ENGINEERED DISSIPATION IN OPEN QUANTUM ACOUSTIC SYSTEMS

By

Joseph M. Kitzman

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Physics – Doctor of Philosophy

2023

ABSTRACT

Quantum acoustic systems, in which superconducting qubits are coupled to quantized me-

chanical degrees of freedom, offer a unique paradigm for the investigation of the fundamental

properties of phonons as well as the development of quantum technologies based on phonons.

This thesis presents several experiments that demonstrate the interaction between a super-

conducting transmon qubit and a high-frequency surface acoustic wave (SAW) resonator.

We use classical modeling techniques to design the electrical conductance of the SAW res-

onator, which encodes the spectral properties of the phononic modes hosted by the SAW

device. These spectral features are then investigated by leveraging the extreme sensitivity

of the qubit to its local environment, and highlight the complex mode structure of surface

phonons in our experiments. Furthermore, by considering the full phononic density of states,

we identify interference effects between resonant and lossy phonon modes, which act as the

primary source of decoherence for the qubit. These phonons that are not confined within

the SAW cavity act as an engineered dissipation channel, which we use advantageously as

a mechanism for the dynamical quantum state preparation and stabilization of high-purity

qubit states. The results of this thesis highlight the versatility of quantum acoustics sys-

tems in both the strong and weak coupling regimes, and emphasize the ability to engineer

phononic interference and dissipation in the quantum regime.

ACKNOWLEDGMENTS

Throughout the course of graduate school, I have had the opportunity to work with many

wonderful people. First and foremost, I would like to thank my advisor Johannes Pollanen for

the vital role he has played in my scientific career. Johannes is a great scientist and has been

an absolutely fantastic mentor, and I’m grateful to have had the opportunity to learn from

him. Much of what I know about experimental physics can be attributed to the tremendous

environment Johannes has worked to create at MSU. I’ll remember our time working together

very fondly. I would like to thank the remaining members of my committee: Mark Dykman,

Morten Hjorth-Jensen, Shannon Nicley, and Jaideep Singh for their continued guidance and

insight. Each of my committee meetings was a wonderful learning experience with many

thought-provoking discussions.

I am also greatly appreciative of Dr. Justin Lane, who taught me the day-to-day skills

I needed to be a scientist. Our times figuring stuff out in the lab, moving ladders around,

and carefully soldering on the fridge are memories I will always treasure. Throughout the

process of my Ph.D., I needed a lot of help from some really great colleagues. I would like to

thank Dr. Heejun Byeon, Dr. Liangji Zhang, and Dr. Kostyantyn Nasyedkin for welcoming

me into the group and helping me out whenever I had silly questions, which was pretty

often. I would also like to thank the current members of our lab: Camille Mikolas, Austin

Schleusner, Camryn Undershute, Jingyu Zhang, and Dr. Pranaya Rath, all of whom I am

very lucky to call both friends and colleagues. Thank you to Dr. Niyaz Beysengulov for

his friendship and guidance the entirety of my time as a graduate student.

I’m lucky to

have overlapped with some other great people in the physics department. To Eric Goodwin,

Michael Gottschalk, Jacob Watkins, and many others- thanks for the great discussions over

the years. You all made graduate school an awesome experience. I also owe a special thanks

iii

to Kim Crosslan for keeping me sane when I needed a break from work and needed to chat,

it was always much appreciated! Thank you to Dr. Baokang Bi for keeping the clean room

running smoothly and answering my many fabrication questions throughout the years.

I

would also like to thank Dr. Reza Loloee for all of the assistance he provides not only to

our lab, but to all the labs in the basement. I owe a special thanks to Cathy Cords, Jessica

Dellinger, and Susie Harwood-Brown for keeping the department running smoothly. I would

also like to thank Rob Bennett, Tom Palazzolo, Jim Muns, Tom Hudson, Thomas Rubley,

and Mike Schoen for helping me out in the machine shop and making sure that I always left

with the same amount of fingers that I came in with.

I have been extremely fortunate to collaborate with some excellent scientists outside of

MSU. In particular, I owe a great deal of thanks to Prof. Kater Murch at Washington

University in St. Louis. Kater is an expert in all things regarding superconducting qubits,

and his willingness to answer any and all questions I could come up with really helped push

forward the open quantum systems experiments in my thesis. Additionally, I would like

to thank Dr. Patrick Harrington and Max Drimmer for helping me out when I had many

questions about simulations.

I am thankful for the support of my parents Mark and Sandy Kitzman. None of this would

have been possible without their encouragement to dream big. I’m grateful for everything

that you guys have done for me for the past five (and more realistically 27) years. To Dawn

and Darrell Slag, I will always be appreciative of your kind words and interest regarding my

work.

Finally, I owe a great deal of thanks to my wife Dezi. Your unwavering encouragement

and optimism regarding my research over the last five years has meant the world to me.

From late night trips into the lab to check on the fridge, to listening to (really rough) first

iv

versions of all of my talks, I’m eternally grateful you have been by my side through it all. I

truly couldn’t have completed this long journey without your support. Thank you.

The projects described in this thesis were partially supported by the National Science

Foundation via grant number ECCS-2142846 (CAREER) and the Cowen Family Endowment

at MSU.

v

TABLE OF CONTENTS

Chapter 1. Applied and Fundamental Quantum Phononics with

Superconducting Qubits . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Transduction and transmission of quantum information with mechanical ex-
citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Generation of non-classical phononic states . . . . . . . . . . . . . . . . . . .
1.4 Phononic sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Quantum simulation with open quantum acoustics . . . . . . . . . . . . . . .
1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2. Superconducting Quantum Circuits

. . . . . . . . . . . . . . . . .
2.1 Quantization of the LC oscillator
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Quantum mechanical circuits
2.3 Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Aside: alternative derivation of Josephson energy . . . . . . . . . . .
2.3.2 Making a two-level system . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Jaynes-Cummings Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Resonant regime
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Dispersive regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Microwave cavities for cQED . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.1 Aside: experimental preparation of coherent states

Chapter 3. Measurement in Circuit Quantum Electrodynamics . . . . . . .
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The cryostat and input/output lines . . . . . . . . . . . . . . . . . . . . . . .
3.3 High-Power cQED readout . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Cavity spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Qubit spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Coherent control
3.7 Quantum state tomography . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 FPGA based measurement setup . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4. Surface Acoustic Wave Devices . . . . . . . . . . . . . . . . . . . .
4.1 Stress and strain in solids
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Two types of waves: transverse and longitudinal . . . . . . . . . . . . . . . .
4.3 Surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Piezoelectric surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
4.5 Construction of a SAW resonator
4.5.1 Coupling of modes and the P-matrix . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
4.5.2
4.5.3 Bragg reflection gratings . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
4.5.4 Complete model of a SAW resonator

Interdigitated transducers

vi

1
1

3
5
8
10
12

14
14
17
19
24
27
28
33
33
36
39
41

42
42
43
46
48
49
51
56
60

63
63
65
68
74
76
77
79
83
85

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

Chapter 5. Open Quantum Systems . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2 Lindblad master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Driven-dissipative qubit
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Driven-dissipative harmonic oscillator . . . . . . . . . . . . . . . . . . . . . .
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88
88
89
94
96
98

Chapter 6. Phononic Bath Engineering of a Superconducting Qubit . . . . 100
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1
6.2 SAW + qubit experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3 Numerical calculation of Lindblad dynamics . . . . . . . . . . . . . . . . . . 110
6.4 Phononic open quantum system results and discussion . . . . . . . . . . . . 111
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 7. Interference and Scattering in Multi-Mode Quantum Acoustic

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3 SAW resonator design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.5 Coupling to transverse bulk phonons
. . . . . . . . . . . . . . . . . . . . . . 132
7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Chapter 8. Vantablack Shielding of Superconducting Qubits . . . . . . . . . 144
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.1
8.2 Experimental details
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

APPENDIX

Flip-chip device assembly procedure . . . . . . . . . . . . . . 176

vii

Chapter 1

Applied and Fundamental Quantum

Phononics with

Superconducting Qubits

1.1

Introduction

Vibrational excitations are often first introduced via one-dimensional (1D) toy models of

masses connected by springs [1]. These toy models are often presented in an entirely clas-

sical context and do not offer insight into the world of quantum mechanics. When treated

quantum mechanically, many mechanical oscillations have a harmonic response, meaning

that they can be mapped onto a simple quantum harmonic oscillator. However, experimen-

tal implementation of simple harmonic oscillators typically does not allow for the study of

highly quantum mechanical states of the oscillator such as squeezed states or states with

negative Wigner functions [2–4]. Since all the transitions in a harmonic oscillator have the

same energy (frequency), classical drive signals to the system simultaneously drive every

transition within the oscillator, leading to a coherent state [5]. In order to truly access the

quantum nature of a mechanical system, it is often useful to introduce the concept of a

hybrid system, in which two (or more) quantum systems are coupled together [6, 7]. In this

1

thesis, we show that by engineering the interaction between a quantum two level system

(in our case a superconducting qubit) and a mechanical harmonic oscillator in a framework

often called quantum acoustics [8, 9], it is possible to investigate mechanical excitations in

the quantum limit. Quantum acoustics largely borrows from the field of circuit quantum

electrodynamics (cQED) [10, 11] to engineer strong coupling between the qubit and bosonic

modes of the mechanical oscillator [12], as well as infer properties of the mechanical oscil-

lator via measurements of the qubit [13] (see Fig. 1.1).

In particular, this thesis focuses

Figure 1.1: Schematic of various quantum acoustics architectures. As described in the
main text, a superconducting qubit (center) is used as a tool for investigating the quantum
mechanical behavior of phononic systems such as surface acoustic waves, bulk phonons,
phononic crystal resonators, and suspended membranes, just to name a few.

on quantum acoustics systems based on elastic waves propagating along the surface of a

piezoelectric crystal, called surface acoustic waves (SAWs), which have many exciting ap-

plications within quantum acoustics [14–16] and beyond [17–19]. This chapter will focus

on introducing quantum acoustic systems with a broad perspective on the potential appli-

cations involving the integration of phonons with quantum information systems as well as

2

|0⟩|1⟩Phononic crystalsSurface acoustic wavesBulk phononsSuspended membranesQubitfundamental physics results within the field of quantum acoustics. The chapters that follow

will introduce the necessary background information for cQED systems and qubits (chapters

2 and 3), SAWs (chapter 4), and open quantum systems (chapter 5) before presenting the

main experimental results on engineered open quantum acoustics systems (chapter 6) [15]

as well as measurement of the interference of surface phonons via the absorption spectrum

of a superconducting qubit (chapter 7) [16].

1.2 Transduction and transmission of quantum infor-

mation with mechanical excitations

Mechanical excitations can potentially be used in quantum memory schemes [20, 21], quan-

tum information can be imprinted on the phononic transmission of a SAW resonator [22,23],

itinerant phonons can be used to communicate between spatially separated qubits [24–26],

and phononic resonators are a vital part of many proposed microwave-to-optical transduction

protocols [27]. In this section, the latter two examples associated with quantum communi-

cation using phonons will be described in detail.

Phonons can be leveraged as a useful means to store or transduce quantum information in

several different ways. In order to use phonons as a direct means of quantum communication,

it is important to engineer quantum circuits to create single-phonon excitations to maintain

the quantum nature of a given qubit state.

If one were to create a qubit state with a

well-defined phase (i.e. a superposition state), the conversion of this quantum state to a

classical mechanical state would destroy the relevant phase information of the qubit. The

continued advancement of superconducting circuits has enabled the development of quantum

acoustics systems to experimentally realize a single phonon Fock state [4,28] able to preserve

3

this phase information. By engineering a quantum acoustics system in which single surface

phonons can be realized, these itinerant excitations can carry quantum information from one

point to another. In fact, phase information of the state of a superconducting qubit can be

transmitted via these itinerant surface phonons, and has been shown to enable to the creation

of two-qubit Bell states mediated by SAW phonons [24]. Furthermore, the interference of

quantum information encoded in these phonons can be leveraged to implement quantum

erasure experiments and perform “which path” measurements using phonons [29]. By further

optimizing the phononic communication channel, the excitations can be unidirectionally

excited and used to perform interferometric measurements of superconducting qubits [25,

30]. Additionally, phononic beamsplitters have recently been introduced into multi-qubit

experiments and allows quantum acoustics architectures with direct comparisons to linear

optical quantum computing protocols [26]. Although experimental progress using itinerant

phonons as carriers of quantum information has been successful thus far, it remains to be

seen if acoustic losses can be sufficiently reduced to scale linear quantum acoustics-based

processors to large qubit number.

Even if phonons are not used as direct carriers of quantum information, they are still

useful in other potential quantum computing applications by enabling certain microwave to

optical transduction schemes. Quantum state transduction is important for a multitude of

reasons, including protecting quantum signals from thermal noise, as well as the inclusion of

low-loss optical fibers to preserve the integrity of the transmitted quantum signal over kilome-

ter scale distances [31]. For example, nanomechanical transducers and resonators have been

implemented to couple optical photons and microwave signals via piezoelectric optomechan-

ical crystals, achieving coherent signal transfer between microwave and optical fields [32–34].

It has been shown that integrating optomechanical systems thermally cooled to near their

4

quantum ground state allows for bi-directional transduction with minimal added noise to the

quantum signal [35,36]. Even when the mechanical system of interest is not cooled to near its

ground state, it is still possible to engineer highly efficient transduction schemes (near 47%

conversion efficiency) while maintaining minimal optomechanical backaction [27,37–39]. The

integration of optical-based readout and state transfer with superconducting circuits repre-

sents an important milestone for integrated quantum networks, with further experimental

implementation of multi-qubit processors interconnected via optical signals as the next im-

portant step.

1.3 Generation of non-classical phononic states

Mechanical systems are often physically macroscopic and able to host a large number of

excitations, opening the door for potential applications in quantum information processing,

high-precision sensing, and metrology. By coupling these macroscopic phononic systems to

superconducting circuits in the quantum regime, it is possible to investigate their quantum

mechanical properties and attempt to realize these potential applications. Typically, me-

chanical oscillators are linear in nature, making them difficult to operate with a well-defined

number of excitations. However, other linear systems, such as electromagnetic resonators,

have been successfully operated in the so-called “quantum regime”, where quantum mechani-

cal states of these resonators have been initialized and measured by introducing a Josephson

junction as an effective non-linear circuit element [40, 41].

In fact, these efforts were an

important step to understanding the quantum mechanical properties of microwave photons.

This section aims to outline recent progress toward creating truly quantum mechanical states

of vibrational excitations in analogy to the progress made using microwave photons.

5

The first step for the quantum operation of a mechanical resonator is to prepare a known

oscillator state. The simplest well-known state we can imagine making is the quantum

ground state of the oscillator. If the energy required to excite the oscillator is much higher

than the ambient thermal energy of the local environment, the oscillator will be passively

cooled to near its quantum ground state by the fact that it is in thermal equilibrium with

its environment. In particular, it has been shown that under these conditions it is possible

to passively cool a mechanical resonator to low excitation number and measure the quan-

tum mechanical depolarization and decoherence times of a single phonon by interfacing the

resonator with a superconducting qubit [42]. In contrast to these high-frequency oscillators,

mechanical resonators with a transition energy compared to the thermal energy of their local

environment can also be initialized to well-defined quantum states via protocols such as laser

cooling [43], sideband cooling [44], and thermal cooling enabled by quantum reservoirs [45].

Additionally, linear low-frequency optomechanical systems have also been cooled passively

using nuclear demagnetization [46].

Oftentimes, engineering the coupling of these mechanical degrees of freedom with other

quantum systems provides additional experimental tools with which to control the phononic

system. In particular, quantum control of phonons has been shown by coupling a mechanical

harmonic oscillator to optical cavity modes [47,48] as well as superconducting qubits [4,28],

which is the hybrid system of interest for the remainder of this section. These optimized

hybrid systems open up the possibility of resolving and characterizing quantized phononic

states that can be leveraged for the full quantum control of mechanical systems. When the

hybrid system enters the regime where the coupling rate between the qubit and the oscillator

exceeds the decoherence rate of either system (the so-called strong coupling regime), quan-

tum states can be coherently transferred between the qubit and the mechanical oscillator.

6

Since superconducting qubits are sensitive to electrical excitations, hybrid quantum acous-

tics systems often rely on piezoelectric materials, which enable the conversion from electrical

energy to vibrational energy, and vice versa. Using hybrid systems based on piezoelectric

materials, single phonon Fock states can be prepared and measured in phononic resonators

based on bulk phonons [3,28], mechanical waves on the surface of piezoelectric materials [4],

and engineered phononic crystals [49, 50].

Recently, quantum phenomena such as entangled states of mechanical oscillators and

Schr¨odinger cat states of motion in relatively massive mechanical oscillators have been of

great interest, due to their potential applications in quantum information processing. By

using a superconducting qubit that is simultaneously coupled to two separate phononic

crystal resonators the qubit can be used as a tool to create a Bell state between the two

mechanical resonators [50]. This operation opens the door for bosonic quantum computing

using phononic degrees of freedom [51]. Via tomographic reconstruction of the joint phonon

distribution between the mechanical oscillators, it can be shown that the inclusion of su-

perconducting qubits in mechanical system enables deterministic control of multiple bosonic

degrees of freedom in potential quantum processors [50].

In quantum acoustics architectures based on bulk phonons, it has been shown that it

is possible to create superpositions of phononic coherent states with definite phase differ-

ence [52]. Physically, this type of quantum state corresponds to atomic lattice oscillations

within the bulk of the material having opposite phases. In order to engineer precise phase

and amplitude control over these Schr¨odinger cat states it is necessary to couple the phonon

modes of interest to a superconducting qubit and leverage the resonant regime of the Jaynes-

Cummings Hamiltonian [53, 54], in contrast to most systems that are typically operated in

the dispersive regime [10, 11]. The generation of phononic cat states represents a massive

7

advance in the fundamental understanding of phononic modes at the quantum level as well

as important potential applications in quantum information processing, for example using

these bosonic modes themselves as a qubit [55].

1.4 Phononic sensing

A key ingredient in the previous section was the use of qubit measurement to gain information

about the state of the mechanical oscillator, which was made possible due to a designed

interaction between a qubit and the mechanical modes. In contrast, the aim of this section

is to focus on using information about the oscillator state to gain information about the local

phononic environment. In order to provide context for the interaction between the qubit and

mechanics, we will discuss the dispersive interaction between the two modes [10,11]. As will

be described in the next chapter, when the frequency difference (∆) between the qubit

frequency and mechanical resonator frequency is large relative to their coupling (g (cid:28) ∆),

the Jaynes-Cummings Hamiltonian takes on the dispersive form:

H ≈ ωm ˆm† ˆm + (ω01 + 2χ)ˆσz/2,

(1.1)

where ωm is the frequency of the mechanical oscillator, with creation (annihilation) operator

ˆm† ( ˆm), and ω01 is the qubit frequency, which acquires a dispersive shift of 2χ for each

occupied level of the mechanical resonator. This dispersive shift is due to the ac-Stark effect

and has been widely observed in cQED systems [56], where the qubit acquires a frequency

shift that depends on the number of photons in a microwave resonator. More recently,

this effect has been observed in circuit quantum acoustodynamics (cQAD) systems [3, 16,

8

28, 49, 57], as the population of phononic modes modifies the qubit absorption spectrum in

accordance with Eqn. 1.1.

In the strong dispersive regime, the frequency shift imparted on the qubit per excita-

tion is larger than both the qubit and resonator’s decay rate. This allows for the precise

determination of the resonator’s state, i.e. its statistical distribution over the phononic Fock

basis. An important feature of this dispersive readout technique is that the measurement

Hamiltonian commutes with the resonator’s operators, allowing for quantum non-demolition

(QND) measurements of the mechanical resonator. Not only are QND measurements inter-

esting from a fundamental physics perspective, as they allow one to gain information about

a quantum state without destroying it, but QND measurements are also essential to most

quantum error correction (QEC) schemes [58].

Since the first experimental demonstration of the strong dispersive regime in cQAD [49,

57], dispersive readout has proven to be a powerful experimental tool to sense the phononic

environment coupled to the qubit. Dispersive readout techniques have been combined with

classical drives for quantum phononic state preparation techniques (as discussed in the last

section of this chapter) to perform Wigner tomography of a mechanical resonator [3, 50, 59].

In other experiments, a superconducting qubit was tuned to simultaneously be in the strong

dispersive regime with two mechanical resonators. This allowed for the qubit to act as a

sensor for both resonators simultaneously and was recently demonstrated to dispersively

readout entanglement between the two mechanical oscillators [50]. Even in the situation

where the qubit or mechanical resonator decoherence is larger than the dispersive shift (the

weak dispersive regime), we can still rely on the qubit to sense its phononic environment. For

instance, this was recently leveraged to measure a phononic Fano resonance [60,61], where a

single, strongly resonant mode of a phononic resonator interferes with a continuum of lossier

9

phononic modes of the same resonator resulting in a characteristically asymmetric phonon

spectrum [16]. The latter results are discussed in chapter 7 of this thesis.

Two-level systems (TLSs) introduce an additional source of environmental decoherence

for qubits and mechanical resonators [62, 63]. Like any resonant system coupled to its sur-

roundings, a TLS can decay into the larger environment via phononic channels. While

TLSs can in some cases have longer coherence times than that of a qubit or mechanical

system [64, 65], a lack of control over TLS properties limits easy investigation of TLSs in an

experimental setting [66,67]. Nevertheless, a more complete understanding of the emergence

and decay of TLSs can provide insight into minimizing decoherence of future resonant sys-

tems and remains a topic of active research. Specifically, integrating superconducting qubits

with mechanical resonators has been shown to provide a promising means for investigating

the decay of phonons into a bath of resonant, or near resonant TLSs [68].

1.5 Quantum simulation with open quantum acoustics

Physical quantum systems that are capable of simulating complicated phenomena dictated

by quantum mechanics are referred to as quantum simulators [69,70]. Advances in the scaling

of quantum devices and engineered systems has produced significant growth in the devel-

opment of quantum simulators, based on a wide variety of different quantum systems [71].

Quantum acoustic devices are a novel platform for quantum simulation, particularly in the

simulation of non-Markovian and open quantum systems, which are never truly isolated

from their environments. Oftentimes, the environment or “bath” is an unwanted source of

decoherence for the quantum system of interest.

In simple cases, open quantum systems

can be modeled via the Markovian master equation [72], however, if the system exhibits

10

finite memory effects, a modified non-Markovian master equation is needed to model the

system dynamics [73,74]. In these types of systems, the current state of the quantum system

depends on the past states of the system, indicating a “memory”. These types of memory

effects have been both predicted and observed in cQED systems [75–77] and are now under

investigation in cQAD systems. Because phonons travel significantly slower than photons

(approximately 5 orders of magnitude slower at the same frequency), quantum acoustics

systems are an ideal platform for studying open quantum systems with memory effects. By

coupling a mechanical system to a superconducting qubit over relatively large distances, it

has been predicted, and verified experimentally, that non-Markovian effects can dominate

the evolution of a superconducting qubit [78, 79]. Extensions of experiments that couple

superconducting qubits to mechanical resonators in a regime in which the finite memory

effects of the bath are a dominant effect open the door to non-Markovian bath engineering.

By engineering the Hamiltonian in such a way, it could be possible to fully investigate the

impact of phonon transit time and interference, at the level of single phonons, on qubit

dynamics. The flexibility of quantum acoustics systems in this regard offer a very exciting

avenue for the fundamental investigation of non-Markovian systems.

In addition to the ability to tailor the Hamiltonian of quantum acoustics systems, hybrid

quantum phononic systems also offer the ability to control the dissipation of a supercon-

ducting qubit with a high level of precision using the spectral properties of the phononic

resonator [4, 15]. Well developed cQED theories about noise, amplification, and measure-

ment [80] have allowed for breakthroughs in dissipation engineering with superconducting

qubits coupled to their electromagnetic environment.

It has been shown that coupling a

transmon qubit to a frequency dependent impedance allows for the control of the dissipation

experienced by the qubit [81]. Designed dissipation engineering in the field of cQED has

11

proven to elucidate the understanding of the light-matter interaction in superconducting

circuits [82–90]. By leveraging controlled dissipation, it has been shown that it is possible

to create systems in which effective non-Hermitian systems can be simulated using cQED

architectures [91–94].

By coupling a superconducting qubit to an effective electrical impedance produced by

a mechanical resonator, it is possible to implement an effective frequency dependent loss

channel that dictates the decay of a transmon qubit [95]. This enables experiments utilizing

engineered phononic dissipation channels to simulate open quantum acoustic systems to pro-

duce dissipative state preparation and dynamical state stabilization [15], as will be discussed

in chapter 6. By continuing to optimize the designed dissipation experienced by the qubit via

its coupling to a phononic bath, future experiments could be used to control the dissipation

of higher transmon levels. By implementing a system in which different qubit levels have

vastly different loss rates, it could be possible to create an effective two-level system with de-

cay out of the qubit manifold, allowing for the simulation of an effective non-Hermitian [96]

open quantum acoustics system. Combining the ability to design both the loss spectrum

and the Hamiltonian to a high degree of precision in quantum acoustics systems makes these

qubit-phonon hybrid systems an ideal playground for understanding both superconducting

circuits and vibrational excitations in the quantum regime.

1.6 Conclusion

This chapter has served as a broad introduction on the current state of play in the field of

quantum acoustics. Engineered phononic degrees of freedom can be used to transit quantum

information via itinerant phonons, or phonons can be used as an intermediary to enable

12

microwave-to-optical transduction schemes. When coupled to nonlinear elements such as

superconducting qubits, the phononic degrees of freedom can be controlled at the level of

single phonons. This quantum control can be used to generate highly non-classical phononic

states including Fock states, entangled states, and Schr¨odinger cat states. Additionally, the

coupling to the qubit can be designed such that the oscillator’s excitation spectrum can be

imparted onto the qubit, allowing for the sensing of phononic decay or interference. Finally,

due to the ability to fine-tune the interaction between the qubit and the mechanical modes as

well as the phononic excitation spectra, the field of quantum acoustics is an ideal candidate

for quantum simulation of both non-Markovian and open quantum systems, and the results

presented in this thesis break the foundational ground in this effort.

13

Chapter 2

Superconducting Quantum Circuits

Interfacing high-frequency mechanical resonators cooled to their quantum ground state with

quantum circuits offers an exciting platform for quantum memory applications [20], linear

quantum computing [26], as well as quantum communication enabled via phonons [24, 25].

In order to operate these phononic modes in the quantum regime, we rely on the framework

of circuit quantum acoustodynamics (cQAD) [8], an analogue of circuit quantum electro-

dynamics (cQED) [10, 11]. Furthermore, cQED also provides a means with which to both

manipulate and measure the state of our qubits. This chapter aims to develop the necessary

theoretical background for cQED systems, with important analogues to cQAD systems. The

topics discussed in this chapter provide the background knowledge for engineering the inter-

action between quantum two-level systems and bosonic degrees of freedom (either photonic

or phononic depending on the experiment).

2.1 Quantization of the LC oscillator

A quantum two-level system, or qubit, is an integral component for the results of this thesis.

In this chapter, we will build an understanding of how we can experimentally realize such a

quantum system and how we can manipulate and measure the state of this system. Addi-

tionally, analogs of these techniques will become increasingly important for the experiments

described in chapters 6 and 7 when we consider a qubit strongly coupled to various types of

14

Figure 2.1: A parallel LC circuit. The node flux φ(t) is defined relative to ground. In this
simple circuit, the energy oscillates between the electric and magnetic fields.

mechanical excitations.

A simple resonating electrical circuit one can imagine constructing in a laboratory consists

of a parallel LC circuit where the stored energy in the circuit oscillates between charge

imbalance across the capacitor and currents passing through the inductor. An example of a

circuit can be seen in Fig. 2.1. In order to investigate how energy is stored in the circuit,

we begin by constructing the Lagrangian of the circuit as defined by the node fluxes in the

system as shown in Fig. 2.1.

φ(t) =

(cid:90) t

−∞

V (t(cid:48))dt(cid:48)

(2.1)

In Eq. 2.1, V (t(cid:48)) represents the voltage relative to ground at the node at time t(cid:48). We can use

fundamental circuit relations to see that the voltage across the inductor is given by V = L ˙I,

which allows us to write the node flux as a function of the current through the inductor,

φ = LI. The energy stored in the inductor can then be written in terms of the node flux

rather than in terms of the current:

EL =

LI2 =

1
2

φ2
2L

.

(2.2)

15

Similarly, by expressing the voltage across the capacitor in terms of the time derivative of

the node flux (V = ˙φ), we can write the energy stored across the capacitor plates as:

Ec =

CV 2 =

1
2

C ˙φ2.

1
2

(2.3)

Having expressed the energy stored in each component of the circuit, we can identify the

node flux as an effective coordinate in a Lagrangian treatment of the problem.

In this

framework, we can think of the energy stored in the inductor as an effective potential energy

(the energy only depends on the effective coordinate φ), and the energy stored across the

capacitor pads as an effective kinetic energy (this energy depends on the time derivative of

the coordinate, and therefore resembles a velocity!) With this analogue in mind, we can

construct the Lagrangian L for the circuit:

L = Ec − EL =

C ˙φ2 −

1
2

φ2
2L

.

(2.4)

In order to investigate the energy spectrum of this system, it is necessary to carry out a

Legendre transformation of Eq. 2.4 to obtain the Hamiltonian.

