Generation and Efficient Measurement of Single Photons
Using Superconducting Circuits
by
W. F. Kindel
B.S., The Ohio State University, 2007
M.S., University of Colorado at Boulder, 2012

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
2015This thesis entitled:
Generation and Efficient Measurement of Single Photons Using Superconducting Circuits
written by W. F. Kindel
has been approved for the Department of Physics

Konrad W. Lehnert

Prof. Murray Holland

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
Kindel, W. F. (Ph.D., Physics)
Generation and Efficient Measurement of Single Photons Using Superconducting Circuits
Thesis directed by Prof. Konrad W. Lehnert

In this thesis, I demonstrate and evaluate an on-demand source of single propagating microwaves photons. Working in the context of a quantum network, nodes are connected via propagating, nonclassical states of the electromagnetic field. As such, preparing and detecting propagating
quantum states is an essential task. I work with one particular node consisting of a microfabricated,
effective two level system coupled to a microwave resonator and study its ability to produce propagating nonclassical states, such as single photon states. In principle, states generated by this node
could be sent to other such nodes. However, I send them into a Josephson parametric amplifier(JPA) to characterize the source.
In particular, I discuss how to design and couple the two components that form my source:
a fixed frequency transmon qubit and a 3D superconducting waveguide cavity. I demonstrate theability to control of the dynamics of this combined system and implement a single photon generation protocol, which utilizes a single microwave control field that is far detuned from the photon
emission frequency. To characterize the generation, I perform tomography on the propagating photon state to determine its density matrix ρ. I perform repeated JPA-backed, linear measurementsof the propagating state. Based on the histograms of my measurements, I infer a maximum single
photon component ρ11 = 0.36 ± 0.01. I characterize the imperfections of the photon generationand detection, including detection inefficiency and measurement backaction. I find that within
uncertainty my measurements match my expectation.

Dedication

In memory of Ethan Townsend.

v

Acknowledgements

I would like to thank everyone who has helped, supported, advised, and learned alongsideme throughout my time at the University of Colorado. To credit properly would take far too
many pages than is allotted here, so can only highlight a few of the many acknowledgments due. I
will start by thanking the two people who have worked directly with me the most: Hsiang-Sheng
Ku and Michael Schroer. I started my first project with Hsiang-Sheng Ku. I thank him for histhoughtful questions and insights as we tried to understand quantum mechanics of the field, andall of his efforts completing that experiment. For my second project, this thesis work, I owe a great
debt to Michael Schroer. His knowledge of hardware, software, physics, and his constant drive to
make progress laid the groundwork for my photon measurements. It was a great pleasure working
alongside both of you.
I would like that my advisor Konrad Lehnert. He gave me the opportunity complete thisproject while providing key insights into the physics of the experiment. He has supported the
directions I wanted to take my research and has taken to time to teach me not just about thescience, but also about how to communicate and work as a scientist. He has taken his role as amentor seriously, to the benefit of myself and all his students and researchers. Thank you, Konrad.
Next, I would like to thank all of those whom I have worked with in the Lehnert Lab creatinga productive, and yet at times silly, working environment. I thank all the current group members:Pete Burns, Ben Chapman, Xizheng Ma, Tim Menke, Dan Palken, Adam Reed, Eric Rosenthal,
Lucas Sletten, Andrew Higginbotham, Maxime Malnou, Bad Moores, Jeremie Viennot and Juan
Montoya– nearly all of whom help me edit this document in some capacity– and former members

vi
who have worked by my side: Jennifer Harlow, Francois Mallet, Reed Andrews, Joe Kerckhoff,
Gerwin Koolstra, Brad Mitchell, and Tauno Palomaki. I would like to thank Jennifer Harlow andFrancois Mallet for teaching me about microwaves when I first stated. Jennifer, Adam Reed andTauno Palomaki for the years together in lab and outside of lab exploring Colorado. Reed Andrews,
as they say, it has been one heck of a ride. Ben Chapman, you are relatively new in lab, but you
have championed the conquest of the JILA Cup. I trust it will not be lost under your tenure.
Jeremie Viennot and Xizheng Ma, it has been fun playing with qubits in lab and learning the ropesof climbing outside. As for the rest of you, a thanks is due. Working with all of you, you havehelped me with day to day problems and collective are truly the reasons I have made progress.
I would like to thank those who have supported me during my time in Boulder, of which
there are far too many to name. And of course, thank you to my family for supporting me andputting up with me over these Boulder years.

vii

Contents

Chapter
1 Introduction

1

1.1

Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Experiment overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.1

Photon source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.2

Photon detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Other single photon generation protocols . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3.1

Photon like coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3.2

Purcell limited decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3.3

Frequency tunable qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.3.4

Higher order transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.3

2 Quantum mechanics of photon generation and measurement
2.1

2.2

Circuits as harmonic oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9
9

2.1.1

Transmission line decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2

Quadratures

2.1.3

States of the field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

A two level system coupled to a cavity: CQED . . . . . . . . . . . . . . . . . . . . . 15
2.2.1

Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2

Dispersive limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

viii
2.2.3
2.3

System dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Measuring a quantum state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1

Parametric amplifier measurements . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.2

Wigner function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.3

Homodyne measurement of quantum states . . . . . . . . . . . . . . . . . . . 25

2.3.4

Heterodyne measurement of quantum states . . . . . . . . . . . . . . . . . . . 25

2.3.5

Inefficient measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Qubit and cavity design
3.1

3.2

3.3

30

The qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.1

From linear to non-linear circuits . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.2

Cooper pair box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.3

CPB dynamics in phase basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1.4

CPB solutions in phase basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.5

Transmon qubit limit of the CPB . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.6

Transmon qubit design constraints . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.7

Qubit fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1

Idealized cavity modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.2

HFSS calculation of cavity modes . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.3

Cavity fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.4

Cavity input and output coupling. . . . . . . . . . . . . . . . . . . . . . . . 48

Coupled system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.1

Effective dispersive Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.2

Dipole coupling model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Qubit and cavity’s characterization and dynamics
4.1

56

Cavity spectroscopy and qubit readout . . . . . . . . . . . . . . . . . . . . . . . . . . 56

ix

4.2

4.3

4.1.1

Qubit readout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1.2

Two-tone spectroscopy of the qubit . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1.3

Single shot readout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Control pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1

Qubit pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.2

Blue sideband pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Qubit characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.3.1

Qubit lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.2

Qubit dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Photon generation and detection
5.1

5.2

5.3

Photon generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.1

Blue sideband measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.1.2

Cavity photon detection with qubit . . . . . . . . . . . . . . . . . . . . . . . . 81

Heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2.1

Two quadrature filter and photon variance . . . . . . . . . . . . . . . . . . . 85

5.2.2

Comparison to qubit state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2.3

Density matrix reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Optimized photon generation and homodyne detection . . . . . . . . . . . . . . . . . 96
5.3.1

Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3.2

Homodyne detection and tomography . . . . . . . . . . . . . . . . . . . . . . 100

5.3.3

Characterizing photon generation and efficient homodyne detection . . . . . . 108

6 Conclusions and outlook
6.1

78

115

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

xBibliography

118

Appendix
A Description of photon generation and homodyne detection apparati

124

xi

Tables

Table

xii

FiguresFigure
2.1

The Wigner function for a single photon state (n = 1).

. . . . . . . . . . . . . . . . 24

2.2

Homodyne measurement of a single photon: single quadrature probability distribution for the n = 1 Fock state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3

Heterodyne measurement of a single photon: two quadrature probability distribution
for the n = 1 Fock state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1

The circuit schematic of a Cooper pair box (CPB). . . . . . . . . . . . . . . . . . . . 33

3.2

The energy levels of the CPB in the large charging energy Ec limit (EJ /Ec = 0.2). . 35

3.3

The energy levels of the CPB where EJ = Ec . . . . . . . . . . . . . . . . . . . . . . . 39

3.4

The energy levels of the CPB in the transmon limit (EJ /Ec = 50). . . . . . . . . . . 40

3.5

The circuit schematic of a transmon qubit.

3.6

The center shows a 500 by 200 nm Josephson junction where two Aluminum layers. . . . . . . . . . . . . . . . . . . . . . . 42

overlap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.7

An image of the cavity used in the experiment. . . . . . . . . . . . . . . . . . . . . . 46

3.8

The TE011 mode of a cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.9

Results from an HFSS calculation of the coupling rate for various pin lengths. . . . . 50

3.10 The reflection response and fit used in setting the coupling rate κ1 of the cavity. . . 52
3.11 The qubit coupled to the cavity by placing it at the center of the cavity. . . . . . . . 52
4.1

Transmission spectroscopy schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . 57

xiii
4.2

The power dependent cavity spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3

Two-tone spectroscopy schematic.

4.4

Two-tone spectroscopy on the state of the qubit. . . . . . . . . . . . . . . . . . . . . 63

4.5

Histograms of the measured readout tone’s voltage (y-axis) at various readout tone

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

powers (x-axis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6

The results from the Rabi measurement adjusting pulse amplitude and frequency. . . 68

4.7

The results of the blue sideband measurement, as a function of pulse amplitude andfrequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.8

The results from the blue sideband measurement with fixed pulse amplitude. . . . . 71

4.9

Measurement of the qubit |ei state lifetime. . . . . . . . . . . . . . . . . . . . . . . . 73

4.10 Ramsey measurement of the qubit dephasing as I adjust the pulse frequency. . . . . 75
4.11 Ramsey measurement of the qubit dephasing with the drive detuned 1 MHz. . . . . 76
5.1

Energy level schematic for photon generation with the blue sideband. . . . . . . . . . 80

5.2

Excited state probability during a blue sideband drive (marks) fit to a offset exponentially decaying sinusoid (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3

Detecting cavity photons created by a blue sideband π-pulse using photon number
dependent qubit π-pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4

The two heterodyne filter functions f1 (t) (red) and f2 (t) (blue) along with theirptemporal profile f12 (t) + f22 (t) (black dotted line) . . . . . . . . . . . . . . . . . . 87

5.5

Measurement of var(Xi ) (above) and Pe (below) in response the a blue sideband
pulse drive as the frequency and duration of the pulse is adjusted. . . . . . . . . . . 89

5.6

Histograms of two quadrature measurement of the photon state (top) and the vacuum
state (bottom) plotted as the Q-function (probability density). . . . . . . . . . . . . 92

5.7

A line cut through the center of the measured single probability density (markers)
with the best fit to density matrix elements (solid line). . . . . . . . . . . . . . . . . 94

xiv
5.8

Reconstructed density matrix elements |ρij | of the measured photon state using heterodyne detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.9

Simplified schematic for photon generation. . . . . . . . . . . . . . . . . . . . . . . . 97

5.10 Energy level diagram for photon generation and calibration. . . . . . . . . . . . . . . 98
5.11 Timing diagram for the experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.12 Histogram of a quadrature measurement set of single photons (narrow blue bars)
and the no-photon control (wide green bars) with the JPA gain at 29 dB. . . . . . . 103
5.13 Quantifying backaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.14 Diagonal density matrix elements determined from fitting the quadrature histogram
over a range of JPA gains from 17 dB to 33 dB. . . . . . . . . . . . . . . . . . . . . . 107
5.15 The values of g2 (0) computed from the density matrix elements. . . . . . . . . . . . 109
5.16 Thermal sweep data for the JPA operated at 20 (green squares), 25 (blue circles)
and 30 dB (red crosses) gains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.17 The measurement efficiency ηm is determined from a thermal sweep at three JPA
gains (circles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.18 The fidelity of the measured density matrix with respect to: an ideal single photon (blue squares), and the density matrix I expect to measure given pure photon
generation in the cavity (black points). . . . . . . . . . . . . . . . . . . . . . . . . . . 114
A.1 The microwave schematic for the experiment. . . . . . . . . . . . . . . . . . . . . . . 125

Chapter 1

Introduction

In this thesis, I address a quantum node, or building block in the framework of a quantum
network. Specifically, I look at how these nodes transmit information. Many physical platforms
exist for building nodes of a quantum network, from trapped ions [1, 2] to Nitrogen vacancy
centers [3] and electron spins [4, 5]. In this chapter, I describe my experimental platform forrealizing a quantum node, which is a particular type of superconducting circuit, and introduce theconcept of a quantum network. Then, I provide an overview of the entire thesis. Lastly, I describe
my experimental approach to address this problem and compare mine to other approaches withsuperconducting circuits.
With the advent of superconducting circuits used as quantum technologies, the prospectof processing quantum information has become less remote. In particular, the circuit quantum
electrodynamics (CQED) concept has seen many recent successes. This architecture consists of asuperconducting qubit strongly coupled to a cavity [6, 7]. The cavity is an electromagnetic resonator, which, in the absence of coupling to the qubit, acts as a harmonic oscillator. Experimentally,
one can drive this cavity and the average number of photons increase linearly with drive power.
Although the qubit can also be considered an electromagnetic oscillator, it behaves intrinsicallydifferently than the cavity. When driven, its response is highly nonlinear. It can only absorb one
unit of energy, or quanta. As its state can only be measured in two different values, it is a two-levelsystem. This gives rise to two powerful properties. First, because it has only two possible states,
it can be used to encode a bit of information. Furthermore, because it behaves quantum mechani-

2cally, it can be prepared in a quantum superposition of these two states, and thus is referred to asa quantum bit (qubit) [8]. Second, its nonlinear response to an electromagnetic field drive can beused as a resource to prepare and readout nonclassical states of the cavity field (states of microwave
light that go beyond a classical description). For example, the cavity has been prepared in states ofquantum coherent superpositions of photon, with detectible interferences [9]. Moreover, quantum
coherent superpositions of the classical coherent state have been prepared, called Schrodinger cat
states in a nod to Schrodinger’s notorious cat [10].
Together, these qubit-cavity systems can be viewed as a building block. In a broader context,
one can think of a quantum network where qubit-cavity systems are nodes that perform elementary
operations [11]. In order to have these nodes working together to process quantum information,
one has to couple them coherently. In other words, one must have channels connecting the nodes
to propagate quantum information among them. In a local approach, physically separated qubitscommunicate via a cavity bus [12]. Using this bus, the states of the two qubits can be tied together
in what is known as entanglement. Harnessing this entanglement between two physically separated
qubits, the quantum state of one qubit is destroyed and recreated in the second qubit without
measurement, in a process known as quantum teleportation [13]. In the approach I consider, thecommunication channel takes the form of a transmission line along which electromagnetic waves
can propagate. This approach allows for arbitrary connectivity among quantum modules such as
QCED systems. Commutation among these nodes via propagating modes along transmission lines
can entangle the nodes, forming a larger quantum system or quantum network [14].
In addition to their utility in establishing quantum networks, propagating non-classical states
can be an interesting resource in various other contexts. In particular, in the field of quantum micromechanics, a current challenge is to prepare profoundly nonclassical states of motion, such a single
phonon states or Schrodinger cat states. Mechanical systems can be coupled strongly to propagating
microwave modes, enabling their preparation in coherent states [15, 16]. But by themselves, theylack sufficient intrinsic nonlinearity to evolve spontaneously into a profoundly nonclassical state.
One path is to prepare a profoundly nonclassical propagating microwave field and transfer that3
state to a mechanical oscillator.
Given the importance of these propagating modes in forming a quantum network and increating non-classical states of motion, I focus on generating one of the simplest nonclassical states:
a single photon. The core of this thesis studies how a single photon can be created in the cavity
using the qubit, how this photon’s wavefuntion can slowly move out of the cavity and into thetransmission line, and how I can detect this photon.
In the optical domain, photon detectors are available and routinely used to detect single
photons. These detectors register the presence or absence of a photon with a binary output signal;i.e., they click as one photon is measured. At microwave frequencies, such detectors are not available
because the energy carried by a single photon is too small to create an electron-hole pair in asemiconductor. What is available are voltage amplifiers or linear detectors. Due to the small powerof these single photon signals, stringent noise requirements are placed on these detectors. Meetingthese requirements is impossible with commercially available amplifiers. I therefore use a homemade amplifier that achieves near-quantum limited amplification 1 . This amplifier is a parametricamplifier called the Josephson Parametric Amplifier (JPA).

1.1

Thesis overview
I begin by introducing the concept of photon generation and detection experiment within amodular architecture of a quantum network. In doing so, I introduce the two key elements of theexperiment: the qubit-cavity system, which generates single propagating photons, and the JPA,
which acts as a detector. I specifically discuss the concept of photon generation and detection andcontrast my work with other photon generation experiments in superconducting circuits. In Chapter
2, I introduce a quantum mechanical formalism for circuits, including models for CQED system andtheir dynamics, and the quantum mechanics of measurement. In Chapter 3, I discuss designinga qubit-cavity system within the constraints of the photon generation demands. In Chapter 4
I experimentally characterize the qubit-cavity system and demonstrate its basic operations. In
1

The added noise is close to what is required by quantum mechanics [17]

4Chapter 5, I describe single photon generation and detection. Finally, in Chapter 6 I discuss
extending my generation from single photons to flying qubits, and discuss integrating the photon
generation with other quantum modules, such as mechanical systems.

1.2

Experiment overview

1.2.1

Photon source

In designing the photon source, I consider parameters such as the itinerant photon’s frequency
and bandwith to ensure compatibility with the target module. So that I am not limited by qubit
lifetimes, I choose to make the source module from most coherent, yet fixed frequency qubits. Thesesystems consist of a transmon qubit embedded in the center of a superconducting cavity [18].
These so called 3D transmons are both simple to fabricate and have lifetimes (T1 ≈ 100 µs)
compatible with narrowband modules such as: electromechanical devices [16, 19], microwave to
optical convertors [20], and other 3D transmon systems; i.e., bandwidths (inverse lifetimes) that
do not exceed 1 MHz. However, in this style of CQED system that I use, neither the qubit nor thecavity have tunable resonance frequencies; thus, the only controls available are microwave fields.
Because I am limited to control fields, I am interested in a single transition that creates a
photon in the cavity. In CQED systems, the qubit or two level system breaks the harmonicityof transitions, and thereby allows deterministic photon generation protocols to be implemented
by driving these transition [21]. The transition I am interest in is known as a blue sideband
transition (Section 2.2.3.2). This transition can be used to generate a single photon while exciting
the qubit. It is a two photon process, roughly at the average of the qubit and cavity frequency
ωblue ≈ (ωc + ωq )/2. Hence, this transition is off resonance with both the qubit and the cavity.
Experimentally, blue sideband pulses have been utilized in CQED systems, demonstrating
their ability to generate cavity excitations or photons. The blue sideband transitions were initiallyspectroscopically studied in these QCED system [22]. Generating cavity excitations (photons)
was indirectly demonstrated through an experiment that entangled two qubits [23]. In Ref. [23]’s5experiment, the blue sideband creates a photon as an intermediate step while entangling the two
qubits. By verifying the entanglement between the two qubits indirectly, this work shows that blue
sideband pulses can create photons in QCED systems.
Given the experimental demonstrations of the blue sideband pulse and long coherence timesof QCED systems, I combine these two elements to form an elementary protocol for generating
itinerant photons. First, I use a control field at the blue sideband transitions to generate a single
photon in the cavity. Then, the photon in the cavity decays through a strongly coupled output
port, creating the propagating photon.

