ABSTRACT
Title of dissertation: NONLINEAR OPTICS QUANTUM
COMPUTATION AND QUANTUM
SIMULATION WITH CIRCUIT-QED
Prabin Adhikari, Doctor of Philosophy, 2014
Dissertation directed by: Dr. Jacob M. Taylor
Joint Quantum Institute
University of Maryland
Superconducting quantum circuits are a promising approach for realizations of
large scale quantum information processing and quantum simulations. The Joseph-
son junction, which forms the basis of superconducting circuits, is the only known
nonlinear non-dissipative circuit element, and its inherent nonlinearities have found
many different applications. In this thesis I discuss specific implementations of these
circuits. I show that strong two-photon nonlinearities can be induced by coupling
photons in the microwave domain to Josephson nonlinearities. I then propose a
method to simulate a parent Hamiltonian that can potentially be used to observe
fractional quantum Hall states of light. I will also explore how superconducting cir-
cuits can be used to modify system-bath couplings to emulate a chemical potential
for photons. Finally, I consider the limitations of devising a scheme to couple su-
perconducting circuits to trapped ions, and consider the challenges for such hybrid
approaches.
NONLINEAR OPTICS QUANTUM COMPUTATION AND
QUANTUM SIMULATION WITH CIRCUIT-QED
by
Prabin Adhikari
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2014
Advisory Committee:
Dr. Mohammad Hafezi, Chair
Dr. Jacob M. Taylor, Co-Chair/Advisor
Dr. Frederick C. Wellstood, Co-Chair
Dr. Eite Tiesinga
Dr. Dionisios Margetis
c© Copyright by
Prabin Adhikari
2014
Acknowledgments
The past seven years at the University of Maryland have been intellectually
stimulating, challenging, and very rewarding. During my time here, I have learned
a lot of physics and acquired many skills necessary for scientific research. More
importantly, I feel that I have acquired the skills for reasoning and critical thinking,
which are important in many areas of life besides the academic world. I have also
gotten to know many interesting and hardworking people from whom I have ben-
efited greatly. I am very grateful to everyone I have been in touch with for their
unwavering support in making this possible.
First of all I would like to thank my advisor Dr. Jacob Taylor for his mentor-
ship. He has consistently encouraged me to work hard and supported me throughout
my time as a graduate student. He has provided me everything I needed– from logis-
tical resources, scientific advice and moral support, to grant support for participation
in very illuminating conferences and schools. I am very grateful for the wonderful
group outings, kickball games, and informal meetings both inside and outside the
University that enabled our research group as a whole to communicate and dissemi-
nate ideas. Furthermore, I want to acknowledge the support of the Physics Frontier
Center, funded by the National Science Foundation, without which my research
would not have been possible.
I also want to thank Dr. Mohammad Hafezi whose support has been pivotal in
my research, and my understanding of physics. It has been an honor to collaborate
both with Dr. Taylor and Dr. Hafezi. I have also had the opportunity to do
ii
independent study with Dr. Eite Tiesinga from the JQI, and Dr. Frank Gaitan
from LPS. During my work with them, I learned a lot of new physics besides my
immediate research area. I would like to offer many thanks to them for their valuable
time and feedback.
I am also indebted to a great mentor and a friend of mine, Dr. William
Wallace, who supported me throughout my undergraduate years at St. John’s Uni-
versity. Without him I would not have made it to the University of Maryland. I also
want to thank Dr. Robert Finkel for the guidance he provided me when I was an
undergraduate. I will always cherish the wonderful memories from the four valuable
years at St. John’s University.
I have developed deep friendships during my time at Maryland. First, I want
to acknowledge my office mate Ranchu Mathew for his camaraderie. I will always
cherish the interesting scientific and philosophical discussions that I have had with
him. I also want to thank my past and present group colleagues Dvir Kafri, Xun-
nong Xu, Haitan Zhu, Steve Ragole, and Vanita Srinivasa for their friendship and
their advice on scientific matters. Additionally, I want to thank my experimentalist
friends Varun Vaidya, Rajibul Islam, Elizabeth Goldschmidt, and Baladitya Suri
for enlightening me with experimental knowledge. Outside the University, I want
to acknowledge my friends Jesus Anguiano, James Creznic, and Pradeep Subedi for
their friendship over the past decade.
Finally, I want to thank all my family members in Nepal for their love. Despite
the great distance separating us, my family has been a source of unwavering encour-
agement throughout my life, especially while in America. Specifically, I am very
iii
grateful to my parents Kamal and Kalpana and my two younger brothers Prabal
and Prajesh for their love and support.
My journey over the past seven years would not have been possible without
my wife Clare. I thank her for supporting me all these years with patience and
understanding, and working hard for our future.
iv
Table of Contents
List of Tables vii
List of Figures viii
1 Introduction to Quantum Computation and Quantum Simulation 1
1.1 Church-Turing Thesis and Classical Computers . . . . . . . . . . . . 1
1.2 Universal Quantum Computation . . . . . . . . . . . . . . . . . . . . 4
1.3 Introduction to Quantum Simulation . . . . . . . . . . . . . . . . . . 9
2 Superconducting Circuits and Quantized Hamiltonians 12
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Isolated Josephson Junction . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Quantum Description of an Isolated Junction . . . . . . . . . . . . . 16
2.4 Josephson Junctions Connected to External Circuits . . . . . . . . . . 17
2.4.1 Charge Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2 Flux Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Circuit-QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Nonlinear Optics Quantum Computing with Circuit-QED 31
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Outline of Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 The Circuit Model and Hamiltonian . . . . . . . . . . . . . . . . . . . 36
3.4 Linearization and Quantization . . . . . . . . . . . . . . . . . . . . . 40
3.5 Diagonalization of Linear Hamiltonian . . . . . . . . . . . . . . . . . 44
3.6 Subspace Hamiltonian and Two-Photon Nonlinearity . . . . . . . . . 47
3.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.8 Adiabatic and Non-Adiabatic Loss . . . . . . . . . . . . . . . . . . . 50
3.9 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 51
v
4 Circuit-QED Implementation of the Pfaffian State Parent Hamiltonian 60
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Parent Hamiltonian for the Pfaffian State . . . . . . . . . . . . . . . . 63
4.3 Implementation of Magnetic Field . . . . . . . . . . . . . . . . . . . . 65
4.4 Three-Body Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5 Optimization of Parameters . . . . . . . . . . . . . . . . . . . . . . . 69
4.6 Experimental Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.7 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 A Chemical Potential for Photons 87
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 General Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Circuit-QED Implementation of Parametric Hamiltonian . . . . . . . 91
5.4 Input-Output Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.5 Correlation Functions and Thermal Spectrum . . . . . . . . . . . . . 100
5.6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 103
6 Dynamics of an Ion Coupled to a Superconducting Circuit 107
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Model and Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3 Linearization of the Parametric Oscillator . . . . . . . . . . . . . . . 112
6.4 Time-Dependent Quantum Harmonic Oscillator . . . . . . . . . . . . 114
6.5 Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.6 Derivation of the Interaction . . . . . . . . . . . . . . . . . . . . . . . 117
6.7 Sideband Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.8 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 120
7 Conclusions 122
8 Appendix 124
8.1 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.2 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.3 Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.4 Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Bibliography 137
vi
List of Tables
2.1 The various regimes in which a shunted Josephson junction can operate. 18
vii
List of Figures
1.1 Representation of a quantum system and a quantum simulator . . . . 10
2.1 An SIS junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 A “Mendeleev table” of superconducting circuits . . . . . . . . . . . . 19
2.3 Schematic of a Cooper pair box . . . . . . . . . . . . . . . . . . . . . 20
2.4 Energy levels of a Cooper pair box . . . . . . . . . . . . . . . . . . . 21
2.5 A flux qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Energy levels of various inductively shunted junctions . . . . . . . . . 29
2.7 An LC circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1 Implementation of a two-photon phase gate . . . . . . . . . . . . . . 33
3.2 Implementation of two-photon nonlinearity . . . . . . . . . . . . . . . 35
3.3 Energy levels of the system and schematic of the phase shift protocol 36
3.4 The circuit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 Contour plots of g1 and g2 . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Coupling along the trajectory . . . . . . . . . . . . . . . . . . . . . . 54
3.7 Dressed energy levels of the coupled system and two-photon nonlinearity 55
3.8 The detuning δ and qubit nonlinearity . . . . . . . . . . . . . . . . . 56
3.9 Comparison of the analytical and numerical results . . . . . . . . . . 57
3.10 Comparison of the analytical and numerical frequencies . . . . . . . . 58
3.11 Static and dynamic losses . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Energy levels of the system in the presence of a two and a three-body
interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Implementation of Hp . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 A Josephson junction shunted by an inductive loop . . . . . . . . . . 67
4.4 Plots of U2 and U3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5 Plot of U3 at points where U2 < 5× 10−4 . . . . . . . . . . . . . . . . 78
4.6 Two coupled hybrid superconducting circuits . . . . . . . . . . . . . . 78
4.7 Bosonic constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.8 Potential and wavefunctions . . . . . . . . . . . . . . . . . . . . . . . 80
4.9 Energy levels and wave functions . . . . . . . . . . . . . . . . . . . . 80
4.10 Energy spectrum with nonlinearity . . . . . . . . . . . . . . . . . . . 81
viii
4.11 Matrix elements of φ . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.12 The matrix element 〈0|φ |1〉 . . . . . . . . . . . . . . . . . . . . . . . 83
4.13 Analytical and numerical comparison of the normalized eigenfuctions 83
4.14 Oscillations in H1 and H2 . . . . . . . . . . . . . . . . . . . . . . . . 84
4.15 Oscillations in the three excitation manifold . . . . . . . . . . . . . . 85
4.16 Matrix element 〈m| Nˆ |n〉 for m 6= n . . . . . . . . . . . . . . . . . . . 85
4.17 Plot of the first few matrix elements of Nˆ . . . . . . . . . . . . . . . 86
5.1 Schematic of the system-bath coupling . . . . . . . . . . . . . . . . . 93
5.2 The spectral coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3 The approximate spectral coefficients . . . . . . . . . . . . . . . . . . 105
5.4 The bandwidth of thermal spectrum . . . . . . . . . . . . . . . . . . 106
6.1 Schematic of an ion coupled to an rf-SQUID . . . . . . . . . . . . . . 109
6.2 The effective parametric circuit . . . . . . . . . . . . . . . . . . . . . 114
8.1 A plot of the third order correlation . . . . . . . . . . . . . . . . . . . 127
8.2 A schematic of a transmission line connected to external circuits . . . 129
ix
Chapter 1
Introduction to Quantum Computation and Quan-
tum Simulation
1.1 Church-Turing Thesis and Classical Computers
The key assumption from the modern theory of classical computation can be
summarized by the Church-Turing thesis: Any algorithmic process can be efficiently
simulated using a Turing Machine [1]. A Turing Machine is a device that consists
of a tape, a read-write head, and a set of instructions from an alphabet. The tape
is divided into cells and each cell carries one symbol from the alphabet. The read-
write head moves along the tape according to a given set of instructions and changes
symbols on the tape as it moves. The output of the Turing Machine are the contents
of the tape when the instructions have been completed.
Alan Turing proposed the Church-Turing thesis in a seminal 1936 paper [2].
This thesis cannot be proven, but is supported by empirical evidence. This thesis
essentially claims that what we think of as algorithms can be completely captured
1
by a Turing Machine. Intuitively, an algorithm is simply a set of steps to be carried
out for any given input in order to get a desired output. At the same time Turing’s
thesis advisor Alonzo Church introduced a universal model of computation called
the lambda calculus. He and Turing then showed that the Turing machine and the
lambda calculus were equivalent in their capabilities [3].
Not long after Turing’s paper, the first computers constructed from electronic
components were developed. John von Neumann developed a simple theoretical
model for putting together in a practical fashion all the components required for a
computer to be fully capable as a Turing Machine [4]. Although the early computers
were slow and bulky by today’s standards, after the development of the trasistor by
John Bardeen, Walter Brattain, and Will Shockley in 1947 [5,6], computer hardware
began a long process of becoming faster and more powerful. Computer power has
grown at an amazing pace ever since, so much so that this growth was codifed by
Gordon Moore in 1965 in what is now known as Moore’s law. This “law” states that
computer power will double for constant cost every 18 months. Today, a small hand-
held cell phone has much more computing power than all but the fastest computers
of a few decades ago.
It is clear, however, that we cannot make computing devices arbitrarily small
without altering the physical basis of computation. Traditional or so-called classical
computers are based on Boolean or classical logic. As computer hardware is made
even smaller, quantum effects can become significant. The classical physics that
we use in the macroscopic world starts to break down. For example, in quantum
physics, a system may not have a definite classical state. For example, one may
2
not be able to say whether an electron has a spin up, corresponding to a classical 0
(say), or spin down corresponding to a classical 1 (say). That is, an electron spin,
which is a simple quantum system, can be in a superposition of both up and down
at the same time, and if one measures the spin, the results 0 or 1 are obtained with
equal probability.
Not only that, there is a limit to how many transistors can be fabricated
within a given planar area. The Intel Core I7 processors of today have more than
2.2 × 109 transistors. This type of computation requires energy, and hence, with
many components operating at high computing speeds, one has to start considering
the dissipation of energy [7]. This can be characterized by Landauer’s principle
which states that for every bit of information erased, an entropy S = kB ln 2 is
generated, where kB is Boltzmann’s constant [8]. The power density of modern day
computers is around 130 W cm−2. For comparison, the power density at the surface
of the sun is around 6000 W cm−2 [9]. But the temperature at the surface of the sun
is around 5778 K. A computer on the other hand, needs to be maintained at room
temperature. This suggests that this type of computers cannot be made arbitrarily
fast.
One potential solution to these problems may be offered by quantum compu-
tation, which as the name suggests, uses quantum logic. Quantum computation is
reversible in principle, and seems to offer exponential speedup over classical comput-
ers. However, as Richard Feynman pointed out in 1982 [10], typical quantum sys-
tems cannot be simulated by classical computers using efficient algorithms. Roughly
speaking, an efficient algorithm is one which runs in time polynomial in the “size” of
3
the input. In contrast, an inefficient algorithm typically requires exponential time.
One well-known example from mathematics is the problem of factoring a number
into its prime factors. The fastest known classical algorithms require a time for
obtaining the factor, that is exponential in the number of bits in the binary rep-
resentation of the input. In contrast, there are some quantum algorithms that are
much faster than their classical counterparts [11] including Grover’s search algorithm
and Shor’s factoring algorithm. Grover’s search algorithm [12] can search for an en-
try in an unsorted database consisting of N elements in O(
√
N) time, while Shor’s
factoring algorithm [13] can factorize a number N in time poly(logN). However,
despite these successes, it is not clear for what general class of problems quantum
algorithms can perform better than their classical counterparts. It should also be
kept in mind that many problems can be solved today by classical computers, as
efficiently as they could be solved by a quantum computer, if it existed.
1.2 Universal Quantum Computation
In classical computation there are various Boolean operations likeAND, NOT ,
OR, NAND etc. However, a two-bit gate such as AND or OR and a single-bit gate
such as NOT are sufficient to perform all classical logic operations [14]. We call
a set of such gates universal. An analogous result holds for universal quantum
computation. In order to do universal quantum computation, it is sufficient to be
able to perform all single-qubit gates and the controlled-not two-qubit gate [15,16].
All unitary operations on arbitrarily many qubits can then be constructed from a
4
polynomial number of these gates.
One approach to universal Quantum Information Processing (QIP) which I will
discuss more in the first part of the thesis, involves the use of single photons. An
advantage of using single photons to do QIP is that many critical techniques are well
developed in quantum optics. Also, photons interact weakly with the surroundings
resulting in slow decoherence, which is essential for quantum computing. Optical
photons, for instance, can be experimented with at room temperature, unlike, say
ion traps, which require low temperatures for best operation [17]. However, one
of the major obstacles to using single photons for QIP is that two photons will
not interact with each other unless they are in a nonlinear medium. This makes
it difficult to implement two-qubit gates using photons. Reliably producing single
photons on demand and detecting them also remains a major challenge [18].
A single-qubit gate is a unitary operator U that acts on a single-qubit state
|Ψ〉 = α |0〉q +β |1〉q ∈ H, where α and β are arbitrary complex numbers on the unit
circle and |0〉q and |1〉q are logical basis states of the Hilbert spaceH. In principle, in
a photon system any single-qubit gate can be created using linear optical elements
such as beam-splitters and phase-shifters. More precisely, any unitary operator U
acting on a single qubit can be decomposed into rotations about the Z and Y axes
on the Bloch Sphere [1] modulo a phase factor. That is:
U = eiαRz(β)Ry(γ)Rz(δ). (1.1)
Here the rotation operators are Rz(β) = e−i
β
2 σz and Ry(γ) = e−i
γ
2 σy with σy and σz
being the Pauli matrices. A rotation by angle θ about the X axis can be written
5
as Rx(θ) = Rz(pi2 )Ry(−θ)Rz(−pi2 ). Thus one can perform arbritrary rotations about
the X, Y , or Z axes, and one can perform any arbitary single-qubit operation.
To understand how to build two-qubit optical gates, I next consider the dual
rail basis with |0〉q ≡ |01〉 and |1〉q = |10〉, where a single photon can be in one of
two modes [1]. Let us analyze the effect of a φ phase shift. Consider a qubit state
|Ψ〉 = α|0〉q + β|1〉q that transforms to
α|01〉+ βeiφ|10〉
= ei
φ
2 (e−i
φ
2α|01〉+ eiφ2 β|10〉)
= ei
φ
2 e−i
φσz
2 (α|0〉q + β|1〉q)
= ei
φ
2RZ(φ)(α|0〉q + β|1〉q)
= ei
φ
2RZ(φ)|Ψ〉 (1.2)
One can see that upto a global phase, this phase shift is equivalent to a rotation
about the Z axis.
In a similar manner one can show that rotations of −2θ about the Y axis can
be achieved by using a beam splitter tilted at angle θ. In general, an ideal lossless
beam splitter can be represented by a unitary transformation [1]
U(θ, φ) =




