Quantum Control over a Low Frequency MechanicalOscillator Using a Superconducting Qubit
by
Xizheng Ma
B.S., William Marsh Rice University, 2013
M.S., University of Colorado, 2017

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2020

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This thesis entitled:
Quantum Control over a Low Frequency Mechanical Oscillator Using a Superconducting Qubit
written by Xizheng Ma
has been approved for the Department of Physics

Prof. Konrad W. Lehnert

Prof. John D. Teufel

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Ma, Xizheng (Ph.D., Physics)
Quantum Control over a Low Frequency Mechanical Oscillator Using a Superconducting Qubit
Thesis directed by Prof. Konrad W. Lehnert

The ability to access a broad range of quantum states with mechanical oscillators has many
applications and is an enduring ambition in the fields of opto- and electromechanics. However,as mechanical oscillators are linear at the quantum scale, arbitrary quantum control over themrequires an extrinsic nonlinearity. In this thesis, I aim to provide this extrinsic nonlinearity andestablish quantum control over the motion of a suspended aluminum disk by coupling it strongly
to a superconducting qubit. To this end, I design and fabricate a device, when operated at its
maximum coupling strength, could enable phonon-number-resolved measurements and arbitrary
quantum control over the mechanical motion. However, limited by unclear reasons that break thequbit readout, I operate at approximately a quarter of the maximum coupling strength.
Nevertheless, at this smaller coupling strength, I demonstrate the preparation of a nonGaussian nonclassical state of motion. Because of the large coupling achieved in this work, thesideband transitions are phonon-number-sensitive, selectively altering the phonon populations in
only a few Fock states. Using these phonon-number-sensitive sideband transitions, I dissipatively
stabilize the mechanical oscillator into a highly-energized sub-Poissonian state, where its energy
fluctuations are below the classical limit. This result represents a major step toward the long-time
ambition of accessing a broad range of nonclassical states with macroscopic mechanical oscillators.
Moreover, requiring neither number-resolution nor coherent control of the qubit, the dissipative
stabilization technique demonstrated in this work also provides an accessible path toward preparing
nonclassical states in other harmonic oscillators coupled to qubits.

Dedication

In Loving Memory of My Grandfather, Zhitian Mav

Acknowledgements

I would like to thank everyone who has helped, supported, and advised me throughout myyears at CU Boulder. I cannot imagine finishing this journey without any of you. However, toproperly acknowledge everyone would be take far too many pages than is allotted here. Instead, I
make a meager attempt to highlight just a few:
To start, I must express my gratitude to Jérémie Viennot. You showed me on a daily basis
what a true experimental physicist should be like. It was great fun and honor to have worked so
closely with you for four years, and I shall be forever indebted to you for the great many things
you have taught me.
I am extremely lucky to have worked alongside many talented physicists during the past
years. In particular, I would like to thank my collaborator Shlomi Kotler for the many enlightening
discussions that always propelled the project forward; Will Kindel for showing me how qubits
and JPA work; Daniel Palken for teaching me all about statistics; and Lucas Sletten and Robert
Delaney for bearing with my countless “dumb questions”. More broadly, I would like to thankevery member of the Lehnert lab, you have all helped and supported me in one way or another,making this journey not just rewarding but also fun.
Of course, this journey would have been possible without the support from my advisor Konrad
Lehnert. I could not ask for a better mentor. I am always amazed by your ability to provide brilliant
insights on all facets of physics, from experimental details to profound physical meanings. More
importantly, you are always available to your students. It must have taken an incredible amountof dedication.

vi
Finally, I would like to thank my friends and family for sharing all my frustration and elationof the past seven years. To my parents, thank you for your support and encouragement. To mywife, Sylvia Bao, your love and sweetness drove me to finish this thesis.

viiContents

Chapter
1 Introduction
1.1

1

Thesis contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Theory of a mechanical oscillator coupled to a qubit

11

2.1

Mechanical oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3

2.2.1

Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2

A charge-sensitive qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3

Qubit decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.4

The transmon limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Qubit-mechanics interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1

Coupling a mechanical oscillator to a charge-sensitive qubit . . . . . . . . . . 27

2.3.2

The Rabi Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.3

Dispersive limit and an effective quadratic interaction . . . . . . . . . . . . . 30

2.3.4

System dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.5

Beyond the Rabi Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Hybrid system design

44

3.1

Choosing a CPB qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2

Collapse voltage and maximum coupling rate . . . . . . . . . . . . . . . . . . . . . . 46

viii
3.3

Improving the maximum coupling rate with a series capacitance

3.4

Using symmetry to protect against the noise on the dc-line . . . . . . . . . . . . . . 53

3.5

Coupling the qubit to a readout cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.6

3.5.1

Cavity formed by a microwave resonator . . . . . . . . . . . . . . . . . . . . . 61

3.5.2

Qubit-cavity interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5.3

Qubit readout through the cavity . . . . . . . . . . . . . . . . . . . . . . . . . 65

Design choices for the fabricated device and expected performance . . . . . . . . . . 67
3.6.1

Choosing a drum size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6.2

Parasitic capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.6.3

Estimating the maximum voltage . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.6.4

Final designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Characterizing the hybrid system
4.1

4.2

. . . . . . . . . . . 49

75

Cavity-enabled qubit readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.1

Inferring the qubit state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.1.2

Setting the qubit to its sweet-spot . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1.3

Selecting the qubit parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1.4

Qubit characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Turning on the qubit-mechanics coupling. . . . . . . . . . . . . . . . . . . . . . . . 81

4.2.1

Motion induced qubit spectrum broadening . . . . . . . . . . . . . . . . . . . 84

4.2.2

Determining the qubit-mechanics coupling rate . . . . . . . . . . . . . . . . . 84

4.2.3

Break-down of the qubit readout at large dc-voltage . . . . . . . . . . . . . . 89

5 Extracting phonon distribution from qubit spectrum

93

5.1

Understanding the mechanical state using the qubit spectrum . . . . . . . . . . . . . 93

5.2

Simulating quantized motion with classical gate-charge modulation . . . . . . . . . . 94
5.2.1

Qubit spectrum under classical gate-charge modulation . . . . . . . . . . . . 96

5.2.2

Qubit spectrum for a phonon Fock state . . . . . . . . . . . . . . . . . . . . . 98

ix

5.3

5.4

5.2.3

Classical vs. quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2.4

Effects of charge noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.2.5

Finding the qubit spectrum for phonon Fock states . . . . . . . . . . . . . . . 115

The reconstruction procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3.1

Reconstruction using Lucy-Richardson deconvolution . . . . . . . . . . . . . . 116

5.3.2

Bootstrapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Thermal and displaced thermal states . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6 Manipulating the phonon distribution using sideband transitions

127

6.1

Ac-dither sideband transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.2

Sideband transition’s phonon number sensitivity . . . . . . . . . . . . . . . . . . . . 130

6.3

Dissipatively squeezing the oscillator’s energy . . . . . . . . . . . . . . . . . . . . . . 134
6.3.1

Modelling the time evolution of the hybrid system under sideband drives . . 137

7 Energy squeezing below the classical limit
7.1

142

Preparation and measurement of a nonclassical state of motion . . . . . . . . . . . . 142
7.1.1

Measuring the qubit spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7.1.2

Extracting the nonclassical phonon distribution . . . . . . . . . . . . . . . . . 143

7.1.3

Bounding the Fano factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.1.4

Relating the Fano factor to other measurements of nonclassicality . . . . . . 149

7.2

Mechanical recoil caused by qubit transitions . . . . . . . . . . . . . . . . . . . . . . 151

7.3

Limitations on the achieved nonclassicality

8 Conclusion. . . . . . . . . . . . . . . . . . . . . . . 152
159

x
Bibliography

161

Appendix
A Device fabrication

174

B Measurement setup and filtering

176

B.1 Filtering the dc-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.2 The cavity input line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
B.3 JPA amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
C An approximate PSF

179

D Mechanical damping rate measurement

181

xi

TablesTable
3.1

Estimated essential performance parameters of the final designs . . . . . . . . . . . . 73

4.1

Essential performance parameters of the device at Vdc = 6 V. . . . . . . . . . . . . 92

xii

FiguresFigure1.1

Quadratic electromechanical coupling. . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2

Qubit-mechanics coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.1

Josephson junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2

A Cooper-pair box qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3

Fictitious field picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4

Voltage-noise-induced qubit relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5

A transmon qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6

A coupled qubit-mechanical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.7

Longitudinal vs. transverse coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.8

Frank-Condon description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.9

Decoherence of the qubit-mechanical system . . . . . . . . . . . . . . . . . . . . . . . 36

2.10 Spectrum of a simple driven dissipative system at r = 0 . . . . . . . . . . . . . . . . 41
3.1

Electrostatic instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2

Improving the maximum qubit-mechanics coupling rate with a series capacitance . . 50

3.3

The series capacitance improves the maximum qubit-mechanics coupling rate . . . . 52

3.4

Creating strong qubit-mechanics coupling with a symmetric arrangement . . . . . . 55

3.5

Maximum achievable qubit-mechanics coupling rate in the symmetric arrangement . 60

3.6

Adding a microwave cavity to the qubit-mechanics device . . . . . . . . . . . . . . . 62

xiii
3.7

Layout of the device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.8

Parasitic capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.9

The device used in this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1

Cavity phase shift as a function of gate charge . . . . . . . . . . . . . . . . . . . . . 77

4.2

Qubit parity jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.3

Time-domain qubit characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4

Steady-state qubit spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5

Motion-induced broadening of the qubit spectrum . . . . . . . . . . . . . . . . . . . 85

4.6

Detecting the mechanical resonance via qubit spectrum . . . . . . . . . . . . . . . . 87

4.7

Measuring χm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.8

Observing a 3rd parity position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.9

Investigating the 3rd parity position . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.1

Measured qubit response under classical gate-charge modulation . . . . . . . . . . . 95

5.2

Driving a phonon-number-resolved system . . . . . . . . . . . . . . . . . . . . . . . . 100

5.3

Dynamics in the phonon-number-resolved regime . . . . . . . . . . . . . . . . . . . . 101

5.4

Driving blue sideband transitions in a phonon-number-sensitive system . . . . . . . . 103

5.5

Sideband spectrum in a phonon-number-sensitive system . . . . . . . . . . . . . . . . 104

5.6

Transition rates under classical modulation vs. quantized motion . . . . . . . . . . . 107

5.7

Noise-induced asymmetric qubit lineshape . . . . . . . . . . . . . . . . . . . . . . . . 110

5.8

A noisy gate-charge modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.9

Finding Pe (ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117|ni

5.10 Reconstruction vs. iteration number . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.11 Finding iterm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.12 Histogram of RSMEtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.13 Reconstruction bootstrapping and confidence interval. . . . . . . . . . . . . . . . . 123

5.14 Thermal and displaced-thermal mechanical state . . . . . . . . . . . . . . . . . . . . 126

xiv
6.1

Ac-dither sideband spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.2

Phonon-number-sensitive sideband transitions . . . . . . . . . . . . . . . . . . . . . . 131

6.3

Demonstrating the phonon-number-sensitivity of a blue sideband drive . . . . . . . . 132

6.4

Demonstrating the phonon-number-sensitivity of a red sideband drive . . . . . . . . 133

6.5

Energy squeezing: idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.6

Energy squeezing: experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.7

Energy squeezing: theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6.8

Finding the sideband driving parameters . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.1

Measurement sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.2

Measurement-induced changes in phonon distribution

7.3

Extracting the nonclassical phonon distribution . . . . . . . . . . . . . . . . . . . . . 147

7.4

Bounding the Fano factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.5

Relating the Fano factor to negativity in the Wigner function . . . . . . . . . . . . . 151

7.6

A spurious cooling effect from driving ac-dither sideband below the qubit: theory . . 155

7.7

A spurious cooling effect from driving ac-dither sideband below the qubit: experiment156

7.8

A spurious heating effect from driving ac-dither sideband above the qubit . . . . . . 158. . . . . . . . . . . . . . . . . 145

B.1 The full measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
D.1 Measuring γm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Chapter 1

Introduction

Throughout history, mechanical systems have proved useful as tools that advanced our understanding of the physical world. In 1851, the procession of a Foucault pendulum succinctly
demonstrated the rotation of the earth. For a long time, torsional oscillators were the most sensitive force detectors: in 1798, they were used in the Cavendish experiment [1] to achieve the firstmeasurement of the gravitational constant. In 1909, their inability to measure a force in the Eötvösexperiment demonstrated the equivalence of inertial mass and gravitational mass; even today, they
are still used to test the validity of Newtonian gravity at the microscopic scale [2]. For centuries,
mechanical oscillators have also provided the most reliable methods to measure time: pendulums
were used as reliable clocks since the mid-17th century, while spring-loaded mechanical oscillators
allowed for the creation of portable watches and chronometers. More recently, modern forms of
mechanical oscillators, such as quartz tuning forks, have become the standard in both keeping timeand sensing forces. Perhaps the most salient example of the importance of mechanical systems inphysics was the experimental observation of gravitational waves in 2015 [3]. By continuously monitoring the separation between freely suspended test masses, LIGO was able to observe a minute
relative motion (with an amplitude less than 1/1021 of the initial separation) at frequencies around
100 Hz induced by the gravitational strain and detect the merger of a binary system of black-holesa billion light-years away.
Necessitated by such minute effects, the development of gravitational wave detectors motivated the understanding of quantum effects on measurements. The famous Heisenberg uncertainty

2principle states that a completely accurate measurement of an oscillator’s position will kick it witha completely random force such that it’s position at a later time would be impossible to predict.
Although pioneers in quantum physics understood this compromise between measurement and disturbance as early as 1925 [4], prior experiments with macroscopic objects neither required nor
achieved precision at the quantum scale. However by the 1970s, this was no longer the case withthe development of gravitational wave detectors. This development evolved around two different
paths, where the oscillator’s position is detected using either electricity or light. Early experimental
efforts envisioned that a gravitational strain will drive motion in a massive bar-resonator [5, 6, 7],
whose motion can be converted into electrical signals using either piezo-electricity [6] or capacitivetransducers [8]; on the other hand, LIGO [9, 10] uses laser interferometry to detect the relative motion between two freely suspended masses. As experiments steadily improved detector sensitivity,
both paths — despite their different approaches — were equally confronted with hard questions:when will quantum effects limit the sensitivity of the detector? Is it possible to leverage quantum
mechanics and build detectors beyond the classical limit? To answer these fundamental physics
questions, from the two approaches to build gravitational wave detectors emerged the parallel fieldsof circuit electromechanics [11, 12] and cavity optomechanics [13].
The first answers to these questions came in 1980, when two theoretical papers [14, 15]
discussed the limitation on the sensitivity of continuous position measurement using electromagnetic
fields. Later known as the standard quantum limit, this limitation arises because the quantum
fluctuations in the field also create fluctuating forces on the oscillator, and is consistent with
the Heisenberg uncertainty principle. Moreover, the authors pointed to a way to surpass such
limitations — by preparing the mechanical oscillator in nonclassical states of motion.
The experimental progress of coaxing mechanical oscillators into nonclassical states is muchslower. The majority of electro- and optomechanical experiments use the radiation-pressure interaction between the mechanical motion and the energy of an ancillary electrical or optical cavity.
As demonstrated in the pioneering experiment of 1970 [16], this interaction can provide an additional damping or anti-damping to the oscillator’s motion to change its dynamical behavior. It

3
was envisioned that this phenomena, termed “dynamical backaction”, can be leveraged to control
the mechanical motion and prepare it into nonclassical states [17, 18]. However, to reach this goal
took a concerted experimental effort. A major obstacle is cooling the mechanical oscillator to its
ground state using the radiation-pressure interaction. Because the mechanical oscillators in theseexperiments often have low resonant frequencies (< 100 MHz) [13], even at the base temperaturesof dilution refrigerators (∼ 25 mK), they have high thermal occupations that can obscure quantum
signatures. Using electricity [19] and light [20], two experiments in 2011 finally achieved the active
cooling of macroscopic mechanical oscillators to their ground state. Since then, mechanical oscillators have been prepared into quadrature squeezed states [21, 22, 23], where the uncertainty in one
quadrature is below vacuum fluctuation. Entanglement has also been created between mechanicaloscillators and other systems, such as microwave [24] or optical [25] fields or a separate macroscopic
mechanical oscillator [26, 27, 28], allowing for the construction of highly sensitive force detectors
beyond the classical limit [29, 30, 31].
In addition to building highly sensitive detectors, the prospect of preparing nonclassical statesof motion inspired new applications for macroscopic mechanical oscillators. However, to fully exploit their potential, one needs access to a broad range of nonclassical states. States characterized
by sub-Poissonian energy distributions, such as Fock states, have been analyzed for their ability
to improve the sensitivity of gravitational wave detectors [15, 14]. Macroscopically distinguishable
superposition states of motion [32, 33] are envisioned to test wavefunction collapse models [34]. Mechanical oscillators are also versatile tools in quantum information, for the construction of quantum
computers and quantum networks. Because of their long coherence times, mechanical oscillators
can be used as quantum memories [35, 36]. Because they couple easily to both electrical and opticalsystems, they can be used as quantum transducers between different frequency domains [37]. Furthermore, they can be used as amplifiers for the quantum information stored in its own motion [38],or in a propagating field [24]. Although Gaussian nonclassical states such as quadrature squeezed
states can be useful in certain tasks [39], any process that only involves Gaussian states can be efficiently simulated using classical computers [40]. Instead, many advanced quantum protocols, such

4
as entanglement distillation [41, 42, 43] and Shor’s algorithm [44], explicitly call for non-Gaussian
resources.
Because macroscopic mechanical oscillators are generally linear at the quantum scale, accessing such non-Gaussian states requires an extrinsic nonlinearity such as a nonlinear source [45] ordetector [25, 46]. In standard opto- and electromechanical systems based on the radiation-pressureinteraction, driving the system at the difference frequency of the ancillary-cavity and mechanicalresonances initiates a beam-splitter interaction that coherently exchanges the mechanical phonons
with the cavity photons [35]. This linear interaction is leveraged in the 2017 experiment [45] tocapture and store a propagating superposition state of zero and one photons in the motion ofa macroscopic mechanical oscillator, creating a non-Gaussian mechanical state using a nonlinear
source. Instead of a nonlinear source, a nonlinear detector such as a photon counter [47, 48] canalso create non-Gaussian nonclassical states in the mechanical oscillator using the “heralding” technique: the detection of a single photon in a correlated photon-phonon pair heralds the presence ofa single phonon in the mechanical oscillator [25, 46]. Such correlated phonon-photon pairs can becreated in opto- and electromechancial systems by driving at the sum frequency of the mechanicaloscillator and ancillary cavity [24]. Alternatively, the extrinsic nonlinearity can also be a strong
nonlinear interaction.
The radiation-pressure interaction is inherently nonlinear, and can in theory [49, 50, 51]
create non-Gaussian nonclassical states in the mechanical oscillator. Such proposals call for an
extremely large coupling such that the radiation-pressure force from a single cavity photon will
displace the mechanical oscillator by more than its zero-point fluctuation xzp . To reach this socalled “single-photon strong-coupling” regime requires the single-photon coupling rate g0 to belarger than both the mechanical frequency ωm and the cavity decoherence rate κ. However, thenature of the radiation-pressure interaction dictates that g0 is intrinsically small. To illustrate
this, I consider a standard electromechanical system, where the mechanical motion x̂ alters the
microwave resonance frequency ωc of a LC-circuit by changing the separation between the twoelectrodes of the capacitor. The mechanical modulation of the small single photon energy ~ωc5stored in the circuit creates a parametric coupling of rate g0 = (dωc /dx)xzp ≈ ωc (xzp /x0 ), where
x0 is the static separation between the capacitor electrodes. Limited by physical constraints suchas the Casmir force, the ratio xzp /x0 is typically small (∼ 10−7 ) [52], and consequently in mostexperiments g0  κ  ωm . Despite a variety of proposals and efforts to enhance g0 , such as
mediating it through a qubit [53, 54] or modulating it at ωm [55], no experiment so far has comeclose to reaching the single-photon strong-coupling regime. Instead, most experiments operatewith a large cavity drive, enhancing the coupling strength but yielding a linear interaction [56, 57]
between the mechanical position and cavity field.
However, the drive-enhanced coupling can remain nonlinear if the cavity energy is coupled
to the position-squared, instead of just position, of the oscillator motion [58, 59]. This “quadratic
coupling” can be generated using the electromechancial setup [14] of Fig. 1.1: two identical LC
resonators are coupled by a common inductance Lc at rate J = ωc Lc /2L and their resonant frequencies ωR,L are modulated with opposite phase by the anti-symmetric motion of the mechanicaloscillator xzp dωR,L /dx = ±g0 . Consequently, the normal modes of this circuit anti-cross at x = 0
with a splitting 2J, and sense the square of the mechanical displacement with quadratic couplingstrength x2zp d2 ω± /dx2 ≈ ±g02 /2J. By selecting on the measured mechanical position squared, an
experiment in 2016 [60] was able to generate non-Gaussian, albeit classical, bimodal states of motion in a nanostring. Accessing non-Gaussian nonclassical states of motion, however, is prevented
because a residual linear coupling emerges under a large cavity drive, and causes accelerated mechanical decoherence [61, 60]. Despite early doubts [61], this limitation can be overcome by active
feedback [62, 60] or better designs [63, 64, 65] without requiring the single-photon strong-coupling
regime. Yet, quadratic coupling schemes for opto- and electromechanics have not yielded nonclassical states in mechanical oscillators.
A strong nonlinear coupling can also be created by replacing the harmonic cavity with ananharmonic device, such as a superconducting qubit. These superconducting qubits have propelledprogress in quantum science, such as preparing arbitrary quantum states in microwave cavities [66,67], and their integration with mechanical oscillators was heavily anticipated for the prospects of6

Figure 1.1: Quadratic electromechanical coupling± (x), are modulated with opposite
a, The capacitance of two mechanically compliant capacitors, Cm
phase by the anti-symmetric motion of the mechanical oscillator (blue). The two separate LC
resonators are coupled by a common inductance Lc at a rate J = ωc Lc /2L0 . b, The resonant
frequencies of the uncoupled LC resonators (dashed) depend linearly on mechanical displacement x
with opposite phase, xzp dωR,L /dx = ±g0 , where xzp is the zero-point displacement of the mechanicaloscillator. The normal modes of the circuit (solid), however, anti-cross at x = 0 with splitting 2Jand sense the square of mechanical displacement with quadratic coupling strength x2zp d2 ω± /dx2 ≈±g02 /2J.

7
creating a broad range of nonclassical states of motion [68, 69, 70, 71, 72, 73, 74]. However, it has
been experimentally challenging to create a coupling strength large enough to achieve this goal.
One strategy to strongly couple mechanical motion to qubits involves piezo-electricity: whenmotion deforms the crystal lattice of a piezoelectric material, such as quartz or AlN, accumulated
charges in the material create an electric field that drives the qubit and vice versa. In order toincrease the strength of this electric field, piezoelectric materials often have high stiffness, andcreate mechanical oscillations at high frequencies from hundreds of megahertz to many gigahertz.
This approach was first demonstrated in a 2010 experiment, where the high-frequency 6 GHz
dilational-mode of a bulk-acoustic resonator was resonantly coupled to a tunable superconducting
qubit. During the years of my Ph.D., this approach experienced tremendous success and has developed into a distinctive field known as “quantum acoustics”. Through resonant coupling, thepreparation of phonon Fock states has been demonstrated using both surface-acoustic-wave [75]and bulk-acoustic-wave resonators [36]. In the dispersive coupling limit, where the qubit and mechanical frequencies are far apart, phonon-number-resolved measurements have also been reported
on both surface-acoustic-wave resonators [76] and phononic-crystal defect resonators [77], but the
preparation of phonon Fock state is prevented by a limited mechanical decoherence rate. In theseexperiments, the investigated mechanical motion are confined to a small class of elastic waves in
stiff piezoelectric material. Although such mechanical oscillators can be useful for special tasks inquantum computing, they are much less general-purpose compared to the more conventional mechanical oscillators. For example, because of the high stiffness, they are not sensitive force detectors;
because of their typically small zero-point motions (∼ 10−16 m) [78], they are less macroscopic andthus less useful for testing wave-function collapse models [34].
Without using piezo-electricity, conventional, center-of-mass motions of low-frequency macroscopic mechanical oscillators can be coupled to qubits using the schematic shown in Fig. 1.2 . Insteadof leveraging the particular material properties to transduce mechanical motion to electrical signals,
a dc-voltage Vdc can transduce the motion of a macroscopic mechanical oscillator to a positiondependent charge-source by embedding the oscillator in a mechanically compliant capacitor. When

8

Figure 1.2: Qubit-mechanics coupling
The dc-voltage Vdc transduces the mechanical motion (blue) into a position-dependent chargesource ac cross open terminals A and B. This charge-source controls the electrostatic energy of acharge-sensitive qubit and couples the mechancial motion to the qubit energy.

this charge-source drives a charge-sensitive qubit, such as a Cooper-pair box (CPB) qubit [79, 80],
it couples the mechanical position to the qubit energy with a strength controlled by Vdc .
However, the experimental progress in creating nonclassical states of motion using this coupling scheme has been slow. Using a nano-beam resonator, a 2009 experiment [81] first createdthis type of qubit-mechanics coupling and observed a shift in mechanical frequency when the qubit
frequency is changed — an entirely classical effect. Limited by a large qubit decoherence rate, theexperiment is unable to control the mechanical motion using the qubit. To combat this problem,
later experiments [82, 83] attempted to reduce the qubit’s charge-sensitivity by going toward thetransmon limit [84], which reduces the qubit decoherence due to charge-noise but also reduces thequbit-mechanics coupling rate. In 2013, the phonon-number-dependent Stark shift of the qubit
frequency was first observed [82]. However, because of the small coupling rate, this effect is only
visible at very large phonon numbers and cannot distinguish the quantization of mechanical energy. In the same experiment, motional sidebands that simultaneously excite the qubit and change
the phonon occupation were also observed. However, these sidebands are only observable at large
phonon numbers where their effects on the motion are insignificant. Moverover, the small qubitmechanics coupling rate impedes the selective manipulation of the phonon occupation in each Fockstate. Instead, driving these sideband transitions at a particular frequency would indistinguishably

9address hundreds of phonon Fock states. Despite the progress made in these early demonstrates,
it remains an experimental challenge to prepare nonclassical states of motion in macroscopic mechanical oscillators using qubits.
In this thesis, I aim to pursue the coupling scheme shown in Fig. 1.2, and establish quantum
control over the motion of a suspended aluminum disk using a superconducting qubit. Instead of
exchanging coupling strength for improvements in the coherence time, I aim to create a stronger
coupling between the qubit and mechanical oscillator by using a CPB qubit. I design and fabricate a device, when operated at the maximum dc-voltage, could enable phonon-number-resolved
measurements and arbitrary quantum control over the mechanical motion. However, limited byunclear reasons that break the qubit readout, I operate at approximately a quarter of the maximum
dc-voltage.
Nevertheless, at this smaller coupling strength, I demonstrate the preparation of a nonGaussian nonclassical state of motion [85]. Because of the large coupling achieved in this work,
the sideband transitions are phonon-number-sensitive, selectively altering the phonon populations
in only a few Fock states. This phonon-number-sensitivity can be observed by either cooling themechanical oscillator close to its ground state using a red sideband transition that removes phonons
from the mechanical oscillator, or by preparing a highly non-thermal phonon distribution using blue
sideband transitions that add excitations to the oscillator [86]. Using these phonon-number-sensitive
sideband transitions, I dissipatively stabilize the mechanical oscillator into a highly-energized subPoissonian state [85], where its energy fluctuations are below the classical limit [87]. This result
represents a major step toward the long-time ambition of accessing a broad range of nonclassicalstates with macroscopic mechanical oscillators by strongly coupling it to a superconducting qubit.
Moreover, requiring neither number-resolution nor coherent control of the qubit, the dissipative
stabilization technique demonstrated in this work also provides an accessible path toward preparing
nonclassical states in other harmonic oscillators coupled to qubits.

10

1.1

Thesis contents
Chapter 2 introduces both the mechanical oscillator and the superconducting qubit, andprovides a theoretical description of the coupled qubit-mechanics system. Chapter 3 details thedesigning of the hybrid system, and explains the reasoning behind many design choices, such as
using a CPB qubit. It also introduces a microwave resonator that couples only to the qubit,
allowing for its control and readout. Chapter 4 demonstrates the control and readout of thequbit using the microwave resonator, and provides an initial experimental characterization of thecoupled qubit-mechanics system by measuring the qubit-mechanics coupling rate. It also shows
the break down of the qubit readout at large Vdc and provides a plausible explanation for thislimitation. Chapter 5 describes the extraction of mechanical energy distributions from measured
qubit spectra, and demonstrates the procedure on thermal and displaced-thermal mechanical states.
Chapter 6 shows the manipulation of phonon population using motional sideband transitions. It
demonstrates the phonon-number-sensitivity of these sideband transitions and provides a theoreticalmodel to predict their effects. Using these sideband transitions, it also demonstrates the dissipative
stabilization technique, capable of simultaneously adding energy to and extracting entropy from a
massive mechanical oscillator. Finally, Chapter 7 uses the this technique to create a non-Gaussian
nonclassical state of motion whose uncertainty in energy is below the classical limit. Chapter 8
looks to future experiments and advancements that could follow the work of this thesis.

Chapter 2Theory of a mechanical oscillator coupled to a qubit

In order to provide a consistent mathematical language and theoretical background for the
rest of this thesis, this chapter theoretically describes the interaction between a mechanical oscillator
and a superconducting qubit. Sec.2.1 begins by providing a quantum description for mechanicaloscillators. Sec.2.2 introduces the superconducting qubit, which is an anharmonic oscillator formed
by a nonlinear circuit. Finally, Sec.2.3 describes the interaction between the two systems when acoupling is created using the schematic shown in Fig. 1.2.

2.1

Mechanical oscillators
Mechanical oscillators are perhaps the most tangible examples of harmonic oscillators. Inclassical mechanics, the oscillator motion is described by its position and momentum {x, p}, which
can be simultaneously specified. Canonically depicted as a mass m attached to a spring of constantk, the oscillator is subject to a restoring force F = −kx when displaced from its equilibrium position
by a distance x. The evolution of this system is specified by the classical Hamiltonian,
Hm =

p2
1
+ kx2 .
2m 2

(2.1)

With an initial displacement of xi , the position of the mass oscillates around the equilibriump
position according to x(t) = xi cos(ωm t), where ωm = k/m is the resonant angular frequency ofthe mechanical oscillator.
Quantum mechanically, the state of the mechanical oscillator is more naturally described by
its energy. The dynamical variables from classical mechanics are replaced by quantum operators

12x̂ and p̂, which obey the commutation relation [x̂, p̂] = i~. They can be expressed in terms of thephonon creation â† and annihilation operators â,Here, the zero-point motion xzp =x̂ = xzp (â† + â),(2.2)

p̂ = imωm xzp (â† − â).

(2.3)

p
~/2mωm corresponds to the magnitude of position fluctuation

when the mechanical oscillator is in its energy ground state, h0| x̂2 |0i = x2zp . Written with phonon
operators, the quantum Hamiltonian analogous to Eqn.(2.1) is
Hm =

p̂2
1
1
+ kx̂2 = ~ωm (n̂ + ),
2m 2
2

(2.4)

where n̂ = â† â is the phonon number operator. The eigenstates of the Hamiltonian are phonon
Fock states |ni, with evenly spaced energies En = ~ωm (n + 1/2) such that all transitions between
adjacent phonon states are degenerate.
This degeneracy prevents the access to individual Fock states with a classical drive. Instead,classically driving a mechanical oscillator addresses all energy levels at the same time, and preparesit into a classical state — a coherent state. Such a state is denoted by |αi and satisfies â |αi = α |αi,where α is a complex number that describes the strength of the classical drive. In the Fock basis,a coherent state is described by a Poisson distribution,|αi = e

−

|α|2
2

∞
X
α
√ |ni ,
n!
n=0

(2.5)

with equal mean and variance in energy or phonon number, var(n) = hni = |α|2 .
Over time, the energy stored in a damped mechanical oscillator by the external drive willslowly decay into an external bath at a rate γm . The external bath can be thought of as a large
collection of harmonic oscillators at various frequencies, which can exchange energy with the mechanical oscillator. Because we do not keep track of the bath states, energy appears as lost when
it flows from the mechanical oscillator into the bath. To model this disspative process, I write themaster equation of the mechanical oscillator in the Lindblad form [88],dρ= −i[Hm , ρ] + Lm (ρ),
dt

(2.6)

13
where ρ =Pm,n ρmn |mi hn| is the density matrix of the oscillator, and Lm is the Lindbladian. Fora thermal bath at temperature T ,Lm (ρ) = γm (nb + 1)D[â](ρ) + γm nb D[â† ](ρ),(2.7)

where nb = [exp(~ωm /kB T ) − 1]−1 is the bath-photon number at the mechanical frequency ωm ,
and D[Ô](ρ) = 12 2ÔρÔ† − ρÔ† Ô − Ô† Ôρ is the Lindblad superoperator. Starting from any
statistical mixture of phonon Fock states, the initial density matrix is diagonal and commutes with
Hm . Subsequently, because Eqn.(2.7) couples diagonal elements only among themselves and not to
off-diagonal elements, this initial state evolves according to,
1 d
P (n) = nb nP (n − 1) + (nb + 1)(n + 1)P (n + 1) − [nb (n + 1) + (nb + 1)n] P (n),
γm dt

(2.8)

where P (n) = ρnn = hn| ρ |ni is the probability of having n phonons in the mechanical oscillator. In
steady state where the mechanical oscillator reaches equilibrium with the thermal bath, its phonon
distribution is described by a thermal distribution,
Pth (nth , n) =nnth,
(nth + 1)n+1(2.9)

where,
nth = hni = nb = [exp(~ωm /kB T ) − 1]−1 ,(2.10)

is the mean phonon occupation number. In this work, I use a mechanical oscillator with ωm =2π × 25 MHz. Operating in a dilution refrigerator with a typical temperature of 25 mK (kB T /~ ≈2π × 0.5 GHz), its thermal occupation is nth ≈ 20 phonons.