It is then important to

recall that the Lagrangian is a function of both position and velocity (L = L(q, ˙q)), while

the Hamiltonian is a function of coordinate and momentum (H = H(q, p)). The conjugate

momentum to the coordinate φ is then calculated as

p =

∂L
∂ ˙φ

= C ˙φ = CV,

(2.5)

where we immediately recognize the conjugate momentum as just the charge stored across the

capacitor pads Q = CV . As such, we will refer to this stored charge across the capacitor pads

16

interchangeably with the momentum for this particular circuit. Having correctly determined

the conjugate momentum for the LC oscillator, we can now calculate the Hamiltonian:

H = ˙φ

∂L
∂ ˙φ

− L =

C ˙φ2 +

1
2

φ2
2L

.

(2.6)

Remembering to express the Hamiltonian as a function of the momentum Q rather than

the “velocity” ˙φ, we arrive at our final expression for the classical Hamiltonian of the LC

oscillator:

H =

Q2
2C

+

φ2
2L

.

(2.7)

Importantly, Eq. 2.7 is quadratic in both the momentum and coordinate. As such, it is

reasonable to expect a response from the system resembling that of a harmonic oscillator.

Recalling that the resonant frequency for a simple LC oscillator is given by ω = 1/

√

can re-express Eq. 2.7 as

H =

Q2
2C

+

1
2

Cω2φ2.

LC, we

(2.8)

As seen above, this Hamiltonian is reminiscent of a standard harmonic oscillator, with the

capacitance serving as a substitute for the mass.

2.2 Quantum mechanical circuits

Since our experiments ultimately require quantum mechanical circuits, an important exten-

sion of the analysis involves promoting the variables representing charge and flux in the

previous section to quantum operators rather than classical variables. Since the charge was

found to be the conjugate momentum to the “position” φ, once the continuous variables are

promoted to quantum mechanical operators, they will inherit the canonical commutation

17

relation [ ˆφ, ˆQ] = i(cid:126). Because we found that the classical LC oscillator exhibited harmonic

response, it is natural to express the quantum mechanical operators as linear combination

of harmonic oscillator ladder operators ˆa and ˆa†. Including prefactors that represent the

quantum mechanical fluctuations of both charge and phase, the operators become:

ˆφ = φZPF

(cid:16)

ˆQ = −iQZPF

ˆa + ˆa†(cid:17)
(cid:16)

ˆa − ˆa†(cid:17)

(2.9)

,

where the ladder operators maintain the fundamental commutation relation [ˆa†, ˆa] = 1. Fur-

thermore, we can express the quantized Hamiltonian in terms of these new ladder operators

as

ˆH = (cid:126)ω

(cid:18)

ˆa†ˆa +

(cid:19)

.

1
2

(2.10)

Additionally, we assume that the zero point fluctuations of both charge and phase are

given by the expressions which depend on the characteristic impedance of the oscillator

Z0 = (cid:112)L/C :

φZPF =

QZPF =

(cid:114) (cid:126)Z0
2
(cid:126)
2Z0

(cid:115)

.

(2.11)

What does all of this analysis tell us? Although we have carried out a thorough investigation

of a simple electromagnetic resonator, this system does not function as a two-level system,

but rather it is a system with infinitely many evenly spaced energy levels ∆E = (cid:126)ω, where

√

ω = 1/

LC. However, this analysis has allowed us to learn how to quantize simple circuits

(a detailed description of more complex circuits can be found in Ref. [97]). The following

18

Figure 2.2: Top: Schematic of a Josephson junction. Two superconducting islands (grey; S)
are separated by a thin insulating layer (white; I). Cooper pairs can tunnel across the junction
through the insulator. Bottom: Circuit representation of a Josephson junction. The junction
responds as a nonlinear inductor with inductance LJ . There is some intrinsic capacitance
CJ associated with the insulating layer that is represented by the box surrounding the cross.

section will introduce a modification to the circuit described above and will introduce the

necessary non-linearity needed to create a well-defined quantum two-level system, which

turns out to be relatively straightforward to implement experimentally.

2.3 Superconducting qubits

In order to create the required non-linearity into our circuits so that we can independently

address any two individual energy levels, we require the incorporation of a Josephson junc-

tion [98, 99]. As will be shown in this section, for the experiments relevant to this thesis,

a Josephson junction can be effectively thought of as an inductor with a non-linear voltage

response to a time-varying current. This non-linearity will be a key building block neces-

sary in our experiments. A Josephson junction consists of two superconducting islands that

are separated by a thin insulating layer as shown in Fig. 2.2. The superconducting charge

carriers (called Cooper pairs) can tunnel across this thin layer, weakly connecting the two

superconducting islands. The superconducting state at any position (cid:126)r in the material can

19

be described by an order parameter ψ((cid:126)r) =

√

np exp[iφ((cid:126)r)]. The density of Cooper pairs is

represented by np, and the phase of the order parameter is φ((cid:126)r). In a Josephson junction, the

superconducting order parameter will have phases φL and φR in the left and right supercon-

ductors, respectively, as shown in Fig. 2.2. The famous result of Josephson [98] relates the

difference in phase (φ ≡ φL − φR) between the two superconductors to the current flowing

through the junction:

I = Ic sin φ,

(2.12)

where Ic is the critical current of the junction, which is defined as the maximum super-

conducting current the junction can support before the superconducting state is destroyed.

If a constant voltage is applied across the junction, the phase difference takes on a time

dependence given by:

dφ
dt

=

2eV
(cid:126) .

(2.13)

Together, Eqns. 2.12 and 2.13, form the so-called Josephson relations. Up to this point, the

Josephson junction was promised to be important due to its nonlinear inductance, which we

can obtain from these equations. We can calculate the inductance of a Josephson junction

by taking the time derivative of Eqn. 2.12:

˙I = ˙φIc cos φ,

(2.14)

which can then be related to the voltage across the junction via V = L ˙I. Combining

Eqns. 2.13 and Eqns. 2.14 allows us to solve for the phase-dependent inductance

L(φ) =

LJ
cos φ

,

20

(2.15)

where we have introduced the magnetic flux quantum Φ0 =

(cid:126)
2e and the characteristic Joseph-

son inductance LJ =

Φ0
2πIc

. Eqn. 2.15 has a nonlinear dependence on the phase difference

across the junction (and therefore the current flowing through the junction). Often, we oper-

ate devices consisting of two Josephson junctions in parallel. This particular device geometry

is referred to as a DC Superconducting QUantum Interference Device (SQUID). If the two

junctions have the same critical current Ic, the effective critical current of the entire loop is

found to be [100]:

Ic,loop = 2Ic cos

πΦext
Φ0

,

(2.16)

where Φext is the applied external magnetic flux threading the loop (see Fig. 2.3). As we

will see in the following sections, this means that by applying an external flux through a

SQUID loop will allow us to tune the resonant frequency of our qubits. If the SQUID loop

consists of two junctions whose critical currents are not identical (as is typically the case in

realistic devices), the effective critical current of the loop is:

Ic,loop = (I1 + I2) cos

(cid:115)

1 + a2

πΦext
Φ0

(cid:18)

tan

(cid:19)2

,

πΦext
Φ0

where I1 and I2 are the critical currents of each junction in the loop, and a =

(2.17)

I1−I2
I1+I2

is the

parameter that quantifies the asymmetry in the critical currents between the junctions in

the loop.

At this point, it will be extremely useful to consider what kind of Hamiltonian will

describe a Josephson junction. As the phase φ across the junction changes in time, the

change in the junction energy is given by the integral of the electrical power dissipated in

21

Figure 2.3: Top left: Transmon qubit with a single Josephson junction. The equivalent
circuit is pictured below. As we will describe later in this thesis, much of the capacitance
comes from the large dipole antennae attached to our qubits. Top right: Transmon qubit
in a SQUID geometry. The area of this particular loop is 16 µm2. An external magnetic
field can thread the loop and vary the effective critical current of the device. The equivalent
circuit is pictured below.

the junction over time [100]:

(cid:90)

∆E =

I(t)V (t)dt,

(2.18)

Where the current flowing through the junction is I and the voltage across the junction is

V . The integrand of Eqn. 2.18 may be related to the flux Φ by the relation V dt = dΦ, and

the flux may be related to the phase φ by Φ = φ

2π Φ0 (note: this φ not the same as the node

flux as defined in Eqn. 2.1). By substituting these relations into Eqn. 2.18, and using the

Josephson relation for the current (Eqn. 2.12), the junction energy can be determined as

22

follows:

(cid:90)

∆E =

(cid:18)

Ic sin φ d

Φ0

(cid:19)

φ
2π

(2.19)

=

= −

(cid:90)

IcΦ0
2π
IcΦ0
2π

sin φ dφ

cos φ.

In order to simplify the expression for ∆E, we introduce the Josephson energy, EJ =

IcΦ0
2π ,

which depends on the critical current of the junction, and is therefore tunable with an

external flux in the case of a SQUID loop. This allows us to express the change in the

junction energy as a function of phase:

E(φ) = −EJ cos φ.

(2.20)

In addition to the tunneling of Cooper pairs through the junction, a full description of

the system also involves taking into account the electrostatic potential energy due to the

imbalance of Cooper pairs on either superconducting island, which is given by n. This

electrostatic energy can be written in terms of the Cooper pair number operator ˆn [101]:

ˆUES = 4Ec

(cid:0)ˆn − ng

(cid:1)2 ,

(2.21)

where the total capactive charging energy Ec depends on the intrinsic capacitance of the
junction CJ as well as an external shunting capacitance CS as Ec = e2
2CT

, and the total

capacitance is CT = CS + CJ , and e is the electron charge. The parameter ng represents

the effect of an electric field bias to the system, and can be used to tune the parameters of

23

the system via an external gate. Alternatively, ng can represent spurious coupling to the

local electrostatic environment near (or within) the junction. Putting Eqns. 2.20 and 2.21

together, we arrive at the total Hamiltonian for the system as:

ˆH = 4Ec

(cid:0)ˆn − ng

(cid:1)2 − EJ cos ˆφ.

(2.22)

Equation 2.22 describes a circuit known as the Cooper pair box. Early generation qubit

devices were based on the Cooper pair box in a parameter regime where EJ ∼ Ec. However,

in this regime unwanted fluctuations in the offset charge ng lead to significant fluctuation in

the energy spectrum of the system, making these circuits somewhat difficult to work with

experimentally [12, 56, 102]. To avoid these issues, a seminal theoretical work in the field

of superconducting qubits explored the parameter range EJ (cid:29) Ec, and found that this

parameter regime largely protects the energy spectrum of the qubit from fluctuations in

ng [103]. This range of device parameters is referred to as the transmon regime and devices

within this parameter range of EJ and Ec are called transmon qubits. The transmon qubit

is the workhorse of the results of this thesis and is currently one of the most promising qubits

for modern superconducting qubit architectures [104–108].

2.3.1 Aside: alternative derivation of Josephson energy

In this subsection, we present an alternative derivation of the Josephson energy that does

not use the Josephson relations, but rather begins with a phenomenological Hamiltonian

describing the tunneling of Cooper pairs through the junction, leading to the same result as

Eqn. 2.20. This derivation is presented in full detail in Ref. [109].

Consider a Josephson junction as in Fig. 2.2. If the left superconductor has nL Cooper

24

pairs and the right superconductor was nR Cooper pairs, we can consider the difference

n ≡ nL − nR in Cooper pair number across the junction. Rather than associating the state

of the system with a definite number of Cooper pairs on either superconducting island, we

can think of a state |n(cid:105) that only depends on the Cooper pair difference n. The number

operator for this state can then be written as:

ˆn =

n=∞
(cid:88)

n=−∞

n |n(cid:105) (cid:104)n| ,

(2.23)

which acts on number states as ˆn |m(cid:105) = m |m(cid:105). Then, if a Cooper pair tunnels from the

right side of the junction to left, the state |n − 1(cid:105) becomes |n(cid:105), since nL has increased by

one. The inverse process (tunneling left to right) similarly brings the state |n(cid:105) to |n − 1(cid:105). If

both tunneling events occur with the same probability, we can write down the Hamiltonian

for these processes as:

ˆH = −

EJ
2

n=∞
(cid:88)

n=−∞

(|n − 1(cid:105) (cid:104)n| + |n(cid:105) (cid:104)n − 1|) ,

(2.24)

where the energetic cost for a tunneling event to occur is the Josephson energy EJ . We note

that Eqn. 2.24 resembles a nearest-neighbor hopping interaction between the states |n(cid:105) and

|n − 1(cid:105).

Alternatively, rather than working with the Hamiltonian in the charge basis (consisting

of the states |n(cid:105)), it is useful to introduce a new basis of states (the so-called phase basis).

States in this basis are related to the states |n(cid:105) by:

|φ(cid:105) =

n=∞
(cid:88)

n=−∞

einφ |n(cid:105) .

25

(2.25)

Conversely, the charge basis states are related to the phase basis states via the continuous

Fourier transform:

|n(cid:105) =

1
2π

(cid:90) 2π

0

dφe−inφ |φ(cid:105) .

(2.26)

Inserting Eqn. 2.26 into Eqn. 2.24 and identifying the integral representation of the Kroneker

delta function: δθ,θ(cid:48) = 1
2π

(cid:82) 2π
0

exp[i(θ − θ(cid:48))x]dx, we can express Eqn. 2.24 in the phase basis:

ˆH = −

EJ
2

1
2π

(cid:90) 2π

0

eiφ + e−iφ(cid:17)
(cid:16)

dφ

|φ(cid:105) (cid:104)φ| .

In order to simplify Eqn. 2.27, we introduce the following operator:

ei ˆφ =

1
2π

(cid:90) 2π

0

dφ eiφ |φ(cid:105) (cid:104)φ| ,

(2.27)

(2.28)

and by recasting Eqn. 2.27 in terms of this operator, and combining the resulting exponential

terms, we are left with a simplified version of Eqn. 2.24 in the phase basis:

ˆH = −EJ cos ˆφ,

(2.29)

exactly as we had in Eqn. 2.20. Note that this model allows us to think of creating systems

in which pairs of Cooper pairs can tunnel across the junction (then the model would connect

the states |n(cid:105) and |n − 2(cid:105)), allowing for further protection of the system from certain types

of decoherence [110, 111].

26

2.3.2 Making a two-level system

In the transmon regime, since the offset charge ng does not impact the energy spectrum of

the system, we often write the Hamiltonian of Eqn. 2.22 neglecting the offset charge. In

order to transform Eqn. 2.22 into something that resembles a two-level system, we Taylor

expand the cosine in Eqn. 2.22 to fourth order:

ˆH = 4Ecˆn2 − EJ

(cid:18)

1 −

ˆφ2 +

1
2

1
4!

ˆφ4 + · · ·

(cid:19)

,

(2.30)

where we have assumed that we are in the transmon regime by neglecting ng. Furthermore,

we rewrite the number and phase operators in terms of bosonic raising and lowering operators:

ˆφ =

(cid:18) 2Ec
EJ

(cid:19)1/4 (cid:16)

ˆa† + ˆa

(cid:17)

ˆn =

i
2

(cid:18) EJ
2Ec

(cid:19)1/4 (cid:16)

ˆa† − ˆa

(cid:17)

.

(2.31)

(2.32)

As seen in Eqn. 2.31, the prefactor of the phase operator is related to the zero point fluctua-

tions of that phase. These zero point fluctuations decrease with increasing EJ /Ec, justifying

the use of the Taylor expansion of the cosine term. Substituting Eqns. 2.31 and 2.32 into

Eqn. 2.30, we can rewrite the transmon Hamiltonian in terms of raising and lowering oper-

ators [11]:

ˆH = ωqˆa†ˆa −

Ec
2

ˆa†ˆa†ˆaˆa,

(2.33)

where the qubit frequency ωq =

√

8EJ Ec − Ec has been introduced in terms of the Joseph-

son energy and the capactive charging energy. Recalling the derivation of Eqn. 2.19, the

Josephson energy EJ depends linearly on the junction critical current Ic, which, in the case

27

of a SQUID loop, depends on the external flux. This allows us to dynamically tune the

Josephson energy EJ and therefore the qubit transtion frequency ωq.

Additionally, we can verify that Eqn. 2.33 does in fact give us two individual energy levels

that we can address experimentally. We compute the first and second transition frequencies

of the transmon as:

ω01 = (cid:104)1| ˆH |1(cid:105) − (cid:104)0| ˆH |0(cid:105) = ωq

(2.34)

ω12 = (cid:104)2| ˆH |2(cid:105) − (cid:104)1| ˆH |1(cid:105) = ωq − Ec.

From Eqn. 2.34 we can see ω01 (cid:54)= ω12, which allows us to isolate the lowest two energy levels of

the transmon as a qubit. Of course, there are additional energy levels (see Fig. 2.4), but each

these levels are anharmonically spaced due to the nonlinearity of the Josephson junction.

Furthermore, the anharmonicity in the transmon limit is simply given by the capactive

charging energy Ec, which is a geometric property of the qubit that can be controlled at the

level of the device design.

2.4 Jaynes-Cummings Hamiltonian

Now that we have seen how to create a physical circuit in which we can isolate any two energy

levels, how do we interact with this system? Borrowing from the framework of cavity quan-

tum electrodynamics, in which an atomic transition is coupled to the light field in an optical

cavity [112], we couple the qubit (which has a microwave frequency transition frequency)

to an electromagnetic cavity having a fundamental frequency also in the microwave regime.

How do these two objects interact? The transmon, which we can think of as an artificial

28

Figure 2.4: Effective potential energy versus phase difference provided for a Josephson junc-
tion. The cosine term deviates from the quadratic potential, and produces eigenvalues with
non-equal spacing as indicated by the black dashed lines. For visual clarity, only the lowest
three energy levels have been labeled.

atom, has a dipole moment that can be described in terms of a vector (cid:126)d01 coupled to the

Pauli raising and lowering operators ˆσ+ and ˆσ− as (cid:126)d = (cid:126)d01 (ˆσ+ + ˆσ−). The quantized zero

point electric field in the cavity (cid:126)EZPF is coupled to photon raising and lowering operators ˆa

and ˆa† as (cid:126)E = (cid:126)EZPF

(cid:16)

ˆa + ˆa†(cid:17)

for a single cavity mode. The interaction between the qubits

dipole moment and the cavity field can be written as in classical electrodynamics:

ˆHint = −(cid:126)d · (cid:126)E

(2.35)

By inserting the functional form for the dipole moment and the electric field, we can describe

the qubit-photon interaction in terms of the Pauli operators for the qubit and the electric

field raising and lowering operators, and rewriting the prefactors as the coupling strength g

between the qubit and photons:

ˆHint/(cid:126) = g

(cid:16)

ˆaˆσ+ + ˆa†ˆσ+ + ˆaˆσ− + ˆa†ˆσ−

(cid:17)

.

(2.36)

29

The Hamiltonian of the non-interacting system can easily be described as just the Hamilto-

nian of the qubit plus the Hamiltonian of the cavity:

ˆH0/(cid:126) =

1
2

ωq ˆσz + ωcˆa†ˆa,

(2.37)

where the first term represents the energy of the quantum two-level system and the second

term represents the energy in the cavity, and we have neglected the zero point fluctuations

of the cavity. In order to simplify the description of the composite system, it is useful to

move to the interaction picture, where the operators evolve via their free time evolution:

ˆσ+(t) = ˆσ+(0)eiωqt and ˆa†(t) = ˆa†(0)eiωct and similarly for their Hermitian conjugates. By

looking at the time evolution of each term in Eqn. 2.36 we identify “fast oscillating” terms,

which oscillate at a frequency of ωq + ωc, as well as slowly varying terms, which oscillate

at a frequency of ωq − ωc. In time, the quickly oscillating terms will average to zero, and

thus we can often neglect them when considering the slower dynamics of the qubit-cavity

system. This is an example of the rotating wave approximation, a standard tool in quantum

optics [113]. After making the rotating wave approximation, the full Hamiltonian can then

be written as the famous Jaynes-Cummings Hamiltonian [53, 54]:

H/(cid:126) =

1
2

ωq ˆσz + ωcˆa†ˆa + g

(cid:16)

ˆσ+ˆa + ˆσ−ˆa†(cid:17)

.

(2.38)

Eqn. 2.38 is a central building block vital for understanding the experiments in this thesis.

As we will see, the interaction between the qubit and the cavity will allow for control and

measurement of the qubit state via transmission through the microwave cavity. Additionally,

an analogue of Eqn. 2.38 will be used in chapter 7 to measure the properties of phononic

30

coherent states coupled to a qubit.

In order to investigate the structure of Eqn. 2.38, it is useful to look at the energy

spectrum of this Hamiltonian. How large is the Hilbert space of this system? Well, the qubit

can have either zero or one excitations, and the cavity can host many photons. Solving this

problem seems rather daunting, however the interaction term conserves the total number

of excitations shared between the qubit and the cavity.

Intuitively, we can think of the

term ˆσ+ˆa as converting a cavity photon into a qubit excitation and the term ˆσ−ˆa† as doing

the exact opposite process (converting a qubit excitation into a cavity photon). Because

of this fact, the Hamiltonian is block-diagonal and we can write down the a 2 × 2 matrix

that describes the energy exchange between the states |n, e(cid:105) and |n − 1, g(cid:105), where n is the

number of cavity photons and e (g) represents the excited (ground) state of the qubit. In

this framework, the 2 × 2 Hamiltonian is






ˆH/(cid:126) =

(n − 1)ωc +

ωq
2

√

g

n

√

g

n

nωc −

ωq
2




 .

(2.39)

Diagonalization of this Hamiltonian can be performed by introducing the mixing angle θn

for each subspace with n excitations

tan 2θn =

√
2

ng

∆

,

(2.40)

where ∆ = ωq −ωc is the detuning between the qubit and cavity frequencies. The eigenstates

of the system represent the hybridization of the qubit and cavity modes, and can be written

as

31

Figure 2.5: Left: energy eigenvalues of the Jaynes-Cummings Hamiltonian with a single
excitation (n = 1). When the qubit and cavity modes are tuned near resonance (∆ = 0),
they hybridize and an anticrossing appears. Right: Spectroscopic measurement of a coupled
qubit-cavity system as the resonant frequency of the qubit is tuned through the bare cavity
frequency using an external flux created by supplying a current to a small solenoid. The
dashed black lines are a fit to the Jaynes-Cummings Hamiltonian. In this particular mea-
surement, we extract a coupling strength of g/(2π) ≈ 75 MHz.

|n, +(cid:105) = sin θn |n, g(cid:105) + cos θn |n − 1, e(cid:105)

(2.41)

|n, −(cid:105) = cos θn |n, g(cid:105) − sin θn |n − 1, e(cid:105) .

In Eqn. 2.41, the label +(−) refers to the state with higher (lower) energy. The energy

eigenvalues of the Hamiltonian in Eqn. 2.38 can be computed to be:

(cid:18)

(cid:15)n,± = ωc

n −

(cid:115)

±

ng2 +

(cid:19)

1
2

(cid:19)2

,

(cid:18)∆
2

(2.42)

with corresponding ground state energy (cid:15)0 = −ωq/2.

As introduced in Section 2.3, we often use a flux-tunable transmon qubit. In this case,

we can actually tune the resonant frequency of the qubit through the frequency of the cavity,

and use this measurement to extract the coupling between the two systems. As shown in

32

Fig. 2.5 this is predicted by the energy spectrum of the Jaynes-Cummings Hamiltonian and

a representative measurement of the qubit-cavity system confirms this coupling.

2.4.1 Resonant regime

When the frequencies of the qubit and the cavity are nearly degenerate, i.e. ωq (cid:39) ωc, the

mixing angle between the eigenstates of the Jaynes-Cummings Hamiltonian approaches the

limit θn → ± π

4 (depending on the sign of ∆). In this limit, the two system eigenstates in

Eqn. 2.41 approach equally weighted superposition states of qubit and cavity modes. By

setting ∆ = 0 in Eqn. 2.42, we see that these two energy levels are split by 2

√

ng, which

implies that energy is “traded” back and forth between qubit and cavity excitations at a rate

g. Of course, in order for the two systems of interest to exchange quanta in this coherent

fashion, the coupling rate g must be the dominant rate in the system and significantly exceed

the decay rate of both the qubit and the cavity modes. Since both components of the hybrid

system are equally important in this resonant regime, the decay of the hybridized modes are

given by (γ + κ) /2, where γ is the decay of the bare qubit mode and κ is the decay of the

bare cavity mode.

2.4.2 Dispersive regime

In order to leverage the cavity as a tool for measuring the qubit state, we focus on the

so-called “dispersive regime” of circuit quantum electrodynamics [10, 11, 103, 114]. In this

regime, the detuning between the qubit and the cavity is relatively large (g/|∆| (cid:28) 1). Rather

than directly diagonalize Eqn. 2.38, one can use perturbation theory to find corrections to

the energy to second order in Eqn. 2.38. Several excellent derivations of this procedure can

33

be found in Refs. [10, 115, 116]. By expanding Eqn. 2.38 in this fashion, we arrive at the

dispersive Jaynes-Cummings Hamiltonian:

ˆH/(cid:126) =

(cid:18)

ωc +

(cid:19)

g2
∆

ˆσz

ˆa†ˆa +

ωq
2

ˆσz.

(2.43)

Importantly, we see that the effective frequency of the cavity has picked up an additional

term that depends on ˆσz, meaning that the resonant frequency of the cavity will change by
an amount 2χ ≡ 2 g2

∆ when the qubit is driven from its ground to excited state (or vice-versa).

The parameter χ is called the dispersive shift. This qubit state-dependent cavity frequency

shift is a tool that we will exploit experimentally in order to determine the state of the

qubit via measurement of the cavity transmission. Since Eqn. 2.43 is vital to the

experiments presented in this thesis, it is useful to gain some intuition regarding how the

interaction between the cavity and qubit can be recast this way. Earlier, we mentioned that

this Hamiltonian comes about via perturbation theory, so it is instructive to look at the

energy shifts in second order perturbation theory, which are given by:

(2)
n,p =

E

(cid:88)

(cid:88)

n(cid:54)=n(cid:48)

p(cid:54)=p(cid:48)

(cid:10)n(cid:48), p(cid:48)(cid:12)
(cid:12) g

(cid:16)

ˆσ+ˆa + ˆσ−ˆa†(cid:17)

|n, p(cid:105)

(cid:12)
(cid:12)
(cid:12)

(0)
n,p − E

E

(0)
n(cid:48),p(cid:48)

2

(cid:12)
(cid:12)
(cid:12)

.

(2.44)

In Eqn 2.44, the indices n and n(cid:48) represent the cavity photon excitation number, and the

indices p and p(cid:48) take on the values ±1, representing the qubit “parity” (physically, this

is just the ground or excited state of the qubit). The numerator in Eqn. 2.44 represents

the interaction strength between unperturbed states of the system and the denominator

is equal to the energetic differences of the unperturbed states that are interacting. Be-

cause the Jaynes-Cummings Hamiltonian can be represented as a block-diagonal matrix, we

34

would only expect coupling between the levels |n, g(cid:105) and |n − 1, e(cid:105). As seen in Eqn. 2.39,

the coupling elements are simply given by the off diagonal elements g

√

n. Therefore for

either possibility of the qubit state ˆσz, the magnitude of the numerator of Eqn. 2.44 sim-

ply becomes ng2. The denominator, depending on the state of interest, takes the value

of ±∆. For example, if the unperturbed state is |n, g(cid:105), the denominator simply becomes

En,1 − En−1,−1 = (cid:0)nωc − ωq/2(cid:1) − (cid:0)(n − 1)ωc + ωq/2)(cid:1) = −∆, and vice versa for the state

|n − 1, e(cid:105). As such, we can write the perturbing Hamiltonian in second order as [117]:

ˆH(2) = 2χˆa†ˆaˆσz.

(2.45)

This perturbation can be grouped with terms in the free Hamiltonian (Eqn. 2.37) to express

the frequency of the cavity as “dressed ” by the state of the qubit. The effect on both

the cavity amplitude and its phase response can be seen in Fig. 2.6. However, in practice,

the transmon qubit is not a simple two level system. Additional bound states within the

confining potential are also present and slightly modify the dispersive shift χ. To illustrate

this we follow Ref. [103] and define the total dispersive shift χ as having contributions from

two partial dispersive shifts caused by the interaction between neighboring transmon levels:

χ = χ01 −

χ12
2

,

(2.46)

where χij = g2

ij/(ωij − ωc), and the coupling strengths between the cavity field and the

transmon levels scale according to gn,n+1 ≈ g

√

n + 1 [114], where g is the coupling rate

from the Jaynes-Cummings Hamiltonian. Simplifying Eqn. 2.46 under these assumptions

yields the expression for χ applicable to transmon-based circuit quantum electrodynamics

35

Figure 2.6: Left: amplitude response of a cavity dispersively coupled to a qubit. The state of
the qubit dresses the resonant frequency of the cavity. Right: Phase response of the cavity
under the same conditions as the left panel.

experiments:

χ = −

g2
∆

(cid:18) α

∆ − α

(cid:19)

,

(2.47)

where α = Ec is the anharmonicity of the transmon.

2.5 Microwave cavities for cQED

At this point in the thesis, we have described how we might theoretically realize a quantum

two-level system and its interaction with a harmonic oscillator. Because this is a thesis on ex-

perimental physics, we need to learn how to put these ideas into practice in a meaningful way.

