INTEGRATING SUPERCONDUCTING QUBITS WITH QUANTUM FLUIDS AND
SURFACE ACOUSTIC WAVE DEVICES
By
Justin R. Lane
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics - Doctor of Philosophy
2021
ABSTRACT
INTEGRATING SUPERCONDUCTING QUBITS WITH QUANTUM FLUIDS AND
SURFACE ACOUSTIC WAVE DEVICES
By
Justin R. Lane
Superconducting qubits, mesoscopic superconducting circuits with a single quantum co-
herent degree of freedom, have emerged as both a promising platform for quantum compu-
tation and a versatile tool for creating hybrid systems with quantum mechanical degrees of
freedom. In this dissertation, we report on several experiments investigating the coupling of
superconducting 3D transmon qubits to two different systems: superfluid helium and piezo-
electrically actuated surface acoustic waves (SAWs). We report the first measurements of
superconducting qubits in the presence of superfluid helium, studying the spectroscopic and
decoherence properties of this combined system. We analyze the spectroscopic properties of
this composite system using the framework of circuit quantum electrodynamics, and in the
presence of superfluid helium we observe modest increases in the pure dephasing time. We
attribute this to improved thermalization of the microwave environment via the superfluid,
raising hopes that thermalization mediated by superfluid helium may be a resource for ex-
periments employing superconducting circuits. We also present ongoing work developing a
new capacitive coupling scheme for creating hybrid superconducting qubit-SAW resonators.
The tools developed to model and implement this experiment lay the groundwork for future
experiments to achieve robust coupling between 3D transmon qubits and surface acoustic
wave devices. Finally, we present results describing the first measurements of SAW induced
transport in exfoliated graphene devices.
ACKNOWLEDGMENTS
I am of the conviction that it takes a village to raise a scientist. Because of that, I cannot
possibly thank everyone who contributed to my Ph.D. in one way or another. However try
my best to acknowledge everyone who was pivotal to the process.
First off, I’d like to thank the person who was most important to this process: my
advisor, Johannes Pollanen. Johannes is an amazing scientist and role model, and I have
been incredibly lucky to have had his guidance for the past 6 years. Johannes has given
me more trust and academic freedom than a grad student could reasonably ask for, and has
always been available and willing to lend help when I need it (which is a lot.) He has been
an excellent teacher of both science and the process of doing science, and yet somehow has
always made me feel like a colleague rather than a student. I would be nowhere near the
scientist I am today without his guidance, patience, and friendship, and can only hope to
one day be as good a mentor to someone else.
In addition to Johannes, would also like to thank the members of my guidance committee:
Prof. Dean Lee, Prof. Chris Wrede, Prof. Norman Birge and Prof. Mark Dykman. I am
particularly in debt to Mark and Norman, who have been both willing to share a great deal
of knowledge and expertise with me and excellent at doing so.
On top of the senior scientists at MSU who have guided me for the past several years, I am
lucky to have had the opportunity to collaborate with a group of excellent physicists outside
of MSU. Chief among those I have to thank is Prof. Kater Murch at Washington University
in St. Louis, without whom this dissertation would not have been possible. Kater’s expertise
in superconducting qubits, and his willingness to share that expertise, has been absolutely
foundational to the work presented here. I also must thank Prof. Erik Henriksen, also at
iii
Wash. U., who played a pivotal role early in my Ph.D. in guiding the graphene experiments
that appear in this thesis. Thank you also to the grad students at Wash. U. who I have had
the pleasure to collaborate with over the years: Dr. Dian Tan, Daria Kowsari, and Dr. B.
“Nero” Zhou.
I have been fortunate to have the opportunity to work with many amazing scientists
in LHQS over the past 6 years. I am indebted to all of the postdocs who I have had the
opportunity to work with, and who have lent me much of the guidance that has made
this dissertation possible: Dr. Mazin Khasawneh, Dr. Kostya Nasyedkin, and especially
Dr. Niyaz Beysengulov. Special thanks must also be given to the original LHQS crew,
Liangji Zhang and Dr. Heejun Byeon, who I’ve been lucky to call colleagues and friends ever
since the lab was an empty room.
Some of the best advice I have ever been given is that folks both younger and smarter
than you will appear in your life, and rather than being bitter about it you should take every
opportunity to learn as much as possible from them. With that in mind, I am indebted to
Joe Kitzman, Camille Mikolas, and Steve Hemmerle, who have made the last several years
both an excellent learning experience and a lot of fun. I would also like to thank the small
army of undergraduate students who have passed through LHQS in my time: Anna Turnbull,
Josh Milem, Taryn Stefanski, Evan Brook, Gabe Moreau, and Brennan Arnold.
There are many other people in the PA department for whom I am better off having
known. I’d like to thank co-organizers of the QuIC seminar, who out of shared interest
and determination made the seminar a realty: Ryan LaRose, Ben Hall, Niyaz Beysenguov
(again) and Jacob Watkins. I am also indebted to many members of the department support
staff, who have been an invaluable source of help during my time at MSU. Special thanks to
Dr. Reza Loloee, Dr. Baokang Bi, Kim Crosslan, Cathy and Jessica Cords, and to the shop
iv
guys: Tom Palazzolo, Jim Muns, Tom Hudson and Rob Bennett.
I, like roughly everyone else on Earth, had my life turned upside-down by the pandemic,
and have been lucky enough to have a great group of friends who have been a source of
comfort from afar during this past year: thank you to Eli Clausen, Travis Schess, Gabe
Wascher, Drew Jordan, Zach Campbell, and Nick and Kelly Jaroz for your support and
friendship from a distance during this past, fairly arduous year. Of all of these folks, I am
particularly indebted to Nick, who 9 years ago was the first person to convince me that a
normal person such as myself could build a career out of intellectual curiosity. I would also
like to thank Rob Elder and Kolten LaBarre, who in addition to being great housemates
made the first and most uncertain months of the pandemic much more enjoyable.
I am deeply indebted to my family: my parents Monica and Rich Lane, my sister Meghan
and her husband Joe Carlson. Without your support for the past 6 (and, frankly, 28) years,
none of this would have been possible. Finally, I would like thank Kailey Draves, who has
put up with me more than any other person during this process. You’re ok kid.
The projects described in this dissertation were partially supported by the National
Science Foundation under grant numbers DMR-1708331, DMR-2003815 and DMR-1810305,
and by the generous support of the Jerry Cowen endowment.
v
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 More is different (and surprisingly still quantum) . . . . . . . . . . . . . . . 2
1.2 Quantum acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 A short outline of this dissertation . . . . . . . . . . . . . . . . . . . . . . . 5
Chapter 2 Superconducting circuits and circuit QED . . . . . . . . . . . . 6
2.1 A brief history of superconducting quantum circuits . . . . . . . . . . . . . . 7
2.2 The building blocks of quantum circuits . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Quantizing the LC oscillator . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Josephson junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.3 The Cooper pair box . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.4 Energy scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.5 The transmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.6 Where did the charge noise go? . . . . . . . . . . . . . . . . . . . . . 27
2.3 Circuit quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Putting together the building blocks of cQED . . . . . . . . . . . . . 30
2.3.1.1 Driving the system . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1.2 Coupling a qubit and a harmonic oscillator . . . . . . . . . 32
2.3.2 The Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . . 34
2.3.3 The strong dispersive limit and readout in cQED . . . . . . . . . . . 37
Chapter 3 Superconducting qubits in the lab . . . . . . . . . . . . . . . . . . 40
3.1 The 3D transmon geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.1 Electromagnetic modes in 3D cavities . . . . . . . . . . . . . . . . . . 41
3.1.2 3D transmon qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Filtering and shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.1 Cryogenic shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.2 Input/output line filtering . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.3 Amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.4 Possible improvements . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Spectroscopic measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.1 “Punchout” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.2 Two-tone spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Time-resolved measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.1 General time-resolved setup . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.2 Coherent control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.3 Punchout-based readout . . . . . . . . . . . . . . . . . . . . . . . . . 69
vi
3.4.4 T1 , T2 and T2e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Chapter 4 Integrating superfluids with superconducting qubit systems . 77
4.1 Superfluid helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Decoherence and noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.1 Common noise sources in superconducting qubits . . . . . . . . . . . 85
4.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.1 Gas handling system and fill line . . . . . . . . . . . . . . . . . . . . 88
4.3.2 Superfluid helium leak tight sample cell . . . . . . . . . . . . . . . . . 92
4.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4.1 Cavity and qubit spectroscopy . . . . . . . . . . . . . . . . . . . . . . 94
4.4.2 Qubit Relaxation and Decoherence . . . . . . . . . . . . . . . . . . . 100
4.4.2.1 Qubit energy relaxation . . . . . . . . . . . . . . . . . . . . 102
4.4.2.2 Qubit dephasing . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4.2.3 Long timescale fluctuations in qubit coherence properties . . 107
4.4.3 Residual excited state population . . . . . . . . . . . . . . . . . . . . 109
4.5 Discussion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.5.1 Coupling to superfluid acoustic modes? . . . . . . . . . . . . . . . . . 114
Chapter 5 Surface acoustic waves and surface acoustic wave devices . . . 117
5.1 Elastic waves in solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.1.1 Rayleigh Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 Piezoelectricity and piezoelectric Rayleigh waves . . . . . . . . . . . . . . . . 129
5.2.1 Equations of motion and Boundary conditions . . . . . . . . . . . . . 130
5.2.2 Sketch of Piezoelectric Rayleigh waves . . . . . . . . . . . . . . . . . 132
5.2.3 Relevant parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3 SAW devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3.1 Interdigitated transducers . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3.2 Coupling of Modes and the P-matrix . . . . . . . . . . . . . . . . . . 140
5.3.3 A second look at transducers . . . . . . . . . . . . . . . . . . . . . . 143
5.3.4 Acoustic Bragg mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.3.5 SAW resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Chapter 6 Quantum acoustics using surface acoustic waves . . . . . . . . . 150
6.1 Mediating the coupling between SAWs and qubits . . . . . . . . . . . . . . . 152
6.2 Device design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.2.1 Modeling the experiment . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2.1.1 Finite element modeling . . . . . . . . . . . . . . . . . . . . 157
6.2.1.2 Extracting the Butterworth-van Dyke equivalent circuit . . 160
6.2.1.3 Classical circuit simulations . . . . . . . . . . . . . . . . . . 162
6.2.2 Optimizing the SAW device parameters . . . . . . . . . . . . . . . . . 165
6.3 Device fabrication and assembly . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.4.1 Devices on ST-X quartz . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.4.1.1 Uncontrolled substrate spacing . . . . . . . . . . . . . . . . 173
vii
6.4.1.2 Mirror reflectivity . . . . . . . . . . . . . . . . . . . . . . . 175
6.4.1.3 IDT position between the mirrors . . . . . . . . . . . . . . . 176
6.4.1.4 Coherence times . . . . . . . . . . . . . . . . . . . . . . . . 178
6.4.2 Devices on LiNbO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Chapter 7 Acoustoelectric transport in graphene . . . . . . . . . . . . . . . 183
7.1 Graphene: a brief introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.2 Surface acoustic waves and two-dimensional electron systems . . . . . . . . . 186
7.2.1 SAW attenuation and velocity shift . . . . . . . . . . . . . . . . . . . 187
7.2.2 Acoustoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.3 Acoustoelectric effect in exfoliated graphene . . . . . . . . . . . . . . . . . . 193
7.3.1 Cold-finger and sample stage . . . . . . . . . . . . . . . . . . . . . . . 194
7.3.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.3.3 SAW delay line characterization . . . . . . . . . . . . . . . . . . . . . 201
7.3.4 Graphene acoustoelectrics . . . . . . . . . . . . . . . . . . . . . . . . 202
7.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.4.1 Graphene acoustoelectrics in a magnetic field . . . . . . . . . . . . . 207
7.4.2 ∆v/v measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.4.3 Measurements of 2DESs using SAW resonators . . . . . . . . . . . . . 209
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
APPENDIX A A note on the rotating wave approximation . . . . . . . . . . . . . 212
APPENDIX B Other cQED experimental techniques . . . . . . . . . . . . . . . . 215
APPENDIX C Fabrication Recipes . . . . . . . . . . . . . . . . . . . . . . . . . . 222
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
viii
LIST OF TABLES
Table 3.1: RF circuitry components for the circuit quantum electrodynamics exper-
imental setup, along with part numbers and suppliers. For some compo-
nents, we use multiple variants of the same component. For these cases,
the variable parameter is marked in bold, and the corresponding change in
the part number is matched to the variable in the component description.
Some of the specific products here are no longer manufactured, however as
of writing this, every manufacturer listed either still makes the part listed
or makes an updated version of it. . . . . . . . . . . . . . . . . . . . . . . 57
Table 4.1: Spectroscopic parameters of the cavity/qubit system both in the presence
and absence of superfluid helium. ωc , δω, ω01 , and ω12 are measured values,
while EJ , EC and g01 are extracted by solving the generalized Jaynes-
Cummings Hamiltonian constrained by measured spectroscopic parameters. 96
Table 5.1: Relevant material parameters for common SAW substrates, reproduced
from Chapter 4 of Ref. [160] and Chapter 9 of Ref. [167], both of which
contain more complete tables of materials/parameters. . . . . . . . . . . . 135
Table 6.1: Summary of the design parameters, phenomenological parameters, and de-
rived parameters used in the model of the example SAW resonator consid-
ered in § 6.2. All design parameters are defined in fabrication. . . . . . . . 164
Table 6.2: Summary of the target design parameters, phenomenological parameters,
and derived parameters for the device on ST-X quartz experimentally tested
and described in § 6.4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Table 6.3: Transmission characteristics of a 3D cavity with/without LiNbO3 , extracted
from the data plotted in Fig. 6.13 . . . . . . . . . . . . . . . . . . . . . . 180
ix
LIST OF FIGURES
Figure 2.1: A simple LC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Figure 2.2: The Josephson junction and the Cooper pair box . . . . . . . . . . . . . 15
Figure 2.3: Eigenvalues of the Cooper pair box Hamiltonian . . . . . . . . . . . . . . 20
Figure 2.4: A mechanical analogue to the Cooper pair box Hamiltonian . . . . . . . 24
Figure 2.5: Schematic of circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Figure 2.6: Cavity-qubit avoided crossing . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 2.7: Cavity dispersive shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Figure 3.1: 3D electromagnetic cavities . . . . . . . . . . . . . . . . . . . . . . . . . 43
Figure 3.2: Cavity decay rate κ vs. pin depth . . . . . . . . . . . . . . . . . . . . . . 45
Figure 3.3: The 3D transmon qubit geometry . . . . . . . . . . . . . . . . . . . . . . 46
Figure 3.4: Cryogenic shielding for superconducting qubit experiments . . . . . . . . 50
Figure 3.5: Filtering and amplification of the input-output lines . . . . . . . . . . . . 53
Figure 3.6: Qubit “punchout” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Figure 3.7: Two-tone spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Figure 3.8: General time resolve measurement setup . . . . . . . . . . . . . . . . . . 64
Figure 3.9: Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Figure 3.10: IQ mixer modulation and demodulation . . . . . . . . . . . . . . . . . . 71
Figure 3.11: IQ plane measurement and binning . . . . . . . . . . . . . . . . . . . . . 72
Figure 3.12: Measuring T1 , T2 , and T2e . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Figure 4.1: Noise and decoherence of a qubit . . . . . . . . . . . . . . . . . . . . . . 82
Figure 4.2: Gas handling system and cold trap . . . . . . . . . . . . . . . . . . . . . 89
x
Figure 4.3: Superfluid Helium + superconducting qubit setup . . . . . . . . . . . . . 91
Figure 4.4: Superfluid leak-tight sample cell, iteration #1 . . . . . . . . . . . . . . . 93
Figure 4.5: Cavity and qubit spectroscopy is the presence of superfluid helium . . . . 95
Figure 4.6: Simplified circuit model for finding ∆g01 . . . . . . . . . . . . . . . . . . 98
Figure 4.7: Description of time-resolved and averaged measurements . . . . . . . . . 101
Figure 4.8: T1 , T2 and ω01 as a function of temperature with/without superfluid helium102
Figure 4.9: Long timescale fluctuations of coherence parameters with/without super-
fluid helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Figure 4.10: Excited state population of the qubit as a function of temperature with/without
superfluid helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Figure 4.11: Proposals for future superfluid-superconducting qubit experiments . . . . 115
Figure 5.1: Stress and strain in elastic media . . . . . . . . . . . . . . . . . . . . . . 119
Figure 5.2: SAW propagation in an isotropic medium . . . . . . . . . . . . . . . . . 127
Figure 5.3: Interdigitated transducers for SAWs . . . . . . . . . . . . . . . . . . . . . 137
Figure 5.4: The P-matrix for SAW devices . . . . . . . . . . . . . . . . . . . . . . . 141
Figure 5.5: Effect of reflections on IDT conductance . . . . . . . . . . . . . . . . . . 143
Figure 5.6: Bragg Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Figure 5.7: COM model of a SAW resonator . . . . . . . . . . . . . . . . . . . . . . 147
Figure 6.1: Schematic of the proposed experiment . . . . . . . . . . . . . . . . . . . 153
Figure 6.2: Finite element simulations of the paddle structure . . . . . . . . . . . . . 158
Figure 6.3: Butterworth-van Dyke equivalent response of a SAW resonator . . . . . . 160
Figure 6.4: Extracting the Hamiltonian parameters from classical circuit simulations 162
Figure 6.5: Finite element simulations of the paddle structure . . . . . . . . . . . . . 164
Figure 6.6: Mask for aligning the flip-chip device stack . . . . . . . . . . . . . . . . . 167
xi
Figure 6.7: Assembling the flip-chip stack . . . . . . . . . . . . . . . . . . . . . . . . 168
Figure 6.8: Verifying coupling of the transmon-like mode to the electromagnetic cavity 171
Figure 6.9: Spectroscopy of the ST-X Quartz quantum acoustics device . . . . . . . 172
Figure 6.10: Dependence on the substrate spacing z . . . . . . . . . . . . . . . . . . . 174
Figure 6.11: Dependence of the SAW response on the grating reflectivity . . . . . . . 175
Figure 6.12: Dependence of gm on the IDT position . . . . . . . . . . . . . . . . . . . 177
Figure 6.13: 3D cavity transmission in the presence of LiNbO3 . . . . . . . . . . . . . 180
Figure 7.1: DC transport measurements of graphene . . . . . . . . . . . . . . . . . . 186
Figure 7.2: SAW attenuation and velocity shift vs. 2DES conductivity . . . . . . . . 189
Figure 7.3: Cold-finger design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Figure 7.4: Acoustoelectrics sample stage . . . . . . . . . . . . . . . . . . . . . . . . 197
Figure 7.5: Graphene acoustoelectrics device schematic . . . . . . . . . . . . . . . . 199
Figure 7.6: SAW reflection and acoustoelectric voltage vs. drive frequency . . . . . . 202
Figure 7.7: Acoustoelectric effect: charge carrier polarity dependence . . . . . . . . . 204
Figure 7.8: Gate-tunabe acoustoelectric effect . . . . . . . . . . . . . . . . . . . . . . 205
Figure 7.9: Power dependence of the Gate-tunable acoustoelectric effect . . . . . . . 206
Figure A.1: The rotating wave approximation . . . . . . . . . . . . . . . . . . . . . . 213
Figure B.1: Excited state population measurement . . . . . . . . . . . . . . . . . . . 216
Figure B.2: Spin-locking setup and measurement . . . . . . . . . . . . . . . . . . . . 219
Figure B.3: Spin-locking spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Figure C.1: Description of the Dolan bridge junction fabrication process . . . . . . . 229
Figure C.2: SAW device fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Figure C.3: Examples of SAW IDT dosing effects . . . . . . . . . . . . . . . . . . . . 233
xii
Chapter 1
Introduction
Nearly a century after Schrödinger wrote down his famous equation, largely solidifying the
formalism of quantum mechanics still in use today, the fact remains that we live in a classical
world. Everyone agrees that quantum mechanics is living under the hood of every physical
system, but at the scale of us humans it is completely washed out. This suppression of
quantum physics is nearly complete: you don’t need to understand quantum mechanics to
play baseball, or to build a car, or even to put a person on the moon, even though all of
these activities are centered around the motion of objects whose constituents are supposed
to be quantum mechanical. A pitcher need not worry about the Heisenberg uncertainty
relationship when throwing a curve ball, an engineer doesn’t consider entanglement when
designing a crank shaft, and Houston didn’t consider the possibility of the lunar lander in a
superposition of landing on the Moon and Mars.
The emergence of our classical, macroscopic world from the underlying “quantum-ness”
of its constituent particles remains an important question in fundamental physics and active
area of research. Conversely, one may ask how far can we push the concept of “quantum-
ness”? Can we engineer a “macroscopic” system that obeys the laws of quantum mechanics?
How far can we depart from the realm of single particles and still maintain an object that
obeys the Shrödinger equation?
1
1.1 More is different (and surprisingly still quantum)
Phil Anderson, in his landmark 1972 paper [1], coined the term “more is different” to annun-
ciate the idea that the study of emergent complexity, rather than simply being an application
of the laws governing the system’s constituents, is itself a fundamental scientific enterprise.
Anderson’s main thesis is that reductionism does not imply constructionism; that the emer-
gent degrees of freedom of a complex system cannot, in general, be a priori predicted from
the laws governing the underlying constituents. In the 50 years since, “more is different” has
become somewhat of a mantra of condensed matter physics, which can be summarized as
the study of the emergent degrees of freedom in many-body systems whose constituents are
(usually) electrons and ions interacting electromagnetically.
But is there any reason to believe that these many-body degrees of freedom should behave
quantum mechanically? For example, the vibrations of ions in a crystal may be quantized
into phonons, which may be thought of as “particles” in the same way photons are quanta of
the electromagnetic field. While phonons are invaluable in understanding many properties
of condensed matter systems, is it valid to think of them as quantum mechanical particles,
like photons? Indeed, our experience tells us that sufficiently large many-body systems,
baseballs and crankshafts and spaceships, tend to behave classically. Can the collective
motion of trillions of atoms in a crystal be in a superposition state? Can a many-body
degree of freedom be entangled with another particle?
The answer to this question is, perhaps surprisingly, a resounding yes. In a ground-
breaking experiment in 2010, O’Connell et al. [2] demonstrated coherent coupling of a di-
latational mode of a microelectromechanical (MEMS) device to a superconducting quantum
bit, or qubit. As we shall see in Chapters 2 and 3, superconducting qubits are mesoscopic
2
circuits (typically ∼ 1 µm−1 mm) with a quantum coherent degree of freedom defined by the
rigid motion of the superconducting condensate1 . Originally designed to be the fundamental
computing element of a quantum computer, these devices may be engineered into a wide
variety of geometries, with fine control of the associated electric field profile. O’Connell et
al. used this flexibility to create a superconducting qubit device that coupled strongly to a
MEMS resonator via the piezoelectric effect. The authors were able to demonstrate coher-
ent exchange of a single quanta between the qubit and the dilatational mode of the MEMS
resonator, and even measure the phase coherence of a superposition state of the resonator,
an unambiguous verification of the quantum nature of the collective mode.
1.2 Quantum acoustics
In the years since, the model of leveraging the quantum coherence of superconducting circuits
to study and manipulate acoustic degrees of freedom at the single quantum level has blos-
somed into a research program in its own right. Superconducting qubits have been employed
to control several different types of acoustic devices at the single quantum level, including
MEMS structures [2, 3, 4] , bulk acoustic wave resonators [5, 6], and surface acoustic wave
devices [7, 8, 9, 10, 11]. Many impressive experimental results have been demonstrated,
including single phonon splitting of the qubit spectrum [4, 12], Wigner function negativity
of the acoustic resonator [10, 13], squeezed states of sound [14] and erasure of which-path in-
formation of a propagating phonon [15], demonstrating unambiguously the quantum nature
of these collective excitations.
1 Superconducting qubits are themselves a profound answer to the question of how large can we make
a system that behaves like a single quantum degree of freedom. As we’ll see, the answer turns out to be
“visible to the naked eye.”
3
This field of research has been branded “quantum acoustics” in analogy to it’s electro-
magnetic parallel quantum optics. In quantum optics, we typically consider the interaction
a quantum system (such as an atom) with electromagnetic radiation; from this perspec-
tive, quantum acoustics is the interaction of a single quantum system (a superconducting
qubit) with acoustic radiation. The fundamental difference between quantum acoustics and
quantum optics is that the speed of sound is 104 − 105 times slower than the speed of light
(meaning that the wavelength of sound at a given frequency is 104 − 105 times smaller than
corresponding electromagnetic radiation.) This disparity in time/length opens up regimes of
study that aren’t accessible in traditional quantum optics: for example, one may enter the
“giant atom” regime, where the size of the single quantum object (the qubit) is much larger
than the wavelength of the radiation it interacts with [16, 17].
Not too long ago, the question of engineering a many-body quantum system was purely
academic. However, in parallel with the rise of experimental quantum information processing
systems and nascent quantum computers, physicists have already started to ask how quantum
acoustics may be leveraged in these applications. Again, the low propagation speed/small
wavelength are the primary resource of quantum acoustics. ∼GHz frequency acoustic res-
onators may be fabricated with high quality factors and small spatial extent, raising the
possibility of dense bosonic quantum memories [18]. The wavelength of sound at telecom
frequencies (where qubits are typically operated) may also be engineered to match common
optical wavelengths, opening up an avenue to coherent microwave-to-optical conversion [19].
4
1.3 A short outline of this dissertation
This dissertation represents, for lack of a better term, a grab bag of experiments, motivated
by the recent progress in quantum acoustics, that lays the foundation of a similar research
program at MSU. We will begin with a theoretical description of superconducting qubits in
Chapter 2, followed by a survey of the experimental capabilities of our lab in Chapter 3. In
Chapter 4, we will report on experiments investigating the compatibility of superconducting
circuits with another system known to host many interesting mechanical excitations: super-
fluid helium. We will then spend Chapters 5 and 6 discussing surface acoustic waves (SAWs),
how we might leverage our experimental capabilities to create novel geometries for quantum
acoustics with SAWs, and finally report on preliminary experimental results along this front.
Chapter 7 catalogues an experiment from earlier in my Ph.D. using SAWs to study an en-
tirely different many-body system: graphene. While this experiment doesn’t directly relate
to superconducting qubits and quantum acoustics, much of the experimental knowledge of
SAW techniques in our lab was developed during this experiment. I also believe there is am-
ple room for overlap between SAW experiments studying low dimensional electron systems
and many other ideas outlined in this thesis, which I will attempt to motivate in the future
work section of Chapter 7.
5
Chapter 2
Superconducting circuits and circuit
QED
The extraordinary rise of computing power over the past 50 years is a direct product of
the massive scalability and miniaturizability of complementary metal oxide silicon (CMOS)
integrated circuits. The top-down fabrication of metal-oxide transistors via photolithography
has allowed for device complexity to scale more or less exponentially, to the point where a
high-end phone today has on the order of 10 billion transistors.
Ever since the infancy of quantum computing, the search for a physical platform on which
to perform quantum computations has been predicated on this requirement of massive scal-
ability. Naturally, the success rendered by the top-down fabrication of classical integrated
circuits has often percolated this discussion: is it possible to process quantum information us-
ing circuits fabricated with the same photolithographic techniques used in CMOS integrated
circuits? From this line of thought, superconducting circuits have emerged as a promising
candidate for quantum computing. In this chapter, we survey several fundamental elements
of superconducting circuits, and discuss the circuit quantum electrodynamics (cQED) archi-
tecture for manipulating and reading out qubits based on superconducting circuits.
6
2.1 A brief history of superconducting quantum cir-
cuits
In 1911, H. Kamerlingh Onnes discovered that, upon cooling down to liquid helium temper-
atures, the electrical resistance of mercury completely disappeared [20]. Since then, many
metals have been found to transition into a superconducting state below some critical tem-
perature Tc , characterized by no dc resistance and complete expulsion of external magnetic
fields [21]. While the study of superconductivity has been active for over 100 years, this
brief discussion will be constrained to to several theoretical and experimental breakthroughs
important to the realization of superconducting quantum circuits.
In 1950, Ginzburg and Landau (GL) proposed a phenomenological theory of supercon-
ductivity, where the superconducting state is characterized by a complex order parameter
ψ(r), interpreted as a pseudo-wave function describing the superconducting state [22]. Seven
years later, Bardeen, Cooper and Schrieffer (BCS) proposed a fully microscopic theory where
paired electrons, called Cooper pairs, condense into a macroscopic ground state [23]. One
hallmark of the BCS theory was the prediction that single particle excitations above the
ground state are gapped, with an energy gap ∆BCS comparable to the thermal energy at
the superconducting transition temperature kB Tc . This gap parameter, which in general
may be a complex function of the crystal momentum k (and thus r) was soon shown [24] to
be directly proportional to the GL order parameter ψ(r), justifying the phenomenological
nature of the GL theory.
In 1962, Josephson observed [25] that when two superconductors are connected by a
tunnel barrier, there may be a discontinuity in the phase of the complex order parameter
across the barrier. This observation led to prediction of the so called Josephson effect, which
7
was shortly thereafter observed by Anderson and Rowell [26]. The current and voltage
across these tunnel barriers, called Josephson junctions, may be written in terms of the
phase difference ϕ = ϕ2 − ϕ1 :
I = Ic sin ϕ (2.1a)
dϕ 2eV
= (2.1b)
dt ~
Here, Ic is the critical current of the Josephson junction: the maximum amount of super-
current that may pass through the junction before the onset of dissipative current. Josephson
junctions essentially represent a new superconducting lumped circuit element with a non-
linear current-voltage relationship. The nonlinearity of the Josephson junction forms the
backbone upon which superconducting quantum circuits are built.
While superconductivity is generally described as a macroscopic quantum phenomenon,
in 1980 Leggett argued that the macroscopic effects of superconductivity, such as the Joseph-
son effect, can be viewed as arising from the product of many single Cooper pair (two par-
ticle) wave functions rather than a coherent superposition of one many-body wave function
[27]. In the same manuscript, Leggett proposed that truly macroscopic quantum effects may
be observed in a superconducting ring interrupted by a Josephson junction (an RF super-
conducting quantum interference device, or SQUID) by leveraging the nonlinearity of the
Josephson junction. It is possible to write the potential energy of a SQUID loop as a function
of the phase difference across a junction, and to have a potential landscape where more than
one local minimum in the phase variable may exist. It was argued that such a “phase parti-
cle”, a macroscopic state of many Cooper pairs, may be able to undergo quantum tunneling
8
between two potential wells, an inherently quantum phenomenon.
This prediction was followed up by a series of seminal experiments by Clarke et al. showing
that the phase difference across a Josephson junction could, in fact, act as a quantum
“phase particle” in a potential well, with quantized energy levels and a finite tunneling
probability into an adjacent well [28]. This experiment, and many that followed, opened up
the possibility of using Josephson junction-based superconducting circuits to engineer and
manipulate macroscopic quantum-mechanical degrees of freedom. Coherent manipulation
of a macroscopic quantum variable in a Josephson junction circuit was demonstrated by
Nakamura et al. [29], ushering in the era of superconducting qubits, where the many-body
quantum states of superconducting circuits could be coherently manipulated and read out.
2.2 The building blocks of quantum circuits
The analysis of superconducting quantum circuits necessitates the combination of quantum
mechanics (generally written in the language of Hamiltonians and operators) and circuit
analysis (generally written in terms of currents and voltages). Combining these languages is
somewhat counterintuitive at first, so before we discuss Josephson junction-based qubits and
superconducting circuits it will be useful to build intuition by considering a more familiar
case: a lossless LC circuit. Much of the following discussion is based in piecemeal fashion
off Ref. [30] and chapter 3 of Ref. [31]. A detailed discussion on circuit quantization may be
found in Ref. [32].
9
C L
Figure 2.1: A simple LC circuit: A parallel LC circuit. We have defined two nodes: one
at ground, and one at node flux relative to ground φ(t)
2.2.1 Quantizing the LC oscillator
Consider a parallel LC circuit, like the one shown in Fig. 2.1. For reasons that will become
apparent shortly, instead of describing the circuit in terms of voltages and currents, we
will describe the circuit in terms of the node flux φ(t) of the nodes (i.e., wires connecting
elements) of the circuit
Z t
φ(t) = V (t0 )dt0 (2.2)
−∞
Here, V (t0 ) is the voltage of the node relative to ground: we are explicitly constructing φ(t)
such that dφ(t)/dt = φ̇ = V (t). For the parallel LC circuit in Fig. 2.1, we have two nodes:
one at ground potential, and one node “above” ground potential at φ(t). From elementary
circuit analysis, we know that the voltage across the inductor is V = LI˙ = φ̇ → LI = φ
for a static inductance, where L and I are the inductance and current across the inductor
respectively. We also know that the energy stored in the inductor and the capacitor may be
written as
10
1 1
UC = CV 2 = C φ̇2
2 2 (2.3)
1 2 1 2
UL = LI = φ
2 2L
From this point of view, the energy in the capacitor looks like the kinetic energy of
a particle with position φ, and the energy in the inductor looks like the potential energy.
Keeping this convention in mind, we write down the Lagrangian of the system and find the
conjugate momentum Q to our coordinate φ:
1 1 2
L = UC − UL = C φ̇2 − φ (2.4)
2 2L
∂L
Q= = φ̇C = V C (2.5)
∂ φ̇
which is, of course, the charge on the capacitor plates. Our convenient choice of φ as a
coordinate has given us a set of physically meaningful conjugate variables! We can then
Legendre transform the Lagrangian and write down the Hamiltonian for the circuit in terms
of the two conjugate variable φ and Q:
Q2 φ2
H = Qφ̇ − L = + (2.6)
2C 2L
which, since there is no explicit time dependence, is simply the sum of the energies UC and
UL . With the system Hamiltonian written in terms of a set of conjugate variables, we may
now safely promote the Hamiltonian/variables to operators with a cannonical commutation
relationship [φ, Q] = i~. If we recall that the resonant frequency of a simple LC oscillator is
ω0 = (LC)−1/2 , the fully quantum mechanical Hamiltonian may be written as
11
1 2 1
H= Q + Cω02 φ2 (2.7)
2C 2
We recognize this as exactly the Hamiltonian of a 1D harmonic oscillator, with the mass
replaced by the capacitance, and x and p replaced with φ and Q respectively. Following
the standard treatment of a quantum harmonic oscillator, we construct a set of raising and
lowering operators for our circuit:
φ = φZP F (a + a† ) (2.8a)
Q = −iQZP F (a − a† ) (2.8b)
[a† , a] = 1 (2.8c)
We can then write down the Hamiltonian in terms of these raising and lowering operators
1
H = ~ω0 a† a + (2.9)
2
with the familiar result for the energy E = ~ω0 (N + 1/2). The prefactors to the right
hand sides of Eqns 2.8a and 2.8b are the magnitudes of the zero point fluctuations of φ and
Q respectively, and are given by:
s
~Z0
φZP F = (2.10a)
2
s
~
QZP F = (2.10b)
2Z0
12
s
L
Z0 = (2.10c)
C
where Z0 is the characteristic impedance of the oscillator. Why do we call them the zero
point fluctuation magnitudes? In the ground state of the harmonic oscillator it is easy to
show that
hφi = φZP F h0|(a† + a)|0i = 0
(2.11)
φ2 = φ2ZP F h0|(a†2 + a2 + aa† + a† a)|0i = φ2ZP F
D E
We therefore see that the zero point variance of φ is ∆φ2 = φ2 − hφi2 = φ2ZP F . The
D E
same may be shown for Q.
Wait a minute. Is this valid? Compared to what we normally consider “quantum” scale
systems, capacitors and inductors are macroscopic objects: even a ∼ µm scale piece of metal
has order 1012 electrons. How can we talk about a single quantum degree of freedom arising
from this hopelessly many-body system?
The answer arises from the physics of superconductors: in a superconductor, single parti-
cle excitations above the superconducting ground state are gapped by an energy ∆BCS . The
existence of the superconducting gap decouples the motion of the condensate from micro-
scopic degrees of freedom, and allows current in a superconductor to flow with vanishingly
small dissipation. In the context of an LC oscillator, this means that we may straightfor-
wardly fabricate superconducting resonators with very low decay rates κ, or equivalently
very high quality factors Q = ω0 /κ. In the language of particle physics or quantum optics,
one might say that the line-width of a superconducting resonator1 can be made extremely
1 That being said, we can and do use high-Q (∼ 103 − 104 ) resonators from normal conductors such as
copper. While the utility of such a resonator will ultimately be limited by resistive loss, they too can be
modeled by a single quantum degree of freedom.
13
narrow compared to the level spacing, such that individual energy levels may easily be re-
solved. The “quantum harmonic oscillator” is really describing the rigid oscillations of the
entire superconducting condensate, shielded from decaying into the many-body environment
by the superconducting gap.
2.2.2 Josephson junctions
As with classical circuits, capacitors and inductors have finite utility on their own. Even
to make a single qubit, we must combine them with some nonlinearity: if we didn’t, all we
could make are harmonic oscillators with evenly spaced energy levels. Without some sort
of anharmonicity, we could never address the transition between two energy eigenstates, say
the ground state |0i and first excited state |1i, without accidentally climbing the harmonic
oscillator ladder (|2i , |3i , ...). To build an analogy to systems we learned about in intro-
ductory quantum mechanics, instead of building a circuit that makes harmonic oscillators,
we would like to build a circuit that makes an “artificial atom” [33]: a non-linear quantum
system with anharmonicity. If the anharmonicity in the energy level spacing is large enough,
we can constrain ourselves to two levels to make an effective qubit.
When working with superconducting circuits, we realize this required anharmonic os-
cillator by replacing geometric inductors with Josephson junctions. A Josephson junction
contains a thin insulating layer that splits the superconducting circuit but leaves them weakly
coupled (Fig. (2.2a)). The split in the superconductor also splits the complex order param-
eter, and the order parameter on either side of the junction may have a different phase.
However, the insulating barrier is thin enough that Cooper pairs may tunnel across the
junction, weakly connecting the two sides.
Take the idealized picture shown in Fig. (2.2a): we have two (identical) superconducting
14
(a) S I S (b) LJ
CS CJ
LJ CJ V
Figure 2.2: The Josephson junction and the Cooper pair box: (a) Schematic of a
Josephson junction: two superconductors (S, light blue) are separated by a thin insulating
barrier (I, white), across which Cooper pairs may tunnel. Josephson junctions act as nonlin-
ear inductors (inductance LJ ) and are typically represented by either a cross, or a cross in a
box which signifies the inductance LJ in parallel with some small but finite capacitance CJ
associated with the barrier. (b) The “Cooper pair box” circuit. Here, we’ve explicitly sep-
arated out the Josephson inductance and capacitance, and have added a shunt capacitance
and a “voltage source.” Note that the red part of the circuit is only connected to the rest
of the circuit via the junction, and can only exchange integer numbers of Cooper pairs with
the rest of the circuit via Josephson tunneling.
islands separated by an insulating barrier forming a Josephson junction. If the temperature
is much lower than the superconducting gap kB T 2∆BCS , all single particle excitations
are exponentially suppressed and the only free parameters in our system are nL and nR , the
number of Cooper pairs on the left/right islands respectively. Since charge is conserved, we
may simply consider the difference in Cooper pair number n = nL − nR , and label the state
of the system as |ni. We define the Cooper pair number operator:
n=∞
n= (2.12)
X
n |ni hn|
n=−∞
E E
which acts on a Cooper-pair number eigenstate n n0 = n0 n0 . Let’s build a phenomenolog-
ical Hamiltonian that captures the physics of our idealized system: a Cooper pair can tunnel
from left to right (taking |ni to |n − 1i), or from right to left (taking |n − 1i to |ni). Say
that there’s some energy EJ associated with tunneling across the junction. A Hamiltonian
15
that describes the coherent tunneling of Cooper pairs can be written down as:
E n=∞
HJ = − J |n − 1i hn| + |ni hn − 1| (2.13)
X
2 n=−∞
The first term describes a Cooper pair tunneling from left to right (n = nL −nR decreases
by 1) and the second a Cooper pair tunneling from right to left. The sum in the Hamiltonian
must run from −∞ to ∞, since we may have a disequilibrium of Cooper pairs biased to the
right side of the junction just as easily as to the left side. More abstractly, this Hamiltonian
describes (discrete) translations in the number n, and therefore we should expect the basis
that diagonalizes this Hamiltonian to be the (discrete) Fourier conjugate to |ni. Indeed, if
we plug in the ansatz wave function
n=∞
|ϕi = einϕ |ni (2.14)
X
n=−∞
we find
E X 0
HJ |ϕi = − J |n − 1i hn| + |ni hn − 1| ein ϕ n0
E
2 0
n,n
E
=− J ei(n+1)ϕ |ni + ei(n−1)ϕ |ni
X X
2 n n
= −EJ cos ϕ einϕ |ni
X
n
HJ |ϕi = −EJ cos ϕ |ϕi (2.15)
Since we said ϕ looks a lot like the Fourier conjugate to n, we could guess that we could write
down |ni in the |ϕi basis by taking an inverse Fourier transform of ϕ. Since n is discrete, ϕ
16
is only defined mod 2π:
Z 2π Z 2π n0X=∞ 0 n0X
=∞
dϕe−inϕ |ϕi ei(n −n)ϕ n0 δn,n0 n0
E E
= dϕ = 2π
0 0
n0 =−∞ n0 =−∞
1 Z 2π
|ni = dϕe−inϕ |ϕi (2.16)
2π 0
So we see, in fact, that our guess was correct. Given that ϕ is the Fourier conjugate of the
Cooper pair number n, they form a set of canonically conjugate variables. Since n = Q/2e,
where e is the electron charge, we see that ϕ is directly proportional to the node flux φ across
the junction that we defined earlier
φ
ϕ= mod 2π
Φ0
(2.17)
~
Φ0 =
2e
where Φ0 is the reduced superconducting flux quantum. Reminding ourselves of the definition
of φ in terms of the voltage between two nodes, this equivalence leads to the second Josephson
relationship Eqn. 2.1(b). The equivalence between the node flux φ and the phase difference
ϕ leads to a physical interpretation of Josephson junctions that will guide our discussion:
Eqn. 2.15 implies that a Josephson junction acts like a non-linear inductor whose energy,
rather than being quadratic in the flux ϕ across the element, is ∝ cos ϕ.
2.2.3 The Cooper pair box
The above picture of a Josephson junction was oversimplified: we failed to take into account
the electrostatic energy associated with a charge imbalance between the two sides of the
17
Josephson junction. Examining Figure 2.2(a), we see that in addition to being a nonlinear
inductor, a Josephson junction consists of two metal electrodes that will have some mutual
capacitance CJ . Given our prior work quantizing the LC oscillator, it will behoove us to
consider electrostatic energy of Cooper-pair disequilibrium in terms of the charging of the
capacitor formed by the two superconducting leads.
Consider the setup described in Figure 2.2(b): here, we’ve added in a shunt capacitor
CS which completely isolates the charge carriers in the red conductor (the “Cooper pair
box”) from the charge carriers in the rest of the circuit except by tunneling through the
junction. We also have a voltage source in the black part of the circuit, which serves to
bias the red island relative to the black part of the circuit via the shunting capacitor. This
voltage source may either describe a deliberately applied electric field or the microscopic
electrostatic environment of the junction.
In this circuit, we see that any tunneling events will cause a buildup of charge on the
capacitor plates. If we write CΣ = CS +CJ , we can write down the electrostatic contribution
to the Hamiltonian that comes from the capacitance CΣ
U = 4Ec (n − ng )2 (2.18)
e2
Ec =
2CΣ
Cg V
ng = −
2e
Here, EC is the charging energy of one electron, i.e. the electrostatic energy associated with
taking an electron from one side of the capacitor and moving it to the other (the factor of
18
4 comes from the fact that we’re dealing with Cooper pairs that have q = 2e.) We have
encapsulated the voltage source within an offset charge ng : a change in V (either deliberately
or by microscopic fluctuations) will shift the equilibrium charge distribution, and thus modify
the Hamiltonian. Note that while the number of Cooper pairs on the island is quantized (by
virtue of our capacitor plates being isolated except through the Junction, which only integer
numbers of Cooper pairs can tunnel across), ng is a continuous variable.
Taking into account both the electrostatic energy and the Josephson tunneling energy,
we write down the full Hamiltonian for our first prototype qubit: the Cooper pair box (CPB)
HCP B = 4Ec (n − ng )2 − EJ cos ϕ (2.19)
If we write the number operator in the phase basis, n = i∂/∂ϕ, the time independent
Schödinger equation for this Hamiltonian becomes a differential equation called Mathieu’s
equation, and is exactly solvable in terms of Mathieu functions [34]. Plotted in Figure 2.3(a)
are several low-lying eigenvalues of the CPB Hamiltonian as a function of of the offset charge
ng for the ratio of Hamiltonian parameters EJ /EC = 1.
The CPB was, in fact, the first superconducting quantum bit in the sense that it was
the first superconducting circuit in which it was demonstrated that a coherent superposition
of two macroscopic states of the system could be prepared [29]. The nonlinearity of the
Josephson junction shows up clearly in Figure 2.3(a): the energy levels are far from equally
spaced at all values of ng . Examining Figure 2.3(a), one sees the remnants of the parabolic
dependence of the energy bands on offset charge for each integer value of Cooper pair. In
the absence of tunneling, each parabola would be completely uncoupled to the adjacent
parabola, however the CPB spectrum has a strong avoided crossing at ng = 1/2 caused
19
Figure 2.3: Eigenvalues of the Cooper pair box Hamiltonian: Eigenvalues of the
Cooper pair box Hamiltonian, as a function offset charge ng for various ratios EJ /Ec . As
the ratio EJ /Ec goes up, charge dispersion is exponentially suppressed, but anharmonicity
is also suppressed. Reproduced from Ref. [34].
by the Josephson tunneling term. In Ref. [29], coherent oscillations between Cooper-pair
number eigenstates were observed by nonadiabatically tuning the gate voltage from a point
far from ng = 1/2, where the CPB was likely to be in the |ni = |0i state, to ng = 1/2 where
√
the eigenstates of the CPB Hamiltonian are (|0i + |1i)/ 2. This was a proof of principle
that coherent manipulations could be executed on the two lowest energy levels of the CPB.
This, coupled with proposals to read out the charge eigenstates of the CPB using a single
electron transistor [35], was at the time a promising route of pursuing solid state quantum
computing.
20
2.2.4 Energy scales
Up until this point, we have written down energies EJ and EC without considering the scale
of these energies. To progress further, we will need to consider the characteristic energy
scale of Josephson junction circuits. This will also define a characteristic temperature at
which superconducting circuit experiments must be performed, and prelude some of the
experimental considerations we will encounter in Chapter 3.
The only energy scale that has entered the picture so far is the superconducting gap2
∆BCS (T = 0) ≈ 1.76kb Tc , where Tc is the critical temperature of the superconductor
[21]. Pragmatically, coherent control of a qubit system with |0i → |1i transition energy
(E1 − E0 ) = ~ω01 is achieved by applying coherent electromagnetic radiation at frequency
ω01 to the circuit. This presents a hard upper bound to the characteristic energies of the
system ~ω01 2∆BCS , else the control signals at ω01 will carry enough energy per-photon
to break Cooper pairs and cause dissipation [21]. Fortunately, it is fairly straightforward to
engineer Josephson junctions such that EJ ≤ ∆BCS .
For commonly used aluminum, ∆BCS /h ≈ 40 GHz, and thus superconducting circuits
made from aluminum should be operated at frequency 80 GHz. Modern telecommunica-
tion technology has made microwave electronics in the 3 − 10 GHz range readily available
at reasonable expense, and engineering microwave setups at these frequencies is somewhat
more forgiving than at higher frequencies. Thus, we will quote the “typical” energy of a
superconducting qubit ω01 /(2π) ∼ 5 GHz.
In addition to anharmonicity, another prerequisite for coherently operating a quantum
circuit is that the system of interest can be reliably initialized in a known fiducial state,
2 The temperatures we work at in this thesis are all much less than the superconducting transition temper-
ature Tc of aluminum, where ∆BCS (T ) ≈ ∆BCS (0). In light of this, I’ll drop the T dependence of ∆BCS ,
and quote only the zero temperature value.
21
and that spurious excitations over the course of an experiment are rare. Both of these
prerequisites are satisfied if the temperature of the experiment is low enough such that
kb T ~ω01 . In this case, thermal excitations of the qubit are exponentially suppressed in
temperature, and initializing the qubit into the |0i simply becomes a matter of waiting for
it to thermalize with its environment. Converting to temperature, a transition frequency
of 5 GHz corresponds to T ≈ 240 mK: superconducting qubit experiments require rather
cold temperatures! Fortunately, commercially available dilution refrigerators routinely reach
base temperatures < 10 mK, and thus an ambient temperature that satisfies kb T ~ω01 is
fairly easy to achieve.3
2.2.5 The transmon
A major problem with early implementations of the CPB is apparent from Figure 2.3: for low
EJ /EC the qubit frequency ω01 is a strong function of the offset charge ng . The information
in a qubit is not only encoded in the relative amplitudes of |0i and |1i, but also in the relative
√
phase. Recall how a quantum two-level system initially in |ψ(0)i = (|0i + |1i)/ 2 evolves
as a function of time
1
|ψ(t)i = √ (|0i + e−iω01 t |1i) (2.20)
2
If ω01 is constant, this phase factor evolves deterministically, and we may account for it
by working in the frame rotating with the qubit state vector. However, we see that, for low
EJ /EC , ω01 is not constant: it depends strongly on the offset charge ng , and changes by
large values over a fraction of a Cooper pair. As stated previously, the offset voltage V also
3 As we shall see in future chapters, cooling everything down to the nominal temperature of the cryostat
isn’t so straightforward.
22
serves to model the local electrostatic environment of the CPB. It is clear then that small
fluctuations in the electrostatic environment change the transition frequency ω01 , and thus
cause the relative phase of |0i and |1i to evolve nondeterministically. This implies that CPB
qubits are extremely sensitive to charge noise.
From Fig. 2.3, it is abundantly clear that a solution to this problem is to work in the
large EJ /EC regime, where the charge dispersion is suppressed. We can accomplish this by
making the shunt capacitance CS very large compared to the intrinsic Josephson junction
capacitance CJ , driving the capacitive charging energy EC down and the ration EJ /EC up.
A CPB in the parameter range EJ /EC ≈ 50 − 70 is called a “transmon” qubit [34], and will
be the main superconducting circuit featured in this thesis.
One immediate drawback we can from see Fig. 2.3 is that in the large EJ /EC regime the
anharmonicity becomes weaker: as we increase EJ /EC , the circuit begins to look more and
more like a harmonic oscillator. To understand why transmons have low anharmonicity, we
can map the Hamiltonian Eqn. (2.19) onto a more familiar problem: a rigid pendulum of
length l and mass m experiencing a gravitational force g [34, 31]. The Hamiltonian for this
system is
L2z
Hpend = − mlg cos ϕ (2.21)
2ml2
Here, ϕ becomes the angle that the pendulum makes with respect to it’s equilibrium
position (consistent with our demand that ϕ only be defined mod 2π), and (ignoring the
offset charge for now) n = Lz /~ becomes the conjugate (angular) momentum.
In this model, EC = ~2 /8ml2 is inversely proportional to the pendulum’s moment of in-
ertia, and EJ = mgl is the magnitude of the gravitational potential energy. In the transmon
23
Figure 2.4: A mechanical analogue to the Cooper pair box Hamiltonian: (a) An
analogue to the Cooper pair box Hamiltonian: a rigid pendulum of length l and mass
m living in a gravitational field. To account for the offset charge ng , we also give the
pendulum some fictitious charge q and tack on a vector potential to the Hamiltonian by
threading a flux Φt though a tube at the pendulum’s hinge. (b) Josephson-junction cosine
potential (light blue) and eigenstates (eigenenergies dotted, wave functions solid) compared
to a harmonic oscillator potential (dashed orange) and eigenenergies (dotted orange). In the
transmon regime, large EJ makes the circuit act like an “almost harmonic oscillator” (i.e.
a quantum pendulum in the small angle regime) and exponentially suppresses tunneling to
into neighboring wells (a complete loop of the pendulum) through which charge noise enters.
Inspired by Figure 3 of [34].
regime EJ /EC 1, the mechanical Hamiltonian is dominated by the gravitational potential
energy term, and we expect the rotor to remain in the almost harmonic small angle regime.
In the small angle limit, we may expand the cosine in the CPB Hamiltonian
ϕ2 E ϕ2
H = 4EC n2 − EJ (1 − + O(ϕ4 )) ≈ 4EC n2 + J (2.22)
2 2
where we’ve dropped both the constant offset and all of the anharmonic higher order terms.
