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Photonic and Phononic Band Gap Engineering for Circuit Quantum Electrodynamics and Quantum Transduction
Citation
Banker, Jash Haren
(2022)
Photonic and Phononic Band Gap Engineering for Circuit Quantum Electrodynamics and Quantum Transduction.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/jrf3-gx27.
Abstract
The ability to pattern materials at the wavelength and sub-wavelength scale has led to the concept of photonic crystals and metamaterials - artificially engineered structures that exhibit electromagnetic properties not found in conventional materials. Such engineered structures offer the ability to slow down and even inhibit the propagation of electromagnetic waves giving rise to a photonic band gap and a sharply varying photonic density of states.
Quantum emitters in the presence of an electromagnetic reservoir with varying density of states can undergo a rich set of dynamical behavior. In particular, the reservoir can be tailored to have a memory of past interactions with emitters, in contrast to memory-less Markovian dynamics of typical open systems. In part 1 of this thesis, we investigate the non-Markovian dynamics of a superconducting qubit strongly coupled to a superconducting metamaterial waveguide engineered to have both a sharp spectral variation in its transmission properties and a slowing of light by a factor of 650. Tuning the qubit into the spectral vicinity of the passband of this slow-light waveguide reservoir, we observe a 400-fold change in the emission rate of the qubit, along with oscillatory energy relaxation of the qubit resulting from the beating of bound and radiative dressed qubit-photon states. Further, upon addition of a reflective boundary to one end of the waveguide, we observe revivals in the qubit population on a timescale 30 times longer than the inverse of the qubit’s emission rate, corresponding to the round-trip travel time of an emitted photon. With this superconducting circuit platform, future studies of multi-qubit interactions via highly structured reservoirs and the generation of multi-photon highly entangled states are possible.
While microwave frequency superconducting circuits are near ideal testbeds for quantum electrodynamics experiments of the type discussed in part 1, microwave photons are not well suited for transmission of quantum information over long distances due to the presence of a large thermal background at room temperature. Optical photons are ideal for quantum communication applications due to their low propagation loss at room temperature. Coherent transduction of single photons from the microwave to the optical domain has the potential to play a key role in quantum networking and distributed quantum computing. In part 2 of this thesis, we extend the notion of band gap engineering to the optical and acoustic domain and present the design of a piezo-optomechanical quantum transducer where transduction is mediated by a strongly hybridized acoustic mode of a lithium niobate piezoacoustic cavity attached to a silicon optomechanical crystal patterned on a silicon-on-insulator substrate. We estimate an intrinsic transduction efficiency of 29% with <0.5 added noise quanta when our transducer is resonantly coupled to a superconducting transmon qubit and operated in pulsed mode. Our design involves on-chip integration of a superconducting qubit with the piezo-optomechanical transducer. Absorption of stray photons from the optical pump used in the transduction process is known to cause excess decoherence and noise in the superconducting circuit. The recovery time of the superconducting circuit after the optical pulse sets a limit on the transducer repetition rate. We fabricate niobium based superconducting circuits on a silicon substrate and test their response to illumination by a 1550 nm laser. We find a recovery time of ~ 10 μs, indicating that a repetition rate of 10 kHz should be possible. Combined with the expected efficiency and noise metrics of our design, we expect that a transducer in this parameter regime would be suitable to realize probabilistic schemes for remote entanglement of superconducting quantum processors. We conclude by discussing some of the challenges associated with fabricating niobium superconducting qubits and lithium niobate piezoacoustic devices on silicon-on-insulator substrates and provide initial steps towards realizing our transducer design in the lab.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Superconducting circuits; piezo-opto-mechanics; non-Markovian physics; quantum transduction
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Painter, Oskar J.
Thesis Committee:
Vahala, Kerry J. (chair)
Faraon, Andrei
Minnich, Austin J.
Painter, Oskar J.
Defense Date:
20 October 2021
Record Number:
CaltechTHESIS:01222022-151042739
Persistent URL:
DOI:
10.7907/jrf3-gx27
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Description
DOI
Journal article adapted for chapters 2, 3, and 4.
ORCID:
Author
ORCID
Banker, Jash Haren
0000-0002-2130-0825
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
14484
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Deposited By:
Jash Banker
Deposited On:
26 Jan 2022 01:46
Last Modified:
08 Nov 2023 18:50
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Photonic and Phononic Band Gap Engineering for Circuit
Quantum Electrodynamics and Quantum Transduction

Thesis by

Jash Haren Banker

In Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2022
Defended October 20, 2021

Jash Haren Banker
ORCID: 0000-0002-2130-0825

iii

ACKNOWLEDGEMENTS

It has been a long and bumpy road through grad school since I first arrived in
Pasadena in 2015. I would not have made it this far had it not been for the help I
received from others along the way.
To my advisor Oskar Painter, thank you for accepting me into your group when I
was a third year Ph.D student looking for a new home. Your approach to research,
the level of rigour you demand of all your students and your deep scientific insights
have all been instrumental in shaping my scientific abilities.
None of the work in this thesis would have been possible without the support and
help of the members of the Painter group to whom I owe many thanks. To Alp
Sipahigil, who was a second advisor to me: Alp, you were a rare combination of
amazing scientific aptitude coupled with a genuine desire to mentor young graduate
students. You provided guidance and encouragement when I needed it most for
which I will always be grateful. To Vinicius Ferreira, I am glad we worked together
on the metamaterial project. The first half of this thesis would not have been possible
without you. Thank you for making my transition to the Painter group smooth and
for providing a friendly ear when I needed to vent my frustrations. I hope I have
been able to do the same for you. To Andrew Keller, for teaching me all my
nanofabrication skills that have served me so well over the years. Your patience and
calm demeanor made you a great teacher. To Mohammad Mirhosseini, for guiding
me through my first project and publication. I am glad I got to work with you and
learn from you. To my office mates—the Transducers, the second half of this thesis
would not have been possible without you. It’s a shame COVID-19 robbed us of
the opportunity to work together for a longer period of time. To Srujan Meesala
and David Lake, I learned a lot about performing experiments in a careful and
organized way from both of you and am grateful for your advice on everything from
experiments to presentations and job applications. To Steven Wood, as someone
who worked extensively in the cleanroom you understood the challenges involved
in fab heavy projects and provided great advice (though your advice about fruit is
suspect). To Piero Chiappina, I enjoyed our time working together on fab and on
the transducer manuscript. I see a bright grad school career (and potential surfing
career) ahead of you. To Michael Fang, you were instrumental in helping me make
my decision to join the Painter group. It has been a pleasure working alongside
you to keep our cleanroom running. To Hengjiang (Jared) Ren and Jie (Roger)

iv
Luo, I enjoyed all our chats ranging from science to PubG. To Eunjong Kim and
Xueyue (Sherry) Zhang, I was always inspired by your work ethic, your scientific
skill, and your excellent presentations that taught me a lot. To Mahmoud Kalaee,
Greg MacCabe, Clai Owens, and Mo Chen, we did not work together much but you
were all always super helpful and ready to share your experience. To Sameer Sonar,
Utku Hatipoglu, Andreas Butler, and Gihwan Kim, as young grad students you guys
injected energy into the group. Your enthusiasm was infectious and I have enjoyed
watching you learn and succeed and accomplish so much so fast. To Barry Baker the Fabulous One, thank you for keeping the labs running like clockwork. You may
now make graduation jokes. To the QPG staff, Kamal Yadav, Harry Liu, and Matt
McCoy, having you looking after our labs has made doing research so much easier.
Thank you for doing such a great job.
To Guy DeRose, Bert Mendoza, Matt Hunt, Nathan Lee, Alex Wertheim, Tiffany
Kimoto, Jennifer Palmer, and all the other KNI staff present and past, thank you for
keeping KNI running so smoothly. It has been a blessing to have access to such an
advanced and well maintained nanofabrication facility.
To Christy Jenstad and Jennifer Blankenship, thank you for being the ‘work moms’
for all of us in the department. You provided a safe space for us whether we needed
to scream or sob. The bowl of candy always helped.
To all the friends I made along the way, thank you for the memories and experiences
you have given me. You have enriched my life far beyond any science I learned in
the lab. A special thank you to the Yangji crew. To Michelle Cua, for providing
me with unwavering support and ice wine. To Pai Buabthong, for being one of the
calmest people I know. Pai, hanging out with you was as relaxing as meditation.
To Joeson Wong for everything from help on problem sets to driving me around in
your Prius to my first In-n-Out burger (double-double animal style is undoubtedly
the way to go). To Peishi Cheng for our discussions on politics and for reminding
me that it’s ok to care about things outside of the lab. To Samuel Loke for gumbo,
spaghetti bolognese, summer barbeques and game nights. To Claire Taĳung Kuo
for the pineapple cakes. To Kai Matsuka, I wish I had gotten to know you earlier. I
blame Michelle. To Angeliki Laskari for being a kindred soul who shared my taste
for 80s music and my obsession with the Boss. On days when I wanted to hide on
the backstreets, the music we shared lifted my spirits and made me dance - even if I
was just dancing in the dark.

To Jeannette, you may be right, I may be slightly crazy, but without you I would
have gone completely insane. Thank you for always being there for me and dealing
with my moods and my ups and downs. I look forward to our new beginning in
Munich.
Last but not least, to my parents who have been with me on this journey not just for
6 years, but for the last 28 years. No amount of words can express my gratitude for
all you have done, so I just have two: thank you.

ABSTRACT

The ability to pattern materials at the wavelength and sub-wavelength scale has
led to the concept of photonic crystals and metamaterials - artificially engineered
structures that exhibit electromagnetic properties not found in conventional materials. Such engineered structures offer the ability to slow down and even inhibit
the propagation of electromagnetic waves giving rise to a photonic band gap and a
sharply varying photonic density of states.
Quantum emitters in the presence of an electromagnetic reservoir with varying
density of states can undergo a rich set of dynamical behavior. In particular, the
reservoir can be tailored to have a memory of past interactions with emitters, in
contrast to memory-less Markovian dynamics of typical open systems. In part 1 of
this thesis, we investigate the non-Markovian dynamics of a superconducting qubit
strongly coupled to a superconducting metamaterial waveguide engineered to have
both a sharp spectral variation in its transmission properties and a slowing of light
by a factor of 650. Tuning the qubit into the spectral vicinity of the passband of this
slow-light waveguide reservoir, we observe a 400-fold change in the emission rate
of the qubit, along with oscillatory energy relaxation of the qubit resulting from the
beating of bound and radiative dressed qubit-photon states. Further, upon addition
of a reflective boundary to one end of the waveguide, we observe revivals in the
qubit population on a timescale 30 times longer than the inverse of the qubit’s emission rate, corresponding to the round-trip travel time of an emitted photon. With
this superconducting circuit platform, future studies of multi-qubit interactions via
highly structured reservoirs and the generation of multi-photon highly entangled
states are possible.
While microwave frequency superconducting circuits are near ideal testbeds for
quantum electrodynamics experiments of the type discussed in part 1, microwave
photons are not well suited for transmission of quantum information over long distances due to the presence of a large thermal background at room temperature.
Optical photons are ideal for quantum communication applications due to their low
propagation loss at room temperature. Coherent transduction of single photons
from the microwave to the optical domain has the potential to play a key role in
quantum networking and distributed quantum computing. In part 2 of this thesis,
we extend the notion of band gap engineering to the optical and acoustic domain
and present the design of a piezo-optomechanical quantum transducer where trans-

vii
duction is mediated by a strongly hybridized acoustic mode of a lithium niobate
piezoacoustic cavity attached to a silicon optomechanical crystal patterned on a
silicon-on-insulator substrate. We estimate an intrinsic transduction efficiency of
29% with <0.5 added noise quanta when our transducer is resonantly coupled to a
superconducting transmon qubit and operated in pulsed mode. Our design involves
on-chip integration of a superconducting qubit with the piezo-optomechanical transducer. Absorption of stray photons from the optical pump used in the transduction
process is known to cause excess decoherence and noise in the superconducting circuit. The recovery time of the superconducting circuit after the optical pulse sets a
limit on the transducer repetition rate. We fabricate niobium based superconducting
circuits on a silicon substrate and test their response to illumination by a 1550 nm
laser. We find a recovery time of ∼ 10 𝜇s, indicating that a repetition rate of 10
kHz should be possible. Combined with the expected efficiency and noise metrics
of our design, we expect that a transducer in this parameter regime would be suitable to realize probabilistic schemes for remote entanglement of superconducting
quantum processors. We conclude by discussing some of the challenges associated
with fabricating niobium superconducting qubits and lithium niobate piezoacoustic
devices on silicon-on-insulator substrates and provide initial steps towards realizing
our transducer design in the lab.

viii

PUBLISHED CONTENT AND CONTRIBUTIONS
[1] V. S. Ferreira∗ , J. Banker∗ , A. Sipahigil, M. H. Matheny, A. J. Keller, E. Kim,
M. Mirhosseini, and O. Painter. “Collapse and revival of an artificial atom
coupled to a structured photonic reservoir”. In: Phys. Rev. X 11.4 (2021),
p. 041043. doi: 10.1103/PhysRevX.11.041043.
J.B. participated in the conception of the project, design and fabrication of
the device, and performance of the measurements.
[2] E. Kim, X. Zhang, V. S. Ferreira, J. Banker, J. K. Iverson, A. Sipahigil,
M. Bello, A. González-Tudela, M. Mirhosseini, and O. Painter. “Quantum
electrodynamics in a topological waveguide”. In: Physical Review X 11 (1
2021), p. 011015. doi: 10.1103/PhysRevX.11.011015.
J.B. contributed to the design of the metamaterial waveguide.
[3] M. Mirhosseini, A. Sipahigil, M. Kalaee, and O. Painter. “Superconducting
qubit to optical photon transduction”. In: Nature 588.7839 (2020), pp. 599–
603. doi: 10.1038/s41586-020-3038-6.
Extended Data Figure 7 reprinted in this thesis, by permission from Springer
Customer Service Centre GmbH, ©2020.
[4] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D.
Oliver. “A quantum engineer’s guide to superconducting qubits”. In: Applied
Physics Reviews 6.2 (2019), p. 021318. doi: 10.1063/1.5089550.
Figure 1 reprinted and referenced in this thesis, with the permission of AIP
publishing.
[5] A. J. Keller, P. B. Dieterle, M. Fang, B. Berger, J. M. Fink, and O. Painter.
“Al transmon qubits on silicon-on-insulator for quantum device integration”.
In: Applied Physics Letters 111.4 (2017), p. 042603. doi: 10 . 1063 / 1 .
4994661.
Figure 1a. reprinted and referenced in this thesis, with the permission of
AIP publishing.
[6] J. Chan, A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, and O. Painter.
“Optimized optomechanical crystal cavity with acoustic radiation shield”.
In: Applied Physics Letters 101.8 (2012), p. 081115. doi: 10 . 1063 / 1 .
4747726.
Figure 2 adapted and referenced in this thesis, with the permission of AIP
publishing.

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . viii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Part 1: Superconducting Metamaterials for Circuit Quantum Electrodynamics in the Non-Markovian Regime . . . . . . . . . . . . . . . . . 5
Chapter I: Background: Superconducting Qubits . . . . . . . . . . . . . . . . 6
1.1 Superconducting Qubit Basics . . . . . . . . . . . . . . . . . . . . . 6
1.2 Qubit Frequency Tuning . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Qubit State Preparation . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Qubit Readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Qubit Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter II: Collapse and Revival of an Artificial Atom Coupled to a Structured
Photonic Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Slow-Light Metamaterial Waveguide . . . . . . . . . . . . . . . . . 17
2.3 Non-Markovian Radiative Dynamics . . . . . . . . . . . . . . . . . 22
2.4 Time-Delayed Feedback . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Chapter III: Utilization of Metamaterial Waveguide for 2D Cluster State Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Chapter IV: Appendix: Details of Device Design, Fabrication, Measurement
Setup, and Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 Fabrication and Measurement Setup . . . . . . . . . . . . . . . . . . 33
4.2 Capacitively Coupled Resonator Array Waveguide Fundamentals . . 36
4.3 Physical Implementation of Finite Resonator Array . . . . . . . . . . 45
4.4 Disorder Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 Modeling of Qubit Q1 Coupled to the Metamaterial Waveguide . . . 50
4.6 Modeling of Qubit Coupled to Dispersion-less Waveguide in Front
of Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Part 2: Quantum Transduction . . . . . . . . . . . . . . . . . . . . . . . . 64
Chapter V: Background: Cavity Optomechanics and 1D Optomechanical
Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1 Cavity Optomechanics Hamiltonian . . . . . . . . . . . . . . . . . . 66
5.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3 1D Optomechanical Crystals . . . . . . . . . . . . . . . . . . . . . . 70

Chapter VI: Design of a Wavelength-Scale Piezo-Optomechanical Quantum
Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2 Piezo Cavity Design . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3 Optomechanics Design . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.4 Full Device Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.5 Efficiency and Added Noise . . . . . . . . . . . . . . . . . . . . . . 84
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Chapter VII: Niobium based Transmon Qubits on Silicon Substrates for Quantum Transduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.2 Qubit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.3 Device Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.5 Qubit Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.6 Qubit Response to Optical Illumination . . . . . . . . . . . . . . . . 95
7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Chapter VIII: Towards Fabricating Niobium based Qubits on Silicon-onInsulator Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.1 Fabrication Challenges . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.2 Development of a Sputtering Process for Niobium Thin Films . . . . 103
8.3 Development of an Etching Process for Niobium Thin Films . . . . . 105
8.4 Protection of Nb Surface from VHF Attack . . . . . . . . . . . . . . 106
8.5 Quality of Sputtered and Etched Niobium Thin Films . . . . . . . . 107
8.6 Etch of VHF Release Holes . . . . . . . . . . . . . . . . . . . . . . 108
8.7 Proposed Fabrication Process for Niobium Transmon Qubits on SOI . 109
Chapter IX: Appendix: Development of an Etch Process for Lithium Niobate
on Silicon-on-Insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.1 Dry Etch Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.2 Wet Etch Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.3 Proposed Hybrid Etch Process . . . . . . . . . . . . . . . . . . . . . 118
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

LIST OF ILLUSTRATIONS

Number
Page
1.1 Circuit model and energy diagram of a quantum harmonic and anharmonic oscillator. Figure reproduced from [15] . . . . . . . . . . . 8
1.2 Example layout of a superconducting qubit chip . . . . . . . . . . . . 10
1.3 Capacitively coupled LC oscillator model of qubit coupled to readout
resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Example CW spectroscopy data . . . . . . . . . . . . . . . . . . . . 12
1.5 Standard pulsed measurements of superconducting qubits . . . . . . 14
2.1 Microwave coupled resonator array slow-light waveguide . . . . . . . 19
2.2 Artificial atom coupled to a structured photonic reservoir . . . . . . . 21
2.3 Non-Markovian radiative dynamics in a structured photonic reservoir 24
2.4 Time-delayed feedback from a slow-light reservoir with a reflective
boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Schematic of the measurement chain inside the dilution refrigerator . 34
4.2 Comparison to tight-binding model and bandwidth-delay trade-off
for capacitively coupled resonator array . . . . . . . . . . . . . . . . 40
4.3 CAD layout of boundary resonators and transmission spectrum of a
resonator array with impedance matching . . . . . . . . . . . . . . . 45
4.4 Disorder analysis of capacitively coupled resonator array . . . . . . . 49
4.5 Master equation numerical simulations of the qubit-slowlight waveguide system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6 Circuit model simulations of the qubit-slowlight waveguide system . 59
4.7 Markovian to Non-Markovian crossover . . . . . . . . . . . . . . . . 62
5.1 Canonical cavity optomechanical system consisting of a Fabry-Perot
optical cavity with a movable mirror . . . . . . . . . . . . . . . . . . 66
5.2 Schematic of a driven optical cavity coupled to a mechanical mode . 68
5.3 1D optomechanical crystal. Figure reproduced from [152] . . . . . . 71
6.1 Schematic of piezo-optomechanical transducer . . . . . . . . . . . . 75
6.2 Design of phononic shield . . . . . . . . . . . . . . . . . . . . . . . 76
6.3 Design of piezoacoustic cavity . . . . . . . . . . . . . . . . . . . . . 78
6.4 Design of the optomechanical cavity . . . . . . . . . . . . . . . . . . 80
6.5 Optomechanical cavity mode structure . . . . . . . . . . . . . . . . . 81

xii
6.6
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
8.1
8.2
8.3
8.4
8.5
8.6
9.1
9.2
9.3
9.4
9.5

Full piezo-optomechnaical transducer design . . . . . . . . . . . . . 83
Pulse sequence and laser induced quasiparticle recovery for a quantum transducer device. Figure reproduced from [10] . . . . . . . . . 89
Optical image of Niobium transmon qubit on silicon . . . . . . . . . 90
Experimental setup for optical tests on Nb transmon qubits . . . . . . 94
𝑇1 and 𝑇2∗ time constants of a niobium qubit on silicon . . . . . . . . 95
Nb readout resonator spectroscopy under optical illumination . . . . 96
Spectroscopy of Nb qubit under optical illumination . . . . . . . . . 96
Recovery of niobium qubit after laser illumination . . . . . . . . . . 98
Dependence of qubit population and decoherence on peak optical power 99
Dependence of qubit population on repetition rate and optical pulse
duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Ramsey measurement interrupted by a laser pulse . . . . . . . . . . . 100
Fabrication process for Al transmon qubits on SOI . . . . . . . . . . 103
Tuning the stress of a sputtered Nb film . . . . . . . . . . . . . . . . 104
He FIB image of etched Nb on Si . . . . . . . . . . . . . . . . . . . 107
Protection of Nb surface using ALD alumina . . . . . . . . . . . . . 108
Optical image and measured quality factor of Nb lumped element
resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Proposed fabrication process for Nb transmon qubits on SOI . . . . . 110
LN piezo box etched using an Ar+ ion based ICP-RIE dry etch . . . . 116
Dry etched LN piezo-acoustic cavity showing Si damage . . . . . . . 117
Wet etched LN ‘boxes’ . . . . . . . . . . . . . . . . . . . . . . . . . 119
LN piezo box etched using a combination of wet and dry etches . . . 120
Proposed hybrid etch process for LN . . . . . . . . . . . . . . . . . . 121

xiii

LIST OF TABLES
Number
Page
7.1 Qubit design parameters . . . . . . . . . . . . . . . . . . . . . . . . 90
8.1 Nb sputtering parameters . . . . . . . . . . . . . . . . . . . . . . . . 103
8.2 Nb etching parameters . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.3 Etch rates of various materials in the Nb etch . . . . . . . . . . . . . 106
8.4 ALD alumina etching parameters . . . . . . . . . . . . . . . . . . . 106
8.5 VHF release holes etching parameters . . . . . . . . . . . . . . . . . 109
9.1 LN dry etch parameters . . . . . . . . . . . . . . . . . . . . . . . . 115

INTRODUCTION
The propagation of electromagnetic waves through a macroscopic medium is governed by Maxwell’s equations which in the absence of free charges or currents can
be written as:
∇ · 𝜖E = 0
∇×E=−

𝜇H
𝜕𝑡

∇ · 𝜇H = 0
∇ × H = 𝜖E
𝜕𝑡

(1)

where 𝜖 is the permittivity and 𝜇 is the permeability. If the wavelength of the
electromagnetic wave is much larger than the atoms of the media, the microscopic
details of the medium are averaged out and the medium is characterized by the two
macroscopic parameters 𝜖 and 𝜇. Periodically patterning the medium at the deep
sub-wavelength scale can alter the effective 𝜖 and 𝜇 of the material allowing us
to engineer ‘meta-materials’ which can have novel electromagnetic responses not
found in ‘conventional’ materials such as a negative refractive index, chirality and
the existence of photonic bandgaps.
A related concept is that of photonic crystals. In the theory of solid state physics,
crystals are periodic arrangements of atoms in a lattice which give rise to a periodic
potential experienced by the electrons. This periodic potential gives rise to an
electronic band structure which in certain cases can have a band gap (such as
in a semiconductor). In a similar manner, photonic crystals are periodic structures
patterned at the wavelength scale giving rise to a periodically modulated permittivity
𝜖 (r) = 𝜖 (r + R). By substituting this periodically modulated permittivity into
Maxwell’s equations, one can solve for a photonic band structure in analogy to the
electronic band structure [1]. Crucially, it is possible to engineer the band structure
to create a photonic band gap where photons of a certain frequency cannot propagate.
This same idea can be extended to the acoustic domain where periodic patterning at
the wavelength scale gives rise to a periodic modulation of the elastic modulus and
the mass density leading to an acoustic band structure which can host acoustic band
gaps.
In this thesis, we explore two applications of this idea of band structure engineering
which represent two distinct projects I pursued during my time in the Painter group.

As such, the thesis is divided into two fairly independent parts united by the common theme of using wavelength and sub-wavelength scale periodically patterned
structures to engineer photonic and phononic bandgaps.
In Part 1 of this thesis, we develop a superconducting, microwave frequency metamaterial platform for ‘non-Markovian’ circuit Quantum Electrodynamics (cQED)
experiments. A canonical example of a ‘Markovian’ process is the spontaneous
emission of an atom coupled to the fluctuating electromagnetic vacuum [2]. In this
process, information in the form of energy flows from the atom into the electromagnetic reservoir. However, by carefully engineering the electromagnetic reservoir, it
is possible to introduce ‘non-Markovian’ memory effects where there is information
back-flow from the reservoir into the atom [3–6]. By utilizing a capacitively coupled array of identical superconducting LC resonators with a deep sub-wavelength
lattice constant, we develop a one-dimensional (1D) finite bandwidth slow-light
waveguide that exhibits sharp band edges and a flat, nearly ripple-free passband
where the group velocity of microwave photons is reduced by a factor of ∼ 650.
This 1D slow-light waveguide forms a 1D electromagnetic reservoir to which we
couple a superconducting qubit that acts as an ‘artificial’ atom. By performing
spectroscopic field measurements as well as time dependent measurements of qubit
dynamics, we demonstrate that our qubit-metamaterial waveguide platform is deep
in the non-Markovian regime. Further, by utilizing the slow-light nature of our
waveguide, we demonstrate time-delayed feedback of microwave photons emitted
by the qubit. This time-delayed feedback can be used to generate two-dimensional
photonic cluster states which have been proposed as a universal resource for measurement based quantum computing [7]. Our measurements allow us to estimate
the attainable fidelity and scale of photonic cluster states that can be generated using
this metamaterial platform.
All of the experiments described in Part 1 use a superconducting circuit platform. The high degree of control offered by superconducting circuits to engineer
qubit-waveguide couplings, tune qubit frequencies in-situ and perform single qubit
operations at timescales much faster than the coherence time of the qubit make this a
powerful test bed for performing quantum optics experiments. More broadly, these
same features make superconducting qubits very promising for quantum computing.
Recent landmark demonstrations have established superconducting quantum circuits
as a leading platform for quantum computing and simulation [8]. However, the superconducting circuit platform has some drawbacks too. Superconducting circuits

encode quantum information in microwave-frequency photons ( 𝑓 ∼5GHz). To keep
these circuits in the quantum ground state, they must be cooled to millikelvin (mK)
temperatures (𝑘 𝑏 𝑇
ℎ 𝑓 ). Further, microwave photons suffer from propagation
losses as high as 1dB/m [9]. The large thermal background at room temperature
and high propagation loss of microwave photons makes transmitting quantum information between remote superconducting processors a challenging task. In contrast,
optical photons (f∼200THz) have a negligible thermal background at room temperature. Optical fibers exhibit propagation losses as low as 0.2dB/km [9]. As a
result, optical photons are naturally suited for low loss, long distance transmission
of quantum information. The complementary properties of these two systems have
spurred interest in transducers that can coherently convert quantum information
between microwave and optical frequencies. Such transducers would enable optically connected networks of remote superconducting quantum processors analogous
to classical networks underlying the internet and large-scale supercomputers with
optical interconnects.
In Part 2, we present the design of a wavelength scale piezo-optomechanical quantum
transducer device optimized for high conversion efficiency and low added noise. In
our scheme, transduction between the microwave and optical frequency photons is
mediated by an intermediate mechanical mode which couples to microwave photons
via the piezoelectric effect and to optical photons via a parametric optomechanical
interaction. It is crucial that the optical and mechanical mode involved in the
transduction process have high coherence. Here we utilize the idea of photonic
and phononic (optomechanical) crystals to engineer optical and acoustic bandgaps.
By introducing carefully engineered defects that interrupt the periodicity of our
photonic/phononic crystals, we can create highly localized optical and acoustic
modes that are well isolated from the environment due to the presence of bandgaps
and also exhibit strong optomechanical and piezoelectric couplings. Our design
is based on a lithium niobate on silicon-on-insulator platform. We theoretically
analyze the expected efficiency and noise metrics of our transducer design. We
also discuss limitations to the repetition rate of our transducer device arising from
optically generated quasiparticles in the superconducting qubit during operation of
the transducer. Previous work from our group using a similar piezo-optomechanical
transducer with aluminum superconducting qubits was limited to a repetition rate
of ∼ 100 Hz due to the relatively long quasiparticle lifetime in aluminum (∼ ms)
[10]. A superconductor with shorter quasiparticle lifetime is niobium [11, 12]. We
experimentally investigate the optical response of niobium based superconducting

qubits and find a qubit recovery time on the order of ∼ 10 𝜇s after optical illumination.
Our results suggest that switching from aluminum to niobium based superconducting
qubits could allow a 100-fold increase in the repetition rate over previous work
[10]. In the last two chapters of Part 2, we discuss some practical fabrication
related challenges associated with developing niobium qubits on silicon-on-insulator
and also lithium niobate piezo-mechanical devices on silicon-on-insulator—both of
which are necessary for realization of our quantum transducer design. We provide
initial steps towards addressing these fabrication challenges that may hopefully
benefit future generations of graduate students in bringing our transducer design to
life in the lab.

PART 1: SUPERCONDUCTING METAMATERIALS FOR
CIRCUIT QUANTUM ELECTRODYNAMICS IN THE
NON-MARKOVIAN REGIME

In Part I of this thesis, we are interested in studying non-Markovian dynamics of
an atom strongly coupled to a highly structured one-dimensional electromagnetic
reservoir. Instead of working with actual atoms, we will use a superconducting
circuit platform where superconducting qubits will behave as ‘artificial atoms’. The
basics of the superconducting qubit platform will be introduced in Chapter 1. Since
superconducting circuits are designed and fabricated on chip, they can be engineered to a very high degree offering precise control over the circuit parameters. In
particular, we can pattern these circuits at the deep sub-wavelength scale allowing
us to realize microwave frequency metamaterials that can give rise to novel photonic bandstructures. In Chapter 2, we will develop a superconducting metamaterial
platform that allows us to realize a one-dimensional finite bandwidth waveguide
where the group velocity of propagating microwave fields is drastically reduced.
This metamaterial waveguide will serve as our one dimensional electromagnetic
reservoir. We will then show experimental results demonstrating non-Markovian
dynamics of a superconducting qubit strongly coupled to this ‘slow-light metamaterial waveguide’. A potential application of our metamaterial platform for generating
two dimensional cluster states of microwave photons will be discussed in Chapter
3. Further details about the design, fabrication, and theoretical modelling of our
system will be presented in the Appendix at the end of Part 1 (Chapter 4).