1.2.2

Photon detection

Rather than using a second CQED module as the target, I use a quantum module which acts
as a detector of the propagating photon: namely, the JPA. This quantum module preforms a reversible and deterministic transformation on the propagating mode, amplifying one quadrature anddeamplifying, or squeezing, the orthogonal quadrature [24, 25]. I choose the JPA because Josephson
junction based parametric amplifies (like the JPA) are among the most efficient microwave amplifiers and have been successfully integrated with QCED systems. Their development has enabled
30 fold efficiency improvements over the best commercial low noise amplifiers (HEMTs) because
JPAs add only a small fraction of vacuum fluctuations on top of amplified signals [17]. JPAs are
the natural choice for measuring the propagating, single photon state because they have been used
to perform efficient measurement of propagating nonclassical states [26, 27, 28, 29]. Moreover, similar Josephson junction based parametric amplifiers have already been integrated with qubit-cavity
systems to perform qubit readout [30, 31, 32, 33]. In these systems, the parametric amplifiers
have improved qubit detection to nearly ideal quantum non-demolition (QND) [34]. Simultaneously demonstrating high efficiency and compatibility with CQED modules, they have been used
to measure multiple quantum trajectories of a qubit [35].
A JPA is a nonlinear LC resonator that can act an amplifier when pumped with strong rf
power. The pump can either be at a single frequency that defines the center of the JPA band,6
or it can consist of two tones where the average of the two defines the center of the JPA band.
Additionally, JPAs can be operated in one of two modes (Section 2.3.1). In one mode, heterodyne
mode, the JPA amplifies a tone regardless of the tone’s phase. In the second mode, homodyne
mode, the JPA is phase sensitive. The JPA amplifies one quadrature of signal that is in phase witha reference. For technical reasons, when I use the JPA as a homodnye detector, I double pump,
and when I use the JPA as a heterodyne detector, I use one pump tone. To operate the JPA as a
homodyne detector, the center of the JPA’s band must be centered on the band of signal. In thiscase, I double pump the JPA because I do not want the strong pump to resonate with the cavity
which defines the photon frequency. To operate the JPA as a heterodnye detector, the signal bandis detuned from the center of the JPA’s band. Because the single pump is already detuned, I use
a single pump.

1.3

Other single photon generation protocols

1.3.1

Photon like coherent states

As a stepping stone to nonclassical communication between modules, communication between
CQED modules has been studied using classically described coherent states. In Yin and Wenner’s
experiments [36, 37], the coupling between the cavity and the transition line is controlled dynamically. By adjusting the coupling, a propagating coherent state with an average photon numbern̄ = 1 is captured in the cavity of a CQED module. The captured photon can be released andsent to another module. Though this work utlizes classical states, it demonstrations that CQED
modules are capable of catching and releasing states whose energy is the same as nonclassical statessuch as single photon states.

1.3.2

Purcell limited decay

Initial work on photon generation using superconducting qubits by Houck [38] relied oncoupling the qubit directly to an output mode. In this simple protocol, a pulse at the qubit frequency

7
excites the qubit. The qubit then decays through the output port creating a propagating photon at
the qubit frequency. In this method, the pulse is at the same frequency as the photon and couples
to the output mode. As a result, the photon and the pulse are neither spectrally or temporally
isolated during the pulse. Therefore, the single photon cannot be completely distinguished fromthe control field, partially spoiling the single photon character of the generated state. Furthermore,because the pulse is at the same frequency as the photon, the large control pulse may also excite
the target module.
Similar to Houck, my protocol for generating photons is simple in that it relies on a single
calibrated control pulse. But, because I use a blue sideband pulse, the control field is detuned
from the single photon. The photon is emitted at the cavity’s resonance frequency ωc , whereas thecontrol pulse is approximately at the average between cavity and qubit (ωq + ωc )/2, giving thephoton spectral isolation.

1.3.3

Frequency tunable qubits

There are several demonstrations of photon generation with qubit-cavity systems that do
not use control fields, but rather rely on frequency tunable qubits [39, 36]. When the qubit istuned to the cavity frequency, excitations oscillate between the the qubit and cavity [40, 6]. By
controlling the interval that the two are on resonance, then excitation can be swapped from thequbit into the cavity. Thus, by preparing the qubit in the excited state with a qubit pulse, andthen tuning the qubit to swap the excitation into the cavity, a single photon is created in thecavity. By strongly coupling the cavity to an output mode, a propagating photon is created. In
one particular example, Eichler [39] generated propagating arbitrary superpositions of zero and one
photon Ψ = A |0i + B |1i, which can be thought of as a propagating or flying qubit.
However, in the type of qubit-cavity system that I adopt and is discussed in Ref. [18], thequbit is not easily tunable. To create a more coherent qubit, this style of qubit-cavity system
places the qubit in the center of a superconducting cavity making it difficult to apply the magnetic
flux used to tune qubits because the superconducting cavity shields magnetic fields. Therefore,

8
protocols relying on frequency tunable qubits cannot be easily implemented.
My work is further distinguished by its detection. In Eichler’s work [39], the photon generation protocol was repeated billions of times to overcome the inefficient HEMT measurements
in order tomographically reconstruct to an inference of generated state. In my approach, I use
an efficient JPA to measure the photon state. I perform the tomography on the measured state,
without artificially removing the added noise of the measurement.

1.3.4

Higher order transitions

More recently, Pechal [41] has demonstrated photon generation using only control fields
detuned from the photon. Thus, this protocol is compatible with 3D transmons. It used multiple
transitions including higher level states of the qubit making it more complicated than my single
pulse protocol. Pechal also removes the added noise from inefficient measurements, rather than
implementing an efficient measurement of the photon state.

Chapter 2

Quantum mechanics of photon generation and measurement

To establish a mathematical language and a consistent notation with which to describe photon
generation and detection experiments, I review a portion of the quantum optics formalism. Because
I am interested in generating and measuring states of cavities (resonators) and transmission lines, I
begin by discussing the quantum mechanics of these modes, which are harmonics oscillators. Then,
I look at examples of common classical and nonclassical states of these modes. Next, I cover a
simple quantum mechanical model for a qubit coupled to cavity, and discuss a method for photon
generation in an ideal limit. Lastly, I cover the framework for representing and measuring thesequantum states. I discuss both how to operate the JPA to measure states, and how to calculate
the expected probability distribution of these measurements for particular quantum states.

2.1

Circuits as harmonic oscillators
Modes of cavities, resonators and transmission lines behave as harmonic oscillators. To

understand these modes and the states of these modes, I start by writing the Hamiltonian of aharmonic oscillator
Hho = ω(a† a + 1/2)

(2.1)

in terms of the unitless raising a† and lowering a ladder operators. Here and for the rest of the
document, I have omitted the ~’s. It is therefore understood that angular frequencies must be
multiplied by ~ to recover units of energy. The energy eigenstates called Fock states, or photon
states, I label as |0i, |1i, ... |ni. These eigenstate have energy hHho i = ω(n + 1/2) and thus form

10
a harmonic spectrum. The creation ladder operator a† acting on eigenstate state |ni results ina† |ni =√n + 1 |n + 1i .

(2.2)

Thus, some combinations of these ladder operators acting on the ground state, the vacuum state|0i, can generate any pure quantum state. For a more detailed description of mode quantization,
consult a Quantum Optics text book such and Gerry and Knight’s [42].

2.1.1Transmission line decay

In this experiment, I am interested in measuring states propagating down transmission linesthat are generated by the cavity. Specifically, I want to think about: what propagating mode doesthat cavity evolve into. To understand this, I consider how incoming and outward propagatingstates of the transmission line relate to the internal state of the cavity. Because I can represent
pure quantum states with combinations of ladder operators acting on the vacuum state, I consider
how operators of the cavity a relate to operators of incoming and outward propagating states ain
and aout , respectively. Considering a cavity coupled to a transmission line with power decay rate
κ, these relations are:√κȧ = − a + κain2√
aout = κa − ain

(2.3)
(2.4)

according to input/output formalism [43]. Consequently, to find the cavity’s output at all times, I
must solve Eqn. 2.3:
a(t) = a(0)e

t
−κ
2

+

√

−κ
t
2

Z t

κe

κ

e− 2 z ain (z)dz.

(2.5)

0

I am not interested in a continuum of output modes, but a single mode propagating down
the transmission line. Therefore, I define a particular output mode by considering the temporalprofile of excitations decaying out of the cavity. Because amplitude decays out of the cavity at a
rate κ/2, I define the temporal profile as
fout (t) =√κtκe− 2 Θ(t)

(2.6)

11
in terms of the Heaviside step function Θ(t). With fout (t), I define the output mode as
Z ∞Z
Aout =

fout (t)aout (t)dt =√κtκe− 2 aout (t)dt

(2.7)

0

Using Eqns. 2.4 and 2.5, I write out the output mode as being equal toZ ∞Aout = a(0)κe−κt dt +0

Z ∞γ 3/2 e−κt

Z t

κe− 2 z ain (z)dzdt −

0

0

Z ∞

√

κt

κe− 2 ain (t)dt.

(2.8)

0

This equation simplifies to
Aout = a(0).

(2.9)

Thus, the state of the output mode Aout defined in Eqn. 2.7 is the same as the state of the cavity
a at t = 0. Consequently, if the cavity is prepared in a single photon state at t = 0, then the
output mode Aout is in the single photon state. Therefore, in this experiment, I am interesting in
generating and measuring propagating modes similar to the mode Aout defined in Eqn. 2.7.

2.1.2

Quadratures

Next, I construct operators that are tied closely to linear measurement. I construct thesequadrature operators Y1 and Y2 out of the ladder operators a and a† by forming linear superpositions. They are defined as
Y1 =

1
†
2 (a + a )(2.10)Y2 =

1†i2 (a − a ).

(2.11)

This definition of Y1 and Y2 is in the lab frame. In this frame, the operators a and a† oscillate at
ω as seen from the Heisenberg’s equationȧ = i[Hho , a] = −iωa.

(2.12)

Therefore,a(0)eiωt(2.13)a† (t) = a† (0)e−iωt(2.14)

a(t) =

12
Because a and a† oscillate in the lab frame, Y1 and Y2 oscillate. I define quadratures X1 and X2
that do not oscillate by transforming into in a frame that rotates at ω:X1 =cos(ωt)Y1 + sin(ωt)Y2 =

X2 = − sin(ωt)Y1 + cos(ωt)Y2 =1(a(0) + a† (0))21(a(0) − a† (0)).
i2(2.15)(2.16)For the remainder of this document, quadratures refer to these quadratures in the rotating frame.
Looking at the electric field operator of a mode, we can gain intuition about these two quadratures.
At a particular location in the mode and aligned with the polarization, the electric field operatoris [42]E = 2E0 (X1 cos(ωt) + X2 sin(ωt)) ,(2.17)

where E0 is a constant with units of electric field. Thus, we can think of X1 as the cosine and theX2 as the sine component of an oscillating field.
These quadratures X1 and X2 are the operators that the JPA measures. When the JPA
operates phase sensitively in homodyne mode, it measures X1 , X2 or a linear combination of thetwoXθ = X1 cos(θ) + X2 sin(θ).

(2.18)

When the JPA operates phase insensitively in heterodyne mode, it measures both quadratures.
However, both quadratures X1 and X2 cannot be determined with arbitrary accuracy because of
their commutation relationi[X1 , X2 ] = .
2(2.19)The measurements of X1 and X2 have an uncertainty relationshiph(∆X1 )2 i h(∆X2 )2 i ≥Thus, two-quadrature measurements must add noise.

1.
16(2.20)

13
2.1.3

States of the field

Having introduced the types of modes I am working with, and the operators I can measure, I
highlight a few classes of classical and non-classical states that I work with. Additionally, I discuss
some relevant properties of these states including quadrature expectation value and variance.
The state I want to generate and detect is the single photon state |1i. It is one of the Fockstates |ni. These states have on average a quadrature value ofhXθ i = 0(2.21)

for any |ni. They have a quadrature variance1
var(Xθ ) = hX2θ i − hXθ i2 = (2n + 1).
4(2.22)

Thus, quadrature measurements of a photon states |ni can determine n through their variance (orhigher cumulants), but not their average.
Another state I encounter in this experiment is a thermal distribution of states. This distribution, the thermal state, is a statistical mixture of pure Fock states characterized by a Boltzmann
distribution. Because it is a statistical mixture of pure states, it must be described by a density
matrix ρ rather than a single state vector. The dentistry matrix for a thermal state ρth (n̄) isparametrized in average photon occupation n̄ asρth (n̄) =

X
n

n̄n
|ni hn| .
(n̄ + 1)n+1(2.23)

Here, the average photon occupation isn̄ =1
exp(~ω/kB T) − 1(2.24)

in terms of the temperature T and the Boltzmann constant kB . Because the thermal state is nota pure photon or number state, it has nonzero number variance. Its number variance isvar(n) = Tr(ρth n2 ) − Tr(ρth n)2 = n̄ + n̄2 .

(2.25)

14
Like the pure Fock states, the thermal state has an average quadrature value
Tr(ρth Xθ ) = 0.

(2.26)

Its quadrature variance isvar(X) = Tr(ρth Xθ 2 ) =

1 n̄
+ .
4 2

(2.27)

Quadrature measurements of thermal states also manifest themselves in the variance of the field,
highlighting the need for a more sophisticated approach to distinguishing the n = 1 photon state forother states. Consequently in Subsections 2.3.3 and 2.3.4, I compute the probability distribution
for one and two-quadrature measurement of the signal photon state.
The last major class of states I encounter in this experiment is the classically described
coherent state. Quantum mechanically, I can generate a coherent state by taking the eigenvaluesof the ladder operator a. Thus,a |αi = α |αi ,(2.28)

where |αi is the coherent state and α is the coherent state’s complex amplitude. In the Fock basis,the coherent state is
|αi =

n
1
2 α
e 2 |α| √ |ni .
n!
nX(2.29)

The coherent state has an photon number variancevar(n) = n̄,(2.30)n̄ = hα| n |αi = |α|2 .

(2.31)

where

It has a quadrature variancevar(Xθ ) =equal to the vacuum state.

14(2.32)

15

2.2

A two level system coupled to a cavity: CQEDThe qubit is an important element for my photon generation experiment that is not described

as a harmonic oscillator. Instead, it is a two level system. The qubit is coupled to a cavity to fromthe QCED system. To utilize and understand this system I must write out a quantum mechanicalframework for it [21]. To do so, first I write out the Hamilton for the system including the direct
coupling between the qubit and cavity. Then, I look at a useful limit, the dispersive limit, where
the cavity and qubit affect each other, but do not spontaneously exchange excitations. Finally, I
look at dynamics of the system including a method of photon generation.

2.2.1

Hamiltonian

By coupling a cavity mode to a qubit, I can create a system with richer dynamics. The qubit
has resonant frequency ωq and bare Hamilton
Hq = ωqσz2(2.33)

where σz is a Pauli matrix. It has two eigenstates labeled |gi and |ei for ground and excited,
respectively When an anharmonic 3 level system is considered, the third level is labeled |f i.
Coupling the qubit to a cavity via the dipole interaction yields a coupling that is proportional
to the E field of the cavity dotted with that of the qubit. In other words,
Hint = g(σ+ + σ− )(a† + a),(2.34)

where g is the coupling strength and a the field operator for the cavity. The Hamiltonian for thesystem becomesHRabi = ωqknown as the Rabi Hamiltonian.

σz+ ωc a† a + g(σ+ + σ− )(a† + a)2

(2.35)

16
2.2.2

Dispersive limit

I want to design the system so I can gain information about the qubit from the cavity mode’s
fields while minimally disturbing the qubit. Following Ref. [44], this condition can be achieved bygoing into the limit where the two level system’s transition is detuned from the cavity mode bygreater than the coupling between the two, that is|ωq − ωc | = |∆| > g.

(2.36)In this limit, called the dispersive limit, an excitation in the two level system shifts the frequencyof the cavity, yet at the same time excitations do not exchange between the cavity and the qubit.
In the dispersive limit, the Hamiltonian becomes
Hdis = ωqσzσz+ ωc a† a + 2χ( a† a),2
2(2.37)

2

where χ = g∆ and ∆ = ωq − ωc . The value of the dispersive shift 2χ can be determined by treating
the interaction listed in Eqn. 2.33 as a perturbation using second order perturbation theory.
Regrouping the a† a terms in Eqn. 2.38, I get an effective cavity frequency ωc0 that is a functionof the qubit stateωc0 = ωc + 2χσz.
2

(2.38)

Therefore, the frequency of the cavity contains information about the state of the qubit. Morespecifically, detecting the 2χ dispersive shift of cavity resonances detects the state of the qubit. Thisdispersive shift forms the basis for a type of qubit readout I use in the my experiment. Similarly, I
can regroup σ2z terms to get an effective qubit frequency ωq0 that depends on the number of cavity
photonsωq0 = ωq + 2χa† a.

(2.39)

Therefore, the frequency of the qubit contains information about the number of photons in thecavity.

17
2.2.3

System dynamics

Now, I describe a method of using oscillating fields to drive dynamics in the qubit-cavity
system, and control the state of the qubit. First, I consider driving fields to effect the dynamics
of the qubit. These fields alter the Hamiltonian of the qubit to drive transitions and allow us toperform arbitrary operations on the qubit itself. Then, I use a quantum mechanical treatment of
the qubit-cavity system with drive fields to construct an effective Hamiltonian for the blue sideband
transition, that jointly creates or destroys single excitation of the qubit and cavity.

2.2.3.1

Rabi oscillations

To control the qubit, I follow closely Ref. [42] and consider the qubit coupled to an oscillating
drive through a dipole interaction. I treat the drive classically, where the drive adds an oscillatingterm to the Hamiltonian of the qubit. To understand the dynamics, I solve the equations of motion
for the ground |gi and excited |ei states coefficients Cg (t) and Ce (t).
As result of the of the drive, I get an interaction Hamiltonian
Hint (t) = V0 cos(ωd t)(2.40)

where V0 = −d(σ− +σ+ )E is the magnitude of coupling dipole energy, and ωd is the drive frequency.
The dipole moment of the qubit is d and the electric field of the drive is E. In this case, the timedependent wavefuntion for the qubit is|ψ(t)i = Cg (t) |gi + Ce (t)eiωq t |ei .

(2.41)To arrive at the dynamics of the system I apply the Schrodinger equationi |ψ̇(t)i = H(t) |ψ(t)i(2.42)

to |ψ(t)i listed in Eqn. 2.41. I use the two level system Hamilton with the interaction term in
Eqn. 2.40. I get a set of two differential equations in terms of Cg (t) and Ce (t) listed as:
1Ċg = −idE (ei(ωq +ωd )t + ei(ωq −ωd )t )Ce2
1 i(ωq +ωd )tĊe = −idE (e+ ei(ωq −ωd )t )Cg .
2(2.43)(2.44)

18
Because I am driving a qubit near its resonance, the ωq + ωd term oscillates much faster than theωq − ωd term. Due to the fast oscillations, the dynamics of the ωq + ωd terms are averaged over,and I can therefore drop these terms in what is called the rotating wave approximation. Solving
the differential Eqns. 2.43 and 2.44 with boundary conditions Cg (0) = 1 and Ce (0) = 0, I get
Ce (t) =where ∆ = ωq − ωd and Ωr =√iEd i∆t/2
Ωr t
e
sin(
),
Ωr
2

(2.45)∆2 + d2 E 2 . The exited state probability is Pe (t) = |Ce (t)|2 .

Therefore,
Pe (t) =

(Ed)2
Ωr t
sin2 ().
2Ωr2(2.46)

This simplifies when driving on resonance (∆ = 0) to
Pe (t) = sin2 (

Ωr t
).
2

(2.47)

Eqns. 2.47 and 2.45 demonstrations that a microwave drive can control the state of the qubit.
Starting from any known pure state of the qubit, I can use an on resonance drive to prepare thequbit in any desired pure state Ψ = A |gi + B |ei.

2.2.3.2

Sideband drives

Now I consider the whole qubit-cavity system’s response to drives that effect both the qubit
and cavity states. I can drive transitions that jointly create excitations of the cavity and qubit, or
swaps them. These transition are named as if the qubit modulates the cavity frequency (|ωc ± ωq |).
The transition on the blue (higher frequency) side of the cavity (ωc + ωq ), which jointly creates
excitation in the cavity and qubit, is called the blue sideband. The transition on the red (lower
frequency) side of the cavity (|ωc − ωq |), which swaps excitations between the cavity and qubit,
is called the red sideband. For my style of qubit, the transmon qubit, these transitions can only
occur as a two photon process. Therefore they are achieved by using two photons each of energy|ωc ± ωq |/2.
To construct the sideband dynamics I closely follow Ref. [45]. I consider two drives acting on
the qubit and add them into the Rabi Hamiltonian. I then perform three unitary transformations on

19
the Hamiltonian keeping only the lowest order terms to simplify the Hamiltonian. Finally, I set thedrive frequency to the sideband transition (ωc + ωq )/2 and perform a rotating wave approximation
to yield an effective Hamiltonian for the sideband drive. The effective, sideband Hamiltonian clearlyshows the blue sideband transition.
To start, the two drives have a Hamiltonian
Hdrive =XΩj (a† + a) cos ωd,j t.