Fabrication of Josephson junctions is relatively easily done using standard nanofabrication

procedures (see Ref. [118]), and in section 2.4 we derived the Jaynes-Cummings Hamiltonian

under the assumption that we would couple the dipole moment of the qubit to the electric

field in a microwave cavity. Our next task is to create a physical system that behaves like

a quantum harmonic oscillator and provides the necessary dipole coupling mechanism for

the Jaynes-Cummings Hamiltonian. Our lab uses the standard “3D” geometry, in which

the qubit is actually housed inside a three dimensional microwave cavity [119], rather than

coupling the devices on the same chip, as pioneered in Ref. [12]. To create a sufficiently large

36

Figure 2.7: (a) Image of a fully assembled 3D cavity. The microwave input/output SMA
ports are attached to the cavity and a superconducting solenoid responsible for tuning the
qubit frequency can be seen. (b) Finite element method simulation of the normalized spatial
voltage profile Vnorm of the TE101 mode of the microwave cavity. The resonant frequency of
this mode is ωc/(2π) = 6.98 GHz. The voltage antinode at the center of the cavity allows
the electric field to couple to the dipole moment of the qubit. (c) Image of a transmon qubit
housed within the bottom half of a microwave cavity. The qubit dipole antenna of dimension
250 µm × 500 µm are visible. (d) Image of an “open” cavity. The inset inside the bottom
half of the cavity allows us to secure the substrate containing the qubit within the cavity.

dipole moment to experimentally realize significant coupling g, we fabricate large aluminum

antenna paddles in series with the Josephson junction as seen in Fig. 2.7(c). These antenna

paddles are also responsible for creating a large shunting capacitance, ensuring that we can

achieve the correct value for Ec and subsequently reach the transmon regime. In order to

create a physical system that both behaves like a quantum harmonic oscillator and can house

the qubit, a slot of an appropriate dimension is milled out of two halves of bulk conducting

material (typically either copper or aluminum). To estimate the resonant frequency of such

a cavity, we can approximate the cavity as a 3D rectangular box with perfectly conducting

walls. For a cavity having a volume V = Lx × Ly × Lz, where Lx,y,z specify the dimension

37

of the cavity, the resonant electromagnetic frequencies are [120]:

ωnlm = c

(cid:115)(cid:18) nπ
Lx

(cid:19)2

(cid:19)2

+

(cid:18) lπ
Ly

+

(cid:19)2

,

(cid:18) mπ
Lz

(2.48)

where the indices n, l, and m are integer values that index the mode number of the cavity.

As seen in Fig. 2.7, the microwave cavities we use are macroscopic (you can hold them

in your hand!) How can we think of something this large as a single quantum object such

as a harmonic oscillator? We often use the lowest frequency mode (the TE101 mode of the

cavity) as our mode of interest. By ensuring that the hybrid system is operated at sufficiently

low temperatures ((cid:126)ω (cid:29) kbT ), where ω is the relevant frequency of both the qubit and the

cavity modes, the hybrid quantum system should be passively cooled near its ground state

and we can think of the microwave cavity as a collection of harmonic oscillators each with

some transition frequency ωk, which only depends on the geometry of the cavity. For a

sense of scale, the relationship between temperature and frequency can be approximated

as 5 GHz ≈ 240 mK. If the quality factor of each cavity mode is sufficiently large (or,

equivalently, the linewidth of each mode is sufficiently small), we can think of each mode k as

an independent harmonic oscillator that do not interact with each other, and the Hamiltonian

can be written as:

(cid:88)

ˆH =

ωkˆa†

kˆak.

k

(2.49)

By ensuring that the frequency of both the qubit and cavity mode of interest are largely

detuned from the other cavity modes, we can think of our microwave cavity as a single

mode cavity. Although this approximation holds very well for the results of this thesis, the

presence of these additional cavity modes will contribute to both qubit dephasing and energy

decay [81, 121]. Additionally, the qubit within the cavity slightly modifies the spatial profile

38

of all cavity modes [122].

2.5.1 Aside: experimental preparation of coherent states

As we apply a classical drive signal to our microwave cavity, all of the transitions in the

harmonic oscillator are simultaneously driven, leading to a coherent state.

In this sub-

section, we provide a brief refresher on coherent states of the harmonic oscillator and how

they can be realized in the lab. The wavefunction for a harmonic oscillator in a coherent

state is given by:

|α(cid:105) = e−

|α|2
2

∞
(cid:88)

n=0

αn
√
n!

|n(cid:105) .

(2.50)

Here |n(cid:105) are the Fock states of the resonator, and |α|2 = (cid:104)n(cid:105) is the mean excitation number.

It is then straightforward to calculate the probability of the resonator having m excitations:

| (cid:104)m|α(cid:105) |2 =

|α|2
2

e−

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

∞
(cid:88)

n=0

αn
√
n!

δmn

2

(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)

.

(2.51)

Carrying out this calculation, one finds the probability of having m excitations in the res-

onator, given a coherent state with n mean excitations:

Pn(m) = e−n nm

m!

(2.52)

The result, Eqn. 2.52, is very important, it shows that populating a resonator with a coherent

state leads to a Poisson probability distribution, which in general, is asymmetric about the

mean. Coherent states can be generated via the following displacement operator :

ˆD(α) = exp(αˆa† − α∗ˆa).

(2.53)

39

By using the well known operator identity [123]: e

re-write the displacement operator as

ˆAe

ˆB = e

ˆA+ ˆBe− 1

2 [ ˆA, ˆB], it is possible to

ˆD(α) = e− 1

2 |α|2

eαˆa†

e−α∗ˆa.

(2.54)

From this point, it is possible to identify the following properties of the displacement oper-

ator [5]:

ˆD†(α) = ˆD−1(α) = ˆD(−α)

(2.55)

ˆD†(α)ˆa ˆD(α) = ˆa + α

ˆD†(α)ˆa† ˆD(α) = ˆa† + α∗.

Furthermore, by applying the identities listed in Eqns. 2.55 to the vacuum state |0(cid:105), it is

possible to identify the displacement operator as the one that creates a coherent state from

the vacuum:

|α(cid:105) = ˆD(α) |0(cid:105) .

(2.56)

Finally, we turn to the functional form of Eqn. 2.53, we see that the argument is (up to a

phase) the same as that of the cavity electric field described in Section 2.4, indicating that

displacing a harmonic oscillator in phase space with a time varying electric field will generate

a coherent state! This detail will be an important point for the key results of chapter 7.

40

2.6 Conclusion

This chapter has introduced circuit quantum electrodynamics (cQED). We began by de-

scribing a quantum mechanical representation of the LC oscillator. We then discussed the

Josephson junction as a means for creating non-linear circuits, and this allowed us to create

a quantum mechanical two level system. We then introduced the Jaynes-Cummings Hamil-

tonian and explored how to measure the state of the qubit via a microwave cavity. The next

chapter will describe the experimental measurement techniques of cQED using the key ideas

presented in this chapter.

41

Chapter 3

Measurement in Circuit Quantum

Electrodynamics

3.1

Introduction

The previous chapter introduced the theoretical framework of circuit quantum electrody-

namics (cQED), allowing us to couple a superconducting qubit to a microwave cavity. This

chapter aims to expand upon the theoretical background outlined in the previous chapter

and introduce several experimental techniques for characterization of the cQED and circuit

quantum acoustodynamics (cQAD) systems studied in this thesis. We will begin by dis-

cussing the cryogenic system and the key electronic components in our experimental setup.

We then describe basic spectroscopic measurements of the hybrid qubit/cavity system. These

spectroscopic measurements serve as a calibration for the more complicated measurements

of the qubit in the time domain, which are important for the open quantum systems work of

chapter 6. We describe the latest “upgrade” to the experimental measurement capabilities in

our lab, i.e. integrating our superconducting qubit measurements with a field programmable

gate array (FPGA) measurement setup allowing for increased control of the system while

simultaneously increasing the rate at which we can perform measurements. Finally, we note

that a complete description of the original measurement setup can be found in the thesis of

42

J. R. Lane (Ref. [118]).

3.2 The cryostat and input/output lines

We typically operate our superconducting qubits and microwave cavities with resonant fre-

quencies in the GHz regime (a small aside:

it is possible to operate some special types

of superconducting qubits at much lower frequencies, even as low as tens of MHz, see

Refs. [124–127]). In order for these GHz-frequency systems to be passively cooled to near

their quantum ground state, the systems must be cooled to very low temperatures relative

to their transition frequencies (recall that 5 GHz (cid:39) 240 mK). To ensure near-unit probabil-

ity that both the qubit and the cavity are thermally initialized to their ground state, our

lab employs a cryogen-free BlueFors LD-400 dilution refrigerator having a nominal lowest

temperature below 10 mK (in practice, we often introduce enough experimental heat load to

the cryostat that the minimum temperature is closer to 15 mK). Even so, we typically infer

the effective temperature of the cQED system via measurements of the residual excited state

population of the qubit (see Refs. [128,129] for a description of this experimental technique),

and find that the qubit-cavity system is at a higher temperature than the thermometer at

the lowest temperature stage (the mixing chamber) of the cryostat to which the experiment

is mounted [130, 131]. Nonetheless, operating the hybrid system at these low temperatures

serves as an effective starting point for our experiments, largely initializing (on average) all

the components of the hybrid system near their quantum mechanical ground states.

At this point, we have emphasized that we will need to implement coupling between

a qubit and a microwave cavity to carry out the experiments relevant to this thesis. We

have not, however, described how we get the relevant GHz-frequency drive signals into the

43

cryostat or how we will preserve the “quantum” integrity of the output signals. To explain

the measurement wiring in the cryostat we will trace the path from room temperature, down

the input line to the hybrid system, and back out through the measurement output line (see

Fig. 3.1). On the input line, we attenuate the signal by 60 dB (six orders of magnitude in

the power of the signal). By strongly attenuating the input control signal, we also attenuate

any higher frequency electromagnetic black-body radiation from higher temperature stages

within the cryostat. This thermal radiation can exist at frequencies comparable to the reso-

nant frequency of the cavity modes, which can lead to spurious population of photons in the

cavity and thus destroy the phase information of the qubit [121, 132]. This electromagnetic

radiation can also exist at frequencies comparable to the superconducting gap of aluminum

and can therefore lead to qubit depolarization via the breaking of Cooper pairs [133]. We

distribute the attenuation over three different stages within the cryostat to minimize resistive

heating in the attenuator at any one stage. These attenuators also provide a thermal contact

between the inner pin of the coaxial cable and ground. In addition to attenuating the input

signal, we also use two different types of filters at the mixing chamber stage of the cryostat

to further condition the drive signal applied to the qubit and cavity. We use commercially

available low pass filters (LPFs) (K&L Microwave 6L250-00088), which strongly filter any

signals at frequencies higher than the operating frequency of a given experiment (typically

a cut-off frequency of 8 GHz is sufficient for our purposes). In order to further protect the

system from spurious high-frequency photons, we also connect homemade epoxy-based filters

(called “Eccosorb” filters) based on the recipe in Ref. [134] in series with the input line.

In order to protect the system from unwanted radiation coming down the output line,

we protect the experiment with two cryogenic microwave circulators (Raditek RADC − 4 −

8 − Cryo − 4 − 77K − S23 − 1WR − MS − b) connected in series and thermally anchored

44

to the mixing chamber. Each circulator provides approximately 30 dB of isolation from

counter-propagating signals, leading to a total of 60 dB of attenuation for signals directed

down the measurement chain towards the experimental cell. Interestingly, there have been

several recent advances towards integrating on-chip non-reciprocal circuit elements for further

protection from noise via the output line [135, 136].

Since we significantly attenuate the input excitation signal and would like to measure the

small signals produced by the experiment, we need to incorporate low-noise amplification

to increase the size of the output of the measurement circuit in order to have signals of

large enough amplitude to reliably measure. An amplifier will not only amplify the signal

at its input, but also any noise “on top” of that signal. Even worse, the amplifier will

also add additional noise to the output signal. This added noise is often characterized

by an effective noise temperature and it can be shown that for a chain of amplifiers in a

circuit, the total noise temperature for the amplification chain depends most strongly on

the noise temperature of the first amplifier in the chain [80] (by first we mean the first

amplifier that the signal reaches coming from the experiment). The first amplifier in our

measurement chain is a commercially available High Electron Mobility Transistor (HEMT)

amplifier made by Low Noise Factory (LNF) (LNF − LNC03 14A) mounted on the 4K

stage of the cryostat that provides (cid:39) 42 dB of gain with an effective noise temperature of

approximately 3.5 K over its functional bandwidth (0.3-14 GHz). We further amplify the

signal at room temperature using two low-noise high frequency amplifiers (Minicircuits ZX60-

83-LN-S+) for a total gain of (cid:39) 86 dB. Refs. [118, 130] provide a more detailed description

of the experimental circuitry we typically use in our experiments. Since the first amplifier in

the amplification chain is in some sense the most important (due to its contribution to the

total noise temperature), the addition of a quantum limited amplifier would be of great use

45

Figure 3.1: Schematic of the cryostat microwave wiring showing the relevant filtering, atten-
uation, and amplification.

for our lab. Josephson parametric amplifiers (JPAs) [137–139] and traveling wave parametric

amplifiers (TWPAs) [140,141] are becoming of increasing importance for high-fidelity, single

shot measurement in cQED systems. Our lab has recently purchased a commercially available

JPA (Raytheon BBN-PS2-JPA-DEVICE-QEC) that should (in principle) offer a minimum

gain of 20 dB over an instantaneous bandwidth of 300 MHz while adding minimal noise to

the output signal (the quoted noise temperature of this JPA is approximately 300 mK). At

the time of the writing of this thesis, calibration of the JPA in our system has yet to be

done, but will offer significant advances in the readout fidelity in our experiments.

3.3 High-Power cQED readout

Having described how we will get signals into and out of our experimental system, we will

introduce the basic measurements we carry out when cooling down a sample for the first time.

Often, the first question we ask upon reaching base temperature is: “Is the qubit alive?”

By this, we really mean: “Is there a properly functioning Josephson junction whose dipole

46

Eccosorb300K50K4KStillCold PlateMXCRoom temperature amplifiers0 dB0 dB0 dB0 dB0 dB0 dB-20 dB-20 dB0 dB-20 dBHEMTAmplifierAttenuatorcQED systemK&L FilterLow pass filterCirculatorInputOutputK&L FilterEccosorbK&L FilterEccosorbFigure 3.2: Transmission through the microwave cavity at low power (Pin = −30 dBm (blue))
compared to the “bare” cavity transmission observed at high power (Pin = −10 dBm (red)).
In this measurement the measurement power is taken to be the input power at the top of
the cryostat. As the cavity becomes populated with many photons, the resonant frequency
of the cavity shifts from the dressed cavity frequency to the bare cavity frequency.

moment is coupled to the cavity electric field?” In order to answer this simple, yet important,

question we typically first do what we colloquially call a “punch-out” measurement [142]. In

this type of measurement, we monitor the transmission through the microwave cavity using a

vector network analyzer (VNA) (Agilent N5230A PNA-L Network Analyzer). As we increase

the amplitude of the input signal, we measure a shift in the resonant frequency of the cavity

from ωc + χ at low measurement power to ωc at larger measurement power, where ωc is the

bare cavity frequency and χ is the dispersive shift due to the coupling between the qubit

and cavity. As we increase the amplitude of the input signal, a model of the hybrid system

has two solutions that destructively interfere, creating a dark state in the transmission of

the signal. As the power increases to a critical power (this critical power depends on the

state of the qubit) the system supports a “bright state” at the bare cavity frequency ωc.

The key concept that makes this method of qubit state readout possible is that the onset

of the bare cavity response (the aforementioned bright state) in Fig. 3.2 occurs at a lower

power if the qubit is in the state |1(cid:105) versus the state |0(cid:105) [142–144]. This means that if we

47

6.946.956.96Frequency (GHz)806040S12 (dB)Bare CavityDressed Cavitysend in a high-power microwave signal to the cavity at the bare cavity frequency ωc at near

the nominal “punch-out” power, the transmission through the cavity will be large when the

qubit is in state |1(cid:105) as compared to the transmission through the cavity when the qubit is in

state |0(cid:105). If the qubit is not functioning, the resonant frequency of the cavity is independent

of the input power.

3.4 Cavity spectroscopy

Many of the important experiment we carry out involve a transmon qubit in a SQUID loop

geometry (see Section. 2.3), and we need to determine if the qubit responds to an external

tuning magnetic field. Therefore, our next calibration measurement is typically to measure

the response of the cavity at low drive powers (such that the cavity frequency is dressed

by the qubit) as we tune the resonant frequency of the qubit with a magnetic flux. To

supply this flux though the SQUID loop of the transmon, we wind a copper-clad niobium

(TC (cid:39) 9K) twisted pair wire around the microwave cavity 30 times. This twisted pair is then

thermalized at each stage within the cryostat and connected to a Keithley 2400 SourceMeter

at room temperature. Based on the coil geometry (the number of times the wire is wound

around the cavity and the area of the SQUID loop), we estimate that we thread a single flux

quantum through the SQUID loop in this geometry every Iwire (cid:39) 125 mA, where Iwire is

the DC current through the coil sourced at room temperature. We then set the current to a

particular value and measure the response of the cavity before stepping to the next current.

This allows us to tune the qubit through its entire range of possible frequencies (determined

by the geometry of the system and the maximum Josephson energy EJ ). As we tune the

qubit resonance through the resonance of the cavity, we fit the data to a model of coupled

48

oscillators and extract the coupling rate between the qubit and the cavity (see Fig. 3.3).

Figure 3.3: Transmission through the microwave cavity as an external flux tunes the resonant
frequency of the qubit. As the detuning ∆ between the qubit and cavity modes approaches
zero, an avoided crossing between the modes is observed. The spectrum is fit to a model
of coupled resonances (black dashed line) to extract the qubit-cavity coupling rate, which is
g/(2π) = 75 MHz in this case.

3.5 Qubit spectroscopy

Having described the effect of the qubit state on the cavity, we next turn our attention

to a more direct measurement of the qubit spectrum using a method known as two-tone

spectroscopy [56]. This form of spectroscopy allows us to infer the absorption spectra of

the qubit by leveraging the dispersive interaction between the qubit and the cavity. In this

measurement protocol we continuously monitor the transmission through the cavity at the

dressed cavity frequency ωc +χ on the VNA while simultaneously sending in another variable

frequency signal into the cavity using a continuous wave (CW) microwave source (Berkeley

Nucleonics Model 845). When this second tone is resonant with the qubit frequency, the

49

603003060Flux Tuning Current (mA)4.744.784.82Probe Frequency (GHz)504030S12 (dB)qubit state will oscillate between ground and excited states, and the cavity frequency will

shift depending on the state of the qubit [11]. On the timescale of our measurement (we

normally average the cavity transmission at a given qubit probe frequency for ∼1 second),

the qubit is, on average, in the excited state with probability of 1/2. This leads to an

average shift of the resonant frequency of the cavity and therefore a drop in transmission

at the frequency ωc + χ. This dip in cavity transmission allows us to measure the qubit

frequency ωq to high precision. In fact, measurement of the qubit spectra can allow us to

learn about the population of the cavity modes, since the dispersive interaction between the

qubit and the cavity also impacts the frequency of the qubit (see Eqn. 2.43). This feature is

the crux of the surface phonon interference measurements presented in chapter 7.

If we increase the power of the signal driving the qubit transition in this two-tone mea-

surement, we can excite the two photon transition of the transmon state from |0(cid:105) → |2(cid:105). The

frequency at which this transition occurs is ω02/2 (since we require two photons to actually

drive this transition). Assuming the device is operated sufficiently far into the transmon

regime (EJ /Ec (cid:29) 1) this allows us to extract the anharmonicity α = 2 × (ω02/2 − ω01).

Fig. 3.4 shows a representative measurement of both the ω01 and ω02/2 transitions (note:

this two-photon transition should be forbidden by selection rules, but can be observed in

cQED systems by using large drive powers [114]). Finally, if the qubit is strongly interacting

with another quantum system (such as an acoustic resonator mode), we can tune the fre-

quency of the qubit through its resonance and extract the coupling strength between qubit

and this additional system. As we will see in chapter 6 we use this protocol to measure

the coupling between the qubit and a surface acoustic wave mode in our hybrid quantum

acoustic system (see Fig. 6.3).

50

Figure 3.4: Left: measurement of the qubit ground to excited state transition via two-tone
spectroscopy. The qubit frequency is extracted from the dip in the spectrum. Right: mea-
surement of the two photon transition from the state |0(cid:105) to |2(cid:105). From these measurements,
we can extract an anharmonicity α/(2π) (cid:39) 320 MHz for this particular device.

3.6 Coherent control

Up to this point in this chapter we have described only continuous wave (CW) measurements.

In these types of experiments, the measurement time is often much longer than the qubit

lifetime, and therefore we are unable to directly measure either the direct depolarization or

decoherence of the qubit in time. In this section, we will modify the measurement circuit

such that we can perform measurements in the time domain.

In particular, rather than

measuring the spectrum of either the qubit or cavity on the VNA, we use an arbitrary

waveform generator to synthesize pulses to a high precision in the time domain and quickly

measure the output signal from the device with an analog-to-digital converter (ADC). The

crux of this measurement circuit is an arbitrary waveform generator (AWG) with sampling

rate 1 GS/s (yielding a minimum time resolution of 1 ns for pulses). Our lab uses a four

channel AWG (Tabor Electronics WX2184C), which we often just refer to as “the Tabor”.

We use microwave frequency IQ mixers to modulate the microwave signals we send into the

cryostat. An IQ mixer is a four-port device with three inputs, we use two of the inputs to the

51

4.264.34.34Probe Frequency (GHz)Transmission (a.u.)01/(2) = 4.32 GHz4.14.154.2Probe Frequency (GHz)02/(4) = 4.16 GHzIQ mixer as low-frequency modulation signals, which we call I(t) and Q(t). Additionally, we

apply an signal using a CW microwave source at a frequency ω to the local oscillator (LO)

port of the mixer (the 3rd input port) LO(t) = A cos (ωt). Two output signals from the

Tabor at frequency ωm are sent into the I and Q ports of an IQ mixer nominally 90 degrees

out of phase with each other, yielding the following time dependence:

I(t) = Amod cos (ωmt + φ)

Q(t) = Amod sin (ωmt + φ),

(3.1)

(3.2)

where Amod is the amplitude of the pulse and φ is the phase shift of the oscillatory signals

in the pulses. Under these conditions with the signals going into the I, Q, and LO ports, it

may be shown that the signal out of the RF port will be:

RF (t) = A × Amod cos ((ω + ωm) t + φ),

(3.3)

where A is the amplitude of the signal into the LO port. Eqn. 3.3 shows that we can control

the relative amplitude, frequency, and phase of the output of an IQ mixer if we have precise

control over the parameters of the pulses going into the I, Q, and LO ports of the mixer.

As such, the signals from the Tabor allow us to quickly synthesize pulses with well-defined

values for these parameters.

The typical mode of operation for time domain experiments in our lab is to set the LO

source at a frequency ωq −ωm, where ωq is the qubit frequency and ωm is the modulation fre-

quency on the Tabor output (we typically use a modulation frequency of ωm/(2π) = 150 MHz,

although it certainly is possible to use other values). We then tune the parameters of the

52

signals into the I and Q ports of the IQ mixer while monitoring the signal out of the RF

port on a spectrum analyzer (Agilent 8592B) in order to verify that the signal sent into the

cryostat only has significant spectral components at the frequency ωq. By uploading a file

containing information to encode a pulse sequence to the Tabor having the appropriate pulse

parameters (i.e. the length, amplitude, or phase of the pulse in Eqn. 3.3), we may perform

measurements of the system as a function of these pulse parameters.

A first calibration measurement of the qubit state in the time domain is the oscillation

between its ground and excited state, called Rabi oscillations [145]. By applying a microwave

signal at frequency ωq to the qubit and measuring the probability that the qubit is in its

excited state as a function of the length of the pulse, we can verify that we are able to drive

the qubit from its ground to excited state (and measure this oscillation via the transmission

through the microwave cavity). We repeat the pulse sequence many times (typically we

average on the order of 1000 measurements) and bin the measured transmission through the

microwave cavity in a histogram, allowing us to discriminate between the qubit ground and

excited state (see Fig. 3.5). A representative Rabi oscillation for a transmon in our lab is

shown in Fig. 3.5.

Measurement of Rabi oscillations also serves as an important calibration for subsequent

experiments: for a given set of drive parameters we are able to measure how long it takes our

microwave signal to drive the qubit from the ground state to excited state (or vice versa).

We refer to a pulse that drives the transition |0(cid:105) → |1(cid:105) as a π-pulse, since this corresponds

to rotation of π radians of the qubit state vector on the unit sphere (also called the Bloch

sphere).

Having calibrated the π-pulse for a given set of drive parameters, we are then able to

measure the coherence properties of the qubit in the time domain. By applying a π-pulse to

53

Figure 3.5: Left: Representative Rabi oscillation between the qubit ground and excited state.
By fitting the data (blue dots) to a decaying sinusoid (dashed grey line), we extract a Rabi
frequency of Ω/(2π) = 30 MHz in this case. Right: Representative distribution of readout
signals from the system. In this particular measurement we prepare the qubit in its ground
state, measure the transmission through the cavity and repeat the process, preparing the
qubit in its excited state. The vertical dashed line represents the threshold value for the
cavity transmission. We attribute all signals to the left of the dashed line as corresponding
to the qubit in its ground state and all measurements to the right of the dashed line as the
qubit excited state.

the qubit and preparing the |1(cid:105) state, we are able to measure the probability that the qubit

remains in this state as a function of time. Colloquially, this type of measurement asks the

question “If I give the qubit a single quanta of energy, how long, on average, does the qubit

remain in its excited state?” By fitting the resulting (typically) exponential decay of this

measurement as a function of time, we can extract the qubit energy relaxation time, which

is referred to as T1. See Fig. 3.6(a) for a representative T1 measurement and corresponding

fit.

Figure 3.6: (a) Representative qubit energy decay measurement. The relevant pulse sequence
is shown in the inset. (b) Representative Ramsey decay measurement. The microwave pulse
sequence is shown in the inset. For ease of fitting, we often detune the drive pulses from the
qubit transition frequency, leading to the oscillation seen in the data.

54

0100200Drive Duration (ns)0.40.60.8P(e)/(2)=30MHz300Signal (a.u.)05001000Counts|e|gIn order to measure the phase coherence of the qubit, we apply what we call a π/2-

pulse, which is identical to a π-pulse, except having half the duration (additionally we

could, in principle, keep the length of the pulse the same and divide the amplitude of the

pulse by 2). This change to the pulse duration then creates a superposition state of the

qubit |ψ(cid:105) = 1/

√

(cid:16)

2

|0(cid:105) + eiφ |1(cid:105)

(cid:17)

, where the phase φ is set by the phase of the pulse (see

Eqn. 3.3). After preparing a superposition state, we let the state evolve for a variable time

before applying another π/2-pulse with the same phase as the first and measure the state

of the qubit. If these pulses are detuned from the qubit frequency ωq by an amount ∆, the

resulting measurement yields a decaying sinusoid having frequency ∆. The decay constant of

the sinusoid is called the Ramsey decay time, T2R (see Fig. 3.6(b)). In general, throughout

the process of this measurement, the qubit state can lose both phase information as well as

emit energy (i.e. undergo a T1 process). Together, the two decay times T1 and T2R allow us

to calculate the pure dephasing time Tφ as:

Tφ =

(cid:18) 1
T2R

−

1
2T1

(cid:19)−1

.

(3.4)

Note that in order for the dephasing time Tφ to be a positive number, it is physically

required that 2T1 ≥ T2R. Dephasing processes occur at a rate Γφ = 1/Tφ and are produced

by variations in the qubit frequency ωq in time. If these variations are reversible in time, we

could actually maintain the phase information in the qubit state for a longer duration. The

addition of a π-pulse centered between the two π/2-pulses in a Ramsey decay measurement

“refocuses” low-frequency noise and actually increases the measured decay constant. This

is an example of a Hahn echo measurement, a technique developed for nuclear magnetic

resonance (NMR) experiments [146] where the “echo” represents the refocusing of the qubit

55

state subject to the aforementioned time reversible noise. We often call the resulting decay

constant from a Hahn echo measurement T2,echo, which is related to the Ramsey decay

constant as T2,echo ≥ T2R. However, in our devices, we often find that T2,echo (cid:39) T2R,

indicating that this type of low-frequency, time reversible, phase noise is not the dominant

source of dephasing. This echo experiment can be thought of as applying a filter function

in the frequency domain, effectively shaping the noise power spectrum experienced by the

qubit. By applying many echo pulses to the system throughout a measurement of the phase

coherence, one can effectively filter out low frequency noise in the system (this frequency

cutoff is related to the number of echo pulses), a technique known as a Carr-Purcell-Meiboom-

Gill (CPMG) measurement [147].