2.2

Superconducting qubits
A superconducting qubit is an LC-circuit with a strong nonlinearity. Similar to a mechanicaloscillator, a linear LC-circuit also behaves like a harmonic oscillator with an evenly spaced photon
energy En . However, by replacing the inductor with a nonlinear element, an anharmonicity is
introduced that lifts the degeneracy of neighboring-photon transitions. When this anharmonicity

14
is large enough, the nonlinear circuit can be treated as an effective two-level system, also known asa quantum bit (qubit).

2.2.1

Josephson junction

The nonlinear inductors in such circuits are provided by Josephson junctions. A Josephson
junction consists of two superconducting electrodes separated by an insulating layer (Fig. 2.1).
In each superconducting electrode, charges bind together to form Cooper-pairs of charge 2e. All
Cooper-pairs in one electrode form a common quantum state characterized by its quantum phase.
Because the wavefunctions of macroscopic quantum states from the two superconducting electrodes
overlap in the Josephson junction, Cooper-pairs can tunnel across the insulating layer. This tunneling of discreet Cooper-pairs creates a periodic current across the junction,
I(t) = Ic sin φ(t),(2.11)

where φ is the phase difference between the two electrodes. The critical current Ic is the maximum
superconducting current that can flow through the junction, given by the Ambegaokar-Baratoff
relation [89],
Ic =

π∆
tanh
2eRN



∆
2kB T


,

(2.12)

where RN is the normal state resistance of the tunnel junction, and ∆ is the superconducting gapof the electrodes. The tunneling Cooper-pairs also create a voltage potential,
V (t) =

Φ0 d
φ(t),
2π dt

(2.13)

where Φ0 = h/2e is the superconducting flux quantum. Taken together, this Josephson effectcreates an nonlinear, phase-dependent inductance

−1
1 d
LJ (φ) =I(t)= LJ0 / cos φ,
V (t) dt

(2.14)

where LJ0 = Φ0 /2πIc is the characteristic Josephson inductance. The tunneling Cooper-pairs also
store energy in the junction,Z
E(φ) =

dt0 V (t0 )I(t0 ) = EJ cos φ,(2.15)

15

Figure 2.1: Josephson junctions
a, A Josephson junction is created by connecting two superconducting electrodes (grey) with aninsulating layer (red). b, The Josephson junction can be schematically represented by the symbol“X”. The junction is characterized by the phase difference φ across the two electrodes: the instantaneous current I flowing through the junction is related to sin φ by the critical current Ic ; andthe voltage drop V across the junction is proportional to the time derivative of φ. Together, they
store an energy EJ cos φ, where EJ is the Josephson energy. c, To provide in-situ control over thisJosephson energy, a Josephson junction is often split in two, such that an external magnetic flux
ΦB piercing the superconducting loop can change EJ according to Eqn.(2.18).

16
where EJ = Ic Φ0 /2π is the Josephson energy. Quantum mechanically, this corresponds to a
Hamiltonian [90],HJ = EJ cos φ̂,(2.16)

where φ̂ is the quantum operator for the phase difference across the junction.
Alternatively, the tunneling of Cooper-pairs can be understood phenomenologically in the
charge basis |ni 1 , associated with n̂ number of Cooper-pairs having crossed the junction. In thisbasis, the tunneling connects adjacent charge states with a Hamiltonian [79],
HJ =

EJ X
(|ni hn+1| + |n+1i hn|) .
2(2.17)n∈Z

Evidently, the Josephson energy characterizes the amount of overlap between wavefunctions ofadjacent charge states within the junction.
To provide in-situ control over the Josephson energy, a Josephson junction is often split in
two as shown in Fig. 2.1(c). The energy of the split Josephson junction can be controlled by an
external magnetic flux piercing the superconducting loop ΦB . Assuming identical junctions eachcharacterized by a Josephson energy EJ /2, the overall Josephson energy is given by [80],ΦB
EJ (ΦB ) = EJ cos π
.
Φ0
2.2.2

(2.18)

A charge-sensitive qubit

By embedding the nonlinear Josephson junction in the circuit as shown in Fig. 2.2(a), one
creates a qubit [79, 80]. The circuit consists of a superconducting island suspended from ground bya total capacitance CΣ = CJ +Cg , where Cg is the gate-capacitance, and CJ is the stray capacitance
between the two superconducting islands of the Josephson junction. The biasing gate-voltage Vg
sets a preferred amount of excess gate-charge on the island, ng = Cg Vg /2e in units of Cooper-pairs.
However, the actual excess charge on the island is a quantized, integer number of Cooper-pairs n̂.
This gives rise to an electrostatic energy stored on the island by the gate-voltage,
Hel = 4Ec (n̂ − ng )2 ,1bold to differentiate from phonon Fock states in mechanical oscillators(2.19)

17

Figure 2.2: A Cooper-pair box qubit
a, A Cooper-pair box (CPB) qubit is created by connecting a suspended superconducting island(red shaded box) to a superconducting reservoir through a Josephson junction. The biasing gatevoltage Vg sets a preferred amount of excess gate-charge on the island, ng = Cg Vg /2e in units of
Cooper-pairs, where Cg is the gate capacitance. However, the actual number of charges n̂ on theisland is a quantized, integer number of Cooper-pairs. The gate voltage thus stores an electrostatic
energy on the island, proportional to the CPB charging energy Ec and (n̂ − ng )2 . b, As a functionof ng , the electrostatic energy for a charge state |ni (i.e., n Cooper-pairs on the island) is describedby a parabola (dashed black lines). Because the Josephson junction allows Cooper-pairs to tunnel
across the junction, it couples adjacent charge states at rate EJ /~, where EJ is the Josephsonenergy. This coupling lifts the energy degeneracy at the charge degeneracy points ng = 1/2, andresults in discrete energy levels (solid lines). In this figure, I assume Ec = EJ . The CPB qubit
is formed by the lowest two energy levels |gi (blue) and |ei (red). √At ng = 1/2, the qubit states
are given by a superposition of charge states |e(g)i = (|0i ± |1i)/ 2. c, The qubit frequency isgiven by the difference in the ground and excited state energies, and is highly dependent on thegate-charge (Eqn.(2.24)).

18
where Ec = e2 /2CΣ is the electrostatic charging energy to add a single electron to the island. In theabsence of the Josephson junction, this system is described by a family of parabolas corresponding
to the energy of a capacitor with a fixed number of charges on it (dashed lines in Fig. 2.2(b)).
By allowing Cooper-pairs to tunnel across the junction and reach ground, the Josephson junction
couples adjacent charge states. This coupling results in discrete energy levels as shown with solidlines in Fig. 2.2(b) that are highly anharmonic.
In the so called “Cooper-pair box (CPB) regime” where EJ . Ec , the high level of anharmonicity allows the nonlinear circuit to be treated as an effective two-level system, acting as a qubit.
In this limit, the lowest two energy eigenstates of the circuit belong to the subspace spanned bycharge states {|0i , |1i}. At the point of charge degeneracy ng = 1/2, the degeneracy between thetwo charge states is lifted by the Josephson energy. Consequently, the ground |gi and the excited √|ei energy eigenstates of the qubit are given by the charge superposition states |0i ± |1i / 2.
This effective two-level system is most conveniently described in the language of a spin-1/2
particle. Working around the charge degeneracy point in the CPB regime, the Pauli operators canbe approximately written as,σ̂z0 = 2(n̂ − 1)

(2.20)

σ̂x0 = |ni hn+1| + |n+1i hn| .
After removing the constant terms, the Hamiltonian of this charge-sensitive qubit is given by
HCPB = Hel + HJ =

1
4Ec (1 − 2ng )σ̂z0 + EJ σ̂x0 .
2(2.21)

This resembles the Hamiltonian of a spin with dipole moment µ = 1 subject to a fictitious magnetic~ = EJ x̂0 + 4Ec (1 − 2ng )ẑ 0 (Fig. 2.3). The quantization of energy happens along the net
field of B
magnetic field axis ẑ, rotated about ŷ 0 axis by an mixing angleθ0 = arctan

EJ
4Ec (1 − 2ng )
.

(2.22)In this new coordinate system, the Hamiltonian of the qubit is simply
1
Hq = ~ωq σ̂z ,2(2.23)

19

Figure 2.3: Fictitious field picture
The qubit Hamiltonian HCPB resembles that of a spin with dipole moment µ = 1 subject to a
~ = EJ x̂0 + Eel ẑ 0 , where the electrostatic energy Eel = 4Ec (1 − 2ng ).
fictitious magnetic field of B
The quantization of energy happens along the net magnetic field axis ẑ rotated about ŷ 0 axis by an
mixing angle θ0 .

where
ωq =

1
~

q

[4Ec(1 − 2ng )]2 + EJ2(2.24)

is the qubit frequency. The eigenstates of this two-level system areσ̂z |gi = − |gi ,(2.25)σ̂z |ei = |ei .

2.2.3

Qubit decoherence

When the qubit is coupled to the outside world, a variety of factors in its environment will
degrade its performance. These can be characterized in two categories: relaxation and dephasing.
Relaxation is the process in which the qubit state is changed by its interaction with the environment,
whereas dephasing doesn’t involve a change in the qubit’s excitation. Instead, dephasing causes
noise in the relative phase between different quantum states such that a superposition state will lose
its quantum coherence. Both decoherence processes can be modeled by writing its Lindbladian [91]Lq (ρ) = Γ↓ D[σ̂− ](ρ) + Γ↑ D[σ̂+ ](ρ) +ΓφD[σ̂z ](ρ),
2(2.26)

20
where ρ =P

ρij |ii hj| is the density matrix of the qubit, Γ↑↓ is the environmental excitation ori,j=g,erelaxation rate of the qubit, and Γφ is the qubit’s pure dephasing rate.

2.2.3.1

Relaxation

Random fluctuations in a coupled environmental parameter can cause qubit relaxation when
they are resonant with the qubit frequency. For example, to control the gate voltage of a CPB qubit,
we connect the qubit to an external voltage source. In this process, the qubit is also introduced toan outside electrical environmental impedance Z(ω), as shown in Fig. 2.4. Given by the JohnsonNyquist formula, the voltage noise δV from the impedance is characterized by a two-sided noise
spectral density
Sv (ω) =2~ω
Re [Z(ω)] ,1 − e−~ω/kB T(2.27)and is related to a noise in the gate charge by δng = Cg δV /2e. When resonant with the qubit
transition, this noise can excite (↑) or relax (↓) the qubit at rates [92]Γv↑↓ =



4Ec Cge~22

hg| n̂ |ei Sv (∓ωq ).

(2.28)In steady state, the qubit will be polarized with a mean excitation
Pess =

Γv↑
Γv↑ + Γv↓.

(2.29)

Consequently, to have a qubit naturally relax to its ground state in a dilution refrigerator at 25 mK,
it is desirable to work with qubit frequencies ωq > 2π × 3 GHz, such that ~ωq  kB T . In this work,
I use a qubit at frequency ωq ≈ 2π × 3.9 GHz. In this limit, the voltage noise from the impedance
would not excite the qubit (Γv↑ = 0), and an excited qubit relaxes to its ground state at rate
Γv↓ = 2~ωq Re Z(ωq )

4Ec Cge~22

hg| n̂ |ei .

(2.30)

This voltage-noise-induced qubit relaxation can be generalized to any circuit capacitively
coupled to any type of qubits [93]. Specifically, in the case of a qubit coupled to a single-mode
electrical cavity, this leads to the well-known Purcell effect [94], where the qubit relaxes through21

Figure 2.4: Voltage-noise-induced qubit relaxation
By connecting it to an external voltage source Vg , the qubit is also introduced to an outside electricalenvironmental impedance Z(ω). The voltage noise δV arising from the impedance can drive qubit
transitions, and cause qubit relaxation.

the cavity at rate,
Γκ↓ =



gc
ωq − ωc2κ,(2.31)

where gc is the qubit-cavity coupling rate, ωc is the cavity frequency, and κ is the coupling rate of
the cavity to the outside world.
Another source of qubit relaxation is dielectric loss. When the electric field of the qubit passes
through the dielectric materials surrounding it, such as the substrate material or the insulating layer
between the junctions, energy can be lost in the process. The resulting loss rate is proportional
to the loss tangent of the dielectric material. When the electric field passes through multiple
materials, the contribution to the dielectric relaxation rate from each material is weighted by their
participation ratios.
The overall qubit relaxation rate [92]
Γ1 = Γ↑ + Γ↓sums over all sources of relaxation, and corresponds to the rate for the qubit to return to its
thermal equilibrium. For a gigahertz-frequency qubit thermalized to the base temperature of adilution refrigerator, Γ↑ ≈ 0, and Γ1 ≈ Γ↓ .

22
2.2.3.2

Dephasing

The qubit dephasing is generally understood as fluctuations of the qubit frequency, whichcauses the qubit to accumulate a random phase. Qubit relaxations will dephase the qubit at a rate
Γ1 /2 [91]. In addition, there can be fluctuations in the qubit frequency during the the course of one
qubit relaxation, labeled as pure dephasing rate Γφ . This is typically caused by noise at frequencies
much lower than the qubit frequency. Finally, if the fluctuations occur at even longer timescales,
varying from experiment to experiment, they create an ensemble dephasing rate Γ∗φ . Because they
are not intrinsic to a single experiment, this last type of ensemble dephasing can be reduced bycertain experimental techniques, such as using spin-echo protocols or post-selecting on the qubit
frequency. The overall qubit dephasing rate is given by the sum of all these rates,
Γ2
Γ1=+ Γφ + Γ∗φ .
22(2.32)A major source of dephasing for a CPB qubit is charge noise. Because of the strong dependence of qubit frequency on gate-charge (Eqn.(2.24)), a small fluctuation in ng or Vg can lead tolarge changes in ωq and dephase the qubit. To reduce this detrimental effect, a CPB qubit is often
operated at the charge degeneracy point, where the qubit energy become first-order insensitive tocharge. Nevertheless at this “sweet-spot”, charge noise leads to a dephasing rate Γφ through thecurvature of the charge-dispersion [84],Γφ ∝ ∂ 2 ωq /∂ng 2= (8Ec )2 /EJ .

(2.33)

ng =1/2

A more devastating source of dephasing comes from quasiparticles. These are single-electronlike excitations residing in a superconductor, created by either thermal excitations or other nonequilibrium processes such as the absorption of electromagnetic irradiation from cosmic rays or
higher-temperature environments [95]. At the base temperature of a dilution refrigerator, thequasiparticle population is dominated by the non-equilibrium processes. In aluminum-based superconducting qubits, the quasiparticle density is typically found to be within the range of 0.1 –10 µm−3 [70, 96, 97]. When a single quasiparticle tunnels across the Josephson junction, it will

23change the gate charge by one electron. For a CPB qubit operating at the charge degeneracy,
the tunnelling of a quasiparticle with bring the qubit to ng = 1, and drastically change the qubit
frequency from EJ /~ to 4Ec /~. One solution to alleviate this problem is to reduce the probabilityof these tunneling events by reducing the number of available quasiparticles. This can be achievedby simply limiting the size of the junction electrodes. Alternatively, the quasiparticle density canbe reduced by using quasiparticle traps composed of either normal metal [98, 99] or magnetic
vortices [97, 100, 101].

2.2.4

The transmon limit

Going toward the transmon regime (i.e., EJ  Ec ) [84] offers another solution to the problemsof both quasiparticle and charge-noise-induced qubit dephasing. In this limit, the normal modeenergies of the nonlinear circuit become increasingly insensitive to charge. Instead of being thesuper-position of charge states |ni, the eigenstates of the circuit in this transmon regime are better
described by the quantum phase φ̂ of the superconducting island, and their wavefunctions are given
by Mathieu functions [102].
However, the decreased charge-sensitivity comes at the cost of a reduced anharmonicity.
Instead of a simple two-level system, the nonlinear circuit in the transmon regime is often approximated as a weakly anharmonic oscillator,
H=

p
1
Ec
8Ec EJ (b̂† b̂ + ) −(b̂ + b̂† )4 ,2
12(2.34)

where b̂, b̂† are the annihilation and creation operator for the harmonic oscillator approximating
the circuit, and are related to the original phase and Cooper-pair operators according to [103],

EJ 1/4 1 †√ (b̂ − b̂),
n̂ =
8Ec
2

1/4
1
8Ec
√ (b̂† + b̂).
φ̂ = iEJ
2(2.35)In this limit, the transition frequency ωj,j+1 between neighboring eigenstates of the oscillator |ji ↔

24Figure 2.5: A transmon qubit
By making EJ  Ec , the qubit enters the transmon regime, and its energy becomes effectively
insensitive to the gate-charge. Here, I plot the energy landscape of a transmon qubit with EJ =100Ec as a function of ng . The energy is normalized to Ec , and chosen such that the lowest energy
state |0i has energy E0 = 0 at n√g = 0. The lowest two energy levels can be treated as a qubit, witha qubit frequency ωq = ω01 ≈ 8EJ Ec /~. However, the insensitivity to gate-charge comes at thecost of a lowered anharmonicity, ω12 − ω01 ≈ Ec /~, small compared to ω01 .

25|j + 1i is different from ωj+1,j+2 by a weak anharmonicity,ωj+1,j+2 − ωj,j+1 =Ec1p ω01 =
8Ec EJ .
~~(2.36)

Nevertheless, when this anharmonicity is larger than the spectral width of an external drive resonantwith ω01 , the nonlinear circuit can be treated as an effective two-level system consisting only of thelowest two energy levels, with a qubit frequency given byωq = ω01 =1p
8Ec EJ .
~(2.37)

Because the charge-sensitivity decreases faster than anharmonicity [84], nonlinear circuits in thetransmon regime can become effectively insensitive to charge fluctuations while remaining anharmonic enough to be considered a qubit.
It is important to note here that being insensitive to charge does not mean the qubit cannot
be coupled to charge. Indeed, numerous experiments [47, 67, 45] have demonstrated the coupling
between transmon qubits and electric fields that drive the gate-voltage Vg . Similarly, transmon
qubits can also be coupled to mechanical oscillators through charge, as I show in the next section.

2.3

Qubit-mechanics interaction
Having introduced both building blocks, I now discuss a coupled system between a super-

conducting qubit and a mechanical oscillator. By altering the separation between two electrodes,
the mechanical motion can modulate ng of a charge-sensitive qubit through a position-dependentgate-capacitance,
Cg = Cm (x) =0
Cm,
1 + x/x0(2.38)

where x is the position of the oscillator, x0 is the static separation between the electrodes, and
0 = C (0) is the static gate-capacitance. Consequently, the motion alters the qubit energy toCmmcreated a coupled qubit-mechanical system. In this section, I explore theoretically the effects ofthis coupling.

26

Figure 2.6: A coupled qubit-mechanical system
A mechanical oscillator (blue) held by a spring of constant k is suspended above another electrodeto create a capacitor. When the oscillator is displaced from its static position x0 by an amount x,
it changes the separation between the two capacitor electrodes, and creates a position-dependentcapacitance Cm (x). When this capacitance is used as the gate capacitance of a charge-sensitive
qubit, an applied dc-voltage Vdc converts the mechanical motion into a modulation on the qubit’sgate-charge ng . Consequently, this motion-induced ng modulation couples the mechanical motion
to the qubit’s electrostatic energy.

27
2.3.1

Coupling a mechanical oscillator to a charge-sensitive qubit

To start, I find the Hamiltonian for a coupled system when the qubit is in the CPB regime.
By applying a dc-voltage Vdc , the motion of the mechanical oscillator is converted into a positiondependent gate-charge, ng (x) = Vdc × Cm (x)/2e. This creates an interaction,
HI = x̂∂ng (x)∂H = 8Ecxzp (â + â† )(n̂ − ng ),∂x∂x(2.39)

where the mechanical position x̂ = xzp (â+â† ) is coupled to the Cooper-pair number operator n̂. For
small motional amplitudes such that the charge modulation is smaller than a single Cooper-pair,
the CPB charge basis can be restricted to two adjacent charge states {|0i , |1i}. Because a qubit in
the CPB regime has a strong anharmonicity, it can be considered as an effective two-level systemformed by the lowest two energy levels. Defining σ̂z to align with the its energy quantization axis
(Eqn.(2.20)-(2.23)), I find an interaction Hamiltonian,
HI = x̂∂ng (x)∂H = 4Ecxzp (â + â† ) σ̂z cos θ0 − σ̂x sin θ0
∂x
∂x


= ~gm â + â† σ̂z cos θ0 − σ̂x sin θ0 ,(2.40)

where θ0 = arctan[EJ /4Ec (1−2ng )] is the mixing angle between the charging energy and Josephsonenergy, andgm =∂ng (x)1
4Ec 0 xzp
4Ec
xzp =
C
Vdc
~
∂x
2e~ m x0

(2.41)

is the single-phonon qubit-mechanics coupling rate.
Within this Hamiltonian, there are two types of couplings: longitudinal and transverse, as
shown in Fig. 2.7. Away from the charge degeneracy, the longitudinal coupling~gm cos θ0 â + â† )σ̂z ,(2.42)

z = g cos θ dominates.
where mechanical motion changes the qubit energy with a coupling rate gmm0

As expected from the qubit energy’s charge dispersion (Fig. 2.2), this coupling changes sign atevery half-integer gate-charge. Because a positive mechanical displacement x increases the separation between the electrodes and decreases the gate capacitance, it reduces the gate-charge ng .

28

Figure 2.7: Longitudinal vs. transverse couplingz , blue) and
There are two types of qubit-mechanics coupling in a CPB qubit: longitudinal (gmxtransversal (gm , red). Away from the charge degeneracy point, the longitudinal coupling, wherez . Because a positive
mechanical motion changes the qubit energy, dominates with coupling rate gmzdisplacement reduces the gate charge, gm is positive between 0 < ng < 1/2 and negative between
1/2 < ng < 1. At the charge degeneracy, the qubit energy becomes first-order insensitive to gatez vanishes. Instead, the coupling rotates entirely to a transversal coupling, where thecharge, and gm
mechanical motion drives the qubit excitation.

29
Consequently, the smaller ng leads to a higher qubit energy between 0 < ng < 1/2 and a positivez , and vice versa between 1/2 < n < 1. However, at the charge degeneracy, the qubit energy
gmg

become in first order insensitive to gate-charge, and the longitudinal coupling vanishes. Instead,the coupling rotates entirely to a transversal coupling, where the mechanical motion drives thequbit excitation at rate gm .
Similarly, the charge insensitivity of a transmon qubit does not prevent it from being coupled to mechanical motion, but merely rotates the coupling to a transversal one that drives qubit
excitation. To see this, I begin by rewriting Eqn.(2.39) in the uncoupled qubit eigenbasis,
HI = 2~gm (â + â† )(n̂ − ng )= 2~

Xgi,j |iib b hj|(â + â† ),(2.43)

i,j

where |iib is the i-th eigenstate of the uncoupled nonlinear oscillator that acts as a qubit, gm isgiven by Eqn.(2.41), and the matrix elementgi,j = gm b hi|n̂ |jib .

(2.44)

Approximating the transmon qubit as a weakly anharmonic oscillator according to Eqn.(2.34), theabove matrix elements are, to first-order,
gj,j = 0,gj,j+1 = gj+1,j = gm
gj,j+l ≈ 0

∀

EJ
8Ec

1/4 rj+1,
2(2.45)|l| > 1,

showing that the mechanical motion couples to a transmon qubit only transversely, driving excitations between adjacent qubit states. For clarity in this chapter, I restrict the qubit to the lowest
two energy states, and write the transmon-mechanics interaction using Pauli matrices asx
HI = ~gm(â + â† )σ̂x ,(2.46)

wherex
gm= g01 =



EJ
32Ec

1/4

4Ec 0 xzp
CVdc
2e~ m x0

(2.47)

30
However, because a transmon qubit is only weakly anharmonic, it is highly inaccurate to ignore
the mechanical coupling to higher-level qubit states. Indeed, as I will discuss in the next chapter(Sec.3.1), these couplings can strongly diminish the observable effects of a transmon-mechanics
interaction.

2.3.2

The Rabi Hamiltonian

The coupled qubit-mechanical system is described by a Rabi Hamiltonian either by using atransmon qubit or by working at the sweet-spot of a CPB qubit,
1xHRabi /~ = ωq σ̂z + ωm â† â + gmσ̂x (â† + â),2

(2.48)

x is given by either Eqn.(2.41) for a CPB qubit or Eqn.(2.47) for a transmon qubit. If
where gm

the qubit and the mechanical oscillator were resonant, the energy would slosh between the two
x . However, because we work with gigahertz-frequency qubits and megahertz-frequency
at rate gm

mechanical oscillators, the large difference in their resonant frequencies impedes the spontaneoustransfer of energy.

2.3.3

Dispersive limit and an effective quadratic interaction

Instead, the qubit energy become sensitive to the square of the mechanical displacement,
creating an effective quadratic coupling. To see this, I apply an unitary transformation [104],
 x

x
gm
gm
†
†
Ûdisp = exp(âσ̂+ − â σ̂− ) +(â σ̂+ − âσ̂− ) ,∆Σ(2.49)

with ∆ = ωq − ωm , and Σ = ωq + ωm , to remove the effects of spontaneous energy exchange.
Contrary to the standard dispersive transformations employed in circuit quantum-electrodynamics(cQED) [47, 105, 106, 107], Eqn.(2.49) accounts for the counter-rotating terms â† σ̂+ and âσ̂− inthe Rabi Hamiltonian because ωm  ωq . Keeping to the first order of gm /∆ and gm /Σ, I find,1
1
Hdisp = ~ωm â† â + ~ωq σ̂z + ~χm σ̂z (â + â† )2 ,2
2(2.50)

31
where
χm =

x )2
(gm
(g x )2+ m∆
Σ

(2.51)

x , performing
includes the Block-Siegert shift [108]. For large qubit frequencies such that ωq  gmthis dispersive transformation doesn’t alter the eigenstate of the coupled system [109]. Because
a qubit excitation is much faster than the mechanical dynamics, the position can be regarded asstationary under a sudden qubit excitation. The physics is clear if the Hamiltonian is written as


1
p̂2
1 2 1
2χm
σ̂z x̂2 ,Heff = ~ωq σ̂z ++ kx̂ + k
2
2m 2
2
ωm

(2.52)

where ωm and ωq are the uncoupled mechanical and qubit frequency, m and k are the mass andthe spring constant of the mechanical oscillator, and x̂ = xzp (â† + â), and p̂ = imωm xzp (â† − â) are
the oscillator’s position and momentum operators respectively.
The quadratic coupling can be understood as a qubit-state controlled mechanical spring
constant,2χmk(σ̂z ) = k 1 +
σ̂z .
ωm

(2.53)

A change in qubit state simultaneously alters both the mechanical frequency
ωm (σ̂z ) =pk(σ̂z )/m

(2.54)Zm (σ̂z ) =p(2.55)and the mechanical impedancek(σ̂z )m.

These two changes corresponds to two different physical phenomena.
First, the dispersive shift of the mechanical frequency leads to a phonon-number-dependent
Stark-shift of the qubit resonance,
ωq|ni = (ωq + χm ) + 2χm n,(2.56)

where n is the number of phonons in the mechanical oscillator. When this dispersive shift is large
enough, one can address individual phonon numbers using the qubit, as I will discuss in Sec.2.3.4.

32Second, the impedance determines the spatial scale of the mechanical wavefunction. Thischanging impedance allows for prominent sideband transitions, where a qubit excitation also causes
the creation or annihilation of mechanical phonons. To understand this effect, I introduce theFrank-Condon description.

2.3.3.1

Frank-Condon description

As shown in Fig. 2.8, the mechanical spatial wavefunction for a phonon Fock state |ni isgiven by [110],1
ψ(x, n, σz ) = √
2n n!



± =
where σz = ±1 is the qubit state, Zm

±
Zm
π~

1

4

 s
± 2
Zm
x /2~

e

Hn  x


±
Zm
,
~

(2.57)

pk(±1)m is the qubit-state-dependent mechanical

impedance, and the functions Hn (z) are Hermite polynomials.
As such, a qubit excitation can connect otherwise orthogonal mechanical states, according to
the Frank-Condon principle [111, 112]. The probability of such a transition is given by the overlap
in the spatial wavefunctions,Z ∞Pn,m ∝ψ ∗ (x, n, −1)ψ(x, m, 1)dx,(2.58)−∞

where Pn,m is the probability of observing transition |g, ni → |e, mi. Because of the symmetry in themechanical potential, only transitions that change the phonon occupation by an even number are
expected. When m = n, the qubit transition preserves the mechanical phonon number. However,when m > n, a blue sideband transition is realized, where phonons are added while exciting thequbit. Vice versa when m < n, a red sideband transition is realized, which cools the mechanicaloscillator while exciting the qubit.

2.3.3.2

System eigenstates

To provide a quantitative understanding of the interaction and be able to make predictions,
I directly diagonalize the Hamiltonian of Eqn.(2.50) to find the system eigenstates. To do this, I33

Figure 2.8: Frank-Condon description
A qubit excitation causes a sudden change of the mechanical potential (parabolas) and a non-zero
overlap between spatial wavefunctions (lines) of different mechanical states. Because of symmetry,
this process only connects an initial state |n, gi (shaded green) with states of the same mechanicalparity |n, ei (shaded red), creating or annihilating phonons by pairs.

34
introduce an unitary transformation [113, 114],




1
2
†2
Ŝ r(σ̂z ) = exp r(σ̂z )(â − â ) ,2(2.59)

where for a small dispersive shift χm  ωm 2 ,1χmχm σ̂zr(σ̂z ) = arctanh≈σ̂z ≡ rσ̂z .
2ωm + χm σ̂z2ωm(2.60)

Because Ŝ † (r)Ŝ(r) = Ŝ(−r)Ŝ(r) = 1̂, Ŝ † (r) = Ŝ(−r). This unitary transformation diagonalizes the
Hamiltonian to,
Hsq = ~Xωn− |gib |nib b hn| b hg| + ωn+ |eib |nib b hn| b he|(2.61)nwhere |gib , |eib , and |nib are the eigenstates of the uncoupled qubit and mechanical oscillator
respectively. The eigenvalues are given by11
ωn± = nωm ± (ωq + 2χm n) = nωm ± ωq|ni ,22(2.62)|ni

where ωq is the phonon-number-dependent qubit frequency given by Eqn.(2.56). The eigenstatesof the diagonalized Hamiltonian |gib |nib and |eib |nib are related to the the eigenstates of the
original Rabi Hamiltonian Eqn.(2.50) through the unitary transformation Ŝ r(σ̂z ) ,|g, ni = Ŝ(rσ̂z ) |gib |nib = |gib Ŝ(−r) |nib ,(2.63)|e, ni = Ŝ(rσ̂z ) |eib |nib = |eib Ŝ(r) |nib .
Compared to the uncoupled mechanical oscillator, the excited-state qubit stiffens the mechanicalspring, and squeezes the oscillator energy along its position axis. Similarly, the qubit ground stateloosens the mechanical spring, and anti-squeezes the oscillator energy along its position axis.

2.3.4

System dynamics

Having diagonalized the Hamiltonian, I now proceed to describe the time evolution of thecoupled qubit-mechanical system. This is given by the master equation,dρ = −i[H, ρ] + Lqm ρ,dt
2(2.64)In this work, I explore a qubit-mechanical system with a maximum coupling rate gm ≈ 2π×80 MHz (see Sec.4.2.2).
Before this work, no experiment [81, 82, 83] has come close to such a large coupling rate value. Nevertheless, with aqubit frequency ωq ≈ 2π × 3.8 GHz, this large coupling rate only corresponds to a dispersive shift χm ≈ 2π × 3.2 MHz,much less than ωm ≈ 2π × 25 MHz.

35
where ρ =Pi,j ρij |ii hj| is the density matrix, with eigenstates |ii and |ji given by Eqn.(2.63).

