Converting quantum information to mechanical motion
byAdam P. Reed
B.S., The Ohio State University, 2011
M.S., University of Colorado, 2015

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
2017This thesis entitled:
Converting quantum information to mechanical motion
written by Adam P. Reed
has been approved for the Department of Physics

Konrad Lehnert

John 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
Reed, Adam P. (Ph.D., Physics)
Converting quantum information to mechanical motion
Thesis directed by Prof. Konrad Lehnert

Mechanical systems that combine motion and electricity are often used to process information.
They are employed as compact clocks, filters, and sensors in almost all modern electronic devices.
Yet these devices are limited to processing classical information. To exploit mechanical systems
in emerging quantum communication and computation technologies, such systems must process
fragile quantum bits of information. In this thesis, I experimentally demonstrate the conversion ofquantum bits encoded in electrical signals to the motion of a micron-scale mechanical resonator.
This capability is crucial for harnessing mechanical systems as memories for quantum signals, oras converters of information between electronic quantum processors and telecommunications light.
Beyond quantum information processing, this work opens up the possibility to test quantum theory
in objects of an unprecedented mass scale.

iv

Dedication

To my family

v

Acknowledgements

I’d like to thank my PhD mentor Konrad Lehnert who worked with me since I started graduate
school in 2011. Konrad taught me many things about low-temperature physics and microwave
engineering, and more importantly he showed me how to grow as a scientist. He also taught me
how to effectively communicate scientific ideas to my peers and the community. Furthermore, he
was willing to consider new ideas and ways to solve any sort of challenge that emerged during mytime in graduate school. All of these qualities will serve me well beyond the work in his lab.
In addition to Konrad, I’ve been very grateful to have John Teufel as my second mentor
during graduate school. John was always willing to take the time to chat with me when I waslearning about electromechanics or trying find a direction for my thesis work. His patience andpositive attitude was a constant source of encouragement that kept my spirits high.
I’d also like to thank Cindy Regal, Dave Wineland, and Zoya Popovich for serving on my
PhD defense committee. In particular, I’d like to thank Dave Wineland for providing thoughtfulrevisions to my thesis.
I’m very grateful for our theory collaborators at NIST. In particular, I’m indebted to Karl
Mayer for his contributions to this thesis. Karl was always very patient when I asked silly theory
questions, and he provided thorough yet easy to understand answers. He also took the responsibility
to greatly improve upon my initial data analysis methods. Additionally, I’d like to thank Scott
Glancy and Manny Knill for carefully reading our manuscript and providing valuable feedback.
I’d like to thank many other collaborators at NIST. Florent Lecocq has kindly shared his
knowledge of quantum circuits with me. He also mentored me while I was designing my thesis

viexperiment, and provided valuable feedback regarding my very rough manuscript drafts. ManuelCastellanos-Beltran helped me understand the limitations of JPAs, and taught me the nuances ofcommercial microwave test equipment. Kat Cicak helped me with fabrication and related work in
the NIST clean room. Additionally, I’d like to thank Leila Vale and Martin Sandberg for showing
me how to process Josephson junctions at NIST.
At JILA, I’d like to thank Michael Schroer and Will Kindel for getting me started in the
superconducting qubit world. Michael showed me how to set up a qubit experiment from scratch,and how to execute many important qubit measurements. Will not only developed the qubit-based
photon source used in this thesis, but also taught me how make my own source and integrate itwith our electromechanics experiments.
It has been a pleasure working with many other current and former Lehnert lab members.
Jen Harlow taught me how to analyze microwave circuits when I first joined the lab. She also
thoughtfully mentored me while I was searching for a thesis topic. Reed Andrews showed me how
to attack problems from new angles, and taught me effective troubleshooting skills both in the clean
room and in the lab. Tauno Palomaki was always very patient when I asked him endless questions
about his experiment. Brad Mitchell joined me in our efforts to adopt and understand waveguidemicrowave cavities for use in our lab. Lucas Sletten and Xizheng Ma patiently pushed forward ourefforts to operate cavities and qubits with long coherence times. Rob Delaney spearheaded new
ways to operate our electromechanical device as an amplifier.
I’d also like to thank everyone else who has supported me in the Lehnert lab: Andrew
Higginbotham, Brad Moores, Max Malnou, Jeremie Viennot, Joe Kerchoff, Hsiang-Sheng Ku, PeteBurns, Dan Palken, Ben Chapman, Eric Rosenthal, Max Urmey, Nicole Ehrlich, Tim Menke,
Mehmet Anil, and Gerwin Koolstra. In many ways, my lab mates have helped me out on a dayto-day basis and have served as a source of encouragement during my time in graduate school.
I’ve also had the pleasure to learn optics knowledge from Bob Peterson, Tom Purdy, and Ben
Yu in the Regal lab.
JILA has a unique and amazing set of technical shops, and I benefited from their expertise.

vii
The members of the electronics shop have helped me design and build many custom pieces ofelectronics. The members of the machine shop not only built many things for our lab and provided
their technical support, but also taught me many important machine shop skills.
Additionally, members of the computing services have helped me out on numerous occasions
when I had to deal with the agonies of software. In particular, I’d like to thank the computer
wizard JR for helping me connect up the network of computers used in our experiment.
I’d like to thank Mark Carter for going out of his way to ensure that our experiments always
had enough liquid helium to keep them running. In the extraordinary event that liquid helium was
not available, Ted Stauffer at NIST and Dave Erikson at JILA went out of their way to help me
safely move a Dewar of liquid helium from NIST to JILA just to keep my experiment running.
In addition to my colleagues in Boulder, I’m very grateful have the opportunity to collaborate
with Rob Schoelkopf and his team at Yale. This extraordinary team is world-class and friendly.
Matt Reagor got me off the ground with the collaboration, and was always willing to patiently
answer my numerous questions about operating transmon qubits. Luke Burkhart fabricated the
awesome transmon qubit used in this thesis, and answered many questions about it. Wolfgang Pfaff
provided his expertise in creating flying microwave quantum states. Chris Axline shared with me
many important experimental details. Luigi Frunzio enthusiastically helped me troubleshoot manyof the issues we had when shipping transmon qubits from Yale to JILA.
Outside the lab I’ve had the good fortune to discuss my experiments with Karl Mayer, Bryce
Bjork, Steven Moses, and Rabin Paudel. And we enjoyed home-brewing many beers together, too.
Lastly, I’d like to thank my family for their support and encouragement as I pursed my interests in experimental physics. My mom and dad were very patient as I built trebuchets, lasers, Tesla
coils, rockets, and many other things that could have killed me when I was young. My life-long partner Li-Wei Hung not only introduced me to powerful statistical analysis methods when I was trying
to understand my data, but more importantly, she also supported me throughout the inevitable
ups and downs of graduate school. Her love and support drove me to finish this thesis.

Contents

Chapter
1 Introduction
1.1

1

Thesis contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Theory of continuous- and discrete-variable quantum information
2.1

2.2

2.3

2.4

7
8

Harmonic oscillators as continuous-variables systems . . . . . . . . . . . . . . . . . .

9

2.1.1

Classical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.1.2

Quantum description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3

Quantum states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.4

Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Qubits as discrete-variable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1

Representation on the Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2

Rotations on the Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 15

A qubit coupled to a harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1

Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2

Dispersive regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.3

Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.4

Blue sideband transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Stationary and propagating modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1

Input-output theory for a one-port resonator . . . . . . . . . . . . . . . . . . 20

ix
2.4.2
2.5

Resonator state evolves into a propagating mode . . . . . . . . . . . . . . . . 20

Quantum state tomography of a propagating mode . . . . . . . . . . . . . . . . . . . 21
2.5.1

Homodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5.2

Heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5.3

Method of maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Theory and characterization of the electromechanical device
3.1

3.2

Theory of the electromechanical device . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1

Heisenberg-Langevin equations of motion . . . . . . . . . . . . . . . . . . . . 27

3.1.2

Time-dependent dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.3

Operation as a linear amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.4

Frequency-tunable device using electrostatic actuation . . . . . . . . . . . . . 39

Characterization of the electromechanical device . . . . . . . . . . . . . . . . . . . . 42
3.2.1

Circuit parameters and frequency tuning . . . . . . . . . . . . . . . . . . . . . 43

3.2.2

Electromechanical coupling and mechanical occupation . . . . . . . . . . . . . 44

3.2.3

Mechanical relaxation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Design and characterization of the photon source
4.1

4.2

4.3

26

47

Design of the photon source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.1

Qubit design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.2

Cavity design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1.3

Coupled qubit and cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Characterization of the qubit and cavity . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1

Cavity spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2

Measuring the qubit state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.3

Qubit characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Generation of propagating single photons . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3.1

Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

x
4.3.2

Generation and measurement protocol . . . . . . . . . . . . . . . . . . . . . . 66

4.3.3

Quadrature amplitude extraction . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.4

Density matrix reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Conversion of propagating single photons to mechanical motion
5.1

Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.1

5.2

5.4

5.5

Sub-optimal capture of single photons . . . . . . . . . . . . . . . . . . . . . . 74

Electromechanical device as a low-noise amplifier . . . . . . . . . . . . . . . . . . . . 77
5.2.1

5.3

73

Amplification of thermal states . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Two protocols used to characterize the capture process . . . . . . . . . . . . . . . . . 81
5.3.1

Calibration and capture protocols . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3.2

Quadrature amplitude extraction . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.3

Scaling the quadrature amplitudes . . . . . . . . . . . . . . . . . . . . . . . . 84

Capture, storage, and amplification of single photons . . . . . . . . . . . . . . . . . . 85
5.4.1

Tomographic procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.2

Tomographic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.3

Characterizing nonclassicality . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4.4

Characterizing the storage time . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Characterization of the capture process . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.5.1

Average fidelity characterizes process . . . . . . . . . . . . . . . . . . . . . . . 91

5.5.2

Classical bound on the average fidelity . . . . . . . . . . . . . . . . . . . . . . 92

5.5.3

Conversion of qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.5.4

Model of the capture process . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 Conclusion

98xi
Bibliography

101

Appendix
A Measurement network

120

A.1 Electromechanical device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.2 Transmon qubit and microwave cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.3 Actuation line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A.4 Arbitrary microwave pump and signal generation . . . . . . . . . . . . . . . . . . . . 122
A.5 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A.6 Strong pumps saturate the JPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
B Error analysis

126

B.1 Density matrix estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B.2 Average fidelity calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

TablesTable
3.1

Parameters of the electromechanical device . . . . . . . . . . . . . . . . . . . . . . . 43

4.1

Parameters of the cQED system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

B.1 Systematic errors on the estimated density matrix elements . . . . . . . . . . . . . . 128
B.2 Estimated density matrices used to calculate the average fidelity . . . . . . . . . . . 131

Figures

Figure1.1

Schematic of an electromechanical device. . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Schematic of an early electromechanical device . . . . . . . . . . . . . . . . . . . . .

3

1.3

A source of quantum information coupled to a mechanical resonator . . . . . . . . .

6

2.1

A mass connected to a spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92.2

Diagram of a Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3

Energy level diagram of the qubit-cavity system . . . . . . . . . . . . . . . . . . . . . 18

2.4

Evolution of a stationary mode into a propagating mode . . . . . . . . . . . . . . . . 19

2.5

Plots of Q-functions for ideal vacuum and single-photon states . . . . . . . . . . . . 23

3.1

Diagram of a cavity optomechanical system . . . . . . . . . . . . . . . . . . . . . . . 28

3.2

Diagram of the coupled propagating modes, electromagnetic resonator, and mechanical resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3

Diagram depicting the capture, storage, release of a propagating field . . . . . . . . . 34

3.4

Model of a two-mode parametric amplifier . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5

Two-electrode model of a tunable capacitor . . . . . . . . . . . . . . . . . . . . . . . 40

3.6

A three-electrode tunable capacitor incorporated in a microwave circuit . . . . . . . 41

3.7

Schematic and micrograph of the electromechanical device . . . . . . . . . . . . . . . 42

3.8Frequency tuning of the electromechanical device . . . . . . . . . . . . . . . . . . . . 42

xiv
3.9

Electromechanical coupling rate and equilibrium occupation of the aluminium membrane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.10 Mechanical relaxation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1

Schematic of a Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2

Schematic of a transmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3

Schematic and picture of the qubit in a cavity . . . . . . . . . . . . . . . . . . . . . . 52

4.4

Cavity spectroscopy at room temperature . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5

Cavity transmission as a function of probe power and frequency . . . . . . . . . . . . 57

4.6

Histograms of single-shot measurements using the JPA . . . . . . . . . . . . . . . . . 59

4.7

Single-shot measurements of the qubit state using the electromechanical device as
an amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.8

Calibration of the qubit control pulses . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.9Time-domain measurements of the qubit coherence times . . . . . . . . . . . . . . . 63

4.10 Diagram of the photon generation apparatus . . . . . . . . . . . . . . . . . . . . . . 66
4.11 Single photon generation and detection . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.12 Histograms of measured quadrature amplitudes for single photons. . . . . . . . . . 72

5.1

Diagram of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2

Picture of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3

Initial capture protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.4

Sub-optimal capture of propagating coherent signals and single photons . . . . . . . 78

5.5

Protocol for amplifying thermal states . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.6

Amplified thermal states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.7

Calibration and capture protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.8

Adjustable gain of the mechanical amplifier . . . . . . . . . . . . . . . . . . . . . . . 83

5.9

Correlated transmon qubit and mechanical states . . . . . . . . . . . . . . . . . . . . 86

5.10 Capture, storage, and amplification of single propagating photons . . . . . . . . . . . 87

xv
5.11 Evolution of a mechanical thermal state . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.12 Black box model of the conversion process . . . . . . . . . . . . . . . . . . . . . . . . 91
5.13 A set of linearly independent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.14 Conversion of propagating qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.15 Model of the capture process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
A.1 Detailed schematic of the measurement network . . . . . . . . . . . . . . . . . . . . . 121
A.2 Temporal envelopes of the microwave pumps and signals . . . . . . . . . . . . . . . . 124
B.1 Bootstrapped histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.2 Bootstrapped histograms of the calculated g (2) function . . . . . . . . . . . . . . . . 130
B.3 Bootstrapped histograms of the average fidelity . . . . . . . . . . . . . . . . . . . . . 131

Chapter 1

Introduction

Motion contains information. In 1797, this simple idea enabled Henry Cavendish to accurately
measure – for the first time – Earth’s mass using mechanical motion[1]. His approach was to measure
the gravitational interaction between two lead balls. One of the balls was freely suspended, andits motion was measured using light. Cavendish used the suspended ball to convert the force ofgravity to mechanical motion. From motion, he gained information about the mass of Earth.
Light also contains information. And remarkably, it can be converted to motion. Even a century before Cavendish’s famous experiment, Johannes Kepler made the observation that a comet’s
tail points away from the sun[2]. Put in another way, the deflection of the comet’s tail contains
information about the direction of the sun’s rays[3]. This strange behavior was not rigorously explained until 1873 when Maxwell developed the theory of electromagnetism, which predicted thatlight can move objects[4]. By 1901, Maxwell’s prediction was quantitatively verified by experimentsthat measured the displacement of vanes mounted on a spindle as they were illuminated with thegentle touch of light[5, 6]. However, these experiments suffered from thermal effects that obscured
motion due to light.
By the 1970s, the invention of the laser led to a new era of manipulating objects using light.
Because the laser provided high intensity light, it could easily trap and accelerate freely suspended
dielectric spheres micrometers in diameter[7]. This advance inspired proposals aimed at harnessing
light to trap and cool the motion of single atoms[8, 9, 10]. Such ideas were utilized in experimentsthat used laser light to damp the motion of a trapped atom[11, 12], which enabled cooling of an

2

Figure 1.1: Schematic of an electromechanical device. A mechanical element forms part of acapacitor (C) which is shunted by an inductor (L). These two elements form an electrical resonant
circuit that is inductively coupled to a transmission line. Propagating electrical signals (green) areinjected into a port.

atom to its lowest possible energy state of motion[13]. This technique of laser cooling is now the
workhorse of many atomic physics experiments in laboratories around the world, including ones
here in Boulder at the National Institute of Standards and Technology (NIST) and JILA.
Apart from laser cooling, converting information between mechanical motion and light wasstudied in the context of improving the sensitivity of early gravitational wave detectors. Initialefforts in the 1960s made detectors using massive vibrating bars coupled to electrical circuits[14,15, 16]. While controversial, such efforts inspired a new type of gravitational wave detector thatused light to measure motion[17, 18]. The basic idea was to measure the disturbance caused by agravitational wave on suspended masses in a laser interferometer. This led to the development of theLaser Interferometer Gravitational Wave Observatory (LIGO) which would – because of immensely
heroic and resilient efforts – directly measure the amplitude and phase of a gravitational wave forthe first time[19]. This stunning result provided a surprising amount of information about black
holes billions of light years away from Earth[20].
The development of gravitational wave detectors – spanning over 40 years – led to fantastic
progress in the understanding of how light and motion fundamentally affect each other. Experimentsin the 1970s revealed that light shined on a mirror could not only move it, but also damp or
amplify its motion[21]. As experimenters pushed the sensitivity of interferometers, it was discoveredthat quantum mechanics may limit how much information can be gained from motion[22, 23,24]. From these investigations emerged the parallel fields of cavity optomechanics and circuit
electromechanics[25, 26, 27]. Their aim is to advance fundamental science and technology by

3

Figure 1.2: Schematic of an early electromechanical device. A microwave cavity (mwr) is covered bya flexible metal diaphragm (D) which forms a small capacitive gap (d). Mechanical displacements∆x of the diaphragm modulate microwave fields in the cavity. Image reproduced from Ref. [23].

harnessing light or electricity to manipulate mechanical motion. In this thesis, I will focus on
circuit electromechanics.
A schematic of a circuit electromechanical system is shown in Fig. 1.1. The circuit is formed by√an inductor (L) and capacitor (C) which resonate at the frequency defined by ω0 /2π = 1/(2π LC).
The upper plate of the capacitor is a flexible element which is free to vibrate. Because the capacitance depends on the separation of the two plates, motion of the upper capacitor plate will modulate
the electrical resonance. By injecting electrical signals into the circuit and measuring its response,
information about the motion of the capacitor plate can be gained.
The initial goals of circuit electromechanics were to develop methods for measuring small
mechanical displacements of macroscopic objects. In the early 1980s, an early electromechanicaldevice consisted of a centimeter-sized bulk niobium cavity with a flexible diaphragm[23]. This
device relies on the parametric coupling between motion and electricity in which vibrations of
the flexible diaphragm modulate the resonant frequency of the cavity, as diagrammed in Fig. 1.2.
Remarkably, the level of performance achieved was better than any optical methods for measuring displacements at the time. By the 1990s, advances in microfabrication techniques – due tothe booming silicon industry – led to a drastic miniaturization of mechanical elements to thenanometer and micrometer scale. These capabilities enabled experiments to construct sensitive
detectors by coupling micrometer- and nanometer-scale mechanical resonators to single-electron4transistors[28, 29, 30], atomic point contacts[31, 32], superconducting circuits[33, 34, 35], piezoelectric materials[36, 37], and even single-electron spins[38]. Apart from improving detectors, theoretical proposals emerged[39, 40, 31, 41, 42, 43, 44, 45, 46] that aimed to use circuit electromechanics
to observe quantum states of motion in a macroscopic object for quantum information processing
and foundational tests of quantum theory. A prerequisite to accessing quantum states of motion isthe ability to cool mechanical motion to its quantum ground state.
Cooling mechanical motion to its quantum ground state requires minimizing undesired thermal effects that will obscure quantum behavior. For a mechanical resonator that oscillates at ωmand is at a temperature T , reaching the ground state of motion requires reducing the resonator’s
thermal energy kB T so that it is far below the mechanical quanta of energy ~ωm where kB is
the Boltzmann constant and ~ is the reduced Planck constant. In 2010, experimenters cooleda piezoelectric mechanical resonator that oscillates at ωm /2π ≈ 6 GHz to an environment temperature of T < 100 mK. This low temperature enabled the single 6-GHz mechanical mode tooccupy its quantum ground state approximately 93% of the time[37]. Instead of simply reducing
the environment temperature of the resonator, one can use optical or microwave fields to cool the
motion to its quantum ground state. This idea spawned intense efforts both theoretically[47, 48]
and experimentally[49, 50, 51, 34, 52] to develop methods – borrowed from early experimentswith trapped ions[13] – for reaching the quantum quantum state of motion in a macroscopic object.
When I started graduate school in 2011, two experiments reached this regime. One experiment usedmicrowave fields to cool the motion of a micrometer-scale aluminum membrane in a microwavefrequency superconducting circuit[53]. A separate experiment used optical fields to cool the motionof a nanometer-scale crystal in a photonic cavity[54].
The ability to prepare mechanical motion in its ground state enabled experiments to probe
profoundly quantum phenomena in macroscopic objects. One of the most basic phenomena is
the inherent fluctuations of a mechanical resonator even while in its ground state. These zeropoint fluctuations have been observed in optomechanical[55] and electromechanical[56, 57] devices,and then manipulated to produce squeezed states with displacement fluctuations less than the5resonator’s zero-point motion[57, 58, 59]. Furthermore, the zero-point motion of a mechanicalresonator was entangled with a propagating microwave field.[60].
These experiments enable electromechanical devices to serve as general-purpose signal processing elements that could be used in emerging quantum communication[61, 62] and modular
quantum computing architectures[63]. In a single electromechanical device, the parametric coupling between motion and electricity can be rapidly varied. For example, the ability to suddenly
turn off the interaction between electricity and motion allows the state of a propagating field tobe converted to, and trapped in, the motion of the resonator[64]. From this ability emerges fantastic practical applications of the electromechanical device, such as an on-demand memory[64],a dynamic signal processing element[65], and a pulsed amplifier[60, 66] for propagating quantum
signals. Other experiments have also demonstrated the ability to frequency convert or continuously
amplify microwave fields using an electromechancial device[65, 67, 68].
To exploit the aforementioned signal-processing capabilities of an electromechanical device
in a general quantum information processor, one must work with states that have non-Gaussianstatistics such as a superposition of zero and one photons[69]. These states may enable information
processing that cannot be performed classically[70]. Although Gaussian states can be harnessed forcertain quantum information tasks[71], any process using only Gaussian states can be simulated
efficiently on a classical computer[72]. But in the regime that electromechanical devices operate,
the equations that describe the coupling are linear[27]. This linearity ensures that a Gaussian stateof the microwave field or mechanical resonator will never evolve into a non-Gaussian state.
Accessing non-Gaussian mechanical states requires either a nonlinear detector such as a single
photon counter or a source of non-Gaussian states. In optical systems, single photon counters havebeen used to generate and measure single-photon states[73]. This technique relies on ‘heralding’
the presence of a single photon generated by a source of correlated two-photon pairs. In an optoand electromechanical device, correlated pairs of mechanical quanta – phonons – and photons canbe produced[60]. Because the pairs of phonons and photons are correlated, detection of a single
photon corresponds to the presence of a single phonon occupying the mechanical resonator[74]. For

6Figure 1.3: A source of quantum information coupled to a mechanical resonator. The source
consists of a qubit with a ground |gi and excited state |ei. Electrical signals (green) propagate ina transmission line (black). Such signals encode the qubit’s state as superpositions of zero and one
photons where ϕ is the phase of the qubit.

an electromechanical device, correlated single photons can in principle be resolved using a superconducting qubit[44, 75, 66]. Similarly, the use of a nonlinear detector has been recently exploredin optomechanical systems[76, 77]. If a nonlinear detector is not used, the mechanical resonatorcan be coupled to a source of non-Gaussian states. This approach enabled an experiment in 2010
to take the first pioneering step towards deterministically converting quantum information froma non-Gaussian source to mechanical motion[37]. Specifically, the source was a superconducting
qubit which strongly coupled to a piezoelectric mechanical resonator. However, the few-nanosecond
lifetime of the mechanical resonator prohibited tomographic measurements of its state.
The strategy I pursue in this thesis is to exploit a superconducting qubit as a source of
propagating photons that carry quantum information to an electromechanical device[78]. A diagramof the experiment concept is shown in Fig. 1.3. In this approach, the qubit and mechanical resonator
are in two separate modules. As such, each module can be readily characterized and separately
optimized to reach state-of-the-art levels of performance[79, 53]. Communication from the qubit to
the mechancial resonator occurs naturally through the use of the propagating microwave-frequency
photons. In the optical domain, the use of propagating photons has been considered as a route
towards producing non-Gaussian mechanical states[80, 81, 82]. The benefit to working in the
microwave domain is that superconducting qubits can deterministically produce arbitrary quantum
states[83, 84, 85], which can also be encoded in propagating microwave photons[86].
After encoding the quantum state in a propagating photon, it can be converted to the state

7of the mechancial resonator. As an initial test, I use the electromechanical device to capture,
store, and amplify single photons generated by a superconducting qubit. To efficiently measure themechanical state, I developed a new way of operating the electromechanical device as a low-noise
amplifier. I exploit this capability in order to determine the density matrix of the mechanical state.
I find that the quantum state can be stored on a timescale exceeding 100 µs, an improvement ofover four orders of magnitude compared to previous work that demonstrated the storage of a nonGaussian state in an electromechanical device[37]. To characterize how the capture process affects
arbitrary propagating qubit states, I use the electromechanical device to capture superpositionsof zero and one photons. The degree to which this process preserves quantum information isquantified by the average fidelity[87], which I find to be Favg = 0.83+0.03−0.06 where the limits are
the 90% confidence interval. This level of performance exceeds the fidelity achievable using onlyclassical resources, indicating that the electromechanical device is suitable for the transduction ofquantum information.

1.1

Thesis contents
Chapter 2 provides a brief introduction to quantum information encoded in continuous anddiscrete variables. This chapter is not meant to provide a rigorous derivation of the many rich topics
in quantum information theory, but rather it introduces ideas that are relevant for subsequent chapters. Chapter 3 describes how incorporating a mechanical resonator in an electrical circuit forms
an electromechanical device, which can be used to manipulate propagating microwave fields. Additionally, it describes the characterization of the electromechanical device. Chapter 4 describes how
a circuit quantum electrodynamics (cQED) system forms a source of propagating single photons.
Additionally, it briefly discusses characterization of the source. Chapter 5 demonstrates the ability
to faithfully convert the propagating single photons to motion of the mechanical resonator. Lastly,
Chapter 6 provides an outlook for future experiments that may utilize this conversion capability.

Chapter 2Theory of continuous- and discrete-variable quantum information

Emerging quantum processors and quantum communication networks require the ability tomanipulate information encoded in the state of a quantum system[88, 89]. There are numerous
physical implementations of such quantum systems that include photons[90], ions[91], nuclear andelectronic spins[92, 93, 94], ultracold atoms[95], and superconducting circuits[96]. Among thesevastly different systems is a description of quantum information in terms of either a continuous ordiscrete variable[71, 69]. This dichotomy is analogous to how classical information can be encoded
as either an analog or digital signal. To efficiently process and distribute information, virtually
all modern computation and communication devices utilize both analog and digital encoding[97].
Similarly, emerging quantum technologies may rely on a combination of discrete- and continuousvariable quantum systems.
This chapter presents a brief overview of continuous- and discrete-variable systems that are
relevant for subsequent chapters. Section 2.1 introduces classical and quantum harmonic oscillators,
which are continuous-variable systems. Section 2.2 introduces qubits, which are discrete-variable
systems. Section 2.3 describes how incorporating a qubit into an electromagnetic resonator can be
exploited to generate single photons. Section 2.4 describes how quantum states are generated instationary modes, and then transmitted using propagating modes in a transmission line. Section 2.5
describes how to characterize a propagating mode using quantum state tomography.

9

Figure 2.1: A mass connected to a spring.

2.1

Harmonic oscillators as continuous-variables systems
A continuous-variable system encodes quantum information in states that can have a con-tinuum of eigenvalues. In this thesis, I employ two such systems: a mechanical resonator and a
microwave field. The mechanical resonator can be described in terms of its position and momentum,
whereas the microwave field can be described in terms of its amplitude and phase quadratures[98].
Although these two systems are physically quite different, they can both be modeled as harmonic
oscillators.

2.1.1

Classical description

In classical mechanics, the prototypical harmonic oscillator is a mass connected to a spring
as diagrammed in Fig. 2.1. The object has a mass m and it experiences a linear restoring force
F = −kx where k is the spring constant. Because of this force, the object is harmonically bound.
The complete state of the object at time t is described by its position x(t) and momentum p(t) =mv(t) where v(t) is the object’s velocity. Both of these variables are governed byṗ(t) = −kx(t)ẋ(t) =

p(t)
m

(2.1)

which are a set of coupled differential equations written in the Hamiltonian form[99]. If the massis initially displaced by x(0), solutions to the above equations reveal that its position oscillates atpan angular frequency of ω = k/m according to x(t) = x(0) cos(ωt).
Hamilton’s equations explicitly couple position and momentum for the spring-mass system.

10
To write a pair of equations that are decoupled, complex field amplitudes are used to describe thestate of the system. Specifically, appropriate linear combinations of Eqns. 2.1 yieldȧ(t) = −iωa(t)where√
a(t) =

mω
2



ip(t)
x(t) +mω

and ȧ∗ (t) = +iωa∗ (t)√∗and a (t) =

mω
2

(2.2)



ip(t)
x(t) −mω

(2.3)

are the complex field amplitudes[100]. These amplitudes have been normalized such that |a(t)|2describes exactly half of the system’s total energy, given by H = T + V where T = p2 /(2m)and V = ω 2 x2 /2 are the object’s kinetic and potential energies, respectively. The utility of thisapproach is that the field amplitudes a and a∗ are classical analogs of the annihilation and creation
operators used in the formalism of the quantum harmonic oscillator.

2.1.2

Quantum description

The quantum analog of a classical harmonic oscillator is described by its total energy, as
discussed in Ref. [101]. Briefly, the quantum harmonic oscillator has a total energy given by the
HamiltonianĤho =p̂2
1+ mω 2 x̂2
2m 2

(2.4)

where x̂ and p̂ are operators that satisfy the canonical commutation relation given by[x̂, p̂] = x̂p̂ − p̂x̂ = i~(2.5)

where ~ is Planck’s constant h divided by 2π. Using this commutator relation, the Hamiltonian for
the quantum harmonic oscillator can be cleverly written as1†Ĥho = ~ω a a +
2

(2.6)

where
r
a=

mω
2~



ip̂
x̂ +
mω

†

and a =

r

mω
2~



ip̂
x̂ −
mω


(2.7)

are the ‘annihilation’ and ‘creation’ operators, respectively. As written, they satisfy [a, a† ] = 1.

11
Importantly, the quantum harmonic oscillator has an infinite set of discrete energy states{|ni} for any integer n ≥ 0. Such states are defined byĤho |ni = En |ni(2.8)

where En = ~ω(n + 1/2) is the energy of each state. A given state can be generated using(a† )n|ni = √ |0i
n!

(2.9)

where |0i is called the ‘vacuum state’ which is the state with the lowest allowed energy. Although
the vacuum state contains no excitations (n = 0), its average energy h0| Ĥho |0i = ~ω/2 is not zero.
To describe the time evolution of the quantum harmonic oscillator, I use the Heisenberg
representation[102]. In this representation, the creation and annihilation operators obey the Heisenberg equations of motion given byȧ(t) =

i[Ĥho , a(t)]~(2.10)and the Hermitian conjugate equation for a† (t). The solutions to these equations area(t) = a(0)e−iωt

and a† (t) = a† (0)eiωt .

(2.11)In order to remove the free evolution of the amplitudes, it is natural to move into a rotating frameof reference. By making the transformation a(t) → a(t)e−iωt , the solutions are a(0) and a† (0).
In an experiment, the operators a(0) and a† (0) are not directly measured. Instead, one can
employ a nonlinear detector (such as a single-photon counter) to measure the average occupancyha† (0)a(0)i of the oscillator. Apart from nonlinear detection, one can employ a linear detector (suchas a linear amplifier) to sense linear combinations of a(0) and a† (0) given by1
X = (a + a† )2

and Y =

1(a − a† )2i(2.12)

which are written in a rotating frame at ω. The above quantities are ‘quadrature operators,’
and they are essentially dimensionless parameters for the position and momentum of a harmonic
oscillator. In Section 2.5, I discuss two detection methods used to measure them.