3.7 Quantum state tomography

A standard measurement of a superconducting qubit will give us the expectation value of the

qubit state in the z -basis (i.e. we measure (cid:104)σz(cid:105)). However, this measurement does not provide

any information about the phase of the qubit state vector. In order to learn something about

this phase one can apply a π/2-pulse about either the x or y-axis of the Bloch sphere to

re-project any component of the qubit vector to one of the poles of the Bloch sphere. In order

to fully characterize the qubit state vector, we often repeat a given measurement three times,

one measurement without an extra π/2-pulse, and two more measurements with the extra

π/2-pulse about either the x or y-axis of the Bloch sphere. Together, these measurements

allow us to reconstruct both the amplitude and phase of the qubit state vector, and this

measurement technique is known as quantum state tomography. An example pulse sequence

of state preparation and tomography measurement is shown in Fig. 3.7.

56

Figure 3.7: Representative pulse sequence for quantum state tomography. By applying a
π/2 rotation about the x-axis (y-axis) of the Bloch sphere, we project the y-component (x-
component) of the qubit state vector onto the z-axis of the Bloch sphere and subsequent
measurement in the z basis allows us to measure (cid:104)σy(cid:105) ( (cid:104)σx(cid:105)). To measure (cid:104)σz(cid:105), no additional
rotation is required.

The re-projection pulse is turned off to measure (cid:104)σz(cid:105). In order to measure (cid:104)σx(cid:105), the tomog-

raphy pulse should have relative phase 90◦ (assuming that the x-axis of the Bloch sphere

is defined as driving the system with relative phase 0◦). And finally to measure (cid:104)σy(cid:105), the

tomography rotation pulse should have a relative phase of 0◦, assuming the same phase

convention.

Measurement fidelity calibration

Unfortunately, we never actually measure the qubit with perfect fidelity. This leads to

an arbitrary signal difference between the qubit excited and ground states. For a simple

measurement of the qubit coherence time this does not matter, it only leads to a reduction

in the overall amplitude or contrast of the signal and does not impact the extracted coherence

time. However, for a tomography measurement, we hope to learn about the dynamics of the

expectation values of the qubit projection along a given axis of the Bloch sphere, which

should be bounded by ±1. In order to rescale the experimental data to conform to these

57

readout(π2)x⟨σy⟩readout(π2)y⟨σx⟩bounds, an additional calibration of the measurement fidelity is needed.

In general, one

needs to calibrate the measurement fidelity along six different axes of the Bloch sphere

(|e(cid:105) , |g(cid:105) , |+x(cid:105) , |−x(cid:105) , |+y(cid:105), and |−y(cid:105)). This is done by preparing each of these states and

re-projecting it to a pole of the Bloch sphere where a standard measurement of the qubit

state gives the relative measurement contrast for that state.

A representative measurement outcome for this calibration protocol is shown in Fig. 3.8:

Figure 3.8: Representative calibration of all three Bloch sphere axes

For the calibration data shown in Fig. 3.8, we can see that the signal for the “excited state”

is approximately 0.8, and the signal for the ground state is near 0.4.

In order to rescale

experimental data given this measurement fidelity we scale (for a given axis) the data using

the following equation:

Scaled Data = 2 × (Raw Data − Si) /Di,

(3.5)

where Si is the mean of the contrast between the excited and ground state for the ith axis (in

our example here it would be Si = (0.8 + 0.4)/2 = 0.6) and Di is the difference in contrast

58

051102153204253304Sequence Number0.40.50.60.70.8Signal (a.u.)Z CalibrationX CalibrationY Calibrationbetween the two states, here Di = 0.8 − 0.4 = 0.4. The additional factor of 2 comes from

the fact that our data acquisition software measures a probability (bounded between 0 and

1) and we are interested in expectation values (bounded between -1 and +1).

Phase correcting quantum state tomography data

In order to correct tomography data such that we can discuss the dynamics of the qubit

expectation values one must scale the results according to the prescription outlined in the

previous section. In the case of a resonant drive, this is quite simple (the state preparation

pulse is either in phase with or 90◦ out of phase with the tomography pulses). In the case

where the state preparation pulse is detuned from the qubit frequency, there will be unwanted

phase accumulation because the drive is not stationary relative to the qubit (i.e. the frame

rotating with the drive is no longer the same as the frame rotating with the qubit).

In

order to correct for this unwanted phase accumulation, the tomography pulses must have

(cid:48)
an adjusted phase of φ

= φ − (360◦ × ∆ × t). For example, in order to measure (cid:104)σx(cid:105), the

phase of the tomography pulse should be φ

(cid:48)

= 90◦ − (360◦ × ∆ × t), and likewise for (cid:104)σy(cid:105),

(cid:48)
φ

= 0◦ − (360◦ × ∆ × t). Here, ∆ is the detuning between the drive and the qubit frequency

and t is the length of time between the beginning of the pulse sequence and the time at which

the tomography pulses are turned on.

It is important to make sure that the sign of ∆ is

correct, but this can be easily verified, and changed, by adding this correction to the phase

of the tomography pulses rather than subtracting it if necessary. When the calibration is

done correctly the evolution of tomography data as a function of ∆ will be smooth in ∆

rather than something that looks periodic. If the data looks periodic in ∆, that should be a

clear sign that the phase correction is not correct and the relative rotating frames are wrong.

59

3.8 FPGA based measurement setup

Although the Tabor-based measurement setup has been highly effective as a measurement

tool in our lab, it does have some limitations. Oftentimes, the bottleneck for our mea-

surement speed actually comes from the time it takes to upload pulse sequences from the

measurement software to the Tabor itself. When this is happening, no measurements are

being done, the instrument is simply communicating with the measurement software. Ad-

ditionally, the Tabor only has four fully programmable outputs. Since we use IQ mixers

for pulse shaping, this means we can only have complete control over two different input

signals without making the measurement circuit unnecessarily complicated. Another com-

plication in this measurement setup is that as we tune the qubit frequency, we must update

the frequency of the local oscillator for the qubit signal, leading to changes in optimal pa-

rameters for the signals I(t) and Q(t) on the Tabor output. This means that as we tune

the qubit frequency, the IQ mixers can become “unbalanced” leading to significant spectral

components at frequencies other than ωq within the input signal. To improve the existing

qubit measurement setup in our lab, we have recently switched measurement hardware to

an experimental setup using a Field Programmable Gate Array (FPGA) board (Xilinx Zynq

UltraScale+ RFSoC ZCU216) running the open-source Quantum Information and Control

Kit (QICK) software [148]. We often refer to this new measurement setup simply as “the

QICK board”. Although all of the experiments presented in this thesis were performed

using the Tabor-based measurement circuit, the remainder of this section serves as a brief

summary of the new measurement system, which will be used moving forward.

The QICK board minimizes the need for extra equipment such as IQ mixers or an ex-

ternal clock to synchronize multiple instruments in the experimental setup. The board is

60

equipped with 16 digital-to-analog converter (DAC) ports with a maximum sampling rate

of 9.85 GS/s (almost an order of magnitude faster than the Tabor). Additionally, the board

has 16 analog-to-digital (ADC) ports, with a maximum sampling rate of 2.5 GS/s. Due to

this lower sampling rate on the ADC side of the board, we cannot reliably measure signals

at the cavity frequency (approximately 5 GHz) directly. In order to make sure that we can

impart information about the qubit state onto a much lower frequency signal, we use a stan-

dard double balanced mixer (Minicircuits ZMX-10G+) to mix a low phase noise LO signal

(generated via a SignalCore SC5511A CW microwave source) with a low-frequency pulse

synthesized from a DAC port on the QICK board (we typically use pulses with frequency

ωDAC/(2π) = 150 MHz again, although this choice of frequency is simply due to habit from

the previous measurement setup). The frequency of the mixer output is then ensured to

be at the cavity frequency by correctly detuning the frequency of the LO signal from the

cavity frequency such that ωc = ωLO ± ωDAC. Note that the mixer will produce signals

at two frequencies, we typically use a standard band-pass filter from Miniciruits to ensure

that we filter out one sideband and only send signals at the frequency ωc into the cavity.

Conveniently, since the DAC ports have a maximum sampling rate near 10 GS/s, we directly

synthesize the qubit drive tones on the board, apply the appropriate band-pass filters and

amplification, and send the signal into the cryostat. A huge advantage for this measurement

setup is that the phase of the qubit signal is controlled via the software communicating with

the QICK board, negating the need for IQ mixers to set the phase of the qubit (or cavity)

drive signals. The output signal from the cryostat is then filtered, amplified, and mixed with

the same LO signal to ensure that the input to the measurement (ADC) port on the board

can be referenced to the signal applied via the DAC port. Fig. 3.9 shows an image of this

measurement setup, and Ref. [148] provides a detailed description of the microwave circuitry.

61

Figure 3.9: Left: Image of the FPGA-based measurement setup. The QICK board is housed
within a rack-mounted drawer from Thorlabs and the relevant microwave circuitry is assem-
bled on the optical breadboard above. Right: Close-up image of the FPGA board itself.
The relevant microwave input-output signals attach to the baluns on the upper part of the
image.

62

Chapter 4

Surface Acoustic Wave Devices

Creating resonant structures for elastic waves on the surface of, and within the bulk of,

piezoelectric materials is a key concept that runs throughout much of this thesis. Mechan-

ical excitations of various forms have been shown to couple to superconducting circuits [9]

including bulk [3,28,52,59,149–153] and surface phonons [8,15,16,22–24,26,29,57,154–156], as

well as phonons constrained to reside within engineered defects [21,32–36,49,50,68,157–159].

This chapter begins with the fundamental notions of stress and strain within a material and

derives the solution for elastic waves from these principles. We then construct a resonant

structure out of piezoelectric surface acoustic waves (SAWs) and use a numerical modeling

technique known as coupling of modes to design the spectral features of SAW resonators

to the level of high precision needed for the hybrid quantum acoustic devices studied in

subsequent chapters.

4.1 Stress and strain in solids

This discussion is inspired by the excellent Refs. [118, 160–162], which discuss both classical

and quantum applications of SAW resonators. Elastic waves within the bulk, and on the

surface, of materials have two important quantities, stress and strain. We begin by consid-

ering a material consisting of many “particles” (these particles aren’t necessarily atoms, but

rather volumetric fractions of the composite material that are much smaller than all other

63

more macroscopic length scales of the material). Then the displacement, (cid:126)u, of each particle

from its equilibrium position is determined by the components of the strain tensor Sij:

Sij((cid:126)x) =

1
2

(cid:18) ∂ui
∂xj

+

(cid:19)

.

∂uj
∂xi

(4.1)

For a three dimensional space, the strain tensor can be represented as a 3×3 matrix. However,

Eqn. 4.1 is unchanged under the exchange of the indices i and j, indicating that the matrix

representation of the strain tensor is symmetric, constraining the number of independent

matrix elements from nine to six.

In order to describe the internal forces in a material subjected to some amount of strain,

we introduce the components of the stress tensor Tij. The force in the ith direction per unit

area on a plane with corresponding normal vector ˆn is given by:

Fi = Tijnj,

(4.2)

where we have introduced the Einstein summation notation and implicitly suppressed the

summation over the index j. This notation will be used for the remainder of this chapter.

Similar to the strain tensor in Eqn. 4.1, it may be shown that the stress tensor is also

symmetric under exchange of indices, Tij = Tji. If the stress is sufficiently small, a deformed

material will return to its equilibrium state once the deforming force is removed. This is the

definition for the elastic response of a material. This is reminiscent of a damped spring mass

system, and we can therefore think of a generalization of Hooke’s Law that relates stress and

strain:

Tij = cijklSkl,

(4.3)

64

where cijkl is a component of the fourth-rank stiffness tensor. The stiffness tensor, with

all symmetries and thermodynamic constraints considered, consists of 21 independent en-

tries [160]. Depending on additional symmetries of a material, it is possible to further

constrain the number of independent entries in the stiffness tensor (down to as few as two

for isotropic materials) [160].

Having introduced both the notion of stress and strain (or equivalently, internal forces

and displacements within a material), we turn our attention to the equation of motion for

a particle within the material. Assuming that the particle is well-approximated by a cube

with characteristic side length δ and mass density ρ, one may calculate the net force on the

particle in the ith direction as [160]:

Fi = δ3 ∂Tij
∂xj

.

(4.4)

This net force is related to the acceleration of the particle in the ith direction (∂2ui/∂t2) via

Newton’s Second Law:

ρ

∂2ui
∂t2 =

∂Tij
∂xj

,

(4.5)

giving an equation of motion for the particle in terms of the stress, which is related to the

strain via Eqn. 4.3.

4.2 Two types of waves: transverse and longitudinal

As mentioned in Section 4.1, the simplest physical geometry of a material (an isotropic

material) corresponds to a stiffness tensor with two independent entries. For this type of

65

material, we can write the stiffness tensor as [163]:

cijkl = λδijδkl + µ (cid:0)δikδjl + δilδjk

(cid:1) ,

(4.6)

where the Kronecker delta has the important properties δij = 1 for i = j and δij = 0 for

i (cid:54)= j. Additionally, we have introduced the two positive constants λ and µ, called the Lam´e

constants, which are related to the elastic moduli of the material [164]. We insert Eqn. 4.6

into Eqn. 4.3 to write the stress tensor as

Tij = λδij∆ + 2µSij,

where we have simplified the diagonal elements of the strain tensor as:

∆ ≡ Sii =

∂ui
∂xi

.

(4.7)

(4.8)

Furthermore, this allows us to write the equation of motion for our particle (Eqn. 4.5) as:

ρ

∂2ui
∂t2 = (λ + µ)

∂∆
∂xi

+ µ∇2ui,

(4.9)

where ∇2 = ∂2
∂x2
i

.

At this point, our goal is to find solutions to Eqn. 4.9. A good ansatz is plane waves with

angular frequency ω and wavevector (cid:126)k:

(cid:126)u((cid:126)x, t) = (cid:126)u0 exp[i

(cid:16)

(cid:17)
ωt − (cid:126)k · (cid:126)x

],

(4.10)

66

where the physical displacement of the particle is related to Re ((cid:126)u ((cid:126)x, t)). Each time derivative

of Eqn. 4.10 provides a factor of iω, while spatial derivatives provide factors of −ikj. Plugging

our ansatz into Eqn. 4.9 and simplifying for the equation of motion for the vector (cid:126)u0, we

arrive at:

ω2ρ(cid:126)u0 = (λ + µ)

(cid:16)(cid:126)k · (cid:126)u0

(cid:17) (cid:126)k + µ|(cid:126)k|2(cid:126)u0.

(4.11)

There are two interesting classes of solutions to Eqn. 4.11. The first solution refers to waves

which satisfy (cid:126)u · (cid:126)k = 0, indicating that the polarization of the wave is perpendicular to its

propagation. These types of waves are referred to as transverse or shear waves. The second

class of solutions corresponds to the polarization and propagation vectors being parallel, and

are called longitudinal waves. For transverse waves, Eqn. 4.11, simplifies to:

ω2ρ(cid:126)u0 = µ|(cid:126)kt|2(cid:126)u0,

which relates the wavevector for transverse waves, (cid:126)kt, to the material properties

|(cid:126)kt|2 =

ρω2
µ

,

(4.12)

(4.13)

and the corresponding wave velocity is given by: vt = (cid:112)µ/ρ. This allows us to rewrite

Eqn. 4.9 for transverse waves as:

¨(cid:126)ut = v2

t ∇2(cid:126)ut.

(4.14)

Since we used the fact that (cid:126)u · k = 0 for transverse waves, we know that they will satisfy

(cid:126)∇ · (cid:126)ut = 0. In the case of a longitudinal wave, Eqn. 4.11 can be rewritten as:

ω2ρ(cid:126)u0 = (λ + 2µ) |(cid:126)kl|2(cid:126)u0,

(4.15)

67

indicating that we can write the wavevector for longitudinal waves as

|(cid:126)kl|2 =

ρω2
λ + 2µ

,

(4.16)

with corresponding longitudinal wave velocity vl = (cid:112)λ + 2µ/ρ. Similar to Eqn. 4.14, we

can write down a wave equation for longitudinal waves in terms of their velocity:

¨(cid:126)ul = v2

l ∇2(cid:126)ul.

(4.17)

Since for longitudinal waves the polarization and propagation are parallel, longitudinal waves

satisfy (cid:126)∇ × (cid:126)ul = 0. Recall that the Lam´e coefficients are defined as positive numbers,

indicating that the velocity of longitudinal waves is greater than the velocity of transverse

waves.

4.3 Surface waves

We now turn our attention to surface waves and this section follows a treatment based on

Refs. [160, 165]. Section 4.2 focused on elastic waves in an infinite bulk medium. In this

section, we focus on the surface elastic waves important for most of the results in this thesis.

Rayleigh waves (also often called surface acoustic waves or SAWs) are named after Lord

Rayleigh [166] and propagate on the surface of a substrate, constraining the wave equation

that governs particle displacement. We begin by considering a medium that is infinite in

two dimensions and half-infinite in the third direction (see Fig. 4.1). For the purpose of this

derivation, we consider a wave propagating parallel to the x-axis. The wave equation that

dictates the particle displacement is given by Eqn. 4.9, and the presence of the surface of the

68

Figure 4.1: Schematic of a surface acoustic wave on the plane z = 0 at the surface of a
semi-infinite medium. The SAW propagation and wave vector are parallel to the x-axis.
The wave properties are invarant along the y-axis. The medium is defined as the volume
with z < 0 with vacuum filling the volume z > 0.

medium imposes the boundary condition that there are no forces at the plane where z = 0.

Since the wave properties depend on both x (direction of propagation) and z (enforces the

boundary condition), the (x,z) plane is quite important and is referred to as the sagittal

plane.

We find surface waves that satisfy Eqn. 4.9 by considering linear combinations of lon-

gitudinal and transverse waves as discussed in Section 4.2. Since we are seeking a solution

constrained to the surface of the material, the solution should decay into the bulk. Addition-

ally, we will assume that the material has translational symmetry along the y-direction, and

so the solution should be independent of the y-coordinate, imposing the constraint ∂(cid:126)u

∂y = 0.

We are seeking a wave with a single frequency component that propagates along x, which

means that both the longitudinal and transverse components of our solution should also

propagate along x. The assumptions listed above allow us to use the following ansatz for a

solution for each wave component:

|(cid:126)u| = ei(βx−ωt) f (z),

(4.18)

69

where β is the wave number and ω is the angular frequency of the wave. The function f (z)

will depend on the polarization (transverse versus longitudinal) and dictates how the wave

will decay into the bulk. Inserting Eqn. 4.18 into the corresponding wave equation (Eqn. 4.14

for transverse waves or Eqn. 4.17 for longitudinal waves) we find the following differential

equation for f (z) for both types of waves:

(cid:32)

β2 −

d2f
dz2 =

(cid:33)

f.

ω2
v2
l,t

(4.19)

Since the wave needs to decay into the bulk of the material, we require β2 − ω2/v2

l,t > 0.

If we did not impose this constraint, the solutions for f (z) would oscillate within the bulk,

which is nonphysical for surface waves. Solutions to Eqn. 4.19 then take the form:

(cid:32)

(cid:115)(cid:18)

f (z) ∝ exp

±z

β2 −

(cid:19)(cid:33)

.

ω2
c2

(4.20)

The next question we must answer is which of the two solutions in Eqn. 4.20 to choose. If

we were to choose the solution with the minus sign, the magnitude of f (z) would increase as

z → −∞. This would not correspond to a wave decaying into the bulk, so we must take the

solution with the plus sign. In order to simplify the solution for the displacement, we follow

convention and introduce the damping coefficient in the z-direction as κl,t =

(cid:113)

β2 − ω2/v2
l,t.

With this simplification, we can write the solution to the wave equation as:

|(cid:126)u| ∝ ei(βx−ωt) exp(cid:2)zκl,t

(cid:3).

(4.21)

Recall that at the beginning of this section, we said that we were seeking a solution that is

a combination of longitudinal and transverse waves. At this point, we can take Eqn. 4.20

70

and determine the correct linear combination of transverse and longitudinal waves. We use

the fact that longitudinal waves satisfy (cid:126)∇ × (cid:126)ul = 0, so if the wave has no component in the

y-direction this simplifies to ∂

∂z ul,x − ∂

∂x ul,z = 0. We insert Eqn. 4.20 into this expression

to find:

κlul,x − iβul,z = 0,

(4.22)

where we have used the damping coefficient corresponding to the longitudinal wave velocity

(cid:113)

β2 − ω2/v2

l . We identify solutions to Eqn. 4.22 (up to the overall wave amplitude A)

κl =

as:

(cid:126)ul = A (β ˆx − iκl ˆz) e−i(ωt−βx)+κlz.

(4.23)

To investigate the transverse component of the wave, we recall that for transverse waves

(cid:126)∇ · (cid:126)ut = 0. For a wave invariant along y, this simplifies to ∂

∂xut,x + ∂

∂z ut,z = 0. Inserting

Eqn. 4.20 yields the following equation:

iβut,x + κtut,z = 0,

(4.24)

where we have used the damping coefficient corresponding to the transverse wave velocity

(cid:113)

κt =

β2 − ω2/v2

t . Solving Eqn. 4.24 gives the transverse wave component as:

(cid:126)ut = B (κt ˆx − iβ ˆz) e−i(ωt−βx)+κtz,

(4.25)

again with an overall wave amplitude B. Now that we have an expression for both the

longitudinal and transverse wave components, we are able to take the boundary conditions

into consideration. Since the medium ends at z = 0, the boundary conditions require that

71

Figure 4.2: Real part of the z-component of the displacement field Re(uz((cid:126)r)) for v2
Both the x and z-axes are normalized by the SAW wavelength λ.

l = 2v2
t .

the stress at z = 0 must be equal to zero. We can relate the spatial derivatives to the stress

via Eqn. 4.7 and find three differential equations for (cid:126)u from this boundary condition:

Txz|z=0 =

Tyz|z=0 =

∂ux
∂z
∂uy
∂z

+

+

Tzz|z=0 = v2
l

∂uz
∂z

∂uz
∂z
∂uz
∂y
(cid:16)

+

= 0,

= 0,

v2
l − 2v2
t

(4.26)

(4.27)

(4.28)

(cid:17) ∂ux
∂x

= 0.

Since the solutions for (cid:126)u are independent of y, we have excluded the y-dependence in

Eqn. 4.27. The total solution for (cid:126)u is then (cid:126)u = (cid:126)ul + (cid:126)ut as defined in Eqn. 4.23 and Eqn. 4.25

for longitudinal and transverse waves, respectively. Simplifying the differential equations

(Eqns. 4.26, 4.27, and 4.28) yields the following set of linear equations for the overall ampli-

72

tudes A and B:

A (2βκl) + B
l + β2(cid:17)
κ2

A

(cid:16)

(cid:16)

t + β2(cid:17)
κ2

= 0

+ B (2βκt) = 0.

(4.29)

We solve Eqn. 4.29 by creating a matrix of the coefficients A and B and setting its determi-

nant to zero. After a tremendous effort, we arrive at:

(cid:32)

2 −

v2
v2
l

(cid:33)2

(cid:115)

= 4

1 −

(cid:115)

1 −

v2
v2
t

v2
v2
l

,

(4.30)

where we have introduced the SAW velocity as v = ω/β.

In order to gain some insight

regarding Eqn. 4.30, we rescale the SAW velocity relative to the transverse wave velocity as

v = ξvt, and simplify Eqn. 4.30 as:

(cid:16)

2 − ξ2(cid:17)2

(cid:113)

= 4

1 − ξ2

(cid:115)

1 − ξ2

(cid:19)2

.

(cid:18) vt
vl

(4.31)

Importantly, Eqn. 4.31 indicates that the SAW velocity is independent of its frequency, it

only depends on the quantity vt/vl. Eqn. 4.31 can be solved numerically for ξ given the

ratio vt/vl, and is often within the range ξ ≈ 0.875 − 0.955 based on realistic material pa-

rameters [118]. Additionally, we are able to solve for the ratio of the overall amplitudes A
1 − ξ2(cid:17)

and B of longitudinal and transverse waves, respectively as B/A = (cid:0)ξ2 − 2(cid:1) /

(cid:112)
2

(cid:16)

.

Furthermore, one can inspect the total solution for SAW waves based on Eqns. 4.23 and 4.25

and identify a relative phase difference between the ˆx and ˆz-components of the wave, in-

dicating that the individual “particles” follow elliptical displacement paths, as displayed in

Fig. 4.2.

73

4.4 Piezoelectric surface waves

Now that we have built a general understanding of how surface waves behave, we turn

our attention to a more complicated form of Rayleigh waves - namely piezoelectric surface

acoustic waves. In a piezoelectric medium, the underlying crystalline inversion symmetry is

broken, and the presence of an external electric field can generate strain in the material and

vice versa. In a medium that does not support piezoelectricity, the electric displacement field

(cid:126)D in the medium is related to the electric field (cid:126)E via the well known relation (cid:126)D = (cid:15) (cid:126)E, where (cid:15)

is the dielectric tensor. Additionally, we can relate the stress and strain in a non-peizoelectric

medium (Eqn. 4.3) via the stiffness tensor. However, if the material is piezoelectric, the

relation for both the electric and strain fields are modified [167]:

Di = (cid:15)ijEj + eijkSjk

Tij = cijklSkl − ekijEk,

(4.32)

(4.33)

where eijk is component of the piezoelectric tensor. Similar to the treatment in the previous

section, we can use Eqn. 4.32 and Eqn. 4.33 to find an equation of motion for a particle in

such a material. In the case of a piezoelectric material, this equation of motion is much more

complicated than the isotropic case [160]:

ρ

∂2ui
∂t2 = ekij

∂2Φ
∂xj∂xk

+ cE

ijkl

∂2uk
∂xj∂xl

,

(4.34)

where Φ is the electric potential and the superscript E on the stiffness tensor indicates that

the electric field is held constant. In addition to the mechanical boundary conditions imposed

74

by Eqns. 4.26, 4.27, and 4.28 we now additionally impose that (if the material is insulating)

there are no free charges and thus we have an additional electrical condition that (cid:126)∇ · (cid:126)D = 0

throughout the medium. We can relate this boundary condition to find another differential

equation for the system [160]:

(cid:15)S
ij

∂2Φ
∂xi∂xj

− eijk

∂2uj
∂xi∂xk

= 0,

(4.35)

where the superscript S on the component of the dielectric tensor indicates that the strain is

held constant. Together Eqn. 4.34 and Eqn. 4.35 yield a system of four coupled differential

equations for three spatial displacement components ui as well as the electrostatic poten-

tial Φ. Although these equations can be solved analytically in some cases, a full solution is

outside the scope of this thesis.

Next, we consider two special cases for the electrical surface boundary condition. If the

surface of the piezoelectric medium is metallized, the electric field parallel to the surface

must be zero.

If the surface is not metallized, it is often assumed that there are no free

charges and there will be a non-zero electric potential above the surface of the medium.

In the case of a metallized surface, the acoustic wave will have a modified velocity vm

compared to the free surface velocity vf . In order to quantify the strength of the interaction

of the acoustic wave with electrical environment we introduce the normalized velocity change

K2 = 2 (cid:0)vf − vm

(cid:1) / (cid:0)vf

(cid:1). Materials with large values of K2 are considered to be strong

piezoelectrics while in weaker piezoelectric materials the value of K2 is quite small and the

interaction of the acoustic and electric fields is minimized. To provide context for this velocity

shift, the primary results of this thesis utilize a particular cut of piezoelectric lithium niobate

referred to as YZ -lithium niobate, which has a relatively large value of K2 = 4.5%, much

75

larger than that of other commonly used piezoelectric materials such as quartz (K2 = 0.11%)

or gallium arsenide (K2 = 0.06%) [168].

4.5 Construction of a SAW resonator

In order to physically construct a resonant structure for the SAW waves derived in Sec-

tion 4.3, we need to understand the key ingredients for both exciting and reflecting SAW

excitations. SAW resonators consist of periodic metallic structures fabricated on the surface

of a piezoelectric substrate so that the electromechanical properties of the device are coupled

as in Eqn. 4.32 and Eqn. 4.33. The physical periodicity of the metallic structures defines the

wavelength λ of the SAW excitations. The operating frequency of the device is then given

by f = v/λ, where v is the speed of sound on the surface of the substrate. We use the well-

known coupling of modes (COM) technique [160] to numerically calculate the response of

each component of the device as well as the response of the composite resonator as a whole.

By defining both fabrication and material parameters for the device and substrate and using

the COM modeling method, we are able to calculate the electromechanical properties of the

device or its constituent parts. This section will be broken into three subsections; the first

will describe the “P-matrix” techniques that are required for COM analysis, the second part

will describe the modeling techniques we use for acoustic transducers, and the final subsec-

tion will describe how we model the Bragg mirrors that form the effective reflecting walls of

the acoustic cavities.

76

4.5.1 Coupling of modes and the P-matrix

We use the standard P-matrix approach to model SAW devices [160]. A P-matrix for a SAW

device is a 3 × 3 matrix that represents an acoustic structure comprised of two acoustic ports

(Ai,j) and one electrical port (I, V ). In this picture each acoustic port represents either left

or right moving acoustic waves. We can then write down the P-matrix as:










A1,out

A2,out

I



















=

P11 P12 P13

P21 P22 P23

P31 P32 P33




























A1,in

A2,in

V

(4.36)

The matrix elements relating the two acoustic ports (port 1 and port 2) contains information

about how incoming and outgoing acoustic waves are scattered off of the structure and can

be thought of in very close analogy to the scattering matrix S of a microwave circuit. If the

structure is “passive” (meaning there are no electrical connections) matrix elements relating

port 1 and port 2 are the only non-zero matrix elements of the P-matrix. Bragg mirrors

and empty surfaces are passive elements in this fashion. The P-matrix element P11 relates

the incoming and outgoing mechanical waves from port 1 and is therefore often the relevant

P-matrix element that identifies reflecting properties.