Chapter 1

BACKGROUND: SUPERCONDUCTING QUBITS

This chapter will be a brief introduction to superconducting qubits. It will cover
the essential concepts needed to follow the rest of this thesis. For a more extensive
review, please see [13–15].
1.1

Superconducting Qubit Basics

Just like a classical bit encodes information in the form of a 0 or 1, a quantum bit
(qubit) encodes quantum information using two levels which we call the ground (|𝑔i
or |0i) and excited (|𝑒i or |1i) states. A superconducting qubit can be thought of
as a two level system formed using superconducting electrical circuits. Since these
circuits are designed and fabricated on a chip, they offer a large degree of control
over their circuit parameters. One of the simplest electrical circuits we can conceive
of is an LC oscillator (Fig. 1.1a). The Hamiltonian describing this circuit is
𝐻=

𝑄 2 𝜙2
2𝐶 2𝐿

(1.1)

where Q is the charge, 𝜙 is the flux, C is the capacitance, and L is the inductance.
Quantum mechanically, we can promote the charge and flux variables to quantum
operators, and this Hamiltonian can be shown to reduce to that of a quantum
harmonic oscillator (QHO),
𝐻ˆ = ~𝜔(𝑎 † 𝑎 + )

(1.2)

This Hamiltonian has a number of equally spaced energy levels which are 𝜔 = √ 1
𝐿𝐶
apart (Fig. 1.1b). In order to have an effective two level system to serve as our
qubit, we need to isolate two of the energy levels. This can be achieved by replacing
the linear inductor of the LC oscillator by a non-linear element like a Josephson
junction (Fig. 1.1c). A Josephson junction is a pair of superconductors separated
by a thin insulating layer. The voltage and current are determined by the Josephson
relations:
~ 𝑑𝜙
2𝑒 𝑑𝑡
𝐼 = 𝐼𝑐 sin(𝜙)

𝑉=

(1.3)

The energy stored in the Josephson junction can then be written as
𝐸𝐿 =
𝑉 𝐼𝑑𝑡
~𝐼𝑐
cos(𝜙)
2𝑒
= −𝐸 𝐽 cos(𝜙)
=−

(1.4)

We can now write the Hamiltonian of this non-linear LC oscillator as
𝑄ˆ 2
− 𝐸 𝐽 cos 𝜙ˆ
2𝐶
𝜙ˆ2 𝜙ˆ4
𝑄ˆ 2
− 𝐸𝐽 1 −
+ ...
2𝐶
2 24

𝐻=

(1.5)

As can be seen from the Taylor expansion in the second line of Eq.1.5, our nonlinear oscillator now has higher order terms in addition to the quadratic term—this
introduces anharmonicity into our system, and we get multiple energy levels which
are no longer equally spaced as can be seen in Fig. 1.1d. Now if we operate this
circuit at frequencies close to the |0i → |1i transition frequency then all the higher
transitions are detuned and can be ignored. Thus by replacing the linear inductor
with a non-linear Josephson junction, we have reduced our LC circuit to an effective
two-level system that acts as our quantum bit. For the bulk of this thesis, we will
ignore the higher energy levels and treat our superconducting qubit as having only
two states. However it is important to remember that these higher levels exist, and
we will on occasion make use of them (particularly in Chapter 2 when we discuss a
protocol for generating cluster states).
The particular type of superconducting qubit used in this thesis is the transmon qubit
𝑒2
[16]. The transmon qubit is characterized by a large 𝐸𝐸𝐶𝐽 ratio (∼ 100), where 𝐸𝐶 = 2𝐶
is the charging energy and 𝐸 𝐽 is the Josephson energy defined in Eq. 1.4. The large
𝐸𝐽
𝐸𝐶 ratio makes the transmon less susceptible to charge noise. For a transmon qubit,
the |𝑔i → |𝑒i transition frequency is given by 𝜔 𝑞 = ( 8𝐸 𝐽 𝐸𝐶 − 𝐸𝐶 )/~ while the
anharmonicity is given by 𝛼 = −𝐸𝐶 /~. The sensitivity of the qubit to charge noise
drops exponentially as a function of 𝐸𝐸𝐶𝐽 while the anharmonicity drops linearly in
𝐸𝐶 . As a result, by operating at large 𝐸𝐸𝐶𝐽 the transmon qubit can be made insensitive
to charge noise without compromising too much on the anharmonicity.
1.2

Qubit Frequency Tuning

To make our qubit frequency tunable, we replace the single junction shown in the
circuit of Fig. 1.1c. with a pair of junctions that form a superconducting loop

Figure 1.1: Circuit model and energy diagram of a quantum harmonic and anharmonic oscillator. a. Circuit for a parallel LC-oscillator (quantum harmonic oscillator, QHO), with inductance
𝐿 𝑟 in parallel with capacitance, 𝐶𝑟 . b. Energy potential for the QHO, where energy levels are
equidistantly spaced ~𝜔𝑟 apart. c. Circuit for a superconducting qubit, with the non-linear Josephson junction (𝐿 𝐽 , 𝐶 𝐽 ) in parallel with capacitance 𝐶𝑠 . d. Energy potential for the qubit, where
energy levels are no longer equidistantly spaced allowing us to isolate the two lowest levels |0i and
|1i that form the computation subspace. Figure reproduced from Ref. [15]

called a SQUID loop (Superconducting Quantum Interference Device). The pair
of junctions acts as a single junction with an effective critical current that depends
on the external flux threading the squid loop as 𝐼𝑐,𝑒 𝑓 𝑓 = 2𝐼𝑐 cos 𝜋 𝜙𝜙𝑒𝑥𝑡
where
is the magnetic flux quantum. Since the Josephson energy and hence the
𝜙0 = 2𝑒
qubit frequency depends on the critical current, we can tune the frequency of our
qubit by changing the flux threading the SQUID loop. This can be done using an
on-chip ‘Z-line’ (also called ‘flux-bias line’) or an off chip superconducting coil. In
the Z-line approach, an on-chip coplanar waveguide is shorted to ground in close
vicinity to the SQUID loop as shown in Fig. 1.2b. The inductive coupling between
this bias line and the SQUID loop allows us to change the flux threading the SQUID
loop (and hence the qubit frequency) by applying DC currents on the Z-line. Since
this line is a microwave line, it is also possible to rapidly change/modulate the qubit
frequency by applying time varying currents to this line.

1.3

Qubit State Preparation

To excite the qubit and prepare it in various superposition states, we use an on-chip
‘XY line’ that takes the form a microwave coplanar waveguide which is routed close
to the qubit capacitor and is terminated in an open circuit near the qubit capacitor as
shown in Fig. 1.2b. The capacitive coupling between the open end of the XY line
and the qubit capacitor allows us to drive the qubit using microwave pulses that are
resonant with the qubit frequency. By adjusting the length, amplitude, and phase of
the microwave pulse, one can prepare the qubit in any superposition of the |𝑔i and
|𝑒i states. Typical pulse times are on the ∼10 ns timescale.
1.4

Qubit Readout

To readout the state of the qubit, we use a standard technique called dispersive readout. In this technique, the qubit is coupled to a microwave frequency readout resonator. The joint qubit-resonator system can be described by the Jaynes-Cummings
Hamiltonian,
𝜎ˆ𝑧
(1.6)
𝐻ˆ = ~𝜔𝑟 𝑎ˆ † 𝑎ˆ − ~𝜔 𝑞 + ~𝑔 𝑎ˆ 𝜎ˆ+ + 𝑎ˆ † 𝜎ˆ−
The first term in Eq. 1.6 represents the readout resonator (RR) with frequency 𝜔𝑟 ,
the second term represents the qubit as a two-level system with frequency 𝜔 𝑞 , and
the last term represents a qubit-RR interaction with a coupling strength ‘g’. We
design the readout resonator to be detuned from the qubit at a detuning 𝜔 𝑞 − 𝜔𝑐 = Δ.
In the dispersive limit given by |𝑔| < |Δ|, we can approximate the Hamiltonian of
Eq. 1.6 as
𝜎ˆ𝑧
𝑔2
𝐻ˆ = ~𝜔𝑟 𝑎ˆ † 𝑎ˆ − ~𝜔 𝑞 − ~ 𝜎ˆ𝑧 𝑎ˆ † 𝑎ˆ
2
𝑔2
𝜎ˆ𝑧
= ~ 𝜔𝑟 − 𝜎ˆ𝑧 𝑎ˆ † 𝑎ˆ − ~𝜔 𝑞

(1.7)

It is clear from the second line of Eq. 1.7 that the readout resonator acquires a
qubit state dependent frequency shift. We define the dispersive shift 𝜒 = 2𝑔Δ as the
difference in the two readout resonator frequencies (corresponding to the two qubit
states 𝜎𝑧 = ±1). In practice, since the qubit has higher levels which contribute to
the dispersive shift, this formula is modified as
𝜒=

2𝑔 2
Δ 1 + Δ𝛼

(1.8)

where 𝛼 is the anharmonicity of the qubit. To perform qubit readout, we simply interrogate the readout resonator and its frequency allows us to infer the state of the qubit.

Physically, we implement our readout resonator as either a transmission line resonator as shown in Fig. 1.2a. or a lumped element resonator. The readout resonator
is capacitively coupled to the qubit with some coupling capacitance 𝐶𝑔 . A circuit
model for a readout resonator capcitively coupled to a qubit is shown in Fig. 1.3. In
this model, we can treat the qubit as simple LC oscillator. For a given set of designed
circuit parameters, the coupling strength ‘g’ (and hence the dispersive shift) can be
calculated as
𝐶𝑔
𝑔=
𝜔𝑟 𝜔 𝑞
(1.9)
2 (𝐶𝑞 + 𝐶𝑔 )(𝐶𝑟 + 𝐶𝑔 )
The readout resonator is also capacitively coupled to an on-chip coplanar waveguide
that allows us to interrogate the frequency of the readout resonator and infer the state
of the qubit. We call this coplanar waveguide the ‘readout line’ and it is shown in
Fig. 1.2a.
a.

b.
GND Plane

GND Plane

Readout Claw

Qubit
Z Line

Qubit Capacitor

Readout Line

SQUID

CPW Readout
Resonator
XY Line
XY Line

Z Line

Figure 1.2: Example layout of a superconducting qubit chip. a. Sample qubit chip layout
indicating the qubit, CPW readout resonator, qubit control lines, and readout line. b. Zoom into
qubit region showing the qubit capacitor, SQUID loop, XY line, Z line, and the readout claw for
coupling the qubit to the readout resonator.

Figure 1.3: Capacitively coupled LC oscillator model of qubit coupled to readout resonator.

11
1.5

Qubit Characterization

Now that we have covered the basic concepts related to the design and control of superconducting qubits, we will briefly review some basic single qubit measurements
that will be used frequently throughout this thesis. Since superconducting qubits
operate in the 𝑓𝑞 ∼ GHz frequency range, they need to be cooled down to cryogenic
temperatures 𝑇 << ℎ 𝑓𝑞 . All the measurements in this thesis were carried out at the
base temperature (∼ 10 mK) of a dilution refrigerator. Typical qubit characterization
proceeds as follows:
Continuous Wave (CW) Measurements
We begin our measurements by applying continuous wave (CW) signals to the readout and XY lines. The goal of these initial measurements is to quickly determine
the frequencies of the readout resonator and superconducting qubit.
Locate the Readout Resonator
The first step is locating the readout resonator. We use a Vector Network Analyzer
(VNA) to measure the readout resonator in transmission (𝑆21 ) or reflection (𝑆11 ).
We show an example of a reflection measurement (𝑆11 ) of a lumped element readout
resonator in Fig. 1.4a. The dip in the reflection spectrum gives us the frequency of
the readout resonator.

Locate the Qubit
Next we apply a continuous wave (CW) microwave signal on the XY line at frequencies near the expected qubit frequency. As we sweep the frequency of the XY
tone, we continue using the VNA to monitor the readout resonator frequency. When
the XY tone is resonant with the qubit frequency, it drives the qubit which causes a
dispersive shift in the readout resonator frequency and shows up as a change in the
VNA spectrum as shown in Fig. 1.4b. We can repeat this measurement as we apply
a DC current on the flux line until we have our qubit tuned to the desired frequency.

Pulsed Measurements
In the continuous wave measurements described above, we are constantly driving
the qubit into a mixed state. To get greater control over the qubit state, we employ
pulsed XY drive signals that allow us to prepare the qubit in any superposition of
|𝑔i and |𝑒i. An arbitrary waveform generator (AWG) is used to synthesize finite

12
RF L

H 5H GR

5HVR

F LQ

H 4X L ) H XHQF

) H XHQF

Qubit dip

Readout Resonator dip
91 ) H

H F

91 ) H XHQF

Figure 1.4: Example CW spectroscopy data. a. Example dataset from a broad VNA sweep to
loacate the readout resonator. The resonator shows up as a dip in the VNA measurement b. Example
dataset from a two-tone spectroscopy measurement to locate the qubit frequency.

length pulses at an intermediate frequency (IF) of ∼ 100 MHz. This IF signal is
then upconverted to RF frequencies using an IQ mixer. The upconverted RF signal
is used to excite the qubit via the XY line. To perform readout, a similarly upconverted RF signal is used to interrogate the readout resonator and the reflected or
transmitted signal from the readout resonator is downconverted to IF frequencies
and then demodulated to DC using a digitizer. This technique allows us to measure
both the phase and the amplitude of the readout signal.
Rabi Measurement
We now proceed to calibrate the pulse lengths needed to prepare the qubit in various
states. The pulse sequence and a sample dataset for this measurement is shown in
Fig. 1.5a. where we apply a XY pulse of a variable duration to the qubit followed
by readout. As we sweep the duration of the XY pulse for a fixed pulse amplitude,
we see Rabi oscillations of the qubit. The maxima of the Rabi curve gives us the
duration (𝑡 𝜋 ) of the XY pulse required to put the qubit in the |𝑒i state. A pulse of
half the duration (𝑡 𝜋/2 ) can be used to create an equal superposition of |𝑔i and |𝑒i.
T1 Measurement
Once we have identified the 𝜋-pulse duration for a given pulse amplitude, we proceed to characterize the lifetime (T1 ) of the qubit. This is the rate at which a qubit in
state |𝑒i relaxes back to state |𝑔i. A T1 measurement involves applying a 𝜋 pulse to
prepare the qubit in state |𝑒i, and then waiting for some variable time delay 𝜏 before
reading out the state of the qubit. We repeat this measurement multiple times as we
sweep the time delay 𝜏 and measure the probability of finding the qubit in the state

13
|𝑒i as a function of 𝜏. The T1 pulse sequence and an example T1 dataset is shown
in Fig. 1.5b. We can fit the data to an exponential decay of the form 𝑃𝑒 = 𝑒 −𝜏/𝑇1 to
extract the qubit lifetime 𝑇1 .
T∗2 Measurement
Apart from the lifetime, another important metric for a qubit is its decoherence
time (T∗2 ). This is the time for which a qubit is able to stay phase-coherent. A T∗2
measurement begins by applying a 𝜋2 pulse to the qubit. After a variable time delay
𝜏, we apply another 𝜋2 pulse and readout the state of the qubit. In the absence of
decoherence, the two 𝜋2 pulses separated by a time 𝜏 should behave as 𝜋 pulse and
put the qubit in state |𝑒i. However, if the qubit experiences decoherence in the time
delay 𝜏 between the two 𝜋2 pulses, then the final qubit state may be rotated away
from |𝑒i. Sweeping the time delay 𝜏 allows us to extract the decoherence rate of
the qubit. The pulse sequence employed for this measurement is called a Ramsey
pulse sequence and is shown in the inset of Fig. 1.5c. along with a sample dataset.
To allow for greater fitting accuracy, in practice this measurement is performed at
a small detuning (𝛿) from the qubit frequency giving rise to the Ramsey fringes
seen in the data. We fit the data to 𝑃𝑒 = 𝑒 −𝜏/𝑇2 cos (𝛿𝜏 + 𝜙) and extract the qubit
decoherence time T∗2 .

(xcited 6tDte PRSulDtiRn Pe

RDEi Curve
1.0

0.8

Readout

0.6
0.4
0.2

𝜋-pulse duration

0.0
0.0

0.2

0.4
0.6
Pulse DurDtiRn τ (μs)

0.8

T1 decay

1.0

fit
data

XY
0.8

Readout

0.6

1.0

T1 fit

2.03 ± 0.06 μs

0.4
0.2
0.0

τ (μs)

T2* decoherence
fit
data

1.0
T2* fit

0.8

1.55 ± 0.05 μs

0.6
0.4

𝜋/2

0.2

Readout

0.0

τ (μs)

Figure 1.5: Standard pulsed measurements of superconducting qubits. a. Rabi pulse sequence
(inset) and example Rabi dataset indicating the 𝜋-pulse duration b. 𝑇1 pulse sequence (inset) and
example 𝑇1 dataset c. 𝑇2∗ pulse sequence (inset) and example 𝑇2∗ dataset.

15
Chapter 2

COLLAPSE AND REVIVAL OF AN ARTIFICIAL ATOM
COUPLED TO A STRUCTURED PHOTONIC RESERVOIR
[1] V. S. Ferreira∗ , J. Banker∗ , A. Sipahigil, M. H. Matheny, A. J. Keller, E. Kim,
M. Mirhosseini, and O. Painter. “Collapse and revival of an artificial atom
coupled to a structured photonic reservoir”. In: Phys. Rev. X 11.4 (2021),
p. 041043. doi: 10.1103/PhysRevX.11.041043.
2.1

Introduction

Spontaneous emission by a quantum emitter into the fluctuating electromagnetic
vacuum, and the corresponding exponential decay of the emitter excited state, is
an emblematic example of Markovian dynamics of an open quantum system [2].
However, modification of the electromagnetic reservoir can drastically alter this
dynamic, introducing “non-Markovian” memory effects to the emission process,
a consequence of information back-flow from the reservoir to the emitter [3–6].
A canonical example of this, considered in early theoretical work [17–19], is the
behavior of a quantum emitter whose natural emission frequency lies close to the
gap edge of a photonic bandgap material [20, 21] where a sharp transition of the
photonic density of states (DOS) occurs. Inside the bandgap the emitter sees a
reservoir devoid of electromagnetic states, while just outside of the bandgap lies
a continuum of states. This structure of the photonic bandgap reservoir leads to a
strong dressing of the emitter, and a resulting emission dynamics modified by the
interplay between bound and radiative emitter-photon resonant states [22–26].
More recently, theoretical studies have explored how a structured reservoir with nonMarkovian memory alters the entanglement within a quantum system coupled to such
a reservoir [27–29]. This has led to the paradigm of reservoir engineering, where
non-Markovianity is a quantifiable resource for quantum information processing
and communication. Theory work from this quantum information perspective has
shown that long-lived reservoir correlations can be used for the generation and
preservation of entanglement [30, 31] and quantum control [32] of a quantum
system, enhancement of the capacity of quantum channels [33], and the synthesis
of exotic many-body quantum states of light from single emitters [7].
In practice, observation of non-Markovian emission phenomena can be achieved

16
by strongly coupling an emitter to a single-mode waveguide—a one-dimensional
(1D) reservoir with a continuum of states. Waveguides which break continuous
translational symmetry, or which host resonant elements within the waveguide, are
of particular interest in this regard owing to the structure in their spectrum [34–36].
For example, an array of coupled resonant elements leads to a constriction of the
1D continuum of guided modes to a transmission band of finite bandwidth, with
sharp transitions in the photonic DOS occurring at the bandedges as in a photonic
bandgap material.
Spectral constriction of the waveguide continuum, and the concomitant frequency
dispersion, can also result in the slowing of light propagation which enables observation of additional non-Markovian phenomena. For instance, by placing a reflective
boundary (mirror) on one end of a slow-light waveguide, a fraction of the emitter’s
radiation can be fed back from the waveguide reservoir to the emitter at significantly delayed timescales [37–39]. The non-Markovian regime is reached when
𝜏d Γ1D > 1, where Γ1D is the emitter’s emission rate into the waveguide and 𝜏d is
the round-trip travel time of an emitted photon. Theoretical studies have shown
that such non-Markovian delayed feedback in a 1D waveguide reservoir can lead to
revivals in excited state population of an emitter as it undergoes spontaneous emission decay [37, 40–45], realization of stable bound states in a continuum (BIC) [46,
47], and enhanced collective effects including multipartite entanglement and superradiant emission from emitters interacting via a common waveguide channel [29,
48–52]. This deceptively simple mechanism of time-delayed feedback can also
be used for the generation of multi-dimensional photonic cluster states by a single
emitter, and has been proposed as a means for generating the universal resource
states necessary for measurement-based quantum computation [7].
Superconducting microwave circuits incorporating Josephson-junction-based qubits
[53, 54] represent a near-ideal test bed for studying the quantum dynamics of emitters interacting with a 1D continuum [55, 56]. In comparison to solid-state and
atomic optical systems [57–60], superconducting microwave circuits can be created
at a deep-sub-wavelength scale, giving rise to strong qubit-waveguide coupling far
exceeding other qubit dissipative channels. This has enabled a variety of pioneering
experiments probing qubit-waveguide radiative dynamics, employing waveguide
spectroscopy [39, 61–63], time-dependent qubit measurements [64–67], and analysis of higher-order field correlations [68, 69]. Recent experiments have also explored
the coupling of superconducting qubits to acoustic wave devices, demonstrating the

17
capability of these systems to produce significant time-delayed feedback and remote
entanglement of qubits [63, 67].
In this work, we present the design and characterization of an all-electrical slowlight waveguide consisting of a chain of coupled lumped-element superconducting
resonators patterned on a Silicon microchip. We demonstrate that this compact,
low-loss microwave waveguide has sharp bandedges, and a passband with group
delay of 55 ns per centimeter over an 80 MHz bandwidth. Through the addition
of strongly coupled Xmon-style superconducting qubits [16, 70] to the slow-light
waveguide, we are able to realize a quantum emitter-reservoir system operating deep
within the non-Markovian limit. Spectroscopic measurement of the coupled system
shows the emergence of dressed qubit-photon resonant states near the bandedges of
the constricted passband of the waveguide [18, 19, 62]. Using non-adiabatic tuning
of the qubit emission frequency, we also measure the time-dependent dynamics
of the qubit excited state population when it is resonant at different points across
the bandgap and passband of the waveguide. We directly observe non-exponential,
oscillatory radiative decay of the qubit, which modeling indicates is a result of
the interference of the pair of bound and radiative dressed qubit-photon states that
exist on either side of the bandedge of the slow-light waveguide [22]. Further, by
terminating one-end of the slow-light waveguide with a reflective boundary, we
explore the effects of time-delayed feedback on the qubit emission as it emits into
the passband of the slow-light waveguide. In this regime, we observe multiple, wellresolved revivals in the qubit excited state population, and explore the cross-over
between Markovian and non-Markovian emission dynamics through in situ tuning of
the qubit coupling to the waveguide. From this series of measurements, we estimate
the achievable fidelity of entangling a number of photon pulses emitted at different
round-trip times of the waveguide, and find that the demonstrated qubit-waveguide
system is a promising platform for the sequential generation of multi-dimensional
photonic cluster states as described in the theoretical proposals of Refs. [7, 71–73].
2.2

Slow-Light Metamaterial Waveguide

In prior work studying superconducting qubit emission into a photonic bandgap
waveguide [64], we employed a metamaterial consisting of a coplanar waveguide
(CPW) periodically loaded by lumped-element resonators. In that geometry, whose
circuit model simplifies to a transmission line with resonator loading in parallel to
the line, one obtains high efficiency transmission with a characteristic impedance
approximately that of the standard CPW away from the resonance frequency of

18
the loading resonators, and a transmission stopband near resonance of the resonators. The spectral characteristics of the metamaterial in Ref. [64] were studied
via spontaneous emission lifetime and lamb-shift measurements of a weakly coupled superconducting qubit, which revealed information about the local DOS at the
qubit frequency that were consistent with the metamaterial’s engineered dispersion.
In contrast, here we seek a waveguide with high transmission efficiency, slow-light
propagation within a transmission passband, and considerably stronger qubit coupling to the waveguide’s guided modes. The stronger coupling renders the Born
approximation inapplicable in such a system, where the effect of the qubit’s interaction with the photonic reservoir takes on significantly more complexity than simply
a decay rate dependent solely on the DOS at the qubit’s frequency. Furthermore,
the increased propagation delay gives rise to non-Markovian memory effects in the
waveguide-mediated interactions between qubits, for which the waveguide degrees
of freedom can no longer be be traced out, as in Ref. [65] for instance.
Large delay per unit area can be obtained by employing a network of sub-wavelength
resonators, with light propagation corresponding to hopping from resonator-toresonator at a rate set by near-field inter-resonator coupling. This area-efficient
approach to achieving large delays is well-suited to applications where only limited
bandwidths are necessary. However, realizing such a waveguide system in a compact chip-scale form factor requires a modular implementation that can be reliably
replicated at the unit cell level without introducing spurious cell-to-cell couplings.
In optical photonics applications, this sort of scheme has been realized in what are
called coupled-resonator optical waveguides, or CROW waveguides [74, 75]. Here
we employ a periodic array of capacitively coupled, lumped-element microwave
resonators to form the waveguide. Such a resonator-based waveguide supports a
photonic channel through which light can propagate, henceforth referred to as the
passband, with bandwidth approximately equal to four times the coupling between
the resonators, 𝐽. The limited bandwidth directly translates into large propagation
delays; as can be shown (see Appendix 4.2), the delay in the resonator array is
roughly 𝜔0 /𝐽 longer than that of a conventional CPW of similar area, where 𝜔0 is
the resonance frequency of the resonators.
An optical and scanning electron microscope (SEM) image of the unit cell of the
metamaterial slow-light waveguide used in this work are shown in Fig. 2.1a. The
cell consists of a tightly meandered wire inductor section (𝐿 0 ; false color blue)
and a top shunting capacitor (𝐶0 ; false color green), forming the lumped-element

20 µm

φx

... L0

Cg
C0 ...

4.8

(L0C0)-1/2

(L0(C0+4Cg))

-1/2

-20
-40
-60

200

Group Delay (ns)

Transmission (dB)

100 µm

4.6

100 µm

100

4.65

4.7

4.75

Frequency (GHz)

Figure 2.1: Microwave coupled resonator array slow-light waveguide. a. Optical image of a
fabricated microwave resonator unit cell. The capacitive elements of the resonator are false colored
in green, while the inductive meander is false colored in blue. The inset shows a false colored SEM
image of the bottom of the meander inductor, where it is shorted to ground. b. Circuit diagram
of the unit cell of the periodic resonator array waveguide. c. Theoretical dispersion relation of the
periodic resonator array. See Appendix 4.2 for derivation. d. Transmission through a metamaterial
slow-light waveguide spanning 26 resonators and connected to 50-Ω input-output ports. Dashed
blue line: theoretical transmission of finite array without matching to 50-Ω boundaries. Black line:
theoretical transmission of finite array matched to 50-Ω boundaries through two modified resonators
at each boundary. Red line: measured transmission for a fabricated finite resonator array with
boundary matching to input-output 50-Ω coplanar waveguides. The measured ripple in transmission
is less than 0.5 dB in the middle of the passband. e. Measured group delay, 𝜏𝑔 . Ripples in 𝜏𝑔 are
less than 𝛿𝜏𝑔 = 5 ns in the middle of the passband.

microwave resonator. Note that these delineations between inductor and capacitor
are not strict, and that the meandered wire inductor (top shunting capacitor) has
a small parasitic capacitance (parasitic inductance). The resonator is surrounded
by a large ground plane (gray) which shields the meander wire section. Laterally

20
extended ‘wings’ of the top shunting capacitor also provide coupling between the
cells (𝐶𝑔 ; false color green). Note that at the top of the optical image, above each
shunting capacitor, we have included a long superconducting island (𝐶𝑞 ; false color
green); this is used in the next section as the shunting capacitance for Xmon qubits.
Similar lumped-element resonators have been realized with internal quality factors of
𝑄 𝑖 ∼ 105 and small resonator frequency disorder [64], enabling propagation of light
with low extinction from losses or disorder-induced scattering [76]. The waveguide
resonators shown in Fig. 2.1a have a bare resonance frequency of 𝜔0 /2𝜋 ≈ 4.8 GHz,
unit cell length 𝑑 = 290 𝜇m, and transverse unit cell width 𝑤 = 540 𝜇m, achieving
¯ = ( 𝑑𝑤)/(2𝜋𝑣/𝜔0 ) ≈ 1/60, where 𝑣 is the
a compact planar form factor of 𝑑/𝜆
speed of light in a CPW on a infinitely thick silicon substrate.
The unit cell is to a good approximation given by the electrical circuit shown in
Fig. 2.1b, in which the photon hopping rate is 𝐽 ∝ 𝐶𝑔 /𝐶0 [13]. We chose a
ratio of 𝐶𝑔 /𝐶0 ≈ 1/70, which yields a delay per resonator of roughly 2 ns. Note
that we have achieved this compact form factor and large delay per resonator while
separating different lumped-element components by large amounts of ground plane,
which minimizes spurious crosstalk between different unit cells. Analysis of the
periodic circuit’s Hamiltonian and dispersion
q can be found in Appendix 4.2, where

the dispersion is shown to be 𝜔 𝑘 = 𝜔0 / 1 + 4 𝐶𝑔0 sin2 (𝑘 𝑑/2). Figure 2.1c shows
a plot of the theoretical waveguide dispersion for an infinitely periodic waveguide,
where the frequency of the bandedges of the passband are denoted with the circuit
parameters of the unit cell.

For finite resonator arrays, care must be taken to avoid reflections at the boundaries
that would result in spurious resonances (see Fig. 2.1d, dashed blue curve, for
example). To avoid these reflections, we taper the impedance of the waveguide by
slowly shifting the capacitance of the resonators at the boundaries. In particular, we
modify the first two unit cells at each boundary, but in principle, more resonators
could have been modified for a more gradual taper. Increasing 𝐶𝑔 to increase
the coupling between resonators, and decreasing 𝐶0 to compensate for resonance
frequency changes, effectively impedance matches the Bloch impedance of the
periodic structure in the passband to the characteristic impedance of the inputoutput waveguides [77]. In essence, this tapering achieves strong coupling of all
normal modes of the finite structure to the input-output waveguides by adiabatically
transforming guided resonator array modes into guided input-output waveguide
modes. This loading of the normal modes lowers their 𝑄 such that they spectrally

Figure 2.2: Artificial atom coupled to a structured photonic reservoir. a. False-colored optical
image of a fabricated sample consisting of three transmon qubits (Q1 ,Q2 ,Q3 ) coupled to a slowlight metamaterial waveguide composed of a coupled microwave resonator array. Each qubit is
capacitively coupled to a readout resonator (false color dark blue) and a XY control-line (false
color red), and inductively coupled to a Z flux-line for frequency tuning (false color light blue). The
readout resonators are probed through feed-lines (false color lilac). The metamaterial waveguide path
is highlighted in false color dark purple. b. SEM image of the Q1 qubit, showing the long, thin shunt
capacitor (false color green), XY control-line, the Z flux-line, and coupling capacitor to the readout
resonator (false color dark blue). c. SEM zoom-in image of the Z flux-line and superconducting
quantum interference device (SQUID) loop of Q1 qubit, with Josephson Junctions and its pads false
colored in crimson. d. Transmission through the metamaterial waveguide as a function of flux.
The solid magenta line indicates the expected bare qubit frequency in the absence of coupling to the
metamaterial waveguide, calculated based on the measured qubit minimum/maximum frequencies
and the extracted anharmonicity. The dashed black lines are numerically calculated bound state
energies from a model Hamiltonian of the system; see section 4.5 for further details. e. Zoom-in
of transmission near the upper bandedge, showing the hybridization of the qubit with the bandedge,
and its decomposition into a bound state in the upper bandgap and a radiative state in the continuum
of the passband.

overlap and become indistinguishable, changing the DOS of a finite array from
that of a multi-mode resonator to that of finite-bandwidth continuum with singular
bandedges. Further details of the design of the unit cell and boundary resonators
can be found in Appendix 4.3.
Using the above design principles, we fabricated a capacitively coupled 26-resonator
array metamaterial waveguide. The waveguide was fabricated using electron-beam
deposited aluminum (Al) on a silicon substrate and was measured in a dilution refrigerator; transmission measurements are shown in Fig. 2.1d,e, and further details
of our fabrication methods and measurement set-up can be found in Appendix 4.1.
We find less than 0.5 dB ripple in transmitted power and less than 10% variation in
𝑑𝜙
the group delay (𝜏𝑔 ≡ − 𝑑𝜔
, 𝜙 = arg(𝑡 (𝜔)), where 𝑡 is transmission) across 80 MHz
of bandwidth in the center of the passband, ensuring low distortion of propagating signals. Qualitatively, this small ripple demonstrates that we have realized a
resonator array with small disorder and precise modification of the boundary resonators. More quantitatively, from the transmitted power measurements we extract a

22
standard deviation in the resonance frequencies of 3 × 10−4 × 𝜔0 (see Appendix 4.4).
Furthermore, we achieve 𝜏d ≈ 55 ns of delay across the 1 cm metamaterial waveguide, corresponding to a slow-down factor given by the group index of 𝑛𝑔 ≈ 650. We
stress that this group delay is obtained across the center of the passband, rather than
near the bandedges where large (and undesirable) higher-order dispersion occurs
concomitantly with large delays.
2.3

Non-Markovian Radiative Dynamics

In order to study the non-Markovian radiative dynamics of a quantum emitter, a
second sample was fabricated with a metamaterial waveguide similar to that in the
previous section, this time including three flux-tunable Xmon qubits [70] coupled
at different points along the waveguide (see Fig. 2.2a-c). Each of the qubits is
coupled to its own XY control line for excitation of the qubit, a Z control line
for flux tuning of the qubit transition frequency, and a readout resonator (R) with
separate readout waveguide (RO) for dispersive read-out of the qubit state. The
qubits are designed to be in the transmon-limit [16] with large tunneling to charging
energy ratio (see Refs. [64, 78] for further qubit design and fabrication details).
As in the test waveguide of Fig. 2.1, the qubit-loaded metamaterial waveguide is
impedance-matched to input-output 50-Ω CPWs. In order to extend the waveguide
delay further, however, this new waveguide is realized by concatenating two of the
test metamaterial waveguides together using a CPW bend and internal impedance
matching sections. The Xmon qubit capacitors were designed to have capacitive
coupling to a single unit cell of the metamaterial waveguide, yielding a qubit-unit
cell coupling of 𝑔uc ≈ 0.8𝐽.
In this work, only one of the qubits, Q1 , is used to probe the non-Markovian emission
dynamics of the qubit-waveguide system. The other two qubits are to be used in a
separate experiment, and were detuned from Q1 by approximately 1 GHz for all of
the measurements that follow. At zero flux bias (i.e., maximum qubit frequency), the
measured parameters of Q1 are: 𝜔𝑔𝑒 /2𝜋 = 5.411 GHz, 𝜂/2𝜋 = (𝜔 𝑒 𝑓 − 𝜔𝑔𝑒 )/2𝜋 =
−235 MHz, 𝜔𝑟 /2𝜋 = 5.871 GHz, and 𝑔𝑟 /2𝜋 = 88 MHz. Here, |𝑔i, |𝑒i, and | 𝑓 i
are the vacuum, first excited, and second excited states of the Xmon qubit, with
𝜔𝑔𝑒 the fundamental qubit transition frequency, 𝜔 𝑒 𝑓 the first excited state transition
frequency, and 𝜂 the anharmonicity. 𝜔𝑟 is the readout resonator frequency, and 𝑔𝑟
is the bare coupling rate between the qubit and the readout resonator.
As an initial probe of qubit radiative dynamics, we spectroscopically probed the

23
interaction of Q1 with the structured 1D continuum of the metamaterial waveguide.
These measurements are performed by tuning 𝜔𝑔𝑒 into the vicinity of the passband
and measuring the waveguide transmission spectrum at low power (such that the
effects of qubit saturation can be neglected). A color intensity plot of the measured
transmission spectrum versus flux bias used to tune the qubit frequency is displayed
in Fig. 2.2d. These spectra show a clear anti-crossing as the qubit is tuned towards
either bandedge of the passband (a zoom-in near the upper bandedge of the passband
is shown in Fig. 2.2e). As has been shown theoretically [22, 23], in the single
excitation manifold the interaction of the qubit with the waveguide results in a
pair of qubit-photon dressed states of the hybridized system, with one state in the
passband (a delocalized ‘continuum’ state) and one state in the bandgap (a localized
‘bound’ state). This arises due to the large peak in the photonic DOS at the bandedge
(in the lossless case, a van Hove singularity), the modes of which strongly couple
4 /4𝐽) 1/3 , resulting in a
to the qubit with a coherent interaction rate of ΩWG ≈ (𝑔uc
dressed-state splitting of 2ΩWG . This splitting has been experimentally shown to be
a spectroscopic signature of a non-Markovian interaction between an emitter and a
photonic crystal reservoir [61, 62]. Further details and discussion can be found in
Appendix 4.2 and section 4.5.
The dressed state with frequency in the passband is a radiative state which is responsible for decay of the qubit into the continuum [19]. On the other hand, the state with
frequency in the gap is a qubit-photon bound state, where the qubit is self-dressed by
virtual photons that are emitted and re-absorbed due to the lack of propagating modes
in the waveguide for the radiation to escape. This bound state assumes an exponenÍ
tially shaped photonic wavefunction of the form 𝑥 𝑒 −|𝑥|/𝜆 𝑎ˆ †𝑥 |vaci, where |vaci is
the state with no photons in the waveguide, 𝑎ˆ †𝑥 is the creation operator of a photon
in unit cell at position 𝑥 (with the qubit located at 𝑥 = 0), and 𝜆 ≈ 𝐽/(𝐸 𝑏 − 𝜔0 )
is the state’s localization length. In the theoretical limit of an infinite array, and in
absence of intrinsic resonator and qubit losses, the qubit component of the bound
state does not decay even though it is hybridized with the waveguide continuum; a
behavior distinct from conventional open quantum systems. Practically, however,
intrinsic losses and the overlap between the bound state’s photonic wavefunction and
the input-output waveguides will result in decay of the qubit-photon bound state.
In complement to spectroscopic probing of the qubit-reservoir system, and in order
to directly study the population dynamics of the qubit-photon dressed states, we
also performed time-domain measurements as shown in Fig. 2.3. In this protocol

Figure 2.3: Non-Markovian radiative dynamics in a structured photonic reservoir. a. Pulse
sequence for the time-resolved measurement protocol. The qubit is excited while its frequency is
0 )
250 MHz above the upper bandedge, and then it is quickly tuned to the desired frequency (𝜔𝑔𝑒
for a interaction time 𝜏 with the reservoir. After interaction, the qubit is quickly tuned below the
lower bandedge for dispersive readout. b. Intensity plot showing the excited state population of the
qubit versus interaction time with the metamaterial waveguide reservoir as a function of the bare
qubit frequency. c. Line cuts of the intensity plot shown in (b), where the color of the plotted curve
matches the corresponding horizontal dot-dashed curve in the intensity plot. Solid black lines are
numerical predictions of a model with experimentally fitted device parameters and an assumed 0.8%
thermal qubit population (see Appendix 4.5 for further details).