12
2.1.3

Quantum states

An infinite set of states whose dynamics approach that of a classical harmonic oscillator are
called coherent states. Such states are denoted by |αi and satisfy a |αi = α |αi, where the eigenvalue
α is a complex number[98]. This number relates to the average occupation of a coherent state byhni = ha† ai = |α|2 . In terms of the measured scatter in the quadrature amplitudes X and Y , acoherent state yields a variance of Var(X) = hX 2 i − hXi2 = 1/4 quanta and Var(Y ) = 1/4 quanta,
where ‘quanta’ refers to a single excitation of the harmonic oscillator.
In contrast to a coherent state, the state |ni has no analogous representation in the formalismof the classical harmonic oscillator[98]. This ‘number state’ has n quanta, each with energy ~ω.
Although a number state has a well-defined number of quanta, its phase (the conjugate variable)
is maximally uncertain. Regardless of n, a number state yields quadrature amplitudes X and Y
that have a mean of zero. However, their total variance of Var(X) = Var(Y ) = (2n + 1)/4 quanta
is non-zero for any integer n ≥ 0.
To represent a statistical mixture of pure number states, one employs the density matrix ρ.
For a single pure number state, the corresponding density matrix is ρ = |ni hn|. However, if an
ensemble of number states is prepared each with a statistical probability P (n), the density matrix
becomesρ=X

P (n) |ni hn|(2.13)nwhere n labels each number state in the ensemble[98]. To calculate the expectation value of anobservable, such as a quadrature operator X, one takes hXi = tr(ρX) where tr(· · · ) denotes a trace
operation.
The density matrix provides a convenient description of an ensemble of number states held
at a non-zero temperature T . The density matrix ρth that describes such a system has probabilities
that follow a Boltzmann distribution. As such, they are given by
P (n) =

hnin
(hni + 1)n+1(2.14)

where hni = (exp(~ω/kB T )−1)−1 is the average occupation[98]. When measured in the quadrature

13basis, a thermal state yields quadrature amplitudes X and Y that have a mean of zero. However,the quadrature variance is the same as that of a pure number state with Var(X) = tr(ρX 2 ) =(2hni + 1)/4 quanta and similarly for the Y amplitude.

2.1.4

Decoherence

Any harmonic oscillator in the laboratory will inevitably experience dissipation. For example,
such dissipation could be a result of a frictional force −cẋ(t) which causes the oscillator to loose
energy where c is a constant coefficient. Because of the fluctuation-dissipation theorem[103], thisdamping is also a source of a fluctuating force F0 (t) that drives the oscillator. To include these two
effects in the description of the classical harmonic oscillator, one can write Eqns. 2.1 asṗ(t) = −kx − cẋ(t) + F0 (t)(2.15)

where F0 is the fluctuating variable that obeys a Gaussian distribution with mean zero and has acorrelation function given by hF0 (t)F0 (t0 )i = 2ckB Tenv δ(t − t0 ) where Tenv is the temperature of theoscillator’s environment and h· · · i represents an ensemble average[104]. This noise term ensuresthat the oscillator comes into thermal equilibrium with its environment. Eqn. 2.15 is a specificcase of the Langevin equation, which was originally developed to describe the Brownian motion ofa particle immersed in a fluid[105].
To introduce dissipation in a quantum harmonic oscillator, one can write Eqns. 2.10 in theform of the Langevin equation. In the presence of dampening κ and a fluctuating field a0 (t), thefield operators evolve according toȧ(t) = −iωa(t) −√κa(t) + κa0 (t)2(2.16)

where ha†0 (t)a0 (t0 )i = N δ(t − t0 ) is the correlation function for the fluctuating field a0 (t) andN = (exp(~ω/kB Tenv ) − 1)−1 is the occupation of the environment. Here, ha†0 (t)a0 (t0 )i is normalized to have units of photons/sec. Eqn. 2.16 and the Hermitian conjugate equation for a† (t)constitute the Heisenberg-Langevin equations of motion for a quantum harmonic oscillator[106].

14
These equations provide a full description of how a(t) and a† (t) evolve for the oscillator as its energy
decays because of dissipation.
In terms of the density matrix ρ, the quantum master equation provides a full description of
the dissipative quantum harmonic oscillator. This approach takes the quantum harmonic oscillator
as the system of interest and couples it to an external environment. The evolution of ρtot (t) for
this total system is governed by the von Neumann equation, an equivalent form of the Schrödinger
equation, given byiρ̇tot (t) = − [Htot , ρtot (t)]~(2.17)

where the total Hamiltonian Htot includes the system, its environment, and their mutual interactions. To calculate the state of the dissipative quantum harmonic oscillator, one performs apartial trace over the environment ρ = trenv (ρtot ). By modeling the environment as a collectionof harmonic oscillators, one can derive a master equation for the system of interest. Numerous
derivations exist[107, 108, 109], and so I simply state the result given by
 N 

i
(N + 1)  †dρ= − [Hsys , ρ] −
κ a aρ + ρa† a − 2aρa† − κ aa† ρ + ρaa† − 2a† ρa
dt
~
2
2

(2.18)

where N is the average occupation of the environment, and κ is the rate of energy exchange between
the environment and the oscillator. The first term on the right hand side describes the evolutionof the system of interest described by Hsys , and the remaining two terms account for energy decay
and random thermal jumps in the oscillator’s energy states.

2.2

Qubits as discrete-variable systems
A discrete-variable system encodes quantum information in states that form a discrete andfinite set. The prototypical discrete-variable system is a quantum bit (qubit), which is a fundamental unit of quantum information[70]. For this thesis, I utilize a qubit to generate quantum
superpositions of harmonic oscillator states that encode the qubit’s state.

15

Figure 2.2: Diagram of a Bloch sphere. The red arrow denotes a pure qubit state |ψi parameterizedby ϕ and θ. The qubit’s ground and excited states are denoted by |gi and |ei . For a pure state, thearrow has unit length.

2.2.1

Representation on the Bloch sphere

A single qubit is a two-state system, typically taken to be the ground |gi and excited |eistates of an atom. Because of quantum mechanics, these states can exist in a superposition state|ψi = a |gi + b |ei where a and b are complex amplitudes such that |a|2 and |b|2 yield the probabilityof occupying the |gi and |ei states, respectively. Because the square of these amplitudes correspondto probabilities, they must satisfy |a|2 + |b|2 = 1. This normalization condition removes one degreeof freedom, and so the qubit’s state vector can be represented as|ψi = cos(θ/2) |gi + eiϕ sin(θ/2) |ei(2.19)

where θ and ϕ parametrize the state’s relative populations and their phase, respectively. Importantly, these parameters can be conveniently mapped onto spherical coordinates for a ‘Bloch sphere’
as diagrammed in Fig. 2.2.

2.2.2

Rotations on the Bloch sphere

Alterations to the qubit’s state correspond to rotations on the Bloch sphere. For example,
if the qubit state is |ψi = |gi, it can be rotated by θ = π using σx |ψi = |ei where σx is a Paulimatrix. In general, an arbitrary rotation is represented by a 2 × 2 unitary matrix U . Any such
matrix can be written as exp(iθH/~) where H is a Hermitian matrix that is a linear combinationof the Pauli matrices. This operator corresponds to a rotation on the Bloch sphere. For example,

16
to rotate the qubit state about the x axis by an angle of θ, one would use exp(−iθσx /2).

2.3

A qubit coupled to a harmonic oscillator
To deterministically generate arbitrary quantum states of a harmonic oscillator, it mustcouple to a discrete-variable system. The archetypal qubit and harmonic oscillator system is anatom with two energy levels coupled to a single mode of an electromagnetic resonator. Here, I
refer to a ‘qubit’ as one bit of quantum information, and I call the ‘atom’ the physical object that
initially encodes the qubit. For the harmonic oscillator, the resonator is typically taken to be an
optical cavity, and its lower-frequency counterpart is a microwave cavity or circuit. When the atom
is excited, it can decay in such a way as to create a single excitation of the cavity mode–that is,a single photon. By preparing the atom in an arbitrary superposition state, this qubit can betransferred to the cavity mode manifest as a superposition of zero and one photons.

2.3.1

Jaynes-Cummings model

The Jaynes-Cummings model describes the interaction between an atom and a single quantized mode of a cavity[98, 109]. This fundamental interaction has been realized using real atoms[110],
quantum dots[111, 112], and superconducting circuits[113, 114]. Regardless of the physical system,
the Hamiltonian for the atom is that of a qubit: Hq = ~ωq σz /2 where ωq is its ground to excited state transition frequency. In contrast to the qubit, the cavity mode oscillates at ωc andis a quantized harmonic oscillator with the Hamiltonian Hho = ~ωc (a† a + 1/2) as described in
Section 2.1.2.
By coupling the atom’s dipole operator dˆ to the electric field operator Ê of the cavity, these
two systems can interact with each other. Here, I take the dipole moment to be the matrix element
ˆd = he|d|gi,
and I assume d is real so that dˆ = σx d. Because E ∝ (a† + a), the resulting dipole
interaction −dˆ· Ê is described by the Hamiltonian Hint = ~gσx (a† + a) where g is the coupling rate
between the two systems.
To obtain the Jaynes-Cummings model from Hint , the system must satisfy |ωc −ωq |  ωc +ωq .

17
In this regime, the rotating wave approximation is valid and so one can neglect rapidly oscillating
terms at ωq + ωc . Using this approximation and σx = σ− + σ+ , one obtains the Jaynes-Cummings
Hamiltonian
HJC = ~ωc a† a + ~ωqσz~g+ (aσ+ + a† σ− )22(2.20)

where σ+ (σ− ) excites (de-excites) the atom and I shifted the energy reference by removing theterm ~ωc /2. If the atom and cavity are detuned by |ωq − ωc |  g, they are resonant with each otherand thus can directly exchange energy. The system can also be engineered to enter the dispersive
regime in which the atom and cavity are not resonant with each other.

2.3.2

Dispersive regime

To extract information from the qubit while minimally altering its state, the qubit-cavity
system must enter the dispersive regime. In this regime the qubit’s transition frequency and cavity
frequency are far-detuned from each other with a detuning of ∆ = ωq − ωc and their coupling ratesatisfies g  |∆|. By expanding the Jaynes-Cummings Hamiltonian to second order in g/∆, one
obtains the dispersive Hamiltonian[115]Hd /~ = ωc a† a + ωqσzσz
+ 2χ a† a2
2(2.21)

where χ = g 2 /∆ is the dispersive shift. The energy spectrum of the dispersive Hamiltonian isdepicted in Fig. 2.3 assuming ∆ < 0 as is the case for the cQED system discussed in Chapter 4.
The dispersive Hamiltonian reveals how the presence or absence of a single photon in thecavity substantially alters the qubit’s transition frequency and reciprocally, how the qubit’s state
affects the cavity frequency[116]. By collecting terms with factors of σz /2, one finds that the qubit’s
frequency ωq → ωq + 2χn where n is the number of photons in the cavity. As such, the qubit’stransition frequency is a function of how many photons are in the cavity. Furthermore, by collecting
terms with factors of a† a, one finds that the cavity frequency depends on the qubit state accordingto ωc → ωc + χhσz i where hσz i = ±1 depending on the qubit’s state. As such, one can determine
the qubit state by measuring the response of the cavity.

18

Figure 2.3: Energy level diagram of the coupled qubit-cavity system. The dashed lines correspondto the eigenstates of the uncoupled Hamiltonian, where the cavity photon states are denoted by|ni and the qubit levels are denoted by |gi and |ei. When the qubit and cavity are coupled in thedispersive regime such that |ωq − ωc |  g, the energy levels shift as indicated by the solid lines.

2.3.3

Rabi oscillations

In the presence of a drive field, the qubit’s state |ψ(t)i can undergo ‘Rabi oscillations’ between
its two levels. The time evolution of |ψ(t)i is given by the unitary operator U (t) = e−iHt/~ where
H is the system’s Hamiltonian. If the center frequency of the drive is near the qubit’s transition,
the system’s interaction Hamiltonian is Hint = 12 ~Ω~n · ~σ where ~n is a unit vector that specifies therotation axis on the Bloch sphere, ~σ is a vector of the Pauli matrices, and Ω is the coupling rate
between the drive field and the qubit.
To understand how the drive affects the qubit state, consider the simple case in which ~n = x̂.
The corresponding unitary operator is U (t) = cos(Ωt/2)I2 − i sin(Ωt/2)σx where I2 is the 2 × 2
identity matrix.1 Suppose the qubit is initially prepared in its |gi state. After the drive is on fora time t = T , the probability of the qubit occupying the excited state is
Pe (T ) = | he| U (T ) |gi |2 = sin2 (ΩT /2).

(2.22)

Because the qubit’s population oscillates between its ground and excited states, the qubit can beprepared in any superposition state |ψ(t)i = a(t) |gi + b(t) |ei using a pulsed drive field with an
1

To derive this expression for a 2 × 2 matrix A that satisfies A2 = I2 , use the identity
exp(iθA) = cos(θ)I2 + i sin(θ)A where θ is a real angle[102].

19

Figure 2.4: Evolution of a stationary mode into a propagating mode. a, The diagram depicts asingle stationary mode a(t) coupled to the modes ain (t) and aout (t) that propagate in a transmission
line. The stationary mode oscillates at ω0 and couples to the propagating mode at a rate of κext .
Additionally, the stationary mode couples to the environment at a rate of κ0 . At time t = 0, theamplitude of the stationary mode is a(0). b, The plot depicts aout (t) emitted from the stationarymode. This field can be measured using either homodyne or heterodyne detection (see Section 2.5).

appropriate duration T and coupling rate Ω. For instance, the choice of T = π/Ω drives the qubit
into its excited state. Such a drive is called a ‘π-pulse’ because the qubit’s state rotates by θ = π.

2.3.4

Blue sideband transition

The blue sideband transition provides a mechanism for creating an interaction between thequbit and the cavity[117, 118]. Although these two systems do not directly exchange energy inthe dispersive regime, driving the blue sideband transition at ωb = ωc + ωq creates a simultaneous
excitation of both the qubit and the cavity. For the transmon-style qubit used in this thesis(see Chapter 4), the blue sideband transition occurs only as a two-photon process[115, 117]. The
simplest method of accessing this transition is to drive it using a strong, detuned tone at ωb /2.
When driven in this manner, the qubit-cavity system oscillates between the joint states |gi |0i and|ei |1i. As such, the blue sideband transition enables the creation of a single cavity photon[119].
By coupling the cavity to a transmission line, the photon will decay into a propagating mode.

2.4

Stationary and propagating modes
A quantum state stored in an electromagnetic resonator is confined to a stationary mode.

This mode is essentially a standing wave because it does not, on average, propagate energy in
any net direction. To communicate information contained in the quantum state to a physically-

20
separate system of interest, one must release the energy stored in the stationary mode. This releasecan be achieved by coupling the electromagnetic resonator to a transmission line as depicted inFig. 2.4. If this coupling is constant, the stationary mode will evolve into a propagating mode in
the transmission line at a rate set by the bandwidth of the resonator.

2.4.1

Input-output theory for a one-port resonatorTo mathematically describe a single stationary mode of an electromagnetic resonator, I model
it as a harmonic oscillator with dissipation. As such, the evolution of the complex field amplitudea(t) for this mode is governed by Eqn. 2.16. In a frame rotating at the center frequency of theresonator ω0 , the equation for a(t) becomes√κȧ(t) = − a(t) + κext ain (t)2(2.23)

where κ = κext + κ0 is the total energy decay rate of the resonator, which includes the external
coupling rate κext to the transmission line and the unwanted ‘internal’ decay rate κ0 of energy toan unmeasured port. The external coupling is set by the resonator design, whereas the internal
loss rate is determined by its fabrication details[120]. Here, I assume ain (t) is in the vacuum state,and so ha†in (t)ain (t0 )i = 0 and hain (t)i = 0 as discussed in Section 2.1.4.
2.4.2

Resonator state evolves into a propagating mode

Upon energizing the resonator, its field evolves into a mode that propagates in the transmission line. The amplitude of this propagating field is described by the input-output relation
aout (t) =√κext a(t) − ain (t) where a(t) is defined in Eqn. 2.23. To simplify its solution, I assumeκ0  κext and so
aout (t) =√κa(0)h(t) + κ · (h ? ain )(t) − ain (t)(2.24)

where h(t) = exp(−κt/2), the ? denotes the convolution operation,2 and a(0) is the initial field
amplitude in the resonator.
2

Its definition is (h ? ain )(t) =Rt0h(t − τ )ain (τ ) dτ .

21
To extract a(0) from the continuum of modes in the transmission line, one must define aparticular temporal envelope for the mode of interest[121]. Ideally, for a decaying field emitted
from the resonator, this envelope is given by f (t) =√κe−κt/2 Θ(t) where Θ(t) is the Heavisidefunction. By weighting aout (t) with f (t), one obtains the time-independent output modeZA=f (t)aout (t) dt = a(0)(2.25)

which does not contain the noise term aout (t). Because A = a(0), tomography on mode A reveals
the quantum state of the cavity mode at t = 0.

2.5

Quantum state tomography of a propagating mode
To characterize the state of a propagating mode, I employ quantum state tomography[122].

The goal of quantum state tomography is to determine the density matrix ρ that completely
specifies the quantum state of interest. For a propagating mode, one does not directly measure ρin an experiment. Instead, it can be reconstructed from a set of quadrature amplitudes X and Y
measured using either homodyne or heterodyne detection[123]. The set is constructed by performing
repeated measurements on an ensemble of identically prepared states. From this set, one extracts
ρ by using the statistical method of maximum likelihood.

2.5.1

Homodyne detection

Homodyne detection yields a single quadrature measurement of a field. In the optical domain,a beamsplitter combines the field to be measured and a field called the local oscillator that provides
an adjustable phase reference. Specifically, this phase reference defines an angle φ between thelocal oscillator and the field to be measured. At the beamsplitter’s two output ports, a pair ofphotodetectors produce a signal that is proportional to the field’s quadrature amplitude X. In
the microwave domain[121, 124], one utilizes a phase-sensitive linear amplifier to measure thequadrature amplitude of a given field. Because phase-sensitive amplification can ideally add zero
noise during the amplification process, it enables efficient single-quadrature measurements of a

22
microwave field[125].
By making repeated single-quadrature measurements of identically-prepared quantum states
described by ρ, one samples from its marginal distribution. For a given local oscillator phase, thisprobability density function is given by
Pr(X|φ) = hX| U † (φ)ρU (φ) |Xi(2.26)


where U (φ) = exp −iφa† a accounts for the aforementioned phase reference[123]. To obtain acomplete reconstruction of ρ, one must rotate θ throughout the entire phase space of the quantum
state. This technique is analogous to medical imagining in which a doctor reconstructs a threedimensional picture of a person’s tissue using a collection of two-dimensional images acquired froma known set of angles.

2.5.2

Heterodyne detection

In contrast to homodyne detection, heterodyne detection yields a set of simultaneous quadrature measurements. These joint measurements provide enough information for complete quantum
state reconstruction. In the optical domain, heterodyne detection is performed by splitting a stateof interest on a beamsplitter and measuring a single orthogonal quadrature of each output mode
relative to a common local oscillator[123]. The microwave equivalent of this technique is phaseinsensitive linear amplification[124]. This approach amplifies a signal’s quadratures amplitudes
regardless of its phase relative to the local oscillator.
For quantum-limited heterodyne detection, performing simultaneous measurements of both
quadratures adds at least half a quantum of noise[126]. This noise is accounted for in the probability
density for obtaining a pair of quadrature values X and Y , given by the Husimi Q-function[127]Q(α) =

1hα| ρ |αiπ(2.27)

where α = X + iY and Q(α) is normalized to unity such that 0 ≤ Q(α) ≤ π −1 for all complex
values of α. If ρ is in the vacuum state, the Q-function is Gaussian distribution given byQ(α) =

1 −|α|2e.
π(2.28)

23

Figure 2.5: Plots of Q-functions for ideal vacuum and single-photon states. To clearly represent
the distributions, the plots show a linecuts through the Q-functions for Y = 0 assuming ρ is in
the a, vacuum state and b, single-photon state. The vacuum state follows a Gaussian distribution,
whereas the single-photon state does not.

A plot of a linecut through this function is presented in Fig. 2.5a. In contrast, if ρ is a single-photon
state, the distribution is given byQ(α) =

1 −|α|2 2
e
|α| .
π

(2.29)

The above distribution is not Gaussian as shown in Fig. 2.5b, and so one refers to ρ in this case asa ‘non-Gaussian’ state.

2.5.3

Method of maximum likelihood

The method of maximum likelihood converts a set of measured quadrature amplitudes into adensity matrix ρ. This technique is a general statistical method used to estimate the parameters ofan assumed model, given a set of observed data. For quantum state tomography, the parameters tobe estimated are the elements of ρ and the statistical model is the assumed probability distributionthat the measured quadrature amplitudes are expected to obey.
The goal of maximum likelihood tomography is to find the density matrix which maximizes
the likelihood function for a given set of measurements. In general, given a set of N measurement
outcomes {xk }, each outcome is described mathematically by a positive-operator valued measure(POVM) element Ek [128]. If the system is in the state given by density matrix ρ, the probabilityof observing measurement outcome xk is Pr(xk ) = tr(ρEk ). The probability of obtaining the entire
data set is then the product of the probabilities of each measurement outcome. This product defines

24
the likelihood functionL=Ytr(ρEk ).

(2.30)kMaximizing L may not be computationally efficient because ρ can be a large matrix. Instead, an
equivalent approach is to maximize the log-Likelihood function given byL=X

log(tr(ρEk ))(2.31)kwhere the logarithm converts the product into a sum.
A robust method for maximizing the log-likelihood function is the RρR algorithm[129]. The
algorithm begins with the maximally mixed state ρ(0) = I/d as the initial estimate, where I is
the identity matrix and d is the Hilbert space dimension. With each iteration, ρ is then updated
according toρ(i+1) = N R(ρ(i) )ρ(i) R(ρ(i) ),(2.32)

where N is an overall normalization to ensure that tr(ρ) = 1 at each step, and R(ρ) is a matrix
given by3
R(ρ) =

1 X Ek.
Ntr(ρEk )(2.33)k

The RρR transformation leaves the maximum-likely state unchanged, and while the algorithm is notguaranteed to converge[130], it does so in almost all practical cases, including the cases consideredin this thesis.
The choice of Ek in Eqn. 2.33 depends the level of added noise in the detection method. If
the detection employs the heterodyne method and is quantum-limited, the probability density for
obtaining a pair of quadrature values (Xk , Yk ) is given by the Husimi Q-function[127]:
Pr(Xk , Yk ) = Q(Xk , Yk ) =1
1
hαk | ρ |αk i = tr(ρ |αk i hαk |)ππ(2.34)

Here |αk i is a coherent state with αk = Xk + iYk . From the above equation, note that the POVM
elements for the measurement outcomes are projections onto coherent states:
Ek =
3

1
|αk i hαk | .
πA heuristic way to understand the R matrix is that it is a derivative of log L with respect to ρ.

(2.35)

25
However, if the added noise in the detection method is not quantum-limited, the POVM
elements are no longer projections onto coherent states. For this case, the POVM operators areprojections onto displaced thermal states:
Ek =1
D(αk )ρth D† (αk ).
π(2.36)


where D(α) = exp αa† − α∗ a is the displacement operator and ρth is a thermal state with thermal
occupancy nth . To derive Eqn. 2.36, note that because the added noise of the amplifier is notquantum limited, the probability density for obtaining quadrature values {(Xk , Yk )} is no longer
the Q-function of the input microwave mode but rather the convolution of the Q-function with theadded thermal noise:Z
Pr(X, Y ) =0 2

dα0 Q(α0 )e−|α−α | /2nth

Z

1
0 2
dα0 tr(ρ α0 α0 )e−|α−α | /2nth
π
Z
1
0 2
= tr dα0 ρ α + α0 α + α0 e−|α | /2nthπZ
10 −|α0 |2 /2nth00D† (α)= trρD(α)
dα α α eπ=(2.37)The third equality is from the change of variables α0 → α + α0 , and the integral in the fourth lineof Eqn. 2.37 is equivalent to a thermal state ρth with mean excitation number nth . To see how athermal state emerges in this calculation, note that the P-representation for a thermal state is[123]Zρth =d2 α |αi hα| P (α)(2.38)

0 2

where P (α) = e−|α | /2nth is taken from the fourth line of Eqn. 2.37. Thus, we find
P (αk ) =where Ek is POVM element for a coherent state.

1
tr(ρEk )π(2.39)

Chapter 3

Theory and characterization of the electromechanical device

An electromechanical device converts information between the motion of a mechanical resonator and a microwave field. This device can be exploited in two ways. First, it can capture
the state of a microwave field propagating in a transmission line. Once captured, the state ismechanically-stored and later released. This ‘capture and store’ capability enables the electromechanical device to be utilized as an on-demand memory for microwave qubits[64]. Second, theelectromechanical device can amplify a feeble signal with quantum-limited noise performance.
While similar types of amplifiers are aggressively being developed for measuring superconducting
circuits[131, 132], the electromechanical device provides the benefit of directly amplifying a signalthat is contained in either a microwave field or the mechanical resonator. This benefit enables amicrowave or mechanical quantum state to be efficiently characterized as shown in Chapter 5.
In this chapter, I introduce a theoretical understanding of the electromechanical device andits initial experimental characterization. Section 3.1 establishes a mathematical framework that
describes how the device converts information between the microwave and mechanical domains. As
part of this section, I show how a tunable capacitor enables the electromechanical device to enhance
its operation bandwidth. In Section 3.2, I characterize the parameters of the electromechanicaldevice that are discussed in the theory section.

27

3.1

Theory of the electromechanical device
To combine electricity with mechanical motion, the electromechanical device utilizes an

inductor-capacitor (LC) circuit with a variable capacitor. Charge q sloshes back and forth between the inductor and capacitor, oscillating at about seven billion times a second (a microwave
frequency). One plate of the capacitor is free to move, and so any displacement x alters the
position-dependent capacitance. Because part of the capacitor is flexible, its motion is affected by
the electrostatic force
Fes ∝ q 2 .

(3.1)

From this inherent nonlinearity emerges the ability to damp or amplify the motion of the flexible
capacitor plate, or to control its spring constant.
In this section, I mathematically describe these phenomena. Although the classical dynamics
of the electromechanical device can be calculated by taking the charge on the capacitor plate q andthe plate displacement x as dynamical variables[133], its quantum dynamics are more naturally
modeled using the well-described framework of input-output theory[134, 109, 108]. First, I introduce
the Heisenberg-Langevin equations of motion that model linear electromechanics. Next, I show how
the inherent time-dependence in these equations enables the electromechanical device to capture,
store, and amplify a signal. Lastly, I show how the capacitor plate can be electrostatically actuatedin order to make the center frequency of the electromechanical device tunable.

3.1.1

Heisenberg-Langevin equations of motion

As pointed out in Section 2.1.4, the Heisenberg-Langevin equations of motion provide a framework to describe a harmonic oscillator with dissipation. Because both the mechanical resonator andLC circuit can be modeled as dissipative harmonic oscillators, the Heisenberg-Langevin equations
can describe the dynamics of the electromechanical device. In this section, I start by modeling theclassical dynamics of the electromechanical device. Specifically, I formulate a set of HeisenbergLangevin equations that describe the position of the mechanical resonator and how it couples to28

Figure 3.1: Diagram of a cavity optomechanical system. Two mirrors form an optical cavity, one ofwhich is harmonically bound. An input field bin probes the cavity, and a circulating field b builds up
in the cavity. The reflected outgoing field bout contains information about the state of the system.

the complex field amplitude of the circuit mode.1

I conclude by replacing the amplitudes with

operators in order to describe the quantum dynamics of the electromechanical device.
Formally, the electromechanical device is equivalent to an optical cavity with one of its mirrors
harmonically bound as diagrammed in Fig. 3.1. The cavity is analogous to the LC circuit, and themechanically-compliant mirror is analogous to the flexible capacitor plate. Because the circuit isaddressed by a transmission line, the cavity also is addressed by a propagating mode. This model ofa ‘cavity optomechanical’ system provides a description of how the mirror’s motion and dynamics
of the cavity field affect each other.
Motion of the mirror alters the resonant frequency of the cavity. When the mirror moves by
a displacement x, the resonant frequency of the cavity shifts by Gx where G = ∂ω0 /∂x denotes theshift in the cavity resonant frequency per change in displacement of the mirror. This shift of theresonant frequency is due to the parametric coupling between the mirror’s motion and the cavity
mode. For the cavity depicted in Fig. 3.1, G ≈ ω0 /L where L is the separation between the twomirrors and ω0 is the cavity’s resonant frequency[27]. However, for an electromechanical circuit,G ≈ ω0 /2d where d is the static separation between the capacitor plates[136].
To model the dynamics of the cavity field b(t), I incorporate the parametric coupling mechanism into the equations of motion for a one-port cavity. Although one of the cavity mirrors
is mechanically-compliant, the other mirror is partially transmissive so that energy in the cavity
1This complex field amplitude is a properly normalized variable for the electric field in the circuit mode as
discussed in Ref. [135].

29
decays out at a rate of κext . The corresponding equation for the cavity is√κḃ(t) = −i (ω0 + Gx(t)) −b(t) + κext bin (t)
2

(3.2)

where κ is the total linewidth of the cavity and bin is the amplitude of an input drive field.
Because the cavity field exerts a force on the mirrors, the displacement x(t) of the mechanicallycompliant mirror depends on b(t). The force per photon is ~G, and for |b(t)|2 photons in the cavity
the total force is F = −~G|b(t)|2 . By including F , along with the restoring force of the spring andmechanical dissipation, the equation of motion for the position becomes2ẍ(t) = −κm ẋ(t) − ωmx(t) − ~G|b(t)|2 /meff + ηm (t)/meff(3.3)

where meff is the effective mass of the mechanical resonator[27], ωm its resonant frequency, andηm (t) is a fluctuating force that drives mechanical resonator, as discussed in Section 2.1.4. The
motion of the mechanical resonator is damped because of the dissipative force term with ẋ(t) whereκm is the mechanical resonator’s intrinsic damping rate.
The equations of motion for a cavity optomechanical system are weakly nonlinear in the field
amplitude. The scale of the nonlinear interaction is set by G and the zero-point fluctuations of thepmechanical resonator xzp = ~/(2meff ωm ). In most experiments to date [27], the cavity decay rate
is overwhelming with κ  Gxzp and so the inherent parametric coupling is quite feeble. However,by pumping the cavity with a strong field, one can boost the coupling rate Gxzp by many ordersof magnitude.
In order pump the cavity, it is driven by an input field of the form bin (t) = b̄in e−iωp t . This
field induces an average of |b̄|2 circulating cavity photons, which impart a static displacement x̄
on the mechanical resonator. The equilibrium field b̄ and static displacement x̄ are calculated bysolving Eqns. 3.2 and 3.3 with ḃ(t) = ẍ(t) = ẋ(t) = 0, which yield solutions2
√

b̄ =

κext
b̄in
−i∆ + κ/2and x̄ = −

~G
|b̄|2
2
meff ωm(3.4)

2
2
The relation for x̄ can also be derived by using Hooke’s Law x̄ = −k−1 Fp Nc where k = ωmmeff is the spring
constant, Fp = ~G is the force per photon, and Nc = |b̄|2 is the number of photons in the cavity.