(2.48)

j=1,2

I add the driving terms to the Rabi Hamiltonian (Eqn. 2.35) HRd = HRabi + Hdrive . I then performthe first unitary transformation†††− iU2dU̇2d
HRd U2d
HQCD = U2d(2.49)

built from two displacements corresponding to the two drives
U2d = Πj=1,2 D(αj (t)),(2.50)

where D(α) is the displacement operator, with αj (t) chosen to eliminate direct drive couplings by
settingα̇j (t) = −iωc αj (t) − iΩj e−iωd,j t .

(2.51)

By applying a rotating wave approximation on the drive, the transformed Rabi Hamiltonian becomes
HQCD = ωqXσz+ ωc a† a + g(σ+ + σ− )(a† + a) +Ωj (σ+ e−iωd,j t + σ− eiωd,j t ).
2

(2.52)

j=1,2

Next to remove the direct coupling between the qubit and drive, I apply a second unitary transformation on the Hamiltonian of the form H 0 = Uβ† HQCD Uβ − iUβ† U̇β with
Uβ = exp(βσ− − β ∗ σ+ ),(2.53)Ωj
1 X
eiωd,j t .
2
ωq − ωd,j(2.54)

for
β=

j=1,2

20
Applying Uβ to the Hamiltonian in Eqn. 2.52, I get the Hamiltonian
H 0 = ωq0σz
+ ωc a† a − g(σ+ + σ− )(a† + a) − g(β ∗ + β)(σz − β ∗ σ+ − βσ+ )(a† + a).
2(2.55)

Now, I preform the third and final unitary transformation on the Hamiltonian, the dispersive
transformation
Udis = exp[g(a† σ− − aσ+ )],ωq − ωc(2.56)

which eliminates spontaneous qubit-cavity transitions. By applying this transformation to the
Hamiltonian in Eqn. 2.55, I get an effective Hamiltonian for a qubit cavity system with two off
resonance drives
Heff = ωq0

X
Ω0k
σz
Ωk
†
+ ωc a† a − g(β ∗ + β)σz +g
0 (a + a )
2
ω
−
ω
ω
−
ω
q
q
k
k
0
k,k =1,2× [(ei(ωk −ωk0 )t + e−i(ωk +ωk0 )t )σ− + (e−i(ωk −ωk0 )t + e−i(ωk +ωk0 )t )σ+ ]. (2.57)
This Hamiltonian fully describes a qubit-cavity system that has two drives.
I now consider drives at the blue sideband transition to get the effective blue sideband Hamiltonian. I set both drives to the same frequency such that 2ωk(0 ) = ωq00 + ωc in Eqn. 2.57. The
frequency ωq00 is the shifted qubit frequency with valueωq00 = ωq +

D E
Ω21
Ω22
+
+ 2χ a† a .
ωq − ω1 ωq − ω2

(2.58)

Then, I preform two last rotating wave approximations on Eqn. 2.57 at the qubit and cavity
σz†frequency by performing rotations defined by Uq = e−iωq 2 t and Uc = e−iωc a at and dropping thefast oscillating terms. At last, I get the Hamiltonian for the blue sideband transition
Hblue =

Ω21
Ω22 g †
(a σ+ + aσ− ).
ωq − ω1 ωq − ω2 4

(2.59)

Written more simply
Hblue = Ωblue (a† σ+ + aσ− ).

(2.60)For the system starting in |g, 0i state (where the labels are |qubit state, cavity photon numberi),Hblue causes the system to oscillate between |g, 0i and |e, 1i states at rate Ωblue . Therefore, I can21
drive the blue sideband transition to prepare the system in the |e, 1i state, and thus create a single
photon in the cavity.

2.3

Measuring a quantum state
Now that I have introduced a quantum mechanical description of circuits and single photon

generation, I discuss the framework for performing measurements on the photon state. I briefly
describe the transformation that the JPA performs to amplify states. Then, I move on to the theoryof measuring a quantum state. Before discussing specific measurements, I introduce a quadrature,or phase space, representation of quantum states called the Wigner function. I form expectationsof single quadrature and two quadrature measurements of quantum states. Finally, I introduce amodel that incorporates loss or inefficiency into these measurements.

2.3.1

Parametric amplifier measurements

The JPA is a microwave amplifier that amplifies propagating fields over a tunable frequency
JPA andband. In an ideal sense, the JPA performs transformations of two incident quadratures X1,in
JPA in a frame that rotates at the center frequency ω
X2,in
JPA of the JPA’s band. These transformations

amplify one quadrature while deamplifying the other to create two outgoing quadratures. This
JPA with phase set by the pump tone(s). After
amplification occurs on an arbitrary quadrature Xin

the quadrature is amplified in the ωJPA frame, the measurement is completed by demodulating
JPA by ω
JPA in the rotating
the JPA’s output Xout
JPA . This demodulation turns measurements of Xin
JPA in the lab frame which do not oscillate in time, and therefore canframe to measurements of Xin

be directly digitized.
Regardless of whether the JPA operates in homodyne or heterodyne mode, in the ωJPA
JPA as the amplified
frame the JPA’s fundamental transformations are the same. By defining X1,in22
quadrature, the JPA’s transformations are summarized as [24]JPA
JPA
= GQ X1,in
X1,out(2.61)JPA
X2,out=(2.62)1JPA
GQ X2,inwith quadrature gain GQ .
The mode of operation (homodyne or heterodyne) is determined by whether or not the signals
are at ωJPA . For the JPA to operate in homodyne mode, the signal or the center of the signal’sband must be at ωJPA . In this case, the quadratures in the signal’s frame X1sig and X2sig and thequadrature that describe the JPA’s transformation X1JPA and X2JPA are in the same rotating frame,
JPA , it directly amplifies X sig . After demodulation at ω
ωJPA . Thus, when the JPA amplifies X1,in
JPA ,1

the digitized output Vm (t) is a direct measurement of the signal’s quadrature X1sig
Vm (t) = GQ X1sig .

(2.63)For the JPA to operate in heterodyne mode, the signal must be detuned from ωJPA . For
detuning ∆, the quadratures that describe the signal X1sig and X2sig are in a frame rotating at
ωJPA + ∆. The JPA still performs its amplifying transformation on the quadratures in a frame thatrotates at ωJPA . In terms of the quadratures in the signal’s frame, I write the JPA’s input in the
ωJPA frame assin(∆t)X2sig(2.64)JPA
X2,in= − sin(∆t)X1sig + cos(∆t)X2sig .

(2.65)

JPA
X1,in
=

cos(∆t)X1sig +Then, the JPA’s outputs are
JPA
X1,out=JPA
X2,out=GQ (∆) cos(∆t)X1sig + sin(∆t)X2sig


sig
sig
1
−
sin(∆t)X+
cos(∆t)X,
1
2
GQ (∆)(2.66)(2.67)

where GQ (∆) is the JPA’s quadrature gain ∆ from the center of the JPA’s band. After demodulating the JPA’s amplified quadrature at ωJPA , I find the digitized output to be
Vm (t) = GQ (∆) cos(∆t + φ)X1sig + GQ (∆) sin(∆t + φ)X2sig .

(2.68)

23
Thus, the JPA’s one amplified quadrature contains information about both of the signals quadratures. By fitting the output Vm (t) to a function C1 cos(∆t+φ)+C2 sin(∆t+φ) in terms of constants
C1 and C2 , I can extract measurements of two quadratures X1sig and X2sig based on the values C1and C2 , respectively.
By considering multiple rotating reference frames, we can understand the origin of the 1/2
quanta (one vacuua) of added noise heterodyne measurements must introduce. At the JPA’s input,
both fields oscillating at ωJPA +∆ and fields oscillating at ωJPA −∆ are amplified and demodulated at
ωJPA causing Vm (t) to oscillate at ∆. Consequently, at best amplified vacuum noise (that oscillated
at ωJPA − ∆) is added to the amplified signal (that oscillated ωJPA + ∆) in the digitized output
Vm (t), both oscillating at ∆. The amplified vacuum is then indistinguishable from the amplified
signal. Thus at a minimum, vacuum noise is added to the amplified signal.

2.3.2

Wigner function

To visualize quantum states in phase space, I use a quasi-probability distribution function of
the two quadratures called the Wigner function W (X1 , X2 ). If X1 and X2 were classical coordinates,
then the Wigner function would become a true joint probability distribution of the quadratures.
However, X1 and X2 are operators that have an uncertainly relation. They cannot be simultaneously measured without adding noise, therefore one cannot directly measure the Wigner function
from quadrature measurements. Because W (X1 , X2 ) is not a classical probability distribution, it
is possible for W (X1 , X2 ) to have negative values, further distinguishing it from a true probability
distribution.
The Wigner function is a valid representation of the density matrix. From a density matrix
ρ the Wigner function is calculated by
1
W (X1 , X2 ) =π~Ze−i2yX2 hX1 − y| ρ |X1 + yi dy.

(2.69)As an example, I plot the Wigner function for a single photon (Fig. 2.1). The nonclassical natureof the single photon state is seen in the negative values of the Winger function at the origin. Now

24

Figure 2.1: The Wigner function for a single photon state (n = 1).

25that I have the Wigner function representation of quantum states, I can use it to calculate expected
probability distributions for one and two quadrature measurements.

2.3.3

Homodyne measurement of quantum states

Because single quadrature, or homodyne, measurements can ideally be performed without
the addition of noise, these can be directly calculated from the Wigner function without convolving
any additional Gaussian noise. To calculate the single quadrature probability distribution of aquantum state, I integrate it’s Winger function over the orthogonal quadratureZ
P (X1 ) =W (X1 , X2 )dX2 .

(2.70)As an example, I plot the P (X1 ) for a single photon state (Fig. 2.2). This probability distribution
is non-Gaussian, which distinguishes it from Gaussian probability distributions created by vacuum,
thermal, displaced thermal, squeezed, and coherent states. Therefore measuring these homodyne
and heterodyne probability distributions can form a method for identifying the n = 1 photon statein this experiment.

2.3.4

Heterodyne measurement of quantum states

A two quadrature, or heterodyne, measurement adds at least vacuum noise because the twoquadratures are canonically conjugate variables. The two-quadrature probability distribution caneither be calculated by convolving the Wigner function with vacuum noise or calculated directlyfrom ρ. To calculate directly, I can calculate the Husimi-Q function
P (X1 , X2 ) = Q(X1 , X2 ) = hX1 + iX2 | ρ |X1 + iX2 i /π.

(2.71)

Alternatively, I can calculate the distribution from the Winger function by performing the convolution
2
P (X1 , X2 ) =πZ

W (Y1 , Y2 )e−2|X1 +iX2 −Y1 −iY2 | dY1 dY2 .

(2.72)

26

Figure 2.2: Homodyne measurement of a single photon: single quadrature probability distribution
for the n = 1 Fock state.

27
As an example, I plot the P (X1 , X2 ) for a single photon state (Fig. 2.3). Just like the single
quadrature measurements, this probability distribution is non-Gaussian, and therefore distinguishable from Gaussian probability distributions.

2.3.5

Inefficient measurement

Measurement inefficiency and loss can be incorporated by using a model where a beam splitter
replaces some of the signal with vacuum fluctuations. In this model, one of the beam splitters inputs
is the density matrix ρin that I am measuring and the other is the vacuum state |0i h0|. I measure
one of the outputs, and the other output represents degrees of freedom I do not measure. The
power transmission η of ρin to the measured output represents the efficiency of the measurement.
To calculate the measured density matrix, I first perform a beam-splitter transformation on the
input modes. Then, I sum over, or trace over, the mode I do not measure.
in
I define the beam splitter transformation in terms of two input modes labeled ain
1 and a2and output modes labeled aoutand aout
1
2 . I inject my quantum state in input mode 1 and measure
output mode 1. The beam splinter transformation is defined as

 


out
in
 a1   cos ϑ i sin ϑ   a1 

=


in
aout
i
sin
ϑ
cos
ϑ
a
2
2

(2.73)

for a beams splitter with power transmission η = cos2 ϑ. I can perform the beam splitter transformation by either directly substituting the ladder operators according to Eqn. 2.73, or by applying
a unitary transformation operator††

Ubs = eiθ(a1 a2 +a1 a2 ) .

(2.74)

From the input modes (ρin in mode 1 and vacuum in mode 2), I apply the unitary transformation
to get the output
†
in
ρout
12 = Ubs ρ1 ⊗ |02 i h02 | Ubs .

(2.75)

Because I only measure output mode 1, I trace over output mode 2 to get the measured density

28

Figure 2.3: Heterodyne measurement of a single photon: two quadrature probability distribution
for the n = 1 Fock state.

29
matrixρmeas = tr2 (ρout
12 ) =

X

hn2 | ρout
12 |n2 i .

(2.76)n2With this framework, I can calculate the density matrix for measuring a single photon withefficiency η. In this caseρin = |1i h1| .

(2.77)

Then, the two-mode, output density matrix isρout
12 = η |1, 0i h1, 0| − ipη + η 2 (|1, 0i h0, 1| − |0, 1i h1, 0|) + (1 − η) |0, 1i h0, 1| .

(2.78)

After tracing over mode 2, I get the measured density matrix
ρmeas = η |1i h1| + (1 − η) |0i h0| ,(2.79)

where |1i h1| is the density matrix of a single photon. Thus, measuring a single photon withefficiency η yields a mixed state with a single photon component ρ11 = η. Probability distributions
for homodyne and heterodyne measurements can be computed for ρmeas using Eqns. 2.70 and
2.72.

Chapter 3

Qubit and cavity design

The central resource for photon generation is the qubit-cavity system. To experimentallyrealize this idealized system, I create and couple two elements. The first element is a nonlinear LC
circuit, which acts as the qubit. The second element, the cavity, is a single mode of a 3D waveguidecavity. In this Chapter, I first discuss the design and fabrication of the individual pieces, thendiscuss the details of combining and coupling them.

3.1

The qubit
The qubit is an LC circuit that contains a strong nonlinear element. The qubit is formed byconnecting a structure known as a Josephson junction to a capacitor. The Josephson junction is anelectrically nonlinear inductor, thus forming the LC circuit. In superconducting circuits, Josephson
junctions are the only means of implementing a strong nonlinearity with low dissipation [24, 46].
They form the basis of all types of superconducting qubits (flux, phase, charge, transmon) [47, 48,
49, 50]. A Josephson junction is formed by weakly connecting two regions of superconductivity [51].
The strength of the nonlinearity is set by a single parameter: the junctions critical current
I0 , which is the maximum supercurrent that can tunnel across the junction. The nonlinearity of
the Josephson junction can be seen through its Hamiltonian
HJ = EJ (1 − cos

Φ
),
Φ0

(3.1)

where Φ is the flux through the junction. The constant Φ0 = π~e is the magnetic flux quantum,

31
where e is the charge of an electron, and
EJ =

I0 Φ0
2π

(3.2)

is the Josephson energy [52]. Because the HJ depends on the cosine of Φ, this term is nonlinear.
Thus, I0 sets EJ and therefore the strength of nonlinearity. I can show HJ specifically corresponds
to a nonlinear inductance by Taylor expanding ΦΦ0 to get
" 
#
 4
Φ 2
Φ
HJ = EJ−+ ... .
Φ0
Φ0
In comparison, the Hamiltonian of a linear inductor is 2Φ
HL = EL
,
Φ0

(3.3)

(3.4)

where EL = Φ0 /(2L). Therefore, HJ is the Hamiltonian of an inductor with higher order ΦΦ0 terms.
3.1.1

From linear to non-linear circuits

In order for the nonlinear LC circuit to act as a qubit, it must be an effective two levelsystem that I can control. To create an effective two level system from a non-linear circuit, thecircuit resonance frequency shift per excitation must be larger than the circuits linewidth or decayrate [53]. As a starting point, let’s first consider a linear circuit. In a linear LC resonator, theHamiltonian is
HLC = EC (Q2 ) + EL (

Φ 2
) ,
Φ0

(3.5)

where Q is the charge, and Ec is the charging energy. This Hamiltonian is equivalent to a harmonic
oscillator
HLC =with energy levels spaced√p1
2EC EL (a† a + ),
2

(3.6)

2EC EL . Because all the transitions are harmonic, it is impossible toaddress a single transition without inadvertently addressing other transitions, and therefore, itcannot act as a two level system. If I introduce a weak non-linear term that shift the frequency by
K for each exciton added to the circuit, the Hamiltonian becomes1
HJP A = ω(a† a + ) − Ka† aa† a.
2

(3.7)

32
If the circuit has a linewidth that is larger than the non-linearity term (κ > K), then the circuitchanges its resonance frequency by more than the linewidth only when there are many excitationsin the circuit (as is the case for the JPA). Therefore, it is impossible to address single transitions
because the difference in adjacent energy levels is smaller than their linewidths.
By using a strong nonlinearity, I enter a regime that is fundamentally different. For my qubit,
the non-linearity is quantified by the difference between the first transition |gi to |ei and the nexttransition |ei to |f i, which is |αK | = 0.25 GHz. The linewidth is κq = 26 kHz. Because |αK | ismuch greater than κq , it is possible to treat this system as a two level system. Moreover, I candrive the qubit with a fast pulse and not excite any higher order transitions as long as the pulse’s
spectral width is smaller than |αK |. Thus, |αK | sets how fast I can control the system’s dynamics
with it still behaving as an effective two level system [54].

3.1.2

Cooper pair box

My qubit combines a Josephson junction and capacitor to form what is called a Cooper pairbox (CPB). Specifically, I work in a special case of the CPB that is insensitive to charge noise
called a transmon qubit. The CPB is formed by taking a Josephson junction and connecting it
in parallel to a capacitor. A gate is capacitively coupled to the CPB to provide tunability of thesystem. If an explicit gate is not present in the design, parasitic capacitance still forms an effective
gate [52, 55, 47, 56]. A circuit diagram is shown in Fig. 3.1, where the cross indicates a Josephson
junction.
I form the CPB Hamiltonian by combining the Hamiltonian for a Josephson junction (Eqn. 3.2)
with the Hamiltonian for a capacitor with a gate to control its charge. The CPB Hamiltonian isthen
HCPB = 4Ec (n − ng )2 − EJ cos φ.

(3.8)

The first term is the charging energy, corresponding to the energy stored in the capacitors. The
capacitors and Josephson junction form an island as shown in Fig. 3.1. The dimensionless charge(n − ng ) on the island is set by the the dimensionless gate voltage ng and the operator n corre-

33

Figure 3.1: The circuit schematic of a Cooper pair box (CPB).

34
sponding to the number of Copper pairs on the island. One unit of dimensionless charge (in units
2e corresponding to one Cooper pair) with total capacitance CΣ has a charging energy equal to
4Ec =

(2e)2
.
2CΣ

(3.9)

The second term, the energy of the Josephson junction, is written in terms of the macroscopic
phase difference across the junction φ = ΦΦ0 . Because the number of Cooper pairs on the islandchanges when they tunnel across the junction, phase and number operators are dependent. The
phase and number operator have a commutation relation[φ, n] = i,(3.10)

which allows the CPB Hamiltonian (Eqn. 3.8) to be solved in either number n, or phase φ basis.
Before I go into the phase basis to understand the transmon limit, I briefly look at the
number basis to gain some intuition about the solutions. In the number basis, we think of thesolutions as number eigenstates with a coupling term introduced by the Josephson energy. I canPwrite the Josephson energy term as, Hj = − E2J n (|ni hn + 1| + |n + 1i hn|). In this basis, the
CPB Hamiltonian becomes
HCPB = 4Ec (n − ng )2 −EJ X(|ni hn + 1| + |n + 1i hn|)
2 n(3.11)[52]. For large Ec /EJ , the left term (Ec ) dominates the Hamiltonian. As a consequence, the number
states form parabolas over gate voltage as shown in Fig. 3.2 as dashed lines. These number states
are coupled by EJ causing the charge eigenstates to couple near degeneracies. Thus, the HCPB ’s
eigenstates form oscillatory bands shown as the solid color lines.
The transitions are anharmonic and can form an effective two level system or qubit. As
shown in Fig. 3.2, the |gi (red) to |ei (blue) transition has a different energy than the |ei (blue)
to |f i (green) transition for nearly every value of ng . By tuning the gate voltage, I can tune
these transitions, therefore tune the frequency of the qubit. However, as a direct consequence ofthe qubits tunability with gate voltage, fluctuation in gate voltage or charge will cause the qubit
frequency to fluctuate, thus dephasing the qubit.

