Quantum Acoustics with Multimode Surface Acoustic
Wave Cavities
byLucas R. Sletten
B.A., St. Olaf College, 2015
M.S., University of Colorado, 2018

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2021
Committee Members:
Konrad W. Lehnert, ChairJohn Teufel
James Thompson
Murray Holland
Zoya Popovic

ii
Sletten, Lucas R. (Ph.D., Physics)
Quantum Acoustics with Multimode Surface Acoustic Wave Cavities
Thesis directed by Prof. Konrad W. Lehnert

Control over the quantum state of macroscopic mechanical oscillators promises advances in
the understanding of fundamental physics as well as the development of new quantum technologies. Surface acoustic waves are an attractive mechanical system in a quantum context as they canbe designed to interact with many popular quantum platforms, including superconducting qubits.
Moreover, surface acoustic waves are already a mature commercial technology widely used in classical signal processing where the slow speed of sound (km/s) means structures with long delays andtherefore fine frequency features can be engineered in chip-scale geometries.
In this thesis, I will describe how quantum acoustics with surface acoustic waves coupled
to superconducting qubits can leverage these long delays to build cavities with high densities ofresonant modes and qubit-phonon interactions precisely tailored in the frequency domain to suitexperimental demands. I first demonstrate resonant interaction between a transmon qubit and amulti-mode surface acoustic wave resonator where the qubit-resonator coupling strength exceeds
not only the decay rates of the qubit and resonator but also the spacing between resonant modesof the cavity. As a natural extension of this result, I describe how intentional shaping of the qubitcavity coupler in real space leads to a desirable frequency-dependent interaction strength. Thishybrid system can achieve interaction strengths large enough for the single-phonon Stark shift toexceed the relevant dissipation rates, leading to the resolution of phonon number states in the qubit
spectrum. I will conclude by evaluating the prospects for improved qubit and acoustic performancethat would enable a host of experiments, in particular showing that the dominant acoustic loss
mechanism, phonon diffraction, can be eliminated by implementing focusing acoustic cavities.

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Acknowledgements

The work contained in my thesis was not performed in a vacuum. I have been truly fortunate
to have had help from many amazing people. Their broad expertise, thoughtful guidance, andready assistance made my research possible and my time in Boulder so enjoyable.
First, thanks are in order for my committee members John Teufel, James Thompson, Murray
Holland, and Zoya Popivic, for giving me their valuable time and invaluable feedback, and also foragreeing to another multi-hour Zoom meeting. I sincerely hope my defense is the last one they
have to do remotely. I would like to thank Zoya in particular, as she taught me and a substantial
contingent of Lehnert lab members the fundamentals of microwave engineering that underlies somuch of the work in this thesis.
This thesis involved a significant amount of microfabrication, and so I would like to thank the
Keck lab staff, Dave Alchenburger and Mark Carter, and the more extensive BMF staff, for theirpatience and expertise. Travis Autrey deserves special thanks for helping me develop my process
and learn the tools in the BMF cleanroom and for the fruitful discussions about focusing SAW
cavities.
I also want to thank everyone in the Lehnert group. The lab dynamic was always friendly
and collaborative, filled with lively conversations about technical difficulties, exciting results, andthe occasional JILA cup adventure. I need to thank Adam Reed for his training during my firstyear, showing me first-hand how to measure quantum systems, run a dilution refrigerator, and how
to be a successful graduate student. I owe a special thanks to Brad Moores, who was right by myside through the core of my PhD and whose work and ideas pervade this thesis. I also must thankiv
Jeremie Viennot for his contagious and effective mix of keen skepticism, scientific enthusiasm, andhigh tolerance for questions. Lastly, I want to thank Brendon Rose for helping push me across thefinish line and who, along with Alec Emser and Pablo Aramburu-Sanchez, have brought fresh ideas
and energy to the project that will without a doubt take them to many exciting places.
Of course, I owe a great deal to Konrad. He has been an excellent advisor, providing countlessinsights and priceless guidance. In particular, he has a practiced knack for transcribing his deep
knowledge base into concise models, which he so clearly enjoys sharing (and re-sharing), and thatwill forever shape my view of the world.
Finally, my family deserves the biggest thanks of all. Ellyn is simply the best, and I wouldn’t
have been able to make it here without her support, particularly in this last year when the worldchanged so drastically. My parents and brother and sister are wonderful people who have shaped
me into who I am and supported my psychological well-being through their continual love andencouragement.
I am sure this list is incomplete and inadequate, but to everyone who has helped me along
the way: thank you so much!

Contents

Chapter
1 Introduction

1

2 Surface Acoustic Wave Resonators

10

2.1

2.2

2.3

Elastic Waves in a Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1

Stress and Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2

Transverse and Shear Waves

2.1.3

Rayleigh Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.4

Incorporating Anisotropy and Piezoelectricity . . . . . . . . . . . . . . . . . . 15

. . . . . . . . . . . . . . . . . . . . . . . . . . . 12

SAW resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1

Bragg Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2

Fabry-Perot Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.3

Cavity Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.4

Interdigitated Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Externally Coupled Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.1

Input-Output Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Quantum Acoustics with Superconducting Qubits

39

3.1

Quantum Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2

Superconducting Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.1

Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

vi
3.2.2
3.3

3.4

Transmon Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Jaynes Cummings Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1

Resonant Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.2

Dispersive Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.3

Multi-mode Jaynes Cummings Model . . . . . . . . . . . . . . . . . . . . . . 52

3.3.4

Transmon Qubit-SAW Cavity Coupling Strength . . . . . . . . . . . . . . . . 53

Waveguide Quantum Acoustodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Strong Multi-mode Coupling in Quantum Acoustodynamics
4.1

4.2

4.3

58

Device Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.1

Acoustic Cavity Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1.2

Transmon Qubit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.3

Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Device Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.1

Acoustic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.2

Resonant Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2.3

Qubit Measurements in the Dispersive Regime . . . . . . . . . . . . . . . . . 77

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Resolving Phonon Number States in a Multi-mode SAW Cavity

85

5.1

Multi-mode Strong Dispersive Regime . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2

Split IDT Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3

5.2.1

Split IDT Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2.2

Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3.1

SAW cavity Avoided Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3.2

Interaction with Unconfined Phonons . . . . . . . . . . . . . . . . . . . . . . 95

5.3.3

Number Splitting Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 100

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5.3.4

Qubit Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Towards Multi-phonon Entanglement: Low-frequency, Focusing Acoustics

109

6.1

Measurement Based Multi-mode Entanglement . . . . . . . . . . . . . . . . . . . . . 109

6.2

Possible Design Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3

6.4

6.2.1

Aluminum Nitride on Sapphire . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.2

Indium Bonded Flip Chip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2.3

Lower Frequency: Improved Coherence . . . . . . . . . . . . . . . . . . . . . . 116

6.2.4

Fluxonium Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Stable Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3.1

Cavity in an Isotropic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.3.2

Anisotropic Wave Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.3.3

Distributed Mirrors and Curved IDTs . . . . . . . . . . . . . . . . . . . . . . 126

Testing Stable Cavities on Quartz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.4.1

Cavity Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.4.2

Focusing Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.4.3

Resonator Vibrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4.4

High-Q Focusing Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7 Conclusion and Outlook

136

Bibliography138

Tables

Table

Figures

Figure2.1

Rayleigh Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2

Mirror Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3

Bragg Grating Reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4

Bragg Grating Reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5

IDT Equivalent Circuit

2.6

Cavity Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1

Josephson Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2

Transmon Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3

Jaynes-Cummings Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4

Jaynes-Cummings Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5

Avoided Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6

Dispersive Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.7

Acoustically Coupled Transmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1

Qubit in SAW Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2

Cavity Design Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3

SAW Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4

Dolan Bridge Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.5

Junction-IDT Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

x
4.6

Device Micrograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.7

Wiring Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.8

Acoustic Cavity Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.9

Qubit-Cavity Avoided Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.10 Multi-mode Avoided Crossings Model . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.11 Model Eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.12 Qubit Flux Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.13 Transmon Anharmonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.14 Qubit Linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.15 Phonon Dispersive Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.1

Multimode Dispersive Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2

Split IDT Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3

Readout Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.4

Bandage Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5

Split Qubit SEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.6

Wiring Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.7

Qubit-SAW Avoided Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.8

Broadband Qubit Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.9

Qubit Decay Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.10 Phononic Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.11 Spectrum at ωz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.12 Qubit Coherence at ωz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.13 Phonon Number Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.14 Power Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.15 Room Temperature Wiring Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.1

Multimode Entanglement by Measurement . . . . . . . . . . . . . . . . . . . . . . . . 110

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6.2

Configuration for Multi-phonon Entanglement . . . . . . . . . . . . . . . . . . . . . . 112

6.3

Aluminum Nitride on Sapphire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4

Flip Chip Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.5

Fluxonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.6

Fluxonium Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.7

Focusing Cavity Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.8

Quartz Slowness Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.9

Coupling Near Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.10 Focusing Cavities Photograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.11 Focusing SAW Cavity Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.12 Transverse Mode Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.13 Transverse Mode Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.14 Vibrometry Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.15 Focusing Cavity Spectrum at 20 mK . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Chapter 1

Introduction

Modern science, having mastered the strange rules of quantum mechanics that govern ourworld, has endeavored to engineer increasingly complicated quantum systems to achieve specifictasks [1]. The impressive early successes of these efforts combined with the transformative potentialof quantum technologies, particularly in efficient processing of information, has begun to generate
substantial scientific and even commercial interest. A diverse array of physical platforms is underdevelopment, from individual trapped atoms to superconducting electrical circuits. Each platform
possesses unique advantages and produces their own lessons, each strengthening the wider enterprise. In this thesis, I will describe research into the quantum behavior of mechanical oscillatorsin the form of surface acoustic waves (SAWs). I will describe my work developing SAW systems
in a quantum context, with a focus on what applications these mechanical systems could find indeveloping quantum technologies as well as what general lessons can be learned from this particular
platform.
Long before the development of quantum mechanics, mechanical oscillators were at the forefront of revolutionary technologies and scientific understanding. These systems can be describedwith simple models, designed to meet a wide range of specifications, and measured with great
precision. For example, pendulum clocks revolutionized the keeping of time, and their widespread
adoption in the mid-1700’s brought about the high degree of human coordination required to begin
the Industrial Revolution [2]. Foucalt, a French scientist in the mid-1800’s, used a simple extensionof these pendulum clocks to prove Earth’s rotation in a concise and widely disseminated exper-2
iment. Torsional oscillators, where the gravitation restoring force is replaced with the generally
weaker force provided by the twisting of a wire, can be used as incredibly sensitive sensitive force
sensors. By measuring the attractive force between two known masses, this type of oscillator wasused to make the first measurements of Newton’s gravitational constant [3], as well as the most precise measurement to date [4], two experiments spanning over 200 years and 4 orders of magnitudein precision.
In the last century, the ascent of microelectronics led to a new class of commercial applications
for mechanical systems. Several transduction strategies exist to integrate mechanical components
with electronics. Through such integration, motion can be precisely driven and recorded. The
most direct method of transduction is using the piezoelectric effect, where the stress in a material
is directly coupled to its electric polarization, but a range of other methods exist each specialized
for various applications. The scope and prevalence of technologies based on electro-mechanicalsystems is hard to overstate, with applications ranging from the microphones and speakers present
in all cell phones to sonar and medical ultrasound technology that produce images of objects
using reflections of high frequency (> 1 kHz) acoustic waves. Acoustic excitations in piezoelectricmaterials themselves are widely used in commercial technologies. Waves in the bulk of piezoelectricmaterials can be confined to form mechanical oscillators whose resonant frequency can be measured
with great precision. The most popular modern timepieces, used widely in wrist watches andcomputers, operate by counting the oscillations of a quartz tuning fork. Similar quartz resonators
are used to make micro-balances that are sensitive enough to detect a single additional layer ofmetal adhered to its surface [5].
Elastic materials support a special type of wave that is strongly confined to the surface.
These waves, known either as known as Rayleigh waves [6] or more generally as surface acoustic
waves, interact strongly with the surface topography, and, on a piezoelectric material, the detailsof the conversion between acoustic waves and electrical signals depend directly on the geometryof the transducer, an effect discussed further in Ch. 2. Moreover, the slow speed of sound (km/s)
compared to light means acoustic wavelengths are short compared to typical devices (mm) even at

3
RF frequencies. Thus, devices built using piezoelectric Rayleigh waves can realize a wide range inbehavior by leveraging modern lithographic techniques to generate complicated transducer geometries and surface topographies with great detail and precision. Crucially, devices can be designedwith length scales spanning sub-wavelength to hundreds of wavelengths long at RF frequencies,
resulting in fine control over the realized frequency response. SAW devices are a mature commercial technology, used widely as oscillators, sensors, and filters, but particularly in applications
where large delays and specific dispersion responses are required, such as pulse compression filtersin radar and band-pass filters in telecommunications [7]. As discussed later in this chapter, thesesame attributes make SAWs attractive candidates for use in future quantum technologies.
The fundamental excitations of mechanical systems, termed phonons, were known to behave
according to quantum mechanics since the early 1900’s. The measurements of heat capacity ofsolids at low temperatures, shown to follow a T 3 temperature dependence, were well explainedby Debye [8]. In his model, mechanical excitations in the solid are described by a collection of
independent quantum harmonic oscillators, each representing a normal mode of the solid withfrequency ω, with a high frequency cut-off corresponding to when the wavelength is twice theatomic lattice spacing. As a material approaches zero temperature, more modes are cooled to theirquantum ground state, as their energy per excitation exceeds the available thermal energy kB T ,and no longer contribute to the materials specific heat, thus providing evidence that mechanicalexcitations are indeed quantized.
Continued use of mechanical systems at the forefront of scientific discovery created deeper
interest in the quantum behavior of motion, particularly into the interaction between motion andlight. The now successful efforts by LIGO to measure gravitational waves using a Michelson interferometer first drew attention to the quantum behavior of motion and its potential technologicalrelevance [9]. While LIGO is predominantly an optical measurement, its sensitivity depends directly on the motion of the mirrors that form the optical interferometer. To detect the strain
generated by a passing gravitational wave from an interstellar signal of interest, which is of order10−23 , the mirrors must be separated by a large distance (km), but also held exceptionally still and4then measured with incredible precision (am).
Reaching these unprecedented levels of precision required to observe gravitational waves
raised concerns about the role of quantum effects. In the 1980’s, researchers concluded that quantum
mechanics does place a limit on the ability to measure the position of a mechanical oscillator using
coherent laser light [10, 11]. Increasing the power in the light field increases the resolution of themeasurement, but the increased flux of photons also acts as a stochastic drive on the oscillator
through radiation pressure, an effect termed “measurement backaction.” Eventually, increasing
the power leads to an increased position uncertainty as the radiation pressure drives position
fluctuations of the mirrors that exceed the measurement resolution. LIGO employs enormous
powers (100 kW) to achieve the necessary strain resolution, and at these powers measurement
backaction, an effect of the quantum nature of mechanical motion, constitutes a significant limit
on LIGO’s sensitivity [12]. Already, the experiment has incorporated a non-classical light source
where the quantum fluctuations are redistributed in phase space, called squeezed light, with ademonstrable increase in sensitivity [13].
The interest in the quantum behavior of macroscopic mechanical systems, first investigatedin the context of gravitational waves, has since garnered broad scientific interest. The quantizedcollective excitations of mechanical systems have a range of properties that makes them attractive
for fundamental science and advancing quantum technologies. For one, how quantum decoherenceof mechanical oscillators might result from gravitational interactions, a question at the intersection of the famously incompatible theories of general relativity and quantum mechanics, could be
investigated by preparing a massive phononic system in a sensitive quantum state and monitoring
its decay [14]. Mechanical systems experience many forces besides gravity, and their performanceas general sensors could be improved using quantum enhanced metrology [15]. The wide range ofquantum platforms that can interact with motion makes mechanical systems promising intermediary elements in quantum transduction schemes [16, 17, 18]. And, as mechanical energy cannot
radiate into free space, vibrational modes can be engineered to have high quality factors [19, 20],
making them attractive for storing and processing information [21, 22]. Additionally, potential

5quantum technologies built around phonons would benefit from the simple fact that sound travels significantly slower than light. The slowness of sound is widely used by surface acoustic wave
devices to construct delay lines and other filtering devices. Long delays can be used to create
multi-mode acoustic resonators [23] that hold potential for processing quantum information but
in a small fraction of the space required using electromagnetic excitations [24]. These multi-mode
resonators have potential for serving as registries in a quantum random access memory [25, 26, 27],
a promising architecture for quantum information processing and a main motivating goal in thisthesis.
In the last decades, experimental efforts have made great strides towards control of mechanical oscillators at the quantum level. One of the first obstacles towards quantum control is the large
thermal occupancy of mechanical oscillators at accessible temperatures. Mechanical oscillators
typically have resonant frequencies below 100 MHz, so even dilution refrigerators with base temperatures of 20 mK correspond to a significant number of thermal excitations. However, throughradiation pressure interactions with either microwave or optical fields linearized around around astrong pump, phonons can be swapped with higher frequency photons. This swapping interaction
results in a mechanical occupancy near that of the electromagnetic resonator, which corresponds
to an effective temperature well below the thermal environment. Cooling to the quantum groundstate, where less than one thermal phonon is present on average, was achieved for a range of mechanical oscillators in the last decade [28, 29, 30]. Beyond cooling, experiments were able to prepare
mechanical squeezed states [31, 32] as well as entangle the state of the mechanical resonator withpropagating photons [33, 34] and other stationary oscillators [35]. Impressively, LIGO has recently
reported that they can cool the motion of a 10 kg mirror from room temperature to an averageoccupancy of 10.8 phonons, corresponding to an effective temperature of 77 nK [36].
Despite successes in cooling and controlling mechanical motion, schemes built on linearizedradiation pressure interactions have struggled to create non-classical states of motion. The linear
nature of the effective interaction means input Gaussian states must transform to output statesthat are also Gaussian. Some success has been achieved by either using a non-classical input state6[22] or by deploying a non-linear measurement, such as photon counting [34, 37]. A more generalapproach is to replace the electrical or optical cavity with an non-linear element.
For example, the fundamentally non-linear internal electronic states of atomic ions confinedin an electromagnetic trap can be coupled to their harmonic motion [38]. The interaction between
harmonic motion and the non-linear, atomic degree of freedom is realized using precisely tuned
pulses of laser light. These trapped ion systems successfully generated an array of non-classicalstates of the ions motion, including number states, squeezed states, and so-called Schrodinger cat
states [39, 40]. More recently, superpositions of vastly different Fock states have been created [41].
These superposition states are of great interest to quantum metrology as they reach the maximum
sensitivity for a given number of excitations. While potentially useful as force sensors, ion motion
is quite distinct from the acoustic technology widely employed in classical information processing.
Another powerful example of quantum control using direct coupling to a non-linear element
can be drawn from the the maturing field of circuit quantum electrodynamics (cQED) [42, 43, 44].
In cQED, microwave frequency electromagnetic cavities interact strongly with superconducting
qubits, quantum systems formed by the collective electronic excitations of microwave circuits witha non-linearity provided by Josephson junctions. The state of the qubit can be determined throughits interaction with the electromagnetic cavity [45]. Conversely, using the qubit for control andreadout, nearly arbitrary quantum states can be created and measured in the inherently linear
electromagnetic cavities [46, 47, 48, 49]. The long lifetimes and large Hilbert space of these electromagnetic resonators has generated significant interest in their usefulness as quantum registers[50, 51, 52, 53].
The value of cQED in efforts to control the quantum state of a mechanical oscillator extends
beyond being a useful example. In contrast to the internal states of atomic qubits, the non-linearities
in superconducting qubits can be coupled directly to common types of mechanical excitations
through a variety of methods, many with direct analogies to how microelectronics are integrated
with mechanical systems in commercial settings. In a seminal work, a thin film bulk acoustic wave
resonator was coupled to a superconducting phase qubit through piezoelectricity such that the two

7system exchanged excitations in about 4 ns when tuned on resonance [54]. Dilution refrigerator
temperatures were sufficient to cool the high frequency mechanical mode, with a resonance justbelow 6 GHz, to its quantum ground state. By swapping excitations with the phase qubit, groundstate cooling of the motion could be directly verified. Moreover, a single excitation in the phase
qubit could be swapped into the mechanical oscillator, placing the macroscopic oscillator in a single
phonon number state. Although limited by poor coherence times of below 10 ns, this impressive
result was the first demonstrations of basic quantum control of a macroscopic mechanical oscillator.
Building on this result, piezoelectric Rayleigh waves are a particularly rich system to consider
in a quantum context. To a significant degree, surface acoustic waves are a slow analog of lightin two dimensions. This analogy makes many physical models developed for trapped atoms in
optical cavities or superconducting qubits coupled to microwave light readily applicable to theacoustic case. However, departures from this analogy lead to many potential unique applicationsfor SAWs. For one, SAWs are able to transduce information between various platforms of quantum
information processing technologies [55, 56, 57, 58]. Additionally, the interaction between surface
acoustic waves and superconducting qubits can be modelled on commercial SAW transducers, andthe many-wavelength extent of these coupling elements differs greatly from the standard dipole
coupling in circuit and cavity QED. This spatially extended interaction gives fine frequency-domain
control over the qubit-phonon interaction [59]. Furthermore, acoustic resonators take up a small
fraction of the volume of their electromagnetic counterparts, allowing them to reach deeper into
multi-mode regimes while occupying less fridge space. Reflectors for SAWs are narrow-band, andthus these devices naturally have simultaneous access to both cavity and waveguide QED physics[60, 61, 62]. Finally, initial work has already demonstrated that propagating SAW phonons can be
successfully coupled to a superconducting transmon qubit [63].
The work in this thesis investigates the prospects for surface acoustic waves as a platform for
future quantum technologies, focusing in particular on the potential of multi-mode acoustic cavities
in processing information in a hardware efficient manner [26]. I present results demonstrating strong
coupling between a superconducting qubit and a multi-mode SAW resonator, highlighting the high8density of modes available in acoustic platforms [64]. Then, I show that by tailoring the qubitphonon interaction, the Stark shift of individual phonons can be resolved in the spectrum of asuperconducting qubit [65]. This ability to count phonons demonstrates their quantum nature andis a key milestone towards simultaneous quantum control over an array of surface acoustic wave
resonances [46]. Furthermore, the strategy described is extendable to massively multimode cavities
with high densities of strongly coupled resonant modes. I conclude with a proposed experiment
to entangle multiple modes of a SAW cavity through joint, projective measurement. Progress
towards several improvements are described that would make this experiment possible, including
the demonstration of coherent, focusing SAW resonators.
Concurrent with the research performed with this thesis, the field of quantum acoustics grew
rapidly. High overtones of a bulk acoustic wave (BAW) resonator were strongly coupled to atransmon qubit [66], and multi-phonon Fock states were created and measured through resonant
exchange with the qubit [67]. These BAW resonators can also be interfaced with optical light[68], suggesting a path to microwave to optical conversion. For SAWs, an initial demonstrationof dispersive interactions between a superconducting qubit and a radio-frequency resonator [69]was quickly followed by demonstrations of quantum control of SAW excitations [70]. Further work
simulated distant entanglement of two superconducting qubits using the long travel time of SAW
wave packets [71], as well as a phononic quantum erasure where two-phonon interference disappears
when their which-path information is measured [72]. A one dimensional beam was coupled strongly
to a superconducting qubit using a piezoelectric thin film, and the qubit spectrum showed thediscrete quantized excitations of the beam [73]. While not exactly quantum acoustics, incorporatingsuperconducting non-linearities into a radiation-pressure type devices described earlier enabled
number-sensitive electro-mechanics [74] and the preparation of sub-Poissonian states of motion[75].
The structure of the thesis is as follows: an introduction to surface acoustic wave resonators
is given in Ch. 2, followed by their quantum description and an introduction to superconducting
qubits in Ch. 3. Experimental results demonstrating strong, multi-mode coupling are presented

9in Ch. 4 and phonon number splitting are given in Ch. 5. Finally, work towards improved device
performance by moving to lower frequencies and implementing focusing cavities is summarized in
Ch. 6 before conclusions and an outlook are presented in Ch. 7.

Chapter 2

Surface Acoustic Wave Resonators

Surface acoustic waves (SAWs) are a type of elastic wave that is confined to the surface ofa material. These waves come in many varieties, the simplest and the type used in this thesis
are named after Lord Rayleigh who discovered their existence in 1885 [6]. Surface acoustic waves
became technologically relevant in the mid 1960’s with the invention of the interdigitated transducer(IDT) [76], a structure capable of efficiently converting signals between the electronic and acoustic
domain through the piezoelectric effect. Importantly, this transducer is formed from a patterned
thin metallic film and as such is compatible with modern lithographic techniques, enabling the
production of SAW devices at scale. Such devices have become widely used for filtering; thefilter performance is determined by the transducer geometry, which can be controlled with highprecision [7]. Crucial to their commercial success is the slow speed of sound that can be used togenerate long delays in a compact geometry. These long delays can also be used to create multimode cavities with potential uses in quantum information processing, as the density of resonantmodes is proportional to the cavity round trip time. The following chapter introduces the Rayleighwave solution, describes the components required to construct SAW cavities and the techniques todesign their performance, and concludes with presenting a useful model for describing multi-mode
resonators.

11

2.1

Elastic Waves in a Solid
An analytical description of Rayleigh waves is important to understanding the devices dis-cussed in this thesis and the nature of their limitations. First, I will derive Rayleigh wave solutionsin an isotropic medium, as this derivation shows the underlying physics and produces the generalwave behavior without introducing the full complexity of piezoelectric, anisotropic media. The
following derivation is drawn from [7].

2.1.1Stress and Strain

Acoustic waves in a solid are described by two vector quantities: stress and strain. Strain
characterizes the internal displacement u of constituent ‘particles’ of from their equilibrium position,1
Sij (x1 , x2 , x3 ) =
2



∂uj
∂ui
+
∂dxj
∂dxi


.

(2.1)

Stress Tij , on the other hand, characterizes the internal forces in a material. The force in the ith
direction per unit area on a plane with normal n is,fi =X

Tij nj .

(2.2)

jFor sufficiently small amplitudes, stress and strain are related by the stiffness tensor, cijkl ,
Tij =

XX
k

cijkl Skl .