30

Figure 3.2: Diagram of the coupled propagating modes, electromagnetic resonator, and mechanicalresonator. The propagating modes bin (t) and bout (t) couple to the electromagnetic resonator at
a rate of κext . The resonator mode is b(t), which couples to its environment at a rate of κ0 .
Additionally, the resonator couples to the mechanical resonator at a rate of g. The mechanicalresonator couples to an environment at a rate of κm .

where ∆ = ωp − ω0 − Gx̄ has a dependence on b̄ via x̄. This dependence is the analog to the Kerr
effect, and it leads to phenomena such as bistability[137, 138, 139].
Dynamic solutions are obtained by examining fluctuations of the field amplitude b(t) andposition x(t) about their average values. To simplify the equations of motion, I assume steady statesolutions exist[140, 137, 27] and they are of the form b(t) → (b̄ + b(t))e−iωp t and x(t) → x̄ + x(t).
By inserting these replacements into Eqns. 3.2 and 3.3 and assuming |b(t)|  |b̄|, the linearized
equations of motion areḃ(t) = (i∆ − κ/2)b(t) − iGb̄x(t) +2ẍ(t) = −ωmx(t) − κm ẋ(t) −√κext bin (t)~Gb̄b∗ (t) + b̄∗ b(t) + ηm (t)/meff
meff

(3.5)

where the parametric coupling is enhanced because every instance of G is scaled by b̄ or b̄∗ . Thus,
the simple increase of the pump amplitude boosts the parametric coupling.
To highlight the similarity between the dynamics of the cavity and mechanical resonator,
I recast Eqns. 3.5 in terms of the mechanical resonator’s field amplitude x(t) = xzp (c(t) + c∗ (t))
and the pump-enhanced coupling g = Gxzp b̄. In a frame rotating with the pump at ωp , these two

31
substitutions yieldḃ(t) = (i∆ − κ/2)b(t) − ig(c(t) + c∗ (t)) +√κext bin (t) +ċ(t) = (−iωm − κm /2)c(t) − i (gb∗ (t) + g ∗ b(t)) +√√κ0 b0 (t)κm c0 (t)(3.6)

where the terms with b0 and c0 are the fluctuating fields that drive the cavity or the mechanical resonator (see Section 2.1.4), respectively, and the magnitude squared of such terms is normalized tohave units of photons/second. As pointed out in Section 2.4.2, the propagating mode bin (t) is normalized so that |bin (t)|2 has units of photons/second whereas |b(t)|2 and |c(t)|2 are in units of quanta.
Lastly, b(t) couples to bin (t) and bout (t) by the input-output relation bout (t) =√κext b(t) − bin (t).

This equation, together with Eqns. 3.6, describes the linearized behavior of the optomechanicalsystem shown in Fig. 3.1. A schematic representation of these equations is shown in Fig. 3.2.
To describe the quantum dynamics of an optomechanical system[48], one promotes the field
amplitudes to operators (e.g., take b → b̂) and demands that they satisfy the canonical commutation
relations (e.g., require [b̂, b̂† ] = 1). This procedure results in the Heisenberg-Langevin equationsof motion, which provides a quantum description for the cavity optomechanical system shown in
Fig. 3.1. These equations of motion (i.e., the quantized version of Eqns. 3.6) can be generated[141]by starting with the Hamiltonian (here, I drop the hats on the operators)






1
1
†
†
H = ~ω0 b b ++ ~ωm c c ++ ~g0 b† b c + c†
2
2

(3.7)

where g0 = Gxzp is the single-photon coupling rate. In the above expression, the first two terms
describe the harmonic modes of the cavity and resonator, and the last term is the coupling between
the field in the cavity and the mechanical resonator.
After writing Eqns. 3.6 in terms of field operators, an asymmetry emerges in the correlation functions for the noise operators. For instance, the fluctuating field that drives the mechanical resonator obeys hc†0 (t)c0 (t0 )i = nm δ(t − t0 ) where nm is the occupation of the mechanical resonator’s environment. Because of the commutator relations, the field operator also obeys
hc0 (t)c†0 (t0 )i = (nm + 1)δ(t − t0 ). Such correlation functions account for quantum noise present in32
the system[131]. To recover a classical description of the system, one simply takes ensemble averages, e.g., take c0 (t) → hc0 (t)i where h· · · i denotes an ensemble average.

3.1.2

Time-dependent dynamics

The coupling g = Gxzp b̄ factor in Eqns. 3.6 depends on the time-dependent pump b̄. By varying its strength b̄ in time, the coupling becomes g → g(t). This time-dependent coupling provides asimple way to rapidly vary the parametric interaction between a propagating microwave field andmechanical motion. By rapidly varying this interaction, two key capabilities emerge. First, thestate of a propagating microwave field can be captured and stored in the electromechanical device.
Second, it can amplify both the propagating field and the motion of the mechanical resonator.

3.1.2.1

Optimal capture of a propagating microwave field

A propagating microwave field parametrically couples to motion of the mechanical resonator.
To capture the state of the propagating microwave field, one utilizes a pump that is detuned below(red-detuned) the microwave circuit resonance with detuning ∆r = −ωm . For this detuning, I eliminate the evolution of the field amplitudes in Eqns. 3.6 by using the replacements b(t) → b(t)e−iωm t
and c(t) → c(t)e−iωm t to move into a frame rotating with the detuned pump. The dynamics are
simplified by making the rotating wave approximation, which neglects terms oscillating at 2ωmbecause they oscillate significantly faster than any other rates in the equations of motion. In orderfor this approximation to be valid, the condition 4ωm  κ emerges which places the system in the‘resolved sideband’ regime[142]. In this regime, Eqns. 3.6 reduce to√κḃ(t) = − b(t) − ig(t)c(t) + κext bin (t)2κmċ(t) = − c(t) − ig ∗ (t)b(t)
2

(3.8)

where I set c0 (t) = b0 (t) = 0 for brevity.
Eqns. 3.8 can be generated by using the following Hamiltonian
Hr = ~g(bc† + cb† ).

(3.9)

33
This interaction is similar to a beamsplitter Hamiltonian in that it coherently swaps states of the
microwave field and the mechanical resonator. Crucially, the time-dependent coupling g(t) enables
the beamsplitter’s transmission to be rapidly varied. In order to transfer information between a
propagating field, bin (t), and the mechanical resonator, the coupling rate g must be slower thanthe energy decay of the circuit κ. For g < κ/2, the dynamics of the circuit follow b(t). Specifically,
the evolution of the field in the circuit is ḃ(t)  b(t)κ/2 and so its dynamics can be adiabaticallyeliminated[143, 144].
I make this approximation by setting ḃ(t) = 0 and so Eqns. 3.8 reduce to a single differential
equation given by1ċ(t) = −
2



4|g|2
κm +
κ



√
2ig ∗ κextc(t) − √√ bin (t).
κκ

(3.10)

The above equation describes how a propagating field couples to mechanical motion. To describe
the converse situation, one must use bout (t) =√κext b(t) − bin (t) together with Eqns. 3.8. Aftermaking the adiabatic approximation, one obtains
 √

√
κext
2ig κext
bout (t) = −√ c(t) + 2 √ − 1 bin (t)κκκ(3.11)

where bout (t) depends on the motion of the mechanical resonator.
Crucially, Eqns. 3.10 and 3.11 reveal that the rate at which energy is transferred between
motion of the mechanical resonator and the propagating field is simply Γ(t) = 4|g(t)|2 /κ as depicted
in Fig. 3.3a. In terms of Γ(t), these two equations reduce top1ċ(t) = − (Γ(t) + κm )c(t) − e−iψ ηΓ(t)bin (t)2pbout (t) = eiψ ηΓ(t)c(t) + (2η − 1)bin (t)(3.12)

where η = κext /κ is the coupling efficiency and ψ = Arg(−ig) is the phase of the pump. Eqns. 3.12
concisely describe how a propagating field interacts with the mechanical resonator.
To understand how Eqns. 3.12 reveal the ability to cool the mechanical resonator’s motion,
consider capturing a noisy field bin (t). For this case, I assume that this field is emitted from a34

Figure 3.3: Diagram depicting the capture, storage, release of a propagating field. a, The diagram
depicts the propagating modes bin (t) and bout (t) coupled to the mechanical mode c(t) at a rateof Γ. In this case, the circuit mode was adiabatically eliminated and the mechanical dissipation
was neglected. b, The plots depict the coupling Γ and propagating modes as a function of time. At
time t < t1 , the electromechanical device captures an input field bin (t) and converts it to mechanicalmotion. During storage, Γ = 0. After storage, the coupling pulses on at t > t2 and converts theenergy in the mechanical mode back into a propagating field.

source of Johnson noise, and that the mechanical resonator is driven by a noisy field c0 (t). The
noise properties of these two fields are succinctly described by[131]hb†in (t)bin (t0 )i = ne δ(t − t0 )and hc†0 (t)c0 (t0 )i = nm δ(t − t0 )(3.13)

where ne and nm are the average occupations for the electrical and mechanical mode, respectively,and δ(t − t0 ) is a delta function. The solutions to Eqns. 3.12 contain integrals that consume thedelta functions, and for constant coupling Γ(t) → Γ they reveal that the mechanical resonator’soccupation hc† (t)c(t)i decays in time:hc† (t)c(t)i = hc† (0)c(0)ie−(Γ+κm )t +κmnm
Γ + κm(3.14)

where c(0) is the initial state of the mechancial resonator. Additionally, I assumed that the microwave mode is in a ‘cold’ vacuum state such that ne = 0, and the mechanical mode is ‘hot’ withnm  1. The above expression shows that by capturing a cold microwave field, one can cool themechanical resonator to its quantum ground state provided that Γ  κm nm . In this regime, theelectromechanical device is said to be ‘quantum-enabled’ because it can process a quantum signal
with low-added noise.
Beyond capturing a noisy field, one can efficiently capture an arbitrary input field bin (t). For

35
this task, one must choose an optimal Γ(t). If bin (t) has a temporal envelope that rises exponentiallyin time as depicted in Fig. 3.3b, then Γ(t) is simply a piecewise constant function[64]. However,efficient capture of a field with an arbitrary temporal envelope requires modulating Γ(t) beyond
that of a piecewise constant function.
For η ≈ 1, efficient capture of an input field is equivalent to minimizing the reflected energy.
To determine the form of Γ(t) required to minimize the reflected energy[144], one can use theRT
Euler-Lagrange equation to calculate the stationary point of the reflected energy, 0 |bout (t)|2 dt,
where bout (t) is a functional of Γ(t) given by Eqns. 3.12. The result of this minimization procedureyieldsΓ(t) = b (0)2
in

Γ(0)

eκm t |bin (t)|2Rt+ 0 eκm t0 |bin (t0 )|2 dt0(3.15)

where Γ(0) is the coupling at t = 0. This expression shows that simply by modulating Γ(t), one
can efficiently capture an arbitrary bin (t). I use this capability to efficiently capture fields emitted
from the photon source discussed in Chapter 4.

3.1.2.2

Pulsed amplification of a microwave field and mechanical motion

To amplify a propagating field and motion of the mechanical resonator, one utilizes a bluedetuned pump with detuning ∆b = +ωm . For this detuning, Eqns. 3.6 can be recast in terms of
Γ(t) = 4|g(t)|2 /κ where g(t) is the coupling between the field in the circuit and the mechanicalresonator due to the blue-detuned pump. As in the previous section, I can reduce Eqns. 3.6 top1ċ∗ (t) = Γ(t)c∗ (t) + ηΓ(t)bin (t)e−iψ2pbout (t) = ηΓ(t)eiψ c∗ (t) + (2η − 1)bin (t)(3.16)

where η = κext /κ is the coupling efficiency and ψ = Arg(−ig) is determined by phase of theblue-detuned pump.
To generate the equations of motion used to obtain Eqns. 3.16, one employs the Hamiltonian
Hb = ~g(b† c† + bc).

(3.17)

36
This interaction is similar to that of a two-mode squeezer Hamiltonian[145, 98] which amplifies[44]the energy in the two coupled modes. For Eqns. 3.16, the mechanical and microwave modes arecoupled through Γ(t). While Γ(t) is pulsed on, the circuit emits a propagating field with a temporalenvelope that rises exponentially at a rate of Γ/2. Crucially, the state of the emitted field depends
on both the states of the mechanical resonator and the input field before amplification.
To see this dependence, I solve Eqns. 3.16 using a piecewise constant coupling Γ(t) → Γ thatis pulsed on at time t = 0. For this coupling, the output field ispbout (t) = c∗ (0) ηΓ · eiψ h(t) + Γη · (h ? bin )(t) + (2η − 1)bin (t)(3.18)

where h(t) = exp (Γt/2) and ? denotes the convolution operation. This field depends on both theinitial state of the resonator at t = 0 and the input field while the coupling is pulsed on.
For bin (t) = 0 at t ≥ 0, a field emerges from the circuit that contains the initial state of theresonator. This state can be prepared by capturing a propagating field while t < 0. After capture,
Γ(t) is pulsed on and the output field ispbout (t) = c∗ (0) Γη · eiψ h(t)(3.19)

which has a conjugated phase relative to that of the mechanical resonator.
If instead a field bin (t) is injected while the coupling is pulsed on, a pulse emerges from thecircuit that depends on the state of the input field. Consider a propagating field that emerges froma cavity. This field has a temporal envelope given approximately bybin (t) =where√

γBe−γt/2 Θ(t)(3.20)

R∞

2
2−∞ |bin (t)| dt = |B| is the total energy contained in the field and γ is its power decay rate.

This field is chosen to be similar to what is emitted by the photon source. While the coupling ispulsed on, the expression for the output field takes on a simple form when γ = Γ. In this case,
Eqn. 3.18 reduces tobout (t) = 2pηΓB sinh (Γt/2) + (2η − 1)bin (t)(3.21)

37

Figure 3.4: Model of a two-mode parametric amplifier. The electrical mode b interacts with themechanical mode c through the two-mode squeezer interaction S(r) where r is the squeezing parameter. The output mode is bout . In terms of density matrices, the input electrical and mechanicalstates are ρe and ρm , respectively, and the output state is ρout . The cross indicates the output
mechanical mode is not measured.

The above expression shows that the temporal envelope of the output field is approximately a rising
exponential, and the phase of the output field is independent of the blue-detuned pump’s phase.

3.1.3

Operation as a linear amplifier

The two-mode squeezer interaction introduced in the previous section enables the electromechanical device to function as a phase-insensitive parametric amplifier. Such an amplifier increases
the amplitude of a signal regardless of its input phase. For an ideal, phase-insensitive parametricamplifier, the absolute minimum amount of added noise is half a quantum[126]. Because the added
noise can in principle be limited by Heisenberg’s uncertainty principle, this type of amplifier iswell-suited for measuring a signal at the quantum level[121, 124]. When operating the electromechanical device as an amplifier, the signal to be measured is either a quantum state of the input
microwave field or the mechanical resonator.
A model of the parametric amplifier realized using the electromechanical device is diagrammed in Fig. 3.4. This model has two input modes b and c that represent the microwave
field and mechanical resonator, respectively. For the amplifier’s output, we measure the outgoingmicrowave field bout . This field relates to the amplifier’s two inputs through the two-mode squeezer
operator S defined bybout = S † bS = b cosh(r) − c† sinh(r)(3.22)

38
where S(r) = exp r(bc − b† c† ) and r is the squeezing parameter[146, 147, 148]. Crucially, thisparameter determines the gain of the parametric amplifier.
To model how the amplifier transforms a quantum state, I describe its input and output
in terms of density matrices. Specifically, the input electrical and mechanical states are taken tobe ρe and ρm , respectively, and the output is ρout . Ideally, one of the two input modes is in its
quantum ground state. Prior to executing the two-mode squeezer operation, the second input mode
is in the quantum state of interest. For example, it could be a single-photon state. The result of
the two-mode squeezer operation is a joint entangled state Sρe ⊗ ρm S † where ⊗ denotes a tensor
product. Because I only measure the outgoing electrical mode, the resulting output state isρout = trm Sρe ⊗ ρm S †(3.23)

where trm denotes a partial trace over the mechanical mode.
Experimentally, one does not measure ρout directly. Instead, it can be determined by using
quantum state tomography as discussed in Section 2.5. In this approach, I simultaneously measure
the quadrature amplitudes X and Y of bout . Repeated measurements of these quadrature amplitudes are sampled from the Husimi Q-function that describe the output state. This function is
given by
Qout (α) =1hα| ρout |αiπ(3.24)

where α = X + iY . In order to determine the input state of interest from the above expression, it
must be related to the Q-function of that state.
Conveniently[126], the output Q-function is a scaled version of the input Q-function when r  1.
If the primary mode contains the signal of interest and the secondary mode is in its quantum groundstate, the output Q-function is simply√
Qout (α) = Qin (α/ G)/G
where

√

(3.25)

G = cosh(r). This result demonstrates that obtaining Qout is equivalent to Qin up to ascale factor. Having obtained Qin , one determines the input state ρin using the method of maximum

39
likelihood parameter estimation discussed in Section 2.5.3. Importantly, ρin could either be ρe or
ρm because either the electrical or mechanical mode can be regarded as the primary mode.
The expression given by Eqn. 3.25 is only true if the secondary mode is in its quantum groundstate. When this is not the case, the input Q-function of the signal becomes convolved with thenoise in the second mode. Suppose the input signal is in the primary mode, and the secondarymode is in a thermal state with a mean occupation of nth . Then the output Q-function is√
Qout (α) ∝ (Qin ? N ) (α/ G)(3.26)

where ? denotes a convolution operation, N is a Gaussian distribution with a variance of nth ineach quadrature, and the proportionality is made exact by enforcing proper normalization of Qout .
From this expression, one can determine ρin from Qout simply by taking the thermal occupation of
the secondary mode into account.

3.1.4

Frequency-tunable device using electrostatic actuation

The previous two sections showed how a single electromechanical device can mode-convert andamplify a signal. But these capabilities are limited to a narrow bandwidth if the device is fixed infrequency. By making the device tunable in frequency, it can manipulate a signal centered anywherein its operation band. This tunability enables the electromechanical device to be compatible withthe fixed-frequency photon source discussed in Chapter 4.
The center frequency of the electromechanical device is inherently tunable because it employsa variable capacitor. By applying a force that pulls the flexible capacitor plate towards its otherplate, the capacitance and thus resonant frequency of the device can be made tunable. A simple
way to generate this force is by imposing a static voltage across the two plates, which results in an
attractive electrostatic potential between them. This design is inherently limited in its tunability
because the electrostatic force can cause the flexible capacitor plate to collapse and stick to the otherplate. A way to circumvent this limitation is to introduce a third electrode into the design[149].

40

Figure 3.5: Two-electrode model of a tunable capacitor. a, The diagram shows a capacitor whose
upper plate is a flexible membrane. A source imposes a voltage Vdc across the two electrodes of thecapacitor. b, The plots show the total potential energy Utot of the capacitor as a function of themembrane deflection, where Um and Ue are the mechanical and electrostatic contributions to thetotal energy. When the applied voltage Vdc is less than the pull-in voltage Vp , the total potential
has a stable local minimum near x = 0. But once Vdc > Vp , the local minimum near x = 0 vanishes
and so the membrane collapses towards the lower plate.

3.1.4.1

Two-electrode model for static tuning

To understand how a static voltage can tune a capacitor, consider the simple model shownin Fig. 3.5a. The upper plate of the capacitor is a flexible metallic membrane, and the lower plateis a fixed electrode. These two plates are separated by a distance d. In order to deflect the flexible
plate by a displacement x, an external source imposes a constant voltage Vdc across the two plates
and thus creates an attractive electrostatic force between them. This force is described by the
2 /2 where the minus sign is due to the work performed
electrostatic potential energy Ue = −C(x)Vdc

by the source in order to keep Vdc constant. As the electrostatic force deflects the flexible plate, it
experiences a restoring force that is approximately harmonic given by Um = kx2 /2 where k is theeffective spring constant of the membrane. As such, the net force that the membrane experiences
is described by the total energy Utot = Ue + Um . This energy is minimized when the two plates
stick together at x = d. However, the mechanical restoring force of the membrane enables a stablelocal minimum to exist near x ≈ 0 as shown in Fig. 3.5b.
Once the electrostatic force overwhelms the mechanical restoring force, the membrane col-

41

Figure 3.6: A three-electrode tunable capacitor incorporated in a microwave circuit. The diagramshows a variable capacitor (C) shunted by an inductor (L). A third electrode electrostaticallyactuates the flexible membrane. This electrode is recessed so that d0 > d.

lapses and sticks to the other plate. This effect occurs when the minimum near x ≈ 0 vanishes due
to Vdc exceeding a certain ‘pull-in’ voltage Vp as shown in Fig. 3.5b. Both the capacitor’s geometry
and membrane’s spring constant set the scale for the pull-in voltage. Regardless of this scale, theequilibrium position of the membrane when Vdc = Vp is x = d/3.
The inherent pull-in effect limits the smallest gap achievable in a two-electrode capacitor to
be no more than 2d/3. As such, if one were to incorporate this capacitor into an inductor-capacitorp(LC) microwave circuit, its resonant frequency ωLC could not be reduced below 2/3ωLC or about
80% of its initial value. In practice, parasitic capacitance in the circuit will provide a further
reduction in this dynamic range. To enhance the dynamic range of the tunable capacitor, its
design must be altered.

3.1.4.2

Benefit of a third electrodeThe addition of a third electrode can avoid the pull-in effect that limits the tuning of a twoelectrode capacitor. A schematic of this approach is depicted in Fig. 3.6. The variable capacitor
is shunted by an inductor in order to form an LC circuit. To tune its center frequency, a third
electrode creates an electrostatic force on the flexible membrane. The key to the design is a recessed
third electrode. This design choice enables the membrane’s deflection to approach x → d, and so
the circuit’s resonant frequency can in principle assume ωLC → 0. In practice, this tuning is limited
by effects such as the Casmir force and bound charges on the two electrodes[65].

42

Figure 3.7: Schematic and micrograph of the electromechanical device. a, The diagram shows a
simplified schematic of the electromechanical device. The upper plate of the capacitor is a thin
(100nm) tensioned membrane, and beneath it is a microwave electrode (center) and an actuationelectrode (annular region). A voltage source imposes a voltage Vdc between the actuation electrodeand the membrane. The circuit elements have parameters of L ≈ 10 nH and C ≈ 40 fF, thereby
creating an inductor-capacitor (LC) circuit that resonates at a microwave frequency. Microwave
power is delivered to the circuit via a port connected to a transmission line, which inductively
couples to the circuit at a rate of κext . b, A false-color scanning electron micrograph of thealuminium membrane and nearby circuitry. The circuit (gray) is fabricated on a sapphire substrate(blue).

Figure 3.8: Frequency tuning of the electromechanical device. a, The plot shows the measured
center frequency ωe of the microwave circuit as a function of the applied voltage Vdc . b, The plot
is similar to a but for the mechanical resonant frequency ωm .

3.2

Characterization of the electromechanical device
In order to operate the electromechanical device, it is cooled to a temperature of less than

25 mK in a dilution refrigerator so that its aluminium circuitry superconducts. A diagram andpicture of the electromechanical device is shown in Fig. 3.7. Design and construction methods ofthe device are discussed elsewhere[150, 149].

43
Table 3.1: Parameters of the electromechanical device. All parameters were measured with anapplied dc voltage of Vdc = 5.83 V.
SymbolDescription

Value and unitsωLC /2πκLC /2πκext /2π
ωm /2πκm /2πnm
g0 /2πCircuit resonant frequency
Circuit decay rate
Circuit decay rate into the transmission line
Mechanical resonant frequency
Mechanical decay rate
Average occupancy of the mechanical oscillator
Electromechanical coupling

7.283 360 GHz
3 ± 0.1 MHz
2.59 ± 0.01 MHz
9.345 MHz14.5 ± 1 Hz
42 ± 2283 ± 14 Hz

Parameters of the electromechanical device are determined by measuring its microwave response. The device has one port, which provides electrical access to the LC circuit. A microwave
probe with an amplitude Vin is injected into the port. After the probe interacts with the device,
the reflected outgoing field Vout is then measured. This measurement technique enables one todetermine the response of the circuit and properties of the mechanical resonator. The parameters
for the electromechanical device are presented in Table 3.1.

3.2.1

Circuit parameters and frequency tuning

The measured response of the device to a microwave probe tone reveals its circuit parameters.
By injecting a microwave tone into the device and comparing it to the reflected signal, one obtains asingle scattering parameter S11 (ω) = Vout /Vin that characterizes the device response at a particularfrequency[151]. This probe is then swept over a range of frequencies that encompass the circuit’s
bandwidth. Fitting the resulting measurements of S11 (ω) enables one to extract the resonant
frequency of the circuit (ωLC ), its external coupling rate (κext ), and total coupling rate (κLC ).
Parameters of the electromechanical device are presented in Table 3.1.
To verify the frequency tuning of the electromechanical device, we measure its response whilevarying the voltage Vdc applied to the actuation electrode. For both positive and negative voltages,
the resonant frequency of the circuit is reduced as shown in Fig. 3.8a. At the lowest achievable
resonant frequency, the circuit tunes by more than 600 times its total bandwidth. At Vdc ≈ 17 V,

44

Figure 3.9: Electromechanical coupling rate and equilibrium occupation of the aluminium membrane. The black points show the average energy of the mechanical resonator (in units of quanta)
as the refrigerator’s base temperature was varied. Error bars show the standard error of the meanfrom five consecutive measurements. The solid line represents the expected mechanical occupation
nm assuming the resonator is in equilibrium with its environment at temperatures between 30 mK
and 100 mK and an electromechanical coupling rate g0 /2π = 283 ± 14 Hz.

an instability emerges that causes the membrane to stick to the electrode beneath it as evidencedby ωm → 0 as shown in Fig. 3.8b. However, when the applied voltage is reset to Vdc = 0, the circuit
returns to within ∼ 5% of its initial center frequency. Operation of the device past the point of
instability is not recommended.

3.2.2

Electromechanical coupling and mechanical occupation

The electromechanical coupling g0 as defined by Eqns. 3.7 is determined by measuring the
microwave response of the electromechanical device. A microwave pump is injected into the device,
and motion of the aluminum membrane modulates the pump and produces modulation sidebandson the reflected pump. The difference between the upper sideband and pump frequency is theresonant frequency of the membrane (ωm ), whereas the magnitude of the sidebands indicate thecoupling strength between the microwave circuit and the aluminum membrane. The amount of
power in the sideband (Pm ) at a frequency ωm above the injected microwave pump is proportional
to the amount of power in the injected microwave pump (Pc ):Pm
g 2 x2
κ2ext
1
= 0 2
2
Pc
2 xzp ∆2 + (κext − κ/2) (∆ + ωm )2 + (κ/2)2(3.27)

45

Figure 3.10: Mechanical relaxation rate. The plot shows the measured effective mechanical damping
Γeff = κm + Γr as a function of the pump strength Γr normalized to the intrinsic mechanicallinewidth κm . The solid line is a fit to the data that yields κm /2π = 14.5 Hz.

where ∆ is the difference between the pump frequency and the microwave circuit’s resonant frequency, xzp is the zero-point fluctuations of the membrane, and x2 describes the motion of themembrane. For a membrane in equilibrium with the thermal environment provided by the dilution
fridge, x2 = 2x2zp nm , where nm ≈ kB T /~ωm is the average phonon occupation of the membrane
for dilution refrigerator base temperature T .
To determine g0 , we vary the refrigerator temperature and monitor the ratio Pm /Pc . We
express this ratio as an effective phonon occupation, nm . At high temperatures (30-100 mK), themembrane is in thermal equilibrium with the refrigerator as evidenced by the linear relationshipbetween T and nm shown in Fig. 3.9. From this relationship, we extract an electromechanicalcoupling of g0 /2π = 283 ± 14 Hz. At the refrigerator base temperature of approximately 19 mK,
the phonon occupation number saturates to nm = 42 ± 2. Although this occupation is well above
the ground state, it can be further reduced by actively cooling the motion of the membrane as
pointed out in Section 3.1.2.1.

3.2.3

Mechanical relaxation rate

In order to determine the intrinsic mechanical relaxation rate κm , we measure the ringdown
time of the aluminium membrane. We probe the membrane’s motion by injecting a tone with a

46
detuning ∆r = −ωm into the circuit. To coherently drive the membrane’s motion, we briefly pulse
on an additional tone centered at the circuit’s resonance. These two tones create a time-varying
force that drives the motion of the membrane at its resonant frequency of ωm . Immediately after thedrive tone pulses off, we measure the amplitude decay of the probe’s upper modulation sideband.
Because this tone also damps the motion of the membrane (see Section 3.1.2.1), we monitor
the ringdown decay time as a function of the probe strength as shown in Fig. 3.10. A fit to thisdecay yields the effective mechanical damping Γeff = κm + Γr where Γr is the damping rate thatthe probe induces. For Γr  κm , the intrinsic mechanical relaxation rate is obscured becauseΓeff ≈ Γr . At low damping, the ringdown measurements reveal that Γeff asymptotes to the intrinsic
mechanical relaxation rate.

Chapter 4

Design and characterization of the photon source

Photons are ideal carriers of quantum information because they can rapidly and efficiently
propagate in cables or optical fibers[152, 153]. At microwave frequencies, photons are electricalsignals[116] that propagate in a transmission line. Such photons are crucial for emerging applications in quantum information processing using superconducting circuits. As such, single-photon
sources at microwave frequencies are actively under development[154, 155, 121, 156, 124, 157, 158,
159, 119, 160, 86]. Here, I realize a photon source by coupling a quantum bit (qubit) to one modeof a microwave cavity. The qubit enables a single photon to be created in the cavity mode, whichthen decays into a mode that propagates in the transmission line. Crucially, this source enables thecommunication of quantum information because it can generate propagating qubits[161] encoded
as superpositions of zero and one photons.
In this chapter, I describe the design and characterization of an on-demand photon sourcethat is compatible with the electromechanical device. Section 4.1 describes the design of the photon
source, which consists of a cQED system formed by a qubit in a cavity. Section 4.2 presents thecharacterization of these two elements. Lastly, Section 4.3 demonstrates that the cQED system
can efficiently emit single photons with spectral and temporal content that is compatible with theelectromechanical device.

48

4.1

Design of the photon source
The electromechanical device processes narrow-bandwidth signals at microwave frequencies,

as discussed in Section 5.3. To ensure compatibility of the electromechanical device with a single
photon source, the source must produce propagating microwave-frequency photons with a sufficiently narrow spectrum. As such, entering this regime requires a highly coherent circuit quantum
electrodynamics (cQED) system[115, 96]. This system is a superconducting quantum circuit realized by embedding a transmon qubit in a high-quality microwave cavity[79].

4.1.1

Qubit design

The qubit is formed by an inductor-capacitor (LC) circuit[162, 163]. To operate it as aqubit, the circuit must have an anharmonic energy spectrum so that individual transitions can beaddressed. Such an anharmonic energy-level spacing is achieved by making the circuit nonlinear.
The linear part of the circuit is described by the Hamiltonian
H=

Q2
Φ2
+
2C
2L

(4.1)

where Q is the charge on the capacitor and Φ is the flux through the inductor. This Hamiltonianis the same as a harmonic oscillator with
H = ~ω0

1
a a+
2
†


(4.2)

√
where ω0 = 1/ LC and a is a linear combination of charge and flux operators[133]. Because all
the energy levels are equally-spaced by ~ω0 as discussed in Section 2.1, this circuit cannot act as aqubit. However, it can be made nonlinear by writing Eqn. 4.2 as
H = ~ω0

1
a a+
2
†


−~

K † 2
(a a)
2

(4.3)

where K is the shift in the resonant frequency for each excitation in the oscillator. This parametercharacterizes the circuit’s anharmonanicity at the quantum level[164], and the effect is a direct
analog to the optical Kerr effect[165]. If the nonlinearity in the circuit is sufficiently strong such

49

Figure 4.1: Schematic of a Josephson junction. a, The diagram depicts two superconducting
electrodes (S) separated by an insulating layer (I). This device forms a Josephson junction. b, A
circuit model of a Josephson junction with inductance LJ and capacitance CJ . The circuit symbol
for a Josephson junction with the parasitic capacitance is shown on the right.

that |K| is much larger than the effective linewidth of the circuit, then it is possible to individually
address two energy levels and treat them as a qubit. Entering this regime with a circuit requires a
Josephson junction, which functions as a nonlinear inductor that does not introduce dissipation ordephasing into the circuit.