35

Figure 3.2: The energy levels of the CPB in the large charging energy Ec limit (EJ /Ec = 0.2). The
charging energy states of Cooper pair number state (black dash lines) are coupled by the Josephsonenergy EJ , forming the system eigenstates (solid lines).

36
3.1.3

CPB dynamics in phase basis

Because the CPB is potentially sensitive to charge noise, I consider the transmon limit where
the eigenstates become insensitive to charge at the expense of decreasing the anharmonicity. To
understand this charge insensitive limit, I closely follow Ref. [57]. I first write the equations that
describe the dynamics of the CPB in the phase basis. Next, I cast the equation of motion in a form
with analytic solutions. Then I write the energy eigenstates in terms of a characteristic equation
for these differential equations using the CPB parameters. Lastly, I look at these solutions in the
charge insensitive limit.
I solve for the eigenstates by solving the Schrodinger equation in the phase basis. Because of∂
the commutation relation in Eqn. 3.10, I can write the number operator as n = −i ∂φ. Then, theHamiltonian becomes
HCPB = 4Ec (−i∂− ng )2 − EJ cos φ.
∂φ(3.12)

Then, the Schrodinger equation is[4Ec (−i∂− ng )2 − EJ cos φ]Ψ(n, φ) = ECPB Ψ(n, φ)∂φ(3.13)

I cast this equation in the form of a particular differential equation known as the Mathieu equation,
which has the form:d2 y+ [a − 2q cos(2x)]y = 0.
dx2

(3.14)

Because the solutions of the Mathieu equation are known in terms of special functions, then the
eigenstates and energies of the differential equation are also known in terms of these special functions.
I begin by changing variables in the Schrodinger equations (Eqn. 3.13) to suppress its ng
dependence. I write the equation in terms of h(φ) with Ψ(φ) = h(φ)eing φ . So, then4Ec eing φ (−i∂ 2) h(φ) − EJ cos φeing φ h(φ) = ECPB eing φ h(φ).
∂φ

By canceling eing φ and simplifying I get:


∂2
ECPB
EJ
h(φ) ++
cos φ h(φ) = 0.
∂2φ
4Ec
4Ec

(3.15)(3.16)

37
To write the simplified differential equation (Eqn. 3.16) as a Mathieu equation, I start by
∂
∂x ∂
1 ∂
substituting 2x = φ, noting that ∂φ= ∂φ∂x = 2 ∂x . Thus,∂2
ECPB EJ+
cos 2x)h(2x) = 0.
h(2x) + (
2
∂ x
Ec
Ec

(3.17)

Lastly, I substitute h(2x) = g(x), so then
∂2
ECPB EJ+
cos 2x)g(x) = 0.
g(x) + (
2
∂ x
Ec
Ec

(3.18)In the standard form, this becomes:∂2g(x) + (a − 2q cos 2x)g(x) = 0∂2x(3.19)

Ψ(φ) = Ψ(2x) = g(x)ei2ng x = g(φ/2)eing φ ,(3.20)

where,

a=

ECPB
,
Ec

(3.21)

EJ.
2Ec(3.22)andq=−

In Eqn. 3.19, I have written the CPB Hamiltonian and equations of motion in terms of Mathieu
equations [57, 58]. Next, I solve this equation to find the system’s eigenenergies.

3.1.4

CPB solutions in phase basis

By solving the Mathieu equation, I arrive at the solution of the CPB for all values of charging
energy Ec , Josephson energy EJ , and dimensionless gate voltage ng . The Mathieu equation has
eigenvalues a parametrized by q, defined for the CPB in Eqns. 3.21 and 3.22, respectively. Additionally, the eigenvalues are parametrized in terms of a constant ν (not found in the Mathieu equation),
which sets the periodicity or phase gained from a π offset of the corresponding eigenstate. The
constant ν is written formally as g(x + π) = eiνπ g(x). These eigenvalues, Mathieu characteristic
value, are conventionally expressed as aν (q). The values of aν (q) are tabulated for ν and q andbuilt into computational software such as Mathematica.

38
EJ
Because EνCPB (q) = Ec aν (q) and q = − 2E
, then the CPB’s energy eigenstates arec

ECPB = Ec aν (−

EJ
)
2Ec

Because I know EJ and Ec , my task is to find ν. I start by stating that the wavefunction Ψ isperiodic in φ and note the restrictions it places on ν.
Ψ(φ + 2π) = Ψ(φ) = Ψ(φ)ei(−l2π)
where l is an integer. Changing variables from φ to x (φ = 2x), I findΨ(2x + 2π) = Ψ(2x)ei−(l2π) .
Writing Ψ in terms of g(x) from Eqn. 3.20 (Ψ(2z) = g(z)e2ng z ), I haveg(x + π)e2ng (x+π) = g(x)e2ng x ei(−l2π) .
So then,g(x + π) = g(x)e−2π(ng +l) .
Comparing with the definition of ν, I find thatν = −2(ng + l)

(3.23)

where l is an integer. For a given ng , the ground state is whichever state corresponds to the valueof l that minimizes the eigenvalue. Please note that ±ν yield identical eigenvalues. Combining
these, I find CPB’s energy eigenvalues
ECPB = Ec aν (q) = Ec a−2(ng +l) (−

EJ
)
2Ec

(3.24)

in terms of the standard solutions to the Mathieu equation. Here in Eqn. 3.24, I have the energy
levels for a CPB for any Josephson energy EJ , charging energy Ec and gate voltage ng [57, 58].

39

Figure 3.3: The energy levels of the CPB where EJ = Ec . Here, the system is largely anharmonic
and the energy transitions depend on the gate charge.

3.1.5

Transmon qubit limit of the CPB

Now that I solved the CPB eigenenergies in terms of EJ , Ec and ng , I investigate the eigenenergy values. Specifically, I look at an eigenenergy’s charge dependence and anharmonicity for avariety of values of EJ /Ec . I find that as the ratio of EJ /Ec increases, the energy levels becomeless anharmonic, and depend less on the value of gate charge ng . In particular, I find that the
charge dependence falls faster than the anharmonicity. The relative reductions allow for a parameter regime (EJ /Ec ≈ 50) where the charge dispersion is effectively eliminated, while someanharmonicity remains. A CPB in this regime is call a transmon qubit [57].
From the CPB solution (Eqn. 3.24), I can write out the energy levels as a function of gatecharge for various ratios EJ /Ec . The energy levels are plotted in Fig. 3.3 for a ratio EJ /Ec = 1.
We see that the energy of the transitions are both anharmonic and depend on the gate charge ng .
For comparison, when EJ /Ec = 50 the charge dispersion has completely disappeared as shown in
Fig. 3.4. The anharmonicity has decreased, but is still clearly present. For example, using thevalues for EJ /Ec = 50 shown in Fig. 3.4, a charging energy of Ec = 2π × 0.23 GHz gives a qubit
frequency of E01 = 2π × 4.4 GHz and have anharmonicity of E12 − E01 = αK = −2π × 0.25 GHz.

40

Figure 3.4: The energy levels of the CPB in the transmon limit (EJ /Ec = 50). Here, the system isless anharmonic and the energy transitions do not depend on the gate charge.

At this anharmonicity, the system is an effective two level system for dynamics happening slower
than characteristic time τ = 1/(2π|α|) = 0.6 ns. Because this style of qubit is insensitive to charge,
it does not dephase in the presence of charge noise. Considering these advantages, I use a transmon
style qubit in my qubit-cavity system.
For all values of the charging energy Ec and Josephson energy EJ , I can use the Mathieu
characteristic value aν (q) (Eqn. 3.24), to calculate the qubit frequency and anharmonicity. To avoid
calculations using Mathieu characteristic value, I list some approximations from Refs [57, 58]. In
the transmon limit, the anharmonicity is approximated by the charging energy
αK ' −Ec .

(3.25)

Also in this limit, the qubit frequency is approximated by the geometric mean of the charging
energy and Josephson energy
E01 'p
8Ec EJ .

(3.26)

41
3.1.6

Transmon qubit design constraints

Just like the Cooper pair box, the transmon qubit is a nonlinear LC resonator with thenonlinearity arising from a Josephson junction. Since the transmon’s large EJ /Ec ratio means it
is insensitive to static charge, then the gate is removed to simplify the circuit schematic to justtwo elements as shown in Fig. 3.5. Here, the capacitor plays two roles: it provides capacitance forthe nonlinear LC circuit, and it gives this qubit mode a dipole moment, which allows the qubit tocouple to a cavity.
In addition to being a transmon, there are several constraints on the design of the qubit.
First, I want the qubit to be compatible with JPAs and other microwave devices that operate in
the 4 to 8 GHz range, so
2π × 4 GHz < E01 'p
8Ec EJ < 2π × 8 GHz.

(3.27)

Second, I want the performance of the system to be robust to fast dynamics on the order of ananosecond so,
Ec > 2π × 100 MHz.

(3.28)

Third, the values of the charging energy and Josephson energy should be compatible with state of
the art qubit fabrication techniques [59]. Lastly, the qubit should be designed with a significant
dipole moment so it can be strongly coupled to a cavity. As the qubit is located in a cavity that
forms a pristine microwave environment, the dipole only couples to the modes of the cavity anddoes not radiate into the continuum of free space.
Satisfying all of these constraints, I use a qubit that has Ec = 2π × 288 MHz or total
capacitance CΣ = 67 fF (from Eqn. 3.9). The qubit has a Josephson energy EJ = 2π × 9.60 GHz
or a critical current I0 = 0.019 µAmps (from Eqn. 3.2). These values EJ and Ec are determinedfrom the qubit spectroscopy described in Section 4.1.2. This Josephson energy corresponds to alinear inductance of Lj0 = 17 nH (From Eqn. 3.4). From the Mathieu characteristic value aν (q)(Eqn. 3.24), these energies give the qubit a frequency of E01 = 4.4 GHz×2π, anharmonicity ofαK = −2π × 350 MHz and a ratio EJ /Ec = 33. Note that |αK | converges to −Ec in the limit of42

Figure 3.5: The circuit schematic of a transmon qubit.

43
large EJ /Ec . For the ratio EJ /Ec = 33, the Mathieu characteristic values (Eqn. 3.24) are necessary
to accurately calculate the transmon transitions.
To achieve these qubit parameters, the qubit is laid out according to Fig. 3.5, with a planar
capacitor paddle forming a dipole antenna and a junction in the center. Each half of the paddle is
a 400 µm by 400 µm planar square of conductor forming half of the capacitor. These two halves
are separated by 100 µm. The junction is located along the conductor connecting the two halvesof the capacitor. As seen in Eqn. 3.1 and 3.2, an ideal junction is singularly parametrized by thecritical current I0 . In practice, the critical current is determined during fabrication.

3.1.7

Qubit fabrication

The qubit was fabricated and designed by Martin Sandberg in David Pappas’ group at NIST
Boulder labs. The superconducting qubit is a hybrid TiN/Al design fabricated on high resistivity,
intrinsic silicon (ρ >20 kΩ-cm). The qubit’s capacitors are titanium nitride (TiN), which arefabricated from TiN grown on silicon substrates. The growth process results in a buffer regionof SiN in between the Si substrate and the titanium nitride which is believed to help suppress
undesirable microscopic two level systems at the interface. The only component not made out of
TiN is the Josephson junction which is an aluminum-aluminum oxide junction fabricated using
shadow evaporation techniques [59].
To make the Josephson junction, two regions of aluminum are overlaid over some junction
area with a thin layer of aluminum oxide in between, separating them and forming a tunnel barrier.
The junction is formed by depositing a layer of aluminum in vacuum, then oxidizing the aluminum
surface by letting oxygen in the vacuum chamber. Lastly, a second layer of aluminum is deposited
on top of the first, forming a junction. An image of a 500 by 200 nm Josephson junction fabricatedby the author in the JILA Clean Room is shown in Fig 3.6.

44

Figure 3.6: The center shows a 500 by 200 nm Josephson junction where two Aluminum layers
overlap.

45

3.2

CavityThe second element required for the qubit-cavity system is the cavity. In my case, the cavity

is a hollow rectangular box with superconducting (Aluminum 6061) boundaries as shown in Fig. 3.7.
The cavity has many modes, but I am only interested in coupling to the fundamental mode, while
keeping the higher order modes spectrally distant from the qubit, allowing those modes to beignored.
The cavity is essential for the experiment as it provides inputs and outputs to the system.
Moreover, the cavity is critical to itinerant photon generation by providing a temporary mode forthe photon and by defining the photon’s coupling to the microwave lines which sets the photon’s
bandwidth. I want to operate the qubit-cavity system in the dispersive regime (Section 2.2.2) sothat the qubit can be indirectly measured and controlled, while at the same time, cavity excitations
do not cause qubit excitation. This indirect coupling via the cavity isolates the qubit from thecontinuum of modes in the transmission line. In the dispersive limit, the detuning between the qubit
and the cavity is much larger than their coupling. I would like strong coupling (g ≈ 2π × 100 MHz),
therefore the detuning must be |∆|  100 MHz. Because I know the qubit is at 4.4 GHz, I design
the cavity to have a resonance frequency at roughly 5.8 GHz.

3.2.1

Idealized cavity modes

To understand how to design the cavity with the correct resonance frequency and how its
fields will couple to the qubit, I analytically solve an idealized model of the cavity that ignores thequbit and the coupling ports. The modes are solutions to the electromagnetic wave equation(∇2 + µ∂2
)E = 0,∂t2(3.29)

where µ is the permeability,  is the permittivity and E is the electric field. The solutions are
subject to boundary condition of a perfect conductor. In other words, the parallel component of
the electric field is zero E || = 0 at the boundary of the box cavity. The cavity is aligned in Cartesian
coordinates with dimensions lx , ly and lz (for the respective axes) with a corner of the cavity at

46

Figure 3.7: An image of the cavity used in the experiment.

47
the origin. To solve this case, I align the cavity with lx as the smallest dimension. Because lx isthe smallest dimension, I start by assuming the solutions are separable with the field polarized in
the x̂ direction. I can write the electric field E as
E = Ex (x)Ey (y)Ez (z)Et (t)x̂.

(3.30)

I solve for E with perfectly conducting boundaries in planes defined at x = 0, lx , y = 0, ly andz = 0, lz . Solving Eqn. 3.29 for E, I get the x̂ polarized solutions:E = Ẽ0 cos






my π
mx π
mz π
x sin
y sin
z e−iωt x̂,
lx
ly
lz

(3.31)

where
r
my 2
mx 2
mz 2
ω = cπ (
) +(
) +(
) .
lx
ly
lz

(3.32)

Here, c is the speed of light, and mx , my and mz are indices. The indices my and mz are integers≥ 1, and the index mx is an integer ≥ 0. Using these indices, the modes are labeled Emx my mz . The
fundamental mode is TE011 and has frequencysω011 = cπ11( )2 + ( )2 .
lylz

(3.33)

The cavity used in the experiment and shown in Fig. 3.7. has lx = 4.0 mm (0.157”), ly =48.26 mm (1.90”) and lz = 28.85 mm (1.136”). According to this simplified model that ignoresthe qubit, ω011 = 2π × 6.057 GHz. From room temperature measurements of the cavity without
the qubit chip, the cavity resonance is ω011 = 2π × 6.052 GHz. I use this fundamental mode tocouple to the qubit. According to Eqn. 3.31, the fundamental mode is polarized in the narrow (x̂)
dimension and has no variation across that dimension. Because I want to couple the qubit as adipole to the cavity’s electric field, I align the qubit’s dipole moment with the electric field (x̂) to
maximize the coupling. In the y and z dimensions (the two larger ones of the cavity), the field is
maximum at the center and vanishes at the edges. Accordingly, I place the qubit in the center ofthe cavity in the y-z plane to maximize the qubit-cavity coupling.

48

Figure 3.8: The TE011 mode of a cavity. The equipotential contours of the mode are shown.

3.2.2

HFSS calculation of cavity modes

The exact solution of an ideal cavity works well for describing an empty cavity, but a moresophisticated method is needed to account for qubit and coupler perturbations. A 3D high frequency
field solver called HFSS is used to calculate the modes in this more realistic case. The electric field
contours of fundamental mode TE011 is shown in Fig. 3.8.

3.2.3

Cavity fabrication

The cavity is formed by milling out two pieces of Aluminum 6061 which are shown in Fig. 3.7.
The two halves come together at a seam defined by the plane of the qubit chip. The cavity usedin this experiment was machined by the author in the JILA Staff Shop. The cavity includes a slot
used to hold the qubit chip in its center along with openings to mount couplers.

3.2.4

Cavity input and output couplingI add two couplers to create an input and an output. The combination of both allows fortransmission spectroscopy of the system and readout of the qubit state. The input coupler allows
an input drive to control the dynamics of the qubit-cavity system. Of critical importance, the49output coupler facilitates the decay of a cavity photon into the propagating photon I desire. The
cavity’s decay rate κ sets the bandwidth of the propagating photon, whereas the ratio of output
decay rate to total cavity decay rate κout /κ sets the likelihood the cavity photon decays throughthe output coupler.
There are several considerations to choosing the output coupling rate. I want the photon
bandwidth to be smaller than the JPA’s bandwidth of around 1 MHz at around 25 dB of gain.
Conversely, I want the coupling to be large enough so that the decay to the output mode is thedominant decay rate. That is, the output coupling rate is much larger than the internal loss rateand input coupling rate. However, the input coupling rate cannot be arbitrarily small because I
use this port to control the system. Considering all of these factors, I chose 300 kHz as the target
for the output coupling, dominating over the 30 kHz target for the input.
I couple into the cavity by inserting pins into the cavity aligned with the electrical fieldpolarization. For each coupler, its center pin is connected to the center conductor of microwave
co-axial cables. The cavity coupling is estimated by HFSS simulations but is set experimentally byadjusting the length of coupling pins until the desired coupling is measured.
In HFSS, I estimate the coupling rate of a cavity (similar to the cavity used in the experiment)
to pins connected to a coplanar waveguide. Specifically, I calculate the loss rate of the cavity’s
eigenmode when the only source of loss is coupling to the transmission line. In this case, thecalculated loss rate is the coupling rate. I compute the coupling for a variety of pin lengths tounderstand how the coupling rate scales. The pins are positioned to match the polarization of theelectric field, and 1/4 up the cavity’s height lz and 1/4 across ly the cavity’s width as indicatedin Figs 3.7. Thus the coupling is lower than if their couplers were at the field maximum in thecenter, but should not affect the scaling with pin length. Though the exact value of the coupling
depends on the location of the coupler, the scaling with pin length can be applied to many cavity
geometries.
The results of the calculation are plotted in Fig. 3.9. Here, positive pin length corresponds
to the distance that the pin extends into the cavity. Negative distance corresponds to the distance

50

Figure 3.9: Results from an HFSS calculation of the coupling rate for various pin lengths.