H = HRabi + Hd is the overall system Hamiltonian, containing the Rabi Hamiltonian of Eqn.(2.48)
and contributions from any external drive Hd . Although this master equation contains all dynamics
of the coupled system, it provides little insight. Instead, in this section, I attempt to achieve amore intuitive understanding by analyzing three extreme cases. Based on these understandings, inSec.6.3.1, I will write a simplified master equation more suitable for simulating the experiments inthis work.

2.3.4.1

Dissipative dynamics

First, I consider a system without any external drives to explore the dissipative dynamics.
Because HRabi commutes with ρ, time evolution only arises from the system’s interaction with its
environmental bath, illustrated in Fig. 2.9 and described by the Lindbladian Lqm [115],







X Γφ 
X
X
X
ij
ij
ij
Lqm (ρ) =D |ii hi| (ρ)+Γq D |ii hj| (ρ)+Γm D |ii hj| (ρ)+Γm D |ji hi| (ρ).
2
i

i,j>i

i,j>i

i,j<i

(2.65)
The first term describes the dephasing of the qubit, but is decoupled from the mechanical oscillator.
The second terms describes the system relaxation through the qubit channel,2Γij
q = Γ1 |hi| σ̂x |ji| ,(2.66)

where Γ1 is the qubit decay rate. Because kB T  ~ωq , the qubit contains no thermal excitation, andthe system can only relax from a higher-energy eigenstate |ji to a lower-energy state |ii (j > i).
Given the much larger qubit frequency (ωq  ωm ), this relaxation process is dominated by therelaxation of the qubit state accompanied by a possible change of phonon number, |e, ni 7→ |g, mi.
Because σ̂x operates on the uncoupled qubit eigenbasis |gib and |eib , the relaxation rates can befurther evaluated using Eqn.(2.63),∗ 2
Γij
q = Γ1 |αmn | ,(2.67)∗
αmn≡ hg, m| σ̂x |e, ni = b hm|Ŝ † (−2r) |nib

(2.68)

where |ii = |g, ni, |ji = |e, mi, and36

Figure 2.9: Decoherence of the qubit-mechanical system
A qubit in its excited state relaxes to the ground state at rate Γ1 . In the coupled qubit-mechanicalsystem, because a decaying qubit causes a sudden change of mechanical spring constant, it can connect different mechanical states at rates Γ1 |αm,n |2 , proportional to the overlap between mechanicalspatial wavefunctions of |e, mi and |g, ni (see Similar to Sec.2.3.3.1). Because of symmetry, thisrate is only non-zero between states with phonon numbers that differ by an even value. The qubitmechanics interaction also alters the mechanical relaxation, such that the mechanical decay rate
becomes qubit-state dependent γm (σ̂z ) = e−2σ̂z γm . This relaxation process connects adjacent mechanical states of the same qubit excitation. Finally, all states also experience qubit dephasing,which is unaffected by the qubit-mechanics interaction. This dephasing process is not illustrated
in this figure.

37
is the same overlap in spatial wavefunctions between different mechanical states as discussed in
R∞ ∗∗ |2 =
Sec.2.3.3.1, |αmn−∞ ψ (x, m, 1)ψ(x, n, −1)dx. Consequently, a qubit relaxation can alsochange the phonon occupation by an even number.
Finally, the last two terms describe the dissipative dynamics from the mechanical channel.
In addition to relaxation, the environmental bath can also excite the qubit-mechanical system byadding thermal phonons into the mechanical oscillator (because kB T  ~ωm ). The rates of theseprocesses are given by,

Γij
m =



2


γm (nth + 1) hi| X̂ |ji

j > i (relaxation)

2γm nth hj| X̂ |ii

i > j (excitation),(2.69)

where X̂ = â+â† , and nth is given by Eqn.(2.10). Because X̂ operates on the uncoupled mechanicaleigenstates |nib , the inner product in the above equation can be evaluated,hg, n| X̂ |g, mi = e2r
he, n| X̂ |e, mi = e√√mδm,n+1 + m + 1δm,n−1 ,−2r√mδm,n+1 +√(2.70)
m + 1δm,n−1 ,where δm,n is the Kronecker delta. Thus, this last term in the Lindbladian connects adjacentmechanical states under the same qubit excitation with a qubit-state dependent mechanical decayrate γm (σ̂z ) = e−2σ̂z γm .

2.3.4.2

Driven dynamics without dissipation

Next, I consider the dynamics of the system without any dissipation, but driven by a coherent
signal applied to the qubit. The coherent qubit drive can be characterized by the Hamiltonian,
Hd (t) = 2~ΩR σ̂x cos (ωd t),(2.71)

where ΩR is the Rabi rate due to the external drive, and ωd is the drive frequency. Instead of
directly solving the master equation, I go into the interaction picture with
HRabi = ~Xn

ωn− |g, ni hg, n| + ωn+ |e, ni he, n| ,(2.72)

38
where ωn± are the eigenenergies given by Eqn.(2.62). The resulting interaction Hamiltonian isV(t) = e−iH0 t/~ Hd (t)eiH0 t/~Xi∆dmn t∗−i∆dmn t|e, mi hg, n| ,|g, ni he, m| + αmn e
αmn e
= ~ΩR

(2.73)

n,m

where αmn = b hm|Ŝ(−2r) |nib is the overlap between different motional states under a qubit flip
(Eqn.(2.68)), and+∆dmn = ωm− ωn− − ωd = ωq|ni + (m − n)(ωm + χm ) − ωd(2.74)

is the detuning between the qubit drive and transition |g, ni ↔ |e, mi.
Assuming an initial state |g, ni, the interaction Hamiltonian of Eqn.(2.73) identifies two typesof resonant conditions. First, whenωd = ωq|ni ,(2.75)

the system will oscillate between states |g, ni ↔ |e, ni at a rateΩn,0 = ΩR |αnn | = ΩR b hn|Ŝ(−2r) |nib ,(2.76)

exciting the qubit while preserving the mechanical phonon number. A second type of resonancecorresponds to sideband transitions. Whenωd = ωq|ni + 2l(ωm + χm )(l ∈ Z6= )(2.77)

the system oscillates between states |g, ni ↔ |g, n + 2li at a sideband rate ofΩn,2l = ΩR |αn+2l,n | = ΩR b hn + 2l|Ŝ(−2r) |nib ,(2.78)

adding 2l phonons into the mechanical oscillator while exciting the qubit.
An interesting figure of merit in the above driven response of the qubit-mechanical system isthe ratio χm n/ωm . A change in the qubit excitation alters the mean mechanical energy by1
δkhx̂2 i = 2~χm n,
2(2.79)

39
where n is the initial phonon number. When this value is larger than the energy of two phonons
2~ωm , a qubit transition is likely to alter the phonon number |αnn |2 . 0.5. Thus, the limiting
condition
χm
n&1
ωm

(2.80)

signifies the entry into a regime where sideband transitions become dominant. In contrast to
standard cQED systems [47, 105, 106, 107] where the large cavity frequency suppresses thesesideband transitions, the small ωm means that a qubit-mechanical system can display prominent
sideband transitions even at a small phonon number, as I observe in Chapter 5.

2.3.4.3

A driven dissipative system

Finally, I consider the effect of the same coherent drive in a dissipative system. In particular,
I am interested in the probability of finding the qubit in the excited state as a function of ωd . These
qubit spectra Pe (ωd ) are extremely useful monitors of the mechanical phonon distribution.
For simplicity, here I only consider the simple case of r = χm /ωm ≈ 0 such that αmn = δmn .
In this limit, the qubit dynamics become decoupled from the mechanical dynamics, and theirinteraction appears only as shifts in ωq and ωm . With exactly n phonons in the mechanical oscillator,
the driven qubit-mechanical system oscillates between states |g, ni and |e, ni at rate ΩR . For aweak qubit spectroscopic drive (ΩR  Γ1 , Γ2 ), the effective two-level system spanned by states{|g, ni , |e, ni} (Fig. 2.10(a)) reaches steady state with a characteristic time scale given by the qubit
lifetime T1 = 1/Γ1 . Because mechanical relaxation typically occurs at a much slower rate comparedto the qubit decay γm  Γ1 , the driven two-level system can reach steady-state before the phonon
number changes. This steady-state qubit spectrum with exactly n phonons in the mechanicaloscillator is described by a Lorentzian,
Pe|ni (ωd ) =Ω2R Γ2 /4Γ1∆2n + (Γ2 /2)2 + Ω2R Γ2 /2Γ1

(AΓ2 /2)21=,
2 ∆2n + (Γ2 /2)2 (1 + A2 )(2.81)

40|ni

where ∆n = ωq − ωd is the detuning between the drive frequency and the qubit resonance, andr
A = ΩR2
Γ1 Γ2(2.82)

is an unitless parameter describing the transition rate. This Lorentzian is centered at frequency|niωq with a maximum probability of finding the qubit excited state
1 A2,
2 1 + A2

Peres =

(2.83)

and a full-width at half-maximum (FWHM) broadened by the drive power,
Wspec = Γ2p
1 + A2 .

(2.84)

When r 6= 0, the qubit excitation and relaxation can both change phonon numbers in themechanical oscillator, which complicates the qubit spectrum significantly. I defer a more detailed
discussion on this topic to a later chapter (see Sec.5.2.2). Nevertheless, from Sec.2.3.4.2, I can
predict that the qubit spectrum will contain excitation peaks not only at the qubit resonance, butalso at the sideband transition frequencies given by Eqn.(2.77).
|ni

Regardless of the value of r and the specific shape of the qubit spectrum Pe (ω) for phonon|ni

Fock state |ni, because the qubit resonance ωqis phonon-number-dependent (Eqn.(2.56)), eachphonon in the mechanical oscillator will shift its corresponding spectrum by an amount 2χm n.
Consequently for a statistical mixture of phonon Fock states in the mechanical oscillator, the overall|niqubit spectrum is given by a convolution between the phonon distribution P (n) and Pe (ω),
Pe (ω) =X

P (n) × Pe|ni (ω).

(2.85)n

When Γ2 < 2χm , the qubit spectrum produced by a weak spectroscopic drive becomes phonon|ni

number-resolved, where Pe (ω) from adjacent photon numbers are individually resolvable (Fig. 2.10(c)).
In other words, this phonon-number-resolved regime allows a weak qubit drive to address transitions corresponding to individual phonon Fock states. Conversely, when Γ2 ≥ 2χm , individual|ni

photon peaks cannot be resolved directly in the qubit spectrum. Nevertheless, because of ωq ,

41

Figure 2.10: Spectrum of a simple driven dissipative system at r = 0
a, In the simple case of r = 0, the driven qubit-mechanical system can be simply modeled by|ni
a sum over many effective two-level systems spanned by {|g, ni , |e, ni}, each at frequency ωqgiven by Eqn.(2.56). A resonant qubit drive excites this effective two-level system at rate ΩR .
The qubit decays to its ground state at rate Γ1 , and dephases at rate Γ2 /2 b, Starting with an
initial state |g, ni, the steady-state qubit spectrum is given by a Lorentzian centered at detunning|ni∆n = ωq − ωd , with a FWHM Wspec broadened by the drive strength ΩR according to Eqn.(2.84).
To plot this spectrum, I assumed the following parameters: ΩR = 0.1Γ1 , Γ2 = 3Γ1 , and the qubit
detuning is normalized to the minimum qubit FWHM Γ2 . c, When the mechanical state is composedby a statistical mixture of Fock states, the overall qubit spectrum is given by a convolution between
the qubit spectra of different phonon Fock states and the phonon distribution. As an example, here
I plot the overall qubit spectrum (green) for a mechanical state with P (n − 1) = P (n) = P (n + 1) =1/3, with the same parameters in b and 2χm = 2Γ2 . In this phonon-resolved-regime, individualphonon Fock states result in distinct qubit excitation peaks, and are easily resolvable.

42both phonon-conserving transitions (Eqn.2.75) and sideband transitions (Eqn.2.77) are phononnumber-sensitive. In this limit, the phonon distribution in the mechanical oscillator can be ex|ni

tracted through a deconvolution procedure with knowledge of both Pe (ω) and Pe (ω), as will be
discussed in Chapter 5. Moreover, a qubit drive can address a section of the phonon population
with a characteristic width of
W = 2χm /Γ2(2.86)

phonons. The power of the resulting phonon-number-sensitive sideband transitions will be exploredin Chapter 6 & 7.

2.3.5

Beyond the Rabi Hamiltonian

Finally, I consider what happens when the gate-charge deviates from the sweet-spot of a CPB
qubit. Although it is desirable to operate at the charge degeneracy point when working with a CPB
qubit, charge noise — arising either from noise on the voltage bias or impurities in the material —can lead to deviations in the gate-charge δng . This introduces a residual σ̂z coupling between thequbit energy and mechanical oscillator position. According to Eqn.(2.40), the overall Hamiltonianat ng = 1/2 + δng is
1xz
H0rsd = ωm â† â + ωq σ̂z + gmσ̂x (â + â† ) + gmσ̂z (â + â† ).
2(2.87)

wherex= gm sin θ0 ,gm

(2.88)zgm= gm cos θ0 ,with θ0 given by Eqn.(2.22).
The residual σ̂z coupling can have important effects because it describes the coupling between
qubit energy and mechanical position at the first order; whereas the σ̂x coupling discussed above
describes the second order coupling of position to energy. Thus, even a small coupling rate cansignificantly alter the qubit-mechanical system. To quantify its effect, I apply consecutively thefollowing unitary transformations to diagonalize the Hamiltonian under the assumption of a small

43z  gx ≈ g ,deviation from the charge degeneracy point, gm
m
m

rsd
Hdiag
= Ŝ †


 
 


 
 

†
†
†
rsd
r(σ̂z ) D̂ α(σ̂z ) R̂ θ(X̂) Ûdisp H0 Ûdisp D̂ α(σ̂z ) R̂ θ(X̂) Ŝ r(σ̂z ) , (2.89)

where





χzm X̂ 2
1≈ 1̂
R̂ θ(X̂) = exp − i σ̂y arctanz X̂ + χ X̂ 2
2
ωq /2 + gmm
x g z 1 + 1  χ , andis the qubit rotation operator with X̂ = â + â† , χzm = gmmm ∆Σ(2.90)






†
∗
D̂ α(σ̂z ) = exp α(σ̂z )â − α (σ̂z )â ,
z σ̂
gm
gz
z
≈ − m σ̂z ≡ β σ̂z
α(σ̂z ) = −ωm + 4χm σ̂zωm(2.91)

is the qubit-state-dependent displacement operator. Similar to Eqn.(2.63), the eigenstate of H0rsd
are given by,|g, ni = Ŝ(rσ̂z )D̂(β σ̂z ) |gib |nib = |gib Ŝ(−r)D̂(−β) |nib ,(2.92)|e, ni = Ŝ(rσ̂z )D̂(β σ̂z ) |eib |nib = |eib Ŝ(r)D̂(β) |nib .
Consequently, the overlap between different mechanical states under a qubit transition is given byrsd
αmn= he, m| σ̂+ |g, ni = b hm|D̂ † (β)Ŝ(−2r)D̂(−β) |nib .

(2.93)

Therefore, as the residual σ̂z coupling breaks the symmetry in the system, a qubit excitation canconnect all mechanical states. Starting from state |g, ni, the coherent qubit drive of Eqn.(2.71) will
drive oscillations between the initial state and state |e, n + li at frequencies
ωd = ωq|ni + l(ωm + χm )(l ∈ Z)(2.94)

with transition rates
rsdΩn,l = ΩR |αn+l,n|.

(2.95)

Chapter 3

Hybrid system design

In pursuit of reaching the phonon-number-resolved regime, I choose to use a CPB qubit tocouple strongly to an anti-symmetric mode of a mechanical oscillator. In this chapter, I explain thereasons behind this choice, as well as other design considerations. In Sec.3.1, I explain the choiceof a CPB qubit over a transmon as the better candidate for reaching a strong dispersive coupling
rate χm larger than the qubit dephasing rate Γ2 /2. However, because the qubit-mechanics couplingarises from an external dc-voltage Vdc applied across the mechanically compliant capacitor, themaximum qubit-mechanics coupling rate is limited by a maximum dc-voltage Vmax , beyond which
the capacitor plates collapse under an electrostatic instability. This limitation is explained inSec.3.2, and improved upon in Sec.3.3 by introducing an additional capacitance in series withthe mechanically compliant capacitor. In Sec.3.4, I discuss another complication because of theapplied dc-voltage: voltage fluctuations on Vdc , composed of both thermal and quantum fluctuation,
can drive motion in the mechanical oscillator, and cause an increased mechanical relaxation. By
coupling the qubit to the anti-symmetric, second mode of the mechanical oscillator, I create asymmetry that protects the mechanical motion from noise on the dc-voltage. In addition to themechanical oscillator, in Sec.3.5, the qubit is coupled capacitively to a microwave resonator toallow for its control and readout. Taken together, these choices allow me to design a series of
qubit-mechanical systems in Sec.3.6 that, when operated at the maximum voltage, could reach thephonon-number-resolved regime.

45

3.1

Choosing a CPB qubit
In this work, I choose to use a CPB qubit instead of a transmon because the transmon can

only be coupled effectively to mechanical oscillators that have a resonant frequency ωm close to thequbit frequency ωq . Consequently with the low-frequency mechanical oscillators (ωm  ωq ) that
I’m interested in, a transmon-mechanics system will have a very small dispersive coupling rate χm .
At its core, the reason for the vanishing χm lies with the weak anharmonicity of a transmon qubit.
Because of this weak anharmonicity, it is inaccurate to simplify a transmon qubit to consist of only
the lowest two energy levels as I did in the previous chapter (see Sec.2.2.4 & 2.3.3). Instead, thecoupling between the mechanical oscillator and higher transmon levels will contribute to diminish
the resulting dispersive shift χm .
Indeed, it can be shown [116] that a coupled transmon-mechanics system has a dispersive
shift χm that approaches zero in the limit of ωm /ωq → 0. By modeling the transmon qubit as aweakly anharmonic oscillator, the effects of mechanical coupling to higher transmon levels can be
captured by writing the overall Hamiltonian,1Ec1x
H/~ = ωq (b̂† b̂ + ) −(b̂ + b̂† )4 + ωm (â† â + ) + gm(â† + â)(b̂† + b̂).
2
12~
2

(3.1)

In this equation, the first two terms describes the weakly anharmonic oscillator that is the transmon√(Sec.2.2.4) with a qubit frequency ωq = 8EJ Ec /~ and anharmonicity Ec /~  ωq . The third termis the mechanical oscillator, with frequency ωm . Finally, the last term corresponds to the transmonmechanics coupling with coupling rate (see Sec.2.3.1)
x
gm
=



EJ
32Ec

1/4
gm ,

(3.2)

∂ng (x)
1
4Ec
xzp .
~
∂x

(3.3)

where
gm =

Without the second, anharmonic term in Eqn.(3.1), the Hamiltonian simply describes two coupledharmonic oscillators, whose normal modes can be found through direct diagonalization. Treating
the anharmonic term as a perturbation (because Ec /~  ωq ) and rewriting it in the basis of

46
these normal modes [117, 116], a phonon-number-dependent frequency shift on the transmon-mode(cross-Kerr) can be identified, which corresponds to the dispersive shift χm . Assuming a small
x /ω , this dispersive shift ismechanical frequency ωm  ωq and keeping to the second-order in gm
qfound to be [116]x )22Ec x 2 ωm
Ec (gmχm ≈ 4 (gm) 2 2 ≈4~∆ Σ~ωq ωq



ωm
ωq

2

(g x )2
=2 m
ωq

r

Ec
2EJ



ωm
ωq

2
(3.4)

where Ec  EJ , and I approximate ∆ = ωq − ωm ≈ ωq and Σ = ωq + ωm ≈ ωq . Comparing thisresult to the dispersive shift found when I only keep the lowest two energy levels of the transmon
qubit (Eqn.(2.51)),
χm =

x )2
(gm
(g x )2(g x )2+ m ≈2 m ,
∆
Σ
ωq

(3.5)

the detrimental effect of the mechanical coupling to higher transmon levels is clear. This detrimental
effect can be alleviated by increasing the qubit’s anharmonicity.
To this end, it is beneficial to use a CPB qubit, whose strong anharmonicity makes it valid
to keep only the lowest two qubit energy levels. In this limit, the CPB-mechanics coupling rate gm
is given by Eqn.(3.3), and the resulting dispersive shift
χm =

2
g2
gm
+ m,
∆
Σ

(3.6)

is no longer diminished by the small mechanical frequency ωm .

3.2

Collapse voltage and maximum coupling rate
Although gm seems to be linearly enhanced by increasing the external dc-voltage Vdc (see,for example, Eqn.(2.41)), it is not possible to reach an arbitrarily large coupling rate by simply
increasing Vdc . Instead, the coupling rate is limited by a maximum dc-voltage Vmax , beyond which
the capacitor plates collapse under an electrostatic instability.
To understand this instability, consider with the simple picture as shown in Fig. 3.1(a): Vdc is
0 and separation
applied across a mechanically compliant vacuum-gap capacitor with capacitance Cm

x0 between the two capacitor plates at Vdc = 0 V. This voltage creates an electrostatic force that47
pulls the two plates closer, but is impeded by a restoring force from the spring. To find the static
separation xv under dc-voltage Vdc , I write the potential energy of the suspended electrode at any
separation xs ,1
U (xs ) = −
2



1
0 x0
2
Cm
Vdc
+ k(x0 − xs )2 ,
xs
2

(3.7)

where the first term is the electrostatic energy Eel stored in the capacitor, and the second term isthe potential energy Ek stored in the spring. The static separation xv is found when the total force
experienced by the suspended plate is zero,
F (xv ) = −∂U (xs )
= 0.
∂xs
xs =xv

However, as Fig. 3.1(b,c) shows, when Vdc is increased beyond a maximum voltage 3/2 sk
2Vmax =
x0
,
0
3
Cm

(3.8)

(3.9)

the above equation has no real-positive solution, and the capacitor plates collapse.
This maximum voltage therefore leads to a maximum qubit-mechanics coupling rate gmax
for the basic coupled-system discussed in Sec.2.3.1 (reproduced in Fig.3.1(d)), where a Josephson
junction is inserted in series between the mechanically compliant capacitor and ground. Although
0 across the Josephson junction because of thethere can be a small voltage-drop on the order of e/Cm
0 drops almost entirely across thequantized nature of Cooper-pairs, the large dc-voltage Vdc  e/Cm

mechanically compliant capacitor [79]. Thus, the maximum voltage Vmax and the static separation
xv found above remain accurate with the addition of the Josephson junction. Small displacement xof the suspended electrode around its static position xv alters the CPB gate-capacitance accordingto0
Cg (x) = Cm (xv , x) = Cm

x0
1
1
v
= Cm
,
xv 1 + x/xv
1 + x/xv(3.10)

v = C 0 x /x is the static capacitance at dc-voltage V . This position-dependent capaciwhere Cm
v
dc
m 0

tance is then converted into a position-dependent gate-charge to create a qubit-mechanics couplingof rategm (Vdc ) =

v x
4Ec
d
e Cm
zp
xzp Vdc Cm (xv , x) =
Vdc ,
2e~
dx
~ CΣ x v(3.11)

48

Figure 3.1: Electrostatic instability
0 is created by suspending a top plate witha, A mechanically compliant vacuum-gap capacitor Cma spring constant k at a distance x0 above a bottom electrode. A dc-voltage Vdc applied acrossthis capacitor creates an electrostatic force that pulls the two plates together, but is resisted by the
restoring force from the spring. b, As a function of the plate separation xs , the potential energyof the suspended plate is plotted for three values of Vdc . c, The overall force experienced by theplate is in turn given by the slope of these potentials. For a small voltage Vdc = V2 (mustard line),
two equilibrium positions can be identified at the dashed lines where the overall force is zero. At
this voltage, the static separation between the plates xv is found at the stable equilibrium position.
However, when Vdc is further increased beyond a threshold V3 > Vmax (red line), no equilibrium
position can be found, and the electrostatic instability in the potential causes the suspended plate
to collapse. d, A CPB qubit can be created by placing a Josephson junction in series between themechanically compliant capacitor and the ground. In such a device, Vdc couples the mechanicalmotion x around the static separation xv to the qubit energy. The maximum coupling rate gmax isfound at the maximum voltage Vdc = Vmax

49v dependents on V , and C = C v +C . At the point of collapse (Vwhere both xv and Cm
ΣJdcdc = Vmax ),mthe static separation xv reaches its minimum value of2
xmin = x0 .
3

(3.12)v,
Using the expression for both Vmax and xmin and assuming a small stray capacitance CJ  Cmthe maximum qubit-mechanics coupling rate for this simple system is given by,
s
s


0
3Cm
2k
e
2k
e
xzp
≈ xzp
.
gmax =
0
0
0~ 3Cm + 2CJ
3Cm
~
3Cm

(3.13)

0 is linearly dependent on the area of the capacitor plate, this equation
Because the capacitance Cmshows that to maximize the qubit-mechanical coupling rate, it is beneficial to use a smaller mechan0 until C becomes non-negligible. However,
ical oscillator (i.e., smaller plate area) to reduce CmJ

this increased coupling rate comes at the cost of a larger charge-noise-induced qubit dephasing, andmay be counter-productive to reaching the strong-dispersive limit, as I will discuss in Sec.3.6.

3.3

Improving the maximum coupling rate with a series capacitanceBeyond reducing the size of the mechanical oscillator, inserting a static capacitance C0 inseries between the dc-voltage and the mechanically compliant capacitor (Fig. 3.2) can also increase
the maximum coupling rate gmax . At first glance, this may seem counter-productive because theaddition of C0 reduces the participation ratio of the mechanically compliant capacitor in the CPB
gate-capacitance, and therefore reduces the qubit-mechanics coupling rate gm per external dcvoltage. However, by converting the dc-voltage source into an effective charge source, this static
capacitance can greatly delay the onset of the electrostatic instability and increase the maximum
coupling rate gmax by (at best) a factor of approximately two.
To demonstrate the improvement of adding C0 , I start by quantifying the effect of diluting the mechanical participation ratio in the CPB gate-capacitance. In this new system, thegate-capacitance Cg (x) is position-dependent and given by the series capacitance of C0 and the

50

Figure 3.2: Improving the maximum qubit-mechanics coupling rate with a series capacitance
By inserting a static capacitance C0 in series between the dc-voltage and the mechanically compliant
capacitor, I increase the maximum coupling rate gmax by increasing the maximum voltage Vmax
that can be applied before the mechanically compliant capacitor collapses.

51
mechanically compliant capacitor Cm ,
 2
vC
v C2
x
Cm (xv , x)C0
Cm
Cm
x
0
0
Cg (x) =
≈ v
+
+O
,
2
v
Cm (xv , x) + C0
Cm + C0 (Cm + C0 ) xv
xv

(3.14)

where I have expanded around a small motional amplitude x  xv . Because the CPB gate-voltage
is simply Vdc , the coupling rate is found using Eqn.(3.3),
gm (Vdc ) =

xzp
4Ec
C0
d
e
Vdc ,
xzp Vdc Cg (x) ≈v
2e~dx~ Cm + C0 x v(3.15)

where I have again assumed a small stray capacitance CJ such thatCΣ = CJ +C v C0Cm (xv , x)C0
≈ vm.
Cm (xv , x) + C0
Cm + C0(3.16)

Comparing Eqn.(3.15) and Eqn.(3.11), it is clear that adding C0 reduces the qubit-mechanics
coupling rate per external dc-voltage.
However, C0 can also delay the onset of the electrostatic instability, and greatly increase themaximum voltage Vmax that can be applied. As discussed above, for a large Vdc  e/Cg , very little
voltage will be dropped across the Josephson junction. Ignoring the junction, I again write thepotential energy of the circuit when the motional electrode is suspended xs above the bottom one,U (xs ) = −0C
1 x 0 Cm
102
+ k(x0 − xs )2 .
Vdc0
2 x0 Cm + xs C0
2(3.17)

From this potential, the static separation xv can be found for any dc-voltage. When this separationis reduced to a minimum value of1
xmin =
3



C0
2− m
C0


x0 ,

(3.18)

Vdc reaches its maximum value  03/2 s2 CmkVmax =
+1
x0
0
3 C0
Cm

(3.19)

beyond which the instability in the potential will collapse the mechanically compliant capacitor.
0 ≥ 2C , x
Interestingly when Cm0min can become negative! Physically, this means the vacuum-gap

52

Figure 3.3: The series capacitance improves the maximum qubit-mechanics coupling rate
The maximum achievable qubit-mechanics coupling rate gmax (λ) is plotted as a function of the
0 /C , normalized to g (0). Assuming an ideal case (blue line) where thecapacitor ratio λ = Cm0mstray capacitance CJ = 0, gmax monotonically increases until the two capacitor plates can bestatically brought into
√ contact at λ = 2 (dashed black line), giving a maximum enhancement on
gmax by a factor of 3. However, this maximum enhancement is diminished by a nonzero CJ that0 ). Additionally, the maximum
reduces the CPB charging energy Ec (mustard line for CJ = 0.1Cm
enhancement is further reduced if the capacitor plates cannot be brought into contact, but insteadcollapse at a minimum separation xcas because of other factors such as the Casimir force (red line
0 , and x
assumes CJ = 0.1Cm
cas = 10 nm with an initial separation of x0 = 50 nm).

53of the mechanically compliant capacitor can be statically reduced until its two plates come into
contact at a final voltage√
Vtouch = 2



0
Cm
C0

s


x0

k
.
0
Cm

(3.20)

To find the maximum achievable qubit-mechanics coupling rate gmax , I substitute Vdc in
Eqn.(3.15) by the expression of Vmax (Eqn.(3.19)) if there exists an electrostatic instability, or by
the expression of Vtouch (Eqn.(3.20)) if the plates can smoothly come into contact. Expressed as a
0 /C ,
function of the capacitance ratio λ = Cm
0


√


 1 + λgmax,0λ≤2gmax (λ) =√ 3gmax,0λ>2,(3.21)

where gmax,0 = gmax (0) is the maximum coupling rate without adding C0 found in the previous
section, Eqn.(3.13). This equation shows the optimal λ = 2 can result in a maximum improvement√
on gmax by a factor of 3, and a maximum improvement of χmax by a factor of 3. In reality,
however, this improvement is limited by other factors such as a non-negligible stray capacitance
CJ , as I show in Fig. 3.3. Moreover, although C0 can prevent the mechanically compliant capacitor
from collapsing from the electrostatic instability, it offers no protection against the Casimir force
collapse at a minimum separation xcas ∼ 10 nm [52]. Consequently, the optimal λ appears at asmaller value, and the maximum coupling rate is reduced accordingly. More details on the modelingof the Casimir force is discussed in Sec.3.6.

3.4

Using symmetry to protect against the noise on the dc-line
Although the large external dc-voltage is crucial for creating a strong qubit-mechanics inter-action, it also makes the mechanical oscillator more susceptible to noise on this dc-line. This noise,composed of both thermal and quantum fluctuation, can drive motion in the mechanical oscillator
and cause an increased mechanical relaxation. To combat this problem, I use the anti-symmetric,
second mode of the suspended capacitor plate as the mechanical oscillator, and couple it to a CPB

54
qubit. This symmetric arrangement (Fig. 3.4) has the benefit of simultaneously allowing for a large
qubit-mechanics coupling rate, and protecting the oscillator from noise on the dc-voltage.
To understand this design choice, I first quantify the effect of noise on the mechanical oscillator when an external dc-voltage Vdc is applied [118]. The Johnson-Nyquist noise caused by the
impedance Z(ω) of the dc-line not only drives random motion on the oscillator, but also induces anextra source of damping. In thermal equilibrium, these two effects are related through the quantum
fluctuation-dissipation theorem (i.e., detailed balance), and can be expressed as a function of Z(ω).
To see this, I consider the simple picture discussed in Sec.3.2, and write the overall noisy voltage
applied on the suspended capacitor plate as,
Vtot = Vdc + Vn ,(3.22)

where Vn is a small fluctuation on the applied dc-voltage Vdc , with a random phase and an amplitude
characterized by the double-sided noise power spectrum (see Sec.2.2.3)
Sv (ω) =2~ω
Re [Z(ω)] .
1 − e−~ω/kB T(3.23)

This voltage fluctuation Vn exerts a noisy force
Fv =

v
Cm
Vdc Vn + O(Vn2 )xv

on the oscillator, whose double-sided power spectrum is
 2
2
Cm
SF (ω) =Vdc Sv (ω).
xv

(3.24)(3.25)

When this noisy force is resonant with the oscillator frequency ωm , it can add (Γ+ ) or remove (Γ− )phonons from the oscillator’s motion at rates
Γ± =1SF (∓ωm ),2m~ωm(3.26)

where m is the mass of the oscillator. In thermal equilibrium at temperature T , these noise-induced
transition rates obey detailed balance, which determines the oscillator’s thermal distribution as
discussed in Sec.2.1 and dictates the relationship~ωm
SF (ωm ) = exp
SF (−ωm ).
kB T(3.27)

55

Figure 3.4: Creating strong qubit-mechanics coupling with a symmetric arrangementa, Using the anti-symmetric, second mode of the suspended capacitor plate as the mechanical oscillator, I create a symmetry in the qubit-mechanics device that protects the oscillator against noiseon the dc-line. Meanwhile, this arrangement allows for the creation of a strong qubit-mechanics
coupling rate: a voltage Vdc converts the anti-symmetric motion of the oscillator into a voltage
Vg (x) across the open terminals A and B. b, The Thevenin equivalent representation of the circuit
seen by the junctions is a Cooper-pair box qubit, with a mechanical-position dependent gate chargeng (x) = Vg (x) × Cg /2e.