This is exactly the quantum harmonic oscillator we considered in §2.2.1, with n = Q/(2e)
and ϕ = φ/Φ0 , and we may recycle the machinery from that discussion:
Q2 φ2
H≈ + (2.23a)
2CΣ 2LJ
24
Φ20
LJ = (2.23b)
EJ
Here, we reintroduce the variable CΣ and introduce the Josephson inductance LJ . We
then write down the Hamiltonian as ~ωJ (a† a + 1/2), with the raising and lowering operators
defined exactly as in section 2.2.1. We may just copy and paste the results we already
obtained:
1 1q
s
ωJ = = 8EJ EC (2.24a)
LJ CΣ ~
s s
LJ ~ EC
ZJ = = 2 (2.24b)
CΣ e 2EJ
s !1
~ZJ 2EC 4
φZP F = = Φ0 (2.24c)
2 EJ
s !1
~ EJ 4
QZP F = = 2e (2.24d)
2ZJ 32EC
Eqn. (2.24a) defines the so called “Josephson plasmon frequency”, which is simply the
frequency of harmonic oscillation of a Josephson junction modeled as a parallel capacitance
and inductance. ZJ is the characteristic impedance of the junction: for a sense of scale,
~/e2 ≈ 4 kΩ. Eqns. (2.24c) and (2.24d) are the magnitude of the zero point flux and charge
fluctuations respectively, with the proper normalization factors to convert φ ↔ ϕ and
Q ↔ n pulled out. From these relations, we can immediately read off that, for EJ /EC 1
the zero point fluctuations in the charge are larger than a Cooper pair, and the zero point
fluctuations in the phase are small. Since this entire argument hinges on the small angle
25
approximation, it is comforting that φZP F is less than unity. It is interesting to note that,
even though the transmon is derived from the CPB “charge” qubit4 , the Cooper pair number
n clearly isn’t a good quantum number. The physical qubit states are the plasma oscillations
of the superconducting condensate across the Josephson junction.
The small anharmonicity is, however, crucial to our ability to use transmons as a qubit.
To add the anharmonicity back into the equation, consider the O(ϕ4 ) term as a perturbation
to the harmonic oscillator Hamiltonian:
!4
E ϕ4 1 ΦZP F †
H(4) =− J = −EJ (a + a) (2.25)
4! 24 Φ0
where I’ve used the rules laid out in Section 2.2.1 to write down ϕ in terms of the raising and
lowering operators. After some algebra, we reduce this to terms diagonal in the unperturbed
Hamiltonian5 :
Ec † †
H(4) = − (a a aa + 2a† a) (2.26)
2
Then the total Hamiltonian, to the level of first order perturbation theory, is
EC † †
H ≈ (~ωJ − EC )(a† a + 1/2) − a a aa (2.27)
2
The first excited state energy is renormalized to
q
~ω01 ≈ 8EJ EC − EC (2.28)
4 Superconducting qubits tend to be classified by their good quantum numbers, i.e. what basis the Hamil-
tonian is diagonalized in during normal operation. The CPB is a charge qubit in that it is intended to be
operated in a parameter regime where the Hamiltonian is diagonalized in the charge basis (away from the
avoided crossings). Phase and flux qubits are also common classifications.
5 Terms non-diagonal in the unperturbed Hamiltonian may be ignored because of the rotating wave ap-
proximation, see Appendix A. This is only valid if EC ~ωJ , which is always true in the transmon regime.
26
and the second term gives us the anharmonicity α = ω12 − ω01 ≈ −EC . As stated, before,
the desired EJ /EC ∼ 50 ratio is yielded by increasing the shunt capacitance CS to drive
down EC . Typically CS ∼ 70 fF, while the intrinsic Josephson capacitance is typically
. 1 fF. Fortunately, for CΣ ≈ 70 fF, EC /h ≈ 300 MHz, and thus for a transmon qubit
with ω01 /(2π) ∼ 5 GHz, the anharmonicity remains an appreciable fraction of the transition
frequency, and sufficiently large to address only the two-level subspace |0i and |1i.
2.2.6 Where did the charge noise go?
To build intuition for why transmon qubits are insensitive to charge noise, let’s go back to
the quantum pendulum analogue. Remember that, when we made the analogy with the
pendulum, we ignored the offset charge ng in the CPB Hamiltonian. To add the charge
offset into the mechanical analogue, we may give the rotor a fictitious charge q and thread a
magnetic flux Φt through a tube at the hinge of the pendulum. To be clear, the magnetic field
in this toy model only exists inside the tube: the rotor (with its fictitious charge positioned
at the end of the pendulum) never experiences any Lorentz forces. However, the presence of
the magnetic flux will cause a nonzero vector potential in the vicinity of the charge6 :
A~ = Φt ~z × ~r (2.29)
2πr
The presence of the vector potential takes p~ → p~ − q A, ~ which takes the angular momentum
to
~ z ⇒ ~ − i ∂ − q Φt
!
~ z = (~r × p~)z − q(~r × A)
L (2.30)
∂φ 4πe Φ0
6 It’s worth repeating this for clarity, since things can get confusing here: the charge q on the pendulum is
a completely fictitious charge living within the picture of the rotor analogue, with no relation to the actual
charges in the superconductor. It exists solely to interact with the fictitious flux Φt
27
If we plug the new angular momentum into Eqn. (2.21) and compare it to the original Cooper
pair box Hamiltonian, we immediately see that
ng ∝ Φt (2.31)
This is curious. Classically, the magnetic flux completely confined to the tube at the hinge
of the rotor could never interact with the charge at the end of the pendulum. However,
quantum mechanically, we know that a charged particle that traverses around a magnetic
flux may pick up an Aharanov-Bohm phase, even if it never experiences a Lorentz force [36].
Therefore, the only way for the particle to possibly be influenced by the flux Φt is for it to
completely traverse the loop. Since a simple phase doesn’t effect the spectrum of a quantum
system, the only way the qubit spectrum can depend on Φt is via interference between the
path where the rotor didn’t do a loop and the path where it did.
In the transmon regime EJ EC , however, we have seen that at low excitation number
our rotor will be well confined to small angles: the “force of gravity” is strong and the “mass”
of the rotor (moment of inertia) is large. In order for the rotor to undergo a so called phase
slip (a loop around the flux that changes the equilibrium value of φ by 2π), the particle (with
mass ∝ 1/EC ) must tunnel over a barrier whose height is proportional to EJ . However, the
probability of tunneling over a large potential energy barrier is exponentially suppressed in
the barrier height (EJ EC ), and since the only way the energy eigenvalues can depend
on ng is via interference between paths with different winding numbers around the hinge,
the transmon’s dependence on the offset charge ng is also exponentially suppressed [34].
Transmon qubits were first proposed in Ref. [34] as a charge qubit capacitively coupled to
a coplanar waveguide resonator, and were quickly used to experimentally demonstrate (at the
28
time) impressive results in the dispersive regime of circuit quantum electrodynamics [37, 38,
39]. Since then, material and design parameters have improved to the point where transmon
qubits commonly have coherence times approaching or exceeding 100 µs [40, 41, 42, 43, 44].
As such, the transmon qubit has become a workhorse platform for demonstrating proof
of principal quantum computing applications [45, 46, 47, 48, 49, 50, 51, 52, 53, 54] and for
conducting “microwave quantum optics” experiments [55, 56, 57, 58, 59, 43]. While transmon
qubits have rapidly advanced in coherence to the point where many complex protocols can
be performed on them, it is clear that further suppression of loss mechanisms is needed
for scalable quantum computing applications, and research on decoherence mechanisms in
superconducting qubits remains an active field. We will discuss decoherence in greater detail
in Chapter 4.
2.3 Circuit quantum electrodynamics
While making the transmon insensitive to charge noise greatly increases possible coherence
times, it also invalidates early proposals for control and readout mechanisms of the CPB,
which relied heavily on the different electrostatic configurations of various Cooper pair num-
ber states. The transmon qubit is, however, exponentially insensitive to its local electrostatic
environment, rendering these schemes useless. Therefore, we need a new scheme to read out
and manipulate transmon qubits.
Circuit quantum electrodynamics (cQED) [31, 33, 60, 61, 62, 63, 64], is an experimental
protocol which allows us to manipulate and read out charge insensitive qubits while simulta-
neously protecting the qubit from what would be its dominant decay channel: spontaneous
decay into the electromagnetic vacuum. In cQED, we place the qubit into a high finesse
29
microwave cavity with resonant frequency ωc ∼ ω01 . This has two advantages: the cavity
shields the electromagnetic environment of the qubit, greatly reducing the density of states
available for the qubit to decay into [65, 40], while the confined discrete electromagnetic mode
of the cavity strongly couples to the qubit, allowing for fast manipulation and readout.
2.3.1 Putting together the building blocks of cQED
In essence, cQED is the combination of two of the basic components we’ve already built up:
a harmonic oscillator (which we’ll model as an LC circuit) and a qubit (which we’ll take
to be a transmon.) The model here roughly follows [66, 64, 67]: in real experiments, the
harmonic oscillator is replaced with an electromagnetic cavity, which has infinite discrete
modes (see Chapter 3 for more details.) However, if the detuning ∆ between the qubit and
the fundamental mode of the cavity (|∆| = |ω01 − ωc |, where ωc is the cavity frequency) is
much smaller than the detuning between the qubit and any other mode, the single mode
model considered here is an excellent approximation.
2.3.1.1 Driving the system
We begin by considering the circuit in Fig. 2.5 composed of a transmon qubit (blue) capac-
itively coupled to both a driving field (black) and a harmonic oscillator (orange.) To start
off, let’s ignore the orange harmonic oscillator and see how the driving field couples to the
qubit. In this sub-circuit, we identify two nodes with node fluxes φ1 and φD . We can write
down the Lagrangian for this subcircuit with ease by adding up the capacitive and inductive
energies component-wise
1 1 1 2
L ≈ CΣ φ̇21 + CD (φ̇1 − φ̇D )2 − φ (2.32)
2 2 2LJ 1
30
CD Cg
CJ
Cs Ch Lh
LJ
Figure 2.5: Schematic of circuit QED: A circuit schematic of the circuit quantum eletro-
dynamics (cQED) protocol. A qubit (blue) is coupled to a microwave frequency electromag-
netic cavity, modeled as an LC oscillator (orange). We apply microwave qubit control and
readout pulses through an external drive (modeled as a voltage source in black.) Here, we
take the qubit to be a transmon qubit, and the coupling to the cavity/drive to be capacitive.
where, for now, we’ve dropped the (small) nonlinearity of the Josephson junction potential.
The Lagrangian at first appears to be a function of two dynamic variables, however we notice
immediately that φ̇D = V (t), and thus φD is completely constrained by the applied input
voltage. To find the canonical momentum to φ1 , let’s set VD (t) = 0 and write down the
Hamiltonian
Q21 φ2
H≈ + 1 , Q1 = φ˙1 (CΣ + CD ) (2.33)
2(CΣ + CD ) 2LJ
The canonical charge (and the capacitive energy) is renormalized by the presence of
the coupling capacitance, as we should expect. Now, the terms in the Lagrangian that
incorporate the presence of the drive are
1 1
Ld = CD φ̇2D − CD φ̇D φ̇1 = CD VD (t)2 − CD VD (t)φ̇1 (2.34)
2 2
The first term contains no dynamic variables, and we may ignore it. The second term
31
encapsulates the effect of the drive on the qubit: if we assume weak coupling to the drive,
we may write it down in terms of the canonical charge and add it back into the Hamiltonian
as a perturbation
Q21 φ2 CD
H= + 1 − V (t)Q1 (2.35)
2(CΣ + CD ) 2LJ CΣ + CD D
As we have done multiple times now, we quantize this Hamiltonian in the harmonic
† †
oscillator basis, taking Q1 = −iQZP F (aq − aq ), where aq (aq ) is the raising (lowering)
operator for the transmon circuit. Doing so gives us
1
H = ~ωq (aq† aq + ) + (t)(aq − aq† ) (2.36)
2
where ωq = (LJ (CΣ + CD ))−1/2 ≈ (LJ CΣ )−1/2 for weak coupling to the driving and
iQZP F CD VD (t)
(t) = (2.37)
CΣ + CD
is the amplitude of the drive. This analysis gives us a mathematical framework by which
we may analyze how an applied (classical) electric field7 couples to a qubit (or a harmonic
oscillator.) In fact, this framework of driving the qubit with classical electric fields is the
basis of coherent control, where we use classical fields to manipulate the state of a quantum
system. We will discuss coherent control more in Chapter 3.
2.3.1.2 Coupling a qubit and a harmonic oscillator
We now turn to the case of the qubit coupled to the harmonic oscillator. The analysis here is
exactly the same, however instead of a classical drive, we note that even when the harmonic
7 In the literature, the applied field term is typically written down as ∝ a† + a . The sign change is simply
q q
a convention that depends on whether we took Q or φ to be the canonical coordinate.
32
oscillator is in its ground state, there are zero point voltage fluctuations. To see this, we can
simply take the time derivative of the flux operator φh using the Heisenberg equations of
motion [31]
dφh i Q
Vh (t) = = [H, φh ] = h = −iVZP F (ac − ac† ) (2.38)
dt ~ Ch
s
Q ~Z0
VZP F = ZP F = ωc (2.39)
Ch 2
†
where ac (ac ) are the raising (lowering) operators for the harmonic oscillator cavity mode,
and ωc = (Lh Ch )−1/2 is the harmonic oscillator resonant frequency. These zero point voltage
fluctuations will act on the qubit in the exact same way as the drive voltage VD (t). Thus,
even when the Harmonic oscillator is in the ground state, there will be some interaction
between the qubit and the harmonic oscillator. Adding back in the Josephson non-linearity,
we may write down the composite transmon + harmonic oscillator + coupling Hamiltonian
in the same form as Eqn. 3.12, replacing the applied voltage with the zero-point voltage
fluctuations of the cavity
1 1 Cg
H = ~ωq (aq† aq + ) + O(φ41 ) + ~ωc (ac† ac + ) − V (t)Q1 (t)
2 2 CΣ + Cg h
(2.40)
1 1
= ~ωq (aq aq + ) + O(φ1 ) + ~ωc (ac† ac + ) − ~g(aq − aq† )(ac − ac† )
† 4
2 2
where I have absorbed all the capacitances and zero-point fluctuations into the constant
g known as the vacuum Rabi constant. We now make two approximations:
1. We invoke the nonlinearity of the transmon circuit to constrain ourselves to the qubit
33
manifold, i.e the transmon |0i and |1i states. When we think of quantum two-level
systems, the canonical example that comes to mind is a spin-½ system. We there-
fore import the language of spins to model the effective two-level transmon qubit, and
replace the harmonic oscillator-like operators with Pauli operators: the qubit rais-
†
ing operator aq becoms σ + , the lowering operator aq becomes σ − , and the number
†
operator aq aq + 1/2 maps onto σ z /2.
2. We invoke the rotating wave approximation and say that, if g ωq , ωc counter-rotating
† † †
terms such as ac aq (or, equivalently, ac σ + ) will oscillate quickly over the timescale
of system evolution, and will thus average out to zero (see Appendix A for a more
complete discussion of the rotating wave approximation). We can therefore drop the
†
ac σ + and ac σ − terms.
Upon making these approximations, we arrive at the famous Jaynes-Cummings Hamil-
tonian, describing the interaction of a two-level atom with an electromagnetic mode:
~ωq 1
HJC = σ z + ~ωc (ac† ac + ) + ~g(ac† σ − + ac σ + ) (2.41)
2 2
2.3.2 The Jaynes-Cummings model
The Jaynes-Cummings (JC) model is a powerful theoretical tool for exploring the interaction
of electromagnetic radiation with superconducting qubits. When deriving it, we already
got a taste of how we may manipulate qubits with coherent electromagnetic drives (see
Chapter 3 for a more complete discussion of coherent control.) We will now see how, within
the framework of the JC model, we may also measure the state of the qubit.
The Jaynes-Cummings Hamiltonian consists of three terms: the bare qubit energy, the
34
bare cavity (harmonic oscillator) energy, and the interaction term. We may think of the
†
interaction term as the action of the qubit exchanging an excitation with the cavity (ac σ − )
and it’s Hermitian conjugate, the cavity exchanging an excitation with the qubit (ac σ + ).
The interaction term conserves the total number of quanta in the system, so total excitation
number N = −σ z /2 + a† a is a constant of the motion and we may subtract it off. Doing so
yields
~∆
H0 = HJC − ~ωc N = σ z + ~g(a† σ − + aσ + ) (2.42)
2
where ∆ = ωq − ωc is the cavity/qubit detuning. Since excitation number is conserved, the
Hamiltonian is block diagonal with 2 × 2 blocks connecting the states |n, 0i and |n − 1, 1i,
where the quantum numbers are the resonator excitation number and the qubit state re-
spectively. We can write down the block of the Hamiltonian that acts on states in the space
{|n − 1, 1i , |n, 0i} as a 2 × 2 matrix
√
∆/2 ng
H0 /~ = (2.43)
√
ng −∆/2
This Hamiltonian is straightforward to diagonalize, and yields two coupled qubit-photon
eigenstates
|+, ni = cos θn |n − 1, 1i + sin θn |n, 0i (2.44a)
|−, ni = − sin θn |n − 1, 1i + cos θn |n, 0i (2.44b)
√
1 −1 2g n
θn = tan (2.44c)
2 ∆
with energy levels
35
q
E±,n = ±~ ng 2 + (∆/2)2 (2.45)
There are several important features of this result to highlight:
1. There are only two parameters that characterize the system: the cavity/qubit detun-
√
ing ∆, and the cavity/qubit coupling g n, which depends on the number of quanta
(photons) in the cavity.
2. When ∆ = 0 (i.e. the cavity and the qubit are on resonance), θn = 45◦ , and the
eigenstates become equal and orthogonal superpositions of |n − 1, 1i and |n, 0i. These
√
states are not degenerate: they undergo an avoided crossing and are split by 2~g n,
as shown in the Fig. 2.6. These states are neither qubit nor cavity states: they are
hybridized states, consisting of both cavity and qubit in equal weight. When there is
only one excitation in the system (n = 1), we can interpret this splitting as a quantum
of energy being freely exchanged between the harmonic oscillator and the qubit at a
rate 2g, hence the name Vacuum Rabi splitting.
√
3. In the limit where the ∆ >> g n, the eigenstates of the Hamiltonian are almost pure
qubit/cavity states, but not quite. We take this to mean that for example, a qubit
excitation spends some time as a photon in the cavity. Along the same lines, the energy
eigenvalues of the Hamiltonian are almost the bare qubit and cavity energies ωq and
ωc , but not quite. We call these shifted eigenstates “dressed” states.
36
(a) (b)
Figure 2.6: Cavity-qubit avoided crossing: (a) Energy eigenvalues of the Jaynes-
Cummings Hamiltonian plotted as a function of the cavity/qubit detuning ∆ for n = 1.
The bare qubit/cavity energies are plotted as dashed lines And the JC Hamiltonian eigen-
states as colored lines: the more red the line is, the more “cavity-like” the eigenstate is, and
the more blue the more “qubit-like”. Near ∆ = 0, the eigenstates become equal and orthog-
onal superpositions of the cavity/qubit, and the eigenvalues undergo an avoided crossing.
(b) Spectroscopy of a coupled electromagnetic cavity + tunable transmon qubit system. As
the flux bias is tuned, the transmon resonant frequency passes through the (constant) cavity
mode, and the two systems undergo an avoided crossing. In this data, g/(2π) ≈ 85 MHz.
2.3.3 The strong dispersive limit and readout in cQED
We now turn our attention to the dispersive regime of the cQED, where the qubit and cavity
are detuned by much more than their interaction strength (g/∆ << 1) and the eigenstates
of the JC Hamiltonian are the “dressed” cavity and qubit states. In this regime, rather than
exactly diagonalizing the JC Hamiltonian, it is useful to expand the Hamiltonian in powers
of g/∆. Expanding to second order (see [31] for a derivation), we find
g2 1
~
HJC,disp ≈ ~ ωc + σ z (a† a + ) + ωq σ z (2.46)
∆ 2 2
This Hamiltonian looks like the uncoupled cavity/qubit energies, except the cavity energy
has been shifted by an amount that depends on the state of the qubit. More precisely, the
37
2χ
Transmission amplitude Transmission phase
2χ
ωc ωc
Measurement Frequency Measurement Frequency
Figure 2.7: Cavity dispersive shift: A signal transmitted through an electromagnetic
resonator typically takes a Lorentzian shape as a function of frequency, centered on the
resonant frequency ωc with lindwidth κ. If the qubit induced dispersive shift χ & κ, the
effective resonant frequency of the cavity ωc ±χ will shift appreciably and we may distinguish
the state of the qubit by measuring the amplitude (a) or phase (b) of the signal transmitted
through the cavity. Typically, the phase signal as a higher sensitivity [68, 69].
cavity resonant frequency will be shifted from its bare frequency by χ = ±g 2 /∆ if the qubit
is in its ground/excited state. If the cavity linewidth κ is small enough such that χ ≥ κ, the
amplitude (or phase) of a near-resonance microwave signal transmitted through (or reflected
off) the cavity will change appreciably depending on the state of the qubit, as shown in the
Figure 2.7. We also require χ ≥ γ1 , the decay rate of the qubit, or else the qubit will decay
faster than the measurement time τ ∼ 1/χ required to discern between the two states of the
cavity. If these two conditions are met, the signal transmitted through the cavity near the
resonant frequency ωc imparts information about the state of the qubit, and we may use the
38
transmission through the cavity to measure the state of the qubit.8 .
With the nonlinearity required to constrain ourselves to a two-state system given by the
Josephson junction, and the ability to manipulate and measure this qubit system given by
the cQED architecture, incredible progress has been made in using cQED as a basis for
quantum information processing. The transmon, first realized in 2007 [34, 38, 72], is now a
workhorse device for large number of quantum computing experiments [73] including a proof
of principle computation demonstrating the ability of a superconducting quantum processor
to perform a computation faster than the world’s most powerful classical supercomputer
[54]. These experiments and demonstrations all employ what are fundamentally scaled up
versions of the circuit presented in § 2.5: superconducting transmon qubits, controlled and
read out via cQED.
8 In fact, dispersive measurements in cQED are sufficient for executing quantum nondemolition (QND)
measurements of the qubit energy eigenbasis. A QND measurement is a “textbook” ideal measurement: the
back action of the measurement on the system being measured only projects the system into the observable
eigenstate, such that the system could repeatedly be measured producing the same result over and over
again [70]. When discussing QND measurements, it is convenient to divide the full device into a “system”
with Hamiltonian Hsys which we wish to measure, a “meter” with some Hamiltonian Hmeter with which
we will measure the system, and an interaction term Hint that describes the interaction between the system
and the meter [70, 71]. In this division, Htot = Hsys + Hmeter + Hint and the condition for a QND
measurement of some observable A is that [Htot , A] = 0. We can make this division easily in Eqn. 2.46,
identifying the qubit as the “system”, the bare Harmonic oscillator as the “meter”, and the dispersive shift
as the interaction between them. Since [HJC,disp , σ z ] = 0, a measurement of the cavity will project the
qubit into an eigenstate of the Hamiltonian, satisfying the condition for a QND measurement.
39
Chapter 3
Superconducting qubits in the lab
In our description of superconducting qubits and circuit quantum electrodynamics in Chap-
ter 2, our discussion largely centered around ideal circuits, with difficulties like loss or deco-
herence only referenced tangentially. Arguably the most important advancement of quantum
information science in the past 15 years has been the rapid development of experimental
methods and technologies by which we can take these idealisms and bring them into the lab.
In this chapter, we tether ourselves back to reality and discuss the devices and experimental
setup by which we build and manipulate systems that (at least approximately) are described
by the physics of Chapter 2. We will also give a survey of basic experimental methods by
which we can characterize these systems.
3.1 The 3D transmon geometry
The main type of qubit employed in this work is the so called “3D transmon” qubit. In
the 3D transmon geometry, the qubit and cavity system of cQED are largely the same as
in Chapter 2. However rather than a lumped element LC oscillator, the role of the linear
electromagnetic oscillator is played by a macroscopically large waveguide cavity resonator,
typically a slot milled out of a copper or aluminum box. Fig. 3.1(a) shows a schematic of
such a 3D electromagnetic cavity, and Fig. 3.1(b) shows a picture of the bottom half of one
of these boxes.
40
3.1.1 Electromagnetic modes in 3D cavities
If we take the walls of the cavity resonator to be perfectly conducting surfaces (a good
approximation for copper, and an excellent approximation for aluminum well below it’s
superconducting transition temperature of 1.2 K), we may approximate the structure as a
3D rectangular box subject to the boundary conditions that the tangential components of
the electric field E
~ must go to zero at the walls of the box. These boundary conditions
force solutions to the electromagnetic wave equation to be built out of discrete Fourier
components, each of which will have an associated resonance. A box with dimensions Lx , Ly
and Lz bounded by conducting walls (see Fig. 3.1) admits classical electromagnetic modes
at frequencies [74]
v
u !2 !2 !2
u mπ nπ lπ
ωmnl = c t
+ + (3.1)
Lx Ly Lz
where m, n and l are the integer mode numbers, and c is the speed of light in the cavity
that depends on the dielectric constant of the material inside of the cavity. If the quality
factor of each mode Q = ω/κ 1, each mode may be modeled as a noninteracting harmonic
oscillator that, when quantized, will act as a single electromagnetic harmonic oscillator
that can couple to the qubit. Upon quantizing, we may write down the Hamiltonian of
the electromagnetic field in the cavity as a sum of discrete, uncoupled harmonic modes
†
HEM = n ~ωn an an [75]. We typically work in the regime Lx & Lz Ly , such that the
P
lowest frequency, or fundmental, mode is the T E101 mode, with m, l = 1, n = 0. We choose
this mode to be the “cavity” mode from cQED: since ω01 /(2π) is typically 4 − 8 GHz, the
41
cavity mode is generally engineered to be in the 5 − 9 GHz range1 , meaning the long axes of
cavity are roughly 2−3 cm. The electric field profile for this mode is plotted in Fig. 3.1(c): to
maximize qubit and cavity coupling, the qubit should be as close to the center of the cavity
as possible (where the electric field is maximal) and the capacitor pads that provide coupling
to the cavity should be aligned with the electric field (to maximize voltage fluctuations across
the junction, see § 3.1.2.)
It is reasonable to ask whether or not the assumption that we may ignore the higher
modes is valid. The presence of higher modes will renormalize the spectrum and coupling
between the qubit and the fundamental mode [78], and may contribute to qubit dephasing
(caused by fluctuating mode occupation [76]) and depolarization (via decay into the finite
density of states associated with those modes, i.e. Purcell emission [65].) However, there
are several reasons these concerns are alleviated in our experiments. If Lx ≈ Lz , the second
q
lowest frequency modes (T E102 and T E201 ) will be of order (5/2)f0 , see Eqn. 3.1. For
a cavity with f0 ∼ 7 GHz, this places the next-lowest modes around ∼ 11 GHz, rendering
coupling to a qubit in the 4-6 GHz range small. Furthermore, the electric field associated with
these modes (and the next-next highest T202 mode) will be close to zero at the center of the
cavity, greatly reducing the coupling to a qubit placed there, see Fig. 3.1 [41]. Additionally,
since ~ × 2π × 11 GHz > 500 mK ∼ 10 mK, the dilution refrigerator base temperature,
if the measurement lines are properly filtered and the photon bath in these modes is well
thermalized, all cavity electromagnetic modes will be in their quantum ground state with
all-but unit probability. Note that if the microwave lines that couple to the experiment are
1 We use the fundamental mode of the cavity for several reasons. Since the wavelength of light at microwave
frequencies is ∼cm scale, using higher modes would require a rather large resonator for their frequency to be
similar the qubit frequency. The fundamental mode of a distributed resonator will also, in general, have the
highest quality factor of all the modes [74, 76, 77]. There is also a significant increase in Purcell emission
when the qubit frequency ω01 is positioned between two cavity resonances, so it is generally advisable to
fabricate qubits with a slightly lower frequency than the fundamental cavity mode, see Ref. [65].
42
Figure 3.1: 3D electromagnetic cavities: (a) A CAD model of a typical 3D electromag-
netic cavity used as the linear electromagnetic resonator in our lab. The qubit sits on a
silicon substrate in the center of the cavity, and we address the cavity via two 50 Ω lines that
antenna couple to the cavity mode. (b) Picture of a copper cavity. The cavity we use is cut
in half so we may load and unload qubit devices in the center. (c) Finite element simulation
of the magnitude of the electric field for the 4 lowest frequency modes of the cavity. When
the qubit is positioned at the center of the cavity, it sits at the antinode of (i.e. is strongly
coupled to) the lowest frequency T E101 mode, and at a node of (weakly coupled to) the 3
next lowest frequency modes. For the all modes shown here, E ~ is parallel to ŷ.
improperly filtered, this assumption breaks down and the higher frequency modes can drive
significant dephasing [76].
In order to manipulate or read out the state of the cavity/qubit system, we must have
some sort of external coupling to the cavity resonator. We achieve this coupling with two
microwave connectors (typically SMA) terminated to conducting antennas, which extend into
the cavity and provide coupling between the microwave lines and the cavity. We quantify
the strength of coupling between the cavity and microwave feedlines via the cavity decay
rate κ which is the sum of internal (lossy) contributions and external loss into the feedlines
43
κ = κin + κext (3.2)
The total decay rate κ is readily extracted from cavity transmission measurements (see
section 3.3.1) as the full width at half max (FWHM) of the Lorentzian cavity response
[75]. The external coupling κext is readily controlled by the length of the input/output
antenna. Typically, signal-to-noise is limited by the signal we can collect from the cavity,
and not by the power we can apply to the cavity. Therefore we use an asymmetrically
coupled cavity, with the input microwave line weakly coupled (short antenna) and the output
(measurement) microwave line strongly coupled (long antenna) to maximize cavity decay into
the measurement line. Plotted in Fig. 3.2 is a measurement of κ versus the length of the
output coupling antenna. In these measurements, the length of the input antenna is fixed
and short enough such that, in the absence of the output pin, κ is dominated by internal
losses (i.e., κ ≈ κin for short antenna lengths.)
3.1.2 3D transmon qubits
The large mode volume of 3D microwave cavities presents an intrinsic problem when we
consider coupling to a qubit: in order for the cavity-qubit coupling g to be large, the zero
point voltage fluctuations2 of the cavity VZP F must efficiently induce voltage fluctuations
across the Josephson junction. In a 3D microwave cavity, these voltage fluctuations occur
over ∼ cm length scales and thus the voltage gradient is small: if the transmon circuit
is microscopic/mesoscopic, a large voltage will not build up across the junction, and the
2 In Chapter 2 we derived V p
ZP F = ωc ~Z0 /2 by considering the zero point fluctuations across an LC
oscillator. The final result, however, only contains the oscillator frequency and characteristic impedance, and
is true for any distributed electromagnetic mode as well. The impedance of a 3D cavity with no dielectric is
simply the impedance of free space Z ≈ 377 Ω.
44
6
4
κ (MHz)
2
0 1 2 3
output antenna length (mm)
Figure 3.2: Cavity decay rate κ vs. pin depth: Measured decay rate κ of a 3D microwave
cavity as a function of the output antenna length with the input antenna length fixed. The
length is referenced relative to the inner surface of the microwave cavity (and thus may be
negative if the pin is recessed from the cavity wall.) These measurements were taken in a
polished copper cavity at T ≈ 300 K.
cavity-qubit coupling will be suppressed.
The solution to this problem is to fabricate a large qubit [40]. A 3D transmon (see
Fig.3.3) consists of a Josephson junction connected to two coplanar ∼ mm scale pads. These
pads perform a double purpose: first, they act as the shunt capacitance CS which brings
the qubit into the transmon regime. The “typical” geometry3 shown in Fig.3.3(a), when
fabricated on silicon, provides a shunt capacitance CS ≈ 70 fF, which gives Ec ≈ 270 MHz.
Second, the pads increase the spatial extent of the qubit, increasing the efficiency with which
voltage fluctuations in the cavity induce fluctuations across the junction, and thus increasing
3 This geometry is by no means unique: in principle, any geometry with ∼ mm spatial extent that provides
the desired capacitance will suffice, see Ref. [79] for some examples. In fact, later in this thesis we will discuss
some devices with much different geometry than this. The capacitances of these structures may be readily
calculated using commercial finite element modeling software, such as COMSOL.
45
(a) (b)
Figure 3.3: The 3D transmon qubit geometry: (a) Schematic of a “typical” 3D trans-
mon geometry, inspired by the design in Ref. [40]. The qubit consists of two aluminum
pads, typically fabricated on high resistivity silicon, connected by an aluminum/aluminum
oxide/aluminum Josephson junction. The two pads simultaneously provide the shunt ca-
pacitance (≈ 70 fF for this design) to bring the device into the transmon regime and also
allow the qubit to couple strongly to the large volume 3D cavity. Bottom: picture of a 3D
transmon as it sits in an aluminum cavity. The pads are easily visible with the naked eye.
(b) The two pads are either connected by a single junction (left) or by a “split junction”
SQUID loop (right), which allows us to change the effective Josephson energy (and thus the
qubit frequency) by applying an external dc magnetic flux. Bottom: scanning electron mi-
croscope images of a single junction (left) and SQUID loop (right) fabricated by double-angle
evaporation.
the cavity-qubit vacuum Rabi splitting g. This makes for an interesting situation, where a
quantum system that hosts a collective, coherent degree of freedom we can manipulate is
visible to the naked eye!
While the whole qubit is ∼ mm scale, the Josephson junction that provides the nonlin-
earity is only ∼ 100 × 100 nm2 . Junctions are fabricated using the Dolan bridge, double-
angle evaporation process [80]; a full description of the fabrication process is given in Ap-
pendix C. When designing an experiment, perhaps the most critical parameter to consider is
√
the transmon |0i → |1i transition frequency ~ω01 = 8EJ EC − EC . As mentioned above,
46
EC = e2 /2C is controlled by the geometry of the (macroscopic) pads, and is thus fairly
easy to design and fabricate with a high degree of accuracy. The Josephson energy, however,
is more subtle: it is proportional to the critical current of the junction EJ = ~Ic /2e, and
depends on the thickness of the oxide layer and the area of the junction. Since we have no
way of readily measuring the oxide layer thickness, and the junction is small enough that
typical variations in the fabrication may cause appreciable changes in the junction area, it is
difficult to a priori predict EJ for a given design. It is, however, possible to infer the critical
current of the junction before cooling the device down: the Ambegaokar-Baratoff relation-
ship predicts that the critical current of a Josephson junction is inversely proportional to
the normal-state resistance of the junction Rn [21]
π∆BCS (0)
Ic = (3.3)
2eRn
where ∆BCS (0) = 170 µeV is the T = 0 superconducting gap of aluminum. We may
therefore determine the critical current, and thus EJ , of the junction simply by measuring the
resistance of the junction at room temperature4 . These tests are crucial for quick feedback
in the fabrication process, and also for verifying that the junctions are working, since small
junctions are notoriously easy to destroy with static discharge.
Even with normal state resistance testing at our disposal, variations in the junction
fabrication process make it difficult to obtain a desired EJ with high precision. We nor-
mally have variations in EJ of ∼ 10% between nominally identical junctions fabricated on
the same silicon wafer. State-of-the-art industrial processes have yielded junction critical
current variations approaching 0.5-1% [81, 82], however these processes are untenable with
4 We measure the normal state resistance of these junctions using a home built “probe station”. Instruc-
tions on how to use the probe station may be found in the lab shared drive. A shortcut I find useful when
probing qubits is EJ /h ≈ 134 GHz / (Rn in kΩ)
47
our fabrication capabilities. There is, however, a simple way to tune the critical current in
situ: we can modify the design of the qubit slightly, replacing the single junction with two
junctions in parallel, enclosing some area A and forming a loop (see Fig. 3.3(b).) For the
purpose of this dissertation this geometry, called a superconducting quantum interference
device (SQUID), acts like a single junction with a critical current that may be modulated
by threading the loop with a magnetic flux Φ = B ~ For a symmetric SQUID, consisting
~ · A.
of two junctions each with critical current Ic , the total critical current of the SQUID loop is
given by [21, 83]
Φ
IS = 2Ic cos (3.4)
Φ0
where, once again, Φ0 = ~/2e is the reduced superconducting flux quantum. The effective
Josephson energy EJ is modified proportionally, allowing us to change EJ (and thus the
qubit frequency) by applying a dc magnetic field. We apply this magnetic field by driving
a current through a coil of superconducting wire that winds around the cavity, as shown in
Fig. 3.4. Note that, since superconductors expel magnetic fields, in order to “flux tune” the
qubit the cavity must be made of a metal that is not superconducting at 10 mK: copper is
by far the best option, since it is an excellent thermal conductor at low temperatures, and
low-magnetic impurity oxygen free high conductivity (OFHC) copper is readily available at
reasonable prices.
While the extra handle of in situ EJ tunability is experimentally handy, it also opens
up another noise channel, by which the transmon may decohere, as flux noise may readily
modify the qubit frequency. To mitigate flux noise, the area A of the SQUID loop should
be kept as small as possible, while still being large enough to tune the qubit by passing a
48
reasonable current through the coil. In our lab, we find that A = 16 µm2 is sufficient such
that a 30 turn coil can provide a flux quantum to the loop by passing ≈ 130 mA through
the coil.
3.2 Filtering and shielding
As stated in Chapter 2, the nonlinearity of the Josephson Junction is a necessary but insuffi-
cient requirement for operating the cavity-qubit system in the quantum regime: we also need
both the cavity and the qubit to be initialized with high probability in the ground state, and
for few thermal excitations to decohere the system. This requirement imposes the constraint
~ω01 , ~ωc kb T : for a sense of scale, 5 GHz ≈ 240 mK. This means that the temperature at
which these experiments operate at must be of order 10’s of mK. Nominally, these tempera-
tures are straightforward to achieve in commercial dilution refrigerators: all superconducting
qubit experiments in our lab are conducted on a BlueFors LD-400 cryogen-free dilution re-
frigerator with a nominal base temperature of 7 mK. However, the temperature read out by
the thermometer on the mixing chamber plate of a dilution fridge does not necessarily reflect
the temperature of a thermal bath coupled to the cavity-qubit system. Spurious excitations
in the cavity/qubit system, as well as external noise that couples to a qubit, are a major
source of decoherence.
Spurious excitations and decohering noise may come from the ambient environment of
the experiment, or through the microwave feedlines with which we manipulate/measure the
system. Thus, quite a bit of work has gone into optimizing the shielding of the cavity/qubit
system, as well as filtering on the input/output lines. The setup in our lab, detailed below,
is fairly standard, though there is some room for improvement.
49
Figure 3.4: Cryogenic shielding for superconducting qubit experiments: Left:
schematic of the 3D cavity as it sits in the copper shielding box and Cryoperm magnetic
shielding. Center: Fully assembled, shielded cavity on the lab bench. Visible is the Cryop-
erm shielding (silver), the hermetic copper box and the input/output coaxial cables. Right:
partially disassembled box, displaying how the 3D cavity is connected to the microwave/flux
lines. Also visible in this picture is the IR absorbing material (black surface on the bottom
of the lid) that coats the inside of the copper box.
3.2.1 Cryogenic shielding
It has been shown that superconducting qubit devices may be susceptible to stray infrared
radiation and microwave inside the cryostat [84, 85]. To shield the experiment from stray
photons in the cryostat, we place the cavity inside a copper “box”, which is hermetically
sealed with indium o-rings to prevent thermal radiation from leaking in through the seams
(see Fig. 3.4.) In newer iterations of the experimental setup, the input/output lines, as
well as the flux tuning line, are fed through the lid of the box via brass flanges, which are
also hermetically sealed with indium o-rings. The inside walls of the box are coated with
a home-made infrared absorber dubbed “Berkeley black” [86] consisting of Stycast 2850FT
epoxy (73% by weight5 ), carbon lampblack (7% by weight) and 175 nm diameter glass beads
(20% by weight.) This IR absorbing material is visible on the lid of the box in the right-most
5 The reference cited calls for Catylist 24LV, however we used Catylist 9, and changed the ratios such that
the total weight of the epoxy added up to 73%.
50
picture in Fig. 3.4.
In addition to shielding from stray IR, superconducting qubit devices (especially split-
junction tunable devices) are susceptible to magnetic flux noise. To avoid noise from stray
magnetic fields in the lab, and to cool the device in as low a magnetic field as possible to
avoid trapped vortices, we further shield the cavity/box with a high magnetic permeability
metal (trade name “Cryoperm”) as seen in the middle panel of Fig. 3.4.
3.2.2 Input/output line filtering
An inescapable consequence of the input/output pins needed to control/read out the state
of the system is added noise and loss. In fact, a completely lossless system would be of little
interest to us: we want photons in the microwave cavity to decay into the transmission line
connected to the output pin, from which we may infer the state of the system! While we
must incur this penalty to be able to manipulate the system, we would like to filter away
as much unwanted noise as possible. As such, we extensively filter the input and output
microwave lines; this filtering is detailed in Fig. 3.5.
The input/output microwave lines may be a major source of noise in our experiments.
Without proper filtering, the coax cables may act as a waveguide that brings high-frequency
electromagnetic radiation from higher temperatures to mixing chamber plate at ∼ 10 mK.
Radiation in the 5-20 GHz range may populate the lower frequency modes of the cavity,
driving dephasing [76, 87, 88], while higher frequency (> 100 GHz) radiation may break
Cooper pairs in the qubit and cause depolarization [42]. Furthermore, the coaxial structure
of the microwave lines presents a problem: it is difficult to thermalize the inner conductor of
the coax, since it is galvanically disconnected from the outer (ground) conductor, and thus
51
cannot transfer heat to the dilution refrigerator through conduction electrons6 [87, 90].
To alleviate these concerns, at every stage of the cryostat there is an “attenuator” on
both the input and output lines. These are either real attenuators, in the form of voltage
dividers that both attenuate signals from higher stages of the cryostat and provides galvanic
(and thus thermal) contact to ground, or so called “0 dB attenuators”, which are micro-
strip heat exchangers in attenuator packaging that provide thermal connection to ground
without significant attenuation. Note that we use real attenuators on the input line, and
thus by necessity must attenuate the control signals we send into the fridge. Attenuating
the signal also causes resistive heating: we therefore distribute the attenuators across the
different cryostat stages such that the majority of the resistive dissipation occurs at higher
temperature stages (with more cooling power) while the qubit is shielded from potentially
harmful thermal radiation from the 4K and still (∼ 800 mK) plates [91].
In addition to attenuation, we also use two types of lowpass filters on both the in-
put and output lines to further attenuate high frequency radiation. We use a commercial
lumped-element low-pass filter to attenuate radiation above the operating frequency of the
cavity/qubit system. We have observed that these filters greatly increase T2 of the qubit,
likely because they filter away radiation that populates the higher-frequency modes of the
cavity, which is known to be a major source of dephasing [76]. We also use home-made filters
(based on the design in Ref. [92]) that utilize a lossy-magnetic epoxy (trademarked under
the name “Eccosorb”) to attenuate high-frequency signals. Since the Eccosorb filters do not
contain lumped element components, they do not have detectible re-entrant modes at higher
frequencies, and have been shown to strongly attenuate signals up to 100 GHz (i.e. photons
6 Thermal conduction from phonons in solid insulators becomes vanishingly small at dilution refrigerator
temperatures [89]
52
meas.
Input
Vacuum can flux bias
50 K
0 dB 0 dB
(5)
4K
-20 dB
HEMT 0 dB
Still
-20 dB 0 dB (6)
Cold
0 dB 0 dB
Plate
MXC -20 dB 0 dB
LPF LPF Ecco (4)
(2)
Ecco
(1)
Attenuator (1) Circulator (4) (3)
LPF Low-pass filter (2) Amplifier (5)
Ecco Eccosorb filter (3) DC block (6)
Figure 3.5: Filtering and amplification of the input-output lines: Left: schematic
diagram detailing the filtering and amplification along the input/output lines of the super-
conducting qubit setup. In this schematic, each component is thermalized to the cryostat
plate corresponding to the box the component resides in (for example, the HEMT amplifier
is thermalized to the 4K stage.) Right: pictures of one of each componenet as they reside
on the dilution refridgerator. The top picture, focusing in on the HEMT amplifier, is taken
at the 4K stage, while the bottom picture is taken below the mixing chamber plate (MXC.)
Note that the amplifier at room temperature is a different model than the one shown (see
Table 3.1.)
53
with enough energy to break Cooper pairs in aluminum.)
While we may attenuate the input lines to protect the cavity/qubit system, the signals we
would like to measure are rather weak, and attenuating them by ∼ 60 dB would render them
undetectable. To protect the system from radiation coming down the output line without
attenuating the desired signal, we use a series of microwave circulators. A circulator is a
three port device which uses a ferrite core to break reciprocity: an ideal circulator would
allow signals entering one port to propagate only to the port counter-clockwise (i.e., if ports
were labeled 1, 2 and 3 in a counterclockwise fashion, signals from port 1 could propagate to
port 2 but not port 3.) In reality, the circulators we use provide ≈ 30 dB of isolation from
backwards-propagating signal, so we use two circulators in series to achieve robust isolation.
The unused port of the circulator is capped off with a 50 Ω terminator.
3.2.3 Amplification
In addition to protecting the cavity/qubit system from decohering noise, we also need to
amplify the signal that encodes the state of the system. Amplifying a signal also inevitably
adds noise to the signal, on top of amplifying any noise that was previously being carried
along with the unamplified signal [93]. To understand how the noise added by an amplifier
is quantified, it is useful to recall the Johnson-Nyquist relation for thermal noise: the power
spectral density (PSD) of noise emitted into a transmission line by an impedance matched
load is proportional to the temperature of the load [93, 94, 71]
S[ω] = kb T (3.5)
The noise PSD is defined as the power per unit bandwidth of the noise: if we had a
54
detector with bandwidth B that detected signals only between ωa and ωa + B, the total
power of the noise that would be detected would be P = ωωaa +B dωS[ω] = kb T B. Impor-
R
tantly, Johnson-Nyquist noise is “white noise”, i.e. it has no frequency dependence7 . This
conveniently allows us to define the noise temperature of an amplifier: imagine an amplifier
with power gain G impedance matched to a transmission line that carries a signal with noise
at temperature Ts . If the amplifier added no additional noise, the noise PSD of the output
signal would simply be given by
Sout [ω] = Gkb Ts (3.6)
However, it can be shown [93] an amplifier must add additional noise to the signal: a real
amplifier with noise temperature Tn instead outputs a signal with a noise PSD given by
Sout [ω] = Gkb (Ts + Tn ) (3.7)
Thus, amplifiers are typically characterized by their power gain G and their noise temperature
Tn . What happens, however, if we must amplify the signal again? We will, of course, pass the
signal through another amplifier, which will also add noise to the system: the twice-amplified
noise PSD will be given by
Sout,2 [ω] = G2 kb (G1 (Ts + Tn1 ) + Tn2 ) = Gchain kb (Ts + Tchain ) (3.8)
7 This model breaks down at low temperature and high frequencies, where ~ω becomes comparable to
kb T . A full quantum treatment of the noise of the load yields the result S[ω] = ~ω/2 coth ~ω/2kb T [93].
At high temperatures, this approaches the Johnson-Nyquist result, however as T → 0 it bottoms out at
~ω/2, the zero point fluctuations of the electromagnetic field at ω. Using frequency and temperature scales
that are common in this thesis yeilds a situation where the Johnson-Nyquist theory breaks down: A 5 GHz
signal coming from the MXC plate of the cryostat, held at 10 mK, will have an “effective noise temperature”
Tef f = S[ω]/kb = ~ω/2kb ≈ 125 mK, much hotter than the physical temperature of the load.
55
where Gchain = G1 G2 is the total amplification of the chain of amplifiers, and the effective
noise temperature of the amplifier chain is given by
T
Tn,chain = Tn1 + n2 (3.9)
G1
We could in principle8 amplify the signal ad nauseam by stringing together more ampli-
fiers: it is relatively easy to see that, for a chain of m amplifiers, the gain of the chain will
be G1 G2 ...Gn and the effective noise temperature of the chain will be
T T
Tn,chain = Tn1 + n2 + n3 + ... (3.10)
G1 G1 G2
This relationship tells us that for a chain of amplifiers that have large power gain Gi , the
effective noise temperature of the chain (and, thus, overall noise added by the amplifiers) is
usually dominated by the first amplifier in the chain. Thus, the first stage of amplification
is by far the most important when it comes to optimizing signal to noise. The first stage of
amplification on our experimental setup is provided by a high electron mobility transistor
(HEMT) amplifier at the 4K stage, with a gain of ≈ 36 dB (which corresponds to an absolute
power gain G ∼ 4000) and a noise temperature of ≈ 3.5 K at typical signal frequencies. The
second amplifier (at room temperature) has a noise temperature of ∼ 110 K, indicating that
we are, in fact, in the limit where the amplification noise is dominated by the first amplifier.
3.2.4 Possible improvements
As stated above, the experimental setup in our lab is fairly standard for supeconducting
qubit experiments, and is sufficient for the purposes of this thesis. There is, however, ample
8 In practice we really, really couldn’t.
56
Component description Supplier/manufacturer Part number
Cryogenic attenuator
XMA Corporation 2082-6418-XX-CRYO
(-XX dBm)
Cryogenic compatible Richardson RFPD/API
8039
inner/outer dc block Tech
50 Ω terminator XMA Corporation 2001-6113-00
Lumped element low-pass
K&L Microwave 6L250-00088
filter
Circulator Raditek RADC-4-8-Cryo-4-77K-
S23-1WR-MS-b
HEMT amplifier Low Noise Factory LNF-LNC03_14A
IQ mixer (operating
Marki Microwave IQ-XXYYLXP
frequency XX - YY GHz)
Room temperature
Minicircuits ZX60-83LN-S+
RF-amplifier
Splitter/Combiner Minicircuits ZFSC-2-10G
dc amplifier board Texas Instruments DEM-OPA84XU
dc op-amp Texas Instruments OPA847
Quasi-dc low-pass filter Minicircuits BLP-21.4+
Table 3.1: RF circuitry components for the circuit quantum electrodynamics experimental
setup, along with part numbers and suppliers. For some components, we use multiple variants
of the same component. For these cases, the variable parameter is marked in bold, and
the corresponding change in the part number is matched to the variable in the component
description. Some of the specific products here are no longer manufactured, however as of
writing this, every manufacturer listed either still makes the part listed or makes an updated
version of it.
57
room for improvement on the filtering/shielding in future experiments. Here, we outline
some known weaknesses in the current setup, as well as some recent literature that suggests
improvements that could be made.
One place for immediate improvement is in the magnetic shielding: when doing exper-
iments using flux tunable transmons, we can see shifts in the qubit frequency when the
magnet on the other dilution refrigerator in the lab (roughly 3-4 m away) is varied. One
way the magnetic shielding could be improved is by using an superconducting shield (such
as aluminum) in addition to the Cryoperm shield [95]. There is also a push in some groups
to remove all magnetic material from the experimental stage: this would require replacing
all stainless steel screws with brass screws. Additionally, recent experiments [42] have also
indicated that the concentration of superconducting quasiparticles may be significantly re-
duced if an additional set of Eccosorb filters on the input/output lines are placed inside of
the hermetic copper box.
On the other hand, recent experiments have also shown that quasiparticle bursts in su-
perconducting islands could be caused by long time-scale thermalization in the cryostat [96].
The proposed improvements outlined above largely involve adding more superconducting
material (which acts as a thermal break, as the superconducting condensate doesn’t carry
entropy) or insulators, which, without electronic heat conduction, cool inefficiently at di-
lution refrigerator temperatures. If long timescale thermalization is found to be a limiting
source of decoherence once other sources are efficiently eliminated, it may be of great interest
to optimize the thermalization of many individual parts of the experimental setup, or to re-
duce the thermal load of the experiment by judiciously choosing design geometries/materials
with a low specific heat when possible.
58
3.3 Spectroscopic measurements
We now turn to a survey of some basic experimental techniques used in our lab. These
experimental techniques largely fall into two categories: spectroscopic experiments, where we
experimentally identify energy eigenstates of the system, and time-resolved measurements,
where we apply pulses to control the system Hamiltonian as a function of time. We begin
with a description of several common spectroscopic measurement techniques, from which we
extract the relevant Hamiltonian parameters such as the cavity frequency ωc or the qubit
|0i → |1i transition frequency ω01 .
3.3.1 “Punchout”
The very first experiment one typically does upon cooling down a new sample is verifying
that a functioning qubit is, in fact, coupled to the microwave cavity. We typically accomplish
this by measuring the transmission though the cavity at different driving powers. At very
low power, the cavity transmission amplitude is well described by a Lorentzian centered on
the dispersively shifted cavity frequency ωc ± χ (depending on whether the ω01 is lower or
higher than ωc .) As the drive power increases, a semiclassical model of the system predicts
a region of bistability with two solutions that destructively interfere, creating a “dark state”
response, followed by the abrupt onset of a Lorentzian response at the bare cavity frequency
ωc at some critical power where the bistable region vanishes [97, 98, 99].
The data plotted in Fig (3.6), taken in our lab by monitoring the transmission through
the microwave circuit using an Agilent N5230A vector network analyzer (VNA), agree with
the transmission predicted by this model very well. We colloquially refer to the onset of the
bright state (occurring at f ≈ 5.775 GHz and PVNA ≈ -8 dBm in Fig. 3.6(b)) as “punchout”:
59
(a) VNA (b)
300 K
DF
HEMT
10 mK
Ecco LPF
LPF
Ecco
(c)
Circulator Ecco Eccosorb filter
LPF Low-pass filter Amplifier
Attenuator
Figure 3.6: Qubit “punchout”: (a) Schematic of the measurement setup employed when
doing a “punchout” measurement. (b) Transmission through a 3D cavity plotted a function
of both VNA power and cavity frequency. At low power, we observe a Lorentzian spectrum
centered at the dressed cavity frequency ωc + χ. At intermediate powers, the transmission at
the dressed frequency is suppressed and a dark state forms in agreement with theory [97, 98].