4.1.1.1

A Josephson junction provides the nonlinearity

The key nonlinear element in a superconducting qubit is a Josephson junction[166]. It is
formed by connecting together two superconducting electrodes with a weak link[167]. A single
Josephson junction is depicted in Fig. 4.1a. In each superconducting electrode, electrons bind
together to form Cooper pairs with charge 2e. All the Cooper pairs in each electrode are in
the same quantum state and can described by a single wave function with a phase. Because thewave functions overlap in the junction, Cooper pairs tunnel through the weak link. Crucially, thecollective behavior of all the Cooper pairs in the Josephson junction are parameterized by thedifference in the phases δ between the wave functions for each electrode.
A pair of constitutive relations describe the flow of Cooper pairs through the junction[168,169]. Without any voltage applied across the electrodes, a supercurrent will flow according to
Is = Ic sin(δ)(4.4)

where Ic is the maximum supercurrent the junction can support before it switches to a dissipative

50
state. If a voltage V is applied across the electrodes, the phase evolves according to
V =

Φ0 ∂δ
2π ∂t

(4.5)

where Φ0 = h/2e is the magnetic flux quantum.
The constitutive relations for a Josephson junction reveal that it is a nonlinear circuit element.
By using the expression for the voltage drop across an inductor, together with Eqn. 4.5, one findsthat the inductance isL=

LJ
cos(δ)(4.6)

where LJ = Φ0 /Ic is the Josephson inductance and Ic is its critical current. Importantly, Ic
is the sole parameter that determines the strength of the nonlinear inductance. In addition to an
inductance, the junction has capacitance CJ from the electrodes. These two elements form a circuit
as shown in Fig. 4.1b. This circuit is analogous to an LC resonator with a resonant frequency of
r
1
,
(4.7)
ωp =
LJ CJ
called the ‘plasma frequency’ because charge sloshes back and forth in a way that is similar to thatof a bulk plasma oscillation[170]. For δ  1, this frequency can be expressed as~ωp =p
8EJ EC(4.8)

where the Josephnson energy EJ = Φ0 Ic /2π is the characteristic energy stored in the Josephson
inductance, and the charging energy EC = e2 /2CJ is the energy required to charge the capacitor
with one electron.
Both the charging energy and Josephson energy determine the energy spectrum of the nonlinear circuit shown in Fig. 4.1. The nonlinear contribution to the Hamiltonian is from the Josephsonenergy of the junction given by
HJ = −EJ (1 − cos δ) .

(4.9)

By expanding HJ to fourth order, linear and nonlinear terms for the inductance emerge. The
nonlinear term relates to the anharmonicity K in Eqn. 4.3 by
EJ 4
K
=
δ
2
4!

(4.10)

51

Figure 4.2: Schematic of a transmon. a, A Josephson junction is shunted by an external capacitance C. b, Incorporating a Josephson junction in a coplanar capacitor realizes a transmon qubit.
The metal paddles above and below the junction form its shunting capacitance, and they provide
the circuit with an effective dipole moment.

This term sets the scale of the nonlinearity, and is responsible for making the circuit’s energy
spectrum anharmonic.

4.1.1.2

Introduction to the transmon qubit

The transmon is a charge qubit with improved coherence times[171, 172]. It consists of a
Josephson junction shunted by an external capacitance, as shown in Fig. 4.2. This capacitance is key
to the circuit design because it reduces the sensitivity of the qubit’s transition frequency to charge
fluctuations on the junction. As such, the transmon suppresses the undesired influence of charge
fluctuations on the qubit’s coherence times[173]. These times have increased from nanoseconds[174,175, 176] to microseconds[172, 177] because of the transmon circuit design.
Operating a charge qubit in the transmon regime requires EJ /EC  1, which is well-describedin the literature[171]. Briefly, operating in this regime enables the variation in the circuit’s energy
level spacing to be suppressed exponentially in EJ /EC . As such, the transition frequency used forthe qubit is approximately independent of any charge noise. Fortunately, the anharmonicity decreases only as a weak a power in EJ /EC , and it remains sufficiently anharmonic for selective control
over individual transitions. In the transmon regime, the anharmonicity is given by |Kq | ≈ −EC .

4.1.1.3

Transmon qubit parameter constraints

The EJ and EC parameters for my qubit design are constrained for three reasons. First, inorder to have a highly coherent qubit, we want to operate it deep in the transmon regime with

52

Figure 4.3: Schematic and picture of the qubit in a cavity. a, A schematic depicts the transmon
qubit incorporated in a superconducting cavity. The cavity has dimensions set by Ly < Lx < Lz .
b, A picture of the assembled cavity with two copper cables connected to the ports. c, The picture
shows the launcher made from a non-magnetic coaxial cable (UT-85C-LL Microcoax). d, Optical
microscope image of a single-junction transmon qubit fabricated on a sapphire substrate. The
chip rests inside the cavity near a region indicated by the dotted box. e, A scanning electron
micrograph of the single-junction transmon qubit. Layers of aluminium (Al), aluminium oxide(AlOx ), and aluminium form the junction. Micrograph courtesy of Luke Burkhart.

EJ /EC ∼ 50. This regime enables the qubit to have coherence times that exceed 10 µs, which
are readily achievable using standard superconducting circuit designs[173]. Second, the qubit’s√transition frequency ωq /2π ≈ 8EJ EC /h must be sufficiently detuned from the cavity so that the
coupled qubit-cavity system can be in the dispersive limit[178, 116]. As part of entering this limit,
we choose a qubit frequency to be in the range of approximately 5.5 − 6.5 GHz. Third, we want todrive individual transitions with fast, transform-limited pulses. In terms of the anharmonicitiy[171],the minimum pulse duration is τp ∼ |2πK|−1 . For τp = 10 ns, the required anharmonicity is|K|/2π > 10 MHz. These three conditions are satisfied by using EJ /h ∼ 15 GHz and EC /h ∼300 MHz.

4.1.2

Cavity design

The qubit is embedded in a superconducting cavity, as shown in Fig. 4.3a. This system enables
a narrow-band and efficient photon source for two reasons. First, the cavity protects the qubit fromunintended interactions with the environment. As such, the qubit can have coherence times thatare a hundred times longer than chip-based implementations[79, 179, 180]. Such long coherence

53
times are necessary in order to generate photons with high spectral purity[119]. Second, the cavity
provides a well-controlled electromagnetic environment with low dissipation. This environment
couples to a transmission line, thereby enabling single photons in the cavity to evolve into a modethat propagates in the transmission line as discussed in Section 2.4.
The geometry of the cavity determines its electromagnetic mode structure. For the idealized
cavity depicted in Fig. 4.3, its geometry is essentially that of a rectangular waveguide shorted atboth ends with two metal plates. This ‘waveguide cavity’ has mode frequencies given bys  2  2
ω`nm
` 2
m
n
= cπ
+
+
2π
Lx
Ly
Lz

(4.11)

where `, n, m ≥ 0 are integers for each mode and c is the speed of light[170, 151]. Although thecavity supports more than one mode, we couple the dipole moment of the qubit to the fundamental
TE101 mode with `, n = 1 and m = 0. This coupling is maximized by placing the qubit in the centerof the cavity[79]. For a cavity with Lx = 50 mm and Lz = 25 mm, the fundamental mode resonatesat ω101 /2π ∼ 7 GHz. Because the transmon circuit perturbs the electromagnetic environment, finite
element software packages are used to accurately predict the mode spectrum of the cavity[181].
Two ports provide access to the electromagnetic environment of the cavity[120]. In order
to excite either the qubit or cavity, a weakly-coupled port delivers energy to the cavity at a rateof κ2 . The second port of the cavity is its output, which decays into a transmission line at a rate ofκ2  κ1 . These two couping rates contribute to the total linewidth of the cavity κc = κ0 + κ1 + κ2
where κ0 represents any undesired ‘internal loss’ in the cavity. In order to realize an efficient photon
source, it is important minimize κ1 and κ0 because the probability that a cavity photon decays into
the output port is set by κ2 /κc . This ratio is the ‘coupling efficiency’ of the cavity.
The coupling rate for each port is set by two design constraints. First, the total linewidth ofthe cavity must be smaller than the bandwidth required to efficiently capture single photons using
the electromechanical device. As discussed in Section 5.3, this bandwidth is . 100 kHz. Second,
the internal loss of the cavity must be small compared to κ2 . For a rectangular cavity1 cooled to
1

The cavity was made out of 6061 aluminium alloy and it contained a qubit sample fabricated on a sapphire
substrate.

54

Figure 4.4: Cavity spectroscopy at room temperature. a, Measured cavity response of the stronglycoupled port in reflection. The cavity contains the chip with the transmon qubit. A fit (solidline) yields the total linewidth κc /2π = 3.2 MHz and the decay rate of the strongly-coupled port
κ2 /2π = 70 kHz. The top plot shows the measured power and the bottom plot shows the measured
phase response. b, Measured transmitted power. A fit (solid line) yields the total linewidth andκ1 /2π = 600 Hz with κ2 determined in a held fixed.

less than 25 mK, we experimentally found that the internal loss rate ranges from approximately
5 kHz to 10 kHz. Given these constraints, we set κ2 in the range of 50 kHz to 100 kHz. Because
we cannot make the input coupling rate arbitrarily small, we set κ1 < 1 kHz.
By injecting microwave power into the cavity and measuring its response at room temperature, we determine the coupling rate for each port. Initially, the reflected power off the stronglycoupled port of the cavity is measured. A fit to the amplitude and phase response yields κ2 andκc . Next, the transmitted power through the cavity is measured. Knowing κ2 and keeping it fixedduring the fit yields κ1 to within measurement error. Example measurements for this procedure
are presented in Fig. 4.4.

4.1.3

Coupled qubit and cavity

The qubit couples to a single mode of the cavity. This system can be treated as two harmonic
oscillators with a perturbation introduced by the transmon circuit[181]. The harmonic oscillator
modes for the cavity and transmon circuit are described by a and b, which acquire their nonlinearity from the fourth-order term in the expansion of the Josephson energy relation discussed in

55
section 4.1.1.1. By keeping terms up to fourth-order in δ, the Hamiltonian for the cQED system isH
Kc † 2 Kq † 2≈ ωc a† a + ωq b† b −(a a) −(b b) − 2χa† ab† b~2
2(4.12)

where Kc (Kq ) is the anharmonicity of the cavity (qubit) mode and χ is the ‘cross-Kerr’ interactionthat causes a state dependent shift between the qubit and cavity mode[164].
In order to understand how the cross-Kerr interaction affects the cavity response, I write
Eqn. 4.12 in terms of the Pauli spin operator σz . First, I restrict the transmon circuit to its groundand excited states by taking b† |1i = 0 and neglecting terms with factors of (b† b)2 . In this truncatedHilbert space, the Pauli spin operate emerges as b† b → σz /2. Second, I neglect terms with factorsof (a† a)2 because Kc is small compared to the linewidth of the cavity. Applying these substitutions
to Eqn. 4.12 yields
Hσz= ωq + (ωc + χσz ) a† a~2

(4.13)

where the terms have been rearranged so that its form is of the dispersive limit for the JaynesCummings Hamiltonian[110, 116]. As pointed out in Section 2.3.2, the effective resonant frequencyof the cavity is ωci = ωc + χσz where i = {g, e} is a function of the qubit in either its ground or
excited states |gi and |ei, respectively. Measuring this effective cavity resonance provides a way tocharacterize both the qubit and cavity.

4.2

Characterization of the qubit and cavity
Before we can operate the cQED system as a single-photon source, we first determine thecavity parameters discussed in Section 4.1.2 and the qubit parameters defined by Eqn. 4.13. We thencalibrate a set of control pulses used to readout and control the qubit state. Lastly, we confirm thatthe coherence times of the qubit are sufficiently long for the photon generation protocol discussed
in Section 4.3. The parameters for the cQED system are presented in Table 4.1.

56
Table 4.1: Parameters of the cQED system.
SymbolDescription

Value and units

ωc /2πωg /2πωe /2πκ1 /2πκ2 /2πκc /2πχ/2πωeg /2πωsb /2π
Kq /2πT1
T2?

Cavity resonant frequencya (high power)
Cavity resonant frequencyb (qubit in ground state)
Cavity resonant frequencyb (qubit in excited state)
Weakly coupled cavity port decay rate
Strongly coupled cavity port decay ratec
Total cavity decay ratec
Dispersive shift
Ground to excited state transition frequency
Blue sideband transition frequency
Qubit anharmonicity
Energy relaxation time
Coherence time

7.276 781 GHz
7.290 156 GHz
7.283 360 GHz< 200 Hz
50 kHz60 kHz−3.413 MHz
5.652 MHz
2 × 6.462 GHz
380 MHz∼ 60 µs∼ 14 µs

a Measured at 104 fW (see Fig. 4.5a).
b Measured at 10−4 fW (see Fig. 4.5b).
c In this chapter, κ /2π = 70 kHz. For the experiment in Chapter 5, it was reduced
2

to 50 kHz.

4.2.1

Cavity spectroscopy

Upon cooling down the cQED system, the nonlinearily of the qubit reveals itself in the powerdependent cavity response[182, 183]. To measure its response, we inject a pulsed probe tone into
the weakly-coupled port of the cavity and monitor the transmitted amplitude from the output port.
This scheme measures the scattering parameter |S21 | = |V2 /V1 | where V1 and V2 are the amplitudes
for the input and output voltage waves, respectively. To observe the nonlinear response of thecavity, we measure |S21 | as a function of the probe power, as shown in Fig. 4.5a. At high powers,
the transmission peaks are at the bare resonant frequency of the cavity ωb /2π = 7.277 GHz. As
the probe power is reduced, the cavity response becomes highly nonlinear because of the dispersiveinteraction between the cavity and the qubit.
At low probe powers, the response is linear and it reveals the qubit state-dependent dispersive
shift of the cavity. When the qubit is in its ground state |gi, the transmission peaks at ωcg = ωc − χ
where ωcg /2π = 7.290 GHz. There is faint transmission centered at ωce /2π = 7.283 GHz because ofresidual occupation of the qubit’s excited state |ei. One can determine the dispersive shift of the

57

Figure 4.5: Cavity transmission as a function of probe power and frequency. a, The plot shows
the measured cavity transmission (log magnitude) as a function of the probe frequency and power.
The probe power is referenced to the input port of the cavity. At high probe power (> 10 fW), theresponse of the cavity is centered at its bare resonant frequency of ωb /2π = 7.277 GHz (indicatedby the arrow). At low probe power (< 10−3 fW), the cavity response is linear and it exhibits two
peaks corresponding to the qubit in its ground and excited states |gi and |ei, respectively. b, The
plot shows the measured transmission at low power (at 10−4 fW) with the qubit prepared in either
its |gi or |ei states. The solid line indicates a Lorentzian fit with a linewidth of κc /2π = 80 kHz.
c, The plot shows a line cut of the cavity transmission at ωb as the probe power is varied. Whenthe probe power exceeds a sufficiently high power, the transmission exhibits a sharp jump that
corresponds to the emergence of the bare cavity resonance. The onset of this sharp jump depends
on the state of the qubit, which was initialized in either its |gi or |ei state.

cavity χ by taking ωcg − ωce = −2χ. For the data presented in Fig. 4.5b, the visibility of these two
resonant features reveals that χ/2π = −3.5 MHz. Additionally, by probing the low-power responseof the cavity centered at ωcg , we determine its total linewidth κc /2π = 80 kHz and output port
coupling rate κc /2π = 70 kHz. Having determined the parameters that characterize the responseof the cavity, we can now exploit its nonlinear response to measure the qubit state.

4.2.2

Measuring the qubit state

In this thesis, we employ two qubit-state measurement schemes. The first scheme enables a
simple and robust measurement protocol that can determine the qubit state with high fidelity[182].
For this scheme, a high-power tone probes the cavity state to destructively determine the qubit
state. In contrast, the second scheme probes the linear response of the cavity at low powers and58
can be used to project the qubit state via measurement[184, 185, 186].
In order to obtain a comparable signal-to-noise ratio using the low-power scheme, it requiresa relatively low-noise amplifier. Such amplifiers using Josephson junction-based devices have beendeveloped for qubit readout[187, 188, 189, 119]. However, integrating these amplifiers with theelectromechanical device comes at the cost of additional complexity to the measurement protocol.
We overcome this challenge by using the electromechanical device itself as a low-noise amplifier.
This approach is unique to my experiment because it removes the complexity of the JPA and isreadily integrable with the conversion experiment presented in Chapter 5.

4.2.2.1

High-power readout

The high-power readout scheme uses the nonlinear dispersive interaction to project the qubit
state onto the classical state of the cavity. This approach effectively harnesses the cQED systeman amplifier, and does not require the use of a subsequent low-noise amplifier such as the JPA. To
infer the qubit state, a strong probe tone measures the cavity response in transmission[182]. The
probe tone is centered at the bare resonant frequency of the cavity ωb . When the power of theprobe exceeds a critical value, the onset of a high-transmission ‘bright state’ emerges as shown in
Fig. 4.5a.
Because the power level at which this onset occurs depends on the qubit state, this effectcan be used to determine the initial state of the qubit as shown in Fig. 4.5c. To measure thequbit’s state, one must choose a probe power that maximizes the difference in transmission for
the two states. Crucially, the difference in transmission can be much larger than the noise in thedetector used to measure the probe signal. While this readout scheme is relatively straight forward
to implement, it is not a non-demolition measurement2 because the high-power probe induces asubstantial photon population in the cavity that drives the transmon qubit out of its |gi and |ei
subspace. In order perform a non-demolition measurement of the qubit’s state, one must use alow-power probe and a nearly quantum-limited amplifier to detect it[191].
2

A non-demolition measurement leaves the qubit in a state that is unaffected by subsequent measurements[22, 190].

59

Figure 4.6: Histograms of single-shot measurements using the JPA. Each single-shot measurement
outcome is the in-phase quadrature I of the transmitted probe signal through the cavity. The
probe signal was pulsed on for 10 µs and it induced ∼ 10 circulating photons in the cavity. a, The
qubit was left in thermal equilibrium with the environment and its state was measured. The redsolid line indicates a fit to a sum of two normal distributions, which reveals a residual excited state
population of P (e) = 2%. The gray solid line indicates the threshold Ith used to discern the two
qubit states. b, When approximately half the qubit population is driven to its excited state |ei,
the separation between the two distributions becomes evident.

4.2.2.2

Low-power readout

The low-power readout scheme relies on the qubit state-dependent dispersive shift of thecavity. To infer the qubit state, a weak probe tone measures the cavity response in transmission[178,192]. This response is probed by either measuring the amplitude or phase of the transmitted tone.
Because my cQED system operates in the regime where the dispersive shift is χ  κc , the ability
to discern the qubit state is optimized by centering the probe frequency at the cavity’s response
and measuring the transmitted amplitude[193]. For example, consider centering the probe at ωce .
When the qubit’s state is |ei, the cavity response peaks at ωce and the probe will be transmitted.
Conversely, when the qubit’s state is |gi, the cavity will be at ωcg and the probe will not betransmitted. Thus, this scheme can discriminate the qubit states by probing the cavity response.
Employing a Josephson parametric amplifier (JPA) at the front end of the detector provides
a way to perform efficient single-shot measurements of the qubit state. To access the single-shot
regime, the measurement duration must be substantially shorter than the excited to ground state
relaxation time of the qubit. The signal-to-noise of this measurement is determined by its duration,
power level, the cavity bandwidth, and the detector efficiency[186]. Because the JPA enhances thedetector efficiency, the measurement duration and power level can be significantly reduced and thus

60

Figure 4.7: Single-shot measurements of the qubit state using the electromechanical device as anamplifier. Each single-shot measurement outcome is the magnitude squared I 2 + Q2 of the in-phase(I) and quadrature (Q) components of the transmitted probe signal through the cavity. a, The
histogram shows the distribution of measurement outcomes when the qubit was either driven or
not using a control pulse. The gray solid line indicates the threshold Ith used to discern the two
qubit states. b, The plot of the cumulative probability indicates a readout contrast of 60% fordiscerning the qubit’s two states.

enables single-shot high-fidelity qubit readout.
The ability to effectively discriminate between qubit states is quantified by collecting a repeated set of single-shot measurements. The outcome of each single-shot measurement is plottedin a histogram as shown in Fig. 4.6. Each outcome is the in-phase amplitude I of the transmitted
probe through the cavity. If the probe is not transmitted, then I = 0. Thus, when the qubit
state is |gi, we expect a normal distribution of measurement outcomes centered at I = 0 as shownin Fig. 4.6a. However, when we drive approximately half the qubit population to the |ei state,
an offset emerges because approximately half the measurement outcomes are from trials in which
the probe was transmitted through the cavity. This case is shown in Fig. 4.6b. Because the two
distributions are resolvable, we define a threshold Ith that separates the outcomes. If an outcome I
is larger than Ith , we assign the outcome to the |ei state; conversely for I < Ith . This thresholding
enables one to extract the probability that the qubit occupies its ground or excited states from a
set of measurements.
Apart from using the JPA, low-power and single-shot measurements of the qubit state areenabled by harnessing the electromechanical device as an amplifier. The theory of operating it as
an amplifier is discussed in Section 3.1.3. To readout the qubit state, we probe the response of the61cavity using pulsed signals as with the readout method using the JPA. However, in this case, we
use the electromechanical device to amplify the readout pulse. We then collect a set of single-shotmeasurements and plot their outcomes in a histogram as shown in Fig. 4.7a. To quantify the ability
to discern the qubit states, we calculate the maximum difference between the cumulative probability
curves as shown in Fig. 4.7. This difference defines the readout contrast, which we calculate to
be about 60%. This contrast level is comparable to what is obtained in the JPA-backed readout
scheme using my cQED system.

4.2.3

Qubit characterization

The qubit readout methods discussed in the previous section enable characterization of thetransmon qubit parameters summarized in Table 4.1. This section presents a characterization of
the qubit’s energy spectrum, control pulses used to alter its state, and its coherence times.

Energy spectrum
By employing either of the two qubit readout schemes presented in Section 4.2.2, we determine
the anharmonicity Kq by measuring the qubit’s spectrum. It is measured by driving the qubit witha control pulse, and then reading out the qubit state over a range of frequencies as shown in
Fig. 4.8a. This qubit spectroscopy reveals two peaks centered at ωge and ωgf /2, which correspondto the |gi → |ei and |gi → |f i transitions. Because the anharmonicity is defined as Kq = ωge − ωefwhere ωge , it can be written as Kq /2 = ωgf /2−ωge . The data in Fig. 4.8a shows that Kq = 380 MHz.
As discussed in Section 4.1.1.3, this amount of anharmonicity in the qubit spectrum is necessary
in order to drive the qubit using fast (10 ns) control pulses.
Additionally, measuring the qubit spectrum provides information about the Josephson energyof the qubit. The Josephson energy EJ can be expressed in terms of transition frequencies by
calculating the eigenenergies of the qubit spectrum using perturbation theory[171]. The result of
this analysis yields
2
1 ωge
EJ ≈ −
8 EC(4.14)

62

Figure 4.8: Calibration of the qubit control pulses. a, The plot shows the qubit excited state
probability P (e) as a function of the drive detuning. At low drive power, the qubit |gi → |eitransition is centered at ωq . This power level is the reference (0 dB). At a higher drive power(20 dB), the qubit transition broadens and the |gi → |f i transition emerges centered at ωef /2.
These two frequencies are used to extract the anharmonicity Kq . b, When the qubit is driven at
ωq , the qubit population oscillates between the |gi and |ei states. c, Similarly, when the qubit isdriven at ωbs /2, the qubit population oscillates as the cQED system oscillates between the |gi |0i
and |ei |1i states.

where EC ≈ −Kq is the charging energy. Using this result and the data presented in Fig. 4.8a, we
calculate that EJ ≈ 14 GHz. Thus, the spectrum reveals that the qubit is deep in the transmon
regime with EJ /EC ∼ 37  1.

Calibrated control pulses
Having measured the qubit’s spectrum, we calibrate the amplitude and duration of a pulsed
tone used to control the qubit state. Specifically, these two parameters must be appropriately
calibrated so that we can deterministically place the qubit in its excited state |ei or the superposition√
state 1/ 2 |gi + eiϕ |ei where ϕ is the phase of the qubit. To calibrate the control pulse, we firstcenter the tone at ωq . We then choose a fixed amplitude and vary its duration τq . As shown in
Fig. 4.8b, the qubit population oscillates between its |gi and |ei states. These oscillations provide
evidence that the qubit is indeed an addressable two-level system. By setting τq at half the first
oscillation period, the qubit population can be driven from the |gi to |ei state. This calibration
sequence realizes a ‘π-pulse’ because we can effectively rotate the qubit by an angle of θ = π on
the Bloch sphere (see Section 2.2). Simply by adjusting the duration (or amplitude) of the pulse,

63

Figure 4.9: Time-domain measurements of the qubit coherence times. a, Measured probability
P (e) of the qubit excitated state |ei population as a function of delay time ∆τ between a π pulse
and a qubit measurement pulse. A fit to the data (solid line) yields T1 = 145 ± 3 µs. b, Similar toa but P (e) is plotted as a function of the delay ∆τπ/2 between two π/2 pulses. A fit to the data(solid line) yields T2 = 14 ± 1 µs.

we can rotate the qubit by θ = π/2 thereby placing it in the superposition state where the pulse
phase determines ϕ.
In order to generate single photons, we use a pulsed tone that can control the blue sideband
transition |gi |0i → |ei |1i as introduced in Section 2.3.4. To deterministically place the qubit andcavity in the joint state |ei |1i, we must calibrate an appropriate control sequence. Because theblue sideband transition is driven via a two-photon process[118], we center the tone at half thefrequency of the blue-sideband transition ωbs /2. After the amplitude of the pulse is set, we varyits duration τb and measure the qubit state as shown in Fig. 4.8c. The qubit population varies
because the cQED system oscillates between the states |gi |0i and |ei |1i. As τb increases, photons
decay out of the cavity at a rate of κc and so the oscillations saturate as the qubit settles to its |eistate as described in Ref. [118]. In order to create an effective π-pulse on this transition, we set τbat half the first oscillation period. Using this calibrated pulse, the qubit can be put in the |ei |1ithereby creating a single photon in the cavity.
To generate a superpostion of cavity photon states, we employ a two-step protocol[116,121, 194]. In the first step, a π/2-pulse prepares the qubit and cavity in the superposition state√
1/ 2 |gi |0i + eiϕ |ei |0i . In the next step, a π-pulse on the blue-sideband transition swaps the
population in the |gi |0i and |ei |1i states without affecting the population in the |ei |0i state. The

64√
qubit and cavity are now in the joint state 1/ 2 |ei |1i + eiϕ |ei |0i . Because the |ei state can befactored out, the photon state is unentangled with the cavity. Thus, the cavity is prepared in the√superposition state 1/ 2 |0i + e−iϕ |1i which decays out of the cavity into a propagating mode of
the transmission line. In order to prepare this state, the protocol’s duration must be substantiallyless than the qubit’s coherence times.

Coherence times
We employ a two-step protocol to determine the qubit’s relaxation time from the |ei to |gi
states. This characteristic time is called T1 and it describes energy loss of the qubit. In order toimplement the photon generation protocol discussed in Section 4.3, the qubit’s lifetime must satisfy
T1  κ−1
c . The protocol used to measure T1 starts with preparing the qubit in its |ei state usinga π-pulse. After a delay time of ∆τπ , the qubit state is measured. By varying ∆τπ , the qubit’s
excited state probability P (e) decays as shown in Fig. 4.9a. A fit to the data3 yields T1 = 145 µs.
Because of this relatively long decay time, care must be taken to ensure a wait time > 5T1 between
each execution of the protocol so that the qubit has time to relax to its |gi state with a probability
that exceeds 99%.
In addition to determining the qubit’s T1 time, a three-step protocol is used to measure its
dephasing time. This characteristic time is called T2? and it describes the random drift in the phasedifference between the qubit states |gi and |ei in a superposition during an ensemble measurement.
In order to generate coherent superposition states, the qubit’s dephasing time must satisfy T2?  τbwhere τb is the duration of the blue sideband pulse. The protocol used to measure T2? starts with√

preparing the qubit in the superposition state 1/ 2 |gi + eiφ |ei using a π/2-pulse. After a delay
time of ∆τπ/2 , another π/2-pulse is applied to the qubit. By making repeated measurements of
the qubit state as ∆τπ/2 is varied, one finds that the qubit’s excited state population oscillates
3

For measurements on the same sample, the lowest relaxation time we measured was approximately 60 µs.
Fluctuations in the T1 time at the level of a factor of two have been well-documented[195], and may be explained bydrifts in the quasiparticle density or occupation of two-level systems in the junction.

65
according to
P (e) ∝ cos ∆d · τπ/2 exp −τπ/2 /T2?

(4.15)

where ∆d = ωq −ωd is the detuning of the drive centered at ωd from the qubit transition frequency.4
This method is commonly referred to as ‘Ramsey interferometry’ and is often the workhorse of
precision spectroscopy experiments[196]. For the data presented in Fig. 4.9b, the fit to Eqn. 4.15
yields T2? = 14 µs. This timescale is sufficient to prepare a superposition state using a π/2-pulsethat is of order 100 ns in duration.

4.3

Generation of propagating single photons
In this section, we provide a rudimentary evaluation of the photon source. We demonstratethat the source efficiently generates single photons, and we show that their bandwidth is sufficiently
narrow enough to be compatible with the electromechanical device.

4.3.1

Experimental apparatus

We create an experimental apparatus that enables the detection of propagating signals emitted from the cQED system. The main components of the apparatus are shown in Fig. 4.10. Qubit
control and readout signals are injected into the weakly-coupled port of the cQED system. Its
output port strongly couples to a transmission line that connects to a detector. The performanceof the detector is characterized by two parameters: the efficiency ηd and gain Gd . The efficiency
ranges from 0 (no measurement) to 1 (ideal measurement). The gain is a quantity that relates
voltage fluctuations at the output of the detector to those at its input.
In order to efficiently measure single photons, the detector employs a Josephson parametricamplifier (JPA) as a sensitive preamplifier. We operate the JPA as homodyne detector in order tohave the highest possible measurement efficiency, as pointed out in Section 2.5. Because we wantto measure single photons emitted at the frequency ωce when the qubit is in its excited state, we
4This expression assumes T2?  T1 as is the case in this thesis.

66

Figure 4.10: Diagram of the photon generation apparatus. Qubit control and measurement pulses
are injected into the input of the photon source. The photon source consists of a cQED system.
A Josephson parametric amplifier (JPA) serves as the sensitive preamplifier for the detector which
has an efficiency ηd and gain Gd .

operate the JPA with its gain centered at ωce . The output of the detector is demodulated and thevoltage trace V (t) is recorded.
The experimental apparatus presented in this section is similar to the one previously constructed in our lab[119]. However, we did not include a switch with a calibration noise source
because of two reasons. First, we did not want to add any excess insertion loss between the cQED
system and the detector, which would reduce the fidelity of the propagating single-photon state.
Second, we want to avoid any stray infrared light that could enter in through the switch. Suchlight may degrade the coherence times of the qubit[197].