To understand the electrical properties of a given SAW structure, we turn our attention

to the P33 matrix element. If we consider the case where there are no incoming surface waves,

we can relate the current to the voltage via I = P33V . This relationship allows us to identify

the matrix element P33 as the electrical admittance, Y (ω), of the resonator. The real part of

this quantity is the conductance G(ω), while the imaginary part gives the susceptance B(ω).

77

Additionally, reciprocity leads to several more relations between elements of the P-matrix:

P21 = P12

P31 = −2P13

P32 = −2P23.

(4.37)

(4.38)

(4.39)

As will be described in the following sections, SAW resonators are comprised of three key

components. Firstly, acoustic transducers are responsible for converting electrical excitations

to mechanical excitations and vice versa. This indicates that for these structures the primary

P-matrix element of interest will be P33. Second, Bragg mirrors are responsible for the

reflection of acoustic excitations, and therefore for mirror structures we will primarily be

interested in the quantity P11 (the P22 matrix element would also suffice, since for symmetric

resonators we would expect P11 = P22). Finally, free propagation space between components

of a resonator will allow for an increased effective acoustic cavity length, decreasing the free

spectral range of the resonator and allowing us to design the number of resonant modes

the device hosts. The primary advantage of utilizing this type of modeling is that it allows

us to calculate the P-matrix for a given structure and cascade many structures together to

calculate the P-matrix for the composite device using a method known as coupling of modes.

This method is similar to the ABCD matrix multiplication method for 2-port microwave

systems [169], although the transduction between electrical and mechanical signals makes

the calculations relatively more complicated. A complete description of this procedure can

be found in Ref. [160]. Since finding analytic solutions for complicated acoustic resonators

can prove to be a tedious calculation, it is convenient to use numerical methods [118]. The

remainder of this chapter will focus on the device parameters and subsequent considerations

78

one needs to take into account when modeling an acoustic resonator. We will build a “toy

resonator” on YZ -lithium niobate as an example. By varying the parameters of the device

we will see how the electrical response of the resonator changes. The relevant material

properties of the resonator are listed in Table 4.1, and can be found in Ref. [168].

Parameter
vsound
K2

Physical quantity
Speed of sound (at room temperature)
Piezoelectric Coupling Coefficient

Value
3488 m/s
4.5%

Table 4.1: Summary of material parameters for a SAW resonator on YZ-lithium niobate

4.5.2 Interdigitated transducers

Before using P-matrix techniques to identify the admittance of an acoustic interdigitated

transducer (IDT), we will construct a minimal model that allows us to gain some qualitative

understanding of important properties of these structures, which are used to excite SAWs.

Let’s begin by considering a physical model of a transducer as a linear array of metal strips

with spatial periodicity λ along the x-direction having a voltage on each strip that alternates

between +1 (in some normalized voltage unit) and ground, i.e. V = 0 (see Fig. 4.3). For

the purposes of this model, each metal strip of the IDT will be considered to be infinitely

thin and infinite along the y-direction. We can then write down the total voltage profile as

a sum of delta functions spaced along the x-direction:

V ∝

Np−1
(cid:88)

n=0

δ (x − λn) ,

(4.40)

where Np is the number of finger pairs in the transducer. The upper bound of the sum only

runs to Np − 1 rather than Np because we choose to start the sum at n = 0 rather than

79

n = 1. We Fourier transform Eqn. 4.40 to identify the wavevector response for this system

(also called the transfer function):

H(k) ∝

Np−1
(cid:88)

n=0

e−iλkn =

sin (cid:0)Npkλ/2(cid:1)
sin (kλ/2)

e−i(Np−1)kλ/2.

(4.41)

Where we have summed the geometric series (cid:80)N −1

n=0 xn =

1 − xN (cid:17)
(cid:16)

/ (1 − x). Additionally,

the wavevector is related to the speed of sound k = 2πf /vs. Furthermore, the transducer

will have a central frequency f0 related to the speed of sound and the periodicity of the

structure f0 = vs/λ. Near f0, we use the approximation sin(kλ/2) (cid:39) kλ/2. These relations

and approximation allow us to write down the transfer function for the IDT model as a

“sinc” function centered around the frequency f0:

H(f ) (cid:39) Np|sinc (cid:0)Npπ(f − f0)/f0

(cid:1) |,

(4.42)

where sinc(x) ≡ sin(x)/x. In order to study the electrical properties of a transducer, we

consider its total frequency-dependent admittance:

Y (ω) = iωC + Ya(ω),

(4.43)

where C is the geometric capacitance of the interdigitated transducer structure and the

admittance Ya(ω) is a complex admittance describing the transduction between acoustic and

electrical excitations. In the case where all of the dissipated electrical power is converted to

acoustic waves, it can be shown that the real part of Ya(ω) is related to the transfer function

(Eqn. 4.42) as Re(Ya(ω)) ≡ Ga(ω) ∝ |H(ω)|2. In addition to the real part, the admittance

also contains an imaginary part [170]. Taking the full complex admittance into account, we

80

Figure 4.3: (a) Minimal model of an acoustic transducer. Alternating electrodes are biased
at V = +1 relative to ground. The spatial periodicity of the electrodes is λ. In this geometry
the surface acoustic wave will propagate along the x-direction. (b) Schematic of the circuit
representation of an acoustic transducer. An alternating voltage is biased across the IDT
relative to ground to excite SAWs. The width of the SAW beam is given as w and the
length of the transducer is L. (c) Schematic of the same circuit with the IDT represented
as three parallel circuit elements. The geometric capacitance of the structure is C, and the
electromechanical response of the transducer consists of a real conductance Ga(ω) and an
imaginary susceptance Ba(ω).

can represent the SAW IDT as a lumped electrical circuit consisting of 3 parallel components

(see Fig. 4.3(c)):

Y (ω) = iωC + Ga(ω) + iBa(ω),

(4.44)

where both the conductance and susceptance can be written as:

Ga(ω) = Ga0 sinc2 (X)

Ba(ω) = Ga0

(cid:18) sin (2X) − 2X
2X2
(cid:18) Npπ(ω − ω0)
ω0

(cid:19)

.

X =

(cid:19)

(4.45)

(4.46)

(4.47)

Calculation of the prefactor Ga0 is quite involved and leads to Ga0 (cid:39) 1.3K2N 2

p ωW Cp,

where the parameters K2, NP , W,and Cp are the piezoelectric coupling coefficient, number

81

of finger pairs, SAW beam width, and capacitance per IDT finger pair per unit length,

respectively [170].

The previous simplified model of an acoustic transducer was missing a key ingredient-

namely internal reflections within the transducer structure were not taken into account.

Fortunately for us, the coupling of modes method allows us to take into account these

internal reflections and solve for more realistic values of Ga(ω) and Ba(ω). A representative

calculation for Ga(ω) and Ba(ω) using the coupling of modes method is shown in Fig. 4.4

with λ = 1 µm, L = 25 µm, the reflection probability per finger |r| = 2%, and W = 50 µm.

Figure 4.4: Conductance and susceptance of an IDT structure calculated using the coupling
of modes method. The skew in the device resonance arises from internal reflections within
the transducer (r = 2% per finger pair).

We can identify some of the physical quantities that are important for the design and

construction of an acoustic transducer. Fig. 4.5 displays the log-scale magnitude of the P33

matrix element for an IDT structure as the length L of the structure is varied (see Fig. 4.3).

Moreover, as the length is increased, the periodicity of the lobes of the sinc function decrease

in frequency, consistent with the prediction from Eqn. 4.45. As the length of the transducer

increases, the impact of internal reflections within the IDT structure become more apparent,

82

3.03.54.0Frequency (GHz)100102030Conductuance (mS)Ga()Ba()with the transducer response becoming heavily skewed towards lower frequencies.

Figure 4.5: Left: Log-scale conductance of the IDT calculated with coupling of modes.
As the transducer length is increased, the width of the response in frequency drastically
decreases. Middle: Conductance of the transducer with a width L = 10 µm. The width of
the central peak is of order 100 MHz. Right: Conductance of the transducer with a width
L = 100 µm. The width of the central peak is of order 10 MHz.

4.5.3 Bragg reflection gratings

Figure 4.6: Left: amplitude of P11 for a SAW mirror structure. In this example, the center
of the mirror stop band is near 3.45 GHz, with the reflection rapidly decaying outside of the
mirror stop band. Right: phase response of P11 for the same mirror structure.

Now that we have built a framework that allows us to design a device that transduces

electrical energy into surface phonon acoustic energy, we next introduce Bragg reflection

gratings. These gratings will allow for the strong reflection of phonons at certain frequencies,

allowing us to construct a resonator. By fabricating many strips of metal having a spatial

83

33.54Frequency (GHz)104070100Transducer Length (m)505ln(Re(P33)) (mS)33.54Frequency (GHz)-6-303ln(Re(P33)) (mS)L = 10 m33.54Frequency (GHz)-3036ln(Re(P33)) (mS)L = 100 m33.54Frequency (GHz)00.51|P11|33.54Frequency (GHz)0Phase P11 (rad)Figure 4.7: Left: Amplitude of P11 for the mirror structure from Fig. 4.6 with varying
|P11| with a reflectivity of rm = 0.6%. The stop-band is
mirror reflectivity rm. Center:
quite narrow, and the maximum reflectivity is significantly less than unity. Right:
|P11|
with a reflectivity of rm = 0.6%. Compared to the center panel, the mirror reflectivity is
much more broad and the maximum reflectivity is near unity.

periodicity λmirror that are electrically disconnected from each other, we can leverage the

finite reflectivity per electrode to form an acoustic “mirror” structure. Since the reflectivity

per electrode is quite low (approximately 1% per electrode), we often require on the of

order several hundred gratings to construct a Bragg mirror with near-unit total reflectivity.

The frequency range over which SAWs are faithfully reflected is often referred to as the

mirror stop-band (∆f ), which is directly proportional to the reflectivity of the electrodes

∆f /f ≈ 2|rm|/π [160]. As described in Section 4.5.1, the relevant P-matrix quantity for

a mirror structure is P11. The magnitude and phase of the quantity P11 is displayed for a

mirror structure on YZ -lithium niobate with λmirror = 1.01 µm, length Lmirror = 100 µm,

and reflectivity per grating rm = 2% is shown in Fig. 4.6. Since the stop-band width depends

intimately on the reflectivity of the mirrors, it is important to study the properties of P11

for a mirror structure as a function of the reflectivity. As shown in Fig. 4.7, as we increase

the reflectivity of the mirrors, we observe a corresponding increase in the width of the mirror

stop-band. This is quite important since the reflectivity of the mirrors is not a geometrically

84

33.54Frequency (GHz)12345Mirror Reflectivity (%)0.20.40.60.8|P11|33.54Frequency (GHz)00.51|P11|rm = 0.6%33.54Frequency (GHz)00.51|P11|rm = 5%defined quantity, but rather a somewhat difficult to control fabrication parameter of the

device [171]. Since the spectral shape of the mirror reflectivity is quite complicated (i.e. it is

not a simple boxcar function), it is often useful as thinking of the acoustic cavity as spatially

distributed over the length of the mirror structure with an effective length Leff as described

in the following section.

4.5.4 Complete model of a SAW resonator

Now that we understand how to both excite and reflect phonons, we have the complete

framework to construct a composite SAW resonator consisting of both and IDT structure

surrounded by Bragg mirrors on either side so that phonons generated in the center of the

device are strongly reflected by the Bragg mirrors, forming an acoustic cavity as shown in

Fig. 4.8. An example calculation is done below using the pieces we have already constructed

in the previous two sections. We take the response of the IDT structure in Fig. 4.4 and

place it between two Bragg mirrors as seen in Fig. 4.6 with zero free propagation distance

between the IDT and mirrors (Lfree = 0). The total electrical conductance for this resonator

is shown in Fig. 4.8. For completeness, we also plot |P11| for the mirrors and Re(P33) for

the IDT on the same plot to identify salient features of both the IDT and mirrors on the

response of the composite resonator.

The effective length of the SAW cavity is comprised of three parts. The first part is the

distance (on average) waves will propagate into the mirrors. Since each grating reflects only

a small percentage of the wave, one can estimate the penetration depth of the wave into the

mirrors as LP (cid:39) |rm|λmirror/2. The last length scale that is important in defining the total

length of the effective SAW cavity is the free propagation distance between the IDT and the

mirrors (see Fig. 4.8). Taking each of these parameters into account, we can estimate the

85

Figure 4.8: Left: geometric sketch of a complete SAW resonator. The IDT (red) is re-
sponsible for the transduction of electrical excitations into phononic excitations. The Bragg
mirrors (blue) are responsible for reflecting these excitations. The free spectral range of the
resonator is determined by the speed of sound as well as the effective length of the cavity
Leff = 2LP + L + Lfree. Right: Conductance for a SAW resonator calculated via coupling
of modes with Lfree = 0. There is a single confined mode near 3.45 GHz. The peak of the
IDT conductance is within the mirror stop-band, leading to an enhancement of the resonator
conductance within the mirror stop-band.

effective length of the SAW cavity as Leff = 2LP + L + Lfree, where we have included the

factor of 2 to account for the mirrors on either side of the IDT.

The spacing in frequency between confined SAW modes (called the free spectral range)

in this type of resonator is given by: ∆fF SR = vs/ (2Leff). As the free propagation distance

Lfree is increased the free spectral range decreases. When the free spectral range becomes

significantly smaller than the width of the mirror stop-band, multiple resonant modes can

exist within the mirror stop band, leading to a multi-mode acoustic cavity [8, 57, 155].

In order to gain intuition for how important the free propagation distance Lfree is for

the response of the resonator, we plot Re(P33) for the model SAW resonator from Fig. 4.8

as Lfree is varied with all other parameters held constant. As the free propagation distance

is increased, we see the device hosts an increasing number of resonant modes. By specifi-

cally designing Lfree for a given resonator design, we are able to produce resonant spectra

containing a custom-made number of resonant acoustic modes.

86

Figure 4.9: Left: Log-scale conductance of the SAW resonator in Fig. 4.8 with varying free
propagation distance Lfree. Center: resonator conductance with Lfree = 0. This resonator
supports a single confined surface phonon mode. Right: resonator conductance for the same
SAW resonator but with Lfree = 50 µm. This resonator supports five confined mechanical
modes.

4.6 Conclusion

This chapter has focused on the design of surface acoustic waves beginning with the concept

of elastic waves in a solid. By constraining the wave to the surface we have found the general

structure of surface acoustic waves. We have used the coupling of modes machinery to model

each constituent part of a SAW resonator and explored how the physical parameters of each

part of the resonator impact the response of individual components of the resonator as well as

the composite response of the resonator as a whole. We build on these ideas in the following

chapters by precisely designing the conductance of a high-frequency SAW resonator that will

be used in our quantum acoustic experiments.

87

33.54Frequency (GHz)02550Free Propagation Length (m)505ln(Re(P33)) (mS)33.54Frequency (GHz)-8-4048ln(Re(P33)) (mS)Lfree = 0 m33.54Frequency (GHz)-6-3036ln(Re(P33)) (mS)Lfree = 50 mChapter 5

Open Quantum Systems

5.1

Introduction

When we learn quantum mechanics for the first time, an emphasis is placed on closed quan-

tum systems. What do we mean by “closed” in this context? For the purposes of this thesis,

a closed quantum system is a system whose dynamics follow the rules of quantum mechanics,

and the system is completely isolated from its environment. In such a closed system, the

Hamiltonian ( ˆH) can be identified and the evolution of the wavefunction of the system |ψ(t)(cid:105)

follows the results from the Schr¨odinger equation depending on the initial state of the system

|ψ(0)(cid:105) and the Hamiltonian ((cid:126) = 1):

|ψ(t)(cid:105) = exp(−i ˆHt) |ψ(0)(cid:105) .

(5.1)

Note, that Eqn. 5.1 assumes that the Hamiltonian ˆH is time-independent.

In the case

of a time-dependent Hamtiltonian the solution to Schr¨odinger’s equation becomes slightly

modified:

(cid:34)

|ψ(t)(cid:105) = T←exp

−i

(cid:35)
dt(cid:48) ˆH(t(cid:48))

|ψ(0)(cid:105) ,

(cid:90) t

t0

(5.2)

where the arrow indicates that products of time-dependent operators are ordered such that

their arguments in time increase from right to left.

88

Now let’s imagine a more complicated example. Take our quantum system of interest

(a qubit is actually the simplest case!), and let’s try to write down the Hamiltonian that

describes the qubit plus the local microwave environment plus all of the defects in the system

and how strongly they couple to the qubit. What do these interactions look like at the level

of the Hamiltonian? How large is the Hilbert space for this system? As one can see, this

problem ranges from very difficult to highly impractical. As such, when we are interested

in the evolution of a quantum system that is interacting with its environment, we need

to introduce an new framework for analyzing systems that interact with many degrees of

freedom. This is an example of what we consider an open quantum system, where the

interactions between the system of interest and the environment are sufficiently strong that

they cannot be ignored. For this type of system, rather than study the wavefunction |ψ(cid:105) we

need to work with the reduced density matrix ρ. As we will see in this chapter, by considering

both coherent and dissipative effects in the system, we are able to construct an equation of

motion for the qubit’s reduced density matrix, which allows us to study the dynamics of

certain types of open quantum systems.

5.2 Lindblad master equation

In this section, we present a partial derivation of the Lindblad Master Equation [72], which we

will use in our analysis of open quantum systems. This section is inspired by Refs. [172–176].

We begin by considering the density matrix ρ for a quantum system. The density matrix

has several important properties, listed below:

• Unit trace: T r(ρ) = 1.

• Hermiticity: ρ† = ρ.

89

• Positive eigenvalues: λ ≥ 0 ∀ λ of ρ.

The last two bullet points define the density matrix ρ to be a positive semidefinite matrix.

The density matrix can be constructed as as a weighted sum over many different pure states

of the quantum system |ψi(cid:105):

(cid:88)

ρ =

pi |ψi(cid:105) (cid:104)ψi| .

i

(5.3)

In Eqn. 5.3, the sum is over each pure state |ψi(cid:105) and is weighted by the probability pi. Note

that the first bullet point listed above constrains the probabilities in Eqn. 5.3 such that

(cid:80) pi = 1. The density matrix can be used to as a tool to probe the operators acting on the

system via the equation (cid:104) ˆO(cid:105) = T r(ρ ˆO) for a given system operator ˆO. In order to see this,

we consider the expansion of T r(ρ ˆO):

T r(ρ ˆO) =

=

=

(cid:88)

i,j

(cid:88)

i,j

(cid:88)

i

(cid:10)ψj

(cid:12)
(cid:12)ψi

(cid:11) (cid:104)ψi| ˆO (cid:12)

(cid:12)ψj

(cid:11) ,

pi

(5.4)

δi,jpi (cid:104)ψi| ˆO (cid:12)

(cid:12)ψj

(cid:11) ,

pi (cid:104)ψi| ˆO |ψi(cid:105) ,

(cid:88)

=

pi(cid:104) ˆO(cid:105)i = (cid:104) ˆO(cid:105).

i

Additionally, the density matrix also informs us of the purity of the state via the quantity

P = T r(ρ2), which will be an important metric for the open quantum systems work of chapter

7. The purity of a maximally mixed state is given by the minimum purity Pmin = 0.5, while

the maximum state purity is given by a pure state with Pmax = 1. A state is called a pure

state if it can be represented by a sum of basis states and has P = 1, while a state is called

a mixed state if it can only be represented by a statistical mixture of pure states [177].

90

In order to study how the density matrix changes in time, we consider the full Hamiltonian

ˆH of a quantum system interacting with its environment:

ˆH = ˆHS ⊗ ˆIE + ˆIS ⊗ ˆHE + ˆV.

(5.5)

In Eqn. 5.5, the first term describes the Hamiltonian of the system in isolation, ˆHS, and we

take the tensor product of ˆHS with the identity operator for the environment such that the

operator acts on the total system and environment Hilbert space. The second term describes

the Hamiltonian of the environment ˆHE and the tensor product with the identity operator

for the system. The third and final term represents the interaction between the quantum

system and its environment. The density matrix that describes the composite interacting

system is ρT . In order to describe the redcued density matrix for the system dynamics (ρ),

we simply trace over the environmental degrees of freedom: ρ = T rE(ρT ). In the interaction

picture, the equation of motion for the density matrix ρT is given by

˙ρT = −i[ ˆV(t), ρT ].

Integrating Eqn. 5.6 and substituting the solution back into itself yields the equation:

˙ρT = −

(cid:90) t

0

[ ˆV(t), [ ˆV(t(cid:48)), ρT (t(cid:48))]] dt(cid:48).

(5.6)

(5.7)

We assume T rE([ ˆV(t), ρT (0)]) = 0 via a choice of the interaction Hamiltonian, allowing us to

set the constant of integration to 0. At this point it is useful to trace over the environmental

91

degrees of freedom in order to write an equation of motion for the reduced density matrix ρ:

˙ρ(t) = −

(cid:90) t

0

(cid:16)

(cid:17)
[ ˆV(t), [ ˆV(t(cid:48)), ρT (t(cid:48))]]

T rE

dt(cid:48).

(5.8)

In order to further simplify Eqn. 5.8, we need to make several assumptions about the inter-

actions between the system and the environment. By assuming that there are no initial

correlations between the system and the environment, then ρT (0) = ρ(0) ⊗ ρE(0),

where ρE is the density matrix describing the environment. Furthermore, we can make the

Born approximation [178], assuming that the coupling between the system and its environ-

ment is weak (| ˆV| (cid:28) | ˆH0|) we can rewrite the total density matrix as ρT (t) = ρ(t) ⊗ ρE(t).

Along the same lines, the Born approximation also allows us to consider the environment

as a large “bath” for the system, by which we mean the environmental dynamics do

not substantially change the dynamics of the system in time, i.e. ρE(t) = ρE.

Furthermore, by assuming that there is no flow of information from the environment

back to the system (the Markov approximation) [172] we can replace ρ(t(cid:48)) in the in-

tegrand of Eqn. 5.8 with ρ(t). Additionally, if we assume that the system state does

not change significantly on time scales relevant to the environment decay, we can

rewrite Eqn. 5.8 as:

˙ρ(t) = −

(cid:90) ∞

0

(cid:16)

(cid:17)
[ ˆV(t), [ ˆV(t − t(cid:48)), ρ(t) ⊗ ρE]]

T rE

dt(cid:48).

(5.9)

Eqn. 5.9 is known as the Markovian master equation, and the assumptions that led us from

Eqn. 5.8 to Eqn. 5.9 (highlighted in bold text) are known as the Born-Markov approxi-

mations.

In order to further simplify Eqn. 5.9, we will expand Eqn. 5.9 in terms of the

92

Figure 5.1: Depiction of an open quantum system. The system (blue) is described by
a reduced density matrix ρ, has “system” operators ˆO, and is subject to a Hamiltonian
ˆH. The environment also has its own reduced density matrix, operators, and Hamiltonian
(ρE, ˆOE, ˆHE). The goal of this chapter is to develop a framework to disentangle the two
from each other so that we may study the system independent from the environment.

interaction potential, and it will be possible to further simplify the Markovian master equa-

tion into Lindblad form (see Refs. [172–175] for details), where the dynamics of the density

matrix is comprised of two primary parts:

˙ρ = −i[ ˆH, ρ] +

γkD[ ˆLk].

(cid:88)

k

In Eqn. 5.10, we have introduced the super-operator D[ ˆLk]ρ = ˆLkρ ˆL†

k − 1

2 { ˆL†

k

(5.10)

ˆLk, ρ} that

describes dissipation in the system. Each rate γk provides the appropriate weight to the

corresponding “jump” operators ˆLk, which are responsible for driving the quantum system

between states.

Although Eqn. 5.10 seems to be incredibly compact, in principle we could consider many

jump operators ˆLk. Additionally, Eqn. 5.10 describes the evolution of a matrix, not a single

vector. In order to solve this equation for a particular system given set of jump operators, it

93

ρ,̂O,̂HSρE,̂OE,̂HÊV̂Vis sometimes useful to “vectorize” Eqn. 5.10 (see Ref. [179] for more details). The remainder

of this chapter will focus on providing several simple examples of solutions to Eqn. 5.10 in

order to gain some intuition for the effects of drive and dissipation on a quantum system.

5.3 Driven-dissipative qubit

Since many of the results of this thesis are important within the context of quantum two

level systems, it is an extremely useful example to consider the dynamics of a qubit under the

presence of drive and dissipation. We can consider the Hamiltonian of the driven-dissipative

qubit as:

ˆH = −

ωq
2

ˆσz + Ω cos (ωdt)ˆσx.

(5.11)

In Eqn. 5.11, the qubit energy splitting is given by ωq, and the drive is parameterized by its

frequency ωd and amplitude Ω. We can transform this Hamiltonian to the frame rotating
with the drive via ˆH → ˆU ˆH ˆU † + i ˆU † ˙ˆU , where ˆU = exp(−ωdtˆσz/2).

In this frame, the

Hamiltonian becomes time-independent and is therefore much easier to analyze:

ˆH =

∆
2

ˆσz +

Ω
2

ˆσx,

(5.12)

where we have introduced the detuning ∆ = ωd − ωq. In order to discuss dissipation in this

system, we consider two jump operators. To describe energy relaxation, we use the operator

ˆσ− = ˆσx − iˆσy, along with corresponding rate γ1 (the subscript “1” is to indicate that this

describes a T1 process as discussed in chapter 3). To account for loss of phase information,

we use the operator ˆσz along with corresponding rate γφ/2 (this follows the convention of

pure dephasing as described in chapter 3). Under this type of drive and dissipation, we can

94

Figure 5.2: Left: Tomography components of a driven-dissipative qubit. Each component
oscillates in time before decaying to a maximally mixed state. The trajectory of each com-
ponent is set by solutions to Eqn. 5.13. Right: Bloch sphere representation of the data from
the left panel.

write Eqn. 5.10 as:

˙ρ(t) = −i

(cid:20)∆
2

ˆσz +

Ω
2

(cid:21)

ˆσx, ρ(t)

+ γ1D[ˆσ−] +

γφ
2

D[ˆσz].

(5.13)

Since we are considering only a two-level system in this case, it is possible to solve Eqn. 5.13

analytically. However, for simplicity, we employ Qutip [180], a Python package which is

optimized to find numerical solutions to the Lindblad equation. By solving Eqn. 5.13 for

ρ(t), we can calculate trajectories of each qubit vector component (cid:104)ˆσi(cid:105) for i ∈ x, y, z. As

seen in Fig. 5.2, the qubit state vector traces out a particular path within the Bloch sphere

and eventually decays to the center of the Bloch sphere. In this particular simulation, we

have set Ω = −∆ and have scaled the rates corresponding to all jump operators so that

Ω = |∆| = 4γ. In this parameter regime, the drive is dominant and we are able to see the

qubit state vector trace out multiple oscillations before it descends to the center of the Bloch

sphere as seen in Fig. 5.2.

95

5.4 Driven-dissipative harmonic oscillator

Although the many of the results of this thesis focus on superconducting qubits, the Lindblad

master equation is useful for a variety of quantum systems. Specifically, within the context of

quantum acoustics systems, it may behoove us to study the dynamics of harmonic oscillator

states as described towards the end of Ch. 6. In this section we consider solutions to the

Lindblad equation for a quantum harmonic oscillator. In the absence of an external drive

applied to the system, we take the Hamiltonian of the oscillator to be:

ˆH = ˆa†ˆa.

(5.14)

At the level of this example, we will only include two jump operators ˆa and ˆa†. In the case of

an electromagnetic oscillator (like our microwave cavities), these jump operators correspond

to photon loss and photon gain, respectively. We relate the rate of each jump operator

to the average population of a thermal bath, so that the rate of photon loss is given by

γ− = κ(1 + nth), and the rate of photon gain is given by γ+ = κnth. In this case, Eqn. 5.10

becomes:

˙ρ(t) = −i[ˆa†ˆa, ρ(t)] + κ(1 + nth)D(ˆa) + κnthD(ˆa†).

(5.15)

At this point, Eqn. 5.15 looks rather intimidating to solve analytically, so we continue to

utilize Qutip for numeric simulations. By initializing the oscillator state in a coherent with

n = 4, we can solve for the density matrix ρ at a given point in time and use it to investigate

physical quantities (operator expectation values) of interest for the oscillator. For example,

we can calculate quantities like the average photon number as n = T r(ρˆn), as well as the

probability of having n photons in the oscillator as ρnn, as the nth diagonal density matrix

96

element. As shown in Fig. 5.3, when the oscillator is initialized in a coherent state, the

photon number distribution is Poissonian at t = 0 as expected [5]. In this particular we

have defined the environment as a thermal bath with average energy equal to the transition

energy of the harmonic oscillator. As the system interacts with its dissipative environment,

the mean phonon number decays from the initial value to n = 1, where the system is in

thermal equilibrium with its environment, which is an external parameter we set in this

particular simulation.