56Moreover, because the noise-induced force is more likely to remove phonons (because SF (ωm ) >SF (−ωm )), it damps the oscillator’s motion at rateγv =1[SF (ωm ) − SF (−ωm )] .
2m~ωm

(3.28)

Using Eqn.(3.25), Eqn.(3.27), and assuming ~ωm  kB T , the damping rate can be related to theenvironmental impedance Z(ω) according to,
 v
2
2
 v
Cm
Sv (ωm ) + Sv (−ωm )Cm
Re [Z(ωm )]
γv =
Vdc
Vdc
=
.
xv
4kB T mxv
m(3.29)

This voltage-noise-induced damping rate contributes to the overall mechanical relaxation rate,
γm = γv + γ0 ,(3.30)

where γ0 is the intrinsic mechanical relaxation rate associated with the material, geometry andtension [119] of the mechanical oscillator.
In order for γv to not significantly increase the overall mechanical relaxation rate, a stringent
requirement is placed on the allowed resistive impedance Re [Z(ωm )] on the dc-line. To give senseof scale, consider a typical circular mechanically compliant vacuum-gap capacitor [57, 19] of 15 µm
diameter, 48 pg mass with an initial separation x0 ≈ 50 nm, and spring constant k ≈ 190 N/m.
0 ≈ 31 fF is increased toUsing the results from Sec.3.2, I can estimate the initial capacitance Cmv ≈ 46 fF at the maximum voltage V
approximately Cm
max ≈ 2.1 V. According to Eqn.(3.29), in orderfor γv to be much smaller than the intrinsic damping rate of approximately γ0 = 2π × 32 Hz [19]when operating at the maximum voltage, the resistive impedance at the mechanical frequency needs
to be much smaller than 1 Ω3 ! Although this can be achieved using a carefully designed filter, it
is a rather demanding design.
A better way to protect the mechanical oscillator from noise on the dc-voltage is to engineer
a symmetry that naturally reduces the random force it experiences. To this end, instead of thefundamental mode of the suspended plate, the mechanical oscillator corresponds to the second, antisymmetric mode, where motion displaces two halves of the suspended plate in opposite directions
3

I have also optimistically assumed that the voltage noise is well thermalized to its environment. Any out-ofequilibrium voltage noise will only make this worse

57
away from the static position xv to create two position-dependent capacitances,
v,±
Cm
(x) =

v /2
Cm
.
1 ± x/xv(3.31)As shown in Fig. 3.4(a), each half of the mechanically compliant capacitor is connected to groundby a series capacitance C0 /2 or Cc /2. The Josephson junction connects the two legs of the circuit
in a Wheatstone-bridge manner, and as I will show shortly, creates a CPB qubit that couples tomotion in the anti-symmetric mode. Intuitively, the protection offered by symmetry can be deduced
by realizing that when the suspended plate is stationary at static separation xv , the symmetry from
C0 = Cc means the voltage drops across the two halves of the plate are identical, and cannot drive
the oscillator with opposite phase.
To go beyond this intuition, I start by finding the noisy force the drum experiences at static
separation xv and dc-voltage Vdc for any values of C0 and Cc . This can be achieved by finding
the Thevenin equivalent circuit seen by the Josephson junction as shown in Fig. 3.4(b). For small
motion x  xv , the equivalent voltage Vg (x) is the voltage difference between open terminals A
and B,


 2
v
v
v
(C0 − Cc )Cm
C0 Cm
Cc Cm
x
x
Vg (x) =Vtot ++
Vtot + O
,
v
v
v
2
v
2
(C0 + Cm )(Cc + Cm )(C0 + Cm )(Cc + Cm ) xv
xv

(3.32)

where Vtot = Vdc +Vn is the noisy voltage applied on the device. Similarly, the equivalent capacitanceis found by replacing the voltage source with a short circuit, 2v )(C + C v )v(C0 − Cc )Cmxx(C0 + Cmcm
Cg (x) =
+
+O
.
v)
v)x
2(C0 + Cc + 2Cm
2(C0 + Cc + 2Cm
xvv(3.33)

Ignoring the effect of the Josephson junction, when the mechanical oscillator is stationary (x = 0)
at static separation xv , it experiences a noisy force of amplitude
Fv =v )2 (2C C + C C v + C C v )(C0 − Cc )(Cm
0 c
0 m
c m
Vdc Vn + O(Vn2 )v )2 (C + C v )2 x2(C0 + Cm
cv
m(3.34)

58
Similar to Eqn.(3.29), this noisy force results in a mechanical damping of rate2v )2 (2C C + C C v + C C v )(C0 − Cc )(Cm
Re [Z(ωm )]
0 c
0 m
c m
γv =
Vdc
v
2
v
2
2(C0 + Cm ) (Cc + Cm ) xv
m
2  v
2
Cm
1
1
Re [Z(ωm )]
1
−
Vdc
=
2
v
2
v
4 (1 + Cm /C0 )xvm
1 + (1 + Λ)Cm /C0




2
2
v /C
v
Cm
Cm
Re [Z(ωm )]
0
2
Vdc
≈
Λ
+
O(Λ
)
,
v /C )3(1 + Cmxm0v(3.35)

where I expanded to the first order in the asymmetry between the two arms
Λ=C0 − Cc.
Cc(3.36)

Clearly, when Λ = 0, the random force vanishes, and fluctuations on Vdc has no effect on theoscillator. In reality, I intentionally preserve a small asymmetry of approximately Λ ≈ −0.09 toallow for coherent driving of the mechanical oscillator using the combination of Vdc and an ac-signal
at ωm , both applied on the dc-line (see Sec.4.2.2, Sec.5.4 and Appendix.B). Nevertheless, assuming
0 /C = 2, Λ = −0.09, and the same oscillator parameters as discussed above, the quasi-symmetryCm0of the circuit reduces the voltage-noise induced mechanical damping by more than three orders ofmagnitude at the point of collapse4 .
This symmetric design not only protectes the mechanical oscillator against voltage fluctuations on Vdc , but also has the additional benefit of further increasing the maximum qubitmechanics coupling rate gmax achieved in the previous section. When the two-halves of the capacitor plate oscillate in opposite phase around the static position xv , the changing capacitancepushes the charges — deposited on them by Vdc — across the Josephson junction to create a qubitmechanics coupling. The rate of this coupling can be found from the position-dependent gate-chargeng (x) = Cg (x)Vg (x)/2e, where Vg (x) and Cg (x) are given by Eqn.(3.32) and Eqn.(3.33) respectively.
For the simple case of Cc = C0 , I find the qubit-mechanics coupling rate to be,
4Ec∂
xzp ng (x)
~
∂x
v
v
xzp
xzp
e
C0 Cm
e 2C0 Cm
=
V
≈
Vdc ,
dc
v
2
v
~ CΣ (C0 + Cm ) xv
~ (C0 + Cm ) xv

gm (Vdc ) =4(3.37)

0
Because Cm/C0 = 2, there is no electrostatic instability. Instead, I assume the mechanically compliant capacitor
will collapse under the Casimir force at separation xcas ≈ 10 nm, which corresponds to a dc-voltage of Vdc ≈ 11 V0(see footnote 5). To reach γv = 2π × 32 Hz with the same oscillator parameters (Cm= 31 fF and initial separation
x0 = 50 nm), I find Re [Z(ωm )] ≈ 2.3 kΩ, more than three-orders of magnitude larger than the 1 Ω found above.

59
where I have assumed a small stay capacitance CJ such that10CΣ = Cg (x) + CJ ≈ (C0 + Cm).
2

(3.38)One should take care when comparing this qubit-mechanics coupling rate to the ones found in previous sections (such as Eqn.(3.15)): because this symmetric arrangement uses a higher frequency√
mode, the zero-point motion xzp ∝ 1/ ωm is reduced accordingly. Compared to using the fundamental mode of a circular plate, using its second, anti-symmetric mode increases ωm by a factorof 1.59 [120], and reduces the xzp by a factor of 0.8. Nevertheless, accounting for this reduction
in xzp , the symmetric arrangement can still create a larger maximum coupling rate than thatfound in Sec.3.3. Using the expression for xmin and Vmax found previously5 , I find the maximum
achievable qubit-mechanics coupling rate in this arrangement as a function of the capacitance ratio
0 /C ,
λ = Cm0

gmax (λ) =0.8 × √3λ gmax,0
1+λ

λ≤2


√


0.8 × 2 2gmax,0λ>2,(3.39)

where gmax,0 is given by Eqn.(3.13) assuming CJ = 0, and the factor of 0.8 comes from the reduced
xzp discussed above. At λ = 0, the Josephson junction is shorted to ground on both nodes throughthe series capacitance C0 /2, and the coupling rate vanishes. Similar to the discussion in Sec.3.3,
the maximum improvement on the coupling rate is limited by other factors such as the presence ofa stray capacitance CJ or the capacitor collapse because of Casimir force, as shown in Fig. 3.5.

3.5

Coupling the qubit to a readout cavity
To control and read out the qubit state, it is coupled capacitively to a microwave resonator(henceforth referred to as the cavity to avoid possible confusion with the mechanical oscillator)
in the dispersive limit. In Sec.3.5.1, I introduce the cavity, which is formed by a transmission
5

The static position xv and the collapse voltage Vmax can be found by treating the Josephson junction asa short circuit. This treatment is valid because very little voltage drops across the Josephson junction when theplate is stationary. In doing so, I find a similar circuit to the one discussed in the previous section, but with C0 in
Fig. 3.2 replaced by (C0 + Cc )/2. Thus with this replacement, expressions for xv , xmin and Vmax found in Eqn.(3.17– Eqn.(3.20)) remains accurate.

60

Figure 3.5: Maximum achievable qubit-mechanics coupling rate in the symmetric arrangement
The maximum coupling rate gmax (λ) achievable with the symmetric arrangement is plotted as a
0 /C assuming C = C . This coupling rate is normalized
function of the capacitance ratio λ = Cm
00c
to gmax,0 achieved with the basic arrangement discussed in Sec.3.2. In both the ideal case (blue line)0 ), the maximum enhancement on g
and CJ 6= 0 (mustard line for CJ = 0.1Cm
max is achieved whenthe mechanically compliant capacitor can be statically brought into contact at λ = 2 (dashed blackline), with a maximum enhancement of approximately 3. However, this maximum enhancement isdiminished of the capacitor plates collapse at a minimum separation xcas because of factors other
0 , andthan the electrostatic instability, such as the Casimir force (red line assumes CJ = 0.1Cm
xcas = 10 nm with an initial separation of x0 = 50 nm).

61
line resonator and behaves like a harmonic oscillator. By placing this cavity in series between thecapacitance Cc /2 and the electrical ground (Fig. 3.6), a qubit-cavity coupling is created as explainedin Sec.3.5.2. However, there is a trade-off between having a large qubit-cavity coupling rate gc andhaving a large qubit-mechanics coupling rate gm . Thus, in Sec.3.5.3, I optimize the qubit readout
in the dispersive limit such that I minimize gc while allowing for single-shot qubit readout with one
cavity photon.

3.5.1

Cavity formed by a microwave resonator

The cavity is formed by a half-wave co-planer waveguide (CPW) resonator. Designed tohave a fundamental frequency of ωc = 2π × 5 GHz and a characteristic impedance Z0 = 50 Ω,
the resonator6 has a total length ξ = 12.76 mm and can be characterized by a distributed modelwith capacitance per unit-length Cξ = 153 pF/m and inductance per unit-length Lξ = 394 nH/m.
Although this CPB resonator consists of many modes, here I’m interested only in its fundamental
mode, which behaves like a harmonic oscillator with Hamiltonian
Hc = ~ωc ĉ† ĉ,(3.40)

where ĉ† and ĉ are the cavity’s photon creation and annihilation operators. Similar to how amechanical oscillator’s phonon operators â† and â relate to its position x̂ (see Sec.2.1), these photon
operator are related to the resonators voltage according toV̂ = Vzp (ĉ† + ĉ),(3.41)

where [103]
s
Vzp =

~ωc
ξCξ

(3.42)

is the zero-point voltage fluctuation.
Access to the electromagnetic environment of the cavity is provided by two ports on eitherside of it. From these ports, energy can enter of leave the cavity at rates κin or κout , determined
6

The CPW resonator is formed by a 10 µm wide center conductor, which is separated from the lateral groundplanes by a gap 5 µm wide. These patterns are created by depositing 100 nm thick aluminum on a sapphire substrate.

62

Figure 3.6: Adding a microwave cavity to the qubit-mechanics device
To control and read out the state of the qubit, it is coupled capacitively to a microwave resonator(orange), placed in series between the capacitance Cc /2 and the ground, in the dispersive limit.
The resonator is formed by a half-wave co-planer waveguide (CPW) resonator with a total lengthξ and represented as a distributed LC circuit with capacitance per unit-length Cξ and inductance
per unit-length Lξ . Control and readout of the qubit state is achieved with a voltage Vcav applied
to the resonator’s center conductor through an input-port defined by capacitance Cin . Finally, theresonator is connected to the amplifier chain through its output-port defined by Cout .

63by the capacitance values of the coupling capacitors Cin and Cout respectively [121]. In Sec.3.5.3,
I explain how these coupling rates are chosen. In addition to these ports, energy in the cavity canalso leak to the environment at a rate κloss . Thus, the overall rate at which energy leaves the cavityisκ = κin + κout + κloss .

3.5.2(3.43)

Qubit-cavity interaction

The cavity is placed in series between the capacitance Cc /2 and the electrical ground as
shown in Fig. 3.6. While this arrangement allows for a strong qubit-cavity interaction as I show
shortly, the addition of the cavity doesn’t change the qubit-mechanics interaction. At frequencies
close to dc or ωm , the cavity appears as a large shunt capacitance Cshunt = ξCξ ≈ 2 pF to ground.
Because this capacitance is much larger than any reasonable choice of Cc (a few femto-Farads),
node C in Fig. 3.6 can be treated as if it were grounded, and I recover Fig. 3.4(a) discussed in theprevious section.
To understand the qubit-cavity coupling, I treat the cavity as a voltage source [103], definedby the voltage difference Vc between its center-pin and the ground plane. This voltage is composedof a classical dc-voltage Vset set by the dc-component of Vcav 7 , and a quantum voltage V̂ due tothe photons inside the cavity,
Vc = Vset + V̂ .

(3.44)I then find the Thevenin equivalent circuit seen by the Josephson junction. Ignoring the mechanicalmotion (x = 0), the equivalent gate-voltage Vg is the voltage difference between open terminals A
and B,
Vg = Vdcv (C − C )
Cm
Cc
0
c
+
V
+
V̂
.
set
v
v
v
(Cm + Cc )(Cm + C0 )Cc + Cm(3.45)

Here I find that, in addition to the dc-voltage applied on the mechanical oscillator Vdc , the dcvoltage on the cavity Vset can also be used to bias the CPB gate-voltage. Indeed as discussed in7dcdc
Vset = Vcav
Cin /(Cin + Cshunt ), where Vcav
is the dc-component of Vcav

64
Sec.4.1.2, Vset is used experimentally to reset the CPB gate-charge to the sweet-spot before eachexperiment. In the rest of this section, I consider the simple case of Cc = C0 , and simplifies theabove equation to,
Vg = Vset + V̂C0.
v
C0 + Cm(3.46)

Similarly, the equivalent capacitance is found by replacing both voltage sources with short circuits,
1v).
Cg = (C0 + Cm
4

(3.47)

Combining these two equations, I find the electrostatic energy of this CPB qubit (Eqn.(2.19)),

Cg Vg 2
Hel = 4Ec n̂ −2e 2 
 C
C0
C0 Vset 20= 4Ec n̂ −+V̂ 2Vset + V̂ +n̂V̂ ,
8e
8e
4e(3.48)

where n̂ is the Cooper-pair number operator. In this equation, the first term is the original electrostatic Hamiltonian under a classical dc-voltage. The second term corresponds to the extra
electrostatic energy stored in the qubit capacitor by the cavity photon, independent of the qubit
state. Finally, the third term couples the electrostatic energy of the qubit to the cavity field. Transforming n̂ to qubit Pauli operators σ̂z and σ̂x (see Sec.2.2.2), I find the qubit-cavity interaction
Hamiltonian,Hint = ~gc ĉ + ĉ† )(cos θ0 σ̂z − sin θ0 σ̂x ,(3.49)

where θ0 = arctan[EJ /4Ec (1−2ng )] is the mixing angle between the charging energy and Josephsonenergy (Eqn.(2.22)),
gc =

4Ec
C0
e
4C0
Vzp C0 =
Vzp ≈
g
v
v c0
2~e
CJ + (C0 + Cm )/4 ~C0 + Cm(3.50)

is the qubit-cavity coupling rate, and gc0 = eVzp /~ ≈ 2π × 315 MHz according to Eqn.(3.42). For
the rest of this chapter, I assume that the CPB qubit is operated at the charge-degeneracy pointng = 1/2 so that θ0 = 0.
There is a trade-off between having a large qubit-cavity coupling rate gc and having a large
qubit-mechanics coupling rate gm . While increasing gm corresponds to increasing Vdc applied on

65
the mechanical oscillator and thereby reducing the vacuum-gap xv of the mechanically compliantv and decreases g . Thus, the question arises: what
capacitor, doing so simultaneously increases Cm
cis the minimum gc , such that I can maximize gm while still able to read out the qubit? To answer
this question, I calculate the measurement rate of the qubit state via the cavity.

3.5.3

Qubit readout through the cavity

To read out the qubit, the qubit-cavity system is operated in the dispersive limit, with acoupling rate gc much smaller than the qubit-cavity detunning ∆c = ωq − ωc such that energy will
not spontaneously transfer between the two. This type of interaction is already discussed in thecontext of qubit-mechanics interaction in Sec.2.3.3, but can be greatly simplified here due to thelarge ωc . Because Σc = ωq + ωc  ∆c , the qubit-cavity dispersive shift is approximately,
χc =

gc2
g2
g2
+ c ≈ c.
∆c Σc∆c(3.51)

Furthermore, because the large ωc strongly suppresses the sideband transitions at small cavity
photon number nc (see discussion around Eqn.(2.80)), the qubit-state-dependent cavity-impedance
has little effect (see Eqn.(2.55)). Consequently, the qubit-cavity interaction merely causes a qubitstate-dependent cavity frequency
ωc0 (σ̂z ) = ωc + χc σ̂z .

(3.52)

By determining the frequency of the cavity, the state of the qubit is measured.
The frequency of the cavity is determined by measuring the transmission phase ϕc of a
microwave signal at frequency ωc . Applied on the cavity input-port with a field strength c (corresponds to the ac-component of Vcav in Fig. 3.6), the probe signal enters the cavity at rate κin andpopulates it with a coherent state characterized by a mean photon number [88],hnc i =

κin 2c.
χ2c + (κ/2)2(3.53)

Because of the qubit-cavity interaction, the coherent state acquires a phase shiftϕc (σ̂z ) = arctan2χc σ̂zκ,(3.54)

66
dependent on the qubit state. Carrying this information, the coherent state then leaves the cavity
from the output-port with a probability κout /κ, and become amplified and recorded. As such, I
acquire information on the qubit state at rate [122],
Γmeas =

2hnc iχ2c κout.
χ2c + (κ/2)2(3.55)

Depending on the type of measurement [123], the qubit state is determined after a time
tmeas ≥

11

ηmeas Γmeas(3.56)

for a two-quadrature measurement, andtmeas ≥

11
ηmeas 2Γmeas(3.57)

for a single-quadrature measurement, where ηmeas is the measurement efficiency.
With this understanding, I can now design the qubit-cavity system to achieve efficient qubit
readout while maintaining a large qubit-mechanics coupling rate. First, the energy in the cavity
should predominantly leave from the output-port at rate κout  κin , κloss . Next, for a given phonon
number hnc i and dispersive shift χc , the measurement rate is maximized when κ = 2χc . Finally,
I desire to achieve single-shot readout of the qubit state, meaning tmeas should be smaller than
the qubit decay time T1 = 1/Γ1 . Assuming a two-quadrature readout scheme, a near perfectmeasurement efficiency (η ≈ 1) and taking κ = 2χc , this requires
Γmeas = hnc iκout ≥ Γ1 .

(3.58)

With these in mind, I now minimize the qubit-cavity coupling rate. Assuming a qubit decay rate
Γ1 ≈ 2π × 1 MHz (see next section) and hnc i ≈ 1, I design the cavity with an output coupling rateκout = 2π × 1.55 MHz (Cout ≈ 13.9 fF), and an input coupling rate κin ≈ 2π × 171 kHz  κout
(Cin ≈ 4.6 fF). Assuming a small environmental coupling rate κloss on the order of 2π × 10 kHz
for the superconducting aluminum CPW resonator [124], the overall cavity loss rate κ ≈ 2π ×1.58 MHz determines the optimal qubit-cavity dispersive shift χc = 2π × 0.79 MHz. Choosing a

67
qubit frequency at the charge degeneracy point ωqb = EJ /~ = 2π × 4 GHz, the minimum qubitcavity coupling rate needed to achieve single-photon qubit readout with hnc i = 1 is thereforeq
gc = χc (ωc − ωqb ) = 2π × 28.1 MHz.

3.6

Design choices for the fabricated device and expected performance
From the preceding discussions, I have arrived at a circuit schematics of the mechanics-qubit-

cavity hybrid system as shown in Fig. 3.6 (reproduced in Fig. 3.7(a)). To implement this circuit,the device is laid out as shown in Fig. 3.7(b). The mechanical oscillator is formed by an aluminum
drumhead (blue), similar to those used in previous electromechanical experiments [19]. It is suspended above two aluminum electrodes (green and yellow) to realize two mechanically compliant
capacitors. These electrodes then extend out and are connected by the Josephson junctions. This
device is finally embedded between the center-pin (orange) and one ground plane (grey) of the
CPW resonator. In this section, I decide on the exact geometry of this device.

3.6.1

Choosing a drum size

The first question is: how large should the drumhead be? In Sec.3.2, I have shown that itis beneficial to reduce the size of the suspended plate in order to increase the maximum voltage
Vmax and thereby increase the maximum qubit-mechanics coupling rate. However, the reduction of
the plate size causes a larger charge-noise-induced qubit dephasing rate Γφ , which, as a function of
the plate size, increases faster than χmax . Consequently, because I aim to reach a strong dispersive
qubit-mechanics interaction, where χm is larger than the overall qubit dephasing rate Γ2 /2 =
Γ1 /2 + Γφ , I should reduce the drumhead size to increase χm until Γφ becomes the dominant sourceof qubit dephasing.
To see this, I assume the suspended plate has an area A. This area is related linearly to the
0 ∝ A. Additionally, assuming a constant plate thickness,mechanically compliant capacitance Cm

68

Figure 3.7: Layout of the device
To implement the circuit schematic of a (identical to Fig. 3.6), the device is laid out as shown inb. The mechanical oscillator is formed by an aluminum drumhead (blue). It is suspended above
two aluminum electrodes (green and yellow) underneath to realize two mechanically compliant
capacitors. These electrodes then extends out, and are connected by the Josephson junctions. This
device is finally embedded in between the center-pin (orange) and one ground plane (grey) of the
CPW resonator to form capacitors C0 /2 and Cc /2.

69
the plate area also changes the mass of the mechanical oscillator and therefore its zero-point motion,
s
r
r
~
~
4 1
√
xzp =
=
∝
,
(3.59)
2mωm
A
2 mk
where k is the spring constant of the mechanical oscillator. Combined, χm at Vmax and at the
charge degeneracy point depends on this area according to (see Eqn.(3.13)),
χmax ≈ 2

2
x2zp
~gmax
∝ 0 ∝ A−3/2 .
EJ
Cm(3.60)On the other hand, assuming a small stray capacitance CJ from the junctions, the CPB charging
energy depends inversely to A because,
Ec =

1
1
e2
∝ 0 ∝ .
CΣ
Cm
A

(3.61)

Recalling from Sec.2.2.3 that the dephasing rate of a CPB qubit because of charge noise is proportional to Ec2 (Eqn.(2.33)),
Γφ ∝ (8Ec )2 /EJ ∝1,
A2(3.62)

I findχmax √∝ A.
Γφ(3.63)

Indeed, once the overall qubit dephasing rate Γ2 /2 is dominated by Γφ , I should stop further
reducing the plate area A, because its detrimental effect (i.e., increased Γφ because of charge noise)
exceeds its benefit (i.e., increased χm ).
Thus, to determine the size of the plate, I compare the expected qubit relaxation rate Γ1 tothe expected Γφ because of charge noise. The qubit relaxation is likely dominated by the dielectricloss on the dirty surfaces of the mechanically compliant capacitors: because of the fabrication
process used to create the suspended drumhead, residuals of silicon nitride are likely to be left on
the surfaces between the plates of the mechanically compliant capacitor (see Appendix.A). Indeed
in a previous electromechanical experiment [46], where a linear LC-circuit is coupled to a similar
drumhead, this dielectric loss is identified as the dominant source of internal loss for the LC circuit,and is observed to happen at a rate of approximately 2π × 1 MHz when one photon is inside the

70
LC-circuit. Assuming a similar level of cleanness on the capacitor surfaces, I therefore estimate thequbit relaxation rate will be approximately Γ1 ≈ 2π × 1 MHz. In comparison, assuming a typical√charge noise intensity of 10−3 − 10−4 e/ Hz at 10 Hz [125] and EJ = Ec , the charge-noise-induced
qubit dephasing is estimated to be a few hundred kilohertz [84], comparable to Γ1 /2.
In the end, because I cannot exactly predict Γφ and Γ1 , I choose three different sizes ofthe drum, each with diameter 6 µm, 7.6 µm and 9.5 µm. Assuming a static separation x0 =0 value of approximately 5 fF, 8 fF, and 12.4 fF
50 nm at Vdc = 0 V, these sizes each have a Cm

respectively. These capacitance values are chosen such that when operated at their maximum
voltage (see next sub-section), they would each correspond to a EJ /Ec ratio of approximately 1,
1.5, and 2 respectively. In other words, while the 6 µm drumhead device can create the largest χm ,the 9 µm drumhead device will have the smallest Γφ .

3.6.2

Parasitic capacitance

Having decided on the size of the drumhead, I can now specify the rest of the device geometry.
For example, by specifying the shape and size of the upper arm connecting the Josephson junction
to one bottom drumhead electrode (yellow in Fig. 3.7), as well as the distance from this arm to thecenter-pin of the cavity, the capacitance Cc /2 is determined. However, in addition to this desired
capacitance, this upper arm also creates other parasitic capacitances as illustrated in Fig. 3.8(a),which include a parasitic capacitance to the ground Cct , a parasitic capacitance to the other arm
Ct , and a stationary capacitance Cr to the suspended drumhead. By simulating in Sonnet, I canfind these capacitance values for different geometries.
Including these parasitic capacitances in the circuit schematic as shown in Fig. 3.8, I canrealistically predict the performance of a given device. The stray capacitance CJ from the Josephson
junction is estimated to be approximately 1 fF for a typical junction area of 0.01 µm2 [126].
Following similar procedures discussed in previous sections, the qubit-mechanics and qubit-cavity
coupling rates can be found by finding the Thevenin equivalent circuit seen by the Josephson
junction.

71

Figure 3.8: Parasitic capacitance
To realistically predict the performance of the designed device, I need to consider the effects ofthe parasitic capacitances. For example, in the layout of a, one arm of the CPB qubit (yellow)
is connected to other components of the device through the following capacitances: an intentional+ (x) to the top
Cc /2 to the center-pin of the cavity (orange), an intentional motional capacitance Cm
plate (not represented), a parasitic Cr to the dc-line, a parasitic Cct to the ground plane (grey), anda parasitic Ct to the other CPB arm (green). This device is represented by the circuit schematic
of b.

72
3.6.3

Estimating the maximum voltage

Finally, I find the dependence of the vacuum-gap xv on the external dc-voltage Vdc in thepresence of the parasitic capacitances. Similar to previous discussions, I treat the Josephson junction as a short circuit and write down the potential energy U (xs ) in the circuit when the top
drumhead plate is suspended xs above the bottom electrodes. To compute this separation in arealistic manner, I also include the Casimir force potential [127],
Ucas (xs ) = −1 ~cπ 2 A,
3 240x3s(3.64)

where c is the speed of light and A is the plate area. Numerically, I can find xv for any Vdc bysolving,
F (xv ) = −∂ U (xs ) + Ucas (xs)= 0.
∂xs
xs =xv(3.65)

To find the point of collapse, I look at how Vdc softens the effective spring constant keff at thestatic position xv . To do this, I approximate the potential around xv as a second order polynomial,
whose coefficient corresponds to keff /2. From previous electromechanical experiments using similar
drumhead devices [19, 57], I estimate an initial effective mechanical spring constant k0 = 190 N/m
at Vdc = 0 V. The instability sets in when keff approaches zero as Vdc is increased. In this work,
I estimate the maximum voltage Vmax and the minimum vacuum-gap separation xmin by choosing
an arbitrary, but pessimistic, value of minimum spring constant kmin = 120 N/m. Once keff reaches
kmin , I consider the drumhead to be at the verge of collapse, and assign the applied Vdc to be Vmax .

3.6.4

Final designs

For each drum size, I optimize the geometry of the device such that when operated at the
maximum voltage Vdc , the qubit-mechanics coupling rate gm is maximized while the qubit-cavity
coupling rate is approximately gc ≈ 2π×31.5 MHz. The estimated essential performance parameters
for the final designs are presented in Table 3.1. Assuming a qubit relaxation rate Γ1 ≈ 2π × 1 MHz,and a qubit dephasing rate because of charge noise Γφ ≈ 2π × 1 MHz, the overall qubit dephasing

73
rate is approximately Γ2 /2 = Γ1 /2 + Γφ ≈ 2π × 1.5 MHz. Indeed, when operated at Vmax , the final
designs should allow the device to reach the strong dispersive limit, where χm > Γ2 /2.
For the rest of this work, I will be working with the device with a drum diameter of 7.6 µm.
Pictures of this device is shown in Fig. 3.9.
drum diameter
maximum voltage, Vmax
maximum qubit-mechanics coupling rate, gmax /2π
maximum qubit-mechanics dispersive shift χmax /2π 8qubit-cavity coupling rate at Vmax , gc /2π

6 µm
22.1 V
133.2 MHz
8.9 MHz31.7 MHz

7.6 µm20.5 V
120.3 MHz
7.2 MHz32.6 MHz

9.5 µm18.7 V
100.7 MHz
5.1 MHz32.4 MHz

Table 3.1: Estimated essential performance parameters of the final designs

8

Evaluated at the point of charge degeneracy assuming EJ /~ = 2π × 4 GHz

74

Figure 3.9: The physical device used in this worka, Optical micrograph of the device chip, showing the cavity (a CPW resonator), input and output
ports and the dc-line. b, One stage of the on-chip LC filter, composed of an inductance in series
and a waffled capacitance to ground (see Appendix.B). c, Zoom on the area where the qubit andmechanical oscillator are embedded. The input coupling capacitance of the resonator is visibleon the left. d, Scanning electron micrograph of the oscillator and the qubit. The center pin andground of the cavity are also visible. e, Zoom on the Josephson junctions (JJ). The split junction
allows for in-situ tuning of EJ using an external magnetic flux, and the long and skinny wires
connecting the junction to the rest of the circuit could contribute to a long parity-switching time(see Sec.4.1.3). f, Viewed at an angle, the mechanical oscillator (blue) is suspended above the twobottom electrodes (green and yellow) to form the mechanically compliant capacitor.

Chapter 4

Characterizing the hybrid system

Having explained the theory and design of the hybrid system, I now start discussing theexperimental aspects of this work. In this chapter, I describe the basic characterization of thehybrid system. With the mechanical oscillator decoupled from the qubit at Vdc = 0 V, I explain in
Sec.4.1 how control and readout of qubit state is achieved experimentally using the microwave cavity(i.e., the CPW resonator). With this ability, in Sec.4.2, I increase Vdc to 6 V to observe the firstindication of having achieved a strong qubit-mechanics interaction, and extract the corresponding
dispersive coupling rate χm . Although this device could reach the strong dispersive limit (2χm > Γ2 )if it were operated at its designed maximum dc-voltage of 21 V, a breakdown of the qubit readout
occurs when Vdc is increased beyond 6 V. This breakdown is explained in Sec.4.2.3. Consequently,
the reminder of this work will be performed at Vdc = 6 V, where the dispersive shift 2χm is smallerthan, but a significant fraction of, Γ2 . For reasons that will become clear in later chapters, I namethis regime the “phonon-number-sensitive regime”.