4.3.2

Generation and measurement protocol

We generate single photons by adapting a protocol developed previously in our lab[119].
This protocol creates a photon in the cavity by driving the blue-sideband transition |gi |0i → |ei |1iwith a π-pulse as diagrammed in Fig. 4.11a. The cavity photon then evolves into a mode that
propagates in the transmission line. For each repetition of the protocol, we measure a voltage trace
V (t) using the homodyne detector discussed in Section 4.3.1. For a control experiment, we executethe protocol but replace the π-pulse on the blue-sideband transition with a π-pulse on the qubit’stransition. This sequence prepares the cQED system in the state |ei |0i instead of |ei |1i. As such,the cQED system does not emit a photon.
To increase the photon generation fidelity, we measure and condition on the qubit’s state[198].
Prior to generating a photon in the cavity, we measure the state of the qubit using a ‘pre-select’67
readout pulse. Because we want the cQED system to be initialized in the state |gi |0i, we selectthe measurements in which the qubit is in its |gi state. This selection removes approximately 10%of the measurement outcomes in which the qubit occupies its excited state. After generating asingle photon in the cavity, the cQED system must be in the state |ei |1i in order for the photon tobe emitted at the excited state cavity frequency ωce . Thus, we ‘post-select’ the data in which thequbit is in its |gi state. This selection removes approximately 60% of the measurement outcomes inwhich the qubit decays from its |ei to |gi state, or in instances when the blue-sideband transition
is not driven because of infidelity in the π-pulse.

4.3.3

Quadrature amplitude extraction

In order to characterize propagating signals at the single-photon level, we express measured
signals in terms of photons per second, or quanta, referred to the input of the detector. For eachvoltage trace V (t) measured at the detector, we extract a single quadrature amplitude X. Thisquadrature amplitude describes one particular mode of the propagating signal measured at thedetector. To define a particular mode, we use a mode-matching function f (t) that weights each
V (t) in order to produce an optimum estimate of the quadrature amplitude[121]. We choose f (t) sothat it matches the temporal envelope of the photon signal measured at the detector. Specifically,
the form of f (t) depends on the bandwidth of the emitted single-photon signal (set by the cavity
linewidth of the cQED system) and the bandwidth of the detector (set by the JPA). As such, we
use a function defined by




0



f (t) = h(ωLPF , t) ?
1 − e−κ(t−t0 )/2





 e−κ(t−t0 )/2t ≤ t0t0 < t ≤ Tbs(4.16)

t > Tbs

where ? denotes a convolution operation, h is the finite impulse response function of a low-passfilter, ωLPF is the cutoff frequency of the low-pass filter, t0 is the start time of the blue-sideband
pulse, Tbs is its duration, and κ is the power decay rate of the signal. For the bandwidth of the
low-pass filter, we choose it to be ωLPF = κJPA /2, where κJPA is the bandwidth the JPA. A plot

68

Figure 4.11: Photon generation and detection. a, A timing diagram depicts the protocol used togenerate propagating photons. The π-pulse (black) at time t ≈ 70 µs either generates a photon
or excites the qubit for the reference vacuum measurement. Qubit readout pulses (pre-select andpost-select) are used to condition the measurement on the qubit state. b, Measured voltage signals
hV (t)i at the detector averaged over 4,798 individual time traces with the π-pulse frequency either
at the blue sideband transition frequency (black) or the qubit’s frequency (gray). c, The plot issimilar to b but for the voltage variance of the time traces. d, The plot shows the filter function
f (t) used to extract the quadrature amplitude for each time trace. The filter function has a power
decay rate of 83 kHz.

69
of the mode-matching function is shown in Fig. 4.11d in which we used κJPA /2π = 2 MHz andTbs = 250 ns.
After determining f (t), we calculate the quadrature amplitude for a given voltage trace using
the method of least-squares parameter estimation[199], where the parameters to be estimated are
the amplitude, Vq , of f (t) and the amplitude, Vdc , of the background windowing function b(t). We
use a background function to account for the offset in each V (t) that drifts during the acquisitionof a complete data set. The function b(t) is a piecewise-constant function that is non-zero during
the photon generation protocol except when the qubit state is measured. To obtain these twoparameters from each discretized voltage trace {Vk } (containing N samples), we minimize the costfunctionC=

N
X[Vk − (Vq fk + bk Vdc )]2(4.17)

k=1

with respect to Vq and Vdc . The result of this minimization yeilds
!
!
N
N
N
X
X
X
V` · b` ·bm · fm
Vq =Vk · fk −k=1`=1(4.18)

m=1

where fk and bk are discretized versions of the mode-matching and background functions, respectively. Eqn. 4.18 yields an uncalibrated quadrature amplitude for each measured voltage trace.
To calibrate the set of quadrature amplitudes, we execute the control sequence in which no
photons are generated. This calibration assumes the convention that one quadrature of a signal
in the vacuum state has a variance of Var(X) = 1/4 quanta as discussed in Section 2.1.3. The
uncalibrated quadrature amplitudes are binned in a histogram, and this data is fit to a normal
distribution with the gain Gd of the measurement apparatus as the only free parameter. The
center of the distribution is zero, and its variance is set to be 1/4 for a vacuum state. We use
the gain determined from the fit to scale the quadrature amplitudes obtained when photons weregenerated. The calibrated quadrature amplitudes are presented in Fig. 4.12.

70
4.3.4

Density matrix reconstruction

To characterize the quantum state emitted from the photon source, we reconstruct the state’s
density matrix ρ. The density matrix provides a complete description of the state by taking into
account both quantum and statistical uncertainty in the measurement. In the number basis, thematrix element ρnn conveniently describes the probability of detecting n photons. For an ideal
single-photon state, ρ11 = 1 with all other elements equal to zero. However, imperfections in thegeneration and measurement process yield ρ11 < 1.
We estimate ρ by performing a microwave version of homodyne tomography, as discussed in
Section 2.5. This procedure relies on making an ensemble of repeated single-quadrature measurements on a set of identically-prepared quantum states emitted from the photon source. For eachmeasurement, the outcome is plotted in a histogram. Because we measure in the quadrature basis,
we expect the data to follow the marginal distribution
Pr(X) = hX| ρ |Xi .

(4.19)In the photon-number basis, the density matrix is given byρ=X

Pn |ni hn|(4.20)nwhere Pn = ρnn is the probability to be in the state |ni for an integer n ≥ 0. This form of ρ enables
Eqn. 4.19 to be written in the number basis:
Pr(X) =

X

ρnn |hX|ni|2 .

(4.21)n

By calculating the eigenstates of the quantum harmonic oscillator in the quadrature basis, one finds1hX|ni = √2n n!

 1/4√22
e−X Hn ( 2X)π(4.22)

where Hn are the Hermite polynomials[101].
After we generate a histogram of the calibrated quadrature amplitudes, we perform a fit
using Eqn. 4.21 in a three-photon number basis. Without conditioning on the qubit state, the fit

71
to the marginal distribution yields ρ00 = 0.61 ± 0.01, ρ11 = 0.36 ± 0.02, and ρ22 = 0.02 ± 0.02
(the ρ33 element is indistinguishable from statistical noise). In order to increase the single photon
generation efficiency, we condition on the state of the qubit. This conditioning purifies the statebecause we discard measurements in which we expect no photon to be generated. For this case,the results of the fit yield ρ00 = 0.48 ± 0.01, ρ11 = 0.49 ± 0.01, and ρ22 = 0.03 ± 0.02. A histogramof the measured quadrature amplitudes is shown in Fig. 4.12.
We model the measured state by taking into account two sources of infidelity. First, because
the cavity has internal loss, the probability of a photon decaying out of the port coupled to thetransmission line is ηc = κ2 /κc = 0.87 ± 0.03. Second, the detector has an efficiency that is not
ideal. In a separate cooldown of the apparatus, we used a calibrated noise source and found that
this detector has ηd = 0.60 ± 0.02. Using these two parameters, we model the expected state asρm = (1 − ηc ηd ) |0i h0| + ηc ηd |1i h1|(4.23)

which is a statistical mixture of a pure vacuum and single-photon state[194]. This model predicts
ρ11 = 0.52 ± 0.02 which is in reasonable agreement with what is measured.
The results in this section show that the cQED system can generate propagating single
photons efficiently and with a narrow bandwidth. As such, the next step is to interface the cQED
system with the electromechanical device discussed in Chapter 3.

72

Figure 4.12: Histograms of measured quadrature amplitudes for single photons. The plots show
the plotted probability distribution Pr(X) as a function of quadrature amplitude X scaled in units√of quanta. The solid line indicates a fit to expected distribution for a diagonal density matrix
with a three-photon Fock basis. a, Control experiment (‘no photon’) in which no photons weregenerated. The solid line is a normal distribution with a variance of Var(X) = 1/4. b, Experiment(‘photon’) in which photons were generated. A fit to the measured distribution yields the diagonal
density matrix elements ρ00 = 0.48 ± 0.01, ρ11 = 0.49 ± 0.01, and ρ22 = 0.03 ± 0.02.

Chapter 5

Conversion of propagating single photons to mechanical motion

An electromechanical device converts the state of a propagating microwave field to mechanicalmotion, as pointed out in Chapter 3. By using the single-photon source presented in Chapter 4,
the field can be prepared in a superposition of zero and one photons that encodes a quantum bit(qubit) of information. In this chapter, I combine these two capabilities and demonstrate that theelectromechanical device can capture a state that encodes quantum information. Once captured,it can be mechanically stored for over a hundred microseconds and then retrieved on-demand. Inorder to measure the state of the input field or mechanical resonator, I exploit the electromechanicaldevice as a nearly quantum-limited amplifier as described in Chapter 3. Using this capability, I
characterize the capture process and find that it can preserve the information contained in anarbitrary qubit better than what is possible using only classical resources.
This chapter describes the experimental results of converting propagating single photons tomechanical motion. Section 5.1 describes the experimental apparatus. Section 5.2 demonstrates
that the electromechanical device can function as a low-noise amplifier. Section 5.3 presents two
protocols used to characterize the conversion process from propagating microwave fields to mechanical motion. Section 5.4 demonstrates the ability to capture, store, and amplify single photons.
Section 5.5 presents the conversion of qubits and a characterization of the capture process.

74

5.1

Experimental apparatus
The electromechanical device interfaces with a single-photon source through a network of

microwave-frequency components. The main components of the network are shown in Fig. 5.1a,
and a picture of the experimental apparatus is shown in Fig. 5.2. Microwave pumps and test
signals are injected into the network using a directional coupler (the microwave equivalent of abeamsplitter), and the photon source is pumped via a separate port. The photon source consistsof the cQED system discussed in Chapter 4. This source emits a signal that is routed to theelectromechaincal device using a circulator, which also provides a sufficient level of isolation toprotect the cQED system from the strong pumps used to control the electromechanical device(see Section 3.1.1). After the signal interacts with the electromechanical device, it is routed to a
detector. Once detected, the signal is downconverted in frequency with a demodulation mixer andthen digitized. Data collected consist of digitized voltage waveforms V (t). Detailed descriptions of
the detector and remaining components of the measurement network are presented in Appendix A.
A simplified frequency-domain representation of the experiment is shown in Fig. 5.1c. The
photon source emits a signal centered at ωce when the qubit occupies its excited state. To capture
the propagating signal at this fixed frequency, the electromechanical device is tuned into precise
resonance with ωce using a voltage bias Vdc . This bias is applied to a separate port of the electromechanical device as shown in Fig. 5.1b and as discussed in Section 3.1.4. To create either thebeamsplitter or two-mode squeezer interactions discussed in Section 3.1.1, microwave pumps are applied below and above the center frequency ωLC of the electromechanical device by the mechanicalresonant frequency ωm .

5.1.1

Sub-optimal capture of single photons

As an initial test of the beamsplitter interaction between a single photon and the mechanicalresonator, we implement the two protocols diagrammed in Fig. 5.3. Both of these protocols employ
the red-detuned pump to capture a propagating field. For the protocol diagrammed in Fig. 5.3a,

75

Figure 5.1: Diagram of the experiment. a, A simplified schematic shows the electromechanicaldevice connected to a photon source consisting of a transmon qubit in a microwave cavity. Pumps(arrows) used to create the capture (red) and amplification (blue) interactions are injected into thetransmission line. b, False-color micrograph of the electromechanical device where aluminium films(gray) are deposited on a sapphire substrate (blue). The spiral forms the inductor and the disc is
the mechanically-compliant capacitor. Micrograph acquired with assistance from Paul Blanchard
at NIST. c, Pumps are detuned below and above the LC circuit’s resonant frequency ωLC by themechanical resonant frequency ωm . Using a voltage bias Vdc , the LC circuit’s resonant response(dashed magenta) is tuned to match the much narrower resonance of the microwave cavity (green
solid) at the frequency ωce .

the propagating field consists of a large-amplitude test signal generated by a microwave synthesizer(see Appendix A.4). Importantly, we tailor this test signal to have spectral and temporal content
similar to that of a single-photon signal emitted from the cQED system. Such a signal has a center
frequency given by ωLC and a temporal envelope of the formbin (t) ∝ Θ(t)e−κc t/2(5.1)

where κc is the bandwidth of the cQED system and Θ(t) is the Heaviside step function. Because
the red-detuned pump is not optimally shaped to capture a decaying signal (see Section 3.1.2.1),
the efficiency of the capture process is sub-optimal. For the protocol diagrammed in Fig. 5.3b, nolarge-amplitude test signal is employed. Instead, a single photon is generated by driving the qubit’s

76

Figure 5.2: Picture of the experiment. It is mounted to the based stage of a dilution refrigerator
that provides an environment temperature of Tenv < 25 mK. The magnetic shield for the cQED
system is not shown.

77

Figure 5.3: Initial capture protocol. a, The timing diagram depicts the protocol used to capture alarge-amplitude coherent input test signal (green). A constant red-detuned pump is on during theentire duration of the protocol (bottom). b, The diagram depicts a protocol similar to the one in
a but without employing the test signal. Instead, a qubit control pump drives the blue sideband
transition in order to generate a single photon.

|gi |0i → |ei |1i transition using a π-pulse, as discussed in Section 2.3.4.
Conversion results for both a large-amplitude test signal and a single photon are presented
in Fig. 5.4. For the experiment in which the test signal is employed, we measure an average voltage
trace at the detector as shown in Fig. 5.4a. When the pump’s detuning is ∆r = −0.1ωm the
propagating signal is not captured. Instead, it is captured when ∆r = −ωm but with an efficiencyof less than 50%. (This sub-optimal capture is remedied in Section 5.3.) To model the suboptimal capture, we numerically integrate Eqns. 3.6. Using the parameters outlined in Table 3.1,
we find good agreement with the model and data. For the experiment in which a single photon is
employed, we measure the variance of voltage traces as shown in Fig. 5.4b. The response is similar
to that of the test case with a large-amplitude signal, indicating partial capture of a single photon.
Additionally, this initial capture experiment reveals that the strong, red-detuned pump adversely
affects the detector (discussed in Appendix A.6). To circumvent this problem, we developed a way
to operate the electromechanical device as an efficient pre-amplifier for the detector.

5.2

Electromechanical device as a low-noise amplifier
Signals at the single-photon level can be efficiently measured by operating the electromechan-ical device as a low-noise amplifier. A description of how it functions as an amplifier is discussed

78

Figure 5.4: Sub-optimal capture of propagating coherent signals and single photons. a, The plotshows voltage signals V (t) measured at the detector averaged over 102,400 repetitions of each
protocol. Large amplitude (> 106 photons/sec) coherent signals with a power decay rate of 60 kHz
are injected into the electromechanical device. A constant red-detuned pump is also injected into
the electromechanical device. The strength of the pump is Γr /2π ≈ 60 kHz. The top curve(blue) was acquired when the capture pump was at a detuning of ∆r = −0.1ωm where ωm is
the mechanical resonant frequency. The bottom curve (orange) was acquired when the capture
pump was red-detuned at ∆r = −ωm , the decaying signal is captured with a sub-optimal efficiency.
The dotted line corresponds to the expected behavior using independently measured parameters
for the electromechanical device. b, Propagating single photons were generated using the cQED
system and routed to the device. The plot shows the measured voltage variance σV2 (t) averaged
over 819,200 repetitions of each protocol. Top and bottom curves were acquired under similar
conditions to those in a. Note: an arbitrary offset is applied to the top curves (blue) in each plotin order to display them on the same plot as the bottom curves (orange).

79

Figure 5.5: Protocols used to amplify thermal states of mechanical motion. a, The mechanicalresonator is left in thermal equilibrium with its environment prior to pulsing on the blue-detuned
amplification pump. A signal with a rising temporal envelope emerges from the electromechanical device. b, A red-detuned pump cools the motion of the mechanical resonator. When theblue-detuned pump pulses on, a signal with a rising temporal envelope emerges from the electromechanical device. The amplitude of the emitted signal is smaller than the amplitude in a because
the resonator was cooled, and so it started with an occupation near its ground state.

in Section 3.1.3. In this section, we demonstrate that the electromechanical device can amplify
thermal states, and that its noise performance approaches the quantum limit[126].

5.2.1

Amplification of thermal states

As a first test of the electromechanical device’s performance as an amplifier, we use it toamplify thermal states of mechanical motion. By measuring thermal states of two different temperatures, we can determine the mechanical resonator’s occupation nth after it is cooled. Thisapproach is analogous to a Y -factor measurement[151] used to determine the noise temperatureof an amplifier by measuring Johnson noise emitted from a resistor held at two or more known
temperatures. For the electromechanical amplifier, we can either cool the motion of the resonatoror leave it in equilibrium with the environment.
In the first set of measurements, we execute the protocol shown in Fig. 5.5a. For this protocol,
the mechanical resonator is left in thermal equilibrium with its environment for a duration of atleast 30/κm where κm is the mechanical resonator’s linewidth. We then pulse on a blue-detuned
pump to both amplify the state of the resonator and convert it into a propagating microwave field.
The detector measures this output field, which we express in terms of its quadrature amplitudes X
and Y (see Section 5.3.2). These amplitudes have a total variance described by Var(Sh ) where thesub-script denotes ‘hot’ because the resonator was not cooled using the red-detuned pump.

80

Figure 5.6: Amplified thermal states. a, The figure shows scatter plots of measured quadrature√amplitudes S = X + iY of signals at the detector where S is in units of quanta. Without coolingthe mechanical mode, the thermal motion of the mechanical resonator is amplified while the bluedetuned pump is pulsed on. The quadrature amplitudes (Sh , hot) are shown in the ‘without cooling’plot. Cooling the mechanical mode results in a reduced scatter in the quadrature amplitudes (Sc ,cold) as shown in the ‘with cooling’ plot. b, The plot shows the ratios of the total variance ofthe quadrature amplitudes without cooling, Var(Sh ), and with cooling, Var(Sc ), as r = Γb τb wasvaried. For these measurements, Γb /2π = 60 kHz while τb was varied from 5 µs to 55 µs. Oncer  1, the plotted ratio approaches the expression given by Eqn. 5.2. Error bars correspond to thestatistical uncertainty in estimating the variance of 2,048 measurements.

In the second set of measurements, we execute the protocol shown in Fig. 5.5b. For thisprotocol, we cool the mechanical resonator to nearly its quantum ground state by using the reddetuned pump. We then turn on the amplification interaction and measured signals at the detector.
For these measurements, we obtain a total variance Var(Sc ) where the sub-script denotes ‘cold’
because the resonator was cooled using the red-detuned pump. Once r  1, the ratio of thevariances approaches
Var(Sh )nm + 1=.
Var(Sc )nth + 1

(5.2)

If the mechanical mode is ideally in its ground state, then nth = 0 and Eqn. 5.2 yields nm + 1 where
nm is the mechanical resonator’s equilibrium occupation that is discussed in Section 3.2.2. We
observe that the ratio in Eqn. 5.2 is slightly reduced from nm + 1 as shown in Fig. 5.6. We attribute
this reduction to a small residual occupation in the mechanical mode that is nth = 0.09±0.01 quanta.
Such a low occupation enables the electromechanical device to function as a nearly quantum-limited
amplifier.

81

5.3

Two protocols used to characterize the capture process
Exploiting the electromechanical device as an amplifier enables characterization of the capture

process. Because the process maps states of fields at the input of the electromechanical device tothe resonator, we must determine the input state and compare it to the captured state. To thisend, we have developed ‘calibration’ and ‘capture’ protocols that are used to determine the input
and captured states, respectively. Timing diagrams of the protocols is presented in Fig. 5.7a.
The two protocols are initially tested with coherent signals whose center frequency and bandwidth are chosen to match those created by the cQED system, as shown in Fig. 5.7b. To optimally
capture the input field, the red-detuned pump is modulated as prescribed by Eqn. 3.15. For a fieldwith a decaying temporal envelope of the form given by Eqn. 5.1, optimally capturing it requiresΓr (t) =

γe−γt
Θ(t)
1 − e−γt + γ/Γr (0)(5.3)

where γ is the power decay rate of the signal to be captured and Γr (0) is the coupling rate at t = 0.
For a high capture efficiency, Γr (0)  γ. As discussed in Section 3.1.2.1, the maximum attainable
coupling rate is limited by the circuit’s bandwidth Γr (t) < κLC . For our system, the decaying signal
has γ/2π = 60 kHz, and Γr (0)/2π ≈ 1 MHz. Using these parameters, we optimally capture signals
with decaying temporal content. For the data presented in the lowest plot in Fig. 5.7, the fractionof the input signal’s energy reflected off the LC circuit is 4.7%, which is in good agreement withthe predicted value of 4.5% obtained by numerically integrating Eqns. 3.6.

5.3.1

Calibration and capture protocols

To determine the field at the input of electromechanical device, we use the calibration protocol. We implement it by applying the blue-detuned pump coincident with the input field. In
this case, the electromechanical device functions as a linear phase-preserving amplifier whose input
and output are the incident and reflected microwave fields, respectively. These pulsed fields havedifferent envelopes; nevertheless, they are related by an energy gain of cosh2 (r/2) where r = Γb τband τb is the pump’s duration as discussed in Section 3.1.3. If we regard the input of the amplifier

82
as the incident microwave field, the fluctuations of the resonator’s motion are the source of theamplifier’s added noise, reaching the quantum limit[126] if the resonator is in its ground state[53].
After obtaining the input state, we use the capture protocol to determine the resonatorstate. We first apply the red-detuned pump coincident with the input field. Once it is captured, wethen apply the blue-detuned pump to amplify the resonator’s state. In contrast to the calibration
protocol, the amplifier’s input can now be regarded as the state of the resonator. The output isstill the reflected field, but the added noise is due to the vacuum fluctuations of the incident field.
When interpreted this way, the electromechanical device functions as a linear phase-conjugating
amplifier with an energy gain of sinh2 (r/2).
The gain is inherently tunable by adjusting r = Γb τb . For the data presented in Fig. 5.7b, WeR∞obtain the gain by calculating the total energy of the amplified signal, Eout = −∞ hV (t)i2 dt whenR∞
the pumps were on. We then normalize Eout to the energy of the input signal, Ein = −∞ hV (t)i2 dt
when the pumps were off. During these measurements, Vdc = 0 and the LC circuit was tuned outof resonance with the input signal. In Fig. 5.8, we plot the energy gain as a function of Γb . The
model prediction was generated by discretizing and numerically integrating Eqns. 3.6 for the total
input and output energies as Γb was varied. For the model prediction, we used the parameters
outlined in Table 3.1. We find that the behavior of the amplification process (for large amplitude
coherent signals) agrees well with the model.

5.3.2

Quadrature amplitude extraction

In order to characterize signals at the single-photon level, we express them in terms of a pairof quadrature amplitudes X and Y . During the amplification portion of either the calibration or
capture protocol, the electromechanical device emits a signal that is routed to the detector. Thissignal has a known temporal envelope given by f (t) = exp (Γb t/2) as discussed in Section 3.1.2.2
and is mixed down to a frequency of ωIF /2π = 1 MHz. We digitally sample the mixed down signal
at a rate of Rs = 5 MHz and form a discrete set of time-stamped voltages {tk , Vk } where tk = k · Ts
and Ts = Rs−1 is the sampling period. Because the signal’s spectral and temporal content is known,

83

Figure 5.7: Calibration and capture protocols. a, Timing diagrams depicting the input coherent
signal (green) of amplitude Vin and the pulse shapes of the pumps, Γr (t) and Γb (t), used to create
the capture or amplification interaction. For the calibration protocol (top), the amplification pumpis coincident with the signal pulse. The capture protocol (bottom) has a timing diagram similar tothe calibration protocol, but the input signal is coincident with a capture pulse that is temporally
shaped for optimal capture of the signal. At t = 30 µs, the mechanical state is amplified andconverted back into a microwave field. b, The plots show the voltage signals, V (t), measured at
the detector and averaged over 500 repetitions of each protocol when the pumps were either off or
on. During amplification, Γb /2π = 60 kHz which results in a gain of 53 dB.

Figure 5.8: Adjustable gain of the mechanical amplifier. The plots shows the measured energy gainas r = Γb τb was varied. The measurements were acquired using the capture protocol. For thesemeasurements, τb = 30 µs was held constant while Γb /2π was varied from 15 kHz to 55 kHz. The
solid line indicates the expected performance obtained by numerically integrating Eqns. 3.6 withall parameters determined separately.

84
we can define a pair of quadrature amplitudes[65]
r
X=
r
Y =

where C =

N
P

N

2Ts X
Vk f (tk )cos(ωIF tk )
GC
2Ts
GC

k=1
N
X

Vk f (tk )sin(ωIF tk )

(5.4)

k=1

|f (tk )|2 , N is the total number of samples, and G is the gain that relates quantum

k=1

fluctuations at the input of the electromechanical device to voltage fluctuations at its output.

5.3.3

Scaling the quadrature amplitudes

To scale the quadrature amplitudes, we determine G using reference measurements froma noise source of known variance. For each repetition of the calibration and capture protocols
discussed in Section 5.3, we make a reference measurement in which vacuum fluctuations of the
microwave field are injected into the electromechanical device. For this reference, the input to theelectromechanical device is Johnson noise emitted from a 50 Ω load that is thermally anchored tothe base stage of the dilution refrigerator held at a temperature T . For T < 25 mK and near thefrequency ωLC , the fluctuations in the microwave fields emitted from the load approach that of avacuum state with an average occupation of approximately 0.06 quanta[131]. As such, this input
microwave mode can be approximated as an ideal vacuum state described by the density operator
ρ0 = |0ih0|.
For the reference data set, we acquire a set of uncalibrated quadrature amplitudes that have
2 . If both the microwave and mechanical modes of the electromechanicala total voltage variance σVdevice were in pure vacuum states, then the voltage fluctuations measured at the detector would
have a total variance that corresponds to 1 quanta. For this case, one would calculate G so that2 = 1. However, even after cooling the mechanical resonator, we observe that it is in a weakG −1 σV

thermal state with an average occupation nth . We take into account this estimate by calculating G
so that2
G −1 σV= 1 + nth .

85
We then use G to scale the quadrature amplitudes (defined by Eqns. 5.4) for both data sets to have√units of quanta. To minimize systematic errors in the estimate of G due to potential drifts during
the measurement, we alternate between the vacuum reference and single photon measurements
every 512 executions of each protocol.

5.4

Capture, storage, and amplification of single photons
The calibration and capture protocols enable the capture, storage, and amplification of a

propagating microwave field as shown in Fig. 5.7. To verify that these two protocols can process
a non-Gaussian quantum state, we prepare the propagating field in a mixed vacuum and singlephoton state (see Section 4.3). Once the field is either directly amplified or captured, its state must
be determined. This task requires the ability to perform quantum state tomography on both the
input field and the mechanical resonator.

5.4.1

Tomographic procedure

By operating the electromechanical device as a low-noise amplifier as discussed in Section 5.2,
we can perform full quantum state tomography on both the input microwave field and the mechanical resonator. For each repetition of the two protocols depicted in Fig. 5.10, we record a voltage
signal, V (t), at the detector during the amplification portion of each protocol. For each voltage
record, we use Eqns. 5.4 and extract a pair of quadrature amplitudes, X and Y , for the state ofeither the resonator or input field. By making repeated measurements of V (t), we obtain a set ofquadrature amplitudes. We then use this information to extract a density matrix ρ via the methodof maximum likelihood state tomography discussed in Section 2.5.3 and in Ref. [121]. We refer tothe states of the input microwave field and of the mechanical resonator as ρe and ρm , respectively.

5.4.2

Tomographic results

After developing a tomographic procedure for determining the states of the input field andmechanical resonator, we test the calibration and capture protocols (see Section 5.3) using single

86

Figure 5.9: Correlated transmon qubit and mechanical states. The plot shows the probability ofthe qubit occupying the excited state P (e) and the ρ11 component of the mechanical state’s density
matrix as a function of the qubit’s blue sideband drive detuning ∆bs = ω − ωbs .

photons generated by the cQED system. To avoid obscuring the single-photon signals with thermal
fluctuations of the mechanical resonator, we employ sideband cooling to prepare the mechanicalresonator near its quantum ground state of motion[53]. We execute this cooling step prior to thestart of each protocol, thereby cooling the resonator’s motion to a remaining occupancy of approximately 0.1 quanta (see Section 5.2.1). In addition to cooling, we perform a reference measurementin which no single photons are generated. This reference measurement enables proper scaling of√
X and Y in units of quanta as discussed in Section 5.3.3. Additionally, it serves as a control
to verify the scatter in X and Y increases in the case that single photons are employed. Such anincrease is expected for single-photon signals (see Section 2.1.3).
As a first test of the capture protocol with single photons, we verify that the qubit state andreconstructed mechanical state ρm are correlated. Because we drive the transition |gi|0i → |ei|1i to
produce a single photon, the qubit excited state probability P (e) and the single-photon componentof ρm should depend on the sideband drive detuning ∆bs = ω − ωbs . Specifically, we expect that a
detuning of ∆bs = 0 will maximize the single-photon component of ρm . As shown in Fig. 5.9, thesetwo quantities are indeed correlated.
Next, we execute the calibration and capture protocols together with the tomographic procedure discussed in Section 5.4.1. For both protocols, the tomography yields density matrix estimates

87

Figure 5.10: Capture, storage, and amplification of single propagating photons. a, The diagramshows the calibration protocol where the green decaying sinusoid represents an input microwave
field in the state ρe . Prior to injecting an input state, the mechanical mode described by ρm iscooled close to its quantum ground state. A red cross indicates a mode was not measured. b,
Histograms of 512,000 measured quadrature amplitudes X and Y for the input state are plottedsuch that the histograms form a discretized and normalized joint probability distribution Pr(X, Y ).
The labels ‘no photon’ and ‘photon’ indicate whether single photons were generated or not. c, Thediagonal elements of ρe are obtained using a maximum likelihood state tomography (Supplemental
Information). d, The diagram shows the capture protocol where ρm,i is the initial mechanicalstate and τs is an adjustable storage time. After storage, the mechanical mode is in the final
state ρm,f . During amplification, the electrical mode is in a vacuum state ρ0 . e, The figure shows
the difference of the ‘photon’ and ‘no photon’ histograms acquired using the capture protocol,
highlighting the phase symmetric character of a single phonon state. f, Diagonal elements of ρmobtained at τs < 3 µs. g, The diagonal elements of ρm decay toward their thermal equilibrium values
as a function of τs . A model (solid lines) of ρm yields a characteristic storage time of τm = 137±6 µs
as discussed in Section 5.4.4.

88containing significant elements only on the diagonals as shown in Fig. 5.10c. In particular, we findthat the probability of detecting a single photon is [ρe ]11 = 0.33+0.02−0.01, where the limits are the
90% confidence interval as discussed in Appendix B. After capture, the probability of a single
phonon occupying the mechanical mode is [ρm ]11 = 0.26+0.01−0.02. In addition to performing tomography on the mechanical mode, we confirm that the captured state is nonclassical and that it can bemechanically-stored for an adjustable time.

5.4.3

Characterizing nonclassicality

To distinguish the captured state from a thermal or coherent state, we calculate the degree(2)of second-order coherence gm for the mechanical state. This quantity is often used to characterize
the statistical properties of photons emitted from a source[98]. For example, an ideal single photon
source has ρ11 = 1 with all other elements equal to zero. This state yields g (2) (0) = 0. In general,g (2) (0) =hn̂2 i − hn̂i
hn̂i2

(5.5)

where n̂ is the number operator. In terms of the density matrix elements, Eqn. 5.5 becomesPn(n − 1)ρnn
g (2) (0) = n
(5.6)
2
P
nρnn
n

where ρnn = hn|ρ|ni. Here, we use Eqn. 5.6 and the density matrix elements presented in Fig. 5.10f(2)

to calculate gm= 0.89+0.05−0.17 where the limits are the 90% confidence interval as discussed in(2)

Appendix B. For comparison, a thermal or coherent state of motion yields gm ≥ 1.