(2.3)

l

This relationship is Hooke’s law generalized to a three dimensional solid. While this relationship is
simple, the generalization comes with a significant amount of added complexity: even incorporating
symmetry and thermodynamic constraints, there are 21 independent elements in this stiffness
tensor. Particular material symmetries can further reduce the number of elements. For example,
cubic crystals are described by 3 independent elements, while isotropic materials are described byjust 2 [7].
The dynamics of the system are determined by considering Newton’s second law. In a material
with density ρ, an infinitesimal cube of length δ has mass ρδ 3 . In the presence of non-zero stress,

12
the cube will experience a net force proportional to the area per side and the change in stress acrossthe cube. Newton’s second law then gives the equation of motion,ρ∂ 2 ui X ∂Tij
.
=
∂t2
∂xj

(2.4)

j

Combined with 2.1, this equation relates acceleration to curvature and is a familiar wave equation,
albeit in a complicated three dimensional space.

2.1.2

Transverse and Shear Waves

In a general isotropic medium, symmetry simplifies the stiffness tensor cijkl to only two
independent constants. The stiffness tensor takes the form,
cijkl = λδij δkl + µ(δik δjl + δil δjk ),(2.5)

where µ and λ are known as Lamé constants and δij is the Kronecker delta. This simplified stiffness
tensor yields stresses,
Tij = λδij ∆ + 2µSij ,where ∆ =

(2.6)

P ∂ui

i ∂xi . The equation of motion is,ρ∂ 2 uj∂∆= (λ + µ)+ µ∇2 uj .
2
∂t
∂xj

(2.7)

This equation supports two types of waves in an infinite medium. To see this, consider a plane
wave solution with frequency ω and wavevector k,u = u0 exp[j(ωt − k · x)],(2.8)

where u0 is a constant amplitude vector. Substituting this wave into equation 2.7 gives,ω 2 ρu0 = (λ + µ)(k · u0 )k + µ|k|2 u0 .

(2.9)

There are three vector quantities in this equation, and two of them point along u0 . To have nontrivial solutions, there are two possibilities: either k · u0 is zero and thus k is perpendicular to u0 ,or that k is parallel to u0 and all the terms have the same direction.

13
In the first case, when k · u0 = 0, the waves are transverse or shear waves. From equation 2.7,
these shear waves have wavevector kt given by,|kt |2 = ω 2 ρ/µ,and are dispersionless with velocity Vt =(2.10)pµ/ρ.

In the second case, with the wave-vector k parallel to the displacement u0 , the waves are longitudinal in character. Similarly to the transverse case, equation 2.7 means that these longitudinal
waves have wavevector kl given by,|kl |2 = ω 2 ρ/(λ + 2µ),and velocity Vl =(2.11)p(λ + 2µ)/ρ. Note that since the Lamé constants µ and λ are always positive,

longitudinal waves always have a faster speed of sound than transverse waves.

2.1.3

Rayleigh Waves

To find Rayleigh waves solutions, a surface must be considered. Without loss of generality,
assume the isotropic medium occupies only the space below the plane at x3 = 0 with vacuum forall x3 > 0. Here, I will derive a Rayleigh wave propagating in the x1 direction with no variation in
the x2 direction, satisfying both the equation of motion [Eq. 2.7] as well as the boundary condition
that there are not any forces on the free surface at x3 = 0.
The heart of the surface wave solution found by Rayleigh is to add together a longitudinal
and a shear wave. By symmetry of the isotropic material, these two partial waves must havedisplacements only in the plane spanned by the direction of propagation and the surface normal,
known as the sagittal plane, so their wavevectors have just two components: one in the direction of
propagation x1 and the other into the surface x3 . The two waves must have the same wavevector
β in the direction of propagation x1 . This fixes the x3 components of the wavevectors to be,
T 2 = ω 2 /Vt2 − β 2 ,L2 = ω 2 /Vl2 − β 2 ,(2.12)

14
where T is the x3 component of the transverse wavevector and L is the x3 component of thelongitudinal wavevector. For the waves to be confined to the surface, they must be evanescent inthe −x3 direction, and so both L and T must be positive and imaginary.
The unit vector of the waves displacement for both shear and longitudinal cases can be
determined by using that longitudinal waves have ul is parallel to k, while for transverse waves
they are perpendicular. Thus,ul = (1, 0, L/β)A exp[−j(βx1 + Lx3 )],(2.13)

ut = (1, 0, −β/T )B exp[−j(βx1 + T x3 )],(2.14)

where A and B are constants. These solutions are no longer strictly plane waves, but are still valid
solutions satisfying the equations of motion and are bounded in the given half-space.
The remaining constants and wavevectors can be determined by applying the boundary conditions. The free surface condition, i.e. T13 = T23 = T33 = 0, can be applied to equation 2.6 andthe partial wave solutions above to give a pair of linear relations between the values of A and B.
For there to be a non-trivial solution, the determinant of the relations must be zero,(T 2 − β)2 + 4β 2 L = 0.

(2.15)

This equation then gives the relationship between the Rayleigh velocity VR and the transverse andlongitudinal phase velocities,q(2 − VR2 /Vt2 )2 = 4 (1 − VR2 /Vt2 )(1 − VR2 /Vl2 ).

(2.16)

This equation has only one solution, thereby determining VR . The Rayleigh velocity is near but
slightly below the transverse velocity, with the ratio VR /Vt set by the ratio Vl /Vt .
Once VR is known, the displacements can be found for the full Rayleigh wave,
u1 = [γ exp(aβx3 ) − exp(bβx3 )] exp[j(ωt − βx1 )],(2.17)

u3 = j[γa exp(aβx3 ) − b−1 exp(bβx3 )] exp[j(ωt − βx1 )],(2.18)

15
where a = −jL/β, b = −jT /β and γ = (2 − VR2 /Vt2 )/(2ab) are all positive and real. The displacements for the Rayeleigh solution on fused quartz (Vl = 5968 m/s, Vt = 3764 m/s, VR = 3409 m/s)
are shown in 2.1. The relative phase between u1 and u3 means that the motion of each particle iselliptical. The decay into the substrate has two terms; a slow term from the shear partial wave,
which has a velocity closer to that of the Rayleigh wave, and a fast term from the longitudinal
partial wave. Interestingly, the two decaying terms have opposing sign, and this can result in achange of sign of the displacement in the bulk compared to the surface.
Rayleigh waves can essentially be treated as one dimensional phenomena as they are invariant
in the x2 direction and strongly confined in the x3 direction. The majority of this thesis uses aquasi-1D description, despite the reality that any device will have a finite extent in the transverse
direction. Phenomena outside this assumption almost always require only a 2D description. For
example, the diffraction of acoustic waves requires considering SAWs only in the plane of thesurface, while scattering of waves from surface perturbations into the bulk of the material requiresconsidering only waves in the sagittal plane.

2.1.4

Incorporating Anisotropy and Piezoelectricity

Surface acoustic waves are more relevant to technology when the medium is piezoelectric. The
coupling of stress and electric field mean that acoustic waves can be transduced to electromagnetic
ones, enabling a wide range of commercial and quantum applications. However, piezoelectricity
requires a breaking of inversion symmetry, and considering this anisotropy greatly complicates theanalysis.
In a piezoelectric material, the internal stresses are not only due to strain but also electricfield. In the regime where this effect can be approximated as linear, the stress becomes,
Tij =

XX
k

l

cE
ijkl Skl −X

ekij Ek ,(2.19)kwhere Ek is the electric field, ekij is the piezoelectric tensor, and the superscript E on the stiffness
tensor indicates the electric field is assumed to be constant. This modified equation for stress leads

16

Figure 2.1: The strain of a Rayleigh wave in the sagittal plane on fused quartz, shown as a grid in(a) and a vertical line cut in (b). Particles move in ellipses as the u1 and u3 displacements are outof phase, and the grade of the ellipse is different on the surface compared to the bulk as the sign of
u1 changes. The magnitude of the strain decays quickly into the bulk, with most of the mechanicalenergy stored within λ of the surface.

to a more complicated equation of motion,X∂ 2 ui X X∂2Φ
∂ 2 uk
ρ 2 =
ekij
+
cE
ijkl
∂t
∂xj ∂xk∂xj ∂xljk!
,(2.20)lwhere Φ is the electric potential, and it is assumed that the electric fields vary slowly enough intime to be governed by electrostatics. This assumption is valid as sound travels much more slowly
than electromagnetic excitations.
This equation of motion combined with the assumption that the material is an insulator

17with ∇ · D = 0 gives four equations for ui and Φ. These equations generally have four distinct
solutions. One of these solutions is electrostatic in nature, essentially reducing to Laplace’s equation.
The other three solutions are of more interest, as each is a dispersionless acoustic wave withsome general characteristics reminiscent of the isotropic case. In most cases, one solution closely
resembles a longitudinal wave with displacement nearly parallel to its wavevector, while the othertwo resemble shear waves with two different polarizations and phase velocities. Additionally, theunderlying anisotropy means the waves have phase velocities that depend on propagation direction.
This angular dependence leads to many important effects, including modified diffraction and beamsteering where the fronts of constant phase are not perpendicular to the flow of energy.
Similar to the isotropic case, surface waves can be constructed from these solutions by adding
them together. However, the added complexity precludes analytical results in most cases, instead
requiring computational methods. The reduced symmetry permits surface waves to have displacements in all three directions, despite having no variation out of the sagittal plane. Here, we areconcerned with the class of solutions with displacements only in the sagittal plane, which are called
piezoelectric Rayleigh waves and typically occur on surfaces and directions involving high degrees
of symmetry. These waves share many properties of Rayleigh waves on an isotropic material, including the characteristic penetration depth of about one wavelength. Modified diffraction andbeam steering arising from crystal anisotropy are considered more in Ch. 6.
One key difference from the isotropic, non-piezoelectric case is that for a piezoelectric Rayleighwave the electrostatic boundary condition must be specified in addition to the mechanical one.
There are two cases to consider: when the surface is free and when it is has a thin metallic layer
present. In the free-surface case, it is assumed there are no free charges and there will in general
be a non-zero potential in the vacuum above the surface. In the metallized case, the electric fieldparallel to the surface is shorted out by a thin conductor whose mechanical properties are assumed
to be negligible. Crucially, this metallized boundary condition results in velocity vm that is slower
than the free-surface velocity vf . The normalized difference K 2 = (vf − vm )/(2vf ) is a direct
measure of how strongly the wave interacts with electrical disturbances at the surface and will be

18
used heavily later in the work. The factor of two is included by convention.

2.2

SAW resonators
Surface acoustic waves are by nature propagating fields. To use these acoustic fields to store

and process quantum information, they must be confined to form resonant modes. The resonators
we use here are analogous to optical Fabry-Perot cavities where a pair of mirrored surfaces reflect
and confine light. However, any localized element that strongly reflects SAWs also scatters energy
out of the cavity, typically into shear bulk modes. To circumvent this unwanted scattering, SAW
reflectors are formed by a periodic array of weak perturbations, such as metal strips or etchedgrooves, that are highly reflective only in aggregate.

2.2.1

Bragg Grating

Periodic arrays of surface perturbations, known as distributed Bragg gratings, are capableof efficiently reflecting incoming waves but require a more complicated description than localizedreflectors. The periodicity inherent in the grating results in a dispersive behavior that both limits
the bandwidth over which the grating is reflective and alters the phase of scattered waves.
A Bragg grating is defined by its periodicity p, complex reflectivity per element rs , andnumber of elements Nm . These parameters combine to determine the gratings reflection coefficientΓ. To begin, I will first consider waves propagating in an infinite grating. These waves travel either
right or left, and are uncoupled in the space between the grating elements. Consider the wave
amplitudes at half-way between elements n and n − 1 as shown in Fig. 2.2(a). Reflection from themirror element couples the leftward travelling waves b to the rightward travelling waves c. The
relationship between these four waves is given by two linear equations,bn−1 = (rs cn−1 + ts bn ) exp(−jkp),(2.21)

cn = (rs bn + ts cn−1 ) exp(−jkp).

(2.22)

where ts is the transmission through the elements. The waves on the left can be expressed in terms

19of those on the right by first re-arranging the first equation for bn and substituting this into thesecond equation,rs1
cn−1 + bn−1 exp(jkp)(2.23)
ts
ts
1
rs
cn = cn−1 exp(−jkp) + bn−1 ,(2.24)
ts
ts
pwhere power conversation is assumed (ts = 1 − |rs |2 ). Solutions in the grating will have thebn = −

form cn = cn−1 exp(−jγp). Plugging this into the above equations relates γ to k and the grating
parameters,cos(γp) =cos(kp).
ts(2.25)

When ts is close to unity, meaning each element is weakly reflective, this equation generally hasa real solution for γ that is approximately k. However, when kp is near a multiple of π, thencos(kp)/ts is larger than one and γ must have an imaginary part even for ts close to one. The
real and imaginary part of γ as a function of kf are shown in Fig. 2.2(b). A non-zero imaginary
component to the wave-vector implies that waves decay exponentially in the grating, and energy
conservation then implies that waves incident on the grating with frequencies inside this stop bandmust be completely reflected.
We are concerned primarily with the first stop-band of the grating. This band has center
frequency f0 = vs /(2p) with a width ∆m given by the range in k where Im(γ) is non-zero,∆m2= sin−1 (|rs |).
f0
π(2.26)All incident waves in this band are perfectly reflected, so that |Γ| = 1. However, the phase of Γchanges with frequency, shifting by π over the mirror bandwidth. This reflected phase, θm , is givenby the relation,
θm = θr + tan−1k − k0,−α(2.27)

where θr = ±π/2 is the phase of single element reflections, k is the wave vector outside the grating,pk0 = π/p is the Bragg wave-vector, and α = |rs |2 /p2 − (k − k0 )2 is the imaginary part of the
wave vector inside the grating. The phase shift is approximately linear near the center of the

20

Figure 2.2: (a) A simple model of a mirror, considering fields travelling left and right as well as
exiting and entering a given mirror element, can be used to determine the disperision of a SAW
in a Bragg mirror. (b) The mirror propagation constant γ in an infinite grating can be foundby relating these four fields to each other and looking for a periodic solution. This propagation
constant acquires an imaginary part for a range in k space around π/p determined by the reflectivity
per mirror element.

band, resulting in a delay τm = −dθm /dω = (4f0 |rs |)−1 . Therefore, at the center of the stop-band,
the Bragg grating functions as a perfect, localized reflector positioned at Lp = vs τm = p/(2|rs |)inside the reflector. However, the dispersion in the mirrors means that the penetration length Lp
is frequency dependent and increases dramatically near the mirror band edge.
Finite gratings have reflections coefficients of magnitude below one as there is non-zero transmission through the mirror stack. At the center of the stop-band, the total reflection is given by,|Γ| = tanh(Nm |rs |).

(2.28)

Therefore, the array reflects incoming waves effectively when Nm |rs |  1. Waves with wave-vectors
near the edge of the stop-band penetrate further into the mirror, and those waves with an imaginary
part that is small compared 1/Nm will be partially transmitted through the mirror. The reflectedmagnitude and phase from several finite Bragg gratings are depicted in Fig. 2.3.

21

Figure 2.3: The reflection coefficient of a finite Bragg grating near its center frequency approaches
unity when the number of elements is sufficiently large. Additional elements increase the magnitudeof the reflection coefficient near the edges of the mirror stop-band. The phase of the reflection
coefficient changes by π over the mirror bandwidth with an approximately constant delay τm near
the center of the stop-band.

Bragg gratings for SAWs are most often implemented either with shallow grooves etched into
the surface or metal strips deposited on top. To avoid significant scattering to bulk modes, rs istypically between 1 − 3%, resulting in bandwidths that are ∼ 1% of f0 and penetration lengths of∼ 35p. The magnitude and sign of the reflectivity depend on the exact parameters of the element.
The performance of various grating types have been studied on common materials to develop
phenomenological models [7, 77]. Several strategies have been pursued to suppress conversion tobulk waves, including so called “hiccup” cavities where the mirrors and periodic structure used

22
for transduction discussed in Sec. 2.2.4 are directly adjacent to each other, offset only by a small
phase shift (or “hiccup”) to reduce the geometric discontinuity at the mirrors, [78] as well having
a grating everywhere with a smoothly tapered periodicity [79] to create the cavity.

2.2.1.1

Numerical Modelling

The analysis above assumes power conservation and works well for simple geometries. The
mirror behavior including loss, which usually results from scattering of surface waves into unconfinedbulk modes, or more complicated geometries involving multiple periodicities must be determined
numerically. Here, I describe a matrix formulation from Ref. [80] designed for cascading many
circuit components together and can handle arbitrary geometries of several hundred elements easily.
While this model is concerned with voltages and currents of microwave circuits, it applies readily tothe mechanical displacement and pressure in a surface acoustic wave and is perhaps more intuitive
than the P-matrix formalism commonly used in SAW analysis [7].
The voltages and currents at the input and output of an arbitrary two-port network can be
related with a 4-element matrix, inventively called the ABCD matrix. The relationships are given
by,
  
 
V
A
B
1  V2  1  1
 =
 .
I1
C1 D 1I2(2.29)

These ABCD matrices are useful because they can be easily cascaded. If another arbitrary two
port-network is in series with the first, then the matrix corresponding to both elements in series
is simply the product of the two matrices. The behavior of any series of two-port networks withQ
matrices Mi can be described by a matrix Mt = i Mi . This final matrix includes any complicatedseries of internal reflections and can be turned into final scattering parameters using simple formulas[Eq. 2.31]. Calculating the response of complicated combinations of two-port networks then becomes
determining the individual ABCD matrices and multiplying them together, a computationally easytask.
For calculating the reflection from a Bragg mirror, there are 2 matrices to consider: propa-

23
gation in free space and propagation in the mirror element. The ABCD matrix for propagation
over length li in a medium with velocity vi and impedance Zi is,cos(ωl
/v
)
j
sin(ω
/v
)
∗
Z
i i
i i
i

Mi = 
,
j sin(ωli /vi )/Zicos(ωli /vi )(2.30)

where ω is the frequency of the wave. Losses can be included by adding an imaginary component to
the propagation term ωli /vi . In this model, reflections are generated from an impedance mismatchat the interface between the region of free space with impedance Zf and the mirror element withimpedance Zm . Typically, the acoustic impedances in the two regions are difficult to measure.
However, the measured mirror bandwidth for a uniform grating can be used to infer the reflectivity
per element, which then constrains the impedances to match |rs | = (Zm − Zf )/(Zm + Zf ).
Once the component matrices are multiplied together, the final ABCD matrix Mt determines
the reflection coefficient of the entire stack Γ,Γ=At + Bt /Zf − Ct Zf − Dt.
At + Bt /Zf + Ct Zf + Dt

(2.31)The model prediction for Γ is only sensitive to the relative difference between impedances, and so
the only inputs needed are the reflectivity per element and center of the mirror bandwidth.
This technique is most useful when describing non-uniform gratings. Such a structure holds
promise in engineering a mirror with a built-in frequency-dependent delay. This extra dispersion
leads to resonant cavities with non-uniform mode spacings which could be used for building an
acoustic quantum random access memory [26]. The simplest implementation of this is a two-tone
mirror, with a short segment at the mirror stack with a shifted center frequency from the rest. The
short segment adds a frequency dependent delay, while the rest of the stack provides the majorityof the confinement.

2.2.2

Fabry-Perot Cavities

An acoustic resonator analogous to optical Fabry-Perot cavities can be formed by placing two
Bragg mirrors on axis with each other separated by length L. A wave propagating on axis inside

24

Figure 2.4: Two Bragg gratings can form a Fabry-Perot type resonator. The penetration of waves
into the distributed mirrors results in an effective cavity length Le = L + 2Lp .

the cavity with frequency inside the mirror stop-band will be confined between the two mirrors,
as shown in Fig. 2.4. Resonance occurs when the frequency of the wave is such that it acquires aphase of 2π in a round trip.
To find the resonant condition, one must consider two sources of phase. The dominant
phase is acquired when the wave traverses the separation between the two mirrors, where it picks
up a phase θL = ωL/vp where vp is the phase velocity of the wave. The wave also acquires aphase θm when it is reflected off the Bragg grating, given by Eq. 2.27. Near the center of themirror stop-band, the mirror phase increases as approximately τm ω resulting in an effective cavity
length Le ≈ L + 2λ0 /(4|rs |). This approximately constant cavity length leads to an even spacing
between resonant modes of ws ≈ 2πvs /(2Le ). However, dispersion in the mirrors leads to a fasteraccumulation of phase with frequency and consequently reduced mode spacing near the edge of themirror bandwidth.
Once the resonant frequencies ωm of the cavity are determined, the standing wave of the25
acoustic modes can be defined explicitly. The surface potential φm (x) for mode m is piecewise,




x≤0
exp[2σx] sin(k0 x − θm )φm (x) ∝ sin(km x − θm )(2.32)
0≤x≤L







(−1)m exp[2σ(L − x)] sin(k (L − x) − θ ) L ≤ x0where σ =mp|rs |2 /p2 − (k − k0 )2 is the decay constant inside the mirrors, km = ωm /vs is the

wavevector in free space, and k0 = π/p is the Bragg wavevector. The symmetry of the cavity about
its center at x = L/2 implies that resonant modes have either either nodes or anti-nodes at thecavity center. While the mirror phase spatially shifts the modes relative to each other near the
mirror, the resonant modes near the center of the mirror band are only shifted by a small phase,resulting in the anti-nodes of most modes occurring close together near the mirror. The standing
waves near the mirror are discussed further in Ch. 6.
Commercial applications usually use SAW resonators that only support a single resonancewithin the mirror bandwidth. These cavities are short, with ωs < 2ω0 |rs |/π. In this limit, thelength must be chosen carefully so that a resonant mode occurs at the center of the mirror band
[7, 70]. This thesis is more concerned with multi-mode cavities where the free spectral range issmaller than the mirror bandwidth, ωs < 2ω0 |rs |/π. These cavities support many modes, regardlessof the exact cavity length.

2.2.3

Cavity Lifetime

The time it takes for a phonon confined to a cavity to decay is a key metric for evaluating
its performance. The energy in an undriven cavity decays exponentially with characteristic decay
time τ . Resonators are often measured using spectroscopy, and their Lorentzian linewidth κ arerelated to this decay time τ as κ = 1/τ where κ is in units of angular frequency. To calculate
the linewidth measured in standard frequency units most commonly used in the lab, the lifetime
must be multiplied by 2π. Cavity spectroscopy is discussed further in Sec. 2.3.1. The overall loss
rate κ is composed of several independent channels κi that all remove phonons with some rate, and

26
κ=

P

i κi . In multi-mode cavities, it is useful to classify these losses into two types: loss that occurs

every oscillation and loss that only occurs every round-trip. Two dimensionless figures of merit areused to quantify these processes: the characteristic number of oscillation periods defines the quality
factor Q = ω0 /κ, and the number of round trips defines its finesse F = ωs /κ. This distinction
is useful when extrapolating to different cavity geometries: if the dominant process occurs once
per round trip and therefore limits finesse, then making the cavity longer will improve the quality
factor, while a cavity with a dominant Q-limiting process has the same linewidth regardless of cavity
length. In optical cavities, scattering from mirrors is almost always dominant as light travelling in
vacuum is practically dissipationless, and therefore finesse is the important, geometry-independent
figure of merit.
There are several well known loss mechanisms of surface acoustic wave cavities. Any featureson the surface of the material make the Rayleigh solutions imperfect and result in scattering toother acoustic modes, such as bulk modes or unconfined surface waves. This scattering occurs
primarily at mirror elements or other patterning of the surface such as any metallic electrodes,
but also at any defects the waves encounter. For common interfaces, like Al electrodes or grooves
on quartz, phenomenological parameters exist to give approximate values for this scattering [55].
As mentioned at the end of section 2.2.1, several strategies exist for suppressing bulk scattering,
however none eliminate the trade-off between mirror bandwidth and mirror finesse. Unsurprisingly,
unwanted blemishes on the surface should be avoided, and high quality material and surfaces are ofparamount importance. However, even on an unpatterned and unblemished surface, finite viscosityof the material leads to damping of acoustic waves, a property often characterized as material Q
that is strongly dependent on temperature [23].
Another loss mechanism is transmission through the Bragg mirror. At the center of the stopband, several hundred elements with rs ∼ 1% is sufficient to render this source of loss negligible.
However, modes near the band-edge can still have significant leakage through the mirror. Evenin an infinite but slightly lossy grating, modes near the band edge can be lossier due to increased
time spent propagating inside the mirrors. These losses can be modelled effectively using ABCD

27
matrices.
A dominant source of dissipation in the cavities considered in this thesis is diffraction losses
resulting from the finite transverse extent of the cavity combined with the use of parralel reflectors.
The width of the mirrors W leads to a diffraction limited Qd given by
5π
Qd =|1 + γ|



λ0
W

2
(2.33)

where γ describes the modified diffraction on the surface due to material anisotropy [81], discussed
further in Sec. 6.3.2. Interestingly, careful choice of materials can make γ ≈ −1, leading to SAWs
that do not diffract to first order. As other considerations frequently limit the cavity aperture W ,deploying a minimally diffracting material is an active area of research [82]. On GaAs, the substrateused in Ch. 4 and Ch. 5, γ = −0.537, while on ST-X quartz, which is used in Ch. 6, γ = 0.378 [81].
Another effect of the cavities finite transverse extent is the existence of modes with nodes
in the transverse direction. In typical plane-parallel geometries, these transverse modes are only
weakly confined by the reflectors and are generally low finesse [83] with frequencies that can beapproximated by considering the acoustic cavity as a rectangular waveguide. Several strategies
have been used to suppressed transverse modes by changing the geometry of the transducers thatare described in the following subsection. [84, 85, 86, 87, 88].

2.2.4

Interdigitated Transducers

Transduction between electrical and acoustic signals, crucial to measuring the cavities introduced in the previous section and generally to SAW technologies, is performed by a structure
called an interdigitated transducer (IDT). The behavior of these transducers is of fundamental
importance in designing devices, and so an effective analysis for understanding their design andbehavior is necessary.
Generally, interdigitated transducers are an array of electrodes on a piezoelectric half-space.
Typically, they are periodic and contain just two electrodes with many interleaved fingers, resembling two hands meshed together. A voltage applied between the electrodes generates strain in the28
substrate and therefore launches a travelling surface wave. Conversely, a travelling SAW carries
with it some spatially varying voltage. As a wave traverses the IDT, this voltage is imposed acrossthe IDT electrodes. These structures are usually many wavelengths long to couple preferentiallyto waves on the surface with minimal excitation of bulk waves.