At high powers, we observe an abrupt onset of a high transmission Lorentzian peak at the
bare cavity frequency ωc . (c) A more typical “punchout” measurement, where we only record
the transmission at low and high powers to extract ωc , χ and the quality factors at these
powers. Note that this data corresponds to a different cavity than the data presented in (b).
the high power signal “punches out” the qubit influence on the cavity transmission, and the
cavity responds at its classical geometric resonant frequency with χ = 0. A back-of-the-
envelope calculation indicates that the power required to punch out the qubit corresponds
to a cavity photon occupation of ∼ 105 , which agrees well with the number quoted in
Ref. [99].
In a typical experiment, rather than mapping out the full cavity response as a function of
power, one only measures the low and high power tranmissions (see Fig. 3.6(c)). This yields
60
the cavity resonant frequency ωc , the dispersive shift χ and the low and high power quality
factors of the cavity, which may be extracted from a Lorentzian fit to the data or, more
quickly, Q−1 = 2(ω0 − ωP/2 )/ω0 where ωP/2 is the frequency at which half the maximum
power is transmitted.
3.3.2 Two-tone spectroscopy
Upon verifying that a functional qubit is in the cavity, we need to find its |0i to |1i tran-
sition frequency ω01 . We generally accomplish this through two-tone spectroscopy [100],
which takes advantage of the qubit state dependent dispersive shift imparted on the cavity
frequency. We use the VNA to continuously monitor the transmission through the cavity at
the dressed cavity frequency ωc ± χ. At the same time, we sweep the frequency of another
tone impinging on the cavity. When this variable frequency tone matches ω01 , we Rabi drive
the qubit, changing its average excited state population from P1 ≈ 0 to P1 ≈ 0.5. Since two-
tone measurements typically take place on ∼second time scales while Rabi drive frequencies
are typically in the ∼MHz range, the Rabi drive manifests itself as a splitting of the cavity
transmission into two peaks (one dispersively shifted by the qubit in the |1i state, and one
dispersively shifted by the qubit in the |0i state.) This causes the previously on-resonance
transmission measurement to drop in magnitude. Thus, we measure ω01 by recording the
frequency at which we see a dip in the cavity transmission.
There are several things one should keep in mind while doing a two-tone measurement:
• The linewidth of the qubit is ultimately limited by T2−1 , however width of the trans-
mission dip may be power broadened by either the cavity or qubit drive tones [68, 101].
To obtain as accurate data as possible, one should use a small power for both tones
61
(a) Trig in (b)
VNA
Func
out
300 K
(c)
DF
HEMT
10 mK
Ecco LPF
LPF
Circulator Ecco Eccosorb filter
Ecco LPF Low-pass filter Amplifier
Attenuator Microwave source
Figure 3.7: Two-tone spectroscopy: (a) Schematic of measurement setup for two-tone
spectroscopy. (b) When the drive tone of the variable microwave source is resonant with
ω01 , the qubit is Rabi driven and the spectral weight of the cavity at ωc + χ drops (red)
relative to the case when the qubit is in the ground state (blue). (c) We vary the frequency of
the microwave source near the expected qubit frequency, and monitor the cavity transmission
at the dressed frequency (arrow in (b)), recording ω01 when the transmission drops.
(keep in mind that this comes with degraded signal to noise.)
• If the linewidth of the qubit at a given drive power γef f < χ, it is possible to see
photon number resolved peaks in the qubit spectrum that arise from cavity photons
dispersively shifting the qubit frequency [37, 68].
• At sufficiently high variable drive power, it is possible to drive the two photon transition
to the second qubit excited state (|0i to |2i.) The drive frequency at which this occurs
is ω02 /2, which along with ω01 provides ω12 , a value which is useful to have for many
experiments/models.
62
3.4 Time-resolved measurements
Spectroscopic measurements are generally good diagnostic tools, however time-resolved mea-
surements are a far more powerful technique for manipulating and characterizing the system.
In a time-resolved measurement, we drive the system for a fixed amount of time, let it evolve
freely for a fixed amount of time, and then at the end measure the state of the system to
see how it has evolved. In this manner, we can investigate how different components of the
system interact with each other, or how each subsystem interacts with its environment. This
section will describe the general measurement scheme used for time-resolved measurements
in this thesis, and then provide a (by no means exhaustive) sample of common time-resolved
measurement protocols.
3.4.1 General time-resolved setup
The heart of the time-resolved measurement setup is the Tabor WX2184C arbitrary waveform
generator (hereon referred to as “the Tabor”.) The Tabor has four outputs, each of which may
output a sequence of arbitrary waveforms with a resolution of 1 GS/s. We mix these outputs
with continuous wave microwave tones to create the GHz pulses that manipulate/read out
the system. To initiate the start of a waveform, we trigger the Tabor with a voltage pulse
from the SRS DG535 digital delay generator. The Tabor outputs the waveform, steps to
the next waveform in the sequence, and then idles until the next trigger pulse. By this
method we may step through sequences of waveforms, which is necessary if we want to
make measurements as a function of the pulse parameters (delay time, amplitude, etc...)
The Tabor also has lower voltage resolution “marker” outputs which may be used to trigger
other components in the circuit at a specified time during the pulse sequence. We use the
63
Demodulation
LO
I Q
ADC
RF
Pulse prep.
300 K
LO LO DR
HEMT
I Q I Q
RF RF
10 mK
Ecco LPF
LPF
Tabor Ecco
DG535
LO
Circulator Ecco Eccosorb filter Splitter/combiner I Q IQ mixer
RF
Quasi-DC
LPF RF low-pass filter Amplifier Variable attenuator low-pass filter
Attenuator Microwave source Variable phase shifter
Figure 3.8: General time resolve measurement setup: A schematic diagram of the
most general measurement setup employed when doing time-resolved measurements in this
thesis.
marker output to trigger the analog to digital converter (ADC) to start recording data.
3.4.2 Coherent control
As alluded to in Chapter 2, the conduit by which we manipulate delicate quantum systems
is coherent electromagnetic radiation: a driving electric field oscillating at ωd ∼ ω01 adds a
64
new term into the qubit Hamiltonian that we have control over. To see this, let’s rewrite
the Hamiltonian of a transmon qubit in the presence of a (classical) electric field that we
derived9 in § 2.3.1:
1
H= ~ωq (aq† aq + ) + O(φ4 ) + (t)(aq + aq† ) (3.11)
2
†
where, as a reminder, aq , aq are the qubit energy level raising and lowering operators,
and O(φ4 ) contains the nonlinearity of the Josephson junction that allows us to operate
the transmon circuit as a qubit. Recall also that (t) is a term that encapsulates both
the (classical) applied voltage V (t) and the coupling between the transmon and the voltage
source. To model a coherent field impinging on the qubit, we will take the applied voltage
to be a constant-amplitude sinusoidal drive at frequency ωd , i.e. V (t) = V0 cos (ωd t + φ).
We can simplify our lives by encapsulating the amplitude of the sinusoid V0 along with the
coupling in a single constant A ∝ V0 , and writing the Hamiltonian as:
1
H= ~ωq (aq† aq + ) + O(φ4 ) + A cos(ωd t + φ)(aq + aq† ) (3.12)
2
We now once again take the qubit approximation, saying the nonlinearity is large enough
that we may constrain ourselves to the {|0i , |1i} manifold10 , and switch from the language
†
of raising and lowering operators to the language of spins by sending ap to σ + and ap to σ − .
In this approximation, the above Hamiltonian takes the familiar form of the “Rabi problem”
9 In a footnote in § 2.3.1, I mentioned that the choice of the drive term ∝ (a − a† ) vs ∝ (a + a† ) is
q q q q
simply a matter of convention. To be consistent with the literature, in this section I have opted to work with
†
the dependence in the form ∝ (aq + aq ): in the two-level approximation this is equivalent to changing the
driving term in the Hamiltonian from σ y to σ x , which, as can be seen from Eqn. 3.17, we may accomplish
by adding in an arbitrary phase shift to the drive.
10 While we use |0i and |1i as our canonical example, it is important to remember that the results presented
here are valid for any two states with sufficient nonlinearity to prevent leakage into adjacent states. For
example, we may just as well Rabi drive between the |1i and |2i states.
65
~ω01
HRabi = − σ z − A cos (ωd t + φ)σ x (3.13)
2
If |ωd − ω01 | << ω01 , we may again invoke the rotating wave approximation and solving
for the time dependence of the state probability amplitudes using first-order time-dependent
perturbation theory. Doing so yields the famous Rabi oscillation result [36], where |0i and |1i
q
state amplitudes C0 and C1 oscillate back and forth at a rate ΩR = (ωd − ω01 )2 + A2 /~2
that depends both on the drive amplitude and the detuning between the drive frequency ωd
and the qubit frequency ω01 (see Fig 3.9 (a-b).) More succinctly, the excited state probability
as a function of time is
A2
P1 (t) = |C1 (t)|2 = sin2 (ΩR t/2), (3.14)
(~ΩR )2
It is more instructive to work with the Hamiltonian in the frame rotating with the drive. We
change basis by applying a unitary operator that “undoes” the rotation caused by the drive
ω
−i 2d σ z t
U= e (3.15)
To transform the Hamiltonian, we apply this unitary to the time-dependent Schrödinger
equation and solve for an effective Hamiltonian in the rotating frame
∂
U i~ |ψi = UH|ψi
∂t
∂ ∂U
i~ (U|ψi) − |ψi = UHU† U|ψi
∂t ∂t
(3.16)
∂ ∂U †
i~ U|ψi = UHU + i~ † U U|ψi
∂t ∂t
∂U †
H̃ = UHU† + i~ U
∂t
66
If we plug our Hamiltonian 3.13 into the above equation, after doing some algebra and
invoking the rotating wave approximation, we find
(ω01 − ωd ) A
H̃Rabi = −~ σz − cos(φ)σ x + sin(φ)σ y (3.17)
2 2
The implication of Eqn. 3.17 is that, in the frame rotating with the drive, we may control
the quantization axis by the qubit-drive detuning ω01 − ωd and the phase of the drive φ
(several different example conditions are shown in in Fig. 3.9(c-e).) Thus, it is important for
us to have both frequency and phase sensitive control of the drive tone.
Returning to the measurement setup shown in Fig. 3.8, we employ microwave frequency
IQ mixers to mix the low-frequency modulation signal from the Tabor with the high frequency
continuous-wave tone from the microwave source. An IQ mixer is a 4-port of mixer that,
given a tone into the Local Oscillator (LO) port A1 cos(ωt) and signals I(t) and Q(t) into
the I and Q ports respectively, outputs (ideally)
RF = A1 (I(t) cos(ωt) − Q(t) sin(ωt)) (3.18)
which is visualized in Fig. 3.10. Here, I stands for “in phase” and Q stands for “quadrature.”
Modulating the signal using an IQ mixer is particularly important, since it allows us to control
the phase of the microwave qubit control pulse. If we choose I(t) = A2 cos(ωmod t + φ) and
Q(t) = A2 sin(ωmod t + φ), we can use the elementary trigonometric identities
sin(a) sin(b) = (cos(a − b) − cos(a + b))/2
(3.19)
cos(a) cos(b) = (cos(a − b) + cos(a + b))/2
67
Figure 3.9: Rabi oscillations: (a) A plot of the qubit excited state probability as a function
of both qubit-drive detuning and drive time, exhibiting the characteristic Rabi “chevron”. (b)
An experimental measurement of the Rabi chevron. (c-e) Changing the drive-qubit detuning
ω01 − ωd and the drive phase φ controls the Bloch sphere axis around which Rabi oscillations
precess. The quantization axes (arrows) and procession path (lines) are plotted for (c)
ω01 − ωd = 0, φ = 0 (corresponding to the solid blue
√ line in (a)), (d) ω01 − ωd = 0, φ = π/2
(dashed red line in (a)), and (e) ω01 − ωd = A/ 3, φ = 0 (solid green line in (a).) Bloch
spheres rendered using the QuTiP package [102].
to show that
RFφ = A1 A2 cos[(ω + ωmod )t + φ] (3.20)
i.e. the phase of the intermediate frequency I(t) and Q(t) signals is imparted onto the high
frequency signal out of RF. We can therefore define the axis of Bloch sphere rotation simply
by setting the relative phase two sinusoidal waveforms from the Tabor.
68
To physically implement qubit control pulses, we typically set the continuous wave mi-
crowave signal into the LO port to ω01 − ωmod , and modulate this tone with digitized
sinusoidal pulses from the Tabor oscillating at a frequency of ωmod ≈ 150 MHz. One must
also “tune” the Tabor to compensate for phase delays in the cables leading to the IQ mixer
and imperfections in the IQ mixer, or else tones at the lower sideband (ω01 − 2ωmod ) and
LO frequency (ω01 − ωmod ) may leak through the mixer into the experiment. Typically,
we adjust the dc offset and relative amplitude/phases of the two Tabor channels leading
to the IQ mixer to maximize the upper sideband of the mixed signal (at ω01 ) and mini-
mize leakage. It is also prudent to remember that mixers are nonlinear elements, and that
driving them at excessively high power runs the risk of introducing appreciable signals at
ω01 ± 2ωmod , ω01 ± 3ωmod , etc... A safe upper bound is ∼ 7 dBm into the LO port.
3.4.3 Punchout-based readout
As mentioned in Chapter 2, we may measure the energy eigenstate of the qubit with the
dispersive readout technique, monitoring the qubit-state dependent shift in the cavity fre-
quency [60, 61, 103, 68, 34]. While this is entirely possible to do in our lab with only a
commercially available high mobility electron transistor (HEMT) amplifier and room tem-
perature amplification, getting high readout fidelity with this method requires optimization
of the ratio χ/κ [68, 103, 69]. There turns out to be a more straightforward way to measure
the energy eigenstate of the qubit based on the “punchout” mechanism described in Section
3.3.1.
The onset power of the bright state (Fig. 3.6(b)) depends on the state of the qubit
[97, 98, 99], meaning that when the qubit is in the |1i state the power required to “punchout”
is slightly lower than when the qubit is in the |0i state. We can utilize this fact to make a
69
high fidelity measurement of the qubit: if we apply a measurement tone to the cavity at the
bare cavity frequency ωq and at a power just below the bright-state threshold when the qubit
is in |0i, the cavity will “punchout” only when the qubit is in the |1i state. We then map the
qubit |0i (|1i) states to the low (high) transmission dark (bright) states respectively, which
have sufficiently high contrast in the IQ plane to easily be amplified by a HEMT.
In order to map the tone transmitted though the cavity onto the IQ plane, we once again
employ an IQ mixer, except this time as a demodulator. That is to say IQ mixers also work in
reverse: if we feed a tone A1 cos (ωt) into the LO port and a tone I(t) cos (ωt) − Q(t) sin (ωt)
into the RF port, the output of the I port will be I(t) and the output of the Q port will
be Q(t). Here, I(t) and Q(t) are slowly varying (with respect to the carrier frequency ω)
envelope functions which encode the measured state of the qubit. In this manner, we may
reconstruct the position of the microwave pulse on the IQ plane, where the I axis represents
the part of the signal in-phase with the LO reference pulse (see Fig. 3.10 for more details.)
Going back to the experimental setup (Fig 3.8), we split a tone at the cavity frequency ωc
into two components. One component feeds into the LO port of the demodulating IQ mixer,
where it acts as a reference for the amplified signal coming out of the experiment. The other
component feeds into another IQ mixer, which functionally acts as a microwave switch11 ,
allowing signal to pass through when we want to measure the state of the qubit. We then
send the output signal through a variable attenuator, which allows us to fine-tune the power
of the cavity interrogation tone to just below the “punchout” threshold. The in-line variable
phase shifter is optional: we may introduce a phase to make binning the I-Q data in the
software somewhat easier.
11 For the purposes of the “punchout” readout technique, the IQ mixer here is actually somewhat overkill:
we could (and sometimes do) just use a standard microwave switch.
70
Figure 3.10: IQ mixer modulation and demodulation: A microwave tone at the carrier
frequency ω is sent into the LO port. If we run the mixer as a demodulator, we send a
carrier pulse into the RF port (top, blue) which is composed of in phase (orange, bottom)
and out of phase (green, bottom) components relative to the LO signal. If the LO and RF
signals are at the same frequency, we record quasi-dc “boxcar” pulses out of the I and Q
ports which encode I(t) and Q(t). We use this information to reconstruct the position of
the signal on the IQ plane (background.) We may also run the process in reverse, feeding in
I(t) and Q(t) to create the desired RF output.
After pulse preparation, the signal is fed into the input line of the experiment, and
passes through the input+cavity+output with isolation/amplification as described in § 3.2.
The signal is then demodulated using the protocol described above. After one more round
of amplification and filtering, the outputs of the I and Q ports are recorded using a two
channel analog-to-digital converter (ADC), and the two outputs are plotted in real time as
a point on the IQ plane. This constitutes a single measurement of the qubit: after many
71
(a) (b)
Figure 3.11: IQ plane measurement and binning: (a) Example of a series of measure-
ments of the system while Rabi driving the qubit, plotted on the IQ plane. Each point in
phase space represents a single punchout measurement of the cavity/qubit system. We bin
the measurements by simply defining a cutoff, labeling all points left of the cutoff as |0i
measurements and all points right of it |1i measurements. (b) Probability of measuring |0i
(point left of the cutoff) as a function of the length of the Rabi drive tone.
measurements, if the experiment is tuned up correctly the points on the IQ plane will form
two “blobs”, corresponding to measurements of the qubit in |0i and |1i, as in Fig. 3.11(a).
We then bin the data and calculate the probability P0 of the qubit being in the excited
state |0i at a given time step. In this manner, we can reconstruct the time dependence of
expectation values of the system, allowing us to reconstruct system dynamics/decoherence.
An example, is shown in Fig. 3.11(b): here, we apply a variable length tone near the qubit
frequency. As the tone length gets longer, the qubit undergoes Rabi flopping from |0i to |1i
and back, which we can clearly resolve with a fidelity of ∼ 80%.
3.4.4 T1 , T2 and T2e
Having described the experimental setup, we now turn to some of the simplest time-resolved
measurements we can do: measuring the energy relaxation and decoherence times of a qubit
T1 and T2 , as well as the spin-echo time T2e . The first step in each of these experiments is to
72
prepare the measured qubit in the ground state. Since ~ω01 kB T , this task is as simple as
waiting for time t T1 in between repeated measurements. We rarely measure T1 > 30µs,
so a repetition rate of several kHz is sufficiently slow to ensure that the qubit has relaxed
back to its ground state.
With the qubit in |0i, we next apply a πx -pulse, a microwave control pulse that rotates the
Bloch vector by π radian about the x axis, which puts the qubit in the excited state |1i, wait
for delay time τ , and then measure with a tone at the cavity frequency (Fig. 3.12(a), top.)
According to the Bloch-Redfield equations for the evolution of the density matrix (see dis-
cussion in § 4.2), we should expect the excited state population to decay exponentially as a
function of τ with a characteristic time constant T1 . In order to measure T1 , we repeat the
experiment varying the delay time τ , and then repeat this sequence of experiments many
times. After sufficiently many experiments (typically ≥ 1000), we can calculate the proba-
bility that the qubit was found in the excited state as a function of the delay time P1 (τ ), and
fit that function to a decaying exponential to extract T1 (see Fig. 3.12 (a) for an example.)
Measuring T2 constitutes a similar process: we start off with the qubit in |0i, except this
time we apply a πx /2 pulse, that is to say a microwave pulse that rotates the qubit π/2 radians
√
about the x axis of the Bloch sphere. This pulse prepares the qubit in |ψi = 1/ 2(|0i + |1i).
We then let the qubit evolve for a variable time τ , apply another πx /2 pulse and measure.
This process is visualized in Fig. 3.12 (b).
The time evolution of a superposition state has two separate components: a deterministic
component, which causes the sinusoidal oscillations, and non-deterministic dephasing which
causes the amplitude decay. Recall from Eqn. 3.17 that we accomplish a rotation about the
x-axis of the Bloch sphere by setting the control pulse phase to φ = 0. This implies we
must track the phase of each pulse relative to the first pulse in order to rotate about the
73
Rabi Rabi Rabi
cavity cavity cavity
(a) time (b) time (c) time
Figure 3.12: Measuring T1 , T2 , and T2e : Pulse sequences and typical data sets for (a) T1 ,
(b) T2 , and (c) T22 measurements. In a given measurement, the pulse amplitude/width is
fixed, and the waiting time τ is varied between different measurement runs. The sequence
is then repeated, and we plot the probability of finding the qubit in the excited state P1
(red) and fit it to a decaying exponential ((a) and (c), black) or a decaying exponential
superimposed on a sinusoid ((b), black) to extract the respective coherence times. We may
also extract the qubit-drive detuning ωof f = |ω01 − ωd | from the T2 fit (b).
desired axis of the Bloch sphere: this is another way of staying that we implicitly work in
the frame rotating with the drive pulse. From Eqn. 3.17 we may read off that when we aren’t
applying a microwave pulse (A = 0), the effective Hamiltonian of the system will be that
of a qubit whose transition energy is ω01 − ωd , meaning the superposition will experience
coherent evolution according to
1
|ψ(t)i = √ (|0i + e−i(ω01 −ωd )t |1i) (3.21)
2
i.e., the state vector will deterministically process around the z-axis of the Bloch sphere at
frequency ω01 − ωd . The action of the second πx /2 rotation will depend how much the Bloch
vector has rotated: if we apply the second pulse immediately (i.e. τ = 0) it will send the
qubit to |1i as if we had just applied a π pulse. If we however wait for the state vector to
rotate by π about the z axis (half a period π/(ω01 − ωd )) the qubit will be projected back
into |0i instead. Thus, the measured excited state probability P1 will oscillate sinusoidally
74
with frequency (ω01 − ωd )/(2π). Since we know the frequency of the drive signal to high
precision, this means that the transition frequency of the qubit ω01 is also imparted on the
T2 measurement data.
The qubit will also undergo non-deterministic evolution, which when averaged over many
runs will result in the amplitude of the deterministic oscillations decaying exponentially in
τ with a characteristic time constant T2 (again, see § 4.2.) Thus we fit the P1 (τ ) extracted
from a T2 experiment to a decaying exponent superimposed on a sinusoid to extract T2 and
ω01 to high precision.
Pure qubit dephasing (at a rate Γφ = Γ2 −Γ1 /2, see § 4.2) may be thought of as stochastic
fluctuations in the qubit resonant frequency ω01 , caused by elastic (energy conserving) in-
teractions with the environment, which modify the rate at which the qubit precesses around
the Bloch sphere [75, 104, 34, 62]. Since these processes are elastic, they are in principle
reversible: if we were able to run time “backwards” we would be able to undo the evolution
caused by these interactions (as well as any deterministic evolution over the same interval.)
We cannot, of course, reverse time. However, if during a T2 measurement at time τ /2 we
apply a πy pulse (Fig. 3.12(c)), we will invert the position of the Bloch vector about the
y-axis of the Bloch sphere sending the relative phase φτ /2 accumulated during the first half of
the experiment to −φτ /2 . If the phase change was caused by sources that are slow compared
to the experiment time12 , we should expect the accumulated phase in the second half of the
12 It is reasonable to ask what we mean by “slow with respect to the experiment.” The addition of a π
y
pulse can be modeled as adding a filter function to the spectral density Sz [ω] of noise that causes fluctuations
in the qubit energy ω01 [105, 106, 62]. The filter function, in this case, pushes the frequency range sampled
by the experiment up, which is generally beneficial since many sources of noise exhibit a 1/f -like spectrum.
One may add an arbitrary number of π pulses to the sequence, a technique that was also first developed
in NMR called a Carr-Purcell-Meiboom-Gill (CPMG) sequence. In general, the more pulses you add, the
higher the frequency range you sample. In fact, it is possible to utilize the fact that the sampled Sz [ω]
depends on pulse number to work backwards and reconstruct Sz [ω] using a series of CPMG measurements
with different numbers of pulses, as was done in [106].
75
to also change by φτ /2 , rendering the total phase φτ = 0. This phase refocusing experiment,
known as a “spin-echo” experiment in nuclear magnetic resonance (NMR) where it was first
developed [107], “undoes” deterministic phase evolution and the phase evolution caused by
slowly varying noise sources, which removes the sinusoidal pattern in T2 data and may ex-
tend the dephasing time. This type of experiment is so common that the decay constant
observed during it gets its own name: T2e (which is shorthand for T2 -echo.) T2e should be
≥ T2 , however since we cannot undo spontaneous decay by refocusing the Bloch vector, we
are still limited by T2e ≤ 2T1 .
76
Chapter 4
Integrating superfluids with
superconducting qubit systems
When we built a description of quantum circuits in Chapter 2, we used the fact that the
superconducting gap shielded the qubit from single-particle excitations to sweep the micro-
scopics under the rug and write down phenomenological Hamiltonians in terms of macro-
scopic circuit parameters like the EC and EJ . We also required that the system be cooled
to dilution refrigerator temperatures, where kb T ~ω01 , to exponentially suppress any
thermal excitations to the system. Then, in Chapter 3, we dedicated quite a bit of time
to discussing strategies to mitigate decohering noise and spurious excitations that limit our
ability to perform complex control operations on the system. Ideally these steps would be
enough to protect superconducting qubits from decohering noise.
In reality, however, the microscopic details of the Hamiltonian become important for
decoherence. Microscopic fluctuations may cause fluctuation in the circuit parameters, which
may in turn cause fluctuations in ω01 and dephase the qubit. Microscopic quantum systems
with transition energies near ~ω01 may provide a density of states for the qubit to decay
into, and these systems will also fluctuate in unpredictable ways. Additionally, various
degrees of freedom that couple to the qubit may be poorly thermalized to the cryostat,
with background excited state occupancies much larger than naively expected from the
77
mixing chamber temperature TM XC ≈ 10 mK. Indeed, the MXC thermometer is at best
a proxy for the temperature of electrons in the copper MXC plate: it cannot be assumed
that the temperature of a given thermal bath mechanically connected to the MXC is at this
temperature.
From this perspective, it would be advantageous to have some way of locally thermalizing
various heat baths that couple to the system. Looking to other areas of low temperature
physics, liquid helium is commonly employed as a passive thermalizing agent for experiments
requiring extremely low temperatures. For example, helium immersion cells are used in the
study of two-dimensional electron systems in the quantum Hall regime to more efficiently
thermalize these systems at milli-Kelvin temperatures [108, 109]. However similar techniques
have not been employed in the setting of superconducting circuits.
There is also growing fundamental interest in studying the mechanical motion of su-
perfluid helium at the quantum limit. Recent experiments and proposals have investigated
superfluid helium as a platform for optomechanical experiments [110, 111, 112, 113], or as
a substrate for an electron motional qubit [114, 115, 116, 117, 118, 119], where details of
the superfluid surface mechanics are important to understanding the decoherence of the
proposed qubit. More in line with the main topic of this dissertation, one may also ask
whether it is possible to coherently control mechanical excitations of the superfluid at the
single quantum level, similar to what has been achieved in solid-state mechanical resonators
[2, 10, 5, 4]. When considering both the study and control of mechanical excitations at the
single quantum level, the flexibility of superconducting circuits offers a distinct advantage
to engineering systems that preferentially couple to specific modes.
With both these motivations in mind, this chapter details an experiment systematically
studying the effects of superfluid helium on a “standard” 3D-transmon superconducting
78
qubit/cavity system. The main focus of the experimental results is understanding how the
superfluid helium influences the spectral and decoherence properties of the cavity/qubit
system. At the end of the chapter, we will loop back around and discuss how one may or
may not be able to use superconducting circuits to study/manipulate mechanical excitations
in quantum fluids. The results in this chapter were also reported in Ref. [120].
4.1 Superfluid helium
The story of superfluid helium once again begins with H. Kamerlingh Onnes, who in 1908
became the first person to liquify the noble gas helium1 by cooling it to 4.2 K. In fact, the
technical accomplishment of liquifying helium was what allowed Kamerlingh Onnes to study
the resistance of low-temperature metals, leading to the discovery of superconductivity:
to this day, liquid helium is indispensable as a cryogenic refrigerant. Every method for
cooling bulk matter below 10 K uses helium in some fashion [89]. Since most conventional
superconductors also require these temperatures to operate, industrial or scientific processes
that use superconductors, from MRI machines to particle accelerators, are also completely
dependent on refrigeration from liquid helium.
From a physics perspective, however, liquid helium has far more interesting properties
than a low evaporation temperature. For starters, at atmospheric pressure liquid helium
never freezes: it remains a liquid all the way down to absolute zero. This astonishing fact
may be understood in in terms of zero point fluctuations in the motion of helium atoms.
1 Helium has two stable isotopes: 4 He, which is largely formed in Earth’s crust by alpha decay of heavy
radioactive elements and extracted from natural gas, and much less abundant 3 He, which is usually produced
artificially by tritium decay. These isotopes have very different low temperature properties, owning to their
significant mass difference and also the fact that 3 He is a fermion while 4 He is a boson. However, 3 He only
exists in trace quantities in natural helium reserves, and therefore we will focus exclusively on 4 He in this
chapter, unless noted otherwise.
79
Helium atoms are very light, and the van der Waals interaction between two helium atoms
is very weak. In a hypothetical helium solid, an individual helium atom would be held in
place by a 3D harmonic oscillator potential. However, because of the light mass and weak
intra-particle interaction, the zero point energy fluctuations EZP F = 3~ω/2 ∝ m−1/2 would
far outstrip the attractive van der Waals potential, rendering such a solid unstable [121].
Even at the lowest temperatures, helium does not form a solid unless pressurized above
∼25 atmospheres!
However, upon cooling to low enough temperatures, helium does not remain a typical
fluid. In the late 1920’s, it was discovered that the specific heat of liquid 4 He exhibits a
dramatic and narrow peak as a function of temperature around Tλ = 2.17 K, indicating the
existence of a phase transition at this temperature. The following decade saw a series of
experiments building evidence that, below Tλ , liquid helium does not behave like a normal
fluid, but rather exhibits many strange phenomena, especially when flowing through tight
constrictions such as a thin capillary tube or the microscopic channels in a tightly packed
powder [122]. These strange observations were eventually reconciled by modeling liquid
helium below Tλ as a sum of two component fluids: a normal fluid with density ρn and a
superfluid with density ρs , obeying
ρHe = ρn + ρs (4.1)
where ρHe is the total density of liquid helium. The superfluid component flows with precisely
zero viscosity, and carries no entropy, while the normal component behaves more or less like
a normal fluid (i.e. with finite viscosity and entropy.) This two-fluid model [123] explains
the behavior of the fluid flow through tight constrictions: the non-viscous superfluid may
80
easily flow through such a “superleak”, while the viscous normal component is impeded so
much that the flow is effectively zero.
Superfluidity turns out to be closely related to Bose-Einstein condensation, where a
macroscopic number of atoms occupy the ground state of the system. The theoretical de-
scription of superfluid helium is, however, more complicated than that of Bose-Einstein
condensation owning to the strong intra-particle interaction of the densely packed helium
atoms [121]. For the purposes of this dissertation, however, it is sufficient to note that well
below the transition temperature Tλ , the normal fluid density is empirically found to be
[121]
ρn ∝ T 4 (4.2)
Thus, at the temperatures relevant to superconducting circuits, ρn ≈ 0 and ρs ≈ ρHe .
For the experimental portion of this chapter, we will also ignore the direct exchange of
excitations of the qubit with mechanical modes of the helium. In this approximation, the
superfluid becomes, for all intents and purposes, an extremely low loss dielectric2 that may
conduct heat away from various thermal baths coupled to the cavity/qubit system [89, 125].
4.2 Decoherence and noise
Much of the experimental result below focuses on the decoherence properties of the qubit,
and how the presence of superfluid helium modifies this decoherence. Strictly speaking,
decoherence is the process by which the system we care about (the qubit) interacts with
2 High power measurements in the context of accelerator physics have yielded an upper bound on the loss
tangent of liquid helium tan δ = Im{}/ Re{} < 10−10 at ∼ 1.5 K [124].
81
Figure 4.1: Noise and decoherence of a qubit: Left: depolarizing noise (sometimes
called transverse noise) of a qubit may be thought of as spurious environmental interactions
that kick the Bloch vector (red dashed arrow) in a direction transverse to the deterministic
evolution of the qubit (black dashed arrows.) This noise is introduced by terms in the qubit
Hamiltonian ∝ σ x or σ y . Right: Dephasing noise (sometimes called longitudinal noise) of a
qubit may be thought of as spurious environmental interactions that kick the Bloch vector
along the equator, causing shifts relative to the deterministic phase evolution of the qubit.
This noise is introduced by terms in the qubit Hamiltonian ∝ σ z .
and becomes entangled with its environment, a many-DOF system that we have no hope
of tracking. From this perspective, we can think of entanglement as information loss into
the environment: the fluctuations in the environment take a well characterized quantum
state and, over some timescale, kick it around such that the information we once had about
the state is now useless. Mathematically, this is modeled as a pure state (a state that may
be written down as a wave function) undergoing non-unitary evolution and decaying into a
mixed state (a state which we’re forced to write as a density matrix, since we have incomplete
knowledge of it.)
The process of non-unitary evolution is often modeled by a “master” equation, or Lind-
bladian, however the decay of a quantum two-level system may often be described by the
Bloch-Redfield approach [126, 127]. In the Bloch-Redfield picture, the non-unitary evolution
82
(i.e. decoherence) of a quantum two-level system characterized by two rates, the depolar-
ization rate Γ1 and the dephasing rate Γ2 . The depolarization rate, often quoted as the
reciprocal of the depolarization time T1 = Γ−1 1 , describes how quickly we lose information
about the relative magnitudes of a |0i and |1i in a superposition state. Information loss
about the magnitudes comes about because of inelastic scattering, i.e. the qubit exchanging
excitations with its environment. Since rates add, Γ1 is given by
Γ1 = Γ↑ + Γ↓ (4.3)
where Γ↑ is the rate at which the qubit is spuriously excited by the environment and Γ↓ is
the rate at which the qubit in the excited state loses an excitation to its environment. In
the limit ~ω kB T , spurious excitations are exponentially suppressed (Γ↑ Γ↓ ) and Γ1 is
given by Fermi’s golden rule3 [104, 34, 93, 62]:
2π 2
Γ1 ≈ Γ↓ = h0|Ĥ 0 |1i ρ(~ω01 ) (4.4)
~
where here, ρ(~ω01 ) is the density of states at the qubit transition energy and Ĥ 0 is a
perturbative term in the Hamiltonian induced by some noise source. The energy decay
rate is also called the longitudinal decay rate, since the only perturbative terms that cause
relaxation are terms ∝ σ x , σ y , i.e. terms that rotate the Bloch vector longitudinally relative
to the quantization axis.
The dephasing rate Γ2 quantifies how quickly we lose information about the relative
phases of a |0i and |1i superposition state. Recall the argument we made in Chapter 2
3 Fermi’s golden rule may be equivalently formulated in terms of a fluctuating quantity λ that couples to
the qubit via a perturbative Hamiltonian term H0 = Aλσ x . In this formulation Γ↓,↑ = (A/~)2 Sλλ [±ω01 ],
where Sλλ [ω] is the autocorrelation function of the noisy prefactor λ at frequency ω [93].
83
when discussing the original justification for the transmon qubit: an equal superposition
√
state evolves as a function of time |ψ(t)i = 1/ 2(|0i + e−iω01 t |1i). If ω01 is constant,
we can account for this dynamic phase by working in the frame rotating with the qubit,
however if ω01 fluctuates as a function of time, the phase will evolve non-deterministically,
and we will lose information about the system. We may also lose phase information about
the system from depolarizing processes: the dephasing rate Γ2 is the sum of a contribution
from depolarization and a “pure dephasing” rate Γφ quantifying fluctuations in ω01 :
1
Γ2 = Γ1 + Γφ (4.5)
2
Note that even with little pure dephasing (Γφ → 0), we are still limited by Γ2 ≥ Γ1 /2 (or,
equivalently, T2 ≤ 2T1 .) Noise that causes changes in ω01 effectively adds a perturbation to
the qubit Hamiltonian ∝ σ z . Therefore, Γ2 is also called the transverse decay rate, since
terms ∝ σ z rotate the Bloch vector along its transverse (equatorial) axis.
When ~ω01 >> kb T (Γ1 ≈ Γ↓ ), the Bloch-Redfield approach predicts that the density
matrix ρ of a qubit with the initial state |ψ(t = 0)i = α |0i + β |1i will evolve as [104, 62]
1 + (|α|2 − 1)e−t/T1 αβ ∗ eiω01 t−t/T2
ρ= (4.6)
α∗ βe−iω01 t−t/T2 |β|2 e−t/T1
i.e. the excited state probability ρ11 will decay exponentially with a time constant T1 , while
the off-diagonal terms will decay exponentially4 with a time constant T2 .
4 This simple exponential decay is conditional on the noise source driving decoherence being weak enough
to be treated perturbatively and not having long timescale correlations. This second condition is violated
for common 1/f noise, where Sλλ [ω] ∝ 1/ω n . Violation of this assumption can result in a non-exponential
decay envelope [104], however all the data in this chapter fit a decaying exponential reasonably well and we
will ignore these subtleties.
84
4.2.1 Common noise sources in superconducting qubits
To facilitate some of the discussion below, we offer a brief survey of noise sources that
commonly drive decoherence in superconducting circuits.
Charge noise: In Chapter 2, we briefly discussed charge noise, i.e. the uncontrolled
fluctuation of the microscopic electrostatic environment of the qubit. These fluctuations
may be caused a number of sources: shot noise in nearby conductors, fluctuations in charged
lattice defects in the substrate, fluctuations in adsorbed surface molecules, etc. However,
since transmon qubits (and most modern superconducting qubits for that matter) are expo-
nentially insensitive to charge noise, we will ignore charge noise for the present discussion.
Flux noise: In the case of the flux tunable transmon (or other flux tunable qubits, such
as flux qubits or fluxonium), fluctuations in the magnetic flux Φext threading the SQUID
loop may cause cause fluctuations in ω01 , driving dephasing. Flux noise is thought to be
caused by both macroscopic sources (such as current fluctuations in the flux-tuning coil) and
microscopic sources (such as spins at the metal-insulator surface [128].) In the experimental
data presented below, we use a fixed frequency transmon, and thus flux noise is of little
concern for us.
Dielectric loss: Like all classical integrated circuits, superconducting circuits must be
fabricated on a dielectric substrate. All real dielectric materials absorb some fraction of an
impinging electric field: this absorption may be modeled by making dielectric constant
imaginary, where the Im{} quantifies the electrical loss of the material. A common figure
of merit of a dielectric is the loss tangent: tan(δ) = Im{}/ Re{}. The two most common
substrates superconducting circuits are fabricated on, high resistivity silicon and sapphire,
have very low tan(δ) < 10−6 [79], though there is evidence that sapphire is the better of the
85
two [129]. In addition to the host substrate, interfaces (say, between metal and substrate
or metal and air) are though to host thin, amorphous layers of oxides and organic materials
that contribute significantly to loss [130, 79, 129, 131].
On the quantum level, dielectric loss is likely caused by a large ensemble defects and
impurities that effectively create a bath of quantum two-level systems (TLS) [132]. Though
any one of these TLSs is weakly coupled to the qubit mode, they effectively create a density
of states for the qubit to decay into. These ensemble losses are, however, largely a function
of the fabrication details. Since our experiment provides a comparison of one device subject
to systematically different conditions, we will assume these details are constant over the data
presented below.
Two-level systems: In a qualitatively different regime, individual TLSs near or in the
Josephson junction may strongly couple to the qubit degree of freedom, driving decoherence
[133]. If a TLS is inside the Josephson tunnel barrier, a shift in the state of the TLS may
locally change the microscopic details of the tunneling barrier, slightly modifying the critical
current and thus EJ . These TLSs are thought to cause dephasing [134], and have been
blamed for discrete shifts in the qubit frequency that persist over long timescales [40].
Additionally, strongly coupled TLSs with transition frequencies near ω01 provide a density
of states for the qubit to decay into. It has recently been shown that these TLSs, which
themselves lack the transmon’s protection from quasistatic charge noise, undergo random
walks in frequency space, moving in and out of resonance with the qubit on ∼hour timescales
[135]. As the TLS moves in and out of resonance, an environmental interaction channel is
effectively turned on and off, which is thought to cause long timescale changes in the qubit
coherence properties that pose a problem for benchmarking many-qubit systems [136, 137,
138]. These fluctuations may be impacted by the presence of thermalizing superfluid helium
86
[125], and strongly coupled TLSs are of interest to us in this study.
Photon shot noise: In Chapter 2, we saw that when the cavity and qubit are disper-
sively coupled, the effective resonant frequency of the cavity depends on the state of the
qubit. Conversely, we could take Eqn. 2.46 and regroup terms:
1 g2 †
~
HJC,disp ≈ ~ωc (a† a + ) + ωq + a a σ z (4.7)
2 2 ∆
From this perspective, it looks like the effective frequency of the qubit depends on the
cavity photon number occupancy [68, 37]. The dispersive shift, written in this way, im-
plies that if the cavity photon population is fluctuating (i.e., if the cavity mode is poorly
thermalized), these fluctuations will also drive dephasing in the qubit [76, 87, 88, 90].
Quasiparticles: In § 2.2.4, we placed a great deal of emphasis in working in conditions
where electronic excitations above the superconducting ground state (referred to as super-
conducting quasiparticles, or simply “quasiparticles” in the literature) are suppressed. The
expected density of quasiparticle excitations should be ∝ e−∆BCS /kB T : for aluminum at
10 mK, this factor is vanishingly small. However, many experiments [139, 140, 141, 142] have
recorded quasiparticle densities many orders of magnitude higher than the naive prediction
from BCS theory, such that loss from quasiparticles (sometimes referred to as “quasiparticle
poisoning”) is a significant limiting factor in qubit coherence.
The exact origin of these elevated quasiparticle levels is still subject to some debate. It is
agreed upon that improper filtering may introduce pair-breaking photons from warmer parts
of the cryostat, and we discussed several ways to mitigate these effects in Chapter 3. However,
even with highly efficient filtering, there is a growing body of evidence that nonequilibrium
quasiparticles may be generated in significant quantities by decay of radioactive isotopes in
87
the experimental setup [143], cosmic rays impinging on the substrate [95, 144], or potentially
“heat leaks” in the cryostat that take a very long time (∼days-months) to thermalize [96].
Quasiparticles may also be excited by/convert into high frequency phonons in the underlying
substrate [145], suppressing recombination and lengthening the lifetime of deleterious excita-
tions. Suppressing quasiparticle sources, and figuring out how to thermalize these excitations
more efficiently, is an open area of research in the field.
4.3 Experimental setup
4.3.1 Gas handling system and fill line
Introducing a controllable level of superfluid helium into an experiment introduces several
experimental complications that must be solved. First and foremost, since superfluid helium
has zero viscosity, it will leak through more or less any porous structure, and through many
crevices that normal fluid would not leak through. Identifying and preventing superleaks
is an age-old pastime of low temperature physicists, and the experimental setup must be
carefully designed to minimize the risk of superleaks developing.
Additionally, one must consider the thermalization of superfluid helium, normal liquid
helium, and helium vapor at the various stages of the cryostat. Care must be taken to
ensure that the helium thermalizes to the base temperature of the cryostat (∼ 10 mK.) On
the other hand, the superfluid film flow will cause a thin superfluid layer to form on the
inside of any surface exposed to helium below the superfluid transition temperature. Since
superfluid helium is a good thermal conductor, the film flow effect puts us in danger of
thermally connecting the lower stages of the cryostat, which may result in unwanted heat
flow into the MXC.
88
(a)
Bypass
To Helium
cylinder
To fridge
Pump
LN2 cold
Auxillary trap Valve
port
Helium Pressure
Reservoir gauge
(b)
Figure 4.2: Gas handling system and cold trap: (a) Schematic of the helium gas han-
dling system. The red valve represents a compound shutoff-throttling valve to controllably
introduce helium into the sample cell. All helium introduced into the cell should be passed
through the cold trap. (b) Cut-away of the cold trap. Two copper tubes are fed into a
stainless steel cylinder with brass caps hard soldered on the end. The cold trap is filled
with activated charcoal, which has a high surface area that adsorbs atmospheric gasses when
chilled to liquid nitrogen temperatures. The copper tubes are filled with a copper mesh to
act as another adsorbing surface and to prevent charcoal debris from exiting the cold trap.
We must also be extremely careful to prevent the introduction of non-helium gas into
the fill line. Even at the 50 K stage of the cryostat, all common atmospheric gasses will
have solidified: if there is a leak to atmosphere, one runs the danger of blocking the fill line
and leaving no escape route for the liquid helium already in the fridge, which could lead to
extreme pressure buildups.
In Figure 4.2 (a) we present a schematic diagram of the gas handling system used to
controllably and safely introduce helium into the experiment. We introduce helium into the
experiment from a helium reservoir with a known volume though the throttling valve (red,
Pfeiffer EVN 116), which precisely controls the flow rate. The pressure gauge on the helium
reservoir gives us a rough estimate of how much helium has been introduced into the fridge.
89
When filling the experimental volume with helium, one should always fill through the liquid
nitrogen cold trap (Fig 4.2 (b)), which provides a large, cold surface area for atmospheric
gasses that may have leaked into the gas handling system to adsorb onto. It is also best
practice keep the cold trap cold for the duration of the experiment and open to both the fill
line and the (emptied) reservoir: this will hopefully prevent atmospheric gas that leaks into
the system from blocking the fill line, and in the event of a pulse tube shutoff provides a safe
path to the reservoir for the helium to expand into when it boils and exits the cell.
After filtering through the liquid nitrogen cold trap, the helium enters the fridge though a
series of hermetically sealed tubes that constitute the helium fill line. The fill line must also
thermalize the helium at each stage of the dilution refrigerator in transit to the sample cell, a
hermetically sealed 3D microwave cavity (see Fig. 4.3 for a schematic of the fill line.) Inside
of the fridge, the capillary line is a 1/16” diameter stainless steel capillary everywhere with
the exception of two sections: the section between room temperature and the 50 K plate
(made of 1/8” brass tube5 ), and the section between the cold plate and mixing chamber
plate, where a 0.017” CuNi capillary is used to minimize heat flow between these two plates
by a superfluid film within the capillary. The incoming helium is thermalized at five points:
twice by mechanically clamping the fill capillary to the 50 K and 4 K plates of the cryostat,
and by a passing the helium through a copper sinter heat exchanger at each of the still plate
(800 mK), cold plate (100 mK), and mixing chamber plate (MXC). In this configuration,
we find that the lowest temperature of the dilution refrigerator is not substantially changed
upon filling the 3D cavity with superfluid 4 He.
For ease of installation and repair, the fill line is broken up at several points in the fridge.
The first connection inside the vacuum can, made between the 1/4” copper tube feeding
5 We also put copper mesh in the end of the brass section to make a secondary cold trap
90
To ADC (TR)
q c
LO
I Q
RF
To VNA (CW)
300 K
50 K
HEMT
4K
-20 dB 0 dB
800 mK
-20 dB 0 dB
100 mK
-20 dB 0 dB
MXC
-10 dB 0 dB
Ecco Ecco
Legend
-20 dB Attenuator Microwave Valve
Source
Eccosorb low LO Sinter heat
Ecco pass filter
I
RF
Q IQ mixer sinter exchanger
Splitter Crimp heat
Circulator exchanger
Amplifier DC block
Figure 4.3: Superfluid Helium + superconducting qubit setup: Schematic diagram
of the superfluid helium + superconducting qubit experimental setup.
91
into the fridge and the 1/8” brass tube, is a brass Swagelok joint. Every other joint in the
fridge is either a soft solder joint or a joint consisting of two brass flanges sealed with an
indium o-ring (see Fig. 4.3). It is imperative to thoroughly leak check each joint with a
helium leak-checker upon first making the connection. If there is a superleak inside of the
fridge during a cool down, superfluid helium that has escaped from the fill line will flow to
the warmest point in the fridge, evaporate and thermally short every stage of the fridge to
room temperature and causing the fridge to warm up very quickly. This is, needless to say,
a scenario worth avoiding. Note that the flow impedance through the sinter heat exchangers
is large, rendering it difficult to leak check the parts of the line that come after them. It is
therefore advisable to leak check the sample cell before one puts it on the fridge.
4.3.2 Superfluid helium leak tight sample cell
The experiment consists of a single-junction transmon circuit, fabricated by Kater Murch’s
group at Washington University in St. Louis, housed in a rectangular 3D aluminum mi-
crowave cavity (see Fig. 4.4(a).) The two halves of the cavity are hermetically sealed with a
conventional indium wire o-ring typically used for making superfluid leak tight joints. The
external microwave coupling to the cavity is provided via two hermetically sealed 50 Ω assem-
blies (see Fig. 4.4(b).) These assemblies consist of commercial hermetic GPO feedthroughs6
in which the room temperature rubber o-ring has been replaced with a cryogenic indium seal
[146, 147]. These GPO feedthroughs are seated in custom brass flanges that are themselves
sealed to the body of the cavity with indium o-rings.
Coupling pins (Fig. 4.4(b)(5)) are soldered to the inner portion of each assembly to
provide microwave signals to the cavity. The coupling pin is made out of 16 gauge copper
6 Gilbert Engineering part # 0119-783-1
92
(a)
(1)
(2)
(b) (3)
(4)
(5)
Figure 4.4: Superfluid leak-tight sample cell, iteration #1: (a) Picture of the hermet-
ically sealed 3D superconducting microwave cavity. Visible are (1) the helium fill capillary
and flange and (2) the two microwave coupling ports. (b) Exploded rendering of the custom
microwave coupling assembly. The hermetic GPO feedthrough (4) sits in a brass flange (3),
and is sealed with an indium o-ring in between both the feedthrough and the flange, and the
flange and the wall of the 3D cavity. A 50 Ω impedance matched copper pin (5) is soldered
into a GPO “bullet” connector and extends to the inner wall of the cavity and provides
coupling to the TE101 fundamental mode of the cavity.
wire, which is substantially thicker than the inner pin of a standard GPO connector. To
fit the pin into the GPO connector, one end was etched in ferric chloride for ∼20 minutes
to decrease its diameter. A small Teflon cylinder fits around the port to match7 the pin to
50 Ω. The end of the pin on the input port is flush with the cavity wall, while the pin on
the output port protrudes into the cavity to increase coupling to the output. The microwave
ports are connected to a standard filter/amplifier chain (see Fig 4.3.) The cavity is thermally
anchored to the mixing chamber plate of the dilution refrigerator, and helium is introduced
via a hole on the top of the cavity that connects to a stainless steel fill capillary through a
custom brass flange, which is itself hermetically sealed to the cell with an indium o-ring.
7 This was likely superfluous, since the cavity itself is not impedance matched to 50 Ω. More recent
experiments have dropped the Teflon.
93
4.4 Experimental results
4.4.1 Cavity and qubit spectroscopy
To characterize the effect of adding superfluid 4 He we first perform continuous wave spec-
troscopy of the cavity/qubit coupled system, both when the cavity is empty and under
vacuum, and when it is filled with superfluid helium. Using a vector network analyzer we
characterize the cavity response by measuring the microwave transmission (S21 ) through the
measurement circuit as a function of frequency. At high power (∼ −80 dBm power injected
into the cavity), the measured response is Lorentzian and peaked at the classical cavity fun-
damental frequency [98, 97, 99] fc = ωc /2π, shown as the blue (dark gray) traces in Fig.
4.5(a). The change in the speed of light caused by the presence of a dielectric of relatively
√
permittivity should shift the bare cavity frequency from fc → fc / . Indeed, we find that,
when helium is added to the cavity, the fundamental frequency fc shifts from 6.93480 GHz
to 6.75395 GHz (see Table 4.1), corresponding to an effective cavity dielectric constant of
= 1.054, which agrees well with that of superfluid helium He = 1.057 [148, 149]. We also
note that the quality factor of the microwave resonator is not significantly affected by the
presence of helium, consistent with the findings in Refs. [150, 110, 148].
The hybrid cavity/qubit system may be described by the generalized Jaynes-Cummings
Hamiltonian (JCH), which takes into account the higher excited states of the transmon
circuit |ii:
ĤJC = ~ωc a† a + gi,i+1 |ii hi + 1| a† + h.c. .
X
ωi |ii hi| + ~ (4.8)
X
i i
In Eq. 1 a† and a correspond to the microwave cavity photon creation and annihilation
94
(a)
-30
S21 (dB)
Bare cavity
-40 Dressed cavity
Full Empty
-50
6.8 6.9
Frequency (GHz)
(b)
-55
S21 (dB)
-60
Empty
-65 Full
5.0 5.1 5.2
Frequency (GHz)
Figure 4.5: Cavity and qubit spectroscopy is the presence of superfluid helium: (a)
Measured cavity transmission S21 as a function of frequency when the cavity is empty (right)
and full of superfluid helium (left). Depending on the level of the applied microwave power
we can measure both the cavity resonance dressed by the qubit in its ground state (red/light
gray, P ≈ −120 dBm) or the bare cavity resonance (blue/dark gray, P ≈ −80 dBm.) (b)
Two-tone spectroscopy of the qubit immersed in liquid helium (red/light gray) and in vacuum
(blue/dark gray), offset vertically for clarity. Right of the dotted line, a low power tone is
applied to excite the qubit (Pq ≈ −120 dBm) from its ground state |0i to its first excited
state |1i, while to the left of the line a high power (Pq ≈ −90 dBm) tone is applied to induce
a two photon transition from |0i to |2i. The dips in the transmission correspond to qubit
excitation frequencies ω01 (right) and (ω01 + ω12 )/2 (left).
operators respectively, and h.c. stands for hermitian conjuugate. In the transmon regime
[34], the uncoupled qubit frequencies ωi are determined by the Josephson energy EJ , the
√
charging energy EC , and the cavity/qubit couplings constants gi,i+1 ≈ g01 i + 1. In this
limit, Eqn 4.8 is therefore determined by EJ , EC , the ground-to-first excited state vacuum
Rabi splitting g01 , and the cavity frequency ωc .