cos θ −eiφ sin θ
e−iφ sin θ cos θ



 . (1.3)
The input and output modes are related by aˆ†l →
∑
m Umlaˆ
†
m with l,m ∈ {1, 2}
representing the two modes of the beam splitter. Using this relation one can show
that a beam splitter with φ = 0 will convert an incident state |Ψ〉 according to
|Ψ〉 = α|0〉q + β|1〉q → RY (−2θ)(α|0〉q + β|1〉q) = RY (−2θ) |Ψ〉 . (1.4)
6
Since an arbitrary single-qubit rotation can be decomposed into rotations about the
Y and Z axes, this implies that all single-qubit operations can be achieved by linear
optics alone.
A two-qubit gate is a unitary operation which acts on a state |Ψ〉 of two qubits.
An example of a two-qubit gate is a controlled-not (CNOT) gate [1]. A CNOT gate
performs the following operation on the basis states {|00〉 , |01〉 , |10〉 , |11〉} of two
qubits:
|00〉 → |00〉, |01〉 → |01〉, |10〉 → |11〉, |11〉 → |10〉. (1.5)
Another example of a two-qubit gate is the controlled-phase (CP) gate [1] which
implements a controlled pi phase shift.
|00〉 → |00〉, |01〉 → |01〉, |10〉 → |10〉, |11〉 → eipi|11〉. (1.6)
As noted earlier, it is not possible to perform these gates using linear optics alone.
Nevertheless, one of the first proposals for quantum computation in this modality
was the quantum optical Fredkin gate [19]. The gate used single-photon optics
and the Kerr effect, which occurs in media with an intensity dependent refractive
index [20]. A Fredkin gate is a three-qubit gate which can be used to do controlled-
swap operations. The first qubit is the control qubit and the second and third are
target qubits respectively.
The Hamiltonian for the Kerr effect is [21]
HI = −~χaˆ†1aˆ1aˆ†2aˆ2, (1.7)
where χ is a coupling constant that depends on the third-order nonlinear suscepti-
7
bility of the material. Here, aˆ1 and aˆ2 are the annihilation operators corresponding
to two input modes of light entering a Mach-Zender interferometer. By using a
Mach-Zender interferometer with a nonlinear medium in either arm, one can im-
plement the Fredkin gate. However, in practice there are two major problems with
this approach. The nonlinearities at the single-photon level in optical systems are
too small to create a large phase (pi), and crystals with high nonlinearities exhibit
appreciable absorption.
Although there has been tremendous progress on both theoretical and experi-
mental fronts, the subject of quantum information processing remains an active area
of research. Besides photons, many other physical sytems have been considered for
the physical realization of quantum computers including ions in a trap [22], quantum
dots [23], nitrogen vacancy (NV) centers in diamond [24], and superconducting de-
vices [25]. Each system has its own advantages and drawbacks. David DiVincenzo
suggested that a sucessful physical implementation of a quantum computer must
satisfy certain criteria. The ones that are important for superconducting qubits can
be summarized as [26]:
1. A scalable physical system with well characterized qubits.
2. The ability to initialize the state of the qubits to a simple fiducial state like
|000 . . .〉.
3. Decoherence times which are much longer than the gate operation times.
4. A universal set of quantum gates.
8
5. A qubit-specific measurement capability.
1.3 Introduction to Quantum Simulation
I have already remarked that quantum systems cannot by efficiently simulated
by classical computers. It can even be a difficult problem to study quantum systems
with a few tens of particles. Because a system of N spin-half particles has a Hilbert
space of dimension 2N , for N = 40, a 4 TB classical memory register is needed
to store the elements of the Hamiltonian matrix. For N = 300, the dimension
of the Hilbert space is more than the number of particles in the entire observable
universe! This exponential explosion is unavoidable unless approximation methods
are used, or the system contains symmetries that can be exploited. However, good
approximations are not always available and are not always reliable.
Feynman proposed a possible way around this problem by using the features
of quantum mechanics. I quote Feynman [10]: “Let the computer itself be built
of quantum mechanical elements which obey quantum mechanical laws.” Quantum
systems have the capacity to contain an exponentially large amount of information
without using an exponentially large amount of physical resources, thus making it
a natural tool to perform quantum simulation. The storage capacity of N qubits,
for example, is exponentially larger than that of N classical bits. As was shown by
Seth Lloyd more than a decade after Feynman’s proposal, a quantum computer can
indeed act as a universal quantum simulator [27].
To simulate a given quantum system with a Hamiltonian Hsys, one can con-
9
|(0)i |(t)i
| (0)i | (t)i
U
U 0
Quantum'System'
Quantum''Simulator'
Prepara2on' Measurement'
Evolu2on'
Figure 1.1: Schematic representation of a quantum system and a corresponding
quantum simulator.
struct another quantum system which can be accurately initialized and controlled
with a Hamiltonian Hsim [28]. The initial quantum state |φ(0)〉 evolves to |φ(t)〉
via the unitary transformation U = e−iHsyst. The quantum simulator evolves from
the prepared state |ψ(0)〉 to |ψ(t)〉 via U = e−iHsymt. The simulator is designed
such that there is a mapping between the simulator and the simulated system, in
particular, the mappings |φ(0)〉 ↔ |ψ(0)〉, |φ(t)〉 ↔ |ψ(t)〉, and U ↔ U ′, and is
assumed to be controllable, as depicted by the colored arrows in Figure 1.1. The
controllabilty of the quantum simulator is very important. The information about
the quantum system can only be extracted through measurements of the quantum
simulator. Although the basic idea underlying a quantum simulator is very simple,
implementation of a universal quantum simulator remains highly non-trivial.
10
Outlook and Overview of Thesis
The importance of making advancements in the field of quantum science to
harness its many potentials cannot be overstated. In this thesis, I will explore some
problems in quantum computation and quantum simulation, dealing mainly with
hybrid quantum systems. Hybrid systems, as the name suggests, are systems which
are constructed from a combination of subsystems, sometimes with widely different
properties [29]. Hybrid systems take advantage of the desirable features of each
subsystem. For example, microwave photons in a cavity or in a transmission line
coupled to superconducting qubits comprise a hybrid quantum system [30], and
so do trapped-ions coupled to superconducting qubits. Similarly, nanomechanical
resonators magnetically coupled to electron spins also form a hybrid system [31].
In the following chapter, I review the basic physics of superconducting circuits
and circuit quantization. In the third chapter, I will show how strong two-photon
nonlinearities have been attained by coupling photons in the microwave domain to
superconducting circuits. These nonlinearites will then be used to create two-photon
CP gates in the dual-rail basis. In the fourth chapter I construct a parent Hamil-
tonian with excitations that exhibit many interesting properties, and in the fifth
chapter I propose an architecture to emulate a chemical potential for light. In the
sixth chapter, I investigate the potential of coupling trapped-ions to superconduct-
ing circuits. Finally, in the seventh chapter I briefly conclude by summarizing my
main findings. The eighth chapter includes the appendices.
11
Chapter 2
Superconducting Circuits and Quantized Hamilto-
nians
2.1 Introduction
Superconductivity was discovered in 1911 by H. Kamerlingh Onnes [32, 33]
only three years after he liquefied helium which gave him the refrigeration technique
required for cooling to a few degrees Kelvin. He observed that, when cooled, the
electrical resistance of metals such as mercury, lead, and tin vanished completely in
a small temperature range at some critical temperature Tc, which is characteristic
of the material. Once a current was set up in a superconducting ring for instance,
the currents were observed to flow without measurable decrease for an entire year.
However, for decades, a fundamental understanding of this phenomenon was
absent, until Ginzburg and Landau introduced a phenomenological theory now
known as the Ginzburg-Landau (GL) theory of superconductivity [34] in 1950.
This theory concentrated entirely on the superconducting electrons rather than the
12
quasi-particle excitations in the system. In particular, they proposed a complex
pseudowave-function ψ as an order parameter within Landau’s general theory of
second-order phase transitions, with the local density of superconducting electrons
ns given by ns = |ψ(x)|2.
A microscopic theory of superconductivity came seven years later in 1957 when
Bardeen, Cooper and Schrieffer proposed what is now known as the BCS theory [35].
Their basic idea was that the interaction between electrons resulting from virtual
exchange of phonons is attractive when the energy difference between the electron
states involved is less than the phonon energy. This leads to the formation of bound
states of electrons called Cooper pairs when the thermal energy kBT is less than
an energy scale 2∆, which is typically on the order of 10−3 eV for conventional
low-Tc superconductors. The Cooper pairs are responsible for conductivity without
dissipation.
2.2 Isolated Josephson Junction
In 1962, Josephson made the remarkable prediction [36] that a zero-voltage
supercurrent can flow between two superconducting electrodes (S) separated by
a thin insulating (I) barrier. This type of structure is now called a Josephson
junction. If the insulating layer is thin enough, Cooper pairs have a small but
nonvanishing probability (p ∼ 10−5 – 10−3) of penetrating from one electrode to
another via quantum tunneling through the energy barrier created by the insulator
[37]. The Cooper pairs in the superconducting electrodes have no net spin, and a
13
SS 2
1
Figure 2.1: An SIS junction with phases χ1 and χ2 on the two electrodes.
pair, therefore, obeys Bose-Einstein statistics.
For typical insulators in use currently, when the thickness d of the insulating
layer is roughly (d ∼ 10−9m), the net current I flowing through the contact —the
Josephson junction —contains a significant supercurrent IS [37]. The supercurrent
is a direct function of the phase difference
φ = χ1 − χ2, (2.1)
where χ1 and χ2 are the phases of the condenstate wavefunctions inside the two
superconducting electrodes. This function is 2pi-periodic, and in the simplest case,
is sinusoidal [37]. That is
I ≡ IS = Ic sinφ. (2.2)
Here, the critical current Ic is a constant that is determined by the shape and
structure of the Josephson junction. It is the maximum current that can flow in the
superconducting state. When a voltage V is applied across the junction, the phase
φ evolves according to the ac Josephson relation [37]
dφ
dt
=
2eV
~
. (2.3)
14
One can define an effective inductance of a junction LJ from the relation
V = LJ
dI
dt
. (2.4)
From (2.2) the rate of change of current is
dI
dt
= (Ic cosφ)φ˙ = Ic cosφ
2eV
~
. (2.5)
Combining (2.4) and (2.5) one gets
LJ =
~
2eIc cosφ
=
Φ0
2piIc cosφ
, (2.6)
where Φ0 = h/(2e) is the superconducting flux quantum. One can see that LJ
becomes arbritrarily large as φ→ pi/2. The corresponding energy UJ stored in the
junction by virtue of its nonlinear inductance can be found as follows:
UJ =
∫ t
t0
I(t)V (t)dt
=
∫ t
t0
Ic sinφ(t)V (t)dt
=
∫ φ
φ0
Ic sinφ
~
2e
dφ
=
~Ic
2e
(cosφ0 − cosφ)
= −Φ0Ic
2pi
(cosφ− cosφ0). (2.7)
From this result, I can define the Josephson energy EJ by
EJ =
Φ0Ic
2pi
. (2.8)
Ignoring the constant term in (2.7), one gets UJ = −EJ cosφ.
A real Josephson junction will have a capacitance CJ between its electrodes
and an associated charging energy defined as EC = (2e)2/(2CJ). This is the energy
required for a Cooper-pair to tunnel through the junction.
15
2.3 Quantum Description of an Isolated Junction
When the quantum fluctuation of the junction is large compared to thermal
fluctuations, which occurs when kBT ≤ ~ω, with ω the characteristic frequency
of the junction, and T the temperature, a quantum description is required. One
characteristic frequency of the junction is the plasma frequency ωp given by
ωp =
√
2eIc
~CJ
=
√
2ECEJ
~
. (2.9)
The basic principles of quantum mechanics state that variables like φ, V ,
and I that describe the junction classically, cannot simultaneously be known quan-
tum mechanically. Therefore, in the quantum regime, one needs to introduce non-
commuting conjugate pairs. One pair is given by Nˆ = Qˆ/(2e) and φˆ, with the
commutation relation [φˆ, Nˆ ] = i. Here, Nˆ describes the number of Cooper pairs
tunneling through the junction and φˆ is the phase of the junction. The commuta-
tion relation implies that ∆φˆ∆Nˆ ≥ 1/2. The Hamiltonian of the junction can then
be written as
H =
Qˆ2
2CJ
− EJ cos φˆ = ECNˆ2 − EJ cos φˆ. (2.10)
Note that sometimes the charging energy is defined as EC = e2/(2CJ) so that the
kinetic term becomes 4ECNˆ2. In the coordinate or phase representation φˆ = φ and
Nˆ = −i∂/∂φ. This leads to
H = −EC
∂2
∂φ2
− EJ cosφ. (2.11)
In the limit ~ωp  EJ , the energy levels of the system are localized at the
bottom of the cosine well. The minima of the potential occur at φn = 2npi where
16
n ∈ Z. One can explore quantum fluctuations φ˜ = φ − φn around this minima so
that
U = −EJ
(
1− φ˜
2
2
)
. (2.12)
The Hamiltonian is reduced to that of a harmonic oscillator with frequency ωp.
Thus, the lowest energy levels of the junction become approximately linear and are
given by
En =
(
n+
1
2
)
~ωp. (2.13)
2.4 Josephson Junctions Connected to External Circuits
So far I have only discussed an isolated junction. However, one can realize
many different architectures by connecting Josephson junctions to external circuits.
These externally shunted junctions can have very different properties, but they all
have one property in common. That is, these systems can exhibit large nonlinearities
depending on the choice of parameters. The nonlinearities in the junctions enable
one to approximate some circuits as a qubit or two-level system, while others can be
treated as a harmonic oscillator with a moderate or large anharmonicity. The various
regimes in which a shunted junction operate are outlined in Table 2.1. The ratio
EJ/EL is approximately equal to the number of minima in the potential landscape,
while the ratio EJ/EC is roughly equal to the number of energy levels per well
around each minima. This is outlined in a “Mendeleev” table of superconducting
circuits in Figure 2.2 [25].
17
Regime Energy Relationships
Phase EC  EL < EJ
Fluxonium EL < EC < EJ
Flux EC < EL < EJ
Hybrid EC < EL ∼ EJ
Transmon EC  EJ
Charge EJ  EC
Table 2.1: The various regimes in which a shunted Josephson junction can operate.
2.4 Charge Qubit
The first such architecture, a Cooper pair box, is an example of a charge qubit
(Figure 2.3). A Josephson junction with Josephson energy EJ and a capacitance