2.2.4.1

SAW Green’s Function Approach

An analytical description of IDTs, drawn from [7], is based on Green’s functions. Here, we
focus on the electric potential φ(x, ω) of a SAW, but all the dynamical quantities are related andcould be used with the same effect. Surface waves are sourced by surface charge density σ(x, ω) on
the electrodes of the IDT. This charge density in principle has two components: an electrostatic
component σe (x) from the applied voltage on the electrodes and a piezoelectric component σa (x, ω)
generated by SAWs. Here, I take the quasi-static approximation where the surface charges canbe calculated ignoring acoustic excitations and are therefore governed by electrostatics, a safe
assumption on materials with weak piezoelectricity such as GaAs and quartz. The surface potentialof the acoustic wave is given by,
φa (x, ω) = GS (x, ω) ∗ σe (x),(2.34)

where ∗ indicates convolution with respect to x. The acoustic Green’s function is,
GS (x, ω) = j

K2
exp(−jkf |x|),2∞(2.35)

where K 2 is a common metric for the piezoelectric coupling strength to a given piezoelectric wavepdefined earlier, ∞ = 0 + 11 33 − (13 )2 is an effective dielectric constant, and kf = ω/vf . If aregion is considered to one side of all the source electrodes, the absolute value of x in the Green’s
function can be replaced by x, and the acoustic potential is given by,
φa (x, ω) = j

K2
σ e (kf ) exp(jkf x).
2∞(2.36)

where σ e (k) is the Fourier transform of σe (x). The acoustic potential is proportional to the Fouriertransform of the charge density, which is defined by the geometry of the IDT. This means that the29spatial profile of the IDT translates directly into the frequency profile of the transduction. The
speed of sound provides the scaling, which at a few km/s implies that MHz frequency resolution
can be achieved on the mm scale.

2.2.4.2

IDT AdmittanceThe current response to an applied voltage Vt determines an IDTs performance as a transducer. The ratio of this current to the applied voltage is known as the admittance, and it is useful
to separate this admittance into three separate components,
Yt (ω) = Ga (ω) + jBa (ω) + jωCt ,(2.37)

where Ga (ω) and Ba (ω) are the real and imaginary parts of the admittance resulting from interaction with SAWs, while jωCT is the electrostatic part of the admittance from the capacative natureof the IDT. This capacative part can be found directly by determining the current flow from theelectrostatic charge density σe (x) = V ρe (x),
Ie = jωVZW

ρe (x)dx= jωV Ct .

(2.38)

x

The acoustic part can be found in a similar fashion. To determine the acoustic part of thecurrent, the applied voltage is replaced with the acoustic potential resulting from the drive, so thatthe acoustic current in a given electrode is Ian = ρen (x)φa (x, ω). Then,
Ia = − jωW

Z ∞X

P̂n ρen (x)φa (x, ω)dx

(2.39)−∞ nZ ∞= −jωWρe (x)φa (x, ω)dx.

(2.40)−∞

The acoustic potential is given by Eq. 2.36 but it is valid only in regions to one side of all electrodes.
Using this restricted solution is sufficient to calculate the acoustic conductance,
Ga (ω) = ωWK2
|ρ (kf )|22∞ e(2.41)

where ρe (k) = σ e (k)/Vt . The solution to φa over all space is tedious to derive because of theabsolute value of x in the Green’s function but this solution is required to determine the acoustic

30
susceptance. The conductance and susceptance are the real and imaginary parts of an analytic
function as guaranteed by causality, so they can be related by Kramers-Kronig relations [89]. This
then gives the gives the acoustic susceptance,
Ba (ω) = −ωWiK2 h|ρe (kf )|2 ∗ kf−12π∞(2.42)

where ∗ represents convolution with respect to kf .

2.2.4.3

Common IDT Geometries

Typical IDT geometries permit the charge distributions ρe (x) to be accurately approximatedwith simple expressions. The primary assumption that leads to convenient calculations is thatthe electrodes are periodic and edge effects are negligible, so that each element in the array has
the same charge distribution. The charge distribution can then expressed as a combination ofan element factor ρf (x) describing the charge on individual electrodes and an array factor F (x)to account for the electrode repetition and connectivity. The total charge density ρe (x) is the
convolution ρf (x) ∗ F (x), and thus the Fourier transform ρe (x) is the product of the transforms,
ρe (k) = ρf (k)F (k).
The array factor F (x) is given by a sum of delta functions at each electrode location,
F (x) =XP̂n δ(x − xn ),(2.43)nwhere xn are the positions of the electrodes and P̂n = 1 if the electrode is connected to the driven
electrode and is zero if connected to the grounded one. Later, it is convenient to define the afrequency-dependent factor that is normalized,
A(k) = F (k)/Np .

(2.44)For an IDT with electrodes of alternating polarity of pitch p with Np periods,
F (k) =

sin(Np kp)
Np sin(X)≈,sin(kp)X

where X = πNp (ω − ω0 )/ω0 and the approximation is accurate for frequencies near f0 .

(2.45)

31
IDTs with alternating polarity between electrodes, known as single-finger IDTs, often sufferfrom high reflectivity at the same frequency they efficiently launch sound. Each metal strip in
the transducer perturbs the waves passing underneath, resulting in a weak reflection. For singlefinger IDTs, each element is spaced by a half-wavelength at its center frequency, so that thesesmall reflections add coherently; the IDT then behaves as a Bragg grating with a center frequency
matching the frequency where it effectively transduces sound. Using thin metal and only a small
number of finger periods can diminish the IDTs reflectivity. Reflections can be suppressed further
if the metal electrodes are recessed into the surface, thereby reducing the geometric discontinuity[84, 88] but requiring a more complicated fabrication process. Alternatively, the IDT geometry
can be modified so that the electrodes have polarity that alternates in pairs (i.e. 0,0,1,1,. . . ). This
doubling up of electrodes results in a mechanical Bragg frequency that is twice the frequency where
the IDT launches efficiently SAWs. This geometry, known as a split-electrode IDT, is a powerfultool for suppressing reflections from the IDT electrodes, but comes at the cost of a capacitor
with half the pitch, increasing both the capacitance for a given transducer and the demands onlithography. These two common types of IDTs are depicted in Fig. 2.5. The array factor for the
split finger IDT is unchanged near its center frequency.
The Fourier transform element of the factor for an infinite array of electrodes, while closedform, is still somewhat complicated. This factor is given by,
ρf (β) = ∞

2 sin πsPm cos(∆)
P−s (− cos ∆)for m ≤βp
≤ m + 1,2π(2.46)

where ∆ = πa/p, a is the metallization ratio, P−s is a Legendre function and Pm is a Legendre
polynomial, m is the Bruillion zone of interest (equivalently, the harmonic), and s = βp/(2π − m),so that 0 ≤ s ≤ 1. The metallization ratio a, which greatly effects the elemental charge density, isthe fraction of the pitch occupied by an electrode. While the array factor is usually more importantin determining the frequency response of the IDTs, it is important to consider the elemental charge
density, as for standard geometries there are certain harmonics that are allowed by the array factor
but the elemental charge density is near zero. In particular, single-finger IDTs with a = 0.5 do not32

Figure 2.5: Typical IDT geometries are either (a) single-finger or (b) split-electrode. (c) These IDTs
are well-modeled by a frequency-independent capacitance in parallel with an acoustic conductanceGa and susceptance Ba . (c) The calculated admittance and conductance (in milli-Siemens) are
shown for an IDT with either Nt = 100 or Nt = 60 periods on GaAs with a 4 GHz center frequency
and W =25 µm.

launch at the third harmonic.
For most applications, the array factor will be the dominant source of the frequency dependence of the IDT. For a single-electrode IDT with N periods of electrodes of pitch p, at frequencies
near ωc = 2πvs /(2p),Ga (ω) = Ga0

sin X
X

2
,

(2.47)

and, by Hilbert transform,
Ba (ω) = Ga0

sin(2X) − 2X.
2X 2

(2.48)

33
The prefactor Ga0 comes from the elemental charge distribution evaluated near ωc . For both single
and split finger IDTs, this term is given by,
Ga0 ≈ 1.3ωc Cs W K 2 NP2 .

(2.49)

The acoustic susceptance and conductance for a typical transducer on GaAs are shown in Fig. 2.5.
The term Cs is the capacitance per unit finger length and is related to ∞ . For a split-electrode
IDT, the capacitance is given by,Ct =For a single-finger IDT, the capacitance is√√

2Np W Cs .

(2.50)

2 smaller.

A common use of IDTs is to make a delay line by launching sound from one transducer andreceiving it at the other. The slow speed of sound means the delay can be several microseconds
long on a mm sized chip. However, SAW delay lines are often quite lossy. On most materials, theIDT launches sound in both directions with equal amplitude, resulting in half the signal being lostat launch and by reciprocity the same at reception. On a weak piezoelectric material and at highfrequencies, it is difficult to make the IDT impedance close to 50 Ω that is the typical characteristic
impedance of microwave transmission lines, resulting in large reflections at the IDT. With enoughfinger periods, the conductance can be made close to 1/50 Siemens, but the associated capacitance
begins to short out the launch.

2.3

Externally Coupled Cavities
The properties of acoustic structures, such as Fabry-Perot type cavities, can be probed using

purely electrical techniques by using an IDT to convert an electromagnetic probe tone to an acoustic
wave, which then interacts with the structure of interest before being converted back to electricity
and measured.
For measuring Fabry-Perot type SAW resonators, two general geometries are possible. In thefirst geometry, the mirrors are made to be semi-transparent and an IDT is positioned on each sideof the cavity. Transmission from one IDT to the other will increase when the driving frequency

34matches an acoustic resonance, where energy builds up inside the cavity. This style of measurement
is analogous to how optical cavities are probed, but poses two challenging tasks in the acoustic
domain: the IDT must efficiently transduce SAWs into free space, and the mirrors must have justthe right reflectivity to let enough energy through to facilitate measurement without significantly
increasing the cavity loss rate. Alternatively, an IDT could be placed inside the cavity, as shownin Fig. 2.6. In this geometry, the IDT launches sound directly into the cavity. On resonance,
the drive rings up the cavity, and the relative amplitude of the signal leaking back out the IDT
increases, resulting in a change in reflection on resonance discussed more thoroughly in Sec. 2.52.
In this thesis, I will focus primarily on the second geometry with the IDT inside the cavity. This
design benefits from resonant enhancement of the IDT launch efficiency and no constraints on thereflectivity of the mirrors.
The behavior of the IDT changes significantly when placed inside a resonator. The SAWs
that are launched do not simply propagate away, but are confined by the mirrors. The interactionof the IDT with these confined waves confined can be modelled using the strategies like ABCD
matrices. However, a simpler and more intuitive approach is to assume the cavity supports high-Q
resonant modes and consider how the IDT interacts with these discrete resonances.
An IDT connected to a transmission line and placed inside the cavity behaves in many ways
like an additional source of dissipation for the resonant modes. A confined wave that traverses the
IDT generates an electrical signal that then travels down the transmission line, never to return.
This interaction decreases the lifetime of a phonon in the cavity. In the limit that the cavity finesse
is much higher than one, this loss rate is given by,κe = 4.4Ga (ω)Zc

vs
ζ(x)2
L

(2.51)

where Zc is the characteristic impedance of the transmission line connected to the IDT, Ga (ω) is
the acoustic conductance given in Eq. 2.41, and ζ(x) characterizes the spatial alignment between
IDT and the acoustic standing wave. Essentially, the IDT extracts fractional energy Ga Zc every
round-trip time L/vs for an IDT with electrodes at the antinodes of a given standing wave where35
ζ(x) = ±1. Since IDTs are distributed, SAW cavities are many wavelengths long, and the Bragg
gratings result in frequency dependent boundary conditions, ζ(x) can be difficult to predict for agiven mode. Symmetry can be helpful; for example, at the cavity center all even modes have an
an anti-node and all odd modes have a node. The most robust method is to compare the IDT
electrodes to the standing waves described in equation 2.32.
The “loss” caused by energy entering a transmission line is special because, while the effecton the overall cavity lifetime is the same as other sources of dissipation, the energy extracted by
the transmission line is not really lost as it can be measured. This type of loss is referred to as
an external coupling rate. In addition to being measurable, external coupling rates are usually set
intentionally when designing resonators. In contrast, internal losses are typically unwanted, arisingfrom various dissipation mechanisms described in section 2.2.2.

2.3.1

Input-Output Theory

The internal and external coupling rates of a general resonator, acoustic or electric, canbe extracted by measuring the scattering of signals from external coupling ports as a functionof frequency. The response can be derived from input-output theory as described below. This
formalism is useful not only in its simplicity, but also because it extends readily to a quantum
mechanical descriptions.
Let a(t) describe the state of the phonon field inside the resonator normalized such that|a(t)|2 ~ω is the energy circulating in the cavity. The rate of change of a(t) depends on the various
cavity coupling rates as well as the incident fields. In a frame rotating with the incident field,ppκ
ȧ(t) = − a(t) + i∆a(t) + k1 a1,inc (t) + k2 a2,inc (t)2(2.52)

where κ1,2 are two external coupling rate, κ = κ1 +κ2 +κi is the total cavity linewidth with internal
loss rate κi , ∆ is the detuning between the incident field and the cavities resonant frequency, andainc (t) is the incident field on port 1 or 2, normalized so that |a1,inc (t)|2 ~ω is the incident poweron port 1 of the cavity. To make these equations describe a quantum systems, all of the field terms

36
a’s must become operators, and an additional noise term κi finc has to be included to account forunavoidable quantum fluctuations from cavity dissipation.
The steady state internal field generated in response to a constant incident drive is found by
setting ȧ(t) = 0. Then,√a(t) =

κ1 a1,inc.
κ/2 − i∆(2.53)The magnitude of a(t) gives the number of circulating phonons,n = |a(t)|2 =

κ1
Pin
.
2
2
∆ + (κ/2) ~ω0(2.54)

where Pin is the power incident on port 1. Intuitively, a resonant drive results in an intracavity
phonon number equal to the number of phonons arriving in a cavity lifetime multiplied by ratio of
external to total coupling rates.
The reflected field a1,out is composed of both the promptly reflected incident signal a1,inc
as well as the field leaking out from the cavity, such that a1,out = a1,inc −√κ1 a. The reflection

coefficient, defined as the ratio of the output field to the input, is given by,
R=1−

κ12i∆ + (κi + κ2 ) − κ1=.
κ/2 − i∆2i∆ + (κi + κ2 ) + κ1

(2.55)

This reflection coefficient depends strongly on the ratio κ1 /κ. When κ1  κ, the reflected field
is composed mostly of promptly reflected signal, and its phase is always between −π/2 and π/2.
In this regime the port is called “under-coupled”. When κ1 ≈ κ, the port is “over-coupled” andon resonance the reflected signal is predominantly energy leaving the cavity. Crucially, measuringreflection from an overcoupled port contains a full 2π phase phase wrap across resonance. Finally,if κ1 ≈ κ/2, then the cavity is critically coupled, and on resonance, the promptly reflected signal
perfectly cancels the field leaking out of the cavity, resulting in a vanishing reflection coefficient.
The magnitude of the reflected signal is invariant under the exchange κ1 ↔ κ2 + κi . In theextremes of over- and under-coupled regimes, |R| ≈ 1. Thus, if only the amplitude of the reflectedsignals is considered, it is impossible to tell if the given response is under- or overcoupled, as shown
if Fig. 2.6(b). Measuring the phase of the cavity response is the key to differentiating over- and37

Figure 2.6: (a) The response of an IDT placed inside an acoustic Fabry-Perot cavity can be characterized in terms of internal and external coupling rates. Note that the right IDT (red) couples
weakly as the cavity standing wave has nulls at the IDT finger positions. (b-c) The complex reflection off each port and transmission through the cavity are plotted for κ1 = 4κ2 = 40κi . In
reflection, the phase information is crucial as the difference between the over- and under-coupledport cannot be distinguished in magnitude alone.

under-coupled resonances. When considering only reflection, the external coupling out the second
port can be regarded simply as an additional source of loss.
The transmission from one port to the other is given by,
√
κ1 κ2
T =
.
κ/2 − i∆(2.56)The phase of the transmitted signal changes by π across resonance regardless of the coupling rates.
The magnitude of transmission reaches unity on resonance when κ1 = κ2 and κi = 0. These38
conditions mean both ports 1 and 2 are critically coupled, so that no power is reflected, and as no
power is lost, then it all must be transmitted. The transmitted power is often the most accessible
quantity experimentally. It is given by,|T |2 =

4κ1 κ2.
2
κ + 4∆2(2.57)The transmitted power falls to half of its max at ∆ = κ/2, and so κ is referred to as the full-widthat-half-max, or FWHM, and is the convention used here for specifying a linewidth.
The general results from input-output theory apply easily to describing multi-mode SAW
resonators. Including multiple resonant modes is a fairly-straight forward extension of the model,
requiring an additional cavity field for each mode, but all with the same input and output fields.
The modes are independent, so each has its own linear differential equation as in Eq. 2.52. The
output fields have contributions from each resonance,
aout = ainc +X√κm am ,

(2.58)mwhere m indexes the resonant mode with detuning ∆m , external coupling κe,m , and total linewidthκm . The multi-mode reflection is given by,
R=1−

κe,m
.
κ
/2
− i∆m
m
m

X

(2.59)The multi-mode transmission is just a sum of the transmissions for each resonant mode, with eachconsecutive mode increasing in phase by π. For most SAW resonators, the spacing between purely
longitudinal modes is small compared to their linewidths. In this case, only one term in the sum
is significant, and the single-mode case is good approximation. However, resonant modes with
non-zero transverse mode number can be nearly degenerate with their corresponding longitudinalmode, and the multi-mode reflection case must be considered.

Chapter 3

Quantum Acoustics with Superconducting Qubits

Investigating quantum acoustics with surface acoustic waves requires a quantum descriptionof the resonators described in the previous chapter. Additionally, a non-linear element is needed toescape the correspondence principle which limits linear quantum systems with classical control toreplicate classical physics. In this chapter, I introduce a quantum description of acoustic resonatorsand superconducting qubits then conclude with a description of the coupled qubit-cavity system.

3.1

Quantum Harmonic Oscillator
The simple harmonic oscillator is a canonical model in physics and in quantum mechanics

in particular. While introduced as a mass on a spring, the description is applicable to a diverse
range of physical systems including the resonant modes of surface acoustic wave cavities. The
model consists of mass m connected to a spring with stiffness k such that a displacement from its
equilibrium position x results in a restoring force F = −kx. The Hamiltonian for the system is,H=

1 2 k 2p̂ + x̂ ,
2m
2

(3.1)

where p̂ = m dx̂
dt is the momentum. The quantum nature of this Hamiltonian comes from the commutation relation between position and momentum, [x̂, p̂] = i~. This relation can be manipulated
to define the creation and annihilation operators â and â† ,
r
â =

mω
2~



i
p ,
x+
mω

(3.2)

40
and [â, â† ] = 1. Using these operators, the Hamiltonian can be rewritten as,1†
H/~ = ω â â +
,
2

(3.3)

where â† â is the number operator. This Hamiltonian has eigenstates |ni such that â† â|ni = n|niwith energies En = ~ω(n+1/2). Crucially, the energy difference between all eigenstates is identical.
It is this equal spacing that precludes creating non-classical states of motion in harmonic
oscillators with classical drives alone. Such a classical drive excites multiple transitions in theoscillator simultaneously, creating a coherent state |αi that follows a Poissonian distribution in thenumber basis,
∞

X
αn
√ |ni.
|αi = exp −|α|2 /2
n!
n=0

(3.4)

Coherent states are eigenstates of the annihilation operator, â|αi = α|αi with average excitation
number |α|2 = n. Importantly, the behavior of a quantum harmonic oscillator in a coherent state
appears classical, albeit with unavoidable noise of quantum origins.
Another important state in the simple harmonic oscillator is the thermal state. When inthermal equilibrium with some bath of temperature T , the phonon probability distribution is givenby,
P (nth , n) =nnth,
(nth + 1)n+1(3.5)

where nth = [exp(~ω/kB T )−1]−1 is the mean phonon number. To be cooled to its quantum groundstate, the probability of zero phonons P (nth , 0) must be near one. This condition is reached whenthermal energy kB T is be small compared to the transition energy ~ω. At temperatures reachedin modern dilution refrigerators of near 25 mK, mechanical oscillator must have frequencies larger
than 500 MHz to be cryogenically cooled to their ground state.
Electromagnetic cavities are also well described as simple harmonic oscillators. Electromagnetic waveguide cavities and LC oscillators are of particular interest here for their role in superconducting circuit design and measurement. The waveguide cavity is described in more detail as thereadout resonator in Sec. 5.2.1.1. The LC circuit is composed of a parallel inductor and capacitor,p√and is characterized by its resonant frequency ω = 1/ LC and impedance L/C. A quantum

41
treatment can be developed by considering the charge on the capacitor Q̂ = CV and the flux acrossthe inductor Φ̂ = LI, where V is the voltage and I is the current. The Hamiltonian for this systems
is,
H=

Q̂2
Φ̂2
+
,
2C
2L

(3.6)

with commutation relation [Q̂, Φ̂] = i~. The clear analogy between this system and the simple
Harmonic oscillator means they have similar quantum behavior.

3.2

Superconducting Qubits
Superconducting qubits consist of low-loss microwave circuits with an inherent non-linearitythat makes their spectrum anharmonic. While these circuits contain many energy levels withvarying levels of complexity, the unequal transition frequencies allow individual pairs of levels to beconsidered exclusively. These pairs can be modelled as a two-level system and mapped to a spin- 12
particle. The Hamiltonian for such a two-level system is,1
H/~ = − ωq σ z .
2

(3.7)

where ωq is the qubit frequency and σ z is the Pauli matrix whose eigenstates have spin oriented
along the z-axis, either pointing up or down. Here, I will refer to spin up as the ground state,denoted by |gi, and spin down as the excited state, denoted by |ei. Similar to the case of theharmonic oscillator, this system will be primarily in the ground state if ~ωq  kB T .

3.2.1

Josephson Junctions

The element that makes superconducting qubits possible is the Josephson junction [90]. Thiselement is formed by a superconductor interrupted by a thin insulating layer as shown in Fig. 3.1(a).
In the superconductor, electrons form two electron bound states, known as Cooper pairs, and arecharacterized by a superconducting phase φ. When the insulating interruption is thin enough,typically around a few nm, Cooper pairs can tunnel across the barrier. Crucially, a discrete a phasedifference δ can develop between the two superconducting islands. The voltage and current flowing

42

Figure 3.1: (a) A Josephson junction formed by two superconducting leads with an insulating
barrier thin enough for Cooper pairs to tunnel across. The difference between the superconducting
phases φ1 − φ2 = δ and the critical current (IC ) determines the voltage and tunnelling current andvoltage across the junction. (b) Josephson junction schematic, including small (several fF) junction
capacitance. (c) A DC squid is formed by two junctions in a loop threaded by flux Φa and acts as
an effective (d) single junction with flux dependent critical current.

across the junction are given by the Josephson relations:
I = Ic sin(δ)

(3.8)

Φ0 dδ
,
2π dt

(3.9)

V =

where Ic is the junction’s critical current and Φ0 = h/2e is the flux quantum.
These relations mean that a Josephson junction acts as a non-linear inductor. To see this,differentiate Eq. 3.8 with respect to time, and then substitute the resulting expression for dδ/dt
into Eq. 3.9. This substitution gives,V =Lj0 dI1dI
Φ0=.
2π Ic cos(δ) dtcos(δ) dt(3.10)

Φ0
The term Lj0 = 2πI
is defined as an effective Josephson inductance when δ = 0. The tunnellingcof Cooper pairs across the junction stores energy much in the same was an inductor, however the
relationship between δ and I means that this Josephson inductance changes with current. The

43
tunnelling energy of these electrons is equal to EJ (δ) = EJ cos(δ) where EJ = Ic Φ0 /2π. In addition
to an inductance, junctions have an inherent capacitance CJ between the two superconducting
islands that depends strongly on its geometry. With the junctions and superconducting circuits weare interested in here, this capacitance is negligible.
A widely used element, known as a DC superconducting quantum interference device (SQUID),
consists of two Josephson junctions connected end to end in a loop [Fig 3.1(c)]. In the case thatthese two junctions have the same critical current Ic , the DC squid acts much like a single Josephson
junction with critical current 2IC that decreases depending on the magnetic flux threaded throughthe SQUID loop. The effective critical current of the full loop is given by [91],ΦaIL = 2Ic cos π.
Φ0(3.11)

where Φa is the applied flux. In this way, Josephson junctions can act as tunable, loss-less inductors.
Asymmetry between two junctions may need to be included when more accuracy is needed,
particularly as current fabrication methods lead to variation in junction energies on the order of
5 − 10%. In the case of two junctions with critical currents I1 and I2 , the effective critical currentof the loop is given by,

s


Φa
Φa
2
2
IL = (I1 + I2 ) cos π,
1 + d tan πΦ0Φ0(3.12)

2
where d = II11 −I+I2 quantifies the junction asymmetry [92]. The effect of the asymmetry is small nearapplied flux near zero. However, near Φ = π/2, the asymmetry prevents IL from approaching zero.
Many useful electrical circuits can be made based on Josephson junctions. A diverse group of
superconducting qubits have been realized using the non-linearity of the junctions integrated withdifferent linear elements to design circuit Hamiltonians with desirable properties [43]. The nonlinearity of Josephson junctions are also widely used to implement and 4 wave mixing in quantum
limited parametric amplifiers [93, 94, 95, 96, 97]

44

Figure 3.2: (a) The transmon qubit is formed by shunting a Josephson junction or DC squid witha large capacitance C such that EJ /EC  1. (b) The transmon acts as a flux particle in a cosine
potential. The weakening potential at larger flux results in a negative anharmonicity α.

