with the nB-th

beatnote of beam B, along with the net AOM frequency shift, an atomic transition with

energy 2π fatom would be addressed by these two beatnotes as given by,

fatom = n frep ±∆ f , (2.78)

where n = nA −nB.

46



A detailed derivation of Raman transitions using pulsed lasers can be found in Jonathan

Mizrahi’s thesis [127]. In Chapter 3 I would describe specific examples for a given experi-

ment.

2.4.1 4-photon Stark shift

An additional complication of using a pulsed laser is the presence of off-resonant beat-

notes that do not satisfy Eq.2.78. These off-resonant beatnotes effectively shift the qubit

levels, creating undesired σ z terms in the Hamiltonian. These shifts can arise from off-

resonant beatnotes in each beam as well as from beatnotes of cross-beam interference.

The amplitude of the four-photon Stark shifts scales as ∼ Ω2/∆[121], where Ω is the two-

photon Rabi frequency (Eq. 2.32) and ∆ is the detuning from the virtual excited state |e⟩

(Fig. 2.9). We can minimize the Stark shift from each beam by optimizing the polarization

in the ”lin-perp-lin” configuration as described in subsection 2.3.2. The cross-beam Stark

shift is nullified by creating an effective imbalance of the beam intensity between the two

beams. However, fluctuations in polarization or laser intensity can cause varying Stark shift

noise, leading to decoherence during entangling operations. For a comprehensive review of

different Stark shift processes and their mitigation, please refer to [114]. Another method

to minimize the four-photon Stark shift is to tune the laser repetition rate. A more detailed

description of this approach can be found in Laird Egan’s thesis [102].

47



Chapter 3: Phase-Stable Quantum Simulation in A Cryogenic Trapped-

Ion Apparatus

Trapped-ion quantum simulators have proven effective in studying interacting spin

models, high-energy physics models, quantum chemistry, and quantum optimization prob-

lems. However, scaling these simulators to accommodate larger system sizes remains

challenging. Conventional room-temperature experiments at ultra-high-vacuum (UHV)

pressures ( 10−11 Torr) are limited by chain lifetime issues due to background gas colli-

sions, restricting typical system sizes to around 50 atoms. One way to mitigate this issue

is to cool the vacuum system to cryogenic temperatures, achieving extreme-high-vacuum

(XHV) conditions. This is the primary motivation behind developing a cryogenic trapped-

ion quantum simulator [98]. This chapter describes the cryogenic apparatus used in the

Cryo-qsim lab of Monroe Group, including discussions on ion vibration caused by the

cryo-cooler and strategies for generating a phase-insensitive entangling operation to miti-

gate these vibrations.

3.1 Overview of the Cryogenic Quantum Simulator

The trapped ion apparatus utilised a macroscopic segmented blade-trap enclosed in a 4K

environment, which routinely trapped about 100 171Yb+ions. The cryostat (SHV-4XG-

15˙UHV) used in this experiment is a closed-cycle Gifford–McMahon cryostat manufac-

tured by Janis Inc. (fig. 3.1a). Almost all of the trapped-ion “toolboxes” employed here,

48



including trapping potentials, CW laser operations, and coherent qubit manipulations, are

common across other trapped-ion quantum simulators. This apparatus has successfully pro-

duced results including QAOA optimization[83], confinement phenomena in high-energy

physics [78], many-body dephasing[84], and EIT cooling using a ‘tripod’ scheme[85]. For

extensive details of the apparatus and some of the experiments please refer to[97, 98, 131].

3.2 The “Bad” Vibrations

Although the cryogenic vacuum system significantly facilitated trapping and maintain-

ing long chains of over 100 ions, the requirement for coherent laser control and efficient

readout of atomic levels constrained most experiments to approximately 40 ions. One of

the primary limitations on coherent control arises from the vibrations of the cryo-cooler.