CJ is connected to a gate voltage Vg through a gate capacitance Cg. The Cooper
pair box (CPB) is made of an electrode of the Josephson junction and an electrode
of the gate capacitor and the superconducting lead connecting them.
The Lagrangian L of the system is
L =
1
2
CJV
2
J +
1
2
Cg(Vg − VJ)2 + EJ cosφ. (2.14)
Recall the relation
VJ = −
~
2e
φ˙. (2.15)
18
101
100
102
103
104
105
0
1/2 11/8 1/41/16
EJ/EC
EL/EJ
Flux Qubit
Hybrid Qubit
Phase Qubit
Fluxonium
Cooper Pair Box
Transmon
Quantronium
≈
Figure 2.2: A “Mendeleev table” of superconducting circuits. I will consider flux
and hybrid circuits (italicized).
The kinetic energy is
T =
1
2
(
~
2e
)2
CJ φ˙
2 +
1
2
Cg
(
Vg +
~
2e
φ˙
)2
=
1
2
(CJ + Cg)
(
~
2e
)2
φ˙2 +
~
2e
CgVgφ˙, (2.16)
where the constant term CgV 2g /2 has been dropped. The conjugate momentum is
pi =
∂L
∂φ˙
=
(
~
2e
)2
(CJ + Cg)φ˙+
~
2e
CgVg. (2.17)
The Hamiltonian is
H = piφ˙− L
=
1
2
(2e)2
CJ + Cg
(N −Ng)2 − EJ cosφ, (2.18)
where Ng = CgVg/(2e), and Q = −2eN is the number of Cooper pairs on the island.
19
+QJ
QJ
EJ , CJ
Qg
+Qg Cg
Vg
Ig
Figure 2.3: Schematic of a Cooper pair box. The parts inside the dashed box
comprises the Cooper pair island whose excess charge corresponds to the qubit
degree of freedom. Ig is the current flowing through the circuit.
Let EC = (2e)2/(Cg + CJ). Then the quantum Hamiltonian is
H =
1
2
EC(Nˆ −Ng)2 − EJ cos φˆ. (2.19)
The charge qubit operates in the regime where EC  EJ . This means phys-
ically that the tunneling between states with different N is suppressed and N is
therefore a good quantum number. This assumption breaks down when N ∼ Ng
where the tunneling energy dominates over the Coulomb energy.
From the commutation relation [φˆ, Nˆ ] = i, one gets φ = i∂/∂N , and e±iφˆ |N〉 =
|N ∓ 1〉. Therefore, the Hamiltonian in the {|N〉} basis can be written as
H =
∑
N∈Z
EC
2
(N −Ng)2 |N〉 〈N | −
EJ
2
(|N〉 〈N + 1|+ |N + 1〉 〈N |). (2.20)
Ng is a controllable parameter. If Ng is set in the vicinity of (N + 1/2) with N ∈ N,
then states |N〉 and |N + 1〉 have almost degenerate energies. All other states have
20
Figure 2.4: Energy levels of a Cooper pair box for ωC/ωJ = 10. The ellipse denotes
the region where the ground and the first excited state are nearly degenerate, at
Ng = 1/2. These states form a qubit.
higher energies and can be ignored. Then
H =
EC
2
[
N2g |N〉 〈N |+ (1−Ng)2 |N + 1〉 〈N + 1|
]
−EJ
2
[|N〉 〈N + 1|+ |N + 1〉 〈N |]
= −1
2
Bzσz −
1
2
Bxσx, (2.21)
where Bz = EC/2(1 − 2Ng), Bx = EJ , and I have ignored the constant terms. σx
and σz are the familiar Pauli matrices. Note that because of the discrete nature of
N , φ is compact and the wavefunction ψ(φ) is the same as ψ(φ+ 2npi) for n ∈ Z.
2.4 Flux Qubit
The simplest flux qubit is an rf-SQUID. It consists of a Josephson junction
connected by a superconducting loop with inductance L as depicted in Figure 2.5.
21
EJ , CJ
+QJ
QJ
x L
Figure 2.5: A flux qubit with flux Φx threading the inductive loop.
The Hamiltonian of a flux qubit in the φ representation is
H = −EC
∂2
∂φ2
− EJ cosφ+
1
2
EL(φ+ φx)
2, (2.22)
with EC = (2e)2/(2CJ).
In contrast to the Cooper pair box, φ is now non-compact and is defined on
all of R. This is because N = q/(2e) is no longer discrete as the presence of the
inductive loop allows continuous charge to move from one side of the junction to
the other. In the limit where EJ > EC > EL, the device is called a fluxonium.
Similarly, if EJ < EL and there is only one minimum in the potential, it is called
a hybrid superconducting circuit. Figure 2.6 shows the energy levels of inductively
shunted junction for various energy regimes. If EL  EJ , the potential can be
well approximated around the minima by a harmonic well. Then the circuit can be
regarded as a harmonic oscillator with a nonlinear perturbation.
In experiments the noise characteristics of superconducting circuits have to
be taken into account. Qubits are characterized by two times denoted T1 and T2.
The relaxation time T1 is the time required for the qubit to relax from the first
excited state to the ground state. This process involves energy loss. The dephasing
22
time T2 is the time over which the phase difference between two eigenstates become
randomized. Both relaxation and dephasing can be theoretically described using a
model where the system is weakly coupled to the quantum noise produced by the
environment [38,39]. This approach predicts that energy relaxation from the excited
to the ground state occurs due to the spectral density of the noise at the frequency
difference of the two states. The dephasing rate, by contrast, has two contributions
so that
1
T2
=
1
2T1
+
1
Tφ
. (2.23)
The first contribution arises from relaxation processes whereas the second contribu-
tion, also called pure dephasing arises from other energy-conserving processes. One
can distinguish the dephasing time T2, which is an intrinsic timescale for decoherence
of a single qubit, from another timescale T ∗2 , which is the result of measurements on
an ensemble of such qubits [40]. It is the case that T ∗2 < T2.
At the present time, Cooper pair boxes discussed above suffers from excessive
charge noise. By decreasing the ratio of EC to EJ one can form a so called transmon
which is much less sensitive to charge noise [41]. Similarly, flux noise has an adverse
effect on flux qubits [42, 43]. However, over the past decade, clever engineering as
well as the understanding of sources of noise [44,45] has led to improvement in qubit
lifetimes by almost six orders of magnitude [25,46].
23
2.5 Circuit-QED
Circuit Quantum Electrodynamics (circuit-QED) [47] borrows techniques from
the field of atomic cavity Quantum Electrodynamics (QED), which investigates the
interaction of matter with light at the quantum scale [48]. When confined to a cavity,
the radiative properties of an atom differ fundamentally from the atom’s radiative
properties in free space [49]. Spontaneous emission is inhibited if the cavity has
characteristic dimensions which are small compared to the radiation wavelength,
and enhanced if the cavity is resonant. Surprisingly, superconducting circuits which
are macroscopic entities containing billions of atoms can behave very much like a
single atom, albeit with more tunable properties. They can be used to reach the
so called strong coupling regime where the coupling between the excitations in the
circuit and light in the cavity exceed the corresponding decay rates, leading to a
coherent periodic exchange of a single photon on resonance [30].
In the preceding sections, I wrote down the quantum or circuit-QED Hamilto-
nian for various shunted Josephson junctions. In this section, I provide a prescrip-
tion for deriving such Hamiltonians. The reader can refer to [50] for an introductory
treatment of circuit quantization. For a more advanced treatment, especially of cir-
cuits involving mutual inductances, the reader is advised to consult [51].
To derive a quantum Hamiltonian for a circuit, I first derive the classical
Hamiltonian in the lumped element limit. This limit is appropriate when the cir-
cuit components are much smaller than the electromagnetic wavelengths in the fre-
quencies of interest. I will assume that each circuit component has two terminals,
24
although it can have more. Every two terminal component b has a voltage vb(t)
across it and a current ib(t) through it. In the classical case, these two variables are
used to describe a circuit. However, for the purpose of circuit quantization, one can
define the flux and charge variables,
Φb(t) =
∫ t
−∞
vb(t
′) dt′, (2.24)
qb(t) =
∫ t
−∞
ib(t
′) dt′. (2.25)
One can assume that at t = −∞ all voltages and currents are zero, and that any
static bias fields such as magnetic fluxes imposed on the inductors are assumed to
be switched on adiabatically from t = −∞ to t = 0.
One can work with two types of components. The first is of the capacitive
type which satisfies a general (possibly nonlinear) relation
vb = f(qb). (2.26)
The second is of the inductive type which satisfies
ib = g(Φb). (2.27)
A Josephson junction, for example, would satisfy a nonlinear equation of the induc-
tive type.
Every circuit can be regarded as a graph with the terminals being represented
by the nodes of the graph and the elements connected across the terminals by the
edges. A spanning tree of a graph is a union of edges of the graph that contains all
the nodes but does not contain any loops. The branches of the spanning tree are
called tree branches. All other branches are called chords. Each chord is associated
25
with a unique loop that is formed when it is added to a tree. With these basic ideas,
one can follow a series of steps that leads to a classical Hamiltonian of a circuit.
The steps are as follows:
1. Represent the circuit as a network or graph of two-terminal capacitors and
inductors.
2. It is helpful to simplify the circuit using the standard rules of series and parallel
components.
3. Choose the ground node of the circuit. The remaining nodes of the graph are
called active nodes.
4. Choose a spanning tree T of the graph that contains all the capacitors and as
few inductors as possible.
5. Introduce a node flux for each active node n as the time integral of the voltage
along a path on T from the node to the ground. That is,
φn(t) =
∑
b
SnbΦb =
∑
b
Snb
∫ t
−∞
vb(t
′) dt′. (2.28)
Snb is 0 if the path on T from the ground to n does not pass through b.
Otherwise it is ±1 depending on the orientation of the path.
6. Write the kinetic T and potential energy V of the components in terms of the
node fluxes, and their time derivatives. For a branch b connecting two nodes
n and n′, the branch voltage vb is the time derivative of the branch flux Φb,
i.e. vb = Φ˙b. The branch flux is Φb = φn− φ′n + Φ˜l(b), with Φ˜l(b) = 0 for b ∈ T .
26
Otherwise, Φ˜l(b) is the externally-applied magnetic flux through the loop l(b)
that is produced by adding b to T , i.e. the unique loop associated with the
chord b.
7. Form the Lagrangian
L = T (φ1, φ˙1, . . . , φN , φ˙N) = T − V. (2.29)
8. Define the canonical momenta
qn =
∂L
∂φ˙n
. (2.30)
9. Perform the Legendre transformation to get the Hamiltonian
H(φ1, q1, . . . , φn, qn) =
N∑
i=1
qiφ˙i − L. (2.31)
10. To quantize the circuit, promote the canonical variables to operators that
satisfy
[φˆi, qˆj] = i~δij. (2.32)
For an introductory example, I consider a simple LC circuit driven by a possi-
bly time-dependent external flux Φx(t) as depicted in Figure 2.7. The ground node
(black dot) has flux φg and the only active node (red dot) has flux φ. I denote the
flux of the branch comprising the inductor by ΦL, and for the branch comprising
the capacitor by ΦC . The tree (in green) is chosen so that it includes the inductor.
One can let φg = 0 since the ground flux is arbitrary. According to the sixth rule
ΦL = φ, (2.33)
ΦC = φ+ Φx(t). (2.34)
27
The kinetic and potential energies T and V are
T =
1
2
CΦ˙2C =
1
2
C(φ˙+ Φ˙x(t))
2, (2.35)
V =
φ2
2L
. (2.36)
From this, one can construct the classical Lagrangian
L = T − V = 1
2
C(φ˙+ Φ˙x(t))
2 − φ
2
2L
. (2.37)
The canonical momentum is
q =
∂L
∂φ˙
= C(φ˙+ Φ˙x(t)) =⇒ φ˙ =
q
C
− Φ˙x(t). (2.38)
The classical Hamiltonian is then obtained by a Legendre transformation giving
H = qφ˙− L = q
2
2C
+
φ2
2L
− qΦ˙x(t). (2.39)
The Hamiltonian can be quantized by promoting q → qˆ and φ→ φˆ with [φˆ, qˆ] = i~.
2.6 Conclusions and Outlook
Superconducting circuits with Josephson junctions possess an intrinsic nonlin-
earity which enables the creation of qubits or anharmonic oscillators. In this chapter
I showed that they can have widely varying properties. The rapidly improving life-
times of superconducting circuits has advanced their potential use for large scale
quantum information processing. In this chapter, I also showed a general prescrip-
tion for obtaining quantum Hamiltonians with linear elements. This will pave the
way for the derivation of quantum Hamiltonians of nonlinear circuits.
28
(a)
(b)
(c)
Figure 2.6: The first few energy levels for φx = 0.1 of (a) a hybrid circuit with
ωJ/ωC = 5, ωL/ωC = 10, (b) fluxonium with ωJ/ωC = 10, ωL/ωC = 0.05, and (c)
flux qubit with ωJ/ωC = 15, ωL/ωC = 2.
29
Figure 2.7: An LC circuit
An LC circuit with only one active node (red dot). The spanning tree is in green.
30
Chapter 3
Nonlinear Optics Quantum Computing with Circuit-
QED
3.1 Motivation
In Chapter 1, I introduced universal quantum computation with photons.
Specfically, linear optics quantum computing (LOQC) has proven to be one of the
conceptually simplest approaches to building novel quantum states and proving the
possibility of quantum information processing. This approach relies on the robust-
ness of linear optical elements, but implicitly requires an optical nonlinearity [52–55]
as linear optics alone is insufficient to implement universal quantum gates. Unfortu-
nately, progress towards larger scale systems remains challenging due to the limits
to optical nonlinearities, as well as the measurement of single photons [18,56].
In this chapter I explore how recent advances in circuit-QED in which opti-
cal and atomic-like systems in the microwave domain are explored for their novel
quantum properties [57], provides a new paradigm for quantum computing with
31
photons [30,58,59], which, in contrast to LOQC, is deterministic. Specifically, using
superconducting nonlinearities in the form of Josephson junctions and the related
quantum devices such as flux and phase qubits [60,61], key elements of my approach
have been realized: the creation of microwave photon Fock states [59, 62–64], con-
trollable beam splitters [59, 65], and single microwave photon detection [66, 67]. In
many cases, the photons stored in a transmission line-based resonator or inductor-
capacitor resonator have much better coherence times than the attached supercon-
ducting qubits [68–70]. This suggests that the main impediment to photon based
quantum computing is the realization of appropriate photon nonlinearities to en-
able two-qubit gates like two-photon phase gates, which are sufficient for universal
quantum computation [15,52].
The key element of a two-photon phase gate is a two-photon nonlinear phase
shifter. It imparts a pi phase on any state consisting of two photons, while leaving
single photon and vacuum states unaffected. A deterministic approach to achieve
such photon nonlinearity is based on the Kerr effect [68, 71–73]. In the context of
circuit-QED, in Ref. [73], a four level N scheme using a coplanar waveguide res-
onator and a Cooper pair box is used to arrange for EIT [74] to generate large Kerr
nonlinearities. In this chapter I demonstrate a different approach to photon nonlin-
earity. I explore the possibility of using a dc-SQUID [75] to implement a nonlinear
coupling between qubit and resonator, which, through an adiabatic scheme, enables
a high fidelity, deterministic two-photon nonlinear phase shift in the microwave do-
main. Along with the nonlinearity, I envision using dynamically controlled cavity