5.4.4

Characterizing the storage time

After capturing single photons, we vary the storage time τs and test the ability to mechanicallystore a non-Gaussian state. The results of this test are shown in Fig. 5.10f. To quantify the storage
time, we use a master equation formalism to model the evolution of ρm with the characteristic
storage time τm as the only free parameter. The master equation is discussed in Section 2.1.4 and89
in Ref. [108]. In the number basis, it becomes a set of coupled ordinary differential equations given
bydPn= −κ [(N + 1)nPn (t) + N (n + 1)Pn (t) − (N + 1)(n + 1)Pn+1 (t) − N nPn−1 (t)] ,
dt

(5.7)

where Pn = hn|ρm |ni describes the probability of observing n excitations in the mechanical modeand N = nm is the average occupation of the mechanical resonator’s environment. Because N  1
as discussed in Section 5.2.1, we make the approximation N + 1 ≈ N . This approximation allows
one to parametrize Eqn. 5.7 using the decoherence rate, γm = κN , as a single parameter. By
discretizing and numerically integrating Eqn. 5.7, we obtain Pn (t). In this model, the mechanicaldecoherence rate γm is a free parameter and the estimated density matrix elements of the initial state
are used to fix Pn (0). Simultaneous fits to the inferred diagonal elements presented in Fig. 5.10g−1 = 137 ± 6 µs, which isyields a characteristic storage time for a mixed photon state of τm = γm

about ten times longer than the time used to capture the input photon state.
As a control experiment, we do not capture and store single photons. For this experiment, we
execute the protocol depicted in Fig. 5.10 of the main text but without producing single photons
using the cQED system. The results of this experiment and fits to the model are shown in Fig. 5.11.
To model the data, we assume the mechanical resonator is initially in a thermal state and remains
thermal but with increasing average occupation
hni =

X

nPn

n

as it equilibrates with the thermal bath. In this case, Eqn. 5.7 can be simplified by usingXddPn (t)hn(t)i =
n(t)
dt
dt
n

(5.8)

d
hn(t)i = −κ (hn(t)i − N ) .
dt

(5.9)

and Eqn. 5.7 to obtain

Using the solutions to this rate equation, and the thermal distribution
Pn =

hnin
,
(hni + 1)n+1(5.10)

90

Figure 5.11: Evolution of a mechanical thermal state. The plot shows the first three diagonal
density matrix elements of the extracted mechanical state ρm as a function of the storage timeτs . The plot is similar to the one in Fig. 5.10g, but for the control experiment in which no single
photons were generated and captured. The solid lines indicate a model of a thermal state that uses
τm as a free parameter. The fit yields τm = 137 ± 8 µs. For comparison, the dashed line indicatesthe evolution of P1 for the initial mixed phonon state presented in Fig. 5.10g. In this case, a single
photon was captured and stored in the mechanical resonator.

We simultaneously fit the estimated density matrix elements shown in Supplementary Fig. 5.11 andextract τm = 137 ± 8 µs, which agrees with the storage time obtained using single photons.

5.5

Characterization of the capture process
Having demonstrated the ability to capture single photons, we now characterize how thecapture process affects an arbitrary qubit state. In general, the process of converting a quantum
state from a propagating photon to motion of the mechanical resonator can be thought of as a
‘black box’ with an input and output[70], as depicted in Fig. 5.12. This black box takes an input
electrical state ρe and converts it to an output mechanical state ρm . These two states are related by
the quantum operation E where ρm = E(ρe ). Ideally, the black box does not corrupt the converted
quantum state. In this case, E is the identity matrix and so ρm = ρe . However, in any realexperiment, there is loss and noise that corrupt the conversion process and so E will not simply be
the identity matrix. To characterize the quality of the map E, we use the average fidelity.

91

Figure 5.12: Black box model of the conversion process. The input (green) is described by thestate ρe . A process E converts ρe to mechanical motion described by the output state ρm .

5.5.1

Average fidelity characterizes process

In general, a quantum process is described mathematically by a completely-positive trace
preserving (CPTP) map
E : ρ → E(ρ).
The average fidelity of a process is a measure of how well that process preserves quantum information. It is defined asZdΨ hΨ| E(|Ψi hΨ|) |Ψi

Favg =
whereR(5.11)dΨ = 1. That is, the average fidelity gives the average overlap of the output of the processwith the initial state, averaged over all pure states. A related quantity is the entanglement fidelity,P
which is defined as follows: let |φi = d12|ii |ii denote a maximally entangled state. Here, {|ii}iis an orthonormal basis and d is the dimension of the Hilbert space on which the process E acts.
Then the entanglement fidelity is
Fe = hφ| (E ⊗ I)(|φi hφ|) |φi .

(5.12)The entanglement fidelity measures how well a system’s entanglement with another system is preserved when the quantum process acts only on the first system. The entanglement fidelity can becalculated by expanding Eqn. 5.12 in terms of its basis states:
Fe =

1 X
hi| hi| (E(|ii hj|) ⊗ |ii hj|) |ji |ji .
d2(5.13)

ij

Because hi|ji = δij , the above expression simplifies to
Fe =

1 X
hi| E(|ii hj|) |ji .
d2
ij(5.14)

92

Figure 5.13: A set of linearly independent states. Converting this set of states is sufficient todetermine the average fidelity of the conversion process that acts on states of dimension d = 2.

The utility of the entanglement fidelity is that it is easy to calculate, and there is a simple formula
relating the average fidelity to the entanglement fidelity [87]:Favg =

d × Fe + 1
d+1(5.15)

Because quantum processes act as linear operators on the space of operators, the fidelity can be
determined if the outputs E(ρi ) are known for each of a set of input states ρi that span the spaceof operators on Hilbert space. For a Hilbert space of dimension d, d2 linearly independent density
matrices are required[128]. A set of four linearly independent states for d = 2 are shown Fig. 5.13.

5.5.2

Classical bound on the average fidelity

There is a bound of Favg = 2/3 for converting a single qubit state using only classical resources. This bound is achieved as follows. Suppose we have two people, Alice and Bob, who want
to share quantum information with each other. Imagine Alice has a qubit prepared in state ρA ,which she wishes to send to Bob through a classical communication channel (e.g., a telephone). If
Alice and Bob share a maximally entangled Bell state, then this transmission of quantum information can be achieved via a teleportation protocol[200]. However, the use of such an entangledstate would constitute a quantum resource and is thus not allowed in this consideration. The best
Alice can do then is measure the qubit in some basis, and report the outcome to Bob, who thenprepares his qubit in the eigenstate corresponding to Alice’s measurement outcome. The quantum
operation describing this process isE(ρ) = P0 |0i h0| + P1 |1i h1|(5.16)

93
where P0 = h0| ρA |0i is the probability that Alice measures |0i, and simlarly for P1 . To calculate
the average fidelity, I insert this process into Eqn. 5.14:
Fe =1X
hi| E(|ii hj|) |ji
4
ij

=

1X
4

hi| h0|ii hj|0i |0i h0| + h1|ii hj|1i |1i h1|) |ji

ij

1
= .
2
Using Eqn. 5.15 and the above result, we find that Favg = 2/3. This bound can be generalized
by considering the problem of guessing the state of a qubit given an optimal measurement on
an ensemble of N identical copies of that qubit[201]. In this case, maximum possible fidelity is(N + 1)/(N + 2), which equals 2/3 for an ensemble consisting of a single qubit, N = 1.

5.5.3

Conversion of qubits

Qubits are encoded as superpositions of zero and one photons that propagate in a transmission line connected to the electromechanical device. As discussed in Section 5.5.1, the process of
capturing a propagating state is described by a map E between incident and captured states whose
quality is characterized by the average fidelity Favg . To determine Favg , it is sufficient to capture
a set of states that includes a single photon state and superpositions of zero and one photons as
pointed out in Section 5.5.1. we can create superposition states by first preparing the transmon

qubit in the superposition √12 |gi + eiϕ |ei , with varying phase ϕ, as shown in Fig. 5.14a.
To perform tomography on the qubit as shown in Fig. 5.14b, we measure the qubit’s excited state population P (e) using the electromechanical device as a low-noise amplifier (see Section 4.2.2.2). Achieving full state tomography can be achieved by first preparing the qubit in asuperposition state and then measuring its Pauli components[202]. To determine the qubit’s −σzcomponent, we simply measure P (e). Next, to determine the −σy component, we apply a π/2 rotation about the x-axis and then measure P (e). Lastly, we obtain the +σx component by applying
a π/2 rotation about the y-axis and then measuring P (e).

94
By driving the transition |gi|0i → |ei|1i, we can transfer the superposition state from thetransmon to the cavity and then let the cavity state evolve into the propagating field. Operating
the capture protocol on this set of states shows that the phase of the qubit state is converted tothe motion of the mechanical resonator as shown in Fig. 5.14c.
To perform tomography on the input and captured states, we follow the procedure illustrated
in Fig. 5.10 and determine both ρe and ρm . To calculate the fidelity of the capture process in thisexperiment, the estimated input and output density matrices are truncated to a Hilbert spacedimension of d = 2. The input states are obtained by estimating the density matrix of the input
microwave field after executing the calibration protocol. Similarly, the output states are obtainedby estimating the density matrix of the mechanical resonator state after executing the capture
protocol. A set of basis states is formed by using the following input states: a vacuum state, amixed vacuum and single-photon state, and mixed superposition states with phases chosen from theset {0, π/2}. We expand operators |ii hj| in this basis. The average fidelity Favg is then calculatedusing the known input and output states presented in Table B.2 together with Eqns. 5.14 and 5.15.
The input and captured density matrices provide enough information to calculate Favg = 0.83+0.03−0.06
for arbitrary qubit states. Crucially, the average fidelity exceeds 2/3, the highest possible fidelity
for transferring qubits using only classical resources as pointed out in Section 5.5.1.

5.5.4

Model of the capture process

The capture process maps the state of a propagating microwave field to the motion of themechanical resonator. This process can be viewed as implementing a beamsplitter interaction between the microwave and the mechanical modes of the electromechanical device[64]. A rudimentarymodel of the capture process therefore involves sending the state of the microwave field throughone port of a beamsplitter and the state of the mechanical mode through the other input port.
The transmissivity of the beamsplitter models imperfect capture of the propagating mode. For an
ideal capture of the propagating microwave mode, there would be no reflected microwave power
off of the electromechanical device. After tracing over the output microwave mode, the mechanical95

Figure 5.14: Conversion of propagating
qubits. a, The transmon qubit was prepared in the super1iϕ√
position state 2 |gi + e |ei with a phase ϕ chosen from the set {0, π/2, π, 3π/2}, as denoted by
the green arrow plotted on a Bloch sphere. b, The plot shows measured Pauli component amplitudes, hσk i, of the qubit state as a function of its phase where k = {x, y, z}. Single-shot readoutof the qubit state was achieved by using the electromechanical device as a nearly quantum-limited
amplifier. The solid lines are fits with the readout contrast of 60% as the only free parameter. c,
Difference between the ‘photon’ and ‘no photon’ histograms (similar to Fig. 5.10e) show the mechanical quadrature amplitudes, X and Y , as the transmon qubit’s phase was varied. d, The plotshows the argument of the off-diagonal density matrix element, ρ01 , for both the input microwave,
ρe , and captured mechanical, ρm , states as a function of ϕ. The mechanical state changes linearlyin ϕ, indicating that the conversion process is coherent. The apparent opposite dependence of ϕof the input and captured states is a result of the phase conjugate amplification of the mechanicalstate compared to the direct amplification of the input microwave state (solid lines indicate theexpected behavior).

96

Figure 5.15: Model of the capture process. The diagram depicts the three modes and the two
beamsplitter interactions used to model the capture process. The input state of the electrical modeis ρe and the ancilla state ρan describes the state of the additional electrical mode. The mechanicalmode is in the thermal state ρth . The electrical modes couple via a beamsplitter interaction B1
and the states ρ0 and ρth couple via another beamsplitter interaction B2 . A red cross denotes apartial trace.

resonator would contain the state of the input propagating microwave mode.
A limitation of this model is that it does not take into account the non-zero occupancy ofthe mechanical mode that is observed in this experiment. Ideally, an input vacuum state of the
microwave field to a beamsplitter with perfect efficiency would result in a vacuum state of themechanical mode. This model would describe the ideal cooling of the mechanical mode to its
quantum ground state. However, we observe a small residual occupation in the mechanical mode
nth . A possible source of this residual occupation is the internal loss of the LC circuit due to asurface layer of two-level system fluctuators[203]. This internal loss couples the microwave mode
to a thermal bath with a non-zero occupation.
We incorporate imperfect capture efficiency and internal loss of the LC circuit into a heuristic
model, as shown in Fig. 5.15. This model contains two beamsplitters, B1 and B2 , which model theinternal loss of the LC circuit and the capture efficiency, respectively. The input electrical state ρeand an ancillary electrical thermal state ρan are sent through a beamsplitter B1 , and the reduced
state of the output electrical mode is then mixed with a thermal state of the mechanical mode ρth
on beamsplitter B2 . Explicitly, this complete process isρout = tre (B2 ρ0 ⊗ ρth B2† )(5.17)

where ρ0 = tran (B1 ρan ⊗ ρe B1† ). The ancilla state ρan is taken to be a thermal state whose thermal
occupancy is determined implicitly by the requirement that a vacuum input state ρe = |0i h0|
yields a thermal intermediate state ρ0 = ρth with mean occupancy nth = 0.1 quanta, as discussed

97
in Section 5.2.1. In the Heisenberg picture[102, 98], a beamsplitter transforms the annihilation
operators for its two input modes a1 and a2 as B † a1,2 B = a01,2 , where
  
 
0
a1  cos θ − sin θ a1 
 =
 .
0
a2
sin θ cos θ
a2(5.18)

Under this convention, θ = 0 gives the identity whereas θ = π/2 corresponds to perfect capture.
The beamsplitters in the model are thus each parametrized by an angle θi where i = {1, 2}, which
in turn is related to the reflection coefficient Ri = sin2 θi .
For the data presented in Fig. 5.7, the fraction of energy reflected from the electromechanicaldevice is approximately 5%. As such, we use R2 = 0.95 for B2 . Note that a low amount of reflectedenergy corresponds to a large value of the beamsplitter reflection coefficient, because the reflection
coefficient models the efficiency of mode swapping. As discussed in Chapter 3, the fractional
resonator energy loss is measured to be approximately 0.14 and so we use R1 = 0.14 for B1 .
This cascaded beamsplitter model provides a way to predict the captured mechanical statesfrom the known input states obtained using the calibration protocol. For example, using themixed single photon input state described in the main text, the model predicts [ρm ]00 = 0.67,[ρm ]11 = 0.27, and [ρm ]22 = 0.05, whereas the 90% confidence intervals for these quantities obtained
from tomography and bootstrapping are [0.68, 0.69], [0.24, 0.27], and [0.05, 0.06], respectively. The
model developed in this section is also used to estimate the bias in the average fidelity calculation.
This bias is due to truncating the input and output density matrices to dimension d = 2, as
discussed in Appendix B
The mathematical description of the capture process should be viewed as a heuristic model.
This model relies on two free parameters, the measured circuit loss and capture efficiency, whichwere measured independently and may vary for each experimental run. Nevertheless, to the degree
that this model accurately describes the process, it provides prospects for improving the capture
process fidelity. For example, it is conceivable that reducing the circuit loss, increasing the capture
efficiency, and minimizing the occupation of the mechanical mode should yield a capture processthat approaches unit fidelity.

Chapter 6

Conclusion

In this thesis, I describe the conversion of signals that encode quantum information to the
motion of a micrometer-sized mechanical resonator. Such macroscopic resonators are appealingfor use in quantum science and technology because they can interface diverse physical systems
that include signals at optical and microwave frequencies[27], superconducting qubits[37], and evensingle electronic spins[38]. Yet it has been an outstanding challenge to interface mechanical motion
with a propagating signal comprised of a single photon, which is essential in emerging quantum
communication and modular quantum computation architectures. To address this challenge, I use
the mechanical system discussed in Chapter 3 to capture single photons emitted from the cQED
system described in Chapter 4. Importantly, the cQED system generates superpositions of zeroand one photons that can encode a single quantum bit of information – that is, a qubit.
Converting propagating qubits to mechanical motion is a crucial step towards storing more
complex states for use in protocols that require the feedforward of information, such as certain
quantum teleportation[204] and quantum error correction schemes[205, 206]. For example, teleporting a quantum state utilizes shared entanglement[200]. To generate entanglement, one can use
the cQED system to produce the Bell state√|ψi = (|gi |1i + |ei |0i)/ 2
where the first (second) ket represents the transmon qubit (propagating photon) state[207]. By
converting the Bell state to the motion of the mechanical resonator, its motional state can be entangled with the transmon qubit. To verify entanglement, one must use quantum state tomography

99
to determine the density matrix of the joint qubit and mechanical resonator system[208]. As partof this tomographic procedure, one must determine not only the hσz i component of the qubit, butalso its hσx i and hσy i components. Determining these components requires a protocol that includes
a π/2 pulse to rotate the qubit[202] followed by a measurement pulse to probe the qubit statedependent shift of the cavity[164, 209]. A standard approach to efficiently probing the cavity stateis through the use of a cQED system that employs a separate qubit-readout resonator[180, 86]. To
efficiently measure the mechanical resonator’s state, it must be reconstructed using quantum state
tomography as pointed out in Section 2.5.
Efficient tomography of the mechanical state underscores the need for harnessing the mechanical resonator itself as part of a low-noise amplifier. In this thesis, I use the mechanical resonator
to realize a phase-insensitive amplifier. As pointed out in Section 2.5.2, this type of amplifier inevitably adds 1/2 quanta of noise while simultaneously detecting both quadratures of a signal. In
contrast, achieving ideally noiseless detection requires the use of a phase-sensitive amplifier[131].
A route towards this goal is to use the actuation electrode (see Section 3.1.4) to parametricallydrive the mechanical resonator at twice its resonant frequency. Such an approach has been used
to amplify thermal states of a mechanical resonator[210], but has not yet been used to amplify
non-Gaussian quantum states. By pulsing the parametric drive, it may be possible to achievephase-sensitive gain that exceeds 30 dB. To add nearly no noise during this amplification process,it must be performed on a timescale that avoids unwanted mechanical decoherence. Furthermore,voltage fluctuations from the actuation line must be minimized so that they do not incoherently
drive the mechanical resonator.
Apart from demonstrating phase-sensitive mechanical amplification of a quantum state, anoutstanding challenge is to faithfully convert it to telecommunications light. Such light can efficiently distribute quantum information over distances that can easily exceed a few kilometers.
In contrast, microwave systems can only distribute quantum signals over tens of meters[211]. To
achieve conversion between microwave and optical light, emerging devices couple a mechanicalresonator to both microwave circuitry and an optical cavity[212, 213, 214, 215]. Yet the device

100currently under development in our lab is restricted to a narrow bandwidth of ∼ 10 kHz. To interface the narrow-bandwidth photon source presented in Chapter 4 with this device, its bandwidth
must be boosted by at least two orders of magnitude. Furthermore, achieving high-fidelity transduction requires that the added noise during the conversion process must be much less than 1/2
quanta[216]. Progress is already underway to combat this undesired added noise[217]. To this end,
microfabrication advances provide prospects for boosting mechanical lifetimes to the minute timescale[218, 219].
Beyond advancing quantum information processing, this thesis takes a step towards probing
quantum phenomena in a fundamentally new regime[220]. In particular, recent proposals contend
that macroscopic mechanical resonators may serve as a testbed for quantum theory in massive
objects[221, 82, 222, 223, 224, 225, 226, 227, 228]. Yet many of these proposals face the extraordinary challenge of accessing superpositions of macroscopically distinguishable mechanical states. By
converting superposition of zero and one photons to the motion of a macroscopic mechanical resonator, its center-of-mass motion can be delocalized over a distance of order the mode’s zero-point
motion xzp . To create more delocalized states of motion, the next step is to convert superpositions
with larger average photon numbers such as Schrödinger cat states[85]. Such an experiment isdaunting because the mechanical cat state can be easily thwarted by decoherence. But if carriedout successfully, it may provide an empirical way to falsify speculative conjectures that quantum
theory somehow breaks down for sufficiently massive objects.

Bibliography[1] H. Cavendish, “Experiments to determine the density of earth,” Philosophical Transactionsof the Royal Society of London, vol. 88, pp. 469–526, 1798.
[2] J. Kepler, Tychonis Brahei Dani Hyperaspistes. Frankfurt: Apud G. Tampachium, 1625.
[Online]. Available: https://archive.org/details/tychonisbraheida00kepl[3] M. Köhler, T. Minato, H. Kimura, and I. Mann, “Radiation pressure force acting oncometary aggregates,” Advances in Space Research, vol. 40, no. 2, pp. 266 – 271, 2007.
[Online]. Available: www.sciencedirect.com/science/article/pii/S0273117707005376[4] J. C. Maxwell, “A treatise on electricity and magnetism,” vol. 1, 1873.
[5] P. Lebedew, “Untersuchungen uber die druckkrafte des lichtes,” vol. 311, no. 11, pp.
433–458, 1901. [Online]. Available: http://dx.doi.org/10.1002/andp.19013111102[6] E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat andlight radiation,” Phys. Rev. (Series I), vol. 13, no. 11, pp. 307–320, Nov. 1901. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevSeriesI.13.307[7] A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev.
Lett., vol. 24, pp. 156–159, Jan 1970. [Online]. Available: http://link.aps.org/doi/10.1103/
PhysRevLett.24.156[8] T. Hänsch and A. Schawlow, “Cooling of gases by laser radiation,” Optics Communications,vol. 13, no. 1, pp. 68 – 69, 1975. [Online]. Available: //www.sciencedirect.com/science/
article/pii/0030401875901595[9] D. J. Wineland and H. Dehmelt, “Proposed 1014 δν < ν laser fluorescence spectroscopy on
Tl+ mono-ion oscillator,” Bull. Am. Phys. Soc., vol. 20, p. 637, 1975.
[10] A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett., vol. 40, pp.
729–732, Mar 1978. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.40.729[11] D. J. Wineland, R. E. Drullinger, and F. L. Walls, “Radiation-pressure cooling of boundresonant absorbers,” Phys. Rev. Lett., vol. 40, pp. 1639–1642, Jun 1978. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.40.1639[12] W. Neuhauser, M. Hohenstatt, P. Toschek, and H. Dehmelt, “Optical-sideband cooling ofvisible atom cloud confined in parabolic well,” Phys. Rev. Lett., vol. 41, pp. 233–236, Jul
1978. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.41.233

102[13] F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, “Laser cooling to thezero-point energy of motion,” Phys. Rev. Lett., vol. 62, pp. 403–406, Jan 1989. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevLett.62.403[14] J. Weber, “Observation of the thermal fluctuations of a gravitational-wave detector,”Phys. Rev. Lett., vol. 17, pp. 1228–1230, Dec 1966. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.17.1228[15] ——, “Gravitational radiation,” Phys. Rev. Lett., vol. 18, pp. 498–501, Mar 1967. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevLett.18.498[16] ——, “Gravitational-wave-detector events,” Phys. Rev. Lett., vol. 20, pp. 1307–1308, Jun
1968. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.20.1307[17] M. E. Gertsenshtein and V. I. Pustovoit, “On the detection of low frequency gravitational
waves,” JETP, vol. 43, August 1962.
[18] R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Quarterly
Progress Report, Research Laboratory of Electronics, no. 105, pp. 54 – 76, 1972. [Online].
Available: https://dspace.mit.edu/bitstream/handle/1721.1/56271/RLE QPR 105 V.pdf?
sequence=1#page=38[19] B. P. Abbott et al., “Observation of gravitational waves from a binary blackhole merger,” Phys. Rev. Lett., vol. 116, p. 061102, Feb 2016. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.116.061102[20] ——, “Localization and broadband follow-up of the gravitational-wave transient
GW150914,” The Astrophysical Journal Letters, vol. 826, p. L13, Jul. 2016. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevLett.20.1307[21] V. Braginsky, A. B. Manukin, and M. Y. Tikhonov, “Investigation of dissipative ponderomotive effects of electromagnetic radiation,” Journal of Experimental and Theoretical Physics,
vol. 31, no. 5, p. 829, 1970.
[22] V. Braginsky and Y. I. Vorontsov, “Quantum-mechanical limitations in macroscopic
experiments and modern experimental technique,” Soviet Physics Uspekhi, vol. 17, no. 5, p.
644, 1975. [Online]. Available: http://stacks.iop.org/0038-5670/17/i=5/a=R02[23] V. Braginsky and V. D. Popel’nyuk, “Small-vibration detector for a gravitation antenna,”
Pis’ma Zh. Eksp. Teor. Fiz., vol. 33, pp. 423–425, 1981.
[24] C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,”Phys. Rev. Lett., vol. 45, pp. 75–79, Jul 1980. [Online]. Available: http://link.aps.org/doi/
10.1103/PhysRevLett.45.75[25] M. Poot and H. S. van der Zant, “Mechanical systems in the quantum regime,” Physics
Reports, vol. 511, no. 5, pp. 273 – 335, 2012, mechanical systems in the quantum regime.
[Online]. Available: http://doi.org/10.1016/j.physrep.2011.12.004[26] Y. Greenberg, Y. A. Pashkin, and E. Il’ichev, “Nanomechanical resonators,” Physics-Uspekhi,
vol. 55, no. 4, p. 382, 2012. [Online]. Available: http://stacks.iop.org/1063-7869/55/i=4/a=
A03

103
[27] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,”
Rev. Mod. Phys., vol. 86, pp. 1391–1452, Dec 2014. [Online]. Available: http://link.aps.org/doi/10.1103/RevModPhys.86.1391[28] K. Schwab, “Spring constant and damping constant tuning of nanomechanical resonators
using a single-electron transistor,” Applied Physics Letters, vol. 80, no. 7, pp. 1276–1278,
2002. [Online]. Available: http://dx.doi.org/10.1063/1.1449533[29] M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, “Approaching the quantum limitof a nanomechanical resonator,” Science, vol. 304, no. 5667, pp. 74–77, 2004. [Online].
Available: http://science.sciencemag.org/content/304/5667/74[30] A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A. Clerk, M. P. Blencowe, and K. C.
Schwab, “Cooling a nanomechanical resonator with quantum back-action,” Nature, vol. 443,
no. 7108, pp. 193–196, Sep. 2006. [Online]. Available: http://dx.doi.org/10.1038/nature05027[31] A. N. Cleland, J. S. Aldridge, D. C. Driscoll, and A. C. Gossard, “Nanomechanicaldisplacement sensing using a quantum point contact,” Applied Physics Letters, vol. 81,
no. 9, pp. 1699–1701, 2002. [Online]. Available: http://dx.doi.org/10.1063/1.1497436[32] N. E. Flowers-Jacobs, D. R. Schmidt, and K. W. Lehnert, “Intrinsic noise properties ofatomic point contact displacement detectors,” Phys. Rev. Lett., vol. 98, p. 096804, Mar
2007. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.98.096804[33] J. D. Teufel, C. A. Regal, and K. W. Lehnert, “Prospects for cooling nanomechanical motion
by coupling to a superconducting microwave resonator,” New Journal of Physics, vol. 10, no. 9,
p. 095002, 2008. [Online]. Available: http://stacks.iop.org/1367-2630/10/i=9/a=095002[34] J. D. Teufel, J. W. Harlow, C. A. Regal, and K. W. Lehnert, “Dynamical backaction of
microwave fields on a nanomechanical oscillator,” Phys. Rev. Lett., vol. 101, p. 197203, Nov
2008. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.101.197203[35] C. A. Regal, J. D. Teufel, and K. W. Lehnert, “Measuring nanomechanical motion witha microwave cavity interferometer,” Nature Physics, vol. 4, pp. 555 – 560, 7/2008 2008.
[Online]. Available: http://www.nature.com/nphys/journal/v4/n7/full/nphys974.html[36] C. R. Locke, M. E. Tobar, E. N. Ivanov, and D. G. Blair, “Parametric interaction of theelectric and acoustic fields in a sapphire monocrystal transducer with a microwave readout,”
Journal of Applied Physics, vol. 84, no. 12, pp. 6523–6527, 1998. [Online]. Available:
http://dx.doi.org/10.1063/1.369023[37] A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero,
M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, andA. N. Cleland, “Quantum ground state and single-phonon control of a mechanicalresonator,” Nature, vol. 464, no. 7289, pp. 697–703, Apr. 2010. [Online]. Available:
http://dx.doi.org/10.1038/nature08967[38] D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, “Single spin detection by magnetic
resonance force microscopy,” Nature, vol. 430, pp. 329–332, 2004. [Online]. Available:
http://www.nature.com/nature/journal/v430/n6997/full/nature02658.html

104[39] A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Entanglement and decoherence of a
micromechanical resonator via coupling to a cooper-pair box,” Phys. Rev. Lett., vol. 88,
p. 148301, Mar 2002. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.88.
148301[40] I. Martin, A. Shnirman, L. Tian, and P. Zoller, “Ground-state cooling of mechanicalresonators,” Phys. Rev. B, vol. 69, p. 125339, Mar 2004. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevB.69.125339[41] L. Tian, “Entanglement from a nanomechanical resonator weakly coupled to a single
cooper-pair box,” Phys. Rev. B, vol. 72, p. 195411, Nov 2005. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevB.72.195411[42] E. Buks and M. P. Blencowe, “Decoherence and recoherence in a vibrating rf
squid,” Phys. Rev. B, vol. 74, p. 174504, Nov 2006. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevB.74.174504[43] L. F. Wei, Y.-x. Liu, C. P. Sun, and F. Nori, “Probing tiny motions of nanomechanicalresonators: Classical or quantum mechanical?” Phys. Rev. Lett., vol. 97, p. 237201, Dec
2006. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.97.237201[44] A. A. Clerk and D. W. Utami, “Using a qubit to measure photon-number statistics of adriven thermal oscillator,” Phys. Rev. A, vol. 75, p. 042302, Apr 2007. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevA.75.042302[45] J. Hauss, A. Fedorov, C. Hutter, A. Shnirman, and G. Schön, “Single-qubit lasing andcooling at the rabi frequency,” Phys. Rev. Lett., vol. 100, p. 037003, Jan 2008. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevLett.100.037003[46] D. Wahyu Utami and A. A. Clerk, “Entanglement dynamics in a dispersively coupled
qubit-oscillator system,” Phys. Rev. A, vol. 78, p. 042323, Oct 2008. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevA.78.042323[47] I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state coolingof a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett., vol. 99, p. 093901,
Aug 2007. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.99.093901[48] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted
sideband cooling of mechanical motion,” Phys. Rev. Lett., vol. 99, p. 093902, Aug 2007.
[Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.99.093902[49] A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, “Radiationpressure cooling of a micromechanical oscillator using dynamical backaction,” Phys. Rev.
Lett., vol. 97, p. 243905, Dec 2006. [Online]. Available: http://link.aps.org/doi/10.1103/
PhysRevLett.97.243905[50] S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, B. Hertzberg, K. Schwab,
D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a micromirror by radiationpressure,” Nature, 2006. [Online]. Available: http://www.nature.com/nature/journal/v444/
n7115/abs/nature05273.html