25
(illustrated in Fig. 2.3a), we excite the qubit to state |𝑒i with a resonant 𝜋-pulse on
the XY control line, and then rapidly tune the qubit transition frequency using a fast
current pulse on the Z control line to a frequency (𝜔0𝑔𝑒 ) within, or in the vicinity of,
the slow-light waveguide passband. After an interaction time 𝜏, the qubit is then
rapidly tuned away from the passband, and the remaining qubit population in |𝑒i is
measured using a microwave probe pulse (RO) of the read-out resonator which is
dispersively coupled to the qubit. The excitation of the qubit is performed far from
the passband, permitting initialization of the transmon qubit whilst it is negligibly
hybridized with the guided modes of the waveguide. Dispersive readout of the qubit
population is performed outside of the passband in order to minimize the loss of
population during readout. Note that, as illustrated in Fig. 2.3a, the qubit is excited
and measured at different frequencies on opposite sides of the passband; this is
necessary to avoid Landau-Zener interference [79].
Results of measurements of the time-domain dynamics of the qubit population as
a function of 𝜔0𝑔𝑒 (the estimated bare qubit frequency during interaction with the
waveguide) are shown as a color intensity plot in Fig. 2.3b. In this plot, we observe a
400-fold decrease in the 1/𝑒 excited state lifetime of the qubit as it is tuned from well
outside the passband to the middle of the slow-light waveguide passband, reaching
a lifetime as short as 7.5 ns. Beyond the large change in qubit lifetime within the
passband, several other more subtle features can be seen in the qubit population
dynamics near the bandedges and within the passband. These more subtle features
in the measured dynamics show non-exponential decay, with significant oscillations
in the excited state population that is a hallmark of strong non-Markovianity in
quantum systems coupled to amplitude damping channels [80, 81].
The observed qubit emission dynamics in this non-Markovian limit are best understood in terms of the qubit-waveguide dressed states. Fast (i.e., non-adiabatic)
tuning of the qubit in state |𝑒i into the proximity of the passband effectively initializes it into a superposition of the bound and continuum dressed states. The observed
early-time interaction dynamics of the qubit with the waveguide then originate from
interference of the dressed states, which leads to oscillatory behavior in the qubit
population analogous to vacuum-Rabi oscillations [82]. The frequency of these
oscillations is thus set by the difference in energy between the dressed states. The
amplitude of the oscillations, on the otherhand, quickly decay away as the energy in
the radiative continuum dressed state is lost into the waveguide.
All of these features can be seen in Fig. 2.3c, which shows plots of the measured time-

26
domain curves of the qubit excited state population for bare qubit frequencies near
the top, middle, and bottom of the passband. Near the upper bandedge frequency,
we observe an initial oscillation period as expected due to dressed state interference.
Once the continuum dressed state has decayed away, a slower decay region free of
oscillations can be observed (this is due to the much slower decay of the remaining
qubit-photon bound state). Finally, around 𝜏 ≈ 115 ns, there is an onset of further
small amplitude oscillations in the qubit population. These late-time oscillations can
be attributed to interference of the remaining bound state at the site of the qubit with
weak reflections occurring within the slow-light waveguide of the initially emitted
continuum dressed state. The 115 ns timescale corresponds to the round trip time
between the qubit and the CPW bend that connects the two slow-light waveguide
sections.
In the middle of the passband, we see an extended region of initial oscillation and
rapid decay, albeit of smaller oscillation amplitude. This is a result of the much
smaller initial qubit-photon bound state population when tuned to the middle of
the passband. Near the bottom of the passband we see rapid decay and a single
period of a much slower oscillation. This is curious, as the dispersion near the
upper and lower bandedge frequencies of the slow-light waveguide is nominally
equivalent. Further modelling has shown this is a result of weak non-local coupling
of the Xmon qubit to a few of the nearest-neighbour unit cells of the waveguide.
Referring to Fig. 2.1c, the modes near the lower bandedge occur at the X-point of the
Brillouin zone edge where the modes have alternating phases across each unit cell,
thus extended coupling of the Xmon qubit causes cancellation-effects which reduces
the qubit-waveguide coupling at the lower frequency bandedge. Further detailed
numerical model simulations of our qubit-waveguide system via a tight-binding
model and a circuit model, as well as the correspondence between the observed
dynamics and the theory of spontaneous emission by a two level system near a
photonic bandedge [22], are given in Appendix 4.5.
2.4

Time-Delayed Feedback

In order to further study the late-time, non-Markovian memory effects of the qubitwaveguide dynamics, we also perform measurements in which the end of the waveguide furthest from qubit Q1 is terminated with an open circuit, effectively creating
a ‘mirror’ for photon pulses stored in the slow-light waveguide reservoir. As illustrated in Fig. 2.4a, we achieve this in situ by connecting the input microwave
cables of the dilution refrigerator to the waveguide via a microwave switch. The

27
position of the switch, electrically closed or open, allows us to study a truly open
environment for the qubit or one in which delayed-feedback is present, respectively
(see Appendix 4.1 for further details).
Performing time-domain measurements with the mirror in place and with the qubit
frequency in the passband, we observe recurrences in the qubit population at one
and two times the round-trip time of the slow-light waveguide that did not appear
in the absence of the mirror (see Fig. 2.4b). The separation of timescales between
full population decay of the qubit and its time-delayed re-excitation demonstrates an
exceptionally long memory of the reservoir due to its slow-light nature, and places
this experiment in the deep non-Markovian regime [37]. The small recurrence levels
as they appear in Fig. 2.4b are not due to inefficient mirror reflection, but rather
can be explained as follows. Because the qubit emits towards both ends of the
waveguide, half of the emission is lost to the unterminated end, while the other half
is reflected by the mirror and returns to the qubit. In addition, the exponentially
decaying temporal profile of the emission leads to inefficient re-absorption by the
qubit and further limits the recurrence (see, for instance, Ref. [83, 84] for details).
These two effects can be observed in simulations of a qubit coupled to a dispersionless and loss-less waveguide (pink dotted line; for more details, see Ref. [41] and
Appendix 4.6). The remaining differences between the simulation and the measured
population recurrence (blue solid line) can be explained by the effects of propagation
loss and pulse distortion due to the slow-light waveguide’s dispersion.
We also further probed the dependence of this phenomenon on the strength of
coupling to the waveguide continuum by parametric flux modulation of the qubit
transition frequency [85] when it is far detuned from the passband. This modulation
creates sidebands of the qubit excited state, which are detuned from 𝜔𝑔𝑒 by the
frequency of the flux tone 𝜔mod . By choosing the modulation frequency such that
a first-order sideband overlaps with the passband, the effective coupling rate of the
qubit with the waveguide at the sideband frequency was reduced approximately by
a factor of J12 [𝜖/𝜔mod ], where 𝜖 is the modulation amplitude and J1 is a Bessel
function of the first kind (𝜖/𝜔mod is the modulation index). Keeping a fixed 𝜔mod ,
we observe purely exponential decay at small modulation amplitudes. However,
above a modulation amplitude threshold we again observe recurrences in the qubit
population at the round-trip time of the metamaterial waveguide, demonstrating a
continuous transition from Markovian to non-Markovian dynamics (see Appendix
4.6 for further comparisons between this data and the theoretical model of Ref. [41])

Figure 2.4: Time-delayed feedback from a slow-light reservoir with a reflective boundary a.
Illustration of the experiment, showing the qubit coupled to the metamaterial waveguide which is
terminated on one end with a reflective boundary via a microwave switch. b. Measured population
dynamics of the excited state of the qubit when coupled to the metamaterial waveguide terminated in
a reflective boundary. Here the bare qubit is tuned into the middle of the passband. The onset of the
population revival occurs at 𝜏 = 227 ns, consistent with round-trip group delay (𝜏d ) measurements
at that frequency, while the emission lifetime of the qubit is (Γ1D ) −1 = 7.5 ns. The magenta curve is
a theoretical prediction for emission of a qubit into a dispersionless, lossless semi-infinite waveguide
with equivalent 𝜏d and Γ1D (see Appendix 4.6 for details). c. Population dynamics under parametric
flux modulation of the qubit, for varying modulation amplitudes, demonstrating a Markovian to
non-Markovian transition. When the modulation index (𝜖/𝜔mod ) is approximately 0.4 we have
Γ1D (𝜖) = 1/𝜏d ; the corresponding dynamical trace is colored in blue.

29
2.5

Conclusion

In conclusion, by strongly coupling Xmon qubits to a 1D structured photonic reservoir consisting of a metamaterial slow-light waveguide, we are able to probe the
non-Markovian dynamical regime of waveguide quantum electrodynamics. In this
regime, we observe non-exponential qubit spontaneous decay near the bandedges of
the slow-light waveguide, attributable to interference resulting from the splitting of
the qubit state into a radiative state in the passband and a bound state in the bandgap
region of the metamaterial waveguide. Moreover, by placing a reflective boundary
on one end of the waveguide, we observe recurrences in the qubit population at
the round-trip time of an emitted photon, as well as a Markovian to non-Markovian
transition when varying the qubit-waveguide interaction strength.
The demonstrated ability to achieve a true finite-bandwidth continuum with timedelayed feedback opens up several new research avenues for exploration [38, 40–
52, 86]. As a straightforward extension of the current work, one may probe the
qubit-waveguide-mirror system in a continuous, strongly-driven fashion, and use
tomography to study photon correlations in the output radiation field [38]. This
output field, with expected photon stream of high entanglement dimensionality, has
a direct mapping to continuous matrix product states which can used for analog
simulations of higher-dimension interacting quantum fields [86, 87]. With technical advancements in the tomography of microwave fields [69, 88], and realization
of single-microwave-photon qubit detectors [89–91], the basic tools for characterization of these entangled photonic states and their quantum many-body-system
analogues are now available.
Looking forward even further, the use of the multi-level structure of the transmon
qubit, in conjunction with a second distant qubit side-coupled to the waveguide as
a switchable mirror, can be used to generate 2D cluster states [7]. This system is
capable of entangling consecutively emitted photons as well as photons separated
in time by the round-trip waveguide delay, 𝜏d , thus achieving a 𝑁 × 𝑀 2D cluster
state where 𝑁 is limited by the number of non-overlapping photons that can fit in
the slow-light waveguide and 𝑁 · 𝑀 is limited by the coherence time of the emitter.
With our achieved device parameters, we estimate that a 3 × 3 2D cluster state
could be generated with fidelity greater than 50% (see Ref. [7] and Chapter 3 for
further details). Realistic improvements in 𝜏d and 𝑇2∗ could increase the size of the
state by at least an order of magnitude, with even further improvement possible
via incorporation of compact high kinetic inductance superconducting thin-film

30
resonators or acoustic delay lines [67, 92]. Additionally, by controlling the number
of reflections a photon undergoes before exiting the metamaterial waveguide, cluster
states of 3D or higher entanglement dimensionality can be generated, enabling the
realization of fault-tolerant measurement-based quantum computation schemes [7,
73, 93].
The essential paradigm of our experiment, consisting of a single artificial atom
coupled to a waveguide with a long propagation delay and sharp spectral cutoffs,
could in principle be achieved in other solid-state and atomic optical system, such as
trapped atoms coupled to a nanofiber or defect centers coupled to photonic crystal
waveguides [57–60]. The challenge with such modalities, however, is achieving
a large coupling of the emitter to the guided modes of the waveguide relative
to its decay rate as well as the propagation delay of the waveguide. From an
application standpoint, however, the optical domain is of great interest due to the
mature technology in single-photon detectors, photonic integrated circuits for linear
and nonlinear optics, and optical fibers for long range communication.

31
Chapter 3

UTILIZATION OF METAMATERIAL WAVEGUIDE FOR 2D
CLUSTER STATE GENERATION
We envision leveraging the large time delay and sharply varying photonic DOS
of a slow-light metamaterial waveguide, along with the transmon qubit multi-level
structure, to generate a 2D photonic cluster state. Given a typical transmon anharmonicity of 300 MHz, tuning the 𝑒 − 𝑓 transition instead of the 𝑔 − 𝑒 transition
into the middle of the passband would situate the 𝑔 − 𝑒 transition frequency more
than 200 MHz above the upper bandedge in our current waveguide devices. The
corresponding level structure would then consist of two metastable states (|𝑔i and
|𝑒i) and a third level (| 𝑓 i) that is strongly coupled to the waveguide. It has been previously shown that such a ladder-like level structure can be utilized to generate 1D
cluster states of time-bin photonic qubits through a sequential emission process [71,
94, 95].
In addition, the non-Markovian nature of the slow-light waveguide reservoir can be
further exploited to enrich this one-dimensional entanglement to higher dimensions
via time-delayed feedback [7]. In the case of 2D cluster state generation, this
can be accomplished by using a metamaterial waveguide terminated on one end,
coupling an emitter qubit to the terminated end of the waveguide, and using a
second tunable qubit coupled to the output port of the waveguide as a single photon
switchable mirror [96]. This mirror could be periodically switched on and off
in a manner where consecutively emitted photons reflect on the mirror, interact a
second time with the qubit, and subsequently exit through the waveguide’s output
port without additional reflections, with facile access to the photons for subsequent
measurement enabled by matching of the slow-light metamaterial waveguide to a
50-Ω output waveguide. This resource efficient scheme, requiring only two qubits,
entangles photons separated in time by 𝜏d in addition to the 1D entanglement between
consecutively emitted photons, thus achieving a 𝑁 × 𝑀 2D cluster state, where 𝑁 is
limited by the number of time-bin qubits that can fit in the slow-light waveguide and
𝑁 · 𝑀 is limited by the coherence time of the emitter. And remarkably, increasing
the number of qubit-photon interaction events by simply increasing the number of
reflections in the metamaterial waveguide allows for generation of cluster states
with even higher entanglement dimensionality, paving the way for fault-tolerant

32
measurement-based quantum computation [7, 73, 93].
Moreover, leveraging the rapid flux control of the qubit’s transition frequency confers
several additional advantages to the generation of multi-dimensional cluster states.
For instance, it enables selective coupling and de-coupling of the | 𝑓 i state to the
waveguide via control of the detuning of the 𝑒 − 𝑓 transition to the passband,
allowing for high-fidelity manipulation of the emitter’s three level quantum state
separate from photon emission and re-absorption. Additionally, controlling the
qubit-waveguide interaction strength via parametric flux modulation of the qubit
frequency, as discussed in the main text, allows for pulse shaping of the emitted
photons [97–99], which yields multiple benefits. Firstly, the fidelity of the photon
re-absorption process can be significantly improved by shaping the photons to have
a time-symmetric envelope with bandwidth less than Γ1D [7, 83, 84]. This directly
improves the fidelity of the entanglement between time-bin photonic qubits that
occurs via the time-delayed feedback mechanism. Secondly, pulse-shaping allows
for pre-compensation of the waveguide’s residual dispersion near the middle of the
passband [100], preventing broadening and distortion of propagating photons that
could hinder their eventual measurement.
Already with our achieved device parameters of 𝜏d = 227 ns and 𝑇2∗ = 3𝜇𝑠 (measured
at a flux insensitive sweetspot), along with an increased Γ1D by a factor of two,
entangling between individual time-bin qubits can be performed with over 95%
fidelity through the techniques discussed in Ref. [7], allowing for generation of
cluster states of up to ∼ 9 photons. Note that, due to the enhancement of the
qubit-waveguide interaction strength via the slow-light effect [101, 102], doubling
the Γ1D achieved in this work would correspond to only a small increase of ∼ 2 fF
in the capacitive coupling of the qubit to the metamaterial waveguide. Further,
realistic increases in 𝜏d and 𝑇2∗ would increase the size of possible states by at least
an order of magnitude; with ample room for more substantial improvement via
incorporation of even more compact high kinetic inductance superconducting thinfilm resonators for larger delays, and utilization of error protected qubits [103, 104]
or lower-loss superconducting films [105] for higher qubit coherence. And finally,
we note that techniques for tomography of microwave fields [69, 88] and singlephoton detection of microwave photons utilizing superconducting qubits [89–91]
have attained significant maturity over the last decade, enabling characterization of
generated cluster states and their use in measurement-based quantum computation.

33
Chapter 4

APPENDIX: DETAILS OF DEVICE DESIGN, FABRICATION,
MEASUREMENT SETUP, AND MODELING
4.1

Fabrication and Measurement Setup

Device Fabrication
The devices used in this work were fabricated on 10 mm × 10 mm silicon substrates
[Float zone (FZ) grown, 525 𝜇m thickness, > 10kΩ-cm resistivity], following
similar techniques as in Ref. [78]. After standard solvent cleaning of the substrate,
our first aluminum (Al) layer consisting of the ground plane, CPWs, metamaterial
waveguide, and qubit capacitor was patterned by electron-beam lithography of our
resist followed by electron-beam evaporation of 120 nm aluminum at a rate of 1
nm/s. A liftoff process performed in n-methyl-2-pyrrolidone at 80 ◦ C for 2.5 hours
(with 10 minutes of ultrasonication at the end) then yielded the aforementioned
metal structures.
In our qubit device, the Josephson junctions were fabricated using double-angle
electron beam evaporation of 60 nm and 120 nm of Al (at 1 nm/s) on suspended
Dolan bridges, with an intervening 20 minute oxidation and a subsequent 2 minute
oxidation at 10 mbar, followed by liftoff as described above. Note that prior to
the double-angle evaporation, the sample was cleaned by an oxygen plasma treatment and a HF vapor etch. Finally, in order to electrically connect the evaporated
Josephson junctions to the first Al layer, a 6 min argon ion mill was performed to
locally remove surface aluminum oxide around the areas of overlap between the
first Al layer and the Josephson junctions, which was followed by evaporation of
an additional “bandage” layer of 140 nm Al that electrically connected the metal
layers. Asymmetric Josephson junctions were fabricated in all qubits’ SQUID loops
to reduce dephasing from flux noise, with a design ratio of the larger junction area
to the smaller junction area of approximately 6.
Measurement Setup
A schematic of the measurement chain used in this work is shown in Fig. 4.1. Measurements were performed in a 3He/4He dry dilution refrigerator, with a base fridge
temperature at the mixing chamber (MXC) plate of 𝑇f = 12 mK. The waveguide
sample was wire bonded to a CPW printed circuit board (PCB) with coaxial con-

TWPA
Pump

20 dB

RO
Input

Output
300 K
50 K plate
4 K plate

HEMT
Cold plate

MXC plate
6 dB

500
MHz

20 dB

20 dB
20 dB

40 dB

10 dB

20 dB

Metamaterial
IN
XY

4-8
GHz

TWPA
Metamaterial
OUT
RO Output

attenuation DC
block

microwave
switch

filter

isolator

2x2
switch

20 dB 50Ω
dir. term.
coupler

Figure 4.1: Schematic of the measurement chain inside the dilution refrigerator. See Appendix text for further details (“dir” is shorthand for “directional”, and “term.” is shorthand for
“termination”). See Fig. 2.2 for electrical connections at the sample.

35
nectors, and housed inside a small copper box that is mounted to the MXC plate of
the fridge. The copper box and sample were mounted inside a cryogenic magnetic
shield to reduce the effects of stray magnetic field.
Attenuators were placed at several temperature stages of the fridge to provide thermalization of the coaxial input lines, and to reduce thermal microwave noise at
the input to the sample. We used different attenuation configurations for our GHz
microwave lines (Metamaterial IN, XY, RO Input, TWPA pump) as compared to our
flux line (Z), with significantly less attenuation for the latter, for reasons explained
in Ref. [106]. In addition, we included in the flux line a (reflective) low-pass filter,
with corner frequency at 500 MHz, to minimize thermal noise photons at higher
frequencies while maintaining short rise and fall time of pulses for fast flux control.
Also note that the 40 dB attenuation of the “Metamaterial IN” line at the MXC
plate includes a 20 dB thin-film “cold attenuator” [107] to ensure a more complete
reduction of thermal photons in the metamaterial waveguide.
Our amplifier chain at the “Output” line consisted of a travelling-wave parametric
amplifier (TWPA) as the initial amplification stage [108], followed by a CITCRYO412A high mobility electron transistor (HEMT) amplifier mounted at the 4K plate,
and additional amplifiers at room temperature (Miteq AFS3-00101200-42-LN-HS,
AMT A0262). For operation of the TWPA, a microwave pump signal was added to
the amplifier via the coupled port of a 20dB directional coupler, with its isolated port
terminated in 50-Ω. In between the two amplifiers, we have included a reflective
bandpass filter (thermalized to the MXC plate) to suppress noise outside of 4–8 GHz,
and used superconducting NbTi cables to minimize loss from the MXC plate to the
4K plate. We have also included two isolators in between the directional coupler and
the sample in order to shield the sample from the strong TWPA pump, as well as an
isolator in between the TWPA and the directional coupler in order to suppress any
standing waves between the two elements due to spurious impedance mismatches;
our isolators consist of 3 port circulators with the third port terminated in 50-Ω. All
50-Ω terminations are rated for cryogenic operaion and are thermalized to the MXC
plate in order to suppress thermal noise from their resistive elements.
We also employed microwave switches in our measurement chain in order to provide
in situ experimental flexibility in the following manner. As discussed in the main
text, in between the “Metamaterial IN” chain and the metamaterial waveguide, we
have placed a Radiall R573423600 microwave switch. By electrically opening
the switch, we can establish an open circuit at the end of the waveguide furthest

36
from Q1 , effectively creating a mirror for emission from Q1 , and thereby inducing
time-delayed feedback.
In addition, in order to utilize our amplifier chain for either spectroscopic or
time-domain measurements within the same cool-down, we employed Radiall
R577432000 2x2 microwave switches for selective routing of the outputs of the
metamaterial waveguide or the readout waveguide to the amplification chain. With
our switch configuration, we ensured that when routing the readout waveguide output to the amplification chain, the metamaterial waveguide output was connected to
a 50-Ω termination. This allowed us to maintain a 50-Ω environment at the metamaterial output at all times, and thereby ensured that the metamaterial waveguide
remained an open quantum system due to its coupling to the 50-Ω continuum of
modes. By employing two 2x2 switches instead of one, we had the ability to bypass the TWPA amplifier if desired, although ultimately the TWPA was used when
collecting all measurement data presented in Figs. 2.2–2.4.
For spectroscopic measurements, the “Metamaterial IN” and “Output” lines were
connected to the input and output of a ZNB20 Rohde & Schwarz vector network
analyzer (VNA), respectively. For time-domain measurements, GHz excitation and
readout pulses were generated by upconversion of MHz IF in-phase (I) and quadrature (Q) signals sourced from a Keysight M320XA arbitrary waveform generator
(AWG), utilizing a Marki IQ-4509 IQ mixer and a LO tone supplied by a BNC
845 microwave source. Following amplification, output readout signals were downconverted (using an equivalent mixer and the same LO source) and subsequently
digitized using an Alazar ATS9360 digitizer. For all measurements, qubit flux biasing was also sourced from a M320XA AWG, the TWPA pump tone was sourced
by an Agilent E8257D microwave source, and all inputs to the dilution refrigerator
were low-pass filtered and attenuated such that the noise levels from the electronic
sources were reduced to a 300 K Johnson-Nyquist noise level.
4.2

Capacitively Coupled Resonator Array Waveguide Fundamentals

Band Structure Analysis
We consider a periodic array of capacitively coupled LC resonators, with unit
cell circuit diagram shown in Fig. 2.1b. The Lagrangian for this system can be
constructed as a function of node fluxes 𝜙𝑥 of the resonators, and is written as,

𝐿=

Õ 1

𝜙𝑥
𝐶0 𝜙¤2𝑥 + 𝐶𝑔 ( 𝜙¤𝑥 − 𝜙¤𝑥−1 ) 2 −

2𝐿 0

(4.1)

37
Since we seek traveling wave solutions to the problem, it is convenient to work with
the Fourier transform of the node fluxes, defined as

𝜙𝑘 = √

1 Õ

𝜙𝑥 𝑒 −𝑖𝑘𝑥𝑑 ,

(4.2)

𝑀 𝑥=−𝑁
where 𝑀 = 2𝑁 + 1 is the total number of periods of a structure with periodic
boundary conditions, 𝑑 is the lattice constant of the resonator array, and 𝑘 are
the discrete momenta of the first Brillouin zone’s guided modes and are given by
𝑘 = 2𝜋𝑚
𝑀 𝑑 for integer 𝑚 = [−𝑁, 𝑁]. Using the inverse Fourier transform,

𝜙𝑥 = √

1 Õ

𝜙 𝑘 𝑒𝑖𝑘𝑥𝑑 ,

(4.3)

we arrive at the following 𝑘-space Lagrangian

𝐿=

1 ¤ ¤
𝜙𝑘 𝜙𝑘
−𝑖𝑘 𝑑 2
𝐶0 𝜙 𝑘 𝜙−𝑘 + 𝐶𝑔 𝜙 𝑘 𝜙−𝑘 1 − 𝑒
2𝐿 0

Õ 1

(4.4)

where we note that 1 − 𝑒 −𝑖𝑘 𝑑 is equivalent to 4 sin2 (𝑘 𝑑/2). We then obtain the
Hamiltonian via the standard Legendre transformation using the canonical node
¤−𝑘 𝐶0 + 4𝐶𝑔 sin2 (𝑘 𝑑/2) , yielding:
charges 𝑄 𝑘 = 𝜕𝜕𝐿

Õ 1
𝑄 𝑘 𝑄 −𝑘
𝜙 𝑘 𝜙−𝑘
𝐻=
+
2 (𝑘 𝑑/2) + 𝐶
2𝐿
4𝐶
sin

(4.5)

Promoting charge and flux to quantum operators and utilizing the canonical commu
tation relation 𝜙ˆ𝑘 , 𝑄ˆ𝑘 0 = 𝑖~𝛿 𝑘 𝑘 0 , we define the following creation and annihilation
operators:

𝑖 ˆ
𝑚 𝑘 𝜔𝑘 ˆ
𝑎ˆ 𝑘 =
𝜙𝑘 +
𝑄 −𝑘 ,
2~
𝑚 𝑘 𝜔𝑘
𝑚 𝑘 𝜔𝑘 ˆ
𝑖 ˆ
𝑎ˆ 𝑘 =
𝜙−𝑘 +
𝑄𝑘 ,
2~
𝑚 𝑘 𝜔𝑘

(4.6)

where 𝑚 𝑘 = 𝐶0 + 4𝐶𝑔 sin2 (𝑘 𝑑/2) . The resulting dispersion relation, 𝜔 𝑘 , plotted
in Fig. 2.1c is given by,

38
𝜔0
𝜔𝑘 = q
𝐶𝑔
1 + 4 𝐶0 sin (𝑘 𝑑/2)

where 𝜔0 = 1/ 𝐿 0𝐶0 , and

𝑎ˆ 𝑘 , 𝑎ˆ †𝑘 0

(4.7)

= 𝛿 𝑘 𝑘 0 . Expressing the flux and charge
operators in terms of 𝑎ˆ 𝑘 , 𝑎ˆ 𝑘 0 and substituting them into Eq. (4.5), we recover the
second-quantized Hamiltonian in the diagonal k-space basis
𝐻ˆ =

~𝜔 𝑘

+ 𝑎ˆ 𝑘 𝑎ˆ 𝑘 .

(4.8)

Note that, given the translational invariance of the capacitively coupled resonator
array circuit, it was expected that the the Hamiltonian would be diagonal in the
Fourier plane-wave basis (Bloch Theorem).
Also note that, for two capacitively coupled LC resonators, their coupling 𝐽 =
𝜔0
2 (𝐶𝑔 /(𝐶0 + 𝐶𝑔 )) is positive-valued [13] due to the fact that the anti-symmetric odd
mode of the circuit is the lower energy eigenmode. This results in positive-valued
photon hopping terms in the Hamiltonian, which directly lead to a maximum in
frequency at the Γ point and opposite directions of the phase velocity and group
velocity in the structure, as observed in other dispersive media [109–111].
Comparison to Tight-Binding Model
In the limit 𝐶0
𝐶𝑔 , the dispersion is well approximated to first order by a
tight-binding model with dispersion given by 𝜔 𝑘 = 𝜔 𝑝 + 2𝐽 cos (𝑘 𝑑), where 𝐽 =
𝜔0 (𝐶𝑔 /2𝐶0 ) is approximately the nearest-neighbor coupling between two resonators
of the resonator array, and 𝜔 𝑝 = (𝜔0 − 2𝐽) is the center of the passband. The
difference in the two dispersion relations reflects the coupling beyond nearestneighbor that arises due to the topology of the circuit, in which any two pairs of
resonators are electrically connected through some capacitance network dependent
on their distance. The magnitude of these interactions is captured in the Fourier
transform of the dispersion. Consider the Fourier transform for the annihilation
operator of the (localized) mode of the individual resonator located at position 𝑥,
𝑎ˆ 𝑘 = √

1 Õ

𝑎ˆ 𝑥 𝑒 −𝑖𝑘𝑥𝑑 .