3.2.2

Transmon Qubit

A popular type of superconducting qubit, and the one primarily used in this thesis, is formedby shunting a Josephson junction with phase difference δ̂ by a large capacitance with charge Q̂.
This circuit has a Hamiltonian resembling an LC circuit,
H=

 
Q̂2
+ EJ cos δ̂ ,2C

(3.13)

where the cosine potential appears parabolic near its minima before weakening at larger δ. The
transmon qubit is a limit of this Hamiltonian where the Josephson energy EJ greatly exceeds the
charging energy EC = e2 /2C. In this limit, the ground to excited state transition frequency is given√by ωge = 8EC EJ /~. Here, ωge = ωq , but the two indices are included to clearly indicate other transitions are being considered. The higher order terms in cos δ̂ lower the frequency of the next
highest transition, so that ωef = ωeg − Ec /~. The transmon Hamiltonian can be well approximated45
as a Duffing oscillator [92],
H=



p
1
8EJ Ec b̂† b̂ + Ec b̂† b̂ b̂† b̂ − 1 .
2(3.14)

This slightly anharmonic oscillator can be treated as a qubit to the extent that ωge and ωef arespectrally distinct, which is usually a good assumption given that a typical transmon linewidth γ
is below 1 MHz compared to an anharmonicity α = −Ec /~ of several hundred MHz. While the twolowest levels can be readily used as a qubit, many important properties of transmon qubits require
including at least the |f i level during analysis [92].
The transmon limit involves a trade-off between charge noise sensitivity and anharmonicity.
An ideal qubit is insensitive to noise while being very anharmonic. For the transmon Hamiltonian,increasing the ratio of EJ /EC decreases the qubits sensitivity to charge noise but also decreases
its anharmonicity. The sensitivity to noise decreases exponentially with EJ /EC while the anharmonicity only decreases linearly. At least until EJ /EC ≈ 50, the exponentially suppressed noise
sensitivity is well worth the linearly reduced anharmonicity, as charge noise has been a significant
limit in qubit coherence times and a typical transmon anharmonicity of a few hundred MHz is sufficient for reasonably fast qubit operation. This simple design and its insensitivity to charge noise
has led to a vast improvement in coherence times with low circuit complexity, and the state-ofthe-art attempts to build a quantum computer with superconducting circuits use transmon qubits[98, 99, 100].

3.3

Jaynes Cummings Model
A fully quantum treatment of the interaction between a harmonic oscillator and a two-level system was introduced by Jaynes and Cummings [101]. Their simple model, first derived
for atoms coupled to an optical cavity through their electric dipole moment (Fig. 3.3), captures
many phenomena regarding the interaction between light and matter and has become a “standard
model” of quantum optics [102]. While the harmonic oscillator of interest in this thesis is phononic
in nature and the atom is a superconducting circuit, their model is general enough to accurately

46

Figure 3.3: The Jaynes-Cummings model describes a two-level system with frequency ωq and loss
rate γ coherently exchanging excitations at rate g with a harmonic oscillator with frequency ωr andloss rate κ.

describe the system. Indeed, the successful acoustic realization of the Jaynes-Cummings model is
one of the fundamental results of this thesis.
In the Jaynes-Cummings model, an effective two-level system exchanges energy with a harmonic oscillator when the two systems are near resonance through a dipole interaction. The Hamiltonian for the coupled system is given by,1
HJC /~ = ωr â† â + ωq σ z + g âσ + + â† σ − ,2(3.15)

where σ + (σ − ) is the qubit raising (lowering) operator and g is the rate at which the qubit andcavity exchange energy on resonance. As written here, the Hamiltonian ignores the counter-rotating
terms âσ − and â† σ + that do not conserve the total number of excitations in the coupled system.
Known as the rotating wave approximation, this simplification is valid when the coupling strength
g and the detuning ∆ = ωr − ωq between the two systems are small relative to the qubit and cavity
frequencies.
The Jaynes-Cummings Hamiltonian in the rotating wave approximation can be exactly diagonalized because interactions only occur between states with the same number of total excitations.

47
When constructed in matrix form, the Hamiltonian is composed of two-by-two element blocks along
its main diagonal, each block corresponding to a set number of excitations ne ,


√
ω
n
−
ω
/2
n
g
q
e
 r e

Hne /~ = 
.
√
ne g
ωr (ne − 1) + ωq /2(3.16)

These blocks can be diagonalized,
E±,nE /~ = ne ωr ±p
ne g 2 + ∆2 /4,(3.17)

where + (-) refers to the higher (lower) energy eigenstate in the ne excitation manifold. The
corresponding eigenstates are given by,|−, ne i = cos(θn )|g, ni − sin(θn )|e, n − 1i

(3.18)|+, ne i = sin(θn )|g, ni + cos(θn )|e, n − 1i
 √ 
2g ne
1
θn = arctan,2∆(3.19)(3.20)

where n is the number of excitations in the cavity and the effective angle θn characterizes the
mixing between qubit and cavity states.

3.3.1

Resonant Regime

In analyzing this solution, it is useful to consider two distinct cases: the resonant regime
when ∆  g and the dispersive regime when ∆  g. The eigenvalues in each regime are showngraphically in the energy level diagram of Fig. 3.4. At small detuning, θn ≈ π/4 and the two
eigenstates are approximately even superpositions of qubit and cavity modes with energy splitting
2g~, implying excitations are swapped at rate g. This is only true if the system is in the strong
coupling regime, where the coupling rate g exceeds the dissipation rates of the qubit γ and thecavity κ. The hybridized modes inherit half of the qubit’s loss and half of the cavity’s, resulting in
two modes each with a loss rate (κ + γ)/2.
The hybridization between the qubit and the cavity in the single excitation manifold can bewell modelled using the normal modes of two coupled classical oscillators. The non-linear behavior

48

Figure 3.4: The Jaynes-Cummings model can be exactly diagonalized. (a) On resonance (∆ = 0),
the qubit and cavity form hybridized modes with a splitting that grows with total excitation number.
(b) In the dispersive regime (g/∆  1), the normal modes correspond closely to the uncoupledmodes, but with resonant frequencies that shift by 2χ for each quanta in the other sub-system. In
the case shown where ωq < ωr , the dispersive shift χ is negative.

of the Hamiltonian becomes apparent when the number of excitations in the system grows. Each
additional quanta increases the qubit-cavity swap rate in discrete steps, growing as the square rootof the cavity occupancy [103, 67]. Therefore, resonant swapping of excitations while measuring
the exchange rate between a resonant qubit-cavity system performs a measurement of the cavity
occupancy.

49

Figure 3.5: (a) As the qubit is tuned across the resonator, the eigenmodes avoid each other, reachinga minimum spacing 2g when ∆ = 0 and the eigenmodes are even and odd superpositions of qubit
and cavity. (b) In the multi-mode regime where the coupling is on a similar scale to the spacing
between modes g ≈ ωs , the avoided crossings are more complicated.

3.3.2

Dispersive Regime

With increasing detuning, the hybridization between the qubit and the cavity weakens andthe eigenvalues of the coupled system begin to approach their uncoupled values. This avoided
crossing behavior is shown in Fig. 3.5(a). The dispersive regime is reached when the detuning islarge enough that the eigenmodes correspond closely to the uncoupled qubit and cavity modes,
such that θn ≈ g/∆ → 0. In this limit, the Hamiltonian can be approximated using a unitarytransformation,
D = exphg ∆i
σ − a† − σ + a .

(3.21)

Keeping terms to second order in g/∆ of the transformed Hamiltonian Hd = D† HJC D gives thedispersive Hamiltonian,1
Hd /~ = ωr â† â + ωq σ z + χâ† âσ z ,2(3.22)

50
where χ = g 2 /∆ is known as the dispersive shift. This Hamiltonian has eigenvalues,
Eg,n /~ = nωr − ωq /2 − nχ

(3.23)

Ee,n /~ = nωr + ωq /2 + nχ,(3.24)

which can also be found by expanding Eq. 3.17 for small g/∆. This approximation not only assumes
that g/∆  1 but also that the mean photon number does not exceed ncrit = (∆/2g)2 [104]. In
physical systems, it has been observed that the dispersive approximation frequently breaks down
at mean photon numbers well below ncrit .
The effect of the χâ† âσ z interaction term is that excitations in one system shift the frequencyof the other [Fig. 3.6, Fig 3.4(b)]. If the qubit is in ground state, then the cavity frequency isωr,|gi = ωr −χ, while if qubit is in the excited state the cavity transition shifts to ωr,|ei = ωr +χ. The
problem of determining the state of the qubit can then be mapped onto precise measurement of theresonant frequency of a harmonic oscillator. Moreover, since the interaction term commutes with thequbit part of the Hamiltonian, the measurement effect is only to project the qubit into eigenstatesof σ z , and repeated measurements yield the same answer. The ability to repeat measurements

Figure 3.6: In the dispersive regime, the cavity frequency shifts by ±χ from its bare resonant
frequency ωr depending on the state of the qubit. Similarly, the qubit frequency shifts by 2χ forevery excitation in the cavity. If the cavity is displaced with a coherent state of size n, then thequbit spectrum splits into multiple peaks, with weights reflecting the probability distribution ofcavity excitations. The qubit is also dephased by the loss of cavity excitations which effectively
perform measurements of the qubit state.

51
and measure the same result is known as a quantum non-demolition (QND) measurement [105].
Non-QND measurements include photon counting and other schemes where the quantum state is
destroyed during measurement. Additionally, even performing dispersive readout with ∆  g, using
a coherent state with too large of an amplitude degrades the QND nature of the measurements. As
this breakdown amplitude can be quite small, it is imperative to measure the cavity output with a
low-noise amplification chain.
The dispersive Hamiltonian also means that the qubit frequency depends on the number of
excitations in the cavity. In a semi-classical approximation, the qubit transition shifts smoothly,such that ωq,|ni = ωq − 2χn where |α|2 = n. However, if the qubit frequency can be resolved tobetter than χ, then this frequency shift is observed to be discrete, such that ωq,|ni = ωq − 2χn.
This so-called strong dispersive regime demonstrates the quantized nature of the excitations of thelinear cavity and allows the qubit to measure the number of excitations in the cavity. Combinedwith displacement of the cavity mode, such measurements can lead to the full reconstruction of thecavity quantum state [106, 105, 52].
The residual hybridization between the qubit and cavity adds potential decay channels forboth systems, called the Purcell effect. The coupled loss rates of the qubit γ 0 and cavity κ0 are
related to bare linewidths by,
γ0 = γ + g 2κ,∆ g 2
κ0 = κ +γ.
∆(3.25)(3.26)

Although g/∆ is small, this inherited loss can still be significant. Commonly, the cavity loss rate
is made large (∼MHZ) by engineering a large external coupling rate to facilitate readout, and evena small fraction of this loss can exceed the qubits intrinsic decay rate [107]. This effect can be
circumvented by modifying the density of states as a function of frequency: if cavity is coupled toa transmission line that supports propagating modes at ωr but not at ωq , then the qubit sees areduced density of states and its decay rate from the cavity loss rate suppressed [108].
For transmon qubits, including higher levels is required for accurate predictions of dispersive

52
physics. The presence of the |f i level modifies the dispersive shift,
χ=g

2



1
1
−
∆ ∆+α


(3.27)

where α = ωef −ωge [92]. The two terms in this dispersive shift usually have different sign, reducing
its magnitude. However, if g is small compared to α, then the transmon can be tuned such that
ωge > ωr but ωef < ωr . In this configuration, known as the “straddling regime”, the two terms
add constructively and boost the dispersive shift. In practice, this approximation is accurate when
the highest frequency transmon level is below the cavity frequency. When the transmon frequency
is above the cavity, however, higher transmon levels complicate the interaction, particularly when
the cavity occupancy grows beyond a few excitations. This case requires numerical diagonalization
using a truncated energy basis including several photons and several transmon levels, a process
discussed further in Sec. 4.2.3.4.

3.3.3

Multi-mode Jaynes Cummings Model

Many cavities, both acoustic and electromagnetic, support multiple resonant modes. The
multi-mode Jaynes Cummings Hamiltonian is a simple extension,
H/~ =



X
1
ωm â†m âm + ωq σ z +gm âm σ + + â†m σ − ,
2
m
m

X

(3.28)

where âm is the latter operator for mode with frequency ωm and qubit coupling strength gm . Whenthe qubit is far detuned from all of the modes compared to their coupling strengths, then themulti-mode dispersive approximation gives,
H/~ =

X
1
χm âm â†m σ z ,ωm â†m âm + ωq σ z +
2
m
m

X

(3.29)

2 /∆ and ∆ = ω − ω . For most systems, the scale of the interaction strengths
where χm = gmmmm
q

is small compared to the spacing between resonant modes. In this limit, the qubit will be in
the dispersive regime with all modes but one and the analysis from the single-mode case can be
accurately applied. If the spacing between different resonant modes is comparable to the interaction
strength, then near resonance the model must consider possible hybridization between the qubit

53
and more than one mode [24]. The eigenvalues of these multi-mode systems, which are the focusof this thesis and are shown in fig. 3.5(b), must be calculated numerically. Interestingly, in themulti-mode limit, the strong coupling criteria is weakened such that clear avoided crossings nolonger indicate that the coupling strength exceeds the decoherence rates of all sub-systems. In
the multi-mode dispersive limit, the behavior is a simple sum of dispersive shifts from individual
modes. However, when the coupling rate is much larger than the mode spacing, these shifts may
2 /∆ , a regime not accessed in this thesis.
differ significantly from gm
m

3.3.4

Transmon Qubit-SAW Cavity Coupling Strength

To realize the Jaynes-Cummings model with surface acoustic waves coupled to a transmon
qubit, we need to engineer a strong interaction between the two systems. This is achieved byusing an interdigitated transducer on a peizoelectric material as the shunting capacitance in atransmon qubit. Such IDT geometries can have capacitances of several tens of fF, comfortable forthe transmon regime corresponding to Ec of several hundred MHz.
The interaction between the transmon and acoustic waves can be calculated in much in thesame way as that between a transmon with an electric dipole in a resonant electromagnetic cavity.
Such an interaction is driven by the charge operator n̂ which denotes the number of Cooper pairs
transferred between the islands and is related to the transmon effective ladder operators as,

n̂ =

EJ
8EC

1/4 √2 †(b̂ − b̂).
2

(3.30)

The coupling between cavity mode indexed by number m and the transmon transition from |ii to|ji is given by,0~gm = 2βeVrms,m(x)hi|n̂|jiA(ωm )(3.31)

where Vrms,m (x) is the root mean squared zero-point voltage fluctuations of either the electromagnetic or piezo-mechanical resonator at location x, β is the ratio of coupling capacitance to total
capacitance, and A(ωm ) captures the frequency sensitivity of the interaction and is absent in theelectromagnetic case. As β is controlled in design and typically of order unity, three terms deter-

54

Figure 3.7: A transmon qubit that interacts strongly with a confined SAW mode can be formedby shunting an interdigitated transducer with a Josephson junction placed between two Braggreflectors.

mine the coupling rate: the voltage fluctuations, the matrix elements, and the match of the IDT
to the standing wave, both in frequency and position.
For a transmon, the only significant matrix elements are between nearest-neighbor transitions,with symmetry forbidding transitions between states of the same parity due to the lack of odd terms
in the transmon potential [92]. Such nearest neighbor transitions have matrix element,r
j + 1 EJ 1/4
hj + 1|n̂|ji =,2
8EC
where n̂ is the number operator of the island. This matrix element grows as√(3.32)

j + 1 with increasinglevel. The dependence on the transmon energies EJ and EC is weak, and does not strongly effectthe coupling strength.
The main factor in determining the scale of interaction rate is the zero-point voltage fluctuations of the acoustic cavity. In most piezoelectric materials, the dominant form of energy ismechanical. The amplitude of the zero-point position fluctuations are then determined by calculating the displacement of a single phonon in mode m, given by U0,m , with energy ~ωm ,s~
Um (x),
U0,m (x) =2ρLe W vs(3.33)

55
where ρ is the material density, Le is the effective cavity length and W is its width, and vs is the SAW
velocity. These fluctuations depend on position based on the amplitude of the acoustic standing
wave inside the cavity, given by Um (x) [Eq. 2.32]. The magnitude of the voltage fluctuations are
related to this displacement by the piezoelectric properties of the material,
0Vrms,m(x) =epz
U0,m (x)(3.34)

where  is the dielectric constant of the material and epz is the appropriate element of the piezoelectric tensor in stress-charge form that depends on propagation direction. For weak piezoelectricmaterials like quartz and gallium arsenide, epz / is typically on the order of 1 V/nm.
The novel aspect of using an IDT rather than traditional dipole coupling is that the IDT
samples the acoustic field at many many locations separated by many wavelengths. This multiwavelength extent has implications for the strength of resonant interactions by introducing afrequency-dependent factor A(ω) in direct analogy to how the acoustic conductance depends on
IDT geometry in Eq. 2.41. The dominant contribution to this term is the array factor A(ω) found by
Fourier transform of the IDT geometry including the polarity of each finger as in Eq. 2.43 but normalized. As discussed in Ch.5, the coupling can be designed to have a specific frequency dependence
to meet experimental needs [59]. Additionally, such “giant atoms” can be used to explore physics
far outside the dipole coupled regime, including non-Markovian effects such as non-exponential
decay [109].
All together, the coupling strength between an acoustic mode and the ground to excited statetransition of a transmon can be calculated as,
s


epz
~
EJ 1/4~gm = eβA(ωm )Um (x)ρLe W vs 8EC= g0 A(ωm )Um (x),(3.35)(3.36)

where g0 characterizes the mode-independent strength of the qubit-acoustic coupling. For a transmon with most of its capacitance in the IDT and well-aligned to the acoustic standing wave, bothin space and in frequency, then β ≈ 1, A(ωm ) ≈ 1, and Um ≈ 1. In this thesis, the substrate is56
typically GaAs, which has charge-strain piezoelectric tensor element epz = 0.14 C/m2 , dielectric
constant  = 12.50 , density ρ = 5307 kg/m3 , and wave velocity vs = 2860 m/s. For multi-mode
cavities of somewhat narrow width, coupling rates of several MHz are achievable. Surprisingly, thenumber of fingers in the IDT does not significantly affect the coupling rate.

3.4

Waveguide Quantum Acoustodynamics
A tunable qubit coupled to a SAW resonator can interact with a continuum of unconfinedphonon modes by tuning outside the mirror bandwidth. Access to these propagating phonons could
be used to transfer information across a chip, while the distributed nature of the qubit-phonon
coupling could lead to qubit-qubit interactions protected from waveguide losses [110]. Additionally,
the slow travel time of SAWs could lead to novel routing schemes, where the destination of a
wavepacket can be determined and altered during its relatively long transit time [111], or thecreation of cluster states with using time-delayed feedback [112].
The spontaneous emission rate of a transmon qubit to a propagating SAW mode can be calculated semi-classically by considering the transmon qubit as a parallel RLC circuit with capacitanceand inductance set by the EJ and EC and a resistor determined by Eq. 2.41. This gives,
Ga (ωge )Γ10 = ωge
2

r

LJ
1
≈
CΣ
2

r

CT 2
K ωge Ga0 A(ωge )2 ,CΣ(3.37)

where LJ is the equivalent inductance of the Josephson junction, CT is the IDT capacacitance, CΣ isthe total circuit capacitance, and Ga0 is given by Eq. 2.47. Interestingly, the typical anharmonicityof transmon qubits is on the same order as the size of typical SAW stop-bands. Therefore, if ωgelies in the acoustic stop-band, then the ωef lies below the stop-band. The relatively fast decay of
the |f i state into propagating phonon modes state reduces leakage out of the |gi − |ei manifold andcould be used for unconditional resets.
Interactions with the modes at the edge of the mirror stop-band generates a rich array ofphenomenon. When the qubit is inside the mirror stop band but near the edge, it creates localized
phononic modes inside the band gap. If the localized phonon modes between two qubits overlap,

57
they will exchange energy through virtual phonons, emulating interacting spins with a distancedependent interaction that, as the characteristic decay length depends strongly on detuning fromthe band edge, can easily be tuned in-situ [62, 113]. Such systems have been explored using trapped
atoms and transmon qubits near a photonic stop-bands. Implementing such systems with SAWs
is promising as it would avoid the difficulties of trapping atoms and strongly coupling them to a
waveguide as well as the awkwardly long structures needed to create microwave photonic stop-bands[114].

Chapter 4

Strong Multi-mode Coupling in Quantum Acoustodynamics

In this chapter, I will present results from a device demonstrating that surface acoustic wave
cavities can indeed be integrated with superconducting qubits. I describe how these two systems
can interact strongly enough for their coupling rate to exceed the dissipation rates in the system.
Moreover, the interaction rate can also exceed the cavity free spectral range, meaning that phononsand qubit excitations are exchanged faster than a single cavity round trip time. Together, theseresults show that surface acoustic waves cavities can realize an acoustic analog of circuit QED butwith access to large delays that can be used to create multi-mode cavities and frequency-sensitive
qubit-phonon interactions.

4.1

Device Design
The device consists of a transmon qubit positioned in a multi-mode SAW cavity on a GaAs

substrate, as depicted in Fig. 4.1. The qubit is formed by a SQUID loop shunted by an IDT, which
serves both as its capacitance and coupler to surface acoustic waves. The multi-mode acoustic
cavity is defined by two Bragg reflectors made of open aluminum stripes. The cavity is driven
and measured using a second IDT also positioned inside the cavity that is connected to a coplanar
waveguide which carries microwave signals to and from the device. As the acoustic structures are
periodic in nature, the number of parameters needed to describe the device is relatively small.

59

Figure 4.1: Two Bragg mirrors confined SAWs into a cavity. An IDT shunted by a Josephson
junction couples to those standing waves, forming an acoustically-coupled transmon qubit (red).
The system is driven and measured through a second IDT (yellow) coupled to a transmission line.

4.1.1

Acoustic Cavity Design

The SAW cavity design is characterized by its center frequency ωc , mirror-to-mirror length L,
transverse width W , number of periods in the measurement IDT Ne , and reflectivity per element of
the Bragg mirrors rs . The acoustic cavity center frequency is designed to lie in the standard 4-8 GHz
band typical of most cryogenic superconducting circuit measurements. In addition to compatibility
with the microwave architecture built in our laboratory, this frequency range is comfortable fortransmon qubits to operate. Additionally, a dilution refrigerators can cool these high frequency
mechanical oscillators into their quantum ground state. Fabrication considerations focus the design
goal on the lower end of this band despite an effectively higher thermal occupation; on GaAs, surface
acoustic waves travel at about 2880 m/s, so a 4 GHz wave has a length of 700 nm. At this wavelength
the fingers of a split-electrode IDTs, used to improve performance through suppressed reflections[Sec. 2.2.4.3], have widths of λ/8 ≈ 90 nm. These fine features are at the edge of what is achievablewith university-grade electron beam lithography. All of the acoustic structures, including both
IDT and mirror, are designed to have a center frequency at ωc = 2π × 4.25 GHz, corresponding to
λc = 675 nm.
The reflectivity per element of the mirrors is heavily constrained in this design. For SAW

60
systems in general, rs is kept below 3% to suppress scattering to bulk acoustic modes. Additionally,
the IDTs and the mirrors are fabricated in the same lithography step for simplicity, and so the ideal
film thickness for the IDTs also sets the thickness of the metal grooves in the mirrors. The IDTs
must have good electrical performance, which means the aluminum must be greater than 20 nm,
while thicker metal will create more scattering from the IDTs in the cavity even with a splitelectrode IDT design. Therefore, a thickness of 30 nm is chosen. With 50% metallization in the
mirrors, this gives a reflectivity per element of rs ≈ 2% and a mirror bandwidth of 55 MHz. There
are 400 metal stripes in each mirror so that transmission through the mirrors is a negligible sourceof loss across nearly all of the mirror bandwidth.
The length of the cavity is chosen so that approximately 10 modes are supported within
the mirror bandwidth. As the mirrors have a 55 MHz bandwidth, the free spectral range shouldbe ωs = 2π × c/(2Le ) ≈ 2π × 5 MHz corresponding to a cavity length of 300 µm. The effective
cavity length Le is given by the spacing between the mirrors plus twice the SAW penetration
length Le = L + 2Lp where Lp = λc /(4rs ) ≈12 µm. The chosen mirror-to-mirror length L =275 µm
accounts for this effective length increase.
The transverse dimension of the cavity would be hundreds of wavelengths wide if the only
concern were the quality factor of the cavity. Wider cavities lose less energy to diffraction, following
Eq. 2.33. However, as discussed in the next section, the cavity width plays a role in determining
the transmon qubits charging energy. Therefore, a narrow transverse dimension was chosen that
limited the acoustic internal loss rate to κd ≈ 2π×100 kHz.
Lastly, an IDT placed inside the cavity and connected to a transmission line enables driveand measurement of the acoustics with microwave signals. The number of electrode periods in this
IDT is chosen to achieve an external coupling rate that exceeds the diffraction limited loss rate.
With Ne = 100, the bare external coupling rate should be κe,0 ≈ 2π × 230 kHz with a bandwidth ofω0 /Ne ≈ 2π × 40 MHz. This IDT is placed near the center of the cavity where the spatial structureof the many SAW resonances can be well approximated by only two standing wave profiles having
either even or odd symmetry [Fig. 4.3]. The placement of the IDT relative to these two profiles

61

Figure 4.2: The acoustic cavity is designed to have an external coupling rate that exceeds internal
loss over most of the mirror bandwidth. The cavity loss rate is dominated by diffraction until
propagation through the mirrors takes over at the edge of the mirror bandwidth. Material loss
assumes a Q · f0 product of 1015 , and the mirror contribution assumes a loss-per-wavelength of
10−3 for propagation inside the mirrors, which limits the mirror reflection to 99.6% at the centerof the mirror bandwidth.

determines its external coupling rate; if the IDT was exactly centered, then it would couple only
to odd modes (as IDTs with a whole number of periods have odd symmetry about their center).
The fabrication constraints described in the next section limit the position resolution of the IDT
placement to a precision of about ∼ 400 nm ≈ λ/2. This placement error is large enough that adesign cannot intentionally couple preferentially to even or odd modes.
The measurement IDT is connected to a coplanar waveguide (CPW) leading to the drive line
and amplification chain. The waveguide has finite extent ground electrodes as they are writtenwith electron beam lithography where exposing a large area becomes prohibitively time intensive.
Ideally, the IDT would shunt the center conductor to both ground electrodes. However, to avoid
extraneous aluminum leads inside the acoustic cavity, the IDT connects the CPW center to ground62
only on one side, with half of the acoustic cavity located in a gap formed by increasing the spacing
between the center conductor and ground [Fig. 4.6(a)]. The CPW was designed to be near 50 Ω
on GaAs, with a center conductor of 60 um, a gap of 40 um of GaAs with a relative dielectric
constant of 12.5. In hindsight, the finite width of the ground electrodes (100 µm) combined withthe significant kinetic inductance fraction of the 30 nm aluminum film (near 40%) [115], means the
impedance of the line was likely between 65 Ω and 75 Ω.