coupling to implement a 50/50 beam splitter operation to construct a two-photon
32
Figure 3.1: Use of two nonlinear phase shifters (NL), combined with 50/50 beam
splitters, leads to a deterministic two photon phase gate using dual rail logic. The
two photons in the dual rail basis |0〉L |1〉L = |01〉1 |10〉2 of the two qubits become
bunched into a single mode after passing through the first beam splitter, and then
receive a pi phase from one of the two phase shifters. Storage cavities are represented
by blue lines.
phase gate using so-called dual rail photon qubits [1, 59], in which the logical basis
{|0〉L = |01〉 , |1〉L = |10〉} corresponds to the existence of a single photon in one
of two resonator modes (Figure 3.1). My approach takes best advantage of the
relatively long coherence times for microwave photons in resonators, and couples
only virtually to superconducting quantum bit devices, minimizing noise and loss
due to errors in such devices. When combined with the aforementioned techniques
for Fock state generation and detection, along with dynamically controlled beam
splitters, this provides the final element for nonlinear optics quantum computing in
33
the microwave domain.
3.2 Outline of Approach
I now outline the approach. I consider photons stored in a high-impedance
microwave resonator [76] coupled inductively with strength 0 < χ < 1 to a flux
superconducting qubit (SQ) in a dc-SQUID configuration (Figure 3.2). The res-
onator loops around the dc-SQUID which results in a nonlinear cosine dependent
interaction between the resonator and qubit. In this configuration, I get an ef-
fective coupling of the form V ∼ EJ cos(φˆ + φ′x) cos φˆL, where an external flux
φ′x ≡ 2piχΦ′x/Φ0 is applied to the resonator which consequently threads the smaller
loop of the dc-SQUID, Φ0 being the superconducting flux quantum. The qubit phase
variable and the resonator flux are denoted by φˆ and φˆL = 2piΦˆL/Φ0 respectively.
For φ′x ∼ pi/2, one immediately sees a nonlinear coupling between the qubit and
resonator: V ∼ EJ φˆφˆ2L, where two resonator photons can be annihilated to produce
one qubit excitation, analogous to parametric up conversion in χ(2) systems. This
causes the two-photon state of the resonator to couple to the first excited state of
the SQ with strength g2 (Figure 3.3(a)). In essence, in this region, the two-photon
state with detuning δ from the qubit, becomes slightly qubit-like and acquires some
nonlinearity. However, the single-photon state, inspite of its coupling to the first
excitation of the SQ with strength g1, remains mostly photon-like because it is far
detuned by ∆ from this qubit excitation (Figure 3.3(b)). At the end of the proce-
dure, this leads to an additional phase for the two-photon initial state. The coupling
34
φx
φ′x
(a)
C L L1
Resonator   Qubit
φx
φ′x
(b)
Figure 3.2: (a) Implementation of a high-impedance coiled resonator (blue) coupled
to a dc-SQUID (red) with an inductive outer loop. The flux bias lines are in black.
(b) A simple circuit model of the physical implementation.
of the two-photon state to other modes arises via linear coupling at O(g1) and is
assumed to be far detuned.
The noise in the SQ, with a decay rate γ of its first excited state, may slightly
limit the nonlinear phase shift operation. Although the overall system will mostly be
in the photon-like regime with decay rate κ, there will be an additional probability
for it to decay due to its coupling to the lossy qubit. In the limit where |δ|  |g2|
and |∆|  |g1| with |∆| > |δ|, the two-photon nonlinearity goes like g22/δ, and the
two-photon state decays approximately at a rate γg21/∆
2 + γg22/δ
2. Thus, the losses
due to the qubit go like γ/δ provided one allows g1 to become close to g2, which is
possible by controlling φ′x. Hence, at large detuning, one will then be limited only
by κ. In contrast, a Kerr nonlinearity scales like g41/δ
3 and the noise scales like
γg21/δ
2, leading to more loss due to the qubit for large detuning.
35
 Resonator Ground 
1 0 
2 0 
  3 0 
 Qubit Ground
 0 1 
 0 2 
 1 1 
t
t tq 6
bg1g2
(a)
2 Photons
1 Photon
SLOW
Tim
e  [
AU
]
En
erg
y  [
AU
]
φx
φx
(b)
Figure 3.3: (a) Energy levels of the resonator and qubit system along with the
relevant couplings. (b) Top: The suggested flux bias pulse φx to implement the
nonlinear phase shift; a fast but adiabatic sweep and then a very slow variation
of the pulse near the avoided crossing. Bottom: The coupling g2 between the two
photon state and the first qubit excited state leads to a sizeable avoided crossing.
3.3 The Circuit Model and Hamiltonian
First I derive the Hamiltonian of the system using a circuit model. The circuit
model comprising of an LC circuit coupled inductively with strength 0 < χ < 1 to
a dc-SQUID configuration is shown in Figure 3.4. The inductance and capacitance
of the LC circuit are denoted by L and C respectively. The SQUID consists of two
junctions with Josephson energies EJ1 and EJ2 respectively, along with capacitances
36
C L
Tree
TreeCJ1 CJ2LJ1 LJ2 L1
r\
L 
\
x ’
\x
\L \
\11 \12 \21 \22 \3
\
x ’
\1 \2
Figure 3.4: The circuit model.
and inductances CJ1, LJ1 and CJ2, LJ2. The terms ΦL and Φ represent the node
fluxes with the corresponding trees shown in green. The branch fluxes in the LC
circuit are denoted in blue by Φ1 and Φ2 while the branch fluxes in the qubit are
represented in red by the terms Φ11, Φ12, Φ21, Φ22, and Φ3. The outer loop is
threaded by an external flux Φx while the loop in the LC circuit has a flux Φ′x
through it. This in turn threads the smaller SQUID loop, along with the resonator
flux ΦL, giving rise to the inductive coupling.
Following the guidelines on circuit quantization, the branch fluxes can be writ-
ten as
Φ1 = ΦL + Φ
′
x, (3.1)
Φ2 = ΦL, (3.2)
Φ11 = Φ + χΦL + χΦ
′
x = Φ12, (3.3)
Φ21 = Φ = Φ22, (3.4)
Φ3 = Φ + Φx. (3.5)
37
The kinetic energy term T is [50]
T =
1
2
CΦ˙21 +
1
2
CJ1Φ˙
2
11 +
1
2
CJ2Φ˙
2
22
=
1
2
C(Φ˙L + Φ˙
′
x)
2 +
1
2
CJ1(Φ˙ + χΦ˙L + χΦ˙
′
x)
2 +
1
2
CJ2Φ˙
2
≈ 1
2
CΦ˙2L +
1
2
CJ1(Φ˙ + χΦ˙L)
2 +
1
2
CJ2Φ˙
2, (3.6)
where I ignore the time derivative of external fluxes in the adiabatic limit. The
potential energy term is
V =
Φ2L
2L
− EJ1 cos(φ+ χφL + φ′x)− EJ2 cosφ+
1
2
EL(φ+ φx)
2. (3.7)
I have written V in terms of
φx =
2piΦx
Φ0
, (3.8)
φ′x =
2piχΦ′x
Φ0
, (3.9)
φL =
2piΦL
Φ0
, (3.10)
EL =
Φ20
4pi2L1
. (3.11)
The Lagrangian of the system is L = T − V . The canonical momenta of the system
denoted by qL and q are
qL =
∂L
∂Φ˙L
and q =
∂L
∂Φ˙
. (3.12)
From the Lagrangian
q = (CJ1 + CJ2)Φ˙ + χCJ1Φ˙L, (3.13)
qL = χCJ1Φ˙ + (C + χ
2CJ1)Φ˙L. (3.14)
38
Solving these equations for Φ˙ and Φ˙L I get
Φ˙ =
qC − χCJ1(qL − χq)
C(CJ1 + CJ2) + χ2CJ1CJ2
, (3.15)
Φ˙L =
qL(CJ1 + CJ2)− χqCJ1
C(CJ1 + CJ2) + χ2CJ1CJ2
. (3.16)
Using these results I can write the Hamiltonian H = qΦ˙ + qLΦ˙L−L. The quantum
Hamiltonian is then
H =
[
Φˆ2L
2L
+
qˆ2L
2C˜
]
+
[
qˆ2
2C˜J
− EJ1 cos(φˆ+ χφˆL + φ′x)− EJ2 cos φˆ+
1
2
EL(φˆ+ φx)
2
]
−
(
χCJ1
C(CJ1 + CJ2) + χ2CJ1CJ2
)
qˆqˆL, (3.17)
where the effective resonator and junction capacitances are
C˜ =
C(CJ1 + CJ2) + χ2CJ1CJ2
CJ1 + CJ2
χ=0−−→ C, (3.18)
C˜J =
C(CJ1 + CJ2) + χ2CJ1CJ2
C + χ2CJ1
χ=0−−→ CJ1 + CJ2. (3.19)
Let the dressed resonator frequency be ω = 1/
√
LC˜ and Φ(0)L =
√
Lω~/2 be
the width of quantum fluctuations in the resonator flux. Introduce the dimensionless
parameter µ = 2piΦ0L/Φ0. In terms of the quantum of conductance G0 = 2e
2/h and
the characteristic impedance of the resonator Z =
√
L/C˜, I can write µ = 2piG0Z.
Since µ 1, one can expand V in powers of χφˆL ∝ µ. Performing a series expansion
of the term in V proportional to EJ1 to second order in χφˆL I get
− cos(φˆ+ χφˆL + φ′x)
= − cos(φˆ+ φ′x) + χ sin(φˆ+ φ′x)φˆL +
χ2
2
cos(φˆ+ φ′x)φˆ
2
L +O(φˆ
3
L). (3.20)
39
This gives the resonator, qubit, and interaction Hamiltonians
Hr =
qˆ2L
2C˜
+
Φˆ2L
2L
, (3.21)
Hq =
qˆ2
2C˜J
− EJ1 cos(φˆ+ φ′x)− EJ2 cos φˆ+
1
2
EL(φˆ+ φx)
2, (3.22)
VI = χEJ1 sin(φˆ+ φ
′
x)φˆL +
χ2
2
EJ1 cos(φˆ+ φ
′
x)φˆ
2
L −
χCJ1
C˜(CJ1 + CJ2)
qˆqˆL.(3.23)
For simplicity I assume that the junctions are identical with capacitances CJ and
Josephson energies EJ . Then I get
C˜ = C +
χ2
2
CJ , (3.24)
C˜J =
2CCJ + χ2C2J
C + χ2CJ
, (3.25)
along with the simplified terms
Hr =
qˆ2L
2C˜
+
Φˆ2L
2L
, (3.26)
Hq =
qˆ2
2C˜J
− EJ cos(φˆ+ φ′x)− EJ cos φˆ+
1
2
EL(φˆ+ φx)
2, (3.27)
VI = χEJ sin(φˆ+ φ
′
x)φˆL +
χ2
2
EJ cos(φˆ+ φ
′
x)φˆ
2
L −
χ
2C˜
qˆqˆL. (3.28)
3.4 Linearization and Quantization
In the limit where EL  EJ > 2EC , one can linearize the potential
V (φ, φL) =
(
Φ0
2pi
)2 φ2L
2L
− EJ cos(φ+ χφL + φ′x)− EJ cosφ+
1
2
EL(φ+ φx)
2,(3.29)
about the classical minima φc and φLc of the qubit phase and resonator flux respec-
tively. Thus, in the following, I let curly brackets {f(φ, φL)} denote its evaluation
at the classical minima φ = φc and φL = φLc. Hence,
V (φ, φL)→ V (φˆ+ φc, φˆL + φLc), (3.30)
40
where the hats denote quantum fluctuations. I let
V0 = V (φc, φLc), (3.31)
V1 =
{
∂V
∂φ
}
φˆ+
{
∂V
∂φL
}
φˆL. (3.32)
Using a Taylor series expansion of V about the classical minima, I get
V (φˆ+ φc, φˆL + φLc)
= V0 + V1 +
1
2!
[{
∂2V
∂φ2
}
φˆ2 + 2
{
∂2V
∂φ∂φL
}
φˆφˆL +
{
∂2V
∂φ2L
}
φˆ2L
]
+O(φˆ3).
(3.33)
I set the terms
∂V (φ, φL)
∂φ
=
∂V (φ, φL)
∂φL
= 0, (3.34)
and solve for the approximate classical values φc and φLc. Letting χ˜ = 2piχ/Φ0,
φc(φx, φ
′
x) = −φx +
EJ sinφx + EJ sin(φx − φ′x − χφLc)
EL + EJ cosφx + EJ cos(φx − φ′x − χφLc)
, (3.35)
φLc(φx, φ
′
x) =
(
2pi
Φ0
)
EJLχ˜ sin(φx − φ′x)
1 + EJLχ˜2 cos(φx − φ′x)
. (3.36)
Explicitly, the linearized potential is
V2 =
1
2
[EL + EJ cosφc + EJ cos(φc + φ
′
x + χφLc)] φˆ
2
+
1
2
[(
Φ0
2pi
)2 1
L
+ χ2EJ cos(φc + φ
′
x + χφLc)
]
φˆ2L
+χEJ cos(φc + φ
′
x + χφLc)φˆφˆL. (3.37)
The nonlinear correction to the potential has the form
V3 =
1
3!
[{
∂3V
∂φ3
}
φˆ3 + 3
{
∂3V
∂φ2∂φL
}
φˆ2φˆL + 3
{
∂3V
∂φ∂φ2L
}
φˆφˆ2L +
{
∂3V
∂φ3L
}
φˆ3L
]
,(3.38)
41
where the derivatives are given by
∂3V
∂φ3
= −EJ sinφc − EJ sin(φc + φ′x + χφLc), (3.39)
∂3V
∂φ2∂φL
= −χEJ sin(φc + φ′x + χφLc), (3.40)
∂3V
∂φ∂φ2L
= −χ2EJ sin(φc + φ′x + χφLc), (3.41)
∂3V
∂φ3L
= −χ3EJ sin(φc + φ′x + χφLc). (3.42)
The coupling I am interested in is
1
2
{
∂3V
∂φ∂φ2L
}
φˆφˆ2L = −
χ2
2
EJ sin(φc + φ
′
x + χφLc)φˆφˆ
2
L, (3.43)
which annihilates two resonator quanta in exchange for a single qubit excitation.
For conciseness of notation I let
u(φx, φ
′
x) ≡ cos(φc + φ′x + χφLc), (3.44)
s(φx, φ
′
x) ≡ sin(φc + φ′x + χφLc), (3.45)
r(φx, φ
′
x) ≡ sinφc, (3.46)
t(φx, φ
′
x) ≡ cosφc. (3.47)
Define the parameters
L˜−1 =
[
1
L
+ χ˜2EJ cos(φc + φ
′
x + χφLc)
]
=
[
1
L
+ χ˜2EJu
]
, (3.48)
ω =
1
√
L˜C˜
, (3.49)
ωq =
√
ωC [ωL + (t+ u)ωJ ]. (3.50)
Let Nˆ = qˆ/(2e) be the number of Cooper pairs tunneling through the junction with
an effective charging energy E˜C = (2e)2/C˜J . The Hamiltonian of the system is
42
Hˆl = Hˆrl + Hˆql + VˆI where
Hˆrl =
qˆ2L
2C˜
+
Φˆ2L
2L˜
→ qˆ
2
L
2C˜
+
1
2
C˜ω2Φˆ2L, (3.51)
Hˆql =
E˜C
2
Nˆ2 +
1
2
[EL + EJ(t+ u)] φˆ
2, (3.52)
HˆI = −
χ
2C˜
qˆqˆL + χ˜uEJ φˆΦˆL −
χ˜2
2
sEJ φˆΦˆ
2
L. (3.53)
I now introduce the operators {aˆ, aˆ†, bˆ, bˆ†} satisfying [aˆ, aˆ†] = 1 = [bˆ, bˆ†]. Let
ΦˆL =
√
L˜ω~
2
(aˆ+ aˆ†), (3.54)
qˆL = −i
√
~
2L˜ω
(aˆ− aˆ†), (3.55)
φˆ =
√
ω˜C
2ωq
(bˆ+ bˆ†), (3.56)
Nˆ = −i
√
ωq
2ω˜C
(bˆ− bˆ†). (3.57)
The newly defined operators preserve the commutations relations [φˆ, Nˆ ] = i and
[Φˆ, qˆ] = i~. This leads to the quantized Hamiltonians Hˆrl = ωaˆ†aˆ and Hˆql = ωq bˆ†bˆ.
The linear part of the Hamiltonian is
HˆL = Hˆrl + Hˆql −
χ
2C˜
qˆqˆL + χ˜uEJ φˆΦˆL. (3.58)
The quantized nonlinear interaction term is
Vˆnl = −
χ˜2
2
sEJ φˆΦˆ
2
L. (3.59)
Recall the dimensionless parameter µ = 2piΦ0L/Φ0 ≈ 2pi/Φ0
√
L˜ω~/2. I then define
the energies
η1 = χEJµ, (3.60)
η2 =
χ2
2
EJµ
2 =
η21
2EJ
, (3.61)
η3 =
χe
C˜
√
~
2L˜ω
= η1
~ω
2EJ
. (3.62)
43
In terms of the creation and annihilation operators, the linear interaction term is
Vˆl = −
χ
2C˜
qˆqˆL + χ˜uEJ φˆΦˆL
= − χ
2C˜
(2e)Nˆ qˆL + χ˜uEJ φˆΦˆL
= η3
√
ωq
2ω˜C
(aˆ− aˆ†)(bˆ− bˆ†) + η1u
√
ω˜C
2ωq
(aˆ+ aˆ†)(bˆ+ bˆ†). (3.63)
I then make a rotating wave approximation to get
Vˆ (RWA)l = g1(aˆbˆ
† + aˆ†bˆ), (3.64)
where the linear coupling is
g1 = η1u
√
ω˜C
2ωq
− η3
√
ωq
2ω˜C
. (3.65)
Later it will be seen that the nonlinear phase shift operation will require g1 to
vanish. This is only possible if it changes sign. The term u varies as a function of
the external fluxes but |u| ≤ 1. g1 vanishes when
η1u
√
ω˜C
2ωq
= η3
√
ωq
2ω˜C
=⇒ u
√
ω˜C
2ωq
=
~ω
2EJ
√
ωq
2ω˜C
. (3.66)
Since |u| ≤ 1, for g1 to change sign, I require ω˜C ≤ ωqω/(2ωJ). Note that ωq also
depends on the external fluxes. The non-linear coupling in terms of these operators
is
Vˆnl = −η2s
√
ω˜C
2ωq
(aˆ+ aˆ†)2(bˆ+ bˆ†)
RWA−−−→ −η2s
√
ω˜C
2ωq
(aˆ2bˆ† + aˆ†2bˆ). (3.67)
3.5 Diagonalization of Linear Hamiltonian
I had the linear Hamiltonian
HˆL = ωaˆ†aˆ+ ωq bˆ†bˆ+ g1(aˆbˆ† + aˆ†bˆ). (3.68)
44
To diagonalize HˆL I define new operators cˆ and dˆ with
aˆ = µ1cˆ+ ν1dˆ, (3.69)
bˆ = µ2cˆ+ ν2dˆ, (3.70)
such that [cˆ, cˆ†] = 1 = [dˆ, dˆ†] and [cˆ, dˆ†] = 0 = [cˆ, dˆ]. This requires the conditions
|µ1|2 + |ν1|2 = 1 = |µ2|2 + |ν2|2, (3.71)
µ1µ
?
2 + ν1ν
?
2 = 0. (3.72)
The parametrization µ1 = cos θ, ν1 = − sin θ, µ2 = sin θ, ν2 = cos θ satisfies the con-
straints (3.71), (3.72). Substituting the relations into HˆL and setting the diagonal
terms to zero, I get a new Hamiltonian in the normal mode coordinates given by
HˆN = Ω1cˆ†cˆ+ Ω2dˆ†dˆ. (3.73)
The dressed frequencies are
Ω1 = ω +
∆
2
(
1−
√
1 +
4g21
∆2
)
, (3.74)
Ω2 = ω +
∆
2
(
1 +
√
1 +
4g21
∆2
)
. (3.75)
The detuning ∆ = ωq − ω is assumed to be positive. Note that for ∆  |g1|,
Ω1 → ω, and Ω2 → ωq.
In the following discussion, I denote the basis states of the resonator and qubit
system by |m〉⊗|n〉 ≡ |m n〉, where the first and second labels refer to the quantum
number of the resonator and qubit respectively. The eigenstates of the Hamiltonian
in the new basis are number excitations of the cˆ†cˆ and dˆ†dˆ operators. Denoting these
45
kets as
∣
∣C¯D¯
〉
,
|1¯0¯〉 = cos θ |10〉+ sin θ |01〉 , (3.76)
|0¯1¯〉 = − sin θ |10〉+ cos θ |01〉 , (3.77)
|2¯0¯〉 = cos2 θ |20〉+
√
2 cos θ sin θ |11〉+ sin2 θ |02〉 , (3.78)
|1¯1¯〉 = −
√
2 cos θ sin θ |20〉+ cos 2θ |11〉+
√
2 cos θ sin θ |02〉 , (3.79)
|0¯2¯〉 = sin2 θ |20〉 −
√
2 cos θ sin θ |11〉+ cos2 θ |02〉 . (3.80)
Explicitly, the sines and cosines are
sin θ =
1√
2
√
1− ∆√
4g21 + ∆2
, (3.81)
cos θ =
1√
2
√
1 +
∆
√
4g21 + ∆2
. (3.82)
The parameter θ is also given by tan 2θ = 2g1∆−1. When ∆  |g1|, tan 2θ → 0 or
θ → 0. So sin θ → 0 and cos θ → 1. Thus, |1¯0¯〉 → |10〉, |0¯1¯〉 → |01〉, |2¯0¯〉 → |20〉,
|2¯0¯〉 → |20〉. Similarly, when ∆ → 0, θ → pi/4. Hence, cos θ → 1/
√
2 and sin θ →
1/
√
2.
In terms of the normal mode operators,
Vˆnl = η′2(cos2 θ sin θcˆ†2cˆ− cos3 θcˆ†2dˆ− 2 cos θ sin2 θcˆ†cˆdˆ† + 2 cos2 θ sin θcˆ†dˆ†dˆ
+ sin3 θcˆdˆ†2 − sin2 θ cos θdˆ†2dˆ+ h.c.), (3.83)
where I have defined η′2 ≡ η2s
√
ω˜C
2ωq
.
46
3.6 Subspace Hamiltonian and Two-Photon Nonlinearity
I will work in the subspace spanned by the states {|0〉 ≡ |0¯0¯〉 , |a〉 ≡ |1¯0¯〉 , |b〉 ≡
|2¯0¯〉 , |c〉 ≡ |0¯1¯〉}. The Hamiltonian is
H =