4.1

Cavity-enabled qubit readout
The ability to control and read out the qubit state is integral to all experiments in thiswork. Theoretically discussed in Sec.3.5, this ability is enabled by a microwave cavity that isdispersively coupled to the qubit. In this section, I experimentally demonstrate this. In Sec.4.1.1,
I infer the qubit state from the cavity transmission phase, measured at the bare cavity frequency
ωc . In Sec.4.1.2, I describe using a dc-voltage applied on the cavity to set the CPB qubit to its

76sweet-spot (i.e., charge degeneracy point) in order to minimize the charge-noise-induced dephasing(see Sec.2.2.3). Despite this effort, a jump in the qubit’s charge-number parity, caused by thetunneling of a quasiparticle across the junction, will bring the qubit far away from its sweetspot with devastating effects. To ensure the qubit remains at the sweet-spot throughout eachexperiment, I perform post-selection on the qubit parity as I explain in Sec.4.1.3. Finally in
Sec.4.1.4, I characterize the qubit coherence in both time and frequency domains. For clarity, I
work at Vdc = 0 V to decouple the qubit from the mechanical oscillator in this section. However,
all techniques described in this section remain unchanged for all values of Vdc .

4.1.1

Inferring the qubit state

The qubit state is inferred from the phase shift ϕc acquired by a coherent signal transmitting
through the cavity. As discussed in Sec.3.5.3, the cavity is dispersively coupled to the qubit, whereits bare frequency ωc is pulled by the state of the qubit according toωc0 (σ̂z ) = ωc + χc σ̂z ,(4.1)

where χc is the dispersive qubit-cavity coupling rate. Driven at frequency ωc , the coherent signal
populates the cavity with a coherent state of mean photon number hnc i, and acquires a phase shift
ϕc approximately given by [128]ϕc ≈2Re(χc )hσ̂z i,
κ

(4.2)

where κ is the overall energy loss rate of the cavity, and
χc =

gc2
,
−iΓ2 + ωq − ωc(4.3)

is modified by the qubit dephasing rate Γ2 /2. Experimentally, I determine ωc ≈ 2π × 4.76 GHz
and κ ≈ 2π × 2.3 MHz. In Fig. 4.1, I measure this phase as a function of the gate-charge ng whenthe qubit is in its ground state, hσ̂z i = −1. This response can be understood with the help ofFig. 4.1(a): close to the charge degeneracy where ωq < ωc , χc < 0 and the coherent signal acquires
a positive phase shift ϕc > 0. When the qubit frequency crosses the cavity resonance at ng = 0.38

77

Figure 4.1: Cavity phase shift as a function of gate chargea, The qubit frequency (green) is strongly dependent on the gate charge ng , and would cross thecavity (orange) at ng = 0.38 and 0.62 if they were uncoupled. b, Because they are coupled, the qubit
state alters the cavity frequency and shifts its transmission phase ϕc . Here, ϕc is measured whendriving at the uncoupled cavity frequency ωc with the qubit in its ground state, but would havethe opposite sign if the qubit is excited. This measurement of ϕc vs. ng allows the determinationof the sweet-spot. Moreover at a fixed ng , measurement of ϕc enables qubit readout. This figure is
reproduced from reference [85].

78
and 0.62, ϕc changes sign as can be observed in Fig. 4.1(b). When the qubit is in its excitedstate, the phase response in this figure has the opposite sign. If this phase can be measured withenough accuracy to determine the qubit state within one qubit lifetime T1 , I would achieve singleshot qubit readout. However, despite the design considerations (see Sec.3.5.3), such single-shot
capability remains ellusive in this experiment because of both the limited measurement efficiency(ηmeas ≤ 0.67 because κout /κ ≈ 0.67) and the need to measure with a small cavity photon number(hnc i ≈ 0.6, see Sec.4.2.3).
Instead, I adopt an incoherent measurement technique [129, 86, 85], and infer the averageoccupation of the qubit hσ̂z i from the measured phase ϕc . Working at the charge degeneracy point,
the qubit excited state probability is approximately
Pe =

1
1
|gi
(1 + hσ̂z i) ≈ − ϕc /2ϕng =0.5 ,
2
2

(4.4)

|gi

where ϕng =0.5 ≈ 58.5 (deg) is the phase measured at ng = 0.5 with the qubit in its ground state.
4.1.2

Setting the qubit to its sweet-spot

Because there is typically 1/f noise on the gate-charge parameter ng , all experiments have aroutine running in the background that automatically reset it every one or two seconds to ensure
the qubit is always operated close to its sweet spot. The gate-charge is partly controlled by a dcvoltage applied on the cavity input-port (see Sec.3.5.2 and Appendix.B). Fig.4.1 shows the cavity
phase shift ϕc measured as a function of this dc-voltage. Using this measurement, the dc-voltage
is set to the charge degeneracy point, identified at the point of symmetry in the ϕc response [86].

4.1.3

Selecting the qubit parity

In addition to the 1/f noise, ng is also displaced from the sweet-spot in a sudden and discontinuous way when a quasiparticle tunnels across the junction. As discussed in Sec.2.2.3, quasiparticles are single-electron-like excitations that reside in superconductors. When a quasiparticle
tunnels across the junction, it causes a jump in the qubit parity, changing ng by 1/2. For a qubit

79

Figure 4.2: Qubit parity jumps
The qubit parity jumps can be observed by measuring ϕc in time with the qubit in its ground stateat the sweet-spot. I observe no preferred parity, and the qubit is found in either parity positionsin approximately 50% of the time. Conditioned on the parity being in the “good parity” position,
I post-select the data to ensure all experiments are conducted at the sweet-spot. This figure is
reproduced from reference [86].

80
operated at ng = 1/2, a parity jump takes it to the furthest point away from the sweet-spot, anddrastically changes both the qubit-mechanics (if Vdc 6= 0) and qubit-cavity interactions. In Fig. 4.2,
I observe such parity jumps by measuring ϕc in time with the qubit in its ground state.
From similar measurements, I estimate a characteristic parity-switching time of 1 ∼ 10 s [86].
This relatively long [130] switching time is likely determined by the special geometry around theJosephson junctions. As shown in Fig. 3.9(e), the junctions are defined by small aluminum pads
of approximate dimensions 300 × 300 × 100 nm3 , which are only connected to the rest of thesuperconducting device via wires that are 4 µm long and 60 nm wide. These small volumes ensurethat less than a single quasiparticle (see Sec.2.2.3) resides in the immediate surroundings of thejunctions to cause parity jumps. Although large number of quasiparticles reside in the rest of thedevice, they will need to diffuse across the long and skinny wires to reach the junction, making
them less likely to cause parity jumps.
Because of this long switching time, the qubit parity can be determined in a “single-shot”
manner as already demonstrated in Fig. 4.2. Leveraging this, I henceforth determine the qubit
parity at the end of every experiment. Conditioned on observing the “good parity” position, I
post-select the data to ensure all experiments are conducted at the sweet-spot.

4.1.4

Qubit characterization

With the qubit correctly initialized to its sweet-spot, I now characterize it in both time andfrequency domains [86].
To start, I demonstrate the manipulation of qubit state with a large qubit-drive at frequency
ωqb ≈ 2π ×3.8 GHz by inducing Rabi oscillations as shown in Fig. 4.3(a). Keeping the drive strength
constant, I vary its duration Td and measure the cavity transmission phase, which detects the qubit
excited state probability Pe according to Eqn.(4.4). As discussed in Sec.2.3.4.3, the qubit oscillates
between the ground and excited state at a rate ΩR ≈ 2π × 160 MHz, determined by the drive
strength, before reaching a steady-state excitation at longer time. In this figure, the maximum
excited state probability is low because the qubit decays quickly to ground state during the time

81
I measure. Nevertheless, at tπ ≈ 12 ns, the qubit-drive realizes a “π-pulse”, swapping the qubit
state from |gi to |ei.
Using this calibrated qubit-drive, I now measure the qubit lifetime T1 and decoherence time
T2 . To measure T1 , I excite the qubit with a π-pulse and wait a variable delay τ before measuring
the cavity transmission phase, as shown in Fig. 4.3(b). Because Pe decays according to,
Pe (τ ) = Pe (0)e−τ /T1 ,(4.5)

a fit reveals T1 = 1/Γ1 ≈ 260 ns. To measure the qubit decoherence, the variable delay τ is instead
inserted between two consecutive “π/2-pulses”. As a function of the qubit-drive detunning fromthe qubit resonance ∆d,q , Pe oscillates according toPe (τ ) ∝ cos(∆d,q τ )e−τ /T2 .

(4.6)

Fitting the data presented in Fig.4.3(c) using this equation, I find T2 ≈ 85 ns.
Finally, I measure the steady-state qubit spectrum, where the probability of exciting thequbit is measured as a function of the frequency of a weak qubit-drive signal as shown in Fig. 4.4.
This type of spectroscopic measurement will be used extensively in the rest of this work. To ensure
the spectrum is measured in steady-state, the qubit drive is applied 1 µs (≈ 4T1 ) before the start ofcavity phase measurement. Evident from the small maximum Pe (∼ 1%), I use a very weak qubit
drive to to minimize the power broadening of the qubit lineshape (see Sec.2.3.4.3). Fitting thismeasured qubit spectrum with a Lorentzian lineshape, I find a FWHM Γ2 = 2/T2 ≈ 2π × 3.7 MHz,consistent with the time-domain measurement of T2 ≈ 85 ns. Because of low-frequency noise on
the gate-charge, an asymmetry can be observed in this qubit spectrum, where a tail extends towardhigher frequency. This effect will be explored in more detail in Sec.5.2.4.

4.2

Turning on the qubit-mechanics coupling
Having demonstrated the ability to control and read out the qubit state, I now turn on thequbit-mechanics interaction by increasing Vdc to 6 V. In Sec.4.2.1, I observe the first indication

82

Figure 4.3: Time-domain qubit characterizationa, I observe the qubit Rabi oscillation (green dots) by varying the duration Td (x-axis) of a strong
excitation signal driven at the qubit’s resonant frequency and measuring the cavity transmission
phase ϕc (left-axis). This phase is related to the qubit excited state probability Pe (right-axis)
according to Eqn.(4.4). Because of its short lifetime T1 , the qubit decays quickly from |ei to|gi during the measurement, resulting in a reduced maximum Pe . At the vertical dashed line, I
identify the duration for a π−pulse Tπ ≈ 12 ns. b, After exciting the qubit with a π-pulse, I waitfor different delays τ before measuring ϕc (green dots). Fitting this result acording to Eqn.(4.5),
I find T1 ≈ 260 ns (black line). c, To quantify the qubit decoherence, I vary the detunning ∆d,q
between the qubit excitation signal and ωq0 (x-axis), and measure ϕc (color-scale) as a function of
the delay time τ between two consecutive π/2-pulses. Using Eqn.(4.6), I find a characteristic qubit
decoherence time T2 ≈ 85 ns.

83

Figure 4.4: Steady-state qubit spectrum
Sweeping the frequency of a weak qubit drive, I measure the qubit spectrum (green dots) at Vdc =0 V. The reported Pe (y-axis) is inferred from the measured cavity transmission phase ϕc according
to Eqn.(4.4). Fitting the qubit spectrum with a Lorentzian lineshape (black line), I find a FWHM
Γ2 = 2/T2 ≈ 2π × 3.7 MHz, consistent with the time-domain measurement of T2 . The asymmetricqubit lineshape is caused by noise on the gate-charge (see Sec.5.2.4).

84
of having achieved a strong qubit-mechanics interaction: the qubit spectrum is broadened by thethermal motion of the mechanical oscillator. To begin understanding this motion-broadened qubit
spectrum, I find the qubit-mechanics dispersive shift χm ≈ 2π × 0.26 kHz [86] in Sec.4.2.2. Thisshows that, if this device could be operated at its maximum voltage around 21 V, it could reach thestrong dispersive limit (2χm > Γ2 ). However, as I explain in Sec.4.2.3, attempt to further increase
Vdc beyond 6 V is impeded by a break-down of the qubit readout.

4.2.1

Motion induced qubit spectrum broadening

In Fig. 4.5, I measure the qubit spectrum (orange dots) at Vdc = 6 V. Compared to that measured at Vdc = 0 V (green dots), the 6 V qubit spectrum has a pronounced tail that extends towardhigher frequencies. Because each mechanical phonon would shift the qubit frequency by 2χm > 0(see Sec.2.3.3), this tail in the qubit spectrum is consistent with the expected qubit response toa mechanical thermal state, and indicates the presence of a strong qubit-mechanics interaction.
Indeed, as I will show in Sec.5.4, this qubit spectrum reveals a thermal phonon distribution characterized by its mean phonon number hni = nth ≈ 17.7, consistent with the base temperature of thefridge. However, although much information on the mechanical phonon distribution is contained in
the qubit spectrum, extracting it requires that I determine the dispersive qubit-mechanics coupling
rate χm .

4.2.2

Determining the qubit-mechanics coupling rate

To determine χm , I measure how the mechanical oscillator’s resonant frequency is pulled
by its interaction with the ground state qubit as a function of the gate-charge ng . Because ng
controls both the uncoupled qubit frequency ωq and the mixing angle θ0 between the σ̂x and σ̂zqubit-mechanics coupling (see Sec.2.3.1), it strongly influences the oscillator’s dispersive shift,χm =2 sin2 θgm
g 2 sin2 θ00+ m,ωq − ωmωq + ωm

(4.7)

85

Figure 4.5: Motion-induced broadening of the qubit spectrum
At Vdc = 6 V, I measure the qubit spectrum (orange dots). Compared to the spectrum measured
at Vdc = 0 V with the same qubit drive power (green dots), the 6 V spectrum has a pronounced
tail extending toward higher frequencies, indicating a strong qubit-mechanics interaction.

86
where ωm is the uncoupled mechanical frequency. Thus, by measuring the oscillator’s resonant
frequency as a function of ng , one can determine gm and therefore χm . Such a measurement was
first performed in an experiment [81] that coupled a CPB qubit to a silicon nitride beam oscillator.
However, in contrast to that experiment, the mechanical oscillator cannot be independently measured in this work. Instead, I use the qubit qubit-cavity system to infer the mechanical resonancewhile coherently driving motion on the oscillator [86].
The coherent motion is driven by the product of Vdc and a resonant ac-voltage (i.e., coherent
mechanical drive) applied on the same port, as discussed in Sec.3.4. Once the mechanical drive isturned off, the oscillator’s coherent motion survives for a characteristic time scale 1/γm ≈ 1.7 ms(see Appendix.D). During this time, the qubit frequency is Stark shifted by the coherent motion.
Leveraging this effect, I detect the mechanical resonance via the qubit spectrum as shown in Fig. 4.6.
Although this measurement works well at the charge degeneracy point where the qubit FWHM issmall, it fails at other values of ng because of a large charge-noise-induced qubit dephasing.
Instead, I determine the mechanical frequency by observing a motion-induced rectificationof ϕc while keeping the qubit in its ground state. Indeed, the coherent mechanical motion can betreated as a sinusoidal modulation of ng (see Sec.5.2). This modulation not only causes a Stark
shift on the qubit frequency, but also, in turn, rectifies the cavity phase. This can be intuitively
understood by considering the effect of a ng modulation on the phase signal in Fig. 4.1. With asufficiently large coherent mechanical motion, I can observe the rectified ϕc as shown in Fig.4.7 with
Vdc = 6 V. At ng = 1/2, I observe a mechanical resonance at around 25.078 MHz (selecting on thecorrect parity). Meanwhile at ng = 1/4, I observe a mechanical resonance at around 25.294 MHz(both parities are degenerate at this ng position, and I do not select parity). However, close tong = 0 (or 1), I cannot measure a motion-induced phase rectification with a reasonable number ofphonons because the curvature of the phase signal is very small. Nevertheless, with these two data
points and using Eqn.(4.7), I calculate that, at Vdc = 6 V, gm ≈ 2π × 22 MHz9 , which corresponds
9

The coherent motion has approximately 200 phonons, much less than the critical phonon number ωq /4χ2m ≈1.4 × 104 [122] at the charge degeneracy point.

87

Figure 4.6: Detecting the mechanical resonance via qubit spectrum
a, At Vdc = 3 V and ng = 1/2, I measure the qubit spectrum with the coherent mechanical driveoff. b, Parking the qubit drive at the frequency highlighted by the vertical arrow in a, I measure
the cavity phase ϕc as a function of the frequency of the coherent mechanical drive. When thismechanical drive resonates with the oscillator, its drives coherent motion on the oscillator that
Stark shifts the qubit and causes the qubit drive to become off-resonant. Consequently, ϕc goesback to the value corresponding to the qubit in the ground state, hence the observed contrast. The
observed width of the mechanical resonance is broadened by the short duration of the coherent
drive. This figure is reproduced from reference [86].

88

Figure 4.7: Measuring χma, Because the qubit frequency (green) is strongly dependent on the gate-charge ng , the groundstate qubit pulls the mechanical resonance by different amounts. Consequently, measuring themechanical resonance as a function of ng can reveal the qubit-mechanics coupling rate gm and inturn the dispersive shift χm . b, At Vdc = 6 V and ng = 1/2, I measure the cavity phase ϕc after
a 10 µs mechanical drive at varying frequencies. When the mechanical drive comes in resonancewith the oscillator, the resulting coherent motion rectifies the phase signal in Fig. 4.1 and allows
for detecting the mechanical resonance at frequency 25.078 MHz. c, Same as b but measured at
ng = 1/4, showing a significantly less shifted mechanical resonance at frequency 25.294 MHz. This
figure is reproduced from reference [86].

89
to a dispersive coupling rate χm ≈ 2π × 0.26 MHz at the charge degeneracy point.
2 , if the device were operated at its maximum voltage of 21 V, I expect
Because χm ∝ Vdc

a maximum dispersive coupling rate χmax
≈ 2π × 3.2 MHz. Compared to the measured value ofmΓ2 ≈ 2π × 3.7 MHz, this would place the device in the strong dispersive limit (2χm > Γ2 ).

4.2.3

Break-down of the qubit readout at large dc-voltage

However, further increasing Vdc is impeded by a breakdown of the qubit readout: the cavity
phase response becomes bi-stable at the charge degeneracy point for Vdc & 8 V. This bi-stability
can be observed in Fig. 4.8 using a parity measurement similar to that used in Fig. 4.2, but showsa 3rd stable parity position. From similar measurements, I observe that, while the qubit is in thebad parity position for approximately 50% of the time, the probability of finding the qubit in thegood parity position is reduced.
In Fig.4.9, I further investigate this bi-stability of the cavity phase by plotting the histogramof parity measurements as a function of ng . The 3rd parity position first appears at charge positions where the uncoupled qubit frequency approaches the cavity frequency. As I enhance either
the qubit-mechanics interaction by increasing Vdc (Fig.4.9(a)) or the qubit-cavity interaction byincreasing the cavity photon number hnc i (Fig.4.9(b)), I observe both a broadening of the region
over which the 3rd parity position persists, and a reduction in the probability of finding the goodparity. Thus, I speculate the bi-stability of the cavity phase arises because of a strong three-bodyinteraction that first arises when the frequency detuning between the qubit and the cavity comesin resonance with the mechanical frequency. This speculation is partly justified by Fig. 4.9(c),which shows the 3rd parity position does not appear when the qubit frequency is always sufficiently
larger than the cavity frequency. Unfortunately, the maximum EJ /~ of this device is approximately
2π × 5.1 GHz, too close to ωc to allow for dispersive qubit readout. Instead, I operate the qubit
below the cavity at EJ /~ ≈ 2π × 3.8 GHz.
In order to avoid observing this bi-stability of the cavity phase, for the reminder of this work,
I operate the device at Vdc = 6 V and measure ϕc with a small mean photon number hnc i ≈ 0.6.

90

Figure 4.8: Observing a 3rd parity position
At Vdc = 10 V and ng = 1/2, I perform a parity measurement similar to that in Fig. 4.2. In
addition to the usual parity positions, I observe a 3rd stable parity position at ϕc ≈ 5 deg. Similar
measurements show that the bad parity position still appears at approximately 50% of the time,demonstrating this 3rd position arise from a ϕc bi-stability at ng = 1/2.

91

Figure 4.9: Investigating the 3rd parity positiona, I plot the histogram of parity measurements (color-scale vs. y-axis) as a function of ng (x-axis)
for different values of Vdc . The cavity phase is measured at hnc i ≈ 0.6. At Vdc = 0 V, only twoparity positions can be observed. However, at Vdc = 3 V, the 3rd parity position first appears at
charge positions where the uncoupled qubit frequency approaches the cavity frequency. Further
increasing Vdc broadens the region over which the 3rd parity position is observed. At Vdc = 10 V,
the 3rd parity position can be observed even at ng = 1/2. b, Same as a, but keeping Vdc = 6 V whilechanging hnc i. Beyond hnc i ≈ 0.6, I start to observe a broadening of the region over which the 3rdparity position is observed, similar to the effect observed in a by increasing Vdc . c, At Vdc = 10 V
and hnc i ≈ 0.6, I plot the parity histogram while changing the minimum qubit frequency ωq = EJ /~using an external magnetic flux (see Sec.2.2.1 and Fig. 3.9(e)). When the minimum qubit frequency
is sufficiently larger than the cavity frequency (ωq ≈ 2π × 5.08 GHz), the 3rd parity position do notappear.

92
For clarity, I summarize the important performance parameters of this device when operated at
this Vdc in Table 4.1. Although 2χm ≈ 2π × 0.52 MHz is smaller than Γ2 , it is nevertheless asignificant fraction of it with Γ2 /2χm ≈ 7.1. Using this fact, I will show the extraction of thephonon population from the measurement qubit spectrum (Chapter 5) and the manipulation of asmall section of the phonon population with a characteristic width of approximately 7.1 phonons(Chapter 6) in the following chapters.
Parameter
qubit Josephson energy
qubit charging energy
qubit lifetime
qubit decoherence timemechanical frequency
mechanical thermal occupation
mechanical damping rate
cavity frequency
cavity loss rate
qubit-mechanics coupling rate
single-phonon Stark shift
qubit-cavity coupling rate

Value
EJ /~ ≈ 2π × 3.8 GHz
Ec /~ ≈ 2π × 2.9 GHz
T1 ≈ 0.26µs
T2 ≈ 0.08µs
ωm ≈ 2π × 25 MHz
nth = 13 ∼ 20
γm ≈ 2π × 93 Hz
ωc ≈ 2π × 4.76 GHzκ ≈ 2π × 2.3 MHz
gm ≈ 2π × 22 MHz
2χm ≈ 2π × 0.52 MHz (at ng = 1/2)
gc ≈ 2π × 37 MHz

Table 4.1: Essential performance parameters of the device at Vdc = 6 V

Chapter 5

Extracting phonon distribution from qubit spectrum

Being able to measure and understand the phonon distribution is the foundation to any meaningful manipulation of the phonon state. In this chapter, I discuss how the phonon distribution
can be reconstructed from the measured qubit spectrum. In Sec.5.1, I introduce the idea of using
deconvolution to extract the phonon distribution. However, this procedure requires an accurate|nidescription for the expected qubit spectrum Pe (ω) when there is exactly n phonons in the mechan|ni

ical oscillator. In Sec.5.2, I determine Pe (ω) by simulating the mechanical motion with a classicalmodulation on the qubit gate-charge. In Sec.5.3, I describe the Lucy-Richardson deconvolution
procedure used in this work. Finally in Sec.5.4, I validate both the experimentally determined|ni

Pe (ω) and the deconvolution procedure by extracting the phonon populations of thermal anddisplaced thermal states.

5.1

Understanding the mechanical state using the qubit spectrum
The qubit can be leveraged as a meter for measuring and extracting the phonon distribution

in the mechanical oscillator. This is achieved by measuring the qubit spectrum Pe (ω), whereby I
measure the probability of successfully driving the qubit to its excited-state as a function of thefrequency of a weak coherent drive. Recalling the discussion of Sec.2.3.4.3, a statistical mixture ofphonon Fock states in the mechanical oscillator will cause an overall qubit spectrum,
Pe (ω) =Xn

P (n) × Pe|ni (ω),

(5.1)

94
|ni

|ni

given by the convolution between the phonon distribution P (n) and Pe (ω). Because Pe (ω) isdifferent for different phonon numbers n, a deconvolution procedure can be used to invert thisequation and extract the phonon distribution P (n).
Clearly, an accurate extraction of the phonon distribution requires an accurate description|niof Pe (ω). However, this is not easily achieved because of the small mechanical frequency. In
contrast to the simple Lorentzian lineshape discussed in Sec.2.3.4.3, the small ωm significantly|nicomplicates Pe (ω) with sideband transitions. In addition to being close to the qubit frequency,
these sidebands can also be highly prominent even at a relatively small number of phonons n ∼ 100(because of the relatively large ratio of χm /ωm ≈ 0.01, see discussion around Eqn.(2.80)). Moreover,because of these sideband transitions, the measurement of the qubit spectrum is not a quantumnondemolition (QND) measurement of the phonon number. Instead, as I discuss in Sec.5.2.2,
measuring the qubit spectrum can alter the phonon number, further complicating the issue of|nifinding an accurate description of Pe (ω). Without the ability to prepare arbitrary phonon Fock|nistate (because χm < Γ∗2 ), I cannot determine Pe (ω) directly.

5.2

Simulating quantized motion with classical gate-charge modulation|niInstead, I experimentally determine Pe (ω) by using a classical modulation of the qubit
gate-charge at frequency ωm to simulate the effect of motion. Because the qubit-mechanical coupling originates from a oscillator-position dependent gate-charge ng (x) (Sec.2.3.1), the effect of the
motion, when treated classically, is identical to a classical gate-charge modulation in time,ng (t) =1+ δng + nx cos(ωm t),
2

(5.2)

where nx is the modulation amplitude, and δng is an offset from the charge degeneracy point. This
classical modulation is achieved by setting Vdc = 0 V and driving an ac-voltage on the dc-bias line,
which is weakly coupled to the qubit. Varying the amplitude of the ac-voltage, I measure the qubit
response as shown in Fig. 5.1. In this section, I demonstrate this measured qubit response to be inclose correspondence with the qubit response expected from quantized motion (Sec.5.2.3), allowing

95

Figure 5.1: Measured qubit response under classical classical gate-charge modulation
At Vdc = 0 V, I sweep the frequency of a weak qubit-excitation probe signal (y-axis) and measure
the cavity transmission phase ϕc (color-scale) as a function of the classical modulation amplitude
squared n2x (x-axis). The qubit excitations appear as dips in ϕc . In addition to a qubit frequency
shift that is linearly dependent on n2x , the gate-charge modulation at frequency ωm also causes
the appearance of sideband (l 6= 0) peaks. The background phase increases at larger nx because
the gate-charge modulation around ng = 1/2 rectifies the cavity phase response of Fig. 4.1 (seeSec.4.1.1). Data is not collected in the black region because no feature is expected.

96
|ni

me to extract Pe (ω) (Sec.5.2.5). This section (except Sec.5.2.2) closely follows publication [85].

5.2.1

Qubit spectrum under classical gate-charge modulation

To demonstrate this close correspondence, I first find the expected qubit spectrum for aclassical gate charge modulation according to Eqn.(5.2). I will show that, working around thecharge degeneracy point (δng = 0), the driven modulated-qubit can be viewed as a sum of many
two-level systems, each located at frequencies,
1(4Ec )2 2
ωx,2l (nx ) =EJ +nx + 2lωm
~
EJ

(5.3)

(l ∈ Z),= ωqb + δωqx (nx ) + 2lωm

and oscillates at rates given by the Bessel function of the first kind,Ωx,2l (nx ) = ΩR Jl

δωqx (nx )
2ωm


.

(5.4)

Consequently, the qubit spectrum in steady state is a sum of Lorentzians, each given by Eqn.(2.81).
To derive the above equations, I start by expanding the qubit frequency (Eqn.(2.24)) aroundsmall modulation amplitude,~ωq (t) =q
EJ2 + (4Ec )2 (1 − 2ng (t))2(4Ec )2 2
nx cos2 (ωm t) + O(n4x )EJ(4Ec )2 2 (4Ec )2 2
≈ EJ +
nx +nx cos(2ωm t).
EJ
EJ= EJ + 2(5.5)

Here, the first term corresponds to the bare (i.e., without modulation) qubit frequency at the charge
degeneracy point, ωqb = EJ /~. The second term is a static frequency shift of the qubitδωqx (nx ) = ωqx (nx ) − ωqb =(4Ec )2 2
n
~EJ x

(5.6)

that is proportional to the energy in the classical drive. This frequency shift is analogous to theac-Stark shift proportional to phonon numbers (Eqn.(2.56)). Finally, the last term in the above
equation is a frequency modulation of the qubit frequency at 2ωm .

97
Without decoherence, the time evolution of the modulated qubit under a coherent qubit driveis clear in the interaction picture. I begin with the bare Hamiltonian1
H0c = ~ωq (t)σ̂z ,2whose time evolution operator is





Z
i
i t c
c
H (τ )dτ= exp − φ(t)σ̂z ,Û0 (t) = T̂ exp −
~ 0 0
2

(5.7)

(5.8)

where the time ordering operator T̂ is dropped because the Hamiltonian commutes at any time([H0c (t1 ), H0c (t2 )] = 0), andZ thiδωqx (nx )φ(t) =
ωq (τ )dτ = ωqb + δωqx (nx ) t +sin(2ωm t).
2ωm
0

(5.9)

In the interaction picture, the coherent qubit drive of Eqn.(2.71) is transformed to an interaction
Hamiltonian,Vc (t) = Û0c † (t)Hd (t)Û0c (t)i[φ(t)−ωd t]−i[φ(t)−ωd t]= ~ΩR σ̂+ e+ σ̂− e(5.10)


∞X
i∆x,2l t−i∆x,2l t=~Ωx,2l (nx ) σ̂+ e+ σ̂− el=−∞where I have invoked the Jacobi-Anger expansion to find Ωx,2l (nx ) given by Eqn.(5.4), and∆x,2l = ωx,2l (nx ) − ωd .

(5.11)

Thus, when the qubit drive is resonant with this 2l-th transition frequency, the state of the qubit
will oscillate at rate Ωx,2l (nx ). Similar to Eqn.(2.73), these qubit transitions are separated by 2ωmbecause of symmetric gate-charge modulation.
Finally, the dissipation can be added to find the expected qubit spectrum [131, 132]. Fora weak qubit drive ΩR < ωm close to the 2l-th transition frequency ωx,2l (nx ), Eqn.(5.10) can be
simplified to contain only the near-resonant terms. Furthermore, because ωm  Γ2 , the qubit
spectrum from each resonance is easily resolvable and can be calculated separately. The overall
steady-state qubit spectrum is given by the sum,∞Penx (ωd ) =(Ax,2l Γ2 /2)2
1 X

,
2
2 (ω ) + (Γ /2)2 1 + A2∆2l=−∞
x,2l dx,2l(5.12)

98
wherer
Ax,2l = Ωx,2l (nx )

2
.
Γ1 Γ2

(5.13)

A comparison between the above equations and Eqn.(2.81) shows that the driven modulated qubit
resembles a sum over many effective two-level systems, each of which is located at frequency
ωx,2l (nx ) and oscillates at rate Ωx,2l (nx ).

5.2.2

Qubit spectrum for a phonon Fock state

In comparison to the classical case discussed above, I now calculate the qubit spectrum for
an initial Fock state |g, ni. Although this can in general be accomplished by solving the masterequation in Eqn.(2.64), it is extremely calculation intensive and provides little insight. Instead,
I consider three cases for different phonon-number-sensitivities. For simplicity, I will ignore themechanical relaxation process, γm = 0.
In addition to providing a basis of comparison, this calculation also highlights the fact thatmeasuring the qubit spectrum can alter the phonon number. For the rest of this thesis, I refer to thisphenomena of qubit-measurement-induced change in phonon number as measurement backaction.
This measurement backaction happens through either driving sideband transitions |g, ni ↔ |e, mior phonon-altering qubit decay processes |e, ni 7→ |g, mi (m 6= n). Because of the large ratioχm /ωm ≈ 0.01, the phonon-altering qubit decay can have significant impacts even at a small
phonon number (see Eqn.(2.67) and Sec.2.3.4.2). Consequently, in contrast to standard cQED
systems [47, 105, 106, 107], measuring only the qubit spectrum from phonon-conserving transitions
can still demolish the initial state (i.e., it is not a QND measurement of the phonon number).