4.1.2

Transmon Qubit Design

An attractive aspect of the transmon qubit is its simplicity: it is defined by only its charging
energy, EC , its Josephson energy EJ , as well as by how strongly it interacts with the system ofinterest, here confined acoustic modes [Sec. 3.3.4]. Typically, transmon qubits have EJ /EC  50,and so several hundred MHz of anharmonicity. Here, EC is designed to be 230 MHz, which sets
the overall capacitance of the transmon qubit to be 80 fF. This total capacitance then constrains
the product of the cavity width and the number of qubit IDT finger periods Nq . The number of
IDT finger periods should be much larger than one to suppress unwanted excitation of bulk waves;here, Nq = 20. Combined with the capacitance per unit length of IDTs on GaAs of 120 pF/m [81],this sets the IDT and cavity width to be 25 µm.
Using a DC SQUID allows EJ to be decreased by applying an external magnetic field. Inorder to create resonant interactions between the qubit and the acoustic cavity resonances at ωc ,the bare resonant frequency ω0 is designed to be about 20% above the acoustic band at ω0 ≈ 2π × 5
GHz. This corresponds to a bare Josephson energy of EJ = 12.5 GHz. The SQUID loop has anarea of 49 µm2 [Fig. 4.6(d)] with a mutual inductance to an off-chip superconducting coil such that
1 mA of current in the coil corresponds to approximately 1 flux quantum in the SQUID loop.
The transmon IDT is positioned in the cavity at a distance of approximately L/4 from one
mirror. Its coupling is determined by Eq. 3.35, with overall strength g0 ≈ 2π × 6.5 MHz thatchanges from one acoustic mode to another based on the modes frequency and its spatial overlap
with the IDT. As the qubit IDT has only Nq = 20 finger periods, its bandwidth of 200 MHz is63

Figure 4.3: The measurement and qubit IDTs are placed at (a) L/2 and (b) L/4 in the cavity,
respectively. At these high-symmetry points, many cavity standing waves are closely aligned. Thissymmetry degrades away from these points, shown by the spread in the standing waves modes(gray).

large compared to the mirror bandwidth, so its frequency sensitivity can safely be ignored. The
L/4 point in the cavity, though not as high symmetry as the cavity center, still simplifies the modedependence of the coupling rate [Fig. 4.3(b)]. As increasing the mode index by one means the wave
must pick up an additional π phase over the cavity length L, then the wave must pick up a phaseof π/4 by length L/4, so that the standing wave amplitudes follow,
 π
gm = g0 sin m + φq ,
4

(4.1)

where φq accounts for a small displacement from L/4 introduced by fabrication uncertainty.
There are two obvious traits of this transmon design that will significantly reduce its coherence
times compared to the state-of-the-art. First, the piezoelectric substrate required to couple thetransmon to the acoustic cavity will also allow it to couple strongly to the dense spectrum ofphononic modes supported by the chip. This same effect limits coplanar waveguide resonators
fabricated on GaAs to quality factors of several few thousand [116]. The IDT structure does
preferentially couple the transmon to the quasi-1D surface acoustic waves of interest, but phonon

64radiation to other modes will still contribute a significant amount of loss.
Second, the fine pitch of the capacitor stores a high density of electric field energy at thesurface of the dielectric. Interfaces between different materials host impurities, such as defects,
oxides, organic residues, and adsorbed water, that lead to a substantially higher loss tangent atinterfaces than for bulk dielectrics. A study of transmon qubits made from evaporated aluminumon sapphire, a common process for fabricating transmon qubits, approximated the dielectric loss
tangent at the metal-surface interface to be tan(δ) ≈ 2.6 × 10−3 by assuming the interface to be 3
nm thick with relative dielectric constant of 10 [117]. With these assumptions, a 10 micron pitch
capacitor leads to a 0.1% participation ratio and surface limited quality factor Qs ≈ 4 × 105 . For
the λc /4 ≈ 180 nm pitch capacitors used here [Fig. 4.6(c)], the participation of the interface is
6%, limiting the quality factor to Qs ≈ 6 × 103 , or a linewidth of 600 kHz at 4 GHz. This is asignificant loss channel, and is likely optimistic as it is extrapolated from transmons fabricated on asapphire substrate, which almost surely outperforms GaAs. Together, unwanted phonon radiationand significant participation of the surface constitute major limits on the potential performance oftransmon qubits coupled to GHz surface acoustic wave cavities.

4.1.3

Device Fabrication

The device is fabricated on a chip of GaAs using electron beam lithography as standard
optical lithography does not have the resolution to pattern the required fine pitch transducers or
small critical current junctions. After cleaving 6.2 mm square chips from a (001)-oriented wafer of
epitaxial GaAs, the chips were submersed in ammonium hyrdoxide for 300 s to strip the surface
oxide layer, then sonicated in standard solvents. Note that GaAs cleaves along the (011) plane,
which is parallel to the direction that supports clean Rayleigh wave modes.
After this simple cleaning procedure, three metal layers were patterned all using metal evaporation and lift-off processes. The active parts of the device are composed of aluminum, but multiple
layers are required because the Josephson junctions and acoustic structures have incompatible requirements: the junctions require a resist bridge to be formed between two lines several hundred65
nm apart, as described shortly, while the IDTs, also consisting of several lines separated by about
100 nm, would fail if the resist formed bridges. This incompatibility then requires two separate
lithography steps which must be aligned to better than a micron. This alignment is achieved using
a third layer containing alignment markers. Gold is an ideal marker material because it is a common microfabrication material, and it reflects incident high energy electrons in the electron beam
writer much more strongly than GaAs due to its larger atomic number: gold nuclei have 79 protons
compared to gallium’s 31 or arsenic’s 33.
The gold marker layer is deposited first using a stack of 400 nm MMA under 200 nm PMMA.
Both resists are sensitive to incident and electrons, and will develop away in a mixture of MIBK/IPA
if the electron beam deposits a sufficient amount of energy. Crucially, MMA is more sensitive than
PMMA. This elevated sensitivity combined with the spread and back-scatter of the electron beam
enlarges the developed area in the MMA compared to the PMMA, creating an undercut [Fig. 4.4(ab)]. The mark pattern is exposed in an SEM reconfigured as an ebeam writer, controlled using
NPGS software. After exposure, the resist is developed and 100 nm of gold is deposited, with 10
nm of titanium underneath to promote adhesion. The unwanted titanium and gold film sitting on
unexposed resist are lifted off in acetone.
Next, the aluminum layer containing everything except the Josephson junctions is depositedusing a single layer of 200 nm PMMA. Each acoustic structure was aligned to a separate marker
pattern using the SEM, then exposed, developed in MIBK/IPA, and cleaned using a weak oxygen
plasma, before 30 nm of aluminum is deposited. This aluminum was oxidized in the chamber topromote a well-controlled and reproducible oxide layer, before lift-off in warm (40◦ C) acetone forat least 30 minutes. Lift-off of fine features without a bilayer is difficult. With the 30 kV beam
used, an undercut still forms even in just PMMA. For good lift-off, it was crucial to have thegap between the ends of the fingers and the bus bar to match the separation of the fingers. We
believe the metal between fingers lifts off like the peeling of a sticker, and having a uniform width
helped significantly. Sonication was often required to remove tenacious but unwanted metal. The
back-scattering of electrons effects an area much larger than the radius of the electron beam, and66

Figure 4.4: (a) The bilayer of resist creates an undercut and bridges as the bottom layer (MAA)
is more sensitive to than the top layer (PMMA). Horizontal (b) and vertical (c) linecuts show theundercut and bridging used for lift-off generally and Dolan-bridge Josephson junctions in particular.

the applied dose must be adjusted for the density of features in close proximity. This adjustment
is crucial for successful lithography of the fine IDT electrodes and was implemented by segmenting
the design into several sections with a dose compensation determined iteratively.
Lastly, the junction layer is deposited using the same resist stack, exposure, and development
as the marker layer. After development, a brief, weak oxygen plasma is performed to clean resist
residue from the substrate. The evaporation for the junctions involves two angle depositions at±20◦ separated by a several minute oxidation period, where a small amount of O2 (∼ 1 mbar)
is introduced to the deposition chamber. The junction geometry is shown in Fig. 4.4, where a
discontinuity of length G creates a overlap length d = 2h tanh(θ) − 2G where θ is the evaporation
angle and h is the height of the resist. The length and pressure of the oxidation set the thickness of
the insulating barrier, and combine with the junction geometry to determine the junction criticalcurrent.
The junction critical current can be inferred using room temperature resistance measurements. The Ambegaokar-Baratoff relation provides a direct relation between the metals superconducting gap ∆, the critical current IC , and the normal resistance from electrons tunning across thejunction RN , [118]
Ic =

π∆
,
2eRN

(4.2)

67

Figure 4.5: The Josephson junctions are connected capacitively to the IDT using a large (50 µm2 )overlap between the two layers. A single electrode is pulled through the bottom bus bar, whichallows the junctions to shunt the IDT while being placed outside the acoustic beam.

for kB T  ∆. For this device, EJ = 12.5 GHz corresponds to IC = 32 nA and RN = 4 kΩ. Whilethe critical current of a junction can be estimated coarsely from geometry and oxidization parameters, in practice, an array of test junctions are fabricated and measured, varying the oxidation timeand measuring iteratively to hone in on the design value of the normal resistance.
In this design, there is no galvanic connection between the qubit IDT and the Josephson
junctions. The aluminum in the qubit IDT is exposed to atmosphere after deposition and during
the junction processing, and therefore an insulating aluminum oxide layer forms on its surface.
Instead of a galvanic connection, a large overlap is designed between the two layers, which createsstrong capacitive connection [Fig. 4.5]; the overlap area of 50 µm2 and an oxide thickness of 5 nmwith a relative dielectric constant of r = 10 gives a capacitance of 1 pF. At 4 GHz, this capacitanceis 3 Ω, sufficiently small compared to the transmon impedance of 400 Ω to be considered a goodshort. However, this oxide layer has been exposed to atmosphere and is therefore likely to containdefects, water vapor, resist residue, and other contamination that will adversely impact the qubit
coherence.

68

Figure 4.6: (a) False-color SEM image of the device, showing aluminum on GaAs with gold alignment marks. An acoustic cavity is formed by two Bragg reflectors (green) and contains both atransmon qubit (red and blue) as well as a measurement IDT (yellow). The measurement IDT
connects the center conductor of a CPW to ground. Insets show higher magnification images of the(b) mirror elements, (c) qubit IDT, and (d) Josephson junctions, the last of which was deposited
after the full image was acquired.

At the magnification required to write the small features, the SEM has a writing field of
150 µm per side. As this window is much smaller than the device, the SEM stage has to move thesample to expose the full pattern. This stage movement introduces significant positioning errors on
the order of ±1 µm. Each structure (measurement IDT, each mirror, etc...) is written in a separate
writing field and referenced to the gold marker layer. As the marker layer was also written in thesame SEM and required the same stage movement, there is the same relative uncertainty between
each writing window. However, as both the acoustic and junction layers are referenced to the samegold markers, the position error between these two layers is reduced to well below 100 nm.

4.2

Device Measurement
The device is characterized using microwave reflection from the measurement IDT at thebase temperature of a dilution refrigerator. At these temperatures (20 mK), both the transmon

69

Figure 4.7: The acoustic cavity is measured in reflection at the base of a dilution refrigerator. A
Josephson parametric amplifier (JPA) either provides 25 dB of narrow-band gain or acts as an
open, losslessly reflecting incident signals. Off-chip magnetic coils provide uniform magnetic flux
to tune the resonant frequencies of the qubit and the JPA.

qubit and the acoustic modes of the cavity are passively cooled to their quantum ground state. The
reflected microwave signals carry information about the acoustic resonances and their interaction
with the transmon qubit. This section will describe the various microwave measurements that
demonstrate that the device is well described by multi-mode Jaynes-Cummings physics with a“giant atom,” far from the dipole approximation.

4.2.1

Acoustic Response

The acoustic spectrum is measured in reflection using a directional coupler [Fig. 4.7] with thequbit tuned off resonance. Over the 55 MHz mirror bandwidth, there are 11 prominent resonances
spaced by approximately ωs ≈ 2π × 4.8 MHz [Fig. 4.8]. The magnitude of reflection changes
significantly on resonance, indicating that the acoustic modes are near critically coupled. Threemodes have full 2π wraps in phase across their resonances and therefore are slightly over-coupled.

70
We are unable to determine the absolute index of each mode, and instead begin enumeration of themodes at the first visible resonance, which corresponds to approximately the 880th longitudinalmode of the resonator. There is a clear alternating pattern in the external coupling strength of theresonant modes, with the odd modes being more strongly coupled than the even. This is because,in the center of the cavity, the standing wave moves by π/2 phase with each increased mode, andthe external coupling rate is proportional to the intensity of the wave at the IDT [Fig. 4.3(a)].
The resonances become more under-coupled near the top and bottom of the acoustic band.
There are two effects that contribute: the measurement IDT is less sensitive to these frequencies
and the mirrors are slightly less reflective. The large number of fingers in the IDT gives it abandwidth of 40 MHz which reduces the external coupling rate to the outer modes. At the centerof the IDT bandwidth, this corresponds to a coupling rate of about 100 kHz, but this decreases
quickly away from its center frequency. The acoustic modes have internal linewidths that arecomparable to this at the center of the mirror bandwidth, which matches the diffraction limited
loss calculated earlier. However, near the edge of the mirror bandwidth, phonons penetrate further
into the mirrors, resulting in increased loss both from propagation all the way through the mirrors
as well as increased time spent trapped in the lossy mirror region. The combination of decreasing
external coupling and increasing internal loss, shown in Fig. 4.8, significantly decreases the visibilityof modes near the edge of the acoustic bandwidth.
In addition to the 11 prominent resonances, the spectrum shows at least six additional modeson the high-frequency shoulders of every other prominent resonance. These modes have about threetimes the internal loss and about one eighth the external coupling rate, dramatically reducing theirvisibility. These spurious modes are most likely transverse modes that have two nodes along thetransverse direction of the cavity. Although there is little transverse confinement in the cavities,
there are slight velocity differences along the transverse direction associated with the mirrors andthe IDT bus bars which nonetheless provide sufficient confinement to produce visible resonances[83]. The frequencies of these tranvserse modes ωm,t can be estimated by considering the cavity as

71

Figure 4.8: Reflection from the acoustic cavity with the qubit detuned shows 11 prominent resonances in a 50 MHz span. The main resonances are nearly critically coupled, as shown by the
extracted coupling rates, and are accompanied by transverse modes on the high-frequency shoulder.

a rectangular waveguide with the same velocity in all directions,st2
m2
ωm,t ≈ 2π × vs
+
,
4L2e
4W 2

(4.3)

where m is the longitudinal mode number and t is the transverse mode number. For the cavity
with of 25 µm, t = 1 corresponds to tranverse modes about 0.4 MHz above and t = 2 modes about
1.7 MHz above. By symmetry, the odd order transverse modes do not couple strongly to the evensymmetry IDTs. The prediction for t = 2 approximately matches the measured transverse mode
spacing of ∼ 1.5 MHz.

4.2.2

Resonant Interaction

After characterizing the uncoupled acoustic resonances, we investigate their resonant interaction with the transmon qubit by tuning it through the acoustic bandwidth with an applied magnetic
flux while monitoring reflection with a low power probe tone. Two sets of avoided crossings are

72visible as the magnetic field is swept over one flux quantum, as shown in Fig. 4.9. Several featuresof the system are immediately apparent from the measured interaction. First, the clarity and sizeof the avoided crossings indicates that the qubit and cavity exchange energy coherently and at a
rate comparable to the spacing between resonant modes. Second, the qubit interacts with different
modes with different strengths: indeed, there are three acoustic modes that barely couple to thequbit at all, as shown by their small avoided crossings. Lastly, although the uncoupled modes all
interact relatively equally with the measurement IDT, the external coupling rates of the hybridizedmodes vary dramatically; half of the modes become dark, no longer coupling to the measurement
IDT, while the other half remain bright.
The prominent features of the measured avoided crossings can be accurately described by asimple model despite their apparent complexity. As described earlier in Eq. 4.1, the qubit-cavity
coupling rates can be described by a simple 2-parameter model accounting for an overall couplingstrength and an offset accounting for an offset from L/4 of the cavity. The transverse modes
must also be included, as they clearly participate in the avoided crossings. The frequencies of thetransverse modes can be determined from the acoustic spectrum with the qubit detuned, while
the qubit-transverse resonance interaction strength is extrapolated from how the longitudinal andtransverse modes interact with the measurement IDT: from the measured external coupling rate,
the measurement IDT has a 2.8 times stronger spatial overlap with the transverse modes, and so
the same factor is assumed for the qubit-transverse mode interaction strength.
Using these simplifications and measurements of the acoustic resonant frequencies with thequbit detuned, the interaction Hamiltonian between the qubit and the 17 acoustic modes of interest
can be described by just three parameters: g0 , φq , and a fine offset in the flux-dependence of thequbit frequency. In the low-power limit, where on average there is below a single excitation in the

73

Figure 4.9: Applying a magnetic flux tunes the qubit across the acoustic modes. Clear avoided
crossings are observed at ±0.26Φ/Φ0 with a coupling strength on the same scale as the mode
spacing.

system, the Hamiltonian has the form,



ω
g
1
 1




ω
g

2
2
.

H/~ = 
.. 
..

. .




g1 g2 · · · ωq

(4.4)

The eigenmodes of the system at each ωq can be determined by diagonalizing the Hamiltonian

74

Figure 4.10: The measured acoustic response with the qubit in resonance shows complicated crossings with various strengths as well as bright and dark modes. A model describing the reflection
across the crossing with 3 free parameters closely matches the measurements.

above. The best fit between the measured resonant frequencies and the eigenvalues of the model
gives g0 = 2π × 6.5 MHz and φq = −0.1 rad. There are two modes that achieve nearly the
maximum coupling strength, with g4,8 ≈ 2π ×6.48, while three modes have near vanishing coupling,with g2,6,10 ≈ 2π × 650 kHz. The remaining modes have intermediate coupling strengths of either
g1,5,9 ≈ 2π × 4.1 MHz or g3,7,11 ≈ 2π × 5.0 MHz.
The microwave reflection from the cavity can be reconstructed by modelling the internal andexternal coupling rates of the cavity in addition to the eigenvalues. The model presented here
replicates the measured acoustic response well for all acoustic modes and all flux points, as shownin Fig. 4.10. The model requires not just the eigenvalues, but also the eigenmodes of the system.

75
0 i are superpositons of the uncoupled modes |ψ i,
These eigenmodes |ψm
n
0
|ψm
i=

X

cn,m |ψn i,

(4.5)

n

where cn,m are found by diagonalizing the Hamiltonian, and the sum of their squares is one, i.e.
P 2n cn,m = 1. The internal loss rates of the hybridized modes are a sum of the loss rates of eachparticipating mode weighted by its weight in the eigenmode,
κ0i,m =

18X

c2n,m κi,n ,

(4.6)

n=1

where κi,18 = γ is the qubit linewidth. As the new loss rate is just a weighted average of theuncoupled modes and as most modes have similar internal linewidths, the overall loss rates of thecoupled modes closely approximates the uncoupled ones.
The external coupling rate must be considered differently from these internal channels because
there can be coherent interference between external coupling from different modes. Intuitively, theeigenmodes of the system correspond to some hybridization of the acoustic cavity modes plus some
qubit participation. Spatially, these eigenmodes are superpositons of cavity standing waves, whichhave distinct spatial profiles that can have significantly different overlap with the measurement
IDT. Mathematically, these external coupling rates are,
κ0e,m = κ0

17
X

2

cn,m sin(nπ/2 + φi )A(ωn ) ,

(4.7)

n=1

where κ0 is scale of the external coupling rate of the cavity, A(ω) is the frequency dependence of the
IDT [Eq. 2.44], and the sine term describes the amplitude of the SAW at the IDT of mode n, which
is near Le /2 but with offset phase φi ≈ π/4. Near resonance the hybridized modes are mostly
composed of two neighboring acoustic modes, each with approximately equal external coupling
magnitude, but with a sign that flips every other mode, thus creating an alternating bright-darkmode pattern in the measured cavity response.
The defining characteristic of the strong, multi-mode coupling regime is that the eigenmodesof the system are composed of many uncoupled modes. If |gm |  ωs , then the avoided crossings
will be pairwise, with the qubit hybridizing with only one mode at a time. Here, even though

76

Figure 4.11: Focusing on one eigenmode shows that, with the qubit on resonance, there is asignificant contributions from a large number of the uncoupled modes. The eigenmode shown here
is composed primarily of three neighboring acoustic modes; the qubit participates only weakly,
below 7%, across the whole crossing. The predicted frequency closely matches the measurement(dashed).

gm depends strongly on the mode index, the avoided crossings are far from pairwise between thequbit and individual cavity modes. As shown in Fig. 4.11, many modes participate in the avoided
crossing. Interestingly, the qubit participation remains small over the flux range near the center of
the acoustic band because the qubit participates in many eigenmodes at a given flux. Essentially,
the qubit mediates strong coupling between neighboring acoustic modes. This effect also dilutes
the qubit loss over many modes and implies that clear avoided crossings do not directly imply thesystem has exceeded the strong coupling criteria gm > {κm , γ}. Indeed, the full model recreating themeasured spectra is somewhat insensitive to the input qubit linewidth. Verifying the device meetsthe strong coupling criteria requires independent verification that the qubit linewidth is below themeasured coupling strengths.

77
4.2.3

Qubit Measurements in the Dispersive Regime

The system can be tuned into the dispersive regime despite g0 > ωs because the acoustic
reflectors only support resonances over a narrow bandwidth. In the dispersive regime, the acoustic
resonances experience a shift dependent on the state of the qubit, and the qubit state can be
determined by monitoring an acoustic resonance [Sec. 3.3.2]. The dispersive shift is largest, andreadout is easiest, for the most strongly coupled resonances, and here we use m = 8 to read out
the qubit state.

4.2.3.1

Qubit Spectroscopy

Spectroscopy of the qubit is performed by sweeping a drive tone near its resonant frequency
while a second tone is reflected off the acoustic cavity near ω8 . The acoustic resonance is at ω8 − χ
if the qubit is in the ground state and ω8 + χ if it is in the excited state. Changes in the qubit
population from the qubit drive result in changes in reflection from the shift of the cavity frequency.
The JPA is used when measuring qubit spectroscopy as its bandwidth is more than sufficient forthese slow, single-frequency measurements and the improved signal-to-noise makes spectroscopy
significantly faster.
Using this acoustic readout, the qubit frequency can be tracked as a function of flux up to its
maximum frequency near 5 GHz. The measured qubit frequency closely follows the expected flux
dependence of the SQUID inductance [Eq. 3.12] over the measured range shown in Fig. 4.12. The
clean spectrum shown required significant compensation of the drive power as a function frequency,
as the drive more strongly polarized the qubit at small detuning from the cavity. Driving withtoo much power not only broadens the qubit, but extra transitions appeared, complicating thespectrum. At very high drive powers, several two-level system defects were visible. One such twolevel system near 4.8 GHz showed clear Rabi and Ramsey oscillations with coherence times of ordermicroseconds. The nature of these defects is yet undetermined, but we suspect a defect in the dirtyoxide layer sandwiched between the IDT and junction aluminum films.

78

Figure 4.12: The qubit spectrum is measured by monitoring the reflection from acoustic mode m = 8
while sweeping a qubit drive tone. The measured qubit spectrum closely follows the predicted flux
dependence (dashed).

4.2.3.2

Transmon Anharmonicity

Acoustic readout can also be used to measure the anharmonicity of the transmon. Withthe qubit tuned to 4.077 GHZ, just below the acoustic band, applying a strong qubit drive 20
dB above the low-power limit shows the power-broadened ωge as well as the two-photon |gi − |f itransition at ωgf /2 = ωge − α/2 = 3.952, as shown in Fig. 4.13. This gives an anharmonicity of
α = 2π × 250 MHz. To see the |ei − |f i transition, there must be population in the |ei state.
Retaking the spectroscopy with a second pump fixed at ωge indeed shows a dip at ωef /2π = 3.828
GHz, which gives a second measurement α/2π = 249 MHz. This anharmonicity corresponds to aqubit capacitance of 77 fF, within 10% of the design value of 85 fF.
The range of qubit spectroscopy is limited by the resonant interaction of the ωef transition
with the acoustic modes as well as prohibitively large phonon emission rates just outside the mirror
bandwidths. For a transmon qubit, the dispersive regime must also consider the interaction of
the harmonic modes with the ωef transition. These interactions make qubit readout breakdown
between 4.4 GHz < ωq /2π < 4.55 GHz. Additionally, the qubit has a prohibitively large linewidth

79

Figure 4.13: A weak spectroscopy tone shows the qubit transition (blue), while increasing the power
20 dB shows a broadened qubit transition as well as the two-photon |gi − |f i transition (red) ata lower frequency. This gives α = 2π × 250 MHz. This anharmonicity is verified by observing the|ei − |f i transition directly by applying an additional pump tone fixed at ωge (green).

above 10 MHz in the roughly 100 MHz span on either side of the acoustic band where its IDT
efficiently converts the qubit excitation into propagating modes, as discussed more in the following
sub-section.

4.2.3.3

Qubit Linewidth

Qubit spectroscopy can be used to characterize the qubit decay rate γ and the dispersive
shift of any acoustic mode χm as a function of qubit frequency. The decay rate is measured
spectroscopically by using a variable strength qubit drive to measure its linewidth. We observe thequbit linewidths depends on drive power even at the lowest powers that the transition is visible.
To account for this power dependence, we infer the undriven qubit linewidth by fitting the power
dependent linewidth and extrapolating to the zero power limit. This extrapolated linewidth does
not differ dramatically from the lowest measured one.
The low-power qubit linewidth shows clear structure as a function of its frequency [Fig. 4.14].
The qubit interaction with the continuum of phonon modes is mediated by the inderdigitatedtransducer. The IDT has many coupling points over its many-wavelength extent that create a strong

80

Figure 4.14: The qubit linewidth extrapolated to the low-power limit shows clear oscillations as
a function of frequency. This structure closely matches the predicted qubit spontaneous emission
rate to phonons, with a frequency-insensitive offset γ0 = 2π × 1.1 MHz.

frequency sensitivity in the qubit-phonon interaction strength which translates into a frequencydependent qubit linewidth [Eq. 2.41]. The measured linewidths are well fit by the prediction for thequbit emission rate to acoustic modes based on the IDT geometry, which gives a simple sin2 X/X 2
form where X = Nq π(ω − ωc )/ωc dependence, plus a constant offset that accounts for other decay
channels. The overall scale of the emission rate is given by Eq. 3.37, and here is equal to Γ0 = 2π×32
MHz. The best fit for the constant offset gives γ0 = 2π × 1.1 MHz.
The qubit linewidth at the special frequencies were phonon emission from the IDT is prohibited by its geometry has special significance. First, this linewidth can be attributed to thefrequency-independent qubit decay channels that are present when the qubit is tuned through themirror bandwidth where phonon emission into the main SAW channel is suppressed. As this qubit
internal loss rate of γ0 ≈ 2π × 1.1 MHz is well below the qubit-cavity coupling strength, the device does indeed operate in the strong coupling regime. Second, this frequency of minimal loss
demonstrates a unique feature of quantum acoustics with surface acoustic waves: a qubit can be81completely decoupled from the phonons at its resonant frequency to which it could decay while
maintaining a strong coupling to detuned modes. This feature is leveraged successfully in an acoustic device in the following chapter (Ch. 5), but also suggests an alternative to standard Purcell
filters in planar qubit design [119]. Instead of changing the density of states of the transmission
line at the qubit resonant frequency, the qubit could be designed to decouple from modes at its
resonant frequency by engineering interference between multiple points of coupling.