0 0 0 0
0 Ω1 λ1 0
0 λ1 2Ω1 λ2
0 0 λ2 Ω2












. (3.84)
The parameters λ1 = (
√
2 cos2 θ sin θ)η′2 ≡ r1η′2 and λ2 = (−
√
2 cos3 θ)η′2 ≡ r2η′2.
One can use a Schrieffer-Wolf transformation (Appendix A) to find an effective
Hamiltonian
H˜ =
















0 0 0 0
0 Ω1 −
r21η
′2
2
Ω1
+O(η′32 ) O(η
′3
2 ) O(η
′3
2 )
0 O(η′32 ) 2Ω1 +
r21η
′2
2
Ω1
+O(η′32 ) r2η
′
2 +O(η
′3
2 )
0 O(η′32 ) r2η
′
2 +O(η
′3
2 ) Ω2 +O(η
′3
2 )
















. (3.85)
One can use this Hamiltonian to calculate the two-photon nonlinearity Nl. With
δ′ = Ω2 − 2Ω1 > 0, I have
Nl = η
′2
2
(
3r21
Ω1
− r
2
2
δ′
)
≈ −η
′2
2 r
2
2
δ′
= −g
2
2
δ′
, (3.86)
where I have identified η′2r2 with g2. With this nonlinearity and standard parameters,
the two-photon pi phase shift protocol can be implemented in a few hundred ns.
47
3.7 Numerical Results
In addition to my analytical model, I also diagonalize the Hamiltonian of the
system numerically by working in the tensor product space H = Hr ⊗ Hq of the
resonator and qubit. A basis state of H is written as |n〉 ⊗ |q〉 ≡ |n q〉, a tensor
product of the bases for the resonator and qubit spaces. The basis states in the
resonator space are the number excitations |n〉 which are eigenstates of the number
operator nˆ = aˆ†aˆ. The qubit space is written in the basis of qubit wavefunctions
ψq(φ) = 〈φ| q〉. For this purpose, I start with the non-linearized Hamiltonians
Hr =
qˆ2L
2C˜
+
Φˆ2L
2L
, (3.87)
Hq =
qˆ2
2C˜J
− EJ cos(φˆ+ φ′x)− EJ cos φˆ+
1
2
EL(φˆ+ φx)
2, (3.88)
VI = χEJ sin(φˆ+ φ′x)φˆL +
χ2
2
EJ cos(φˆ+ φ
′
x)φˆ
2
L −
χ
2C˜
qˆqˆL. (3.89)
Then I introduce creation and annihilation operators only for the resonator,
ΦˆL =
√
L˜ω~
2
(aˆ+ aˆ†), (3.90)
qˆL = −i
√
~
2L˜ω
(aˆ− aˆ†). (3.91)
In terms of these operators, the three interaction terms can be written as
V1 = η1(aˆ+ aˆ
†) sin(φˆ+ φ′x), (3.92)
V2 = η2(aˆ+ aˆ
†)2 cos(φˆ+ φ′x), (3.93)
V3 = iη3(aˆ− aˆ†)Nˆ . (3.94)
From the potential V2, the nonlinear coupling g2 is seen to be
g2 = η2 〈20| (aˆ+ aˆ†)2 cos(φˆ+ φ′x) |01〉 =
√
2η2 〈0q| cos(φˆ+ φ′x) |1q〉 . (3.95)
48
For my numerical analysis, I first remark that the dressed parameters like
ω˜C and C˜ are nearly equal to their corresponding bare counterparts ωC and C.
I let ~ = 1 and choose ωC/(2pi) = 1 GHz, ωJ/(2pi) = 5 GHz, ωL = 3ωJ , and
ω/(2pi) = 2.225 GHz. The characteristic impedance Z ≈ 449 Ω. I choose a χ = 0.17,
representing an easily achievable mutual inductance, from which follow η1/(2pi) =
400 MHz, η2/(2pi) = 16 MHz, and η3/(2pi) = 89 MHz.
I first plot g1, g2 and other parameters of the system as a function of the
fluxes φx and φ′x. The red marker in Figure 3.5 represents the starting point where
the system is in the photon-like state and the green marker represents the parking
point of the qubit. The starting point is chosen such that g1 vanishes. The parking
point is chosen so that the two-photon nonlinearity is still appreciable (at least a
few MHz) but not to close to the avoided crossing (see more below). At this point,
the system is still mostly photon-like and only marginally qubit-like. The detuning
δ = ωq − 2ω at this parking point is expected to be much larger than g2.
I also plot the dressed energy levels of the system, along with the two-photon
nonlinearity in Figure 3.7. Finally, I compare my numerical results with the analyt-
ical results derived previously. First, I test the accuracy of the nonlinear coupling
η′2r2 (3.86) which I associate with the numerical value derived from the expression
g2 =
√
2η2 〈0q| cos(φˆ + φ′x) |1q〉. I also compare the analytical and numerical values
of g1 = η1 〈0q| sin(φˆ + φ′x) |1q〉 in Figure 3.9. One can see that they are in good
agreement.
49
3.8 Adiabatic and Non-Adiabatic Loss
Now I discuss the effect of loss on the gate. Throughout the operation of the
gate the system remains mostly photon-like. Hence, loss is dominated by the cavity
decay at a rate κ. Apart from κ, for the photon-like state |2¯0¯〉, there are two other
decay channels due to the cavity-qubit coupling. In the limit ∆  |g1|, the linear
coupling g1 leads to a loss that is approximately
γ1 ≡ γg21/∆2 = γg21/(δ + ω)2. (3.96)
Similarly, for |δ|  |g2|, the nonlinear coupling leads to a loss
γ2 ≡ γg22/δ2. (3.97)
Including the cavity decay rate κ, the total decay rate of the two-photon-like state
becomes
Γ(δ) = κ+ γ1 + γ2. (3.98)
Assuming that g2 is time independent for simplicity, adiabaticity of the state
|2¯0¯〉 requires
g22|δ˙|2(δ2 + 4g22)−3  1. (3.99)
One can set this equal to some 2  1 and solve for
τh(δm) = −
1

∫ δm
δi
|g2|
(δ2 + 4g22)
3
2
dδ, (3.100)
which is the time taken to go from |δi|  |g2| at t = 0 to smaller values of detuning
with a minimum δm. The total dynamic loss during the process is given by
Ld(δm) =
2