105[51] O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, “Radiation-pressure
cooling and optomechanical instability of a micromirror,” Nature, vol. 444, no. 7115, pp.
71–74, Nov. 2006. [Online]. Available: http://dx.doi.org/10.1038/nature05244[52] T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk, and K. C.
Schwab, “Preparation and detection of a mechanical resonator near the ground stateof motion,” Nature, vol. 463, no. 7277, pp. 72–75, Jan. 2010. [Online]. Available:
http://dx.doi.org/10.1038/nature08681[53] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D.
Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanicalmotion to the quantum ground state,” Nature, vol. 475, p. 359–363, 7/2011 2011. [Online].
Available: http://www.nature.com/nature/journal/v475/n7356/full/nature10261.html[54] J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher,
M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its
quantum ground state,” Nature, vol. 478, no. 7367, pp. 89–92, Oct. 2011. [Online]. Available:
http://dx.doi.org/10.1038/nature10461[55] A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, and O. Painter,“Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett., vol.
108, p. 033602, Jan 2012. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.
108.033602[56] A. J. Weinstein, C. U. Lei, E. E. Wollman, J. Suh, A. Metelmann, A. A. Clerk, and K. C.
Schwab, “Observation and interpretation of motional sideband asymmetry in a quantum
electromechanical device,” Phys. Rev. X, vol. 4, p. 041003, Oct 2014. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevX.4.041003[57] F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel,“Quantum nondemolition measurement of a nonclassical state of a massive object,”
Physical Review X, vol. 5, no. 4, p. 041037, 2015. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevX.5.041037[58] E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt,
A. A. Clerk, and K. C. Schwab, “Quantum squeezing of motion in a mechanicalresonator,” Science, vol. 349, pp. 952–955, October 2015. [Online]. Available: http://www.sciencemag.org/cgi/doi/10.1126/science.aac5138[59] J. M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää,“Squeezing of quantum noise of motion in a micromechanical resonator,” PhysicalReview Letters, vol. 115, no. 24, p. 243601, 2015. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.115.243601[60] T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanicalmotion with microwave fields,” Science, vol. 342, pp. 710–713, October 2013. [Online].
Available: http://science.sciencemag.org/content/342/6159/710[61] A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms andoptical photons,” Rev. Mod. Phys., vol. 87, pp. 1379–1418, Dec 2015. [Online]. Available:
http://link.aps.org/doi/10.1103/RevModPhys.87.1379

106[62] A. Narla, S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, E. Zalys-Geller, S. O.
Mundhada, W. Pfaff, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Robust concurrent
remote entanglement between two superconducting qubits,” Phys. Rev. X, vol. 6, p. 031036,
Sep 2016. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevX.6.031036[63] C. Monroe, R. Raussendorf, A. Ruthven, K. R. Brown, P. Maunz, L.-M. Duan, andJ. Kim, “Large-scale modular quantum-computer architecture with atomic memory andphotonic interconnects,” Phys. Rev. A, vol. 89, p. 022317, Feb 2014. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevA.89.022317[64] T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W.
Lehnert, “Coherent state transfer between itinerant microwave fields and a mechanicaloscillator.” Nature, vol. 495, pp. 210–214, March 2013. [Online]. Available: http://www.nature.com/nature/journal/v495/n7440/full/nature11915.html[65] R. W. Andrews, A. P. Reed, K. Cicak, J. D. Teufel, and K. W. Lehnert, “Quantum-enabled
temporal and spectral mode conversion of microwave signals.” Nature Communications,vol. 6, p. 10021, May 2015. [Online]. Available: http://dx.doi.org/10.1038/ncomms10021[66] F. Lecocq, J. D. Teufel, J. Aumentado, and R. W. Simmonds, “Resolving the vacuum
fluctuations of an optomechanical system using an artificial atom,” Nature Physics, vol. 11,
pp. 635–639, June 2015. [Online]. Available: http://dx.doi.org/10.1038/nphys3365[67] F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel,“Mechanically mediated microwave frequency conversion in the quantum regime,”
Physical Review Letters, vol. 116, no. 4, p. 043601, 2016. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.116.043601[68] C. F. Ockeloen-Korppi, E. Damskägg, J. M. Pirkkalainen, T. T. Heikkilä, F. Massel,and M. A. Sillanpää, “Low-noise amplification and frequency conversion with a multiportmicrowave optomechanical device,” Physical Review X, vol. 6, p. 041024, 2016. [Online].
Available: https://journals.aps.org/prx/abstract/10.1103/PhysRevX.6.041024[69] U. Andersen, J. Neergaard-Nielsen, P. van Loock, and A. Furusawa, “Hybrid discrete- andcontinuous-variable quantum information,” Nature Physics, vol. 11, no. 9, pp. 713–719, 2015.
[Online]. Available: http://www.nature.com/nphys/journal/v11/n9/full/nphys3410.html[70] I. L. Chuang and M. A. Nielsen, “Prescription for experimental determination of the
dynamics of a quantum black box,” Journal of Modern Optics, vol. 44, pp. 2455–2467, Nov.
1997. [Online]. Available: http://www.tandfonline.com/doi/abs/10.1080/09500349708231894[71] C. Weedbrook, S. Pirandola, R. Garcı́a-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, andS. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys., vol. 84, pp. 621–669, May
2012. [Online]. Available: http://link.aps.org/doi/10.1103/RevModPhys.84.621[72] S. D. Bartlett, B. C. Sanders, S. L. Braunstein, and K. Nemoto, “Efficientclassical simulation of continuous variable quantum information processes.” PhysicalReview Letters, vol. 88, no. 9, p. 097904, 2002. [Online]. Available:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.88.097904

107[73] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, “Quantum
state reconstruction of the single-photon fock state,” Phys. Rev. Lett., vol. 87, p. 050402,
Jul 2001. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.87.050402[74] C. Galland, N. Sangouard, N. Piro, N. Gisin, and T. J. Kippenberg, “Heralded single-phonon
preparation, storage, and readout in cavity optomechanics,” Phys. Rev. Lett., vol. 112,
p. 143602, Apr 2014. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.112.
143602[75] J.-M. Pirkkalainen, S. U. Cho, J. Li, G. S. Paraoanu, P. J. Hakonen, and M. A.
Sillanpää, “Hybrid circuit cavity quantum electrodynamics with a micromechanicalresonator,” Nature, vol. 494, pp. 211–215, Feb. 2013. [Online]. Available: http://www.nature.com/nature/journal/v494/n7436/full/nature11821.html[76] J. D. Cohen, S. M. Meenehan, G. S. Maccabe, S. Gröblacher, A. H. Safavi-Naeini,
F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of ananomechanical resonator,” Nature, vol. 520, pp. 522–525, Apr. 2015. [Online]. Available:
http://www.nature.com/nature/journal/v520/n7548/full/nature14349.html[77] R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant,
M. Aspelmeyer, and S. Gröblacher, “Non-classical correlations between single photons andphonons from a mechanical oscillator,” Nature, vol. 530, pp. 313 – 316, 2016. [Online].
Available: http://www.nature.com/nature/journal/v530/n7590/full/nature16536.html[78] A. P. Reed, K. H. Mayer, J. D. Teufel, L. D. Burkhart, W. Pfaff, M. Reagor, L. Sletten,
X. Ma, R. J. Schoelkopf, E. Knill, and K. W. Lehnert, “Faithful conversion of propagating
quantum information to mechanical motion,” ArXiv e-prints, Mar. 2017. [Online]. Available:
https://arxiv.org/abs/1703.02548[79] H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. R. Johnson,
M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf,“Observation of high coherence in Josephson junction qubits measured in a three-dimensional
circuit QED architecture,” Physical Review Letters, vol. 107, no. 24, p. 240501, 2011.
[Online]. Available: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.107.240501[80] U. Akram, N. Kiesel, M. Aspelmeyer, and G. J. Milburn, “Single-photon opto-mechanics
in the strong coupling regime,” New Journal of Physics, vol. 12, 2010. [Online]. Available:
http://iopscience.iop.org/article/10.1088/1367-2630/12/8/083030[81] F. Khalili, S. Danilishin, H. Miao, H. Müller-Ebhardt, H. Yang, and Y. Chen, “Preparing amechanical oscillator in non-gaussian quantum states,” Phys. Rev. Lett., vol. 105, p. 070403,
Aug 2010. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.105.070403[82] O. Romero-Isart, M. L. Juan, R. Quidant, and J. I. Cirac, “Toward quantum superpositionof living organisms,” New Journal of Physics, vol. 12, no. 3, p. 033015, 2010. [Online].
Available: http://stacks.iop.org/1367-2630/12/i=3/a=033015[83] M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D.
O’Connell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of Fock states in asuperconducting quantum circuit,” Nature, vol. 454, pp. 310–314, 2008. [Online]. Available:
http://www.nature.com/nature/journal/v454/n7202/abs/nature07136.html

108[84] M. Hofheinz, H. Wang, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley, A. D. O’Connell,
D. Sank, J. Wenner, J. M. Martinis, and A. N. Cleland, “Synthesizing arbitrary quantum
states in a superconducting resonator,” Nature, vol. 459, no. 7246, pp. 546–549, 2009.
[Online]. Available: http://dx.doi.org/10.1038/nature08005[85] B. Vlastakis, G. Kirchmair, Z. Leghtas, S. E. Nigg, L. Frunzio, S. M. Girvin, M. Mirrahimi,
M. H. Devoret, and R. J. Schoelkopf, “Deterministically Encoding Quantum Information
Using 100-Photon Schrödinger Cat States,” Science, vol. 342, no. 6158, pp. 607–610, 2013.
[Online]. Available: http://science.sciencemag.org/content/342/6158/607[86] W. Pfaff, C. J. Axline, L. D. Burkhart, U. Vool, P. Reinhold, L. Frunzio, L. Jiang, M. H.
Devoret, and R. J. Schoelkopf, “Schrödinger’s catapult: Launching multiphoton quantum
states from a microwave cavity memory,” ArXiv e-prints, Dec. 2016. [Online]. Available:
https://arxiv.org/abs/1612.05238[87] M. A. Nielsen, “A simple formula for the average gate fidelity of a quantum
dynamical operation,” Physics Letters A, vol. 303, pp. 249–252, 2002. [Online]. Available:
https://arxiv.org/abs/quant-ph/0205035[88] V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev,“The security of practical quantum key distribution,” Rev. Mod. Phys., vol. 81, pp. 1301–
1350, Sep 2009. [Online]. Available: http://link.aps.org/doi/10.1103/RevModPhys.81.1301[89] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien,“Quantum computers.” Nature, vol. 464, no. 7285, pp. 45–53, 2010. [Online]. Available:
http://dx.doi.org/10.1038/nature08812[90] E. Knill, G. Milburn, and R. Laflamme, “A scheme for efficient quantum computation
with linear optics,” Nature, vol. 409, no. January, pp. 46–52, 2001. [Online]. Available:
http://www.nature.com/nature/journal/v409/n6816/full/409046a0.html[91] H. Häffner, C. F. Roos, and R. Blatt, “Quantum computing with trapped ions,” Phys. Rep.,
vol. 469, pp. 155–203, Dec. 2008. [Online]. Available: https://arxiv.org/abs/0809.4368[92] L. M. K. Vandersypen and I. L. Chuang, “Nmr techniques for quantum control andcomputation,” Rev. Mod. Phys., vol. 76, pp. 1037–1069, Jan 2005. [Online]. Available:
http://link.aps.org/doi/10.1103/RevModPhys.76.1037[93] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, “Spinsin few-electron quantum dots,” Rev. Mod. Phys., vol. 79, pp. 1217–1265, Oct 2007. [Online].
Available: http://link.aps.org/doi/10.1103/RevModPhys.79.1217[94] F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Simmons, L. C. L. Hollenberg,
G. Klimeck, S. Rogge, S. N. Coppersmith, and M. A. Eriksson, “Silicon quantum
electronics,” Rev. Mod. Phys., vol. 85, pp. 961–1019, Jul 2013. [Online]. Available:
http://link.aps.org/doi/10.1103/RevModPhys.85.961[95] I. Bloch, “Quantum coherence and entanglement with ultracold atoms in opticallattices,” Nature, vol. 453, no. 7198, pp. 1016–1022, Jun. 2008. [Online]. Available:
http://dx.doi.org/10.1038/nature07126

109[96] M. H. Devoret and R. S. Schoelkopf, “Superconducting Circuits for Quantum Information:
An Outlook,” Science, vol. 339, no. 6124, pp. 1169 – 1173, 2013. [Online]. Available:
http://science.sciencemag.org/content/339/6124/1169[97] H. Baher, Analog and Digital Signal Processing.

Wiley, 1990.

[98] C. Gerry and P. Knight, Introductory Quantum Optics. Cambridge University Press, 2004.
[99] D. Morin, Classical Mechanics.

New York, USA: Cambridge University Press, 2008.

[100] W. H. Louisell, Coupled Mode and Parametric Electronics.
[101] R. Shankar, Principles of quantum mechanics.

Wiley, 1960.

New York, NY: Plenum, 1980.

[102] J. J. Sakurai, Modern quantum mechanics; rev. ed.

Reading, MA: Addison-Wesley, 1994.

[103] J. Weber, “Fluctuation dissipation theorem,” Phys. Rev., vol. 101, pp. 1620–1626, Mar
1956. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRev.101.1620[104] Y. Fukui and T. Morita, “Derivation of the stationary generalized Langevin equation,”
Journal of Physics A Mathematical General, vol. 4, pp. 477–490, Jul. 1971. [Online].
Available: http://iopscience.iop.org/article/10.1088/0305-4470/4/4/011[105] D. S. Lemons and A. Gythiel, “Paul Langevin’s 1908 paper ‘On the Theory of Brownian
Motion’ [‘Sur la théorie du mouvement brownien,’ C. R. Acad. Sci. (Paris) 146, 530–533(1908)],” American Journal of Physics, vol. 65, no. 11, pp. 1079–1081, 1997. [Online].
Available: http://dx.doi.org/10.1119/1.18725[106] M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields
produced in parametric amplification,” Phys. Rev. A, vol. 30, pp. 1386–1391, Sep 1984.
[Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.30.1386[107] G. Lindblad, “On the generators of quantum dynamical semigroups,” Comm. Math. Phys.,vol. 48, no. 2, pp. 119–130, 1976. [Online]. Available: http://projecteuclid.org/euclid.cmp/
1103899849[108] C. Gardiner and P. Zoller, Quantum Noise, 3rd ed. 6020 Innsbruck, Austria: Springer, 2004.
[109] D. Morin, Quantum Optics.

Berlin, Germany: Springer, 2008.

[110] S. Haroche and J. M. Raimond, Exploring the Quantum: Atoms, Cavities, and Photons.
Oxford: Oxford Univ. Press, 2006.
[111] J. P. Reithmaier, G. Sek, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh,
V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum
dot-semiconductor microcavity system,” Nature, vol. 432, no. 7014, pp. 197–200, Nov. 2004.
[Online]. Available: http://dx.doi.org/10.1038/nature02969[112] T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, C. Ell, O. B. Shchekin, andD. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic
crystal nanocavity,” Nature, vol. 432, pp. 200–203, Nov. 2004. [Online]. Available:
http://www.nature.com/nature/journal/v432/n7014/abs/nature03119.html

110[113] E. K. Irish and K. Schwab, “Quantum measurement of a coupled nanomechanicalresonatorcooper-pair box system,” Phys. Rev. B, vol. 68, p. 155311, Oct 2003. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevB.68.155311[114] A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R. Huang, J. Majer, S. Kumar, S. Girvin,
and R. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using
circuit quantum electrodynamics,” Nature, vol. 431, no. 7005, pp. 162–167, 2004. [Online].
Available: http://www.nature.com/nature/journal/v431/n7005/abs/nature02851.html[115] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity quantum
electrodynamics for superconducting electrical circuits: An architecture for quantum
computation,” Physical Review A, vol. 69, no. 6, p. 062320, June 2004. [Online]. Available:
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.75.032329[116] D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta,
A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin,
and R. J. Schoelkopf, “Resolving photon number states in a superconducting
circuit.” Nature, vol. 445, no. 7127, pp. 515–8, feb 2007. [Online]. Available:
http://www.nature.com/nature/journal/v445/n7127/abs/nature05461.html[117] A. Wallraff, D. I. Schuster, A. Blais, J. M. Gambetta, J. Schreier, L. Frunzio,
M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Sideband transitions andtwo-tone spectroscopy of a superconducting qubit strongly coupled to an on-chipcavity,” Phys. Rev. Lett., vol. 99, p. 050501, Jul 2007. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.99.050501[118] P. J. Leek, S. Filipp, P. Maurer, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl,
L. Steffen, and A. Wallraff, “Using sideband transitions for two-qubit operations insuperconducting circuits,” Phys. Rev. B, vol. 79, p. 180511, May 2009. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevB.79.180511[119] W. F. Kindel, M. D. Schroer, and K. W. Lehnert, “Generation and efficient measurementof single photons from fixed-frequency superconducting qubits,” Physical Review A,
vol. 93, p. 033817, 2016. [Online]. Available: https://journals.aps.org/pra/abstract/10.1103/
PhysRevA.93.033817[120] M. J. Reagor, “Superconducting cavities for circuit quantum electrodynamics,” Ph.D. dissertation, Yale University, 2015.
[121] C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, “Experimental state
tomography of itinerant single microwave photons,” Phys. Rev. Lett., vol. 106, p. 220503,
Jun 2011. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.106.220503[122] A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state
tomography,” Rev. Mod. Phys., vol. 81, pp. 299–332, Mar 2009. [Online]. Available:
http://link.aps.org/doi/10.1103/RevModPhys.81.299[123] U. Leonhardt, Measuring the Quantum State of Light.
Press, 1997.

Cambridge: Cambridge University

111[124] C. Eichler, D. Bozyigit, and A. Wallraff, “Characterizing quantum microwave radiation andits entanglement with superconducting qubits using linear detectors,” Phys. Rev. A, vol. 86, p.
032106, Sep 2012. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.86.032106[125] M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W.
Lehnert, “Amplification and squeezing of quantum noise with a tunable josephson
metamaterial,” Nature Physics, vol. 4, pp. 929 – 931, 12/2008 2008. [Online]. Available:
http://www.nature.com/nphys/journal/v4/n12/full/nphys1090.html[126] C. M. Caves, J. Combes, Z. Jiang, and S. Pandey, “Quantum limits on phase-preserving
linear amplifiers,” Physical Review A, vol. 86, no. 6, p. 063802, 2012. [Online]. Available:
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.86.063802[127] U. Leonhardt and H. Paul, “Phase measurement and Q function,” Phys. Rev. A, vol. 47, pp.
R2460–R2463, Apr 1993. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.
47.R2460[128] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th ed.
New York, NY, USA: Cambridge University Press, 2011.
[129] Z. Hradil, “Quantum-state estimation,” Phys. Rev. A, vol. 55, pp. R1561–R1564, Mar 1997.
[Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.55.R1561[130] J. Řeháček, Z. c. v. Hradil, E. Knill, and A. I. Lvovsky, “Diluted maximum-likelihood
algorithm for quantum tomography,” Phys. Rev. A, vol. 75, p. 042108, Apr 2007. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevA.75.042108[131] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction
to quantum noise, measurement, and amplification,” Rev. Mod. Phys., vol. 82, pp.
1155–1208, Apr 2010. [Online]. Available: https://journals.aps.org/rmp/abstract/10.1103/
RevModPhys.82.1155[132] A. Roy and M. Devoret, “Introduction to parametric amplification of quantum signals with
Josephson circuits,” Comptes Rendus Physique, vol. 17, pp. 740–755, Aug. 2016. [Online].
Available: http://www.sciencedirect.com/science/article/pii/S1631070516300640[133] M. Devoret, B. Huard, R. Schoelkopf, and L. F. Cugliandolo, Quantum Machines:
Measurement and Control of Engineered Quantum Systems. Oxford, 2014.
[134] C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum
stochastic differential equations and the master equation,” Phys. Rev. A, vol. 31, pp. 3761–
3774, Jun 1985. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.31.3761[135] H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev., vol. 128,
pp. 2407–2413, Dec 1962. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRev.
128.2407[136] J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W.
Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature, vol.
471, pp. 204–208, Mar. 2011. [Online]. Available: http://www.nature.com/nature/journal/
v471/n7337/full/nature09898.html

112[137] A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, “Optical bistability andmirror confinement induced by radiation pressure,” Phys. Rev. Lett., vol. 51, pp. 1550–1553,
Oct 1983. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.51.1550[138] A. Gozzini, F. Maccarrone, F. Mango, I. Longo, and S. Barbarino, “Light-pressure bistability
at microwave frequencies,” J. Opt. Soc. Am. B, vol. 2, no. 11, pp. 1841–1845, Nov 1985.
[Online]. Available: http://josab.osa.org/abstract.cfm?URI=josab-2-11-1841[139] P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, “Theory of radiation-pressuredriven interferometers,” J. Opt. Soc. Am. B, vol. 2, no. 11, pp. 1830–1840, Nov 1985.
[Online]. Available: http://josab.osa.org/abstract.cfm?URI=josab-2-11-1830[140] C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantumnoise reduction using a cavity with a movable mirror,” Phys. Rev. A, vol. 49, pp. 1337–1343,
Feb 1994. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.49.1337[141] C. K. Law, “Interaction between a moving mirror and radiation pressure: A hamiltonian
formulation,” Phys. Rev. A, vol. 51, pp. 2537–2541, Mar 1995. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevA.51.2537[142] A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolvedsideband cooling of a micro-mechanical oscillator,” Nature Physics, pp. 415–419, 2008.
[Online]. Available: http://www.nature.com/nphys/journal/v4/n5/full/nphys939.html[143] J. Zhang, K. Peng, and S. L. Braunstein, “Quantum-state transfer from light tomacroscopic oscillators,” Phys. Rev. A, vol. 68, p. 013808, Jul 2003. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevA.68.013808[144] J. W. Harlow, “Microwave electromechanics: Measuring and manipulating the quantum stateof a macroscopic mechanical oscillator,” Ph.D. dissertation, University of Colorado, Boulder,
2013.
[145] M. M. Nieto and D. R. Truax, “Holstein-Primakoff/Bogoliubov Transformations and theMultiboson System,” Fortschritte der Physik, vol. 45, pp. 145–156, 1997. [Online]. Available:
https://arxiv.org/abs/quant-ph/9506025[146] C. M. Caves and B. L. Schumaker, “New formalism for two-photon quantum optics. i.
quadrature phases and squeezed states,” Phys. Rev. A, vol. 31, pp. 3068–3092, May 1985.
[Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.31.3068[147] B. L. Schumaker and C. M. Caves, “New formalism for two-photon quantum optics. ii.
mathematical foundation and compact notation,” Phys. Rev. A, vol. 31, pp. 3093–3111,
May 1985. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.31.3093[148] B. L. Schumaker, “Quantum mechanical pure states with gaussian wave functions,”
Physics Reports, vol. 135, no. 6, pp. 317 – 408, 1986. [Online]. Available:
http://www.sciencedirect.com/science/article/pii/0370157386901791[149] R. W. Andrews, “Quantum signal processing with mechanical oscillators,” Ph.D. dissertation,
University of Colorado, Boulder, 2015.

113[150] K. Cicak, D. Li, J. A. Strong, M. S. Allman, F. Altomare, A. J. Sirois, J. D. Whittaker,
J. D. Teufel, and R. W. Simmonds, “Low-loss superconducting resonant circuits using
vacuum-gap-based microwave components,” Applied Physics Letters, vol. 96, no. 9, p.
093502, 2010. [Online]. Available: http://dx.doi.org/10.1063/1.3304168[151] D. M. Pozar, Microwave engineering; 3rd ed.

Hoboken, NJ: Wiley, 2005.

[152] J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,”
Nature Photonics, vol. 3, no. 12, pp. 687–695, 2010. [Online]. Available: http://www.nature.com/nphoton/journal/v3/n12/abs/nphoton.2009.229.html[153] T. E. Northup and R. Blatt, “Quantum information transfer using photons,” Nat Photon,
vol. 8, no. 5, pp. 356–363, 2014. [Online]. Available: http://dx.doi.org/10.1038/nphoton.
2014.53[154] A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. Schreier, B. R. Johnson, J. M. Chow,
L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Generating
single microwave photons in a circuit,” Nature, vol. 449, no. 7160, pp. 328–331, Sep. 2007.
[Online]. Available: http://dx.doi.org/10.1038/nature06126[155] D. Bozyigit, C. Lang, L. Steffen, J. M. Fink, C. Eichler, M. Baur, R. Bianchetti, P. J. Leek,
S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Antibunching of microwave-frequency
photons observed in correlation measurements using linear detectors,” Nature Physics,
vol. 7, no. 2, pp. 154–158, 2011. [Online]. Available: http://dx.doi.org/10.1038/nphys1845[156] Y.-F. Chen, D. Hover, S. Sendelbach, L. Maurer, S. T. Merkel, E. J. Pritchett,
F. K. Wilhelm, and R. McDermott, “Microwave photon counter based on josephson
junctions,” Phys. Rev. Lett., vol. 107, p. 217401, Nov 2011. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevLett.107.217401[157] C. Lang, C. Eichler, L. Steffen, J. M. Fink, M. J. Woolley, A. Blais, and A. Wallraff,“Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at
microwave frequencies,” Nature Physics, vol. 9, no. 5, pp. 1–4, 2013. [Online]. Available:
http://dx.doi.org/10.1038/nphys2612[158] Y. Yin, Y. Chen, D. Sank, P. J. J. O’Malley, T. C. White, R. Barends, J. Kelly,
E. Lucero, M. Mariantoni, A. Megrant, C. Neill, A. Vainsencher, J. Wenner, A. N.
Korotkov, A. N. Cleland, and J. M. Martinis, “Catch and release of microwave
photon states,” Phys. Rev. Lett., vol. 110, p. 107001, Mar 2013. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.110.107001[159] M. Pechal, L. Huthmacher, C. Eichler, S. Zeytinoğlu, A. A. Abdumalikov, S. Berger,
A. Wallraff, and S. Filipp, “Microwave-controlled generation of shaped single photons incircuit quantum electrodynamics,” Phys. Rev. X, vol. 4, p. 041010, Oct 2014. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevX.4.041010[160] Z. H. Peng, S. E. de Graaf, J. S. Tsai, and O. V. Astafiev, “Tuneable on-demand
single-photon source in the microwave range,” Nature Communications, vol. 7, p. 12588,
Aug. 2016. [Online]. Available: http://www.nature.com/doifinder/10.1038/ncomms12588

114[161] K. M. Gheri, K. Ellinger, T. Pellizzari, and P. Zoller, “Photon-Wavepackets as Flying Quantum Bits,” Fortschritte der Physik, vol. 46, p. 401, 1998.
[162] J. Clarke and F. K. Wilhelm, “Superconducting quantum bits,” Nature, vol. 453, no. 7198,
pp. 1031–42, jun 2008. [Online]. Available: http://www.nature.com/nature/journal/v453/
n7198/abs/nature07128.html[163] J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting
http:circuits,” Nature, vol. 474, pp. 589–597, Jun. 2011. [Online]. Available://www.nature.com/nature/journal/v474/n7353/full/nature10122.html[164] G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar,
M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observationof quantum state collapse and revival due to the single-photon Kerr effect.”Nature, vol. 495, no. 7440, pp. 205–9, mar 2013. [Online]. Available:
http://www.nature.com/nature/journal/v495/n7440/full/nature11902.html[165] B. E. A. Saleh and M. Carl Teich, Fundamentals of Photonics. New York, USA: Wiley, 2007.
[166] K. K. Likharev, Dynamics of Josephson Junctions and Circuits. Gordon and Breach Publications, 1986.
[167] T. V. Duzer and C. W. Turner, Principles of Superconductive Devices and Circuits. Prentice
Hall, 1999.
[168] B. D. Josephson, “Possible new effects in superconductive tunnelling,” Physics Letters,
vol. 1, pp. 251–253, Jul. 1962. [Online]. Available: http://www.sciencedirect.com/science/
article/pii/0031916362913690[169] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman lectures on physics. Vol. 3:
Quantum mechanics. Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1965.
[Online]. Available: http://www.feynmanlectures.caltech.edu/III 21.html[170] J. D. Jackson, Classical electrodynamics, 3rd ed.

New York, NY: Wiley, 1999.

[171] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H.
Devoret, S. M. Girvin, and R. J. Schoelkopf, “Charge-insensitive qubit design derived fromthe cooper pair box,” Phys. Rev. A, vol. 76, p. 042319, Oct 2007. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevA.76.042319[172] J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster, B. R. Johnson, J. M.
Chow, J. M. Gambetta, J. Majer, L. Frunzio, M. H. Devoret, S. M. Girvin,
and R. J. Schoelkopf, “Suppressing charge noise decoherence in superconducting
charge qubits,” Phys. Rev. B, vol. 77, p. 180502, May 2008. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevB.77.180502[173] A. A. Houck, J. Koch, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Life after charge
noise: recent results with transmon qubits,” Quantum Information Processing, vol. 8, no. 2,
pp. 105–115, 2009. [Online]. Available: http://dx.doi.org/10.1007/s11128-009-0100-6115[174] V. Bouchiat, D. Vion, P. Joyez, D. Esteve, and M. H. Devoret, “Quantum Coherence witha Single Cooper Pair,” Physica Scripta Volume T, vol. 76, pp. 165–170, 1998. [Online].
Available: http://dx.doi.org/10.1238/physica.topical.076a00165[175] Y. Nakamura, Y. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum
states in a single-Cooper-pair box,” Nature, vol. 398, no. April, p. 786, 1999. [Online].
Available: https://www.nature.com/nature/journal/v398/n6730/full/398786a0.html[176] K. W. Lehnert, K. Bladh, L. F. Spietz, D. Gunnarsson, D. I. Schuster, P. Delsing,and R. J. Schoelkopf, “Measurement of the excited-state lifetime of a microelectronic
circuit,” Phys. Rev. Lett., vol. 90, p. 027002, Jan 2003. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.90.027002[177] R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro,
J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N.
Cleland, and J. M. Martinis, “Coherent josephson qubit suitable for scalable quantum
integrated circuits,” Phys. Rev. Lett., vol. 111, p. 080502, Aug 2013. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.111.080502[178] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and
R. J. Schoelkopf, “Approaching unit visibility for control of a superconducting qubit withdispersive readout,” Phys. Rev. Lett., vol. 95, p. 060501, Aug 2005. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.95.060501[179] C. Rigetti, J. M. Gambetta, S. Poletto, B. L. T. Plourde, J. M. Chow, A. D. Córcoles,
J. A. Smolin, S. T. Merkel, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen,and M. Steffen, “Superconducting qubit in a waveguide cavity with a coherence time
approaching 0.1 ms,” Phys. Rev. B, vol. 86, p. 100506, Sep 2012. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevB.86.100506[180] M. Reagor, W. Pfaff, C. Axline, R. W. Heeres, N. Ofek, K. Sliwa, E. Holland,
C. Wang, J. Blumoff, K. Chou, M. J. Hatridge, L. Frunzio, M. H. Devoret,
L. Jiang, and R. J. Schoelkopf, “Quantum memory with millisecond coherence incircuit QED,” Phys. Rev. B, vol. 94, p. 014506, Jul 2016. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevB.94.014506[181] S. E. Nigg, H. Paik, B. Vlastakis, G. Kirchmair, S. Shankar, L. Frunzio, M. H.
Devoret, R. J. Schoelkopf, and S. M. Girvin, “Black-box superconducting circuit
quantization,” Phys. Rev. Lett., vol. 108, p. 240502, Jun 2012. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.108.240502[182] M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun, D. I. Schuster, L. Frunzio, and
R. J. Schoelkopf, “High-Fidelity Readout in Circuit Quantum Electrodynamics Using the
Jaynes-Cummings Nonlinearity,” Phys. Rev. Lett., vol. 105, p. 173601, Oct 2010. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevLett.105.173601[183] L. S. Bishop, E. Ginossar, and S. M. Girvin, “Response of the Strongly Driven
Jaynes-Cummings Oscillator,” Phys. Rev. Lett., vol. 105, p. 100505, Sep 2010. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevLett.105.100505

116[184] A. Palacios-Laloy, F. Mallet, F. Nguyen, P. Bertet, D. Vion, D. Esteve, andA. N. Korotkov, “Experimental violation of a Bell’s inequality in time with weakmeasurement,” Nature Physics, vol. 6, no. 6, pp. 442–447, Apr. 2010. [Online]. Available:
http://dx.doi.org/10.1038/nphys1641[185] R. Vijay, D. H. Slichter, and I. Siddiqi, “Observation of quantum jumps in a superconducting
artificial atom,” Phys. Rev. Lett., vol. 106, p. 110502, Mar 2011. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.106.110502[186] M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geerlings, T. Brecht,
K. M. Sliwa, B. Abdo, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H.
Devoret, “Quantum Back-Action of an Individual Variable-Strength Measurement,”
Science (New York, N.Y.), vol. 339, no. 6116, pp. 178–181, 2013. [Online]. Available:
http://www.sciencemag.org/cgi/doi/10.1126/science.1226897[187] D. Ristè, J. G. van Leeuwen, H.-S. Ku, K. W. Lehnert, and L. DiCarlo, “Initialization bymeasurement of a superconducting quantum bit circuit,” Phys. Rev. Lett., vol. 109, p. 050507,
Aug 2012. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.109.050507[188] D. Ristè, C. C. Bultink, K. W. Lehnert, and L. DiCarlo, “Feedback control of a solid-state
qubit using high-fidelity projective measurement,” Phys. Rev. Lett., vol. 109, p. 240502, Dec
2012. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.109.240502[189] M. D. Schroer, M. H. Kolodrubetz, W. F. Kindel, M. Sandberg, J. Gao, M. R. Vissers,
D. P. Pappas, A. Polkovnikov, and K. W. Lehnert, “Measuring a topological transition in
an artificial spin-1/2 system,” Phys. Rev. Lett., vol. 113, p. 050402, Jul 2014. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevLett.113.050402[190] V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum Nondemolition Measurement,”
Science, vol. 209, p. 547, 1980.
[191] N. Boulant, G. Ithier, P. Meeson, F. Nguyen, D. Vion, D. Esteve, I. Siddiqi, R. Vijay,
C. Rigetti, F. Pierre, and M. Devoret, “Quantum nondemolition readout using a josephson
bifurcation amplifier,” Phys. Rev. B, vol. 76, p. 014525, Jul 2007. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevB.76.014525[192] R. Bianchetti, S. Filipp, M. Baur, J. M. Fink, M. Göppl, P. J. Leek, L. Steffen,
A. Blais, and A. Wallraff, “Dynamics of dispersive single-qubit readout in circuit quantum
electrodynamics,” Phys. Rev. A, vol. 80, p. 043840, Oct 2009. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevA.80.043840[193] J. Gambetta, A. Blais, M. Boissonneault, A. A. Houck, D. I. Schuster, andS. M. Girvin, “Quantum trajectory approach to circuit QED: Quantum jumps andthe Zeno effect,” Phys. Rev. A, vol. 77, p. 012112, Jan 2008. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevA.77.012112[194] W. F. Kindel, “Generation and efficient measurement of single photons using superconducting
circuits,” Ph.D. dissertation, University of Colorado, Boulder, 2015.
[195] K. L. Geerlings, “Improving coherence of superconducting qubits and resonators,” Ph.D.
dissertation, Yale University, 2013.