4.2.3.4

Acoustic Dispersive Shift

The dispersive shift of the qubit is measured by driving a given acoustic mode, here m = 8,
with a coherent state of varying size while monitoring the qubit spectrum. The qubit linewidth
exceeds the size of the dispersive shift, which means the shift of individual phonons is unresolved,
and in the low phonon number limit the qubit experiences a smooth shift of size 2χ8 n8 where
n8 is the average phonon number in mode 8. For qubit frequencies below the acoustic band,measurements indeed show that the qubit experiences a linear frequency shift with increasing
average phonon number.
A calibrated measurement of χ8 requires precise knowledge of average phonon number corresponding to each drive strength. However, the net attenuation in the lines is not known within ±2
dB, which translates to a large uncertainty in the scaling of the power axis. The dispersive shift
calculated from Eq. 3.27 matches the measured values when the qubit is below the cavity modes
within this uncertainty. The conversion between drive power and average phonon number is thenassumed to produce the calculated dispersive shifts.
While the dispersive shifts as a function of frequency closely follow the prediction below theacoustic band, at higher frequencies there is significant deviation [Fig. 4.15(a)]. The measured qubit
frequency shift as a function of drive power is not linear for many frequencies in this range, andrequires a description beyond the dispersive approximation. The model used here is a generalized
Jaynes-Cummings model that takes into account the lowest four levels of the transmon as well as
phonon occupancies in mode 8 up to 50 excitations. Like the standard Jaynes-Cummings model,

82

Figure 4.15: (a) The extracted dispersive shift for mode m = 8 shows good agreement with thepredicted transmon behavior when ωge < ω8 , but deviates significantly above the cavity. A model
incorporating multi-phonon transitions and higher levels of the transmon captures the behavior
well, but predicts a (b) non-linear behavior with phonon-drive that is indeed measured in theexperiment.

the Hamiltonian in matrix form is block diagonal. In manifold with at least four excitations ne ,each block of the Hamiltonian has the form,
√
2 n − 3g8
E4 + (n − 4)ω8

 √

p


 2 n − 3g8

E3 + (n − 3)ω8
3(n − 2)g8




p
p
 (4.8)Hne /~ = 
3(n − 2)g8 E2 + (n − 2)ω8
(2(n − 1)g8




p


√

2(n − 1)g8 E1 + (n − 1)ω8
ng8 




√
ng8
E0 + nω8where Ei = iωq + i(i − 1)α/2 are the levels of the transmon approximated as a duffing oscillator[Eq. 3.14] indexed by i, and the transmon-cavity coupling matrix elements are approximated to83
grow as√i. The full matrix is diagonalized with a cutoff of 50 phonons for an array of qubit

frequencies.
In the dispersive regime, the eigenvalues of the system E(i, n) can be identified with a given
transmon state i and phonon number n. The qubit transition with n phonons in the cavity is givenby,ωge (n) = E(1, n) − E(0, n),(4.9)

and so dispersive shift per phonon with n is,2χ(n) = ωge (n + 1) − ωq (n).

(4.10)The dispersive shift diverges, and the dispersive approximation becomes invalid, when pairs of
uncoupled energy levels become degenerate. In a standard three mode analysis, the two importantdegeneracies are when the |gi−|ei transition or |ei−|f i transitions are resonant with the cavity, i.e.
E(1,n)=E(0,n+1) or E(2,n)=E(1,n+1). These interactions are both first order as they involve theexchange of one excitation. The presence of non-linear behavior in χ suggests that second orderinteractions also matter. The primary degenaracies to consider are when E(2, n − 2) = E(0, n)
and when E(3, n − 2) = E(1, n). These second order crossings occur when ωge = ω8 + α/2 andωge = ω8 + 3α/2. The former occurs in the straddling regime and is not measured, while the lattercondition is precisely where the measured shifts deviate from the simple prediction.
The model can produce predictions for the dispersive shift given n phonons in the acoustic
mode. The prediction with n = 10 phonons in the cavity closely follows the measured dispersive
shift across the entire flux region measured [Fig. 4.15(a)]. However, as the shift per phonon is notconstant, this misrepresents the qubit response; in many frequency regions, the qubit response to
an increased drive is quadratic, not linear. The model qualitatively matches the measured qubit
frequency in response to a phononic drive [Fig. 4.15(b)]

84

4.3

Conclusion
The device described in this chapter successfully demonstrated that surface acoustic waves

could be strongly coupled to multi-mode acoustic resonators. Moreover, these quantum acoustic
systems are shown to be well described by the familiar Jaynes-Cummings model, but with accessto large delays that miniaturize multi-mode resonators and structure the qubit-phonon couplingin the frequency domain. Combined, these features give surface acoustic wave systems a uniqueability to realize strong, multi-mode coupling, but with the access to specific frequencies where thequbit is dispersive with all resonant modes without significant spontaneous emission to unconfinedmodes.
The device also made clear several drawbacks of coupling superconducting qubits to surface
acoustic waves. The qubit lifetimes are severely limited by dielectric loss from fine pitch of thecapacitor and acoustic loss from radiation into uncontrolled phonon modes. At the same time, thequbit readout speed is limited by the small cavity external coupling rate and the small dispersive
shift. The fast qubit decay compared to the readout time prohibited time-domain measurementsof the qubit state. Additionally, the capacative coupling between the IDT and junctions likely
contributed significant losses from interactions with defects hosted in the dirty oxide layer.

Chapter 5

Resolving Phonon Number States in a Multi-mode SAW Cavity

This chapter discusses the motivation, design, and measurement of a multi-mode acoustic
cavity coupled to a transmon qubit that enters the strong dispersive regime where single phonons
shift the qubit by more than its linewidth.

5.1

Multi-mode Strong Dispersive Regime
The strong dispersive regime provides many strategies for quantum control over linear res-onant modes. For superconducting qubits coupled to electromagnetic resonators, reaching thisregime [46] precipitated a large number of results demonstrating non-demolition photon measurement [105], creation of arbitrary photonic states [48], and interaction between photon modes withmutual coupling to a superconducting qubit [120, 25]. Entering the strong dispersive regime in mechanical systems would enable quantum control of phonons by deploying these similar techniques.
In particular, high-fidelity control over the high density of resonant modes present in SAW
resonators would prove a powerful resource for processing quantum information. Harmonic oscillators are attractive candidates for storing quantum information because they typically have one
dominant source of error (bit flips) [52], which simplifies correction schemes, and their large Hilbert
space provides the freedom to store information in a range of encodings [53]. Electromagnetic resonators at microwave frequencies perform well but are bulky [51], and fitting large numbers of themin a dilution refrigerator is a daunting task. Mechanical resonators occupy a small fraction of thevolume [121], and as demonstrated in the previous chapter, a dozen modes can be easily coupled to86a single superconducting qubit. Entering the strong dispersive regime with this significant numberof modes would provide an alternative and hardware-efficient route to many qubits, where information is stored in the acoustic modes. The dispersive regime is particularly relevant in acoustic
systems because the superconducting qubits typically have poor lifetimes compared to the acoustic
resonances, an asymmetry that likely will grow as acoustic resonators can be exceptionally high-Q[19, 122, 123, 124]. Dispersive operation limits the loss inherited by the acoustic modes from thesuperconducting qubit [26].
There is, however, a tension inherent in trying to reach the strong dispersive regime in a
multi-mode cavity. Consider a system composed of a cavity with resonant modes ωm spaced by ωsthat couple to a qubit with strength gm . To be dispersive, the detuning between the qubit and any
acoustic mode must exceed their coupling strength, such that gm  ∆m for all m. If the couplingstrength is constant between all modes, such that gm = g0 , then since the maximum detuning isωs /2, then g0  ωs /2 [Fig. 5.1(a)]. This limited coupling then constrains the dispersive shift, and asystem must compromise either mode density or coupling strength to achieve dispersive operation.
A finite-bandwidth reflector side-steps this constraint by supporting resonant modes only ina specific frequency range. The qubit can then be dispersive with all resonant modes by operating
at frequencies outside the mirror bandwidth as done in the previous chapter [Ch. 4]. However,once outside the mirror bandwidth, the qubit is exposed to the continuum of propagating modes,
and its lifetime may be limited by spontaneous emission. The device in the previous chapter also
demonstrated that the IDT allows the qubit to decouple from specific phonon modes, shielding itfrom acoustic loss at particular frequencies. However, in that device, the frequencies where emission
was prohibited were too far detuned from the resonant modes for the acoustic dispersive shifts tobe appreciable compared to the qubit intrinsic linewidth.
Another related strategy to enter the strong dispersive regime with a densely multi-mode
resonator is to engineer the coupling strength to the resonant modes to vary. If, for example, gm
is proportional to detuning from a particular frequency with zero coupling, ωz , then large couplingstrengths can be achieved for modes more detuned without sacrificing dispersive operation or mode

87

Figure 5.1: (a) A multi-mode cavity coupled to a two-level system with uniform strength is only
dispersive if g  ωs /2. (b) If the coupling strength is proportional to detuning from the qubit with
sufficiently shallow slow slope (dashed), then there is no such constraint.

density [Fig. 5.1(b)]. This proportionality creates a region of decreased coupling strength amid anarray of strongly coupled modes; the qubit can be tuned into the low-coupling region for dispersive
operation, while maintaining a high-density of resonant modes and a strong coupling rate to therest of the resonances.
A qubit coupled to SAWs through an IDT is capable of realizing this frequency-structured
interaction. The Fourier transform of the IDT geometry sets its frequency response [Eq. 2.41], andso a desired frequency response can be engineered by computing its inverse Fourier transform andshaping the IDT accordingly. And, as the speed of sound is slow, megahertz frequency resolution
can be realized with millimeter geometries.
In a continuous wave context, an IDT can be designed to realize nearly arbitrary frequency
responses. However, when the IDT forms the capacitor of a resonant circuit, either classical or
quantum, the finest frequency resolution is limited by decay time of the resonance. The sharp
features in the continuous wave IDT response depend on interference between spatially separated
electrodes. A resonant circuit is not sensitive to these sharp features if it fully decays in a time scale

88
that is short compared to the travel time between these distant electrodes. This limitation includes
decay into the SAW mode of interest as well as internal losses, so that if one segment of the qubit
IDT radiates a phonon on a time scale faster than the transit time across the whole IDT, even a
perfect qubit with no unintentional losses will not see the continuous wave IDT response. In thisregime, the qubit environment becomes non-Markovian, characterized by information back-flow
resulting in non-exponential decays [125, 109].

5.2

Split IDT Concept
The desired frequency dependent qubit-phonon interaction is realized by splitting the IDT

in half. As in the previous device, a transmon qubit is formed by an IDT shunted by a SQUID
loop and positioned inside a multi-mode SAW cavity. However, in this design, the IDT is split
into two halves each of length D and separated by S [Fig. 5.2(a)]. The frequency domain responseA(ω) is determined by the Fourier transform of the split IDT about its symmetry point [Eq. 2.44].
In close analogy to double slit diffraction, the Fourier transform of these two separated regions is
the product of two terms: a slow sinc envelope centered on ωc with period 2π × v/D and a fastsinusoidal modulation with period 2π × v/S:
A(ω) = sinc[(ω − ωc )D/2v] sin[ωτ /2].

(5.1)

where τ = S/v. Essentially, a long phonon travel time τ between the two IDT halves creates
sharp fringes in the qubit-phonon interaction rate as a function of frequency. In the absence of
mirrors, the qubit spontaneous emission rate is proportional to |A(ω)|2 [Fig. 5.2(b)]. Inside themirror band, the coupling between the qubit and a given acoustic mode is proportional to A(ω)
and the amplitude of the acoustic mode at the center of the split IDT [Fig. 5.2(c)]. If the IDT isperfectly centered,gm = g0 A(ωm ) ≈ g0 sin(ωm τ )(5.2)

where g0 is the scale of the acoustic-qubit coupling strength if the IDT and the SAW standing wave
were perfectly aligned, and the slow sinc is approximated as unity.

89

Figure 5.2: (a) A qubit with a intentionally designed frequency sensitivity is realized by splitting
the IDT in half and embedding it in a SAW resonator. (b) The spontaneous emission rate of
the qubit depends strongly on frequency with direct analogy to a double slit diffraction pattern infrequency space. (c) The strength of the resonant coupling inherits the same frequency dependence.
The slope of the coupling near a zero crossing (red line) sets a floor on how dispersive the interactions
are with nearby acoustic modes.

Dispersive operation with all acoustic modes can be realized regardless of the acoustic density
or maximal coupling strength by careful design of A(ω). If the qubit is tuned to a frequency ωzsuch that A(ωz ) = 0, then it will decouple from phonons at its resonant frequency. Moreover, acavity mode with detuning ∆z from the qubit will couple with strength gz that is bounded aboveby this detuning multiplied by the slope of the coupling strength near ωz , i.e.,|gz | < g0 A0 (ωz )∆z .

(5.3)

The magnitude of the slope at the crossing g0 A0 (ωz ) sets an upper bound on gz /∆z and therefore sets

90
a floor for how dispersive the qubit-cavity interactions can be as shown in [Fig. 5.2(c)]. If the slope
is sufficiently shallow, then all resonant modes will be dispersive, regardless of the mode density
or their exact resonance frequencies. The robustness of the design to the exact frequencies of theacoustic cavity is especially important because these resonant frequencies of acoustic cavities cannot
be determined in design; as in the earlier device [Sec. 4.1.3], there is uncertainty in the positioningof the acoustic structures that exceeds a wavelength, which means that while the mirror bandwidthand free spectral range are well determined, the actual resonant frequencies vary significantly device
to device.

5.2.1

Split IDT Design

The acoustic cavity is centered at the same frequency ωc = 2π × 4.25 GHz as the previous
device, but its area is decreased to increase the coupling to the qubit. Its width of 16 µm limits theacoustic linewidth to approximately 250 kHz, and the length of 125 µm gives a free spectral rangeof ωs ≈ 2π × 10 MHz after accounting for the 16 µm of mirror penetration per side. The transmon
has Nq = 16 finger periods total in its IDT, this results in a bare acoustic coupling strength of 15
MHz.

5.2.1.1

Readout Cavity

To readout and control the qubit, it is strongly coupled to a 3D electromagnetic resonator byattaching antenna paddles to either side of the IDT. The electromagnetic resonator is a metal box
made from a machined slot in two pieces of copper [Fig. 5.3]. Its transverse dimensions of 35 mm
by 35.5 mm are designed to support a fundamental TE01 resonant frequency near ωr = 2π × 6 GHz
while pushing higher modes above 10 GHz. The seam is located at the center of the cavity, whereby symmetry no currents flow, thereby reducing seam losses. The cavity is 5 mm deep, which,with paddles that are each 725 µm long and 150 µm wide, creates a bare coupling strength with the
3D resonator of 180 MHz. The copper cavity also creates a clean and well controlled microwave
environment for the qubit, particularly compared to the complicated environments typical in sample

91

Figure 5.3: The qubit is strongly coupled to a copper waveguide cavity with frequency near 6 GHz.
The qubit state is read out by monitoring the resonant frequency of the cavity.

packaging for devices with planar readout cavities. The qubit rests in a slight recess in one halfcavity and is held in place by indium dots [Fig. 5.3(b)].
Two microwave transmission lines are coupled to the cavity by inserting their center conductors some length into cavity. One pin is farther into the cavity, giving it an external coupling rateof near 2 MHz that dominates the cavity linewidth. This line runs to the microwave measurement
chain. The other pin is recessed somewhat from the cavity mode in a waveguide below cutoff,
reducing its external coupling rate to near 100 kHz. This line is used to drive the cavity and thequbit.

5.2.1.2

Transmon Qubit Design

The transmon is made up of two capacitors; one for coupling to the acoustic cavity andthe other to couple to the readout cavity. This two capacitor design reduces the coupling toboth the 3D resonator as well as the IDT. The total capacitance is approximately 120 fF, giving
EC /h ≈ 160 MHz. The IDT makes up about 40% of this capacitance, and so its coupling is reduced
to g0 ≈ 2π × 5.5 MHz, while the readout cavity coupling is reduced to ge = 2π × 110 MHz. The

92

Figure 5.4:
The two separate depositions of aluminum used to define the acoustic features(red/blue) and the Josephson junctions (yellow) are connected using a third layer of aluminum(green). Immediately before depositing the bandage layer, the sample is cleaned with an argon
plasma to remove the oxide layer (black).

large capacitor paddles also reduce the electrical participation ratio of the GaAs surface. This
would improve the qubit linewidth if this lossy surface was the dominant loss source in the device,
but would do little to improve unwanted radiation into unconfined acoustic modes.
The delay between the two IDT halves is chosen so that, with the qubit tuned to a zerocrossing in the qubit-phonon interaction, all the acoustic modes will be dispersive. This slope isgiven by g0 A0 (ωz ) ≈ g0 τ /2. We chose τ = 9 ns so that g0 τ /2 = 0.14  1. This guarantees thatmodes will be detuned by at least 7.2 times their coupling strength when the qubit is tuned to ωz .
This delay also conveniently results in a period between coupling minima of 110 MHz that matches
the expected mirror bandwidth of near 80 MHz, guaranteeing there will be a maximum in A(ω)
within the mirror band.

5.2.2

Fabrication

The fabrication of this device differs from the previous device only in how the IDT layer isconnected to the junctions. Here, these two layers are galvanically connected by a third layer ofaluminum [Fig. 5.4]. This third layer, called the bandage layer, is deposited on top of both of theother layers. Electrical continuity is achieved by preceding the bandage layer deposition with an

93

Figure 5.5: A false color SEM image showing the aluminum device on the GaAs substrate. Acoustic
reflectors (green) confined phonons to form a multi-mode SAW resonator. The qubit is formed bya pair of Josephson junctions (yellow) shunted by both an IDT (purple) as well as large antenna
paddles (red/blue) that couple it to the readout cavity. The mirrors in the imaged device have half
the separation as the mirrors in the measured device.

argon plasma clean that scrubs away the oxide layer on the underlying aluminum. The bandage
layer is deposited shortly after the argon plasma clean without breaking vacuum. Such an argon
plasma etch would ruin the resist bridges of the Josephson junctions, and so it cannot be done in
the same step as the junctions.
A false color SEM image shows the device in Fig. 5.5. The acoustic cavity is formed by two
Bragg mirrors situated between the paddles that couple the qubit to the readout cavity. An IDT
formed by two 8-period halves split by 27.2 µm is placed at the center of the cavity. The innermost
fingers of each IDT are pulled through the cavity so that the junctions can shunt the IDT easily.
Only one pair of these IDTs is connected to the lower paddle to avoid creating a superconducting
loop. All of the aluminum except the junctions and the bandage layer are deposited in one step.

94

Figure 5.6: The readout cavity is driven through a weakly coupled port, while a strongly coupledport leads to the amplification chain. The JPA has a separate pump port used for direct flux
modulation at 2ωr .

5.3

Measurement
Measurements are performed by measuring transmission through the electromagnetic readout

resonator at the base of a dilution refrigerator. As shown in the wiring diagram [Fig. 5.6], a JPA
downstream from the cavity either provides narrow-band gain when pumped or else acts as anopen and completely reflects incident signals. Magnetic flux applied from a coil attached to thereadout cavity tunes the qubit frequency down from a maximum frequency above the readout cavity.
Sweeping the applied flux reveals an avoided crossing between the transmon qubit and the readout
cavity, which gives both the coupling rate ge = 2π × 114 MHz as well as a good approximation of
the qubit frequency as a function of flux.

95
5.3.1

SAW cavity Avoided Crossings

Away from the avoided crossing with the readout cavity, the dispersive qubit-cavity interaction can be used to perform qubit spectroscopy. The transmission of the readout tone changes asa drive tone polarizes the qubit. Here, measurements of the qubit-acoustic interactions are madeby observing the qubit, instead of the acoustics as in the previous chapter.
As the qubit is tuned across the acoustic band, its spectrum shows pronounced avoided
crossings with a clear frequency-dependent coupling strength [Fig. 5.7(a)]. The eigenmodes of thesystem are found by diagonalizing a multi-mode interaction Hamiltonian as in the previous chapter[Eq. 4.4]. Here, however, each mode has an independent coupling strength, which can be extracted
as the coupling strengths are less than ωs . Fitting this model to the avoided crossings extracts
coupling strengths that vary significantly from one mode to the next [Fig. 5.7(b)]. Three main
effects explain the observed variation. First, neighboring cavity modes spaced by ωs ≈ 2π × 10
MHz have alternating coupling strength because the IDT is approximately symmetric about thecavity center, coupling the qubit strongly to even acoustic modes and weakly to odd ones. Second,
the interference between the IDT halves modulates the coupling proportional to A(ω) as intended;strong coupling with g0 = 2π×5.1 MHz is achieved at the center of the mirror band, but the coupling
vanishes by ±50 MHz in either direction. This is precisely the frequency-dependent structure thatthe split IDT was designed to achieve. Finally, resonant exchange between the qubit and the cavity
modes at the edge of the mirror bandwidth is spectrally unresolved as the coupling rate is reducedby A(ω) and the intrinsic loss of the cavity increases. Transverse modes are also present at slightly
higher frequency than the even longitudinal modes, and couple with a strength about five times
smaller than their associated purely longitudinal modes.

5.3.2

Interaction with Unconfined Phonons

The avoided crossing measurements, while showing the desired mode-dependent interactions,only sample A(ω) discretely and in a small frequency range. The structuring of the qubit-phonon

96

Figure 5.7: (a) As the qubit is tuned across the acoustic modes, pronounced avoided crossings
appear in its spectrum. (b) The extracted coupling strengths show that the qubit couples preferentially to even modes (green) at the center of the mirror bandwidth while decoupling odd modes(blue) as well as from phonons at either end of the mirror bandwidth. Transverse modes (yellow)
were present at the high-frequency shoulder of the even longitudinal modes. A prediction for A(ω)
extracted from T1 measurements fits the data well (dashed).

interaction can be determined with greater clarity by monitoring the qubit spontaneous emission
rate and resonant frequency as it is tuned outside the mirror bandwidth. When exposed to thecontinuum of unconfined phonon modes, the qubit emits a phonon at a rate proportional to |A(f )|2 .
The qubit spectra is recorded as it is tuned over a 1-GHz span, and the flux dependence is subtracted
to emphasize the features arising from acoustic interactions [Fig. 5.8]. At frequencies detuned fromthe mirror bandwidth and its associated avoided crossings, the qubit linewidth oscillates with aperiod of 110 MHz that is consistent with a delay time of 9 ns and an amplitude that decays as
the qubit tunes out of the bandwidth of the IDT halves. Additionally, the qubit frequency itself
deviates from a simple model of its flux dependence, with residuals from a best fit flux dependence

97

Figure 5.8: The qubit spectrum is monitored as it is tuned over a 1-GHz span, and the best-fit flux
dependence of its resonant frequency is subtracted to highlight the acoustic features. Outside thebandwidth of the mirrors, where the qubit hybridization with the acoustic modes creates avoided
crossings, the qubit linewidth and frequency both show clear oscillations generated by its frequencysensitive interaction with propagating phonon modes. A crossing with a mode of unknown origin
occurs at 4.41 GHz.

showing oscillations at 110 MHz with an envelope matching the linewidth variation. This deviationfrom a simple flux dependence arises from the reactive part of the IDT admittance, and is analogous
to the Lamb shift from atomic physics [126, 59]. Both of these effects, investigated in more detailbelow, are well described by modelling the qubit emission of phonons from the IDT as a frequencydependent conductance, which must be accompanied by a frequency dependent reactance from
Kramers-Kronig relations. The spectrum also shows a large avoided crossing with a defect mode
at 4.41 GHz of unknown nature.

5.3.2.1

Phonon emission rate

The qubit energy decay rate is determined by measuring the excited state lifetime (T1 ) in
the time domain [Fig. 5.9]. In contrast to spectroscopic measurements of linewidth, the measured
excited state lifetime is insensitive to dephasing effects as well as broadening from the applied drive.
With the qubit far detuned from the acoustic cavity modes, we observe oscillations of Γ1 = T1−1 infrequency with large amplitude; the loss increases by a factor of 25 above its minimal value within
a 55 MHz span. The loss follows a model that combines a prediction for the phonon emission rate

98
from the IDT and a constant internal quality factor Qi ,ωqωq − ωc 2
Γ0
Γ1 (ωq ) =+[1 − cos(ωq τ )],sinc πNq
Qi
2
ωc

(5.4)

where Γ0 is the qubits maximal loss rate to phonons. The best fit is for Qi = 1.2×104 , Γ0 = 2π ×11
MHz, ωc = 2π × 4.24 GHz, and τ = 9.04 ns. The measured value of Γ0 is close to the predictedvalue using room temperature GaAs parameters of 12.5 MHz.
The minima in the qubit linewidth are caused by destructive interference between a single
phonon emitted from both halves of the IDT, an effect which has close parallels to an atom interfering with its mirror image [127, 128]. The uniformity of the minima cross the IDT strongly
suggests that any residual loss from imperfect interference between the IDT halves is below 100
kHz; the emission from each IDT changes by more than order of magnitude over the measured
span, which would impact the depths of the nulls if the interference between the two halves wereimperfect. This uniformity confirms that A(ω) is real and goes through zero, implying the IDT
halves are indeed symmetric.
The qubit constitutes a so-called “giant-atom” where the intra-IDT delay time approaches
the phonon-emission limited qubit lifetime. As discussed earlier, deep in this regime the qubit
decays fully before a phonon can travel between the IDT halves, leading to non-Markovian physics[125, 109]. The transition to this regime occurs when the product Γ0 τ /2 approaches unity; for thisdevice, Γ0 τ /2 = 0.3. Any potential evidence of non-Markovian physics is obscured by the presenceof the mirrors which structure the phononic environment near the center of the IDT bandwidth,
as well as the shortness of the time scale (9 ns) of the non-exponential features compared to themeasurement time. A small fraction of measured decays did display non-exponential features, butwith time scales long compared to τ . The frequencies associated with these decays were excluded
from the fit [Fig. 5.9(b)].