∫ δi
δm
Γ(δ)
|g2|
(δ2 + 4g22)
3
2
dδ. (3.101)
50
When the detuning is held at δm for a time τs = piδm/g22, the static loss Ls(δm) =
τsΓ(δm). Thus,
Ls(δm) = pi
[
κδm
g22
+
γδm
(δm + ω)2
(
g1
g2
)2
+
γ
δm
]
, (3.102)
and the total time of the protocol is τg = 2τh + τs. Assuming δm  ω, Ls(δm) is
minimized when δm ≈ g2
√
γ/κ. However, the on-off ratio of the photon nonlinearity
goes like |δi/δm|, and a value of δm that makes this ratio at least a hundred is
desirable. For δ ∼ ω, one can make g1 ≈ g2 so that Ls(δ) < κδ/g22 + 2γ/δ. In this
regime Ls is limited by κ, as can be verified from Figure 3.11b. Thus, I optimize my
protocol so that the loss L = Ld + Ls  1. Note that one can minimize the static
loss by increasing g1, which has the effect of increasing g2. However, to retrieve
the photons with high fidelity, g1 should vanish or be comparable to g2 at large
detuning. I note that my protection is only against qubit noise and loss, and comes
at the cost of increased reliance on the cavity quality factor.
The protocol might also be limited by dephasing of the qubit due to flux
noise [42,43,77]. The average slopes of the single and two-photon energy levels with
respect to the reduced flux φx are approximately 50 MHz and 100 MHz respectively,
while the slope of the qubit energy level is at most 1 GHz for the parameters chosen.
However, the exact loss due to dephasing depends on the flux noise amplitude [46,78].
3.9 Conclusions and Outlook
In conclusion, I have demonstrated that by appropriately tuning two con-
trol fluxes, the nonlinear coupling in a system composed of photons in a resonator
51
coupled to a superconducting circuit enables a two-photon nonlinear phase shift op-
eration, with loss at large detuning limited only by the cavity quality factor. The
loss at large detuning can be further suppressed by increasing the strength of the
linear coupling g1, while at the same time assuring that it vanishes for large de-
tuning. This is highly desirable compared to the self-Kerr nonlinearity which leads
to more photon loss due to the noisy qubit at large detunings. Furthermore, my
approach may be adaptable to recent ultra-high quality factor resonators enabling
nonlinear optics quantum computing in a fully engineered system [70].
In the following chapter I consider the problem of simulating a system of
bosons in a lattice in the presence of an artificial magnetic field and three-body
on-site interactions. Again, I make use of the nonlinear nature of superconducting
circuits to achieve this goal.
52
(a)
(b)
Figure 3.5: (a) Contour plot of g1, and (b) g2 as a function of the external fluxes.
53
Figure 3.6: Variation of g1 and g2 along the trajectory defined by the arrows in
Figure 3.5. Note that the starting point denoted by the red cross is chosen so that
g1 is close to zero.
54
(a)
(b)
Figure 3.7: (a) The dressed energy levels of the coupled system. (b) The two-photon
nonlinearity. The detuning on the horizontal axis is the detuning along the arrows
from the red to the green markers. The minimum detuning δm is around −41 MHz.
55
(a)
(b)
Figure 3.8: (a) The detuning δ = ωq−2ω with ωq1 ≡ ωq. (b) The qubit nonlinearity
ωq2 − 2ωq1.
56
(a)
(b)
Figure 3.9: Comparison of the analytical (dashed) and numerical (solid) results. (a)
Plot of g1 = η1 〈0q| sin(φˆ + φ′x) |1q〉 and the analytical value of the same coupling
from (3.65). (b) Plot of g2 =
√
2η2 〈0q| cos(φˆ+ φ′x) |1q〉 and η′2r2 from (3.86).
57
Figure 3.10: The frequencies (in GHz) 2ω in blue, and ωq in red, with the analytical
(dashed) expressions derived from (3.49) and (3.50) respectively.
58
(a)
(b)
Figure 3.11: (a) A plot of the dimensionless dynamic loss Ld for κ = 1 kHZ, γ = 100κ
and 2 = 0.01. The detuning −536 MHz ≤ δm ≤ −41 MHz. (b) The total static
loss Ls in green, and the static loss without the effect of the cavity decay rate κ in
purple.
59
Chapter 4
Circuit-QED Implementation of the Pfaffian State
Parent Hamiltonian
4.1 Introduction
In nature fundamental particles are indistinguishable. For instance, all elec-
trons in the universe are identical in all respects. One cannot put labels on different
electrons to distinguish one electron from another. Therefore, exchanging any two
identical particles in a system should leave all observables in the system invari-
ant. If a system has n identical particles with positions r1, . . . rN and is described
by a wavefunction ψ(r1, . . . , rk, . . . , rl, . . . , rN) (ignoring spin for now), exchanging
particles at positions rl and rk should give
|ψ(r1, . . . , rk, . . . , rl, . . . , rN)|2 = |ψ(r1, . . . , rl, . . . , rk, . . . , rN)|2. (4.1)
Hence, the wavefunctions can differ at most by an exchange phase,
ψ(r1, . . . , rl, . . . , rk, . . . , rN) = eiφψ(r1, . . . , rk, . . . , rl, . . . , rN). (4.2)
60
It is taught in introductory quantum mechanics courses that the only allowable cases
are eiφ = ±1, which refer to bosons and fermions respectively. But this is a naive
viewpoint. While this is true in three and higher dimensions, in two dimensions,
one can have any complex phase. This is because the topology of two dimensions
is very different from that of higher dimensions. Exchanging two particles twice is
equivalent to moving one around another in a closed loop and in three and higher
dimensions, this loop can be deformed continuously to a point without ever crossing
the stationary particle. However, in two dimensions it is not possible to do this
without crossing the stationary particle. This leads to particle statistics which are
neither bosonic, nor fermionic. Frank Wilczek coined the term “anyons” to refer to
particles exhibiting these novel statistics [79].
Anyons exist as excitations in some condensed matter systems [80]. Such
systems have highly non-trivial groundstates that are described as having topological
order. The best studied example is the so called Laughlin state in the fractional
quantum Hall system at filling factor ν = 1/3 [81]. The filling factor ν = N/Nφ
is the ratio of the number of particles to the number of flux quanta in the system.
It carries Abelian anyons with exchange phase φ = pi/3 and electric charge ±1/3.
At filling factor ν = 1/5, a different kind of state is observed. This state, also
known as the Moore-Read state which has the form of a Pfaffian wave function [82]
admits non-Abelian anyons with charge ±1/4. A good practical reason for interest
in detecting and manipulating anyons is for their potential use in realization of
quantum memory that is protected from decoherence. Furthermore, as shown by
Freedman et al. [83] and Kitaev [84], certain types of non-Abelian anyons can be
61
manipulated for the purpose of universal quantum computation, also referred to as
topological quantum computation.
Greiter et al. proposed a parent Hamiltonian with three-body interactions [85]
which yields the so called Pfaffian state as its ground state, and excitations that are
anyons with charge 1/4 and statistical parameter φ = pi/8. Specifically, they con-
sidered fermions in a magnetic field with repulsive three-body contact interactions
of the form
Vi;jk =
∑
triples
δ(2)(zi − zj)δ(2)(zi − zk), (4.3)
where zi = xi + iyi is the complex representation of the position of particle i in
two dimensions. Although the fractional quantum Hall effect occurs for fermions,
bosonic systems with repulsive interactions can exhibit similar behaviors [86–88].
There have been several efforts to generate such Hamiltonians, using ultra-
cold atomic systems, see for instance refs. [89, 90]. However, the elimination of
two-body interactions while preserving the bosonic nature of excitations remains
challenging [91–95], as expected for perturbatively generated three-body terms [96].
In this chapter, I propose a scheme that uses superconducting circuits to achieve
this goal. In particular, I demonstrate how to engineer a three-body interaction
and the synthetic magnetic field required to implement the parent Hamiltonian of
Greiter et al. [85] on a lattice.
62
4.2 Parent Hamiltonian for the Pfaffian State
The parent Hamiltonian can be simulated on a discrete lattice [94]. An impor-
tant question to ask is how the transition from the continuum to the discrete case
modifies the ground state properties of the system. For example, a charged particle
moving in a magnetic field in two dimensions has energies separated into Landau
levels each of which is highly degenerate. However, in the presence of a discrete
lattice, the spectrum becomes the well known Hofstadter butterfly [97].
Specifically, I want to simulate the parent Hamiltonian
Hp = −J
∑
x,y
[
aˆ†x+1,yaˆx,ye
−ipiαy + aˆ†x,y+1aˆx,ye
+ipiαx + h.c.
]
+
∑
x,y
U3
6
aˆ†3x,yaˆ
3
x,y.
(4.4)
The indices x and y refer to different sites on a lattice where the bosons are located.
There are two main ingredients in Hp. The first is the presence of the phase de-
pendent hopping term analogous to that of a charged particle moving in a magnetic
field. The flux acquired by a bosonic particle under the evolution of Hp in moving
around a plaquette is αΦ0 where α is a dimensionless parameter and Φ0 is the su-
perconducting flux quantum. One can equivalently say that the particle acquires
a phase 2piα when moving around a plaquette. The second main ingredient is the
presence of a three-body repulsive on-site interaction of the form aˆ†3x,yaˆ
3
x,y.
In the discrete lattice case, there are two relevant length scales. The first is
the lattice spacing a and the second is the magnetic length lB =
√
~/(qB) in SI
units, where q is the charge of the particle and B the magnetic field in the system.
63
!0 !0
!0
!0 + U2
3!0 + U3
Figure 4.1: Energy levels of the system in the presence of a two and a three-body
interaction.
For an electron with charge q = e, this reduces to lB = 1/
√
piα. In the limit where
lB  a, which corresponds to weak magnetic fields, the system is weakly sensitive
to the discrete nature of the lattice.
In [86] it was argued that fractional quantum Hall physics persists until α . 0.3
for a system of atoms in optical lattices with a similar Hamiltonian as Eq. (4.4)
but with on-site two-body interactions. Their approach was to calculate the overlap
of the numerically calculated ground state of the discrete Hamiltonian with the
Laughlin state [81]. However, as shown in [98] the topological order in these systems
can be characterized by topological invariants such as the Chern numbers even in
the regime where α is larger. The study of topological invariants and topological
order in these systems is beyond the scope of this thesis. Nevertheless, with the
understanding that these Hamiltonians constitute a lot of interesting and important
physics, in the remainder of the chapter, I will focus exclusively on the simulation
of Hp [99].
64
4.3 Implementation of Magnetic Field
First I discuss the implementation of the magnetic hopping terms. There have
been several proposals in the past to engineer such Hamiltonians in the context of
circuit-QED systems [100, 101], and also proposals without breaking time reversal
symmetry in photonic systems [102, 103]. Here, I follow the approach of [59]. I
consider a lattice of three-body resonators coupled to each other using externally
modulated squids, as depicted in Figure 4.2. The three-body resonators (to be
discussed in the following section) are simply hybrid superconducting circuits biased
appropriately, and with suitable parameters. The frequencies of the resonators are
detuned from each other, the red denoting the one with lower frequency ωr compared
to the blue with frequency ωb. On the horizontal connections at ordinates y, the
modulations have phase φp = 2piαy whereas the vertical connections have no phase
difference. The modulation takes place at a frequency ωp = ωb − ωr and amplitude
δφ 1. That is
φx(t) = δφ(cosωpt+ φp(y)). (4.5)
In the rotating frame with the rotating wave approximation, this induces a hopping
Hamiltonian between two modes i and j of the form aˆ†i aˆje
iφp + aˆ†j aˆie
−iφp . The
difference between the present case and Ref. [59] is that there the hopping was
induced between two modes of the same waveguide, while here the hopping is induced
between two modes of different sites.
65
2⇡↵
x(t) =  cos(!pt + p)Hybrid Superconducting Circuits
Figure 4.2: Implementation of Hp.
4.4 Three-Body Resonators
I will now consider a model for the three-body resonators occupying each site
(x, y) in the lattice. Consider a Josephson junction with Josephson energy EJ , and
charging energy EC = e2/(2CJ), shunted by an inductance L in a superconducting
loop as shown in Figure 4.3. This leads to an inductive energy EL = Φ20/(4pi
2L).
Here, Φ0 = h/(2e) is the superconducting flux quantum and h is Planck’s constant.
An external flux Φx ≡ Φ0/(2pi)φx threading through the superconducting loop can
be used to control the energy levels of the system. The Hamiltonian of such an
66
EJ , EC
EL
x
Figure 4.3: A Josephson junction shunted by an inductive loop.
inductively shunted Josephson junction can be written as
H = 4ECNˆ
2 − EJ cos(φˆ+ φx) +
1
2
ELφˆ
2. (4.6)
Here φˆ is the operator corresponding to the phase across the junction and Nˆ is its
conjugate momentum representing the number of Cooper pairs tunneling through
the junction. They obey the commutation relation [φˆ, Nˆ ] = i. In the φ representa-
tion, Nˆ = −i∂/∂φ.
The regime considered in [104] with EJ > EC  EL, the so called fluxonium
regime, leads to a situation where the first energy transition is different from the
second transition and higher. Such nonlinearity leads to the isolation of the first
transition and the system forms a qubit. In contrast, I choose a regime where the
first and the second transitions are degenerate, while the third one is different as
shown in the second of Figure 4.1. This happens when EL ≈ EJ  EC , the so
called hybrid regime [46].
If the Josephson term in Eq. (4.6) were ignored, one would simply have a
67
harmonic oscillator. A harmonic oscillator with frequency ω has a Hamiltonian
H = ~ωcˆ†cˆ. (4.7)
It has a linear spectrum of energy levels. However, the presence of the Josephson
energy EJ induces a nonlinearity. This nonlinearity can be adjusted by tuning the
strength of EJ relative to EL. Depending on the ratio EJ/EL, the potential well
can have many minima like in the fluxonium regime. However, in the regime I
am interested in, EL ≈ EJ so that the potential has only one minimum. This
is important since I want the first two excited states E1 and E2 to be linear and
bosonic. Hence, the bottom of the potential cannot be too different from a harmonic
potential.
In the presence of nonlinearity, the energy spectrum is no longer uniform. In
the subspace consisting of the lowest states {|0〉 , |1〉 , |2〉 , |3〉}, I define an operator
aˆ such that
aˆ |n〉 = √n |n− 1〉 , n ∈ {1, 2, 3}, (4.8)
aˆ |0〉 = 0. (4.9)
Similarly, aˆ† is defined such that
aˆ† |n〉 =
√
n+ 1 |n+ 1〉 , n ∈ {0, 1, 2, 3}. (4.10)
Then in this subspace [aˆ, aˆ†] = 1 and
H = ~ωaˆ†aˆ+
U2
2
aˆ†2aˆ2 +
U3
6
aˆ†3aˆ3. (4.11)
Alternately,
H = ~ωnˆ+
U2
2
nˆ(nˆ− 1) + U3
6
nˆ(nˆ− 1)(nˆ− 2), (4.12)
68
where nˆ = aˆ†aˆ and nˆ |n〉 = n |n〉. From this one can infer that the ground state
energy E0 = 0 and the energy of the first excited state is E1 = ~ω. The energy
of the anharmonic second and third excited states are E2 = 2~ω + U2 and E3 =
3~ω + U3 + 3U2. This leads to the expressions for the nonlinearities
U2 = E2 − 2E1, (4.13)
U3 = E3 − 3(E2 − E1). (4.14)
I emphasize that these properties of the subspace are consistent with the observed
numerical results which will be derived in the following sections.
4.5 Optimization of Parameters
Since I am working in the non-perturbative regime where EJ . EL, optimiza-
tion has to be done fully numerically. There are three external parameters available
for tuning. The first two are the ratios α ≡ EC/EJ and β ≡ EL/EJ which are
fixed during fabrication. The third parameter is the external flux φx. I fix α = 0.05
and vary β and φx. I then plot U2 and U3 as a function of these parameters in the
regime where the potential of Eq. 4.6 has only a single well. The results are shown
in Figure 4.4.
Recall that I am interested in a pure three-body interaction. Therefore, I
seek points in these contours where U2 vanishes. But due to numerical constraints
arising from finite step size, I can only find those points for which U2 < 5 × 10−4.
At first thought, it might seem that all these points are valid bias points for the
construction of a bosonic three-body resonator. After all, these are points where U3
69
is significantly larger than U2 which is close to zero. However, this is not sufficient.
One still has to verify that the three lowest excitations behave bosonically. To do
this, I adopt the following construction.
I couple two hybrid superconducting devices inductively with mutual induc-
tance M as depicted in Figure 4.6. In the general case with devices corresponding
to different parameters, the Hamiltonian of the system can be written as
HC = H1 +H2 + VI , (4.15)
where for i ∈ {1, 2}, in the φ basis
Hi = −4ECi
∂2
∂φ2
− EJi cosφi +
1
2
ELi
(
φi +
χ
∑
j∈{1,2}(1− δij)φxj
2ELi
)2
. (4.16)
The interaction term is
VI =
1
2
χφ1φ2. (4.17)
The charging energy ECi = e2/(2CJi). The inductive energies are given by
ELi = Φ20/(4pi
2Li), where Li are the loop inductances, and the fluxes through the
loops are denoted by φxi. χ is a coupling parameter with units of energy given by
χ =
(
Φ0
2pi
)2 M
L1L2
. (4.18)
I let J = χ/2 so that VI = Jφ1φ2 and from now on, assume that the two circuits
are identical.
I label the three lowest eigenstates of circuit i by {|0i〉 , |1i〉 , |2i〉}. The coupled
Hamiltonian HC can be written in the basis |m n〉 ≡ |m1 n2〉. The key idea is the
following: if one initializes the system in the state |m n〉 where m,n ∈ {0, 1, 2}, then
70
the system evolves in such a way that m and n do not leak into the other excited
manifolds {3, 4, . . . }. To check this, I analyze the system in the sub-manifolds with
m + n ∈ {0, 1, 2, 3}. The basis states in the sub-manifolds are {|00〉}, {|01〉 , |10〉},
{|02〉 , |11〉 , |20〉}, and {|03〉 , |12〉 , |21〉 , |30〉}.
If the parameters are optimum, the Hamiltonian in the subspaces must have
the following form. For the single excitation subspace,
H1 =




ω01 Ω
Ω ω10



 . (4.19)
Obviously, by symmetry ω01 = ω10. In the subspace consisting of the states with
two excitations one must have a Hamiltonian of the form
H2 =








ω02 g1 1
g1 ω11 g1
1 g1 ω20








, (4.20)
where 1  g1. For the subspaces to be highly linear and bosonic, one must have
ω02 = ω20 ≈ ω11 (since U2 ≈ 0) and g1 ≈
√
2Ω, with
√
2 being the usual bosonic
enhancement factor. However, the third subspace must be nonlinear. In the basis
{|03〉 , |12〉 , |21〉 , |30〉}, its Hamiltonian should look like
H3 =












ω03 g2 2 3
g2 ω12 g3 2
2 g3 ω21 g2
3 2 g2 ω30












, (4.21)
with 2, 3  g2, g3. The couplings g2 ≈
√
3Ω and g3 ≈
√
4Ω = 2Ω are also enhanced
by the usual bosonic factors. By symmetry, the energies ω03 = ω30 and ω12 = ω21
71
but crucially ω30 6= ω21. This is the result of the three body term U3. If the coupled
circuits are in the subspaces with two or less total excitations, they remain in these
subspaces. Thus, each hybrid circuit effectively has energy levels {|0〉 , |1〉 , |2〉} with
bosonic characteristics, but separated from |3〉 with a three-body interaction.
With these additional conditions, it is revealed that the minimum value of β
for which bosonic nature is retained is for β = 1.4 corresponding to φx ≈ 2.683 or
φx/(2pi) ≈ 0.427. Increasing β has the advantage that U2 is suppressed. But at the
same time U3 gets smaller. First, I plot the energy levels and wavefunctions of the
system corresponding to these parameters in Figure 4.9.
I claim that the three lowest energy states {|0〉 , |1〉 , |2〉} are very close to the
three lowest states of a harmonic oscillator. I first check this by confirming the
linearity of these levels. In Figure 4.10(a), I have plotted the first few energy levels
of the system as a function of φx. Then in Figure 4.10(b) I plot U2 and U3 as a
function of φx. It is clear that U2 changes sign at two values of φx and at these points
U3 ≈ 15EJ is non-zero, the first of which corresponds to our point of operation.
I also plot the matrix elements | 〈0|φ |2〉 / 〈0|φ |1〉 | and | 〈1|φ |2〉 /(
√
2 〈0|φ |1〉)|
in Figure 4.11. Near the optimum point where U2 = 0, the matrix elements are close
to the matrix elements of the harmonic oscillator operator aˆ + aˆ†. Therefore, one
can guess that
φ =
√
1
2m~ω
(aˆ+ aˆ†), (4.22)
for some effective mass m and some frequency ω. Now the effective mass of the
oscillator can only come from the charging energy EC since it is the only energy
72
scale present in ∂2/∂φ2. Therefore, one can equate
− 1
2m
∂2
∂φ2
= −4EC
∂2
∂φ2
(4.23)
to get m = 1/(8EC).
However, the variation of the frequency ω as a function of the energy scales is
much more involved. That is because V (φ) is not necessarily quadratic at this point
where U2 vanishes. In fact, it has significant terms of O(φ8). So to find ω, I rely on
the numerical results (see Figure 4.9), and it shows that ω ≈ 0.59 GHz. Therefore,
one can guess that
φ =
√
1
2m~ω
(aˆ+ aˆ†) = 2
√
ωC
ω
(aˆ+ aˆ†) ≈ 0.58(aˆ+ aˆ†), (4.24)
and so 〈0|φ |1〉 ≈ 0.58. In Figure 4.12, I plot the matrix element 〈0|φ |1〉 around the
optimum bias point. Therefore, in the subspace of the lowest three energy levels of
the system, the operator φ behaves like aˆ+ aˆ†.
Finally, I plot the wavefunctions for the states |0〉, |1〉, |2〉, and |3〉 and compare
them to the eigenfunctions of a harmonic oscillator with an effective mass m =
1/(8EC) and frequency ω, but shifted to the left towards the minima of the potential
V (φ) at φ0 ≈ −0.634. Letting Λ = ω/(8ωC), the normalized harmonic oscillator
eigenfunctions are given by
ψn(φ) =
1√
2nn!
(
Λ
pi
) 1
4
e−Λ
(φ−φ0)
2
2 Hn
[√
Λ(φ− φ0)
]
, (4.25)
where Hn are the usual Hermite polynomials. I find that the ground, first, and
second excited states are closer to the harmonic oscillator eigenfunctions than the
third excited state as shown in Figure 4.13.
73
I now let J/EJ = 8× 10−4 << U3/EJ and study the dynamics of two coupled
circuits in the subspaces with one, two, and three excitations as depicted in Figure
4.14. If one initializes the system in the states |01〉, it oscillates at frequency Ω
until |10〉 is occupied. A coherent exchange of excitations takes place. Similarly,
initialization in the state |02〉 results in the occupation of states |11〉 and |20〉.
However, in the subspace of three excitations, the initial state |12〉 evolves into the
state |21〉, but crucially, the states |03〉 and |30〉 do not get populated. This is
because of the presence of U3. Hence, the manifold of states {|0〉 , |1〉 , |2〉 , |3〉} can
be described by the Hamiltonian
H = ~ωaˆ†aˆ+
U3
6
aˆ†3aˆ3. (4.26)
4.6 Experimental Issues
Now I consider experimental issues involving the realization and detection of
Pfaffian states in the proposed circuit-QED system. As discussed above the three-
body interaction must be larger than the tunneling J . That is, one should make U3
as large as possible and J  U3. However, U3 is bounded from above as a small
fraction of the Josephson energy EJ . With a Josephson energy of tens of GHz,
one can achieve U3 of a few hundred MHz. Now J/h determines the frequency Ω
at which the coherent oscillations take place. Hence, J/h  T−11 , T−12 where T1
and T2 are the relaxation and decoherence times respectively. According to recent
experiments, T1, T2  10 µs [25]. Therefore, having J/h ≈ 10 MHz assures that
many oscillations take place before coherence is lost.
74
The system is robust against charge noise. To understand the effect of charge
noise in the system, I first plot the off diagonal matrix elements of Nˆ as a function of
φx. The largest matrix elements are those between adjacent levels, i.e. 〈n| Nˆ |n+ 1〉.
The other matrix elements are non-zero but comparatively smaller. In fact, the ma-
trix elements are not much more different that of a transmon as in Figure 7(a)
of [41]. So one can expect the charge noise to be suppressed just as in the trans-
mon. The diagonal matrix elements are suppressed significantly, which is expected
since the eigenstates of the system are not eigenstates of Nˆ (or equivalently charge
eigenstates). This is because EJ  EC , i.e. one is not in the charge regime, and
charge noise will not play a significant role.
The bias flux, however, needs to be controlled precisely. Recall the Hamilto-
nian in the two excitation subspace
H2 =