In addition to shifting the cavity resonant frequency, the presence of dielectric superfluid
will also modify all of the spectroscopic parameters of the coupled qubit/cavity system,
95
Value Empty (GHz) Full (GHz) change (%)
ωc /2π 6.9348 6.7540 -2.62
δω/2π 0.00875 0.00913 4.32
ω01 /2π 5.1914 5.1747 -0.32
ω12 /2π 4.8834 4.8695 -0.28
EJ /h 13.887 13.895 0.06
EC /h 0.2710 0.2690 -0.82
g01 /2π 0.1235 0.1201 -2.8
Table 4.1: Spectroscopic parameters of the cavity/qubit system both in the presence and
absence of superfluid helium. ωc , δω, ω01 , and ω12 are measured values, while EJ , EC and
g01 are extracted by solving the generalized Jaynes-Cummings Hamiltonian constrained by
measured spectroscopic parameters.
which we can characterize with the framework of cQED. At low input microwave power
(∼ −120 dBm, Fig. 4.5(a) red/light gray traces) the cavity resonant frequency is shifted by
the presence of the transmon circuit in its ground state. In the dispersive limit of cQED
[34], |∆| = |ωc − ω01 | g01 , where ω01 is the qubit ground-to-excited state frequency, this
hybridization causes the cavity resonant frequency to shift by an amount δω ≈ g01 2 /∆. We
measure δω for both the empty cavity and the cavity full of superfluid helium, with the
results reported in Table 4.1.
We utilize two-tone spectroscopy [100] to directly measure the excitation spectrum of the
qubit and how it is modified by the superfluid. We use a low power tone (Fig. 4.5 (b), right
of dashed line) to excite the qubit from ground |0i to first excited state |1i, and a higher
power tone to excite a two photon transition from |0i to the second excited state |2i (Fig.
4.5 (b), left of dashed line). From these measurements, we extract the |0i → |1i transition
frequency ω01 and the |1i → |2i transition frequency ω12 for both the empty and full cavity
configurations, and report these values in Table 4.1.
To extract the values of EJ , EC , and g01 , and how they are modified by the dielectric
96
superfluid, we diagonalize the generalized JCH, and fit the eigenvalues ω01 , ω12 , and δω to
the values obtained from our spectroscopy measurements for the case when the 3D cavity is
empty as well as when it is filled with helium. The results are is summarized in Table 4.1.
The small change of EJ in the presence of helium is consistent with variations in EJ
that we observe between cool downs without helium present in the cavity. It has been
reported that these variations result from changes in the microscopic charge configuration in
the Josephson junction oxide barrier [151, 134]. Therefore our results are consistent with EJ
being unmodified by the presence of liquid helium. In contrast we find that the capacitive
charging energy of the qubit decreases by 0.82%. This reduction in EC agrees with the value
of 0.78% obtained from finite element simulations of the system.
A shift in the vacuum Rabi coupling g01 is also induced by the superfluid helium. Qual-
itatively, this shift results from a change in the zero point energy of the cavity and a spatial
redistribution of electric field lines within the cavity/qubit system upon changing the dielec-
tric constant from = 1 → He = 1.057. Quantitatively, we may write the vacuum Rabi
coupling in terms of the fluctuating zero point voltage of the microwave field in the 3D cavity
[34] VZP F ,
g01 = 2eVZP F βh1|n̂|0i, (4.9)
where n̂ is the Cooper pair number operator, and β is a parameter describing the efficiency
of converting voltage fluctuations in the cavity to voltage fluctuations across the junction
of the qubit [34]. We develop a simple model to capture the change of the vacuum Rabi
coupling g01 as a function of the dielectric constant of the environment surrounding the
qubit. Modeling the cavity as a simple LC oscillator (see Fig. 4.6) allows us to write the
97
zero point fluctuations of the voltage in the cavity as [152]
s
~Zc
VZP F = ωc (4.10)
2
√ q
where ωc = 1/ Lc Cc is the resonant frequency and Zc = Lc /Cc is the impedance of
the oscillator. Uniformly filling the cavity with a dielectric will shift the cavity capacitance
from Cc to Cc , and from Eqn. 4.10 one finds that
VZP F ∝ −3/4 (4.11)
Cg
Cc Lc Cq Lq
Figure 4.6: Simplified circuit model for finding ∆g01 : Circuit model for a 3D transmon
circuit (red/gray) coupled to a linear cavity (black) used to estimate the change in the
vacuum Rabi splitting ∆g01 in the presence of superfluid helium.
To understand the functional dependences of β for our experiment, we model the qubit
as a parallel capacitance8 Cq and nonlinear Josephson inductor Lq coupled to the cavity
via capacitance Cg (see Fig. 2). We assume the system is symmetric and that both of the
antenna paddles of the qubit are identical and have the same capacitance Cg to the 3D cavity
walls. β is then given by the voltage that builds up across Cq when a voltage V exists across
the entire circuit,
8 This capacitance includes both the intrinsic Josephson junction capacitance and the shunt capacitance
provided by the antenna paddles of the qubit.
98
Cg2
β= (4.12)
Cg2 + 2Cq Cg
In the case of uniform dielectric filling, the capacitances will all scale uniformly and β will
be unchanged from its vacuum value. We note, however, that the presence of the silicon chip
and the intrinsic Josephson junction capacitance will cause Cg and Cq to scale differently as
a function of dielectric constant of the cavity medium. We perform finite element simulations
using COMSOL to determine how these capacitances change in the presence of helium. We
find that ∆Cq = 0.78% upon filling the cavity with helium, which agrees very well with the
measured shift of 0.82% extracted from the change in the charging energy EC , and that
∆Cg = 1.65%.
We finally note that the transmon excitation number transition matrix element is pro-
portional to the zero point charge fluctuations of the qubit, which in the nearly harmonic
oscillator regime of the transmon circuit may be written as
s !1/4
j + 1 EJ −1/4 1/4
|hj + 1| n̂ |ji| ≈ ∝ EC ∝ Cq (4.13)
2 8EC
We use the simulated shift in Cq to calculate the expected change of the charge number
matrix element. Combining the shift in the matrix element with the predicted shifts in VZP F
and β, we arrive at a predicted shift in the vacuum Rabi coupling induced by the superfluid
in the cavity
∆g01 = −3.2%, (4.14)
which is in good agreement with our measured value of −2.8%.
99
4.4.2 Qubit Relaxation and Decoherence
Successful integration of qubits into quantum fluid experiments or vice vera requires that
the coherence properties of the qubit do not degrade when immersed in superfluid helium.
To characterize these effects, we use standard pulsed techniques to measure the energy (T1 )
and phase (T2 ) relaxation of the qubit as a function of temperature. We use standard
measurement techniques, described in detail in Chapter 3 and in Fig. 4.7, to measure T1
and T2 . When measuring T2 , the pulses are detuned from ω01 by ∼ 300 kHz, allowing us
to extract both the phase relaxation time and the qubit resonant frequency ω01 (see § 3.4.4
for details.) Additionally we conducted preliminary echo experiments for both the empty
and superfluid-filled cavity configurations at the lowest temperature and found that the spin
echo time T2e was not significantly different than T2 .
To account for long timescale fluctuations of the qubit decoherence, we repeat the mea-
surements of both T1 and T2 over a span of 5 hours. In Fig. 4.7(c-d) we plot the energy
relaxation time T1 , the pure dephasing time Tφ = (1/T2 − 1/2T1 )−1 and |ωdrive − ω01 | as
a function of time for the empty (c) and superfluid filled (d) cavity configurations at the low-
est temperature of the dilution refrigerator (10 mK). From Fig. 4.7(c-d) it is clear that the
coherence times fluctuate significantly over the span of several hours (see Fig 4.7(e-f)). To
account for these fluctuations while extracting decoherence information comparable between
the empty and full cavity configurations, at every temperature examined we take data for
5 hours, and take the average and standard deviations of 5 hours of measurement to make
a single datapoint in Fig 4.8. We plot the temperature dependences of T1 (Fig. 4.8(a)), T2
(Fig. 4.8(c), circles) and Tφ (Fig. 4.8(c), triangles) for the case when the cavity is empty
(open symbols) and when it is filled with helium (closed symbols). The data points in Fig. 4.8
100
(a) (b) (e)
0.8 0.8 30
T1
time
0.6 0.6 T
time
20
P(1) P(1)
0.4 0.4 Count
10
0.2 0.2
0 20 40 0 10 20
(µs) (µs) 0 10 20 30
(c) T1/T (µs)
T1
T 310
30 01
T1,T (µs)
(f) 40
20
305 30
10
(kHz)
0 Count 20
(d)
10
30
T1,T (µs)
300
20 0
240 260 280 300 320
10 (kHz)
(kHz)
250
0
0 1 2 3 4 5
Time (hours)
Figure 4.7: Description of time-resolved and averaged measurements: (a) Repre-
sentative measurement of the qubit energy relaxation time T1 inset with a schematic of the
corresponding pulse sequence. We measure the probability P (1) of finding the qubit in the
excited state |1i after a variable delay time τ after exciting it and fit the data to an ex-
ponential function to extract the decay time T1 . (b) A representative free induction decay
measurement, which is fit to a sinusoid superimposed on a decaying exponential function.
From this fit we extract the dephasing time T2 and the drive/qubit detuning |ωdrive − ω01 |.
We interleave a single run of each measurement described in (a) and (b), and then repeat
this for ∼ 150 s to get a single data set to fit to. We repeat this process for 5 hours
to gather statistics on long timescale fluctuations: (c) is a representative measurement of
T1 , Tφ = (1/T2 − 1/2T1 )−1 , and |ωdrive − ω01 | for the empty cavity at T ' 10 mK. (d) Is
a similar measurement run at the same temperature but for the cavity filled with helium.
Note the difference in scale between right axes of (c) and (d). (e) A histogram of the values
of T1 and Tφ plotted in (d): The average and standard deviations of these datasets are what
is reported in Fig (4.8). (f) A histogram of the values of ∆ω01 recorded in (d): to measure
the frequency shift as a function of temperature, we reject data measured during a discrete
“jump” (light) and average over only data in “steady state” values of ω01 (dark).
represent the average value of a set of repeated measurements, while the error bars are the
standard deviation of each set.
101
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Figure 4.8: T1 , T2 and ω01 as a function of temperature with/without superfluid
helium: (a) Qubit energy relaxation time T1 , as a function of temperature, for both the
empty cavity and the cavity filled with superfluid helium. We fit the data to theory (see
Ref. [153]) for quasiparticle limited T1 , using only the superconducting gap of aluminum
and the nonequilibrium quasiparticle density xqp (solid and dashed curves) to extract the
change in quasiparticle density above ∼ 60 mK when the cavity is filled with superfluid (see
main text for discussion.) (b) Qubit frequency shift relative to it’s base temperature value,
(c) dephasing (T2 ) and pure dephasing (Tφ ) times of the qubit as a function of temperature,
for both the empty and full cavity configurations.
4.4.2.1 Qubit energy relaxation
At the lowest temperatures we find that T1 saturates at roughly the same value (∼ 20 µs)
both when the cavity is empty and when it is full of superfluid helium. Whatever mechanism
is limiting T1 at the lowest temperature we can conclude that it is not significantly suppressed
102
by the presence of superfluid helium.9 On the other hand, this result also demonstrates that
the superfluid does not introduce any significant additional mechanisms for qubit energy
relaxation at the temperatures relevant to cQED.
The temperature dependence of T1 when the cavity is not filled with helium (see Fig. 4.8(a))
may be understood as arising from quasiparticles tunneling across the qubit junction [139,
153, 154, 142], limited by an athermal quasiparticle bath below ∼ 140 mK. However, when
the cavity is filled with helium we observe a qualitatively different temperature dependence:
as we increase the temperature above ∼ 60 mK, we find a modest reduction in T1 when
the cavity is filled with superfluid. We posit that this reduction in T1 could be associated
with a higher nonequilibrium quasiparticle density when the cavity is filled with helium. It
is known that quasiparticles may travel long distances between superconducting islands on
a substrate via conversion into phonons [145]. It is possible that at intermediate tempera-
tures, phonons in the superfluid helium may mediate the transfer of quasiparticles between
the superconducting qubit and the superconducting cavity. If the cavity were to have a
higher nonequilibrium quasiparticle density, this additional coupling channel via the super-
fluid could cause the quasiparticle density in the qubit, and the associated relaxation rate
via quasiparticle poisoning, to increase. At lower temperatures this additional source of
quasiparticles would diminish as the Kapitza boundary resistance between the helium and
the cavity/qubit continues to increase [89, 155]. This would lead to an increase in T1 with
decreasing temperature that would ultimately be limited by the same source that is limiting
T1 in the case of the empty cavity.
9 Note that at the large qubit/cavity detuning used in this experiment, the increase in the Purcell emission
rate Γp caused by the helium induced shift of the cavity frequency is negligible compared to the long timescale
fluctuations in the emission rate. Specifically Γp = (g01 /∆)2 κ ∼ (265 µs)−1 for the empty cavity and
∼ (240 µs)−1 for the full cavity, where κ is the cavity linewidth [34].
103
Working within this hypothetical model, we partially fit, down to ∼ 60 mK, the tem-
perature dependent T1 data for the case when the cavity is full of helium to the theoretical
quasiparticle decay rate given in [153] (see solid curve in Fig. 4.8(a)). In this fit the only pa-
rameters are the normalized nonequilibrium quasiparticle density xqp and the superconduct-
ing gap of aluminum ∆BCS . We find ∆BCS ' 160 µeV, which agrees well with the known
bulk value for aluminum, and that an increase in quasiparticle density of ∆xqp = 4 × 10−6
accounts for the observed difference in T1 above 60 mK when the cavity is filled with super-
fluid.
If the helium is mediating the introduction of extra quasiparticles into the qubit at
intermediate temperatures, this increased quasiparticle density should also cause a shift in
the resonant frequency of the qubit [153] relative to its zero temperature value. An increase in
quasiparticle density of ∆xqp ∼ 4 × 10−6 would produce a shift of ∆ω01 /(2π) ≈ −14 kHz in
the qubit frequency. This quasiparticle induced shift in ω01 must, however, be deconvoluted
from shifts in the qubit resonant frequency caused by other sources of noise. We observe
discrete abrupt changes (“jumps”) in the qubit frequency ω01 of the order 25 − 50 kHz
that occur over ∼hour timescales, similar to those reported in other cQED experiments.
We commonly observe (see Fig. 4.7(c)) that ω01 will jump from some steady state value,
stay at the new value for minutes to hours, and then return to the original steady state
value. Discrete changes such as these are commonly attributed to critical current noise in
the Josephson junction [40]. Importantly, we can rule out shifts in the quasiparticle density
as the origin of these jumps in ω01 , as they would be accompanied by an associated shift in
the qubit relaxation rate Γ1 = 1/T1 of order 25 − 12 µs−1 [153], which we do not observe.
Therefore, to extract the temperature dependent quasiparticle induced shift in ω01 shown
in Fig. 4.8(b), we employ a clustering algorithm to bin the measurements of ω01 around the
104
steady state frequency and reject measurements that occur during a discrete change in the
qubit frequency. Fig. 4.7(f) shows an example of this for the data trace in Fig. 4.7(d): the
central (dark) data are accepted while the outlying (light) data that was recorded during
a discrete jump in ω01 is rejected. In Fig. 4.8(b) we plot the resonant frequency of the
qubit as a function of temperature and find that the qubit frequency does in fact shift down
appreciably in this intermediate temperature regime when the cavity is filled with helium.
While this data is consistent with our hypothesis of an increased quasiparticle density at
intermediate temperatures, our current experiment cannot directly confirm this model of
superfluid phonon mediated quasiparticle coupling between the qubit and the 3D cavity. We
will discuss how one may further confirm or refute this hypothesis in § 4.5.
Finally, while the temperature dependence of our T1 data can be predominantly un-
derstood from the perspective of athermal quasiparticle poisoning, there likely are other
mechanisms affecting the qubit energy relaxation. In particular the presence of near reso-
nant two-level system (TLS) defects [136, 137, 138] can be a potential source of decreased
T1 . Further discussion on these potential mechanisms may be found in §4.4.2.3.
4.4.2.2 Qubit dephasing
In contrast to the energy relaxation of the qubit, we find that above 60 mK the pure dephas-
ing time Tφ is the same both when the 3D cavity is empty and when it is full of superfluid.
Upon cooling below ∼ 60 mK we find that the dephasing time modestly improves in the
presence of helium, indicating that qubit dephasing and energy relaxation are dominated
by different mechanisms. Experiments similar to ours are known to be plagued by ther-
mal photon occupations well above the nominal temperature of the mixing chamber of the
dilution refrigerator [118, 88, 87, 90], and fluctuations in the cavity photon number have
105
been identified as a major limitation to the phase coherence of transmon qubits [76, 88, 90].
Additionally, in these previous experiments (see particularly [76, 87, 88]) the temperature
dependence of the dephasing rate is qualitatively similar to that which we observe in our
measurements both with and without helium.
It is known that many components in the microwave circuit are inefficiently thermalized,
and it has been suggested that dissipative components such as attenuators may heat the
microwave environment within the cavity to temperatures well above the dilution refrigerator
temperature [87, 90]. It is possible that the superfluid helium is serving to better thermalize
the microwave environment within the cavity in our experiment. For example, the helium
could be opening an additional channel to cool the microwave circuitry via the central pins of
microwave coupling lines or by directly cooling the 3D cavity walls. The modest improvement
we observe in qubit dephasing when our cavity is filled with superfluid would correspond to
a relatively minor reduction in the thermal photon number in the cavity. In the limit κ χ
where κ is the cavity linewidth and χ is the shift in the qubit frequency per cavity photon,
and the limit n̄th 1 where n̄th is the thermal cavity photon occupancy, we can express
the dephasing rate arising from residual thermal photons in the cavity as [156, 90]
Γφ = n̄th κχ2 /(κ2 + χ2 ) (4.15)
Using this expression we can estimate the temperature of the photon bath Tph ∼ 80 mK
for the empty cavity and Tph ∼ 70 mK for the superfluid filled cavity.10 Finally, we note
that these results are reproducible over multiple cool-down cycles of the cryostat.
10 At both these temperatures, n̄ 1, justifying the use of Eqn. 4.15
th
106
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Figure 4.9: Long timescale fluctuations of coherence parameters with/without
superfluid helium: (a) Standard deviation of the measured datasets of T1 and Tφ for both
empty and full cavities. (b) Normalized covariance of the sets of measured T1 and Tφ .
4.4.2.3 Long timescale fluctuations in qubit coherence properties
In addition to qubit dephasing produced by a fluctuating cavity photon number, several
dephasing mechanisms are known to be important to superconducting qubits. Single junction
transmon qubits, like ours, are known to be insensitive to both charge and flux noise [76], and
the dephasing caused by quasiparticle poisoning is predicted to be negligible compared to the
relaxation induced by quasiparticles (i.e. Γφ,qp Γ1,qp /2 [157].) Another possible source
of dephasing is from TLSs nearly resonant with the qubit: as these TLSs undergo spectral
diffusion, the associated dispersive shifts will also fluctuate, leading to qubit dephasing.
Fluctuations of near resonant TLSs can also create a fluctuating density of states into which
the qubit can decay. If TLSs are a dominant source of both transverse and longitudinal noise
we should expect some correlation between the fluctuations in T1 and Tφ .
107
In Fig. 4.9(b) We plot the normalized covariance of T1 and Tφ
ρT1 ,T = (hT1 Tφ i − hT1 ihTφ i)/(σT1 σT ) (4.16)
φ φ
where ρT1 ,T = 1 corresponds to perfectly correlated values of T1 and Tφ and ρT1 ,T = 0
φ φ
corresponds to T1 and Tφ being completely uncorrelated. We find that when the cavity is
empty there is little correlation between T1 and Tφ , and that this is broadly true also for
the case when the cavity is filled with helium. There are, however, several measurement sets
when the cavity is full of superfluid helium where ρT1 ,T ≈ 0.8, indicating strong correlation
φ
between T1 and Tφ during these measurements. However, the more pronounced systematic
increase in Tφ when the qubit is immersed in helium taken along with the fact that the
correlation between T1 and Tφ does not show any clear systematic temperature dependence
indicates that, while perhaps not a dominant mechanism, TLSs could be playing a relatively
larger role in the qubit energy decay and dephasing in the presence of superfluid helium for
at least a subset of the measurements.
In addition, we also observe significant long timescale fluctuations of the decay and deco-
herence times T1 and Tφ , which are not associated with changes we observe in ω01 . Several
recent studies [136, 137, 138, 135] have attributed the long timescale fluctuations in T1 to
TLSs in proximity to the qubit both spectrally and in real space. As these TLSs fluctuate
in frequency, they potentially provide a time-varying density of states into which the qubit
can lose energy. In Fig. 4.9 we plot the standard deviation σT 1 (σT φ ) of each set of T1 (Tφ )
measurements as a function of temperature and see that, generally, at lower temperatures
the magnitude of the fluctuations is increased when the cavity is full of helium. These results
indicate that unsaturated TLS fluctuators could be playing a role in qubit decoherence at the
108
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Figure 4.10: Excited state population of the qubit as a function of temperature
with/without superfluid helium: Residual population of the qubit excited state |1i as a
function of temperature for both the empty and superfluid filled cavity configurations. Also
plotted is the theoretical Maxwell-Boltzman (MB) distribution, calculated using the energy
levels obtained from spectroscopy. The presence of helium in the cavity has no significant
effect on the |1i population, and the data fit the expected population well with no adjustable
parameters. Inset: expanded view of boxed region. The |1i population for the qubit saturates
at roughly the same value both when the cavity is full of superfluid and when it is empty.
lowest temperatures of our experiment when the cavity is full of helium. However, further ex-
periments optimized for spectral [137, 138] or time domain [135] analysis of these fluctuations
are needed to confirm the role that liquid helium has on TLS thermalization/fluctuation.
4.4.3 Residual excited state population
To further investigate possible thermalizing effects produced by the superfluid helium we
have directly measured the residual qubit excited state population using a method devel-
oped in Refs. [158, 154] (see Appendix B for more details.) The measured population is
plotted as a function of temperature in Fig. 4.10, along with the expected population calcu-
109
lated from a Maxwell-Boltzmann distribution with a partition function truncated beyond the
3rd excited state of the qubit. As shown in Fig. 4.10, the data are in good agreement with
the theoretical population calculated with no adjustable parameters. Apparently, the super-
fluid helium has no significant effect on the residual excited state population of the qubit,
which saturates at 0.5% − 1% at the lowest temperature in both the empty and full cavity
configurations. This is consistent with the known difficulty of effectively cooling 4 He in the
low milli-Kelvin temperature range due to the Kapitza thermal boundary resistance [89, 110]
between superfluid helium and solid materials. Additionally, recent experiments attribute
the majority of the residual excited state population to athermal quasiparticle poisioning
[142]. The saturation of the qubit excited state at roughly the same value independent of
the presence of superfluid helium is therefore also consistent with our measurements of T1
(which we find to also saturate at roughly the same value) being mainly limited by athermal
quasiparticles at low temperatures.
4.5 Discussion and future work
The three separate decoherence metrics (dephasing, depolarization, and residual excited
state population) measured in this experiment seem to point to the same conclusion: in our
experiment, the superfluid helium is only able to assist thermalization of the system down to
∼ 60 − 70 mK. The depolarization time T1 is more or less independent of temperature below
TM XC = 60 mK, indicating the mechanism causing extra depolarization in the presence
of superfluid is frozen out at below that temperature. The pure dephasing time increases
at the lowest temperatures, and attributing the dominant mechanism of dephasing to shot
noise in the microwave cavity yields an effective cavity temperature of Tph ∼ 70 mK in the
110
presence of superfluid versus an effective temperature Tph ∼ 80 mK in the absence. This
indicates that the superfluid is able to thermalize the cavity mode slightly more efficiently,
however the cavity mode is decoupling from TM XC at ∼70 mK. The transmon residual
excited state population saturates at a temperature below this 60 − 70 mK threshold even
without the superfluid: thus, the lack of significant change is consistent with the idea that
the thermalization mediated by the superfluid largely decouples from the system degrees of
freedom at these temperatures.
When thinking about future experiments one could pursue in studying superconduct-
ing qubits in the presence of superfluid helium, the data presented here opens up several
interesting questions that could potentially be addressed with minor adjustments to the ex-
perimental apparatus/protocols. The helium seems to thermally decouple from the system
at 60 − 70 mK, much higher than the base temperature of the dilution refrigerator: is it
possible to modify the experiment such that a lower passive cooling temperature is reached?
Additionally, the experiment performed here is unable to say much about the mechanisms
of decoherence, and how the superfluid impacts these mechanisms. Could we modify the
experiment to gain information on the different mechanisms of decoherence, and how pas-
sive thermalization mediated by the superfluid impacts them? There are several future
experimental steps one could imagine doing to address these questions:
Better superfluid thermalization: This experiment was not necessarily set up for
optimal thermalization. The superconducting aluminum resonator acts as a thermal open
circuit, and the nearest sinter heat exchanger (on the MXC plate) is separated from the
cavity by nearly a meter of stainless steel capillary line. An obvious step one could take is
to use a copper microwave cavity, which will more easily thermalize to the mixing chamber
plate and help to locally thermalize the helium. Another step one could take is to put a sinter
111
heat exchanger in the experimental cell, providing a large surface area for the superfluid to
thermalize to locally. Since the sinter heat exchanger would functionally act as a powder
filter, to avoid microwave loss the sinter should be placed in a volume close to the 3D cavity
but where the cavity mode has minimal amplitude. An example design implementing these
changes is shown in Fig. 4.11(a).
However, the utility of 4 He for passive thermalization may be finite: since there are no
free electrons in insulating 4 He, all thermal conductivity between helium and a solid must
come from phonons in the liquid converting to phonons in the solid. The slow speed of
sound, as well as the phonon population that decreases as T −3 , yields an increasing Kapitza
thermal boundary resistance of 4 He as temperature decreases [89], limiting its utility as
a thermalizing agent at low temperatures. On the other hand, 3 He is known to have a
superior thermal boundary resistance at very low temperatures, likely because the nuclear
magnetic moment of 3 He allows for the exchange of magnetic excitations between the solid
and the fluid [89]. A superconducting qubit device passively cooled by liquid 3 He would
likely thermalize to a much lower temperature, however such an experiment would require
a substantially different design: The aforementioned magnetic moments of 3 He could drive
decoherence in the system, especially if a flux-tunable qubit were employed, and the large
volume of the 3D cavity in this experiment makes simply reproducing the experiment using
3 He prohibitively expensive.
Further understanding decoherence mechanisms: In recent years, interest has
increased in understanding the long timescale fluctuations of the coherence properties of
superconducting circuits, in both the time [136, 137, 138] and frequency [135] domains.
Long timescale fluctuations have been attributed to two-level systems [136, 137, 138] and
quasiparticles bursts from stray background radiation [144, 143], however a general consensus
112
of the limiting factors of decoherence is not necessarily clear. From this perspective, it would
be useful to have an in situ knob, i.e. the presence of absence of thermalization via the
superfluid, that may influence these mechanisms in different ways.
The measurements in the papers cited above largely rely on measuring the decoherence
properties of a superconducting qubit system with a time resolution of ≤ 1 s, either to
raster through qubit frequency in a reasonable time or to build up a power spectral density
of coherence property fluctuations over several orders of magnitude in frequency. In the
data presented in this chapter, during a coherence time measurement run, each individual
measurement in the five-hour average, corresponding to a single point in Fig.4.7(c), takes
about 120 s. This time is a function of the experimental rep-rate and how many sequences
we average to get an individual point, and could be decreased significantly with a higher
rep-rate11 and less averaging. In our lab, we have demonstrated the ability to get reasonable
quality data to fit to and extract decoherence times (as well as precise measurements of ω01 )
in as little as 1-2 seconds. Therefore, it should be fairly straightforward to systematically
investigate the effect of superfluid helium by measuring the spectrum of decoherence fluctu-
ations over several orders of magnitude, or by using a flux tunable qubit in a copper cavity
to raster the qubit in frequency space.
Testing the quasiparticle limited T1 hypothesis: In addition to gleaning informa-
tion about decoherence mechanisms from fluctuations in the coherence properties of a qubit,
it is possible to lightly modify the setup and directly measure the tunneling of supercon-
ducting quasiparticles across the junction. This has been observed using an “offset charge
sensitive” (OCS) transmon, with EJ /EC ∼ 15 − 25 reduced from the normal transmon op-
11 Note that a high experimental rep-rate may artificially increase the residual |1i population, as the qubit
may still be excited from the previous sequence. A good rule of thumb is (rep rate)−1 > 5 max(T1 ).
113
erating regime [140, 142, 42]. In this regime, ω01 retains a small but measurable dependence
on the offset charge ng : since a single superconducting quasiparticle carries charge q = 1e,
a tunneling event will abruptly shift ng by 1/2, shifting the qubit frequency in turn (see
Fig. 2.3.) The quasiparticle parity (i.e. whether a tunneling event has occurred or not) may
then be measured by measuring the evolution of the qubit state-vector referenced to a tone
at the average value of the ω01 band (see Ref. [142] for more details.)
From these measurements, once can infer parity jumps, i.e. quasiparticles tunneling
events, in real time and infer the density of quasiparticles in the metal. Using this tech-
nique, one could unambiguously measure whether or not the superfluid has any effect on the
quasiparticle density. One major drawback of this technique is that it requires high-fidelity
single-shot readout of the qubit state, likely more sensitive than our current measurement
setup. This could be accomplished by adding a quantum limited amplifier, such as a Joseph-
son parametric amplifier, to the amplification chain, however doing so would require modest
upgrades to our setup/experimental capabilities.
4.5.1 Coupling to superfluid acoustic modes?
To conclude the chapter, we note that the increase in pure dephasing time at the lowest
temperatures is indicative that, at least in the setup tested here, there was very little in the
way of coupling to mechanical modes in the superfluid. This result, in retrospect, isn’t so
surprising. In the experiment presented, we made no attempt to optimize the geometry of the
system to preferentially couple to a specific mechanical mode of the system. One may imagine
an experiment where this geometry is optimized: for example, if the transmon capacitance is
provided by a vacuum gap capacitor, as sketched in Fig. 4.11(b), the electric field mode profile
would have a strong overlap with the fundamental acoustic mode of helium trapped between
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Figure 4.11: Proposals for future superfluid-superconducting qubit experiments:
(a) Top: CAD model of an upgraded experimental cell for improving thermalization/studying
the effects of superfluid helium on qubit decoherence. A copper cavity with local sinter heat
exchangers would both better thermally anchor the helium to the cryostat temperature and
allow for flux tunability of the qubit. Bottom: picture of the upgraded experimental cell,
built for future experiments. (b) Sketch of a possible experiment optimizing the overlap of
a qubit mode with an acoustic mode in the superfluid.
the capacitor plates. The envisioned geometry, with appropriate spacing/size to both localize
a superfluid mode in the 2 − 5 GHz range and provide the correct transmon capacitance,
should be achievable with standard vacuum gap capacitor fabrication techniques.
However, even this optimized geometry may not provide enough coupling to the acoustic
mode in the helium. In the absence of piezoelectricity, fluctuations in the helium density
cannot directly induce voltage fluctuations across the capacitor pads, as the zero point
fluctuations of the electromagnetic cavity we considered in Chapter 2 do. The zero point
fluctuations of the helium density will, however, modify the effective dielectric constant of
the system, coupling the mechanical mode to the electrical mode. This situation is, in some
sense, similar to the optomechanical coupling between a mechanical mode and an optical
115
resonator, where shifts in the optical cavity resonant frequency are caused by displacements of
the mechanical oscillator. While not entirely analogous, cavity optomechanics experiments
coupling the motion of helium to an electromagnetic cavity have reported single photon
coupling rates12 ranging from g0 /(2π) = 10−8 Hz for a kHz acoustic resonator overlapping
with an GHz microwave resonator [159, 110] to g0 /(2π) = 3 × 103 Hz for a MHz acoustic
resonator overlapping with an optical resonator [113]. To truly enter the regime of quantum
acoustics, the vacuum Rabi coupling rate gm between the qubit and the mechanical mode
should satisfy gm > max[κm , Γ2 ], where κm is the loss rate of the acoustic cavity. While a low
coupling rate does not preclude experiments demonstrating the coupling of a qubit to a highly
populated mechanical mode, the low single photon coupling rates reported in superfluid
optomechanics experiments make the prospect of superfluid quantum acoustics mediated
only by electrostriction dim. To perform quantum acoustics experiments, we likely need a
method to more strongly couple mechanical strain to electric fields, such as piezoelectricity.
12 Optomechanics experiments may afford to have much lower single photon coupling, since many photons
may be circulating in the linear optical resonator. The strong nonlinearity of qubits heavily limits the number
of photons in the “optical” (qubit) mode, usually to one.
116
Chapter 5
Surface acoustic waves and surface
acoustic wave devices
As indicated in our discussion of superfluid helium, experiments investigating single-excitation
quantum acoustics with superfluid helium seem extremely challenging. In order to couple to
acoustic modes in a substrate, we need a stronger method of mediating the electromechanical
coupling, such as piezoelectricity. How should we go about coupling a quantum circuit to a
mechanical mode using the piezoelectric effect?
Surface acoustic waves (SAWs) are elastic waves that propagate along the surface of
an elastic material, with a displacement amplitude that decays exponentially away from
the surface. In a piezoelectric material, as we shall see, SAWs may easily be excited by
applying an ac voltage at frequency f0 to a periodic structure with periodicity λ = v/f0
where v is the propagation speed of SAWs. Since v ∼ 103 − 104 m/s for typical SAW
materials, at telecommunication frequencies (1-10 GHz), SAW devices have a characteristic
length scale ∼ 0.1−10 µm. These dimensions are easily achievable with modern lithographic
techniques, and moreover the fact that the structures are confined to the surface make them
straightforward to fabricate: often times only a single layer of metal needs to be deposed on
the surface.
The short wavelength (104 − 105 × shorter than the wavelength of light at the same
117
frequency), and ease of fabrication have made SAW devices ubiquitous in modern telecom-
munication electronics. Common components made from SAWs include delay lines, bandpass
filters, and oscillators, and have found applications spanning from gas sensors to garage door
openers [160, 161].
5.1 Elastic waves in solids
Before we discuss piezoelectric surface waves, it will behoove us to recall some properties of
waves in elastic materials. This discussion largely follows that of [160], with some help from
[162]. We’ll start off by thinking about an isotropic elastic medium: a “particle”1 that sits
at rest in it’s equilibrium position ~r may undergo some displacement ~u(~r, t) from ~r (see Fig.
5.1.) When we talk about elasticity, we are interested in deformations of the material, not
overall displacements or rotations (which cause no internal forces and can be simply mapped
onto coordinate changes.) We may satisfy these conditions by constructing the strain tensor
at a point ~r
1
Sij (~r, t) = (∂j ui + ∂i uj ) (5.1)
2
where ∂i = ∂/∂ri . From this form, we can read off that Sij is invariant under overall
displacements (constant offsets in ~u.) We also see that Sij is a symmetric tensor, implying
that there are only 6 independent components of the strain tensor.2
Having assembled an object that describes displacement caused by internal forces, we
1 Here, we’re considering our elastic medium in the continuum limit: the “particle” is much smaller than
the length scale of any relevant deformations (i.e., the wavelength of the acoustic waves we’ll consider) but
much larger than a single unit cell of the underlying crystal.
2 In the literature, this symmetry is often taken advantage of to write S as a 6 component vector
ij
Sij = (S11 , S22 , S33 , S23 , S13 , S12 )T . This representation is called the Voigt notation.
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Figure 5.1: Stress and strain in elastic media: Left: to understand stress and strain in
an elastic medium, we conceptually break up the medium into “particles” located at point
~r when the system is in equilibrium (dashed black cube.) When forces are applied to the
system, the particle at point ~r is displaced from its equilibrium by ~u(~r) (solid black cube,
greatly exaggerated.) Right: A defining feature of elastic media is that adjacent “particles”
may exert forces on each other when displaced from equilibrium. The neighbor (dashed)
adjacent to the x̂ surface of the particle at ~r (solid) may be displaced in any direction, which
will exert a force on the particle at ~r in the displacement direction. The force per unit area
this particle exerts in the î direction is Tix .
now must describe those internal forces: the stress tensor Tij (~r, t) accomplishes this goal.
Consider our “particle” of elastic material as a cube: the side of this cube that is normal to
the x̂ direction experiences a force caused by displacements in the adjacent material (see Fig.
5.1.) Crucially, this force may be in any direction: the Tix component of the stress tensor
is defined as the force per unit area in the î direction acting on the surface of our particle
normal to x̂. Thus, diagonal components of the stress tensor represent compressive forces,
while off-diagonal components represent shear forces. It may also be shown that Tij = Tji ,
and that subsequently the stress tensor also only has 6 independent components. We may
relate stress and strain using a generalized version of Hooke’s law
Tij = cijkl Skl (5.2)
119
where cijkl is the rank-4 stiffness tensor, and summation over repeated indices is implied.3
To write down the equations of motion for the material, it’s convenient to again think of
our “particle” as a cube, this time with explicit edge length δ centered at ~r = (x0 , y0 , z0 ).
Consider the two faces of the cube at x = x0 ± δ/2. Since Tix is the force per unit area in
the î direction on surfaces normal to x̂, the total force on the cube from these surfaces is
∂Tix (x0 , y0 , z0 )
δ 2 Tix (x0 + δ/2, y0 , z0 ) − Tix (x0 − δ/2, y0 , z0 ) = δ 3
∂x
where, in the second equality, we have pulled out a factor of 1/δ and gone to the limit δ → 0.
The forces acting on the surfaces normal to the ŷ and ẑ directions will take similar forms,
and will simply add with the forces on x̂ surfaces. Thus, the total force on the cube in the î
direction is
Fi = δ 3 ∂j Tij
Since the mass of the particle is ρδ 3 , where ρ is the material’s mass density, we can write
down the Newtonian equations of motion as
1
ρüi = ∂j Tij = ∂j cijkl (∂k ul + ∂l uk ) (5.3)
2
where ü = ∂ 2 u/∂t2 .
Up until now, our discussion has been completely general: these equations are valid for
small displacements in any elastic material. It will be instructive for us to break from this
3 Similar to S , T is also commonly written down in the Voigt notation as a 6-component column
ij ij
vector. We can infer from the symmetries of Sij and Tij that cijkl will, at most, have 36 independent
components, since in Voigt notation cijkl can be written a 6 × 6 matrix. In fact, it can be shown that this
6 × 6 matrix itself is symmetric, meaning cijkl has at most 21 independent components. The underlying
crystal symmetry of the material will almost always constrain these components further.
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generality and consider the propagation of waves in isotropic materials, i.e. materials who’s
properties have no dependence on the orientation of the coordinate frame we choose. This
is a curious choice, since by definition piezoelectric materials are anisotropic (they break
inversion symmetry), however there is much intuition to gain from briefly studying waves in
isotropic materials.
The stiffness tensor of an isotropic material may be written down as [160]
cijkl = νδij δkl + µ(δik δjl + δil δjk ) (5.4)
where ν and µ are the Lamé coefficients of the material4 and δij is the Kronecker delta.
Plugging this form of the stiffness tensor into Eqn. 5.2 yields, after some simple tensor
algebra
Tij = νδij Skk + 2µSij (5.5)
We can plug this form into the Eqn. 5.3 to derive the wave equation for waves in an
isotropic material
ρüi = ∂j (νδij Skk + 2µSij )
= ∂j (νδij ∂k uk + µ(∂i uj + ∂j ui ))
= ν∂i ∂k uk + µ∂i ∂j uj + µ∂j2 ui
ρ~u¨ = (ν + µ)∇(
~ ∇ ~ 2~u
~ · ~u) + µ∇ (5.6)
While this is not exactly the familiar wave equation, it does relate the second time derivative
4 It is conventional in the literature to use λ and µ as the Lamé coefficients, however to avoid confusion
with the SAW wavelength I have switched notation to ν and µ.
121
of u to the second spatial derivative of u. It is therefore a reasonable guess that, in the absence
of boundary conditions (i.e. in an infinite isotropic medium), solutions should take the form
of plane waves
~
~u = ~u0 e−i(ωt−k·~r) (5.7)
Plugging this form of the solution into Eqn. 5.6 yields an algebraic equation for the ampli-
tudes of waves in isotropic materials
ω 2 ρ~u0 = (ν + µ)(~k · ~u0 )~k + µ|~k|2~u0 (5.8)
From this, we can pick out two types of elastic waves that may propagate in solids. The first
case is if ~k · ~u0 = 0, i.e. the displacement field is transverse to the wave vector. In this case,
the dispersion relationship of the wave is given by
2
|k~t | = ω 2 ρ/µ (5.9)
q
and we may read off the velocity of these transverse (or shear) waves as vt = µ/ρ. There
are two possible polarizations of transverse waves, in the two directions perpendicular to the
propagation direction. For example, if we take the wave to be propagating in the ẑ direction,
we may write down a general form for transverse waves as ~ut (r, t) = (ux x̂ + uy ŷ)e−i(ωt−kz) .
Since ∇·~
~ ut = 0, it is easy to check that for transverse waves, Eqn. 5.6 reduces to the standard
wave equation with speed of sound vt :
~u¨t = vt2 ∇
~ 2~ut (5.10)
122
A second type of wave is possible when ~k · ~u0 6= 0. In order to have a nontrivial solution
to Eqn. 5.8 satisfying this condition, the wave vector and the displacement field must be
parallel to each other. In that case, (~k · ~u0 )~k = |~k|2~u0 and the dispersion relationship of
these waves is given by
2
|k~l | = ω 2 ρ/(2µ + ν) (5.11)
and we may read off the velocity of these longitudinal (or compression) waves as
q
vl = (2µ + ν)/ρ. Since µ and ν are always positive [160], longitudinal waves in an isotropic
material always propagate faster than transverse waves. Once again, if we take the wave to
be propagating in the ẑ direction, we may write down a general form for longitudinal waves
as ~ul (r, t) = uz ẑe−i(ωt−kz) . Noting that the ∇ ~ × ~ul = 0 and remembering the vector identity
~ ∇
∇( ~ · A) = ∇ ~ 2A + ∇~ ×∇ ~ × A, we once again see that Eqn. 5.6 reduces to the familiar form
of the wave equation, albeit it with a different speed of sound vl :
~u¨t l = vl2 ∇
~ 2~ul (5.12)
5.1.1 Rayleigh Waves
In the previous section, we discussed the two types of plane waves (longitudinal and trans-
verse) that may propagate in the bulk of an isotropic medium, where we have not defined any
boundary conditions. The surface of the substrate will impose a boundary condition, which
will constrain the types of waves that may propagate: namely, since there is no material
above the surface to exert forces on the material at the surface, the stress on the surface
normal to ẑ must be zero, i.e. Txz = Tyz = Tzz = 0. As we will see shortly, SAWs are in fact
123
a linear superposition of transverse and longitudinal waves. These surface waves are named
after Lord Rayleigh who first described them 1885 [163], and we will use the terms “Rayleigh
wave” and “SAW” interchangeably. This derivation largely follows that of [164], and [160].
We’ll start off by building up a reasonable guess for the general form of a SAW, and
then use the boundary conditions to solve for a specific solution. To get started, imagine the
situation presented in Fig. 5.2(a), where an elastic material fills space z < 0 with vacuum in
z > 0. Along the surface at z = 0, a monochromatic surface plane-wave propagates in the x̂
direction. We will assume for this discussion that we have complete translational symmetry
along the ŷ direction5 , and that the displacement field of the SAW obeys both uy = 0 and
∂y ~u = 0.
Our strategy will be to decompose the SAW displacement field into the sum of its trans-
verse and longitudinal components, which each individually obey a wave equations 5.10 and
5.12 respectively, i.e. ~u = ~ul + ~ut . Since we are seeking a monochromatic plane-wave so-
lution, both the longitudinal and transverse components should propagate the same way in
the x̂ direction, and we may write down a general form for the functional dependence of each
individual component of the wave as
|~ul,t | ∝ e−i(ωt−βx) fl,t (z) (5.13)
where β and ω are the same for both the longitudinal and transverse components, and the z-
dependence fl,t (z) depends on the polarization. Inserting each component into its respective
5 SAWs with displacement perpendicular to both the propagation direction and the vector normal to the
surface do exist, and are called shear-horizontal waves. However, they will not concern us in this disertation.
124
wave equation (Eqn. 5.10 or 5.12) yields a differential equation for the z-dependence:
ω2
∂z2 fl,t (z) = β 2 − 2 fl,t (z)
vl,t
from which we infer the functional form of fl,t (z):
v
ω2
u
±κ z
fl,t (z) = , κl,t = (5.14)
u 2
e l,t tβ − 2
vl,t
If the wave is to be confined to the surface, we must impose the requirement that
2 > 0, or else the amplitude will oscillate into the bulk. Note that, since we
β 2 − ω 2 /vl,t
expect the velocity of the SAW to obey v = ω/β, this condition implies v < vl,t , i.e. that
SAWs will propagate slower than either bulk transverse or bulk longitudinal waves. Since, in
our setup, the elastic material occupies half space z < 0, we should take the +κl,t solution,
else the amplitude of the displacement will blow up at z → −∞.
Having ironed out the functional dependence of ~ul,t , we may now write down a specific
vector solution for each component. We start with the transverse component: recall that,
for transverse elastic waves, ∇~ · ~ut = ∂x ut,x + ∂z ut,z = 0. Inserting the solutions 5.13 and
5.14 yields a constraint on the relative amplitudes of the ut,x and ut,z
iβut,x + κt ut,z = 0
from which we may write down the transverse component of the SAW wave up to an overall
constant A
~ut = A(κt x̂ − iβ ẑ)e−i(ωt−βx)+κt z (5.15)
125
Likewise, the longitudinal component of the SAW will satisfy the condition
~ × ~ul = ∂z ul,x − ∂x ul,z = 0. Using this condition, we insert solutions 5.13 and 5.14 to
∇
constrain the relative amplitudes of the ul,x and ul,z
κl ul,x − iβul,z = 0
from which we may write down the longitudinal component of the SAW wave up to an overall
constant B
~ul = B(β x̂ − iκl ẑ)e−i(ωt−βx)+κl z (5.16)
We are now in a position where we can invoke the boundary conditions provided by the
surface, i.e. that the stress on the free surface must be zero. We may use Eqn. 5.5 to write
down three differential equations, one for each boundary condition:
Txz |z=0 = ∂z ux + ∂x uz = 0 (5.17a)
Tyz |z=0 = ∂z uy + ∂y uz = 0 (5.17b)
Tzz |z=0 = vl2 ∂z uz + (vl2 − 2vt2 )∂x ux = 0 (5.17c)
where in Eqn. 5.17c I have dropped the y-dependence and written down µ and ν in terms
of the longitudinal and transverse velocities vl and vc . Clearly Eqn. 5.17b is satisfied by
our decision to work with displacement fields obeying uy = 0 and ∂y ~u = 0. We may
then insert the total displacement field ~u = ~ul + ~ut (where ~ul and ~ut are given by 5.16
and 5.15 respectively) into 5.17a and 5.17c to generate a system of linear equations for the
amplitudes A and B. After a bit of algebra, we arrive at the following set of homogeneous
126
Figure 5.2: SAW propagation in an isotropic medium: (a) Schematic of SAW propa-
gation: we search for surface wave solutions where the surface is normal to the ẑ-direction
and the SAW propagates in plane √ waves along the x̂-direction. (b) Plot of the SAW dis-
placement field ~u(r) for vt = vl / 3. The length/direction of the arrows corresponds to the
displacement amplitude/direction respectively, and color corresponds to real part of the ver-
tical displacement uz . Inset: The (amplitude exaggerated) trajectory of a single “particle”
as a function of time. Over one period of SAW propagation, the particle traces out an ellipse.
This figure was inspired in part by Fig. 2.1 of Ref. [165].
linear equations:
A(κ2t + β 2 ) + B(2βκl ) = 0
(5.18)
A(2βκt ) + B(κ2t + β 2) =0
In order for a nontrivial solution to this set of equations to exist, the determinant of the
matrix composed of the coefficients must be equal to zero. Writing the SAW velocity as
v = ω/β and using the condition 5.14, this gives us an equation for the SAW velocity
127
v v
v2 2 v2 u v2
u u
2 − 2 = 4t1 − t1 −
u
vt vt2 vl2
which becomes more illuminating when we write down the SAW velocity as traverse velocity
in the bulk times some constant of proportionality ξ, i.e. v = ξvt
v
u 2
q
vt
(2 − ξ 2 )2 = 4 1−ξ 1−ξ (5.19)
u
2 t 2
vl
From this, we see that the SAW velocity in an isotropic material is independent of
frequency: it is in fact only dependent upon the ratio vt /vl . While squaring this equation
gives a polynomial equation with multiple roots, we demand that ξ be both positive and
real, and that ξ < 1 (otherwise v > vt , and waves would propagate into the bulk.) These
conditions yield one solution for ξ, which turns out to be ξ ≈ 0.875−0.955 for possible values
of vt /vl [164]. Thus, SAWs propagate with a velocity slightly lower than that of transverse
waves in the bulk. We may also us Eqn. 5.18 to solve for the ratio of the amplitudes A/B =
q
(ξ 2 − 2)/(2 1 − ξ 2 ). Since ξ < 1, this ratio is real, and the longitudinal and transverse
components of the SAW oscillate in phase with each other. Inspecting Eqns. 5.15 and 5.16,
we see that this implies that the x̂ and ẑ components of the SAW displacement oscillate out
of phase with each other, and a “particle” on the surface will follow an elliptical path. The
full solution for ~u(r), with proper relative amplitudes (up to an overall constant) is plotted
in Fig. 5.2(b). The blue dots in the inset track the trajectory of a single “particle”, indeed
tracing out an ellipse as the SAW undergoes a single oscillation.
128
5.2 Piezoelectricity and piezoelectric Rayleigh waves
In certain materials, the application of strain results in the generation of a macroscopic
electric field, and conversely an external electric field may induce strain in the material. This
effect, known as the piezoelectric effect, happens in anisotropic materials whose underlying
structure breaks inversion symmetry, and was first discovered by Jacques and Pierre Curie in
1880 [160]. In a non-piezoelectric medium, the electric displacement D ~ may be related to the
electric field E~ using the familiar formula Di = ij Ej , where ij is the dielectric tensor of the
medium. In piezoelectric materials, we must tack on an additional term to this relationship,
as strain in these materials causes macroscopic polarization fields. Assuming the strain/field
are small enough such that we need to only consider linear terms, the electrical displacement
field in a piezoelectric material may be written down as
Di = ij Ej + eijk Sjk (5.20)
Here, eijk is the rank-3 piezoelectric tensor, which relates the rank-2 strain Sij to the rank-
1 displacement field6 . The converse to this is that electric field may induce stress in the
materials. This is accounted for by generalizing Eqn. 5.2
Tij = cijkl Skl − ekij Ek (5.21)
where ekij is the same piezoelectric tensor as before7 . It is crucial to remember that the
piezoelectric effect occurs explicitly in anisotropic materials, and thus our previous conver-
sation of waves in isotropic materials will not strictly apply here. In reality, the symmetry
6 Keeping with the Voigt notation, it is common to write down e
ijk as a 3 × 6 matrix.
7 In the Voigt notation, the index shuffling amounts to this term containing the transpose of the 3 × 6
piezoelectric matrix.
129
breaking of many crystals makes the analysis of propagating waves rather painful, and often
times only solvable by numerical techniques. Finding an explicit solution for the propagation
of piezoelectric SAWs is beyond the scope of this thesis, and would not contribute greatly to
our understanding of the results anyway. That being said, we will be able to recycle much of
the intuition built in the previous section to build a qualitative description of piezoelectric
Rayleigh waves.
5.2.1 Equations of motion and Boundary conditions
Since sound in elastic materials propagates ∼ 105 × slower than light, it is a good approx-
imation to work with quasi-static electric fields, i.e. we may write the electric field as the
gradient of the scalar electrostatic potential E ~ = −∇ϑ.~ This reduces the degrees of freedom
we have to solve from 6 (3 components of ~u and 3 components of E) ~ to 4. This simplification
allows us to write Newton’s Law (Eqn. 5.3) for the motion of the deformation field in terms
of the electrostatic potential
ρüi = ∂j Tij = ∂j (−ekij Ek + cijkl Skl )
cijkl
= ekij ∂j ∂k ϑ + (∂j ∂k ul + ∂j ∂l uk )
2
ρüi = ekij ∂j ∂k ϑ + cijkl ∂j ∂k ul (5.22)
We will also assume that our substrate is an insulator, so that there are no free charges and
~ ·D
∇ ~ = 0. This provides another constraint to the system
ij ∂i ∂j ϑ + eijk ∂i ∂j uk = 0 (5.23)
130
Equations 5.22 and 5.23 define a system of 4 differential equations, which we may solve
subject to the appropriate boundary conditions to find the fields ~u(r, t) and ϑ associated
with elastic wave propagation in a piezoelectric material.