5.2.2.1

phonon-number-resolvedFirst, I consider the qubit-mechanical system in the phonon-number-resolved regime (2χm >Γ2 ). In this limit, a weak qubit drive close to frequency
ωn,2l = ωq|ni + 2lωm

(5.14)

99
can only excite the transition |g, ni ↔ |e, n + 2li with rate
Ωn,2l = ΩR |αn+2l,n |,(5.15)

but not any other transitions |g, mi ↔ |e, m + 2li for m 6= n. However, the decaying qubit can alter
the phonon number to create a final state inaccessible to the qubit drive. These driven dynamics canbe modeled by a three-level system as shown in Fig. 5.2(a), where |darki represents all states |g, mi(m 6= n) that are inaccessible to the qubit drive. According to Sec.2.3.4.1, state |e, n + 2li decays to|g, ni at rate Γ1 |αn+2l,n |2 and decays to |darki at rate Γ1 (1−|αn+2l,n |2 ). The master equation of thisthree-level system can be solved numerically for different values of |αn+2l,n |2 . Driven at frequency
ωn2l , the time evolution of the probability P|e,n+2li for finding state |e, n + 2li and probability Pdark
for finding state |darki is plotted in Fig. 5.3.
Inspired by the discussion above (Sec.5.2.1), I compare these dynamics to the driven responseof a two-level system located at frequency ωn,2l , which oscillates between states |↑i and |↓i at
rate Ωn,2l (Fig. 5.2(b)). Depending on the strength of Ωn,2l , the evolution of P|e,n+2li is wellapproximated by the evolution of P↑ over a time scale of a few T1 . However, this approximation
fails at larger times as the population slowly accumulates in the dark state |darki, and becomesinaccessible to the qubit drive. Thus, my attempt to measure the initial state |g, ni using qubit
spectroscopy ends up driving it to a different state |g, mi. This measurement backaction preventsthe measurement from being QND even in the case of driving phonon-conserving transitions (l = 0)
when |αn,n |2 6= 1.
Nevertheless, at a drive time Td > T1 but before the effect of measurement backaction becomes
clear (Pdark  1), the qubit spectrum around frequencies ωn,2l is well approximated by the steadystate spectrum of the effective two-level system. Generalizing this result to all l, the overall qubit|nispectrum Pe (ωd ) of the initial state |g, ni can be approximated by a sum over steady-state qubit
spectra of the effective two-level systems∞

Pe|ni (ωd ) ≈(An,2l Γ2 /2)2
1 X

,
2
2
2 1 + A2l=−∞ ∆n,2l (ωd ) + (Γ2 /2)n,2l(5.16)

100

Figure 5.2: Driving a phonon-number-resolved systema, In the phonon-number-resolved regime, the driven qubit-mechanical system can be modelled by|nia three-level system. A qubit drive close to frequency ωn,2l = ωq + 2lωm can only excite transition|g, ni ↔ |e, n + 2li at rate ΩR |αn+2l,n |. Thus, all other states |g, mi with m 6= n appear as darkstates to the qubit drive, and are labeled as |darki. When the qubit decays from |e, n + 2li, it can
2either original state |g, ni at rate Γ1 |αn+2l,n | , or to the dark states |darki at rate
 decay to theΓ1 1 − |αn+2l,n |2 . b, It can be shown that at small driving time Td such that the population
Pdark in the dark state is small, the three-level system in a can be approximated by an effective
two-level system at frequency ωn,2l . In this effective system, the drive excites transitions |↓i ↔ |↑iat rate ΩR |αn+2l,n |, and |↑i decays to |↓i at rate Γ1 . For both a and b, the qubit also experiences
a dephasing at rate Γ2 /2, which is not shown in the figure.

101

Figure 5.3: Dynamics in the phonon-number-resolved regimea, Under a resonant drive ∆n,2l = ωn,2l − ωd = 0, I solve the master equation for both driven
dissipative systems in Fig. 5.2 assuming an initial state |g, ni and the following parameters: ΩR =0.05Γ1 , Γ2 = 3Γ1 . For different values of |αn+2l,n | (different colors), I plot the time evolution of thestates |e, n + 2li (top panel, solid lines), |↑i (top panel, dashed lines), and |darki (bottom panel,solid lines). The driving time Td is normalized to the qubit lifetime T1 = 1/Γ1 . Up to Td = 40T1 ,very little population escapes to the dark state at this weak drive strength, and the dynamics of thequbit-mechanical system is well approximated by the simple effective two-level system of Fig. 5.2(b).
b, Solving the master equation of the qubit-mechanical system at all values of driving frequency,
I find the qubit spectrum P|e,n+2li (ωd ) after different driving times Td = 10T1 (blue), Td = 20T1
(mustard), Td = 40T1 (red). Here, I assume |αn+2l,n |2 = 0.25, and normalize the drive detuning∆n,2l to the qubit spectrum’s minimum FWHM Γ2 (see Sec.2.3.4.3). For reference, I also plot thesteady-state spectrum for the effective two-level system (black) according to Eqn.(5.16). Indeed,because the qubit-mechanical system has experienced little measurement backaction (i.e. Pdark issmall) in these cases, the qubit spectrum is well approximated by Eqn.(5.16). Similar calculations
can be repeated for a larger drive strength ΩR = 0.25Γ1 , and the resulting time evolution (c)and qubit spectrum (d) are plotted. Because the qubit-mechanical system experiences significant
backaction, the resulting qubit spectrum is not well approximated by Eqn.(5.16), with worse qualityof the approximation at larger measurement backaction (i.e., larger Td ).

102
where
r

2
,
Γ1 Γ2

(5.17)∆n,2l (ωd ) = ωn,2l − ωd .

(5.18)

An,2l = Ωn,2l
and

5.2.2.2

phonon-number-sensitive

To further demonstrate the effect of measurement backaction, I now consider a phononnumber-sensitive qubit-mechanical system, where a qubit drive can address the same type of transitions over a section of the phonon population. As an example, I start from an initial state |g, ni
and consider the effect of a qubit drive of strength ΩR close to its two-phonon blue sideband transition |g, ni ↔ |e, n + 2i in the hypothetical scenario shown in Fig. 5.4. When it is resonant withfrequency ωn,2 , the qubit drive will excite the transition with rate Ωn,2 = ΩR |αn+2,n |. Furthermore,
the qubit drive can also excite transition |g, n + 2i ↔ |e, n + 4i at frequency ωn+2,2 = ωn,2 + 4χmwith transition rate Ωn+2,2 = ΩR |αn+4,n+2 |. All other states |g, mi with m 6= {n, n + 2} are inaccessible to the qubit drive, and labeled as |darki. A decaying qubit connects state |e, n + 2i to |g, ni at
rate Γ1 |αn+2,n |2 and to |g, n + 2i at rate Γ1 (1 − |αn+2,n |2 ). Similarly, decays |e, n + 4i 7→ |g, n + 2i
happens at rate Γ1 |αn+4,n+2 |2 and |e, n + 4i 7→ |darki happens at rate Γ1 (1 − |αn+4,n+2 |2 ). I solve
the master equation of this driven dynamics numerically. Approximating |αn+2,n | ≈ |αn+4,n+2 |, I
plot the resulting qubit spectrum at different Td in Fig. 5.5.
When the detuning between the two sideband transitions ωn+2,2 − ωn,2 = 4χm is larger than
Γ2 , |g, n + 2i becomes effectively part of the dark state |darki , and I recover the phonon-numberresolved regime as discussed above. This identifies a characteristic width
W = Γ2 /2χm[phonons](5.19)

over which the qubit drive is effective. In other words, a qubit drive at frequency ωn,2l can excite thetransition |g, mi ↔ |e, m + 2li effectively for |m − n| < W/2, but does little for transitions further

103

Figure 5.4: Driving blue sideband transitions in a phonon-number-sensitive system
Starting from an initial state |g, ni, I consider a qubit drive that can only access two transitions. At
frequency ωn,2 , the qubit drive can excite transition |g, ni ↔ |e, n + 2i at rate ΩR |αn+2,n |; and at
frequency ωn+2,2 = ωn,2 + 4χm , the qubit drive can excite transition |g, n + 2i ↔ |e, n + 4i at rate
ΩR |αn+4,n+2 |. All other states appear as dark states to the qubit drive and are labeled as |darki.
For simplicity, I assume |e, n + 2i can only decay to states |g, ni and |g, n + 2i at rates Γ1 |αn+2,n |2and Γ1 (1 − |αn+2,n |2 ) respectively. Similarly, state |e, n + 4i only decays to either |g, n + 2i at rate
Γ1 |αn+4,n+2 |2 or |darki at rate Γ1 (1 − |αn+4,n+2 |2 ). All states also experiences a qubit dephasingat rate Γ2 /2, which is not plotted. I also assume no mechanical relaxation.

104

Figure 5.5: Sideband spectrum in a phonon-number-sensitive systema, I solve the master equation for the driven dissipative system in Fig. 5.4 assuming the following
parameters: |αn+2,n |2 = |αn+4,n+2 | = 0.25, ΩR = 0.25Γ1 , Γ2 = 3Γ1 , and 4χm = 2Γ2 . Normalizing
the drive detunning (∆n,2 = ωn,2 − ωd ) to the minimum qubit FWHM Γ2 , I then plot the expected
overall qubit spectrum (solid lines) after different values of drive time, Td = 10T1 (blue), Td =20T1 (mustard), Td = 40T1 (red). For comparison, I also plot the expected spectrum according
to Eqn.(5.1) (black line). The overall qubit spectrum Pe (ωd ) has two contributions P|e,n+2i (ωd )(dashed lines) and P|e,n+4i (ωd ) (dotted lines). At this small W = Γ2 /2χm = 1, the qubit drive
has little effect on the population in |g, n + 2i, which appear as if it were part of the dark state|darki. Consequently, the resulting spectrum resembles that found in the phonon-number-resolved
regime (Fig. 5.3(d)). I repeat the same calculation for different values of W = Γ2 /2χm and plot theresulting qubit spectrum: W = 2 (b), W = 4 (c), and W = 20 (d). In these cases, the qubit drive
can excite transition |g, n + 2i ↔ |e, n + 4i effectively, and the measurement backaction appear asa broadening of the overall qubit spectrum accompanied by a shift toward higher frequency.

105
away. In the next chapter, I will explore the effects of such phonon-number-sensitive sideband
drives in more detail.
On the other hand for 4χm < Γ2 , the qubit drive can simultaneously excite both sideband
transitions. At small Td , the population is mostly confined in the sub-space {|g, ni , |e, n + 2i}, andthe overall qubit spectrum (solid lines) is dominated by the probability of finding state |e, n + 2i
(dashed lines). However, at larger Td , the population slowly escapes the sub-space through the
decaying qubit, and can be further excited to state |e, n + 4i (dotted lines). Consequently, because
the measurement increases the phonon number and χm > 0, the overall qubit spectrum broadensand shifts toward a higher frequency. Conversely, if the measurement causes a reduction of thephonon number by driving the red sideband transitions, the overall qubit spectrum will shift towarda lower frequency. These effects are experimentally observed in Sec.7.1.1, and are used to quantify
the amount of measurement backaction the qubit-mechanical system has experienced. Nevertheless,before such effects become prominent, the overall qubit spectrum can be well approximated by
Eqn.(5.16).

5.2.2.3

phonon-number-insensitive

Finally, I consider a qubit-mechanical system that is entirely phonon-number-insensitive(W → ∞), such that a qubit drive does not differentiate between different phonon numbers. Thiscan be understood as an extremely weak coupling between the qubit and the mechanical oscillator.
In this limit, the motion of the mechanical oscillator can be treated classically as was done inSec.5.2.1, and Eqn.(5.16) becomes exact. This trend can also be observed in Fig. 5.5 by imagining
the effect of reducing χm .

5.2.3

Classical vs. quantum

From the above discussions, I find the qubit spectrum for a phonon Fock state can be wellapproximated by the qubit spectrum (Eqn.(5.12)) under a classical gate-charge modulation when
two conditions are simultaneously satisfied:106• The spectroscopic drive time Td needs to be sufficiently short such that the effect of measurement backaction is negligible.
• Under the same resonant condition ωn,2l = ωx,2l , the effective transition rates should bealmost equalΩn,2l ≈ Ωx,2l ,(5.20)

such that the difference between the spectra according to Eqn.(5.12) and Eqn.(5.16) is
negligible.
In Sec.7.1.1, I will experimentally determine the appropriate Td to satisfy the first condition. Here,I demonstrate the second condition can also be satisfied given the achieved χm /ωm ≈ 0.01 in thiswork.
In general, the effective transition rates in a quantum system are not the same as thoseunder a classical gate-charge modulation. Specifically, under a classical modulation, the transition
rates are symmetric around the center qubit peak, Ωx,2l = Ωx,−2l (Eqn.(5.4)). With quantizedmotion, however, the transition rates are asymmetric around the phonon-conserving transitionl = 0. A simple example is the case of an initial state |g, 0i. While phonons can be added into
the mechanical oscillator through blue sideband transitions, no phonon can be extracted from themechanical ground state with a red sideband transition.
In this work, however, the qubit response to quantized motion is well approximated by thequbit response to classical modulation. When the sideband transitions become prominent featuresin the qubit spectrum at n & 100, the asymmetry between the blue and red sideband transitions issmall enough to be neglected. Conversely, when the asymmetry is strong at small phonon numbers,
the qubit spectrum is dominated by the phonon-conserving transition and the sidebands can be
all together ignored. To provide an intuition for this, I calculate the asymmetry by expanding the
sideband rate Ωn,±2 connecting states |g, ni and |g, n ± 2i to first order in χm /ωm (Eqn.(2.78)),sΩn,−2n(n − 1)2
1≈
≈ 1 − + O( 2 ).
(5.21)Ωn,2(n + 1)(n + 2)
n
n

107

Figure 5.6: Transition rates under classical modulation vs. quantized motion
The solid lines correspond to transition rates squared |Ωx,2l |2 as functions of the ng modulation
amplitude squared n2x (bottom-axis) according to Eqn.(5.4). The dots correspond to transition
rates squared |Ωn,2l |2 for a given initial phonon number n (top-axis)
 2 according to Eqn.(2.78). For agiven ratio of χm /ωm a, χm /ωm = 0.01 and b, χm /ωm = 0.2 , nx is related to n by enforcing thesame qubit-frequency shift according to Eqn.(5.22). The quality of approximation using classicalgate-charge modulation degrades for larger χm . This figure is reproduced from reference [85].

108
At n = 100 phonons, this corresponds to a ∼ 2% asymmetry between sideband rates and a ∼ 1%
deviation from the classical case. Indeed a negligible amount compared to the measurement noisein the qubit spectrum.
Going beyond the intuition, a more rigorous numerical comparison can be made between
the transition rates under classical and quantum modulation using Eqn.(5.4) and Eqn.(2.78). To
perform this comparison, I first calibrate the charge modulation amplitude to units of motional|niquanta such that they shift the qubit-frequency by the same amount (ωq = ωqx (nx )),n+1(4Ec )2 2=n ,2
2χm EJ x(5.22)

where n is the motional quanta, and nx is the modulation amplitude of gate-charge. In thisequation, the 1/2 term on the left-hand side corresponds to the qubit lamb-shift [133] inducedby the mechanical vacuum state, but is absent from the right-hand side because the classicalgate-charge modulation is performed when the qubit is decoupled from the mechanical oscillator.
Fig. 5.6 shows numerical comparisons of the transition rates under classical and quantum theory.
Although different for large ratios of χm /ωm (Fig. 5.6(b)), for this work with χm /ωm ≈ 0.01,
Fig. 5.6(a) demonstrates the simulation with classical modulation to be a good approximation for
qubit response to quantized motion in the mechanical oscillator.

5.2.4

Effects of charge noise

So far, the discussion assumes the qubit is always operated at the charge degeneracy point.
However, this is not always possible in the presence of charge noise, which causes deviation δng in
the gate-charge away from the degeneracy point. Indeed, in Fig. 5.1, I observe sideband transition
peaks separated by frequency ωm instead of 2ωm , suggesting deviation from the sweet-spot. In
this section, I explore the effects of a 1/f charge noise on the qubit spectrum, and show theclassical gate-modulation remains a faithful simulation of quantized mechanical motion in thisexperiment (Sec.5.2.4.2). Using the asymmetry in the qubit spectrum at Vdc = 0 V and nx = 0, I
extract a charge noise intensity consistent with typically reported values in Sec.5.2.4.1. Based on

109
this intensity, I theoretically calculate the qubit spectrum in Sec.5.2.4.3 for arbitrary modulation
amplitudes nx . The good correspondence between this calculation and the measured qubit response
demonstrates that the odd order sideband transitions observed in Fig. 5.1 are indeed caused bycharge-noise.
In this section, I model charge noise as a sum of sinusoidal signals at equally-spaced frequencies ωi with random phases φi uniformly distributed within range [0, 2π] [134],X

ne (ωi ) cos (ωi t + φi ).

(5.23)i

The noise amplitude ne (ωi ) is found by integrating the single-sided noise-power spectral density
S2e (ω) over a frequency range centered at ωi ,Z ωi +δωne (ωi ) =dωp
2S2e (ω),(5.24)

ωi −δω

where 2δω = ωi+1 − ωi . Assuming a 1/f noise distribution,
S2e (ω) = ζ 2

2π × 10,ω(5.25)√
where ζ is the noise intensity at 10 Hz in units of Cooper-pairs (2e/ Hz). Thus, the overall timedependence of the gate-charge centered around the degeneracy point is
ng (t) =X
1+ nx cos (ωm t) +ne (ωi ) cos (ωi t + φi ).
2(5.26)i

5.2.4.1

Asymmetric lineshape and charge noise intensity

To explore the effect of this charge noise, I start by finding the noise intensity ζ. This canbe achieved by looking at the qubit lineshape at nx = 0. Because charge noise is dominatedby components at low frequencies and the qubit spectrum is averaged over many experimental
realizations, charge noise result in an asymmetric qubit lineshape (Fig. 5.7). This phenomenon isbest understood phenomenologically by treating the incoherent charge noise of Eqn.(5.23) as a setof stationary charge offsets {δng } away from ng = 1/2. Assuming a Gaussian process for this offset
with mean hδng i = 0 and standard deviation σc , the averaged qubit spectrum with nx = 0 is a sum

110

Figure 5.7: Noise-induced asymmetric qubit lineshape
Using Eqn.(4.4), I convert the phase response at nx = 0 in Fig.5.1 to a qubit spectrum (greendots). Because of the highly averaged measurement protocol, the 1/f charge noise manifests as an
asymmetric qubit lineshape. Modelling the charge noise as a Gaussian random variable, I fit (red)the qubit lineshape with Eqn.(5.27) to find the standard deviation in gate-charge, σc = 0.0071(2e). Alternatively, I fit the qubit lineshape with skewed Lorentzians (blue) without assumptionsof charge noise distribution according to Eqn.(5.41). I use this skewed Lorentzian fit to process themeasured qubit response in Sec.5.2.5. This figure is reproduced from reference [85].

111
of Lorentzians weighted by the corresponding offset probability,"#Zδn2g 
1(AΓncn /2)2p
Pe (ω) =exp − 2dδng ,
2
2σc
2πσc2
dδng
ω − ωqb (δng ) + (Γncn /2)2 (1 + A2 )(5.27)q
where Γncn /2 is the qubit dephasing rate without charge noise, A = ΩR 2/ Γ1 Γncn is the unitq
less parameter describing the transition rate, and ωqb (δng ) = EJ2 + (8Ec δng )2 is the bare qubit
resonance given an offset δng .
Using the above equation, I fit the qubit lineshape in Fig. 5.7 with A, EJ and σc being freeparameters to find a charge-offset uncertainty σc = 0.0071(2e). Given the measurement protocol,√
this σc value corresponds to [135, 134] a 1/f charge noise intensity of ζ/2 ≈ 1.03 × 10−3 e/ Hz at√
10 Hz, consistent with the typical charge noise intensity of 10−3 − 10−4 e/ Hz at 10 Hz [125].

5.2.4.2

Classical gate-charge modulation with charge noiseThis charge noise has two effects on the qubit spectrum when the gate-charge is classicallymodulated: while charge noise with frequency components much less than ωm contributes to an
asymmetric qubit lineshape and cause sideband peaks at odd-orders of ωm , charge noise at ωmcause the classical modulation to become noisy. Nevertheless, the classical gate-charge modulation
remains a faithful simulation of quantized mechanical motion in the presence of these effects.
To see this, I start by writing the overall time dependence of the gate-charge. Instead ofEqn.(5.26), I phenomenologically treat the charge noise with frequency components much less thanωm as a noisy offset charge δng . Keeping only the charge noise at frequency ωm , the time dependenceof the modulated gate-charge is,1+ δng + nx cos ωm t + nm cos (ωm t + φm )2
1
= + δng + nn cos (ωm t − φn ),2ng (t) =(5.28)

where nm = ne (ωm ) and the noisy phase φm is uniformly distributed within range [0, 2π]. Thischarge noise at ωm thus transforms the noiseless nx modulation into a modulation with a noisy
amplitudenn =p
n2x + n2m + 2nx nm cos φm ,(5.29)

112

Figure 5.8: A noisy gate-charge modulation
To clarify Eqn.(5.28), I plot the phase-space picture of the gate-charge modulation. The projection
on the Re-axis is the instantaneous gate-charge value ng (t). Charge noise with frequency components much lower than ωm result in a offset gate charge δng . The noiseless modulation (blue)of amplitude nx rotates around ng = 0.5 + δng at frequency ωm . The charge noise at frequency
ωt extm (red) has noise amplitude nm , and is offset from the noiseless modulation at a noisy angle
φm uniformly distributed between [0, 2π] (red dashed circle). The net result is therefore an overall
modulation (black) with noisy amplitude nn , and a noisy offset angle −φn .

113
and a noisy phaseφn = arctan

nm sin φm,
nx + nm cos φm(5.30)as shown in Fig. 5.8. Thus, nn and φn are random variables found by treating φm as a randomvariable distributed uniformly within [0, 2π].
Similar to Sec.5.2.1, the gate-charge modulation of Eqn.(5.28) causes the qubit frequency toshift by an amount proportional to the modulation power n2n ,24Ec EJ /~223 nn ,
ωqb (δng )(5.31)q
EJ2 + (8Ec δng )2(5.32)

δωqx (nn , δng ) =whereωqb (δng ) =1~

is the bare qubit frequency. Additionally, for a particular δng 6= 0, the gate-charge modulation is
not symmetric around the charge degeneracy point. Consequently, sideband transitions appear at
odd intervals of ωm ,
ωx,l (nn , δng ) = ωqb (δng ) + δωqx (nn , δng ) + lωm(l ∈ Z),(5.33)

consistent with what I observe in Fig. 5.1. A resonant qubit drive of strength ΩR at frequency ωx,lwill drive the corresponding transition at rateΩx,l = ΩRX
2a+b=l δωqx (δng , nn )(8Ec /~)2Ja
Jb
nn δng .
2ωmωm ωqb (δng )

(5.34)To show the classical gate-charge modulation remains a faithful simulation of quantizedmechanical motion in the presence of charge noise, I first show the relative uncertainty in themodulation amplitude of Eqn.(5.29) to be negligible. Using the charge noise intensity ζ foundabove, I integrate S2e (ω) over frequency range [ωm −γm /2, ωm +γm /2] to find nm ≈ 4.02×10−6 (2e).
For a sense of scale, this noise amplitude can be compared to the modulation amplitude n0x ≈|0i

2.7×10−3 (2e) corresponding to the mechanical zero-point motion, found by requiring ωqx (nx ) = ωqaccording to Eqn.(5.22). Indeed, nm is much less than the amplitude of any meaningful gate-charge

114
modulation. Expanding Eqn.(5.29) to first order in nm ,nn = nx + nm cos φm + O(n2m ),(5.35)

I find a relative uncertainty in the modulation amplitudeppvar(nn )nn var (cos φm )≤ 0.001.
=
hnn i
nx

(5.36)

Finally, I compare Eqn.(5.34) to the transition rates Ωn,l identified in Sec.2.3.5 under thesame gate-charge offset and the same static qubit frequency. A similar analysis has been employedin the previous section (Sec.5.2.3) to demonstrate that, when χm  ωm , the classical gate-charge
modulation is a good approximation to the quantized motion at δng = 0. The same is also true forreasonable values of δng 6= 0: neglecting the noise in the modulation amplitude nn = nx and using
the parameter of this work χm /ωm ≈ 0.01, I find the transition rates well approximate each otherfor values of δng within ±3σc ≈ ±0.02 (2e) around zero. I thus conclude the classical gate-charge
modulation to remain a good approximation to the quantized motion for this experiment even in
the presence of charge noise.

5.2.4.3

Expected qubit spectrum due to charge noise

To attribute the appearance of odd-order sideband transitions in Fig. 5.1 to charge noise, I
provide a model that predicts the overall qubit spectrum under the influence of charge noise. Thisis given by a convolution between the probability of finding a particular charge offset δng and its
δncorresponding qubit spectrum Pe g (nx , ω),


δng 2
δn
p
Pe (nx , ω) =exp − 2 Pe g (nx , ω)dδng .
2
2σ
2πσ
dδng
c
c
Z

1

(5.37)

In turn, the qubit spectrum under a specific δng is given by a sum of Lorentzians,δn

Pe g (nx , ω) =(Ax,l Γncn /2)2,2 (ω − ωx,l )2 + (Γncn /2)2 (1 + A2x,l )X1
l(5.38)

115
each located at frequencies ωx,l given by Eqn.(5.33), and driven at a rate that corresponds to,r
2
Ax,l = Ωx,l
Γ Γ
r 1 ncn x (5.39)Xδωq (δng , nx )2(8Ec /~)2
= ΩR
Ja
Jb
nx δng .
Γ1 Γncn2ωmωm ωqb (δng )2a+b=l

Because the qubit drive is applied through the readout cavity at a constant amplitude d , the Rabi
rate is also dependent on the spectroscopic frequency ωΩR (ω) = 2gcd ,ω − ωc(5.40)

where gc is the qubit-cavity coupling rate and ωc is the cavity frequency. Using the extracted
values of Γncn and σc from Fig. 5.7, I plot the numerical result of Eqn.(5.37) in Fig. 5.9(c). In
this figure, only d is left free to match the maximum peak-height in the map of Fig. 5.9(b). The
good agreement between the experiment (b) and theory (c) confirms the odd-order sideband peaks
I observe in Fig. 5.1 can be explained by 1/f charge noise.

5.2.5

Finding the qubit spectrum for phonon Fock states

Having established the qubit spectrum under a classical modulation to be in close correspondence with the qubit spectrum for phonon Fock states, I now convert the experimentally measured
qubit response under classical modulation (Fig. 5.1, reproduced in Fig. 5.9(a)) to the PSF map|ni

Pe (ω) in two steps:(1) I connect the measured cavity transmission phase to qubit excitation probability Pe (ω)
using Eqn.4.4. Instead of directly using the measured phase which contains statistical noise, I instead fit the measured spectrum using a sum of skewed Lorentzians where each Lorentzian describes
one sideband peak or the center qubit peak:ϕc (ω) =(Al Γl /2)2
2
,
2
2
ω−ωl
2
+
(Γ
/2)
1
+
A
l
l
1+L·Sgn(ω−ωl )

X1



l

(5.41)

where ωl is the resonances of the system under the spectroscopic drive, Al describes the corresponding transition rate |Ωx,l |2 , and L captures the asymmetry due to charge noise. In this fit,

116
we also remove the background phase present in Fig. 5.9(a), which increases at larger modulation
amplitude. This background phase arises from time-averaging the cavity phase response of Fig. 4.1
to a classical gate-charge modulation around ng = 1/2 (see Sec.4.2.2).
(2) I find the qubit spectrum at integer phonon numbers by interpolating the fit parameters
extracted from step (1).
I thus create a new map, as shown in Fig. 5.9(b), which I take as the qubit response tomechanical Fock states. In the rest of this chapter, I demonstrate how to use this map in performing
deconvolution procedures and extracting phonon distributions from measured qubit spectra.
|niFinally, two details need to be considered to properly use this extracted Pe (ω). First,
because the qubit lineshape is strongly dependent on the qubit-drive power (Sec.2.3.4.3), I ensure
this power to remain constant for all measurements. Second, I observe the bare qubit frequency
to vary over time (δωqb ∼ 2π × 1 MHz) and between experiments performed at Vdc = 0 V and
Vdc = 6 V (δωqb ∼ 2π × 3 MHz)). I attribute this to a combination of change in the local flux andchange in the qubit’s charging energy (when Vdc is changed). To correct for this error, I extract thebare qubit frequency by measuring the qubit spectrum with the mechanical oscillator in thermal
states before each experiment of interest, and performing a least-squares fit (see Sec.5.4).

5.3

The reconstruction procedure|ni

Having determined Pe (ω), I now discuss the reconstruction procedure of extracting thephonon distribution from measured qubit spectrum. This is achieved using the Lucy-Richardson
deconvolution algorithm [136, 137] discussed in Sec.5.3.1. One way to characterize the uncertainty
in the extracted phonon distribution is by using a bootstrapping method, described in Sec.5.3.2.

5.3.1

Reconstruction using Lucy-Richardson deconvolution

The Lucy-Richardson algorithm is an iterative, Bayesian-based deconvolution algorithm. Often used in astronomy, the algorithm is originally aimed at reconstructing images degraded by
optical aberrations, which spreads a point-source over a wide space according to a point-spread

117

|ni

Figure 5.9: Finding Pe (ω)
a, This is identical to Fig. 5.1, reproduced here for easy comparison. b, I convert the measured
phase response in a to the qubit spectra (color-scale vs. y-axis) at particular mechanical Fock|nistates (x-axis) and find Pe (ω). The cavity transmission phase is fitted (Eqn.5.41), converted tothe probability of exciting the qubit, and interpolated at appropriate qubit Stark shifts (Eqn.2.56).
I use this map to extract phonon distributions. c, Using Eqn.(5.37), I calculate the expected qubit
spectra (color-scale vs. y-axis) under a classical gate-charge modulation. The extracted phonon
number n (x-axis) is related to the classical modulation amplitude nx according to Eqn.(5.22).
To model the charge-noise, a Gaussian distribution with mean hδng i = 0 and standard deviation
σc = 0.0071(2e) is used. The drive strength of the spectroscopic-signal d is a free parameter to makethe maximum Pe found in this calculation to be equal to the maximum Pe found experimentallyin b. The good agreement between b and c confirms the charge-noise as the main source of offset
charge. This figure is reproduced from reference [85].

118
function (PSF). Analogously in this work, detecting a phonon Fock state |ni (point-source) with|ni

the qubit results in an image Pe (ω) spread over a wide range of frequencies. Thus, the PSF is|nigiven by the experimentally determined Pe (ω).
To invert Eqn.(5.1), I use the Lucy-Richardson recurrence formula,
Pi+1 (n) = Pi (n)Xω

Pe|ni (ω) P

Pe (ω)|n0 i
0n0 Pi (n )Pe (ω),(5.42)

where Pi (n) is the deconvolution result after i number of iterations. This process requires an initial
guess for the phonon distribution P0 (n), which, for most of this work10 , is chosen as P0 (n) = 1/N
where N = 200 is the maximum phonon number at which point the Fock space is truncated. This
choice is identical to a Bayes’s postulate [137].
The quality of the deconvolution is strongly influenced by the total number of iterations. In
general, a larger iteration number i means the algorithm will capture finer details in the measured
qubit spectrum. For a noiseless measurement, the extracted phonon distribution will converge to thetrue phonon distribution at large i. However, in the presence of measurement noise, capturing too
many details in the qubit spectrum could mean interpreting measurement noise as actual phonon
distribution (i.e., over fitting). To show this, I simulate an experiment (Fig. 5.10) and find the noisy
qubit spectrum from a thermal mechanical state with nth = 20 by first using Eqn.(5.1) to find thenoiseless qubit spectrum, and then adding noise (see Sec.7.1.3). Terminating the deconvolution
procedure at different iteration numbers, I confirm the dependence of its quality on the iteration
number. While at small i, the procedure hasn’t been repeated enough to deviate far from theinitial guess; at large i, it attempts to capture the measurement noise by introducing noise in theextracted phonon distribution. Because I know the exact phonon distribution in this case, I canquantify the deconvolution quality using the rms-error (RMSEtheory ) between its result and thetrue phonon distribution,s
RMSEtheory (i) =10

1 X[Pi (n) − Pth (nth , n)]2 ,N n(5.43)

see Sec.7.1.2 for the discussion about the special case where an initial guess of Poissonian distribution is used

119
where Pth (nth , n) is the mechanical thermal distribution given by Eqn.(2.9) and nth = 20. As shownin Fig. 5.11(a), the quality of the deconvolution is optimal when RMSEtheory is minimized at the
optimal iteration number iopt = 13. However, iopt cannot be determined universally because it isdifferent for different phonon distributions or for different noise realizations in the qubit spectrum.
Instead, I need to find a consistent protocol to determine a suitable iteration number iterm toterminate the deconvolution procedure without prior knowledge of the true phonon distribution.

Figure 5.10: Reconstruction vs. iteration number
Assuming a thermal mechanical state (b, dashed black line), I synthesize its qubit spectrum (a,green dots) by adding noise to the expected (Eqn.(5.1)) noiseless qubit spectrum (black dashedline). Running the Lucy-Richardson algorithm for i iterations, I extract the phonon distribution(b, solid line) and find its 90% confidence interval (shaded) using non-parametric bootstrapping.
The expected qubit spectra (solid lines) from the extracted phonon distributions are plotted in afor comparison. Also plotted is the initial guess (grey) which is a uniform distribution between 0
and 200 phonons.