ω02 g1 1
g1 ω11 g1
1 g1 ω20








. (4.27)
Preserving the bosonic nature of the system requires that 1  g1. From further
numerical analysis (not presented here), it is evident that U2 being close to zero
is not as strict a requirement as   g1. This second condition requires that one
reduces U3, which is possible by increasing β. However, as discussed before J  U3
but J/h  T−11 , T−12 . Thus, either the flux has to be controlled precisely or the T1
and T2 times of the superconducting circuits have to be improved.
In experimental realizations, one needs to confirm that the system has nonlin-
ear interactions. For this, I suggest a correlation function measurement. Specifically,
75
when a single site with the Hamiltonian (4.11) is driven by a weak coherent field,
U2 and U3 can be obtained from measuring the output correlation functions
g(2) =
〈aˆ†2aˆ2〉
〈aˆ†aˆ〉2 , (4.28)
g(3) =
〈aˆ†3aˆ3〉
〈aˆ†aˆ〉3 , (4.29)
respectively. The details are in Appendix B. Such correlation function measurements
have been successfully achieved in the microwave domain using quadrature ampli-
tude measurements [105]. Alternatively, one can perform nonlinear spectroscopy to
map out the anharmonicity in the energy levels [106].
4.7 Conclusions and Outlook
In conclusion, in this chapter I have demonstrated how one can implement
the parent Hamiltonian for the Pfaffian state using a circuit-QED architecture.
Specifically, I proposed techniques for implementation of magnetic hopping terms
for bosons in a lattice, and the creation of strong pure three-body on-site interac-
tions. This proposal for three-body interactions is notable for its simplicity, and I
have argued that it is experimentally feasible.
76
(a)
(b)
Figure 4.4: A plot of (a) |U2|, and (b) U3 as a function of φx and β.
77
Figure 4.5: Plot of U3 for those points where U2 is optimized to be less than 5×10−4.
The coordinates refer to the pair (φx, β) at these points. Note that the plot is not
meant to be continuous.
M
Figure 4.6: Two identical hybrid superconducting circuits coupled with mutual in-
ductance M .
78
(a)
(b)
Figure 4.7: Contour plots of the norms of (a) g1/(
√
2Ω), and (b) 1/g1. The first
needs to be close to unity and the second needs to be much smaller than unity.
79
Figure 4.8: The first nine energy levels and modulus of the wavefunctions of the
fluxonium circuit for α = 0.05, β = 1.4, and φx ≈ 2.683. The purple curve represents
the potential.
Figure 4.9: The first four energy levels and (unnormalized) wave functions.
80
(a)
(b)
Figure 4.10: (a) The first nine energy levels of the system as a function of φx. (b)
Variation of U2 and U3 with φx. The vertical line represents the operating point
where U2 ≈ 0 but U3 > 0, at φx ≈ 2.683.
81
(a)
(b)
Figure 4.11: (a) The ratios | 〈1|φ |2〉 /(
√
2 〈0|φ |1〉)|, and (b) | 〈0|φ |2〉 |/| 〈0|φ |1〉 |.
82
Figure 4.12: The matrix element 〈0|φ |1〉 whose value is close to the predicted value
0.58.
Figure 4.13: Comparison of the normalized eigenfuctions of the system (solid curves)
with the harmonic oscillator eigenfunctions (dashed).
83
(a)
(b)
Figure 4.14: Dyamics of the (a) single, and (b) two excitation subspaces.
84
Figure 4.15: Oscillations in the three excitation manifold. Transition to |03〉 and
|30〉 are suppressed due to the presence of U3. This indicates the presence of a
three-body interaction.
Figure 4.16: 〈m| Nˆ |n〉 for m 6= n.
85
Figure 4.17: Plot of the first few diagonal matrix elements of Nˆ evaluated to machine
precision.
86
Chapter 5
A Chemical Potential for Photons
5.1 Introduction
In statistical mechanics, a system of fixed volume V , which can exchange
both energy and particles with a reservoir in equilibrium at temperature T can be
described using the grand canonical ensemble [107]. The constraint on the total
number of particles in the system plus reservoir gives rise to a Lagrange multiplier
which is defined as the chemical potential µ. The partition function Z in the grand
canonical ensemble is
Z(µ, V, T ) =
∞∑
N=0
∑
j
e−β(E
(N)
j −µN), (5.1)
where E(N)j is the energy of a configuration of N particles in state j and β = 1/kBT .
In this system, the temperature and number of particles are independent variables.
For an ideal Bose gas, this gives rise to an expected number of particles
〈N〉 =
∑

1
eβ(−µ) − 1 . (5.2)
However, if one considers the thermodynamics of black-body radiation confined
87
in a cavity, one has
〈N〉 =
∑

1
eβ − 1 , (5.3)
i.e. the chemical potential is non-existent. This is because black body radiation
confined in a cavity does not have a fixed number of excitations (photons). The
average particle number does not follow a given conservation law but adjusts itself
to the available thermal energy. This is the essence of the Stefan-Boltzmann law,
i.e. the internal energy of photons per unit volume is given by
U
V
=
pi2
15
(kT )4
(~c)3
, (5.4)
and hence the specific heat is unbounded as T →∞,
cV =
∂
∂T
(
U
V
)
=
4pi2k4T 3
15(~c)3
. (5.5)
As one lowers the temperature of the system, the number of photons decreases to
zero, as they are absorbed by the walls of the confining cavity. Hence, no macroscopic
occupation of the ground state takes place.
Later, it was understood that in the absence of absorbing walls, photons can
acquire a non-zero chemical potential, e.g. photon emission in semiconductor diodes
(LED) [108]. Thus the useful concept of chemical potential started to be applied to
these systems [109–111]. More recently, it was shown that photons can thermalize
with a non-zero chemical potential and form a Bose-Einstein condensate [112–115]
when interacting with a nonlinear medium. A Bose-Einstein condensate (BEC) [116]
is a state of matter where a system comprising of bosons all collapse to the ground
state. This happens when the interparticle separation becomes comparable to the
88
De-Broglie wavelength of the particles, which begin to overlap. The system can
then be described by a single coherent macroscopic wave function. This overlap
of wavefunctions can only happen if the particle number is fixed, and so photons
cannot form a BEC under these conditions. What is required then is an appropriate
method to keep the photon number fixed, and hence generate a chemical potential.
There have been several theoretical proposals for generating chemical potential
of photons [117–119]. Here, I develop a simpler approach than these theories. In
particular, by parametrically coupling a photonic system to a thermal bath, I show
that a photonic system can equilibrate to the temperature of the bath, with a
chemical potential given by the frequency of the parametric coupling. Although it
is possible to apply this scheme to both circuit-QED and optomechanical systems,
I focus on the circuit-QED part.
5.2 General Idea
Consider a system of choice with Hamiltonian HS coupled via λHSB to a bath
with Hamiltonian HB and initial state ρB ∝ e−βHB [120]. I will follow this approach
with one small modification. I replace the coupling λ with a parametric coupling
via λ→ 2λ cosωpt. That is, consider the Hamiltonian
H = HS + 2λ cosωptHSB +HB, (5.6)
with initial conditions ρB ∝ e−βHB . The parametric coupling will enable up and
down-conversion of bath excitations to photons, which will lead to a controlled
chemical potential.
89
To see this explicitly, I will assume that HSB is bi-linear, of the form
HSB =
∑
j
(aˆj + aˆ
†
j)Bˆj, (5.7)
where Bˆj is a bath operator and there exists aˆj, nˆj such that [aˆj, nˆj] = aˆj, as
occurs naturally for photons. This property defines particle numbers nˆj and the
total particle number Nˆ =
∑
j nˆj.
I now consider what happens when the energy scales of the bath are small
compared to ωp, but the energy scales of the system are comparable to it. To do
this, let HS = H ′S +HS,⊥ where HS,⊥ includes all terms in HS that do not commute
with Nˆ =
∑
j nˆj. In this regime, one can move to a rotating frame with the unitary
transformation U = e−itωpNˆ . The transformed system Hamiltonian becomes
U †HSU − i~U †
∂U
∂t
≈ H ′S − ~ωpNˆ , (5.8)
where I have neglected U †HS,⊥U by making the rotating wave approximation (RWA),
requiring ‖HS,⊥‖  ~ωp.
Meanwhile, the bath Hamiltonian remains the same, while the system-bath
coupling terms become
[
aˆj + aˆ
†
j + (e
−2iωptaˆj + e
2iωptaˆ†j)
]
Bˆj ≈
[
aˆj + aˆ
†
j
]
Bˆj. (5.9)
The key approximation is again the RWA to neglect e−2iωptaˆj-type terms, consis-
tent for a bath whose two-point correlation function 〈Bˆi(t + τ)Bˆj(t)〉 has a cutoff
frequency ωc < ωp. Thus, the system bath coupling in the RWA becomes
H ′SB =
∑
j
[aˆj + aˆ
†
j]Bˆj. (5.10)
90
Through this set of transformations, and the rotating wave approximation, one has
a new system-bath Hamiltonian which takes the traditional form
H = H ′S − µNˆ + λH ′SB +HB, (5.11)
where one can identify µ ≡ ~ωp as the chemical potential. For weak coupling λ and
an infinite bath at inverse temperature β, one can expect the system to thermalize
in the long-time limit to a density matrix
ρ ≈ e−β(H′S−µNˆ). (5.12)
5.3 Circuit-QED Implementation of Parametric Hamilto-
nian
I now discuss a circuit-QED architecture to implement the Hamiltonian in
(5.6). I model the thermal bath by a transmission line (TL) which is in thermal
equilibrium at a temperature T by virtue of its interaction with some impedance
Z(ω) which can be modeled by a resistor. For simplicity, I model the photonic
system as a single mode bosonic oscillator. To derive an effective chemical potential
for photons, I couple the photons parametrically to the thermal bath. This allows
photons to exchange energy with the bath.
The parametric coupler consists of a Wheatstone configuration that acts as the
right circuit (Figure 5.1) of the TL. It consists of four identical Josephson junctions
in a Wheatstone bridge configuration [121]. I first derive the Hamiltonian of the
coupler. For this purpose, I assume that each junction of the coupler has a large
91
area, and hence, a large capacitance, so that its charging energy can be ignored. In
this approximation, the Hamiltonian of the Wheatstone bridge is
Hw = −4EJ
[
cos
(
Φx
4ϕ0
)
cos
(
ΨX
2ϕ0
)
cos
(
ΨY
2ϕ0
)
cos
(
ΨZ
2ϕ0
)]
−4EJ
[
sin
(
Φx
4ϕ0
)
sin
(
ΨX
2ϕ0
)
sin
(
ΨY
2ϕ0
)
sin
(
ΨZ
2ϕ0
)]
. (5.13)
Here ϕ0 = Φ0/(2pi), Φ0 = h/(2e) being the superconducting flux quantum. Let
Φx = Φ0/2 be the bias flux through the loop of the Wheatstone bridge. Furthermore,
assume that the mode intensities ΨX ,ΨY ,ΨZ  Φ0. Expanding Hw in ψi = Ψi/Φ0,
i ∈ {X, Y, Z} to third order, one gets
Hw = −2
√
2EJ + µ
(
ψ2X + ψ
2
Y + ψ
2
Z
)
+ λψXψY ψZ . (5.14)
where µ =
√
2EJpi2 and λ = −2
√
2EJpi3 have dimensions of energy.
I assume that the TL is coupled to the mode ΨX = Φ1−Φ2 and the system is
coupled to the mode ΨY = Φ4 − Φ3. The driven mode is ΨZ = Φ1 − Φ3 + Φ2 − Φ4.
The TL will be connected to an external impedance on the left at z = 0 via a
capacitance CL. The external impedance is in thermal equilibrium at temperature
T .
The Lagrangian Ltl of the TL is derived in Appendix C. The mode ΨX can be
written in terms of the TL modes as
ΨX =
∑
ν
ξνϕνL, (5.15)
where I have let ϕνL ≡ ϕν(z = L), i.e. the boundary term. The Lagrangian of the
system is
L = Ltl + LS − VC . (5.16)
92
TLR 
\
\
\ \\x 
L1
System
^X
^Y
Y^Z
Z(t
)
CL
z = 0 z = L
Figure 5.1: A transmission line (blue) is coupled to the mode ΨX . The system
comprising the LC circuit is coupled to the mode ΨY . The mode ΨZ is driven
harmonically at frequency ωp.
The individual terms are
Ltl =
1
2
∑
ν
[
ξ˙2ν − ω2νξ2ν
]
− µ
Φ20
Ψ2X
=
1
2
∑
ν
[
ξ˙2ν − ω2νξ2ν
]
− µ
Φ20
∑
µ,ν
ξµξνϕµLϕνL, (5.17)
LS =
1
2
CΨ˙2Y −
(
1
2L
+
µ
Φ20
)
Ψ2Y , (5.18)
VC =
λ
Φ30
ΨXΨY ΨZ . (5.19)
I assume that ψZ = ΨZ/Φ0 = A cosωpt is a classical drive. I define another canoni-
93
cally conjugate momentum
qY =
∂L
∂Ψ˙Y
= CΨ˙Y . (5.20)
Ignoring the coupling between different TLR modes, the system Hamiltonian is
H = Htl +HS + V , (5.21)
where the individual Hamiltonians are
Htl =
∑
ν
[
bˆ†ν bˆν +
1
2
]
~ων , (5.22)
HS =
[
q2Y
2C
+
(
1
2L
+
µ
Φ20
)
Ψ2Y
]
→
[
aˆ†aˆ+
1
2
]
~ωc, (5.23)
V = λA
Φ20
cosωpt
∑
ν
FνψνLΨY → (A cosωpt)
∑
ν
gν(bˆν + bˆ
†
ν)(aˆ+ aˆ
†).(5.24)
I have defined a coupling
gν =
~λ
2Φ20
ϕνL√
Cωcων
, (5.25)
which has dimensions of energy. I have also introduced quantum operators aˆ and
aˆ† that satisfy [aˆ, aˆ†] = 1. In terms of these operators
ΨY =
√
~
2Cωc
(aˆ+ aˆ†), (5.26)
qY = −i
√
Cωc~
2
(aˆ− aˆ†), (5.27)
where ωc = 1/
√
L˜C with 1/L˜ = 1/L+ 2µ/Φ20.
One can now perform a unitary transformation U = e−iaˆ
†aˆωpt to move into
the frame moving at the pump frequency ωp. The Hamiltonian then undergoes the
94
transformation
H˜ = U †HU − i~U †∂U
∂t
=
∑
ν
[
bˆ†ν bˆν +
1
2
]
~ων +
[
aˆ†aˆ+
1
2
]
~∆
+
A
2
(eiωpt + e−iωpt)
∑
ν
gν(bˆν + bˆ
†
ν)(aˆe
−iωpt + aˆ†eiωpt), (5.28)
where the detuning ∆ = ωc − ωp. Ignoring terms oscillating rapidly with frequency
ωp in the rotated frame, and letting Gν = Agν/2, I arrive at a time independent
Hamiltonian
H =
∑
ν
[
bˆ†ν bˆν +
1
2
]
~ων +
[
aˆ†aˆ+
1
2
]
~∆ +
∑
ν
Gν(bˆν + bˆ
†
ν)(aˆ+ aˆ
†). (5.29)
I note that this Hamiltonian has the same form as the Hamiltonian that arises
in optomechanics where a one-sided cavity with a mechanical oscillator is driven
with a classical field [122]. In that case, the coupling is to a single mechanical mode
µ so that
H =
∑
ν
[
bˆ†ν bˆν +
1
2
]
~ων +
[
aˆ†aˆ+
1
2
]
~∆ +Gµ(bˆµ + bˆ†µ)(aˆ+ aˆ
†). (5.30)
5.4 Input-Output Formalism
Just as in the mechanical case, I assume coupling to a single TL mode µ,
which is in turn coupled to a thermal bath. I also assume that the modes aˆ and bˆµ
have decay rates Γ = κ + κi and γµ. The additional decay rate κ of aˆ arises from
coupling to the transmission line, while κi is the intrinsic decay rate due to other
loss mechanisms in the absence of coupling to the transmission line. The decay rate
95
γµ arises from the coupling of the TL to the thermal bath. I now define the vectors
v(t) =












bˆµ(t)
bˆ†µ(t)
aˆ(t)
aˆ†(t)