117[196] C. J. Foot, Atomic Physics, 1st ed.

Oxford University Press, USA, Feb. 2005.

[197] A. D. Córcoles, J. M. Chow, J. M. Gambetta, C. Rigetti, J. R. Rozen, G. A. Keefe,
M. B. Rothwell, M. B. Ketchen, and M. Steffen, “Protecting superconducting qubits fromradiation,” Applied Physics Letters, vol. 99, no. 18, p. 181906, 2011. [Online]. Available:
http://dx.doi.org/10.1063/1.3658630[198] J. E. Johnson, C. Macklin, D. H. Slichter, R. Vijay, E. B. Weingarten, J. Clarke, and
I. Siddiqi, “Heralded state preparation in a superconducting qubit,” Phys. Rev. Lett., vol.
109, p. 050506, Aug 2012. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.
109.050506[199] G. B. P., Advanced Kalman Filtering, Least-Squares and Modeling: A Practical Handbook.
New York: John Wiley and Sons, 2011.
[200] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters,“Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosenchannels,” Physical Review Letters, vol. 70, no. 13, p. 1895, 1993. [Online]. Available:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.70.1895[201] S. Massar and S. Popescu, “Optimal extraction of information from finite quantum
ensembles,” Physical Review Letters, vol. 74, p. 1259, 1995. [Online]. Available:
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.74.1259[202] M. Steffen, M. Ansmann, R. McDermott, N. Katz, R. C. Bialczak, E. Lucero, M. Neeley,
E. M. Weig, A. N. Cleland, and J. M. Martinis, “State tomography of capacitively shunted
phase qubits with high fidelity,” Phys. Rev. Lett., vol. 97, p. 050502, Aug 2006. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevLett.97.050502[203] J. Gao, J. Zmuidzinas, B. A. Mazin, H. G. LeDuc, and P. K. Day, “Noise
properties of superconducting coplanar waveguide microwave resonators,” Applied
Physics Letters, vol. 90, no. 10, p. 102507, 2007. [Online]. Available:
http://aip.scitation.org/doi/full/10.1063/1.2711770[204] L. Steffen, Y. Salathe, M. Opplinger, P. Kurpiers, M. Baur, C. Lang, C. Eichler,
G. Puebla-Hellmann, A. Fedorov, and A. Wallraff, “Deterministic quantum teleportation
with feed-forward in a solid state system,” Nature, vol. 500, p. 319322, Aug 2013. [Online].
Available: http://www.nature.com/nature/journal/v500/n7462/full/nature12422.html[205] J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank,
J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi,
C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner,
A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in
a superconducting quantum circuit,” Nature, vol. 519, pp. 66–69, Mar. 2015. [Online].
Available: http://www.nature.com/nature/journal/v519/n7541/abs/nature14270.html[206] N. Ofek, A. Petrenko, R. Heeres, P. Reinhold, Z. Leghtas, B. Vlastakis, Y. Liu,
L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J.
Schoelkopf, “Extending the lifetime of a quantum bit with error correction insuperconducting circuits,” Nature, vol. 536, p. 441, 2016. [Online]. Available:
http://www.nature.com/nature/journal/v536/n7617/abs/nature18949.html

118[207] C. Eichler, C. Lang, J. M. Fink, J. Govenius, S. Filipp, and A. Wallraff,“Observation of entanglement between itinerant microwave photons and a superconducting
qubit,” Phys. Rev. Lett., vol. 109, p. 240501, Dec 2012. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.109.240501[208] A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett., vol. 77, pp. 1413–
1415, Aug 1996. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevLett.77.1413[209] R. W. Heeres, P. Reinhold, N. Ofek, L. Frunzio, L. Jiang, M. H. Devoret, and R. J.
Schoelkopf, “Implementing a Universal Gate Set on a Logical Qubit Encoded in an
Oscillator,” ArXiv e-prints, Aug. 2016. [Online]. Available: https://arxiv.org/abs/1608.02430[210] D. Rugar and P. Grütter, “Mechanical parametric amplification and thermomechanicalnoise squeezing,” Phys. Rev. Lett., vol. 67, pp. 699–702, Aug 1991. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.67.699[211] P. Kurpiers, T. Walter, P. Magnard, Y. Salathe, and A. Wallraff, “Characterizing
the attenuation of coaxial and rectangular microwave-frequency waveguides at cryogenic
temperatures,” ArXiv e-prints, Dec. 2016. [Online]. Available: https://arxiv.org/abs/1612.
07977[212] J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanicalcoupling between microwave and optical photons,” Nature Physics, vol. 9, no. 11, pp.
712–716, September 2013. [Online]. Available: http://www.nature.com/nphys/journal/v9/
n11/full/nphys2748.html[213] T. Bagci, A. Simonsen, S. Schmid, L. G. Villanueva, E. Zeuthen, J. Appel, J. M. Taylor,A. Sørensen, K. Usami, A. Schliesser, and E. S. Polzik, “Optical detection of radio waves
through a nanomechanical transducer,” Nature, vol. 507, pp. 81–85, Mar. 2014. [Online].
Available: http://www.nature.com/nature/journal/v507/n7490/full/nature13029.html[214] R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A.
Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave andoptical light,” Nature Physics, vol. 10, no. April, pp. 321–326, 2014. [Online]. Available:
http://www.nature.com/nphys/journal/v10/n4/full/nphys2911.html[215] K. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of
microwave phonons in cavity-optomechanical circuits,” Nature Photonics, vol. 10, no. June,
pp. 489–496, 2016. [Online]. Available: http://www.nature.com/nphoton/journal/v10/n7/
full/nphoton.2016.107.html[216] S. A. McGee, D. Meiser, C. A. Regal, K. W. Lehnert, and M. J. Holland,“Mechanical resonators for storage and transfer of electrical and optical quantum
states,” Phys. Rev. A, vol. 87, p. 053818, May 2013. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevA.87.053818[217] P.-L. Yu, K. Cicak, N. S. Kampel, Y. Tsaturyan, T. P. Purdy, R. W. Simmonds,
and C. A. Regal, “A phononic bandgap shield for high-Q membrane microresonators,”
Applied Physics Letters, vol. 104, no. 2, p. 023510, 2014. [Online]. Available:
http://dx.doi.org/10.1063/1.4862031

119[218] C. Reinhardt, T. Müller, A. Bourassa, and J. C. Sankey, “Ultralow-noise SiN trampoline
resonators for sensing and optomechanics,” Physical Review X, vol. 6, p. 021001, 2016.
[Online]. Available: https://journals.aps.org/prx/abstract/10.1103/PhysRevX.6.021001[219] R. A. Norte, J. P. Moura, and S. Gröblacher, “Mechanical resonators for quantum
optomechanics experiments at room temperature,” Physical Review Letters, vol. 116,
no. 14, p. 147202, 2016. [Online]. Available: https://journals.aps.org/prl/abstract/10.1103/
PhysRevLett.116.147202[220] M. Arndt, M. Aspelmeyer, and A. Zeilinger, “How to extend quantum experiments,”
Fortschr. Phys., vol. 57, 2003. [Online]. Available: http://dx.doi.org/10.1002/prop.200900104[221] W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards quantum superpositionsof a mirror,” Phys. Rev. Lett., vol. 91, p. 130401, Sep 2003. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.91.130401[222] I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and Č. Brukner, “Probing
Planck-scale physics with quantum optics,” Nature Physics, vol. 8, pp. 393–397, May 2012.
[Online]. Available: http://www.nature.com/nphys/journal/v8/n5/full/nphys2262.html[223] T. Hong, H. Yang, H. Miao, and Y. Chen, “Open quantum dynamics of single-photon
optomechanical devices,” Phys. Rev. A, vol. 88, p. 023812, Aug 2013. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevA.88.023812[224] H. Yang, H. Miao, D.-S. Lee, B. Helou, and Y. Chen, “Macroscopic Quantum Mechanics
in a Classical Spacetime,” Physical Review Letters, vol. 110, no. 17, p. 170401, Apr. 2013.
[Online]. Available: https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.110.170401[225] Y. Chen, “Macroscopic quantum mechanics: theory and experimental concepts ofoptomechanics,” Journal of Physics B Atomic Molecular Physics, vol. 46, no. 10,
May 2013. [Online]. Available: http://iopscience.iop.org/article/10.1088/0953-4075/46/10/
104001/meta[226] A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, “Models of wave-function
collapse, underlying theories, and experimental tests,” Rev. Mod. Phys., vol. 85, pp. 471–527,
Apr 2013. [Online]. Available: http://link.aps.org/doi/10.1103/RevModPhys.85.471[227] A. Großardt, J. Bateman, H. Ulbricht, and A. Bassi, “Optomechanical test of the
Schrödinger-Newton equation,” Phys. Rev. D, vol. 93, p. 096003, May 2016. [Online].
Available: http://link.aps.org/doi/10.1103/PhysRevD.93.096003[228] S. G. Hofer, K. W. Lehnert, and K. Hammerer, “Proposal to Test Bell’s Inequality inElectromechanics,” Phys. Rev. Lett., vol. 116, p. 070406, Feb 2016. [Online]. Available:
http://link.aps.org/doi/10.1103/PhysRevLett.116.070406[229] R. G. Rogers, Low Phase Noise Microwave Oscillator Design. Boston: Artech House, 1991.
[230] B. Efron and R. J. Tibshirani, An introduction to the bootstrap, ser. Mono. Stat. Appl.
Probab. London: Chapman and Hall, 1993.

Appendix A

Measurement network

A diagram of the measurement network is shown in Fig. A.1. The network consists of fivemain parts: A transmon qubit is embedded in a microwave cavity (cQED system) with two ports(Fig. A.1a). The cQED system is connected to an electromechanical device, which is mounted tothe base stage of a dilution refrigerator (Fig. A.1b). The center frequency of the electromechanical device is controlled by a voltage bias provided by an actuation line (Fig. A.1c). Microwave
pumps and signals are synthesized and used to control and probe both the cQED system and theelectromechanical device (Fig. A.1d). Microwave signals are measured using a detector (Fig. A.1e).

A.1Electromechanical device
The electromechanical device is mounted to the mixing chamber of a dilution refrigerator andcooled to < 25 mK. The construction and operation of the electromechanical device is described inRef. [65].

A.2

Transmon qubit and microwave cavity
The transmon qubit consists of an Al/AlOx /Al Josephson junction shunted by a supercon-

ducting aluminum coplanar capacitor. This circuit is lithographically fabricated on a single-crystal
sapphire substrate and the coplanar capacitor acts as a dipole antenna. The Josephson inductanceis approximately 11 nH and the total capacitance is approximately 69 fF.
We embed the transmon qubit in a three-dimensional microwave cavity, as described in121

Figure A.1: Detailed schematic of the measurement network. a, Circuit quantum electrodynamics(cQED) system. b, Electromechanical device. c, Actuation line. d, Microwave pumps and signal
synthesis. e, Detector.

Ref. [79]. The cavity is milled from a two pieces of extruded T6061 aluminium and the innersurfaces of the cavity walls are mechanically polished. Two holes that serve as microwave coupling
ports are drilled into the cavity. One of the ports is weakly coupled to a pump line, which we use
to excite either the qubit or cavity. The other cavity port is strongly coupled to a transmission line

122
which is routed to components connected to the electromechanical device.
To reduce stray magnetic fields that can affect the qubit’s coherence times, we use nonmagnetic materials such as copper and brass to construct components that are in close proximity
to the qubit and cavity. For additional magnetic shielding, we enclose the cavity in a Cryoperm
magnetic shield (Amuneal Manufacturing). At room temperature and inside the magnetic shield,
we measure an ambient magnetic field of ∼ 20 mG near the cavity. During operation of theexperiment, the magnetic shield is wrapped in absorptive microwave material (Eccosorb) and athin layer (< 100 µm) of aluminium.
Signals emitted from the cavity are routed using copper cables to the input of two low insertion
loss circulators and a directional coupler. These custom cryogentic and magnetically shielded
circulators (Raditek, Inc.) are used to route signals emitted from the cavity to the electromechanicaldevice, while also providing isolation from the high power (< 1 nW) red- and blue-detuned pumps.
At 300 K and near 7.283 GHz, the isolation of the two circulators connected in series was measured
to be −43 dB. A similar level of isolation was measured at 4 K. Each circulator is specified by the
manufacturer to have an insertion loss of 0.2 dB (at 100 mK). Following the circulators, the outputof the directional coupler is connected to the electromechanical device using a superconducting
niobium-titanium (NbTi) cable.

A.3

Actuation line
The voltage on the actuation line controls the center frequency of the electromechanicaldevice. A stable and low noise voltage source (Yokogawa 7651) provides a constant Vdc during theoperation of the experiment. Filtering on the actuation line is nearly identical to the configuration
described in Ref. [65].

A.4

Arbitrary microwave pump and signal generation
The red- and blue-detuned pumps at ωLC −ωm and ωLC +ωm , respectively, are generated using

two separate microwave synthesizers (Agilent PSG). However, these synthesizers by themselves can-

123
not produce microwave pulses with programmable temporal envelopes as required by the protocols
depicted in Fig. A.2. To generate such pulses, we shape the temporal envelopes of microwave tones
emitted by the synthesizers. To this end, we use a double-balanced mixer (Marki MM1-0625HS)
driven by waveforms with a programmable amplitude provided by an arbitrary waveform generator(Tektronix AWG 5014c). Waveforms generated by the AWG are shown in Fig. A.2. The shaped
microwave pulses have a dynamic range of approximately 50 dB, set by the LO-RF isolation of
the double-balanced mixer. Additional isolation (80 dB) is achieved by pulsing off the microwave
synthesizers when the pumps are not needed. The shaped pulses are then amplified (Mini-Circuits
ZVA-183V) and filtered[229]. Additionally, the pumps have Gaussian-smoothed edges given by acharacteristic time of σt > 200 ns. Such smoothed edges reduce spectral content at ωLC ± ωm that
could drive the mechanical resonator.
The red- and blue-detuned pumps carry enough power that could adversely affect the qubit’s
state. To reduce the pump power incident on the cQED system, we use variable attenuators andphase shifters to create cancellation signals that reduce the pump power incident on the cQED
system by 30 to 40 dB. We monitor and adjust the relative cancellation of the pumps at thedetector.
To test the calibration and capture protocols, we inject a large amplitude coherent signal into
the electromechanical device (shown in Fig. 5.7). This test signal is generated using a microwave
synthesizer, and then shaped using a double-balanced mixer. Its temporal envelope is shown in
Fig. A.2. While executing the protocols for the experiments depicted in Fig. 5.7, the test signal
was not injected into the network.
Qubit pumps and the experiment’s local oscillator (LO) are generated using a set of phasecoherent microwave synthesizers (Holzworth HS9002A) that are modulated on and off using the
AWG. Two channels of the microwave synthesizers are dedicated to producing the experiment’s
LO and a pulse at half the qubit’s blue sideband transition frequency ωsb /2. The remaining twochannels are dedicated to producing microwave pulses (300 ns in duration) at the qubit’s ground toexcited state transition frequency ωge . The phase coherent microwave synthesizers are necessary for124

Figure A.2: Temporal envelopes of the microwave pumps and signals. a, The timing diagram
depicts the calibration protocol. Initially, a red-detuned ‘capture’ pump at ωLC − ωm cools the
motion of the mechanical resonator. After the capture pump turns off, a blue-detuned ‘amplify’
pump at ωLC + ωm is pulsed on for a duration τb . Input signals are coincident with the start of theamplify pump. b, The timing diagram for the capture protocol. For this protocol, the red-detuned
pump is modulated for optimal capture of a signal with a decaying temporal envelope. The input
signals (either generated by the microwave synthesizers or the cQED system) are coincident withthe start of the modulation. The programmable storage time τs is set by the delay between the endof the capture and the start of the amplification pump.

generating coherent propagating microwave fields (emitted from the cQED system) that encode thestate of the qubit. However, these microwave synthesizers have frequency accuracy errors (at the
1 mHz level) that lead to phase drifts relative to the Agilent PSG synthesizers. Additionally, timing
errors between the synthesizers and the AWG lead to additional phase errors. To reduce timing
errors, all synthesizers are set to frequencies that are integer multiples of the protocol repetition
rate (500 Hz). For constant phase drifts due to frequency accuracy errors, we separately measure
and correct for such phase drifts after acquiring a set of measurements.

A.5

Detector
Microwave signals incident or reflected off the electromechanical device are measured using asensitive microwave detector. The detector consists of a Josephson parametric amplifier (JPA), highelectron mobility transistor (HEMT) amplifier, a low noise room temperature amplifier (Miteq with

125
+30 dB gain), a downconverting mixer, and a digitizer. For the protocols presented in Fig. A.2,
we use the electromechanical device as the low-noise preamplifier instead of the JPA. The downconverting mixer is driven by a local oscillator detuned by 1 MHz from ωLC , and the inphase andquadrature channels of this mixer are sampled using a high speed digitizer (AlazarTech ATS 9462).

A.6

Strong pumps saturate the JPA
Strong control pumps for the electromechanical device can saturate the JPA. In order toefficiently capture a signal with a power decay rate of γ < 100 kHz, we use Γr > 1 MHz. For apump detuning of ∆r = −ωm and the parameters presented in Table 3.1, achieving such ∆r requires
a pump power of ∼ 10 nW at the input of the electromechanical device. This power can be reduced
by at least 30 dB using cancellation, which results in a power level of ∼ 10 pW at the input of the
JPA. When operating the JPA near ωLC , its 1 dB compression point is ∼ 30 fW in the presence ofthe red-detuned pump. Thus, we expect the strong pump to saturate the JPA.
In contrast, we do not expect the red-detuned pump to adversely affect the transmon qubit
because it is sufficiently isolated from the strong pump. This isolation is at least 40 dB and isprimarily provided by the circulators placed between the electromechanical device and the cQED
system, as discussed in Appendix A.2. After cancelling the pump, we expect the total reductionin its power to be > 70 dB. Thus, for a ∼ 10 nW pump, we expect the power at the input of thecQED system to be ∼ 1 fW. For a pump detuning of ∆r = −ωm and the parameters presented
in Table 4.1, we expect the circulating power in the cQED system to be < 0.02 photons. For
comparison, non-destructive qubit readout powers are typically at the level of tens of photons. As
such, we do not expect the red-detuned pump to significantly affect the cQED system.

Appendix B

Error analysis

B.1

Density matrix estimation
We use a parametric bootstrap method to analyze the statistical error in the density matrix

elements estimated by the maximum likelihood (ML) tomography. The estimated density matrix ρest is used to generate synthetic data sets, and ML tomography is performed on each of thesynthetic data sets, building a histogram of values for each density matrix element. Example histograms showing the bootstrapped diagonal density matrix elements for the mixed states obtainedusing the calibration and capture protocols are shown in Fig. B.1. These histograms reveal anasymmetry in the bootstrapped distributions, as well as bias in some of the density matrix element
estimates. It is therefore more appropriate to analyze the statistical error in terms of confidence
intervals rather than the standard error. The reported error bars indicate 90% basic bootstrap
confidence intervals obtained from these histograms [230]. The basic bootstrap confidence interval
is obtained as follows: let θ be the estimated parameter and let θlo and θup be the lower and upper
percentile values obtained from the bootstrapped histogram. That is, θlo (θup ) is the value forwhich 5% of bootstrapped values are less than (greater than) θlo (θup ). The differences α = θ − θloand β = θup − θ between the estimated parameter value and the lower and upper percentile values
are then inverted around the estimated parameter θ to obtain the confidence interval [θ − β, θ + α].
For each inferred density matrix we create 1,000 synthetic data sets, with each data set
containing 102,400 simulated measurement outcomes. The ML tomography algorithm is applied
to each synthetic data set with the same number of iterations and Hilbert space truncation as was

127

Figure B.1: Bootstrapped histograms. a, This figure shows the histograms of the ρ00 , ρ11 , and ρ22
density matrix elements obtained via bootstrapping on the estimated state of the input microwave
field in the single photon data set. The red bars indicate the values obtained by ML from theexperimental data. Each histogram is obtained by performing ML on 1,000 synthetic data setswith 102,400 measurements each. b, This figure shows similar histograms, but for the mechanicalresonator state.

used on the experimental data generated by executing either the calibration or capture protocols.
To generate synthetic data sets, we use a Monte Carlo method to sample pairs of joint
quadrature values (Xk , Yk ) from the Husimi Q-function Q(Xk , Yk ) corresponding to the estimateddensity matrix ρest . A uniformly distributed set of random points (Xk , Yk ) all lying within asufficiently large radius from the origin in phase space is generated. The Q-function is computedfor each of these points, and points are then discarded with probability
1−

Q(X, Y )max Q(Xk , Yk )kso that the remaining points are distributed according to the Q-function. For the data sets obtained
from the calibration protocol, the additional thermal noise nth in the mechanical mode must be
taken into account. In this case, the measured joint quadrature values are sampled not from the Qfunction of the microwave mode, but from the Q-function convolved with the added thermal noise

128
Table B.1: Estimated density matrix elements assuming different values of nth in the quadratureamplitude scaling and tomography procedures.
nth[ρe ]00[ρe ]11[ρe ]22[ρm ]00[ρm ]11[ρm ]22

0.08
0.09
0.10
0.11
0.12

0.655
0.652
0.649
0.645
0.642

0.328
0.331
0.334
0.337
0.340

0.014
0.014
0.015
0.015
0.015

0.694
0.689
0.683
0.677
0.672

0.266
0.265
0.264
0.264
0.263

0.038
0.044
0.050
0.056
0.062

in the mechanical mode. We account for this added noise in our Monte Carlo sampling method bysimply adding Gaussian noise to each sample.
Our finite precision in calibrating the thermal noise in the mechanical mode is a source ofsystematic error in the tomographic estimate of density matrix elements. Both the procedure forrescaling histograms as well as the tomography assume a mechanical thermal occupancy of nth = 0.1
quanta. We therefore investigate how the density elements would change if the value of nth weredifferent. We let nth range from 0.08 to 0.12 in steps of 0.01, and for each step we obtain thedensity matrix for both the input electrical and converted mechanical states. The diagonal density
matrices obtained with different values for nth are shown in Table B.1. We find that these density
matrix elements change linearly and by not more than ∼ 0.02 over the range in nth that we explore.
We use Eqn. 5.6 with the diagonal density matrix elements obtained via tomography to(2)calculate gm = 0.89. We use the bootstrap error analysis described in Appendix B.1 to obtain a(2)

90% confidence interval of [0.72, 0.94]. A histogram of bootstrapped values of gm is displayed in
Supplementary Fig. B.2. The density matrix obtained via tomography acts on a truncated Hilbert
space with maximum Fock number n = 15. While the first 3 diagonal elements are presented in(2)

Fig. 5.10, all 16 elements are used in the calculation of gm . These elements rapidly become smaller(2)

with increasing n such that both gm and the bounds of the confidence interval are independent of
the Hilbert space truncation. We find that they converge to within 1% of our reported values oncen > 8.

129

B.2

Average fidelity calculation
The formula for the average fidelity given by Eqn. 5.15 is actually only true if the processmaps all states into a Hilbert space of dimension d, the same dimension as the domain of theprocess. In our experiment, the weak thermal occupation of the mechanical resonator leads to
output states with small but non-zero density matrix elements for n ≥ 2, in which case truncating
the output density matrices to dimension d = 2 leads to a systematic overestimation of the average
fidelity. We can account for this by using a more general expression Favg which is valid in the casewhere the process takes states into a higher dimensional space than its domain:d(Fe + A(E))
d+1

(B.1)

1 X
hi| E(|ji hj|) |ii .
d2

(B.2)

Favg =
where
A(E) =

ij

Eqn. B.1 can be motivated by observing that for a process which does not map states into ahigher dimensional space, Eqn. B.2 is a sum of traces of density matrices (which each have unit
trace) and Eqn. 5.15 is recovered. That Eqn. B.1 is correct for processes which map states outof their domain can be directly verified for simple processes such as the ‘erasure channel,’ givenby E(ρ) = (1 − p)ρ + p |ψex i hψex | , where |ψex i is some external state. For our data the correction
obtained by using Eqn. B.2 rather than Eqn. 5.15 is small, reducing the average fidelity from 0.84
to 0.83.
We obtain a 90% confidence interval on the value of the average fidelity by a bootstrap analysis
similar to the one described in Appendix B.1. We generated 1,000 synthetic data sets, where eachdata set consists of 20,480 measurement outcomes on each of the four input and output statesobtained via tomography. We then run ML on these data sets and compute Favg . A histogramof the results is shown in Supplementary Fig. B.3. Our final result is Favg = 0.83 with a 90%
confidence interval of [0.77, 0.86].
To estimate the systematic error in our average fidelity calculation which results from truncating the input Hilbert space dimension, we employ the model of the capture process described

130

Figure B.2: Bootstrapped histograms of the calculated g (2) function. The figure shows a histogram(2)
of gm for the captured mechanical state, obtained via bootstrapping on the density matrices
estimated by ML tomography. The red bar indicates the value calculated from the estimateddensity matrices. The histogram is obtained by performing ML on 1,000 synthetic data sets with
102,400 measurements each.

in Section 5.5.4. Our model is a process whose average fidelity can be computed exactly; we findmodel = 0.82. We then simulate the entire experiment, using the model instead of the physicalFavg

capture process. Specifically, we use the four known input states to generate synthetic data which
we perform tomography on. We then send the estimated states through the model process anduse the output states to generate synthetic data which we again perform tomography on, before
finally calculating Favg . After 200 repetitions of this procedure we obtain a histogram of averagemodel = 0.01, which is small
fidelities with mean F avg = 0.83. Our estimate of bias is then F avg − Favg

compared to the width of our 90% confidence interval of [0.77, 0.86].

131

Figure B.3: Bootstrapped histograms of the average fidelity. The figure shows a histogram of
average fidelities calculated with 1,000 simulated experiments. Each simulated experiment involves
generating a synthetic data set of 20,480 Q-function measurements from the known input and output
states given in Supplementary Table B.2. ML tomography is then performed on each synthetic data
set and Favg is computed. The red bar indicates the value obtained from the experimental data.

Table B.2: Estimated density matrices used to calculate the average fidelity. The states were
estimated using 500 iterations of the RρR algorithm, which yields estimated density matrices ofdimension d = 16. We present the first 3 × 3 elements of these matrices. The input states ρe were
estimated from 20,480 measurements of the input microwave fields after executing the calibration
protocol. For the capture protocol, measurements of amplified mechanical resonator state yield ρm .
The states labeled with a ‘1’ (‘2’) correspond to a vacuum (single photon) input. Similarly, stateslabeled with a ‘3’ and ‘4’ correspond to the an input mixed state of a superposition of zero and one
photons, with a phase set by ϕ = 0 and ϕ = π/2, respectively.
Label

Input state ρe


1

2

3

4

Mechanical state ρm

0.994 + 0.000i
0.007 + 0.005i
0.005 − 0.026i
0.007 − 0.005i
0.004 + 0.000i−0.000 + 0.001i
0.005 + 0.026i −0.000 − 0.001i
0.001 + 0.000i

0.660 + 0.000i−0.013 − 0.039i 0.030 − 0.010i
−0.013 + 0.039i
0.283 + 0.000i
0.040 − 0.021i
0.030 + 0.010i
0.040 + 0.021i
0.042 + 0.000i

0.826 + 0.000i 0.256 − 0.019i 0.020 − 0.003i
0.256 + 0.019i 0.173 + 0.000i 0.010 − 0.005i
0.020 + 0.003i 0.010 + 0.005i 0.001 + 0.000i

0.775 + 0.000i
0.0451 + 0.294i −0.037 − 0.015i
 0.0451 − 0.294i
0.217 + 0.000i−0.027 + 0.018i
−0.037 + 0.015i −0.027 − 0.018i
0.006 + 0.000i


0.919 + 0.000i
0.005 + 0.001i
0.010 − 0.006i
0.005 − 0.001i 0.0620 + 0.000i −0.016 + 0.001i
0.010 + 0.006i −0.016 − 0.001i
0.017 + 0.000i

0.636 + 0.000i
0.004 − 0.016i −0.034 − 0.008i
 0.004 + 0.016i
0.281 + 0.000i
0.021 − 0.012i −0.034 + 0.008i 0.021 + 0.012i
0.075 + 0.000i

0.763 + 0.000i 0.180 − 0.035i 0.018 − 0.015i
0.180 + 0.035i 0.197 + 0.000i 0.024 − 0.011i
0.018 + 0.015i 0.024 + 0.011i 0.036 + 0.000i

0.759 + 0.000i
0.000 + 0.21i−0.031 − 0.014i
 0.000 − 0.21i
0.234 + 0.000i
0.019 − 0.017i −0.031 + 0.014i 0.019 + 0.017i
0.004 + 0.000i