99

Figure 5.9: (a) The qubit excited state lifetime is measured as a function of flux over a 1-GHz span
excluding inside the mirror bandwidth. The observed decays become significantly longer at specificfrequencies where phonon emission is prohibited by the coherent cancellation between a single
phonon emitted emitted from each half of the IDT (b) The extracted decay times closely match thea model for the T1 time based on a fixed Qi and phonon emission from the IDT. Non-exponential
decays occur occasionally and are excluded from the fit (gray). (c) The qubit relaxation rate
highlights the agreement at short lifetimes and shows the clear analogy to double-slit diffraction.
At the largest Γ1 , two effects contribute to increased variability: the mirrors are slightly reflective
and structure the phonon environment, and the exceedingly fast decays are difficult to measure
precisely.

100
5.3.2.2

Phononic Lamb Shift

The precise measurements of the IDT parameters from the qubit loss rate can be used to
calculate the effective phononic Lamb shift of the qubit. This shift is related to the reactive partof the IDT admittance [Eq. 2.42], and is given by,
δ(ωq ) =ωq − ωc 2
Γ0sin(ωq τ /2).
sinc πNq
4
ωc

(5.5)

While the magnitude of this shift is small (±3 MHz), the oscillations have mean zero over alarge span and are on a much finer frequency scale than the flux dependence. Including junction
anisotropy as in Eq. 3.12, the qubit flux dependence follows,2 #1/4I−I
0
a2 + (1 − a2 ) cos π,Ic"
ωq (I) = ω0

(5.6)

where ω0 is the zero-field qubit frequency, Ic is the coil current required to thread a half flux
quantum through the 50 µm2 qubit loop, I0 is the current offset required to offset ambient fields,
and a is the normalized difference between the junction critical currents. Fitting the measured qubit
frequency [Fig. 5.10(a)], gives ω0 = 2π ×5.718 GHz, Ic = 1.168 mA, I0 =79.2 µA, and a = 0.14. Thisfit cannot capture the fast fluctuations from the phononic Lamb shift; indeed, the residuals fromthe flux dependence match the expectations for the Lamb shift with no free parameters, excluding
regions near avoided crossings [Fig. 5.10(b)]The measurements of the qubit interacting with propagating SAWs provides also provides an
independent inference of the interaction strength between the qubit and the cavity resonances. The
best-fit model for A(ω) from the qubit loss rate can be extended to frequencies inside the mirror
bandwidth. This prediction is shown in Fig. 5.7(b) and shows good quantitative agreement withthe measured coupling strengths.

5.3.3

Number Splitting Measurements

Having characterized the resonant interaction between the qubit and both propagating andcavity phonons, we turn to measuring the dispersive shift of the qubit from cavity phonons. To

101

Figure 5.10: (a) The measured qubit frequency away from avoided crossings (blue) is fit well aflux-dependent model including junction asymmetry (white dashed) excluding points near avoided
crossings (gray). (b) The residuals from the flux-dependent fit show clear oscillations. A prediction
with no free parameters based on the measured phonon emission rate closely matches the residuals,
indicating that these features are a phononic Lamb shift.

achieve the goal of entering the strong dispersive regime, we must confirm both that the qubit
is detuned from all cavity modes by many times its coupling strength, i.e. gm  |∆m |, andthat the dispersive shift χm exceeds both the qubit and acoustic loss rates. Tuning the qubit
to ωz = 2π × 4.318 realizes dispersive operation: the least-dispersive interaction is with modem = 7, with ∆7 /g7 = 8.5  1. In this dispersive regime, the Hamiltonian simplifies to a sum ofdispersive shifts of size χm for each mode, as in Eq. 3.29, where χm must include the third level ofthe transmon, as in Eq. 3.27. The anharmonicity for this device is α = −2π × 190 MHz measured
at ωq = 2π × 4.318 GHz, although this anharmonicity does depend on frequency because of thephononic Lamb shift.
The qubit would be dispersive with all acoustic modes if it was tuned to either the top or

102

Figure 5.11: With the qubit tuned to ωz , it does not hybridize strongly with any acoustic modeand has a linewidth of γ = 2π × 550 kHz. By driving orders of magnitude harder, the small residual
qubit hybridization with the acoustic modes allows them to be driven and read-out. The acoustic
modes have linewitdhs κ1 = 2π × 200 kHz, κ3 = 2π × 250 kHz, κ7 = 2π × 275 kHz,the bottom of the mirror bandwidth. However, the dispersive shift is much larger with the qubit
tuned above the acoustic resonances. This asymmetry is because, with the qubit at the bottomof the mirror bandwidth, the contribution to the dispersive shift from the |ei − |f i transition of
the transmon counteracts the shift from the |gi − |ei transition, while at the top of the mirror
band these two contributions add constructively to create large and positive dispersive shifts. Thisconfiguration is called the straddling regime [92] and the boost it provides to dispersive shifts wasrecently used to enter the strong dispersive regime between a superconducting qubit and a magnon[129].
The measurement of the dispersive shift of a given mode requires that mode to be populatedwith phonons. To displace a target mode, we drive the qubit at a frequency far detuned from its
own transition frequency but resonant with the cavity mode [67]. The slight residual hybridization
turns this qubit drive into a weak cavity drive. Qubit spectroscopy in Fig. 5.11 shows the qubit
transition at ωq = 2π × 4.318 GHz, and, with orders of magnitude higher drive power, the three
acoustic resonances with the strongest qubit coupling at lower frequencies.
The measured qubit linewidth of γ = 2π × 550 kHz is consistent with measurements of energy
decay, intrinsic dephasing, and broadening from the qubit drive tone. The qubit coherence times
are measured in the time domain to be T1 = 415 ns and T2∗ = 705 ns [Fig. 5.12]. The T1 decay

103

Figure 5.12: The qubit T1 = 415 ns and T2 = 715 ns times are measured with the qubit tuned toωz .

contributes the most to the qubit linewidth, at T1−1 = 2π × 360 kHz. As the T2∗ time is nearly twice
T1 , the intrinsic dephasing time Tφ is small, adding only Tφ−1 = 2π × 30 kHz to the qubit linewidth.
Together with the effective Rabi rate from the drive tone (100 kHz) and the finite pulse duration(50 kHz), the expected qubit linewidth is 540 kHz. Additionally, the measured acoustic linewidthsof κm ≈ 2π × 250 kHz agree with the expectations of diffraction limited Q [Eq. 2.33] from theirfinite width of 16 µm.
We measure the dispersive shift of the three strongly coupled modes by varying the populationin these modes while monitoring the qubit spectrum. A 3 µs drive pulse at ωm creates a coherent
state in mode m with nm average phonons. The resulting spectrum, measured with a spectroscopy
pulse concurrent with the acoustic drive, consists of a sum of Lorentzians that each correspond toa phonon number state in the cavity [Fig. 5.13]. These Lorentzians are spaced by 2χm and broaden
with higher phonon number in proportion to κm ; the loss of a phonon effectively measures the qubit
state and therefore causes increased qubit dephasing. As the drive power at each of the three modes

104

Figure 5.13: The qubit spectrum is measured while three different acoustic modes are driven. For
modes m = 5, 7, multiple resolved peaks appear, each corresponding to a different phonon number
state. The fits assume a coherent state distribution and agree well with the measured spectra. The
contribution from each phonon number state to the highest n trace are plotted separately at thebottom of each panel.

is increased, the measured qubit spectrum broadens and shifts upwards in frequency, consistent withthe positive dispersive shifts in the straddling regime [92]. Crucially, several resolved peaks appear
in the qubit spectrum for modes m = 5 and m = 7. Each peak corresponds to a different Fockstate in the phonon number distribution.
To model the measurement, we assume the cavity occupation is Poissonian distributed andfit the average phonon number in each trace. The measured spectra are fit to a sum of Lorentzians,
Pe (ω, n) = C0 + C1nX
max

Pn (n)S(ω, n, n),

(5.7)

n=0

where C0 is an overall offset, C1 an overall amplitude, ω is the spectroscopy frequency, nmax = 6 is105
a cutoff phonon number, and Pn is the weight of Lorentzian S(ω, n, n) with n phonons. These twoterms are given by,Pn (n) = e−nS(ω, n, n) =

nn
n!

and,

γ + κ(n + n).
(ω − (ωq − 2χn))2 + (γ + κ(n + n))2 /4

(5.8)
(5.9)

For the acoustic cavity parameters n, n, χ, and κ, a subscript m is assumed. The measurements
are constrained by independent measurements of the qubit and cavity linewidths as well as thequbits bare resonant frequency. The dispersive shift for a given mode is well constrained from themeasurements without any fitting, particularly for modes m = 5 and m = 7, and is measured
manually and input into the model. The strong acoustic drive creates a small background excitedstate population causing C0 to grow with n. This effect can be measured from the off-resonant
qubit spectrum and subtracted. The drive also pushes the qubit frequency upwards at a rate ofabout 150 kHz per phonon, which can be determined well from the low-occupancy traces.
After inputting the independently calculated parameters and accounting for the unwanted
effects of the strong acoustic drive on the qubit, the spectra are fit by varying n. The resulting fitswith 2χ3,5,7 = 2π × 500, 1050, 890 kHz and κ3,5,7 = 2π × 200, 250, 275 kHz agree closely with themeasurements for all three modes measured. The measured spectra are not well-resolved enough toextract the phonon weights to high precision. However, the good quality of the fits to over n > 1,
combined with linear scaling of n with applied drive power for all three modes [Fig. 5.14] strongly
suggests that the acoustic modes are in a coherent state created by the classical drive. In theabsence of the acoustic drive, the qubit transition is indistinguishable from the ground state n = 0.
This means the total thermal occupancy of the two strongly coupled modes is below nth < 0.02,
corresponding to a temperature below 50 mk.
An unstable avoided crossing with sub-MHz coupling rate appeared intermittently between
4.312 GHz and 4.322 GHz. The frequency and coupling rate of the defect both fluctuated on aseveral-hour timescale. We reject data when the defect was present within a 15 MHz span aroundthe qubit by interleaving independent diagnostics with the Stark-driven spectra and removing

106

Figure 5.14: The extracted average number of phonons from the number splitting measurements
increases linearly with applied power.

defect-present data in post-processing.
The device measured here demonstrates both halves of the wave-particle duality of phonons
in a single device. The granularity of the dispersive shift shows that phonons come in discreteenergy packets. This particle-like nature means phonons can be counted. At the same time, asingle phonon’s wave-like nature means that a phonon emitted from the IDT can interfere withitself, as demonstrated by the high-quality fringes in the qubit decay rate.

5.3.4

Qubit Readout

Three different types of qubit readout, all based on its dispersive interaction with the readout
cavity, were performed depending on the measurement requirements. Qubit spectroscopy as afunction of flux was performed using continuous wave homodyne dispersive readout backed by aflux-pumped JPA [Fig. 5.7, 5.8, 5.10]. For the qubit spectroscopy taken at ωz , a pulsed version ofthis readout was used [Fig. 5.11, 5.12, 5.13]. The long qubit lifetime at this frequency results inhigh-contrast measurement, and the temporal separation of the qubit drive and cavity readout
tones means the qubit is not dephased by the readout dispersive shift. Lastly, for broadband T1
measurements, heterodyne bright state readout was used [Fig. 5.9] [130]. This type of readout iswell-suited for measuring fast qubit decays; if the qubit is excited, the cavity effectively locks into
the bright state for a time exceeding the qubit lifetime. This latching means pulses much longer

107

Figure 5.15: Four generator supply the tones used in the experiment. The JPA pump also supplies
the LO for the experiment using a frequency divider. The relative phase between the readout tone
and the LO is stabilized with a tone that bypasses the fridge and is digitized using the same DAQ.

than the qubit lifetime still give information about the initial qubit state. Still, very fast decay times
are challenging to measure and lead to increased uncertainty for the shortest lifetimes measured.
In both of the homodyne measurements, two different microwave generators supplied the
local oscillator and the readout tone. The generator supplying the local oscillator tone produced
a tone at 2ωr , which was split into a JPA pump and a frequency divider that then fed into the
local oscillator of two mixers. While the two generators share a 10-MHz clock, the relative phasebetween them was prone to drift by about a degree a minute. The phase difference between these
two generators was stabilized using a copy of the readout tone that bypasses the fridge [Fig. 5.15].
The resulting DC voltage between the LO and this bypassed tone was measured and stabilizedusing feedback every 15 s.
In both flux-dependent measurements, the qubit readout and drive parameters had to becalibrated over the flux region of interest in order to cleanly capture the qubit behavior. There were108
three main effects to account for: the frequency-dependent filtering of the qubit drive by the readout
cavity, the dependence of the readout cavity frequency on the qubit frequency, and a frequencysensitive qubit lifetime. The filtering by the readout cavity was assumed to follow a Lorentzian,making correction simple. The qubit lifetime was modelled from preliminary measurements, andthe readout cavity frequency was measured as function qubit frequency.

Chapter 6

Towards Multi-phonon Entanglement: Low-frequency, Focusing Acoustics

The previous two chapters demonstrated the unique ability of surface acoustic waves to leverage long on-chip delays to build multi-mode quantum systems with highly engineerable interactions.
They also showed the challenges of coupling superconducting qubits to acoustic resonators using
piezoelectricity, particularly the significant electrical losses from uncontrolled phonon generationand large participation of the surfaces, along with significant acoustic losses from phonon diffraction
out of unstable, plane-parallel resonators. This chapter describes preliminary work into mitigating
the dominant loss mechanisms by using lower frequency and focusing acoustic cavities as well as a
flip-chip geometry to suppress unwanted phonon radiation. First, I introduce a potential experimentto create multi-phonon entanglement that draws upon the strengths of SAW systems demonstrated
in this thesis and should be within reach once the described improvements are implemented.

6.1

Measurement Based Multi-mode Entanglement
Generating entanglement between the many modes of a SAW cavity would be a significant

step towards using these systems for processing quantum information [131]. Here, I will describe an
intuitive and straightforward way to generate entanglement between multiple cavity modes using
number-resolved phonon measurements. The strategy is based on entangling two qubits whichhave no direct interaction; instead, a joint, projective measurement is designed that conditionallyprojects an initial separable state into an entangled one [132, 133, 134].
As demonstrated in the previous chapter, a single superconducting qubit coupled to a multi-

110

Figure 6.1: In a multi-mode system where χm = χ0 for multiple modes, exciting the qubit with anumber-selective π-pulse at ωq −2χ0 means there is one phonon total between all of the modes. Crucially, no which-mode information is measurable, so that an initial separable state where multiple
phonon modes are displaced is then projected into an entangled state.

mode SAW resonator can act as a phonon counter for several of the acoustic modes simultaneously.
In the strong dispersive regime, this single phonon shift 2χm exceeds the qubit linewidth γ, and theunderlying phonon distribution can be measured using qubit spectroscopy. Similarly, if the qubit
is driven with a π-pulse at frequency ωq − 2nm χm , with spectral width less than χm , then it will
be excited if the cavity has 2nm phonons in mode m. If the qubit is coupled to only a single mode,
then being excited at ω − 2nm χm indicates that there must be nm excitations in that mode.
However, in a multi-mode cavity, the qubit can have the same frequency for multiple different
arrangements of phonons in the cavity. This degeneracy is particularly clear if multiple modes havethe same dispersive shift, i.e. χm = χ0 . Then, if there are nt phonons total, the qubit will havethe same frequency ωq − 2nt χ0 regardless of how the phonons are distributed in the cavity modes[Fig. 6.1]. If the qubit is excited with a π-pulse at ωq − 2χ0 , then there must be one phonon present,
but there is no which-mode information available. This mode-agnostic measurement is in many
ways equivalent to measuring a photon but erasing the which-path information.

111
If the system is prepared in a mixed state where multiple modes have some probability ofpossessing one phonon, then this mode-agnostic number measurement projects the system into an
entangled state. Consider a qubit coupled to two modes where χ1 = χ2 that is prepared in theseparable state,|ψi i = |gi|ααi,(6.1)

where |αi is a coherent state with mean phonon number |α|2 = n. If the qubit is driven with aπ-pulse at ωq − 2χn , then it is measured to be excited if there is one phonon total present. This2

one-phonon case occurs with probability p = 2e−2|α| |α|2 which peaks at |α|2 = 1/2 with p = 1/e.
Conditioned on the qubit being successfully excited, the final state is an entangled state,√|ψf i = |ei

2(|10i + |01i).
2

(6.2)

Extending this to more modes and creating a W -state [135] requires only driving more cavity modes
but the single-phonon case occurs with decreasing probability.
In contrast to other platforms, realizing a system where many cavity modes produce thesame dispersive shift is natural for a qubit coupled to SAW modes. The ideal configuration to
generate similar and large dispersive shifts is to have a dense region of cavity modes with couplingstrength comparable to the mode spacing. This way, when the qubit is detuned from the forestof resonances by many times the coupling strength, all modes have approximately equal detuning.
The dispersive shift can be adjusted to account for the small difference in detuning easily in a SAW
system by modifying the IDT geometry to couple slightly more strongly to modes at the far end of
the mirror bandwidth [Fig. 6.2]. The IDT would also have to be engineered to decouple the qubit
from propagating phonons at its frequency of operation.
In contrast, if the resonant modes are spread out by many times their coupling strength,
then a large variation in detuning would require a large variation in coupling strength, which is not
only difficult to engineer, but results in reduced dispersive shifts. Thus, generating multi-phonon
entanglement through mode-agnostic measurement brings together the two main virtues of SAW
systems: a high density of acoustic modes with g ∼ ωs and excellent frequency domain control of

112

Figure 6.2: Generating entanglement between multiple modes through number-resolved but modeagnostic dispersive measurement requires engineering χm ≈ χ0 and no radiation to unconfinedphonon modes. This configuration can be achieved with with a split-IDT qubit; the frequencydependent coupling shuts off emission to propagating modes at ωz while simultaneously varying
the coupling to the dense forest of modes such that the more detuned resonances have appropriately
stronger coupling.

the qubit-phonon interaction. Furthermore, it builds naturally on the device in the previous chapter[Ch. 5], which showed that entering the strong dispersive regime was possible, and even had twomodes with dispersive shifts that were only 10% different. However, to generate and measure thismulti-phonon entanglement with high fidelity requires significant improvements in both the qubit
and acoustic coherence times, with a goal of κm , γ < χ0 /10. Such improvements are the focus of
the remainder of this chapter.

6.2

Possible Design Improvements
Two main changes drive the design of the next generation of devices. First, we aim to reduce

the unwanted generation of unconfined phonons by removing the components other than the IDTfrom a piezoelectric material. This can be done either with a thin film of piezoelectric material
deposited only where needed, or by using a flip-chip geometry with the IDT on a piezoelectricchip and the rest of the circuitry on a high-performance dielectric such as sapphire. Second,
we move to lower frequencies (ωc ≈ 2π × 500 MHz). At these frequencies, electrical losses are
reduced because the increased pitch of the capacitor results in a smaller participation ratio of lossy

113
interfaces [Sec. 4.1.2], and acoustic losses are reduced because scattering from defects and material
losses should both be suppressed.
The move to lower frequencies has several other significant consequences. Fabrication of
acoustic features can be done optically, as λc ≈ 6 µm. Transmon qubits do not operate comfortably
at these frequencies; their anharmonicity becomes prohibitively small, and readout becomes difficult
as their dispersive shift to a readout cavity in the standard 4-8 GHz band also becomes small.
Fluxonium qubits [136, 137], described shortly [Sec. 6.2.4], have |gi − |ei transitions comfortably
below 1 GHz, but have a large anharmonicity and rich energy level structure that permits highfrequency readout using higher high levels [138].
Fluxonium qubits, however, require much larger EC and therefore much smaller shunting
capacitances than transmon qubits. This reduction in capacitance requires an IDT with narrower
width, which combined with the longer acoustic wavelength, results in stringent constraints on theaperture size of the IDT compared to its wavelength that would cause untenable diffraction losses ina plane-parallel cavity. Therefore, focusing cavities must be used that create a beam waist narrow
enough to couple strongly to a narrow aperture IDT but that do not lose energy to diffraction. The
following subsections will describe each of these improvements individually.

6.2.1

Aluminum Nitride on Sapphire

Aluminum nitride on sapphire is a promising material combination for quantum acoustics
with surface acoustic waves. Sapphire is an ideal substrate for superconducting circuits, and aluminum nitride is a wide bandgap semiconductor and a strongly piezoelectric material that canbe grown on sapphire using several techniques [139], with commercial applications focusing on
ultraviolet LEDs and high-power electronics. Moreover, AlN can be etched away where it is notwanted [Fig. 6.3]. This attractive combination of properties has led to the successful use of AlN
on sapphire to integrate transmon qubits with bulk acoustic wave resonators [67] as well as tocreate high-frequency surface acoustic wave cavities [140, 141], particularly for high temperature
operation [142]. Additionally, the speed of sound is faster than on typical piezoelectric materials by

114

Figure 6.3: A thin film of AlN on sapphire under the IDT strongly couples the qubit and theSAW cavity but this film can be etched away under the junctions and any other circuitry to reduce
unwanted phonon radiation.

approximately a factor of 2, which increases the wavelength at a given frequency, further reducing
electrical and acoustic losses.
However, SAW devices fabricated and measured using C-axis AlN on C-axis sapphire fromtwo different suppliers (500 nm film from Kyma and 350 nm from AIXaTECH) showed a weakpiezoelectric response. From the external coupling rate of one port resonators, measured bothwarm and cold, the extracted K 2 was approximately 10−4 . This value is about 7 times weaker
than GaAs and 30 times weaker than the expected value of 3 × 10−3 . The successful use of AlN
to couple transmons to bulk acoustic waves [66] used material from Kyma, while the results using
AlN for SAWs grew their own material [140, 141]. AIXaTECH claims to produce high quality,
single-crystalline AlN. High quality films are necessary to reduce dielectric losses, which appear
to be limiting the coherence times of state of the art devices [67]. I will note that AIXaTECH
initially sent defective wafers and was slow to ship material and communicate with in general.
We suspect that their AlN was in fact polycrystalline and formed multiple domains that, while
uniformly oriented in the normal direction, had poorly defined in plane rotations. This type ofmaterial would couple strongly to bulk waves and out-of-plane electric fields, but not to in-plane
fields and SAWs. Pursuing this strategy was halted until high performance material for SAW
applications could be obtained.

115
6.2.2

Indium Bonded Flip Chip

Another strategy for reducing loss from uncontrolled phonon radiation is to use two separate
substrates, one piezoelectric for the SAW structures and the other a high-performance dielectric
for all other elements, in a flip-chip configuration. Inductive coupling between a g-mon style qubiton Si and a SAW acoustic resonator on LiBNO3 was successfully used in an array of quantum
acoustics experiments [70, 71, 72]. Such a strategy requires a strong piezoelectric material as theoverall coupling strengths are greatly reduced by the small participation of the IDT capacitor in
the qubit mode.
Reliable and low-loss galvanic connection between the two substrates that allows muchstronger coupling can be achieved using In bumps [143, 144]. Using this galvanic connection,
the junctions and readout resonator could be fabricated on a high-performance substrate such as
sapphire and directly shunted by the IDT fabricated on piezoelectric material [Fig. 6.4]. Indium
bumps on films of indium have been used to make extremely low loss microwave resonators [144],and aluminum films have been connected at DC using indium bumps with a TiN diffusion barrier,
resulting in critical currents of tens of mA [143].
This process for connecting layers at DC using In bumps with a TiN diffusion barrier is currently under development in the Boulder micro-fabrication facility at NIST. Here, we are interested
in using sapphire as the high-quality dielectric material and quartz for the piezoelectric top-chip.
These two materials have similar coefficients of thermal expansion (a 5 mm quartz chip should
contract 1.5 µm relative to a sapphire one [145]), and test resonators comprising a CPW cavity
whose center conductor transfers several hundred times between a quartz and sapphire substrateusing these indium bumps were cooled down and shown to have a Q of several hundred. Translating
this into a two-bump loss rate for a resonator at 500 MHz with the impedance of the fluxonium
qubit described shortly gives a loss rate of 5 kHz. This work is early in its development, and further
refinements, particularly to the pre-bonding treatment of the indium, will likely significantly reduce
this loss.

116

Figure 6.4: A flip chip design removes the junctions and readout resonator from the piezoelectric
substrate using indium bumps to galvanically connect metal layers on separate quartz and sapphirechips. As In and Al make an undesirable intermetallic, TiN is used as a diffusion layer.

6.2.3

Lower Frequency: Improved Coherence

Increasing the wavelength of the acoustics should improve the quality factors of both theacoustic and electrical resonators. The electrical side of the argument is simple: a longer wavelength
reduces the participation of the thin lossy dielectric layer at the surface of the substrate. A factorof 10 increase in the wavelength should decrease the surface participation ratio by a factor of 10.
Assuming the dielectric parameters discussed earlier [Sec.4.1.2] leads to a Q limit of 6 × 104 or alinewidth of 8 kHz.
The improvement in the acoustic domain arises from reduced scattering from imperfections
and reduced visco-elastic losses. If a model with constant Qf product is assumed to describe viscoelastic material loss, then lowering the frequency by a factor of 10 improves the material loss by
the same amount. Loss from scattering from surface roughness and other defects with length scale
below an acoustic wavelength should scale even more favorably, following the λ−4 dependence of117
Rayleigh scattering [146]. These losses, although not believed to be limiting in current devices,
will start to contribute once stable cavities are implemented that eliminate diffraction losses, as
discussed in an upcoming section [Sec. 6.3].
Moving to lower frequency means that thermal occupation will be substantial. A transition
frequency of 500 MHz corresponds to a temperature of 25 mK, near the achievable temperature
in dilution refrigerators, meaning the thermal occupation will be significant. However, numbersensitive side-bands has been shown to be an effective cooling method for thermally occupied
mechanical oscillators [74], and fluxonium, described next, can be efficiently cooled by using sidebands with the lossy and high-frequency readout resonator [138]6.2.4

Fluxonium Qubit

The fluxonium qubit is well suited to couple to low frequency surface acoustic wave cavities.
The fluxonium is formed by shunting a single, small Josephson junction by both a small capacitanceand an array of large junctions that act as a high-impedance linear inductor [Fig. 6.5(a)]. The linear
inductor connects the superconducting islands on either side of the small junction and thereby makes
the circuit insensitive to low frequency charge noise. This circuit achieves the same charge noise
insensitivity as the transmon, but does so without sacrificing anharmonicity.
Excitations of the fluxonium circuit behave like a flux particle that is trapped in a potential
determined by both the linear inductor and the small junction. The Hamiltonian is given by,1
H = 4Ec n̂2 + EL φ̂2 − EJ cos φ̂ − φe
2

(6.3)

where EC is the charging energy, EL is the linear inductor energy, EJ is the Josephson energy, φ̂ isthe phase drop across the inductor, φe is the external flux biasing the loop, 2e×n̂ is the displacement
charge at the capacitor where [φ, n] = i. Fluxonium is defined by the region where EL  EJ and
1 . EJ /EC . 10, which implies that the linear inductance must have an inductance per unit
length of around 104 µ0 . Such a high-impedance inductor, with minimal parasitic capacitance toground, can also be realized using kinetic inductance [147]. Near φe = 0, the low lying excitations

118

Figure 6.5: (a) The fluxonium qubit consists of a Josephson junction shorted by a large linear
inductance and some small capacitance. The large array must have a high inductance per unit
length and is typically made of an array of larger Josephson junctions. An external flux threaded
through the junction-inductor loop (bold) biases the small junction. (b-c) The potential seen by
the flux across the inductor φ and energy spectrum at two values of applied flux φe for EJ /h = 4
GHz, EL /h = 1 GHz, and EC /h = 500 MHz.

are plasma like, and the frequencies and transition dipoles are similar to those of a transmon.
Near flux frustration at φe = π, the potential becomes a degenerate double well potential[Fig 6.5]. The lowest two energy levels are symmetric and anti-symmetric superpositions of excitations in the two wells. The splitting between these two energy levels is determined by the tunnelingthrough the central barrier, and is typically well below 1 GHz. There are additional transition
frequencies which correspond to plasma-like excitations within each well which retain a transmonlike nature and provide substantial anharmonicity. Additionally, their high frequencies and large
matrix elements mean they can be coupled strongly to a resonator for readout.