To this end, I look at the algorithm’s rate of update. First, I look at the rate of update
between adjacent iterations, which I quantify usings
RMSEnext (i) =1 X[Pi+1 (n) − Pi (n)]2 .
N n(5.44)As shown in Fig. 5.11(a), after a period of rapid change, the rate of update slows down significantly
around i ∼ iopt . Thus, the rate of update can be a good proxy for determining iterm . Additionally,
because the rate of update is small around iopt , a small deviation in iterm from the optimal point will

120

Figure 5.11: Finding iterma, Using the same synthesized data shown in Fig. 5.10 and the known thermal distribution with
nth = 20, I determine iopt = 13 by finding the minimum of RMSEtheory (i) (orange). I find the rateof update RMSEnext (i) (black) strongly depends on the iteration number i, and quickly diminishes
when i ∼ iopt . By minimizing the function RMSEdouble (i) (blue), I determine the iteration number
iterm = 14 with which I should terminate the reconstruction. Similarly, I can find iterm = 12 byminimizing RMSEtriple (i) (mustard), and iterm = 10 by minimizing RMSEquadruple (i) (red). b,c, With the iterm and iopt determined in a, I compare their effects on the reconstructed phonon
distributions (c) and the corresponding expected qubit spectra (b). Indeed, because the rate of
update is small at these iterations numbers, the small deviations among them only cause negligible
changes in the extracted phonon distribution.

121

Figure 5.12: Histogram of RSMEtheory
Using a thermal mechanical distribution with nth = 20, I create 105 different synthesized qubit
spectra with different noise realizations. On each synthesized qubit spectrum, I find iterm byminimizing RMSEdouble (i). Using this iterm , I then extract the phonon distribution and compute its
rms-error RSMEtheory (iterm ) with respect to the true distribution. From its histogram (blue), whichlargely overlaps with the histogram of the best achievable rms-error RSMEtheory (iopt ) (orange), I
confirm this method of extracting iterm results in extracted phonon distributions close to the truth.
Different ways of finding iterm such as minimizing RMSEtrip (i) (mustard) or RMSEquad (i) (red)also extract phonon distributions only slightly worse than what can be optimally extracted.

122
not significantly alter the extracted phonon distribution. Because RMSEnext (i) decreases monotonically, I instead determine iterm by finding the minimum of the rate of update when doubling
the iteration numbers
RMSEdouble (i) =1 X[P2i (n) − Pi (n)]2 .
N n(5.45)

Although a similar method can be employed by tripling (RMSEtrip (i)) or quadrupling (RMSEquad (i))
the iteration number to find a slightly different iterm , the small rate of update RMSEnext (i) aroundthese positions means the small deviation in iterm will only incur a negligible amount of change inthe extracted phonon distribution, as shown in Fig. 5.11. To validate this method, I repeat it on
the same thermal phonon distribution with 105 different noise realizations for the qubit spectrum,
find the rms-error between the extracted phonon distribution at iterm and the true distribution
RMSEtheory (iterm ), and plot its histogram in Fig. 5.12. Compared to the histogram of the best
achievable rms-error RMSEtheory (iopt ), the largely overlapped histograms demonstrate that thesemethods of extracting iterm are indeed capable of extracting phonon distributions close to what canbe optimally achieved. Thus, for all deconvolutions in this work, I determine iterm by finding theminimum of RMSEdouble (i).

5.3.2

Bootstrapping

Although a suitable iterm balances between interpreting too few and too many details in
the qubit spectrum as actual phonon distribution, it cannot differentiate measurement noise fromphonon-induced qubit response. Indeed, despite iterm , variations in the noise realization can still
alter the extracted phonon distribution. To quantify this effect, I use non-parametric bootstrapping [138]. As discussed in Sec.4.1.1, all qubit spectra I show in this work are the result of averaging many (100 − 1000) individually acquired traces. Take for example the synthesized dataof Fig. 5.10(a), the noisy qubit spectrum is the result of averaging 916 independent noise realizations. To create a bootstrap sample, I re-sample randomly among these noise realizations withreplacement to find a new set of 916 traces, and then average. Repeating this process for 1000
times, and performing the deconvolution algorithm on those bootstrap samples, I find a histogram

123

Figure 5.13: Reconstruction bootstrapping and confidence intervala, Using the synthesized data of Fig. 5.10(a), I create 1000 bootstrap samples and reconstruct theirphonon distributions using the deconvolution algorithm. I plot the histogram of finding P (n) (colorscale vs. y-axis) at particular phonon numbers n (x-axis). The solid green line corresponds to thereconstructed phonon distribution using the original mean qubit spectrum. b, At a randomly pickedvalue of n = 13 (grey dashed line in a), I plot the histogram for P (13). From this distribution, I
find the 90% confidence interval for P (13) to be within the range [2.46, 2.90]% (dashed black lines).

124of the reconstructed phonon distributions as shown in Fig. 5.13(a). Based on this histogram, I
compute the so-called basic bootstrap 90% confidence intervals for each phonon number, as shownin Fig. 5.13(b). These confidence intervals are always shown as the shaded region around the extracted P (n) corresponding to the original mean qubit spectrum. Thus, a smaller shaded regionmeans the reconstructed distribution is less susceptible to measurement noise. In general, I find thereconstructed P (n) from the original mean qubit spectrum lies within its 90% confidence interval.

5.4

Thermal and displaced thermal states
To validate both the deconvolution procedure and the experimentally simulated PSF map,

I demonstrate the reconstruction of phonon distribution on thermal and displaced-thermal mechanical states. These experiments are performed at Vdc = 6 V. This section follows closely to
publication [85].
First, I measure the qubit spectrum when the mechanical oscillator is in thermal equilibriumwith the dilution fridge as shown in Fig. 5.14(a). Running the deconvolution procedure with the
PSF map found in Sec.5.2, I extract the phonon distribution and its 90% confidence interval.
Because the mechanical oscillator is known to be described by the thermal distribution given by
Eqn.(2.9), I can also perform a least-squares fit on the measured qubit spectrum by replacing P (n)with Pth (nth , n) in Eqn.(5.1). In this least-squares fit, the only free parameters are the mechanicalthermal population nth = 17.7, and the bare qubit frequency (see Sec.5.2.5).
Additional to the thermal state, I can coherently displace the mechanical oscillator by driving
it with the product of Vdc and a resonant ac-signal of amplitude Vcoh applied on the same electrode.
Because the mechanical oscillator starts in a thermal state, this driving procedure prepares it in adisplaced-thermal state characterized by distribution [139],
P (nth , ndisp , n) =

∞
X

nm
th

h
i2
|n−m|
−ndisp min{m, n}! |n−m|
e
n
L
(n
)
,
disp
min{m,n}max{m, n}! disp
(1 + nth )m+1
m=0

(5.46)

where Lln (x) is the associated Laguerre polynomial, nth is the initial thermal population, and2 is the mean phonon displacement due to the coherent drive. As shown inndisp = |α|2 ∝ Vcoh

125
Fig. 5.14(b-f), I measure the qubit spectra at different Vcoh and extract the phonon distributions.
Using Eqn.(5.46) and keeping nth = 17.7 constant, I fit the qubit spectra with ndisp being the only
free parameter. In Fig. 5.14(g), I verify the coherent displacement by plotting the extracted ndisp
2 . The linear fit (dashed line) goes through the origin. The 90% confidence
as a function of Vcohinterval error bars are found from distributions of ndisp , which are extracted by performing theleast-squares fit on non-parametric bootstrap samples.
In Fig. 5.14(a-f), the good agreement between the phonon distributions as well as the associated qubit spectra substantiates the deconvolution procedure and the PSF map. In these dressed
qubit spectra, individual sideband peaks cannot be identified because these features are smearedby the broad phonon distribution associated with the large thermal occupation. Nevertheless, failure to properly account for the sidebands would have led to substantial errors in the extracted
phonon distribution when sideband-induced qubit excitations are erroneously attributed to phonon
populations.

126

Figure 5.14: Thermal and displaced-thermal mechanical state
For different Vcoh , I measure the qubit spectrum (dots) and extract the mechanical phonon distribution (inset). Phonon distributions extracted from deconvolution (solid) and their 90% confidence interval (shaded) are plotted alongside the result of least-squares fit (dashed). a, With
Vcoh = 0 V, the mechanical oscillator is in a thermal state. A least-squares fit to the thermal distribution finds nth = 17.7. b-f, By applying an ac-drive with 25 MHz frequency and amplitude Vcoh
(specified at the generator output), I coherently displace the thermal mechanical state by a meanphonon displacement of ndisp . Keeping the extracted nth from a constant, I perform least-squares
fits assuming displaced thermal distributions to extract ndisp . The extracted parameters are: b,
Vcoh = 100.5 mV and ndsip = 8.0; c, Vcoh = 142 mV and ndsip = 20.0; d, Vcoh = 174 mV andndsip = 30.6; e, Vcoh = 201 mV and ndsip = 53.5; and f, Vcoh = 225 mV and ndsip = 51.9. g, I plot
2 . The error bars correspond to 90% confidence intervals
the extracted ndsip as a function of Vcohof ndsip are found using non-parametric bootstrapping. Dashed blue line is a linear fit that goes
through the origin. This figure is reproduced from reference [85].

Chapter 6

Manipulating the phonon distribution using sideband transitions

With the ability to extract the phonon distribution, I now manipulate it using sideband
transitions that simultaneously alter the qubit state and the phonon population. In order to access sideband transitions that add or remove only one phonon at a time (single-phonon sideband
transitions), I adopt the technique of driving the ac-dither sideband transitions [140], as describedin Sec.6.1. In contrast to that in conventional linear electromechanics [19], the sideband transitions in this work are crucially different in their phonon-number-sensitivity: they only address asmall section of the phonon population with a characteristic width of W ≈ 7.1 phonons [86]. I experimentally demonstrate this property in Sec.6.2. Using these phonon-number-sensitive motional
sidebands, I demonstrate in Sec.6.3 an adiabatic energy squeezing technique [85] where, by trapping the oscillator’s phonon population in between phonon-creating (blue) and phonon-annihilating(red) sideband transitions, I reduce its uncertainty in number space. This technique enables the
preparation of nonclassical mechanical states in Chapter 7.

6.1

Ac-dither sideband transitions
At the charge degeneracy point (ng = 1/2) of the CPB qubit, the symmetry of the Hamilto-

nian means that single-phonon sideband transitions |g, ni ↔ |e, n ± 1i are forbidden to microwave|nidrives applied at frequencies ωd = ωq ± ωm . Although these transitions are clearly observed in thequbit spectrum (see for example, Fig. 7.3), they arise only when charge noise causes ng to deviate
from the degeneracy point, and are thus unreliable resources.

128
Instead, to reliably access these single-phonon sideband transitions, I adopt the techniqueof driving ac-dither sidebands. Proposed in reference [140] for a CPB qubit, this technique drives
sideband transitions by a combination of two microwave signals: one low frequency ac-dither signal
is applied to controllably modulate the CPB gate-charge according to
ng (t) = n0g + ndithercos (ωdither t).
g

(6.1)

This modulation introduces a dynamical σ̂z coupling, which, as discussed in Sec.2.3.5, allows forthe creation and annihilation of odd number of phonons. Consequently, a second microwave signal(sideband drive) applied at frequencies,
ωB (n) = ωq|ni + ωm ± ωdither ,(6.2)

ωR (n) = ωq|ni − ωm ± ωdither ,can drive single-phonon blue or red sideband transitions |g, ni ↔ |e, n ± 1i at respective rates
|ni

√

|ni

√

ΩB =
ΩR =

n + 1ΩB,0 ,(6.3)
nΩR,0 .

≈ 0.05 used in this work,where, for the small dither amplitude nditherg!
8Ec nditherΩRgΩB(R),0 = gm
J1
2 (ωd − ωq )
EJ

(6.4)

is the bare sideband rate, ΩR and ωd are the Rabi rate and frequency of the sideband drive, andJ1 (z) is the 1st order Bessel function of the first kind.
Fig. 6.1 shows a full measured qubit spectrum under an ac-dither at frequency ωdither =2π × 260 MHz with a large spectroscopic power. The center qubit transition is power broadened
and saturated (see Sec.2.3.4.3), with Pe ≈ 0.5 on resonance. Two groups of three satellite peaks
are visible on either side of the main qubit peak, detuned by ωdither . Within each group, I canidentify the single-phonon red and blue sideband transitions. Unless specifically noted, I use theset of sideband transitions below the center qubit transition with ωdither = 2π × 257 MHz for the
rest of this thesis.

129

Figure 6.1: Ac-dither sideband spectrum
Ac-dither sideband transitions can be observed in the qubit spectrum, measured at Vdc = 6 V witha large qubit probe power. The lower (left) set of red and blue sideband transitions are used tomanipulate the phonon populations in this work. This figure is reproduced from reference [86].

130

6.2

Sideband transition’s phonon number sensitivity
In contrast to motional sideband transitions in conventional linear electromechanics that

address all phonon numbers simultaneously [19], the sideband transitions in this work are phononnumber-sensitive, addressing only a small section of the phonon population [86]. Because each|niphonon number Stark shifts the qubit frequency ωq by 2χm , the sideband frequencies in Eqn.(6.2)
are changed accordingly. Consequently, as discussed in Sec.5.2.2, a sideband drive resonant withtransition |g, nSB i ↔ |e, nSB ± 1i is only effective over a characteristic width W = Γ2 /2χm ≈ 7.1
phonons centered about nSB .
To gain an intuitive picture for the effect of driving these phonon-number-sensitive transitions, consider the simplified dynamics of the qubit-mechanical system sketched in Fig. 6.2(a),where the qubit decays are assumed to always conserve phonon numbers. Starting from a thermal
state, because the sidebands in this work are always driven incoherently on a time scale much
longer than the qubit lifetime and dephasing time (T1 ≈ 260 ns, T2 ≈ 80 ns), the resulting mechanical state contains no quantum coherence and is fully described by the diagonal elementsof its density matrix. A blue sideband drive applied at frequency ωB (nB ) drives the transition
|n i|n i 2|g, nB i ↔ |e, nB + 1i resonantly at an effective rate ΓB B = ΩB B
/Γ2 , but also drives its neigh|ni

boring transition |g, ni ↔ |e, n + 1i at a reduced effective rate ΓB , dependent on the ratio of(n − nB )/W (Fig. 6.2(b), see also Sec.6.3.1). Because the qubit decay is the dominant relaxation
process (Γ1  γm ), populations in an initial state of |g, ni is first driven to state |e, n + 1i by the
sideband, and then relaxes to state |g, n + 1i, resulting in an overall addition of one phonon. If the
sideband drive is applied continuously, phonons are repeatedly added until the oscillator reaches a|miphonon number m so far from nB that the corresponding sideband rate ΓBis smaller than the|n ithermal decoherence rate nth γm . At a time scale nB /ΓB B , this process creates a distortion in thephonon distribution around nB with a characteristic width of W , while leaving populations far
away undisturbed, as depicted in Fig. 6.2(c).
Such distortions in the phonon distribution are observed in Fig. 6.3. After driving the ac-

131

Figure 6.2: Phonon-number-sensitive sideband transitionsa, In the simplified dynamics of the qubit-mechanics system, a qubit decay always preserves thephonon number. Starting from an initial state |g, n − 1i, a blue sideband drive excites the tran|n−1isition |g, n − 1i ↔ |e, ni at an effective rate ΓB . Because Γ1  γm , the population in state|e, ni predominantly relaxes to |g, ni. Overall, this process adds one phonon to the oscillator. Similarly, the red sideband drive combined with the relaxation process removes one phonon from theoscillator. b, Because W = Γ2 /2χm ≈ 7.1 phonons, a sideband drive is phonon-number-sensitive.
For example, a blue sideband drive resonant with transition |g, nB i ↔ |e, nB + 1i also drives the|n ineighboring transitions at reduced rates. c, At a time scale nB /ΓB B , the phonon-number-sensitive
blue sideband drive applied on an initial thermal state (dashed orange) creates a distortion in thephonon distribution (solid green), pushing up a section of the phonon population around nB ofcharacteristic width W while leaving phonons far away unaffected. d, Conversely, at a time scale|n inR /ΓR R , a red sideband drive creates a similar distortion in the phonon population, removing asection of the population around nR to lower phonon numbers.

132

Figure 6.3: Demonstrating the phonon-number-sensitivity of a blue sideband drivea, Starting from an initial thermal state (orange dots), I drive the blue sideband at different values of nB and measure the qubit spectrum (dots). Vertical arrows corresponds to the position of|n inB mapped to the Stark shifted qubit frequency (that is: ω = ωq B ). b, Using the deconvolution procedure, I extract the corresponding phonon number distributions (solid lines) and their
90% confidence intervals (shade). The associated qubit spectra (solid lines) are also plotted ina. Compared to extracted initial distribution (orange), it is clear that changing the value of nB
addresses different sections of the phonon population (grey shaded region), demonstrating thephonon-number-sensitivity of the blue sideband drive.

133

Figure 6.4: Demonstrating the phonon-number-sensitivity of a red sideband drive
Starting from a thermal state (orange), I measure the qubit spectra (dots) after applying a red
sideband drive with nR ≈ 8 for a duration of 150 µs (red) and 1.5 ms (black). Inset: The phonon
distribution (solid line) and its 90% confidence interval (shaded) are extracted using the deconvolution procedure. While the red sideband drive removes the phonon population around nR tobelow n = 2 at short time, phonons far away from nR are only affected at a larger time because ofthermal equilibration. This figure is reproduced from reference [86].

134
dither blue sideband transitions at different frequencies, corresponding to nB ranging from 1 to 41, I
measure the qubit spectra and extract the underlying phonon distributions11 using a deconvolution
procedure. Indeed, depending on the value of nB , different parts of the original thermal distributionis altered. As highlighted by the shaded areas, the blue sideband drive takes a small region of thephonon populations around nB and transfers them to higher numbers.
This phonon-number-sensitivity can also be demonstrated with red sideband transitions, but
in the time-domain. In Fig. 6.4, I measure the qubit spectrum after applying an ac-dither redsideband drive at a frequency corresponding to nR ≈ 8. When the red-sideband drive is applied for|n ia relatively short time of 150 µs (∼ nR /ΓR R  1/γm , red), the phonon populations around n = 8
are emptied and transferred to Fock numbers n ≤ 2. Meanwhile, populations at higher n are notaffected because they are outside the number-sensitive region of the sideband drive. However, if
the red sideband drive were left on for a much longer time of about 1.5 ms (∼ 1/γm , black), thesehigher number populations would slowly decay down into the region over which the red sideband
drive is effective and get pushed to even lower n. A similar long-time effect is also observed inFig. 6.3(b): because the blue sideband drive is applied for a long effective time12 , populations far
below nB are affected because they could be thermally excited to the region close to nB .

6.3

Dissipatively squeezing the oscillator’s energy
These phonon-number-sensitive sideband transitions enable a dissipative stabilization tech-nique that can simultaneously add energy to and extract entropy from the oscillator [85]. This
technique can be intuitively understood with the sketches of Fig. 6.5. To add energy to the oscillator, I slowly increase the blue sideband drive frequency ωB (chirping). This adiabatically moves thecenter of the addressed transitions nB up in phonon space and creates a steady-state phonon distribution that contains no population below the final value of nB (Fig. 6.5(a)). To extract entropy from11

The phonon distributions presented in this section are all reconstructed using the approximate PSF discussed
in Appendix.C
12
Instead of letting the mechanical oscillator relax back to a thermal state after each experiment, I pulse on theblue sideband drive for 10 µs every 25 µs, and repeat this sequence for about one minute before resetting ng .

135
the oscillator, I then turn on a red sideband drive centered on the transition |g, nR i ↔ |e, nR − 1i,with nR close to but greater than the maximum value of nB (Fig. 6.5(b)). A Fock state |ni will|ni

be cooled to a lower occupancy if the effective blue sideband transition rate ΓB is slower than the
|ni

|n i

|n i

effective red sideband transition rate ΓR , and vice versa. Thus, under conditions ΓB B > ΓR B and
|n i

|n i

ΓB R < ΓR R , the steady state phonon population becomes trapped in between the two-sideband
drives.

Figure 6.5: Energy squeezing: ideaa, Adiabatically increasing the blue sideband drive frequency (chirping) as a function of time τchanges the center of the addressed sidebands nB . Starting from an initial thermal distribution,
this chirping should empty all phonon population below the value of nB , adding energy to theoscillator. The inset show that the resulting state’s Wigner function13 W(X1 , X2 ) is a narrowp ringaround the (X1 ,X2 ) quadrature space origin with an inner-radius approximately given by nB (τ ).
b, When the blue sideband drive is chirped toward a static red sideband drive centered at nR ,the phonon population should be trapped in between, and become squeezed in energy. Inset: The
Wigner function of the resulting energy squeezed state is non-Gaussian and radially symmetric
about the quadrature space origin, quite distinct from a quadrature squeezed state.

Fig. 6.6 shows an experimental demonstration of this technique. Starting from a thermal
state, I chirp the blue sideband drive for different times τ , corresponding to different values ofnB (τ ), and measure the qubit spectrum. Indeed, at any time τ , the extracted phonon distribution
in Fig. 6.6(a) contains no population below nB (τ ). In Fig. 6.6(b), I demonstrate energy squeezing
by repeating the same experiment, but with a red sideband drive parked at nR = 44. Trapped
between nR and nB (τ ), the phonon population becomes increasingly localized in Fock space (smaller
13

See Sec.7.1.4 for detailed discussion on Wigner function

136

Figure 6.6: Energy squeezing: experimenta, Chirping the blue sideband drive to a final position of nB (τ ) (bottom-axis) by stopping the chirp
at time τ (top-axis), I measure the qubit spectrum (see Fig. 6.8) and extract its phonon distribution(y-axis vs. colorscale) using deconvolution. At any time, the population is empty below the linen = nB (τ ) (dashed blue), confirming the idea explained in Fig.6.5(a). b, Turning on a red sideband
drive parked at nR (dashed red), I repeat the above experiment to observe the phonon population
is indeed trapped between the two sideband drives and become squeezed in energy as nB is chirped
toward nR .

137
uncertainty in phonon number) as nB (τ ) is brought closer to nR . Clearly, these experiments support
the intuitive understanding discussed above.

6.3.1

Modelling the time evolution of the hybrid system under sideband drives

To go beyond the intuition and make realistic predictions, I write down the time evolutionof the qubit-mechanics system. This section follows closely to reference [85] Because the sidebands
are driven on a time scale much longer than the qubit T1 and T2 , the resulting state of the system|ni

is described by the diagonal elements of its density matrix, Pg|niand Pe . Consequently, thetime evolutionary of the qubit-mechanics system is well approximated by a set of classical masterequations, similar to the classical laser rate equations. In general, these master equations are given
by a combination of system dynamics due to sideband drives and decay,
d |ni
d |ni
d |ni
Pg(e) = Pg(e)+ Pg(e)
.
dt
dt
dt
drive
decay

(6.5)

The first term in Eqn.(6.5) describes the dynamics due to sideband driving


d |ni
|ni
|ni
P
= − ΓB (nB ) + ΓR (nR ) Pg|ni
dt g drive
|ni

d |ni
P
dt e drive|ni+ ΓR (nR )Pe|n−1i + ΓB (nB )Pe|n+1i ,


|ni
|ni
= − ΓB (nB ) + ΓR (nR ) Pe|ni
|ni

(6.6)

|ni

+ ΓR (nR )Pe|n+1i + ΓB (nB )Pe|n−1i ,|niwhere Pg(e) is the instantaneous population in the qubit ground (excited) state with n phonons.
|niΓB is the effective blue sideband transition rate between states |g, ni ↔ |e, n + 1i, and is related|ni

to its quantum counterpart ΩB according to

|ni

ΓB =


|ni 2

ΩB

Γ2

.

(6.7)

For an ac-dither blue sideband drive of bare sideband rates ΩB(R),0 (Eqn.(6.4)) applied on resonance|niwith transition |g, nB i ↔ |e, nB + 1i, the effective transition rate ΓB is function of nB , maximized

138
at n = nB but reduced for the other transitions by the frequency detuning 2χm (n − nB ) between
their resonances and the sideband drive,(n + 1)Ω2B,0|ni
ΓB (nB ) =
Γ2

1
1+



n−nB
W/2

2 ,

(6.8)

where W = Γ2 /2χm is the characteristic width of the phonon-number-sensitivity. For a chirped
blue sideband drive, nB is a function of time. The effective red sideband rate is governed by asimilar equation,|niΓR (nR ) =nΩ2R,0
Γ2

1
1+



n−nR
W/2

2 .

(6.9)

Next, I write down the master equations due to qubit and mechanical decay, which corresponds to the second term in Eqn.(6.5). In contrast to the intuitive picture discussed above(Fig. 6.2(a)), the qubit decay in general doesn’t preserve the phonon number. Following the detailed discussion in Sec.2.3.4.1, I write down the following master equations:




d |ni
Pg
= e2r γm nth nPg|n−1i +(1 + nth )(1 + n)Pg|n+1i − (1 + n)nth + (1 + nth )n Pg|ni
dt
decay
X
+ Γ1
|αmn |2 Pe|mi
m



,
d |ni
P
= e−2r γm nth nPe|n−1i +(1 + nth )(1 + n)Pe|n+1i − (1 + n)nth + (1 + nth )n Pe|ni
dt e decay− Γ1 Pe|ni(6.10)
where r = χm /2ωm ≈ 0.005, and αmn = b hm|Ŝ(−2r) |nib is the overlap between different motional
states under a qubit flip. In these equations, I ignore the relaxation process where the mechanicaloscillator losses many phonons to excite the qubit (|g, ni 7→ |e, mi). Because of the large difference
between qubit and mechanical frequency, such process requires the loss of more than 152 phonons(ωq /ωm ≈ 152), and is highly unlikely.
Using the classical master equation of Eqn.(6.5), numerical simulations can be performed to
predict the effects of driving the sideband transitions. To perform these simulations, the mechanicalphonon space is truncated at nmax = 200. At every time step in the simulation, the conservation

139

Pnmax  |ni
|ni
|n
i
|n
i
P
+
P
= 1 is enforced by setting Pe max = 0 and Pg max = 1 −of probability
g
e
n=0

Pnmax −1  |ni
|ni
P
+
P
.
g
e
n=0

Figure 6.7: Energy squeezing: theory
Using the classical master equations, numerical simulations are performed with the same parameters
used in the experiment of Fig.6.6, demonstrating a good agreement between theory and experiment.

Fig. 6.7 displays the results of numerical simulations performed using the experimental parameters found in Fig. 6.6, demonstrating a good agreement between theory and experiment. These
parameters are extracted by performing least-squares fits to the measured qubit spectra, similar toSec.5.4. Specifically:(1) I first find nth of the initial distribution by fitting the qubit spectrum at τ = 0 to thermal
distributions.
(2) Next, I generate a look-up table of phonon distributions with different sideband driving
parameters by numerically solving the classical master equation. Through a convolution
with the PSF map (Sec.5.2.5), this table is converted into a look-up table of expected qubit
spectra.
(3) Using the look-up table, I jointly fit all qubit spectra with only a blue sideband drive but
chirped for different τ in Fig. 6.8(a). Because the chirp rate dnB /dτ ≈ 38.5 phonons/ms is140
determined experimentally, the only two free parameters are the bare blue sideband rate
ΩB,0 = 2π × 170 kHz, and the starting position of the chirp nB (0) = −1.3.
(4) Finally, I jointly fit all qubit spectra with both blue and red sideband drives in Fig. 6.8(b).
Because I use the same blue sideband drive protocol, I keep the fit results from step (3)as constants. The center of transition for the red sideband drive nR = 44 is determinedby the fit result of nB (0) and the frequency detuning between the initial blue and red
sideband drives. A joint least-squares fit to the look-up table finds the only free parameterΩR,0 = 2π × 122 kHz.
It may seem surprising that the sideband drives can move phonon populations so efficiently
given the effective sideband rate Ω2B,0 /Γ2 ≈ 2π × 7.8 kHz is rather comparable to the mechanicalthermal decoherence rate nth γm ≈ 2π × 1.4 kHz. Indeed, this surprising phonon transfer efficiency
comes from the fact that a sideband drive simultaneously drives many transitions [86]. To understand this, consider for instance a two-level system {|0i , |1i}, which is coupled with a hot bathat rate γ such that P (0) = P (1) = 1/2 at thermal equilibrium. Turning on a drive takes the
population in |0i to |1i at a reduced rate Γ = γ. The steady state populations will be P (0) = 1/3
and P (1) = 2/3, demonstrating a very inefficient population transfer. However, the same drive will
already look more efficient if it address a three level system {|0i , |1i , |2i}. At equilibrium with thethermal bath, the populations are given by P (0) = P (1) = P (2) = 1/3. When the drive is on, thesteady state populations are given by P (0) = 1/7, P (1) = 2/7 and P (2) = 4/7. The same goes foreven higher level systems.
In the driven dynamics of Eqn.(6.6), I consider only the effects from the intentionally excited
ac-dither sideband transitions. However, as I discussed in Sec.7.3, the ac-dither sideband drive canalso excites other unintentional sideband transitions, complicating the master equations. Althoughthese unintentional sideband transitions have only negligible effects when the blue sideband drive islocated at small values of nB , they will cause an observable spurious cooling for nB > 50. To accountfor these effects, corrections to the classical master equations will be discussed in Sec.7.3.

141

Figure 6.8: Finding the sideband driving parametersa, Each panel (with the same frequency axis) shows one qubit spectrum (green dots) measured
after chirping the blue sideband drive for a time τ . The extracted phonon distributions from thesequbit spectra are plotted in Fig. 6.6(a). A joint least-squares fit of these qubit spectra to numericalsimulations reveals the blue sideband driving parameters nB = −1.5 and ΩB,0 = 2π × 170 kHz. b,
Similar to a, each panel shows the qubit spectrum after chirping the blue sideband drive for timeτ , but with the red sideband drive on. These qubit spectra corresponds to the phonon distribution
shown in Fig.6.6(b). Keeping the same blue sideband driving parameters, a joint least-squares fit
to numerical simulations reveals the red sideband drive strength ΩR,0 = 2π × 122 kHz. This figure
is reproduced from reference [85].

Chapter 7

Energy squeezing below the classical limit

Finally, I use the energy-squeezing technique to prepare the oscillator in a non-Gaussian nonclassical state of motion [85]. Specifically, I prepare the oscillator in a sub-Poissonian state, where
its energy fluctuation is blow the classical limit [87]. In Sec.7.1, I demonstrate the preparation ofsuch a state, with a large mean phonon number hni = 43 and a sub-Poissonian number fluctuation of approximately 3. In preparing this energy-squeezed state, I clearly resolve the sideband
transitions in the qubit spectrum that were previously obscured by the broad phonon distribution
associated with the large thermal occupation in Sec.5.4. In Sec.7.2, I provide a simple physicalexplanation for these prominent sideband peaks using the Frank-Condon principle [111, 112], anddiscuss their implications for a quantum non-demolition (QND) measurement of phonons. Finally,in Sec.7.3, I discuss the factor that limits the nonclassicality achieved in this work: a spurious
cooling effect mentioned in Sec.6.3.1. This chapter closely follows publication [85].

7.1

Preparation and measurement of a nonclassical state of motion
After optimizing the energy-squeezing protocol discussed in Sec.6.3, I prepare the oscillator ina nonclassical state of motion described by a sub-Poissonian phonon distribution. To demonstrate
this, I infer the phonon distribution from the qubit spectrum. However, as I discuss in Sec.7.1.1,
this inference requires care because measuring the qubit spectrum can alter the phonon distribution. In Sec.7.1.2, I extract the phonon distribution from the measured qubit spectrum using thedeconvolution procedure. Narrower than a coherent state of the same mean phonon number, this ex-

143
tracted phonon distribution is indeed nonclassical, with a Fano factor F = var(n)/hni = 0.257+0.002−0.001
smaller than the classical limit of F = 1 [87]. In Sec.7.1.3, I investigate the statistical significanceof this extracted Fano factor, and determine with 99% confidence that this nonclassical mechanicalstate has a true Fano factor Ftrue ≤ 0.30. Finally, in Sec.7.1.4, I relate the Fano factor to othermeasurements of nonclassicality, such as the negativity in a Wigner function.

7.1.1

Measuring the qubit spectrum

As discussed theoretically in Sec.5.2.2, the sideband peaks observed in a qubit spectrum
correspond to the creation or annihilation of phonons. Consequently, to ensure an accurate inferenceof the phonon distribution, I need to limit the measurement-induced changes in phonon distribution
to a negligible amount. This can be achieved by only driving the qubit probe signal for a short
8 µs duration.
To demonstrate this, I measure the qubit spectrum for different duration Tspec after preparing the mechanical oscillator in a sub-Poissonian state using the energy-squeezing protocol. The
pulsing sequence for this measurement is sketched in Fig. 7.1. Indeed, as Fig. 7.2 shows, when
Tspec is increased beyond 8 µs14 , changes in the qubit spectrum associated with changing phonon
distributions can be observed.