Similar to the isotropic case presented in §5.1.1, we will consider an elastic material
that fills the half-volume z < 0, and search for a plane wave solution propagating in the
x̂ direction8 that is exponentially confined to the material surface z = 0. If the SAW has
wave number β and oscillates at frequency ω, we can guess a general form of the functional
dependence ~u and ϑ
~u = u~0 f (z)e−i(ωt−βx)
(5.24)
ϑ = ϑ0 fϑ (z)e−i(ωt−βx)
We will once impose the mechanical boundary conditions that the components of the stress
Txz = Tyz = Tzz = 0|z=0 . However, we have some flexibility with regards to the electrical
boundary condition. There are two electrical boundary conditions that are important to
consider:
1. A perfectly conducting surface. This situation may arise if, for example, a thin metal
film is deposited onto the surface of the insulating piezoelectric substrate. In this case,
the electrical boundary condition is ϑ|z=0 = 0.
2. If the surface is also insulating, there is nothing stopping the electric field from ex-
tending into the vacuum above the surface. In vacuum, the potential obeys Laplace’s
equation ∇ ~ 2 ϑ = 0, and we may insert the predicted form of the solution (Eqn. 5.24)
8 Note that the coordinate system we define here is not associated the principle axes of the underlying
crystal in any way. We choose a direction of SAW propagation, and rotate the coordinate system (and all
the relevant material parameters cijkl , eijk and ij ) to align that direction with the x̂-axis.
131
into Laplace’s equation to generate a differential equation for the z-dependence of the
field in vacuum:
∂z2 fϑ (z) = β 2 fϑ (z) (z > 0)
This differential equation has solutions fϑ (z) = e±βz , and we choose the −β solu-
tion so the electric field doesn’t blow up at z → ∞. Thus we arrive at a general
feature of piezoelectric surface acoustic waves: in the absence of a conducting layer,
the electrostatic potential (and thus electric field) associated with the wave extends
evanescently above the surface of propagation with a decay constant equal
to the SAW wave number. Equivalently, the decay length is equal to the SAW
wavelength λ/2π.
Since there are no free charges on an insulating surface, we maintain that ∇ ~ ·D~ =0
everywhere, which implies that the ẑ component of D ~ must be continuous across the
surface. Using the form of the solution above, this condition produces the boundary
condition Dz |z=0 = βϑ0 e−i(ωt−βx) .
5.2.2 Sketch of Piezoelectric Rayleigh waves
We are now in a position to sketch the solutions of Equations 5.22 and 5.23 that describe
the propagation of piezoelectric SAWs. As stated previously, the anisotropic nature of piezo-
electric materials makes general solution often only achievable through numerical methods,
which are beyond the scope of this thesis. Therefore, we will only provide a qualitative
overview of the solution, from which we will extract several useful concepts. It should be
noted that there is an analytic solution for SAW propagation in materials in the cubic crystal
system such as gallium arsenide, and a detailed derivation may be found in Ref. [166].
132
In §5.1.1, we constructed the displacement field of a Rayleigh wave as the sum of longitu-
dinal and transverse components, each of which satisfy the equations of motion in the bulk.
In the piezoelectric case, the same general prescription is followed: the deformation field and
the electric potential are built out of partial waves with fixed frequency/wave number that
satisfy the equations of motion in the bulk. To construct these partial waves, we start off
with a reasonable guess for the functional form of the SAW fields
~u = ~u0 eγz e−i(ωt−βx)
(5.25)
ϑ = ϑ0 eγz e−i(ωt−βx)
Since all functional dependence of these fields are contained in exponents, the four differential
equations 5.22 and 5.23 become a set of linear homogeneous equations for the amplitudes of
the waves. These equations may be written as
ux,0 0
0
u
y,0
= (5.26)
W
uz,0 0
0
ϑ0
where W is a matrix determined by the particular form of equations of motion (and, sub-
sequently, by cijkl , eijk , and ij .) In the same manner as with the isotropic case, we claim
that for fixed wave number β, there should be some specific values of γ that allow for these
equations to be satisfied. This is equivalent to saying that a nontrivial solution to Eqn. 5.26
requires det(W ) = 0. The determinant of W gives us an 8th order polynomial in γ, with 8
roots in general, however we only take solutions where Re(γ) > 0, else the fields will blow
up at z → −∞. Generally, 4 roots satisfy this condition, and thus the solution will be a
133
sum of 4 partial waves. Note that while we demanded Re(γ) > 0, unlike in the case of an
isotropic material, γ may be complex, and thus the amplitudes of the displacement/electric
potential may oscillate as well as decay into the bulk of the material.
For each root γm , there will be a particular solution to Eqn. 5.26 for the amplitudes
(m) (m)
~u0 , ϑ0 , defined up to an overall constant. These parameters define the mth partial wave,
and a general solution is a superposition of the 4 partial waves
(m) γz −i(ωt−βx)
~u =
X
Am~u0 e e
m
(5.27)
(m)
ϑ= Am ϑ0 eγz e−i(ωt−βx)
X
m
where Am is the relative amplitude of each of the 4 partial waves. In the same manner
as in §5.1.1, we invoke the appropriate boundary conditions to generate another set of 4
homogeneous linear equations for the coefficients Am , and obtain the velocity of the wave
by setting the determinant of coefficients to zero.
5.2.3 Relevant parameters
Since piezoelectric materials are anisotropic, SAW propagation will depend on the coordinate
system we choose (i.e., how are the surface, which we have taken to be normal to ẑ, and
the SAW propagation direction, which we have taken to be x̂, oriented with respect to
the underlying crystallographic axes?) When designing SAW devices, our goal is often to
maximize the coupling between electrical excitations and mechanical excitation. For a given
material, how should we go about choosing how to orient the surface (also called the crystal
cut, or simply cut) and the propagation direction?
As stated before, there are two electrical boundary conditions to consider: the case of an
134
Material Cut Propagation axis v (m/s) K 2 (%)
LiNbO3 Y Z 3488 4.8
LiNbO3 128◦ -rotated Y X 3979 5.4
Quartz ST X 3158 0.14
GaAs (100) [011] 2864 0.07
Table 5.1: Relevant material parameters for common SAW substrates, reproduced from
Chapter 4 of Ref. [160] and Chapter 9 of Ref. [167], both of which contain more complete
tables of materials/parameters.
insulating surface (where the electric field evanescently extends into the vacuum above the
surface) and the case of a conducting surface (where the ϑ = 0 at the surface.) When the
procedure in §5.2.2 is executed subject to these two boundary conditions, it is typically found
that a SAW propagating along an insulating surface propagates slightly faster than a SAW
propagating along a conducting surface [160]. The relative size of this velocity shift is useful
for quantifying piezoelectric coupling: we define the piezoelectric coupling coefficient for a
given cut/propagation axis as K 2 = 2∆v/v, where v is the insulating surface velocity, vm is
the conducting (metalized) surface velocity, and ∆v = v − vm . For a given material, SAW
cuts and propagation axes are typically chosen to maximize K 2 . Table 5.1 contains useful
properties for some common SAW materials/cuts, reproduced from Refs. [167, 160]9 . When
working with SAW devices, it is conventional to specify both the cut and the propagation
axis along with the material: for example, the quartz cut specified in Table 5.1 is commonly
referred to as “ST-X Quartz”.
9 Note that there seems to be some minor disagreement between these sources: when in conflict, we defer
to Ref. [167]
135
5.3 SAW devices
We now turn to a description of electromechanical devices based on piezoelectric SAWs. As
stated before, these devices are almost all periodic structures with periodicity λ fabricated
on the surface of a piezoelectric substrate, and will operate at or around the so-called center
frequency f0 = v/λ. We have already established that a full solution of SAW propagation on
a surface with translational symmetry is beyond the scope of this thesis, and breaking this
symmetry/adding electromagnetic sources will only complicate the matter. Nevertheless,
there exist powerful tools for describing the electrical response of a SAW device without
directly solving the electromechanical equations of motion. Our strategy will be to use these
techniques to describe the electrical response of a SAW device as a function of frequency,
and map that response onto the response of simple circuit elements.
5.3.1 Interdigitated transducers
The most important component of any SAW device is the interdigitated transducer, or IDT.
Interdigitated transducers are the method by which we convert electrical signals into acoustic
signals (or vice versa): they consist of two electrodes, with interlocked metallic “fingers” of
width a (see Fig. 5.3(a)) deposited on the surface of the substrate. The periodicity of the
IDT is λ = 2a+2b, where b is the spacing between the fingers. One wavelength will consist of
a single finger from either electrode of the IDT, and so we generally characterize the length
of the IDT by counting the number of finger pairs Np . To keep with the convention from
§5.1.1 where SAWs propagate in the x̂-direction, we will define the long axis of the IDTs as
the ŷ-direction: that way, when a voltage is applied across the two electrodes of the IDT the
voltage is spatially periodic along the x̂-direction. The width of the finger overlap w defines
136
(a) (b) +1 0 +1 0 ...
(c)
(d) (e)
Figure 5.3: Interdigitated transducers for SAWs: (a) A schematic of an interdigitated
transducer (IDT) of width w with periodicity λ = 2(a + b). (b) Delta function model
of infinitely thin IDT fingers, which yields the frequency response function like the one
plotted in (c). (c) Frequency response calculated for two different IDTs on ST-X quartz,
with a = b = 1 µm. As we increase the number of finger pairs of the IDT, the response
at the center frequency f0 (≈ 790 MHz) becomes larger, and the bandwidth of the device
becomes more narrow. (d) Full electrical model of an ideal IDT, including real (Ga (f )) and
imaginary (Ba (f )) acoustic impedances, as well as the geometric capacitance CIDT of the
IDT structure. (d) Calculated Ga (f ) and Ba (f ) for the Np = 30 device from (c), with
w = 60µm.
the width of the SAW beam excited by the IDT.
To understand the electrical response of an SAW IDT, let us approximate each finger of
the IDT as an infinitely thin point source, evenly spaced along the x̂-direction [160, 168] (see
Fig. 5.3(b).) We may then write down the applied voltage to the surface as a sum of delta
functions, spaced by λ/2. If we hold one electrode at ground potential and apply a voltage
V = +1 (in arbitrary units) to the other electrode, the applied voltage has an x-dependence
137
of
Np −1
(5.28)
X
ϑ∝ δ(x − λn)
n=0
Fourier transforming this potential gets us the wave vector response (or transfer function)
Np −1
sin Np kλ/2
e−iλkn = e−i(Np −1)kλ/2 (5.29)
X
H(k) ∝
n=0 sin(kλ/2)
PN −1 n
Where to get the second equality we’ve used the identity n=0 r = (1 − rN )/(1 − r)
for a geometric series. Near the center frequency of the IDT, f ≈ f0 , kλ/2 ≈ π, and we may
approximate sin(kλ/2) ≈ −kλ/2. Using the relationships f0 = vs /λ and k = 2πf /vs , we
may write approximate the frequency response as a sinc function centered on f0
sin Np π(f − f0 )/f0
H(f ) ≈ Np (5.30)
Np π(f − f0 )/f0
We can extract some valuable information from the form of this frequency response.
Since the sinc function is maximized when the argument is 0, as expected, the most efficient
conversion of electrical signals into SAWs (and vice versa) will happen at f = f0 . We also
see that as we increase Np , the magnitude of the response function will increase, while the
bandwidth (the frequency range over which |H(f )|/|H(f0 )| is of order unity) will decrease.
Fig. 5.3(c) shows the calculated |H(f )| plotted for two IDT’s with different Np .
We are now in a position to model the SAW IDT as a lumped element electrical compo-
nent, and infer the frequency dependence of this component. When an oscillating voltage V
is applied across the IDT, electrical power will be converted into propagating SAWs: we may
define the real part of the IDT conductance via this dissipated power, i.e. P = Ga |V |2 /2.
138
Likewise, the power of a propagating wave is proportional to the square of the potential
generated by the wave: PSAW ∝ ϑ2SAW . In an ideal case, where all the power dissi-
pated at the IDT is converted into SAWs, these two power are equal to each other, and
we may infer that the frequency dependence of the real part of the acoustic conductance
Ga (f ) ∝ |ϑSAW (f )|2 ∝ |H(f )|2 . It may be shown that the acoustic admittance of the IDT
also has an imaginary component Ba (f ) [167].
The constant of proportionality of Ga depends on the IDT geometry, and needs to be eval-
uated numerically [167, 160]. For a = b, this numerical evaluation leads to a full expression
of the complex SAW IDT conductivity, modeled as 3 parallel lumped element components
(see Fig. 5.3(d)):
YIDT (f ) = Ga (f ) + i(Ba (f ) + 2πf CIDT ) (5.31a)
2
sin X
Ga (f ) = Ga0 (5.31b)
X
sin 2X − 2X
Ba (f ) = Ga0 (5.31c)
2X 2
where CIDT is the unavoidable geometric capacitance of the IDT, and
f − f0
X = Np π
f0
(5.32)
Ga0 ≈ 1.3K 2 Np2 (2πf )wCs
where Cs is the capacitance per IDT finger pair per unit length, and w, Np , f0 and K 2
are the aforementioned beam width, finger pair number, center frequency and piezoelectric
139
coupling coefficient respectively.10 An example calculation of Ga (f ) and Ba (f ) is plotted in
Fig. 5.3(e).
5.3.2 Coupling of Modes and the P-matrix
The above discussion of ideal IDTs ignores the fact that the metal strips that form the
IDT, in addition to transducing SAWs, may also reflect SAWs impinging upon them. The
magnitude of the reflection coefficient |rs | of a single metal strip tends to be small (≈ 0.1−2%,
depending upon the substrate and design [160, 169, 15, 10]), and thus the above analysis is
still fairly accurate if Np |rs |−1 , where only a small fraction of the wave is reflected while
in transit across the device. We will, however, encounter SAW devices that are explicitly
designed to reflect SAWs, such as the Bragg mirrors that form a SAW cavity.
For devices where reflections are non-negligible, we use the coupling of modes (COM)
method (see chapter 8 of Ref. [160] for a more complete discussion) to account for both
transduction and reflections. The COM method, similar to the ideal transducer model
above, assumes that transducers are delta functions that may both reflect and transduce
SAWs. This model is used to set up a system of coupled differential equations relating the
leftward propagating SAW amplitude AL , rightward propagating SAW amplitude AR , and
oscillating current (I) and voltage (V ) across the transducer. The system of equations is then
solved subject to the appropriate boundary conditions to yield a 3-port scattering matrix,
called the P-matrix [160, 167]
10 Since C
IDT = Cs Np w, we may easily compare the relative magnitude of the acoustic admittance and
the capacitive admittance: at f = f0 , the ratio of the acoustic admittance to the capacitive admittance
is Ga (f0 )/2πf0 CIDT ≈ 1.3K 2 Np . Thus we see that, the acoustic admittance doesn’t start dominating
until Np > (K 2 )−1 . For strongly piezoelectric substrates, such as LiNbO3 , this gives us a reasonable value
Np ≈ 20, but for weakly piezoelectric substrates such as quartz or GaAs, the number of finger pairs required
is absurdly large (500-1000), and fulfilling this requirement renders the IDT bandwidth unfeasibly small.
Thus, on these substrates, the IDT response is typically dominated by the geometric capacitance.
140
(a) (b) (c)
(d)
Figure 5.4: The P-matrix for SAW devices: (a) We may model a SAW element as
a 3-port object, with two SAW ports (a left/right port with incoming SAW amplitude
AL,in /AR,in and outgoing amplitude AL,out /AR,out ) and an electrical port with current
I and voltage V . (b) In the absence of any incoming/outgoing SAWs, the only non-zero
component of the P-matrix is P33 , which we recognize as the electrical admittance Y (f ).
(c) In the absence of an electrical connection, the P-matrix collapses down to a two port
scattering matrix describing the reflection of SAWs at the device. (d) The major advantage
of the P-matrix formalism is that several devices may be cascaded into one composite P-
(1) (2)
matrix. These structures may be electrically connected in parallel (such as Pij and Pij in
the example), however we may also cascade elements with no electrical connection (such as
(3)
Pij .)
AL,out P11 P12 P13 AL,in
A
=
P21 P22
P23 AR,in
(5.33)
R,out
I P31 P32 P33 V
Here, AL,in /Ar,in (AL,out /Ar,out ) are the amplitudes of the rightward/leftward SAW
141
propagating into (out of) the structure in question, as shown in Fig. 5.4(a). The P-matrix
has several convenient features that are worth explicitly noting:
1. In the absence of SAWs (AL,in , AR,in = 0), I = P33 V . Therefore, P33 = Y the electri-
cal admittance of the transducer, which we will treat as a lumped element (Fig. 5.4(b).)
In the end, P33 will be the electrical element we insert into our circuit model.
2. P11 , P12 , P21 and P22 describe scattering of SAWs off the structure. This sub-matrix
can be thought of as a scattering matrix for SAWs along the path of a structure. For
example, an ideal non-reflecting transducer will have P12 = P21 = 1 (up to a phase
factor that the SAW picks up traversing the structure) and P11 = P22 = 0.
3. All the elements in the third row/column have to do with transduction (converting
SAWs into an electrical signal/vice versa.) If we have a structure that doesn’t trans-
duce (such as a Bragg mirror that is electrically disconnected from the circuit), these
elements are all zero, and the P-matrix will reduce to the SAW scattering sub-matrix.
The primary advantage of the P-matrix is that multiple structures may be cascaded with
relative ease. This means that, in order to obtain P-matrix elements for a composite SAW
structure such as a resonator (a launching IDT and two Bragg mirrors), rather than solving
the COM equations for the entire structure, we only need the solution for each structure
(Bragg mirrors and IDT) and can cascade the structures together to get a description of
the entire device.11 As such, we’ll individually model Bragg mirrors and interdigitated
transducers, and the concatenate them together (with optional free propagation area) to
model a full SAW resonator.
11 The rules for cascading P-matix elements are described in Chapter 11 of [160]. Python code that
generates P-matrix elements for simple periodic SAW structures and cascades P-matrices automatically may
be found in [170].
142
Np = 20
600 Np = 30
Conductance (µS)
Np = 50
400
200
0
5.10 5.25 5.40
Frequency (GHz)
Figure 5.5: Effect of reflections on IDT conductance: Re(P33 (f )) = Re(Y (f )) = G(f )
calculated for several values of Np , with 1% reflection (solid, rs = −0.01i) and no reflection
(dashed, rs = 0) per transducer electrode. All G(f ) are calculated for a device on ST-X
quartz, with a IDT periodicity λ = 600 nm and finger overlap w = 60 µm. When reflection
is ignored, the transducers have the same sinc function response as derived in the ideal case
in §5.3.1. When we add reflection back in, we see significant distortion of the response as
Np gets larger.
5.3.3 A second look at transducers
To start our analysis of modeling SAW structures with the COM method, we will inspect how
reflections modify the response of an IDT compared to the ideal case considered in §5.3.1.
Since the main metric of interest is the IDT electrical admittance Y (f ), we may make a direct
comparison by solving the COM equations for P33 (f ) with and without reflections taken
into account. Doing so for an ideal, reflection-less transducer12 (i.e. setting the per-grating
reflection |rs | = 0) yields the same sinc function response for Re(P33 (f )) = Re(Y (f )) = G(f )
derived in §5.3.1 (Fig. 5.5, dashed lines.) We see the same characteristic dependence of G(f )
12 Even in the presence of reflections, an effectively reflection-less transducer may be created by using a
double finger structure, where each finger of the IDT consists of two strips of metal instead of one [167, 11, 17].
However, since this reduces the line-width by a factor of two, at high frequencies these double finger structures
become difficult to fabricate.
143
on Np , namely that maximum response G(f0 ) ∝ Np and the bandwidth of the IDT is ∝ Np−1 .
When we repeat the same calculation with reflections included (rs = −0.01i, or 1%
reflection per IDT finger13 ) we observe significant distortion of the sinc function response.
When Np |rs |−1 we expect reflections to not contribute greatly to the response. Indeed,
we see in Fig. 5.5 that for Np = 20 the sinc function response is largely maintained, except
for a minor shift of the center frequency. However, as Np becomes comparable to |rs |−1 ,
the response becomes asymmetric, the bandwidth becomes smaller than in the reflection-less
case, and the peak response becomes larger. This means that a device with strong reflection
will actually have stronger piezoelectric coupling than an otherwise identical non-reflective
device, however these devices must be more carefully designed to account for non-idealities
like frequency distortion.
5.3.4 Acoustic Bragg mirrors
By taking advantage of the finite reflection per electrode, we may fabricate a SAW mirror
by placing electrically disconnected strips of metal in the path of SAW propagation. Each
strip of metal will reflect some small fraction of an impinging SAW: If the half-wavelength
λSAW /2 of an impinging SAW is close to the periodicity of these strips, the reflected waves
will constructively interfere, satisfying the Bragg condition, and SAWs will be coherently
reflected. If the number of strips Nmirror |rs |, the Bragg mirror will reflect impinging
SAWs with a high probability, and we may approximate the structure as a high-finesse
mirror. The frequency range ∆f over which the Bragg condition is met and SAWs are
reflected completely in an infinitely long grating (also called the width of the SAW stop-
13 The phase of the reflection coefficient quantifies what phase the reflected wave will have with respect
to the impinging wave, and will in general be imaginary, with the sign depending on the specific device
parameters [160].
144
band), is determined [160] by the magnitude14 of r: ∆f /f ≈ 2|r|/π.
The P-matrix reflection coefficient P11 calculated by solving the COM equations, and its
magnitude and phase are plotted as a function of frequency in Fig. 5.6(a) and (b) respectively.
Anticipating assembling mirrors and a transducer into a SAW resonator, for this model we
have chosen a λ = 604 nm grating on ST-X quartz, giving f0 ≈ 5.19 GHz slightly lower than
the center frequency of the transducers in Fig. 5.5 to account for distortion. Once again,
the reflection coefficient per strip is set to rs = −0.01i, i.e. 1% of the SAW amplitude is
reflected at each strip. Since the Bragg grating is electrically disconnected, transduction is
excluded. We see from Fig. 5.6(a) that in the stop-band, the SAW is reflected with near unit
probability, while outside the stop-band the reflection coefficient is a complicated function
of frequency.
The phase of the reflected signal evolves smoothly as a function of frequency inside the
stop-band: from this, we can calculate the delay time of a SAW reflecting off the structure
as τd (f ) = −dθP 11 /df (2π)−1 , where θP 11 is the phase of P11 . This relationship gives us a
simplified model of the Bragg mirror: rather than considering the grating as a distributed
mirror, inside the stop band we may think of the Bragg reflector as a point-like mirror
positioned at distance LP (f ) = vτd (f )/2 from the edge of the grating (see Fig 5.7(a).)
5.3.5 SAW resonators
Having developed models for both IDTs and Bragg mirrors, we are now in a position to
model the electrical response of a SAW resonator. A SAW resonator (see Fig. 5.7(a) for a
14 Without doing any calculations, we may gain intuition about this relationship by remembering that
in a Bragg mirror, SAWs are reflected over a finite distance which is roughly inversely proportional to the
reflection per grating. If the SAWs were completely reflected by a single strip (a point-like mirror) there would
be no Bragg condition required for reflection, and SAWs of all frequencies would be reflected. Conversely, as
|r| → 0, the length of a Bragg mirror required to reflect the SAW becomes infinitely long and the range of
frequencies ∆f that meet the Bragg condition in an infinitely long structure approaches zero.
145
20
1.0
0 15
Phase(P11) (rad) Delay time (ns)
0.6
|P11| 10
−π 5
0.2
5.15 5.20 5.25 5.15 5.20 5.25 0 5.15 5.20 5.25
F (GHz) F (GHz) F (GHz)
Figure 5.6: Bragg Mirrors: (a) Magnitude and (b) phase of the reflection coefficient P11 (f )
as a function of frequency calculated for a SAW Bragg mirror. This device, calculated for a
mirror fabricated on ST-X quartz, has 940 reflector gratings spaced by λ/2 = 302 nm (total
device length ≈ 284 µm) and assumes that each grating reflects 1% of an incoming SAW
(rs = −0.01i.) Over the mirror stopband, SAWs are reflected with near unit probability
and the phase of the reflection coefficient evolves smoothly as a function of frequency. (c)
Delay time of a SAW reflected off the mirror τd (f ) = −dθP 11 /df (2π)−1 . From this, we may
calculate the effective penetration length of the mirror LP (f ) = vτd (f )/2.
schematic) consists of an IDT sandwiched between two Bragg mirrors. Quite like an optical
Fabry-Pérot cavity, the resonance condition of a SAW resonator is satisfied when the effective
length of the cavity matches a half-integer number of SAW wavelengths. Unlike its optical
counterpart, the external coupling to the resonator is provided by the IDT, which inherently
lives in the SAW propagation path. While this has the advantage of increasing the coupling
strength (since the SAW may interact with the external circuit over many wavelengths), it
comes with several caveats, which will be explored further in Chapter 6.
To model a SAW resonator, we will take the two structures we have already modeled
(IDTs and Bragg mirrors) and use the cascading properties of the P-matrix to build a
model for a SAW resonator. An example calculation is plotted in Fig. 5.7: it assumes
SAW propagation on ST-X quartz, launched and transduced by an IDT with periodicity
146
(a) (b)
(c)
(d)
Figure 5.7: COM model of a SAW resonator: (a) Schematic of a SAW resonator,
consisting of an IDT (red) for external coupling and two Bragg mirrors (blue) on either side
of the IDT confining emitted SAWs. This structure may be thought of as an acoustic Fabry-
Pérot resonator, with length Lef f determined by the IDT width, the mirror spacing Lf ree
and the effective penetration length of SAWs into the mirror LP . (b) Real and imaginary
parts of the SAW resonator electrical admittance P33 plotted as a function of frequency,
modeled by concatenating the P-matrices of each individual structure, see text for details.
The response plotted in (b) inherits properties from both the transducer admittance P33
and the mirror reflection P11 , as is evident from the zoomed in plot of the conductance
(c) plotted alongside the mirror P11 and transducer conductance Re(P33 ) . However, the
full response is well approximated by an effective series LRC resonance in parallel with the
geometric IDT capacitance CSAW , the so-called Butterworth-van Dyke equivalent circuit,
shown in panel (d). If the free spectral range of the structure is low enough such that more
than one resonance condition may fall within the stop-band, we may model the multiple
resonant structure by adding more LRC resonances in parallel with the structure.
λ = 600 nm, width w = 60 µm, and Np = 25 finger pairs. The mirrors on either side of
the IDT are identical to the structure described in Fig. 5.6. Note that the mirrors have a
periodicity λmirror = 604 nm so that the mirror stop-band is centered on the distorted IDT
response. There are several important caveats to this model worth mentioning:
147
1. To model propagation loss, and make the any resonances have finite width, this model
includes a small imaginary component to the SAW wave vector.
2. Sometimes, for design purposes or to ease fabrication, it makes sense to add some
distance of free SAW propagation Lf ree between the IDT and the Bragg mirrors. To
account for this, rather than cascading 3 P-matrixes together as “Mirror-IDT-Mirror”,
this model actually consists of 5 P-matrices cascaded together to create a more realistic
“Mirror-empty space-IDT-empty space-Mirror” model. In this model, we have chosen
Lf ree = 3 µm.
In Fig. 5.7(b) we plot both the real and the imaginary parts of the cascaded P33 =
Y (f ). The model resonator displays a clear resonance (zero crossing of Im(Y (f )) along
with enhanced Re(Y (f ))) at f ≈ 5.19 GHz. If we zoom in on Re(P33 ) we see that the
fine-scale structure inherits properties of both the Bragg mirrors and the IDT. Inside the
stop-band of the mirrors, the conductance is suppressed (except at the resonant frequency,
where it is enhanced), while outside the stop-band the conductance is a convolution of
the IDT conductance and the mirror reflectance (plotted alongside in orange and dashed
blue respectively.) The fine scale features outside the stop band are, however, somewhat
inconsequential compared to the response of the structure near its resonance. This resonant
response may be accurately modeled as fictitious LRC series circuit in parallel with the
geometric capacitance of the IDT, as shown in Fig. 5.7(d). Thus, the underlying piezoelectric
interaction of the resonator structure may be reduced to a simple LCR resonator, called its
Butterworth-van Dyke equivalent [171]. This approximation greatly simplifies the analysis
of integrating SAW devices with other electronic circuits, such as superconducting qubits,
which will be discussed further in Chapter 6.
148
A hallmark of Fabry-Pérot resonators is that they may host multiple modes, at every
frequency that satisfies the resonance condition that the cavity length Lef f is equal to a
half-integer number of wavelengths. For an ideal point-like mirror, which reflects all wave-
lengths, there are infinitely many resonant frequencies, spaced by the free spectral range
∆fF SR = v/2Lef f . However, our Bragg gratings are not ideal point-like mirrors: as we saw
in § 5.3.4 they will only efficiently reflect over a small stop-band of width ∆f /f ≈ 2|rs |/π.
How many resonant conditions do we expect to satisfy inside a cavity? Will these structures
even necessarily have a resonant frequency?
Since the Bragg gratings reflect roughly |rs | of the impinging SAW per grating, and the
gratings are spaced by ≈ λ/2, a good approximation of the mirror penetration length is
Lp ≈ λ/2|rs |. Therefore, the minimum length of a SAW cavity, is Lef f = 2Lp = λ/|rs |.
For such a cavity, ∆fF SR /f = v/2Lef f f = |rs |/2. Since 1/2 < 2/π, ∆fF SR will be slightly
smaller than the stop-band width, and we expect the cavity to host at least one mode. We
also see that as long as the space between the mirrors is smaller than Lp , the cavity will
usually host only one mode. Of course, if we add in a large free-propagation distance d,
∆fF SR will decrease and the cavity may host many modes [11, 169]. In the case of multiple
resonances, we may model each resonant condition as additional parallel LRC resonance (as
displayed in Fig. 5.7(d) by the elements connected by dashed lines.)
149
Chapter 6
Quantum acoustics using surface
acoustic waves
The wide applicability of surface acoustic waves to microwave telecom frequency electronics,
which superconducting qubits are more or less specifically tailored to be compatible with, has
not gone unnoticed. Not long after O’connell et al. demonstrated coherent coupling between
a superconducting qubit and a MEMS resonator [2], several groups around the world began
reporting results using piezoelectric substrates to mediate coupling between superconducting
qubits, mainly transmons, and surface acoustic waves [7, 8, 9, 10, 11, 17, 12, 172, 15]. Indeed,
transmon qubits have a major advantage when considering interaction with SAWs: the IDT
can play the role of both the SAW transducer and the shunt capacitance that lowers EC and
brings the qubit into the transmon regime.
The slow speed of sound opens up many interesting potential studies/applications of
SAWs in the quantum regime that would not necessarily be accessible with light, or even other
acoustic modes such as bulk acoustic waves or microelectromechanical devices. Importantly,
SAWs propagate on the surface of a substrate, where devices are lithographically defined.
An IDT structure interacts with a SAW over many wavelengths, meaning that in quantum
acoustics with SAWs the “atom” (qubit) is typically much larger than the wavelength of the
radiation interacting with it. This is in stark contrast to quantum optics, where the quantum
150
system (an actual atom or a superconducting qubit) is almost always much smaller than the
wavelength of light. This parameter regime has lead to the prediction and observation of
exotic effects, such as non-exponential decay of a qubit [16, 17] whose decay is dominated
by conversion into SAWs. Additionally, since a qubit may interact with SAWs over multiple
wavelengths, the acoustic coupling rate to a SAW mode may conceivably be made much
larger than its electromagnetic counterpart. It is foreseeable that quantum acoustics devices
may be able to enter the so called ultrastrong coupling regime, where the excitation exchange
rate Γ between the qubit/SAWs is of order ω01 [173], though no experiments exploring the
regime have yet been reported.
In classical microwave electronics, the slow speed of SAW propagation is often used to
create delay lines, where a SAW signal may spend an appreciable amount of time (several µs)
propagating across a substrate before it is transduced again. This pitch-and-catch scheme of
a SAW delay line can foreseeably be used as a resource in quantum devices. Several demon-
strations of employing this delay have already been experimentally recorded, including using
the SAW propagation delay to implement a delayed choice quantum erasure experiment [15]
and using a qubit as a “router” to controllably block propagating SAWs [174]. Additionally,
the small wavelength of SAWs at ∼GHz frequencies have been proposed as a resource for
creating bosonic quantum memories using SAW resonators [18], since a resonator with a high
density of acoustic modes can easily be fabricated with a relatively small spatial footprint.
Quantum acoustics devices have also been proposed for use as coherent microwave-to-optical
converters [175], where the wavelength of the SAW matches the wavelength of a confined
optical mode.
This field of quantum acoustics, using SAWs or other modes, is in its infancy, with
much uncharted territory to be explored both answering fundamental scientific questions and
151
designing potential applications. When thinking about how to start exploring this territory,
we would like to ask “how can we leverage the experimental apparatus built up earlier in this
thesis (i.e. 3D transmon qubits) to start performing quantum acoustics experiments?” In this
chapter, we outline experiments exploring a novel way of mediating the SAW-qubit coupling
in the 3D transmon geometry: a capacitively coupled device in a “flip-chip” configuration.
We will explore how one may model such an experiment, and then report on preliminary
experimental data investigating these devices.
6.1 Mediating the coupling between SAWs and qubits
The first question we may ask is “how can we mediate coupling between SAWs and qubits?”
In light of the previous section, this seems like a somewhat unnecessary question: we already
established that transmon qubits need a shunt capacitance, which can easily be provided by
the IDT. Therefore the most conceptually straightforward method seems to be direct galvanic
coupling, where either side of the Josephson junction is directly connected to either electrode
of the IDT. This design is conceptually straightforward, has been pursued by multiple groups
[7, 11, 9], and in principle offers the strongest coupling, since voltage fluctuations across the
IDTs induce voltage fluctuations across the junction with unit efficiency.
There are, however several drawbacks to this direct galvanic coupling scheme. For in-
stance, the anharmonicity of the transmon is inversely proportional to the IDT capacitance,
and thus there is a limit to the size of the transducer (both finger pair number Np and over-
lap length w) before the anharmonicity gets too small1 . Since, as we saw in Chapter 5, the
1 Here, “too small” is set by the control pulses we send to the device: we would like to remain in the qubit
manifold (i.e. not accidentally excite the transmon to its |2i state) while using pulses of finite width. These
pulses are sinusoidal, however their finite width means they will inevitably have Fourier components at higher
frequencies. We may accidentally drive the |1i → |2i transition if there are significant Fourier components
at frequencies ≥ α. Since control pulses tend to be ∼ 10 ns long, we would at least like α > 100 MHz.
152
(a) (b
(c) (d)
Figure 6.1: Schematic of the proposed experiment: The proposed quantum acoustics
experiment consists of a SAW resonator capacitively coupled to a superconducting qubit
device in a flip-chip configuration (b), read out using a 3D electromagnetic microwave res-
onator (a). (c) The device may be modeled as a superconducting qubit coupled to a SAW
admittance via a capacitive network. (d) We may further approximate the SAW by it’s
Butterworth-van Dyke equivalent circuit, and model the transmon qubit as a weak nonlin-
earity in parallel with a linear circuit to extract the Hamiltonian of the system.
acoustic conductivity of a SAW IDT is proportional to w and NP2 , this puts a limit to the
achievable coupling in practice. Additionally, Josephson junctions are notoriously sensitive
to static discharge, and strongly piezoelectric substrates like LiNbO3 are often susceptible
to discharging events2 and thus unwieldy to fabricate Josephson junctions on. It is, there-
fore, desirable to fabricate the qubit device on a well-known and well-behaved substrate like
silicon, and to couple it to a SAW device on another piezoelectric substrate.
Along these lines, there have been demonstrations of coupling a qubit to a SAW device on
another substrate using a tunable mutual inductance [10, 172, 15]. While these devices have
demonstrated high-coherence and flexibility, the experimental setup is somewhat involved,
2 Lithium niobate is also pyroelectric, i.e. temperature changes can induce polarization of the material.
My experience has been that, when heating devices to cure resist in fabrication, it is possible to hear static
discharge events in the substrate!
153
and incompatible with the 3D transmon geometry, where we aren’t necessarily able to run
bias lines into the cavity to tune individual circuit elements, and the qubit device is explicitly
disconnected from ground. A more simple way to engineer coupling between the SAW and
qubit devices while still fabricating them on different substrates is through a geometric
capacitance, outlined in Fig. 6.1. This coupling is similar to the coupling between a transmon
and a linear LC oscillator described in Chapter 2: a SAW device, which we choose to be
an acoustic resonator in analogy to the electromagnetic resonator of cQED, fabricated on a
piezoelectric substrate is coupled to a Josephson junction device on a silicon substrate. We
then place one device face down on the other: in this “flip-chip” configuration, the two sets
of pads serve a dual purpose: they form the parallel plate capacitance C1 that couples the
SAW device to the qubit, as well as the dipole antenna that couples the qubit to the 3D
electromagnetic cavity we use for manipulation/readout.
6.2 Device design
To understand the design constraints of this experiment, we will need to consider a Hamil-
tonian that describes the coupled qubit-acoustic system. For a first experiment, we would
like to couple a discrete acoustic mode, confined in a SAW resonator, to a superconducting
qubit. This setup is an acoustic analogue to cQED, and therefore it makes sense to bor-
row from the formalism developed in Chapter 2 where a qubit was coupled to a discrete
electromagnetic mode. Therefore, to model this experiment, we choose the by now familiar
Jaynes-Cummings-like Hamiltonian [18]
α † †
† †
H = ~ω01 q† q − q q qq + ~ ωk mk mk + gk (q† mk + qmk ) (6.1)
X
2 k
154
Here q† and q are the raising/lowering operators of the qubit mode, ω01 is the ground to
excited state transition frequency of the transmon qubit, α = EC is the quartic nonlinearity
of the transmon circuit. When designing the experiment, we opt to explicitly maintain
this nonlinear term rather than truncating to the qubit manifold and writing down the
Hamiltonian in terms of Pauli operators, since the transmon capacitance that determines
EC will be a major design parameter for determining if we have enough nonlinearity to
operate the circuit in the transmon regime.
As noted in Chapter 5, a single acoustic resonator may host multiple modes. Thus, the
Hamiltonian Eqn. 6.1 consists of a sum over k acoustic modes, with raising and lowering
†
operators mk and mk , each a linear oscillator oscillating at ωk with a coupling gk to the
transmon mode. In this dissertation, however, we will only be considering geometries with
a single acoustic mode: in this case, the Hamiltonian reduces down to
α † †
H = ~ω01 q† q − q q qq + ~ωm m† m + ~gm (q† m + qm† ) (6.2)
2
Of the four parameters in this Hamiltonian, the SAW resonance is relatively easily con-
strained by λIDT , and ω01 may be tuned in situ by using a split-junction transmon and
applying a magnetic flux. Thus, the main design parameters we must extract from any pro-
posed experimental geometry are the transmon anharmonicity α = EC = e2 /(2Cef f ), where
Cef f is the effective capacitance of the transmon mode, and the qubit-acoustic coupling gm .
The modeling process described below aims to calculate Cef f and gm from a given geometry.
To facilitate this discussion, we will consider an example device throughout this section, with
design parameters and derived values summarized in Table 6.1.
155
6.2.1 Modeling the experiment
As a starting point, let us take the effective circuit diagram of the device outlined in
Fig. 6.1(c). The qubit device on silicon consists of a Josephson junction in parallel with
a capacitance CJ that incorporates both the intrinsic junction capacitance and the capaci-
tance between the paddles. Each paddle forms a parallel plate capacitor C1 with the match-
ing paddles from the SAW device: in the symmetric case, these parallel plate capacitances
are equal. The SAW IDTs have some intrinsic capacitance as well as the capacitance of
the paddle structures connected to the IDT: these add in parallel to the total capacitance
between the two electrodes of the SAW device CSAW . As explained in Chapter 5, the
electrical response of the SAW cavity is encapsulated in a frequency dependent admittance
YSAW (f ) = Ga (f ) + iBa (f ). There will also be some parasitic capacitance Cp between the
non-parallel plate pads, which we would like to minimize. We justify this lumped element
circuit model by noting that electromagnetic radiation < 5 GHz has a wavelength of 6 cm,
and the SAW+qubit device is ≈ 1 mm in its longest dimension.
The comparatively weak nonlinearity of the transmon, such that the Hamiltonian can
approximately be diagonalized in the harmonic oscillator basis, offers a distinct advantage
when designing experiments that involve transmon qubits: when engineering a Hamiltonian,
it is often fairly accurate to break up the transmon into an effective linear inductor in series
with a small, nonlinear element. In this manner, we may model the circuit as a weak
nonlinearity connected to a linear network of capacitors and inductors, which form a system
of linear oscillators, and consider the nonlinearity as a perturbation on this system [78]. We
represent this schematically by splitting the Junction into an inductor LJ = Φ20 /EJ (where
EJ is tunable by an external magnetic flux) along with a parallel nonlinearity (red element
156
in Fig. 6.1(d).) We then quantize the Hamiltonian of the system in the basis of (coupled)
harmonic oscillators.
It was belabored in Chapter 2 that a quantum harmonic oscillator, in the absence of
any anharmonicity, responds with a spectrum equivalent to its classical counterpart. This
is also true for a set of coupled harmonic oscillators [62], i.e. a network of inductors and
capacitors. Thus, the problem of reconstructing the Hamiltonian of a device for a given
design is greatly simplified: all we must do is simulate the classical response of the circuit
(i.e. calculate Y (f ) of the linear network), and find the resonances (where Im(Y (f )) = 0)
and characteristic impedances of these resonances. Equipped with this knowledge, we can
promote these classical oscillators to a set of coupled quantum harmonic oscillators, with
zero point fluctuations described by Eqn. 2.24, and add the transmon nonlinearity back in
as a perturbative term.
Pragmatically we will break this up into several steps: we will use a commercial finite
element solver to numerically solve for the capacitances between the macroscopic electrodes
for a given device geometry. We will then use the coupling of modes model described in
Chapter 5 to construct the Butterworth-van Dyke (BvD) equivalent of the SAW resonator.
This will numerically fill out the circuit diagram shown in Fig. 6.1(d): we will then solve
this circuit for Y (f ) as a function of the tunable linear Josephson inductance LJ . From this
data, we will be able to extract the coupling between the linear modes gm and the effective
capacitance of the transmon-like mode, which will give us α.
6.2.1.1 Finite element modeling
For maximal coupling between the SAW resonator and the superconducting qubit device, we
would like to maximize the capacitance C1 and minimize the parasitic capacitance Cp , see
157
(a) (b)
(c)
"IDT"
"junction"
Figure 6.2: Finite element simulations of the paddle structure: We use a commercial
finite element solver (COMSOL) to solve for the capacitance network between each node in
the device circuit. (a) Ground potential is set at the walls of the cavity, while inside the
cavity we simulate the presence of dielectrics (b) corresponding to the SAW substrate (blue)
and the silicon wafer grey box.) Close to the device, we define a volume (red) with increased
mesh resolution to increase accuracy/efficiency of the simulation. (c) The device consists
of four conducting “ports” (nodes), two on the surface of each substrate. Each node has a
square pad with linear dimension x, and a 5 µm wide lead connecting the pad to the device
(IDT or junction.) The centroids of the pads are separated by d, and the spacing between
the substrates is z.
Fig. 6.1(c). We would also like the device to have strong coupling to the electromagnetic 3D
cavity, so we can perform control and readout on the transmon qubit degree of freedom. To
satisfy these constraints, we choose a geometry outlined in Fig. 6.2(d): both the qubit and
the SAW device consist of square pads with linear dimension x, whose centroids are spaced
by d, connected to the active element by a 5 µm wide lead. These pads, when separated
by a distance z x away from each other in a flip-chip configuration, form a parallel plate
capacitance that defines C1 . The leads that lead to the transmon/SAW devices at the center
of the device are thin, and placed on opposite sides of the device to minimize Cp .
To obtain accurate values for the capacitive network, the grounding of the electric fields
158
must be accurate: when one electrode of the device is raised in voltage relative to the other
electrodes, ground is not at “infinity”, but rather at the walls of the cavity, which are some-
what close to the device relative to its size. Therefore, we encapsulate the simulation with a
conducting boundary condition having the same geometry as the inside of the electromag-
netic cavity (see Fig. 6.2(a).) Additionally, we must define volumes to model the dielectric
host substrates, see Fig. 6.2(b). This is fairly straightforward for materials like silicon and
quartz, which have a single dielectric constant, however LiNbO3 has an anisotropic dielectric
tensor, and so the particular orientation of the SAW device with respect to the underlying
crystal lattice matters for modeling the capacitance3 .
The feature sizes of the device are small (several µm) while the overall volume of the
simulation is large, of order ∼cm3 . Given this disparity, it is benificial to split the element
mesh into two parts: one part close to the device (red volume in Fig. 6.2(b)) with an
extremely fine mesh, and the rest of the volume with a more coarse mesh. This allows for
accurate simulation of the smaller structures without an overwhelming number of elements.
Since the IDT’s are numerous, and have linear dimensions in the ∼ 100 nm range, it would
be unwieldy to simulate them even using this split-mesh approach: we therefore simulate
a dummy structure in place of of the IDT’s, and add the analytic IDT capacitance CIDT
to the total capacitance of the SAW structure CSAW by hand. For the purposes of this
simulation, we also ignore the small geometric capacitance of the Josephson junction.
From our example device summarized in Table 6.1, we have pad size x = 250 µm,
pad spacing d = 1 mm, and wafer separation z = 10 µm. Using these values, we find
3 A summary of physical properties of LiNbO , including the values of the dielectric tensor, may be found
3
in [176], and the Euler angles to rotate axes to common SAW cuts of LiNbO3 may be found in Chapter 4
of [160]. Note that at microwave frequencies, one should use the values of the dielectric tensor at constant
strain S
ij .
159
(a) (b)
Figure 6.3: Butterworth-van Dyke equivalent response of a SAW resonator: (a)
Microscope image of an actual SAW device fabricated on LiNbO3 , with design parameters
corresponding to those listed in Table 6.1. The electrical response of the resonator may
be accurately modeled by the Butterworth-van Dyke (BvD) equivalent circuit: the IDT
geometric capacitance CIDT in parallel with a fictitious LCR resonance. (b) Real and
imaginary parts of Y (f ) = P33 (f ) of the SAW structure depicted in (a), calculated using
the coupling of modes method. Dashed lines: a fit to the admittance of the BvD equivalent
circuit, showing a high degree of accuracy.
C1 = 117.8 fF, Cp = 34.3 fF, CJ = 21.3 fF, and the portion of the SAW capacitance
associated with the pads CSAW − CIDT = 102.6 fF. The large anisotropy between CJ and
the pad proportion of CSAW may be attributed to extremely large values of the dielectric
tensor of LiNbO3 relative to the dielectric constant of silicon.
6.2.1.2 Extracting the Butterworth-van Dyke equivalent circuit
Having extracted the coupling capacitances, we now need to model the SAW resonator to
extract the BvD equivalent circuit. We use the coupling of modes (COM) model, as outlined
in Chapter 5, to generate a P-matrix for each SAW structure and cascade single structure
P-matrices to create a model for the entire SAW resonator. We then fit P33 (f ) = Y (f ) and
extract from the fit BvD equivalent circuit elements that represent the electrical response of
160
the SAW device accurately.
The example resonator from Table 6.1, which has been fabricated and is pictured
in Fig. 6.3 has an IDT with λIDT = 900 nm and Np = 16 finger pairs that have
w = 35 µm overlap. There are 525 Bragg gratings on either side of the IDT, with pe-
riodicity λmirror /2 = 456 nm and no free propagation distance between the IDT and the
gratings (Lf ree = 0.) We model this structure using a Python based COM modeling pack-
age (code may be found in [170]), and inputting a guess for the reflection per unit length
(|rs | = 1.5%, in line with other literature in the field [10]) and a guess for the propagation
loss. Using the process outlined in Chapter 5, we generate a cascaded P-matrix for the device
given these parameters: the real and imaginary parts of P33 (f ) = Y (f ) inside the mirror
stop band are plotted in Fig. 6.3(b).
The main claim of the BvD model is that this structure should be accurately represented
by a series LRC circuit in parallel with the geometric capacitance CIDT . The admittance
across such a structure, shown in Fig. 6.3(a), is given by
Rm (ωCm )2 Cm ω − Cm2 L ω3
m
Y (ω) = + i ωC IDT +
(Rm Cm ω)2 + (1 − Lm Cm ω 2 )2 (Cm Rm ω)2 + (1 − Cm Lm ω 2 )2
(6.3)
where ω = 2πf . We fit the Re(Y (f )) to this function, yielding Rm , Cm , and Lm as
fit parameters. For the model of the SAW resonator described above, Rm = 1.34 Ω,
Lm = 0.338 µH, and CM = 3.95 aF. The fit to both the real and imaginary parts of Eqn. 6.3
are plotted as dashed lines in Fig. 6.3(b), and agree with the numerically derived Y (f ) to a
high degree of accuracy within the stop band of the Bragg mirrors.
161
(a) (b)
Figure 6.4: Extracting the Hamiltonian parameters from classical circuit simu-
lations: (a) Im(Y (f )) of the linear component of the circuit in Fig. 6.1(d), using element
values extracted from FEM/COM modeling and LJ = 12 nH. The zero crossings (red dots)
signify a resonance of the circuit. (b) For each value of LJ , we extract the zero crossings
of Im(Y (f )) (blue dots, red dots correspond to data from panel (a)) and fit to extract the
Hamiltonian parameters.
6.2.1.3 Classical circuit simulations
We are now equipped with the circuit elements that make up the model of our device
outlined in Fig. 6.1(d). The response of the linear segment of the circuit may be calculated
using fundamental circuit analysis techniques, and reduced to a single admittance Y (f ) in
parallel with the non-linear component. In Fig. 6.4(a), we plot Im(Y (f )) as a function of
frequency using the element values obtained from finite element simulations of the capacitive
structure/COM modeling of the SAW structure. For this plot, we choose LJ = 12 nH, which
corresponds to a Josephson energy EJ /h = 13.6 GHz that is easily obtainable experimentally
using a flux tunable transmon. The zero crossings of Im(Y (f )), marked by red dots in
Fig. 6.4(a), signify a resonance of the circuit. We identify two modes of the circuit: the
SAW-like mode, with a resonant frequency approximately equal to that of the bare SAW
resonator ωm /(2π), and a transmon-like mode far detuned from the SAW mode.
162
To extract the Hamiltonian parameters, we run the same simulation using different values
of LJ , and extract the zero crossings to build the spectrum of the linearized circuit as a
function of LJ , plotted in Fig. 6.4(b). We see that the SAW-like mode and the transmon-
like mode behave differently as a function of LJ : the SAW mode is more or less constant
in LJ , while the frequency of the transmon-like mode shifts depends strongly on LJ . In
the uncoupled case, the SAW mode should be at ωm /(2π) for all values of LJ , while the
linearized transmon mode should obey ωq /(2π) = 1/ Cef f LJ , where Cef f is the effective
q
capacitance of the transmon mode. These modes, however, are coupled and undergo an
avoided crossing4 , shifting the frequencies of the coupled system to
ωm + ωq q 2
f± (2π) = ± gm + (∆/2)2 (6.4)
2
where, similar to the discussion in Chapter 2, ∆ = ωm − ωq and gm is the SAW-qubit
coupling we desire. In order to extract gm and Cef f , which gives us α = EC = e2 /2Cef f ,
we fit a single band of the avoided crossing to one of the frequency bands of Eqn. 6.4. A
fit to the data is plotted in Fig. 6.4(b) as a dashed orange line, and is shown to match the
extracted frequencies well. From this fit, we extract that, for this geometry, we should have
gm /(2π) = 40.2 MHz, strong enough coupling to be readily observable in an experiment,
and Cef f = 93.6 fF, large enough to put us in the transmon regime but small enough to
have sufficient nonlinearity to operate the transmon mode as a qubit.
163
Design Parameters Phenomenological Parameters
Substrate 128◦ Y-X LiNbO3 rs −0.015i
d 1 mm Propagation loss 500 Np/m
x 250 µm Derived Parameters
z 10 µm C1 117.7 fF
Np 16 CJ 21.3 fF
λIDT 900 nm Cp 34.3 fF
w 35 µm CSAW 383 fF
λmirror 912 nm f0 4.371 GHz
Ngrat 525 Cef f 93.6 fF
Lf ree 0 gm /(2π) 40.2 MHz
Table 6.1: Summary of the design parameters, phenomenological parameters, and derived
parameters used in the model of the example SAW resonator considered in § 6.2. All design
parameters are defined in fabrication.
Figure 6.5: Finite element simulations of the paddle structure: Color plot: SAW-
qubit coupling gm as a function of finger pair number Np and IDT overlap length w. Con-
tours: Transmon mode capacitance Cef f as a function of finger pair number Np and IDT
overlap length w
164
6.2.2 Optimizing the SAW device parameters
We are now in a position to try and optimize the experimental geometry to maximize coupling
between the SAW resonator and the qubit while maintaining a transmon mode capacitance
that we can work with. For the SAW device, aside from λSAW which sets the operating
frequency f0 , the most important parameters we have control over are the number of finger
pairs Np and the overlap length of the fingers w. How do gm and Cef f depend on these
quantities?
In Fig. 6.5, we plot both gm and Cef f as a function of Np and w, for a device on 128◦ Y-X
LiNbO3 , with λ = 900 nm and a pad geometry that give the capacitance values outlined at
the end of § 6.2.1.1. The first thing to note is that Cef f is only weakly dependent upon the
geometry of the SAW structure: this makes intuitive sense, since the effective capacitance
of several capacitors in series will be dominated by the smallest capacitance. Thus, Cef f is
largely constrained by the macroscopic pad geometry, and changes in CSAW C1 will have
little effect on the overall capacitance of the transmon mode.