The cryogenic apparatus was designed to ensure that the vibrating cryo-cooler was not me-

chanically connected to the vacuum chamber. Instead, a helium exchange gas region served

as a thermal link between the cryo-cooler and the chamber (see the purple-shaded region

in Fig 3.1a). However, the top and bottom parts of the apparatus were mechanically linked

through a rubber bellow (see fig. 3.1a), which, despite its damping properties, transferred

measurable vibrations to the trap. Additional mechanical contraptions were implemented

to further dampen the vibrations induced by the cryo-pump[131]. A careful analysis of the

trap vibrations revealed major in-plane (x-y direction) vibrations at 40 Hz (the cryo-pump’s

pulsing rate) with a peak-to-peak shift of approximately 50 nm. Vibrations along the ver-

tical direction were measured at 1.2 Hz with a peak-to-peak shift of ∼ 40nm [98]. T This

level of vibration was small enough that the ion images on the camera remained stable, a

condition I define as ”good vibration”.

After operating the cryostat for about four years, the rubber bellow deteriorated and

49



z
y

b)
Laser beam ALaser beam B

x

Vacuum chamber

x_
pi

xe
l

y_pixel

ion vibration

c) Slow ion drift

“Good vibration” “Bad vibration”

Figure 3.1: (a): Schematic diagram of the side view of the cryostat. The purple-shaded region is
filled with ultra-high purity helium gas. The lower and the middle part of this region are cooled to
about 4K and 40K, respectively. The helium gas provides thermal contact between the cold head
(top) and the vacuum chamber (bottom) without direct mechanical contact, effectively decoupling
the vibrating cryo pump from the ion trap. (b) Schematic of the Raman laser beam geometry near
the ions. The two laser paths A and B, are controlled by two different AOMs (see fig. 3.3), which are
programmed to drive different Raman transitions. Despite a carefully designed vibration isolation
system (VIS), the ion experiences vibration of the order of a few nanometers (nm) due to the cryo
pump ([98]). The enlarged cartoon of the ion shows its vibration along the laser beam propagation
direction (the y-axis), which introduces a fluctuating phase in the laser-induced transitions. In typi-
cal “good vibration” conditions the ion appears stable on a camera. However, when the performance
of the VIS degrades, the ions can vibrate over a few pixels on the camera and appear defocussed,
referred to as the “bad vibration” condition. Each camera pixel is about 0.163µm [97]. This rela-
tively fast vibration is an extreme situation that the phase-insensitive scheme is designed to tackle.
(c) When the vibration isolation has significantly worsened, apart from the fast vibrations, the ions
also exhibit a slow (∼ 80 min) oscillation along the x− y plane with displacements ranging about
∼40 µm. We attribute this to ice formation inside the helium gas region (purple shade)[97]. During
this motion, the ion moves out of the laser beams and camera alignment. We need to service the
cryostat every ∼1.5 months to restore normal operation.

50



lost its elasticity and damping characteristics. Additionally, the rubber bellow allowed wa-

ter and other impurities to leak through, which then froze at 4K, forming a mechanical

contact between the cold head and the vacuum chamber. When this occurred, we observed

that the image of the ion moves on a camera by a few pixels (∼ 200 nm) at a rate com-

parable to the pulse rate of the cryo cooler ( ∼40 Hz). This condition, which I define as

the “bad vibration” was readily recognized by the ion image appearing defocussed on the

camera (fig. 3.1b).Although the ion could vibrate in all three dimensions, motion along the

y-direction caused phase noise in laser-induced interactions. The phase-insensitive imple-

mentation of interactions discussed in this chapter aims to mitigate such effects.

We replaced the rubber bellows with new silicone bellows as recommended by the

manufacturer. However, we soon realized that the leakage problem was worse with the

new bellows. The mechanical connection between the cryo-cooler and the vacuum chamber

occurred much sooner (about 1.5 months), causing the ion to vibrate and defocus on the

camera (”bad vibration”). During maintenance, we regularly found traces of water in the

VIS region, confirming that the silicone was allowing atmospheric impurities to leak into

the exchange gas region. 1

Interestingly, near scheduled maintenance, when a relatively small amount of impurity

buildup had happened, one could gently push down the top of the cryo cooler (fig. 3.1) and

make mechanical contact between the cold head and the vacuum chamber. This induced

the “bad vibration” condition and one could observe the evident ion vibration on camera.