(4.9)

Substituting Eq. (4.9) into Eq. (4.8), we arrive at the following real-space Hamiltonian,

39
𝐻ˆ = ~

ÕÕ

𝑉 (𝑥 − 𝑥 0) 𝑎ˆ †𝑥 𝑎ˆ 𝑥 0 ,

(4.10)

𝑥0

where 𝑉 (𝑥−𝑥 0) is the distance-dependent interaction strength between two resonators
located at positions 𝑥 and 𝑥 0, and is simply given by the Fourier transform of the
dispersion relation,
𝑉 (𝑥 − 𝑥 0) =

1 Õ
𝜔 𝑘 𝑒 −𝑖𝑘 𝑑 (𝑥−𝑥 ) .
𝑀 𝑘

(4.11)

For example, substituting the tight-binding dispersion 𝜔 𝑘 = 𝜔 𝑝 + 2𝐽 cos (𝑘 𝑑) into
Eq. (4.11) yields 𝑉 (𝑥 − 𝑥 0) = 𝜔 𝑝 𝛿𝑥,𝑥 0 + 2𝐽 𝛿𝑥−𝑥 0,1 + 𝛿𝑥−𝑥 0,−1 , which, upon substitution into Eq. (4.10), recovers the tight-binding Hamiltonian with only nearestneighbor coupling.
In Fig. 4.2a, we plot the magnitudes of nearest neighbor (𝑥 − 𝑥 0 = 1), next-nearest
neighbor (𝑥 − 𝑥 0 = 2), and next-next-nearest neighbor (𝑥 − 𝑥 0 = 3) couplings in the
capacitively coupled resonator array as a function of 𝐶𝑔 /𝐶0 calculated numerically
via the discrete Fourier transform of the dispersion relation. It is evident that for
small 𝐶𝑔 /𝐶0 the nearest neighbor coupling overwhelmingly dominates.
Qubit Coupled to Passband of a Waveguide
The Hamiltonian of a transmon-like qubit coupled to the metamaterial waveguide
via a single unit cell, where only the first two levels of the transmon (|𝑔i , |𝑒i) are
considered, can be written as (~ = 1, 𝑑 = 1),
𝐻ˆ = 𝜔𝑔𝑒 |𝑒i h𝑒| +

𝑔uc Õ † −
𝑎ˆ 𝑘 𝜎
ˆ + 𝑎ˆ 𝑘 𝜎
ˆ+ ,
𝜔 𝑘 𝑎ˆ †𝑘 𝑎ˆ 𝑘 + √
𝑀 𝑘

(4.12)

where 𝜔 𝑘 is given by Eq. (4.7). For an infinite array, the time-independent
Schrodinger equation 𝐻ˆ |𝜓i = 𝐸 |𝜓i has two types of solutions in the single photon
manifold: there are scattering eigenstates, which have an energy within the passband, and there are bound states that are energetically separated from the passband
continuum. We demonstrate this in the following analysis. First, we substitute
into 𝐻ˆ |𝜓i = 𝐸 |𝜓i the following ansatz for the quantum states of the composite
qubit-waveguide system, i.e. for dressed states of the qubit,
|𝜓i = 𝑐 𝑒 |𝑒, vaci +

𝑐 𝑘 𝑎ˆ †𝑘 |𝑔, vaci ,

(4.13)

Coupling (MHz)

104

100

10-3

-1

-2

Bandwidth (GHz)

Delay Per Resonator (ns)

10-4

Cg/C0
Figure 4.2: Comparison to tight-binding model and bandwidth-delay trade-off for capacitively
coupled resonator array a. Magnitude of nearest neighbor, next-nearest neighbor, and next-next
nearest neighbor inter-resonator couplings in an (infinite) capacitively coupled resonator array as a
function of 𝐶𝑔 /𝐶0 ratio. The bare resonator frequency was chosen to be 4.8GHz. b. Magnitude of
delay per resonator and bandwidth of the passband as a function of 𝐶𝑔 /𝐶0 ratio. The bare resonator
frequency was again chosen to be 4.8GHz, and the calculated delays are for frequencies in the middle
of the passband.

where |vaci corresponds to no excitations in the waveguide. Doing this substitution
and subsequently collecting terms, we arrive at the following coupled equations for
𝑐 𝑒 and 𝑐 𝑘 :

𝑔uc Õ 𝑐 𝑘
𝑐𝑒 = √
𝑀 𝑘 𝐸 − 𝜔𝑔𝑒
𝑔uc 𝑐 𝑒
𝑐𝑘 = √
𝑀 𝐸 − 𝜔𝑘

(4.14)
(4.15)

By further assuming that the waveguide supports a continuum of modes (which
is appropriate for a finite tapered waveguide, as described in the main text), the

1 Í

∫𝜋

sum can be changed into an integral 𝑘 → Δ𝑘 𝑘 Δ 𝑘 → Δ𝑘 −𝜋 d𝑘, where Δ 𝑘 =
2𝜋/𝑀. In this continuum limit, 𝐸 can be found by first substituting Eq. (4.15) into
Eq. (4.14) and subsequently dividing both sides by 𝑐 𝑒 , which yields the following
transcendental equation for 𝐸,
𝐸 = 𝜔𝑔𝑒 +
2𝜋

d𝑘

𝑔uc
𝐸 − 𝜔𝑘

(4.16)

where the integral on the right-hand side of Eq. (4.16) is known as the “self-energy”
of the qubit [22, 24, 25]. Note that in the opposite limit of a single resonator (where
𝜔 𝑘 takes on a single value and the density of states 𝜕𝜔
at
𝜕𝑘 becomes a delta–function
2 .
that value), Eq. (4.16) yields the familiar Jaynes-Cummings splitting 𝛿2 + 𝑔uc
Computation of the self-energy for 𝐸 such that 𝐸 > 𝜔 𝑘 or 𝐸 < 𝜔 𝑘 ∀𝑘, i.e.
for energies outside of the passband, yields real solutions for Eq. (4.16). On the
other hand, for energies 𝐸 inside the passband, the self-energy integral contains a
divergence at 𝐸 = 𝜔 𝑘 for real 𝐸 while there is no divergence if 𝐸 is allowed to
be complex with an imaginary component; thus Eq. (4.16) has complex solutions
when Re(𝐸) is inside the passband. While a Hermitian Hamiltonian such as the
one in Eq. (4.12) by definition does not contain complex eigenvalues, it can be
shown that the magnitude of the imaginary component of complex solutions of
Eq. (4.16) gives the decay rate of an excited qubit for a qubit dressed state with
energy in the passband. For further details, we suggest Refs. [19, 24, 25] to the
reader. Thus, the existence of complex solutions of Eq. (4.16) reflect the fact that
qubit dressed states with energy in the passband are radiative states that decay into
the continuum, characteristic of open quantum systems coupled to a continuum of
modes. In contrast, the dressed states with (real) energies outside of the passband
do not decay, and are known as qubit-photon bound states in which the photonic
component of the dressed state wavefunction remains bound to the qubit and is not
lost into the continuum.
For further analytical progress, we consider only the upper bandedge, and make
the effective-mass approximation. This approximation is tantamount to assuming
that the dispersion is quadratic, such that 𝜔 𝑘 ≈ 𝜔0 − 𝐽 𝑘 2 , which is obtained in the
limit of small 𝐶𝑔 /𝐶0 (where 𝜔 𝑘 is well approximated by the tight binding cosine
dispersion) and small 𝑘 (where cos(𝑘) to second order is approximately 1 − 𝑘 2 /2).
This approximation is appropriate when 𝜔𝑔𝑒 is close to the upper bandedge, where
the qubit is dominantly coupled to the Γ-point 𝑘 = 0 modes close to the bandedge due

42
to the van Hove singularity in the DOS, and when the lower bandedge is sufficiently
detuned from the qubit. Complimentary analysis for the lower bandedge can also be
done in the same manner. For a more detailed derivation, see Refs. [24, 112, 113].
Under the effective-mass approximation, the self-energy integral in Eq. (4.16) can be
easily analyzed by taking the bounds of integration to infinity, and is calculated to be
2 /2 𝐽 (𝐸 − 𝜔 ). For 𝜔 = 𝜔 , Eq. (4.16) then has the following two solutions:
𝑔uc
𝑔𝑒

𝐸 𝑏 = 𝜔0 + (𝑔uc
/4𝐽) 1/3 ,

(4.17)

𝐸𝑟 = 𝜔0 − 𝑒𝑖𝜋/3 (𝑔uc
/4𝐽) 1/3 .

(4.18)

These two solutions are indicative of a splitting of the qubit transition frequency
by the bandedge into two dressed states: a radiative state with energy 𝐸𝑟 in the
passband and a bound state with energy 𝐸 𝑏 above the bandedge. The magnitude dif4 /4𝐽) 1/3 , which is the frequency of
ference between the dressed state energies is 2(𝑔uc
coherent qubit-to-photon oscillations for an excited qubit at the photonic bandedge.
For the remainder of the analysis, we focus on the qubit-photon bound state of the
system. The wavefunction of the bound state with energy 𝐸 can be obtained by first
substituting Eq. (4.15) into Eq. (4.13), which yields

|𝜓 𝐸 i = 𝑐 𝑒

𝑔uc Õ
|𝑒i + √
𝑎ˆ †𝑘 |𝑔, vaci .
𝑀 𝑘

(4.19)

The qubit and photonic components of the bound state can be calculated from the
normalization condition of |𝜓 𝐸 i,

|𝑐 𝑒 |

1+
2𝜋

𝑔uc 2
d𝑘
= 1.
𝐸 − 𝜔𝑘

(4.20)

By assuming 𝐸 > 𝜔0 , the integral in Eq. (4.20) is calculated to be equal to
2 /4 𝐽 (𝐸 − 𝜔 ) 3 , which yields the following magnitude for the qubit component
𝑔uc
of the bound state,

1 𝐸 − 𝜔𝑔𝑒
|𝑐 𝑒 | = 1 +
2 𝐸 − 𝜔0

−1

(4.21)

whereas the photonic component is simply d𝑘 |𝑐 𝑘 | 2 = 1 − |𝑐 𝑒 | 2 . We can thus
see that when 𝐸 ≈ 𝜔𝑔𝑒 ≠ 𝜔0 , the qubit is negligibly hybridized with the passband
modes and |𝑐 𝑒 | 2 ≈ 1. On the other hand, as 𝜔𝑔𝑒 → 𝜔0 , we have |𝑐 𝑒 | 2 → 2/3,
indicating that the bound-state photonic component contains half as much energy
as the qubit component when the qubit is tuned to the bandedge.
We can also obtain the real-space shape of the photonic bound state by inserting
Eq. (4.9) into Eq. (4.19), where for a continuum of modes in 𝑘-space, we arrive at
the following photonic wavefunction,

𝑒 −|𝑥|/𝜆 𝑎ˆ †𝑥 |𝑔, vaci ,

(4.22)

up to a normalization constant, where 𝜆 = 𝐽/(𝐸 − 𝜔0 ) and the qubit is assumed to
reside at 𝑥 = 0. We thus find an exponentially localized photonic wavefunction for
the bound state. The localization length 𝜆 increases as 𝐽 increases, indicating that
the bound state becomes more delocalized across multiple resonators as the strength
of coupling between the resonators in the waveguide increases, whereas 𝜆 diverges
as the 𝐸 → 𝜔0 , which is associated with full delocalization of the bound-state as its
energy approaches the continuum of the passband.
Group Delay
Lowering the ratio 𝐶𝑔 /𝐶0 effectively lowers the photon hopping rate 𝐽 between
resonators, and can thus be chosen to significantly decrease the group velocity
of propagating modes of the structure, albeit at the cost of decreased bandwidth
of the passband modes. The group delay per resonator may be obtained from
𝜕𝜔 𝑘
the inverse of the
groupp velocity 𝜕𝑘 , while the bandwidth can be calculated to
be equal to 𝜔0 1 − 1/ 1 + 4𝐶𝑔 /𝐶0 ; both are plotted in Fig. 4.2b. Note that
although the group velocity approaches zero near the bandedge, a traveling pulse at
the bandedge frequency would experience significant distortion due to the rapidly
changing magnitude of the group velocity near the bandedge. At the center of
the passband where the dispersion is nearly linear, however, it is possible to have
propagation with minimal distortion.
Hence, in order to effectively use the coupled resonator array as a delay line, the
coupling should be made sufficiently high such that the bandwidth of propagating
modes (where the dispersion is also nearly linear) is sufficiently high, and the effect
of resonator frequency disorder due to fabrication imperfections is tolerable. After

44
the resonator coupling constraints have been met, the desired delay may be achieved
with a suitable number of resonators. It is thus evident that the ability to fabricate
resonators of sub-wavelength size with minimal frequency disorder is critical to
the effectiveness of implementing a slow-light waveguide with a coupled resonator
array.
An appropriate metric to compare the performance of the resonator array as a
delay line against dispersionless waveguides is to consider the delay achieved per
area rather than per length, in order to account for the transverse dimensions of the
resonators. In addition, typical implementations of delay lines with CPW geometries
commonly require a high degree of meandering in order to fit in a packaged device;
thus the pitch and turn radius of the CPW meandered trace also must be taken
into account when assessing delay achieved per area. However, by making certain
simplifying assumptions about the resonators it is possible to gain intuition on how
efficient the resonator array is in achieving long delays compared to a dispersionless
CPW. For the resonators implemented in the Main text (see Fig. 2.1), the capacitive
elements of the resonator are electrically connected to one end of the meander
while the opposite end of the meander is shunted to ground. This geometry is
therefore topologically similar to a 𝜆/4 resonator, and consequently the lengths of
the meander and a conventional 𝜆/4 CPW resonator will be similar to within an
order of magnitude for conventional implementations (here 𝜆 is the wavelength of
the CPW resonator mode).
Thus, by approximating that a single resonator of the array occupies the same
area as a 𝜆/4-section of CPW, a direct comparison between the delays of the two
different waveguides can be made. In the tight-binding limit, the group delay per
resonator in the middle of the passband is approximately equal to 1/2𝐽, where
𝐽 is the coupling between two resonators of the array. Hence, for N resonators,
array
𝑁/2𝐽
∼ 𝜔0 /𝐽, where 𝜏𝑑 is group delay and 𝑣 is the group velocity of
𝜏d /𝜏dCPW = 𝑁𝜆/4𝑣
light in the CPW. Hence, the resonator array is more efficient as a delay line when
compared to conventional CPW by a factor of approximately 𝜔0 /𝐽 (assuming group
velocity is approximately equal to phase velocity in the CPW). In practice, this
factor will also depend on the particular geometrical implementations of both kinds
of waveguide. For example, for the resonator array described in Fig. 2.1, 𝜔0 /𝐽 ≈ 120
and 𝜏d = 55 ns delay was achieved in the middle of the passband for a resonator
array of area 𝐴 = 6 mm2 . This constitutes a factor of 60 (500) improvement in delay
per area achieved over the CPW delay line in Ref. [66] (Ref. [114]).

C2g

C1g
C2

Transmission

Cg
C1

...

10
10
10

-2
-4
-6
4.65

4.7

Frequency (GHz)

4.75

Figure 4.3: CAD layout of boundary resonators and transmission spectrum of a resonator
array with impedance matching. a. CAD diagram showing the end of the finite resonator array,
including the boundary matching circuit (which in this case includes the first two resonators) and the
first unit cell. b. Corresponding circuit model of the end of the finite resonator array. c. Zoomed-in
SEM images of the first (left) and second (right) boundary-matching resonators. d. Transmission
spectrum of the full resonator array consisting of 22 unit cells and 2 boundary-matching resonators
on either end of the array (for a total of 26 resonators). Measured data is plotted as a red curve and
the circuit model fit is plotted as a black curve. Fit model parameters are given in the text.

4.3

Physical Implementation of Finite Resonator Array

Geometrical Design of Unit Cell
As shown in Fig. 2.1, the unit cell of the resonator array in this work includes a
lumped-element resonator formed from a tightly meandered wire with a large ‘head’
capacitance, and ‘wing’ capacitors which, in addition to providing the majority of
the capacitance to ground, are used to couple between resonators in neighbouring
unit cells. The meandered wire has a 1 𝜇m pitch and a 1 𝜇m trace width for tight
packing. At the top of the meander inductor is the ‘head’ capacitor and a pair of thin
metal capacitor strips which extend to the lateral edges of the unit cell (the ‘wing’
capacitors). The ground plane in between the resonators’ meander inductor and
the lateral wing capacitors acts as an electrical ‘fence’, restricting the meander from
coupling to neighboring resonators via stray capacitance or mutual inductance. This
ensured that the bulk of the coupling between resonators was from the resonators’
wing capacitive elements, thereby facilitating theoretical analysis of the structure
using a simple single resonator per unit cell model. Furthermore, we included
ground metal between the thin metal capacitor traces of neighbouring unit cell wing
capacitors. In this way, the ground planes above and below the resonator array

46
are tied together at each unit cell boundary, thereby suppressing the influence of
higher-order transverse, slot-line modes of the waveguide.
In addition, anticipating integration with Xmon qubits, we incorporated into our
unit cell design a Xmon shunting capacitance to ground, along with pads for facile
addition of Josephson Junctions. This ensured that the addition of a qubit at a
particular unit cell site in the resonator array minimally affected the capacitive
environment surrounding that unit cell, and prevented the breaking of translational
symmetry of the resonator array due to the addition of qubits. The capacitance
between the Xmon capacitor and the rest of the unit cell was designed to be ∼ 2 fF,
yielding a qubit-unit cell coupling of 𝑔uc ≈ 0.8𝐽.
Matching of the Finite Resonator Array to Input-Output CPWs
It has been previously shown that for a finite coupled cavity array, low-ripple transmission at the center of the passband is possible by appropriate variation of the
inter-resonator coupling coefficients for a few of the resonators adjacent to the ports,
effectively matching the finite periodic structure to the input-output ports [115].
In the case of capacitively coupled electrical resonators, modifying the coupling
capacitance in isolation results in a renormalization of the resonance frequency and
thus constitutes a scattering center for propagating light. Thus, concurrent modification of both the coupling capacitance and the shunt capacitance to ground for the
boundary resonators is necessary to achieve low-ripple transmission in the middle
of the passband, as previously shown in filter design theory [116]. By constraining
the total capacitance in each modified resonator to remain constant (and keeping the
inductance constant), the total number of parameters to adjust in order to achieve
low ripple transmission is merely equal to the chosen number of resonators to be
modified, resulting in a low-dimensional optimization problem. A filter design software such as Microwave Office can be used to provide initial guesses on the optimal
circuit parameters with high accuracy, which can then be further optimized.
In the Main text we present results on impedance matching of a resonator array
spanning 26 resonators to 50-Ω CPWs via modification of two resonators at each of
the array-CPW boundaries. The geometrical designs of the boundary resonators are
shown in Fig. 4.3. The number of boundary resonators to modify (2) was chosen
as a compromise between device simplicity and spectral bandwidth over which
matching occurs. In principle, however, more resonators could have been used for
matching of the finite structure to the ports in order to decrease the ripples in the

47
transmission passband near the bandedges. Referring to the notation in Fig. 4.3b,
the targets for the unit cell resonator and boundary resonator elements extracted
from Sonnet [117] electromagnetic simulations were 𝐶2𝑔 = 89 fF, 𝐶1𝑔 = 8.9 fF,
𝐶𝑔 = 6.47 fF, 𝐶2 = 269 fF, 𝐶1 = 351 fF, 𝐶0 = 353 fF, and geometric inductance 𝐿 0 =
2.92 nH. The individual capacitive and inductive elements have parasitic inductance
and capacitance, respectively, and thus were not simulated separately. Rather, circuit
parameters for the three different resonators were extracted by simulating the whole
resonator circuit. We extracted the circuit element parameters from these simulations
by numerically obtaining the dispersion for an infinite array of each of the three types
of resonators via the ABCD matrix method [77]. This yielded 𝜔0 and 𝐶𝑔 /𝐶0 ; 𝐶𝑔 was
obtained from the 𝐵 parameter of the 𝐴𝐵𝐶𝐷 matrix (which contains information
on the series impedance of the unit cell circuit). We found this method of extracting
parameters from simulation to give much higher accuracy when compared to other
approaches, such as simulating unit cell elements separately.
Figure 4.3d shows a plot of the measured transmission spectrum of the fabricated
26 unit cell slow-light waveguide based upon the above design and presented in the
Main text (c.f., Fig. 2.1). A circuit model fit to the measured transmission spectrum
yields the following circuit element parameters for boundary and central waveguide
unit cells: 𝐶2𝑔 = 87.5 fF, 𝐶1𝑔 = 7.3 fF, 𝐶𝑔 = 5.05 fF, 𝐶1 = 352.1 fF, 𝐶2 = 275.5 fF,
𝐶0 = 353.2 fF, and geometric inductance 𝐿 0 = 3.151nH. Based upon this model fit,
we were thus able to realize good correspondence (within 3%) between design and
measured capacitances to ground, while extracted coupling capacitances are systematically lower by approximately 1.5 fF. We attribute the systematically smaller
coupling to stray mutual inductance between neighboring meander inductors, which
tends to lower the effective coupling impedance between the resonators. The slightly
larger fit inductance compared to the design is to be expected as the kinetic inductance of the meander trace was not included in simulation. According to Ref. [118],
for a 1 𝜇m trace width and 120 nm thick aluminum wire, the expected increase in
the total inductance due to kinetic inductance is approximately 5% of the geometric
inductance, in reasonable correspondence to the measured value.
4.4

Disorder Analysis

Fluctuations in the bare resonance frequencies of the lumped-element resonators
making up the metamaterial waveguide breaks the translational symmetry of the
waveguide, and effectively leads to random scattering of traveling waves between
different Bloch modes. This scattering results in an exponential reduction in the

48
probability that a propagating photon traverses across the entire length of the waveguide. Furthermore, if the strength of scattering is large relative to the photon hopping
rate, Anderson localization of light occurs where photons are completely trapped
within the waveguide [76]. Thus, the aforementioned strategy for constructing a
slow-light waveguide from an array of weakly coupled resonators is at odds with the
inherit presence of fabrication disorder in any practically realizable device. Therefore, a compromise must be struck between choosing an inter-resonator coupling
low enough to provide significant delay, but high enough such that propagation
through the metamaterial waveguide is not significantly compromised by resonator
frequency disorder.
Fig. 4.4a shows numerical calculations of the transmission extinction in the metamaterial waveguide as a function of 𝜎/𝐽, where 𝜎 is the resonator frequency disorder.
This analysis was performed for a 50 unit cell waveguide, with 𝐶0 = 353.2 fF,
𝐶𝑔 = 5.05 fF, and 𝐿 𝑖 = 3.101 nH + 𝛿𝑖 . Here, 𝐿 𝑖 is the inductance of the ith unit cell
and 𝛿𝑖 are random inductance variations in each unit cell that give rise to a particular
resonator frequency disorder, 𝜎. These 𝐿 𝑖 were calculated by: (i) determining the
resonator frequencies of each unit cell by drawing from a Gaussian distribution
with mean 𝜔0 and variance 𝜎 2 , and (ii) solving for the corresponding inductances
given the resonator frequencies and a fixed 𝐶0 . Note that we modeled the disorder
as originating from inductance variations, rather than 𝐶0 or 𝐶𝑔 variations, based
on the fact that earlier work showed that disorder in superconducting microwave
resonators was primarily due to variations in kinetic inductance [119]. As we see
in Fig. 4.4a, in order for the average transmission to drop by less than 0.5dB (10%),
the normalized resonator frequency disorder must be less than 𝜎/𝐽 < 0.1 .
In order to quantify the resonator frequency disorder in our fabricated resonator
array, one can analyze the passband ripple in transmission measurements [119] (c.f.,
Fig. 2.1d,e). Given that the effect of tapering the circuit parameters at the boundary
is to optimally couple the normal modes of the structure to the source and load
impedances, the ripples in the passband are merely overlapping low-𝑄 resonances
of the normal modes. Therefore, we can extract the normal mode frequencies from
the maxima of the ripples in the passband, which will be shifted with respect the to
normal mode frequencies of a structure without disorder.
Furthermore, the mode spacing is dependent on the number of resonators and, in
the absence of disorder, follows the dispersion relation shown in Fig. 2.1c where
the dispersion is relatively constant near the passband center and starts to shrink

Disorder Extinction (dB)

0.1

0.2

0.4

0.5

ΔFSR (MHz)

σ/J

0.3

σ (MHz)

Figure 4.4: Disorder analysis of capacitively coupled resonator array. a. Numerically calculated
extinction as a function of disorder. Here, 𝜎 is the disorder in the bare frequencies of the (unit cell)
resonators making up the metamaterial waveguide and 𝐽 is the coupling between nearest-neighbour
resonators in the resonator array. 50 unit cells were used in this calculation, which included tapermatching sections at the input and output of the array that brought the overall passband ripple to
0.01dB. For a given disorder strength, 𝜎, disorder extinction was calculated by taking the mean of
the transmission across the passband for a given disorder realization, and subsequently averaging
that mean transmission over many disorder realizations. Note that the calculated values depend on
the number of unit cells. b. Numerically calculated variance in normal mode frequency spacing as a
function of disorder. See text for details on the method of calculation of ΔFSR . Dashed line indicates
the experimentally measured ΔFSR , which was extracted from the data shown in Fig. 2.1d.

50
near the bandedges. In the presense of disorder, however, this pattern breaks
down as the modes become randomly shifted. Our approach was therefore as
follows. Starting with the fit parameters presented in Appendix 4.3, we simulated
transmission through the metamaterial waveguide for varying amounts of resonator
frequency disorder, 𝜎. For each level of disorder, we performed simulations of
500 different disorder realizations, and for each different disorder realization, we
computed the standard deviation in the free spectral range of the ripples, ΔFSR . This
deviation in free spectral range was then averaged over all disorder realizations for
each value of 𝜎, yielding an empirical relation between ΔFSR and 𝜎.
The numerically calculated empirical relation between variation in free spectral
range and frequency disorder is plotted in Fig. 4.4b. Note that the minimum of
ΔFSR at 𝜎 = 0 is set by the intrinsic dispersion of the normal mode frequencies of
the unperturbed resonator array. As such, in order to yield a better sensitivity to
disorder, we chose to only use the center half of the passband in our analysis where
dispersion is small. From the data in Fig. 2.1d, we calculated the experimental
ΔFSR . Comparing to the simulated plot of Fig. 4.4b, this level of variance in the
free spectral range results from a resonator frequency disorder within the array at
the 1 MHz level (or 2 × 10−4 of the average resonator frequency), corresponding to
𝜎/𝐽 ≈ 1/30. We have extracted similar disorder values across a number of different
metamaterial waveguide devices realized using our fabrication process.
4.5

Modeling of Qubit Q1 Coupled to the Metamaterial Waveguide

In this section, we present modeling of the interaction between Q1 and the metamaterial waveguide. Note that, while we observe dynamics that are due to emission and
propagation of single-photon radiation field states, which are non-classical states of
light, in the single-excitation limit the dynamics of the qubit can also be described by
a classical circuit model, where the qubit is represented by a faux resonator. Thus,
here we share both viewpoints of analysis, and we employ two separate models to
represent our system: a tight-binding model with nearest and next-nearest neighbor
coupling which we analyze via a numerical master equation solver, and a classical
circuit model (shown in Fig. 4.6). We find excellent agreement between the two
models.

51
Tight-Binding Model
System Hamiltonian and Model Formalism
For transient time-domain simulations, instead of using the Hamiltonian presented in
equation 4.12, we instead employ the following tight-binding model (with individual
resonator positions denoted by the indices 𝑥 and 𝑖)

𝐻ˆ = 𝜔𝑔𝑒 |𝑒i h𝑒| +

𝜔𝑥 𝑎ˆ †𝑥 𝑎ˆ 𝑥 + (𝐽𝑥 𝑎ˆ †𝑥 𝑎ˆ 𝑥+1 + 𝐽𝑛𝑛𝑛 𝑎ˆ †𝑥 𝑎ˆ 𝑥+2

𝑥=1

+ ℎ.𝑐) +

𝑔𝑖 𝜎
ˆ 𝑥 𝑎ˆ𝑖† + 𝑎ˆ𝑖

(4.23)

𝑖=1,3,4

where 𝑀 is the number of resonators, 𝜔𝑥 are the frequencies of the individual
resonator modes, and, as discussed in section 4.2, in our parameter regime the
capacitively coupled resonator array Hamiltonian can be well approximated as a
tight-binding Hamiltonian with dominant nearest-neighbor coupling 𝐽𝑥 and small
(∼ 𝐽/100) next-nearest neighbor coupling 𝐽𝑛𝑛𝑛 (which we keep as a constant in the
model for simplicity). In our model, for all unit cells, we set 𝜔𝑥 = 𝜔 𝑝 = 𝜔0 − 2𝐽,
which is the passband center frequency and constitutes the bare resonator frequency
𝜔0 renormalized by its coupling to neighboring resonators; however, for the taper
resonators, we introduce moderate detunings in order to capture the weak reflections
within the slow-light waveguide evidenced by the measured data (see Fig. 2.3).
Further, we include qubit coupling to multiple resonators in the array in our model
with couplings 𝑔𝑖 , where 𝑖 indicates resonator position in the array, in order to capture
both 𝑔uc and the weak non-local coupling of the qubit to a few of the neighboring
unit cells that was evidenced by the measured data.
Going into the rotating frame of the passband center frequency 𝜔 𝑝 and applying the
rotating wave approximation (RWA) to remove counter-rotating terms, we arrive at
the following Hamiltonian

𝐻ˆ = Δ𝑔𝑒 |𝑒i h𝑒| +

𝛿𝑥 𝑎ˆ †𝑥 𝑎ˆ 𝑥 + (𝐽𝑥 𝑎ˆ †𝑥 𝑎ˆ 𝑥+1 + 𝐽𝑛𝑛𝑛 𝑎ˆ †𝑥 𝑎ˆ 𝑥+2

𝑥=1

+ ℎ.𝑐) +

𝑔𝑖 𝑎ˆ𝑖† 𝜎
ˆ − + 𝑎ˆ𝑖 𝜎
ˆ+

(4.24)

𝑖=1,3,4

where Δ𝑔𝑒 = 𝜔𝑔𝑒 − 𝜔 𝑝 and 𝛿𝑥 = 𝜔𝑥 − 𝜔 𝑝 ; see Fig. 4.5a for a visual diagram of the
model. It can be shown that the Hamiltonian in equation 4.24 preserves the number of

Figure 4.5: Master equation numerical simulations of our qubit-slowlight waveguide system.
a. Diagram of tight-binding model used in simulations. Simulation parameters are described in
the text. Note that the next-nearest neighbor coupling 𝐽𝑛𝑛𝑛 , which is present in the model for all
resonators, is omitted from the diagram for readability purposes. b. Simulation of Fig. 2.3b dataset.
Bandedges are highlighted in dashed yellow lines, while dashed black lines are guides to the eye. c.
Scatter plot of the eigenenergies of the Hamiltonian in equation 4.24 with Δ𝑔𝑒 /(2𝜋) = 83 MHz (in
the single excitation manifold) offset by 𝜔 𝑝 . The orange curve is a plot of the dispersion relation
(see equation 4.7). The eigenmode with energy outside of the passband corresponds to the bound
state of the system |𝑏i d. Plot of photonic states of the system as a function of position 𝑥. Top
panel: plot of the photonic wavefunction of the bound eigenstate of the system |𝑏i in open
pÍ red dots;
“norm" indicates that the photonic wavefunction coefficients h𝑥|𝑏i are normalized by
𝑥 |h𝑥|𝑏i| ,
where |𝑥i corresponds to the state
p |01 , 02 , . . . , 1 𝑥 , . . . , 0 𝑀 ; 𝑔i. The solid black line corresponds to
a plot of 𝐴𝑒 | 𝑥−3 |/𝜆 , where 𝜆 = 𝐽/(𝐸 𝑏 − 𝜔0 ) and 𝐴 is a normalization constant. Bottom panel:
plot of the photonic portion of the simulated qubit-waveguide state after 𝑡 = 90 ns. The solid
blue line corresponds to a simulation with initial state |01 , 02 , . . . , 0 𝑀 ; 𝑒i; the dashed black line
𝑚 refers to the scaled density matrix element
corresponds
to a simulation with initial state |𝑏i. 𝜌 𝑛𝑜𝑟
𝑥𝑥
Í
𝜌 𝑥 𝑥 / 10
𝑥=1 𝜌 𝑥 𝑥 . This particular scaling is chosen because it similarly scales the photonic part of
the state within the first 10 resonators of the array, thereby aiding visual comparison between the
blue and dashed black curves. e. Comparison of the dynamics simulated by a modified tight-binding
model of a qubit coupled to a metamaterial waveguide (left), and by population equations of motion
derived in Ref. [22] (right). Refer to (b) for colorbar. Both models assume 𝑔uc /2𝜋 = 19 MHz, as
4 /4𝐽) 1/3 in place of
well as 𝐽/2𝜋 = 33 MHz. See text for description of modified model. We use (𝑔uc
𝛽 for simulations using equations 2.21-2.28 from Ref. [22].

ˆ 𝑁ˆ = 0 with 𝑁ˆ = Í 𝑀 𝑎ˆ †𝑥 𝑎ˆ 𝑥 +𝜎
excitations 𝑁 by noting that the commutator 𝐻,
ˆ +𝜎
ˆ −.
𝑥=1
Consequently, the dynamics of the system can be partitioned into subspaces with
fixed excitation number, and for the purposes of modeling the data in Fig. 2.3
of a qubit’s radiative dynamics in a structured photonic reservoir, we only need to
consider the subspaces of 𝑁 = 0, 1. The Hamiltonian in this reduced subspace can be
computed by explicitly evaluating the matrix elements h𝜙| 𝐻ˆ |𝜙0i between different

53
states {|𝜙i} in the zero and single excitation manifold, and subsequently directly used
in numerical master equation simulations. Finally, while the Hamiltonian in equation
4.24 generates the unitary dynamics of the system, the external loading of the system
to the input/output 50-Ω waveguides is incorporated into the model via dissipation
with rate 𝜅50Ω in the first and last resonators of the array, which is generated in our
master equation simulations via collapse operators which transfer population from
the single excitation states |11 , 02 , 03 , . . . , 0 𝑀 ; 𝑔i and |01 , 02 , 03 , . . . , 1 𝑀 ; 𝑔i to the
(trivial) zero-excitation ground state of the system |01 , 02 , 03 , ..., 0 𝑀 , 𝑔i. Note that
master equation simulations of the qubit’s non-Markovian radiative dynamics are
only possible here due to the fact that we are explicitly simulating all the photonic
degrees of freedom of the slow-light waveguide in addition to the qubit’s degrees
of freedom. A Lindbladian master equation simulation of solely the qubit’s degrees
of freedom, with the photonic degrees of freedom traced out, would not capture
its non-Markovian radiative dynamics. Moreover, a simulation of the entire qubitwaveguide system is only amenable here due to our restriction of the Hilbert space
to its low-energy sector, and would quickly grow intractable if higher number of
excitations were allowed.
Referring to equation 4.24 and Fig. 4.5a, our model assumed the following parameters (2𝜋 factors are omitted for readability): 𝑀 = 50, 𝛿1 = 𝛿50 = 𝛿0 =
−13.9 MHz, 𝛿2 = 𝛿24 = 𝛿27 = 𝛿49 = 𝛿00 − 4.7 MHz, 𝛿25 = 𝛿26 = 𝛿000 =
323 MHz, 𝐽1 = 𝐽24 = 𝐽26 = 𝐽49 = 𝐽 0 = 44.1 MHz, 𝐽2 = 𝐽23 = 𝐽27 = 𝐽48 = 𝐽 00 =
32.47 MHz, 𝐽25 = 𝐽 000 = 349 MHz, 𝐽𝑛𝑛𝑛 = 0.3 MHz, all other 𝐽𝑥 = 𝐽 = 32.52 MHz,
all other 𝛿𝑥 = 0, and 𝜅 50Ω = 169.92 MHz (note that the values of 𝛿000 and 𝐽 000 are very
different from other values in order to accurately capture the circuit of the waveguide’s bend section as discussed in the Main Text). Note that these parameters are
consistent with the circuit parameters of the model shown in Fig. 4.6 that is later
discussed. Furthermore, in the model we coupled the qubit to the first, third, and
fourth resonators of the array (as opposed to just the third resonator), with couplings
𝑔1 = 2.2 MHz, 𝑔2 = 𝑔uc = 26.4 MHz, 𝑔3 = 3.5 MHz. Physically, the coupling to
resonators 1 and 4 was not intentional and was due to parasitic capacitance. We set
𝑔2 = 0 in the model because the second metamaterial resonator was not expected to
parasitically couple to the qubit as strongly as the first and fourth resonator due to the
absence of an interdigitated capacitor or an integrated Xmon shunting capacitance
(see Fig. 4.3 for images of the second resonator of the metamaterial waveguide).
The 𝑔1 and 𝑔4 parasitic couplings were crucial to reproduce some of the subtle
features in the measured data; this will be discussed in detail below.