Changes in CSAW do, however, strongly effect the transmon-SAW coupling gm . Inspect-
ing Fig. 6.5, we observe that in the capacitively coupled configuration, gm monotonically
decreases as a function of w. This may be understood along similar lines as how changes
in the capacitance in the presence of superfluid helium modified g0 in Chapter 4. As we
increase CSAW , voltage fluctuations across CSAW induce voltage fluctuations across the
junction with less efficiency. Thus, increasing w, which from Eqn. 5.32 naively increases the
SAW conductance of a bare IDT actually decreases the SAW-qubit coupling. Increasing Np
also increases CSAW , however from Eqn. 5.32 we know the bare conductance in proportional
4 While more often discussed in the context of quantum mechanics, avoided crossings often appear in
completely classical systems. A system of any two coupled classical oscillators will undergo an avoided
crossing if the resonant frequency of one is tuned through the other.
165
to Np2 . This square dependance wins out, and gm increases slowly as a function of Np .
Thus, it seems that to maximize coupling, we should want to maximize Np while minimiz-
ing w. There are, however, some considerations that prevent us from doing this indefinitely.
The first is that SAWs emitted from a finite length transducer are not precisely a plane-wave:
they undergo some diffraction, and can leak out into the continuum surface unprotected by
the Bragg mirrors/unable to be transduced by the IDTs. For short transducers (small w),
this leakage ultimately limits the quality factor of the SAW resonator. The diffraction limit
of Q is predicted to be ∝ (w/λ)2 with a constant of proportionality that depends on material
parameters [167, 177]. For a first pass experiment, we would simply like the diffraction loss
to be small enough that we don’t need to worry about it being a dominant decay mechanism:
from this perspective, a value (w/λ) ' 40 seems reasonable.
Additionally, distortions from intra-IDT reflections may adversely effect optimal coupling.
While this can presumably be taken into account using the COM theory as was done in
Chapter 5, the exact magnitude of the reflection coefficient rs for a given set of fabrication
parameters seems to be determined empirically in the literature and is not known a priori.
Thus, it would behoove us to be in a regime where the intra-IDT reflections produce minimal
distortion for a wide range of |rs |.
6.3 Device fabrication and assembly
In order to have a predictable acoustic coupling, we must be able to align the paddles that
form the parallel plate capacitors with a fair amount of precision. To do so, we employ
the mask aligner in the class 100 clean room of the Keck Microfabrication Facility in BPS,
which is normally used for aligning substrates to photomasks with high precision for pho-
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(a) (b) Line of
sight port
Line of
To sight port
vacuum
To
vacuum Clear
"Vacuum subsrate
chuck"
Clear subsrate Opaque subsrate
Figure 6.6: Mask for aligning the flip-chip device stack: (a) Exploded view of the
flip-chip alignment mask. When the two plates are flush, they enclose a small trench that
connects two small holes near the center to a hose stub. We tee off the substrate vacuum
line to pump on this volume, which provides enough suction to lift a substrate for alignment
(red arrows) (b) Zoom in on the center portion of the mask. If we carry a clear substrate
with the vacuum chuck, we may use the line of sight port, along with the microscope on the
mask aligner, to align the two substrates relative to each other with high precision.
tolithography. In place of a photomask, we use a custom made “mask” designed to fit on
the 4” photomask plate on the mask aligner, which is actually two steel plates that fit flush
together, see Fig. 6.6. When flush, they enclose a small volume connecting a “vacuum chuck”
(two small holes through the bottom mask) to a hose stub, which fits a hose with the same
diameter as the substrate vacuum hose on the mask aligner. We tee off the substrate vacuum
line to this stub, which provides a strong enough vacuum to pick up a substate. If the sub-
state is clear, we can use the line of sight port through the mask to align a device on the top
(clear) substrate with a device on another substrate under it, which may be opaque. This
is easily done on the mask aligner, which has a built in microscope and 4-axis positioning of
the opaque substrate relative to the clear substrate which is held static on the mask.
To control the vertical distance z between the substrates, we use a spacer fabricated
from hard baked photoresist, as depicted in Fig. 6.7. Hard baked photoresist has several
advantages: photoresist is spun on with a reliable and predictable thickness, and we may
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(a) (b)
Figure 6.7: Assembling the flip-chip stack: (a) Schematics of the devices cut to size, as
well as a rendering of the assembled flip-chip stack. The SAW device is made larger than the
qubit device for ease of assembly/handling. We use hard baked photoresist (maroon) spacers
at the end of the chip to control the spacing between chips, and use soft-baked photoresist
(red) to glue the chips together. (b) Image of the flip-chip stack, as seen through the clear
LiNbO3 substrate.
apply multiple layers to generate a thicker spacer. Additionally, once hard baked, photoresist
is robust to normal substrate cleaning procedures, which allows us to clean surface after
fabricating the spacers. We have fabricated spacer of up to 4 µm, though a thicker layer
is likely possible. A detailed fabrication recipe may be found in Appendix C. We have
observed that the spacer application process destroys the Josephson junction device, and
therefore we fabricate the spacers on the piezoelectric substrate. We have also recently
observed that, while baking the resist spacers, SAW devices fabricated on 128◦ Y-X LiNbO3
almost universally get destroyed by static discharging events. We have, however, been able
to successfully fabricate resist spacers on both ST-X quartz and Y-Z LiNbO3 .
To glue the chips together, we once again employ photoresist [178]: two small dabs of
photoresist are placed on one of the devices (generally the qubit device) such that they will
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land inside of the spacers once the chips brought together, as shown in Fig. 6.7(a). We
then use the mask aligner to align the substrates relative to each other, and bring the chips
into contact until the weight of the alignment mask is supported by the chip stack. The
photoresist dabs act as glue to hold the chips together, and the weight of the mask ensures
that the resist hardens with the chips as close together as possible. We allow the flip-chip
stack to sit in place for ∼ 10 minutes so the resist can harden slightly, and then transfer it
to a hotplate at 110 ◦ C, and let it bake for 3-5 minutes to cure the resist such that the stack
is fairly robust.
6.4 Experimental results
This section details some preliminary experimental results testing the devices we describe
using the model laid out in § 6.2 and fabricate and assemble using the procedure laid out in
§ 6.3. At this point, several experimental runs have been performed, using both ST-X quartz
and LiNbO3 as piezoelectric substrates. We will focus on a single, exemplary experimental
run for each material, and comment on the findings/shortcomings of each.
6.4.1 Devices on ST-X quartz
The first generation of quantum acoustics devices tested were flip-chip stacks with the qubit
device on silicon and the SAW device on ST-X quartz. While quartz is a fairly weak piezo-
electric material, it has a fairly low dielectric constant ( ≈ 3.8) and thus the capacitance
budget is more forgiving. Additionally, it has been shown that, even in fairly weak piezoelec-
tric materials, piezoelectric coupling to bulk phonons in the substrate may be a significant
loss channel that adversely effects delicate quantum circuits [179]. Thus, we initially opted
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Design Parameters Phenomenological Parameters
Substrate ST-X quartz rs −0.01i
d 1 mm Propagation loss 500 Np/m
x 250 µm Derived Parameters
z 4 µm C1 179 fF
Np 25 CJ 9.8 fF
λIDT 600 nm Cp 12.8 fF
w 70 µm CSAW 129.3 fF
λmirror 604 nm f0 5.203 GHz
Ngrat 947 Cef f 75.1 fF
Lf ree ≈ 3 µm gm /(2π) 20.5 MHz
Table 6.2: Summary of the target design parameters, phenomenological parameters, and
derived parameters for the device on ST-X quartz experimentally tested and described in
§ 6.4.1.
to work with ST-X quartz as the piezoelectric substrate, a fairly weak piezoelectric material,
where acoustic losses would hopefully be mitigated.
The targeted design parameters, as well as the circuit/Hamiltonian values derived using
the modeling method from § 6.2 are listed in Table 6.2. Note that the free propagation
length Lf ree is only approximately defied: in order to maximize the number of Bragg grat-
ings/reflectivity of the mirrors, the resonator was lithographically defined in three separate
writing steps (one for the IDT and one for each mirror) with stage translations between each
step. Because stage translations of the scanning electron microscope are only accurate to
several microns, we can only give an approximate value for this number.
In this experiment, we will still use the electromagnetic 3D cavity to manipulate/measure
the state of the coupled cavity/qubit system. One parameter left unconstrained by our model
was the coupling g between the transmon-like mode of the qubit and the 3D cavity mode. We
would like g to be large enough such that we are still in the strong coupling regime between
the 3D cavity and the transmon mode, so that we can control and read out the qubit state
using the same methods as previous chapters. To verify coupling between the 3D cavity
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Figure 6.8: Verifying coupling of the transmon-like mode to the electromagnetic
cavity: Transmission S12 through the 3D electromagnetic cavity with a qubit + ST-X quartz
device loaded in, as a function of both frequency and flux tuning current.
and the transmon mode, we measure the transmission S12 through the cavity, as a function
of applied flux bias. We see characteristic avoided crossings in the cavity spectrum as the
qubit is flux tuned through the cavity resonance, roughly symmetric about zero flux bias,
indicative of strong coupling between the 3D cavity and the flux-tunable transmon qubit
mode. From this data, we may extract the cavity-qubit coupling rate g/(2π) ≈ 90 MHz,
which is of order the cavity-qubit coupling of a “standard” 3D transmon qubit.
Satisfied with the existence of a strongly coupled transmon-like mode in the cavity, we
may now perform two-tone spectroscopy to measure ω01 and ω12 of the transmon-like mode,
from which we may experimentally verify α = EC = ω01 − ω12 . Data measuring these
values are plotted in Fig. 6.9(a-b). This data is taken at flux bias current I = 0: we record
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Figure 6.9: Spectroscopy of the ST-X Quartz quantum acoustics device: Two-tone
spectroscopy measuring the (a) |0i → |1i transition and the (b) two-photon |0i → |2i
transition at I = 0 flux bias current. (c) Two-tone spectroscopy, as a function of flux tuning
current. As the frequency of the qubit changes, we raster the center frequency of the sweep
to keep the |0i → |1i transition in the data range. We observe no obvious avoided crossings
in the expected frequency range of the SAW device.
ω01 /(2π) = 7.170 GHz and α/(2π) = 376 MHz, corresponding to Cef f = e2 /2α = 51.5 fF,
well below the value predicted from the model.
Furthermore, we may perform two-tone spectroscopy as a function of flux tuning current
to extract ω01 as a function of flux. We expect that, as we tune ω01 through the SAW
resonant frequency, we should see an avoided crossing in the data, indicating coupling to the
SAW device. In Fig. 6.9(c), we perform such a measurement: at each value of current, we
raster the center frequency of the two-tone sweep to keep the |0i → |1i transition frequency
within the range of the sweep, while not vastly oversampling. Sweeping from I = 46 mA to
I = 49 mA, which corresponds to ω01 /(2π) ranging from ≈ 5.3 GHz to ≈ 5 GHz, we see no
172
evidence of an avoided crossing, which we would naively expect to be easily visible from the
model outlined above.
Evidently, some portion of the model is deficient in describing the actual experiment in
the fridge. In what follows, we explore some potential posibilities and suggest ways in which
they may be accounted for.
6.4.1.1 Uncontrolled substrate spacing
That the capacitance of the transmon mode Cef f is substantially different from theory
predictions is a strong indicator that spacing between the substrates in the flip-chip stack
may be ill-controlled. Since all electrodes on a wafer are defined with high precision in
lithography, and the pads of the two devices may be reliably aligned relative to one another,
the only major adjustable parameter in determining the geometric capacitance between two
electrodes is the hight z between the substrates.5 To investigate this, we may perform finite
element simulations at several heights z to obtain the values of capacitive network, as a
function of z. Using these values, and all other values outlined in Table 6.2, we calculate the
anticipated Cef f and gm as a function of substrate spacing z, which we plot in Fig. 6.10(a-
b). From these calculations, we expect the experimentally obtained value Cef f = 51.5 fF to
occur if the substrates are z ≈ 10 µm apart.
To investigate if this is in fact the case, we image the edge of the sample in a variable
pressure SEM with tilt control. As is evident in Fig. 6.10(c), too much gluing resist was
applied to this device, and the distance z between the substrates is limited by the glue resist
and not the hard baked bump-bonds (visible as lines on the bottom substrate.) While it is
hard to estimate the distance z from the angle of this image, it is certain that the spacing
5 The substrates may also be tilted with respect to each other, breaking assumption of a symmetric
capacitance network. We plan to explore the effect of substate tilt in the future.
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(a) (b)
(c)
Figure 6.10: Dependence on the substrate spacing z: (a) Transmon mode capacitance
Cef f and (b) acoustic coupling gm as a function of heigh, using the all other parameters
from the model outlined in Table 6.2. Empirically determined mode capacitance marked
with a dashed black line. (c) SEM picture of the flip-chip stack from the edge, indicating
that the spacing of this chip-stack limited by the glue rather than the hard baked spacers.
is > 4 µm. We have taken steps in the fabrication process to more accurately control the
substrate spacing, however in the future it may be necessary to move away from soft baked
resist applied by hand and to a different adhesive, or perhaps more precise substrate bonding
technique such as indium bump bonding.
174
Figure 6.11: Dependence of the SAW response on the grating reflectivity: (a) Real
(blue) and imaginary (orange) parts of the acoustic admittance Y = P33 , calculated using
various values of the per-grating reflectivity. As the reflectivity decreases, the electrical
response of the resonator becomes weaker. (b) gm calculated at various values of |rs | using
all other values defined in Table 6.2.
6.4.1.2 Mirror reflectivity
While the increased substrate spacing z will result in decreased acoustic coupling gm , from
Fig. 6.10(b) we see that the predicted coupling at z = 10 µm is still of order ≈ 13 MHz,
i.e. large enough to be resolved in our data. This is indicative that some assumptions we
made in our model of the SAW resonator were faulty. The most obvious assumptions we
made were in the reflectivity per Bragg grating rs and the propagation loss. Propagation loss
in SAW materials is ill-characterized at low temperature [167], however some studies [169]
have recorded per-grating reflectivity as low as |rs | ∼ 0.2% on ST-X quartz at cryogenic
temperatures. Given the large disparity between this and our guess of |rs | = 1%, we should
ask what effect the reflectivity has on the acoustic coupling.
In Fig. 6.11(a), we plot both the real and imaginary parts of the acoustic admittance as
175
a function of frequency in the vicinity of the SAW resonance for several values of |rs |. We
observe that at low values of |rs |, the acoustic admittance associated with the resonance is
is small, and the magnitude of the admittance grows as |rs | grows. From our analysis in
Chapter 5, we know that as |rs | decreases the effective length Lef f of the SAW Fabry-Pérot
cavity increases, and the IDTs will transduce SAWs/applied voltages over a smaller portion
of the cavity. We also recall that the width of the mirror stop band ∆f decreases as |rs |
decreases, which can be observed Fig. 6.11(a) in as a decrease frequency range over which
fine-scale features of Re(P33 ) are suppressed.
Unsurprisingly, when the acoustic admittance of the resonator is suppressed, the coupling
rate gm to the qubit mode is also suppressed. In Fig. 6.11(b), we plot gm calculated at
several values of |rs |. We see that below |rs | ∼ 1%, the SAW-qubit coupling gm begins
to drop precipitously.6 Note that these calculations are done using all other values of the
model recorded in Table 6.2, i.e. for z = 4 µm, and loss of coupling from low reflectivity will
compound on the loss of coupling caused by unconstrained substrate spacing.
6.4.1.3 IDT position between the mirrors
Low per-grating reflectivity rs could conceivably be accounted for if we could fabricate a very
large array of Bragg gratings and simultaneously engineer the SAW resonance condition to be
near the middle of the mirror stop band to optimize coupling. However, these goals turn out
to be contradictory. Recall that the SAW resonator is written in three electron beam writing
steps, with translations in between each, rendering the free propagation distance Lf ree only
approximate. However, the acoustic coupling gm turns out to be exquisitely dependent upon
6 Since the gratings are finite length, at some low |r | the quality factor of the modeled resonator, previ-
s
ously limited by propagation loss, starts to drop off. This will presumably also decrease coupling, though a
full analysis along with the dependence on the propagation loss is needed.
176
(a) (b)
(c) (d)
Figure 6.12: Dependence of gm on the IDT position: Acoustic coupling is maximized
when the IDT fingers are in phase with the antinodes (a) of the standing wave pattern of
the Fabry-Pérot resonator, and is minimized when the IDT fingers are in phase with the
nodes (b). (c) To investigate this effect, we model the resonator and shift the position of the
IDT inside of the fixed-length resonator by δ. (d) Acoustic coupling gm vs δ. As expected,
coupling is maximized when δ is a half-integer value of λIDT , and minimized at odd-quarter
integer values. Note that near δ/λIDT = 0.25, the response becomes too small to accurately
model. Here, δ = 0 corresponds to the model defined by the values in Table 6.2.
the free propagation distance.
To see this, recall that the resonance condition in a Fabry-Pérot resonator is set when
the total resonator length Lef f equals an integer number of half wavelengths, i.e. it is set
by a standing wave condition. This standing wave will have nodes/antinodes at fixed points
in space, which the IDTs may be in-phase/out-of-phase with, as depicted in Fig. 6.12(a-b).
If the IDT is in-phase with the antinodes, as in Fig. 6.12(a), the voltage induced across the
IDT electrodes will be maximized, and transduction will be maximized. Conversely, if the
IDT is in phase with the nodes, in Fig. 6.12(b), a confined SAW will not cause a voltage
177
will build up across the IDT electrodes, and transduction will effectively drop to zero. This
is outlined in Fig. 6.12(d), where we calculate the acoustic coupling to the transmon mode
gm as a function of IDT offset δ from the center of the resonator. Around δ/λIDT = 0.25
(where the IDT fingers are in phase with the standing wave nodes) the acoustic coupling
drops precipitously.
For small wavelength devices, such as the device cooled down with λIDT = 600 nm,
this presents a difficult fabrication problem: the IDT and mirrors (each write of scale
∼ 50 − 500 µm, with the whole structure close to 1 mm) must be aligned relative to each
other with an accuracy λIDT /4 in order to have predictable acoustic coupling. While this
is perhaps possible, accounting for imperfection in the electron beam lithography calibra-
tion and thermal contractions when cooling down to 10 mK make it a daunting engineering
task. It would be preferable to write the entire structure (IDT + mirrors) in a single write,
however we are only capable of writing finite size structures while maintaining accuracy at
the length scale required to define the IDT/gratings. Pragmatically, this limits the number
of gratings Ng each mirror can have, which will in turn limit the reflectivity of the mirrors
and the quality factor of the SAW resonator, especially for structures fabricated on weak
piezoelectric substrates such as ST-X quartz where |rs | may be low.
6.4.1.4 Coherence times
We have also recorded coherence times T1 and T2 at multiple values of flux tuning current,
ranging from 47-48 mA. Over the range of frequencies measured in this device, we recorded
an average T1 = 1.77 µs, and and average T2 = 388 ns. This device (both SAW resonator
and Josephson junction device) were fabricated completely in-house, and the observed value
of T1 is fairly consistent with other “standard” 3D transmon that were fabricated at MSU.
178
This is consistent with the picture that, in the case of quantum acoustics devices employing
ST-X quartz, the energy relaxation rate is currently limited by our fabrication capabilities.
The low T2 is not terribly surprising: we are in a regime where the qubit frequency is
strongly dependent on the applied flux, meaning the qubit is highly susceptible to flux
noise. Additionally, and perhaps more importantly, the low Cef f means that, in this regime
EJ /EC ∼ 27 i.e. the qubit is in the “offset charge sensitive” transmon regime, where the
transition frequency ω01 experiences a nontrivial amount charge dispersion and fluctuations
in the local electrostatic environment may drive dephasing. Improving flux noise filtering
and increasing the capacitance into the full-fledged transmon regime (EJ /EC ≥ 50) would
likely improve T2 .
6.4.2 Devices on LiNbO3
Given the above considerations, the natural course of action is to move to a material with
stronger piezoelectricity, such as LiNbO3 . This choice was not originally pursued due to con-
cerns that the piezoelectricity would be too strong, such that microwaves in 3D cavity would
be transduced into spurious bulk modes in the LiNbO3 substrate, opening up a loss chan-
nel that would render the experiment inoperable. These concerns were facilitated by early
tests, where the decay rate κ of a 3D cavity increased significantly when a LiNbO3 substrate
was placed inside. Fig. 6.13 shows the transmission S12 through a 3D cavity with/without
a LiNbO3 /silicon flip-chip stack: in the presence of LiNbO3 , κ increases significantly, and
S12,max decreases significantly. However, these concerns are alleviated at low temperatures,
where the loss rate κ significantly decreases, even compared to the no device configuration.
It is unclear why this is case, though there may be some ensemble of defects/contamination
on LiNbO3 that freeze out at low temperatures. A summary of these results is found in
179
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,+*)(%'&
#"
!"
$-. .-" .-. #-"
/()012,
Figure 6.13: 3D cavity transmission in the presence of LiNbO3 : Transmission of
a 3D cavity at room temperature with no device (green), at room temperature with a
LiNbO3 /silicon flip-chip stack (red), and at ∼ 10 mK with a LiNbO3 /silicon flip-chip stack
(blue). Note that the absolute magnitude of the low temperature also encapsulates the ex-
perimental attenuation/amplification, and should therefore not be directly compared to the
magnitude of the other data.
Configuration f0 κ
Room temperature, no device 5.8049 GHz 4.8 MHz
Room temperature, device in 4.7644 GHz 13.3 MHz
10 mK, device in 4.8327 GHz 2.2 MHz
Table 6.3: Transmission characteristics of a 3D cavity with/without LiNbO3 , extracted from
the data plotted in Fig. 6.13
Table 6.3.
We have begun to cool down quantum acoustics experiments that use LiNbO3 in recent
months. While we have not yet cooled down a device in a parameter regime in which we would
expect acoustic coupling7 , we have taken data, qualitatively similar to the data presented
in Figures 6.8 and 6.9, indicating once again the presence of a well defined transmon qubit
mode in the experiment. In this device, we measured α = EC = 236 MHz, corresponding to
7 The most recent experiment had a transmon qubit with an unexpectedly low E , such that ω < ω
J 01 m
even at zero flux bias.
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Cef f = 82 fF, more in line with what we expected from simulations.
We have also measured the coherence times of the transmon-like modes for devices fab-
ricated with LiNbO3 . Preliminary measurements of T1 on devices employing LiNbO3 have
produced T1 ≈ 140 ns, roughly an order magnitude lower than the case of ST-X quartz,
indicating that there is perhaps significant material loss in LiNbO3 at the single photon
level. We stress that this data is preliminary, and that coherence times of superconducting
qubits are known to vary widely for a number of (sometimes ill-controlled) reasons: more
definitive comments on the loss of LiNbO3 would require repeating the experiment several
times. Preliminary measurements of T2 for devices on LiNbO3 have shown T2 ≈ 240 ns, i.e.
close to the relaxation limited value T2 = 2T1 . This is somewhat unsurprising, since for this
device EJ /EC ≈ 48, squarely in the transmon regime where charge noise is exponentially
suppressed. This indicates that there are likely no additional sources of pure dephasing
in LiNbO3 , though we again stress that these values are preliminary and more thorough
measurements are required.
6.5 Future work
While we are still working to demonstrate coupling between SAW resonators and 3D trans-
mon qubits, there are several parallel paths that are worth exploring, several future im-
provements that could be imagined, and several experiments that could be performed once
coupling has been established. We outline a few of them in this section.
As we have seen, a pressing issue in correlating the model to real experiments remains
the uncontrolled phenomenological parameters in the model. For a given quantum acoustics
device, our best bet approach would be to fit any recorded data to the model and extract
181
these parameters, however for design purposes it would be advantageous to know them before
hand. Systematic studies of SAW structures in planar geometries, such as in Ref. [169], could
help determine the per-grating reflectivity and propagation loss at low temperatures. We
have already begun taking steps to perform these studies, and plan to do so in the future.
While not surprising, evidence of suppressed T1 for devices employing LiNbO3 indicates
that devices on strongly piezoelectric substrates may have finite utility. One strategy to
resolve this issue could be to greatly reduce the amount of LiNbO3 in the system: recently,
techniques have been developed to create single crystalline thin film LiNbO3 , with thicknesses
of order several µm bonded to the surface of a silicon wafer. These films have been shown
to host Rayleigh waves [180], and could greatly reduce the available density of states for the
qubit to piezoelectrically decay into, allowing us to maintain both strong coupling and long
coherence times.
An interesting feature of classical SAW devices is the ability to engineer low dissipation
structures with responses that have sharp features as a function of frequency, such as band
pass filters [160]. These sharp frequency response structures may be a valuable resource for
quantum devices. For example, one could imagine creating a SAW based Purcell filter [181],
i.e. a filter that suppresses the density of states for a qubit to decay into. Additionally, the
sharp frequency response of SAW devices could be used to engineer the dissipation environ-
ment of the qubit, preferentially increasing dissipation at some frequencies and suppressing it
at others. This type of “bath engineering” has been used to autonomously prepare arbitrary
states of a qubit system [182] and to create dissipation hierarchies that simulate the response
of non-hermitian Hamiltonians [183].
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Chapter 7
Acoustoelectric transport in graphene
Long before surface acoustic waves were incorporated into quantum circuits, SAWs were
used extensively in condensed matter physics to study systems on or near the surface of
piezoelectric substrates. These techniques were particularly successful in the semiconductor
community, where the AlGaAs/GaAs heterostructures that host two-dimensional electron
systems (2DESs) are naturally fabricated close to the surface of a piezoelectric GaAs sub-
strate. Many interesting properties of the integer and fractional quantum Hall states of
matter have been elucidated by SAW techniques, most notably experimental evidence for
the formation of a composite fermion surface at ν = 1/2 [184, 185].
Much of the motivation for the main focus of this thesis, superconducting qubits and
quantum acoustics, was generated by experiments done earlier in my Ph.D. studying the
interaction of SAWs with another 2DES: exfoliated graphene. This chapter catalogs those
experimental efforts, which were approached with the quantum Hall effect in mind. Thus
while none of the data presented is taken in high magnetic field, much of the motivation will
be given through the lens of previous experiments using SAWs to study the quantum Hall
state, and emphasis will be given to optimizing the experimental setup for high magnetic field
measurements. An introduction to the quantum Hall effect will not be given: the interested
reader may find an excellent introduction in Ref. [186].
While this chapter stands out thematically from the rest of the thesis, I feel that it is
183
important to include it nonetheless. In section 7.4, I will briefly discuss how some of the
ideas developed in this chapter may be integrated with the experimental ideas pertaining
to superconducting qubits and quantum acoustics presented in the rest of this thesis. The
results in this chapter were also reported in Ref. [187].
7.1 Graphene: a brief introduction
Graphene is an allotrope of carbon consisting of a single layer of carbon atoms arranged in a
two-dimensional hexagonal lattice. One may think of of graphene as the single atomic layer
limit of graphite, or conversely think of graphite as many layers of graphene stacked on top of
each other. The electronic properties of graphene were first predicted in 1947 [188], however
it would not be until 2004, over 50 years later, that monolayer graphene was isolated and
its electron transport properties were experimentally confirmed [189]. This long gap elicits
the image of a technologically complex method of isolating graphene, however reality often
turns out to be stranger: graphene was first isolated by mechanically exfoliating graphite
flakes and depositing them onto oxidized silicon substrates using Scotch tape!1 Indeed, to
this day the highest mobility graphene devices, while vastly more complicated than early
devices, are still made by mechanical exfoliation using tape [190].
The ability to mechanically exfoliate and measure electrical transport through an intrinsi-
cally 2D material has given experimentalists a highly flexible toolkit to engineer a wide variety
of experiments studying the 2D electron system formed by the charge carriers in graphene.
The choice of oxidized silicon as a substrate to suspend the graphene flakes was prudent: the
1 A quick internet search pegs the price of a dispenser of Scotch tape at ≈ $2.30. Given that the 2010
Nobel Prize was awarded for the isolation of graphene, by my calculations, Scotch tape is by cost roughly
4 × 108 times more efficient at producing Nobel Prizes than the LHC.
184
oxide layer forms an insulating barrier between the graphene and the highly doped silicon
bulk, allowing for experimentalists to modulate the Fermi energy of the graphene in situ
simply by applying an electrostatic voltage to the conducting host substrate [189]. Quickly
after graphene was isolated, the remarkably high mobility of exfoliated devices led to the
observation of the integer quantum Hall effect in graphene [191, 192]. Further advances in
fabrication techniques, such as encapsulating graphene in mechanically-exfoliated insulating
hexagonal boron nitride [193] have pushed up the mobility of layered 2D devices even further,
allowing for the observation of both odd [194, 195] and even [196, 190] denominator frac-
tional quantum Hall states. Placing multiple exfoliated flakes with crystal axes misaligned,
or “twisted” relative to each other, has produced a bounty of experiments including the
observation of Hofstadter’s butterfly [197] and (possibly unconventional) superconductivity
in bilayer graphene [198].
In addition to its fundamentally 2D nature, the band structure of graphene leads to a
wide variety of interesting physics. Graphene is a Dirac semimetal, with the valence and
conduction bands touching at a single point at the Fermi level. Near the Fermi energy, the
bands are linear as a function of momentum (E = ~kvf where vf is the Fermi velocity), rather
that the typical approximation of quadratic bands at low energies (E = ~2 k 2 /2m∗ , where
m∗ is the effective mass of the charge carriers.) This linear dispersion relation implies that
charge carriers in graphene are effectively massless quasiparticles with an effective “speed
of light” equal to the vf . This massless nature of the low-energy excitations means that
the charge carriers in graphene are better described by the Dirac equation, rather than the
Schrödinger equation, and has lead to a number of observations of “relativistic” phenomena.
Prominent among these is the observation of Klein tunneling, where a massless fermion
paradoxically tunnels through an infinite potential well [199] and the quadratic spacing of
185
4
R (k )
2
-1.0 -0.5 0.0 0.5 1.0
VG (V)
Figure 7.1: DC transport measurements of graphene: Left: Schematic of a gener-
alized DC transport measurement of graphene. Graphene (turquoise hexagons) is isolated
and stabilized on the surface of an oxidized silicon substrate. The bulk of the substrate
(blue) is heavily doped, such that the silicon remains conducting at low temperatures, while
the graphene sample is electrically isolated from the conducting bulk by a several hundred
nanometer thick oxide layer (purple.) The conducting graphene sheet and the bulk silicon
form a capacitor, and applying a DC gate voltage VG to the silicon varies the charge in the
graphene, which modifies the position of the Fermi energy. Right: resistance of the graphene
sheet as a function of gate voltage. At low and high gate voltages, the Fermi energy sits
at positions in the band with a high density of states (left and right insets) resulting in a
low sample resistance. When the Fermi energy is tuned close to the Dirac point, where the
conduction and valence bands meet and the density of states goes to zero (middle inset), the
resistance increases dramatically. Note that the type of charge carrier changes from holes to
electron as the Fermi energy is tuned from the conduction band to the valence band. Data
taken by Liangji Zhang, and reprinted with permission.
Landau levels for massless particles, which extraordinarily has led to the observation of the
integer quantum Hall effect at room temperature [200].
7.2 Surface acoustic waves and two-dimensional elec-
tron systems
As elaborated in Chapter 5, SAWs propagating on piezoelectric substrates carry with them
a co-propagating electric field with the same frequency and wave vector as the SAW, and
this electric field extends evanescently above the surface of the bulk with decay length λ, the
186
wavelength of the SAW. When the charge carriers of a nearby conductor are in the vicinity of
this co-propagating electric field, the SAW and the charge carriers interact, modulating the
SAW propagation and moving the charges. The interaction of SAWs and charge carriers in
two-dimensional electron systems close to the surface of propagation has been employed in
many fundamental studies and proposed devices. Here, we sample two classes of studies com-
monly executed: velocity shift/attenuation studies, where the conducting electrons screen
the SAW and modulate its propagation, and acoustoelectric studies, where a propagating
SAW electric field imparts momentum on the charge carriers in the 2DES.
7.2.1 SAW attenuation and velocity shift
If a SAW propagates in the vicinity of a conductor, the co-propagating electric field will
induce currents in the conductor, leading to Ohmic dissipation. This will attenuate the
SAW as it passes by the conductor (and, as a consequence of the Kramers-Kronig relations,
shift its propagation velocity.) The velocity2 and attenuation per unit length of a SAW
propagating on a piezoelectric substrate with conductivity σ may be written as [201, 202]
2 /2
Kef f
v(ω) = v 1 + (7.1a)
1 + (ω/ωc )2
2
Kef f (ω/ωc )
Γ=k (7.1b)
2 1 + (ω/ωc )2
Here k is the SAW wave vector, v0 is the SAW velocity in the limit where σ → ∞,
f is the effective piezoelectric coupling constant that depends on material parameters
Kef2
2 Unfortunately, the convention within the 2DEG community is to reference the velocity with respect to
the perfectly conducting surface, as opposed to our discussion in Chapter 5 where we referenced it to an
insulating surface. I have decided to keep in line with the quantum Hall literature here, so in this chapter,
velocity shifts caused by a conducting layer at the surface will be positive rather than negative.
187
and geometry3 , and ωc = σ/(1 + 2 ), where 1(2) is the dielectric constant above (below)
the surface. ωc may be thought of as the inverse of a time constant that quantifies how
quickly charges can equilibrate after being perturbed by an external electric field. In the
thin film limit, where the conducting film thickness is much smaller than the SAW wavelength
d k −1 = λ, the effective conductivity of the conducting layer (as sampled by the SAW
electric field) reduces by a factor of dk 1[202]. We may then write
σdk σ k
ωc = =
(1 + 2 ) (1 + 2 )
where σ is the sheet conductivity.
In practice, the SAW frequency is more or less fixed by choice of transducer geometry, so
it is much more convenient to write the velocity and attenuation in terms of the conductivity
of the 2DES, which we can often vary. If we do so, we arrive at
2
Kef f 1
∆v/v = (7.2a)
2 1 + (σ/σm )2
2
Kef f (σ/σm )
Γ/k = (7.2b)
2 1 + (σ/σm )2
where σm = ω/k(1 +2 ) = v0 (1 +2 ) is a material dependent characteristic conductivity.
Here, we have taken the experimentally expedient step of writing down the velocity as the
relative shift referenced to the case of a perfect conductor. Importantly we see that the
velocity shift and attenuation are a function of both the SAW frequency and wave-vector,
implying that these measures probe the frequency and wave vector dependent conductivity
3 For a conducting layer directly on the surface of the piezoelectric, K 2 = K 2 , the piezoelectric coupling
ef f
constant defined in Chapter 5. From Eqn. 7.1a, it may be verified that v(σ = ∞) − v(σ = 0) = Kef 2 /2.
f
188
0.5 0.25
2 )
∆v/v (Keff
0.3 0.15 2 )
Γ/k (Keff
0.1 0.05
10 -2 10 -1 10 0 10 1 10 2
σ(k, ω)/σm
Figure 7.2: SAW attenuation and velocity shift vs. 2DES conductivity: Relative
SAW velocity shift ∆v/v (blue, left axis) and attenuation per unit length normalized by
the SAW wave vector Γ/k (red, right axis) as a function of normalized 2DES conductivity
σ(k, ω)/σm . Both are given in units of the effective piezoelectric coupling constant Kef
2 .
f
σ(k, ω). The normalized values of Γ/k and ∆v/v are plotted in Fig. 7.2.
Velocity shift and attenuation techniques are most sensitive in probing large changes in
the bulk 2DES conductivity around σm , which tends to be low (for a conducting film on the
surface of {110} GaAs, σm ≈ 3 × 10−7 (Ω/)−1 [202]). In the quantum Hall regime, current
is dominated by conducting edge channels, and transport characterization of the insulating
bulk is difficult. Here, SAW techniques have proved particularly valuable, and have been
employed extensively to study high mobility 2DESs in GaAs/AlGaAs heterostructures in the
quantum Hall regime [203, 202, 204, 205, 206, 207]. In particular, the wave vector dependence
on the conductivity proved crucial for providing evidence for the existence of a composite
fermion metal near Landau level filling factor ν = 1/2 [184, 185]. SAW techniques have also
been used in GaAs/AlGaAs heterostructures to study Wigner crystallization at low carrier
density [208, 209, 210], and the 2D metal-to-insulator transition at zero magnetic field [211].
189
7.2.2 Acoustoelectric effect
Conversely, the electric field co-propagating with the SAW may impart momentum on the
charge carriers in the 2DEG, driving a macroscopic DC current [212, 204, 213, 214, 215].
We present a rough sketch of the physical picture that follows that of Ref. [213]. Consider a
SAW propagating in the x-direction with a co-propagating plane-wave electric field
Ex (x, t) = E0 ei(kx−ωt) x̂ (7.3)
If the SAW power isn’t too large, this electric field will generate a local oscillating Ohmic
current density jα (x, t) direction 2DES
jα (x, t) = σαx Ex (x, t) (7.4)
where α = (x, y) is the component of the acoustoelectric current in question and σij is the
(ith , j th ) component of the conductivity tensor of the 2DES. We may equivalently describe
this current as an oscillation of the charge density Ns about its equilibrium value Ns,0
Ns (x, t) = Ns,0 + ∆Ns ei(kx−ωt) (7.5)
Once again, assuming the SAW power isn’t too high, ∆Ns Ns,0 and we may expand the
conductivity as a function of the carrier density
∂σ
σ(x, t) = σ0 + ∆Ns ei(kx−ωt) (7.6)
∂Ns
We may then use the 1D continuity equation ∂(−eNs )/∂t + ∂jx /∂x = 0 write down ∆Ns in
190
terms of the SAW electric field
kσxx σxx
∆Ns (x, t) = − Ex (x, t) = − Ex (x, t) (7.7)
eω v0 e
We may then use this value to calculate the modified conductivity with Eqn. 7.6, and plug
that back into Ohm’s law to find the time averaged current hjα (x)i to first order4 in ∆Ns
* +
∂σαx
hjα (x)i = hσαx (x, t)Ex (x, t)i = σαx,0 + ∆Ns ei(kx−ωt) Ex (x, t)
∂Ns
(7.8)
∂σαx 1 D E
=− σxx,0 Ex2 (x, t)
∂Ns v0 e
Since the time average of Ex (x, t) = 0, we can drop the first term, however the second
term is proportional to Ex2 (x, t), whose time average is not zero: this is the term that
D E
causes macroscopic acoustoelectric currents. In fact, we recognize that σxx,0 Ex2 (x, t) is
the average Ohmic power dissipated by the charge carriers in the 2DES, and may write
down the attenuation of the SAW per unit length as a result of the charge carriers in the
2DES as
1 dI 1D E
hΓi = − = σxx,0 Ex2 (x, t) (7.9)
I dx I
where we have assumed the intensity5 of the SAW may be written as I(x) = I0 e−Γx . We
may then combine Equations 7.8 and 7.9 to write down the time averaged acoustoelectric
current as a function of the SAW attenuation
4 The condition for which this first order approximation is valid, i.e. when we can drop the term ∝ ∆N 2 ,
s
is 2µ|Ex | v0 , where µ is the carrier mobility. This condition is valid in the data presented in this chapter,
where the mobility of the graphene is low, and wavelength of the SAW is large (and thus E ∝ ∇φ is smaller.)
However the same may not be true of high mobility samples (such as those exhibiting the FQHE) at GHz
frequencies.
5 Here, intensity has units power/length.
191
∂σαx 1
hjα (x)i = − hIΓi (7.10)
∂Ns v0 e
In the absence of a magnetic field, the off-diagonal components of the conductivity tensor
disappear, and only remaining component of the time averaged current can be written as
∂σ 1 µ
hjx (x)i = − hIΓi = − hIΓi (7.11)
∂Ns v0 e v0
where µ = Ns eσ is the carrier mobility in the 2DES. Thus, we see that, in the absence
of a magnetic field, the ac electric field associated with the SAW drives a dc current in the
2DES in the same direction as the SAW propagation. The 2DES may also be subject to an
external electric field in the x̂ direction: in this case, we can write the overall DC current in
the sample as
µIΓ
j = σE − (7.12)
v0
Similar to ∆v/v and attenuation measurements, acoustoelectric measurements have been
employed to study in the quantum Hall effect in GaAs quantum wells, facilitated by the
piezoelectricity of the host substrate [212, 204, 213, 214, 216]. More recently, work has
been done using the acoustotelectric effect to pump single electrons, which has opened up
the possibility of using SAWs as a mechanism for coherent control in quantum dot based
quantum computing schemes [217, 218]. Additionally, the observation of plateaus in the
acoustoelectric current through a 1-D channel [219, 220] has raised possibility of using the
acoustoelectric effect as a metrological current standard. While the quantization of current
plateaus in GaAs quantum wells remain insufficient for a current standard [221] there is some
192
hope that a cleaner system, such as electrons on helium [222], may exhibit highly quantized
current plateaus.
7.3 Acoustoelectric effect in exfoliated graphene
In anticipation of doing experiments in the quantum Hall regime, we would like to use ex-
foliated graphene devices, which in general have much higher carrier-mobilities than are
achievable than in, say, chemical vapor deposition (CVD) grown graphene. However, ex-
foliated graphene devices tend to be small (on the order ∼ 10’s of µm or smaller) which
creates an inherit problem for ∆v/v and attenuation measurements: the signal from these
measurements are inherently integrated over the SAW propagation path. We measure the
attenuation of the propagating SAW per unit length, or we measure the accumulated phase
shift of a velocity shifted SAW: either way, small sample sizes severely limit the signal size.
While it may be possible to design experiments around this (see Section 7.4 for some discus-
sion of this), for first-order experiments it is much easier to simply focus on acoustoelectric
measurements, where signals are inherently intensive.
In recent years several theoretical [223, 224, 225, 226] and experimental [227, 228, 229,
230, 231, 232] efforts have been made to extend SAW techniques to study the electronic prop-
erties of graphene. However, even in acoustoelectric studies, experimental challenges arise
when applying these methods to graphene that are absent from similar GaAs experiments.
The most pressing is the need to incorporate a compatible gate electrode for charge carrier
density control. Oxidized silicon, the most common substrate for high quality graphene
devices, is not piezoelectric, and the most strongly piezoelectric materials are also highly
insulating, making gating graphene devices on these substrates difficult.
193
In this chapter, we report the observation of an in situ gate-controlled acoustoelectric
effect in an exfoliated graphene device fabricated on an oxidized silicon substrate. To achieve
this, we use a flip-chip device geometry, qualitatively similar to the quantum acoustics geom-
etry from Chapter 6, where SAWs propagate on a separate piezoelectric substrate which is
flipped upside-down and mechanically clamped to a silicon substrate holding the graphene.
In this geometry, the electric field associated with the SAW evanescently extends above the
LiNbO3 surface, coupling to charge carriers in the graphene. We observe a clear dependence
of this acoustoelectric voltage as we tune the sign of the charge carriers in the graphene.
7.3.1 Cold-finger and sample stage
One of the dilution refrigerators in our lab (a BlueFors LD-400) comes equipped with a 14 T
superconducting solenoid magnet. By virtue of the solenoid not being infinitely long, the
generated magnetic field is spatially inhomogeneous along the magnet bore axis, with a max-
imum field ∼ 41.5 cm below the MXC. In anticipation of quantum Hall experiments, which
necessitate mK temperatures and large magnetic fields, we built a “cold-finger” extension to
the MXC plate from which we could mount exchangeable sample stages that sit at the height
of the maximum magnetic field. A picture of this cold-finger is provided in Fig. 7.3(a).
The cold-finger is designed to minimize heating from the changing magnetic field, caused
by either induced eddy currents or nuclear spin alignment [89], and to minimize the thermal
resistance between the sample stage and the MXC. To efficiently thermally connect the
sample stage and the mixing chamber plate, we used two 1/4”-diameter high-purity silver
rods (Fig 7.3, (4)), which were annealed at 600◦ for 90 hours to remove grain boundaries
194
Figure 7.3: Cold-finger design: (a) Photo of the cold-finger. A copper disk (1) is attached
to the MXC plate, and annealed siver rods (4) provide thermalization to the silver sample
mounting disk (3). Alumina rods (5) provide structural support. Close up views of (b) the
sample mounting disk and (c) the teflon intermediate disk. (d) Mechanism at the interme-
diate disk for compensating for the discrepancy between the thermal expansion coefficients
of alumina and silver upon cooling to cryogenic temperatures.
and lattice defects in the metal and decrease the low-temperature resistivity of the silver.6
Silver is a convenient choice of metal, since it doesn’t superconduct and has a small nuclear
heat capacity at dilution refrigerator temperatures [89]. Since the annealing process removes
grain boundaries, the metal becomes very soft and machining becomes difficult. We therefore
use un-annealed silver (Fig 7.3, (7)) crimped onto the ends of the rods, which are tapped
and screwed onto the MXC mounting disk and sample mounting disks ((Fig 7.3, (1) and (3)
respectively.)
To prevent eddy-current heating when we sweep the magnetic field, we would like to min-
imize the amount of metal in thermal contact to the sample and, when metal is unavoidable,
prevent large conducting loops that pick up large EMF’s. The sample mounting disk, which
6 The quality of a metal as a low-temperature conductor is generally quantified by the residual resistivity
ratio (RRR) defined as ρ[300 K]/ρ[4 K]. This is generally also taken as a quantifier of the purity of the
metal, since at room temperature resistivity is dominated by electron-phonon scattering, while at liquid
helium temperatures resistivity is generally dominated by impurity/grain boundary scattering.
195
must be conducting to thermally anchor the sample, is machined with a notch to break such
a loop. For structural support, we use alumina ceramic rods (Fig 7.3, (5)), capped off with
tapped brass connectors (Fig 7.3, (6)) since alumina is prohibitive to machine. On the sam-
ple mounting disk, near the magnetic field maximum, we instead use caps made of Macor, a
machinable ceramic (Fig 7.3, (8)) to connect the sample mounting silver flange.
When designing structures at cryogenic temperatures, it is critical to take thermal con-
tractions into account. A major problem with the cold-finger scheme presented here is that
the thermalizing silver has a much higher coefficient of thermal expansion than the structural
alumina. If uncompensated, the relative contraction of the silver rods compared to the alu-
mina rods would likely deform or destroy the structure. To circumvent this, we implement
a spring-loaded stress relief mechanism7 (Fig 7.3(d)), where we intentionally design a gap
between one set of alumina rods and the mid-finger teflon disk (Fig 7.3, (2)). At room tem-
perature, a compressed spring on the other other side of the disk provides structural stability
to the rods; as we cool and the silver contracts relative to the alumina the spring decom-
presses, decreasing the overall length of the structural component. This mechanism has the
added benefit that, for any reasonable experimental mass, the resonant oscillation frequency
of such a spring/mass system is far detuned from the pulse tube frequency (∼ 1 Hz) giving
the cold finger intrinsic vibration isolation.
The sample stage, shown in Fig. 7.4 is also machined out of silver, with slots cut into
the large-radius areas to prevent eddy current heating. The sample stage stage is designed
such that 16 DC measurement lines and 4 RF measurement ports may be used in an ex-
periment. Each DC measurement line is connected to cryostat ground via a 470 pF mylar
7 We actually realized this problem after designing the cold finger, and this mechanism was designed on
the spot by Bill Pratt when we asked him for advice. I think he may be a wizard.
196
(1)
(2)
(3)
(5)
(4)
Figure 7.4: Acoustoelectrics sample stage: Picture of the sample stage. (1) We use a
spring loaded mechanism to clamp the SAW device down to the graphene device, which is
mounted on a custom printed circuit board (2) for wire-up. (3) Thin film mylar capacitors
(red components) provide a high frequency short to ground as well as thermalization for the
electrons. (4) The leads are wired up to an 18-pin connector, which connects to the standard
wiring in the fridge. (5) SMA panel-mount connectors at the top of the sample stage (bottom
in this picture) are connected to a strip-line resonator on flexible kapton substrate, which
provides the RF signal to the SAW device.
thin-film capacitor (Quest Components, part No. 192P471X9200.) These capacitors serve
two roles: (1) the wrapped up mylar thin-film provides a large surface area contact between
the electrical leads (and thus the 2DES) and the cold cryostat. This reduced the thermal
resistance between the 2DES and the cryostat while maintaining an infinite DC electrical re-
sistance, alleviating the need for more extreme measures to cool the 2DES such as a helium-3
immersion cell [108]. (2) Along with an in-line 10 kΩ metal-film resistor on each measure-
ment line, the capacitors form a lumped-element low pass filter that attenuates noise above
fLP ∼ (RC)−1 ≈ 200 kHz.
The RF ports are fed in via SMA surface mount connectors press fitted and soldered into
197
the top of the stage. To run the RF signal from the top of the stage to the device mounted
on the bottom, we used a strip of double-sided copper-clad Kapton, with a micro-stip line
etched on one side such that Zstrip ∼ 50 Ω. From there, we used thin gold wires to solder
from the SAW transducers to the RF strip-lines. This was sufficient for the ∼ 300 MHz
SAW excitation signals used in this experiment, however we never tried to quantify losses or
parasitic resonances. At higher frequencies, where both of these problems are exacerbated,
it would likely be simpler and more efficient to run a coaxial cable directly to the level of
the sample head and connect to a surface mount connector on a printed circuit board with
prefabricated strip-lines.
7.3.2 Experimental setup
The graphene device used in this experiment was fabricated by Erik Henriksen’s group at
Washington University at St. Louis. The monolayer graphene sample was mechanically ex-
foliated onto a degenerately doped silicon wafer with a 300 nm SiO2 surface layer. Electrical
leads for conventional and acoustoelectric transport were fabricated such that they crossed
the entire width of the graphene sample perpendicular to the SAW propagation direction
as shown in Fig. 7.5(a). These leads were defined using standard lithographic techniques,
and had a width of 3 µm. Metallization of these electrical contacts was achieved by thermal
evaporation of 4 nm of chromium followed by 70 nm of gold. A voltage applied to the en-
tire silicon substrate was used to gate-control the graphene charge carrier density in situ as
shown in Fig. 7.5(b).
Silicon is not piezoelectric and hence SAW measurements cannot be done using a silicon
substrate alone. Therefore, we employed a flip-chip configuration, where a pair of interdig-
itated transducers (IDTs) for launching SAWs were fabricated onto a separate piezoelectric
198
Figure 7.5: Graphene acoustoelectrics device schematic: Schematic of the experimen-
tal setup. (a) The flip-chip configuration for the acoustoelectric measurements consists of
two devices: a SAW delay line fabricated onto a lithium niobate “black” substrate, and an
exfoliated graphene device on an oxidized silicon chip. A high-frequency signal applied to
one of the interdigitated transducers forming the delay line launches a SAW in the direction
indicated by the white arrow. (b) The SAW delay line is flipped onto and mechanically
clamped to the graphene device, as described in the text. In this configuration the electric
field propagating in tandem with the SAW evanescently couples to the graphene device,
while the silicon back gate may be used to control the graphene charge carrier density in
situ. The resulting acoustoelectric voltage is measured between two electrical contacts that
extended across the entire width of the graphene sample.
substrate that was then mechanically pressed onto the silicon substrate hosting the graphene
device, as shown in Fig. 7.5(b). In our device, aluminum IDTs were fabricated in the form
of a SAW delay line onto a single-crystalline chip of 128◦ Y-X lithium niobate “black” via
conventional photolithography. The IDT pair had a center-to-center spacing of 7 mm with
each transducer composed of 40 pairs of 3 µm wide interdigitated electrode fingers.
A SAW is launched along the surface of the lithium niobate by applying an alternating
199
voltage between the two electrodes of an IDT at the fundamental frequency of the transduc-
ers, f = v/λ ' 332 MHz. This frequency corresponds to a SAW wavelength λ ' 12 µm. The
electric field accompanying the SAW evanescently extends above the surface of the lithium
niobate, meaning the SAW may couple to a 2DES at a height < λ above the lithium niobate
[233, 234]. To effectively couple the SAW to the charge carriers in the graphene, care was
taken to ensure the absence of large debris on either substrate that could become lodged
between them after being flipped into contact. The two chips were mounted in a custom
spring-loaded sample holder to maintain their contact at low temperature and to immobilize
their relative lateral motion. Previous studies [233, 202, 235, 236] of flip-chip assemblies
similar to ours have estimated air gaps substantially smaller than the SAW wavelength in
our experiment. Therefore, while we are not able to directly measure the air gap in our
device, we expect it to be sufficiently small to enable acoustoelectric coupling.
With this experimental setup, low-temperature acoustoelectric measurements were made
between pairs of leads on the graphene by applying an amplitude modulated signal to
one SAW transducer and detecting the correspondingly generated acoustoelectric voltage
Vae using standard ac lock-in techniques. These measurements correspond to the open
configuration in which the total acoustoelectric current density j = 0 (see Eqn. 7.12) and
Vae arises from the charge displacement produced by the SAW in the direction of its prop-
agation. We note that the electrical leads used for detected Vae also allow us to perform
conventional lock-in based low-frequency (13 Hz) transport measurements to characterize
the graphene sample and provide a point of reference when interpreting our acoustoelectric
data. Finally, all data were taken in zero magnetic field and T ' 3.2 K.