This contact could be broken by temporarily pushing the cryo-cooler up and/or temporarily

raising the 4k stage heater temperature to boil off some frozen impurities, thus transitioning

back to the “good vibration” condition. Note that the heater temperature has a limited range

due to the thermal stress limits of the cryostat. Most of the experimental data of this chapter

1Other research groups from the Lincoln Labs also reported worse performance of the silicone bellows
compared to the rubber bellows.

51



were collected close to the maintenance schedule. a better-engineered cryostat should be

used, this situation provided a valuable opportunity to demonstrate the phase-insensitive

scheme in the context of quantum simulation. It is worth noting that the slow oscillations

shown in Fig. 3.1c did not recur once we serviced the VIS and the cryo-pump following

four years of operation. This oscillation was likely due to a large buildup of ice over a long

period inside the VIS region, which was more regularly serviced (∼1.5 months) afterward.

3.3 Phase Noise due to Vibration

The effect of ion vibration can be understood as an unwanted phase fluctuation of the

laser beams. Recall from Chapter 2 that the ion’s motion was represented by a quantum

harmonic oscillator in Eq.2.45. We can generalize this to incorporate the mean position of

the ions (x̄) as follows:

x̂ = x̄+ x0[ae−iωt +a†eiωt ]. (3.1)

The mean position can be absorbed into the effective Hamiltonian in Eq.2.46 as a global

phase ∆kx̄, which adds to the spin phase φS in Eq.??. Therefore, the spin-dependent entan-

gling force is susceptible to the phase instabilities of the optical fields. Specifically, from

Eq.2.68, we note that if the spin phase (φS) fluctuates during an experimental cycle, the

evolution under σ xσ x interaction would not remain phase coherent.

Absolute phase stability is required when qubit levels are addressed with optical fre-

quency laser beams [132]. However, for clock qubits such as 171Yb+, a frequency comb

from an ultrafast mode-locked pulsed laser is used to address the qubit levels [133]. A Co-

herent Paladin Advanced laser, capable of producing up to 24 watts of laser power at 355

nm, was used in this experimental apparatus. Acousto-optic modulators (AOMs) are driven

with radio-frequency (rf) signals to precisely control the phase and amplitude of the beat-

note between different frequency combs. Several instances of implementing phase-stable

52



Path APath B Path APath B

Phase Sensitive Scheme Phase Insensitive Scheme

Optical
waves

Beatnote
 combs

a) b)

Figure 3.2: Beatnotes for Phase-sensitive and phase-insensitive schemes. Frequency combs (solid
wiggly lines) from two beams in path A (black) and B (red and blue) interfere at the ion plane to cre-
ate the radiofrequency (RF) beatnotes (dotted wiggly lines) that drive atomic transitions. Each beam
(A/B) has frequency shifts ( fA/ fB), and phase shifts (φA,φB) imprinted by AOMs. The beam in path
B consists of two frequency combs (red and blue), shifted by two different frequencies fB,r/ fB,b.
(a): In the phase-sensitive scheme, the frequencies are chosen such that the beatnotes driving the
RSB and BSB (dotted red and blue) transitions impart a momentum kick in the same direction.(b)
In phase-insensitive scheme the frequencies are chosen such that the beatnotes driving the RSB and
BSB (dotted red and blue) transitions impart a momentum kick in the opposite direction.

control in trapped ion systems have been demonstrated before [133, 134]. The primary

goal of these works was to design phase-stable single-qubit and two-qubit gates for quan-

tum computing applications.

Here, we demonstrate the a phase-insensitive implementation of the Mølmer-Sørensen

(MS) scheme that realizes an Ising-type Hamiltonian in the far-detuned regime in the pres-

ence of phase noise induc