51
the pin is recessed from the cavity’s wall. Because I want a strongly coupled output that sets thecavity’s decay rate to be in the hundreds of kHz along with a much weaker input, I am interested
in the coupling from tens to hundreds of kHz. These couplings correspond to a pin recessed fromthe cavity wall approximately 0.5 mm to 2.5 mm. In this range of pin length, the coupling scales
exponentially with the coupler length by an order of magnitude with every mm of pin length.
Using my knowledge of the coupling’s scaling gained from the HFSS calculation, I adopt a
procedure where I iteratively measure the coupling rate and adjust the length of the pins. The
coupling is determined by measuring the the frequency response in reflection (S11 ), and then fitting
the response in terms of the coupling rate κ1 and total decay rate κ. For the fit I use|S11 |2 = 1 −

1 2
1 − ( κ−2κκ )2
1 + ( 2∆κ ),(3.34)

where ∆ is the detuning from resonance.
Adopting this procedure, I set the output coupling to κout = 2π × 301 kHz as shown in
Fig. 3.10. From measurements of the cavity when it superconducts, it has a total linewidth κ =2π × 410 kHz. Thus, a photon in the cavity decays through the output modes, κout /κ = 73% of
the time. The input coupler is measured to be κin = 2π × 30 kHz and is located as close to thecenter of the cavity as possible to maximize the cavity mode coupling to the qubit.

3.3

Coupled system
To complete the qubit-cavity system I couple the qubit to the cavity in the dispersive limit(Section 2.2.2). In this limit, the qubit cavity coupling sets how much the state of the qubit pulls
the cavity frequency (2χ in Eqn. 2.37), which is used in qubit readout. Additionally, the it sets
the ability to drive the blue sideband which is used to generate photons (Eqn. 2.60). I place thetransmon qubit at the center of the cavity so the dipole moment length of the qubit (set by its
capacitors) lines up with the polarization of the fundamental mode TE011 , thus causing strong
dipole coupling. The qubit chip placement is shown in Fig. 3.11.

52

Figure 3.10: The reflection response and fit used in setting the coupling rate κ1 of the cavity.

Figure 3.11: The qubit coupled to the cavity by placing it at the center of the cavity.

53
3.3.1

Effective dispersive Hamiltonian

The coupling of a transmon qubit to a cavity is more complicated than the coupling of a true
two level system (as described in Section 2.2) because of the many levels of the transmon. Each of
the many transmon energy levels transitions (not just the |gi to |ei) couple to the cavity and pull
the cavity frequency. Fortunately, it is possible to map the system onto a model based on an ideal
two level system (Eqn. 2.38).
To get the mapping, I take the results from Ref. [57], which treat the transmon as a threelevel system, and ignores the |gi to |f i transition. The mapping to the two level systems model
depends on the qubit’s coupling of the |ei to |f i transition gef to the cavity modes, in addition to
the coupling of the |gi to |ei transition gge . The mapping is:χ = χge − χef /2,(3.35)2gij,
ωij − ωc(3.36)

for effective dispersive shift χ due toχij =qubit transitions ωij , coupling gij and shifted cavity frequency
ωc = ωc0 − χef /2,(3.37)

where ωc0 is the undressed cavity frequency. Therefore, χ depends on the two detunings: (ωge − ωc )and (ωef − ωc ), and the two couplings: gge and gef . Because I know the frequencies of the cavity
and the qubit’s first two transitions, I need to estimate the coupling of the qubit transitions to thecavity to determine the effective dispersive shift of the system (2χ).

3.3.2

Dipole coupling model

The qubit couples to the cavity as an electric dipole in the cavity’s electric field. I consider
a simplified toy model to calculate a simple analytic expression for the coupling. I start by writing
g toy = Udipole = −d • Erms

(3.38)

54
where d is the dipole moment of the qubit and Erms is the root-mean-squared electric field due to thevacuum state. The dipole moment is specific for each transmon transition leading to distinct coupling energies for each transition. That is gge and gef are calculated from dge and def , respectively.
I approximate the dipole moment’s magnitude by treating the length of the transmon’s capacitor
paddles l (Section 3.1.6) as the effective length of the dipole. This treatment overestimates the
dipole moment because it is only valid for charges at the end of the capacitor, separated by l. In
reality there is a distribution of charge along the capacitor, which decreases the dipole coupling.
The charge used in the dipole is computed from the transmon’s change in charge (which is 2e × n
for n Cooper pairs) due to making the transition. Thus, the dipole moment for the i to j transitionisdtoy
ij = l hi| 2en̂ |ji .

(3.39)The dipole moment can be solved by taking results from Ref [57],r
j + 1 EJ 1/4| hj + 1| n̂ |ji | ≈2
8Ec(3.40)

for a transmon qubit. I calculate the electric field by setting the time averaged energy stored in
the fields equal to the energy of the vacuum state. That is,
Z
1
1
~ωc =
E 2 dV2
2 V(3.41)

where E is given by Eqn. 3.31 for the TE011 mode, and V is the volume of the cavity. Solving thisequation, I get the rms value for the electric field at the center of the cavity
r2~ωcErms =.
V(3.42)

I calculate the dipole coupling (Eqn. 3.38) of the transmon transitions by combining Eqns. 3.39,
3.40 and 3.42. I express the coupling in units of angular frequency by dividing by ~. For the qubit
transition I get
toy
gge
≈ el

r

ω
~V



2EJ
Ec

1/4
.

(3.43)For the next transition I get
toy
gef
≈

√

toy
2gge
.

(3.44)

55
With my system’s values (d = 0.9 mm, V = 4.0 mm × 48.26 mm × 28.85 mm, EJ /Ec = 34
toytoyand ω = 2π × 5.8 GHz) the calculated couplings are gge= 2π × 138 MHz and gef= 2π × 195 MHz.

I determine the observed coupling rate from qubit spectroscopy (Section 4.1) getting gge = 2π ×
82 MHz and gef = 2π × 141 MHz. As expected, the toy model overestimates the coupling.

Chapter 4

Qubit and cavity’s characterization and dynamics

Before I can use the qubit-cavity system to produce photons, I need to characterize it todetermine if it is suitable for photon generation. Additionally, I need to demonstrate and calibrate
operations, which control the dynamics of the system in order to implement a photon generation
protocol. Both fully characterizing the system and calibrating control operations depend on readingout the state of the qubit. To begin, I perform a power dependent spectroscopy on the cavity. This
spectroscopy lets me calculate some of the system parameter, and form the basis of qubit readout.
With the qubit readout, I can further characterize the system and calibrate control pulses.

4.1

Cavity spectroscopy and qubit readout
As the starting point for understanding the system, I look at the cavity’s power dependent

spectrum. From this spectrum, I can confirm that I have an effective two level system dispersively
coupled to the cavity. Additionally, I can determine the terms in the system’s effective dispersive
Hamiltonian (Eqn. 2.38), which forms the basis for measuring and projecting the state of qubit.
As shown in Fig. 4.1, I inject a probe tone into the cavity and measure the transmitted
probe’s amplitude with the amplifier chain. Because the cavity acts as a filter, I see a peak intransmission at the cavity’s resonance. By adjusting the power and frequency of the probe tone, we
see the cavity resonance corresponding to the |gi, |ei and |f i states of the transmon and the power
dependence of the cavity frequency (Fig. 4.2). The spectrum is plotted as the scattering parameter
S21 = Vout /Vin in terms of the injected Vin and measured voltages Vout . If we ignore the faint |f i

57

Figure 4.1: Transmission spectroscopy schematic. The cavity has two ports. I inject a coherent
tone in one port and measure the transmitted tone with the second port.

state, at low power the system is well described by the dispersive Hamiltonian (Eqn. 2.38), which
describes a cavity frequency that depends on the state of a two level system. From rearranging theterms in Eqn. 2.38, I get the effective cavity frequency in terms of the qubit state asωc0 = ωc + 2χ

σz
.
2

(4.1)

From the figure, the system has a dispersive shift of χ = −2π × 1.0 MHz and ωc = 2π × 5.866 GHz.
The dispersive Hamiltonian considers a two level system coupled to a cavity. In reality, I haveat least a three level system where I treat the |gi and |ei states as forming a two level system. The
presence of the three levels shifts the cavity frequency from the undressed cavity f0 = 5.8655 GHz
at high power to the dressed cavity fc = 5.8705 GHz at lower power. Moreover, the effective
dispersive shift 2χ, is set by the parameters of the three level system coupled to the cavity as I
described in Section 3.3.1 in Eqns. 3.35, 3.36 and 3.37. I can account for the cavity dressing andeffective dispersive shift by writing out
χ=

2
2
gef
gge
−
= −2π × 1.0MHz,ωge − ωc 2(ωef − ωc )

(4.2)

and
∆ωc = −

2gef

2(ωef − ωc )= 2π × 5.5MHz(4.3)

in terms of the three level system (transmon) parameters. In order to solve for all of the dispersive
three level system parameter described in Eqns. 4.2 and 4.3, I need to implement a more sophisticated spectroscopy to determine the qubit frequency and anharmonicity. Once I have these values,
I can use these relation to determine the qubit-cavity couplings (gge and gef ).

58

Figure 4.2: The power dependent cavity spectra. The cavity transmission is measured adjustinga CW probe tone’s frequency and powers. The probe tone’s power is listed as the approximately
power at the cavity’s input. Low power shows the dispersive shift of the cavity frequency dependingon the state of the transom. Because the transmon is not well thermalized, the |f i states resonate
frequency is visible in addition to the |gi and |ei states. At high power, undressed cavity resonancef0 = 5.8655 GHz is visible.

59
4.1.1

Qubit readout

There are two types of qubit readouts I can implement based on the power dependent cavity
spectrum. One is based on the cavity’s dispersive shift [60]. The second is based on the transmon’s
three levels altering the cavity frequency. From the dispersive Hamiltonian, the cavity has one of
two frequencies depending on the state of the qubit (Eqn. 4.1). Because the resonance frequency ofthe cavity is determined by the state of the qubit, measuring the resonance frequency of the cavity,in effect, measures the state of the qubit. I can use a single tone to measure the cavity’s resonancefrequency either as a transmission amplitude measurement or phase measurement, depending on
the frequency of the readout signal [21, 61].
In transmission dispersive measurement, I place a readout tone at one of the cavity’s resonancestates. Then, the readout tone will be on resonance (and thus transmit) depending on the stateof the qubit. For example, if a readout tone is injected at the cavities excited state resonancefrequency, then it will transmit only if the qubit is in the excited state. If the qubit is in the
ground state or any other higher level transmon state, then the tone will not transmit. Thus, Irealize a qubit readout by measuring the transmission of the injected tone. Likewise, by injecting
the readout tone at the ground state, I can make a measurement that is dependent on the thepopulation of the ground state.
Alternatively, I can measure the phase shift of the probe tone in order to measure the cavity’s
resonance frequency, and thus infer the state of the qubit. By placing a tone in between the two
resonant states of the cavity, the phase of the tone is determined by whether the tone is above or
below the cavity’s of resonate state determined by the qubit. Therefore, by measuring the phase of
the probe tone, I measure the state of the qubit [62, 31, 32, 63]. In this case (where the dispersive
shift is much greater than the cavity decay rate |2χ|  κ and χ < 0 ) if the qubit is in the |gistate, then the tone is below resonance and there is approximately no phase shift. When the qubit
is in the |ei state, the tone is above resonance and there is approximately a π phase shift. By
constructing the measurement so that I am sensitive to this π phase shift in the probe tone, I can60
realize a qubit readout.
As an alternative to the dispersive readout, I can form a measurement based on whether thequbit dresses (shift) the bare frequency of the cavity. As seen from the spectroscopy (Fig. 4.2)
the bare frequency of the cavity emerges at 5.8655 GHz when the probe tone is greater than
−97 dBm at the cavity’s input. Whereas, at lower power the cavity has a frequency fc ± χ/2π =5.8705 ± 0.001 GHz. It turns out that the presences of the undressed cavity state depends on both
the readout tone’s power and qubit state. When the qubit is in the |ei state, the undressed cavity
emerges at a higher power than when the qubit is in the |gi state. I realize an undressed cavity
readout by first setting the readout tone at the undressed cavity state frequency. Second, I set thepower so that the cavity will not transmit if the qubit is in the ground state, and will transmit if
the qubit is in the excited state, or bright state. Thus, I have realized an additional readout called
the bright state readout [64].
The readout can either measure an instantaneous realization of the state of the qubit oraverage over multiple realizations depending on the duration of the measurement compared to thequbit’s lifetime. When performing a measurement much longer than the lifetime of the qubit, themeasurement reflects the average state of the qubit over the duration of the measurement rather
than at any single instance. This type of measurement is called continuous wave (CW) measurement. When the measurement is shorter than the lifetime of the qubit and each measurement
distinguishes the qubit (or cavity) state, then measurement reflects the instantaneous state of thequbit and is called single shot readout.

4.1.2

Two-tone spectroscopy of the qubit

To detect the qubit transitions, I perform a type of spectroscopy where I use one tone toexcite the qubit and the second to detect the change in the qubit state (Fig. 4.3). The readout
tone is tuned to the undressed cavity resonance frequency (bright state), which transmits power
when the qubit is excited (Section 4.1.1). I sweep the frequency and power of the other tone, drive
tone, to detect the qubit transition. I perform this two-tone spectroscopy in CW mode. Because

61

Figure 4.3: Two-tone spectroscopy schematic. Like the transmission spectroscopy, I inject power
in one port and measure the transmission with the second port. In addition to the readout tone, I
inject a drive tone to excite the qubit.

62
I am performing a CW measurement, transmission increases according to the probability that thequbit is in the excited state. When the drive tone is at the qubit transition, then the qubit will
oscillate the excited and ground state. If the rate of oscillations, or Rabi rate, is much larger than
the qubit decay rate, then the qubit will be excited half of the time and I will see a larger signal.
If I decrease the Rabi rate (amplitude of the drive tone) to be comparable to or less than the qubit
decay rate, then the excited state probability deceases with the Rabi rate (Section 2.2.3.1), and I
see a decrease of signal with drive amplitude.
The results of the two-tone spectroscopy are shown in Fig. 4.4. I use a poorly thermalized
cavity different from the one used in photon generation. I see an increased transmission of thereadout tone when the drive is at 4.37 GHz. Thus, the qubit frequency is fge = 4.37 GHz. Because
the qubit is poorly thermalized, we can see the |ei to |f i transition. The poorly thermalized
transmon has a non-negligible probability of being in the |ei state Pe ≈ 15%. If the transmon isin the |ei, then driving the the |ei to |f i transition populates the |f i state, and thereby increase
the time it takes to decay down to the |gi state. As a consequence, driving the |ei to |f i transition
decreases the probability of being in the |gi state and therefore increases the signal transmission.
From this spectroscopy, we see that the |ei to |f i transition is at fef = 4.04 GHz. Using thespectroscopy with the cavity used in the experiment, the qubit frequency is determined to be
fge = 4.398 GHz and the |ei to |f i transmission is fef = 4.052 GHz. These frequencies give ameasurement of anharmonicity αk = ωge − ωef = −2π × 346 MHz.
I can use the results of the two-tone spectroscopy to finish solving for the dispersive transmon
parameters and the solve for the Ec and EJ in the CPB Hamiltonian. Combining the results of thequbit, two-tone spectroscopy with power dependent cavity spectra earlier in this section, I can solve
for the cavity coupling to the |gi to |ei transition gge and the coupling to the |ei to |f i transition
gef . From Eqns. 4.2 and 4.3, I find gge = 2π × 82 MHz and gef = 2π × 141 MHz.
Using the solution for the transmon qubit in terms of the Mathieu characteristic function(Eqn. 3.24) I solve for the values of EJ and Ec . From fq = 4.398 GHz and fef = 4.052 GHz and63

Figure 4.4: Two-tone spectroscopy on the state of the qubit. The transmission of the readout tone
is measured while adjusting the frequency and power of a drive tone. The drive power is specified
at the microwave source. From the spectrum, I see fq = 4.37 GHz and fef = 4.052 GHz.

64
assuming negligible charge dispersion, I numerically solve for EJ and Ec by minimizing



2 






2
EJ
EJ
EJ
EJ
J = Ec a−3 −− Ec a−1 −− ωge + Ec a−5 −− Ec a−3 −
− ωef
2Ec
2Ec
2Ec
2Ec
(4.4)
from Eqn. 3.24. Because the transmon is insensitive to ng , an arbitrarily chosen ng = 0.5 is used.
I get EJ = 2π × 9.60 GHz and Ec = 2π × 288 MHz. These values give us a ratio EJ /Ec = 33.2,
which is below the ideal transmon ratio of 50. As we see later in this Chapter, these values still
give us a lifetime and dephasing rate suitable for photon generation.

4.1.3

Single shot readout

For two reasons the photon generation protocol benefits greatly from single-shot QND readout. First, single shot readout has the greatest possible readout contrast facilitating the calibration
qubit control pulses with high accuracy. Second, a QND measurement projects the system into
the measured state, allowing the system to prepared in the ground state by measurement andpost-selection [65, 66, 67].
Single shot readout can be achieved when the readout’s duration is less than the lifetimeof the qubit, and has the contrast to resolve the two states of the qubit. Given a certain cavity
response depending on the state of the qubit, these requirements place joint constraints on thequbit lifetime, probe power and measurement efficiency. I can demonstrate the conditions used toachieve single shot readout by measuring a qubit that has a steady state excited state probability.
The measurement is an excited state transmission measurement. The qubit measurement is pulsed
for 1 µs, and a JPA is used as a homodyne detector that measures a single quadrature value.
As shown in Fig. 4.5, I adjust the readout power and plot histograms along the y-axis of single
quadrature measurements of the readout tone. At the lowest power, it is impossible to resolve
instances that cavity transmits because the qubit is excited from those instances when the qubit isin the ground state. We see a single Gaussian distribution of measured readout tone values along
the y-axis. As the readout tone’s power is increased to −125 dBm at the cavity, we can start toresolve a two Gaussian peaks. One corresponds to instances the cavity transmits because the qubit

65

Figure 4.5: Histograms of the measured readout tone’s voltage (y-axis) at various readout tone
powers (x-axis). At low power these single shot voltages for a Gaussian distribution set by thenoise floor of the measurement. Above -125 dBm of readout power at the cavity, we can resolve
the transmitted distribution associated with the |ei state apart from the |gi state. By thresholding
at a voltage between the two distribution, I can realize single shot readout.

66
is in the excited state. The other corresponds to instances when the qubit is in the ground state.
In the ground state, the cavity does not transmit and we see a Gaussian peak corresponding to thenoise floor of the measurement. I can set up a single shot readout of the qubit state by choosing a
readout tone power where there are two resolvable distributions and then set a threshold quadrature
measurement between the two. If a measurement of the readout tone above the threshold, then I
assign that measurement to the |ei state. If below, then the |gi state. The success of the single
shot readout is demonstrated in the next section when Rabi oscillationons and control pulses are
discussed.

4.2

Control pulses
Now that I can perform single shot readout on the state of the qubit, I can observe coherent

control of the qubit-cavity system. I demonstrate driving Rabi oscillations, and I calibrate pulsesthat drive the qubit transition (Section 2.2.3.1), which are used for characterizing the system. Then,I demonstrate driving coherent oscillations on the blue sideband transition (Section 2.2.3.2), which
creates photons. Like two-tone spectroscopy (Section 4.1.2), I use both a drive and a readout tone
(Fig. 4.3). Rather than operating both the readout and drive tones in CW mode, as for twotone spectroscopy, I sequentially pulse the two tones. First, I pulse the drive tone at a particularfrequency and amplitude for a set amount of time. Then, I pulse the readout tone to perform thesingle shot on the qubit.

4.2.1

Qubit pulse

Driving the qubit transition creates oscillations between the ground and excited state (Rabioscillations). This is a direct demonstration that our system is a controllable effective two levelsystem. Moreover, the drive on the qubit allows preparation of the qubit into arbitrary states andcan facilitate measurements of the qubit’s lifetime T1 and dephasing time T2 . When I drive thequbit transition, the qubit state will oscillate between ground state |gi and excited state |ei [62].
Starting in the ground state, the excited state probability after being driven for a time td oscillates

67
according to q
td ∆2 + Vd2

Pe (td ) = 2
sin2 
2∆ + Vd2
Vd2(4.5)

from Eqn. 2.46, where ∆ is the drive’s detuning from the qubit transition and Vd is the amplitudeof the drive.
I demonstrate these Rabi oscillations by repeating a protocol where I first drive the qubit fora certain amount of time with a certain power and then, perform a single shot readout. After themeasurement, I wait for the qubit to relax before starting the protocol again. I repeat this protocol
640 times at each power and time to measure the probability of the qubit being in the excitedstate (Fig. 4.6). I see Pe oscillate according to Eqn. 4.5. Observing Rabi oscillations confirms thefrequency of the qubit transition and is a clear demonstration that I am controlling an effective
two level system.