200
7.3.3 SAW delay line characterization
The frequency response of the SAW transducers in our flip-chip device was characterized
using an Agilent N5230A vector network analyzer. Fig. 7.6(a) shows the measured reflection
coefficient S11 of the SAW device as a function of frequency. The resonance in the reflected
power observed at ' 337 MHz is associated with the generation of SAWs in the LiNbO3 . The
slight shift in frequency relative to the expected SAW resonance at 332 MHz is attributable
to an increase in the elastic moduli of LiNbO3 and mechanical strain in the flip-chip device
upon cooling to cryogenic temperatures8 . The primary SAW resonance shown in Fig. 7.6(a)
also exhibits superimposed oscillations as a function of frequency. Examining the inverse
Fourier transform of S11 , shown in the inset of Fig. 7.6(a), reveals a series of decaying peaks
spaced by 3.4 ± 0.1 µs. This time scale corresponds to a SAW propagation distance of
13.3 ± 0.2 mm, which is approximately twice the spacing of the two transducers in our
device. Because the transducers also act to reflect surface acoustic waves, a launched SAW
will propagate across the chip, be reflected by the opposing transducer, and propagate back
to the launching transducer, where its co-propagating electric field will interfere with the
reflected electric signal from the launching transducer as measured by the network analyzer.
We therefore attribute these superimposed oscillations in S11 to the interference between the
signal reflected from the launching transducer and the electric field associated with SAWs
that have propagated to and from the opposing transducer.
8 We note that mass loading alone from the SiO /Si substrate would tend to reduce the resonant response
2
of the SAW delay line. However, since the resonance frequency is shifted slightly higher by ∼ 5 MHz, we
conclude that stiffening of the lithium niobate crystal due to strain and cooling to low temperatures dominate
mass loading by the SiO2 /Si substrate.
201
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Figure 7.6: SAW reflection and acoustoelectric voltage vs. drive frequency: (a)
Frequency dependence of the reflection coefficient (S11 ) of the SAW delay line. Inset: inverse
fast Fourier transform (IFFT) of S11 data. (b) Measured acoustoelectric voltage in the
graphene sample as a function of frequency. This measurement was conducted at 3.2 K with
the sample back gate grounded. To excite the SAW, 0 dBm of microwave power was applied
at the top of the cryostat. Inset: inverse Fourier transform of the acoustoelectric signal.
7.3.4 Graphene acoustoelectrics
Fig. 7.6(b) shows the acoustoelectric voltage measured between two leads 29 µm apart9 on
the graphene device as a function of frequency. The resonance in Vae coincident with the
generation of SAWs centered at ' 337 MHz is associated with the acoustoelectric transport
of charge in the graphene sample. For this data the silicon back gate was held at ground
potential and, as we will describe below, the sample displays p-type conduction typical of
graphene devices on SiO2 exposed to air and polymer resist. The sign of the induced voltage
is consistent with conduction by holes.
The frequency response of Vae shows oscillations superimposed onto the main peak similar
9 Consistent behavior was observed using the other leads on the device.
202
to those observed in S11 . These oscillations are attributable to the modulation in the SAW
amplitude caused by interference between the primary SAW and reflected waves.[221] To
quantitatively understand the nature of these oscillations, we examine the inverse Fourier
transform of Vae , which is shown in the inset of Fig. 7.6(b). Similar to the behavior of S11
we observe a series of decaying peaks spaced by ' 3.4 µs. We attribute these peaks to the
interference of the primary SAW with forward propagating SAWs that have been reflected
by the far transducer and then again by the original transducer. Additionally, we observe
a number of peaks at odd multiples of 1.7 ± 0.1 µs, which correspond to the propagation
distance between the two transducers. We attribute these peaks to modulation in the SAW
amplitude caused by counter propagating waves reflected by the opposing transducer.
To verify that the measured voltage is associated with the charge carriers in the graphene,
we applied a voltage Vg to the silicon back gate to tune the graphene charge carrier type from
holes to electrons. Fig. 7.7(a) shows the acoustoelectric voltage as a function of frequency for
two back gate voltages on either side of the Dirac peak (see Fig. 7.8(a)). With the back gate
grounded (Vg = 0 V, red trace Fig. 7.7(a)), the graphene exhibits conduction due to holes
and the corresponding acoustoelectric signal is positive. By increasing the Vg to +60 V,
electrons become the predominant charge carrier in the graphene and, as expected, the
acoustoelectric voltage correspondingly reverses sign (blue trace in Fig. 7.7(a)). In Fig. 7.8
we show the full gate- and frequency-dependent map of Vae for the device along with the
corresponding gate-dependent two-point resistance of the graphene. In the bottom panel of
Fig. 7.8 we show a constant frequency linecut of Vae at f = 337.45 MHz. As expected, the
sign of acoustoelectric signal changes upon tuning through the Dirac peak at Vg ' 28 V,
validating that Vae is a result of acoustoelectrically induced charge transport in the graphene.
Moreover, in the open configuration of our measurement (where j = 0) Eqn. 7.12 predicts
203
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Figure 7.7: Acoustoelectric effect: charge carrier polarity dependence: (a) Acousto-
electric voltage versus frequency for two gate voltages: Vg = 0 V (hole-doped graphene) and
Vg = 60 V (electron doped graphene) at 3.2 K. For these measurements 0 dBm of microwave
power was used to excite SAWs. (b) Acoustoelectric voltage as a function of power applied
to the SAW circuit, for the same back gate voltages as in panel (a). This data was taken at
a fixed frequency f = 337.45 MHz corresponding to the maximum in the response of the
SAW transducer.
that Vae should be proportional to the SAW intensity I. Therefore, to further verify the
acoustoelectric origin of the signal, we measured Vae as a function of SAW power at a fixed
frequency of f = 337.45 MHz, which corresponds to the peak in SAW resonant response.
The results are plotted in Fig. 3(b). For both hole and electron doping we find that Vae is
linear in the applied SAW power, consistent with Eqn. 7.12 and with previous acoustoelectric
measurements in graphene [227, 229, 230, 231].
In the vicinity of the Dirac peak we observe a marked reduction in the magnitude of
204
Figure 7.8: Gate-tunabe acoustoelectric effect: (a) Low-frequency two-point resistance
of the graphene sample as a function of back gate voltage. (b) Acoustoelectric voltage signal
(taken with 0 dBm of SAW power), plotted as a function of both back gate voltage and
SAW frequency. At gate voltages where the graphene is heavily electron (hole) doped, Vae is
consistently negative (positive). (c) Constant frequency linecut of Vae at f = 337.45 MHz,
indicated by the dashed line in panel (b).
Vae → 0 over a well-defined region in gate voltage. This region is emphasized in Fig. 7.9(a),
where we performed measurements with increasing levels of SAW excitation power to enhance
the magnitude of the acoustoelectric signal. For comparison in Fig. 7.9(b) we show low
frequency transport over the same range of gate voltage while applying a SAW driving signal
of the same power. Over a range of ∼ 20 V, as indicated by the dashed vertical lines, we find
205
2*
3,
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Figure 7.9: Power dependence of the Gate-tunable acoustoelectric effect: (a) Acous-
toelectric signal versus back gate voltage at a SAW frequency f = 337.45 MHz for increasing
levels of SAW power. A well-defined region where Vae = 0 is observed around charge neu-
trality and is associated with heterogenous charge disorder as discussed in the text. (b)
Corresponding low-frequency two-point resistance of the graphene sample as a function of
back gate voltage for the same values of SAW power as shown in (a). The two-point resistance
traces in have been offset by 100 Ω relatively to each other for clarity.
that the average value of Vae ' 0. This range of gate voltage corresponds to a charge carrier
density range of approximately 1.4 × 1012 cm−2 . This region of suppressed Vae is likely
associated with the formation of heterogeneously doped electron-hole “puddles”, which are
known to exist close to charge neutrality due to substrate contamination and impurities [237,
238] but only manifest indirectly in transport measurements as a broadening of the Dirac
peak. In this charge disordered regime one would expect that competing acoustoelectric
signals from roughly equal distributions of electrons and holes to cancel each other and
result in an approximately net zero value of Vae as observed in our measurements. From
the low frequency transport we determine that the field effect mobility of the graphene
206
is µ ' 320 cm2 /Vs on the hole-doped side of the Dirac peak and ' 110 cm2 /Vs on the
electron-doped side. While these values are relatively low for an exfoliated device they are,
however, two orders of magnitude higher than those reported in ion-gated devices used in
recent acoustoelectric measurements [231, 239]. Moreover, the fact that the mobility of
holes in our sample is roughly three times larger than that of electrons naturally explains
the asymmetric position of the region over which Vae ' 0 relative to the Dirac peak. Finally
we note that a reduction in SAW driven current was observed near charge neutrality in
recent measurements of CVD graphene at room temperature [231, 239, 240] using ion gel
gating. In these measurements the authors suggest similarly that electron-hole puddling
is responsible. Our measurements using a different experimental setup and an exfoliated
device at low temperatures are consistent with this interpretation and further highlight that
acoustoelectric methods provide complementary methods for investigating 2DES and reveal
features of the underlying phenomena not directly manifest in conventional transport.
7.4 Future work
There are several directions one could imagine taking the concepts developed in this exper-
iment. We present several of them here:
7.4.1 Graphene acoustoelectrics in a magnetic field
By depositing leads across the graphene and perpendicular to the direction of SAW propaga-
tion, this experiment was designed to measure acoustoelectric current flowing in the direction
of propagating SAW waves, i.e. when the conductivity can be written as a scalar. As is ev-
ident from Eqn. 7.10, when a magnetic field is introduced and the conductivity becomes a
207
tensor, the acoustoeletric current need not flow in the direction of SAW propagation [213].
In-field acoustoelectric measurements have been done in GaAs [212, 204, 213] in a more
traditional Hall bar geometry. It would be fairly straightforward to fabricate a graphene
device with Hall bar geometry, which could potentially allow for these measurements to be
extended into the quantum Hall regime.
7.4.2 ∆v/v measurements
A modern vector network analyzer (VNA) is likely capable of measuring the phase shift
of a microwave signal with a higher precision than the phase-locked loop setup typically
employed in ∆v/v measurements [168, 206]. Using a VNA, it might be possible to do ∆v/v
measurements on a piece of exfoliated graphene. The transducers would need to be made
small, such that the width of the SAW propagation path was comparable to the size of the
exfoliated device (∼ 10s of µm), and the transducers would have to be put relatively close
to one another such that the exfoliated device takes up a significant portion of the SAW
propagation path. For example, for a SAW delay line with two 20 µm wide transducers
spaced by 100 µm, a 20 × 10 µm piece of exfoliated graphene, an eminently reasonable
device size, would cover ≈ 1/10 of the SAW propagation path.
In this proposed experiment, if the microwave environment isn’t carefully engineered to
avoid spurious modes, the close SAW transducers would likely have a large free-space coupling
that completely washes out any SAW coupling. This is, however, a manageable problem,
as displayed by several quantum acoustics experiments with similar SAW propagation path
sizes with a clear SAW propagation signal [7, 17, 174]. We refer the reader to Ref. [241] for
an introduction on engineering experiments to avoid spurious microwave modes.
208
7.4.3 Measurements of 2DESs using SAW resonators
An interesting possibility10 for using SAWs in the quantum Hall regime was inspired by
Ref. [242] where composite fermion cyclotron resonances were resistively detected when they
became commensurate with a piezoelectrically generated periodic electric field in GaAs. In
this paper, the periodic strain field that generates the electric field was created by stripes of
photoresist on the surface of the substrate, which upon cooling to the base temperature of
the cryostat contracted relative to the GaAs substrate. It may be possible to do a similar
resistively detected cyclotron resonance experiment by placing a graphene device in a SAW
resonator. Exciting the SAW resonator at its resonant frequency would set up a stand-
ing wave, which introduces a length scale (the SAW wavelength λ) which may modulate
the transport properties when the radius of the cyclotron orbit of a composite fermion is
commensurate with λ [185, 213].
A particularly tantalizing idea is that if the Bragg mirrors are far enough apart, the
free spectral range of the SAW cavity may easily be engineered to be small enough such
that the cavity hosts several resonant modes with slightly different frequencies/wave vectors
[169]. Each of these resonant modes would subsequently be commensurate with slightly
different composite fermion cyclotron resonances. This would, in principle, allow for one to
investigate cyclotron resonances as a function of frequency on a single sample (albeit over
a small frequency range set by the stop-band of the mirrors, see § 5.3.5) in a resistively
detected manner similar to Ref. [242].
Alternatively, one may be able to infer ∆v/v and Γ from shifts in the real and imaginary
(∝ the linewidth of the resonance) parts of the SAW resonant frequency respectively. A
similar technique was employed in Ref. [243] to study the bath of two-level systems near
10 Thank’s to Leo Li for this idea.
209
the surface of a piezoelectric substrate. Given other topics in this thesis, one could imagine
a situation where these measurements could be enhanced by the incorporation of coherent
quantum circuits coupled such a SAW resonator. For example, fine shifts in the resonant
frequency of the cavity could be detected with high precision by measuring the phase ac-
cumulated by a single phonon in the resonator relative to that of a superconducting qubit
coupled to the resonator.
210
APPENDICES
211
APPENDIX A
A note on the rotating wave
approximation
At several points in this thesis, we have invoked something called the “rotating wave approx-
imation” or RWA. When considering Hamiltonians such as Eqn. 3.13 that contain a coherent
drive term, when we make the transformation to the frame rotating with the drive we will in
general have terms in the Hamiltonian that are a sum of two frequencies and terms that are
a difference of two frequencies. The RWA may be invoked when there is a separation of scale
between the oscillation frequency of the sum (“counter-rotating”) terms and characteristic
rates in the rest of the Hamiltonian: if the counter rotating terms oscillate quickly over the
timescale of the rest of the dynamics, the net effect of these term will average out over many
periods and we can drop them.
For example, consider the derivation of Eqn. 3.17: before invoking the RWA, the Rabi
Hamiltonian in the frame rotating is
ω − ωd A
H̃Rabi = −~ 01 cos(φ)σ x + sin(φ)σ y + e−2iωd t σ + + e2iωd t σ −
σz − (A.1)
2 2
The time dependence of the σ x and σ y terms, the “co-rotating” terms, has been can-
212
celed out, while the counter-rotating terms (σ + and σ − ) have picked up time dependen-
cies ∝ e±2iωd t . The main argument of the RWA is that, from the Shrödinger equation
i~∂t ψ = Hψ the characteristic rates at which the state vector will change are |ω01 − ωd |
and A/~. If ωd max[|ω01 − ωd |, A/~], the counter-rotating terms will oscillate quickly
compared to the evolution of the state vector, and the effect of these terms will average out
to zero.
Figure A.1: The rotating wave approximation: (a) State evolution after invoking the
rotating wave approximation (RWA). Here, we simulate a qubit subject to a resonant Rabi
drive (ωq = ωd ), with ∆ = 0 such that the state vector rotates around the x-axis of the
Bloch sphere. We have chosen ωq /(2π) = 5 GHz and A/(2π) = 0.1 GHz. (b) The same
state evolution without invoking RWA. Small oscillations in the path are averaged out over
time-scales ≈ 1/A, and the state-vector largely takes the same path as if we had invoked the
RWA. Figures made using the QuTiP package [102].
A visualization of this is given in Fig. A.1, where we have numerically solved the
Schrödinger equation for the Hamiltonian Eqn. A.1 with (b) and without (a) the counter-
rotating terms and plotted the corresponding trajectories on the Bloch sphere. For this
simulation, we’ve taken resonant qubit and drive frequencies ω01 /(2π) = ωd /(2π) = 5 GHz
and drive amplitude A/(2π) = 100 MHz. Even for this relatively strong Rabi drive (corre-
213
sponding to a π-pulse duration of ∼ 5 ns), including the counter-rotating terms does little
to effect the overall path of the Bloch-vector.
The RWA is also employed several times in Chapter 2: in those cases, the best way to
think about the RWA is in the Heisenberg picture, where the operators are time dependent.
The raising and lowering operators of the unperturbed harmonic oscillator have time de-
pendencies a† (t) = a† (0)e−iωt and a(t) = a(0)eiωt (exercise: prove this.) Therefore, when
deriving Eqn. 2.27, terms such as a† a† a† a† will oscillate quickly compared to the dynamics
of the rest of the Hamiltonian, and may be ignored. When deriving the JCH (Eqn. 2.41),
† †
dropping terms such as aq ac is only strictly valid when |ω01 − ωc | |ω01 + ωc |, i.e. when
the qubit and the cavity aren’t too far detuned from each other. However, the JCH and
the dispersive regime approximation tend to be valid for detunings that are an appreciable
fraction of the qubit frequency (∆/(2π) ≈ 2 − 3 GHz is very common.)
214
APPENDIX B
Other cQED experimental techniques
In this appendix, we summarize a couple more advanced experimental protocols we have
employed while experimenting on superconducting circuits.
Excited state population
In an ideal scenario, the baseline excited state P1 population of a qubit would follow a
Boltzmann distribution. If this were the case ω01 ∼ 5 GHz ∼ 250 mK, so at typical
dilution refrigerator operating temperatures (∼ 10 − 20 mK) P1 would be exponentially
suppressed to a negligible level1 . However, spurious excitation modes (attributed mainly to
non-equilibrium quasiparticles [141, 154, 142, 42]) may populate the excited state to well
above the level expected by Maxwell-Boltzmann statistics. As such, it is useful to measure
the residual excited state population.
We accomplish this using a method developed in Ref. [158] and further optimized in
Ref. [154], where we measure the ratio of the amplitude of two Rabi drives: one where we
Rabi flop the residual excited state population P1,R , and one where we Rabi flop the residual
ground state population P0,R . To do these two measurements independently, we need a third
energy level that starts off with negligible population to act as the target of the Rabi drive.
Transmon qubits have small anharmonicity, meaning that the |2i state can fulfill this role
1 This is equivalent to saying that Γ , the transition rate from |0i to |1i, is exponentially suppressed.
↑
215
(a)
(b)
Figure B.1: Excited state population measurement: To measure the residual excited
state population, we measure the relative amplitude of two Rabi drives, one (a) where we
Rabi drive the residual excited state population (P1,R , red circles) between the |1i and |2i
states, and one (b) where we first invert the population of |0i and |1i such that we Rabi drive
the residual ground state population (P0,R , green circles) between the |1i and |2i states. At
the end of each measurement, we invert the population of |0i and |1i and measure.
without too much experimental effort.
The first measurement simply consists of driving the qubit at ω12 for a variable time τ ,
which Rabi flops the population P1,R between |1i and |2i, see Fig. B.1(a). To measure, we
apply a π pulse at ω01 , and measure, repeating the measurement while varying τ to build
up P0 (τ ), which should be sinusoidal as predicted by Eqn. 3.14. We fit P0 (τ ) to a sinusoid,
and extract the amplitude A0 .
We then repeat the measurement, except this time before the ω12 Rabi drive we apply a
π pulse at ω01 , inverting P0,R and P1,R . Now, the drive Rabi flops P0,R , the residual ground
state population, between |1i and |2i, see Fig. B.1(b). We repeat the rest of the process,
building up P0 (τ ) and fitting it to a sinusoid, extracting the amplitude A1 . Then, if the
216
qubit |0i and |1i states are the only residually populated states2 , P1,R is simply given by
A0
P1,R = (B.1)
A1 + A0
It was noted in Ref. [154] that since the amplitudes of the sinusoids are really the only
fit parameters we care about in this measurement. every point of the Rabi drive we measure
that isn’t at a maximum or minimum of the sine (τ = nπ/ΩR , where n is an integer) is
functionally dead weight. Thus, a more efficient measurement is to replace the variable
time τ with a variable number of π pulses between |1i and |2i (i.e., a Rabi drive where we
only sample at τ = nπ/ΩR .) With this modified measurement scheme, we can significantly
expedite the data taking process, and average over a far higher number of measurements to
record P1,R at much higher precision (we have measured P1,R down to ∼ 0.5%.)
Spin-locking noise spectroscopy
As outlined in § 4.2 decay (T1 ) measurements are sensitive to the noise spectral density
at ±ω01 , while dephasing (T2 ) measurements sample a wide range of the noise spectral
density. Often times, we want to reconstruct the noise spectral density at frequencies much
lower than ω01 , from mHz to MHz, since noise at these frequencies contribute to qubit
dephasing [105, 106, 62], and since measuring the characteristics of the noise PSD is critical
for feeding into theoretical noise models and engineering experimental setups that minimize
noise [134, 104, 128, 88, 244]. However, these frequencies are not accessible with simple
T1 measurements, the wide range of noise spectrum sampled in a dephasing mechanism
makes it difficult to deconvolute the spectrum from measurements. What we would like is
2 This is a good approximation, given that we typically measure P
1,R ≈ 0.5 − 3%.
217
a measurement that samples a sharp frequency range (like a decay measurement) but can
access these frequencies. Spin-locking measurements [104, 245, 62, 88] are one method of
doing this.
Experimentally, we accomplish spin-locking measurements with something similar to a
T2 measurement: we start off with the qubit in the ground state, and apply a −πy /2 pulse
to rotate the state vector on the onto the x axis of the Bloch sphere (see Fig. B.2(b).) Then,
instead of simply waiting time τ before the next π/2 pulse, during this time we apply a
on-resonant (ω01 − ωd = 0) microwave drive along the x axis (φ = 0 in Eqn. (3.17).) These
conditions render the Hamiltonian in the rotating frame to be
˜ ~Ω
Ĥ = − R σ x (B.2)
2
The state vector we have prepared is in an eignestate of this rotating frame Hamiltonian,
and will thus only undergo non-deterministic evolution (similar to a T1 measurement in the
lab frame.), decaying exponentially at rate Γ1ρ as is depicted in Fig. B.2(b). After time τ ,
we apply another −πy /2 pulse to align the state vector with the z axis of the Bloch sphere
and perform a projective measurement. As above, we repeat this process varying τ , and
repeat this series a sufficient number of times to accurately measure P1 (τ ). Since during
evolution our state vector is in an eigenstate of the rotating frame Hamiltonian, we expect
to only measure non-deterministic exponential state decay. We thus fit P1 (τ ) to a decaying
exponential to extract Γ1ρ . This decay rate is given by [245]
1
Γ1,ρ = Γ1 + Γν (B.3)
2
where Γν is proportional to the noise spectral density at ΩR . To extract Γν , we need to
218
ωq - 150 MHz ωc
(a)
+5 V DC
Circulator
-5 V DC LO
ADC
I Q
LPF Low-pass filter
RF
Attenuator
Ecco Eccosorb filter
LO
Amplifier I Q
300 K
RF
DF
Microwave source
HEMT
Splitter/combiner
10 mK
Ecco LPF
Variable attenuator LPF
Microwave switch
LO
I Q IQ mixer
RF
Tabor Ecco
+5 V DC DC power supply
Resistive combiner
DG535
(b)
Spin-locking
Gates
Measurement
time
Figure B.2: Spin-locking setup and measurement: (a) Modified measurement setup for
spin-locking measurements. To vary the amplitude of the locking Rabi drive (green) while
keeping the control pulse amplitudes (red) constant, we control them using different pairs
of channels on the Tabor. The cavity pulse is then generated using a microwave switch
actuated by a voltage pulse from a Tabor marker channel (blue.) (b) Pulse sequence and
state evolution. We start with the qubit in the ground state, and then apply a −πy /2 pulse
to prepare the state vector along the x axis. We then “lock” the qubit along the x axis
by turning on a Rabi drive, which puts the state in an eigenvector of the rotating frame
Hamiltonian. After time τ , we perform another −πy /2 pulse to put our Bloch vector along
the measurement axis and measure.
219
(a) (i) (ii) (iii) (b)
0.8 0.6
0.6 0.4
P1
0.4
0.2
0.2
0 100 200 0 5 10
Point in sequence (MHz)
Figure B.3: Spin-locking spectroscopy: (a) A typical spin-locking measurement run, were
we measure (i) a Rabi flop to extract ΩR (data span 1 µs), (ii) spin-locking decay to extract
Γ1ρ (data span 15 µs), and (iii) T1 population decay to extract Γ1 (data span 45 µs). We
fit these data to extract the desired parameters, and calculate Γν = Γ1ρ − Γ1 /2. (b) We
repeat this measurement for many Rabi-driving amplitudes to build up a dataset of Γν as a
function of ΩR (the measurement shown in (a) corresponds to the red point in (b).) In this
particular measurement, we see a 1/f -like dependence, with several noise peaks near 5 MHz.
measure Γ1 in parallel so that we can subtract it off of Γ1ρ . We also need to know what the
Rabi frequency of the applied driving tone is, since ΩR determines the frequency of the noise
spectral density we sample in a measurement. A typical measurement strategy is to write
one long sequence that does all three measurements (inversion recovery, Rabi drive, and spin-
locking) back-to-back-to-back, and interlacing these measurements before averaging. This
strategy is useful since a typical spin-locking measurement consists of a loop over many spin-
locking drive amplitudes, and T1 may fluctuate significantly over the typical time required
to take such a measurement (∼several hours.) An example of such a measurement is shown
in Fig. B.3(a), which produces one point in a collection of Γν ’s plotted as a function of ΩR
in Fig. B.3(b).
As stated previously, a typical spin-locking measurement consists of a loop over spin-
locking drive amplitudes. In order to execute such a loop, we must apply both fixed ampli-
tude rotations around the Bloch sphere and a variable amplitude spin-locking drive. This
220
complicates the control pulse programing, since a two axis sweep (one axis varying the spin-
locking time τ and one varying the drive amplitude) would overload the per-channel memory
of the Tabor. To circumvent this problem, we use all four output channels on the Tabor
to program control pulses: two channels (Fig. B.2(a), red) output I(t) and Q(t) for the
fixed Bloch sphere rotations, and two (Fig. B.2(a), green) output I(t) and Q(t) for the
spin-locking drive. We may then modify the amplitude of only the spin-locking channels
without hardwiring the amplitude changes in the control pulse sequence. Since we use all
four channels to control the qubit, we perform the measurement with a microwave switch,
which is actuated with a marker channel on the Tabor (Fig. B.2(a), blue).
221
APPENDIX C
Fabrication Recipes
Substrate cleaning
SAW devices at ∼GHz range frequencies tend to have large (∼ 100’s of µm to mm) areas
covered with small (∼ 100 nm) patterns. Fabricating these structures requires substrates to
be very clean. A recipe I’ve found for cleaning substrates that generally works for preparing
SAW devices:
1. Soak in remover PG on the hotplate for 10 minutes at 80◦ C.
2. Transfer to petri-dish full of acetone using metal tipped tweezers. PG will dissolve most
carbon-tipped tweezer. Transfer again to dilute the PG if you’re paranoid.
3. While in acetone, rub substrate with Ruby stick. Remove from acetone, and rinse with
IPA before acetone evaporates from surface. Blow dry.
4. Sonicate in acetone for 5-15 min. Use the good sonicator in the Class 1000 fume hood.
You can heat up the sonicator if you want, but I somewhat doubt the acetone gets
appreciably warmer if you do.
5. Sonicate in IPA for 5-15 min, again using the good sonicator in the Class 1000 fume
hood.
222
6. Remove sample from IPA and blow dry. I’ve noticed that if, rather than being blown
off by the nitrogen gun, the IPA evaporates off, it will leave a residue that can mess
up fab further down the line, so be diligent when doing this step.
7. Plasma clean in O2 plasma etcher. 300 W for 300 s, sitting in a petri dish covered in
Al foil (new foil if you’re paranoid). The substrate will be hot to the touch after the
etch, so it’s best to bring it to the class 100 room fume hood and let it sit for several
minutes before you deposit resist.
It’s also important to remove organic residues from qubit devices, which can serve as
sources of decoherence. We have not done a systematic study to see how effective this
cleaning method is for preparing qubit devices, though that would be a good thing to do at
some point.
Positive photolithography
Photolithography is the process of defining patterns on a substrate for metallization/etching
by coating the substrate with a light-sensitive polymer and exposing certain areas to UV
light. Depending on the polymer and process, the polymer layer in the exposed area is either
removed (positive process) or retained while the unexposed polymer is removed (negative
process). We outline a standard recipe for the positive photolithography process in our lab.
1. Clean the substrate, using § C as a guide.
2. Spincoat S1813 G2 photoresist onto the substrate, 5000 RPM, 50 seconds.
3. Bake on resist hotplate, 60 seconds 110 ◦ C
223
4. Align photomask relative to substrate. Once aligned, bring the substrate in contact
with the photomask. If you are exposing a large area, you should be able to see Fresnel
rings form between the substrate and the bottom of the photomask.
5. Once satisfied with alignment and contact, expose to UV for 8 seconds
6. Remove substrate from mask aligner and soak in chlorobenzene for ∼5 minutes. This
hardens the upper layer of photoresist, giving an undercut with a single layer process
and making it easier to lift off metal.
7. Develop in 352 developer for 10-25 seconds, DI for 10 seconds, and blow dry. Inspect
under the optical microscope. Depending on how precise you need your features to be,
if the exposed parts aren’t fully developed you can stick the substrate back into 352
for 5-10 seconds (with DI and blow dry afterwords) to develop a little more. But don’t
tell anyone I told you that.
E-beam lithography
In the same way that the resolution of an optical microscope is ultimately limited by the
diffraction of light passing though a small aperture, the feature size possible to define by
photolithography is ultimately limited by diffraction. We expect this limit to be of the same
order as the wavelength of the light in question: in photolithography, we use UV light with a
wavelength of λ ≈ 400 nm. While industrial processes have produced techniques that make
sub-wavelength features possible, for our tabletop mask-aligner we’re limited in practice to
features with linear dimensions > 2 − 3 µm. Therefore, if we want to make smaller features,
we must expose the resist with a smaller wavelength particle. The electrons emitted from a
224
scanning electron microscope (SEM) perform this function for us: we use a modified SEM
to point and shoots the emitted electron beam at the areas of substrate we desire to expose,
“writing” a pattern in the resist that is then developed away to expose substrate. Electron
beam (e-beam) lithography can create much smaller features than photolithography, at the
expense of a much longer write time, since only a small area can be exposed at any given
time.
General bi-layer resist recipe
This a general, bi-layer resist e-beam lithography recipe which is a standard workhorse recipe
in our lab. We use a bi-layer stack of resist with MMA on the bottom and PMMA on the top.
MMA is a “softer” resist than PMMA in that it generally requires a lower dose to expose,
which results in resist undercuts that make lift-off easier. We can also use this bi-layer to
make resist bridges, which are crucial in double-angle evaporation techniques for making
Josephson junctions.
1. If applicable, photolithographically define alignment marks. Marks must be thick
enough to be seen through the resist: 50 nm thick gold alignment marks generally
work.1
2. Spin-coat MMA/EL-9 resist on a clean substrate, 4000 RPM for 45 seconds. If you
want to write many patterns on a single chip, it is advisable to inspect the back of the
substrate after spin-coating, since resist may pool on the back, which will result in the
substrate being tilted during writing and may cause issues with focusing.
1 A “rule of thumb” I’ve found is that the alignment marks should have thickness ≥ 2600/(Z − 14), where
Z is the atomic number of the metal the mark is made out of. I really don’t have any idea how accurate
this rule is, but it produces numbers that have anecdotally worked for me: gold has Z = 79 and thus you
want alignment marks ≥ 40 nm thick, copper has Z = 29 and so to make copper alignment marks you need
≥ 170 nm, while aluminum has Z = 13, and thus you can’t reliably make alignment marks out of aluminum.
225
3. Bake substrate on hotplate, 180◦ C for 10 minutes, or in the big oven at 180◦ C for
1 hour. Remove from hotplate and allow to cool
4. Spin-coat PMMA C2 resist, 4000 RPM for 45 seconds. Again advisable to inspect the
back of the substrate.
5. Bake substrate on hotplate, 180◦ C for 10 minutes, or in the big oven at 180◦ C for
1 hour. Remove from hotplate and allow to cool
6. Secure the device to the SEM stage. Prior to inserting into the SEM/EBL system,
make a small scratch on the corner of the sample to focus on.
7. Insert SEM stage and sample into the vacuum chamber and pump down. Once high
vacuum is obtained, on the NPGS computer set the system to “SEM/imaging” mode.
8. Move to a position away from your substrate, and find a feature (dust/scratch/etc...)
on the surface of the sample stage. Adjust focus/astigmatism until you get a clear
image of the feature at 100,000× magnification.
9. Move to the scratch on the corner of the substrate, and refocus. At this point the
beam should be ready for writing.
10. Switch to “Full EBL” mode and move to desired writing area. If needed, do alignment.
Switching to “Full EBL” prior to moving to the write area will prevent unnecessary
exposure. If you need to write a specific part of the substrate/align, I’ve found it
convenient to make large, very visible alignment marks
226
Superconducting qubit fabrication
Fabrication of superconducting qubit devices in our lab was started not too long before this
thesis was written, and as our local capabilities/knowledge change, the fabrication process
has also been changing. In light of this, this section should be thought of as a snapshot of an
evolving fabrication process. I will detail to the greatest extent possible the current process,
however the reader should be deterred from believing anything here is a definitive (or even
optimal) process.
Substrate preparation
The current iteration of the fabrication process starts off with whole 2” silicon wafers. The
wafers are cleaned using steps 1-5 in the substrate cleaning recipe above (the Ruby stick
cleaning may be omitted.) After sonication in IPA, DI water straight from the tap in the
fume hood is run over the surface of the substrate for ∼ 30 − 60 seconds, and then the wafer
is blow dried.
We then spincoat MMA/PMMA onto the entire wafer, as detailed in steps 2-5 of the
“General bi-layer resist recipe” section. Once this done we score the wafer with a scoring
pen, and cleave it into ∼½”×½” pieces for electron beam lithography. This process will
introduce some detritus onto the surface, however the junction area, which is the only place
that requires high precision fabrication, is small and the probability of a piece of silicon dust
hitting the exact spot a junction is written is rather low.
Dolan bridge junction fabrication process
As stated in Chapter 3 of the main text, the Josephson junction is defined with electron
beam lithography and a two-step evaporation process called the “Dolan bridge” process [80].
227
In the Dolan bridge process, a bi-layer resist stack consisting of a “hard” resist (PMMA/C2)
sitting on a “soft” resist (MMA/EL9) is used. As shown in Fig. C.1(a), a high dose (blue,
∼ 300 µC/cm2 ) is used to expose the hard resist, while a low dose (orange, ∼ 50 µC/cm2 )
exposes the soft resist while leaving the hard resist intact. The resist is then developed in
MIBK:IPA 1:3 for 50 seconds, IPA for 15 seconds, and blow dried. Upon development, in
the places where a low dose was used, the hard resist will remain intact. Near the edges of
the device, this “overhang” is useful for facilitating liftoff, however the important part is the
center, where the MMA layer is cleared out from under a “bridge” of PMMA that will define
the junction. Fig. C.1(b) details these features.
The substrate is then plasma etched in 500 mTorr O2 at 100 W for 20 seconds to remove
residual organics on the exposed surface. After etching, we place the substrate into a thermal
evaporator with the ability to controllably tilt the sample holder, and pump down to low
pressure.2 Once the pressure is low enough, the sample holder is tilted such that the substrate
makes an angle θ1 relative to the evaporation source, with the rotation axes running parallel
to the bridge (tilt axis perpendicular to the bridge), and a film of aluminum is evaporated, see
Fig. C.1(c). After the first evaporation, the device is exposed to a 90/10 Ar/O2 environment,
which oxidized the surface of the aluminum, forming the insulating barrier that will provide
the tunnel barrier for the junction (dark grey in Fig. C.1(d).) The evaporator is then once
again evacuated to low pressure, and a second layer of aluminum is deposited at angle θ2 . As
shown in Fig. C.1(e), if done correctly the two large metallic pads on either side of the bridge
will be connected by only a small area oxide barrier directly under the bridge, highlighted in
yellow. After evaporation, the substrate is placed in acetone overnight, which dissolves the
resist and leaves behind only the pattern of metal on the substrate, see Fig. C.1(f). We then
2 Our thermal evaporator, if operated properly, can reach ∼ 10−7 Torr in ∼3 hours.
228
(a)
PMMA/C2
MMA/EL9
Si substrate
(b)
(c) (d)
(e) (f)
Figure C.1: Description of the Dolan bridge junction fabrication process: Process
flow of the Dolan bridge junction fabrication process. Top-right: a top down view of the
substrate with a resist bridge defined via electron beam lithography. All cross sections (a-f)
run along the red dotted line. In panels (e-f), the part of the oxide layer that defines the
junction is highlighted in yellow. Note that distances are not to scale.
clean the sample by briefly sonicating in acetone/IPA, and then blow drying and inspecting
under the optical microscope.
The Josephson energy EJ of the junction is dictated by both the thickness of the oxide
layer and the size of the overlap. The thickness of the layer will depend on how long and
at what pressure the first metal layer is exposed to O2 : we oxidize for 10 minutes at 3.4
229
Torr 90/10 Ar/O2 . Higher partial O2 pressures, or longer exposure times, will increase the
oxide layer thickness, decreasing the critical current and thus EJ . Since the tunneling rate
will depend exponentially on the barrier thickness, this dependence will likely be nonlinear.
Thus, we haven’t done much optimization on this front, opting to stick with a recipe we
know works.
Using the oxidation parameters described above, and a 100 nm×100 nm junction area,
we fairly reliably fabricate junctions with EJ /h ∼ 8 − 10 GHz. The critical current IC ,
and thus EJ , is proportional to the junction area, so adjustments in the desired EJ can
straightforwardly be made by modifying the bridge dimensions. The main quantity of interest
in calculating the Junction area for a given geometry/set of angle θ1 and θ2 is the height of
the “soft” resist layer. For MMA/EL9 spin-coated at 4000 RPM, the nominal thickness is
330 nm, though we have taken no real attempt to characterize this. As seen in Fig. C.1(e), the
second metallization layer must be deposited on both the area above the first metallization
and directly on the substrate. To ensure galvanic contact between metal deposited on these
areas, it helps to make the second deposition layer appreciably thicker than the first: we
generally deposit 20 nm of aluminum during the first evaporation and 60 nm during the
second.
High-frequency SAW device fabrication
Surface acoustic wave devices with resonant frequencies in the 3 − 5 GHz range can be some-
what difficult to fabricate, as the wavelength, and thus the finger spacing, scales inversely
with frequency. Most common piezoelectric substrates have a SAW propagation velocity
vs ≈ 3000 m/s, so the wavelength of a 5 GHz SAW is λ = f /vs ≈ 600 nm. Since there are
two IDT fingers per wavelength (one attached to each electrode), to fabricate a SAW device
230
Higher dose
Lower dose
Figure C.2: SAW device fabrication: The principle wavelength of SAWs that an IDT can
launch is defined by fabrication, λ = 2(a + b) where a is the width of the IDT fingers and b
is the spacing between fingers. When defining high-frequency SAW IDTs or Bragg mirrors,
proximity dosing from neighboring fingers becomes significant. To compensate for this, it
is helpful to lower the electron beam dose in for the IDTs away from the edges, and up the
dose for the IDTs at the edge and for the sparsely spaced fingers at the edge of an IDT.
with equal electrode (a) and substrate (b) length we need to fabricate strips of 150 nm metal
spaced by 150 nm (a = b = 150 nm) that are many 10’s of microns long.
Clearly, in order to make these devices we have to use EBL. However, a bi-layer recipe is
unavailable to us: a 100 − 200 nm wide strip of bi-layer MMA/PMMA that is many microns
long will almost invariably collapse, since proximity dosing will cause an undercut of order
the width the strip. Thus, in order to make high-frequency SAW devices, we are forced to
use a single layer process.
Below is an outline of a process for reproducibly making SAW devices with λ ≥ 600 nm
using a positive pattern and metallization/liftoff procedure:
1. Clean off the substrate using the procedure outlined in section C. It is important to
be rigorous about the cleaning for these devices, as Bragg mirrors and IDTs tend to
be large structures, and one piece of debris in them can ruin an entire device.
231
2. Spincoat only PMMA following the recipe outlined in section C.
3. If using LiNbO3 or ST-X quartz: deposit a ∼30 nm aluminum discharging layer on top
of the PMMA. The exact thickness doesn’t seem to matter all that much: on quartz
I’ve found that 10 nm is too thin, and I’ve accidentally deposited 60 nm and still was
able to create functional devices. If fabricating on GaAs, this step in not necessary.
4. Following the instruction in section C, insert the substrate into the SEM. Take extra
care focusing the beam, since the minimum feature sizes are small.
5. When designing the NPGS run-file, it is important to take proximity dosing into ac-
count. Electrons that hits the resist/discharging layer can scatter (forward scattering)
and electrons that collide with the substrate can scatter back up to the resist layer
(back scattering.) Either effect causes a finite dose away from the nominal writing
area. Since at high frequencies SAW fingers are close to each other, proximity dosing
plays a major roll. It is useful to raise the dose of leads near the edge of the sample,
and in areas where the leads from only one electrode are (orange areas in Fig. C.2.)
6. In the run file, it is beneficial to make the nominal transducer width in the run-file a
smaller than the desired width, and vary the dose until the desired width is produced.
For example, if the desired λSAW = 600 nm, and the desired metallization ratio is
a = b = 150 nm (see Fig. C.2), a good place to start is a run-file with a = 100 nm
and b = 200 nm. Figure C.3 shows several examples of what typically happens when
varying the dose.
7. Good starting doses: For GaAs, 270 µC/cm2 for inner areas (blue in Fig. C.2) and
320 µC/cm2 for outer areas (orange in Fig. C.2. ) For ST-X quartz, 290 µC/cm2 for
232
Figure C.3: Examples of SAW IDT dosing effects: Left: underdosed SAW IDT. Gen-
erally, when the IDTs are underdosed, lift-off is difficult. Center: optimized dose. Lift-off is
easy, and a ≈ b. Right: overdosed IDT. Clearly in this case, a b, however even though
the space between the IDT fingers is very small, lift-off generally goes well. Eventually, at
too high a dose, lift off will get spotty right before the IDTs simply short together.
inner areas and 340 µC/cm2 for outer areas. These are suggestions that have worked
for me in the past, not hard numbers. Dose will change based on geometry, resist age,
magnification, etc...
8. Once writing is done, remove from EBL. If an aluminum discharging layer was used,
remove the aluminum in a KOH developer such as AZ 300 MIF, rinse in DI and blow
dry. Develop in 3:1 IPA:MIBK for 50 seconds, IPA for 15 seconds, and blow dry.
Inspect under optical microscope.
9. Metallize. I have found that 30 nm of aluminum can reliably be lifted off, but thicker
layers may be possible.
10. After evaporation, place the sample in a beaker of acetone and lift-off over night.
11. Try to remove as much visible metal as possible by pulling it off with tweezers/blowing
the substrate with a stream of acetone (either using a syringe or just from the spray
bottle.)
233
12. Fish out as much metal as possible from the acetone, or pour out most of the acetone
and replenish it several times to get rid as much metal as possible in the beaker. It is
important to always keep the surface of the substrate immersed under acetone until
you’re done with lift-off.
13. Sonicate for 5-10 minutes.
14. Sonication will break up the remaining metal into small particulates. Dilute metal
particulates by again pouring out most of the acetone and replenishing it several times,
again being careful not to expose the substrate to air.
15. While the substrate is still immersed in acetone, rub the area(s) of the substrate that
have SAW devices using a ruby stick3 . Do the rubbing motion along the IDT fingers.
Rub for ≈ 30 seconds.
16. Remove the substrate from the beaker while spraying it with acetone. Before the
acetone on the surface evaporates, rinse the substrate with IPA, and then blow dry.
Hard baked substrate spacers
Below is an outline of the process for making the substrate spacers used in the quantum
acoustics flip-chip experiment. This recipe works well for 4 layers of photoresist, which
renders bump bonds ≈ 4 µm thick.
1. Spincoat S1813 G2 resist, 5000 RPM for 50 seconds
2. Bake substrate at 110 ◦ C for 1 minute
3 If you decide you want to use remover PG to do lift-of, make sure you switch to acetone by this step,
since PG dissolve the tips of Ruby sticks.
234
3. Repeat steps (1) and (2) 3 more times for a total of 4 layers.
4. Expose for 90 seconds
5. Develop in 352 developer for ∼30-60 seconds. Note that it might be a good idea to use
a larger beaker to dilute the photoresist, since 4 layers is a lot of resist. Rinse in DI
for 15 seconds, and then blow dry. Remember that 352 is a NaOH based developer,
and that NaOH also etches aluminum, so it is better to increase the exposure time if
the development is slow.
6. Inspect under the optical microscope. I have found that the last exposed resist to come
off is on the pads of the device, so make sure resist is stripped clean from there.
7. Optional: Oxygen plasma etch at 300 W to remove any residual resist. Plasma etching
will etch the spacers as well, at a rate of ≈ 200 nm/s, so be careful to not over-etch.
Four layers of S1813 G2 nominally gives you 4.6 µm anyways, so if you’re aiming for
as close to 4 µm as possible, etch for 180 seconds.
8. Put on hotplate, which is still at 110 ◦ C. Set the hotplate to 250 ◦ C, and let bake for
1.5 hours.
Note that, when spin-coating on small substrates, resist can pool to significant depth
on the edge of the substrate, which will cause unreliable spacing. My strategy to avoid
this resist is twofold: fabricate multiple SAW devices on a larger substrate, and also leave
some sacrificial area near the edge of the substrate designed to be diced off after spacer
fabrication. For example, I generally fabricate 6 devices on one substrate: this would require
a 14 × 21 mm substrate, however I actually use a 16 × 22mm substrate and cleave the edges
off when dicing the larger substrate to individual device size.
235
BIBLIOGRAPHY
236
BIBLIOGRAPHY
[1] P. W. Anderson. More is different. Science, 177(4047):393–396, 1972.
[2] A. D. O’Connell, M. Hofheinz, M. Ansmann, Radoslaw C. Bialczak, M. Lenander,
Erik Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, John M. Martinis,
and A. N. Cleland. Quantum ground state and single-phonon control of a mechanical
resonator. Nature, 464:697 EP –, 03 2010.
[3] J. M. Pirkkalainen, S. U. Cho, Jian Li, G. S. Paraoanu, P. J. Hakonen, and M. A.
Sillanpää. Hybrid circuit cavity quantum electrodynamics with a micromechanical
resonator. Nature, 494:211 EP –, 02 2013.
[4] Patricio Arrangoiz-Arriola, E. Alex Wollack, Zhaoyou Wang, Marek Pechal, Wentao
Jiang, Timothy P. McKenna, Jeremy D. Witmer, Raphaël Van Laer, and Amir H.
Safavi-Naeini. Resolving the energy levels of a nanomechanical oscillator. Nature,
571(7766):537–540, 2019.
[5] Yiwen Chu, Prashanta Kharel, William H. Renninger, Luke D. Burkhart, Luigi Frun-
zio, Peter T. Rakich, and Robert J. Schoelkopf. Quantum acoustics with supercon-
ducting qubits. Science, 2017.
[6] Mikael Kervinen, Ilkka Rissanen, and Mika Sillanpää. Interfacing planar supercon-
ducting qubits with high overtone bulk acoustic phonons. Phys. Rev. B, 97:205443,
May 2018.
[7] Martin V. Gustafsson, Thomas Aref, Anton Frisk Kockum, Maria K. Ekström, Göran
Johansson, and Per Delsing. Propagating phonons coupled to an artificial atom. Sci-
ence, 346(6206):207–211, 2014.
[8] Riccardo Manenti, Anton F. Kockum, Andrew Patterson, Tanja Behrle, Joseph Ra-
hamim, Giovanna Tancredi, Franco Nori, and Peter J. Leek. Circuit quantum acous-
todynamics with surface acoustic waves. Nature Communications, 8(1):975, 2017.
[9] Atsushi Noguchi, Rekishu Yamazaki, Yutaka Tabuchi, and Yasunobu Nakamura.
Qubit-assisted transduction for a detection of surface acoustic waves near the quantum
limit. Phys. Rev. Lett., 119:180505, Nov 2017.
[10] K. J. Satzinger, Y. P. Zhong, H. S. Chang, G. A. Peairs, A. Bienfait, Ming-Han Chou,
A. Y. Cleland, C. R. Conner, É. Dumur, J. Grebel, I. Gutierrez, B. H. November, R. G.
Povey, S. J. Whiteley, D. D. Awschalom, D. I. Schuster, and A. N. Cleland. Quantum
control of surface acoustic-wave phonons. Nature, 563(7733):661–665, 2018.
237
[11] Bradley A. Moores, Lucas R. Sletten, Jeremie J. Viennot, and K. W. Lehnert. Cavity
quantum acoustic device in the multimode strong coupling regime. Phys. Rev. Lett.,
120:227701, May 2018.
[12] L. R. Sletten, B. A. Moores, J. J. Viennot, and K. W. Lehnert. Resolving phonon
Fock states in a multimode cavity with a double-slit qubit. Phys. Rev. X, 9:021056,
Jun 2019.
[13] Yiwen Chu, Prashanta Kharel, Taekwan Yoon, Luigi Frunzio, Peter T. Rakich, and
Robert J. Schoelkopf. Creation and control of multi-phonon Fock states in a bulk
acoustic-wave resonator. Nature, 563(7733):666–670, 2018.
[14] X. Ma, J. J. Viennot, S. Kotler, J. D. Teufel, and K. W. Lehnert. Non-classical energy
squeezing of a macroscopic mechanical oscillator. Nature Physics, 2021.
[15] A. Bienfait, Y. P. Zhong, H.-S. Chang, M.-H. Chou, C. R. Conner, É. Dumur, J. Grebel,
G. A. Peairs, R. G. Povey, K. J. Satzinger, and A. N. Cleland. Quantum erasure using
entangled surface acoustic phonons. Phys. Rev. X, 10:021055, Jun 2020.
[16] Anton Frisk Kockum, Per Delsing, and Göran Johansson. Designing frequency-
dependent relaxation rates and lamb shifts for a giant artificial atom. Phys. Rev.
A, 90:013837, Jul 2014.
[17] Gustav Andersson, Baladitya Suri, Lingzhen Guo, Thomas Aref, and Per Delsing.
Non-exponential decay of a giant artificial atom. Nature Physics, 15(11):1123–1127,
2019.
[18] Connor T. Hann, Chang-Ling Zou, Yaxing Zhang, Yiwen Chu, Robert J. Schoelkopf,
S. M. Girvin, and Liang Jiang. Hardware-efficient quantum random access memory
with hybrid quantum acoustic systems. Phys. Rev. Lett., 123:250501, Dec 2019.
[19] Mohammad Mirhosseini, Alp Sipahigil, Mahmoud Kalaee, and Oskar Painter. Super-
conducting qubit to optical photon transduction. Nature, 588(7839):599–603, 2020.
[20] H. K. Onnes. Leiden. Comm., 120b, 122b, 124c, 1911.
[21] M. Tinkham. Introduction to Superconductivity. Dover Books on Physics Series. Dover
Publications, 2004.
[22] V. L. Ginzberg and L. D. Landau. On the theory of superconductivity. Soviet Physics
JETP, 20:1064, 1950.
[23] J. Bardeen, L. N. Cooper, and J. R. Schrieffer. Theory of superconductivity. Phys.
Rev., 108:1175–1204, Dec 1957.
238
[24] L. P. Gor’kov. Microscopic derivation of the Ginzberg-Landau equations in the theory
of superconductivity. Soviet Physics JETP, 36(9):1918–1923, June 1959.
[25] B.D. Josephson. Possible new effects in superconductive tunnelling. Physics Letters,
1(7):251 – 253, 1962.
[26] P. W. Anderson and J. M. Rowell. Probable observation of the Josephson supercon-
ducting tunneling effect. Phys. Rev. Lett., 10:230–232, Mar 1963.
[27] A. J. Leggett. Macroscopic Quantum Systems and the Quantum Theory of Measure-
ment. Progress of Theoretical Physics Supplement, 69:80–100, 03 1980.
[28] John Clarke, Andrew N. Cleland, Michel H. Devoret, Daniel Esteve, and John M.
Martinis. Quantum mechanics of a macroscopic variable: The phase difference of a
Josephson junction. Science, 239(4843):992–997, 1988.
[29] Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai. Coherent control of macroscopic quantum
states in a single-Cooper-pair box. Nature, 398(6730):786–788, 1999.
[30] N. K. Langford. Circuit QED - lecture notes. arXiv:1310.1897v1 [quant-ph], 10 2013.
[31] M. H. Devoret, B Huard, R. J. Schoelkopf, and L. F. Cugliandolo. Quantum Machines:
Measurement and Control of Engineered Quantum Systems: Lecture Notes of the Les
Houches Summer School, volume 96. Oxford University Press, 2014.
[32] Uri Vool and Michel Devoret. Introduction to quantum electromagnetic circuits. In-
ternational Journal of Circuit Theory and Applications, 45(7):897–934, June 2017.
[33] R. J. Schoelkopf and S. M. Girvin. Wiring up quantum systems. Nature, 451:664 EP
–, 02 2008.
[34] Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck, D. I. Schuster, J. Majer, Alexan-
dre Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. Charge-insensitive qubit
design derived from the cooper pair box. Phys. Rev. A, 76:042319, Oct 2007.
[35] R. J. Schoelkopf, P. Wahlgren, A. A. Kozhevnikov, P. Delsing, and D. E. Prober. The
radio-frequency single-electron transistor (RF-SET): A fast and ultrasensitive elec-
trometer. Science, 280(5367):1238–1242, 1998.
[36] D.J. Griffiths and P.D.J. Griffiths. Introduction to Quantum Mechanics. Pearson
international edition. Pearson Prentice Hall, 2005.
[37] D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais,
L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf.