54
Dynamical Simulations and Eigenenergy Analysis
Fig. 4.5b shows the simulated dynamics from numerical master equation simulations
as a function of Δ𝑔𝑒 (note that bare qubit frequency 𝜔 𝑝 + Δ𝑔𝑒 is shown in the plot
instead for comparison purposes to Fig. 2.3) with initial state |01 , 02 , . . . , 0 𝑀 ; 𝑒i. It is
evident that there is agreement between Fig. 4.5b and the measured data in Fig. 2.3b,
indicating that our model captures the salient dynamical features of our measured
data. Furthermore, with the Hamiltonian in Equation 4.24, we can numerically
calculate its eigenstates and the eigenenergy spectrum; as an example, the spectrum
when Δ𝑔𝑒 /(2𝜋) = 83 MHz is plotted in Fig. 4.5c. Fig. 4.5c shows a band of
states within the passband, and a state with energy outside of the passband. Because
𝑀 = 50, the Hamiltonian is that of a finite-sized system and the band of states within
the passband represent the normal modes of the finite waveguide structure; however,
in the presence of input/output waveguides, they represent a band of scattering states
that support wave propagation between the input/output waveguides. The state with
energy outside of the band, however, is the bound eigenstate |𝑏i. We calculate
bound state energies as a function of bare qubit frequency Δ𝑔𝑒 , and converting bare
qubit frequency to the physically applied flux through the SQUID loop used to
tune the qubit frequency Φ (via measured qubit minimum/maximum frequencies
and the extracted anharmonicity), we numerically obtain the predicted energy of
the system’s bound eigenstates as a function of flux bias and plot it on Fig. 2.2d
as dashed black lines. As Fig. 2.2d shows, we obtain good quantitative agreement
between the prediction of our model and the spectroscopically measured bound state
energies of the qubit-waveguide system.
In our model, the 𝑔3 coupling primarily sets the coupling of the qubit to the metamaterial waveguide. Its magnitude relative to the 𝐽 between the unit cells, along with
the qubit frequency 𝜔0𝑔𝑒 (Φ), predominantly determines the frequency of oscillations
near the bandedge, as well as the decay rate into the waveguide in the passband. In
the absence of other parasitic couplings, this decay rate is theoretically determined to
2 /𝑣(𝜔0 (Φ)) [112], where 𝑣(𝜔0 (Φ)) is the group velocity of the metamatebe ∼ 𝑔uc
𝑔𝑒
𝑔𝑒
rial waveguide at the qubit-waveguide interaction frequency 𝜔0𝑔𝑒 (Φ). The parasitic
coupling 𝑔4 , however, is necessary to replicate the asymmetry in the dynamics near
the upper and lower bandedges. This is because the lower bandedge modes have an
oscillating charge distribution between unit cells, while the upper bandedge modes
have a slowly-varying charge distribution across the unit cells (which is typical of
1D tight-binding systems). The parasitic coupling of the qubit to the neighboring

55
unit cell therefore has the effect of lowering the qubit coupling to the lower bandedge
modes due to cancellation-effects arising from the opposite charges on neighbouring resonators for lower bandedge modes. On the other hand, coupling of the qubit
to the upper bandedge modes which have slowly-varying charge distributions, is
enhanced.
In addition, in simulations, the onset of oscillations seen at 𝜏 ≈ 115 ns could
be delayed or advanced by increasing or decreasing the number of resonators in
between the qubit and the bend in the metamaterial waveguide model, while it could
be removed altogether by removing the bend section. This indicated that these late
time oscillations are a result of spurious reflection of the qubit’s emission at the
bend, due to the imperfect matching to the 50-Ω coplanar waveguide in between
the two resonator rows (which is manifested in this model through parameters 𝛿000
and 𝐽 000). Note that this impedance mismatch and reflections are amplified near the
bandedges, where the Bloch impedance rapidly changes.
Photonic State Spatial Analysis
In the main text, the observed qubit emission dynamics into the slow-light waveguide
are described in terms of the interplay of the qubit-waveguide dressed states; in
particular, the bound and continuum dressed states of the qubit-waveguide system.
Here we further elucidate this description of our system via our modeling, using as
an illustrative example the dynamics of the system when the qubit is tuned 18 MHz
above the upper bandedge (Δ𝑔𝑒 /(2𝜋) = 83 MHz), corresponding to the brown curve
in Fig. 2.3c.
Firstly, in the main text, we assert that initializing the qubit in state |𝑒i with its
frequency in the proximity of the passband effectively initializes it into a superposition of bound and continuum dressed states. This can be explicitly verified by
first numerically calculating the eigenstates and the eigenenergy spectrum of the
Hamiltonian, as was done for Fig. 4.5c. As previously discussed, the state with
energy outside of the band is the bound eigenstate |𝑏i, and the photonic component
of its wavefunction is plotted in the top panel of Fig. 4.5d. It is evident from
Fig. 4.5d that the photonic component of the bound state wavefuncion is localized
around resonator 3, which is the unit cell that the qubit is predominantly coupled
to. As discussed in section 4.2, the bound state is exponentially localized with
localization length approximately 𝜆 = 𝐽/(𝐸 𝑏 − 𝜔0 ) where 𝐸 𝑏 is the energy of
the bound state; this theoretical photonic wavefunction is plotted in the top panel

56
of Fig. 4.5d with a solid black line, and shows good agreement with the numerically calculated |𝑏i wavefunction plotted in red open dots. Numerically calculating
the overlap between the |01 , 02 , 03 , ..., 0 𝑀 , 𝑒i state and the bound eigenstate yields
|h𝑏|01 , 02 , 03 , ..., 0 𝑀 , 𝑒i| 2 ≈ 0.8, agreeing well with equation 4.21.
Secondly, in the main text we also assert that the amplitude of the early-time oscillations quickly dampen away as the energy in the radiative continuum dressed
state is quickly lost into the waveguide, while the energy in the bound state remains
localized around the qubit, albeit slowly decaying (details of this slow decay are
given in the next paragraph). In order to illustrate this point, in the bottom panel
of Fig. 4.5d we plot the photonic portion of the system’s state at time 𝑡 = 90 ns, at
which point the early-time oscillations have subsided and the qubit can be observed
to be slowly decaying. It is evident that while part of the state is delocalized in the
array, a significant portion is still localized around the qubit location; this portion
corresponds to the bound state portion of the initial state |01 , 02 , 03 , ..., 0 𝑀 , 𝑒i after
time evolution.
Thirdly, in order to understand the slow decay of the qubit following the early-time
oscillations, note that a non-negligible proportion of the bound state wavefunction
is found on resonator 1, the taper resonator directly coupled to output waveguide,
signifying finite overlap between the bound state and the external 50-Ω environment
of the output waveguide. This overlap constitutes the dominant intrinsic loss channel
for the bound state and leads to its slow decay, which in the 𝑡 → ∞ limit results
in the full decay of qubit even if its frequency is tuned outside the passband. Near
the bandedges, it is this loss that results in a slow population decay as compared to
the initial fast dynamics in the data (see top panel of Fig. 2.3c for a clear example),
and results in the feature highlighted by dashed black lines in Fig. 4.5b. This
feature would be flat for an infinite-sized resonator array and there would be partial
“population trapping” [25] of the qubit in the 𝑡 → ∞ limit if its bare frequency was
detuned from the passband and there were no other intrinsic loss channels. Note
that the 𝑔1 coupling between the qubit and the resonator directly coupled to the
50-Ω port is necessary to quantitatively replicate the slow decay rates of the qubit
when its frequency is outside of the passband. In the absence of the 𝑔1 coupling,
this overlap was not sufficiently high in the simulations given the coupling of the
qubit to the metamaterial waveguide (extracted from separate measurements in the
passband). Therefore, this overlap was made larger, while minimizing the increase
to the overall coupling of the qubit to the metamaterial waveguide, by incorporating

57
the small 𝑔1 coupling to the first resonator of the array.
Finally, it can be observed in Fig. 2.3b and Fig. 4.5 that there are differences in
both duration and amplitude between the early-time oscillations and the late-time
oscillations that occur at 𝜏 ≈ 115 ns. This is because, when the qubit frequency is
near the bandedges, the reflected emission is distorted through its propagation in the
metamaterial waveguide due to the significant dispersion near the bandedges. This
results in a spatio-temporal broadening of the emitted radiation, which is evident in
the bottom panel of Fig. 4.5d. The frequencies of both sets of oscillations, however,
are set by 𝑔uc and 𝐽 as discussed in the main text.
Comparison to Paradigmatic Model of Spontaneous Emission Near the Edge
of a Photonic Bandgap
As alluded to in the main text, the early-time oscillations observed in our work are,
qualitatively, a generic feature of the interaction between a qubit and a bandedge in
a dispersive medium, and not merely an attribute of our specific system. In order
to illustrate this point, in Fig. 4.5e, we further compared the initial oscillations to
the theory presented by John and Quang in Ref. [22] of a qubit whose frequency
lies in the spectral vicinity of a bandedge. The model assumed for Ref. [22] was
that of an atom (qubit) with point dipole coupling to an infinite periodic dielectric
environment, whose frequency is in the spectral vicinity of only a single bandedge.
Thus, in order to make a comparison to this theory, we changed the model of our
system described by equation 4.24 and Fig. 4.5a in the following manner: (i) we
removed the parasitic couplings of the qubit to neighboring unit cells, in order to
simplify the coupling to a single point coupling, (ii) we increased the size of the
array and moved the qubit to the middle in order to remove boundary effects from
the dynamics, (iii) we reduced the overall coupling of the qubit to the metamaterial
waveguide so it predominantly couples to only the bandedge it is least detuned from.
Note, however, that the dispersion relation of the waveguide is different than the
dispersion assumed in Ref. [22]. Nonetheless, above the bandedge, we see good
qualitative agreement between the dynamics modeled both by the modified model
and the population equation of motion derived in Ref. [22] (in particular, equation
2.21), with both simulations exhibiting very similar oscillatory decay to what is
observed in Fig. 4.5b and 2.3b. This further confirms our interpretation of the
early-time non-Markovian dynamics in Fig. 2.3 discussed in the Main text: that the
non-exponential oscillatory decay is due to the interaction between the qubit and the

58
strong spike in the density of states at the bandedge.
Circuit Model
In addition to dynamical master equation simulations, we also performed modeling
via classical circuit analysis, where the qubit is represented by a linear resonator; this
is an accurate representation of the qubit-waveguide system in the single-excitation
limit. Time-resolved dynamical simulations were performed with the LTSpice
numerical circuit simulation package, while frequency response simulations were
performed with Microwave Office and standard circuit analysis. Our model, shown in
Fig. 4.6, assumes the following metamaterial waveguide parameters: 𝐶2𝑔 = 92.5 fF,
𝐶1𝑔 = 7.8 fF, 𝐶𝑔 = 5.02 fF, 𝐶2 = 273 fF, 𝐶1 = 351.2 fF, 𝐶0 = 353.2 fF, and
𝐿 0 = 3.099 nH, which were obtained from fitting the transmission through the
metamaterial device shown in Fig. 2.2a with the qubit detuned away (600 MHz)
from the upper bandedge. While in principle there are three independent parameters
for every resonator (capacitance to ground, coupling capacitance, and inductance to
ground), the set of metamaterial parameters above in addition to the qubit parameters
were sufficient to achieve quantitative agreement between simulations and our data.
Our model utilizes a qubit capacitance (excluding the capacitance to the metamaterial
waveguide) of 𝐶Σ = 77.8 fF, which, when assuming 𝐸 𝑐 ≈ −~𝜂, is consistent with
measurements of the anharmonicity that was extracted by probing the two-photon
transition between the |𝑔i and | 𝑓 i states. Furthermore, in the model we coupled
the qubit to the first, third, and fourth resonators of the array, with capacitive
couplings 𝐶1𝑞𝑔 = 0.16 fF, 𝐶3𝑞𝑔 = 1.9 fF, and 𝐶4𝑞𝑔 = 0.25 fF, while 𝐶2𝑞𝑔 = 0 fF, for
reproducing both the dominant and the subtle features in the measured data due to
the same reasons described in the preceding discussion.
Time Domain
Figure 4.6b shows the simulated dynamics of our circuit model as a function of
bare qubit frequency (where the qubit inductance was swept to change the bare
qubit frequency). It is evident that there is agreement between Fig. 4.6b and the
measured data in Fig. 2.3b, indicating that our circuit model captures the salient
dynamical features of our measured data. Moreover, we find excellent agreement
between our circuit model and the tight-binding model presented in the preceding
discussion, which was expected given that the parameters of the circuit model map
nearly directly to the parameters of the tight-binding model. Thus, both models are

Figure 4.6: Circuit model simulations of the qubit-slowlight waveguide system. a. Full circuit
model used in simulations. All inductors were made equivalent, with inductance 𝐿 0 . Parameters are
further discussed in the text. b. Simulation of Fig. 2.3b dataset. Intensity plot is of energy in the fauxqubit resonator normalized by the initial energy; this simulated time-dependent normalized energy
corresponds directly to the qubit’s excited state population measurements of Fig. 2.3b. Simulation
parameters are described in the text. Bandedges are highlighted in dashed yellow lines, while dashed
black lines are guides to the eye. c. Simulation of Fig. 2.2d dataset. Circuit model and simulation
parameters are described in the text. Simulations were done with the aid of the Microwave Office
software package.

appropriate for analyzing the data of Figure 2.3, and the insights into the system
gained from the tight-binding model in the preceding discussion directly carry over
to this circuit model.
Frequency Domain
In addition to time-domain simulations of our circuit model representing the fabricated qubit-waveguide system, in Fig. 4.6c we plot an intensity color plot of the
transmission through the slow-light waveguide as the bare qubit frequency is tuned
across the passband using the circuit model (c.f., the corresponding measurement

60
data plotted in Fig. 2.2d). Note that in order to capture the background transmission
levels as well as the interaction of the qubit with the background transmission, we
included a small direct coupling capacitance of 0.75 fF between the first and last
resonators of the array. These two resonators have the largest crosstalk. This is
due to the large portion of charge contained in the interdigitated capacitors between
the resonators and the input-output waveguides. In simulations without this background transmission, the qubit mode break-up near the bandedge and signatures of
the bound-state outside of the passband were significantly weaker.
In addition, the series capacitance of the boundary resonators coupled to the inputoutput waveguides was made 7 fF higher than the series capacitance of the boundary
resonators coupled to the short CPW section in the bend, which is due to the
proximity of the large bondpads used to probe the waveguides. Our simulations are
in excellent qualitative agreement with the data presented in Fig. 2.2d. They also
capture the spectroscopic non-Markovian features of our data – the repulsion of the
bound state’s energy from the bandedge and the persistence of the bound state even
when the bare qubit frequency overlaps with the passband (see Refs. [61, 62, 112]
for further details).
4.6

Modeling of Qubit Coupled to Dispersion-less Waveguide in Front of
Mirror

In this section, we present the modeling of the time-delayed feedback phenomenon
described in the main text. Here, we employ a dispersion-less waveguide in our
model instead of our slow-light waveguide in order to compare our data to the
dynamics of an ideal scenario where pulse distortion and propagation losses are
absent. We employ a dispersion-less waveguide with equivalent round-trip delay of
𝜏d = 227 ns to the slow-light waveguide. The theoretical model we use is described
at length in Ref. [41]; below we briefly summarize the derivation of the model found
in this reference.
Ref. [41] starts with the following Hamiltonian, where the coupling to different
waveguide modes is now allowed to vary as a function of 𝑘,
𝐻ˆ = 𝜔𝑔𝑒 |𝑒i h𝑒| +

𝑑𝑘𝜔 𝑘 𝑎ˆ †𝑘 𝑎ˆ 𝑘 +

𝑑𝑘𝑔 𝑘 𝑎ˆ †𝑘 𝜎
ˆ − + 𝑎ˆ 𝑘 𝜎
ˆ+ ,

(4.25)

and the same single-excitation ansatz of equation 4.13, but with time-dependent
coefficients 𝑐 𝑒 (𝑡) and 𝑐 𝑘 (𝑡) (and where a continuum of modes is already assumed).

61
Following similar analysis to Appendix 4.2, equations 4.25 and 4.13 are substituted
into the time-dependent Schrodinger equation 𝜕𝑡 |𝜓(𝑡)i = −𝑖 𝐻ˆ |𝜓(𝑡)i, and after
collecting terms and going into the rotating frame of the qubit, the authors arrive at
the following system of coupled differential equations:

𝑐¤𝑒 (𝑡) = −𝑖

𝑑𝑘𝑔 𝑘 𝑐 𝑘 (𝑡),

𝑐¤𝑘 (𝑡) = −𝑖Δ 𝑘 𝑐 𝑘 (𝑡) − 𝑖𝑔 𝑘 𝑐 𝑒 (𝑡).

(4.26)
(4.27)

where Δ 𝑘 = 𝜔𝑔𝑒 − 𝜔 𝑘 . The authors then explicitly integrate equation 4.27 to
obtain a solution for 𝑐 𝑘 (𝑡), and substitute that solution into equation 4.26. In
order to evaluate the resultant equation of motion for 𝑐 𝑒 (𝑡), the authors make the
following assumptions: (i) they assume the dispersion is linearized around the
qubit frequency such that 𝜔 𝑘 = 𝜔𝑔𝑒 + 𝑣(𝑘 − 𝑘 0 ), where 𝑣 is the group velocity,
and (ii) 𝑔 𝑘 = Γ1D 𝑣/𝜋 sin 𝑘𝑥0 , where 𝑥 0 is the qubit position in the waveguide.
The particular form of 𝑔 𝑘 is chosen by asserting that the field assumes a sin 𝑘𝑥
spatial profile such that the field fulfills the boundary condition of being zero at
the waveguide termination; thus the field strength at the qubit is sin 𝑘𝑥0 . With
these expressions for 𝜔 𝑘 and 𝑔 𝑘 , the resultant equation of motion for 𝑐 𝑒 (𝑡) can be
simplified to the following form

𝑐¤𝑒 (𝑡) = −

Γ1D 𝑖2𝑘 0 𝑥0
Γ1D
𝑐 𝑒 (𝑡) +
𝑐 𝑒 (𝑡 − 𝜏d )𝜃 (𝑡 − 𝜏d )

(4.28)

where 𝜏d is the round-trip delay and 𝜃 is the heavyside step function; the first term on
the right-hand side is responsible for the decay of the qubit, while the second term
is responsible for photon re-absorption. Equation 4.28 is finally solved via methods
described in Ref. [120], yielding the following analytic expression for the dynamics
of a qubit excited state population when coupled to a semi-infinite dispersion-less
waveguide:

𝑐 𝑒 (𝑡) = 𝑒

Γ1D 𝑡/2

Õ 1 Γ1D
𝑛!

𝑖𝜙+Γ1D 𝜏d /2

𝑛
(𝑡 − 𝑛𝜏d ) 𝑛 𝜃 (𝑡 − 𝑛𝜏d )

(4.29)

where 𝜙 = 2𝑘 0 𝑥 0 is the round-trip phase gained by the propagating emitted pulse.
Substituting Γ1D /(2𝜋) = 21 MHz and 𝜏d = 227 ns into equation 4.29, we obtain the
magenta curve plotted in Fig. 2.4b. As discussed in the main text, our measured

10-1
10-1

10-1
10-1
10-1

200

400
Time (ns)

600

Figure 4.7: Markovian to Non-Markovian crossover. Replots of the five (white) line cuts of
Fig. 2.4c, with accompanying theoretical predictions for emission of a qubit into a dispersionless,
lossless semi-infinite waveguide. In the theoretical model, 𝜏d was maintained fixed for all simulations,
while the qubit emission rate Γ1D and round-trip phase 𝜙 were allowed to vary as fit parameters in to
capture the effects of the changing flux-modulation amplitude, which not only changes Γ1D but also
causes a residual DC-shift of the average qubit frequency [85], which in turn affects 𝜙. Moreover, a
thermal qubit population of 2.4% was assumed. From top panel to bottom panel, the fit parameters
Γ1D and 𝜙 are, respectively: Γ1D /2𝜋 = 0.17 MHz, 𝜙 = 𝜋/2.6; Γ1D /2𝜋 = 0.6 MHz, 𝜙 = 𝜋/2.6;
Γ1D /2𝜋 = 1.8 MHz, 𝜙 = 𝜋/2.1; Γ1D /2𝜋 = 5 MHz, 𝜙 = 𝜋/2.6. Note that the parameter 𝜙 has
negligible effect for dynamics involving large Γ1D where revival events are clearly discernible, and
for dynamics involving small Γ1D , 𝜙 simply modulates the emission rate. However, for intermediate
Γ1D such as Γ1D /2𝜋 = 0.6 MHz, 1.8 MHz, the shapes of the population dynamics curves are sensitive
to 𝜙.

63
dynamics compare favorably to the ideal scenario of no dispersion-induced distortion
of the traveling emitted pulse, as well as no propagation losses, captured by the model
discussed above. Thus, the limited recurrence observed can be mostly attributed to
emission into the open end of the waveguide, as well as inefficient re-absorption of
the emitted wavepacket due to its exponential shape.
In addition, we have also plotted similar comparisons between this ideal model of
the observed time-delayed feedback phenomenon, and the data shown in Fig. 2.4c.
For this comparison, we choose to plot the five line cuts plotted in white in Fig. 2.4c,
along with comparisons to the theoretical model. The agreement between the two
for all five curves is similar to the agreement observed in Fig. 2.4b. Quantification of
the non-Markovianity of the discussed model under various parameters is presented
in Ref. [37]; however, as the reference notes, there are many competing manners to
quantify non-Markovianity.

PART 2: QUANTUM TRANSDUCTION

As mentioned in the introduction to the thesis, Part 2 will deal with the development
of a quantum transducer device that can convert microwave frequency photons from
a superconducting qubit (f ∼ 5 GHz) into optical photons in the telecommunications
band (f ∼ 200 THz). These optical photons can then be transmitted over long
distances using low loss optical fibers at room temperature opening up prospects
for quantum communication between remote superconducting qubit based quantum
processors. There are many different schemes for quantum transduction including
schemes based on cold atoms [121–123], rare earth ions [124–127], electro-optics
[128–135], and electro/piezo-optomechanics [10, 136–149]. In this thesis, we
develop a piezo-optomechanical quantum transducer device.
Our transducer device can be split into two parts: 1. A piezo-acoustic part that
converts microwave photons from a superconducting qubit into microwave phonons
using the piezo-electric effect. 2. An optomechanical part that subsequently converts
the microwave phonons into optical photons using an optomechanical interaction.
Building on past work done in the Painter group, we utilize a (modified) onedimensional optomechanical crystal (1D-OMC) to generate the optomechanical
coupling. A brief summary of the theory of cavity optomechanics and the basic
concepts underlying 1D-OMC devices developed by past generations of students
in the Painter group is presented in Chapter 5. The design of the piezo-acoustic
cavity is presented in Chapter 6. These two independently designed parts are then
connected together to form our full transducer device. Details of the design of the
full device and analysis of the expected efficiency and added noise of the transducer
are also presented in Chapter 6.
In Chapter 7, we will discuss the repetition rate at which we can operate the transducer device presented in Chapter 6. The repetition rate is limited by the generation
of quasiparticles (breaking of Cooper pairs) in the superconductor when an optical
pulse is applied in close proximity to the superconducting circuit. The quasiparticle relaxation time is a key factor determining the repetition rate of the transducer
device. We will develop niobium (Nb) based transmon qubits (in contrast to the allaluminum (Al) transmon qubits used in Part 1) and test their optical power handling
capability. When exposed to optical illumination, we find that Nb-based transmon
qubits recover on a faster timescale than Al-based transmon qubits and hence are
more suited for incorporation into a quantum transducer device.

65
The final two chapters of Part 2 will deal with some of the nanofabrication challenges
in realizing our transducer device. As will be seen in Chapter 6, our transducer
device is designed on a thin film lithium niobate on silicon-on-insulator platform
(LN on SOI). An etch process for etching lithium niobate on silicon-on-insulator
and the challenges involved is discussed in Chapter 9. Integrating niobium based
superconducting circuits with the transducer device requires fabrication of niobium
qubits on silicon-on-insulator substrates. A fabrication process for niobium based
superconducting qubits on silicon-on-insulator substrates is developed in Chapter 8.

66
Chapter 5

BACKGROUND: CAVITY OPTOMECHANICS AND 1D
OPTOMECHANICAL CRYSTALS

In this chapter, we briefly introduce the theory of cavity optomechanics and the
design of optomechanical devices used in the subsequent chapters. This discussion
will be brief and focused mainly on one-dimensional optomechanical crystals (1D
OMC). For a more complete review of the theory of cavity optomechanics, please
see [150, 151]. For detailed studies of 1D OMC devices carried out in the Painter
group, please see [152–156].
5.1

Cavity Optomechanics Hamiltonian
𝑥' = 𝑥"#$ (𝑏" + 𝑏" % )

Optical cavity (𝑎,
' 𝜔& )
Fixed mirror

Movable mirror
" 𝜔! )
(𝑏,

Figure 5.1: Canonical cavity optomechanical system consisting of a Fabry-Perot optical cavity
with a movable mirror.

The canonical cavity optomechanical system shown in Fig.5.1 consists of a FabryPerot optical cavity where one mirror (of mass ‘m’) is attached to a spring and
free to move. Optical photons entering this cavity exert a radiation pressure force
which causes displacement of the mirror. This in turn changes the effective length
of the Fabry-Perot cavity and modulates the cavity frequency. The Hamiltonian
representing this system can be written as:
𝐻ˆ = ~𝜔𝑐 (𝑥) 𝑎ˆ † 𝑎ˆ + ~𝜔𝑚 𝑏ˆ † 𝑏ˆ

(5.1)

Here the first term represents the optical cavity at frequency 𝜔𝑐 (𝑥) which depends
on the displacement ‘x’ of the movable mirror. The second term represents the

67
mechanical motion of the mirror at frequency 𝜔𝑚 . The optical frequency can be
written as 𝜔𝑐 (𝑥) = 2𝜋 ∗ 𝑐/(2(𝐿 + 𝑥)) where c is the speed of light and L+x is the
effective length of the Fabry-Perot cavity. For 𝑥
𝐿, we can approximate the
frequency of the optical cavity as
𝑐
(1 − 𝑥)
𝜔𝑐 (𝑥) = 2𝜋
2𝐿
(5.2)
= 𝜔𝑜 − 𝜔𝑜
where 𝜔 𝑜 = 2𝜋 ∗ (𝑐/2𝐿) is the bare cavity frequency corresponding
to x = 0.
Identifying the position operator of the mechanical mode as 𝑥ˆ = 𝑥zpf 𝑏 + 𝑏ˆ† where
𝑥 zpf = ~/(2𝑚𝜔𝑚 ) is the zero-point motion of the mechanical oscillator, we arrive
at the following Hamiltonian
†ˆ
𝐻 = ~𝜔 𝑜 𝑎ˆ 𝑎ˆ + ~𝜔𝑚 𝑏 𝑏 − ~𝑔0 𝑎ˆ 𝑎ˆ 𝑏 + 𝑏
(5.3)
where 𝑔0 = 𝜔 𝑜 𝑥 zpf /𝐿. While we derived the Hamiltonian in Eq.5.3 starting for a
Fabry-Perot cavity with a mechanically compliant mirror, it is applicable to a wide
variety of cavity optomechanical systems including the specific nanomechanical 1D
optomechanical crystals we will use in this thesis. In our experiments, we will
typically drive the optical cavity with a laser at frequency 𝜔 𝐿 . We can rewrite the
Hamiltonian in Eq.5.3 in a rotating frame at the laser frequency 𝜔 𝐿 as
†ˆ
𝐻 = −~Δ𝑎ˆ 𝑎ˆ + ~𝜔𝑚 𝑏 𝑏 − ~𝑔0 𝑎ˆ 𝑎ˆ 𝑏 + 𝑏
(5.4)
where Δ = 𝜔 𝐿 − 𝜔 𝑜 . The optical field consists of a large coherent part (𝛼) with
small quantum fluctuations (𝛿 𝑎),
ˆ so we linearize the Hamiltonian by making the
substitution 𝑎ˆ = 𝛼 + 𝛿 𝑎.
ˆ We can then rewrite the interaction part of the Hamiltonian
as
𝐻 𝐼 = ~𝑔0 𝑎ˆ 𝑎ˆ 𝑏 + 𝑏
= ~𝑔0 𝛼 + 𝛿 𝑎ˆ † (𝛼 + 𝛿 𝑎)
(5.5)
ˆ 𝑏ˆ + 𝑏ˆ †

= ~𝑔0 𝑏ˆ + 𝑏ˆ † 𝛼2 + 𝛼𝛿 𝑎ˆ † + 𝛼𝛿 𝑎ˆ + 𝛿 𝑎ˆ † 𝛿 𝑎ˆ
The first term in the last line of Eq.5.5 is proportional to 𝛼2 and corresponds to a
static radiation pressure force exerted on the mechanical oscillator causing a constant
displacement. By redefining the origin of displacement we can ignore this term.
The last term is smaller by a factor of 𝛼, so we will ignore this term too. As a result,
we are left with

𝐻ˆ 𝐼 = ~𝑔0 𝛼 𝛿 𝑎ˆ + 𝛿 𝑎ˆ † 𝑏ˆ + 𝑏ˆ †
(5.6)

68
The full linearized Hamiltonian in the rotating frame is

𝐻ˆ = −~Δ𝑎ˆ † 𝑎ˆ + ~𝜔𝑚 𝑏ˆ † 𝑏ˆ − ~𝑔0 𝛼 𝛿 𝑎ˆ + 𝛿 𝑎ˆ † 𝑏ˆ + 𝑏ˆ †

(5.7)

We identify 𝛼 = 𝑛cav as the square root of the number of photons in the optical
cavity and refer to 𝐺 = 𝑔0 𝑛cav as the optomechanical coupling rate and 𝑔0 as the
single photon optomechanical coupling rate.
5.2

Equations of Motion
𝜅%,$
Optical mode
(𝛿 𝑎,
# −Δ)

𝛿 𝑎#"&

𝜅",$

Mech. mode
) 𝜔! )
(𝑏,

𝜅!

Figure 5.2: Schematic of a driven optical cavity coupled to a mechanical mode.

Using the linearized Hamiltonian of Eq.5.7 and applying input-output formalism to
the case of a driven optical mode coupled to a mechanical mode as shown in Fig.5.2,
we can write down the equations of motion for the optical and mechanical fields as
√
𝜅𝑜
𝛿 𝑎¤̂ = 𝑖Δ −
𝛿 𝑎ˆ + 𝑖𝐺 𝑏 + 𝑏 † + 𝜅 𝑒,𝑜 𝛿 𝑎ˆ𝑖𝑛 + 𝜅𝑖,𝑜 𝛿 𝑎ˆ𝑖𝑛,𝑖
2
(5.8)
¤̂𝑏 = −𝑖𝜔 − 𝜅 𝑚 𝑏ˆ + 𝑖𝐺 𝛿 𝑎ˆ + 𝛿 𝑎ˆ † + √𝜅 𝑏ˆ
𝑚 𝑖𝑛,𝑖
Here 𝜅 𝑜 is the total optical loss rate which we have divided into intrinsic loss 𝜅𝑖,𝑜
and extrinsic loss 𝜅 𝑒,𝑜 as shown in Fig.5.2. 𝜅 𝑚 is the total mechanics loss rate, 𝛿 𝑎ˆ𝑖𝑛
is the input optical field, and 𝛿 𝑎ˆ𝑖𝑛,𝑖 and 𝑏ˆ 𝑖𝑛,𝑖 are quantum noise operators for the
optical and mechanical modes, respectively.

We can solve these coupled equations by Fourier transforming to the frequency
domain which yields
−𝑖𝐺 𝑏[𝜔] + 𝑏 [𝜔] − 𝜅 𝑒,𝑜 𝛿 𝑎ˆ𝑖𝑛 [𝜔] − 𝜅𝑖,𝑜 𝛿 𝑎ˆ𝑖𝑛,𝑖 [𝜔]
𝛿 𝑎[𝜔]
𝑖 (Δ + 𝜔) − 𝜅2𝑜
(5.9)
√
−𝑖𝐺 𝛿 𝑎[𝜔]
+ 𝛿 𝑎ˆ † [𝜔] − 𝜅 𝑚 𝑏ˆ 𝑖𝑛,𝑖 [𝜔]
𝑏[𝜔]
𝑖 (𝜔 − 𝜔𝑚 ) − 𝜅2𝑚

69
Substituting the expression for 𝛿 𝑎[𝜔]
in the expression for 𝑏[𝜔]
in Eq.5.9, it can
be shown that the optomechanical interaction modifies the mechanical frequency as
𝜔0𝑚 = 𝜔𝑚 + 𝛿𝜔𝑚 and also modifies the mechanical loss rate as 𝜅0𝑚 = 𝜅 𝑚 + 𝛾𝑜𝑚 where
𝐺 2 𝜔𝑚
𝛿𝜔𝑚 =
Re
(Δ + 𝜔) + 𝑖 𝜅2𝑜 (Δ − 𝜔) − 𝑖 𝜅2𝑜
(5.10)
2𝐺 2 𝜔𝑚
𝛾𝑜𝑚 = −
Im
(Δ + 𝜔) + 𝑖 𝜅2𝑜 (Δ − 𝜔) − 𝑖 𝜅2𝑜
For the purpose of quantum transduction, we are particularly interested in the
optomechanical damping rate 𝛾𝑜𝑚 at the mechanical frequency (𝜔 = 𝜔𝑚 ). Further,
we make the following assumptions:
1. We assume we are in the sideband resolved regime defined by 𝜔𝑚
𝜅 𝑜 .
For the transducer device considered in this thesis, 𝜔𝑚 ∼ 2𝜋 × 5GHz while
𝜅 𝑜 ∼ 2𝜋 × 500MHz so this assumption is justified
2. We will drive our transducer device with a laser at a frequency (𝜔 𝐿 ) that is
red detuned from the optical resonance by the mechanical frequency. 𝜔 𝐿 =
𝜔 𝑜 − 𝜔𝑚 =⇒ Δ = −𝜔𝑚 .
With these assumptions we find
𝛾𝑜𝑚 =

4𝑔02 𝑛cav
𝜅𝑜

(5.11)

where we have explicitly written the optomechanical coupling 𝐺 in terms of the single photon optomechanical coupling rate 𝑔0 and the intra-cavity photon number 𝑛cav .
Intuitively, we are converting phonons from the mechanical mode into photons
in the optical cavity. This is a parametric process where the laser drive is acting as a
pump to make up for the frequency difference between the phonons and the optical
photons (𝜔 𝐿 = 𝜔𝑚 − 𝜔 𝑜 ). The rate at which we can convert phonons to photons
is given by the optomechanical scattering rate 𝛾𝑜𝑚 which is dependent on the laser
pump power via the intra-cavity photon number 𝑛cav . An important figure of merit
is the optomechanical cooperativity defined as
𝐶𝑜𝑚 =

𝛾𝑜𝑚
𝜅𝑚

(5.12)

Intuitively we can think of 𝛾𝑜𝑚 as the ‘good damping’ rate where phonons in the
mechanical mode are being converted into photons in the optical mode which we

70
can then detect. 𝜅 𝑚 on the other hand is the undesirable ‘bad damping’ rate where
phonons in the mechanical mode are leaking out into the environment and being lost.
We can then understand the optomechanical cooperativity as the ratio of the ‘good
damping’ into the optical mode (𝛾𝑜𝑚 ) divided by the ‘bad damping’ 𝜅 𝑚 . This will
be a very important figure of merit to keep in mind when we discuss the efficiency
of our quantum transducer device in Chapter 6.
5.3

1D Optomechanical Crystals

There are a wide variety of systems that have been used to realize optomechanical
coupling. These range from microscopic systems such as cold atoms coupled to
an optical cavity [157, 158] to macroscopic systems involving suspended mirrors
[159–161]. In our transducer, we will utilize a nanoscale device called a 1D optomechanical crystal (1D OMC). We are interested in coupling microwave frequency
phonons (∼ 5GHz) to telecom band photons (∼ 200 THz). Due to the large difference in the velocity of light and sound, the wavelength of telecom band photons and
microwave phonons is roughly equal (∼ 𝜇𝑚 scale). This is convenient as it allows
us to co-localize microwave frequency mechanical modes and telecom band optical
modes in the same wavelength scale device (∼ 𝜇𝑚). Recall from our expression for
𝑔0 = 𝜔0 𝑥 zpf /𝐿, shrinking the size (L) of our optomechanical system down to the
wavelength scale allows us to achieve large single photon optomechanical coupling.
Here we describe the basic design of a 1D OMC. This topic has been covered in
great detail in previous work from the Painter group so our discussion here will be
brief.
A 1D optomechanical crystal is shown in Fig 5.3 a. It consists of a nano-beam
patterned from a 220 nm thick silicon device layer of a silicon on insulator substrate.
By shrinking the thickness and width of the beam to be sub-wavelength, we get an
effective 1D structure. At either end of the nano-beam, we have the ‘mirror region’
which consists of an array of identical periodically patterned elliptical holes that
act as a metamaterial and are designed to support a simultaneous optical bandgap
centered around 194 THz and acoustic bandgap centered around 5GHz (Fig 5.3 b,c).
(Technically these bandgaps are really pseudo-bandgaps—they act as bandgaps only
for modes possessing a particular symmetry. However, in the ideal situation, modes
of different symmetry do not couple to each other so as long as our mode of interest
possesses the correct symmetry, it can be well isolated inside the pseudo-bandgap.)
In the middle of the nano-beam, we have the ‘defect region’ where we break the
translational symmetry of the mirror region by adiabatically tuning the dimensions

71
a.