119
The coupling strength between the low-lying fluxonium transition and a surface acoustic
wave cavity depends on the |0i − |1i charge matrix element, h0|n̂|1i. This matrix element is highly
sensitive to the circuit parameters and to the applied flux. In one regime, the matrix element can beheavily suppressed, resulting in protection from energy decay [148]. Here, we want to maximize thismatrix element near flux frustration while maintaining a frequency at frustration of approximately
500 MHz [Fig 6.6]. For EC /h = 550 MHz, EL /h = 1 GHz, and EJ /h = 3 GHz, the matrix element
near frustration is near 0.2 while the transition frequency is below 500 MHz. A fluxonium qubit on
sapphire coupled to a waveguide resonator has been successfully fabricated and measured, but has
not been successfully integrated yet with surface acosutic wave resonators.
Realizing these large values of EC means the capacitance must be kept small, therefore requiring a narrow aperture IDT. This small aperture exacerbates the diffraction loss already limiting
our devices, intensifying the motivation to implement focusing reflectors as discussed in the next
section [Sec. 6.3]. The constraints on the IDT geometry can be relaxed by using a substrate witha weaker dielectric constant. To make larger aperture IDTs without pushing EC down or losing
coupling, we switch to quartz, which has r = 3.8 and K 2 = 1.4 × 10−3 , compared to r = 12.5 andK 2 = 7 × 10−4 for GaAs.

6.3

Stable Cavities
Stable resonators are used pervasively in optical technologies. These cavities use curved re-flectors to focus light and can be engineered to have negligible diffraction loss even for a tightly
focusing beam. There are two differences that complicate making analogous resonators for SAWs.
First, unlike free space, phonons have an anisotropic speed of sound that changes with their propagation direction [7]. This anisotropy can be strong, and has significant implication for diffraction[Eq. 2.33] and for the behavior of SAWs confined by curved mirrors. Second, unlike optical mirrors,reflectors for acoustic resonators are distributed, interacting with the acoustic wave over many
wavelengths. I will first introduce the description for stable cavities in an isotropic medium before
discussing these two complications.

120

Figure 6.6: The calculated parameters of fluxonium depend strongly on (a-c) the energies and (d)external flux. At the design goal of EJ /h = 3 GHz, EC /h = 600 MHz, and EL /h = 1 GHz, thematrix element at frustration is 0.2, the ω01 frequency is 500 MHz, and the anharmonicity is 1.7
GHz.

6.3.1

Cavity in an Isotropic Medium

Describing the beam in a stable cavity first requires a brief description of Gaussian optics[149]. A generic stable cavity supports a Gaussian beam with wavefronts that have a position
dependent curvature,R(x, θ) =

x+

x2R
x


,

(6.4)

where xR is the Rayleigh length and x = 0 is the focus of the beam. A cavity is formed by
two mirrors with curvatures R1 and R2 separated by length L but with unknown distance to the121focus, such that the mirrors are positioned at yet unknown positions x1 and x2 with x2 − x1 = L[Fig. 6.7(a)]. The mirrors set boundary conditions, enforcing that waves incident on the mirror
have wavefronts that match the mirror curvature. There are then three equations that determine
the mirror positions relative to the focus as well as the Rayleigh range,x2R
R(x1 ) = x1 +
= R1
x1


x2
R(x2 ) = x2 + R = R2
x2

(6.5)
(6.6)

L = x2 − x1 .

(6.7)

These equations can be inverted to find the beam parameters. These solutions are conveniently
expressed in terms of the g-parameters,
g1 = 1 −L
R1and g2 = 1 −

L
.
R2

(6.8)

With these definitions,
g1 g2 (1 − g1 g2 )L2
(g1 + g2 − 2g1 g2 )2
g2 (1 − g1 )x1 =L
g1 + g2 − 2g1 g2

x2R =

(6.9)
(6.10)(6.11)and x2 = L + x1 . The Rayleigh range is only well defined when 0 ≤ g1 g2 ≤ 1. Indeed, this is thecondition that the reflectors form a stable confined mode [Fig. 6.7(b)]. These beam parameters also
determine the beam waist at the focus,
Lλ
w02 =
π

s

g1 g2 (1 − g1 g2 ),(g1 + g2 − 2g1 g2 )2(6.12)

where w0 is the length at which the beam amplitude falls to 1/e of its peak value. The beam-waistof the cavity increases moving away from the focus,s 2x
w(x) = w0 1 +.
xR(6.13)

122

Figure 6.7: (a) A focusing cavity is defined by the curvature of two mirrors and their separation,
which gives a focal position at x = 0, Rayleigh range xR and beam-waist w0 . (b) The cavity is stable
when 0 ≤ g1 g2 ≤ 1 (shaded), with symmetric cavities along the diagonal (pink) and plano-convex
along g1 = 1, 0 ≤ g2 ≤ 1 (gold).

If the beam-waist at the mirrors, i.e. w(x1 ) and w(x2 ) is small compared to the aperture of themirrors, then diffraction loss, which in a curved cavity is clipping of the Gaussian beam by thefinite aperture of each mirror, will be negligible [149].
Even for multi-mode cavities hundreds of wavelengths long, a wide range in g-parameters
results in beam-waists that are well below 5λ and sufficiently narrow for use in a fluxonium qubit.
However, for the plane-parallel cavities used in the previous two chapters, g1 = g2 = 1, and thebeam waist diverges, and results in the large observed loss rates.
The focusing of the beam creates a phase shift in the wavefronts called the Gouy phase [149],
given by,
ψ(x) = arctan

x
xR


.

(6.14)

Over a distance many times the Rayleigh range, the wave picks up an extra phase of π. Between
the two mirrors, the wave acquires a Gouy phase given by,√ψ(x1 ) − ψ(x2 ) = acos (± g1 g2 ) ,(6.15)

where the sign of arccos argument matches the sign of g-parameters. The Gouy phase acts as an

123
additional source of phase in the resonance condition, and therefore shifts the resonant frequencyof the fundamental longitudinal modes. This overall shift is typically small and difficult to differentiate from a slightly different cavity length. However, the Gouy phase plays the dominant role indetermining the relative spacing between purely longitudinal modes with index m and those with
non-zero transverse mode number t. The transverse modes waves have transverse profile given by√
the same Gaussian waist w(x) times Ht ( 2x/w(x)) where Ht are the Hermite polynomials. Thesewaves pick up an additional Gouy phase of tψ(x), so that the resonances are given by,!
√
acos ± g1 g2πv
ωm,t = m + (t + 1).
πL

(6.16)

The frequency difference between a purely longitudinal mode and its neighboring transverse modes
then precisely measures the Guoy phase, and this spacing determines the product of the two gparameters. This relationship gives a sensitive and experimentally accessible way to measure thecavity g-parameters.

6.3.2

Anisotropic Wave Velocity

The predominant difference between optical cavities and phononic ones is the anisotropy ofthe speed of sound. Piezoelectric materials are necessarily anisotropic, as piezoelectricity requires
the crystal break inversion symmetry. There are two methods to account for this asymmetry: one
applied to focusing SAWs on GaAs based on the angle dependence of the SAW group velocity[150, 151], and another implemented on high-Q focusing bulk acoustic wave resonators based ona paraxial approximation of the angle-dependent phase velocity [152, 122]. The group velocity
method was demonstrated in the most relevant application, and claims to out-perform the paraxial
approximation when attempting to make an IDT with a tight focus, and so this is the method used
to design the focusing cavities discussed here. The paraxial approximation, on the other hand, canmake a wider set of predictions, and preliminary analysis suggests that it can better account forthe observed focusing behavior described further in the following section [Sec. 6.4.2].

124
6.3.2.1

Group Velocity Approach

The concept behind the group-velocity approach to focusing SAWs in an anisotropic medium
can be understood intuitively by considering the strain field generated by a point source. A strain
pulse generated from a point source will radiate away, carrying power outward at vg (θ), the angledependent group velocity [151]. By time reversal, a source that excites a wave-front that matches theradiated pattern will focus back to a diffraction limited point. Thus, these sources, such as mirror
elements, that attempt to focus SAWs at radius R0 should have an angle-dependent curvature,R(θ) = R0 vg (θ)/vg (0).

(6.17)

To determine this angle-dependent group velocity, first the phase velocity vp (θ) of ST quartz isdetermined using COMSOL simulations [Fig. 6.8(a-b)], and converted to group velocity by first
determining the power flow angle,ξ = arctan

dvp 1
dθ vp (θ)

,

(6.18)

which then determines vg = vp / cos(ξ). A difference in phase and group velocity, resulting from anangular dependence in vp , therefore requires power flow that is not orthogonal to the wave fronts[Fig. 6.8(c)]. The diffraction parameter γ used earlier [Eq. 2.33] can be found as γ = dξ/dθ. Notethat a well chosen crystal cut and propagation direction can result in both ξ = 0 and γ = −1,
meaning waves do not diffract to first order [82, 7]. This method for reducing diffraction losses isalso under investigation at this time, with promising initial measurements already taken.
The wavefronts in the cavity are then assumed to follow the angle-dependent curvature,
x2R
R(x, θ) = x +vg (θ)/vg (0).
x(6.19)

Scaling all of the elements to follow this angle-dependence results in a consistent cavity, where theg-parameters are defined using their R0 values.
The written radius of curvature approximated at small angles Rw can differ greatly from
R0 because the second derivative of the group velocity effectively rescales the curvature. At small

125

Figure 6.8: (a) The phase vp and group vg velocities on quartz based on Comsol simulations of vp .
(b) The power flow angle ξ, depicted in (c), as a function of angle. The paraxial approximation isbased on the slope of ξ at θ = 0 (dashed).

angle,
Rw = R0

vg00 (0)1−vg (0)

−1(6.20)

where vg00 (0) is the second derivative of vg with respect to angle evaluated at θ = 0. On quartz,
vg00 (0) = 0.51, which means that the written designs have radii of curvature approximated at small
angle that is nearly twice the effective curvature seen by the acoustic wave.

6.3.2.2

Paraxial Approximation

An alternative approach is to approximate the effect of the anisotropy by an rescaling factorµ by the quadratic term in the angular dependence of the phase velocity. This factor is given byµ = 1 − vp00 (0)/vp (0), which on ST-X quartz gives µ = 0.63 according to COMSOL simulations

126[Fig. 6.8(b)]. This approximation leads to several more analytical expressions given below that
capture how a Gaussian beam is modified by asymmetry, but there is evidence [151] that thisapproximation leads to degraded focusing behavior.
The curvature of the beam is scaled by µ [122], such that,


x2R 1
R(x) = x +.
x µ(6.21)

This simple scaling can then be used to derive that the g-parameters,
g1 = 1 −LµR1and g2 = 1 −

L
,
µR2

where R1 and R2 are the written curvatures, as well as the modified beam waist,s
Lλ
g1 g2 (1 − g1 g2 )w02 =.
µπ (g1 + g2 − 2g1 g2 )2(6.22)(6.23)The other cavity parameters, such as the Rayleigh range and the Gouy phase, depend the gparameters in the same way as the isotropic case.

6.3.3

Distributed Mirrors and Curved IDTs

Stable cavities for SAWs do not have access to localized reflectors. The analysis derived forquasi-1D acoustic reflectors in Sec. 2.2.1 still applies to focusing cavities. In contrast to what is
typically encountered with optical Bragg reflectors, in the case of SAWs the significant penetration
into the mirrors compared to the Rayleigh length means that the elements must change curvature
with position.
The cavities used here are constructed by first choosing the length and g-parameters of thecavity. This design then sets the radius of curvature for all points within the cavity, as well as
those outside, to follow Eq. 6.19. The mirror elements are chosen to follow this curvature, withthe first elements placed such that the distance between the mirrors is equal to the designed
length minus twice the penetration length [Fig. 6.11]. This consistency means that regardless of thepenetration length seen by a wave, the curvature of the element at that length will give the same
beam parameters, only with a different cavity length.

127
The IDT fingers are curved based on their position to follow the same distribution. The widthof both of the mirrors and the IDT is some scalar factor times the beam waist at their position,
following Eq. 6.13. Finally, the spacing between the elements is adjusted to compensate for the
Gouy phase.

6.4

Testing Stable Cavities on Quartz
The performance of stable SAW resonators on quartz was investigated by fabricating anarray of cavities and probing their electrical response. Moving to lower frequency makes opticallithography sufficient to define the features. These techniques are convenient at wafer-scale, which
combined with a microwave probe station enables many devices to be fabricated and measured
quickly. The goal of these tests was to make resonators that focused SAWs to a beam waist ofnear 5λ to be compatible with the constraints on the capacitor size imposed by fluxonium while
suppressing the diffraction losses that limited previous devices.

6.4.1

Cavity Design

The cavities investigated here are primarily two-port plano-convex cavities with one transducer near either reflector. This design is attractive partially because it is simple, as there is only
one curvature that matters, and the achievable beam-waist is quite small. In most designs, the“flat” mirror does in fact have some curvature to move the focal point outside the Bragg stack.
The plano-convex design also offers a stronger coupling to more modes than a symmetric one.
In both designs, the qubit IDT must be located at the focus of the cavity, which is in the center
for a symmetric cavity and near the flat mirror in a plano-convex design. If the qubit is designed
to couple to all of the modes of a symmetric resonator, then it must be shifted from the center√by λ/8, which reduces its coupling by 2. Alternatively, to achieve the same mode density, thecavity could be made twice as long, but this has exactly the same effect on the coupling rate. In a
plano-convex design, the qubit IDT is near the mirrors where the standing waves have nearly thesame phase [Fig. 6.9(a)], particularly for modes near the center of the mirror bandwidth, leading

128

Figure 6.9: (a) The SAW standing wave [Eq. 2.32] near the mirror are shifted by the phase of themirrors, but this shift is small so the anti-nodes are all close together. Modes at the edge of themirror bandwidth (red/blue) have the largest deviation. (b) An IDT√with center (5 + 3/8)λ awayfrom the mirror will couple to all modes strongly compared to the 2 penalty paid for coupling
to all modes of a symmetric cavity by offsetting an IDT at the center of a symmetric cavity λ/8(dashed).

to strong coupling to all of the modes of interest [Fig. 6.9(b)].
This same design with a flat IDT near one mirror could also be realized with a high curvaturedesign with focusing parameters of equal magnitude but opposite sign as the quasi plano-convex
case. While this design would achieve a slightly smaller beam-waist, designing both mirrors withnegative g-parameters requires higher curvature reflectors and results in a cavity stability that ismore sensitive to the crystal anisotropy.

6.4.2

Focusing Tests

We test the focusing behavior of an array of SAW cavities with widely varied curvature bymeasuring the spacing between purely longitudinal modes and their associated transverse resonances. The relative frequency spacing between longitudinal modes and those with non-zero mode
number depends only on the g-parameters of the device [Eq. 6.16].
The fabrication was done using optical lithography. Aluminum was first sputtered onto the129

Figure 6.10: The focusing SAW cavities, shown here resting on a quarter, are fabricated of Al onquartz with etched mirrors. They are probed wafer scale, and the scratches from the probe station
are visible on the bond pads after dicing.

wafer to create a 25 nm film. The aluminum was then etched using a wet-process to define the
IDTs and the transmission lines before the mirror elements were etched into the quartz using aplasma etch. Representative devices are pictured in Fig. 6.10 and shown schematically in Fig. 6.11.
These devices all had the same length, and were designed to produce a somewhat uniform
distribution of transverse mode spacing [Fig. 6.12(a)]. The width of the reflectors was chosen to be
3 times the beam-waist with an expected penetration length of 70 µm from rs = 2.1%. There are
two IDTs in the cavity, both with finger overlap 1.5 times the waist. The narrow IDT width results
in a weak coupling strength that is nonetheless sufficient to measure the acoustic spectra with amicrowave probe station at room temperature.
The measurements show a uniform longitudinal mode spectra between the many devices witha clear migration upwards of the transverse mode as the curvature of the reflectors is increased[Fig. 6.12(b)]. The difference in the depths of the reflection dips on resonance arises primarily from

130

Figure 6.11: The SAW cavities are designed to be near plano-convex with two IDTs, one near the
curved mirror and the other near the flat mirror. The width of the IDTs and mirrors are set to be
proportional to the predicted beam waist. The curvature of the mirrors and length of the cavity
is set by the element Lp into the array, and the “flat” mirror has some non-zero curvature so thatthe focus occurs just in front of its first element.

the differences in external coupling rate as the width of the measurement IDT follows the minimum
expected beam-waist. The transverse modes are low-Q because they have a wider beam-waist thanthe purely longitudinal modes, and there is significant clipping at the finite-width reflectors.
The extracted transverse mode spacing relative to the free spectral range follows the prediction but only if the mirror radii of curvature are rescaled by a factor of 1.4 [Fig. 6.13]. Interestingly,
preliminary calculations indicate that this rescaling is not required if we approximate the mirror elements as circles and then use the paraxial approximation to calculate the effective g-parameters andtransverse mode spacings [Fig. 6.13]. Regardless, the transverse mode spacings strongly indicatethat these SAW cavities are indeed focusing. However, this measurement is only an inference, anddirect measurements of the acoustic standing wave would illuminate many details of the acoustic
modes that are not apparent from cavity spectroscopy.

131

Figure 6.12: (a) An array of 10 devices were designed with varying curvatures such that the expectedtransverse mode frequency spanned a free spectral range. (b) The measured spectra show clear
longitudinal modes that are nearly independent of the focusing parameters across a wide range incurvatures. The transverse modes migrate from the high frequency shoulder for weakly focusing
cavities to the low frequency shoulder for strongly focusing ones (dashed).

132

Figure 6.13: The extracted transverse mode frequency spacings from the previous device closely
follow the designed values, including a phenomenological radius rescaling factor of 1.4 (dashed).
A paraxial approximation with µ = 0.63, where the mirrors are approximated as circular, agrees
well at low curvature but deviates at higher values (solid). The flat mirror curvature R1 changes
between devices to keep the focus in front of the mirror (inset).

6.4.3

Resonator Vibrometry

Vibrometry provides a direct method of imaging the standing wave of the SAW cavities. Thesemethods shine focused laser light on the sample, and acoustic oscillations modulate the reflected
light [153]. Current commercial technology produced by Polytec can resolve pm displacements at
hundreds of MHz with a micron spot size and in a near mm2 field of view. This resolution is
sufficient to directly image the SAW resonators we are interested in, although such multi-mode
cavities have lengths that exceed the measurement field of view.
Polytec measured two acoustic cavities that we fabricated, one more strongly focusing, toprovide a detailed picture of the standing waves in the cavity. The cavity was driven with a chirped
tone during measurement which provides a broadband excitation of the acoustic cavity. The chirp
was slow enough, however, that the acoustic cavity modes could ring up fully while the drive was
swept across resonance. Measurements of several segments of each resonator were taken, each many

133
wavelengths in spatial extent.
Preliminary analysis of this large data set has provided direct confirmation that the cavities
do indeed focus [Fig. 6.14(a)]. The wavefronts are well defined, and the transverse beam profile
is well fit by a Gaussian with a waist that is approximately the predicted size. The resonancethat we had inferred had non-zero transverse mode number from spectroscopy show the expected
Hermite-Gaussian transverse profile with no free parameters for the t = 2 mode [Fig. 6.14(d)]. An
IDT could be designed to decouple from this transverse mode if the SAW amplitude integrated
over the IDT width was zero. Additionally, the curvature of the wave does match the mirror
elements, suggesting no nefarious effects are present from a potentially angle-dependent reflectivity
per element. For the more focusing device not shown here, the somewhat clean spectrum belies thereality that the transverse modes do not go away for strongly focusing devices, but simply become
spectrally indistinct from the transverse modes.
Further analysis of this large data set faces several challenges. Neither device provides clear
simultaneous measurements of both curvature and beamwaist; the wave-fronts in the less focusing
device are too flat to precisely extract a curvature, while the transverse profile in the more focusing
device is a mixture of purely longitudinal and various transverse modes without a well-defined
waist.

6.4.4

High-Q Focusing Cavities

In addition to creating a narrow beam-waist, these focusing cavities were implemented toincrease the quality factor of the acoustic resonators. At room temperature, the internal linewidthof SAW cavities is typically dominated by visco-elasticity. Additionally, the focusing itself will
likely change as the elasticity tensor of quartz changes with temperature [154].
The cryogenic performance of these focusing resonators shows that diffraction loss is indeed
suppressed. A characteristic plano-convex resonator with was cooled down to 20 mK, and its measured spectra shows many high-Q resonances over the mirror bandwidth [Fig. 6.15]. Interestingly,
the Q improves towards the upper edge of the band, an effect likely related to how the strain is134

Figure 6.14: (a) The standing waves in the weakly focusing cavity (g1 = 0.83, and g2 = 0.34) aremeasured at several locations. (b) The measured displacement at 496.70 MHz show clear wavefronts with increasing beam waist and curvature approaching the curved mirror. The measured
displacement of the transverse mode at 496.33 MHz shows clear nodes in the transverse direction.
(c) The measured transmission spectrum through the cavity, with peaks associated with both purely
longitudinal (t = 0) as well as well-separated, 2nd order transverse modes (t = 2). (c) Cuts acrossthe wave-fronts show a Gaussian fundamental mode, whose beam-waist can be used to predict thet = 2 transverse mode profile with good accuracy.

135

Figure 6.15: The transmission spectrum of a highly focusing cavity shows multiple high-Q resonances with linewidths decreasing across the mirror bandwidth to below 2 kHz. Transverse modes
are visible on the low-frequency shoulder of the main, longitudinal resonances but with relatively
small amplitude.

localized relative to the mirror elements [79] as depicted in Fig. 6.9(a). The promincence of thiseffect suggests that the new limit on device coherence is scattering from the mirror elements into
the bulk, although this requires further investigation.
Crucially, the highest frequency resonance has a linewidth of below 2 kHz, which corresponds
to a factor of 100 increase in lifetime and a factor of 10 increase in Q, and indicates that diffraction is
no longer the limiting loss in the cavities. Such a multi-mode, high-Q resonator with a narrow beamwaist is well positioned to be useful immediately in hybrid quantum acoustics experiments withsurface acoustic waves, potentially to generate multi-mode entanglement through joint, projective
measurement.

Chapter 7

Conclusion and Outlook

In this thesis, I demonstrated that superconducting circuits can be coupled to multi-mode
SAW cavities, reaching both the strong, multi-mode coupling regime where the qubit-cavity exchange rate exceeded the cavity mode spacing, and the strong dispersive regime, where the singlephonon dispersive shift exceeds the intrinsic decay rates of the system. These results not only
confirm that quantum acoustics with surface acoustic waves is well described by Jaynes-Cummings
physics, but they also highlight the unique strengths of surface acoustic waves: to create long
delays in a compact space, both to realize high densities of resonant modes and also to precisely
engineer the qubit-phonon interaction strength in the frequency domain. Together, these abilities
make quantum acoustics with surface acoustic waves a promising platform for exploring a deeply
multi-mode analogy of circuit quantum electrodynamics, but with sound.
Future experiments would greatly benefit from improved coherence times. I have shown thatthe dominant acoustic loss mechanism, phonon diffraction loss from an unstable, plane-parallel cavity, can be eliminated by implementing curved reflectors. On the electrical side, moving to lower
frequencies and using a flip-chip geometry should improve the coherence of the superconducting
qubit. Such improvements are likely sufficient to enable a range of exciting experiments, including observation of phonon jumps [105] and entangling multiple phonon modes together througheither 4-wave mixing [26], side-bands [25], or joint-projective measurement [134]. Reconstructing
the multi-mode density matrix will be crucial to verifying this entanglement, and could follow thedemonstrations from cQED [27]. For building a quantum random access memory out of SAW

137
modes, the spacing between resonances must be substantially non-uniform [26], which could beachieved using Bragg stacks with a frequency-dependent delay, or by inserting third reflector centered in the cavity with a frequency-dependent reflectivity to create pairs of resonances with afrequency-dependent splitting.
The outlook for further improvements of the performance metrics is promising. The couplingstrength could be improved by an order of magnitude using a stronger piezoelectric material. This
increased coupling strength could also be traded for increased qubit coherence by reducing theparticipation of the piezoelectric material. However, coupling to surface acoustic waves will always
entail a significant amount of electrical energy stored in the surface where dielectric losses arelargest, and the long-term prospects for improving the device performance are significantly betterfor the acoustic cavities [121]. Identifying the next dominant loss mechanism is of high priority.
If this loss comes from the mirror elements as is currently suspected, then this can be reduced byengineering the strain profile at the mirrors using a tapered geometry or modifying the density ofstates of bulk modes by fabricating the resonators on a Bragg stack into the substrate or suspending
the whole resonator [155].

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