7.1.2

Extracting the nonclassical phonon distribution

To extract the phonon distribution from the measured qubit spectrum, I use the same deconvolution procedure discussed in Sec.5.3, but with one modification: the initial guess for thephonon distribution P0 , instead of a uniform distribution, is assumed as a Poisson distribution.
Fig. 7.3(b) shows the phonon distribution extracted using an uniform initial guess. It reveals the14

Note that Tspec = 8 µs is not a universal condition for avoiding this measurement induced change in phonon
distribution. Instead, it is influenced by both measurement noise and the phonon distribution being measured. Unlessspecifically noted, phonon distributions measured in previous chapters are all performed with the sequence of Fig. 7.1
but a longer Tspec = 100 µs. In those experiments, the phonon distribution is already broad because of the large
nth , and I’m only interested in detecting how the distribution changes as function of other parameters. Thus for
those experiments, I choose to improve the duty cycle of measurements, which use 8 ms repetition rates to allow for
the mechanical oscillator to re-equilibrate, at the cost of a small reduction in the fidelity of the extracted phonon
distributions.

144

Figure 7.1: Measurement sequence
The pulse sequence (not to scale) used to measure the sub-Poissonian state is sketched. The
sequence starts with the preparation of the desired mechanical state by driving the sidebands for aduration τ ≈ 1 ms (blue). As discussed in Chapter 4, after waiting for the qubit to decay back to theground state, I probe the qubit state by simultaneously driving the qubit (green) and measuring
the cavity (orange) phase ϕc for duration Tspec . To ensure the steady-state qubit spectrum ismeasured, the qubit drive is applied 1 µs before the start of the phase measurement. After thequbit has relaxed back to its ground state, I again measure ϕc to readout the qubit parity. Finally,
I wait for more than 6 ms for the mechanical oscillator to thermalize back to its environment before
starting the next measurement cycle.

145

Figure 7.2: Measurement-induced changes in phonon distributiona, After preparing the desired mechanical state using the energy-squeezing protocol, I measure
the qubit spectrum with different qubit probe duration Tspec . An incremental offset of 2% isadded for each trace. The vertical dashed lines correspond to the positions of the qubit excitation
peaks at Tspec = 8 µs. With increasing Tspec , the l = −1 peak shifts toward a lower frequency
because of sideband cooling, and the l = {1, 2} peaks shift toward higher frequencies because of
sideband heating, demonstrating measurement-induced changes in the phonon distribution. With
increasing Tspec , I also observe a broadening of the center qubit peak caused by the damping of themechanical oscillator during measurement. b, I subtract the Tspec = 8 µs qubit spectrum from each
qubit spectrum in a. Structure can be observed in the traces with Tspec > 13 µs at the positions of
the dashed lines. This figure is reproduced from reference [85].

146
expected energy-squeezed phonon distribution around hni = 43, but also finds a small, nonphysicalphonon peak around n = 180. Initial assigned by the uniform guess, these nonphysical population
requires many iterations of the deconvolution procedure to remove. Before this happens, however,
the reconstruction starts to over fit in other parts of the phonon distribution (see Sec.5.3.1). Consequently, at the optimal iteration number, some nonphysical populations are left behind. Indeed,
I confirm this nonphysical feature to be an artifact of the imperfect reconstruction procedure not
only because it disagrees with the physical understanding achieved in Sec.6.3, but also because I
observe similar artifacts when the same reconstruction is performed on simulated experiments withknown phonon distributions (see Sec.7.1.3). Although these artifacts do not obscure the energysqueezed phonon distribution, they could lead to a greatly underestimated nonclassicality. In thiscontext, I remove them by providing a better initial guess — a Poisson distribution.
Fig. 7.3(c) shows the phonon distribution extracted with an Poisson initial guess of meanphonon number hni = 43. Narrower than a Poisson distribution of the same mean phonon number,the mechanical state prepared using the energy-squeezing protocol is indeed nonclassical [87], withan energy fluctuation below the classical limit. I characterize this nonclassical nature with the Fanofactor F = var(n)/hni. For a Poisson distributed state, F = 1, and for a Fock state, F = 0. WhenF < 1, the phonon distribution is nonclassical, energy squeezed, and Fock-like. For the phonon
distribution extracted in Fig. 7.3(c), I find F = 0.257+0.002−0.001 , where the bound is determined by theuncertainty in the bare qubit frequency (see Sec.5.2.5). To determine the statistical significance of
this extracted Fano factor, one option is to use the non-parametric bootstrapping (see Sec.5.3.2)
to investigate the effect of measurement noise. However, such an investigation is unable to reveal
any systematic bias introduced by the deconvolution procedure.

7.1.3

Bounding the Fano factor

Instead, I quantify the confidence in the extracted Fano factor by performing repeated reconstruction procedures on simulated experiments with known phonon distributions. This way, I reveal
simultaneously the effect of measurement noise and any systematic bias from the deconvolution.

147

Figure 7.3: Extracting the nonclassical phonon distributiona, Using Tspec = 8 µs, I measure the qubit spectrum (dots) associated with the energy-squeezed
phonon distribution. The standard error of the mean (SEM) are extracted individually for eachfrequency point (see Sec7.1.3), and plotted as error-bars for every fourth point. b, Using thedeconvolution procedure discussed in Sec.5.3, I extract the phonon distribution (solid line) andits 90% confidence interval (shaded) starting from an uniform initial guess, but find nonphysicalpopulation around n = 180. c, Starting from a Poisson initial guess instead, I remove these artifacts
arising from measurement noise to find a sub-Poissonian phonon distribution (green) characterized
by F = var/hni = 0.257 < 1, narrower than that of a coherent state (dashed black). For comparison,
the associated qubit spectra are plotted in a, where the sub-Poissonian nature of the mechanicalstate is evident in its narrower lineshape. The peaks visible in the qubit spectrum are separatedby ωm , and corresponds to transitions |g, ni ↔ |e, n + li.

148Specifically, this study is performed following a four step procedure:(1) I create a Gaussian distributed phonon distribution characterized by a mean phonon number hni = 43 and a Fano factor Ftrue .
(2) This phonon distribution is converted into a noiseless qubit spectrum by convolving it with
the PSF map found in Sec.5.2.5.
(3) I simulate an experimental realization by adding the appropriate amount of noise andaveraging for the same number of times as the actual experiment (916 times for the data
in Fig. 7.3(a)).
(4) The same reconstruction procedure is performed on the simulated experiment to extract aphonon distribution and Fextract .
Correctly simulating the experiment in step (3) requires an accurate understanding of themeasurement noise. I achieve this understanding by looking at the measurement data corresponding
to Fig. 7.3(a). From the 916 individual recorded qubit traces, I find the standard deviation of themeasurement signal at each frequency point. At each frequency point, the extracted measurement
standard deviation σP (green dot) is plotted in Fig. 7.4(a), and arranged according to the mean
signal at that frequency. I model the dependence of noise on signal amplitude with a second orderpolynomial function (black dashed line).
The four-step procedure is repeated many times for different noise realizations and Ftrue tobuild statistics. Fig. 7.4(b) depicts the result of this statistical study. For each Ftrue , I perform
3000 simulated experiments, each with 916 averages. The extracted Fextract are binned into steps ofδF = 0.01, and the resulting histogram is plotted. The grey solid line corresponds to Fextract = Ftrue ,and demonstrates a systematic bias of the deconvolution that generally finds Fextract > Ftrue . For
the 1216 simulations that resulted in an Fextract within the interval [0.255, 0.265] (black dashedline), the cumulative distribution function (CDF) of Ftrue is plotted in Fig. 7.4(c). Because theexperimentally determined F = 0.257+0.002−0.001 lies within this range, I thus find Ftrue ≤ 0.28 with 95%

149
confidence and Ftrue ≤ 0.30 with 99% confidence for the sub-Poissonian state prepared in Fig. 7.3.

7.1.4

Relating the Fano factor to other measurements of nonclassicality

The Fano factor is but one measurement that can be used to characterize the nonclassicalityof this mechanical state. Indeed, it can be related to other commonly used criteria for describing
nonclassical states, such as the g2 correlation function [141],
g2 (0) = 1 +Var(n) − hni
F −1
=1+
,
2
hni
hni

(7.1)

where g2 (0) < 1 signifies a nonclassical state. Clearly, this condition is automatically fulfilled
by F < 1. However, because the mechanical state resides in a single mode of the oscillator, thecondition g2 (0) < 1 does not imply phonon anti-bunching [142, 141], but only reveals the subPoissonian nature of the phonon distribution. In this context, the g2 (0) function contains the same
information as F , but unhelpfully compresses the nonclassicality of large mean phonon states intoa tiny region of possible values between 1 and 1 − 1/hni. For example, a pure 50 phonon Fock statewith g2 (0) = 0.98 is barely distinguishable from a coherent state with g2 (0) = 1!
Another common measurement of nonclassicality is the negativity in the Wigner function
W [141]. Because any state prepared using the energy-squeezing protocol will contain only diagonal elements (see Sec.6.3), I can easily compute the Wigner function from the extracted phonon
distribution. Doing so with the sub-Poissonian state in Fig. 7.3, I observe a small region of negativity in the Wigner function but with high statistical significance. Unfortunately, the bare qubit
frequency is a kind of nuisance parameter whose uncertainty yields systematic uncertainty in the
average phonon number hni. Although this uncertainty has little effect on F , it causes uncertainty
in the location of the Wigner negativity. If I average over all of the possible Wigner functions
consistent with the possible bare qubit frequencies, the negativity is diminished such that it has
marginal statistical significance.
Instead of further analyzing the data and risk introducing bias, I can relate the extracted
Fano factor to a negativity in the Wigner function under the assumption of a Gaussian number

150

Figure 7.4: Bounding the Fano factor
a, The measurement noise σP is plotted as a function of the mean measured signal. The dashedblack line is a second order polynomial fit, used to model the noise dependence on signal amplitude.
b, I plot the histogram of Fextract (color-scale vs. x-axis) for different Ftrue . Each Ftrue consistsof 3000 different simulated experiments. The solid grey line corresponds to Fextract = Ftrue , andshows that Fextract in general overestimates the Fano factor. c, Using the CDF of Ftrue for Fextract ∈[0.255, 0.256] (dashed black line in b), I find with 95% confidence that Ftrue ≤ 0.28 and with 99%
confidence that Ftrue ≤ 0.30. This figure is reproduced from reference [85].

151
fluctuations, according to reference [143]. In this reference, the authors find that for a Gaussiandistributed sub-Poissonian state, the maximal negativity in the Wigner W function, defined asp
the ratio of min(W)/max(W), is related to its motional amplitude r0 = hni and Fano factor F
according to Fig. 6 of the reference, reproduced in Fig. 7.5. In comparison, the sub-Poissonian
state prepare in this work is approximately Gaussian-distributed around its mean phonon number
hni = 43, and has a Fano factor F ≤ 0.28 with 95% confidence. Under the assumption of a Gaussian
number fluctuation, these values place the sub-Poissonian state at the red-stared position in thisfigure, and indicate a small negativity in its Wigner function.

Figure 7.5: Relating the Fano factor to negativity in the Wigner function
For a Gaussian-distributed sub-Poissonian state, the maximal negativity in the Wigner function(color-axis) can be related to its motional amplitude r0 (x-axis) and Fano factor (y-axis). In
comparison,assuming a Gaussian number fluctuation, the sub-Poissonian state prepared in Fig. 7.3√(r0 = 43 and F ≤ 0.28 with 95% confidence) is located in this figure at the red-stared position,
indicating a small negativity in its Wigner function. This figure by Lörch et al. [143] is reproduced
under CC BY 3.0 license. I have added the red star, its associated labels, and the color-axis label“min(W)/max(W)”.

7.2

Mechanical recoil caused by qubit transitions
In creating the energy-squeezed state, I can now resolve the sideband transitions that werepreviously obscured by the broad phonon distribution associated with the large thermal occupationin Sec.5.4. The center peak in the qubit spectroscopy (l = 0 peak in Fig. 7.3(a)) corresponds to152
the qubit transition that conserves phonon number, where as the satellite peaks at ±2ωm (l = ±2)
are associated with qubit transitions that create and annihilate pairs of phonons. As discussed inSec.2.3.3.1, these sideband transitions arise because the mechanical spring suddenly stiffens undera qubit excitation, causing overlaps between the mechanical spatial wavefunctions that would beotherwise orthogonal. Indeed, these transitions are highly analogous to the vibronic transitions
in molecules, where a fast electronic (qubit) transition connects different slow mechanical states.
When the qubit is located at the charge degeneracy point, the symmetry of the mechanical potential
prohibits the creation and annihilation of odd number of phonons. However, because charge noise
creates a small random bias away from degeneracy, a qubit excitation also exerts a force on theoscillator, as discussed in Sec.2.3.5, displacing it along its position-axis. Consequently, I also observe
peaks at ±ωm associated with single-phonon sideband transitions.
In contrast to the sidebands observed in reference [82] at hni ∼ 104 phonons, these peaks
are easily resolved at the relatively small phonon number hni = 43 because χm /ωm is much larger(see Sec.2.3.3). Consequently, they can substantially alter the oscillator’s phonon distribution, as
evident from the measured-induced change in phonon distribution already discussed in Sec.7.1.1.
The prominence of these sideband peaks also signifies the entry into a regime where the qubit’s
spontaneous decay can detectably alter the phonon distribution (see Sec.2.2.3). In Fig. 7.3(a), I
measure the probability of exciting the qubit when driving at a particular frequency, but each peakmust also correspond to a qubit decay process, driven by the quantum noise in the environment.
From the ratio of the area underneath the l = 0 peak to the total area under all the peaks, I
estimate that a qubit decaying from an initial state of |e, 43i will only preserve the phonon number
with 63% probability! This probability will further diminish if χm were increased to better resolve
individual phonon numbers.

7.3

Limitations on the achieved nonclassicality
According to Sec.6.3, the energy-squeezing protocol can in principle be performed at arbitrar-

ily large hni to create phonon distributions with minimum widths roughly given by the phonon-

153
number-sensitivity, W ≈ 7.1 phonons. Consequently, it is possible to achieve a smaller F , andtherefore a higher nonclassicality, by squeezing at a higher hni.
However, in this work, I only demonstrate the preparation of a nonclassical mechanical state
at a moderate hni = 43, because driving the ac-dither sideband15 introduces an additional unwanted
cooling process (spurious cooling) that significantly alters the phonon population at large phonon
number n > 50. Although this unwanted process does not prevent, but rather helps, achieving
a narrow phonon distribution, as I will demonstrate shortly, I instead operated at a point where
its effect is negligible to best demonstrate the energy-squeezing protocol’s ability of preparing
nonclassical states. Moreover, if I were to tolerate this spurious cooling effect, the maximum hni
achievable is eventually limited by the inaccuracies in the PSF map experimentally extracted inSec.5.2.5, causing nonphysical features in the reconstructed phonon distribution when the subPoissonian state is prepared at hni > 80.
The spurious cooling effect arises because when a blue sideband drive is chirped for time τto position nB (τ ) to empty all phonon populations below, it simultaneously cools the populationsat larger phonon numbers. This additional unwanted cooling process is conceptually illustrated
in Fig. 7.6(a). As mentioned in Sec.6.1, I use the set of ac-dither sidebands below the qubit|niresonance, ωB (n) = ωq ± ωm − ωdither . Meanwhile, the l-th order sideband transition is located at|ni

ωl (n) = ωq +lωm . Thus, an ac-dither blue sideband drive centered on transition |g, ni ↔ |e, n + 1i
can be close to resonance with the 10-th order red sideband transition |g, n + 35i ↔ |e, n + 25i,
creating an additional cooling effect at larger phonon numbers.
Although the resonant 10-th order red sideband transition provides an easy conceptual understanding to this unwanted effect, it is a subdominant effect. Instead, the spurious cooling effectis dominated by other lower order red sideband transitions excited off-resonantly by the ac-dither
blue sideband drive. To understand this, I write the reduced l-th order sideband rate for transition15

For all theory and experiments in this section, ωdither = 260 MHz.

154
|g, ni ↔ |e, n + li due to an ac-dither sideband drive at ωd (see Sec.6.3.1),
Ωln
Γ2

|ni

Γl (ωd ) =

2

1
1+



ωd −ωl (n)Γ2 /2Ω2 rsd 2= R αn+l,n
Γ2

2
(7.2)
1

1+



ωd −ωl (n)
Γ2 /2

2 ,

rsd
where ΩR is the Rabi rate of the sideband drive, and αn+l,nis the overlap given by Eqn.(2.93). At
rsd
smaller l, the increased overlap αn+l,n2more than makes up the larger detuning. Moreover, the|nilarger detuning at smaller l also creates a more uniform sideband rate Γl (ωd ) as a function of n,
resulting in a more efficient manipulation of phonons (see Sec.6.3.1).
In Fig. 7.6(d), I numerically demonstrate the effect of this spurious cooling process. I use the2

rsd
calculated qubit spectrum of Fig. 5.9(c) to find αn+l,n. With the driving parameters extracted

from Sec.6.3.1, I incorporate up to the 10-th order red sideband transitions into the master equationof Eqn.(6.6) and predict the phonon distribution under a chirped ac-dither blue sideband drive.
Compared to Fig. 7.6(c) where the red sideband transitions are ignored, the effect of the spurious
cooling is clear. Indeed, as Fig. 7.7 shows, I am able to experimentally prepare a sub-Poissonian
state with hni ≈ 77 with only the chirped blue sideband drive and its associated spurious cooling
effects.
Conversely when driving the ac-dither sideband above the qubit, a spurious heating effectappears. To demonstrate this effect, I apply an ac-dither red sideband drive above the qubit
frequency at approximately nR ≈ 10 and measure the qubit spectrum. As illustrated in Fig. 7.8(a),this ac-dither red sideband drive is close in resonance with the 9-th order blue sideband transition|g, 39i ↔ |e, n + 48i. This causes the phonon population to bifurcate: for populations close ton = 10, they are cooled to lower occupations, while phonons further above are pushed by the
unwanted blue sideband transitions to even higher numbers. Consequently, the mechanical state
can be regarded as a mixture of two mechanical states, one with a large motion, while the other has
a small motion. Fig.7.8(b) shows that these two states can be differentiated by looking at the cavity
phase ϕc when the qubit is in the ground state (i.e., parity measurement), because their different

155

Figure 7.6: A spurious cooling effect from driving ac-dither sideband below the qubit: theory|0ia, A sideband drive at frequency ωd = ωq − ωdither + ωm is simultaneously resonant with theac-dither blue sideband transition of |g, 0i ↔ |e, 1i and the 10-th order red sideband transition of|g, 35i ↔ |e, 25i. Thus, using this ac-dither blue sideband drive to add phonons to the mechanicaloscillator will cause unwanted cooling at larger phonon number. b, c, I use the master equations
and the extracted driving parameters found in Sec.6.3.1 to calculate the phonon distribution (colorscale vs. y-axis) after chirping the ac-dither blue sideband drive for time τ (top-axis). This process
should empty all phonon populations below the center of the blue sideband drive nB (τ ) (bottomaxis). The spurious cooling process is not considered in b, but it is modeled in c. The unwanted
cooling is visible at nB (τ ) > 50.

156

Figure 7.7: A spurious cooling effect from driving ac-dither sideband below the qubit: experiment
Chirping a blue sideband drive toward a stationary red sideband drive with nR ≈ 82, I energysqueeze the oscillator’s motion and measure the corresponding qubit spectrum (a, green dots). Its
extracted phonon distribution (b, green solid line) and 90% confidence interval (shade) reveals thethe oscillator’s motion is described by a sub-Poissonian distribution with hni ≈ 77. Performing thesame experiment but now with the red sideband drive off, I measure a similar qubit spectrum (a,blue dots with line), also visibly narrower than the expected qubit spectrum from a coherent state(dashed black line). Consistent with the theory prediction in Fig. 7.6, this demonstrates that, atlarge hni, the spurious cooling effect is so strong that a sub-Poissonian state can be prepared solely
with a chirped blue sideband drive.

157
motional amplitudes rectify ϕc by different amounts (see Sec.4.2.2). Selecting on this phase, I canseparate the overall measured qubit spectrum into two different spectra, each corresponding to alarge-motion state or a small-motion state (Fig.7.8(c)). Indeed, while the small-motion state isconsistent with driving ac-dither red sideband transitions, the large-motion state is consistent withdriving the spurious blue sideband transitions.

158

Figure 7.8: A spurious heating effect from driving ac-dither sideband above the qubit|10i
a, An ac-dither red sideband drive is applied at frequency ωd = ωq + ωdither − ωm . This drive
simultaneously resonates with the ac-dither red sideband transition |g, 10i ↔ |e, 9i and the 9-thorder blue sideband transition |g, 39i ↔ |e, 48i. Starting from a thermal state, this will cool phonons
around n = 10 to smaller occupations, while heating populations above to even higher numbers.
b, This bifurcation of the oscillator’s motional amplitude is revealed by a histogram of the cavity
phase ϕc with the qubit in ground state. Indeed, at the good qubit parity position (around ϕc ∼ 60
deg, see Sec.4.1.3), the histogram shows two separate peaks: because different motional amplitudes
rectify ϕc by different amounts (see Sec.4.2.2), the lower peak corresponds to the mechanical statewith small motion, while the higher peak corresponds to the state with large motion. c, Selecting on
this phase, I separate the overall qubit spectrum (green) into two different spectra, corresponding to
the small-motion (red) and large-motion (blue) states respectively. Compared to the initial thermal
state (dashed orange), the spectrum of the small-motion state demonstrates the cooling effect of
the ac-dither red sideband transitions. Finally, I perform the same experiment but turn off theac-dither signal. The resulting qubit spectrum (dashed black) agrees well with the spectrum of thelarge-motion state, demonstrating the spurious heating effect does not depend on the ac-dither.

Chapter 8

Conclusion

In this thesis, I demonstrated the preparation of a non-Gaussian nonclassical state of motion
in a macroscopic mechanical oscillator with a sub-Poissonian phonon distribution. Indeed, it has
been an enduring ambition in the field of opto- and electromechanics to access a broad range ofquantum states, not just limited to Gaussian states. In this context, this work, along with otherdemonstrations of non-Gaussian nonclassical states of motion [25, 45, 36, 75], represents a major
step toward this goal.
To create this sub-Poissonian nonclassical state of motion, I developed a dissipative stabilization technique that can simultaneously add energy to and extract entropy from a massive mechanicaloscillator. Requiring neither number-resolution nor coherent manipulation, this technique provides
a hardware-efficient and accessible path toward creating highly energized Fock-like states in othercQED experiments. Such states can be valuable resources for quantum metrology: they have been
analyzed for their ability to improve the sensitivity of gravitational wave detectors [15, 14] anddemonstrated to resolve small forces on trapped ions [144].
In creating the sub-Poissonian state, I also observed strong sideband peaks in the qubit
spectrum. These peaks reveal that a spontaneous qubit transition is likely to alter the occupationof the harmonic oscillator, in stark contrast with standard cQED experiments, where a qubit-based
photon (or phonon) measurement is regarded as QND in the far dispersive limit [105]. Indeed, thisQND-ness is but an approximation under the assumption of a large oscillator frequency comparedto the dispersive shift times phonon number (ωm  2χm n): a condition easily violated in this work

160
even by the oscillator’s thermal distribution, nth ∼ 15. Instead, in the new regime achieved by thiswork with ωm ∼ 2χm n, the qubit-mechanics system becomes closely analogous to other natural
molecular systems. This could lead to novel approaches of studying and controlling such artificial
systems with methods developed for molecular systems.
Looking forward, a natural next step to this work would be to increase χm by increasing Vdc .
As discussed in Sec.4.2.3, this is prevented by a breakdown of the qubit readout beyond Vdc = 8 V. A
thorough investigation on this breakdown could reveal interesting three-body interactions that may
be leveraged to generate mechanical cat states or tripartite entangled states involving the qubit, themechanical oscillator, and the microwave resonator [74]. Alternatively, solving this problem would,
in principle, allow the device to be operated at a much larger Vdc = 21 V. With 2χm ≈ 2π ×6.4 MHz
larger than the qubit dephasing rate Γ2 , the hybrid system would enter the strong dispersive limit,
where arbitrary quantum control over the mechanical oscillator can be achieved.

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Appendix A

Device fabrication

The fabrication of the device is performed with several aluminium depositions on sapphire.
This process can be categorized into three major steps.
The first step involves the fabrication of the drumhead. This process is similar to the ones
used to create the electromechanical devices in reference [57, 46]. First, a layer of 100 nm Al filmis sputtered on a sapphire substrate, patterned using optical UV lithography, and etched in a Cl
plasma to create the bottom electrodes of the drumhead. Also defined in this layer of Al film isthe bottom electrodes of the waffle capacitor, which forms part of the on-chip dc-line filter. Next,
a layer of 190 nm silicon nitride is deposited and patterned to allow for the suspension of thedrumhead as well as the waffle capacitors on the dc-line filter. Finally, a second layer of 100 nm
Al film is sputtered and patterned to create the top plate of the mechanically compliant capacitor.
Additionally, this second layer of Al film also creates the CPW-resonator, the ground plane, thearms that connect the Josephson junctions to the bottom drumhead electrodes, and the rest of the
dc-line.
The second step in the fabrication process involves the creation of the Josephson junctions.
The Josephson junction is first defined using electron-beam lithography. Using the Dolan-bridge
technique [cite Dolan 1977], the junction is then created by evaporating two layers of Al at ±20
degree angles (25 nm thick followed by 75 nm thick), separated by a thin layer of oxide. Afterward,
the junction is connected to the rest of the circuit with a bandage technique [cite Dunsworth 2017].
Finally, the drumhead and all other suspended structures, such as the waffle capacitors,

175
are released with a SF6 plasma etch [57, 46]. Once released, the suspended drumhead is free todeform because of the stress in the Al film and thermal expansion and contraction. At the basetemperature of the dilution refrigerator (∼ 25 mK), thermal contraction brings the plate separation
to approximately x0 ≈ 50 nm. The Josephson energy EJ of the junction is not changed significantly
during this step.

Appendix B

Measurement setup and filtering

The schematic of the entire measurement setup is sketched in Fig. B.1. The mechanics-qubitcavity device (blue box) is mounted to the base stage of a dilution refrigerator. This device has
three ports. First, the dc-port allows for the application of the external dc-voltage Vdc , whichsets the qubit-mechanics coupling rate. Additionally, a coherent drive applied on this port at themechanical frequency can drive coherent motion in the mechanical oscillator. Also applied on thisdc-port is the dither signal, which is used to drive sideband transitions. The second port of thedevice is the input-port to the microwave cavity. In addition to applying all qubit control andreadout signals, this port is also used to reset the CPB gate-charge to the sweet spot. Finally,
photons in the cavity leaves primarily from the cavity output-port. Carrying information about
the state of the device, these signals are amplified by the combination of a Josephson parametricamplifier (JPA) and a high electron mobility transistor (HEMT) amplifier before being routed toand collected at the room temperature.

B.1

Filtering the dc-line
The filtering of the dc-line is relatively demanding. It must provide a cold radiation envi-

ronment (approximately 25 mK at the base of the fridge) to the mechanical oscillator at 25 MHz,
while allowing microwave drives at such frequencies. The home-made filter (orange box) placed atbase temperature is designed to achieve those goals. It behaves as a low-pass filter at low frequency
with a 500 Hz cutoff. It becomes a capacitive voltage divider at 500 kHz with about 50 dB inser-

177

Figure B.1: The full measurement setup
I sketch the full measurement setup used in this work. Pictures of the on-chip device (blue box)
can be found in Fig. 3.9.

178tion loss up to about 60 MHz, ensuring cold radiation temperature at the mechanical frequency.
Behaving as a reflective attenuator at these frequencies, the filter also ensures the microwave driveat ωm will not create an excessive heat load on the base temperature stage of the fridge. Finally,
seen at its output port, the filter also has a small real impedance at the mechanical frequency
Re [Z(ωm )]  1 Ω to avoid damping the mechanical oscillator (see discussion in Sec.3.4).
In addition to these mechanical considerations, I also ensure the dc-line will not cause anincreased qubit decay rate (see Sec.2.2.3). This is achieved with a lossless, two-stage on-chip filter.
Each stage of this filter is composed of a spiral inductor Lfil ≈ 20 nH shorted to ground by a
waffle capacitor Cfil ≈ 25 pF, as shown in Fig. 3.9(b). Overall, this on-chip filter has a cutoff
frequency around 1 GHz and provides 52 dB of isolation at the qubit frequency ωq ≈ 2π × 4 GHz.
To provide an estimate for the rate of qubit decay Γv1 caused by voltage noise on this dc-line, I
pessimistically assume an input impedance of 50 Ω before the on-chip filter. Using Eqn.(2.30) andassuming Ec /~ ≈ 2π × 3.2 GHz and Cg ∼ 1 fF, I find a negligible Γv1  1 Hz.

B.2

The cavity input line
The microwave input line has standard attenuation and filtering. Another dc-line is added tothe cavity input at the base temperature with a K250 bias-Tee. This allows for fast control over the
CPB gate-voltage with a 10 MHz bandwidth, and is used to set the CPB to its charge degeneracy
point (see Sec.4.1.2).

B.3

JPA amplification
The output of the cavity is first amplified by approximately 16 dB by the JPA. The amplifier

is pumped at a frequency 4 MHz detuned from the cavity probe signal (at ωc ) to perform heterodyne
detection. To ensure very little residual power from the JPA pump would leak into the cavity, thepump is not only detuned many linewidths away from ωc , but is also actively cancelled with acancellation signal.

Appendix C

An approximate PSF

For many early experiments in this work, I use an approximate PSF map to process the measured qubit spectra and extract phonon distributions. This is achieved using the same techniquesof deconvolution or least-squares fitting discussed in Chapter 5. However, instead of the measured|ni

PSF shown in Fig. 5.9(c), I approximate the qubit spectrum Pe (ω) with exactly |ni phonons asa simple skewed Lorentzian that has been Stark shifted by the phonon number,
Pe|ni (ω) =1
2(AΓ2 /2)2
,
2
|ni
ω−ωq
2
2


+ (Γ2 /2) (1 + A )
|ni

(C.1)

1+L·Sgn ω−ωq|niwhere ωq is the phonon-number-dependent qubit frequency given by Eqn.(2.56), A is a parameter
describing the strength of the qubit drive, Γ2 ≈ 2π × 3.7 MHz is the FWHM of the qubit lineshape,and L describes the charge-noise-induced asymmetry in this lineshape (see Sec.5.2.4.3). This approximation is valid for very small phonon numbers n  50 (because ωm /2χm ≈ 50) as can beobserved in Fig. 5.9, but fails at larger phonon number when the sideband peaks become prominent.
Consequently, extracting the phonon distribution using this approximate PSF is prone to mistakes,
where the deconvolution procedure could erroneously attribute sideband-induced qubit excitations
to phonon populations. Indeed, in Fig. [bsb data] where the data is processed using this approximate PSF, the extracted thermal distribution has nonphysical phonon populations around n ∼ 75
because of the sideband peak (l = 1) located at approximately 3.86 GHz. Nevertheless, this approximate deconvolution procedure allowed for a qualitative understanding of the qubit-mechanics

180
interaction in the early stages of the experiment, allowing, for example, the demonstration of the
motional sidebands’ phonon-number-sensitivity (see Sec.6.2).
Among all extracted phonon distributions displayed in this thesis, only Fig. , Fig. , andFig. D.1 are extracted using this approximate PSF. All other distributions are extracted using themore accurate, experimentally determined PSF described in Sec.5.2.

Appendix D

Mechanical damping rate measurement

I measure the damping rate of the mechanical oscillator by driving it out of equilibrium witha red or blue sideband drive and letting it relax to its thermal equilibrium. At different delay times
after turning off the sideband drive, I measure the qubit spectrum, extract the phonon distribution
using either deconvolution or least-squares fit, and find the mean phonon number hni.
Fig. D.1 shows an example of such a measurement after applying a red sideband drive.
Performed at an early stage of this work, this experiment is conducted with a much larger EJ ≈2π × 4.27 GHz, quite different from the usual EJ ≈ 2π × 3.8 GHz used for the majority of this work.
Consequently, this data is processed with the approximate PSF discussed in Appendix.C. Because
the deconvolution procedure could mistake measurement noise in the qubit spectrum to phonon
populations (see for example Sec.7.1.2), it is likely to overestimate hni, especially at small delay
times when hni is small. Instead, I approximate the phonon distributions as thermal at all delay
times, and perform a joint least-squares fit on all qubit spectra at once to find the corresponding
phonon distributions. The extracted hni are plotted in the inset, and fits well with a decaying
exponential back to the thermal occupancy at equilibrium with the fridge.
Similar measurements are also performed by first driving the blue sideband transitions. Overall, I extract a characteristic mechanical decay time 1/γm ≈ 1.72 ± 0.32 ms.

182

Figure D.1: Measuring γm
At Vdc = 6 V, I measure the qubit spectrum (dots) at different delay times (different colors) after
driving a red sideband transition. Assuming the phonon distributions are all described by thermal
distributions, I perform a joint least-squares fit (solid lines) simultaneously on all qubit spectra to
extract the mean phonon numbers hni. Inset: The extracted hni (dots) are plotted as a functionof delay times. An exponential fit (solid line) finds a characteristic damping time of 1.5 ms.