In the presence of an external drive the system Hamiltonian now

Figure 5.3: (a) Photon number distribution at τ κ = 0, where τ is the time during which
the oscillator is subject to dissipation and κ is related to the rate of the operators ˆa and ˆa†.
(b) Photon number distribution at τ κ = 1. (c) Photon number distribution at τ κ = 2. (d)
Evolution of the average photon number n while the system is subject to dissipation.

becomes:

ˆH = ˆa†ˆa + V

(cid:16)

ˆa + ˆa†(cid:17)

,

(5.16)

97

where the strength of the external drive is given by V. The term (ˆa + ˆa†) is the coordinate of

the harmonic oscillator in operator form, and the external drive is reminiscent of an external

force changing the coordinate of our oscillator.

In the following numeric simulations, we

relate the drive strength to the dissipation by setting V = 3

4 κ, so that drive and dissipation

are (almost) of the same magnitude. All other parameters in the simulations in Fig. 5.3 and

Fig. 5.4 are the same across the two sets of simulations. In Fig. 5.4, we now see that rather

than decay to the population of the thermal environment, the average number approaches

a steady state with n > nth due to the presence of the drive. Additionally, the number

distribution starts as a Poisson distribution as in the undriven case, but as the drive and

dissipation act on the system during its evolution, the number distribution is significantly

different than that in Fig. 5.3.

5.5 Conclusion

This chapter introduced the concept of an open quantum system, where interactions between

a quantum system and its environment can lead to the loss of information from the quantum

system. We have presented a derivation of the Lindblad master equation and used two

simple examples to see how the impact of dissipation changes the system dynamics. The

next chapter uses the idea of driven-dissipative systems to leverage the engineered dissipation

of a superconducting qubit as a resource for quantum control.

98

Figure 5.4: (a) Photon number distribution at τ κ = 0, where τ is the time under which the
oscillator is subject to dissipation and κ is related to the rate of the operators ˆa and ˆa†.
(b) Photon number distribution at τ κ = 1. (c) Photon number distribution at τ κ = 2. (d)
Evolution of the average photon number n while the system is subject to both drive and
dissipation.

99

Chapter 6

Phononic Bath Engineering of a

Superconducting Qubit

6.1

Introduction

Phonons, the ubiquitous quanta of vibrational energy, play a vital role in the performance

of many quantum technologies. Coupling to well-defined phonon modes allows for highly-

connected multi-qubit gates in ion trap architectures [181–183] as well as the generation of

entangled states in systems of superconducting qubits [24, 25]. Even when the phonons take

the form of a large dissipative bath, an irreversible flow of heat allows for state initialization

critical to the function of laser systems [184] and the operation of optically active spin

qubits [185, 186]. Conversely, unintended coupling to phonons has been shown to degrade

qubit performance by generating decohering quasiparticles and leading to correlated errors

in superconducting qubit systems [187–189]. Regardless of whether a phononic bath plays

an enabling or deleterious role, it is typically intrinsic to the system and does not admit

specific control over its spectral properties, nor the possibility of engineering aspects of its

dissipation to be used as a resource. Here we show that by precisely designing the coupling

of a superconducting qubit to phononic degrees of freedom enables a novel platform for

investigating the behavior of open quantum systems in which qubit dynamics is determined

100

by phononic loss. By shaping the loss spectrum of the qubit via its coupling to a bath of

lossy piezoelectric surface acoustic wave phonons, we demonstrate preparation and dynamical

stabilization of a family of superposition states when the qubit is subjected to the combined

effects of drive and dissipation. We model this driven open quantum system using the

Lindblad master equation, and find excellent agreement for both the qubit dynamics as

well as its steady state. These experiments highlight the versatility of engineered phononic

dissipation in superconducting circuits. The results advance the understanding of mechanical

losses in open quantum systems and extensions of this work could impact the mitigation of

phonon-induced decoherence in superconducting processors.

Hybrid quantum systems based on the coherent coupling of two or more distinct, but

interacting, systems enable the development of advanced quantum technologies [6], and inves-

tigation into the fundamental properties of complex interacting quantum degrees of freedom.

Hybrid systems based on superconducting qubits, utilizing the experimental tool-kit of circuit

quantum electrodynamics (cQED) [11], are a versatile platform for creating and controlling

heterogeneous quantum systems and investigating their coherent dynamics [7]. Of particular

interest is the ability to leverage the intrinsically strong nonlinearity provided by the qubit to

manipulate collective mechanical and acoustic degrees of freedom and explore new regimes of

circuit quantum optics using GHz-frequency phonons. By engineering strong interactions be-

tween superconducting qubits and mechanical resonators it is possible to study the quantum

limits of high-frequency sound in a wide variety of systems composed of qubits coupled to

bulk phonons [3,28], Rayleigh-like surface waves [4,8,14,154,155,170,190], as well as flexural

modes in suspended structures [191–193]. Impressive experimental results have been demon-

strated using integrated quantum acoustic systems, including single phonon splitting of the

qubit spectrum [49, 57, 59], Wigner function negativity of an acoustic resonator [3, 4, 50, 59],

101

electromagnetically induced acoustic transparency [190], and phonon mediated state trans-

fer [24, 25]. Hybrid quantum acoustics systems that integrate superconducting qubits with

Figure 6.1: (a) Schematic representing a qubit coupled to a bath of SAW phonons. The
qubit (salmon) non-unitarily radiates excitations into a bath of SAW phonons (blue), where
the emission rate is mediated by the electrical conductance of the SAW structure. (b) Image
of the flip-chip qubit-SAW hybrid device mounted in a 3D microwave cavity (left), which
is used for control and readout. False color optical micrograph of the acoustic resonator
(right), consisting of an IDT (green), as well as acoustic Bragg mirrors (red). (c) Equivalent
circuit model of the composite qubit-SAW system. Here CJ is the total capacitance shunting
the Josephson junctions, CC is the capacitance responsible for coupling the qubit and SAW
resonator, which is primarily dictated by the parallel-plate capacitance between the two
substrates, and CSAW is the geometric capacitance of the SAW resonator. The complex
admittance that represents the electro-mechanic response of the SAW resonator is divided
into a conductance Ga(ω), as well as a susceptance Ba(ω).

phononic degrees of freedom typically operate in a domain where the interaction strength

between the two far exceeds the loss rate of either system. In this strong coupling regime,

the emphasis is on the coherent dynamics of the coupled systems rather than on the dissi-

pation presented to the qubit via the phononic bath. However, the ability to create open

quantum acoustic systems in which the qubit is variably coupled to multiple mechanical

degrees of freedom with vastly differing strengths and loss rates opens the door for dissi-

pative state preparation and dynamically stabilized states in the presence of a strong drive

102

250 μm10 μmqubit|g⟩|e⟩SAW phononsCJCC/2Ga(ω)Ba(ω)qubit chipSAW chipCSAWCC/2(a)(b)(c)and customized phononic loss channels. Piezoelectric surface acoustic wave (SAW) devices,

which can be engineered into compact devices with sharp spectral responses [4, 8, 170], are a

promising avenue for engineering highly frequency dependent phononic dissipation channels

for quantum bath engineering protocols, in which the level of surface wave dissipation and

qubit coupling can be precisely designed and controlled (see Fig. 6.1(a)).

The spectral response of the SAW resonator was calculated using coupling-of-modes

(COM) [118]. The periodicity of the SAW interdigitated transducer (IDT) structure is

fabricated to be λIDT = 800 nm. The periodicity of the mirror structure is fabricated to be

slightly larger than the periodicity of the IDT structure (λmirror = 816 nm) to guarantee

a single sharply peaked resonant acoustic mode within the stopband of the mirrors where

phonons are strongly reflected, which produces the confined acoustic resonance at 4.46 GHz.

A list of the complete SAW device parameters are listed in Table 6.1. The last four rows

of Table 6.1 are adjusted to fit the spectroscopy of the hybrid system to the simulated

conductance via coupling-of-modes. The first six rows of Table 6.1 define the geometric

properties of the SAW device and are defined in fabrication, and we calculate the relevant

scattering parameters for each structure in the SAW resonator using coupling-of-modes with

the results shown in Fig. 6.2.

The broad frequency response of the conductance in Fig. 6.2(c) arises from the Fourier

transform of the spatial structure of the SAW transducer. The width of the response is

dictated by the number of finger pairs constituting the transducer, labeled Np in Table 6.1

(see Fig. 6.1) [160]. The SAW device is galvanically connected to two large antenna pads and

coupled to the transmon qubit via a flip-chip technique, forming an effective set of parallel

plate capacitors between the antenna pads on the SAW and transmon chips as depicted in

Fig. 6.2(e).

103

Parameter
λIDT
λmirror
NP
W
Lmirror
LIDT
vsound
η
ri
rm

Physical quantity
Transducer periodicity
Mirror periodicity
Number of finger pairs
Finger pair overlap
Bragg mirror length
Transducer length
Speed of sound
SAW propagation loss
Transducer reflectivity
Mirror reflectivity

Value
800 nm
816 nm
16
35 µm
240.72 µm
12 µm
3638 m/s
500 Np/m
-0.005i
-0.005i

Table 6.1: Summary of relevant parameters for the SAW resonator used in the experiment

6.2 SAW + qubit experiments

We implement a quantum acoustic bath engineering protocol using a hybrid quantum system

consisting of a flux-tunable transmon qubit coupled to a SAW Fabry P´erot cavity fabricated

on the surface of single crystal lithium niobate (see Fig. 6.1 and Fig. 6.2). The complex

admittance that describes the electro-mechanical properties of the SAW device, and tailors

the coupling to the phonon bath, are calculated using coupling of modes [118, 160]. The

qubit and SAW resonator are fabricated on separate substrates and their purely capacitive

coupling is mediated in a flip-chip geometry via antenna pads attached to each device in

the form of a pair of parallel plate capacitors (see Fig. 6.1 and Appendix). For control and

readout, the composite flip-chip device is free-space coupled to the fundamental mode of a

three-dimensional (3D) electromagnetic cavity with a frequency of ωc/(2π) = 4.788 GHz. As

shown in Fig. 6.3(a), we design the SAW resonator spectral response in order to access both

its coherent coupling to the transmon as well as the dissipative qubit-phonon dynamics. The

SAW resonator is engineered to confine a single acoustic mode, which appears as a sharp

peak in the conductance of the resonator near 4.46 GHz (see Fig. 6.3(a)). Near this confined

104

Figure 6.2: (a) Simulated reflection of the Bragg mirror structure in the SAW resonator.
(b) Simulated conductance of the SAW transducer, which is centered between the two acous-
tic Bragg mirrors. (c) Composite conductance of the SAW transducer cascaded with the
mirror structure, leading to a single confined SAW mode near 4.46 GHz and a continuous
conductance associated with the leakage of SAW energy through the Bragg mirrors. (d) Near
the confined mode the SAW resonator conductance can be approximated by a Lorentizian
function (dashed orange fit). The full width at half maximum of this Lorentizian is used to
estimate the energy decay rate of the confined SAW mode. (e) Schematic of the flip-chip
assembly. The antenna pads on either substrate form parallel plate capacitors between the
two devices.

acoustic mode, we fit the simulated conductance to a Lorentzian function and extract the

SAW energy decay rate γSAW/(2π) = 0.6 MHz from the full width at half maximum (see

Fig. 6.2). On either side of this main SAW resonance, phononic energy loss is governed by

a continuum of dissipative SAW states that manifest as frequency ripples in the effective

conductance of the acoustic resonator and correspond to the leakage of surface-phonons out

of the SAW resonator through the acoustic Bragg mirrors (see Fig. 6.1(b)). Measurements

shown in Fig. 6.3(b) reveal how the features of the SAW resonator response are imparted

onto the spectroscopic properties of the hybrid system and enable access to multiple regimes

of circuit quantum acoustodynamics (cQAD) [8]. The qubit and the confined SAW mode

interact with a coupling rate gm/(2π) = 12 ± 0.6 MHz, larger than the loss rate of either

system (γSAW/(2π) = 0.6 MHz and γq/(2π) = 2.67 MHz), which is a hallmark of the

quantum acoustic strong coupling regime. Strong coupling between a resonant SAW mode

and a transmon qubit has been observed in previous experiments [4, 57, 155] and this regime

105

Figure 6.3: (a) Simulation of the SAW resonator conductance Ga(ω) based on the coupling-
of-modes modeling. The resonator is designed to host a single confined SAW mode, which
corresponds to the narrow peak in the device conductance at 4.46 GHz, and the effective elec-
trical conductance of the device mediates the coupling between the qubit and a given SAW
mode. (b) Two-tone spectroscopy of the SAW-qubit hybrid system revealing an avoided
crossing between the qubit and main SAW mode. Black dashed line: fit to the data, indicat-
ing an acoustic coupling of gm/(2π) = 12 ± 0.6 MHz. (c) Spectroscopy line-cuts from (b) at
the positions of the vertical dashed lines. The oscillations in frequency in each scan highlight
the phononic loss channel imparted on the qubit, which arise from the modulation of the
conductance of the SAW resonator and are associated with the leakage of SAW excitations
through the acoustic Bragg mirrors.

serves to demonstrate hybridization between the qubit and SAW modes. More importantly,

the spectroscopy shown in Fig. 6.3(b,c) also reveals signatures of controlled surface phonon

loss arising from the interaction of the qubit with the continuum of SAW modes on either

side of the confined SAW mode. This interaction manifests as a series of dark states in the

qubit spectrum, which appear as horizontal fringes in Fig. 6.3(b), and corresponds to the

acoustic analog of the bad-cavity limit of cQED [11]. In this dissipative regime the dynamics

of the hybrid system is dominated by the loss of phonons from the resonator that have a

frequency outside of the acoustic mirror stop band.

To utilize the frequency-dependent acoustic loss for dynamical quantum state stabiliza-

tion, we consider the effect of phononic decay on qubit decoherence in the presence of a

strong coherent drive of amplitude Ω near resonant with the qubit.

In this regime, the

emission spectrum the system consists of a peak at the drive frequency ωd and two addi-

106

(a)(b)(c)tional sidebands at ωd ± ΩR, where ΩR =

√

Ω2 + ∆2 is the generalized Rabi frequency and

∆ = ωd − ωq is the detuning between the drive and the qubit [194–196] (see Fig. 6.4(a)). In

the presence of a frequency-dependent emission spectrum, one sideband can be suppressed,

leading to preferential emission from the other sideband and non-zero qubit coherence in the

undriven basis for times long compared to the intrinsic lifetime of the qubit [89]. For this

type of bath engineering protocol to work in our device, the SAW admittance must modify

the decoherence rate of the qubit over frequencies comparable to experimentally accessible

values of ΩR. To verify this, we measured the qubit decay rate Γq = 1/T1 across a broad

range of frequencies far-detuned from the main acoustic resonance (see Fig. 6.4(b)) where

phonon leakage through the acoustic mirrors is maximized. At these frequencies, the con-

ductance of the SAW resonator is well approximated by that of the acoustic transducer, as

the reflectivity of the acoustic Bragg mirrors is small, and the total loss of the qubit can be

approximated as:

Γq

(cid:0)ωq

(cid:1) =

ωq
2πQi

+ Γ0 sinc2

(cid:18)

πNp

ωq − ωs
ωs

(cid:19)

,

(6.1)

where Qi = 1.67 × 103 is the qubit internal quality factor, Γ0 = 0.252 ns−1 is the maximum

conversion rate of the qubit excitation into SAW phonons, Np = 16 is the number of finger

pairs in the IDT structure of the SAW resonator, and ωs/(2π) = 4.504 GHz is the central

SAW frequency, which is within 1% of the value predicted from the device fabrication pa-

rameters (see Table 6.1). The first term in Eq. 7.1 describes the decay of the bare qubit

while the second term is associated with qubit energy conversion into SAW phonons. At

ωq/(2π) = 4.001 GHz, where the gradient of qubit loss into SAW phonons is large, the qubit

decay rate varies by a factor of 3.7 over a span of 80 MHz (≈ 2ΩR) and allows us to use

SAW phonon modes for efficient state preparation.

107

Figure 6.4: (a) Top: schematic representing the rotation of the qubit eigenbasis in the
presence of a detuned drive applied to the qubit. Center: representation of pure dephasing
in the dressed basis. Bottom: representation of competing decay rates in the dressed qubit
basis. By tailoring the coupling of the qubit to the frequency-dependent phonon bath, we are
able to control the relative size of γ± as a function of qubit frequency. (b) Measurement of
qubit loss Γq = 1/T1 versus frequency (blue). The red curve is a fit to Eq. 7.1 showing that
the variation in the qubit loss is dictated by conversion into SAW phonons. The green curve
is a result of coupling-of-modes model for the electrical conductance of the SAW resonator
with no fit parameters. The arrow indicates the frequency at which the bath engineering
experiments are performed. The black dashed lines are the endpoints of experimentally
accessible Mollow triplet sidebands. (c) Pulse sequence for investigating the coherence of
the driven-dissipative quantum acoustics system. (d) Tomographic reconstruction of qubit
state evolution at a resonant Rabi frequency of Ω/(2π) = 8.47 MHz and drive detuning
∆/(2π) = − 10 MHz. Dots represent the experimental data while the dashed lines are
solutions to Eq. 6.2 with the same drive parameters, which correspond to γ+ = 3.4 µs−1 and
γ− = 1.3 µs−1. (e) Measurement of the state purity as a function of time. In the combined
presence of phonon loss and drive the purity reaches a value of 0.85 at t = 1 µs, in contrast
to a maximally mixed state represented by the dashed line at P = 0.5. The solid red line is
the expected state purity based on Eq. 6.2.

The dynamics of the reduced qubit density matrix in the combined presence of the drive

108

QubitStatePreparationtdriveΔΩQubitMeasurementTomographyRotation−ΔΩ2θωdΩR|g⟩|˜g⟩|e⟩|˜e⟩|˜g⟩|˜e⟩γ0|˜g⟩|˜e⟩γ+γ−(b)(a)(c)(d)(e)and frequency-dependent SAW loss are described by the Lindblad master equation [72,87,89]:

˙ρ = i[ρ, H] + γ0 cos2 (θ) sin2 (θ)D[˜σz]ρ + γ− sin 4 (θ)D[˜σ+]ρ

+ γ+ cos 4 (θ)D[˜σ−]ρ + γ1D[σ−]ρ +

γφ
2

D[σz]ρ,

(6.2)

where D[A]ρ =

(cid:16)

2AρA† − A†Aρ − ρA†A

(cid:17)

/2. The angle θ is defined by tan 2θ = −Ω/∆ and

represents the rotation of the qubit eigenbasis under the drive (see Fig. 6.4(a)). The operators

˜σ± and ˜σz along with the corresponding rates γ± and γ0 represent transitions between

eigenstates and dephasing in the rotated frame. Dissipation in the lab frame is represented

by the operators σ− and σz along with the rates γ1 and γφ, for qubit depolarization and

dephasing. The transition rates γ± represent competing decay of the qubit into SAW phonons

in the rotated basis, and by tailoring the frequency-dependent phonon bath these rates vary

significantly over the frequency scale 2ΩR as seen in Fig. 6.4(b). In the limit γ± (cid:29) γ∓,

the spectral weight of one sideband of the qubit emission spectrum is suppressed, leading

to dynamical stabilization of a rotating-frame eigenstate. In general, the plane accessible

within the Bloch sphere is controlled by the phase of the drive signal. In the measurements

described below, we set the phase of the drive such that the qubit eigenstates lie in the XZ -

plane of the Bloch sphere. The chosen Rabi frequency and drive detuning further constrain

the qubit eigenstates to a particular axis in this plane.

109

6.3 Numerical calculation of Lindblad dynamics

Given a periodic drive coupling to the σx operator with Rabi frequency Ω, the Hamiltonian

that describes the qubit in the frame rotating at ωd is given by

H =

∆
2

σz +

Ω
2

σx,

(6.3)

where ∆ = ωd − ωq is the qubit-drive detuning. In the presence of the drive, the correspond-

ing qubit eigenstates (|g(cid:105),|e(cid:105)) are rotated to a dressed basis (|˜g(cid:105),|˜e(cid:105)), corresponding to the

eigenvectors of Eq.6.3:

|˜g(cid:105) = cos θ |g(cid:105) − sin θ |e(cid:105)

(6.4)

|˜e(cid:105) = sin θ |g(cid:105) + cos θ |e(cid:105) ,

where the rotation angle of the eigenstates is defined by tan 2θ = −Ω/∆. This allows the

Hamiltonian to be rewritten in the dressed basis as

H =

ΩR
2

˜σz,

(6.5)

√

where ΩR =

Ω2 + ∆2 is the generalized Rabi frequency and ˜σz = sin 2θσx − cos 2θσz. The

evolution of the reduced density matrix in the dressed basis as given by the Lindblad master

equation (Eqn. 6.2). We represent the density matrix as a vector ρ = (ρgg, ρge, ρeg, ρee)T ,

which is evolved in time as ρ(t) = eLtρ(0). The matrix L represents the Lindblad operator

as a 4 × 4 matrix that is exponentiated via the transformation V eDtV −1, where V is the

matrix that diagonalizes L and D is a diagonal matrix containing the eigenvalues of L. We

110

calculate expectation values by taking the trace (cid:104)σx(cid:105) = Tr(σxρ) and similarly for the other

components of the qubit expectation value. Custom Python code for this process can be

found at Ref. [197].

6.4 Phononic open quantum system results and dis-

cussion

We calibrate the magnitude of the resonant Rabi frequency by measuring the excited state

population of the qubit as a function of time over a range of drive amplitudes as shown in

Fig. 6.5(a). The frequency of the resulting oscillatory behavior varies linearly with drive

amplitude as shown in Fig. 6.5(b) and provides a mapping between Rabi frequency and

drive. To demonstrate the phononic bath engineering protocol, we flux bias the qubit to

Figure 6.5: (a) Resonant Rabi oscillations for various drive amplitudes ranging from 0.1V to
1V. As the amplitude is increased, the frequency of oscillation increases. Traces have been
vertically offset for clarity. (b) Extracted Rabi frequencies from the data in panel (a). The
frequency increases linearly with drive amplitude as expected and allows us to interpolate
the resonant Rabi frequency for a given drive amplitude.

ωq/(2π) = 4.001 GHz, where the gradient of the qubit loss varies strongly as a function of

frequency (see Fig. 6.4(b)). By tailoring the drive parameters the qubit emits radiation at

frequencies corresponding to Mollow triplet sidebands [194] at rates governed by the SAW-

phonon induced loss. To calibrate the drive strength, we measure resonant Rabi oscillations

111

as a function of drive amplitude and interpolate the results to obtain a mapping between drive

amplitude to Rabi frequency (see Fig. 6.5). By driving the qubit at a detuning ∆/(2π) = −

10 MHz with strength Ω/(2π) = 8.47 MHz we demonstrate the ability to prepare a qubit state

in the XZ -plane of the Bloch sphere, which we verify using tomographic reconstruction of the

qubit state as a function of driving time tdrive (see Fig. 6.4(c,d)). As shown in Fig. 6.4(d), in

the limit tdrive (cid:29) T1, the qubit density matrix approaches a fixed point determined by the

drive parameters and asymmetric phononic loss. We further quantify this dissipation-enabled

dynamical stabilization protocol by calculating the state purity of the qubit, P = Tr(ρ2),

as a function of time as shown in Fig. 6.4(e). We see that, in the limit tdrive (cid:29) T1, the

state purity reaches P = 0.85, well above the value P = 0.5 of a maximally mixed state.

By including fit parameters that describe the global dephasing rate γφ = 1.48 µs−1 and

global depolarization rate of γ1 = 2.46 µs−1 in the numerical solutions to Eq. 6.2, we find

quantitative agreement with the tomography data shown in Fig. 6.4(d,e).We note that these

optimal fit parameters are in reasonable agreement with the measured values for the bare

qubit decay at this frequency γq/(2π) = fq/Qi (cid:39) 2.4 µs−1 and the pure dephasing rate

γφ,exp (cid:39) 0.93 µs−1.

In the basis dressed by the drive, this dynamically stabilized qubit state is effectively

cooled toward thermal equilibrium via the frequency-dependent emission of energy into

SAW phonons. Because the qubit reduced density matrix is no longer evolving in time

for sufficiently long values of tdrive, the driven-dissipative system has reached its steady

state and the notion of an effective qubit temperature is well-defined. To quantify the ef-

ficiency of this phonon-induced qubit cooling, we consider the driven qubit as a two level

system subject to the Hamiltonian H =

(cid:126)ΩR

2 ˜σz, in equilibrium with an effective thermal

bath, where ˜σz = sin (2θ)σx − cos (2θ)σz. The partition function for this system is

112

Z = 2 cosh

(cid:16) (cid:126)ΩRβeff
2

(cid:17)

, with βeff = 1/kbTeff, where kb is the Boltzmann constant and

Teff is the effective qubit temperature. Over many repeated measurements the average qubit

energy is given by (cid:104)(cid:15)(cid:105) =

(cid:126)ΩR

2 (cid:104)˜σz(cid:105), which can used along with the partition function to re-

late the measured value of (cid:104)˜σz(cid:105) to the effective temperature Teff = −(cid:126)ΩR/(2kb tanh−1 (cid:104)˜σz(cid:105)).

With the qubit in thermal equilibrium with the surface phonon bath, we find an effective

temperature of Teff ≈ 250 µK based on the experimental data in Fig. 6.4(d), which is also the

lowest effective temperature we were able to obtain in our experiments. This low effective

temperature arises from the fact that the driven-dissipative protocol creates and cools an

effective quantum two-level system having an energy splitting ΩR/(2π) ≈ 13 MHz, which

is significantly lower than the frequency of the qubit in the lab frame. This combination of

effective temperature and transition frequency ΩR/(2π) correspond to an excited state pop-

ulation of approximately 10%, equivalent to a transmon qubit with a transition frequency

of 4 GHz at a temperature of 90 mK. Finally we note, that the efficiency of the driven-

dissipative cooling in the current experiment is primarily limited by the relatively large rates

γ1 and γφ of the qubit. In particular, solutions to Eq. 6.2 predict an effective temperature

as low as 85 µK if the global energy decay rate and dephasing of the qubit were improved

by approximately an order of magnitude to γ1 = γφ = 0.1 µs−1. In the combined presence

of drive and phonon loss through the mirrors, the steady-state of the qubit should exhibit

coherence on timescales long in comparison to the intrinsic qubit lifetime in the lab frame.

To investigate this steady-state behavior, we apply a coherent drive to the qubit for a time

tdrive = 3 µs, which is approximately one order of magnitude longer than the measured

depolarization time of the qubit in the absence of drive. By choosing the parameters of

the drive, we control both the rotation of the qubit eigenstates as well as the splitting of

the Mollow triplet sidebands, which modifies their coupling to the lossy phononic bath. We

113

Figure 6.6: (a) By varying both the strength (resonant Rabi frequency) and the detuning
of the state preparation pulse relative to the qubit frequency we are able to demonstrate
dynamical state stabilization that has both positive and negative values of (cid:104)σx(cid:105). (b) Numer-
ical solutions to Eq. 6.2 over the same range of drive parameters as in panel (a). The rates
γ1 and γφ are fit parameters as described in the text, while all other model parameters are
determined empirically. (c) Representative horizontal linecuts from panel (a) along with the
corresponding predictions based on the solution to the Lindblad master equation (panel (b)).

begin by tomographically reconstructing the steady-state expectation value (cid:104)σx(cid:105) = Tr(ρσx)

in the bare qubit eigenbasis as a function of drive parameters and compare with the solutions

to Eq. 6.2 as shown in Fig. 6.6. As we modify parameters of the drive, we observe excellent

agreement between the measured and predicted value of (cid:104)σx(cid:105) of the resulting qubit state,

and reveal a region of zero coherence where the competing loss rates γ± in Eq. 6.2 cancel

each other and (cid:104)σx(cid:105) = 0.

For completeness we also present the measured values of (cid:104)σz(cid:105) and (cid:104)σy(cid:105) in Fig. 6.8(a,c)

along with comparison to Eq. 6.2 in Fig. 6.8(b,d). The qubit reduced density matrix is given

by ρ = 1

2 (1 + σx(cid:104)σx(cid:105) + σy(cid:104)σy(cid:105) + σz(cid:104)σz(cid:105)), where 1 is the 2 × 2 identity matrix.

Each tomography component is calibrated relative to the readout contrast along each axis

of the Bloch sphere and scaled such that the length of the qubit state vector does not exceed

114

(a)(b)(c)Figure 6.7: (a) Measured steady-state purity with the same drive parameters as in Fig. 6.6.
(b) Purity based on solutions to Eq. 6.2 under same drive parameters. (c) Purity of the qubit
steady-state in the XZ -plane of the Bloch sphere across the same range of drive parameters,
where solutions to Eq. 6.2 are overlaid on the experimental data.

unity. The purity of a given state is then calculated to be P = Tr(ρ2) and is shown in

Fig. 6.7(a,c) and is compared with the results of Eq.6.2 in Fig. 6.7(b,c). The systematic

difference between the measured tomography components and those obtained from solutions

to Eq. 6.2 (most evident as small non-zero measured values of (cid:104)σy(cid:105) in Fig. 6.8(c)) likely

arises from systematic errors produced by time-dependent variation of the qubit transition

frequency over the relatively long duration (several hours) of the tomography measurements

as a function of drive parameters. A tomography measurement consists of a π/2 rotation

with a well-defined phase to project either the x or y-component of the qubit state vector

onto the measurement (z ) axis of the Bloch sphere, and subsequent projective measurement.