4.2.1.1

Calibrating pulses

By measuring these oscillations I can calibrate pulses. A pulse which lasts for half the periodof a Rabi oscillation swaps the qubit state from |gi to |ei, or |ei to |gi. This pulse is called a π pulse
because it constitutes π radians worth of Rabi oscillations. A control pulse lasting for π/2 radians√
worth of Rabi oscillations is call a π/2 pulse and pulse will turn a |gi into a (|gi + eiφ |ei)/ 2 statewith the phase φ set by the phase of the pulse.
From a plot of Rabi oscillations similar to Fig. 4.6, I can calibrate a π-pulse by choosing theduration and amplitude that maximizes the swap of the first oscillation. Similarly, I can calibrate a
π/2-pulse, by keeping that same frequency and choosing the duration such that half the population
has swapped. These pulses are important tools because both π and π/2-pulses on the qubit areused to characterize the qubit’s lifetime and dephasing. Furthermore, π-pulses on both the qubit
and blue sideband transition are used to in the photon creation protocol.

68

Figure 4.6: The results from the Rabi measurement adjusting pulse amplitude and frequency. The
pulse amplitude is specified at the microwave source. Rabi oscillations can be seen at the qubit
transition.

69
4.2.2

Blue sideband pulse

Pulses on the blue sideband are used to create photons in the cavity, and are therefore criticalto the photon generation experiment. As we saw in section 2.2.3.2, when I drive the qubit cavity
system at (fq + fc )/2 I get the effective Hamiltonian for the blue sideband drive that a couples
the |N, gi state and the |N + 1, ei state. (The notation convention is |cavity photon number, qubit
statei). Because of this coupling, a drive on the blue sideband will cause oscillations between the|N, gi and |N + 1, ei state. These oscillations are analogous to Rabi oscillations between the |gi
and |ei states for the qubit drive.
I demonstrate these oscillations by using an identical protocol to the Rabi oscillations in theprevious subsection (4.2.1). Starting in the ground state with no photons in the cavity, the systemoscillates between the |0, gi and |1, ei states. I can therefore perform a single shot readout on thestate of the qubit to see the blue sideband oscillations. I adjust the drive frequency and amplitude
to see Rabi like oscillation in Fig. 4.7. Additionally, I see a drive amplitude dependent shift inthe blue sideband frequency. It arises because I populate the cavity with photons and Stark shift
the qubit frequency as seen in the dispersive Hamiltonian (Eqn. 2.38). Because the blue sideband
frequency depends on the frequency of the qubit [fbs = (fq + fc )/2], the blue sideband transition
Stark shifts along with the qubit.
The oscillations become more clearly similar to Rabi oscillation by fixing the amplitude ofthe drive and adjusting both the frequency and duration of the drive (Fig 4.8). In a similar mannerto the previous section, I can define and calibrate a π pulse on the blue sideband. Starting in
the ground state of the system |0, gi, this blue sideband π pulse will create a photon, putting thesystem in the |1, ei state. Thus, the blue sideband π pulse creates a single photon while exciting
the qubit.

70

Figure 4.7: The results of the blue sideband measurement, as a function of pulse amplitude andfrequency. The pulse amplitude is specified at the microwave source. Rabi-like oscillations canbeen seen at the blue sideband transition frequency. These oscillations are distorted by the power
dependence of the blue sideband transition, or Stark shift.

71

Figure 4.8: The results from the blue sideband measurement with fixed pulse amplitude. I adjust
pulse duration and frequency seeing Rabi-like oscillations at the blue sideband transition frequency.

72

4.3

Qubit characterization
Combining the calibrated drive pules with the single shot readout, I can implement protocols

that determine the lifetime T1 and dephasing time T2 of the qubit. The lifetime is the characteristic
time for the qubit to decay from the |ei state to the |gi state. The dephasing rate Γ = 1/T2
describes how fast the phase φ in |gi + eiφ |ei is lost. These parameters must be sufficiently large
for a successful implementation of the photon generation protocol.
The photon creation protocol places constraints on the lifetime of the qubit T1 because it
relies on a blue sideband π pulse. When I create a photon in the cavity with a blue sideband,
I also excite the qubit. Changing the state of the qubit changes the frequency of the cavity andits excitations due to the dispersive Hamiltonian. Because I do not want the photon frequency tochange, I want the qubit to stay in the |ei state until the photon has decayed out of the cavity.
As a result, I want the qubit to have a lifetime that is much longer than the cavity decay timeT1  1/κ. For the system κ = 2π × 410 kHz, therefore I desire T1  0.39 µs.
The dephasing time is another important characteristic of the qubit. The dephasing time T2
must be much longer than the duration of any control operation on the qubit for that operation tohave high fidelity. For the 150 ns pulse length I implement in the photon generation experiment,
T2  0.15 µs is necessary for the qubit’s phase to be well defined over the duration of the qubit
pulse. Additionally, determining T2 is important in the photon generation experiment, because I
use the change in the dephasing time to quantify measurement backaction.

4.3.1

Qubit lifetime

I can measure the lifetime T1 of the qubit by implementing a three-step protocol. Starting in
the |gi state, the protocol begins by first exciting the qubit with a π pulse. Then, I wait some time
tw before I perform single shot readout on the state of the qubit. By repeating this protocol several
times at each tw , I extract the probability that the qubit is in the excited state. By measuring
this probability over a range of wait times, we can see the probability that the qubit is in the

73

Figure 4.9: Measurement of the qubit |ei state lifetime. I excite the qubit and observe it decay.
From the fit, the qubit lifetime is T1 = 10.2 ± 0.3 µs.

74
excited state Pe decays over time. By fitting the decaying excited state probability to a decaying
exponential
Pe (tw ) = Pe (0)e−tw /T1 ,(4.6)

I can extract the lifetime T1 for the qubit [68]. I implement this T1 measurement (Fig. 4.9). Fromfitting the excited state probability to Eqn. 4.6, I get a qubit lifetime T1 = 10.2 ± 0.3 µs. Indeed,
T1  0.39 µs as I desire for the photon generation protocol.

4.3.2

Qubit dephasing

I can measure the characteristic time T2 for the phase φ of a coherent superposition |gi+eiφ |ei
to drift or dephase by implementing a four-step Ramsey spectroscopy protocol. When the dephasetime T2 is measured this way it is denoted as the T2∗ time. In this protocol, I prepare the |gi+eiφ |eistate and observe the stability of its phase φ. In the first step of the protocol, a π/2 qubit pulse
rotates the |gi state to the Ψ(0) = |gi + eiφ0 |ei state, where the phase φ0 is set by the π/2 pulse. In
the second step, I wait some amount of time tw during which the phase may undergo some random
walk or dephase. The last two steps amount to measuring the overlap between this state at t = twand the state at t = 0, or measuring |hΨ(0) |Ψ(tw )i |. The third step rotates Ψ(0) the to |ei state
by applying a second π/2 pulse with the same phase as the first. Lastly, a readout is performed on
the qubit state [68, 62].
When there is no dephasing during the wait time, the two π/2 pulses have the effect of asingle π pulse. In this case, I expect the qubit to be in the |ei state. In the other extreme thequbit completely dephases. In this case, the second π/2 pulse is just as likely to rotate the qubit
to the |ei state as the |gi state. I expect the excited state probability to be Pe = 50%. Thus, as I
increase waiting time, I expect Pe to decrease from 100% to 50% as the qubit dephases. Or written∗

explicitly, Pe = P0 + P1 e−tw /T2 with constants P0 = P1 = 0.5 for the ideal case.
Now, I consider an off resonant drive. When there is no qubit dephasing during the wait
step, the phase of the drive rotates relative to the phase of the qubit at a frequency set by thedifference between the drive frequency and qubit ∆t2 . In this generalized T2∗ sequence, the rotating

75

Figure 4.10: Ramsey measurement of the qubit dephasing as I adjust the pulse frequency. As thequbit dephases at large time, the excited state probability saturates and the contrast in the fringes
is lost.

76

Figure 4.11: Ramsey measurement of the qubit dephasing with the drive detuned 1 MHz. As thequbit dephases the coherent oscillations washout.

77
phase causes Pe to oscillate at that difference frequency. As a result, I expect the excited state
probability to both be decaying and oscillating. Written explicitly, the excited state probability is∗

Pe (tw ) = P0 + cos(∆t2 tw )P1 e−tw /T2 ,(4.7)

where the constant P0 is the steady state excited state probability and P0 + P1 is the initial excitedstate probability. For the ideal case, the constant P0 = P1 = 0.5.
I implement this T2∗ measurement by adjusting the wait time tw and the π/2 pulse’s frequency(and therefore ∆t2 ) in order to measure the qubit dephasing time T2∗ (Fig. 4.10). Most of my Ramsey
measurement are conducted at only a few values of ∆t2 . To demonstration the Ramsey fringes,
the measurement shown in Fig. 4.10 is conducted at 51 values of ∆t2 . However, this particularmeasurement was performed before implementing the high contrast single shot readout. Here, I do
not threshold the single shot measurement voltages to assign the each instance in either the groundor excited state. Instead, I average the raw single shot measurements to plot the average voltage
at each tw and frequency. I fit a subset of data with ∆t2 =1 MHz to Eq. 4.7 where constants P0
and P1 take on values of voltages (Fig. 4.11). I extract the dephasing time T2∗ = 6.2 µs. Thus,
T2  0.15 µs. Therefore, the qubit is coherent enough for the photon generation protocol.

Chapter 5

Photon generation and detection

I now reach the heart of the experiment. In this Chapter, I discuss single photon generationand detection using a superconducting transmon qubit coupled to a cavity. I use the blue sideband
transition of the qubit cavity system to create photons. To analyze the photons, I operate the
JPA in heterodyne (two quadrature) and homodyne (single quadrature) modes. For both detection
methods, I quantify the single photon generation and detection by performing state tomography,
which estimates the density matrix ρ.
To initially verify photon generation, I implement a measurement apparatus designed toprotect and isolate the qubit-cavity system from the amplifiers at the cost of measurement efficiency.
In these measurement, I see several signatures of photon generation and perform tomography on
the propagating photon state using heterodyne detection. Limited by both detection inefficiency
and generation fidelity, I infer a mixed state that corresponds to a single photon 12% of the time(ρ11 = 0.12) of the propagating mode, correcting for heterodyne detection’s 1/2 quanta of added
noise.
Working to optimize photon detection, I implement a homodyne detection scheme that minimizes the loss between the photon source and the JPA. In this case, I reconstruct a mixed statethat is a single photon 36% of the time (ρ11 = 0.36). In this scheme, I quantify a JPA’s backaction
on the qubit-cavity system. I independently estimate the density matrix of the propagating photon
state and the efficiency of my amplifier chain. I find that my photon measurements agree with myexpectations within uncertainty using state fidelity as a metric.

79

5.1

Photon generation
I generate single propagating photons by first generating them in the cavity, and then letting

them decay into a strong coupling output mode. I create the photons in the cavity by performing
a calibrated π pulse on the blue sideband transition of the qubit cavity system, which excites thequbit and creates a photon in the cavity (Fig. 5.1). Once the photon is created in the cavity,
it is strongly coupled to an output mode and decays, creating a propagating photon. I start byinvestigating the blue sideband pulse because it is the key step in the generation protocol.

5.1.1

Blue sideband measurements

I use the dynamics of the qubit-cavity system to confirm that I am addressing the blue
sideband transition. The simplest check I can perform is to look at the Rabi-like oscillation of theblue sideband transition. During a blue sideband drive, the system oscillates between the |0, gi and|1, ei state (Fig. 5.1). Consider the situation when photons decay much faster than the qubit. If
the photon decays when the system is in the |1, ei state, then the system will be in the |0, ei. Until
the qubit decays, the system no longer oscillates because the system does not have a photon to lose
when the qubit flips to the |gi state, and the qubit will be stuck in the |ei state.
In my system, cavity photons decay much faster than the qubit. Photons decay in 1/(2π ×410 kHz) = 0.39 µs, whereas the qubit decays in T1 = 10.2 µs (Section 4.3). Under a continuous blue
sideband drive, the qubit cavity system decays to the |0, ei at a rate set by the cavity decay rate κ =2π ×410 kHz. At larger times, the qubit can decay causing the system to saturate into a mixed statewith the qubit excited half of the time [23]. I measure the excited state probability as a functionof blue sideband pulse length (Fig. 5.2). This measurement protocol is nearly identical to Rabi
oscillation (Section 4.2.1), where I drive the blue sideband transition for a certain time then perform
single shot readout. From fitting these results to an exponentially decaying sinusoid, the oscillations
between ground and excited state decay to the excited state at rate κBS = 2π × (370 ± 10) kHz.
This behavior is consistent with creating decaying single photons. However, I would like a more

80

Figure 5.1: Energy level schematic for photon generation with the blue sideband.

81
rigorous confirmation that the blue sideband π-pulses are creating single photons.

5.1.2

Cavity photon detection with qubit

Now that I have seen indirect evidence that I am creating photons with the blue sideband
drive, I use the system to directly detect cavity photons [69, 70]. Just like the dispersive Hamiltonian
allows the cavity to be used for qubit readout, it also allows the qubit to be used to perform cavity
photon number readout. I use the fact that the qubit frequency depends on the number of cavity
photons to observe photons decaying out of the cavity following a blue sideband π-pulse. From thedispersive Hamiltonian (Eqn. 2.37), I can write the qubit frequency in terms of photon number
ωq0 = ωq + 2χn.

(5.1)

as in Eqn. 2.39. Because the qubit frequency depends on photon number, I can calibrate π-pulsesthat are contingent on photon number. For example, a π-pulse whose frequency is ωq,1 = ωq + 2χ isonly a π-pulse when there is one photon in the cavity. To make qubit π-pulses contingent on cavity
photon number, these pulses must be spectrally narrow compared with the qubit photon number
splitting (2χ). A complete description of this approach can be found in Ref. [71].
To achieve photon number detection, I must do three things: first, I must know the qubit
initial qubit state, then, I apply a photon number dependent π-pulse, lastly, I perform a single shot
readout on the state of the qubit. If the qubit changed state, then a photon number corresponding
to the qubit π-pulse is detected. For example, starting with the qubit in the |ei state, I apply aqubit π-pulse at the single photon qubit frequency (ωq + 2χ). If I measure the qubit in the |gi state,then the π-pulse was successful and there was one photon in the cavity. Otherwise, if the qubit ismeasured in the |ei state, then the cavity does not have one photon.
To detect the decay of photons created by the blue sideband transition out of the cavity, I
implement the following sequence. First, I apply a blue sideband π-pulse to create a photon in
the cavity. Then, I wait for some delay time. Next, I apply a spectrally narrow qubit drive pulse
at some frequency. If this pulse is at the correct frequency, then it will be a π-pulse contingent

82

Figure 5.2: Excited state probability during a blue sideband drive (marks) fit to a offset exponentially decaying sinusoid (solid line). During a blue sideband drive, the qubit oscillates between the|gi and |ei state. As a photon decays from the cavity, the systems saturates to the |ei state.

83

Photon measurement

Photon creation
Blue sideband
Qubit
Readout

Time

Delay time
4.396

4.395

n=0

0.50

4.394

0.45

4.393

0.40

n=1

Ground state probability

Qubit pulse frequency (GHz)

0.55

0.35

4.392

0.1

0.3
0.2
Delay time (μs)

0.4

0.5

Figure 5.3: Detecting cavity photons created by a blue sideband π-pulse using photon number
dependent qubit π-pulses. The |gi state probability is plotted following a second pulse (a qubit
pulse) adjusting the qubit pulse’s frequency and delay after the blue sideband pulse. When thepulse is at 4.3948 GHz or 4.3929 GHz, the pulse is π-pulse contingent on zero or one photon in
the cavity, respectively. Because these pulses rotate the qubit to the |gi, measuring the |gi statemeasures the respective photon number state (when the π-pulse is at 4.3948 GHz for n = 0 and
4.3929 GHz for n = 1). As the delay time increases the photon created by the blue sideband pulse
decays to the n = 0 state.

84
on cavity photon number. Lastly, I perform a single shot readout on the qubit. I repeat thisprocedure roughly 103 times for each combination of delay time and drive frequency to build up
statistics (Fig. 5.3). I adjust the drive frequency across the zero photon and single photon qubit
frequency to detect the |1i state decay to the |0i state. I measure over a range of delay times
to observe the photon decay out of the cavity. At drive frequencies 4.3948 GHz and 4.3929 GHz,
I can resolve the |0i and |1i π-pulses, respectively. With increased delay time, the single cavity
photon probability decreases and the probability of there being zero photon increases. From thismeasurement, I directly observe that the blue sideband π-pulse creates a photon in the cavity, andI see these photons decay.

5.2

Heterodyne detection
To detect the propagating photon, I connect the strongly coupled output to the JPA. The

JPA is operated as a two quadrature, or heterodyne, detector by centering the JPA’s gain ∆ =−2π × 1 MHz detuned from the photon’s frequency. The photon’s frequency is set by the cavity’sfrequency when the qubit is in the excited state (ωc + χ). The center of the JPA’s gain is set by
the frequency of the strong pump tone. As a result of operating the JPA as heterodyne detector
the strong pump tone is 1 MHz detuned from the cavity’s resonance. Because the cavity has alinewidth of κ = 2π × 410 kHz, the the pump is filtered by the cavity.
This measurement strategy minimizes undesirable interaction between the qubit-cavity system and the JPA, but at the cost of reduced measurement efficiency. As a result, the JPA operates
with modest measurement efficiency. In between the cavity and the JPA, there are reactive andattenuation lower pass filters designed to suppress photons whose frequency are above the cavity’s
resonance leaking into the cavity. Additionally, there are 3 non-reciprocal elements called circulators or isolators, which prevents the JPA’s output from leaking back into the cavity. Because of
all these elements, there is 4.6 dB of loss between the cavity and the JPA as measured at room
temperature. As a result, a photon should be absorbed in transit approximately 65% of the time.
In other words, the loss in between the cavity and JPA limits the measurement efficacy to roughly

85ηm ≤ 0.35. Additionally, intrinsic measurement inefficiency and cavity loss further reduces ηm .

5.2.1

Two quadrature filter and photon variance

As a photon decays out of the cavity, the JPA’s measurement yields a time trace Vm (t). As
discussed in Section 2.3.1, a quadrature in a frame rotating at ωJPA (the center of the JPA’s band)is amplified and demodulated to yield the time trace Vm (t). If the input signal were at ωJPA , thenVm (t) would contain a sequence of single quadrature measurements of the signal. In my case, thesignal is at ωJPA + ∆. The quadratures in the frame of the signal oscillate at a rate ∆ with respect
to the quadrature that the JPA measures. Consequently, an input Vin = V0 sin ((ωJP A + ∆)t) going
into the JPA becomes a measured time trace Vm (t) = GV0 sin(∆t + φ0 ), where G is the heterodyne
gain at ωJP A + ∆, and φ0 is a static phase shift in amplification and demodulation process.
In the photon measurements, this time trace Vm (t) must be processed to achieve the twoquadrature measurement. I extract the values of X1 and X2 using two orthogonal filter function
oscillating at ∆. For measurements of a steady oscillating field using the JPA as a heterodyne
detector, the choice of the two filter functions is straightforward. Consider that signal Vin . Its
output Vm (t) oscillates at ∆. For this case, I can construct the two quadrature filters fx1 =N1 cos(∆t) and fx2 = N2 sin(∆t) corresponding to the X1 and X2 quadratures, respectively. The
normalization constants Ni are set so that filter function is normalized over the measurement timeinterval from t0 to t1 . Or,
Z t1

fi2 (t)dt = 1.

(5.2)

t0

Just like the corresponding quadratures, for long measurement intervals the two filter function are
orthogonal, orZ ∞f1 (t)f2 (t)dt = 0.