Mirror
b.

Defect Region

Mirror

Figure 5.3: 1D optomechanical crystal. a. Scanning electron microscope (SEM) image of a 1D
optomechanical crystal cavity. b. mechanical and c. optical band structure for propagation along the
x-axis in the nominal mirror unit cell, with quasi-bandgaps (red regions) and cavity mode frequencies
(black dashed) indicated. In b., modes that are y- and z-symmetric (red bands), and modes of other
vector symmetries (blue bands) are indicated. In c., the light line (green curve) divides the diagram
into two regions: the gray shaded region above representing a continuum of radiation and leaky
modes, and the white region below containing guided modes with y-symmetric (red bands) and
y-antisymmetric (blue bands) vector symmetries. The bands from which the localized cavity modes
are formed are shown as thicker curves. d. The normalized optical 𝐸 𝑦 field and the normalized
mechanical displacement field Q of the localized optical and mechanical modes, respectively. Figure
reproduced from [152]

of the holes. This ‘defect region’ supports both an optical mode at 194 THz inside
the optical bandgap and an acoustic mode at 5 GHz inside the acoustic bandgap.
Conceptually, by periodically patterning holes in the nano-beam, we are modulating
the effective refractive index thus creating a distributed Bragg mirror where we get
constructive interference of multiple reflections which tightly confines the optical
field to the defect region. Similarly, the array of holes is also periodically modulating the mass of the nano-beam giving us an effective ‘acoustic Bragg mirror’ which

72
serves to tightly confine the acoustic field.

We can use finite element simulation methods to calculate the electric field profile
E and the displacement profile Q of the 1D OMC. This allows us to calculate the
single photon optomechanical coupling 𝑔0 which has two contributions,
1. A moving boundary contribution (𝑔0,MB ) similar to the moving end mirror of
a Fabry-Perot cavity which can be calculated as
−1 2
𝜔 𝑜 (Q · n̂)(Δ𝜖E k − Δ𝜖 D⊥ )𝑑𝑆
𝑔0,MB = −
(5.13)
D·E
where Q is the normalized displacement profile on the silicon surface, n̂ is
the surface normal, E k is the electric field parallel to the surface, D⊥ is the
electric displacement field perpendicular to the surface, Δ𝜖 = 𝜖 Si − 𝜖 Air , and
−1 − 𝜖 −1 .
Δ𝜖 −1 = 𝜖Si
Air
2. A photoelastic contribution (𝑔0,PE ) where the strain induced by the mechanical
displacement causes a change in the refractive index. This contirbution can
be calculated as
𝜔 𝑜 𝜖0 𝑛 Si E · [pS] · E 𝑑𝑉
(5.14)
𝑔0,PE =
D · E 𝑑𝑉
where 𝜔 𝑜 is the optical frequency, 𝑛 is the refractive index, E is the electric
field, p is the photoelastic tensor, and S is the strain tensor.
The total single photon optomechanical coupling rate is 𝑔0 = 𝑔0,MB + 𝑔0,PE . The
optical and mechanical mode shapes of a 1D OMC are plotted in Fig 5.3 d. It is
clear that both the optical and acoustic fields are co-localized and confined tightly
to the defect region leading to a large overlap between these fields giving rise to
a large single photon optomechanical coupling rate 𝑔0 . State of the art 1D OMC
devices in silicon have been shown to achieve 𝑔0 ∼ 1.1MHz [152]. The carefully
engineered acoustic and optical bandgaps surrounding our modes of interest help
minimize radiation losses yielding high quality factors for the acoustic and optical
modes. The ease of fabrication and low material loss of silicon further helps to
create modes with very low intrinsic loss. State of the art 1D OMC devices have
demonstrated intrinsic optical linewidths 𝜅𝑖,𝑜 ∼ 500MHz and mechanical linewidths
𝜅 𝑚 ∼ 4kHz [162]. Large 𝑔0 and small 𝜅 𝑜 and 𝜅 𝑚 are essential for maximizing the

73
optomechanical cooperativity and hence the transduction efficiency as we will see
in Chapter 6.

74
Chapter 6

DESIGN OF A WAVELENGTH-SCALE
PIEZO-OPTOMECHANICAL QUANTUM TRANSDUCER
6.1

Introduction

A quantum transducer can be specified as a linear device with a certain conversion
efficiency, added noise level, and repetition rate. Current approaches for microwave
to optical quantum transduction rely on a strong optical pump to mediate the conversion process between single photon-level signals at both frequencies. Increasing
pump power allows for higher conversion efficiency, but due to parasitic effects of
optical absorption in various components of the transducer and the vast difference
in energy scales between optical and microwave frequencies, this often adds more
noise to the conversion process. For applications in the quantum regime, the number of added noise photons per transduced photon should be less than 1. In several
approaches, this trade-off between efficiency and noise has been a key obstacle to
transduction of quantum signals [10, 133, 141, 144, 145, 149]. Recently a piezooptomechanical approach has been used to demonstrate optical measurements of the
quantum state of a superconducting transmon qubit with added noise levels below
1 photon [10]. In this work, we build on this piezo-optomechanical transduction
approach with a design optimized for high efficiency and low noise. The goal of
this design is to achieve performance improvements essential to detect quantum
correlations in transduced photons on reasonable timescales.
Fig. 6.1a illustrates the mode picture of our transduction scheme. An intermediary mode 𝑏ˆ 𝑚 of a nanomechanical oscillator simultaneously couples to microwave
photons from mode 𝑐ˆ𝑞 of a microwave circuit, and to optical photons from mode 𝑎ˆ 𝑜
of an optical cavity. Microwave photons are converted to phonons via a resonant
piezoelectric interaction, and these phonons are subsequently converted into optical photons via a parametric optomechanical interaction. The microwave photonphonon conversion is realized by tuning the circuit frequency 𝜔 𝑞 on resonance with
the mechanical frequency 𝜔𝑚 . The phonon-optical photon conversion is realized
by driving the optical cavity at frequency 𝜔 𝑑 that is red-detuned by exactly the
mechanical frequency s.t 𝜔 𝑑 − 𝜔 𝑜 = −𝜔𝑚 .
We realize the intermediary mechanical mode in the above schematic by connecting a

75
a)

5 GHz

194 THz

Optomechanical
Cavity

Piezoacoustic
Cavity

Figure 6.1: Schematic of piezo-optomechanical transducer. a. Mode schematic for piezooptomechanical transduction. b. Device schematic for the transducer in this work. The device can
be split into two regions, one which couples strongly to microwave electric fields and one which
couples strongly to optical fields. Both are part of the same mechanical ‘supermode’ 𝑏ˆ 𝑚 .

wavelength-scale piezoacoustic cavity and an optomechanical crystal (OMC) cavity
(see Fig. 6.1b). The acoustic modes of these components are strongly hybridized to
form a mechanical ‘supermode’ whose mechanical displacement highly overlaps in
one region with the field of a microwave circuit, and in another region with the field
of an optical cavity. Using physically separate cavities allows us to independently
optimize the piezoacoustic and optomechanical components of the transducer. Our
design is formed from thin-film lithium niobate (LN) on the device layer of a siliconon-insulator (SOI) chip. We define the piezo-acoustic cavity in LN, which has
large piezoelectric coefficients [163]. We define the OMC in silicon, since its large
photoelastic coefficients [164] and refractive index [165] allow high optomechanical
coupling. Well-established nanofabrication processes also allow high optical and
mechanical quality factors for silicon OMCs [152, 162]. For the microwave circuit
in this design, we consider a transmon qubit [16] with electrodes routed over the LN
region to allow for capacitive coupling to the piezoacoustic cavity. The transmon is
patterned using niobium, a standard material platform for realizing high-coherence
qubits [166, 167]. The buried oxide layer underneath these components is etched
away, leaving a suspended silicon membrane as the substrate for our device.
Our design procedure begins with independently optimizing the piezoacoustic and

76
OMC cavities for high 𝑔 𝑝𝑒 and 𝑔𝑜𝑚 , respectively. We design for closely matched
acoustic modes at 5 GHz in both resonators, and for an optical mode at telecom
wavelength (1550 nm). During the design process, it is crucial to maintain a low
acoustic mode density such that the transduction schematic in Fig. 1a using a single
acoustic mode remains valid. Further, since thin film LN has higher microwave
dielectric and acoustic loss than silicon, we aim to minimize the piezo volume in our
device. The two independently optimized cavities are then physically connected,
and the parameters of the resulting hybrid acoustic modes are analyzed. Using this
approach, we design a transducer with expected conversion efficiency at the percent
level while maintaining added noise photons <0.5.
6.2

Piezo Cavity Design
b)

Frequency (GHz)

Silicon
Piezo Box
Electrodes

-5

Figure 6.2: Design of phononic shield a. Schematic illustrating the capacitive routing of the
piezoacoustic cavity to a transmon qubit. The qubit here can be replaced with a microwave resonator
without loss of generality. b. Piezoacoustic cavity geometry, with relevant dimensions defined in
blowout top view. c. Phononic shield unit cell, with relevant dimensions defined. d. Mechanical
bandstructure of phononic shield unit cell in c), with (𝑎, 𝑏 𝑥 , 𝑏 𝑦 , 𝑡 𝑦 ) = (445, 225, 265, 70)nm. We
observe a complete acoustic bandgap in excess of 1GHz around 5GHz. e. Log scale of mechanical
energy 𝑈𝑚 for piezoacoustic cavity mode at 5GHz, normalized to maximum value. We find >4
orders of magnitude suppression for 5 phononic shield periods.

The piezoacoustic cavity consists of a slab of lithium niobate on top of a suspended
silicon membrane patterned in the shape of a box. We work with 100nm thinfilm -Z-cut lithium niobate on top of a 220nm thick suspended silicon device layer.
80nm-thick Nb electrodes run over the top of the slab and are routed in the form of an

77
interdigital transducer (IDT) which capacitively couples the cavity to a microwave
circuit such as a transmon qubit, as seen in Fig. 6.2a.
The box is surrounded by a periodically patterned phononic shield to mitigate
acoustic radiation losses and to clamp the membrane to the surrounding substrate.
The clamps are spaced periodically so that the IDT electrodes are routed over the
top of each clamp, providing a means for electrical routing which is not acoustically
lossy.
The phononic shield uses an alternating block and tether pattern (see relevant dimensions in Fig. 6.2c) consisting of metal electrodes on top of a silicon base. By
tuning the parameters 𝑎, 𝑏 𝑥 , 𝑏 𝑦 , and 𝑡 𝑦 , we achieve a >1GHz acoustic bandgap
centered around 5GHz, the frequency of the mechanical mode of interest (Fig 6.2d).
This yields strong confinement of mechanical energy inside the piezo region for
sufficient number of shield periods, enabling high mechanical quality factors. By
simulating the mechanical energy density across the entire cavity, we find that 5
shield periods provide >4 orders of magnitude suppression of acoustic radiation into
the environment, as shown in Fig. 6.2e.
The dimensions of the piezo box (outlined in Fig. 6.2b) are designed to support
a periodic mechanical mode whose periodicity matches that of the IDT fingers.
This results in high overlap between the electric field from the IDT and the electric field induced by mechanical motion in the piezo box. This overlap gives a
microwave photon-phonon piezoelectric coupling rate which is derived using first
order perturbation theory:
𝜔𝑚
𝑔 𝑝𝑒 = p
D𝑚 · E𝑞 𝑑𝑉 .
(6.1)
4 2𝑈𝑚 𝑈𝑞 LN
Here the integral is taken over the entire LN slab, D𝑚 is the electric displacement
field induced from mechanical motion in the piezo region, and E𝑞 is the singlephoton electric field generated by the transmon qubit across the IDT electrodes.
The fields are normalized to their respective zero-point energies ~𝜔𝑚 /2, yielding
the pre-factor in front of the integral in (1). 𝑈𝑚 is the total cavity mechanical energy,
and 𝑈𝑞 = 12 (𝐶𝑞 + 𝐶IDT )𝑉02 is the total IDT electrostatic energy. We note that the
electrostatic energy is dependent on both the qubit capacitance 𝐶𝑞 and IDT finger
capacitance 𝐶IDT , and therefore the coupling rate scales as (𝐶𝑞 + 𝐶IDT ) −1/2 . For our
calculations in this work, we assume 𝐶𝑞 = 70fF which is a typical value for transmon
qubit capacitance. Replacing the transmon qubit with a high-impedance microwave

78
resonator will allow lower 𝐶𝑞 ∼ few fF [168], and can therefore further increase this
coupling rate. 𝐶IDT is calculated with finite-element electrostatic simulation and is
typically on the order 0.1fF, a small contribution compared to transmon 𝐶𝑞 .
The small value of 𝐶IDT also minimizes the energy participation of the qubit electric
field in the lossy piezo region, given by the ratio 𝜁 𝑞 = 𝐶IDT /𝐶𝑞 ∼ 10−3 . The
contribution of lithium niobate to the qubit loss rate 𝜅 𝑞,𝑖 is then estimated as 𝜁 𝑞 𝜅 𝑞,LN .
Using reported dielectric loss tangents in lithium niobate tan 𝛿 = 2.5 × 10−3 [169]
giving 𝜅 𝑞,LN /2𝜋 = 12.5MHz, we estimate the LN contribution to qubit loss to
be 𝜁 𝑞 𝜅 𝑞,LN /2𝜋 ∼ 10kHz. This contribution is ∼5x smaller than typical loss rates
𝜅 𝑞,SOI /2𝜋 ∼ 50kHz reported in transmon qubits fabricated on SOI [78]. As a result,
the contribution of the piezo cavity to qubit loss is not a limiting factor, and justifies
the on-chip coupling scheme outlined in Fig. 6.2a.
Mode Isolation (MHz)

Frequency (GHz)

IDT Fingers
d)

Frequency (GHz)

-1

Period (nm)

Figure 6.3: Design of piezoacoustic cavity. a. Mode structure for optimized piezoacoustic cavity
design with ( 𝑝, 𝑤 𝑝 , 𝑒) = (783, 423, 118) nm. Mode of interest (red) achieves 𝑔 𝑝𝑒 /2𝜋 of 9.5MHz.
Shaded grey region indicates the mode isolation window with the nearest mode >150MHz away.
b. 𝑔 𝑝𝑒 /2𝜋 and mode isolation for optimized designs with differing number of IDT fingers. We
observe 𝑔 𝑝𝑒 saturating beyond 𝑁 = 4 fingers, and mode isolation decreasing with increasing number
of fingers. c. Mechanical mode shape of red mode in (a). Right shows the in-plane (breathing) and
left shows the out-of-plane (Lamb-wave) components of the optimized mechanical mode. d. Mode
structure and piezoelectric coupling vs. IDT period, with data points colored according to 𝑔 𝑝𝑒 .

An important design consideration with this type of piezoacoustic cavity is the
number of IDT fingers used. For 𝑁 IDT fingers, the length 𝑙 𝑝 of the piezo region is
given by 𝑙 𝑝 = 𝑁 𝑝/2, where 𝑝 is the IDT period. As the number of fingers increases,
the increased size of the piezo box results in a more crowded mode structure (Fig.

79
6.3b), and it is more difficult to isolate a single mechanical mode without coupling
to parasitic modes in the vicinity of the mode of interest. This is of key importance
as these parasitic modes may not hybridize well with the OMC cavity and reduce
overall transduction efficiency. Reducing the size of the piezo region is also important to reduce microwave photon and phonon decoherence, as lithium niobate has
high microwave dielectric and acoustic loss tangents compared to silicon (further
discussion in Sec. 6.4). For these reasons, we choose a 2-finger design to minimize
these effects. The strong piezoelectric nature of lithium niobate allows for 𝑔 𝑝𝑒 values high enough for strong microwave photon-phonon coupling, even in the limit of
2 IDT fingers. We emphasize the small dimensions of the piezoacoustic cavity in
this design in contrast with previous work on piezo-optomechanical quantum transducers [cite]. The benefits of this approach come at the cost of higher sensitivity
of the piezo modes to changes in cavity dimensions. This can have large effects
on hybridization with the OMC cavity and the performance of the final transducer
device, which relies on resonant matching of acoustic modes in both regions. We
show further in Sec. 6.4 that the achievable hybridization between piezo and OMC
modes with this small piezo volume approach is large enough to protect the design
against typical fabrication disorder.
The mechanical mode of interest is periodic with out-of-plane (Lamb-wave) and
in-plane breathing components. The Lamb-wave component of the mode induces
an electric field in the piezo region with high overlap with the IDT electric field,
while the breathing component of the mode enables hybridization with the breathing
mode of the optomechanical crystal to be attached in the full device (see Fig. 6.3c).
The piezo mode can be tuned with three key parameters 𝑝, 𝑤 𝑝 , and 𝑒. 𝑝 is the
periodicity of the IDT fingers and is used to parameterize the piezo box length,
given by 𝑙 𝑝 = 𝑁 𝑝/2 as described earlier. 𝑝 is used to tune the frequency of the
mode of interest (Fig. 6.3d) while maintaining appropriate phase-matching of the
mode periodicity with the IDT fingers. 𝑤 𝑝 is the piezo box width, which can
be increased to increase 𝑔 𝑝𝑒 via larger mode volumes or decreased to reduce the
mode crowding that results from larger box size. Finally, we define a silicon-piezo
buffer parameter 𝑒, which extends the silicon box length/width by an amount 𝑒
compared to the piezo box. This buffer is needed to protect against silicon/piezo
box misalignment in the fabrication process, and acts as an added degree of freedom
for tuning frequency, 𝑔 𝑝𝑒 , and mode isolation. We use numerical optimization to
tune parameters ( 𝑝, 𝑤 𝑝 , 𝑒) to arrive at a design with high piezoelectric coupling
and mode isolation. We employ a Nelder-Mead simplex optimization [170, 171]

80
similar to that described in [152]. After optimization, we obtain a single mechanical
mode with 𝑔 𝑝𝑒 /2𝜋 = 9.5MHz, which is isolated by >150MHz from other mechanical
modes (Fig. 6.3a). We will use this single mode to strongly couple to the modes of
an optomechanical crystal cavity to create the mechanical supermode of Fig. 6.1a.

Frequency (THz)

Phonon
Mirror

Frequency (THz)

Frequency (GHz)

Optomechanics Design

Frequency (GHz)

6.3

Phonon
Waveguide

Figure 6.4: Design of the optomechanical cavity. a. Unit cell geometry, mechanical, and optical
bandstructure of phonon mirror region with (𝑎, 𝑤, ℎ 𝑥 , ℎ 𝑦 ) = (436, 529, 189, 320)nm. Mechanical
bands are color-coded by symmetry, with red (green) corresponding to breathing (Lamb-wave) mode
symmetry classes. Blue bands represent all other symmetries. Mechanical bandgap for breathing
modes and optical bandgap are both highlighted in red. b. Full OMC geometry, with phonon
mirror and phonon waveguide unit cells highlighted. c. Unit cell geometry, mechanical, and optical
bandstructure of phonon waveguide region with (𝑎, 𝑤, ℎ 𝑥 , ℎ 𝑦 ) = (436, 529, 295, 205)nm. Mechanical
bands color-coded as in (a). Breathing mode crosses 5.1GHz resulting in waveguide-like behavior
at the mechanical frequency. The optical bandgap is maintained.

The optomechanical crystal cavity is designed in a similar fashion to previous
work [152], with the crucial change of a modified unit cell design on one side of
the cavity to enable strong mechanical hybridization with the piezo cavity. This
separates the OMC into three distinct regions: a phonon mirror, defect region, and
phonon waveguide (see Fig. 6.4 for details). The phonon mirror unit cell (Fig.
6.4a) is designed to have a simultaneous mechanical and optical bandgap for modes
of certain symmetry classes. In the defect region, the phonon mirror unit cell
transitions to a defect cell designed to co-localize a 5.1GHz mechanical breathing
mode and a 194THz (𝜆 0 = 1550nm) optical mode. The phonon waveguide unit

81
cell (Fig. 6.4c) is mechanically transparent to breathing mode phonons at 5.1GHz,
while maintaining a large bandgap for optical modes. This is achieved by modifying
the ellipticity of the phonon mirror unit cell. We see in Fig. 6.4b that the resulting
mechanical mode is permitted to leak out into the phonon waveguide region, while
the optical mode remains highly localized within the defect region.
a)

c) Phonon

800

Phonon
Waveguide

Mirror

600
400
200
4.9

5.1

5.2

5.3

Frequency (GHz)

Phonon
Mirror

Defect

Phonon
Waveguide

Frequency (GHz)

Defect Hole Index
-1

Figure 6.5: Optomechanical cavity mode structure. a. Resultant mode structure of optimized
OMC design. Highest 𝑔𝑜𝑚 mode (highlighted in red) gives 750kHz optomechanical coupling.
b. OMC mechanical and optical mode shapes, along with unit cell length at each hole in the
defect region. Phonon mirror, phonon waveguide, and defect region are labeled and outlined. c.
Bandgap of different symmetries as a function of hole index along the defect region. Red shaded
region represents the bandgap for breathing mode symmetries, and blue shaded region represents the
bandgap for Lamb-wave mode symmetries.

The optomechanical coupling rate is calculated from the optical frequency shift
arising due to the photoelastic effect [172] and moving dielectric boundaries [173],
giving 𝑔𝑜𝑚 = 𝑔𝑜𝑚,PE + 𝑔𝑜𝑚,MB . For a detailed derivation of both contributions to
𝑔𝑜𝑚 , see [174]. The photoelastic contribution is derived from 1st order perturbation
theory as
𝜔 𝑜 𝜖0 𝑛4 Si E · [pS] · E 𝑑𝑉
𝑔𝑜𝑚,PE =
(6.2)
D · E 𝑑𝑉
where 𝜔 𝑜 is the optical frequency, 𝑛 is the refractive index, E is the electric field, p
is the photoelastic tensor, and S is the strain tensor.
The moving boundaries component is derived similarly as
−1 2
𝜔 𝑜 (Q · n̂)(Δ𝜖E k − Δ𝜖 D⊥ )𝑑𝑆
𝑔𝑜𝑚,MB = −
D · E 𝑑𝑉

(6.3)

82
where Q is the normalized mechanical displacement field, n̂ is the surface normal,
E k is the electric field parallel to the surface, D⊥ is the electric displacement field
−1 − 𝜖 −1 .
perpendicular to the surface, Δ𝜖 = 𝜖Si − 𝜖Air , and Δ𝜖 −1 = 𝜖Si
Air
One may expect the coupling rates in this design to suffer due to the delocalization
of the mechanical mode. However, we find that after a Nelder-Mead simplex
optimization of various OMC dimensions similar to [152], the resulting design
gives multiple modes with high values of 𝑔𝑜𝑚 /2𝜋, with the maximum coupling rate
exceeding 750kHz (Fig. 6.5a). This is comparable to state-of-the-art OMC designs
in silicon which achieve 𝑔𝑜𝑚 /2𝜋 up to ∼1MHz [162].
The radiation-limited optical quality factor 𝑄 𝑜 can be simulated and is found to be
in excess of 106 , corresponding to an intrinsic optical loss rate 𝜅 𝑜,𝑖 /2𝜋 ∼ 200MHz.
However, 𝑄 𝑜 is usually practically limited to ∼ 500,000 (𝜅 𝑜,𝑖 /2𝜋 ∼ 400MHz) [162]
due to optical scattering from surface defects introduced in the fabrication process.
To ensure this limit is reached, we configure the optimization such that 𝑔𝑜𝑚 is
maximized while maintaining 𝑄 𝑜 above ∼ 106 , well above the realistic 𝑄 𝑜 limit.
The total optical loss rate is given by 𝜅 𝑜 = 𝜅 𝑜,𝑖 + 𝜅 𝑜,𝑒 , where 𝜅 𝑜,𝑒 is the decay
rate associated with input coupling. 𝜅 𝑜,𝑒 is controlled with a coupling waveguide
and is typically designed so that 𝜅 𝑜,𝑒 = 𝜅 𝑜,𝑖 . The total optical loss rate is then
𝜅 𝑜 ≈ 2𝜅 𝑜,𝑖 /2𝜋 = 800MHz.
When hybridizing the modes of the piezoacoustic and OMC cavities, we must
consider the relative motional symmetry of the two cavity modes. Our OMC
design contains only breathing motion, whereas the piezo cavity design contains
both breathing and Lamb-wave components. If the OMC bandstructure permits
propagation of 5GHz phonons with Lamb-wave symmetry, then the OMC breathing
mode will hybridize with leaky, delocalized modes of Lamb-wave symmetry. This
can reduce optomechanical coupling and contribute significantly to mechanical
losses in the device. For this reason, the phonon mirror and waveguide dimensions
are chosen such that their bandstructure exhibits a bandgap for modes of Lambwave-like symmetries. This is seen in the mechanics band diagrams of Fig. 6.4,
where we see a bandgap for Lamb-wave-like modes (colored in green) in both the
phonon mirror and waveguide region.
Fig. 6.5c further illustrates this idea by showing the mechanical bandgap of unit
cells across the defect region for both breathing and Lamb-wave-like modes, shaded
in red and blue, respectively. At the phonon waveguide side, the breathing mode

83
bandgap falls below 5GHz, permitting the breathing motion of the piezo mode to
couple strongly to the defect region. However, 5GHz lies inside the Lamb-wave
bandgap, so that Lamb-wave motion from the piezo cavity decays in the phonon
waveguide and does not interact with the defect region.
6.4

Full Device Design
b)

Frequency (GHz)

Period (nm)

Piezo P.R.

-1

Displacement (a.u.)

Frequency (GHz)

Figure 6.6: Full piezo-optomechnaical transducer design. a. Full transducer piezoelectric coupling and mode structure vs. IDT period. Shown are all modes which have either high piezoelectric
or high optomechanical coupling. Data points are colored according to 𝑔 𝑝𝑒 value. b. Full transducer
optomechanical coupling and mode structure vs. IDT period. Shown are the same modes as in
(a), only colored according to optomechanical coupling. c. Mode structure at 𝑝 =800nm, showing
𝑔 𝑝𝑒 , 𝑔𝑜𝑚 , and energy participation ratio in the piezoelectric region. Red shows mode with highest
combined 𝑔𝑜𝑚 and 𝑔 𝑝𝑒 , and lowest piezo participation (<2%). d. Mechanical mode profile of the
mode highlighted in red in (c).

After independently designing the piezoacoustic and optomechanical cavities, we
connect the two as shown in Fig. 6.6d and simulate the resulting hybridized mode
structure. To observe the hybridization of the piezo and optomechanical modes, we
sweep the IDT period in the piezo region to tune the piezo mode through the multiple
optomechanical resonances. We find that over a frequency window >250MHz, there
is a large number of mechanical modes with simultaneous high piezoelectric and
optomechanical coupling rates. The phonon waveguide allows for strong enough

84
mode hybridization that the piezoelectric coupling is distributed across a large
number of modes. As shown in Fig. 6.6c, the mechanical energy participation
in the piezo region 𝜁𝑚 is in the range 1-10%. We find that across the entire
hybridization window, at least one mode can be identified with 𝑔𝑜𝑚 /2𝜋 > 500kHz,
𝑔 𝑝𝑒 /2𝜋 > 1MHz, and 𝜁𝑚 < 5%. In Fig. 6.6c, this mode is highlighted in red
with 𝑔𝑜𝑚 /2𝜋 = 725kHz, 𝑔 𝑝𝑒 /2𝜋 = 2.5MHz, and 𝜁𝑚 < 2%. We will use the
values from this mode to quantify further calculations in this work. In practice, the
frequencies and couplings of these mechanical modes are subject to change due to
multiple sources of fabrication disorder. The multi-mode structure and relatively
large hybridization ensure that the full device is robust to these shifts. While the
exact frequencies and couplings may shift, Fig. 6.6a and 6.6b illustrate that the
qualitative nature of the mode structure remains unchanged for a large range of
frequency shifts. Additionally, the modes are separated far enough in frequency that
their parasitic effect on each other’s transduction efficiencies is minimal.
We may use the simulated piezo participation ratio and radiation loss to estimate the
mechanical decoherence rate 𝜅 𝑚 of our device. There are two dominant contributions
to decoherence in our design. The first is acoustic radiation loss into the surrounding
substrate. This can be simulated and is found to be in the range 𝜅rad /2𝜋 ∼ 1 − 10kHz
for all modes in Fig. 6.6, with 𝜅rad /2𝜋 = 2.3kHz for the mode highlighted in Fig.
6.6c. The second is coupling to two-level systems (TLS), which in both lithium
niobate [175] and silicon [162] has been shown to be the dominant decoherence
mechanism for GHz-frequency acoustic cavities at single phonon level powers and
milliKelvin temperatures. For mechanical piezo participation ratio 𝜁𝑚 , the TLS
induced decoherence rate can be estimated by 𝜅TLS = 𝜁𝑚 𝜅LN + (1 − 𝜁𝑚 )𝜅Si . Using
reported TLS-limited linewidths 𝜅LN /2𝜋 ∼ 100 − 300kHz in lithium niobate [146,
175] and 𝜅Si /2𝜋 ∼ 5kHz in silicon [162], and taking 𝜁𝑚 = 2%, we estimate a TLS
induced decoherence rate of 𝜅TLS /2𝜋 ∼ 10 kHz. The total mechanical decoherence
rate is then estimated to be in the range 𝜅 𝑚 /2𝜋 ∼ 10 − 20kHz.
6.5

Efficiency and Added Noise

To analyze the efficiency and noise of our design, we consider a pulsed scheme for
microwave to optical state transfer on a transmon qubit connected to the transducer
[10]. The qubit is first tuned on resonance with the mechanical mode for a time
𝑡 = 𝜋/𝑔 𝑝𝑒 to complete a microwave photon-phonon swap operation, and subsequently detuned far off-resonance. A red-detuned (𝜔 𝑑 − 𝜔 𝑜 = −𝜔𝑚 ) laser pulse is
then used to upconvert this phonon into an optical photon. The intrinsic efficiency

85
of such a pulsed scheme is simply given by 𝜂𝑖 = 𝜂 𝑝𝑒 𝜂 𝑜𝑚 , where 𝜂 𝑝𝑒 is the piezoelectric photon-phonon swap efficiency, 𝜂 𝑜𝑚 is the optomechanical phonon-photon
conversion efficiency.
𝜂 𝑝𝑒 can be calculated from a master equation simulation of the qubit-mechanics
system. Using 𝑔 𝑝𝑒 /2𝜋 = 2.5MHz, estimated 𝜅 𝑞 /2𝜋 = 60kHz from Section 6.2,
estimated mechanics decoherence rate 𝜅 𝑚 /2𝜋 = 20kHz from Section 6.4, we find
𝜂 𝑝𝑒 = 0.95. The optomechanical readout step determines both 𝜂 𝑜𝑚 and the dominant
noise contribution to the transducer, which arises from optical absorption heating
of the mechanical mode. For a laser pulse duration 𝜏, 𝜂 𝑜𝑚 is given by [10]

𝜂om (𝜏) =

𝛾𝑜𝑚
(1 − 𝑒 −(𝛾𝑜𝑚 +𝜅 𝑚 )𝜏 )
𝛾𝑜𝑚 + 𝜅 𝑚

(6.4)

2 𝑛 /𝜅 is the optomechanical scattering rate, and 𝑛 is the number
where 𝛾𝑜𝑚 = 4𝑔𝑜𝑚
𝑜 𝑜
of intracavity optical photons corresponding to peak power of the optical pulse.
In principle, this efficiency may be unity in the limit 𝜏
1/(𝛾𝑜𝑚 + 𝜅 𝑚 ) and
𝛾𝑜𝑚
𝜅 𝑚 . However, optically-induced heating of the mechanical mode severely
limits 𝜏 in order to maintain <1 added noise photon. This leads to a fundamental
tradeoff between efficiency and added noise resulting from heating dynamics in
optomechanical systems. Maximizing efficiency for a given level of added noise
requires careful choice of pulse duration 𝜏 and optical power 𝑛𝑜 .