239
Resolving photon number states in a superconducting circuit. Nature, 445:515 EP –,
02 2007.
[38] A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. Schreier, B. R. Johnson, J. M.
Chow, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf.
Generating single microwave photons in a circuit. Nature, 449:328 EP –, 09 2007.
[39] J. Majer, J. M. Chow, J. M. Gambetta, Jens Koch, B. R. Johnson, J. A. Schreier,
L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M.
Girvin, and R. J. Schoelkopf. Coupling superconducting qubits via a cavity bus. Nature,
449:443 EP –, 09 2007.
[40] Hanhee Paik, D. I. Schuster, Lev S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears,
B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret,
and R. J. Schoelkopf. Observation of high coherence in Josephson junction qubits mea-
sured in a three-dimensional circuit QED architecture. Phys. Rev. Lett., 107:240501,
Dec 2011.
[41] Chad Rigetti, Jay M. Gambetta, Stefano Poletto, B. L. T. Plourde, Jerry M. Chow,
A. D. Córcoles, John A. Smolin, Seth T. Merkel, J. R. Rozen, George A. Keefe, Mary B.
Rothwell, Mark B. Ketchen, and M. Steffen. Superconducting qubit in a waveguide
cavity with a coherence time approaching 0.1 ms. Phys. Rev. B, 86:100506, Sep 2012.
[42] K. Serniak, S. Diamond, M. Hays, V. Fatemi, S. Shankar, L. Frunzio, R.J. Schoelkopf,
and M.H. Devoret. Direct dispersive monitoring of charge parity in offset-charge-
sensitive transmons. Phys. Rev. Applied, 12:014052, Jul 2019.
[43] M. S. Blok, V. V. Ramasesh, T. Schuster, K. O’Brien, J. M. Kreikebaum, D. Dahlen,
A. Morvan, B. Yoshida, N. Y. Yao, and I. Siddiqi. Quantum information scrambling
on a superconducting qutrit processor. Phys. Rev. X, 11:021010, Apr 2021.
[44] Alexander P. M. Place, Lila V. H. Rodgers, Pranav Mundada, Basil M. Smitham,
Mattias Fitzpatrick, Zhaoqi Leng, Anjali Premkumar, Jacob Bryon, Andrei Vraji-
toarea, Sara Sussman, Guangming Cheng, Trisha Madhavan, Harshvardhan K. Babla,
Xuan Hoang Le, Youqi Gang, Berthold Jäck, András Gyenis, Nan Yao, Robert J.
Cava, Nathalie P. de Leon, and Andrew A. Houck. New material platform for super-
conducting transmon qubits with coherence times exceeding 0.3 milliseconds. Nature
Communications, 12(1):1779, 2021.
[45] J. M. Chow, J. M. Gambetta, L. Tornberg, Jens Koch, Lev S. Bishop, A. A. Houck,
B. R. Johnson, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf. Randomized bench-
marking and process tomography for gate errors in a solid-state qubit. Phys. Rev.
Lett., 102:090502, Mar 2009.
240
[46] L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I. Schus-
ter, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf. Demon-
stration of two-qubit algorithms with a superconducting quantum processor. Nature,
460(7252):240–244, 2009.
[47] Jerry M. Chow, Jay M. Gambetta, Easwar Magesan, David W. Abraham, Andrew W.
Cross, B R Johnson, Nicholas A. Masluk, Colm A. Ryan, John A. Smolin, Srikanth J.
Srinivasan, and M Steffen. Implementing a strand of a scalable fault-tolerant quantum
computing fabric. Nature Communications, 5:4015 EP –, 06 2014.
[48] J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank,
J. Y. Mutus, B. Campbell, Yu Chen, Z. Chen, B. Chiaro, A. Dunsworth, I. C. Hoi,
C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N.
Cleland, and John M. Martinis. State preservation by repetitive error detection in a
superconducting quantum circuit. Nature, 519:66 EP –, 03 2015.
[49] D. Ristè, S. Poletto, M. Z. Huang, A. Bruno, V. Vesterinen, O. P. Saira, and L. Di-
Carlo. Detecting bit-flip errors in a logical qubit using stabilizer measurements. Nature
Communications, 6:6983 EP –, 04 2015.
[50] Nissim Ofek, Andrei Petrenko, Reinier Heeres, Philip Reinhold, Zaki Leghtas, Brian
Vlastakis, Yehan Liu, Luigi Frunzio, S. M. Girvin, L. Jiang, Mazyar Mirrahimi, M. H.
Devoret, and R. J. Schoelkopf. Extending the lifetime of a quantum bit with error
correction in superconducting circuits. Nature, 536:441 EP –, 07 2016.
[51] P. J. J. O’Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends,
J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro,
A. Dunsworth, A. G. Fowler, E. Jeffrey, E. Lucero, A. Megrant, J. Y. Mutus, M. Nee-
ley, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, P. V.
Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis. Scalable quan-
tum simulation of molecular energies. Phys. Rev. X, 6:031007, Jul 2016.
[52] R. K. Naik, N. Leung, S. Chakram, Peter Groszkowski, Y. Lu, N. Earnest, D. C.
McKay, Jens Koch, and D. I. Schuster. Random access quantum information processors
using multimode circuit quantum electrodynamics. Nature Communications, 8(1):1904,
2017.
[53] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink,
Jerry M. Chow, and Jay M. Gambetta. Hardware-efficient variational quantum eigen-
solver for small molecules and quantum magnets. Nature, 549(7671):242–246, 2017.
[54] Frank Arute et al. Quantum supremacy using a programmable superconducting pro-
cessor. Nature, 574(7779):505–510, 2019.
241
[55] Mohammad Mirhosseini, Eunjong Kim, Xueyue Zhang, Alp Sipahigil, Paul B. Dieterle,
Andrew J. Keller, Ana Asenjo-Garcia, Darrick E. Chang, and Oskar Painter. Cavity
quantum electrodynamics with atom-like mirrors. Nature, 569(7758):692–697, 2019.
[56] Brian Vlastakis, Gerhard Kirchmair, Zaki Leghtas, Simon E. Nigg, Luigi Frunzio,
S. M. Girvin, Mazyar Mirrahimi, M. H. Devoret, and R. J. Schoelkopf. Deterministi-
cally encoding quantum information using 100-photon Schrödinger cat states. Science,
342(6158):607–610, 2013.
[57] K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi. Observing single quantum
trajectories of a superconducting quantum bit. Nature, 502:211 EP –, 10 2013.
[58] Z. K. Minev, S. O. Mundhada, S. Shankar, P. Reinhold, R. Gutiérrez-Jáuregui, R. J.
Schoelkopf, M. Mirrahimi, H. J. Carmichael, and M. H. Devoret. To catch and reverse
a quantum jump mid-flight. Nature, 2019.
[59] M. Naghiloo, J. J. Alonso, A. Romito, E. Lutz, and K. W. Murch. Information gain
and loss for a quantum Maxwell’s demon. Phys. Rev. Lett., 121:030604, Jul 2018.
[60] Alexandre Blais, Ren-Shou Huang, Andreas Wallraff, S. M. Girvin, and R. J.
Schoelkopf. Cavity quantum electrodynamics for superconducting electrical circuits:
An architecture for quantum computation. Phys. Rev. A, 69:062320, Jun 2004.
[61] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M.
Girvin, and R. J. Schoelkopf. Strong coupling of a single photon to a superconducting
qubit using circuit quantum electrodynamics. Nature, 431:162 EP –, 09 2004.
[62] D.P. DiVincenzo, Institut für Festkörperforschung, Institut für Festkörperforschung
(Jülich). Spring School, and Jülich Centre for Neutron Science. Quantum Informa-
tion Processing: Lecture Notes of the 44th IFF Spring School 2013. Schriften des
Forschungszentrums Jülich. Forschungszentrum, Zentralbibliothek, 2013.
[63] M. H. Devoret and R. J. Schoelkopf. Superconducting circuits for quantum information:
An outlook. Science, 339(6124):1169–1174, 2013.
[64] Alexandre Blais, Steven M. Girvin, and William D. Oliver. Quantum information
processing and quantum optics with circuit quantum electrodynamics. Nature Physics,
16(3):247–256, 2020.
[65] A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, Jens Koch, J. M. Gambetta,
D. I. Schuster, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. Con-
trolling the spontaneous emission of a superconducting transmon qubit. Phys. Rev.
Lett., 101:080502, Aug 2008.
242
[66] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver.
A quantum engineer’s guide to superconducting qubits. Applied Physics Reviews,
6(2):021318, 2019.
[67] Daniel Sank. Fast, Accurate State Meausrement in Superconducting Qubits. PhD thesis,
University of California Santa Barbara, 2014.
[68] Jay Gambetta, Alexandre Blais, D. I. Schuster, A. Wallraff, L. Frunzio, J. Majer, M. H.
Devoret, S. M. Girvin, and R. J. Schoelkopf. Qubit-photon interactions in a cavity:
Measurement-induced dephasing and number splitting. Phys. Rev. A, 74:042318, Oct
2006.
[69] David Schuster. Circuit Quantum Electrodynamics. PhD thesis, Yale, 2007.
[70] Vladimir B. Braginsky, Farid Ya Khalili, and Kip S. Thorne. Quantum Measurement.
Cambridge University Press, 1992.
[71] Daniel Slichter. Quantum Jumps and Measurement Backaction in a Superconducting
Qubit. PhD thesis, University of California Berkeley, 2011.
[72] J. A. Schreier, A. A. Houck, Jens Koch, D. I. Schuster, B. R. Johnson, J. M. Chow, J. M.
Gambetta, J. Majer, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf.
Suppressing charge noise decoherence in superconducting charge qubits. Phys. Rev. B,
77:180502, May 2008.
[73] Morten Kjaergaard, Mollie E. Schwartz, Jochen Braumüller, Philip Krantz, Joel I.-J.
Wang, Simon Gustavsson, and William D. Oliver. Superconducting qubits: Current
state of play. Annual Review of Condensed Matter Physics, 11(1):369–395, 2020.
[74] David M Pozar. Microwave engineering; 3rd ed. Wiley, Hoboken, NJ, 2005.
[75] D.F. Walls and G.J. Milburn. Quantum Optics. Springer Berlin Heidelberg, 2008.
[76] A. P. Sears, A. Petrenko, G. Catelani, L. Sun, Hanhee Paik, G. Kirchmair, L. Frunzio,
L. I. Glazman, S. M. Girvin, and R. J. Schoelkopf. Photon shot noise dephasing in the
strong-dispersive limit of circuit QED. Phys. Rev. B, 86:180504, Nov 2012.
[77] M. Göppl, A. Fragner, M. Baur, R. Bianchetti, S. Filipp, J. M. Fink, P. J. Leek,
G. Puebla, L. Steffen, and A. Wallraff. Coplanar waveguide resonators for circuit
quantum electrodynamics. Journal of Applied Physics, 104(11):113904, 2008.
[78] Simon E. Nigg, Hanhee Paik, Brian Vlastakis, Gerhard Kirchmair, S. Shankar, Luigi
Frunzio, M. H. Devoret, R. J. Schoelkopf, and S. M. Girvin. Black-box superconducting
circuit quantization. Phys. Rev. Lett., 108:240502, Jun 2012.
243
[79] C. Wang, C. Axline, Y. Y. Gao, T. Brecht, Y. Chu, L. Frunzio, M. H. Devoret, and
R. J. Schoelkopf. Surface participation and dielectric loss in superconducting qubits.
Applied Physics Letters, 107(16):162601, 2015.
[80] G. J. Dolan. Offset masks for lift-off photoprocessing. Applied Physics Letters,
31(5):337–339, 1977.
[81] J M Kreikebaum, K P O’Brien, A Morvan, and I Siddiqi. Improving wafer-scale
Josephson junction resistance variation in superconducting quantum coherent circuits.
Superconductor Science and Technology, 33(6):06LT02, may 2020.
[82] Jared B. Hertzberg, Eric J. Zhang, Sami Rosenblatt, Easwar Magesan, John A. Smolin,
Jeng-Bang Yau, Vivekananda P. Adiga, Martin Sandberg, Markus Brink, Jerry M.
Chow, and Jason S. Orcutt. Laser-annealing Josephson junctions for yielding scaled-
up superconducting quantum processors. arXiv preprint, 2020.
[83] Richard Phillips Feynman, Robert Benjamin Leighton, and Matthew Sands. The Feyn-
man lectures on physics; New millennium ed. Basic Books, New York, NY, 2010.
[84] J M Kreikebaum, A Dove, W Livingston, E Kim, and I Siddiqi. Optimization of
infrared and magnetic shielding of superconducting TiN and al coplanar microwave
resonators. Superconductor Science and Technology, 29(10):104002, aug 2016.
[85] R. Barends, J. Wenner, M. Lenander, Y. Chen, R. C. Bialczak, J. Kelly, E. Lucero,
P. O’Malley, M. Mariantoni, D. Sank, H. Wang, T. C. White, Y. Yin, J. Zhao, A. N.
Cleland, John M. Martinis, and J. J. A. Baselmans. Minimizing quasiparticle gener-
ation from stray infrared light in superconducting quantum circuits. Applied Physics
Letters, 99(11):113507, 2011.
[86] M. J. Persky. Review of black surfaces for space-borne infrared systems. Review of
Scientific Instruments, 70(5):2193–2217, 1999.
[87] Jen-Hao Yeh, Jay LeFebvre, Shavindra Premaratne, F. C. Wellstood, and B. S. Palmer.
Microwave attenuators for use with quantum devices below 100 mK. Journal of Applied
Physics, 121(22):224501, 2017.
[88] Fei Yan, Dan Campbell, Philip Krantz, Morten Kjaergaard, David Kim, Jonilyn L.
Yoder, David Hover, Adam Sears, Andrew J. Kerman, Terry P. Orlando, Simon Gus-
tavsson, and William D. Oliver. Distinguishing coherent and thermal photon noise in
a circuit quantum electrodynamical system. Phys. Rev. Lett., 120:260504, Jun 2018.
[89] Frank Pobell. Matter and Methods at Low Temperatures. Springer-Verlag Berlin Hei-
delberg, 3 edition, 2007.
244
[90] Z. Wang, S. Shankar, Z.K. Minev, P. Campagne-Ibarcq, A. Narla, and M.H. Devoret.
Cavity attenuators for superconducting qubits. Phys. Rev. Applied, 11:014031, Jan
2019.
[91] S. Krinner, S. Storz, P. Kurpiers, P. Magnard, J. Heinsoo, R. Keller, J. Lütolf, C. Eich-
ler, and A. Wallraff. Engineering cryogenic setups for 100-qubit scale superconducting
circuit systems. EPJ Quantum Technology, 6(1):2, May 2019.
[92] D F Santavicca and D E Prober. Impedance-matched low-pass stripline filters. Mea-
surement Science and Technology, 19(8):087001, jun 2008.
[93] A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf.
Introduction to quantum noise, measurement, and amplification. Rev. Mod. Phys.,
82:1155–1208, Apr 2010.
[94] C. D. Motchenbacher and J. A. Connelly. Low-Noise Electronic System Design. John
Wiley and Sons, Inc., 1993.
[95] Lukas Grünhaupt, Nataliya Maleeva, Sebastian T. Skacel, Martino Calvo, Florence
Levy-Bertrand, Alexey V. Ustinov, Hannes Rotzinger, Alessandro Monfardini, Gian-
luigi Catelani, and Ioan M. Pop. Loss mechanisms and quasiparticle dynamics in
superconducting microwave resonators made of thin-film granular aluminum. Phys.
Rev. Lett., 121:117001, Sep 2018.
[96] E. T. Mannila, P. Samuelsson, S. Simbierowicz, J. T. Peltonen, V. Vesterinen, L. Grön-
berg, J. Hassel, V. F. Maisi, and J. P. Pekola. A superconductor free of quasiparticles
for seconds, 2021.
[97] Lev S. Bishop, Eran Ginossar, and S. M. Girvin. Response of the strongly driven
Jaynes-Cummings oscillator. Phys. Rev. Lett., 105:100505, Sep 2010.
[98] Maxime Boissonneault, J. M. Gambetta, and Alexandre Blais. Improved supercon-
ducting qubit readout by qubit-induced nonlinearities. Phys. Rev. Lett., 105:100504,
Sep 2010.
[99] M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun, D. I. Schuster, L. Frunzio, and R. J.
Schoelkopf. High-fidelity readout in circuit quantum electrodynamics using the Jaynes-
Cummings nonlinearity. Phys. Rev. Lett., 105:173601, Oct 2010.
[100] D. I. Schuster, A. Wallraff, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. M. Girvin,
and R. J. Schoelkopf. ac Stark shift and dephasing of a superconducting qubit strongly
coupled to a cavity field. Phys. Rev. Lett., 94:123602, Mar 2005.
[101] Alexandre Blais, Arne L. Grimsmo, S. M. Girvin, and Andreas Wallraff. Circuit quan-
tum electrodynamics. Rev. Mod. Phys., 93:025005, May 2021.
245
[102] J.R. Johansson, P.D. Nation, and Franco Nori. Qutip 2: A python framework for the
dynamics of open quantum systems. Computer Physics Communications, 184(4):1234
– 1240, 2013.
[103] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin,
and R. J. Schoelkopf. Approaching unit visibility for control of a superconducting qubit
with dispersive readout. Phys. Rev. Lett., 95:060501, Aug 2005.
[104] G. Ithier, E. Collin, P. Joyez, P. J. Meeson, D. Vion, D. Esteve, F. Chiarello, A. Shnir-
man, Y. Makhlin, J. Schriefl, and G. Schön. Decoherence in a superconducting quan-
tum bit circuit. Phys. Rev. B, 72:134519, Oct 2005.
[105] Łukasz Cywiński, Roman M. Lutchyn, Cody P. Nave, and S. Das Sarma. How to
enhance dephasing time in superconducting qubits. Phys. Rev. B, 77:174509, May
2008.
[106] Jonas Bylander, Simon Gustavsson, Fei Yan, Fumiki Yoshihara, Khalil Harrabi, George
Fitch, David G. Cory, Yasunobu Nakamura, Jaw-Shen Tsai, and William D. Oliver.
Noise spectroscopy through dynamical decoupling with a superconducting flux qubit.
Nature Physics, 7(7):565–570, 2011.
[107] E. Fukushima. Experimental Pulse NMR: A Nuts and Bolts Approach. CRC Press,
2018.
[108] N. Samkharadze, A. Kumar, M. J. Manfra, L. N. Pfeiffer, K. W. West, and G. A.
Csáthy. Integrated electronic transport and thermometry at millikelvin temperatures
and in strong magnetic fields. Review of Scientific Instruments, 82(5):053902, 2011.
[109] D. I. Bradley, R. E. George, D. Gunnarsson, R. P. Haley, H. Heikkinen, Yu. A. Pashkin,
J. Penttilä, J. R. Prance, M. Prunnila, L. Roschier, and M. Sarsby. Nanoelectronic
primary thermometry below 4 mK. Nature Communications, 7(1):10455, 2016.
[110] L. A. De Lorenzo and K. C. Schwab. Ultra-high Q acoustic resonance in superfluid
4he. Journal of Low Temperature Physics, 186(3):233–240, Feb 2017.
[111] G. I. Harris, D. L. McAuslan, E. Sheridan, Y. Sachkou, C. Baker, and W. P. Bowen.
Laser cooling and control of excitations in superfluid helium. Nature Physics, 12:788
EP –, 04 2016.
[112] L. Childress, M. P. Schmidt, A. D. Kashkanova, C. D. Brown, G. I. Harris, A. Aiello,
F. Marquardt, and J. G. E. Harris. Cavity optomechanics in a levitated helium drop.
Phys. Rev. A, 96:063842, Dec 2017.
246
[113] A. D. Kashkanova, A. B. Shkarin, C. D. Brown, N. E. Flowers-Jacobs, L. Childress,
S. W. Hoch, L. Hohmann, K. Ott, J. Reichel, and J. G. E. Harris. Superfluid brillouin
optomechanics. Nature Physics, 13(1):74–79, 2017.
[114] P.M. Platzman and M.I. Dykman. Quantum computing with electrons floating on
liquid helium. Science, 284:1967, 1999.
[115] M. I. Dykman, P. M. Platzman, and P. Seddighrad. Qubits with electrons on liquid
helium. Phys. Rev. B, 67:155402, Apr 2003.
[116] S.A. Lyon. Spin-based quantum computing using electrons on liquid helium. Physical
Review A, 74:052338, 2006.
[117] D. I. Schuster, A. Fragner, M. I. Dykman, S. A. Lyon, and R. J. Schoelkopf. Proposal
for manipulating and detecting spin and orbital states of trapped electrons on helium
using cavity quantum electrodynamics. Phys. Rev. Lett., 105:040503, Jul 2010.
[118] Fei Yan, Simon Gustavsson, Archana Kamal, Jeffrey Birenbaum, Adam P Sears, David
Hover, Ted J. Gudmundsen, Danna Rosenberg, Gabriel Samach, S Weber, Jonilyn L.
Yoder, Terry P. Orlando, John Clarke, Andrew J. Kerman, and William D. Oliver. The
flux qubit revisited to enhance coherence and reproducibility. Nature Communications,
7:12964 EP –, 11 2016.
[119] Gerwin Koolstra, Ge Yang, and David I. Schuster. Coupling a single electron on
superfluid helium to a superconducting resonator. Nature Communications, 10(1):5323,
2019.
[120] J. R. Lane, D. Tan, N. R. Beysengulov, K. Nasyedkin, E. Brook, L. Zhang, T. Stefanski,
H. Byeon, K. W. Murch, and J. Pollanen. Integrating superfluids with superconducting
qubit systems. Phys. Rev. A, 101:012336, Jan 2020.
[121] J. F and J.F. Annett. Superconductivity, Superfluids and Condensates. Oxford Master
Series in Physics. OUP Oxford, 2004.
[122] Cecil T. Lane. Superfluid Physics. McGraw-Hill Book Company, 1962.
[123] L. Tisza. C. R. Acad. Sci., 207:1035, 1938.
[124] W. Hartung, J. Bierwagen, S. Bricker, C. Compton, T. Grimm, M. Johnson, D. Mei-
dlinger, D. Pendell, J. Popielarski, L. Saxton, and R. York. RF performand of a
superconducting S-band cavity filled with liquid helium. In Proceedings of LINAC
2006, 2006.
247
[125] N. R. Beysengulov, J. R. Lane, J. M. Kitzman, K. Nasyedkin, D. G. Rees, and J. Polla-
nen. Noise performance and thermalization of single electron transistors using quantum
fluids, 2020.
[126] R. K. Wangsness and F. Bloch. The dynamical theory of nuclear induction. Phys.
Rev., 89:728–739, Feb 1953.
[127] A. G. Redfield. On the theory of relaxation processes. IBM Journal of Research and
Development, 1(1):19–31, Jan 1957.
[128] Lara Faoro and Lev B. Ioffe. Microscopic origin of low-frequency flux noise in Josephson
circuits. Phys. Rev. Lett., 100:227005, Jun 2008.
[129] Oliver Dial, Douglas T McClure, Stefano Poletto, G A Keefe, Mary Beth Rothwell,
Jay M Gambetta, David W Abraham, Jerry M Chow, and Matthias Steffen. Bulk
and surface loss in superconducting transmon qubits. Superconductor Science and
Technology, 29(4):044001, 2016.
[130] J. Wenner, R. Barends, R. C. Bialczak, Yu Chen, J. Kelly, Erik Lucero, Matteo
Mariantoni, A. Megrant, P. J. J. O’Malley, D. Sank, A. Vainsencher, H. Wang, T. C.
White, Y. Yin, J. Zhao, A. N. Cleland, and John M. Martinis. Surface loss simu-
lations of superconducting coplanar waveguide resonators. Applied Physics Letters,
99(11):113513, 2011.
[131] A. Bruno, G. de Lange, S. Asaad, K. L. van der Enden, N. K. Langford, and L. DiCarlo.
Reducing intrinsic loss in superconducting resonators by surface treatment and deep
etching of silicon substrates. Applied Physics Letters, 106(18):182601, 2015.
[132] A. K. B. Engebretson and B. Golding. Surface acoustic wave propagation in nb:h at
low temperatures. In 1997 IEEE Ultrasonics Symposium Proceedings. An International
Symposium (Cat. No.97CH36118), volume 1, pages 499–502 vol.1, Oct 1997.
[133] Grigorij J. Grabovskij, Torben Peichl, Jürgen Lisenfeld, Georg Weiss, and Alexey V.
Ustinov. Strain tuning of individual atomic tunneling systems detected by a supercon-
ducting qubit. Science, 338(6104):232–234, 2012.
[134] D. J. Van Harlingen, T. L. Robertson, B. L. T. Plourde, P. A. Reichardt, T. A. Crane,
and John Clarke. Decoherence in Josephson-junction qubits due to critical-current
fluctuations. Phys. Rev. B, 70:064517, Aug 2004.
[135] P.V. Klimov, J. Kelly, Z. Chen, M. Neeley, A. Megrant, B. Burkett, R. Barends,
K. Arya, B. Chiaro, Yu Chen, A. Dunsworth, A. Fowler, B. Foxen, C. Gidney,
M. Giustina, R. Graff, T. Huang, E. Jeffrey, E. Lucero, J.Y. Mutus, O. Naaman,
C. Neill, C. Quintana, P. Roushan, Daniel Sank, A. Vainsencher, J. Wenner, T. C.
White, S. Boixo, R. Babbush, V.N. Smelyanskiy, H. Neven, , and John M. Martinis.
248
Fluctuations of energy-relaxation times in superconducting qubits. Phys. Rev. Lett.,
121:090502, 2018.
[136] Clemens Müller, Jürgen Lisenfeld, Alexander Shnirman, and Stefano Poletto. Interact-
ing two-level defects as sources of fluctuating high-frequency noise in superconducting
circuits. Phys. Rev. B, 92:035442, Jul 2015.
[137] Jonathan J. Burnett, Andreas Bengtsson, Marco Scigliuzzo, David Niepce, Marina Ku-
dra, Per Delsing, and Jonas Bylander. Decoherence benchmarking of superconducting
qubits. npj Quantum Information, 5(1):54, 2019.
[138] Steffen Schlör, Jürgen Lisenfeld, Clemens Müller, Alexander Bilmes, Andre Schnei-
der, David P. Pappas, Alexey V. Ustinov, and Martin Weides. Correlating decoher-
ence in transmon qubits: Low frequency noise by single fluctuators. Phys. Rev. Lett.,
123:190502, Nov 2019.
[139] John M. Martinis, M. Ansmann, and J. Aumentado. Energy decay in superconducting
Josephson-junction qubits from nonequilibrium quasiparticle excitations. Phys. Rev.
Lett., 103:097002, Aug 2009.
[140] D. Ristè, C. C. Bultink, M. J. Tiggelman, R. N. Schouten, K. W. Lehnert, and L. Di-
Carlo. Millisecond charge-parity fluctuations and induced decoherence in a supercon-
ducting transmon qubit. Nature Communications, 4(1):1913, 2013.
[141] J. Wenner, Yi Yin, Erik Lucero, R. Barends, Yu Chen, B. Chiaro, J. Kelly,
M. Lenander, Matteo Mariantoni, A. Megrant, C. Neill, P. J. J. O’Malley, D. Sank,
A. Vainsencher, H. Wang, T. C. White, A. N. Cleland, and John M. Martinis. Exci-
tation of superconducting qubits from hot nonequilibrium quasiparticles. Phys. Rev.
Lett., 110:150502, Apr 2013.
[142] K. Serniak, M. Hays, G. de Lange, S. Diamond, S. Shankar, L. D. Burkhart, L. Frunzio,
M. Houzet, and M. H. Devoret. Hot nonequilibrium quasiparticles in transmon qubits.
Phys. Rev. Lett., 121:157701, Oct 2018.
[143] Antti P. Vepsäläinen, Amir H. Karamlou, John L. Orrell, Akshunna S. Dogra, Ben
Loer, Francisca Vasconcelos, David K. Kim, Alexander J. Melville, Bethany M.
Niedzielski, Jonilyn L. Yoder, Simon Gustavsson, Joseph A. Formaggio, Brent A. Van-
Devender, and William D. Oliver. Impact of ionizing radiation on superconducting
qubit coherence. Nature, 584(7822):551–556, 2020.
[144] Laura Cardani, Francesco Valenti, Nicola Casali, Gianluigi Catelani, Thibault Charp-
entier, Massimiliano Clemenza, Ivan Colantoni, Angelo Cruciani, Luca Gironi, Lukas
Grünhaupt, Daria Gusenkova, Fabio Henriques, Marc Lagoin, Maria Martinez, Giorgio
Pettinari, Claudia Rusconi, Oliver Sander, Alexey V. Ustinov, Marc Weber, Wolfgang
249
Wernsdorfer, Marco Vignati, Stefano Pirro, and Ioan M. Pop. Reducing the impact of
radioactivity on quantum circuits in a deep-underground facility, 2020.
[145] U. Patel, Ivan V. Pechenezhskiy, B. L. T. Plourde, M. G. Vavilov, and R. McDermott.
Phonon-mediated quasiparticle poisoning of superconducting microwave resonators.
Phys. Rev. B, 96:220501, Dec 2017.
[146] A.A. Fragner. Circuit Quantum Electrodynamics with Electrons on Helium. PhD thesis,
Yale University, 2013.
[147] K. Nasyedkin, H. Byeon, L. Zhang, N.R. Beysengulov, J. Milem, S. Hemmerle,
R. Loloee, and J. Pollanen. Unconventional field effect transistor composed of electrons
floating on liquid helium. Journal of Physics: Condensed Matter, 30(465501), 2018.
[148] F. Souris, H. Christiani, and J. P. Davis. Tuning a 3d microwave cavity via superfluid
helium at millikelvin temperatures. Applied Physics Letters, 111(17):172601, 2017.
[149] James S. Brooks and Russell J. Donnelly. The calculated thermodynamic properties
of superfluid helium-4. Journal of Physical and Chemical Reference Data, 6(1):51–104,
1977.
[150] L. A. De Lorenzo and K. C. Schwab. Superfluid optomechanics: coupling of a superfluid
to a superconducting condensate. New Journal of Physics, 16(11):113020, nov 2014.
[151] C. T. Rogers and R. A. Buhrman. Nature of single-localized-electron states derived
from tunneling measurements. Phys. Rev. Lett., 55:859–862, Aug 1985.
[152] M. H. Devoret. Quantum Fluctuations in Electrical Circuits, chapter 10. Elsevier
Science, 1997.
[153] G. Catelani, R. J. Schoelkopf, M. H. Devoret, and L. I. Glazman. Relaxation and
frequency shifts induced by quasiparticles in superconducting qubits. Phys. Rev. B,
84:064517, Aug 2011.
[154] X. Y. Jin, A. Kamal, A. P. Sears, T. Gudmundsen, D. Hover, J. Miloshi, R. Slattery,
F. Yan, J. Yoder, T. P. Orlando, S. Gustavsson, and W. D. Oliver. Thermal and
residual excited-state population in a 3D transmon qubit. Phys. Rev. Lett., 114:240501,
Jun 2015.
[155] J. Pollanen, H. Choi, J.P. Davis, B.T. Rolfs, and W.P. Halperin. Low temperature
thermal resistance for a new design of silver sinter heat exchanger. J. Phys. Con. Ser.,
150:012037, 2009.
[156] A. A. Clerk and D. Wahyu Utami. Using a qubit to measure photon-number statistics
of a driven thermal oscillator. Phys. Rev. A, 75:042302, Apr 2007.
250
[157] G. Catelani, S. E. Nigg, S. M. Girvin, R. J. Schoelkopf, and L. I. Glazman. Decoherence
of superconducting qubits caused by quasiparticle tunneling. Phys. Rev. B, 86:184514,
Nov 2012.
[158] K. Geerlings, Z. Leghtas, I. M. Pop, S. Shankar, L. Frunzio, R. J. Schoelkopf, M. Mir-
rahimi, and M. H. Devoret. Demonstrating a driven reset protocol for a superconduct-
ing qubit. Phys. Rev. Lett., 110:120501, Mar 2013.
[159] Laura De Lorenzo. Optomechanics with Superfluid Helium-4. PhD thesis, California
Institute of Technology, 2016.
[160] D. P. Morgan. Surface Acoustic Wave Devices. Elsevier Ltd., 2nd edition, 2005.
[161] Per Delsing, Andrew N Cleland, Martin J A Schuetz, Johannes Knörzer, Géza Giedke,
J Ignacio Cirac, Kartik Srinivasan, Marcelo Wu, Krishna Coimbatore Balram, Christo-
pher Bäuerle, Tristan Meunier, Christopher J B Ford, Paulo V Santos, Edgar Cerda-
Méndez, Hailin Wang, Hubert J Krenner, Emeline D S Nysten, Matthias Weiß, Geoff R
Nash, Laura Thevenard, Catherine Gourdon, Pauline Rovillain, Max Marangolo, Jean-
Yves Duquesne, Gerhard Fischerauer, Werner Ruile, Alexander Reiner, Ben Paschke,
Dmytro Denysenko, Dirk Volkmer, Achim Wixforth, Henrik Bruus, Martin Wiklund,
Julien Reboud, Jonathan M Cooper, YongQing Fu, Manuel S Brugger, Florian Re-
hfeldt, and Christoph Westerhausen. The 2019 surface acoustic waves roadmap. Jour-
nal of Physics D: Applied Physics, 52(35):353001, jul 2019.
[162] Amir H. Safavi-Naeini and Oskar Painter. Optomechanical Crystal Devices, pages
195–231. Springer Berlin Heidelberg, Berlin, Heidelberg, 2014.
[163] Lord Rayleigh. On Waves Propagated along the Plane Surface of an Elastic Solid.
Proceedings of the London Mathematical Society, s1-17(1):4–11, 11 1885.
[164] L. D. Landau and E. M. Lifshitz. Theory of Elasticity, volume 7 of Course of Theoretical
Physics. Pergamon Press, 2nd edition, 1970.
[165] K. J. Satzinger. Quantum control of surface acoustic wave phonons. PhD thesis, UC
Santa Barbara, 2018.
[166] H. Byeon. Studying electrons on helium via surface acoustic wave techniques. PhD
thesis, Michigan State University, 2021.
[167] Göran Johansson and R. H. Hadfield. Superconducting Devices in Quantum Optics.
Springer, 2016.
[168] L.A. Tracy. Studies of two dimensional electron systems via surface acoustic waves and
nuclear magnetic resonance techniques. PhD thesis, California Institute of Technology,
Sep 2007.
251
[169] R. Manenti, M. J. Peterer, A. Nersisyan, E. B. Magnusson, A. Patterson, and P. J.
Leek. Surface acoustic wave resonators in the quantum regime. Phys. Rev. B,
93:041411, Jan 2016.
[170] J. R. Lane. Coupling of modes analysis for saw devices. https://github.com/
jrlane2/SAW_COM.
[171] K.S. Van Dyke. The piezo-electric resonator and its equivalent network. Proceedings
of the Institute of Radio Engineers, 16(6):742–764, 1928.
[172] A. Bienfait, K. J. Satzinger, Y. P. Zhong, H.-S. Chang, M.-H. Chou, C. R. Conner,
É. Dumur, J. Grebel, G. A. Peairs, R. G. Povey, and A. N. Cleland. Phonon-mediated
quantum state transfer and remote qubit entanglement. Science, 364(6438):368–371,
2019.
[173] Andreas Ask, Maria Ekström, Per Delsing, and Göran Johansson. Cavity-free vacuum-
rabi splitting in circuit quantum acoustodynamics. Phys. Rev. A, 99:013840, Jan 2019.
[174] M. K. Ekström, T. Aref, A. Ask, G. Andersson, B. Suri, H. Sanada, G. Johansson,
and P. Delsing. Towards phonon routing: controlling propagating acoustic waves in
the quantum regime. New Journal of Physics, 21(12):123013, dec 2019.
[175] Vitaly S. Shumeiko. Quantum acousto-optic transducer for superconducting qubits.
Phys. Rev. A, 93:023838, Feb 2016.
[176] R. S. Weis and T. K. Gaylord. Lithium niobate: Summary of physical properties and
crystal structure. Applied Physics A, 37(4):191–203, 1985.
[177] D.L.T. Bell and R.C.M. Li. Surface-acoustic-wave resonators. Proceedings of the IEEE,
64(5):711–721, 1976.
[178] K. J. Satzinger, C. R. Conner, A. Bienfait, H.-S. Chang, Ming-Han Chou, A. Y. Cle-
land, É. Dumur, J. Grebel, G. A. Peairs, R. G. Povey, S. J. Whiteley, Y. P. Zhong,
D. D. Awschalom, D. I. Schuster, and A. N. Cleland. Simple non-galvanic flip-chip inte-
gration method for hybrid quantum systems. Applied Physics Letters, 114(17):173501,
2019.
[179] Marco Scigliuzzo, Laure E Bruhat, Andreas Bengtsson, Jonathan J Burnett, Anita Fa-
davi Roudsari, and Per Delsing. Phononic loss in superconducting resonators on piezo-
electric substrates. New Journal of Physics, 22(5):053027, may 2020.
[180] Thomas Pastureaud, Marc Solal, Beatrice Biasse, Bernard Aspar, Jean-bernard Briot,
William Daniau, William Steichen, Raphael Lardat, Vincent Laude, Alain Laens, Jean-
michel Friedt, and Sylvain Ballandras. High-frequency surface acoustic waves excited
252
on thin-oriented linbo3 single-crystal layers transferred onto silicon. IEEE Transactions
on Ultrasonics, Ferroelectrics, and Frequency Control, 54(4):870–876, 2007.
[181] M. D. Reed, B. R. Johnson, A. A. Houck, L. DiCarlo, J. M. Chow, D. I. Schuster,
L. Frunzio, and R. J. Schoelkopf. Fast reset and suppressing spontaneous emission of
a superconducting qubit. Applied Physics Letters, 96(20):203110, 2010.
[182] K. W. Murch, U. Vool, D. Zhou, S. J. Weber, S. M. Girvin, and I. Siddiqi. Cavity-
assisted quantum bath engineering. Phys. Rev. Lett., 109:183602, Oct 2012.
[183] M. Naghiloo, M. Abbasi, Yogesh N. Joglekar, and K. W. Murch. Quantum state
tomography across the exceptional point in a single dissipative qubit. Nature Physics,
15(12):1232–1236, 2019.
[184] R. L. Willett, R. R. Ruel, M. A. Pallanan, K. W. West, and L.N. Pfeiffer. Enhanced
finite-wave-vector conductivity at multiple even-denominator filling factors in two-
dimensional electron systems. Phys. Rev. B, 47(12):7344, 1993.
[185] R. L. Willett, R. R. Ruel, K. W. West, and L. N. Pfeiffer. Experimental demonstration
of a Fermi surface at one-half filling of the lowest Landau level. Phys. Rev. Lett.,
71:3846–3849, Dec 1993.
[186] David Tong. Lectures on the quantum Hall effect. arXiv preprint: 1606.06687, Jun
2016.
[187] J. R. Lane, L. Zhang, M. A. Khasawneh, B. N. Zhou, E. A. Henriksen, and J. Pollanen.
Flip-chip gate-tunable acoustoelectric effect in graphene. Journal of Applied Physics,
124(19):194302, 2018.
[188] P. R. Wallace. The band theory of graphite. Phys. Rev., 71:622–634, May 1947.
[189] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos,
I. V. Grigorieva, and A. A. Firsov. Electric field effect in atomically thin carbon films.
Science, 306(5696):666–669, 2004.
[190] J. I. A. Li, Q. Shi, Y. Zeng, K. Watanabe, T. Taniguchi, J. Hone, and C. R. Dean. Pair-
ing states of composite fermions in double-layer graphene. Nature Physics, 15(9):898–
903, 2019.
[191] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grig-
orieva, S. V. Dubonos, and A. A. Firsov. Two-dimensional gas of massless Dirac
fermions in graphene. Nature, 438(7065):197–200, 2005.
253
[192] Yuanbo Zhang, Yan-Wen Tan, Horst L. Stormer, and Philip Kim. Experimental
observation of the quantum Hall effect and Berry’s phase in graphene. Nature,
438(7065):201–204, 2005.
[193] L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe,
L. M. Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, and C. R.
Dean. One-dimensional electrical contact to a two-dimensional material. Science,
342(6158):614–617, 2013.
[194] Xu Du, Ivan Skachko, Fabian Duerr, Adina Luican, and Eva Y. Andrei. Fractional
quantum Hall effect and insulating phase of Dirac electrons in graphene. Nature,
462(7270):192–195, 2009.
[195] Kirill I. Bolotin, Fereshte Ghahari, Michael D. Shulman, Horst L. Stormer, and
Philip Kim. Observation of the fractional quantum Hall effect in graphene. Nature,
462(7270):196–199, 2009.
[196] J. I. A. Li, C. Tan, S. Chen, Y. Zeng, T. Taniguchi, K. Watanabe, J. Hone, and C. R.
Dean. Even-denominator fractional quantum Hall states in bilayer graphene. Science,
358(6363):648–652, 2017.
[197] B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young, M. Yankowitz, B. J. LeRoy, K. Watan-
abe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C. Ashoori. Massive
Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure. Science,
340(6139):1427–1430, 2013.
[198] Yuan Cao, Valla Fatemi, Shiang Fang, Kenji Watanabe, Takashi Taniguchi, Efthimios
Kaxiras, and Pablo Jarillo-Herrero. Unconventional superconductivity in magic-angle
graphene superlattices. Nature, 556(7699):43–50, 2018.
[199] C. W. J. Beenakker. Colloquium: Andreev reflection and Klein tunneling in graphene.
Rev. Mod. Phys., 80:1337–1354, Oct 2008.
[200] K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C.
Maan, G. S. Boebinger, P. Kim, and A. K. Geim. Room-temperature quantum Hall
effect in graphene. Science, 315(5817):1379–1379, 2007.
[201] A. R. Hutson and Donald L. White. Elastic wave propagation in piezoelectric semi-
conductors. Journal of Applied Physics, 33(1):40–47, 1962.
[202] A. Wixforth, J. Scriba, M. Wassermeier, J. P. Kotthaus, G. Weimann, and W. Schlapp.
Surface acoustic waves on GaAs/AlxGa1-xAs heterostructures. Phys. Rev. B,
40(11):7874, 1989.
254
[203] A. Wixforth, J. P. Kotthaus, and G. Weimann. Quantum oscillations in the surface-
acoustic-wave attenuation caused by a two-dimensional electron system. Phys. Rev.
Lett., 56:2104, 1986.
[204] A. Esslinger, A. Wixforth, R. W. Winkler, J. P. Kotthaus, H. Nickel, W. Schlapp, and
R. Lösch. Acoustoelectric study of localized states in the quantized Hall effect. Solid
State Communications, 84(10):939, 1992.
[205] I. V. Kukushkin, V. Umansky, K. von Klitzing, and J. H. Smet. Collective modes and
the periodicity of quantum Hall stripes. Phys. Rev. Lett., 106:206804, May 2011.
[206] J. Pollanen, J.P. Eisenstein, L.N. Pfeiffer, and K.W. West. Charge metastability and
hysteresis in the quantum Hall regime. Phys. Rev. B, 94(24):245440, 2016.
[207] B. Friess, Y. Peng, B. Rosenow, F. von Oppen, V. Umansky, K. von Klitzing,
and J. H. Smet. Negative permittivity in bubble and stripe phases. Nat Phys,
doi:10.1038/nphys4213, 2017.
[208] M. A. Paalanen, R. L. Willett, P. B. Littlewood, R. R. Ruel, K. W. West, L. N.
Pfeiffer, and D. J. Bishop. rf conductivity of a two-dimensional electron system at
small Landau-level filling factors. Phys. Rev. B, 45:11342–11345, May 1992.
[209] I. L. Drichko, I. Y. Smirnov, A. V. Suslov, L. N. Pfeiffer, K. W. West, and Y. M.
Galperin. Crossover between localized states and pinned Wigner crystal in high-
mobility n-GaAs/AlGaAs heterostructures near filling factor ν = 1. Phys. Rev. B,
92:205313, 2015.
[210] A. V. Suslov, I. L. Drichko, I. Y. Smirnov, A. F. Ioffe, L. N. Pfeiffer, K. W. West,
and Y. M. Galperin. Study of Wigner crystal in n-GaAs/AlGaAs by surface acoustic
waves. The Journal of the Acoustical Society of America, 138:1938, 2015.
[211] L.A. Tracy, J.P. Eisenstein, M.P. Lilly, L.N. Pfeiffer, and K.W. West. Surface acoustic
wave propagation and inhomogeneities in low-density two-dimensional electron systems
near the metal–insulator transition. Solid State Communications, 137(3):150 – 153,
2006.
[212] A. L. Efros and Yu. M. Galperin. Quantization of the acoustoelectric current in a two-
dimensional electron system in a strong magnetic field. Phys. Rev. Lett., 64:1959–1962,
Apr 1990.
[213] A. Esslinger, R.W. Winkler, C. Rocke, A. Wixforth, J.P. Kotthaus, H. Nickel,
W. Schlapp, and R. Lösch. Ultrasonic approach to the integer and fractional quantum
Hall effect. Surface Science, 305(1):83 – 86, 1994.
255
[214] J. M. Shilton, D. R. Mace, V. I. Talyanskii, M. Y. Simmons, M. Pepper, A. C. Churchill,
and D. A. Ritchie. Experimental study of the acoustoelectric effects in GaAs-AlGaAs
heterostructures. J. Phys. Condens. Matter, 7:7675, 1995.
[215] M. Rotter, A. Wixforth, W. Ruile, D. Bernklau, and H. Riechert. Giant acoustoelectric
effect in GaAs/LiNbO3 hybrids. App. Phys. Let., 73(15):2128, 1998.
[216] J. M. Shilton, D. R. Mace, V. I. Talyanskii, M. Pepper, M. Y. Simmons, A. C. Churchill,
and D. A. Ritchie. Effect of spatial dispersion on acoustoelectric current in a high-
mobility two-dimensional electron gas. Phys. Rev. B, 51:14770–14773, May 1995.
[217] J. Ebbecke, N. E. Fletcher, T. J. B. M. Janssen, H. E. Beere, D. A. Ritchie, and
M. Pepper. Acoustoelectric current transport through a double quantum dot. Phys.
Rev. B, 72(121311(R)), 2005.
[218] M. Kataoka, M. R. Astley, A. L. Thorn, D. K. L. Oi, C. H. W. Barnes, C. J. B. Ford,
D. Anderson, G. A. C. Jones, I. Farrer, D. A. Ritchie, and M. Pepper. Coherent time
evolution of a single-electron wave function. Phys. Rev. Lett., 102:156801, 2009.
[219] J. M. Shilton, D. R. Mace, V. I. Talyanskii, Yu Galperin, M. Y. Simmons, M. Pepper,
and D. A. Ritchie. On the acoustoelectric current in a one-dimensional channel. J.
Phys. Condens. Matter, 8:L337, 1996.
[220] J. M. Shilton, V. I. Talyanskii, M. Pepper, D. A. Ritchie, J. E. F. Frost, C. J. B.
Ford, C. G. Smith, and G. A. C. Jones. High-frequency single electron transport in
a quasi-one-dimensional GaAs channel induced by surface acoustic waves. J. Phys.
Condens. Matter, 8:L531, 1996.
[221] M. R. Astley, M. Kataoka, R. J. Schneble, C. J. B. Ford, C. H. W. Barnes, D. Anderson,
G. A. C. Jones, H. E. Beere, D. A. Ritchie, and M. Pepper. Examination of the surface
acoustic wave reflections by observing acoustoelectric current generation under pulse
modulation. Applied Physics Letters, 89:132102, 2006.
[222] H. Byeon, K. Nasyedkin, J. R. Lane, N. R. Beysengulov, L. Zhang, R. Loloee, and
J. Pollanen. Piezoacoustics for flying electron qubits on helium. arXiv:2008.02330,
Aug 2020.
[223] Peter Thalmeier, Balázs Dóra, and Klaus Ziegler. Surface acoustic wave propagation
in graphene. Phys. Rev. B, 81:041409, Jan 2010.
[224] S. H. Zhang and W. Xu. Absorption of surface acoustic waves by graphene. AIP
Advances, 1(2):022146, 2011.
256
[225] Jürgen Schiefele, Jorge Pedrós, Fernando Sols, Fernando Calle, and Francisco Guinea.
Coupling light into graphene plasmons through surface acoustic waves. Phys. Rev.
Lett., 111:237405, Dec 2013.
[226] Z. Insepov, E. Emelin, O. Kononenko, D. V. Roshchupkin, K. B. Tnyshtykbayev, and
K. A. Baigarin. Surface acoustic wave amplification by direct current-voltage supplied
to graphene film. Applied Physics Letters, 106(2):023505, 2015.
[227] V. Miseikis, J. E. Cunningham, K. Saeed, R. O’Rorke, and A. G. Davies. Acoustically
induced current flow in graphene. App. Phys. Let., 100:133105, 2012.
[228] V. Miseikis. The Interaction of Graphene with High-Frquency Acoustic and Electro-
magnetic Waves. PhD thesis, University of Leeds, 2012.
[229] L. Bandhu, L. M. Lawton, and G. R. Nash. Macroscopic acoustoelectric charge trans-
port in graphene. Applied Physics Letters, 103(13):133101, 2013.
[230] P. V. Santos, T. Schumann, M. H. Oliveira Jr., J. M. J. Lopes, and H. Riechert.
Acousto-electric transport in epitaxial monolayer graphene on sic. Applied Physics
Letters, 102(22):221907, 2013.
[231] Lokeshwar Bandhu and Geoffrey R. Nash. Controlling the properties of surface acoustic
waves using graphene. Nano Research, 9(3):685–691, 2016.
[232] T. Poole and G. R. Nash. Acoustoelectric current in graphene nanoribbons. Scientific
Reports, 7(1):1767, 2017.
[233] A. Schenstrom, Y.J. Qian, M.-F. Xu, H.-P. Baum, M. Levy, and Bimal K. Sarma.
Oscillations in the acousto-electric proximity coupling to a 2d electron gas. Solid State
Communications, 65(7):739 – 742, 1988.
[234] G. S. Kino and T. M. Reeder. A normal mode theory for the rayleigh wave amplifier.
IEEE Transactions on Electronic Devices, 18(10):909–920, 1971.
[235] A. Wixforth, J. Scriba, M. Wassermeier, and J. P. Kotthaus. Interaction of surface
acoustic waves with a two-dimensional electron system in a LiNbO3-GaAs/AlGaAs
sandwich structure. J. Appl. Phys., 64(4):2213, 1988.
[236] Bennaceur Keyan, Benjamin A. Schmidt, Samuel Gaucher, Dominique Laroche,
Michael P. Lilly, John L. Reno, Ken W. West, Loren N. Pfeiffer, and Guillaume Ger-
vais. Mechanical flip-chip for ultra-high electron mobility devices. Scientific Reports,
5:13494 EP –, 09 2015.
257
[237] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H. Smet, K. von Klitzing, and
A. Yacoby. Observation of electron-hole puddles in graphene using a scanning single-
electron transistor. Nat Phys, 4(2):144–148, 02 2008.
[238] M.-Y. Li, C.-C. Tang, D.C. Ling, C.C. Chi, and J.-C. Chen. Charged impurity-induced
scatterings in chemical vapor deposited graphene. J. Appl. Phys., 114:233703, 2013.
[239] S. Okuda, T. Ikuta, Y. Kanai, T. Ono, S. Ogawa, D. Fujisawa, M. Shimatani, K. Inoue,
K. Maehashi, and K. Matsumoto. Acoustic carrier transportation induced by surface
acoustic waves in graphene in solution. Applied Physics Express, 9:045104, 2016.
[240] C. Tang, Y. Chen, D.C. Ling, C.C. Chi, and J. Chen. Ultra-low acoustoelectric atten-
uation in graphene. J. Appl. Phys., 121:124505, 2017.
[241] J Wenner, M Neeley, Radoslaw C Bialczak, M Lenander, Erik Lucero, A D O’Connell,
D Sank, H Wang, M Weides, A N Cleland, and John M Martinis. Wirebond crosstalk
and cavity modes in large chip mounts for superconducting qubits. Superconductor
Science and Technology, 24(6):065001, 2011.
[242] Md. Shafayat Hossain, Meng K. Ma, M. A. Mueed, L. N. Pfeiffer, K. W. West, K. W.
Baldwin, and M. Shayegan. Direct observation of composite fermions and their fully-
spin-polarized fermi sea near ν = 5/2. Phys. Rev. Lett., 120:256601, Jun 2018.
[243] Gustav Andersson, Maria K. Ekström, and Per Delsing. Electromagnetically induced
acoustic transparency with a superconducting circuit. Phys. Rev. Lett., 124:240402,
Jun 2020.
[244] B. G. Christensen, C. D. Wilen, A. Opremcak, J. Nelson, F. Schlenker, C. H. Zimonick,
L. Faoro, L. B. Ioffe, Y. J. Rosen, J. L. DuBois, B. L. T. Plourde, and R. McDermott.
Anomalous charge noise in superconducting qubits. Phys. Rev. B, 100:140503, Oct
2019.
[245] Fei Yan, Simon Gustavsson, Jonas Bylander, Xiaoyue Jin, Fumiki Yoshihara, David G.
Cory, Yasunobu Nakamura, Terry P. Orlando, and William D. Oliver. Rotating-frame
relaxation as a noise spectrum analyser of a superconducting qubit undergoing driven
evolution. Nature Communications, 4:2337 EP –, 08 2013.
258