(5.3)

0

As long as the measurement interval is long compared to the period of oscillations, then the finite measurements are approximately orthogonal. Using these filter functions, the values of the86measured quadratures then become
Z t1
Xi =

fi (t)Vm (t)dt.

(5.4)

t0

Capturing the pulsed photon signal with a heterodyne measurement adds additional requirements. In addition to having two orthogonal filter function with a phase π/2 radians apart, thefilter function must match the temporal envelope of the photon decaying from the cavity. Specifically, for a pulse which has some temporal power profile P(t), I want f12 (t) + f22 (t) = P(t) for
efficient measurement. The temporal profiles is similar the output mode Aout defined in Eqns. 2.6
and 2.7 with one major exception. It is distorted by the JPA’s finite bandwidth.
Shown in Fig. 5.4, are the two quadrature filters matching the X1 and X2 quadratures that
I use in my heterodyne measurements. Taken together, these two filter functions match the power
profile of a state leaking out of the cavity and measured by the JPA. [For comparison to the measured
temporal profile of the power leaking out of the cavity due to photon creation, see Fig. 5.11(c).] The
two filter functions are formed by starting with two decaying sinusoids f10 (t) = e−κ/2t cos(∆t + φ)and f20 (t) = e−κ/2t sin(∆t + φ) for t ≥ 0, both similar to Aout (Eqn. 2.6) oscillating at ∆. The decay
κ is set to the cavity’s decay rate and the detuning ∆ = −1 MHz. The phase φ sets the phase of
the quadrature filters. To account for the bandwidth of the JPA, the two sinusoids are low passfiltered with the cutoff frequency set by the JPA bandwidth (1 MHz for the filter functions shown).
A low pass filter h is used rather than a bandpass because the signal is demodulated at ωJPA . The
results are the filter functions shown in Fig. 5.4 and listed below:
f1 (t) = h(ωLPF , t) ∗ Θ(t)e−κt cos(∆t + φ)

(5.5)

f2 (t) = h(ωLPF , t) ∗ Θ(t)e−κt sin(∆t + φ).

(5.6)

Here, Θ(t) is the Heaviside step function and h(ωLPF , t) is a low pass filter’s impulse response with
cutoff frequency equal to half the bandwidth of the JPA ωLPF = 2π × 0.5 MHz. The symbol (∗)
denotes convolution.
Well defined quadrature filter functions should be orthogonal over their range as definedin Eqn. 5.3. A sine and a cosine functions are approximately orthogonal over a span of many

87

1

f1
f2

0

-1

0

1

2

3

t (μs)

Figure 5.4:
p The two heterodyne filter functions f1 (t) (red) and f2 (t) (blue) along with their temporalprofile f12 (t) + f22 (t) (black dotted line)

88
oscillations (without being constrained to an integer number of periods). Unfortunately for filter
functions listed in Eqn. 5.5 and 5.6 and shown in Fig. 5.4, there are only few oscillations before
the filter decays. Viewed in the frequency domain, this occurs because, the detuning |∆| is notmuch larger than the cavity decay rate κ. As a consequence the two functions are not orthogonal
for an arbitrary phase φ. Fortunately, there is a particular phase φ which makes an orthogonal
pair of filter functions. Because the convolutions in the filter function are performed numerically, I
numerically solve for the orthogonal φ. I use this orthogonal pair of filter functions f1 (t) and f2 (t)(Fig. 5.4) for my heterodyne photon detection with the JPA.
For well behaved arbitrary two-quadrature filter functions, |∆|  κ. Because |∆| is limitedby the JPA bandwidth BWJPA to be amplified, I can cast this condition in terms of the JPA
Bandwidth as:
BWJPA ≥ ∆  κ(5.7)

for well behaved arbitrary two-quadrature filter functions. In part because Eqn. 5.7 is not true
for my heterodyne measurements, I optimized the photon generation and when performing thehomodyne measurement in the next section (Section 5.3). However, I continue with heterodyne
measurements to demonstrate the direct detection of the single photons, before moving on tohomodyne detection and characterizing the entire experiment.

5.2.2

Comparison to qubit state

I use the JPA backed heterodyne measurements with temporal filters to directly observe
photon power exiting the cavity as a result of the blue sideband pulse. As I saw in Section 2.1.3,
creating the single photon state does not shift the average quadrature measurement, but does triple
the noise power, or quadrature variance. Therefore, I expect to see the photon signal in var(Xi ).
Because driving the blue sideband transition causes oscillations between the |0, gi and the |1, ei
states, I expect creating a photon and exciting the qubit to be correlated. Indeed, I demonstrate that
there is an increase in quadrature variance correlated with excited state probability by observing
Rabi-like oscillations in var(Xi ) and Pe in response to a blue sideband drive.

89

1.13

pulse length (ns)var(X1) + var(X2)

250

1
0
250

excited state
probability

pulse length (ns)

80%

0%

0
5.066

5.076
pulse frequency (GHz)

Figure 5.5: Measurement of var(Xi ) (above) and Pe (below) in response the a blue sideband pulse
drive as the frequency and duration of the pulse is adjusted. I observe correlated, Rabi-like oscillations in var(Xi ) and Pe .

90
I implement a protocol similar to the one used to measure Rabi-like oscillation on to blue
sideband. I modify this protocol by alternating between reading out the qubit and directly measuring photons exciting the cavity with the JPA (Fig. 5.5). In this measurement, I apply a fixed
amplitude drive pulse for some duration, at some frequency. Then, I either perform a single shot
readout on the state of the qubit, or I measure Vm (t) with the JPA. From the quadrature measurements, I apply the quadrature filters (Fig. 5.4) to Vm (t) in order to extract both quadrature values
according to Eqn. 5.4. I repeat the protocol 30,720 times at each drive pulse frequency and length,computing the variance var(X1 ) + var(X2 ). For the qubit measurements, I repeat the protocol 3840
times, calculating the probability of being in the excited state.
In both Pe and var(X1 ) + var(X2 ), I see Rabi-like oscillations of the blue side band that are
correlated. These measurement show I can generate and detect photons using the qubit-cavity
system. The fact that the photon power oscillates correlated with Pe indicates that the measured
increase in var(X1 ) + var(X2 ) is due to single photons as the system oscillates between |0, gi and|1, ei state. That is, I directly measure single, propagating photons. Moreover, the measurements
demonstrate I can simultaneously control the generation of single photons with the blue sideband
while detecting the propagating mode with the JPA.

5.2.3

Density matrix reconstruction

The density matrix ρ is the most complete description of the state of a system, including
both statistical and quantum uncertainty. It describes any statistical combinations of pure quantum states. To fully characterize my photon source, I want to determine the density matrix of the
propagating state that the photon source creates. By analyzing the statistics of many two quadrature measurements of identically prepared photon states, I can infer the density matrix of the
propagating photon state. Using this reconstructed density matrix, I can learn about the qualityof my photon generation and detection.
I begin by repeatedly generating single photons. I implement a protocol where I apply
a calibrated π-pulse on the blue sideband. Then, I perform a heterodyne measurement on the91state exiting the cavity with the JPA. I apply the two-quadrature filter functions (Fig. 5.4) tothe measurements to get a pair of measured quadrature values (X1 , X2 ). I repeat this protocol
5,484,800 times to build up statistics on measured quadrature pairs. In Fig. 5.6, I plot the twoquadrature histogram cast as a probability distribution. To act as a calibration or reference, I
make two-quadrature measurements of the vacuum state. I repeat the two quadrature-heterodyne
measurement 5,484,800 times without the blue sideband pulse. These non-pulsed measurements
are also plotted as a histogram in Fig. 5.6. For comparison, the probability distribution for ideal
measurement of a pure photon state is shown in Fig. 2.3.
As discussed in Section 2.3.4, the probability distribution of a two quadrature measurementof a state is that state’s Q-function:
P (X1 , X2 ) = Q(α) =1hα| ρ |αi .
π

(5.8)

Here, α is the complex coherent state amplitude (α = X1 +iX2 ). The Q-function is a representationof the density matrix. By using Eqn. 5.8, I can use the measurements of vacuum to calibrate the x
and y-axes in Fig. 5.6. With the calibrated axes, I can use Eqn. 5.8 to extract the density matrix
elements for the measured photon state.
To facilitate these application of the Q-function, I can rewrite it terms of density matrix ele√P2
ments in a photon basis. Writing the coherent state in a photon basis |αi = e−|α |/2 n (αn / n!) |ni,
I substitute into Eqn. 5.8 getting0Q(α) =

1 −|α2 |/2 X α∗nαn2√ hn| ρ |n0 i e−|α |/2 √ .
e
π
n!
n0 !
0

(5.9)

n,n

Writing the density matrix as a sum over density matrix elements in the photon basis ρ =P
ij |ii ρij hj|, I write the Q-function as a linear superposition of density matrix elements. The

92

0.4

X1

0.3
0.2

0

0.1

Photon Q (X1, X2)

2

-2

-2

0
X2

2

0

X1

2

0.3

0

0.2
0.1

-2
-2

0
X2

2

Vacuum Q (X1, X2)

0.4

0Figure 5.6: Histograms from two quadrature measurement of the photon state (top) and the vacuum
state (bottom) plotted as the Q-function ( probability density). The photon’s distribution is slightly
flattened and broadened compared to vacuum.

93
Q-function now becomes2

P (X1 , X2 ) = Q(α) =ρ00 π1 e−|α |+ρ01 π1 e−|α | α+ρ10 π1 e−|α | α∗(5.10)

2

2

2+ ρ11 π1 e−|α | |α2 |+...

(5.11)

This form of the Q-function shows that the Q-function is linear in density matrix elements ρij .
Moreover, it shows a clear mapping from density matrix elements to phase space (two-quadrature)probability density.
To calibrate the quadratures, I fit the vacuum probability distribution to the probability
distribution of the vacuum state using the measurement chain’s gain as the only free parameter.
The vacuum state probability distribution is given by Eqn. 5.10 when ρ00 = 1 and all other elements
equal zero. Specifically, the measured vacuum probability distribution is fit according to Q00 (Gα)
where G is gain and Q00 is the vacuum state probability distribution. Dividing the measured
quadratures in both the vacuum and photon measurement by G calibrates the quadratures in unitsof quanta.
Once I have the photon probability distribution calibrated in units of quanta, I can fit the
distribution to according to Eqn. 5.10 to extract the desired density matrix elements. The fitting
is a linear fitting to the density matrix elements ρij using a two photon basis (all combinationsof i = 0, 1, or 2 and j = 0,1 or 2). That is, I fit to both diagonal and off-diagonal elements.
A two photon basis is deemed sufficient, because the two component ρ22 is negligible. I apply
the constraint that the trace of the density matrix tr(ρ) = 1. A line cut through the center ofthe photon data and the fit are shown in Fig 5.7. The magnitude of the photon density matrix
elements extracted from the fit are shown in Fig 5.8.
From measured density matrix (Fig 5.8), I can see that I have generated a state whose single
photon element is |ρ11 | = 0.12. I generate and detect a mixed state which is 12% single photon

94

Q (X1,0)

0.4

0.2

0

-2

0
X1

2

Figure 5.7: A line cut through the center of the measured single probability density (markers) withthe best fit to density matrix elements (solid line).

95

Figure 5.8: Reconstructed density matrix elements |ρij | of the measured photon state using heterodyne detection.

96
and reminder is vacuum. This density matrix is consistent with ideal photon generation (ρ11 = 1)
that is measured with 12% efficiency (Section 2.3.5). This efficiency is relative to ideal heterodyne
detection. In other words, it does not include the added vacuum noise all heterodyne detectors
must add. In reality, I likely generated a propagating mixed state ρ = (1 − p) |0i h0| + p |1i h1| whose
single photon component is some p. This mixed state is measured with some efficiency ηm (relative
to ideal two quadrature detection) such that the product pηm = 0.12.
This density matrix confirms that I have successfully generated single photons with the blue
sideband of a qubit-cavity system and detected these photons with the JPA. I have measured amodest signal photon component ρ11 = 0.12. As a consequence, I want to optimize my generationand detection when I move on to homodyne detection.

5.3

Optimized photon generation and homodyne detection
I optimize my detection and operate the JPA in a homodyne detection scheme for two main

reasons. One, homodyne detection adds less noise than heterodyne. Two, using one filter function in
homodyne detection side steps questions of making arbitrary two-quadrature filter function present
in my heterodyne detection. Similar to my heterodyne measurements, I reconstruct the density
matrix ρm of the state I detect to characterize the photon source and detection. In addition, I
independently characterize the system, compare my measurements to my expectation, and findagreement within uncertainty.
I make several changes to optimize the generation and detection. To allow for an independent
characterization of the measurement efficiency, a switch can connect the amplifier chain to a thermal
noise source (Fig. 5.9). I cable the experiment to minimize the loss between cavity and JPA. I movethe filters, remove one circulator, and minimize microwave cable length and number of connectors
to reduce the loss between the cavity and JPA to 1.6 dB as measured at room temperature. (SeeAppendix A for a detailed description of the photon generation and detection apparati.) However,removing the circulator makes the qubit-cavity system less isolated from the JPA, and therefore
more susceptible to amplifier backaction. Additionally, I optimize the quadrature filter function

97

Figure 5.9: Simplified schematic for photon generation. The drive and measurement microwave
generators couple to the input of the qubit cavity system where the measurement tone excites thecavity in order to infer the qubit state, and the drive tone manipulates the state of the qubitcavity system. A switch can connect the amplifier chain to a thermal noise source allowing for an
independent characterization of the measurement efficiency.

and operate the JPA as a homodyne detector. All of which should improve measurement efficiency.
Lastly, I implement a more sophisticated generation protocol to improve the purity the propagating
photon state.

5.3.1

ProtocolTo implement efficient photon generation, I still adopt a protocol based on a blue sideband
π-pulse, but with the addition of qubit measurements. The measurements allow me to select only
the instances when a photon is created, thereby improving the fidelity of photon generation [72].
One measurement occurs before the blue sideband drive pulse and a second occurs after the photon
has decayed [Fig. 5.11(a)]. I select on the initial measurement that project in the |gi state, toinsure that the qubit starts in the |gi state as required for photon generation. This selection
removes approximately 6% of the trials, which are primarily due to the steady-state excited state
population of the qubit. Secondly, I select on instances when the second measurement projects thequbit into the |ei state. Combining both these selections ensures that the blue sideband π-pulse
successfully flips the qubit from |gi to |ei and thereby creates a photon. Additionally, the second
selection ensures that the photon decays at the excited state cavity frequency. The second selection
removes approximately 26% of the instances, primarily due to the qubit decaying between the drive
pulse and the selection. A smaller contribution is due to blue sideband pulse infidelity.

98

Figure 5.10: Energy level diagram for photon generation and calibration.

99

(a)

var(Vm ) (mV2 )hVm i (mV)controldrivemeasure30 (b)
20
10
0

140

(c)

130
120

f(t)

(d)

−6

−4

−2

0

2

4

6

t (µs)

Figure 5.11: Timing diagram for the experiment. (a) A timing diagram for the photon creationand measurement sequence. The drive tone (black) either creates a photon or excites the qubit
for the vacuum calibration. Qubit readout tones (red) are used before and after the drive. (b)The mean voltage hVm i(t) measured from 7000 individual time traces with the drive at the blue
sideband frequency (solid blue) or at the qubit’s frequency (dashed green). (c) The variance of theindividual measurements var(Vm )(t) with the drive at the blue sideband frequency (solid blue) andat the qubit frequency (dashed green). In blue, the photon power can be seen following the blue
sideband pulse at t = 0 µs. For the control in green, no such signal is seen. (d) The mode matching
function f (t) to extract the quadrature measurement of the propagating mode.

100
Because the qubit measurement must preserve the qubit in its ground state for photon creation, the measurement must be quantum non-demolition (QND). Furthermore, because qubit
readout occurs both before and after photon detection is to take place, the qubit readout must becompatible with photon measurement. To satisfy these two requirements, I use a dispersive transmission readout with the JPA [31, 32, 63] (Section 4.1.1). I operate the JPA with its narrow-band
gain centered on the qubit-excited cavity resonance frequency. This arrangement allows the JPA
to be used both for qubit readout and photon homodyne measurement.
To provide an accurate calibration for the experiment, I modify the photon generation protocol the minimal amount that ensures no photons are generated. I replace the π-pulse on theblue sideband transition with a π-pulse on the qubit transition (Fig. 5.10 in green). This control
sequence prepares the qubit cavity system in the |e, 0i state rather than the |e, 1i state; thus, thecavity does not emit a photon. All other aspects of the protocol, including the qubit state and thedata processing procedure are common to both protocols.

5.3.2

Homodyne detection and tomography

To characterize the propagating state, I perform tomography on the output mode of the cavity
to determine its density matrix ρm . In my tomography procedure, I measure a single quadrature of
the cavity output field during the photon generation protocol. I can reconstruct the density matrix
by repeating the generation and measurement protocol many times and at several different valuesof JPA carrier phase. Because I choose to create a single photon, which has a phase independent
density matrix, I unlock the phase between the generation pulse and the JPA carrier thus sampling
all phases uniformly. From a histogram made from many quadrature measurements I extract the
diagonal elements of the density matrix written in the photon number basis. If there were any
off-diagonal elements, these would vanish due to the phase averaging.
I operate the JPA as a homodyne detector with its gain centered on the frequency of thephoton. Like the heterodyne detection, the output of the JPA is mixed down to yield a time trace
Vm (t). Because the JPA’s gain is not detuned from the signal, then its measurement Vm (t) contain

101
information about a single quadrature, where the phase of the quadrature is set by the JPA’s pump.
I form one measurement set by repeating the pulse sequence shown in Fig. 5.11(a) 7000 times (withthe π-pulse applied to the blue sideband). The temporal envelope of the photon can be seen in thevariance of the set var(Vm )(t) at time t = 0 µs [Fig. 5.11(c)]. As expected for a single photon, no
feature is present in hVm i(t) at t = 0.

5.3.2.1

Quadrature measurement

To complete the reconstruction, each Vm (t) must be processed to yield a single quadrature
value of the propagating mode. From this raw data, I make an estimate of one quadrature of themode emitted by the cavity when the drive pulse is applied. I desire a mode matching function f (t),that weights the time average of Vm (t) to produce an optimum estimate of the quadrature value.
For infinite measurement bandwidth, I expect the optimum f (t) to be a decaying exponential pulse
with decay constant κ/2 as defined for output mode Aout (Eqn. 2.7) [73] and a rise time equal tothe duration of the drive pulse (150 ns). Due to the finite JPA bandwidth, I anticipate that f (t)is found by convolving the infinite bandwidth optimum by the effective filter’s impulse responseh(ωLPF , t) [74]. In practice, I write




0
t<0



f (t) = h(ωLPF , t) ∗cos(2πt/Tr ) 0 ≤ t ≤ Tr





 e−(t−Tr )κ/2 Tr < t
as a function of three parameters (rise time Tr , decay constant κ, and low pass filter cutoff
ωLPF = BWJPA /2) and adjust these to minimize the zero-photon contribution of the density matrix extracted from the data set. I perform this determination of f (t) once, using photon creationand calibration data sets that are not used in subsequent analysis. As seen in Fig. 5.11(d), f 2 (t)looks like the photon creation variance [Fig. 5.11(c)], indicating that my optimization of f (t) has
produced a sensible result.
Extracting the quadrature value is complicated by the presence of a dc offset in the Vm (t)which drifts during the acquisition of a full data set. To remove drifts in the dc offset from the

102
quadrature measurement, I perform a linear least-squares fit using a measurement of the background voltage in addition to the mode matching function. For each trace I assign an uncalibrated
quadrature value Vq by minimizing
Z h
i2
J =
Vm (t) − (Vq f (t) + Vdc b̃(t)) dt(5.12)

over Vq where Vdc is the dc voltage and b̃(t) is the windowing function that defines when the
dc offset is measured. The windowing function is a piecewise constant function that is nonzero
during most of a 56 µs interval that includes the photon generation protocol. But during the qubit
measurements, b̃(t) = 0. The result of the cost function minimization is an analytic expression for
an individual quadrature measurement
Z
Vq =

Z
Vm (t)f (t)dt −Z
Vm (t)b̃(t)dtb̃(t)f (t)dt.