The added noise phonons 𝑛𝑚 (𝜏) during optical readout are thought to originate from
optical excitation of material defect states which undergo phonon-assisted relaxation
via the mechanical mode of interest [154, 176, 177]. The timescale 𝜏ℎ for 𝑛𝑚 to
exceed 1 noise phonon depends strongly on 𝑛𝑜 and is found to vary greatly in different
devices. Experiments in low-loss (𝜅 𝑚 . 10kHz) pure silicon OMC devices report
𝜏ℎ ∼ 1𝜇s [156, 162], whereas silicon OMCs integrated in a piezo-optomechanical
transducer with 𝜅 𝑚 = 1MHz report much shorter 𝜏ℎ ∼ 100ns [10]. This suggests the
presence of additional sources of optically induced heating and mechanical damping
in piezo-optomechanical transducers that are potentially correlated. Possible sources
are optical absorption by the IDT electrodes, TLS-limited loss in the piezo region,
and surface defects in the OMC region from additional steps in the transducer
fabrication process. While the dynamics of optically induced heating in piezooptomechanical devices is a subject of future studies, it is clear that a transducer
design aimed at improving optomechanical readout efficiency and noise should make
the acoustic mode involved in the transduction process as silicon-like as possible.

86
In the design presented above, we minimize the dimensions of the piezo cavity so
that most of the energy in the mechanical mode lives in the OMC region. The
estimated mechanical damping rates based on participation ratios of various regions
and calculated optomechanical coupling rates are comparable to those realized
in pure silicon OMCs. Therefore, we may approximate the heating dynamics
of our design as similar to that reported in previous silicon OMC work [156].
Using this heating model, we estimate ∼0.5 added noise photons for a pulse with
𝑛𝑜 = 45 and 𝜏 = 500ns. Using the previously estimated values 𝜅 𝑚 /2𝜋 = 10kHz,
𝜅 𝑜 /2𝜋 = 800MHz, and 𝑔𝑜𝑚 /2𝜋 = 725kHz (s.t. 𝛾𝑜𝑚 /2𝜋 = 120kHz), we estimate
a pulse with 𝑛𝑜 = 45 and 𝜏 = 500ns can achieve 𝜂 𝑜𝑚 ∼ 30%. Combined with
𝜂 𝑝𝑒 = 0.95, we achieve an estimated intrinsic efficiency 𝜂𝑖 ∼ 29%.
There are additional noise sources which we have not considered here such as photodetector dark counts and residual photons from the optical pump pulse. However,
given the measured photon count rates for these noise sources in previous milliKelvin
optomechanics experiments in our group, these noise sources are negligible compared to those from optical absorption heating discussed above.
The total efficiency of our device is given by 𝜂 = 𝜂𝑖 𝜂 𝑘 𝜂 𝑒𝑥𝑡 , where 𝜂 𝑘 = (𝜅 𝑜,𝑒 /𝜅 𝑜 )
determines the fraction of optical photons emitted into the coupling waveguide, and
𝜂 𝑒𝑥𝑡 is the external photon collection efficiency. For 𝜅 𝑜,𝑒 ≈ 𝜅 𝑜,𝑖 (critical coupling)
we have (𝜅 𝑜,𝑒 /𝜅 𝑜 ) ≈ 0.5. In typical optomechanics experiments, 𝜂 𝑒𝑥𝑡 is mainly
determined by the fiber-to-device coupling efficiency, insertion loss of the optical
pump filtering setup, and quantum efficiency of single photon detectors. In our
typical experimental setup we estimate these factors are 0.6, 0.2, and 0.9, respectively
and lead to 𝜂 𝑒𝑥𝑡 ∼ 0.1. The product of all three efficiency estimates above yields a
total transducer efficiency 𝜂 ∼ 1.5 %.
Finally, we consider the expected repetition rate for the transduction sequence in the
pulsed scheme described above. In previous work, this was limited to 100Hz by
the ∼10-ms timescale for quasiparticle (QP) relaxation in the aluminum transmon
coupled to the transducer [10]. We expect that using niobium with QP relaxation
timescale in the ∼ns range [11, 12] will allow for repetition rates in the 10kHz
range (more on this in Chapter 7). At this repetition rate and estimated total
efficiency 𝜂 ∼1.5%, we expect a single photon count rate of ∼ 150Hz and a photon
coincidence rate of order 1Hz. The latter, which is the key figure of merit for second
order intensity correlation measurements as well as heralded remote entanglement
generation, indicates reasonable measurement times in the range of an hour for these

87
experiments.
6.6

Conclusion

We have presented an optimized design for a wavelength scale piezo-optomechanical
transducer suitable for on-chip coupling to a transmon qubit. We have independently
simulated and optimized the design of a piezoacoustic cavity and an optomechanical
crystal cavity, and hybridized their acoustic mode structure in a way which is highly
robust to fabrication disorder. We emphasize that our choice of material platform
and minimized piezoacoustic cavity dimensions allows us to design for high piezoelectric swap efficiency without significantly compromising qubit coherence, and
with optomechanical readout efficiencies comparable to state-of-the-art 1D OMC
devices. With the expected performance from this transducer design, experiments
measuring quantum correlations in photons generated by the transducer as well as
a demonstration of heralded remote entanglement between two transducer devices
should be feasible on reasonable measurement timescales. Finally, we note that the
efficiency, noise, and repetition rate of the above transducer design are expected
to be limited by optomechanical heating rates. Future design improvements can
be made by employing 2D optomechanical crystal cavities, which through better
thermal conductivity to the substrate, have achieved higher optomechanical cooperativities at lower added noise [178]. Replacing the on-chip transmon qubit with
a high-impedance microwave resonator can further increase piezoelectric coupling
strength, and thereby improve the fidelity of the microwave photon-phonon swap
operation. Such a transducer device could be connected to an off-chip qubit in a
modular transducer, which alleviates performance restrictions from co-fabrication
of qubits and transducers.

88
Chapter 7

NIOBIUM BASED TRANSMON QUBITS ON SILICON
SUBSTRATES FOR QUANTUM TRANSDUCTION
7.1

Introduction

In Chapter 6, we discussed the design of our piezo-optomechanical transducer and
the expected conversion efficiency and added noise of our device. Another important
figure of merit of the transducer is its repetition rate. Since our transducer operates
in pulsed mode, the repetition rate is simply the rate at which we can repeat the transduction pulse sequence. The transduction pulse sequence is shown in Fig. 7.1a. It
begins with a XY pulse on the qubit that prepares the qubit in the desired state. Next
the qubit frequency is tuned to be on resonance with the mechanical mode of the
piezo-optomechanical transducer for a time 𝑡swap such that the microwave excitation
in the qubit is exchanged for a phonon in the mechanical mode. The qubit is then
detuned from the mechanical mode. Next a red-detuned optical pump pulse converts
the phonon into and optical photon which is subsequently detected on a photodetector. This optical pulse consists of ∼ 200THz frequency optical photons which have
energy larger than the superconducting gap of typical superconducting materials.
Since our superconducting qubit is in close proximity to our piezo-optomechaical
transducer, absorption of these highly energetic optical photons can break Cooper
pairs in the superconductor creating excess quasiparticles (QPs). Tunneling of these
non-equilibrium quasiparticles across the Josephson junction of a superconducting
qubit creates excess loss in the qubit [179] leading to a transducer dead time until
the excess quasiparticles relax either by recombination to form a Cooper pair with
the emission of a phonon or via an electron-phonon scattering process [180]. This
quasiparticle relaxation time in general is a material and substrate dependent property. For the all-aluminum (Al) qubits on a silicon-on-insulator substrate used in
the transducer device demonstrated in [10], the qubit recovery time after the optical
generation of quasiparticles was on the order of ∼10ms (see Fig. 7.1)b. Other measurements of quasiparticle relaxation in Al based superconducting circuits have also
indicated relaxation times on the order of ∼ ms [181, 182]. Niobium (Nb) is another
commonly used superconducting material with much shorter measured quasiparticle relaxation times (∼ns) [11, 12]. However junctions with a Nb/NbOx/Nb stack
have been found to have poor quality compared to junctions utilizing an AlOx tunnel

89
barrier [183]. In this chapter, we design and fabricate Nb-based transmon qubits
with Al/AlOx/Al tunnel junctions on a silicon substrate and measure their recovery
time when exposed to optical illumination.
a.

Normalized Rabi Contrast

Delay after optical pulse (ms)

Figure 7.1: Pulse sequence and laser induced quasiparticle recovery for a quantum transducer
device. a. Pulse sequence and b. Laser induced quasiparticle recovery for a quantum transducer
device. Figure reprinted by permission from Springer Nature Customer Service Center GmbH:
Springer Nature, Nature, Ref. [10], ©2020

7.2

Qubit Design

An optical image of our Niobium (Nb) based transmon qubit on a silicon substrate
is depicted in Figure 7.2. An interdigitated capacitor patterned from Nb forms
the shunt capacitance of the transmon. A pair of identical Al/AlOx/Al Josephson
junctions form a SQUID loop that allows frequency tuning of the qubit. ‘Bandage’
layers made of Nb electrically contact the Al leads of the Josephson junctions to
the underlying Nb layer. The qubit is capacitively coupled to a lumped element
readout resonator. There is a single microwave co-planar waveguide (CPW) that
is capacitively coupled to both the readout resonator and the transmon qubit and
serves to interrogate the readout resonator as well as excite the qubit. The readout
resonator, CPW, and ground plane are all fabricated from niobium (Nb). Note there
is no on-chip Z-line for flux tuning the frequency of the qubit. Instead an external
DC biased coil placed several millimeters above the chip is used to apply a magnetic
field to the qubit for frequency tuning. The designed qubit parameters are listed in
Table (7.1).
7.3

Device Fabrication

This section lists the fabrication steps for realizing a Nb based transmon qubit with
Al/AlOx junctions on a 1cmx1cm Si chip. The process is split into 3 layers.

90
a.

100um

Readout Resonator
Transmon Qubit

XY + Readout
Line

4um

Bandage

Junctions

Bandage

Figure 7.2: Optical image of Niobium transmon qubit on silicon. a. Optical image of a fabricated
niobium device on silicon consisting of a readout resonator, transmon qubit and XY + readout line.
b. Zoomed-in optical image of the SQUID loop showing the Josephson junctions and the ‘bandage’
layer for electrical contact between the aluminum and niobium layers. Light blue areas are aluminum.
Grey areas are niobium.

Parameter
Designed Value
Qubit Frequency
5.5GHz
Readout Frequency
7.4GHz
Qubit Capacitance
73fF
𝐸 𝐽 /𝐸𝐶
70
Qubit Readout Coupling 80MHz
Qubit XY Coupling
<5kHz
Table 7.1: Qubit design parameters

Layer 1: This layer defines the markers for lithographic alignment, ground plane,
qubit capacitor, readout resonator, and CPWs. All these features are fabricated from
Nb. The steps for this layer are detailed below:
1. Chip Cleaning
• Acetone 5min sonication
• IPA 5min sonication
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min
• 15s dip in 10:1 Buffered HF followed by 2x 10s DI H2 0 rinse

91
2. Spin/Bake
• Pre-bake at 180◦ C for 3min
• Spin ZEP 520a at 3000 rpm for 1 min
• Post-bake at 180◦ C for 3min
3. E-Beam Lithography
• Beam current 50nA
• Fracturing resolution 20nm
• Dose 230 𝜇𝐶/𝑐𝑚 2
4. Development
• ZED N50 for 2.5min
• MIBK for 30s
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min
5. E-beam Evaporation of Nb
• 15s BOE dip just before loading in evaporator to strip native oxide
• Evaporate 150nm thick Nb at 0.4nm/s
6. Lift-off
• NMP at 150◦ C for 2hr
• Acetone 5min sonication
• IPA 5min sonication
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min
• 15s dip in 10:1 Buffered HF followed by 2x 10s DI H2 0 rinse
At this stage, we have used e-beam lithography, e-beam evaporation, and metal
lift-off to define our Nb circuit which forms layer 1.
Layer 2: In the second layer we will define our Al/AlOx Josephson Junctions. We
use a Dolan bridge technique utilizing a bi-layer resist stack and angled e-beam
evaporation to define the junctions. The steps are:

92
1. Spin/Bake
• Pre-bake at 170◦ C for 3min
• Spin Copolymer MMA EL-11 at 2200 rpm for 1 min
• Bake at 170◦ C for 3min
• Spin PMMA 950 A4 at 2200 rpm for 1 min
• Post-Bake at 170◦ C for 3min
2. E-Beam Lithography
• Beam current 1nA
• Fracturing resolution 2nm
• Base Dose 940 𝜇𝐶/𝑐𝑚 2 , relative dose = 0.48
3. Cold Development
• 3:1 IPA:DI H2 0 for 90s at 10◦ C with stirring at 500rpm
• IPA for 10s at 10◦ C with stirring at 500rpm
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 20s
4. Angled E-beam Evaporation of Al with in-situ oxidation
• 15s BOE dip just before loading in evaporator to strip native oxide
• Evaporate 60nm thick Al at 1nm/s, 40◦ angle from normal
• Static oxidation at 10mbar O2 pressure for 20mins
• Evaporate 120nm thick Al at 1nm/s, -20◦ angle from normal
5. Lift-off
• NMP at 150◦ C for 2hr
• Acetone 5min sonication
• IPA 5min sonication
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min
• No BHF dip as BHF reacts with Al. No anhydrous vapor HF (VHF)
either since VHF attacks Nb.

93
Now we have our Nb circuit with Al junctions.
Layer 3: The final layer is layer 3 where we use bandages to electrically connect the
Al layer to the Nb layer. The bandages are essentially rectangles of Nb that connect
the Al and Nb layers and are deposited via e-beam evaporation with an in-situ Ar
mill to remove any native oxide from the surface to make good electrical contact.
The steps are:
1. Spin/Bake
• Pre-bake at 180◦ C for 3min
• Spin ZEP 520a at 3000 rpm for 1 min
• Post-bake at 180◦ C for 3min
2. E-Beam Lithography
• Beam current 50nA
• Fracturing resolution 20nm
• Dose 230 𝜇𝐶/𝑐𝑚 2
3. Development
• ZED N50 for 2.5min
• MIBK for 30s
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min
4. E-beam Evaporation of Nb
• In-situ Ar mill with beam voltage 400V, accelerator voltage 80V and
beam current 21.6mA for 6mins
• Evaporate 150nm thick Nb at 0.4nm/s
5. Lift-off
• NMP at 150◦ C for 2hr
• Acetone 5min sonication
• IPA 5min sonication
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min

94
7.4

Experimental Setup

The measurements of the fabricated Nb-based qubit chip are performed in a dilution
refrigerator at a base temperature of T = 10 mK. The chip is wirebonded to a printed
circuit board (PCB) with multiple 50 Ω co-planar waveguides. Each co-planar
waveguide on the PCB is wirebonded to a corresponding co-planar waveguide on
chip which is used to both excite and readout the qubit. Microwave connectors
soldered to one side of the PCB allow microwave signals to be sent to the device.
The opposite side of the PCB is cut right to the edge of the chip to allow for a
lensed optical fiber to be placed in close proximity to the chip. A 3-axis positioner is
used to precisely position the optical fiber at a height 200um above the plane of the
qubit and a distance of 2mm from the qubit (Fig. 7.3). This geometry is intended
to mimic the scattered light in our transducer device. A 1550nm laser is used to
illuminate the chip via the lensed optical fiber. An acousto-optic modulator (AOM)
allows pulsing of the laser and is used to control both the duration and repetition
rate of the pulse. Microwave signals are generated and read out using the techniques
described in Chapter 1. A digital delay generator is used to maintain synchronization
between the optical and microwave pulses. A hand-wound coil made from Nb-Ti
superconducting wire mounted a few millimeters above the chip is used to generate
a magnetic field to tune the frequency of the qubit. DC current is applied to the coil
using a low noise DC source.

Mixing plate of
Dilution Refrigerator

Lensed
optical fiber
2 mm

b.
Microwave
connectors

Qubit
PCB

Device
under test

Coax cable

Three-axis positioner

Sample stage

3-axis
positioner

PCB
Lensed
Optical
Fiber

Figure 7.3: Experimental setup for optical tests on Nb transmon qubits. a. Schematic and b.
Photograph of the experimental setup on the mixing plate of a dilution refrigerator

95
7.5

Qubit Characterization

We begin by locating the readout resonator by performing a reflection measurement
using a vector network analyzer (VNA). Once we have located the readout resonator,
we locate the qubit, tune the qubit frequency at a desired detuning from the readout
resonator and proceed to characterize the qubit in the absence of optical illumination.
We perform standard lifetime (𝑇1 ) and coherence time (𝑇2∗ ) measurements, the
results of which are shown in Fig 7.4.
71 decay
0.8

fit
data

0.7

4ubit populatioQ

0.6

T1 fit

0.5

1.03 s 0.02 μs

0.4
0.3
0.2
0.1
0.0

7ime (μs)

T2* decohereQce
fit
data

1.0

4ubit 3opulatioQ

0.8

0.6

0.4

T2* fit

1.03 s 0.02 μs

0.2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

TiPe (μs)

Figure 7.4: 𝑇1 and 𝑇2∗ time constants of a niobium qubit on silicon.

7.6

Qubit Response to Optical Illumination

Next we turn our attention to measurements with optical illumination. Since we will
be inferring the qubit response by monitoring the readout resonator spectrum, we
must first characterize the bare readout resonator response under optical illumination.
By sufficiently detuning the qubit from the readout resonator the qubit can be
effectively decoupled from the readout resonator. This allows us to characterize the
optical response of the readout resonator without the influence of the qubit. We
perform spectroscopy on the readout resonator as we apply a 100 ns long, 85 𝜇𝑊
peak power laser pulse at a repetition rate of 10 kHz. The result of this measurement
is shown in Fig ??. Clearly, there is no measurable change in the spectrum of
the readout resonator after the laser pulse. This is as expected since the readout
resonator is made entirely of Nb which has a very fast QP response.
Now we flux bias the qubit to be 320 MHz detuned from the readout resonator. We

Laser

9NA )rHquHnFy (GHz)

9.0
8.5

7.49

8.0
7.5

7.48

7.0
6.5

7.47

6.0
7.46

40
7imH (μs)

Laser pulse applied
at t = 10µs

Readout Amplitude (mV)

7.50

5.5

Figure 7.5: Readout resonator spectroscopy under optical illumination

probe the amplitude of the readout signal as an XY drive tone is swept around the
qubit frequency. When the XY drive frequency is on resonance with the qubit frequency, the qubit is excited and the readout amplitude changes (in our measurement
this appears as a dip in the readout amplitude). We use this technique to monitor
the qubit frequency as we apply a 100 ns laser pulse at a 5 kHz repetition rate and
peak power 240 𝜇𝑊. The result of this measurement is shown in Fig ??.
3000

0.14
0.12

0W dHtuning (kHz)

2000

0.10

1000

0.08
0.06
−1000
0.04
−2000
−3000

0.02

100

150

175

7imH (μs)

Laser pulse applied (t=10µs)

125

0.06

5eadRut Amplitude (mV)

Readout Amplitude (mV)

fit
data
Tr fit

0.05

15.74 s 0.10 μs

0.04
0.03
0.02
0.01
0.00
20

100

120

140

160

180

7ime (μV)

Figure 7.6: Spectroscopy of Nb qubit under optical illumination. a. Spectroscopy of Nb Qubit
under optical illumination. b. Line-cut at 0 detuning showing recovery of the readout amplitude.
Fitting to an exponential yields a recovery timescale of 15.7 𝜇𝑠. Measurements performed with 100
ns laser pulse, 5 kHz repetition rate and 240 𝜇𝑊 peak power.

97
We observe a sharp change in the qubit spectrum as the laser is pulsed. In particular,
at t = 10 𝜇𝑠 when the laser is turned on, there is a downshift of the qubit frequency
and a corresponding increase in the amplitude of the readout signal. As the delay
from the laser pulse increases, the readout amplitude and qubit frequency relax back
to their steady state value. A line cut at 0 microwave detuning shows the recovery
of the readout signal. We find this recovery process fits well to an exponential with
a relaxation time constant of 15.7 𝜇𝑠.
In the above measurement, we are probing the qubit by applying a continuous wave
(CW) microwave signal to the XY drive line. We now utilize pulsed microwave
signals on the XY line to measure the population, energy decay rate (𝛾1 = 2𝜋𝑇
) and
decoherence rate (𝛾2 = 2𝜋𝑇 ∗ ) of the qubit as a delay from the laser pulse. A set of
these measurements performed at a 10 kHz repetition rate with a 100 ns laser pulse
and a peak laser power of 85 𝜇𝑊 is shown in Fig. 7.7.
Focusing on the population measurement, (Fig. 7.7a.), we observe the laser pulse
initially inducing some excess population in the qubit which then relaxes to a steady
state value on a timescale of ∼ 30 𝜇𝑠. We also note that the steady state population
(𝑃𝑒,𝑆𝑆 ) is higher than the thermal population in the laser off case (indicated by the
gray line). Measurements of the energy decay rate (𝛾1 ) and the decoherence rate
(𝛾2∗ ) show a similar fast recovery timescale on the order of ∼ 10 𝜇𝑠 and an excess
∗ ) due to a process slower than the repetition
decoherence rate in the steady state (𝛾2,𝑆𝑆
period (100 𝜇𝑠).
To gain further insight into this slow process, we measure the steady state population
∗ ) of the qubit at a delay of 81 𝜇𝑠 from the laser
(𝑃𝑒,𝑆𝑆 ) and decoherence rate (𝛾2,𝑆𝑆
pulse while sweeping the peak power of the laser pulse. The results are shown in
Fig. 7.8.
We fit the steady state population at each laser power to a Boltzmann distribution
𝑛𝑡ℎ (𝑇𝑒 𝑓 𝑓 ) with an effective temperature 𝑇𝑒 𝑓 𝑓 . We find that this effective qubit
temperature scales linearly with peak laser power in the low power regime 𝑇𝑒 𝑓 𝑓 (𝑃) =
𝑇𝑜 𝑓 𝑓 + 𝛽𝑃, with 𝑇𝑜 𝑓 𝑓 = 126 mK and 𝛽 = 1.6 ± 0.097 mK/𝜇𝑊. We find a similar
linear scaling of the decoherence rate of the qubit as a function of laser power (P),
∗ (𝑃) = 𝛾
𝛾2,𝑆𝑆
2,𝑜 𝑓 𝑓 + 𝛽𝑃 , with 𝑛 = 1.03 ± 0.17 and 𝛽 = 0.53 ± 0.41 kHz/𝜇𝑊 .
We further probe the dependence of the steady state population 𝑃𝑒,𝑆𝑆 as a function
of repetition rate and laser pulse duration Fig. 7.9. We find that the population fits
well to a Boltzmann distribution with an effective temperature that scales linearly in

98
0.4

a.
3opulDtion

Tr fit

lDser off reference
fit
dDtD

30 s 2 μs

0.3

Pe,SS

0.2

Laser

𝑇!

Readout
0.1
0.0

DelDy froP lDser pulse, Td (μs)

0.40
Tr fit

γ1 (0Hz)

0.35

fit
lDsHr off rHfHrHncH rDngH
dDtD

21 s 5 μs

0.30

Laser

0.25

𝑇!

0.20

Qubit XY

0.15

Readout

0.10
0.05

DHlDy from lDsHr pulsH, Td (us)

0.40
Tr fit

γ2* (0Hz)

0.35

fit
lDsHr off rHfHrHncH rDngH
dDtD

12 ± 1 μs

Laser

0.30
0.25

Qubit XY

𝛾!∗ ##

0.20

𝑇!

𝜋/2 𝜋/2

Readout

0.15

DHlDy from lDsHr pulsH, Td (us)

Figure 7.7: Recovery of niobium qubit after laser illumination. a. Qubit excited state population,
b. energy decay rate (𝛾1 ), and c. decoherence rate (𝛾2∗ ) versus delay, 𝑇𝑑 from a 100 ns long laser
pulse (peak power: 85 𝜇W, repetition rate: 10 kHz). Gray line indicates a reference measurement
with the laser off. Dashed line is an exponential fit indicating a recovery time, 𝑇𝑟 . With the laser on,
the steady state population (𝑃𝑒,𝑆𝑆 ) and decoherence (𝛾2,𝑆𝑆
) is higher than the laser off value (gray
line). The right column shows the corresponding pulse sequences used for each measurement.

both repetition rate and laser pulse duration. Our results indicate a trade-off between
power, repetition rate, and pulse duration. For a given pulse duration, we can use
higher peak laser powers by operating at lower repetition rates.
So far we have been focusing on the qubit population and decoherence at long delays
(81 𝜇s) from the laser pulse. We also investigate the effect on the qubit decoherence
rate when the laser pulse is applied during a Ramsey pulse sequence on the qubit.
The pulse sequence for this measurement is shown in the inset of Fig. 7.10. We time
the laser pulse to arrive just after the first 𝜋2 pulse of the Ramsey sequence and probe
the effect of the laser pulse on the decoherence rate of the qubit as a function of peak
laser power (Fig.7.10a.) and repetition rate (Fig.7.10b.). Even at a high repetition
rate of 50 kHz, we find a range of powers upto a few 𝜇𝑊 where the qubit is able to

99
a.
0.30
0.25

3opulaWion in |e⟩ aW Td 81 μs

0.26

fiW
laser off reference range
daWa

0.24
0.22
γ2,* SS (0Hz)

Pe, SS

0.20
0.15
0.10

Toff

126 P.

0.20
0.18
0.16

0.05
0.00

(xcHss dHcohHrHncH aW Td 81 μs
fiW
lasHr off rHfHrHncH rangH
daWa

0.14
100

0.12

102

101
3eak power (μW)

100

101
PHak powHr (μW)

102

Figure 7.8: Dependence of qubit population and decoherence on peak optical power. Dependence
of a. steady state excited state population, 𝑃𝑒,𝑆𝑆 and b. decoherence rate, 𝛾2,𝑆𝑆
of Nb qubit as
a function of peak laser power (P). Experimental sequence is repeated at 10 kHz repetition rate.
Horizontal gray regions indicate laser off values up to one standard deviation. Dashed line in a.
is a fit to 𝑛𝑡 ℎ (𝛽𝑃), where 𝑛𝑡 ℎ (𝑇) is the Boltzmann distribution assuming a two level system with
temperature T. Dashed line in b. is a fit to the expression 𝛾2,𝑜 𝑓 𝑓 + 𝛽𝑃 𝑛 .

0.30
0.25

b.
fit
lasHr Rff rHfHrHncH rangH
data

0.25

fit
laser off reference range
data

0.20
Pe, SS

Pe, SS

0.20
0.15
Toff

0.10

127 m.

0.15
0.10

0.05
0.00

0.30

Toff

127 P.

0.05
100

101
5HpHtitiRn ratH (kHz)

0.00 1
10

102
3ulse duration (ns)

Figure 7.9: Dependence of qubit population on repetition rate and optical pulse duration.
Steady state excited population 𝑃𝑒,𝑆𝑆 as a function of a. Repetition rate (R) (Peak power = 89𝜇𝑊;
pulse duration = 100 ns) and b. Laser pulse duration (D) (Peak power = 9.5 𝜇𝑊; Repetition rate
= 10 kHz). Horizontal gray regions indicate laser off values up to one standard deviation. Dashed
line in a. is a fit to a Boltzmann distribution 𝑛𝑡 ℎ (𝑇𝑒 𝑓 𝑓 ), where 𝑇𝑒 𝑓 𝑓 (𝑅) = 𝑇𝑜 𝑓 𝑓 + 𝛽𝑅, with
𝛽 = 16.9 ± 0.34 mK/kHz. Dashed line in b. is a fit to a Boltzmann distribution 𝑛𝑡 ℎ (𝑇𝑒 𝑓 𝑓 ), where
𝑇𝑒 𝑓 𝑓 (𝐷) = 𝑇𝑜 𝑓 𝑓 + 𝛽𝐷, with 𝛽 = 0.36 ± 0.0017 mK/ns.

maintain coherence after the laser pulse. There is a trade-off between repetition rate
and peak laser power to maintain qubit coherence.

100
0.40

γ2* (0Hz)

0.35
0.30
0.25

b.
fiW
lasHr off
daWa

𝜋/2

Qubit XY

0.35

lasHr Rff
12μW

0.33

31μW

0.30

𝜋/2

γ2* (0Hz)

Laser
Readout

0.20

0.28
0.25
0.23
0.20

0.15
0.10

0.17
10

10
3Hak powHr (μW)

0.15

20
30
40
5HpHtitiRn 5atH (kHz)

Figure 7.10: Ramsey measurement interrupted by a laser pulse. Decoherence rate extracted using
a Ramsey sequence interrupted by a laser pulse as a function of a. Peak laser power (Repetition
rate = 50 kHz) and b. Repetition rate. Horizontal gray regions indicate laser off values up to one
standard deviation. Inset of a. shows the pulse sequence used in both measurements. Pulse duration
was 100 ns for both measurements. Dashed line in a. is a fit to 𝛾2∗ (𝑃) = 𝛾2,𝑜 𝑓 𝑓 + 𝛽𝑃 𝑛 , where P is
peak power. 𝛽 = 1.42 ± 0.32 kHz/𝜇𝑊 𝑛 and 𝑛 = 1.2 ± 0.1.

7.7

Conclusion

In conclusion, our measurements of the population, lifetime and decoherence rate of
a niobium qubit with aluminum/aluminum-oxide/aluminum junctions on a silicon
substrate indicate that these qubits have an initial fast (∼ 10𝜇𝑠) recovery time when
exposed to optical illumination. We also find evidence of a slower process that
creates excess population and decoherence on a ∼ 𝑚𝑠 time scale. Power-dependent
measurements indicate this slower process is likely thermal in origin. We find a
trade-off between power and repetition rate where operating at lower repetition rates
allows higher peak laser powers. Crucially, there is a range of powers upto 10 𝜇W
where we do not have significant excess population or decoherence induced by a 100
ns laser pulse upto a repetition rate of 10 kHz. Given that the transducer device in
[10] operates at 2 𝜇W power with <100 ns pulse lengths, our measurements indicate
that a repetition rate of 10 kHz would be achievable by replacing the all Al qubits
in the device in [10] with hybrid Nb-Al qubits. This would be a 100x improvement
over the repetition rate of 100 Hz reported in [10]. However it is important to keep in
mind that the experiments performed in this chapter involved using a lensed optical
fiber to illuminate the entire qubit chip. In a real transducer device, the lensed fiber
would couple light into an on-chip optical waveguide which would be routed to an
optomechanical crystal cavity that is in close proximity to the qubit. This could
change the effective optical power seen by the qubit. To test this, it is necessary to
pattern optomechanical crystals on chip along with the Nb qubit for which we need

101
to move to a silicon-on-insulator substrate. This poses some fabrication challenges
which form the subject of the next chapter.

102
Chapter 8

TOWARDS FABRICATING NIOBIUM BASED QUBITS ON
SILICON-ON-INSULATOR SUBSTRATES

In the previous chapter, we studied the optical response of niobium (Nb) based
transmon qubits with aluminum/aluminum-oxide/aluminum junctions fabricated on
a silicon substrate. Our results showed a favorable quasiparticle recovery time for
these niobium based qubits compared to the all aluminum transmon qubits used in
[10]. However, as discussed in Chapter 6, our transducer device is designed to be
fabricated on the device layer of a silicon-on-insulator (SOI) substrate. To integrate
our Nb-based transmon qubits with our piezo-optomechanical transducer device, we
need to develop a fabrication process for realizing Nb qubits on SOI. In this chapter
we discuss some of the challenges associated with fabricating Nb-based microwave
circuits on SOI substrates and describe some initial steps towards realizing Nb-based
transmon qubits on SOI.
8.1

Fabrication Challenges

A fabrication process for realizing aluminum (Al) transmon qubits on SOI is outlined in [78] and reproduced in Fig 8.1. This process relies on e-beam evaporation
and metal lift-off to pattern the microwave circuit on the silicon device layer of a
silicon-on-insulator substrate. This is followed by release of the device layer by
etching away the buried oxide layer using anhydrous hydrofluoric acid (VHF) in
vapor form—a step we will refer to as ’VHF release’. It is crucial to etch away the
buried oxide layer everywhere underneath the microwave circuit as the oxide is a
lossy dielectric and can contribute significantly to microwave losses. Since the size
of a typical circuit consisting of a transmon qubit and associated readout resonator
and control lines is on the order of ∼100s of 𝜇𝑚, this results in large (∼mm sized)
released membranes. While this process works well for aluminum on SOI, niobium
films deposited via e-beam evaporation are under considerably higher tensile stress
[184, 185]. This makes release of large area membranes challenging as the high
tensile stress tends to crack and break the membranes. Further, unlike Al, Nb reacts
with anhydrous hydrofluoric acid, and hence must be protected during the VHF
release step.

103
C4F8/SF6
dry etch (*)

Al ground plane
evaporation

junction
evaporation

ion mill, bandage
evaporation

HF vapor
release (*)

SiO2
Si
(c)

Figure 8.1: Fabrication process for Al transmon qubits on SOI. Figure reproduced from [78].
qubit cap.