To obtain the z -component of the state vector, no tomography rotation is required prior to

measurement. In the presence of a detuned drive, as is the case for our measurements at ∆ (cid:54)=

0, the qubit state acquires an additional phase shift of φ = 2π×∆×tdrive, during the detuned

115

(a)Exp.Theory0.60.30.00.30.6x0.30.00.30.60.9z(b)(c)Figure 6.8: (a) Measured values of (cid:104)σz(cid:105) with the same drive parameters as in Fig. 6.6. (b)
(cid:104)σz(cid:105) obtained from solutions to Eq. 6.2 with the same drive parameters as Fig. 6.6. (c)
Measured values of (cid:104)σy(cid:105) with the same drive parameters. (d) (cid:104)σy(cid:105) obtained from solutions
to Eq. 6.2 with the same drive parameters.

drive. To account for this additional phase accumulation, the phase of each tomography

pulse is shifted by the same amount φ prior to projective measurement. Because the phase

of the tomography pulses depend on the detuning in this fashion, fluctuations in ∆ introduce

systematic errors in the measured tomography components. Qubit frequency variations can

be clearly seen in measurements of Ramsey interferometry taken over many hours, as shown

in Fig. 6.9. We observe slow drift in the qubit frequency, indicating the presence of low-

frequency noise in the system as well as discrete “jumps” in the qubit frequency. These

type of discrete jumps are often seen in superconducting qubit experiments and in our case

likely arise from unwanted coupling to dielectric two-level fluctuators [198]. Furthermore,

full tomographic measurement of the qubit state vector allows us to reconstruct the qubit

reduced density matrix and calculate the steady-state purity of the qubit as function of drive

parameters as shown in Fig. 6.7(a), which we also find to be in good agreement with solutions

116

Figure 6.9: Detuning (∆ = ωd − ωq) between the time-varying qubit frequency and a fixed
drive frequency (ωd/(2π) = 4.007 GHz) measured via Ramsey interferometry over 16 hours.
Both slowly varying frequency drift and discrete frequency jumps on the order of 1 MHz are
observed in ωq/(2π). The discrete frequency jumps are likely due to non-resonant coupling
to two-level fluctuators in the qubit chip.

to Eq. 6.2 (see Fig. 6.7(b)). As shown in Fig. 6.7(c) we further investigate the versatility of

this state preparation protocol by plotting the purity as a function of the coordinates of the

qubit state vector in the XZ - plane over the experimentally accessible values of the drive

parameters. We find that the dissipation-enabled state preparation can be used to create

high purity superposition states across a relatively large range of (cid:104)σx(cid:105). However, access to

large negative values of (cid:104)σz(cid:105) is limited in this device by the difference between γ+ and γ−

arising from the slope of the SAW device conductance versus frequency. We find that these

experimental results are also in good agreement with the open quantum systems modeling

based on solutions to Eq. 6.2, which is also displayed in Fig. 6.7(c). We note that the

systematic difference in the span between the experimentally and numerically obtained values

of tomography components in the XZ -plane likely arises from long timescale variation in the

qubit frequency during the relatively long duration (several hour) tomography measurements

(see Fig. 6.9).

Finally, we investigate the thermodynamic properties of the driven dissipative qubit. Im-

portantly, because the qubit cooling procedure depends on the rotation of the qubit eigen-

117

Figure 6.10: Steady-state temperature of the driven-dissipative qubit as a function of drive
strength Ω and drive detuning ∆.

states relative to the undressed basis, we might similarly expect the steady-state effective

temperature Teff to depend on the drive parameters. As shown in Fig. 6.10, the dependence

of the steady-state effective temperature largely follows that of (cid:104)σx(cid:105). Interestingly, we note

that we are able to achieve negative effective temperatures, which correspond to population

inversion, i.e. when the steady-state of the qubit has an appreciable population in the state

|˜e(cid:105). Furthermore, we can re-cast the data presented in Fig. 6.10 as a histogram. As seen

in Fig. 6.11 we measure a distribution of temperatures with an average effective temper-

ature T eff = 1.24 mK. We approximate the width of the distribution as the full-width at

half-maximum ∆Teff = 2.85 mK.

Figure 6.11: Distribution of effective temperatures Teff. We measure a significant portion of
negative effective temperatures, corresponding to population inversion.

118

10203040Rabi Frequency (MHz)1050510Rabi Detuning (MHz)505Teff(mK)1050510Teff(mK)0200400Counts6.5 Conclusion

In this chapter we have described a phononic open quantum system composed of a supercon-

ducting qubit coupled to an engineered bath of lossy surface acoustic wave phonons. This

system enables the investigation of the dynamical and steady-state of superpositions of the

qubit when it is subjected to the combined effects of strong driving and phononic loss. The

lossy SAW environment allows for high-purity qubit state preparation and dynamical stabi-

lization within a plane in the Bloch sphere. Modest improvements to the transducer design

and qubit quality factor offer an avenue to prepare states with purity exceeding 99%. In

particular, this could be achieved by improving the bare qubit lifetime and pure dephasing

rate by an order of magnitude, which would enhance the decay of qubit energy into phonons

rather than the electromagnetic environment. Similarly, modifying the SAW transducer de-

sign to have a sharper spectral response would increase the relative difference of the rates

γ±, improving the efficiency of the phononic bath engineering protocol. Finally we note that

these results also open the door to investigating the non-unitary evolution of quantum states

and effective non-Hermitian Hamiltonians hosting decoherence-induced exceptional points

in open quantum acoustic systems via post-selection protocols [91].

119

Chapter 7

Interference and Scattering in

Multi-Mode Quantum Acoustic

Systems

7.1

Introduction

Quantum acoustic systems offer a unique opportunity to investigate phononic interference

and scattering processes in the quantum regime. In particular the interaction between a su-

perconducting qubit and a phononic oscillator allows the qubit to sense the oscillator’s excita-

tion spectrum and underlying interference effects. In this Chapter we present measurements

revealing Fano interference of a resonantly trapped piezoelectric surface acoustic wave (SAW)

mode with a broad continuum of surface phonons in a system consisting of a SAW resonator

coupled to a superconducting qubit. The experiments highlight the existence of additional

weakly coupled mechanical modes and their influence on the qubit-phonon interaction and

underscore the importance of phononic interference in quantum acoustic architectures that

have been proposed for quantum information processing applications. Additionally, we find

that the composite device geometry supports bulk acoustic excitations and identify the mode

structure and polarization of these spurious vibrational modes.

120

Wave interference is a universal phenomenon manifesting in a wide variety of both clas-

sical and quantum systems ranging from ocean waves to quantum circuits. The spectral

response of these systems encodes the existence of the underlying interference processes,

resonant modes, and their losses. A hallmark example is the Fano resonance [60, 61, 199],

which arises from the interference between a resonantly scattered mode and a continuum

of background states, and leads to a characteristically asymmetric spectral lineshape. Fano

interference has been realized in various fundamentally different quantum systems in which

sharp resonant modes interact with continuum excitations, including atomic and molecular

systems [200, 201], scattering in optical experiments [202, 203], and transport measurements

in quantum dot-based condensed matter systems [204,205]. Regardless of the physical plat-

form, this type of interference manifests as a significant change in the spectral response of

the system, and is therefore an important process to understand when interpreting spectro-

scopic or temporal measurements or assessing device performance. Here we demonstrate, for

the first time, the manifestation of Fano-interference in a quantum acoustics system. This

interference arises from the interaction between resonantly trapped surface acoustic wave

(SAW) phonons with a background of continuous phonon modes in an acoustic Fabry-P´erot

resonator that is cooled to near its quantum mechanical ground state. We infer the SAW

phonon interference by measuring the absorption spectrum of a superconducting transmon

qubit capacitively coupled to the SAW device.

7.2 Experimental setup

Figure 7.1a depicts a schematic of our experiment, which consists of a flux-tunable supercon-

ducting transmon qubit that is capacitively coupled with a SAW device hosting a resonant

121

Figure 7.1: (a) Left: Image of the hybrid system housed in the 3D microwave cavity used for
control and readout. Center: Optical micrograph of the SAW resonator. The large antenna
mediate the coupling between the qubit and SAW resonator. Right: The acoustic transducer
(green) is electrically connected to the antenna pads and phonons are confined within the
(b) Schematic of the experiment. Phononic
acoustic cavity by the Bragg mirrors (red).
Bragg mirrors form a spatially distributed Fabry-P´erot cavity that allows for a coherent
exchange of energy between the qubit and a resonantly confined SAW mode. (c) Top panel:
Measurement of two-tone qubit spectroscopy as the qubit is tuned through the main acoustic
resonance and nearby SAW continuum. Strong coupling is seen between the confined acoustic
mode while weaker coupling is seen between the qubit and the background of acoustic states.
Bottom panel: Simulated mode structure for the SAW device, in which the confined acoustic
mode is housed within a continuum of SAW states.

mode at ωm/(2π) = 4.4588 GHz. The spectral response of the SAW device was precisely

designed using the coupling-of-modes theory to define the electro-mechanical scattering prop-

erties of the device [118, 160], as shown in Fig. 7.1b. The effective electrical conductance of

the SAW device was designed such that the resonantly confined acoustic mode is in close

spectral proximity to a continuum of acoustic states (see Fig. 7.1b and Section 7.3). The

qubit and SAW device are fabricated on separate substrates, with the SAW device on YZ -cut

LiNbO3 and the qubit on high-resistivity silicon with maximum Josephson tunneling energy

EJ,max/h = 19.7 GHz and capacitive charging energy Ec/h = 328 MHz. Both devices are

122

4.404.454.50Frequency (GHz)100101102Conductance (mS)Confined SAW modeContinuousSAW modes4.404.504.354.404.454.50Qubit Frequency (GHz)1050S21(dB)(b)(c)250 μm10 μm(a)Leffgm|0⟩|1⟩galvanically connected to large antenna pads having an area 250 µm × 250 µm that form

a pair of parallel plate capacitors between the two devices when they are assembled in a

flip-chip configuration. The capacitive coupling in this hybrid system ensures that strain in

the piezoelectric SAW substrate induces a voltage across the qubit antennae, allowing for

the exchange of energy between the qubit and SAW phonons (see Fig. 7.1a). The composite

qubit-SAW system is housed in a 3D electromagnetic cavity with fundamental frequency

ωc/(2π) = 4.788 GHz, which is used for qubit control and readout as well as for applying in-

dependent excitation tones to populate the SAW device, and the interaction strength between

the qubit and microwave cavity is measured to be g/(2π) = 75 MHz. A superconducting

coil wound around the cavity provides magnetic flux tunability of the resonant frequency

of the qubit. The qubit absorption spectrum is measured via two-tone spectroscopy, using

the dispersive interaction between the qubit and cavity to determine the qubit state. As the

resonant frequency of the qubit is tuned through the confined acoustic mode, we observe an

avoided crossing of magnitude gm/(2π) = 9.76 ± 0.60 MHz (see Fig. 7.1b). Additional inter-

actions between the qubit and phonon modes are also observed. In particular these features

correspond to interactions between the qubit and phonons that are not strongly confined to

the SAW resonator and therefore couple much more weakly to the qubit, manifesting as a

series of dark states in the qubit spectra.

7.3 SAW resonator design

Although the SAW device used for this set of experiments is the same device as the previous

chapter, we now focus on the response of the phonon resonator in close proximity to the

confined acoustic mode rather than the phonon modes outside of the mirror stop-band. In

123

this section, we take a closer look at the properties of these resonant phonons and identify how

they can satisfy the physical constraints necessary to produce a Fano resonance. A complete

list of the SAW device parameters is presented in Table 6.1. Since the reflectivity of the

mirrors is relatively low per mirror grating, on average acoustic waves penetrate a distance

LP (cid:39) λmirror/2|rm| into the mirrors before being reflected. Based on the parameters in

Table 6.1, we calculate LP = 81.6 µm and the total effective length of the phonon cavity is

Leff = LIDT + 2LP = 175.2 µm. The free spectral range of the SAW resonator can then be

estimated as ∆fFSR = vs/2Leff = 10.4 MHz. The width of the mirror stop band is given
by ∆fmirror = 2|rm|f

= 14.2 MHz [118, 160]. Since the width of the mirror stop band is

π

comparable to the mode spacing, we expect the resonator to host a single strongly confined

acoustic mode as well as other more weakly reflected SAW modes as seen in Fig. 7.2 and 7.5.

In Fig. 7.2 we plot the simulated conductance of the composite resonator along with the

conductance of the IDT structure. On the frequency scale of the confined mode (∼ 1 MHz)

the IDT response is approximately constant creating a continuum of background SAW

phonons.

Figure 7.2: Solid green curve: Simulated conductance of the phononic resonator consisting
of a central acoustic transducer enclosed in Bragg mirrors. Dashed red curve: simulated
conductance of the acoustic transducer alone. The linewidth of the central resonance is
much smaller than the width of the transducer response, which can be approximated as a
continuous background.

124

4.444.464.48Frequency (GHz)050100150200Conductance (mS)ResonatorConductanceTransducerConductance05101520Conductance (mS)7.4 Results and discussion

To utilize the qubit to measure surface phonon excitations, we first calibrate the qubit

response to SAW excitations in the acoustic dispersive limit using the primary confined SAW

resonance. In this regime the qubit frequency and spectral shape depend in an established

and systematic fashion on the excitation number of SAW bosons. This ac Stark shift has

been observed in superconducting qubit systems coupled to both microwave resonators in the

cQED framework [40,56] as well as mechanical oscillators of multiple cQAD architectures [8,

49, 57, 59]. In this dispersive limit, the detuning between the qubit frequency (ωq) and the

resonant SAW mode (ωm) is large compared to gm (gm (cid:28) |∆|, ∆ = ωq − ωm). In particular

we tune the qubit frequency such that ∆/(2π) = −138.6 MHz and EJ /h = 8.5 GHz. In this

regime, we can approximate the Hamiltonian describing the hybrid system as ((cid:126) = 1) [10]

ˆH (cid:39) ωm

(cid:17)
(cid:16)
ˆa†ˆa + 1/2

+

(cid:18)

ωq + 2χmˆa†ˆa +

(cid:19)

g2
m
∆

ˆσz.

1
2

(7.1)

In Eq. (7.1) the SAW degrees of freedom are described by bosonic operators ˆa and ˆa†,

and the qubit is described by the spin 1/2 operator ˆσz. By considering the transmon as a

multilevel artificial atom, the total frequency shift of the qubit frequency per piezophonon,

2χm must take into account multiple partial dispersive shifts and is therefore given by [103]

2χm = −

2g2
m
∆

α
∆ − α

,

(7.2)

where α/h = 328 MHz is the anharmonicity of the transmon. Based on the experimen-

tal parameters of our system the qubit frequency shift per phonon at this detuning is

125

Figure 7.3: (a) Representative qubit spectra at two different applied powers near resonance
with the confined SAW mode. With increasing power the qubit spectrum both shifts in fre-
quency and inherits a spectral lineshape expected from the statistics of the acoustic coherent
state. (b) Measurement of mean phonon number as a function of drive power. The extracted
phonon number follows the expected linear trend (black). (c) Qubit frequency shift versus
extracted mean phonon number. Exact diagonalization considers the qubit as a 5 level atom,
which agrees with the experiment.

2χm/2π = − 0.97 MHz.

By applying a tone that is resonant with the confined acoustic mode we generate a coher-

ent SAW state and subsequent measurement of the resulting qubit spectra allows us to deter-

mine the mean SAW resonator occupancy number. We fit the qubit spectra (see Fig. 7.3a) to

a model [13] consisting of a two level system coupled to a harmonic oscillator coherent state

and extract the average phonon occupation number n. The asymmetry of the qubit spectra

126

4.264.284.304.324.34Qubit Probe Frequency (GHz)0.00.51.0Normalized TransmissionnSAW=0.6nSAW=9.80102030Phonon Number30150Frequency Shift (MHz)gm/(2)=10.15±0.05MHzDataExact Diagonalization(a)(b)(c)04080120160Drive Power (W)0102030Phonon Numberat large excitation numbers (see Fig. 7.3a) arises from the Poisson statistics of the distri-

bution of phonon numbers in the resulting SAW coherent state [5]. As shown in Fig. 7.3b

we find a linear relationship between the drive power populating the SAW resonator and

the extracted mean phonon number. We note that the power reported in our measure-

ments is that applied at room temperature, which is further attentuated by 60 dB in the

cryostat before entering the microwave cavity. Exact diagonalization of a multi-level Jaynes-

Cummings Hamiltonian describing this coupled system allows us to extract the expected

qubit frequency shift as a function of phonon number in the dispersive limit [155] and we

find excellent agreement between this prediction and the measured phonon number as shown

in Fig. 7.3c. In this analysis the coupling strength gm is a fit parameter to the data and we

extract gm/(2π) = 10.15 ± 0.05 MHz, which is in reasonable agreement with the measured

value determined from fitting the avoided crossing between the qubit and SAW modes. Hav-

ing calibrated the frequency response of the qubit to phonons in the confined SAW resonance,

we extend these Stark shift measurements to probe the acoustic environment outside of the

stop-band of the SAW mirrors. We measure the acoustic Stark shift δωq as a function of drive

frequency ωd, and extract the mean phonon occupation n = δωq/2χm over a range of drive

frequencies. As shown in Fig. 7.4a, the measured phonon number has a maximum at the

resonant frequency of the confined acoustic mode ωm/(2π) = 4.4588 GHz, and is strikingly

asymmetric about this peak. The acoustic excitation spectrum hosts a rich structure near

this confined resonance, making it possible for phonons across a range of frequencies to in-

terfere with each other. Because the reflectivity of the mirrors that define the SAW cavity is

relatively low (∼ 0.5% per mirror structure), the reflection process for surface phonons is dis-

tributed over the length of the cavity. This creates a situation in which phononic excitations

that reflect at different spatial positions within the cavity interfere with each other either

127

Figure 7.4: (a) Mean phonon number as a function of SAW drive frequency at a fixed power
of Pdrive = 25 µW. The SAW resonance is asymmetric in frequency and is well-described by
a Fano absorption function. The asymmetry arises from phonons in the confined acoustic
mode interacting with phonons in the SAW continuum. (b) Measurement of the qubit decay
rate over a broad range of frequencies far from the confined acoustic resonance. The fit (red)
is to the analytical expression of the SAW-induced loss due to the central SAW transducer.
The decay rate of the transducer is γIDT/(2π) = 249.7 MHz. (c) Fano asymmetry parameter
q as a function of drive power. The horizontal error bars correspond to the uncertainty in
the fit parameter nmax in Eq. (7.3). The black dashed line indicates q based on the analysis
of two coupled harmonic oscillators as described in the text.

constructively or destructively depending upon their relative wavevectors and propagation

paths. These interference processes produce a Fano lineshape, as phonons in the confined

acoustic mode are subject to differing interference with continuum surface phonons exist-

ing outside of the mirror stop-band. We model the resulting frequency-dependent phonon

128

4.4564.4604.464SAW Drive Frequency (GHz)0510Phonon Numberq = -0.283.43.84.24.6Qubit Frequency (GHz)1031021011(1/ns)IDT102030Maximum Phonon Number0.500.250.00Fano Parameter(a)(b)(c)number as a Fano resonance [60, 61] with an absorption spectra, n(ω):

n(ω) = nmax

(cid:18)

1 + q2 −

(qΓ/2 + ω − ωm)2
(Γ/2)2 + (ω − ωm)2

(cid:19)

+ noff.

(7.3)

Equation (7.3) depends on the linewidth Γ, of the resonantly confined mode, the average

background population of the continuum phonon modes noff, as well as the Fano parameter

q, which describes the level of interference between the confined and lossy surface phonons.

The fit parameter nmax quantifies the maximum phonon number for a given measurement,

which is set by the drive power. The first two terms in Eq. (7.3) do not depend on frequency

and impose the physical constraint that the minimum SAW phonon number is non-negative.

We note that the limit q −→ 0 corresponds to the absence of phonon interference, and in

this limit a Lorentzian response is recovered.

To determine the characteristic level of phonon interference in our device, we use the

qubit to measure the phonon occupation near the confined acoustic mode as a function of

the power used to populate the SAW device with phonons. By fitting each resulting phonon

spectra using Eq. (7.3) we determine the Fano parameter as a function of the maximum mean

phonon number. These results provide a relative measure of the acoustic interference in the

device and are displayed in Fig. 7.4c. In particular, we find that the phonon interactions

are well-described by Eq. (7.3), with a negative Fano parameter, down to the lowest phonon

levels we are able to measure. Additionally, we find that the phonon interference is roughly

constant with q (cid:39) −0.25.

To understand the interference we observe in our measurements, we compare the SAW

phononic system to a minimal classical model of two coupled oscillators that exhibits some

characteristics of a Fano resonance [206]. In this model, the lossy oscillator approximates a

129

continuum over the frequency scale of the confined mode (see Section 7.3) and the interaction

between the two oscillators leads to Fano interference. The loss rate of the confined acoustic

mode, which corresponds to the low loss oscillator, is extracted from the linewidth of the

fit to the resonantly trapped SAW mode in Fig. 7.4a. This linewidth indicates that the

loss of the confined mode is γSAW/(2π) = 630 kHz, corresponding to a SAW quality factor

QSAW (cid:39) 7000. To estimate the loss rate of the significantly broader interdigitated transducer

(IDT) response, which acts as an effective continuum over the scale of γSAW, we measure the

qubit decay rate, Γ1 = 1/T1, as a function of qubit frequency far from the confined acoustic

resonance. In this regime, the qubit loss is proportional to the electrical conductance [15,

81] of the SAW device, which is well described by only the IDT response far from the

confined mode, where the mirror reflectivity is much less than one. In particular we fit Γ1

to a phenomenological form taking into account loss resulting in the transduction of qubit

excitations into phonons that exit the SAW mirrors,

(cid:0)ωq

(cid:1) =

Γ1

ωq
Qi

+ Γ0 sinc2

(cid:18)

πNp

ωq − ωIDT
ωIDT

(cid:19)

,

(7.4)

where Qi = 1.05 × 104 is the qubit internal quality factor (this value is different from the

previous chapter, in this chapter we have defined the qubit quality factor as Qi ≡ ωq × T1i,

where T1i is the intrinsic lifetime of the qubit. The previous chapter uses the convention

Qi ≡ fq × T1i), Γ0 = 0.252 ns−1 is the maximum conversion rate of the qubit excitation

into SAW phonons, Np = 16 is the number of finger pairs in the IDT structure of the SAW

resonator, and ωIDT/(2π) = 4.504 GHz is the central transducer frequency, which is within

1% of the value predicted from the device fabrication parameters. By fitting the qubit

loss to the response given by Eq. (7.4) (see Fig. 7.4b), we are able to extract the overall

130

Figure 7.5: (a) Spectra of the SAW resonator inferred from the phonon-induced qubit Stark
shift as a function of power. As the drive power is increased, two additional spectral features
surrounding the strongly confined mode are visible, which we attribute to SAW modes that
are weakly confined by the Bragg mirrors. (b) Horizontal linecut of panel (a) taken at a
drive power of Pdrive = 160 µW.

loss associated with the central transducer, which we approximate as a Lorentzian with a

width of γIDT/(2π) = 249.7 MHz.

In the limit where the response of the transducers is

broad in frequency compared to that of the confined mode, as is the case in our experiments

(γSAW/γIDT ≈ 0.3%), the Fano parameter can be calculated as [206],

q =

1
γIDT ωSAW

(cid:16)

SAW − ω2
ω2

IDT

(cid:17)

.

(7.5)

Based on the experimental parameters this model predicts q = −0.36, which is indicated

by the black dashed line in Fig. 7.4b. The systematic deviation between the experimentally

determined and predicted values of q could arise from the presence of additional mechanical

modes near the confined SAW resonance. In fact, in our higher power measurements shown

in Fig. 7.5(a,b) we observe evidence for the existence of such modes on either side of the

strongly confined SAW resonance. These modes also exhibit asymmetry, indicating that

131

4.4564.464.464SAW Drive Frequency (GHz)01530|q|/(2) (MHz)(a)(b)4080120160Drive Power (W)250Frequency Shift (MHz)Pdrive=160Wthey are also interfering with the continuous phonon background. Since the calculated free

spectral range of the SAW resonator is comparable to the width of the mirror stop-band, it

is likely that these additional modes correspond to surface acoustic waves weakly confined

within the SAW cavity (see Section 7.3).

7.5 Coupling to transverse bulk phonons

At this point, we have seen how the interaction between different types of phonons is impor-

tant for understanding quantum acoustic systems. Even when specific types of mechanical

excitations are designed (like the surface phonon excitations in the previous section), the

possibility of other, naturally occurring phonon modes exist and can effect device perfor-

mance. Because the experiments described in this thesis use a relatively strong piezoelectric

material (LiNbO3), it is natural for us to consider the possibility of other types of mechanical

excitations within our system as well. This section details how our device geometry sup-

ports transversely polarized bulk phonons that are excited via the large dipole moment of

the device on lithium niobate and how they couple to the cQED system in our experiments.

Furthermore, as we shall see, we are able to perform contact-less excitation and measure-

ment of these acoustic excitations and their coupling to the microwave cavity even at room

temperature.

Quantum acoustics systems, in which superconducting qubits are interfaced with me-

chanical degrees of freedom offer a promising platform for quantum information science,

as mechanical resonators having a small spatial footprint and long coherence times can be

straightforwardly fabricated. By engineering hybrid quantum systems in this fashion, it

possible to design quantum memory protocols [20], implement microwave-to-optical trans-

132

duction schemes [27, 34], and operate the qubit as a sensor [16, 68]. The ability to leverage

mechanical degrees of freedom at the level of single, or few phonons, makes them promising

candidates for quantum information processing, with recent experimental results demon-

strating the creation of phononic Schr¨odinger cat states [52], the joint entanglement of high-

frequency mechanical oscillators [50], and the ability to simulate open quantum acoustic

systems [15]. However, the relatively large and geometrically complex structures typically

used in quantum acoustics devices can host a litany of other mechanical modes that may

have spurious coupling to the electromagnetic degrees of freedom of the quantum circuit.

Identifying and understanding these couplings is vital for high-fidelity control of the device

degrees of freedom, but also opens the door to coupling simultaneously to multiple distinct

types of phononic excitations in a single architecture. On the other hand, when they are

poorly controlled, these modes can act as unwanted decoherence channels for the quantum

systems of interest. Optimization of device geometry allows for the mitigation of these un-

wanted couplings and methods for improving the coherence of quantum acoustic systems. In

this work, we demonstrate a hybrid system that enables the interaction between a microwave

cavity and natural bulk transverse substrate phonons in Y -cut lithium niobate. By integrat-

ing this hybrid electromechanical system with a superconducting qubit, we are able to tune

the cavity frequency via its dispersive interaction with the qubit, enabling measurement of

the coupling between the cavity and high-frequency overtones of the acoustic mode. These

transverse acoustic modes are directly excited via the free-space coupling of the device dipole

to the 3D microwave cavity housing the devices and at low temperature we find that they

exhibit coupling to the qubit-cavity system with an interaction strength comparable to the

decay rate of the hybrid device. Additionally we find that similar devices, in which the qubit

is removed, enable contact-less coupling to these naturally occurring transverse modes with

133

a coupling that persists to room temperature (cf. Ref. [207]).

The experimental results reported here were obtained using a flip-chip hybrid quantum

acoustic device originally designed to investigate the coupling of a flux tunable transmon

qubit, fabricated on silicon, to a surface acoustic wave (SAW) resonator, fabricated on YZ -

lithium niobate [15, 16]. These previous experiments reached the strong coupling regime of

quantum acoustics, with a mechanical coupling strength of g/(2π) ≈ 10 MHz between

the qubit and SAW modes.

In the present manuscript, we report on experimental work

demonstrating the existence of additional bulk phononic modes in the lithium niobate chip

and how their direct coupling to the resonant mode of a 3D microwave cavity is mediated

by the dipole moment of the device. Due to the dispersive interaction between the cavity

and the qubit [11], these bulk acoustic modes also indirectly interact with the qubit. As

shown in Fig. 7.6(a), the qubit and phononic chips are coupled purely capacitively to each

other via large (250 µm)2 antenna pads on either substrate that form a set of parallel plate

capacitors when the devices are vertically aligned and adhered together using standard flip-

chip techniques [208]. The flip-chip stack is mounted in a copper 3D electromagnetic cavity

to control and measure the coupled devices (see Fig. 1 (a)). The antenna pads on the qubit

and phononic chips serve dual purpose, enabling the coupling between the devices as well as

serving as a coupling mechanism between the electric field in the microwave cavity and either

device independently [16]. As we will describe below, these macroscopic antenna pads also

introduce an unintended and surprisingly strong coupling to high-overtone bulk phononic

excitations, with harmonic number beyond 1000, in the lithium niobate substrate.