(5.13)

This expression is applied to each Vm (t) resulting in a set of uncalibrated quadrature measurementsof the propagating mode.
A histogram of this set of measurements is shown as the narrow blue bars in Fig. 5.12. I generate a calibration data set following the same procedure as was used to generate the measurement
set, but using the control protocol (π-pulse on the qubit transition). Indeed, when no photon iscreated, no extra variance is visible in var(Vm )(t) at t = 0 [Fig. 5.11(c)]. Reducing each trace toquadrature values, I find the histogram shown in Fig. 5.12.
To calibrate the quadrature values, I fit the histograms of the calibration data sets to Gaussian
distributions using the gain of the full measurement apparatus as the only fit parameter. But due
to the fact that a small fraction of the JPA output is injected back into the cavity through thefinite isolation of the circulators (determined in Section 5.3.2.2), I do not assume that the cavity isin its vacuum state. Rather, I expect to prepare the cavity in a mixed, squeezed state with n̄  1.
Although the large isolation of the circulators ensures that the cavity’s squeezed quadrature has
almost exactly vacuum variance, the amplified quadrature can have variance measurably larger
than vacuum, particularly at larger values of JPA gain. Because there is a fixed, but unknown

103

0.9
0.8

probability density

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−1.5

−1

−0.5

0

0.5

1

1.5

X
Figure 5.12: Histogram of a quadrature measurement set of single photons (narrow blue bars) andthe no-photon control (wide green bars) with the JPA gain at 29 dB. The histograms are plotted asa probability density (bars) and are fit by the expected distribution for a diagonal density matrix
with a 3 photon Fock basis (solid lines).

104
phase relationship between the quadrature I measure and the squeezed quadrature of the cavity,
I calibrate assuming I measure the squeezed quadrature and assuming I measure the amplified
quadrature. I use these two cases to bound the systematic uncertainty in the calibration of thevacuum variance, where I use the convention that one quadrature of a vacuum state has variancevar(X) = 1/4.

5.3.2.2

JPA backaction

To quantify the cavity photon variance due to JPA backaction, I study the qubit dephasing [75]. As seen in the dispersive Hamiltonian for the qubit cavity system in Eqn. 2.39, thefrequency of the qubit depends on the number of photons in the cavity. Therefore, a varying number of cavity photons (var(n) > 0) dephases the qubit. Because the qubit resonance frequency is
far-detuned from the cavity, I treat the squeezed cavity field as dephasing the qubit with a thermal
distribution. From Ref. [76], in the strong dispersive regime the dephasing rate when there are N
photons in the cavity isΓN = κ [(n̄ + 1)N + n̄(N + 1)] ,(5.14)

where κ is the coupling rate to a thermal bath with average number of cavity photons n̄. Summingover a thermal distribution of N defined by n̄ (Eqn. 2.23), I calculate a photon dephasing rate
Γ = Γ0 + κ[2n̄ + 2n̄2 + O(n̄4 )](5.15)

in terms the intrinsic dephasing rate Γ0 , n̄ and κ. Note that the additional dephasing rate is indeed
proportional to the photon variance Γ − Γ0 = 2κ × var(n) in this case (Eqn. 2.25). Measurementsof the qubit dephasing rate (Γ = 1/T2∗ ) over the range of JPA gains are shown in Fig. 5.13(a).
For the resulting photon distribution in the cavity, I assume that n̄ follows a model characterized by a single isolation parameter L, which characterizes the fraction of JPA output misdirected
into the cavity asn̄ = (1/4)L(GJPA − 1),(5.16)

105

Γ/2π (kHz)

200
(a)

100

0
(b)

n̄

0.1
0.05
0

20

25
JPA Gain (dB)30

Figure 5.13: Quantifying backaction. (a) The qubit dephasing rate (Γ = 1/T2∗ ) is shown (points)
over a range of JPA gains along with a fit (line) to Eqn. 5.15 over the range of JPA gains (maintext). (b) The solid line shows the best estimate of the average photon number in the cavity n̄(uncertainty in dashed lines) due to the JPA at each gain used in the experiment.

106
where GJPA is the JPA gain. I fit this model substituted into Eqn. 5.15 to measurements of Γ,
extracting L = (2.1±0.1)×10−4 (-37 dB) and Γ0 /2π = 40±2 kHz . The isolation is consistent withthe specifications for the two commercial circulators between the cavity and the JPA (Fig. A.1).
From this model, I find the average intercavity photon number due to JPA gain shown in Fig. 5.13(b)plotter over a range of JPA gains.

5.3.2.3

Reconstructing density matrix

Finally, I complete the tomography by fitting the histograms to probability distributions fordiagonal elements of the density matrix in a 3 photon basis yielding one measurement of the density
matrix. A 3 photon basis is sufficient because the three photon component ρ33 is indistinguishable
from 0. Because I am restricting myself to a density matrix where the only the nonzero elements are
diagonal elements ρii , I can treat this state as a probabilistic combination of pure photon states.
As a results, I can write the quadrature probability distribution as a linear combination of theprobability distribution of pure photon statesPρii (X) = ρ00 P0 (X) + ρ11 P1 (X) + ρ22 P2 (X) + ρ33 P3 (X).

(5.17)

I can compute the probability distribution of the pure photon states Pn (X) either from the Wigner
function (Sections 2.3.2 and 2.3.3), or from the wave functions of the Fock states
Pn (X) = Ψ∗n (X)Ψn (X).

(5.18)

Written explicitly, I getq

2 −2X 2πe(5.19)

2 −2X 2 2X
πe

(5.20)

2 −2X 2
(16X 4 − 8X 2 + 1)πe(5.21)

2 −2X 2
(16X 6 − 24X 4 + 9X 2 ).
πe(5.22)

P0 (X) =P1 (X) =4
1
2

P2 (X) =P3 (X) =

2
3

q

q

q

Thus, I fit the probability distributions from each data set to Eqn. 5.17 to extract the diagonal
density matrix elements.

107

0.8

ρ 00

0.7
0.6

ρ ii

0.5
0.4

ρ 11

0.3
0.2
0.1

ρ 22

0
18

20

22

24
26
JPA Gain (dB)

28

30

32

Figure 5.14: Diagonal density matrix elements determined from fitting the quadrature histogram
over a range of JPA gains from 17 dB to 33 dB. The error bars are determined from the standard
deviation of the mean using 8 sets of the quadrature measurements at each gain; (only 4 sets areused at 17 and 33 dB). The systemic uncertainties are indicated with dashed lines.

108
Each density matrix element is determined as the average value of multiple realizations ofeach measurement set. To optimize the photon measurement I find the density matrix elements
over a range of JPA gains from 17 to 33 dB (Fig. 5.14). These density matrix elements are sensitive
to all imperfection in the photon generation and detection. For ideal generation and detection, theρ11 component would be 1 and all others 0. Instead, the single photon component has a peak valueof ρ11 = 0.361 with ± 0.005 statistical and ± 0.005 systematic uncertainties at a JPA gain of 29 dB.
However, at this gain there is a two photon component of ρ22 = 0.027 with ± 0.005 statistical and± 0.015 systematic uncertainties. Considering the two photon generation relative to single photon
generation, I have 2ρ22 /ρ11 ≈ g2 (0) = 0.32 ± 0.07 with a 0.15 to 0.41 systematic uncertainty bound(Fig. 5.15). [For comparison, g2 (0) = 1 for any coherent state.] By decreasing the JPA gain to
17 dB, the two photon component becomes ρ22 = 0.005 ± 0.003 with a single photon component
dropping to ρ11 =0.247 ± 0.004, giving g2 (0) = 0.15 ± 0.08. These results show that an increasein the JPA gain improves the measurement efficiency.
However, as I saw in Section 5.3.2.2, increasing the JPA gain also increases the measurement
backaction. At my peak ρ11 (29 dB of JPA gain), there is an average photon n̄ = 0.041 ± 0.003 inthe cavity due to JPA backaction, likely accounting for most of the two photon component in themeasured density matrix. The deleterious effects of this backaction can be seen in both the larger
systematic uncertainty at larger JPA gain and the increased probability of creating two photons
instead of just one.

5.3.3

Characterizing photon generation and efficient homodyne detection

To understand the limitations of single photon generation and detection, I characterize theexperimental imperfections over a range of JPA gains. In particular, I characterize the internal lossof the cavity and measurement efficiency. This characterization gives a prediction for the state that
I create and an expectation for how efficiently I can measure it. I compare my expectation withmy measurements to validate my understanding of the photon generation process.
I make a prediction of the state I expect to measure if the photon generation protocol created

109

0.5

g2 (0)

0.4

0.3

0.2

0.1

0
18

20

22

24
26
JPA Gain (dB)

28

30

32

Figure 5.15: The values of g2 (0) computed from the density matrix elements. Both the single
standard deviation statistic uncertainty (error bars) and the systematic error bounds (dashed lines)
are calculated from the uncertainties the in density matrix elements. The values of g2 (0) represent
likelihood of detecting two photons compared to one. For a pure photon state g2 (0) = 0 and for acoherent g2 (0) = 1.

110
a pure single photon state in the cavity. First, I form an expression for the density matrix of theoutput mode ρout by accounting for the coupling of the cavity field to unmeasured ports. Dueto the relative coupling rates, a cavity photon has a κout /κ probability of decaying to the output
port, where the coupling rate to the output port is κout /2π = 300 kHz and the total decay rate isκ/2π = 410 kHz. I would therefore expect to generate a propagating state ρout characterized byρout =

κ − κoutκout|0ih0| +|1ih1|
κκ

(5.23)

from Section 2.3.5 with the ratio κout /κ cast as η.

5.3.3.1

Independent measurement characterization

Next, I form an expectation for how well I can measure ρout by independently characterizing
the measurement inefficiency. In order to independently determine the measurement efficiency, I
inject states of known variance into the measurement chain. As I adjust the input variance into
the measurement chain, I determine the additional variance Nadd introduced by the measurement.
The source of known variance is a 50 Ω resistor on a variable temperature stage connected by aswitch to the JPA input (Fig. A.1). I perform the determination of Nadd at three different JPA
gains– 20, 25, and 30 dB (Fig. 5.16). The data are plotted as output-noise power-spectral-density
Sout against input power spectral density Sin (set by Eqns. 2.24 and 2.27). I extract Nadd by fitting
data in Fig. 5.16 toSout = G(Sin + Nadd ),(5.24)

where G is the gain of the measurement chain [26]. These fits yield three values of Nadd plotted asmeasurement efficiency
ηm = (2Nadd + 1)−1(5.25)[77] in Fig. 5.17 (circles).
To find Nadd at other values of the JPA gain, I use a model that decomposes Nadd into
contributions from the JPA itself NJPA and from the remaining measurement NHEMT . Adopting
this added noise model, I interpolate ηm over the range of JPA gains used in this experiment. In

111

Figure 5.16: Thermal sweep data for the JPA operated at 20 (green squares), 25 (blue circles) and
30 dB (red crosses) gains. The data are fit to Eq. 5.24 (solid lines). The input noise source used in
the thermal sweep is a 50 Ω resister whose temperature is adjusted from 79 mK to 900 mK. This
thermal noise power is expressed in units of quanta at 5.8 GHz on the x-axis.

112

0.55

ηm

0.5

0.45

0.4

0.35

18

20

22

24
26
JPA Gain (dB)

28

30

32

Figure 5.17: The measurement efficiency ηm is determined from a thermal sweep at three JPA gains(circles). This quantity is interpolated over the range of JPA gains by fitting the data points toEqn. 5.27 with the uncertainty in dashed lines.

113
this model I assume that NJPA and NHEMT are both constant when referenced to their respective
amplifier inputs. The output power spectral density is thenSout = G(Sin + NJPA +

NHEMT
),
GJPA

(5.26)

where GJPA is the gain of the JPA. From Eqn. 5.26 the added noise can be written as
Nadd = NJPA +

NHEMT
,
GJPA

(5.27)

forming the model I use for interpolation [17]. The three added noises are fit according to Eqn. 5.27.
I find NJPA = 0.39 ± 0.03 and NHEMT = 18 ± 5. Using these added noises, ηm is plotted over therange of JPA gains in Fig. 5.17 (solid line) according to Eqns. 5.25 and 5.27.

5.3.3.2

Measurement comparison

I form an expected density matrix ρexp by considering measurement efficiency and cavity loss.
Following Section 2.3.5, my expectation for the density matrix I should measure for pure cavity
photon generation isρexp = (1 −

κout
κout
ηm )|0ih0| +ηm |1ih1|.
κκ

(5.28)

Finally, I compare my measurement to expectation by computing the fidelity1 of ρm with respect
to ρexp . I find that they are identical (unit fidelity) within uncertainty (Fig. 5.18). This agreementshows that I am able to accurately and independently characterize the measurement inefficiency
and undesired cavity loss. (The measured two-photon component contributes negligibly to theinfidelity.) For comparison, I compute the fidelity of the measured density matrix ρm with respect
to the density matrix of a single photon (Fig. 5.18), which quantifies my combined ability to generateand detect single photons. This fidelity has a peak value of F = 0.600 ± 0.008 also at 29 dB JPA
gain [78].

1

The fidelity of a state A (ρA ) with a state B (ρA ) is F = trp√√ 
ρB ρA ρB .

114

1

fidelity

0.9
0.8
0.7
0.6
0.5
20

25
JPA Gain (dB)30

Figure 5.18: The fidelity of the measured density matrix with respect to: an ideal single photon(blue squares), and the density matrix I expect to measure given pure photon generation in thecavity (black points). Here, the statistical one-standard-deviation uncertainties are plotted as
errors bars. The systematic error plotted in dashed lines is calculated from both the systematic
uncertainty in ηm and in the density matrix elements.

Chapter 6

Conclusions and outlook

The protocol I have demonstrated for generating single microwave photons on demand iswell suited for transferring quantum states to narrow bandwidth signal processing modules, suchas certain types of electro-optic convertors that are under development [20]. In particular, thecompatibility of the protocol with fixed-frequency, highly coherent qubit-cavity systems ensuresthat the photons can be emitted into narrow frequency windows. I have characterized the photon
source and its homodyne detection, and understand their imperfections. Because an ideal single
photon source generates one single photon every time, and never generates two or more, the qualityof the photon generation and detection can be quantified by the probability of measuring a single
photon ρ11 (Fig. 5.14), and the probability of measuring two photons ρ22 , or by g2 (0) (Fig. 5.15). I
have independently characterized the system’s measurement efficiency ηm and cavity loss 1−κout /κ
and find that they account for the measured photon state within uncertainty using state fidelity
as the metric (Fig. 5.18). The non-unity measurement efficiency and non-zero cavity loss yieldpractical, but not fundamental limitations on ρ11 , and may be improved in future implementationsof the single photon source.
These measurements highlight the compromise between backaction and measurement efficiency. My detailed characterization of the protocol reveals an undesirable backaction of the measurement apparatus onto both the qubit and the cavity. It is possible to reduce the backaction
through the use of more commercial isolators (Fig. A.1), however, doing so increases the loss between the cavity and JPA. With advances in low loss isolator elements [79, 80], it may be possible

116
to mitigate this backaction through the use of more isolation while at the same time reducing losses
in detection. Otherwise, a more complicated protocol that pulses on the JPA pumps only whenthe measurement is desired may minimize the backaction. Overall, the measurements presented
here highlight the importance of isolator elements in quantum information processing, and provide
a method to transfer information between qubit-cavity systems and itinerant microwave fields.

6.1

Outlook
There are two simple extensions of this single photon generation and detection experiment.

First, one could use the qubit-cavity system to generate flying qubits or arbitrary combinations of|0i and |1i. To form a flying qubit, the qubit state A |gi + B |ei is mapped onto the propagating
photon state A |0i + B |1i. Second, one could send the propagating photon state to mechanicalmodule.
To generate a flying qubit from the system, one must employ a more complex protocol thanjust a single blue sideband pulse. For example, if one performed a π/2-pulse with blue sideband,√then the system would prepared in the Ψ = 1/ 2 |g, 0i + eiφ |e, 1i . The photon is entangled with
the qubit. As the photon decays to the output modes, this state could transmit entanglement, butdoes not behave like a flying qubit. If only the output mode were measured, then its reduced density
matrix would be found by tracing over the qubit part of the full density matrix. (Section 2.3.5).
The density matrix of the output would then be ρout = 1/2 (|0i h0| + |1i h1|), a maximally mixed
state without any phase information.
One could implement a two step protocol to avoid this entanglement and generate a flying
qubit. In the first step, the qubit state is defined. A qubit pulse prepares the qubit in the Ψ1 =A |g, 0i + B |e, 0i state. In the next step, a π-pulse on the blue sideband swaps the |g, 0i term
to |e, 1i, but leaves the |e, 0i unaffected. After the blue sideband π-pulse, the system is in theΨ2 = A |e, 1i + B |e, 0i state. The photon then decays out as an unentangled, pure state. Because
the qubit is in the |ei state in both terms, |ei can be factored out. The output state then looks likeΨout = B |0i + A |1i, a flying qubit with coefficients A and B swapped compared to the qubit state

117
in the first step. With this protocol, one could create a source of flying qubits to send to other
qubit modules or measure directly with the JPA. However, care must be taken using a common
phase reference for all pulses and any measurements.
After demonstrating the photon source (or potentially a flying qubit source), it could be
used to prepare other modules, such as mechanical circuits, in nonclassical states. Through theuse of coherent control tones, the state of a propagating, incident modes can be mapped onto thephoton state of the a mechanical mode of the oscillator [16]. By sending a single photon into themechanical module and using coherent control tones, a mode of the mechanics could be prepared in
an n = 1 phonon state. Such a demonstration would be landmark achievement of quantum physics
as it would prepare a macroscopic mechanical object in a profoundly nonclassical state.

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Appendix A

Description of photon generation and homodyne detection apparati

The experimental details for the optimized photon generation and homodyne detection (Section 5.3) are summarized by the microwave schematic (Fig. A.1). The experiment is conducted in
an Oxford Triton 200 dilution refrigerator with the qubit-cavity system anchored to the (T < 25mK) base temperature region. The qubit-cavity system is controlled and measured via injecting
tones shown on the left. The strongly coupled output on the right leads to a switch, which either
connects the qubit-cavity system or the thermal load to the measurement chain.
After the switch, circulators route signals into the JPA. The JPA is a nonlinear lumped
element LC resonator pumped by rf power injected using a 20 dB directional coupler. It is operated
with a signal gain from 17 to 33 dB and with an approximate gain bandwidth product of 43 MHz.
The JPA is pumped with two tones in the so called double pump method by modulating a 5.806 GHz
carrier by 240 MHz using an IQ mixer. The carrier is suppressed by the modulation and is further
reduced by a notch cavity filter. The output of the JPA is routed into a HEMT amplifier and room
temperature amplifiers before being mixed down by a second IQ mixer. A copy of the JPA carrier
is used as this mixer’s local oscillator and phase shifted so that the JPA’s amplified quadratureexits the I-port of the mixer. This output is then further amplified and then digitized. An arbitrary
waveform generator determines the protocol timing by triggering the drive tone, the measurement’s
tone, and the digitizer.

125

circ
Tektronix
AWG N3bf

QUINSTAR CTH3f38KC

E8+N&

OPA

switch Radiall RN&3ffy333

measure4K

normal lines

A

OPA8f&3 y3 dB

NbHTi lines

Ecco Eccosorb

Miteq Miteq AMFMy y3 dB

terminator

triggers

25 mK
qubitscavity

drive+3 dB y3 dB

+3 dB

Ecco

switchcirc+3 dB +3 dB
L

I
R
Q

A

+3 dB +3 dB

JPA pump+3 dB

thermalload

modulation

digitizer
OAlazarTech
ATS 7fK+/

HEMT
Miteq Ox+/
yf dB
Ecco

LOHJPA carrier

attenuaters

KuL KL+N3 33388

I
R L
Q

OPA
JPA
...

filtercavityphase shifter

Figure A.1: The microwave schematic for the experiment.