8.2

Development of a Sputtering Process for Niobium Thin Films

Since e-beam evaporation produces highly tensile films,
tech50 μmwe need a deposition
2 μm
ground plane
nique that allows us to control the stress of the deposited film. An alternative to
e-beam evaporation is sputter deposition. Sputtering works by bombarding a metal
target with highly energetic ions (typically Ar ions) to eject target material that then
condenses on the surface of the substrate forming a thin film. Sputtering allows
considerable control over the deposition parameters, which can be tuned to change
the stress of the deposited film. In particular, controlling the sputtering chamber
pressure has been shown to be a convenient way of tuning the stress of Nb thin films
deposited via DC magnetron sputtering using argon ions [186].
We utilize an AJA ATC Orion 8 UHV sputtering system for sputtering our Nb film.
It is routinely capable of achieving low E-9 torr pressures ensuring high quality of
the sputtered film. We perform DC magnetron sputtering using a 2 inch Nb target.
The sputtering parameters are listed in Table 8.1
Parameter
DC Power
Ar Flow Rate
Process Pressure
Substrate Temperature
Substrate Rotation
RF Stage Bias

Value
300 W
20 sccm
Varied to tune stress
Room temperature
10 rpm
Not applied

Table 8.1: Nb sputtering parameters

We tune the process pressure to tune the stress of the deposited film. We utilize
stylus profilometry to measure the bow of a 4 inch test grade silicon wafer before
and after sputter deposition of a 150 nm thick Nb film. We calculate the radius of
curvature from the measurement of the bow as 𝑅 = 𝑟 2 /2𝛿, where 𝑅 is the radius of
curvature, 𝛿 is the bow of the wafer, and 𝑟 is the radius of the substrate. The stress

104
of the deposited film can then be calculated using Stoney’s equation [187, 188].
𝐸 𝑡 𝑠2
𝜎=
(8.1)
6 𝑅 𝑝𝑜𝑠𝑡 𝑅 𝑝𝑟𝑒 (1 − 𝜈) 𝑡 𝑓
where,
𝜎 = stress in deposited film
𝑅 𝑝𝑜𝑠𝑡 = radius of curvature of substrate post deposition
𝑅 𝑝𝑟𝑒 = radius of curvature of substrate pre depostion
𝐸 = Young’s modulus of substrate
𝜈 = Poisson’s ratio of substrate
𝑡 𝑠 = substrate thickness
𝑡 𝑓 = thickness of deposited film

6tress of DeSosited 1b )ilP (03D)

In Fig. 8.2, we plot the stress of the deposited 150 nm Nb film as a function
of sputtering process pressure. Similar to [186], we see an initial compressive stress
at low sputtering pressures, which becomes increasingly tensile as we increase the
process pressure. Crucially, there is a 0 crossing of the stress at low process pressures allowing us to tune the stress of our film to be near 0. In practice, we want our
Nb film to have a small amount of tensile stress to compensate for the compressive
stress in the silicon device layer of our SOI substrate. We target a tensile stress of
∼200 MPa which is achieved at a sputtering pressure of 3.9 mTorr.

600
400

Target stress

200

Tensile Stress

Compressive Stress

−200
−400
−600
−800

6Suttering 3roFess 3ressure (PTorr)

Figure 8.2: Stress of a 150 nm thick Nb film sputtered on a Si substrate as a function of
sputtering process pressure.

105
8.3

Development of an Etching Process for Niobium Thin Films

Sputter deposition produces conformal coatings, that coat the sidewalls of features
which makes metal lift-off difficult. As a result, we need to find an alternative
technique for patterning the sputtered thin films. We utilize reactive ion etching
(RIE) to pattern our sputtered Nb films. In this process, a chemically reactive
plasma (utilizing a fluorine based chemistry) is used to etch the Nb film. Since
our Nb film sits on a thin (220 nm) silicon device layer, care must be taken to
ensure we can stop the etch without over-etching too deep into the silicon device
layer. Further, since our eventual goal is to use these fabrication techniques to
develop an integrated transducer with niobium qubits and a lithium niobate on SOI
piezo-optomechanical device, we need to ensure that the etch selectively etches Nb
without etching lithium-niobate.
We utilize an Oxford Instruments Plasmalab 100 ICP-RIE 380 system to etch our
Nb films. Niobium can be etched in a variety of fluorine based [166, 189, 190]
and chlorine based [167, 191] chemistries. However fluorine and chlorine based
chemistries are known to etch silicon too. To minimize the silicon etch rate, we
utilize a C4 F8 /O2 chemistry. This is inspired by [192] where this chemistry is used
for selectively etching SiO2 while minimizing the etch rate of silicon. Our optimized
Nb etch parameters are listed in Table 8.2. The corresponding etch rates of niobium,
silicon, and lithium niobate (-Z-cut) are listed in Table 8.3. The etch rate of silicon
is about 24 nm/min. This is slow enough that we can afford to over-etch the Nb by
about 1 min and only thin down the Si device layer by ∼ 20 nm. We utilize thin film
ellipsometry as feedback to precisely time the etch and prevent larger over-etching
into the Si device layer.
Parameter
ICP Power
RF Power
𝐶4 𝐹8 Flow
𝑂 2 Flow
Temperature
Process Pressure
Helium Backing Pressure

Value
750 W
150 W
40 sccm
3 sccm
15 C
8 mTorr
4 torr

Table 8.2: Nb etching parameters

Our mask for the niobium etch is 10 nm thick alumina deposited via atomic layer
deposition (ALD-alumina). The alumina mask itself is patterned by a physical argon
ion based ICP-RIE etch utilizing ZEP 520a resist as a mask (parameters listed in

106
Material
Niobium
Silicon
Lithium Niobate
ALD Alumina

Etch Rate
15-25 nm/min
24nm/min
3.4nm/min
0.1-0.2 nm/min

Table 8.3: Etch rates of various materials in the Nb etch

Table 8.4). ALD-alumina exhibits a selectivity of ∼ 100:1 over Nb for our chosen
Nb etch chemistry—making it a suitable masking material (see Table 8.3 for ALDalumina etch rate in C4 F8 /O2 chemistry). It can easily be removed post etching with
a short (∼ 1 min) buffered HF dip.
Parameter
ICP Power
RF Power
Ar Flow
Temperature
Process Pressure
Helium Backing Pressure
ALD Alumina Etch Rate

Value
500 W
50 W
20 sccm
20 C
10 mTorr
4 torr
2nm/min

Table 8.4: ALD alumina etching parameters

We find that our chosen Nb etch chemistry causes some deposition on the sidewalls
of the niobium features. A 10 min long O2 plasma ashing followed by a 30s buffered
HF dip serves to clear up the residue and reveals smooth, vertical sidewalls as shown
in Fig. 8.3.
8.4

Protection of Nb Surface from VHF Attack

Finally, we need to address the issue of anhydrous vapor HF (VHF) reacting with
the Nb surface. To protect the Nb from being exposed to VHF, we must mask the
Nb with a mask that is impervious to VHF. Post the VHF release, we will have a
large suspended membrane which makes removal of the masking material difficult
as any solvent-based mask removal process will crack the membrane due to surface
tension. Ideally we want a mask that does not need to be removed post VHF release.
This mask should be thin enough that it does not greatly affect the quality factor
and frequency of our microwave circuit. At the same time it should be able to
conformally coat the sidewalls or our Nb features and be dense enough (pin-hole
free) to be impervious to VHF.
Atomic layer deposited (ALD) alumina works well as a masking material for this

107

Si
500nm

500nm

Figure 8.3: He FIB image of etched Nb on Si. a. Before and b. after O2 plasma ashing and buffered
HF clean.

purpose. Atomic layer deposition utilizes chemical precursors that sequentially
react with the surface of the substrate in a self-limiting process to grow a thin film
one layer at a time. It is a highly conformal process and can produce high quality,
pin-hole free thin films with very precise control over film thickness.
We grow our alumina thin film in an Oxford Instruments Plasma Technology FlexAL
II system utilizing a plasma assisted process at 300C with a trimethyl aluminum
(TMA) precursor. We find a growth rate of 1.2 Å per cycle for this process. A 15
nm thick alumina film grown in this manner is found to be sufficient to act as a mask
that protects the niobium surface from VHF attack. (Fig 8.4)
8.5

Quality of Sputtered and Etched Niobium Thin Films

Now that we have developed the individual steps for sputtering, etching and protecting the Nb film, we proceed to test the quality of the thin films produced using this
fabrication process. For this purpose, we fabricate lumped element Nb microwave
resonators on a Si substrate. We use an on-chip coplanar waveguide (CPW) to
perform a reflection measurement and extract the intrinsic quality factor (Q𝑖 ) at
cryogenic temperatures (∼10 mK) and low (single photon) power levels. We chose
silicon as our substrate for this test in order to enable comparison to similar e-beam
evaporated resonators fabricated on Si in our lab. We find Q𝑖 ∼ 100,000 which is
comparable to similar lumped element resonators fabricated via e-beam evaporation
in our lab. The fabricated device and measurement results are shown in Fig. 8.5.

108
a.

Before VHF

After VHF

15nm ALD protection

No ALD protection

Figure 8.4: Protection of Nb surface using ALD alumina. Surface of Nb film before (upper panel)
and after (lower panel) exposure to anhydrous vapor HF with a. no ALD protection layer and b.
15nm thick ALD protection layer.
a.

Nb Resonator

CPW

IQtriQsiF 4uality )aFtor (Qi)

500000

400000

300000

200000

100000

20
40
60
AtteQuatioQ oI 0iFrowave 6igQal

Figure 8.5: Optical image and measured quality factor of Nb lumped element resonators. a.
Optical image of a fabricated Nb lumped element resonator on Si using the developed sputter and
etch process b. Measured intrinsic quality factor (Q𝑖 ) as a function of microwave power.

8.6

Etch of VHF Release Holes

In the previous sections, we discussed the development of a sputter and etch process
to fabricate high quality Nb-based superconducting circuits on the Si device layer of

109
an SOI substrate. As mentioned in the introduction to this chapter, we need to etch
away all the buried oxide underneath our Nb circuit to form a suspended Nb on Si
membrane in order to reduce dielectric losses. We do this in a step called ‘release’,
where we flow anhydrous hydrofluoric acid in vapor form (VHF) which selectively
etches away the lossy silicon-oxide without attacking silicon. For this process to
work, the VHF needs to be able to penetrate underneath the silicon device layer
to attack the underlying oxide. To facilitate this, we pattern and etch an array of
∼200 nm diameter holes with 4𝜇m hole spacing in the silicon device layer. These
holes are etched through the alumina, Nb, and Si layers and allow for vapor HF to
penetrate through to the buried oxide layer and etch it away. We do this ‘HF holes
etch’ in a single step using a thick (∼ 800-900 nm) layer of ZEP 520a resist as an
etch mask. First, to etch through the alumina layer, we use the Ar ion alumina etch
with the parameters listed in Table 8.4. Once we have etched through the alumina
layer, we switch to an ICP-RIE etch with SF6 /C4 F8 chemistry (similar to the one
used for aluminum transmon qubits on SOI in [78]). The SF6 /C4 F8 etch serves to
etch through both the Nb and Si device layers. The corresponding etch parameters
are listed in Table 8.5.
Parameter
ICP Power
RF Power
SF6 Flow
C4 F8 Flow
Temperature
Process Pressure
Helium Backing Pressure

Value
1200 W
23 W
16 sccm
40 sccm
15 C
10 mTorr
4 torr

Table 8.5: VHF release holes etching parameters

8.7

Proposed Fabrication Process for Niobium Transmon Qubits on SOI

Having developed a sputter and etch process for Nb thin films and having tested
the ALD alumina protection layer, we conclude this chapter by proposing a detailed
fabrication process for realizing Nb based transmon qubits on SOI substrates. The
process is schematically outlined in Fig. 8.6.
Layer 1: Markers, ground plane, qubit capacitor and CPWs:
1. Chip Cleaning
• Acetone 5min sonication

110
Sputter Nb

C4F8/O2
Etch Nb

JJ Evaporation

VHF Release

SF6/C4F8 Etch release
holes in Nb+Si

Ion mill + Bandages

SiO2
Si

Ar Etch release
holes in alumina

ALD alumina
VHF Protection

Figure 8.6: Proposed fabrication process for Nb transmon qubits on SOI

• IPA 5min sonication
• N2 blow dry
• O2 plasma ash at 150W, 12 sccm O2 flow for 2min
2. Nb Sputter Deposition
• 15s dip in 10:1 Buffered HF followed by 2x 10s DI H20 rinse—this step
etches away the native silicon-oxide on the surface of the substrate and is
done right before loading the chip in the sputterer for a clean oxide-free
Nb-Si interface
• Sputter 150 nm Nb at 3.9 mTorr process pressure and other parameters
as listed in Table 8.1
3. ALD Deposition of Alumina Etch Mask
• Deposit 10 nm alumina at 300◦ C
4. Spin/Bake
• Pre-bake at 180◦ C for 3min
• Spin ZEP 520a at 3000 rpm for 1min
• Post-bake at 180◦ C for 3min
5. E-Beam Lithography
• Beam current 50nA

111
• Fracturing resolution 20nm
• Dose 230 𝜇𝐶/𝑐𝑚 2
6. Development
• ZED N50 for 2.5min
• MIBK for 30s
• N2 blow dry
• O2 plasma ash at 150W, 12 sccm O2 flow for 2 min
7. ICP-RIE Etching of Alumina Etch Mask
• 6 min Ar ion etch of the alumina mask using parameters listed in Table
8.4. The etch rate is ∼ 2nm/min, so we are over-etching by 1 min.
8. ICP-RIE Etching of Nb
This step can be done right after the alumina etch without stripping of the
ZEP 520a resist. The resist will just act as an additional mask on top of the
alumina mask for this step.
• 6-10 min C4 F8 /O2 etch of Nb using parameters listed in Table 8.2.
Use ellipsometry to determine etch stopping point. Aim to stop when
ellipsometer shows 200 nm thick silicon device layer (20 nm over-etch
into Si).
9. Etch Mask Removal
• NMP at 150◦ C for 2hr
• Acetone 5min sonication
• IPA 5min sonication
• N2 blow dry
• O2 plasma ash at 150W, 12 sccm O2 flow for 2min
• 1min dip in 10:1 Buffered HF followed by 2x 10s DI H2 0 rinse—the
previous steps remove any surviving ZEP and then this 1min BHF step
will remove the ALD alumina layer.
At this stage, we have used e-beam lithography, sputter deposition, and reactive ion
etching to define our Nb circuit which forms layer 1.

112
The next layers are Layer 2 (Josephson Junctions) and Layer 3 (Bandage Layer).
The fabrication of these layers is identical to the process outlined in the previous
chapter in section 7.3.

Layer 4: VHF release. In this layer, we will deposit a 15 nm thick ALD alumina
layer for protection of the Nb surface from VHF. We will then pattern and etch
an array of ∼200 nm wide ‘release holes’ with ∼4 𝜇m spacing. These holes are
punched through the alumina, Nb, and the Si device layers to allow anhydrous vapor
HF access to the underlying buried oxide layer. Finally, we flow anhydrous HF in
vapor phase to etch away the buried oxide and create a suspended Si membrane with
the Nb circuit patterned on top.
1. Deposit 15 nm ALD Alumina at 300◦ C
2. Double-Spin/Bake
• Pre-bake at 180◦ C for 3min
• Spin ZEP 520a at 3000 rpm for 1min
• Bake at 180◦ C for 3min
• Spin ZEP 520a at 3000 rpm for 1min
• Post-Bake at 180◦ C for 3min
3. E-Beam Lithography
• Beam current 1nA
• Fracturing resolution 2nm
• Dose 300 𝜇𝐶/𝑐𝑚 2
4. Development
• ZED N50 for 2.5min
• MIBK for 30s
• N2 blow dry
• O2 plasma ash at 150W, 12 sccm O2 flow for 2min
5. ICP-RIE Etching of Release Holes

113
• 8.5 min Ar ion etch of the 15 nm thick alumina layer using parameters
listed in Table 8.4. The etch rate is ∼ 2nm/min, so we are over-etching
by 1 min.
• 8.5min SF6 /C4 F8 etch of the Nb and Si device layers using parameters
listed in Table 8.5. We try to over-etch by ∼50% in this step as the goal
is to make sure we have etched all the way into the buried oxide layer.
6. VHF release

114
Chapter 9

APPENDIX: DEVELOPMENT OF AN ETCH PROCESS FOR
LITHIUM NIOBATE ON SILICON-ON-INSULATOR

In Chapter 6, we discussed the design of our piezo-optomechanical transducer
device based on a lithium niobate (LN) on silicon-on-insulator (SOI) platform.
As explained in Chapter 6, this choice of materials was motivated by a desire to
optimize both the piezo-electric interaction 𝑔 𝑝𝑒 and the optomechanical interaction
𝑔𝑜𝑚 . However, this material platform poses some unique nano-fabrication challenges
related to the etching of thin film lithium niobate without damaging the underlying
silicon device layer of the SOI substrate 1
Lithium niobate is a very chemically stable material, so it is quite challenging to
etch. In the past, a few different processes have been used to etch lithium niobate
mainly for application in nanophotonics. These range from mechanical processes
like dicing [193, 194] and chemical-mechanical polishing (CMP) [195–197] to
wet etching [198–204] and dry etching [205–215]. We are interested in etching
lithium niobate piezo-acoustic cavities with sub 1𝜇m dimensions. In the literature,
mechanical processes for micro-machining LN tend to be limited to feature sizes on
the order of ∼ few 𝜇m which is too large for our purposes. For wet etching lithium
niobate, the most common etchant used is a mix of hydrofluoric acid (HF) and nitric
acid (HNO3 ) [198–201]. Unfortunately, this mixture is also a silicon etchant [216],
so it is also not suitable for our purposes. Undercutting below the mask is another
common issue with wet-etching and makes it difficult to achieve tight dimension
control for sub 𝜇m size features.
9.1

Dry Etch Process

Given the desired small dimensions of our LN piezo-acoustic cavities, we would
like to develop a dry reactive ion etch (RIE) process which exhibits good anisotropy
and allows tight dimension control. Due to its chemical stability however, there
are not many gas chemistries that can etch LN. Fluorine based chemistries have
been employed in the past to etch LN. However, this produces non-volatile lithiumfluoride (LiF) as a by-product which causes severe redeposition issues inhibiting
1 For all of the etches described in this chapter, we use thin film X-cut or -Z-cut LN bonded to a

SOI substrate with a 220 nm thick Si device layer. The bonding is carried out by NanoLN.

115
further etching [206]. In recent years, a purely physical etch based on Ar+ ions
has been employed quite successfully to etch LN [213–215] exhibiting fairly steep
sidewalls and high anisotropy. Inspired by these results, we begin by developing
an Ar+ ion ICP-RIE etch process to pattern LN piezo-acoustic cavities on SOI.
We utilize an Oxford Instruments Plasmalab 100 ICP-RIE 380 system for our etch.
The parameters that we choose to vary are the chamber pressure, ICP power, and
RF power. While the overall dependence of the etch on these three parameters is
complex, we observe two trends: 1. The sidewall angle of the etched LN is steeper
at lower process pressures. 2. For fixed ICP power, the etch rate of LN can be
increased by increasing the RF power. After some exploration of the parameter
space, we arrive at the etch parameters given in Table 9.1. The mask we use
for the dry etch is a thin film chrome (Cr) mask which is patterned using e-beam
lithography, e-beam evaporation, and metal liftoff. We use ∼400nm thick layer of
ZEP 520a e-beam resist for lithography. The evaporation is carried out in a CHA
Mark-40 e-beam evaporator at a rate of 0.2nm/s. We find a 1:1 selectivity of LN to
Cr for our chosen etch parameters. An LN piezo ‘box’ etched using this process is
shown in Fig. 9.1. We are able to achieve a ∼40 degree sidewall angle using this
process. As can be seen from Fig. 9.1, there is fair amount of sidewall roughness in
our piezo ‘box’. The origin of this roughness is not clear, but seems to be influenced
by our choice of masking material. Replacing chrome with PECVD deposited SiO2
seems to greatly improve the sidewall roughness and sidewall angle. Other groups
have reported fairly smooth and steep sidewalls using a hydrogen silsesquioxane
(HSQ) mask for a similar Ar+ ion etch [213, 214].
Parameter
ICP Power
RF Power
Ar Flow
Temperature
Process Pressure
Helium Backing Pressure
LN Etch Rate
Cr Etch Rate
Si Etch Rate

Value
600 W
150 W
20 sccm
21 C
1.5 mTorr
4 torr
20 nm/min
20 nm/min
13 nm/min

Table 9.1: LN dry etch parameters

Using this dry etch process, we fabricate a wavelength scale lithium niobate piezoacoustic cavity as shown in Fig. 9.2a. However, since this dry etch relies on a

116

40° sidewall

Re-deposition

500 nm

Figure 9.1: LN piezo box etched using an Ar+ ion based ICP-RIE dry etch process.

purely physical process where Ar+ ions bombard the surface of the LN to etch it,
the etch is not selective between LN and silicon. As a result, the etch must be timed
precisely to prevent over-etching into the underlying silicon device layer of our SOI
substrate. Since our optimized etch uses fairly high power, it etches silicon at a high
rate of ∼13 nm/min and makes it difficult to prevent significant over-etching into the
silicon. This over-etching damages the surface of the silicon and induces a high level
of roughness in the silicon layer as seen in Fig. 9.2 b,c. We suspect that the cause of
this roughness is micro-masking from redeposited LN during the etch process (Fig.
9.1). Further, the thin film LN as supplied by the vendor is not uniform in thickness.
Even on a 1cmx1cm chip scale, there are thickness variations as high as ∼40nm.
As a result, etching away the LN layer where it is thickest will necessarily involve
significant over-etching in parts where the LN layer is thinner creating unavoidable

117
damage in the silicon layer. Recall from Chapter 6 that our optomechanical cavity
is patterned in the silicon device layer. The quality factor of the optical mode is
extremely sensitive to scattering from defects in the silicon and the large amount of
roughness induced by our LN dry etch in the silicon will almost certainly degrade
our optical quality factors significantly. Thus, while this Ar+ ion dry etch has
shown to be promising for etching nanophotonic devices in LN substrates, it is
not compatible with our LN on SOI platform where our optomechanical device is
patterned in silicon.
a.

b.
LN

Roughness
LN

200nm

Damage
to Si
layer
100 nm

Figure 9.2: Dry etched LN piezo-acoustic cavity showing Si damage. a. LN piezo-acoustic cavity
on SOI patterned via dry etching. b. and c. Zoom-ins showing damage to the Si device layer due to
the physical nature of the dry etch.

9.2

Wet Etch Process

To prevent damage to the silicon (Si) device layer of our SOI substrate, it is important
to develop an etch that is selective between LN and Si. To this end, we started
investigating etchants that could etch LN without attacking Si. As mentioned
earlier, the most commonly used wet etchant for LN is a mix of HF and HNO3 ,
but this etchant readily attacks Si. However [202, 203] showed that concentrated
HF at elevated temperatures is able to etch -Z cut LN. HF does not etch Si and
is commonly used to etch away native silicon oxide from the surface of Si wafers.
However, there are significant safety concerns associated with heating HF to elevated
temperatures. We decided to investigate if room temperature 48% concentrated HF
could selectively etch -Z cut LN without attacking Si. The challenge with using

118
concentrated HF as an etchant is to find a suitable etch mask since most materials are
etched by HF. Gold (Au) is one of the few materials that is inert in HF. Since gold
can be readily deposited using e-beam evaporation, we decided to use a gold mask
for our HF etch. We again used e-beam lithography, e-beam evaporation, and metal
liftoff to pattern our gold mask similar to the process used to pattern the chrome
mask for dry etching. To improve the adhesion of gold to our LN thin film, we first
evaporate a thin ‘adhesion layer’ (∼5-10nm) of chrome (Cr) followed by 200nm of
gold. Such a Cr/Au mask is commonly used in etching of glass using HF. One of the
issues with a Cr/Au mask is the formation of defects or pinholes that allow the HF
to penetrate through the masking layer. These defects are thought to originate upon
cooling of the substrate post e-beam evaporation of the masking layer [217]. Since
the surface of the gold tends to be hydrophilic, the HF can get sucked into these
defects and penetrate through the mask to the underlying LN. To prevent this, we do
our gold deposition in two steps. We first evaporate 100nm of gold and then pause
the evaporation to let the substrate cool. After about 15-30 minutes of a ‘cooldown
period’, we restart the evaporation and deposit the remaining 100nm of gold. In this
manner, any defects generated in the first evaporation are covered up by the second
evaporation making it harder for the HF to penetrate through the masking layer. In
principle, breaking the evaporation into a larger number of steps would increase the
robustness of the mask, but we find a two step evaporation to be sufficient.
Since our etchant (conc. HF) selectively etches the -Z crystal face of LN much faster
than any other crystal face, it is possible to achieve a highly anisotropic etch. To
test our Cr/Au mask and the anistropy of the etch, we try to etch simple LN ‘boxes’
as shown in Fig. 9.3a. It is clear from the figure that this etch is fairly anisotropic
giving sidewall angles as high as ∼50 degrees. Further, the sidewall roughness is
much reduced compared to the dry etch. However, in Fig. 9.3a, we have not yet
etched all the way through the LN to the Si device layer. We find that while the
HF etches the -Z surface of LN quite well and does not attack Si, when the etch
reaches the interface between the LN and the underlying Si, it causes ‘delamination’
of our LN boxes as shown in Fig. 9.3c. We suspect that there is some native oxide
present at the interface between the LN and the Si which is rapidly attacked by the
HF causing the LN box to ‘peel off’.
9.3

Proposed Hybrid Etch Process

In the previous sections, we have seen the advantages and disadvantages of the dry
and wet etch processes for etching LN. The dry etch has the advantage of being

119
a.

Cr/Au Mask

Figure 9.3: Wet etched LN ‘boxes’. a. LN ‘box’ etched using 48% HF. Note that the etch has not
progressed all the way to Si. There is a thin layer of LN left on the surface. b. Optical image of
patterned LN boxes before etching to Si. c. Optical image of patterned LN boxes post etching to Si.
Most of the boxes have ‘peeled off’ and can be seen displaced.

anisotropic and not attacking the interface between LN and Si. However it is not
selective between LN and Si, and hence causes significant damage to the Si surface.
The wet etch using 48% HF on the other hand can be anisotropic and selective
between LN and Si however it rapidly attacks the LN-Si interface. Here we propose
combining these two etches into a ‘hybrid’ etch process. This is a 3-step process. In
the first step, we define a narrow ‘trench’ around where we want our LN box. This
trench is etched most of the way using the wet etch described in Section 9.2, but we
do not etch the trench all the way through to the Si layer. We propose leaving about
20 nm of LN in the trench. By etching the trench most of the way with a wet etch
we are able to get smooth steep sidewalls as seen in Fig. 9.4. At the same time,
stopping the wet etch before it reaches the Si prevents the HF from attacking the
LN-Si interface. Next, we use the dry etch of Section 9.1 to etch through the last 20
nm of LN in the trench and overetch about 10-20 nm into the Si device layer. This
will create some damage to the surface of the Si, but this damage region will be
confined to the trench which can be made as narrow as ∼ 100 nm and is only limited
by the accuracy of alignment during our e-beam lithography. The overetch into Si
in the trench region exposes the LN-Si interface as shown in Fig. 9.4. Subsequently,
we deposit a Cr/Au mask that not only covers the top of our LN ‘box’, but also fills
the trench, and so protects the exposed LN-Si interface. Now we can use the wet
etch to etch away the remaining LN everywhere else on the chip. Since the Cr/Au

120

LN – Si interface

100 nm

Figure 9.4: LN piezo box etched using a combination of wet and dry etches. Smooth sidewalls
of a LN piezo ‘box’ etched using a wet etch most of the way followed by a dry etch to clear the last
∼40 nm of LN and expose the LN-Si interface. Note that this particular sample did not have a trench
defined, but by utilizing a trench the damage to the Si surface can be limited to the trench region.

mask is filling the trench, it covers the LN-Si interface near the LN ‘box’ and we
expect that the box will not delaminate since the HF is unable to penetrate through
the mask and attack the LN-Si interface. In this manner we can utilize a combination
of the above developed etches to etch LN on SOI without creating a large damaged
Si region. The damage is confined to the trench and thus will minimize the effect
on the optical quality factors of the OMC patterned in the silicon. The schematic
of our proposed ‘hybrid’ etch process is shown in Fig. 9.5 Suggested steps for our
proposed hybrid etch process are detailed below. We assume that we are starting
with 100 nm of -Z cut LN on SOI.

121
~20 nm overetch into Si

~20 nm LN left

LN
SiOx
Si

Pattern
Cr/Au
Mask

Dry
Etch

Wet
Etch

Si damage
confined to trench

Mask protects
LN-Si interface

Remove
Mask

Wet
Etch

Pattern
New
Cr/Au
Mask

Remove
Mask

Figure 9.5: Proposed Hybrid Etch Process for LN.

Layer 1: Markers. Markers are used for alignment for subsequent e-beam lithography steps. In our group, we typically use 20𝜇m × 20𝜇m squares of ∼150 nm thick
e-beam evaporated niobium (Nb) as our markers.
1. Chip Cleaning
• Acetone 5min sonication
• IPA 5min sonication
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min
• 15s dip in 10:1 Buffered HF (BHF) followed by 2x 10s DI H2 0 rinse
2. Titanium Conductive Layer for E-beam Lithography
Since LN is an insulating substrate, we need to deposit a conducting layer
to prevent charging effects during electron-beam lithography. This can be
achieved in a variety of ways. There are commercially available conductive
polymers (such as AquaSave) which can be spun on top of e-beam resist and
help disperse charge. An alternative is to lay down a thin layer of metal that
will serve to conduct away charge. This metal layer can be above or below
the resist. We decide to use a thin titanium (Ti) layer underneath the e-beam
resist to reduce charging effects during e-beam lithography. This thin Ti layer
can be easily removed by a short ∼30s BHF dip.
• Deposit ∼5-10 nm of Ti using e-beam evaporation

122
3. Spin/Bake ZEP 520a
• Pre-bake at 180◦ C for 3min
• Spin ZEP 520a at 3000 rpm for 1 min
• Post-bake at 180◦ C for 3min
4. E-Beam Lithography
• Beam current 50nA
• Fracturing resolution 20nm
• Dose 230 𝜇𝐶/𝑐𝑚 2
5. Development
• ZED N50 for 2.5min
• MIBK for 30s
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min
6. E-beam Evaporation of Nb
• 30s BHF dip to remove the Ti conducting layer where the markers will
be deposited
• Evaporate 150nm thick Nb at 0.4nm/s
7. Lift-off
• NMP at 150◦ C for 2hr
• Acetone 5min sonication
• IPA 5min sonication
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min
• 30s dip in 10:1 Buffered HF followed by 2x 10s DI H2 0 rinse - this
removes the remaining Ti conductive layer.
Layer 2: Trench This layer is to define the trench around where we want our LN
‘boxes’

123
1. Fresh Titanium Conductive Layer for E-beam Lithography
• Deposit ∼5-10 nm of Ti using e-beam evaporation
2. Spin/Bake ZEP 520a
• Pre-bake at 180◦ C for 3min
• Spin ZEP 520a at 3000 rpm for 1min
• Post-bake at 180◦ C for 3min
3. E-Beam Lithography
• Beam current 4nA (small features) and 50nA (large features)
• Fracturing resolution 4nm (small features) and 20nm (large features)
• Dose 230 𝜇𝐶/𝑐𝑚 2
4. Development
• ZED N50 for 2.5min
• MIBK for 30s
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min
5. E-beam Evaporation of Cr/Au mask
• 30s BHF dip to remove the Ti conducting layer where the mask will be
deposited
• Evaporate ∼ 5-10 nm thick Cr
• Evaporate 100 nm thick Au
• 15-30 min cooldown period
• Evaporate 100 nm thick Au
6. Lift-off
• NMP at 150◦ C for 2hr
• Acetone 5min sonication
• IPA 5min sonication
• N2 blow dry

124
• O2 plasma ash at 150W, 12sccm O2 flow for 2min
7. Trench Wet Etch
• 2min dip in 48% HF. Etch rate of -Z cut LN in 48% HF is ∼ 40 nm/min
so we are aiming to etch about 80 nm of LN. Followed by 2x 10s DI
H2 O rinse and N2 blow dry.
8. Trench Dry Etch
• 2min dry etch using the parameters listed in Table 9.1. Etch rate of LN
is 20 nm/min so a 2min etch would etch away the remaining 20 nm of
LN and overetch about 15 nm into the Si device layer. We suggest using
the same Cr/Au mask used in the wet etch for this short dry etch but
we have not tested the selectivity of this mask for the dry etch process.
Since it is a short 2min etch we assume the mask will survive the etch.
If it does not a thicker mask might be required.
9. Strip Cr/Au mask
• 3min in Gold Etch TFA (from Transene) at 30◦ C with 400 rpm stirring
followed by 2x DI H2 O rinse and N2 blow dry
• 1min in Chrome Etch 1020AC (from Transene) at 40◦ C with 400 rpm
stirring followed by 2x DI H2 O rinse and N2 blow dry
Layer 3: Final Wet Etch. In this layer, we lay down a fresh Cr/Au mask that fills
the trench defined in Layer 2 and protects the LN-Si interface. We then use a wet
etch to remove the LN everywhere else on the chip.
1. Fresh Titanium Conductive Layer for E-beam Lithography
• Deposit ∼5-10 nm of Ti using e-beam evaporation
2. Spin/Bake ZEP 520a
• Pre-bake at 180◦ C for 3min
• Spin ZEP 520a at 3000 rpm for 1 min
• Post-bake at 180◦ C for 3min

125
3. E-Beam Lithography
In this step, we need to ensure that the write area covers not just the top
of the LN ‘box’, but extends into the trench too. We suggest a pattern that
extends about halfway into the trench area. Since we want this write to line
up precisely with our previously defined trench, the alignment accuracy of
the e-beam tool is essential. Typical misalignment on our e-beam lithography
tool is ∼ ± 50 nm. If we use a trench width > 100 nm, that should be sufficient
to guard against ∼ 50 nm level misalignment.
• Beam current 4nA
• Fracturing resolution 4nm
• Dose 230 𝜇𝐶/𝑐𝑚 2
4. Development
• ZED N50 for 2.5min
• MIBK for 30s
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min
5. E-beam Evaporation of Cr/Au mask
• 30s BHF dip to remove the Ti conducting layer where the mask will be
deposited
• Evaporate ∼ 5-10 nm thick Cr
• Evaporate 100 nm thick Au
• 15-30 min cooldown period
• Evaporate 100 nm thick Au
6. Lift-off
• NMP at 150◦ C for 2hr
• Acetone 5min sonication
• IPA 5min sonication
• N2 blow dry
• O2 plasma ash at 150W, 12sccm O2 flow for 2min

126
7. Final Wet Etch
• 3min dip in 48% HF followed by 2x 10s DI H2 O rinse and N2 blow dry.
This should be enough to etch through the 100 nm thick LN film.
8. Strip Cr/Au mask
• 3min in Gold Etch TFA (from Transene) at 30◦ C with 400 rpm stirring
followed by 2x DI H2 O rinse and N2 blow dry
• 1 min in Chrome Etch 1020AC (from Transene) at 40◦ C with 400 rpm
stirring followed by 2x DI H2 O rinse and N2 blow dry.

127

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