To experimentally investigate the coupling of the microwave cavity to these modes we

measure the microwave transmission S12 through the 3D microwave cavity at a relatively

low input power (Pinput ≈ −110 dBm) as we tune the qubit frequency via an external

134

Figure 7.6: (a) Image of the composite flip-chip stack in the bottom half of the 3D microwave
cavity used to probe the properties of the system. (b) Spectroscopy of the hybrid qubit-
cavity system as the qubit frequency is tuned via the magnetic flux through an external
superconducting coil. The qubit and cavity are coupled with an interaction strength of
g/(2π) = 73 ± 2 MHz (black dashed line). Additional coupling between the hybrid system
and bulk acoustic modes is observed as horizontal features at constant frequency, which are
(c) Spectroscopic data over a tighter
largely independent of the external magnetic flux.
frequency range demonstrating clear avoided crossings between the qubit-cavity system and
the acoustic modes. The fit to the data (dashed black lines) represent a coupled oscillator
model used to extract the coupling strengths (see Fig. 7.7).

magnetic flux. As shown in Fig. 7.6(b), when the resonant frequency of the qubit is tuned

across an entire flux quantum, the qubit and cavity modes undergo an avoided crossing,

which can be used to extract a coupling strength of g/(2π) = 73 ± 2 MHz between the

two modes. As seen in Fig. 1(b) the cavity transmission also exhibits a series of avoided

crossings, indicating coupling to additional modes, which appear as horizontal breaks in

microwave transmission, and are largely independent of external magnetic flux. By zooming

in on several of these additional avoided crossings in Fig. 7.6 (c) the transmission data are

fit to a coupled oscillator model to extract the interaction strength between the qubit-cavity

mode and these additional weaker resonances. In the top panel of Fig. 7.7 we present the

fitted coupling rates gm/(2π) for the five anti-crossings that are within a few line-widths of

the resonant frequency of the microwave cavity. The bottom panel of Fig. 7.7 shows how

the coupling to these modes manifests in the bare cavity transmission when the cavity is

flooded with a large number of photons (Pinput ≈ −60 dBm) [142]. Even when the cavity

135

603003060Flux Tuning Current (mA)4.78254.79004.7975Frequency (GHz)3.94.24.5Transmon Frequency (GHz)4.804.814.82Frequency (GHz)3020100S21(dB)(a)(b)(c)is populated with many photons, and the qubit state no longer dispersively shifts the cavity

resonance, the presence of these additional modes is still strongly imparted on the cavity

spectrum as “notches” in the measured transmission.

Additional acoustic modes in the strongly piezoelectric lithium niobate substrate are a

natural explanation for these additional resonances, which couple to the hybrid qubit-cavity

system. In fact, periodically notched spectra, similar to the one shown in the bottom panel of

Fig. 7.7, have been recently been observed in the transmission of microwave cavities coupled

to high-overtone longitudinal phononic modes in quartz [209]. Additionally, high-overtone

bulk acoustic resonators (HBARs), using a variety of piezoelectric materials, have been shown

to strongly couple to superconducting qubits in quantum acoustics architectures [3, 28, 52,

59, 149, 150, 152, 210]. Furthermore, as we describe below, we use a combination of finite

element modeling (FEM) and additional characterization measurements to verify the nature

of these bulk acoustic modes, their polarization, as well as their coupling mechanism to the

hybrid qubit-cavity system.

As shown in the COMSOL Multiphysics simulations presented in Fig. 7.8(a), the time-

varying electric field within the microwave cavity will impinge on the large-scale aluminum

antenna pads on the surface of the lithium niobate chip. These two pads function as an

electric dipole and the mobile charges in these metallic pads screen the cavity electric field,

locally polarizing the substrate. As a consequence of the piezoelectric effect, this polarization

creates strain within the lithium niobate substrate.

If the frequency of the time-varying

electromagnetic cavity field is resonant with a natural mechanical mode imposed by the

geometry of the substrate and the speed of sound in lithium niobate, the generated strain

will excite acoustic oscillations that will resonate within the bulk of the material. In this

fashion, the dipole antenna on the lithium niobate chip enables the interaction between

136

the cavity mode and bulk phonon modes in the substrate. Furthermore, since the resonant

frequency of the cavity can be weakly flux-tuned via its dispersive interaction with the qubit,

this device architecture allows us to measure the interaction between the cavity mode and

acoustic modes over a frequency range larger than the cavity line-width. To determine the

Figure 7.7: Top: Coupling strength, gm/(2π) between the microwave cavity and five overtone
bulk mechanical modes. Bottom: Measurement of the cavity spectrum at large input power
(Pinput ≈ −60 dBm) demonstrating that the transmission is independent of the qubit state
as described in the main text. Coupling to the acoustic modes manifests as characteristic
“notches” in the cavity transmission.

exact nature of the acoustic modes being excited by this dipole-coupling to the cavity field,

we use COMSOL simulations to find the mechanical eigenmodes of the Y -cut lithium niobate

substrate as well as their polarization [153]. These simulations reveal the existence of both

longitudinal and transverse acoustic modes. In order to model the effect of the cavity electric

field, we apply a time-varying voltage difference across the chip in the x -direction of the

substrate in these simulations (see Fig. 7.8(a)). In the presence of this external electric field,

the symmetry of the combined electromechanical system only supports mechanical modes

that mirror the symmetry of the external electric field, and thus we find that only transversely

polarized mechanical modes can be excited (see Fig. 7.8(b) for the spatial structure of the

137

4.794.804.814.82Frequency (GHz)605040S12(dB)02004006008001000gm/(2) (kHz)n = 5 mode). In particular, the results of the simulations reveal a transverse mode having

fundamental frequency f1 = 3.5784 MHz propagating along the y-axis of the substrate. We

note that the simulations enforce that the mechanical displacement field is equal along the

two edges of the substrate parallel to the x-axis. The fundamental frequency of the transverse

mode, and its harmonics, are related to the speed of bulk transverse sound in lithium niobate

vt and the thickness of the chip:

fn =

nvt
2t

,

(7.6)

where n indexes the harmonic number and t = 500 µm is the thickness of the substrate used

in our experiments and simulations. The speed of transverse sound within the bulk lithium

niobate is related to the material properties of the substrate via Hooke’s Law:

C44 = ρv2
t ,

(7.7)

where C44 = 5.95 × 1010 Pa is the appropriate elastic constant for transverse mechanical

waves in Y -cut lithium niobate and ρ = 4647 kg/m3 is the mass density of the material [211].

Eqn. 7.7 gives the speed of transversely polarized bulk sound as 3578 m/s, consistent with

Ref. [212], and a corresponding free spectral range of ∆fn = 3.578 MHz between adjacent

acoustic modes. When a spatially uniform external electric field (Eext) is applied across

the substrate, as is the case for the cavity field in the vicinity of the dipole antenna on the

lithium niobate chip in our experiments, the piezoelectric coupling gem is proportional to

the overlap integral between the electric and strain fields [209]:

gem ∝

(cid:90)

V

Eext S((cid:126)r) dV,

(7.8)

138

Figure 7.8: a) Top: Schematic indicating the orientation of the time-varying electric field of
the 3D microwave cavity. Bottom: COMSOL simulation of the induced normalized voltage
on the surface of the lithium niobate substrate when the system is driven at the fundamental
frequency of the transverse mode ω/(2π) = 3.5784 MHz. The black squares represent the
antenna pads, and the profile of the induced voltage has a dipole spatial structure. (b) FEM
simulation of the transverse strain Sxy associated with the 5th harmonic of the transverse
bulk acoustic mode.

where S((cid:126)r) is the spatially varying strain field within the chip and the integral is taken over

the entire volume V . The y-dependence of the strain field for transverse phonons is given by

S(y) = S0 sin

(cid:17)

(cid:16) 2πy
λn

, where S0 is the amplitude of the strain. For the nth harmonic of the

transverse mode having wavelength λn = 2t/n, Eqn. 7.8 predicts non-zero coupling for odd-

indexed modes only. This implies that in our experiments the 3D microwave cavity electric

field should couple only to every other transverse phonon mode, leading to an effective

free spectral range 2 × fn (cid:39) 7.16 MHz based on the fundamental frequency determined

from the FEM simulations. This matches (to within 10%) the measured free spectral range

determined from the spacing of adjacent notches in the microwave cavity transmission shown

in Fig. 7.7. Based on the calculated free spectral range of the bulk acoustic modes and the

cavity frequency, we can reliably excite bulk mechanical excitations with mode number

n (cid:39) 1339 that couple to the qubit.

139

E(r,t)0Vnorm-11-202Z (mm)-202X (mm)(a)(b)xySxy1-1Figure 7.9: Room temperature characterization of the transverse acoustic modes. (a) When
the dipole moment of the metallic pads ((cid:126)d) is parallel to the cavity electric field, coupling is
observed between the cavity mode and the acoustic modes of the substrate. (b) Zoomed in
cavity transmission from (a) near the cavity resonance. The coupling to the bulk acoustic
modes is apparent as “notches” in the cavity transmission. (c) When the LiNbO3 substrate
is oriented such that the dipole moment is perpendicular to the cavity field, no coupling
between the two systems is observed. The shift in the cavity frequency between panels (a,b)
and (c) arises from the large anisotropy of the lithium niobate dielectric tensor [211].

In addition to spectroscopy measurements at cryogenic temperatures, we also performed

additional characterization experiments at room temperature to investigate these transverse

bulk acoustic modes, and their coupling to the microwave cavity.

In particular, to fur-

ther elucidate the coupling of the cavity field to the dipole formed by the metallic antenna

pads, we fabricated an additional device having only large antenna pads on a separate Y -

cut lithium niobate chip and loaded it into a second 3D microwave cavity. As shown in

Fig. 7.9(a,b) the appearance of the notches in the cavity transmission clearly persist even

140

4.784.804.824.844.864.88Frequency(GHz)605040S12(dB)E(r,t)d5.145.165.185.205.225.24Frequency (GHz)504030S12(dB)E(r,t)d4.824.834.844.85Frequency (GHz)484440S12(dB)(a)(b)(c)T = 293 Kat room temperature, albeit with a reduced amplitude. These measurements serve to re-

inforce the conclusion that the 3D cavity enables a method for contact-less excitation of

these high-overtone acoustic modes. Additionally, we find that when the sample is rotated

by 90◦ in the cavity, the notches in the microwave transmission disappear indicating that

the coupling to the transverse modes vanishes (see Fig. 7.9(c)). In this configuration the

dipole moment of the device is perpendicular to the cavity electric field and cannot induce a

polarization between the pads, nor the reciprocally generated transverse strain field via the

piezoelectricity of the substrate. This rotation dependence is consistent with the conclusions

we draw from the FEM simulations of the device and further verify that the dipole coupling

between the pads and the cavity field are responsible for exciting the transverse modes of

the lithium niobate chip. Finally, comparing Fig. 7.9(a) with Fig. 7.9(c) clearly shows that

the microwave cavity line-width is significantly broadened when the lithium niobate chip is

oriented such that the antenna pads are aligned to the cavity field. In particular, we find

that in this configuration the cavity quality factor is decreased by a factor of approximately

3.75, from Q (cid:39) 1500 to Q (cid:39) 400, relative to the configuration in which the pads are or-

thogonal to the cavity field. To understand the origin of these additional loss, we tested

bare lithium niobate chips having the same dimensions but without the metallic antenna

pads. Unsurprisingly, in these additional control samples we do not observe any coupling

between the cavity field and bulk acoustic modes regardless of the orientation of the chip

in the microwave cavity (i.e. no cavity notches are observed). Interestingly, however, we

find that the cavity quality factor is Q (cid:39) 1500 regardless of the chip orientation, strongly

indicating that the reduction in the cavity quality factor observed in Fig. 7.9(a,b) arises from

the mechanical loss associated with the transverse phonon modes. Finally we note that the

quality factor of the cavity in the low-temperature measurements presented in Fig. 7.7(b) is

141

Q (cid:39) 2500 indicating a reduction of the mechanical losses when the lithium niobate chip is

cooled to cryogenic temperatures.

We have demonstrated the capacitive dipole coupling between a hybrid 3D cavity system

and high-overtone transverse bulk acoustic modes in Y -cut lithium niobate. These transverse

acoustic modes are excited by the microwave field of the 3D cavity and detected as absorption

features in the cavity transmission. The coupling to the acoustic modes is found to be

comparable to other losses in the hybrid system, reinforcing the importance of understanding

the role of unintended mechanical modes in quantum acoustic systems proposed for quantum

applications. Beyond quantum acoustics, we find that this contact-less method of acoustic

excitation and detection in our devices persists to room temperature, highlighting the utility

and versatility of 3D cavities for characterizing the frequency response and mechanical losses

in piezoacoustic devices in general.

7.6 Conclusion

In conclusion, we have demonstrated the existence of additional phonon modes in our device

and how they couple directly to the qubit or to the microwave cavity. In particular, we have

demonstrated the Fano interference of surface acoustic wave phonons in a hybrid quantum

acoustic device containing a resonantly confined SAW mode embedded in a continuum of

surface phonons. This phononic interference is inferred from qubit-assisted spectroscopy

of the SAW device and we find that it persists down to extremely low excitation number.

The experimental results are in excellent agreement with the functional form of a Fano

resonance and highlight the importance of phononic interference in quantum acoustic devices

proposed for applications in quantum information processing. Additionally, we measure and

142

characterize transversely polarized bulk phonons that strongly couple to the microwave cavity

that houses the hybrid quantum acoustics system. We are able to integrate room temperature

measurements of these bulk phonons with FEM simulations to characterize the additional

excitations within the system. This chapter highlights the importance of considering all

possible types of phononic excitations to more completely characterize quantum acoustic

systems.

143

Chapter 8

Vantablack Shielding of

Superconducting Qubits

8.1

Introduction

This chapter describes a set of experiments not directly related to quantum acoustics. These

experiments were directed at understanding a new material, Vantablack, as a radiation ab-

sorbing coating to improve coherence of cQED systems. Superconducting qubits are among

the state of the art technologies being developed in the pursuit of a functional quantum

computer [104, 213, 214], with current depolarization and dephasing coherence times for

transmon-based processors exceeding 100 µs [215], and recent experiments employing heavy

fluxonium [125] demonstrating coherence times up to 1 ms [127]. One of the primary sources

of decoherence in modern superconducting processors is the presence of non-equilibrium

quasiparticles (broken Cooper-pairs) in the in the superconducting electrodes that form

the circuit [128, 216, 217]. It is well known that stray black-body infrared (IR) radiation,

which has energy greater than the superconducting gap, can break Cooper pairs in the

superconductor, increasing the density of non-equilibrium quasiparticles, and poisoning co-

herence [133, 218, 219]. Additionally, it has been found that effectively shielding the system

from these IR photons improves in the internal quality factor of the superconducting res-

144

onators employed for qubit control and readout [220].

In order to further increase qubit

coherence times it is important to continue to investigate new methods and materials for

shielding superconducting qubits and resonators from unwanted sources of decoherence. In

this chapter, we investigate the performance of a new IR shielding material: Vantablack [221].

Our preliminary results indicate that Vantablack has potential to yield performance beyond

standard coatings when properly applied.

8.2 Experimental details

Vantablack is a coating composed of vertically aligned nanotube arrays, grown via a mod-

ified chemical vapor deposition process, and exclusively developed by Surrey NanoSystems

Ltd [221]. It is one of the darkest substances known, reflecting less than 0.2% of light in

the visible spectrum. These extraordinary absorbing characteristics extend into the infrared

spectrum, where Vantablack reflects less than 0.5% of IR photons [222]. Vantablack is found

in many light absorbing applications, including commercial thermal imaging systems [223]

as well as beam dumps for high power optical experiments [224].

To investigate the performance of this shielding material in the context of superconduct-

ing qubit systems we measure the coherence properties of a single-junction transmon qubit

housed in a three-dimensional (3D) microwave cavity [103]. In cQED experiments, such as

the ones we perform, it is common to enclose the experimental setup in a shielding material

that protects both the control/readout resonator and qubit from stray black-body radiation.

In our experiments, this is achieved by housing the qubit and resonator in a copper cylinder

coated internally with the shielding material, and then thermally anchoring the cylinder to

the mixing chamber of a dilution refrigerator having a base temperature of (cid:39) 10 mK, which is

145

much less than the effective temperature associated with the qubit transition frequency (ap-

proximately 250 mK). To compare the performance of Vantablack to a standard epoxy-based

coating, which serves as a control experiment, we cover the inside of two identical copper

cylinders in either a standard IR coating or Vantablack as shown in Fig. 8.1a. The standard

infrared absorbing coating we use for our control experiment is composed of a mixture of

(by mass) 68% Stycast 2850FT epoxy, 5% Catalyst 24LV, 7% carbon lamp black, and 20%

175 µm diameter glass beads. This particular type of epoxy-based coating is often referred

to as Berkeley Black [225] and has been used in previous microwave circuit experiments to

produce systematic improvements in the quality factor of superconducting resonators [220].

Then, on two-separate cool-downs of the dilution refrigerator, we place the same microwave

cavity and qubit into one of these two copper cylinders and measure the coherence properties

of the qubit.

Both the control and readout of the qubit, which has a frequency ωq/2π = 5.165 GHz,

are mediated via the electric field of the TE101 mode of the 3D electromagnetic resonator

with frequency 6.936 GHz (see Fig 8.1b.) [119]. For a given measurement, microwave pulses

of appropriate length and amplitude are used to manipulate the state of the qubit. After a

specified amount of free-evolution, the state of the qubit is then inferred via measurement of

the transmission through the microwave cavity, which is dispersively shifted by the presence

of the qubit [10]. In particular, we employ a high-fidelity readout protocol based on the non-

linearity of this interaction [142–144]. A more detailed description of the relevant microwave

circuit and qubit measurement protocols can be found in Ref. [130].

146

Figure 8.1: (a) Cylindrical copper housings containing the 3D microwave cavity and super-
conducting qubit. The interior of these cylinders is coated with either a Vantablack (left)
or a Berkeley Black (right) coating. (b) Bottom half of the 3D microwave control/readout
cavity and transmon qubit, which was fabricated on high resistivity silicon.

8.3 Results and discussion

To characterize the effectiveness of the coatings at mitigating loss, we measure the depolar-

ization time T1 and the dephasing (Ramsey) time T ∗

2 of the qubit in the copper housings

covered in Berkeley Black and Vantablack. In Fig. 8.2 we show representative measurements

of both T1 and T ∗

2 for the qubit housed in the Vantablack coated shield as well as the ac-

companying microwave pulse sequences applied to perform these measurements. Because

the measurement of the Ramsey decay time is a phase sensitive measurement, a detuning ∆

between the drive frequency and ωq will lead to a decaying sinusoid rather than a decaying

exponential function. This allows us to extract the magnitude of the detuning between the

drive frequency and the qubit frequency in addition to the dephasing time of the qubit.

147

Figure 8.2: (a) Microwave pulse sequence to measure T1 (top) along with a representative
measurement (bottom). (b) Microwave pulse sequence to measure T ∗
2 (top) and a repre-
sentative measurement (bottom). As described in the text, a detuning ∆/2π = 287 kHz
separated the drive frequency and ωq in these measurements.

It is well known that qubit coherences can fluctuate over long timescales due to inter-

actions with quantum two level systems [226–230] as well as stray radiation from sources

such as cosmic rays [187, 231]. Therefore in order to quantify qubit decoherence, we perform

repeated measurements of both T1 and T ∗

2 to obtain sufficient statistics to understand the

performance of the two coatings relative to one another. Specifically, we interleave T1 and

T ∗
2 measurements over a 16 hour period and each measurement contains 252 points in time

with 100 averages at a repetition rate of 9 kHz. This allows us to extract T1 and T ∗

2 at a rate

of approximately (5 s)−1. From these measurements we extract the pure dephasing time of

the qubit,

Tφ =

(cid:19)−1

(cid:18) 1
T ∗
2

−

1
2T1

(8.1)

and compare the resulting distributions between the two experiments with different shielding

materials. The measured probability distributions for both T1 and Tφ for the qubit system

shielded by Vantablack and Berkeley Black coatings are plotted in Fig. 8.3.

In order to

compare probability distributions of these coherence measurements, we divide each histogram

bin by the total number of experiments to normalize the histograms to have area equal to

one. The average values T1 and Tφ obtained from each distribution are listed in Table 8.1

148

demonstrating a similar level of coherence in both experiments and indicate that Vantablack

is a compatible coating with cQED experiments.

Figure 8.3: Probability distributions of both T1, along with fits to Poisson distributions (red
curves), and Tφ in either the Vantablack coating (a,b) or the Berkeley Black coating (c,d).

Coating
Vantablack

T1 (µs)
22.9 ± 5.6
Berkeley Black 27.5 ± 8.3

Tφ (µs)
29.7 ± 14.9
45.9 ± 41.2

Pe(%)
3.2 ± 0.59
0.5 ± 0.17

Table 8.1: Averaged results from the qubit coherence distributions shown in Fig. 8.3 and
their standard deviations. Pe represents the thermal population of the excited state of the
qubit as described in the main text.

While there does not seem to be a large difference between T1 in either shielding en-

vironment, it is noteworthy that the average depolarization time in the Berkeley Black

environment is slightly longer. Uncorrelated losses of qubit excitations will manifest as a

Poissonian distribution in T1 [232], which we find in both sets of measurements regardless

of coating. Because the distribution of T1 measurements is governed by Poisson statistics,

a larger mean will also lead to a skewed distribution towards higher values of T1, which is

consistent with the slightly higher value observed in experiments using Berkeley Black as a

coating.

149

In order to further investigate the differences in performance between these two coatings

we measure the residual excited state population of the qubit using a method described

in Refs. [128, 129], which can be interpreted as an effective qubit temperature. As shown

in in final column of Table 8.1 we find that the qubit system housed in the Vantablack

coating has a residual excited state population of 3.2% as compared to 0.5% in the Berkeley

Black coating. This maps onto an effective qubit temperature of 72.6 mK (Vantablack) and

46.6 mK (Berkeley Black). Using the principle of detailed balance, we are able to calculated

the rate at which spurious excitations in the system drive the qubit from its ground to excited

state,

PgΓ↑ = PeΓ↓,

(8.2)

where Pg and Pe are the residual population of the qubit ground and excited states, re-

spectively. Γ↓ = 1/T1, and Γ↑ is the rate at which the qubit is driven from the ground to

excited state. We find that in Vantablack, Γ↑ (cid:39) 1.4 × 103/s, and in the Berkeley Black

coating, we find that Γ↑ (cid:39) 0.19 × 103/s. Two primary mechanisms may be responsible for

the difference in spurious excitation rate between the two coatings. First, the qubit tem-

perature between the two experiments could differ slightly, perhaps due to a relatively low

thermal conductivity for Vantablack versus Berkeley Black at mK temperatures. In order

to test such a hypothesis, more extensive studies of the thermal properties of Vantablack

are necessary. Alternatively, and more interestingly, the density of stray photons present in

the Vantablack shielded environment could be higher than in the housing coated in Berkeley

Black, indicating that the performance of Vantablack is inferior to that of Berkeley Black.

These preliminary measurements do not allow us to unambiguously disentangle these two

possibilities and future experiments optimizing the Vantablack coating are needed to identify

150

which may be causing the increased rate of qubit excitation.

In particular, although Vantablack has an extremely low reflectivity at wavelengths in the

IR range, it is known that increasing the thickness and roughness of light-shielding coatings

can further decrease the reflectivity of a coating. In particular, Ref. [225] reports a ∼ 30%

reduction in IR reflectance upon increasing the thickness of commercially available coating

(Chemglaze Z306) from 25 µm to 100 µm. Vantablack is created as a relatively thin coating,

with thickness ranging from 20 µm to 50 µm [222]. Assuming a similar scaling with thickness

would reduce the IR reflectance of Vantablack from 0.5% to 0.35%. For comparison Berkeley

Black has a typical application thickness near 100 µm, or more, with the addition of glass

beads which provide extra scattering sites for stray photons. This leads us to believe that

the addition of extra materials into Vantablack that increase the scattering of stray photons

may improve qubit performance in future experiments using Vantablack as a IR shielding

material.

8.4 Conclusion

In conclusion, we have investigated of the utility of Vantablack as a novel shielding material

on the coherence properties of a superconducting qubit system. We find that Vantablack does

not significantly negatively impact the measured coherence properties of the qubit. It could,

however, lead to a higher effective qubit temperature and therefore higher rate of spurious

qubit excitation. However, future experiments are needed to more completely understand the

potential of this material in improving the state-of-the-art. In particular, future experiments

in which high-frequency microwave and IR radiation can be controllably injected are needed

to systematically study qubit coherence as a function incident power [218]. Similarly, planar

151

superconducting qubit geometries have a much smaller mode volume, and are much more

sensitive to surface losses [119], and may respond more dramatically to changes in the IR

shielding environment. Building upon the initial experiments reported here, these future

experiments can advance the understand of Vantablack as an IR shielding material for cQED

systems based in superconducting circuits.

152

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APPENDIX Flip-chip device assembly procedure

One of the primary technical accomplishments of this thesis was the integration of a “flip-

chip” procedure to implement the capacitive coupling between the transmon qubit and the

high-frequency SAW resonator (see Fig. 6.2). The fabrication of both the qubit and the SAW

resonator follow standard nano-fabrication procedures (the details of which can be found in

the Ph.D. thesis of Dr. Justin Lane in Ref. [118]). Once both the qubit and SAW resonator

are fabricated, the next step needed to create a hybrid cQAD system is to implement the

coupling between the two devices via a flip-chip procedure. This appendix describes how we

adhere the two chips together spatially, allowing for the strong coupling between qubit and

mechanical modes necessary for the primary results of this thesis.

Hard-baked resist spacers

When designing a quantum acoustics device, we want to specify some nominal coupling

between the two chips. We choose to do this by fabricating spacers made of hard-baked

photoresist on the chip containing the SAW wafer (we choose not to fabricate the spacers

on the qubit chip to minimize the chance of accidentally destroying the Josephson junctions

with extra fabrication steps). We spin-coat four individual layers of S1813 resist at a speed

of 5000 RPM for 50 seconds on the wafer containing the SAW chip. With this spin speed,

the nominal thickness of each layer is (cid:39) 1.3 µm, creating a total resist thickness of 5.2 µm.

Between each step of spin-coating, we bake the wafer on a hotplate at 110◦C for 1 minute.

We then expose the resist to UV light for 90 seconds using a standard mask aligner, and

develop the exposed resist in 352-developer for (cid:39) 45 seconds. At this point, the chip should

have a functioning SAW resonator as well as the pattern of developed S1813 resist nominally

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5.2 µm thick. In order to make the resist robust for further fabrication steps, we then set the

hotplate to 250◦C and bake the sample for 90 minutes. This sufficiently hardens the resist

so that it will not easily come off the sample even when exposed to solvents. At this point,

the qubit and SAW chips are ready for the flip-chip procedure.

Flip-chip coupling

Our current flip-chip assembly procedure relies on the use of a custom-made mask (we refer

to it as the “flip-chip mask”) compatible with the mask aligner in the Keck Microfabrication

facility (KMF) cleanroom. The mask consists of a plate with two small vacuum holes (see

Fig. A.1). These holes are connected to the “ball lock” vacuum lines on the mask aligner

by use of a vacuum tee. This allows us to vertically suspend a small chip (we use the chip

containing the SAW resonator) upside down and align the antenna pads on the SAW chip

with the antenna pads on the qubit chip. We use transparent wafers for the SAW chips

(either double side polished lithium niobate or quartz), so that we can see through the SAW

chip to the qubit chip underneath. Moving forward, it may be useful to fabricate the qubit

chip on a transparent substrate (such as sapphire), so that either chip can be used as the

“top chip”. In order to make sure that the two chips actually stick together, we use a fine-tip

needle to place two small drops of resist on the qubit chip (note: essentially no drop of resist

is too small for this step, I’ve often noticed that drops too large in size can both cause a

mess and cause the chips to be spaced very far apart from each other). Once the gluing

resist has been placed on the qubit chip, we mount the qubit chip on the sample stage of the

mask aligner. We then connect the flip-chip mask to the necessary vacuum lines and place

the SAW chip on the flip-chip mask such that the sample is held to the mask via the custom

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Figure A.1: Left: Bottom side of the flip-chip mask. The dimension of the mask is approxi-
mately 4 × 4 inches, which is compatible with our mask aligner. Inset: Zoomed in image
of the line of sight port and the vacuum holes that allow us to suspend a substrate upside
down. Right: Top side of the flip-chip mask. The vacuum tee that connects to the lines
on the mask aligner is indicated on the left side of the image while the line of sight port is
indicated at the center of the mask.

vacuum ports (see Fig. A.1, left panel). We mount the flip-chip mask to the mask aligner

(make sure to have the vacuum tee facing towards you when you mount the mask, otherwise

the tee will interfere with the microscope attached to the mask aligner). We then adjust

the bottom chip (the qubit chip) using the four axis control of the sample stage. Once the

large antenna pads of the qubit and SAW chips are laterally aligned, we move the qubit chip

vertically upwards until it makes good contact with the SAW chip (you can tell the contact

is good by moving the qubit chip downwards, if the resist on the qubit chip is in contact

with the SAW chip, it will bring the SAW chip down with it).

Once the qubit chip and SAW chips are stuck together, bring the newly formed hybrid

system away from the flip-chip mask by moving the sample stage vertically downward. Re-

move the flip-chip mask from the mask aligner. At this point, I find it useful to let the resist

“harden” by letting the hybrid system sit at room temperature for five minutes (we haven’t

systematically investigated whether this improves the reproducibility of the procedure, so

this step may not be necessary).

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The final step of the procedure is to bake the hybrid system so that the resist (acting

as a glue) will harden, ensuring that the system is stuck together. We typically bake the

system with the larger chip (the SAW chip) on the “bottom” of the stack at 110◦C for 5

minutes. Once the hybrid system has been taken off of the hotplate and cooled, it is ready

to be mounted into a 3D cavity and tested in the cryostat.

179