, (5.31)
ξ0(ω) ≡ ξin(t = t0, ω) =












bˆµ(t = t0, ω)
bˆ†µ(t = t0, ω)
aˆ(t = t0, ω)
aˆ†(t = t0, ω)












≡












bˆµ0(ω)
bˆ†µ0(ω)
aˆ0(ω)
aˆ†0(ω)












. (5.32)
I also define the matrices
A =












−iωµ − γµ/2 0 −iGµ −iGµ
0 iωµ − γµ/2 iGµ iGµ
−iGµ −iGµ −i∆− Γ/2 0
iGµ iGµ 0 i∆− Γ/2












, (5.33)
B =












√
γµ 0 0 0
0
√
γµ 0 0
0 0
√
κ 0
0 0 0
√
κ












, (5.34)
C =












√
γµ 0 0 0
0
√
γµ 0 0
0 0 Γ/
√
κ 0
0 0 0 Γ/
√
κ












. (5.35)
96
The Heisenberg equations of motion for the system with Hamiltonian HS can
be written as
d
dt
v(t) = Av(t)−Bξin(t). (5.36)
Let the input field column vector be
ξin(t) =












bˆµ,in(t)
bˆ†µ,in(t)
aˆin(t)
aˆ†in(t)












=
1√
2pi
∫ ∞
−∞
dω e−iω(t−t0)ξ0(ω), (5.37)
where t0 < t is the initial time. This vector consists of components of the noise
affecting the system. Define
v˜(ω) =
1√
2pi
∫ ∞
−∞
eiω(t−t0) v(t) dω. (5.38)
The corresponding inverse relation is given by
v(t) =
1√
2pi
∫ ∞
−∞
e−iω(t−t0) v˜(ω) dω. (5.39)
Equation (5.36) then becomes
1√
2pi
∫ ∞
−∞
(−iω)e−iω(t−t0) v˜(ω) dω
=
1√
2pi
∫ ∞
−∞
e−iω(t−t0) Av˜(ω) dω − 1√
2pi
∫ ∞
−∞
e−iω(t−t0) Bξ0(ω) dω. (5.40)
This implies that
∫ ∞
−∞
dω e−iω(t−t0) [(iωI + A)v˜(ω)−Bξ0(ω)] = 0. (5.41)
Thus, (iωI + A)v˜(ω) − Bξ0(ω) = 0 or v˜(ω) = (iωI + A)−1Bξ0(ω) ≡ MBξ0(ω),
where M = (iωI + A)−1.
97
Let the output field column vector be
ξout(t) =












bˆµ,out(t)
bˆ†µ,out(t)
aˆout(t)
aˆ†out(t)












=
1√
2pi
∫ ∞
−∞
ξ˜out(ω)e
−iω(t−t0)dω. (5.42)
The input-output relations for aˆ and bˆµ are
aˆout(t)− aˆin(t) =
Γ√
κ
aˆ(t), (5.43)
bˆµ,out(t)− bˆµ,in(t) =
√
γµbˆµ(t). (5.44)
I therefore have the input output relations in matrix form as
ξ˜out(ω) = ξ0(ω) + Cv˜(ω)
= ξ0(ω) + CMBξ0(ω)
= (I + CMB) ξ0(ω)
≡ Sξ0(ω). (5.45)
Using this result, one can write the output operators aˆout and aˆ
†
out as




aˆout(ω)
aˆ†out(ω)



 =




ζ¯µ(ω) η¯µ(ω)
η¯∗µ(−ω) ζ¯∗µ(−ω)








bˆµ0(ω)
bˆ†µ0(ω)



+




α¯(ω) β¯(ω)
β¯∗(−ω) α¯∗(−ω)








aˆ0(ω)
aˆ†0(ω)



 ,
(5.46)
98
where the frequency dependent parameters are given by
ζ¯µ(ω) =
4Γ
√
γµGµ (2 (ω + ωµ)− iγµ) (Γ + 2i(∆ + ω))
D1(ω)
,
η¯µ(ω) =
4Γ
√
γµGµ (2(ω − ωµ)− iγµ) (Γ + 2i(∆ + ω))
D1(ω)
,
α¯(ω) = − [(Γ− 2i∆)
2 + 4ω2] + 32G2µωµ(iΓ + 2∆) + 4ω
2
µ[(Γ− 2i∆)2 + 4ω2]
D2(ω)/(γµ − 2iω)2
,
β¯(ω) = −32iG
2
µΓωµ
D2(ω)
, (5.47)
with D1(ω) and D2(ω) given by
D1(ω) =
√
κ[(γµ + 2iω)
2 (4∆2 + (Γ + 2iω)2
)
+ 4ω2µ
(
4∆2 + (Γ + 2iω)2
)
−64∆G2µωµ], (5.48)
D2(ω) = [4∆
2 + (Γ− 2iω)2](γµ − 2iω)2 − 64G2µ∆ωµ
+4ω2µ[4∆
2 + (Γ− 2iω)2]. (5.49)
I am interested in the field inside the cavity and can write




aˆ(ω)
aˆ†(ω)



 =




ζµ(ω) ηµ(ω)
η∗µ(−ω) ζ∗µ(−ω)








bˆµ0(ω)
bˆ†µ0(ω)



+




α(ω) β(ω)
β∗(−ω) α∗(−ω)








aˆ0(ω)
aˆ†0(ω)



 .
(5.50)
The corresponding frequency dependent parameters for the intracavity field are
ζµ(ω) =
4Gµ
√
γµ[Γ− 2i(∆ + ω)][iγµ + 2(ω + ωµ)]
[4∆2 + (Γ− 2iω)2](γµ − 2iω)2 − 64G2µ∆ωµ + 4ω2µ[4∆2 + (Γ− 2iω)2]
,
ηµ(ω) =
4Gµ
√
γµ[Γ− 2i(∆ + ω)][iγµ + 2(ω − ωµ)]
[4∆2 + (Γ− 2iω)2](γµ − 2iω)2 − 64G2µ∆ωµ + 4ω2µ[4∆2 + (Γ− 2iω)2]
.
(5.51)
99
Similarly,
α(ω) =
2i
√
κ[−16G2µωµ + (iΓ + 2(∆ + ω))(γµ − 2i(ω − ωµ)(γµ − 2i(ω + ωµ))]
[4∆2 + (Γ− 2iω)2](γµ − 2iω)2 − 64G2µ∆ωµ + 4ω2µ[4∆2 + (Γ− 2iω)2]
,
β(ω) = − 32iG
2
µ
√
κωµ
[4∆2 + (Γ− 2iω)2](γµ − 2iω)2 − 64G2µ∆ωµ + 4ω2µ[4∆2 + (Γ− 2iω)2]
.
(5.52)
5.5 Correlation Functions and Thermal Spectrum
Now I calculate the correlation functions for the photon intracavity field. One
can write the input field operators in terms of their Fourier components as
aˆin(t) =
1√
2pi
∫ ∞
−∞
dω e−iω(t−t0)aˆ0(ω), (5.53)
aˆ†in(t) =
1√
2pi
∫ ∞
−∞
dω e−iω(t−t0)aˆ†0(ω) =
1√
2pi
∫ ∞
−∞
dω eiω(t−t0)aˆ†0(−ω). (5.54)
Note that
[aˆ0(−ω)]† =
1√
2pi
∫ ∞
−∞
dt eiω(t−t0)aˆ†in(t) = aˆ
†
0(ω). (5.55)
The commutator [aˆin(t), aˆ
†
in(t
′)] = δ(t−t′) gives [aˆ0(ω), aˆ†0(−ω′)] = δ(ω−ω′). Hence,
aˆ†0(−ω′) and aˆ0(ω) are the relevant Bose operators. One can similarly define bˆ†µ0(−ω′)
and bˆµ0(ω).
One has the correlations
〈bˆ†µ0(−ω′)bˆµ0(ω)〉 = δµµδ(ω′ − ω)n1(~ω′), (5.56)
〈bˆµ0(−ω′)bˆ†µ0(ω)〉 = δµµδ(ω′ − ω)(1 + n1(~ω′)), (5.57)
〈aˆ†0(−ω′)aˆ0(ω)〉 = δ(ω′ − ω)n2(~ω′), (5.58)
〈aˆ0(−ω′)aˆ†0(ω)〉 = δ(ω′ − ω)(1 + n2(~ω′)). (5.59)
100
where ni(~ω) = (eβi~ω − 1)−1, βi = 1/kBTi. All other commutators vanish. The
spectrum for the intracavity field can then be calculated to be
〈
aˆ†(−ω′)aˆ(ω)
〉
= η∗µ(ω
′)ζµ(ω)〈bˆµ0(−ω′)bˆµ0(ω)〉+ η∗µ(ω′)ηµ(ω)〈bˆµ0(−ω′)bˆ†µ0(ω)〉
+ζ∗µ(ω
′)ζµ(ω)〈bˆ†µ0(−ω′)bˆµ0(ω)〉+ ζ∗µ(ω′)ηµ(ω)〈bˆ†µ0(−ω′)bˆ†µ0(ω)〉
+β∗(ω′)α(ω)〈aˆ0(−ω′)aˆ0(ω)〉+ β∗(ω′)β(ω)〈aˆ0(−ω′)aˆ†0(ω)〉
+α∗(ω′)α(ω)〈aˆ†0(−ω′)aˆ0(ω)〉+ α∗(ω′)β(ω)〈aˆ†0(−ω′)aˆ†0(ω)〉. (5.60)
I will now assume that the noise influencing the system is at zero temperature
so that n2 = 0, i.e. it is vacuum noise. Then
〈
aˆ†(−ω′)aˆ(ω)
〉
= η∗µ(ω
′)ηµ(ω)〈bˆµ0(−ω′)bˆ†µ0(ω)〉+ ζ∗µ(ω′)ζµ(ω)〈bˆ†µ0(−ω′)bˆµ0(ω)〉
+β∗(ω′)β(ω)〈aˆ0(−ω′)aˆ†0(ω)〉
= η∗µ(ω
′)ηµ(ω)δ(ω
′ − ω)(1 + n1(~ω′)) + ζ∗µ(ω′)ζµ(ω)δ(ω′ − ω)n1(~ω′)
+β∗(ω′)β(ω)δ(ω′ − ω). (5.61)
Since one wants the system to be in thermal equilibrium with the bath in the
steady state, this means that energy will have to be extracted from the bath by
the system and vice versa. Note that the coupling in the Hamiltonian (5.29) has
terms like aˆbˆ†µ + aˆ
†bˆµ and aˆbˆµ + aˆ†bˆ†µ. The first leads to energy exchange with the
conservation of the total number of photons and phonons, while the second leads
to squeezing. It is well known from optomechanics that if ∆ = ωµ for some µ, the
former interaction will dominate [122]. Note that my definition of ∆ here is different
101
from the definition in the literature in the sign. For frequencies ω = ωµ ± δ with
δ  ωµ, one can make further simplifications to
〈
aˆ†(−ω′)aˆ(ω)
〉
. First, I impose the
constraint that
[ |ηµ(ω)|2
|ζµ(ω)|2
]
ω=ωµ
=
γ2µ
γ2µ + 16ω2µ
 1. (5.62)
Similarly, I require
[ |βµ(ω)|2
|ζµ(ω)|2
]
ω=ωµ
=
64G2µκω
2
µ
(Γ2 + 16ω2µ)(γ3µ + 16γµω2µ)
 1. (5.63)
Then in this limit, considering coupling to a single mode µ, I get
〈
aˆ†(−ω′)aˆ(ω)
〉
= ζ∗µ(ω
′)ζµ(ω)δ(ω
′ − ω)n1(~ω′). (5.64)
In this regime, approximately
ζµ(ω) ≈
4iGµ
√
γµ
4G2µ + [Γ + 2i(ωµ − ω)][γµ − 2i(ω − ωµ)]
,
α(ω) ≈ − 2
√
κ(γ − 2i(ω − ωµ))
4G2µ + [Γ + 2i(ωµ − ω)][γµ − 2i(ω − ωµ)]
. (5.65)
The relevant quantity is |ζµ(ω)|2 which I plot in Figure 5.2 with the choice of pa-
rameters ωµ/κ = 300, Gµ/κ = 20, κi/κ = 0.1, γµ/κ = 80. This is the regime where
the constraints (5.62) and (5.63) are satisfied. There is a peak in the approximate
value of |ζµ(ω)|2 at ω = ωµ as can be evaluated from (5.65), provided 8G2µ < γ2µ+ Γ2
as depicted in Figure 5.3(a).
The approximate bandwidth Bw of |ζµ(ω)|2 is shown in Figure 5.4. Over this
bandwidth, |ζµ(ω)|2 is approximately constant and
〈
aˆ†(−ω)aˆ(ω)
〉
≈ 16G
2
µγµ
(4G2µ + γµΓ)2
(
1
eβ~ω − 1
)
. (5.66)
102
Rotating back to the lab frame I get,
〈
aˆ†(−ω)aˆ(ω)
〉
≈ 16G
2
µγµ
(4G2µ + γµΓ)2
(
1
eβ~(ω−ωp) − 1
)
. (5.67)
Thus, the system has an effective chemical potential µ = ~ωp over the bandwidth
Bw.
5.6 Conclusions and Outlook
In this chapter I showed that parametrically modulating the coupling between
a generic system and a thermal bath leads to thermalization of the system. In
particular, for a photonic system, this leads to thermalization with a chemical po-
tential equal to the frequency of the parametric coupling. A similar analysis for
photons coupled to an optomechanical system in thermal equilibrium leads to the
same conclusions.
103
(a) (ωµ, Gµ, κi, γµ)/κ = (300, 20, 0.1, 80).
(b) (ωµ, Gµ, κi, γµ)/κ = (300, 20, 0.1, 80).
Figure 5.2: Plot of the logarithm of the spectral coefficients for ∆ = ωµ = 300κ.
104
(a) (ωµ, Gµ, κi, γµ)/κ = (300, 20, 0.1, 80).
(b) (ωµ, Gµ, κi, γµ)/κ = (300, 20, 0.1, 80).
Figure 5.3: Plot of the approximate values (dotted) of log(|ζµ(ω)|2) and log(|α(ω)|2)
for ∆ = ωµ = 300κ. For positive frequencies, the results are in good agreement.
105
Figure 5.4: Approximate bandwidth Bw of |ζµ(ω)|2.
106
Chapter 6
Dynamics of an Ion Coupled to a Superconducting
Circuit
6.1 Introduction
We have already seen that superconducting circuits are promising candidates
for the implementation of quantum information processing and the study of quantum
phenomena. They have been used to demonstrate strong coupling to a single photon
[30], for the realization of quantum error correction [123], and for the generation of
single-photon Fock states [59] among others. They also form an important aspect
of hybrid quantum systems. In Chapter 3 a flux qubit was used to generate two-
photon nonlinearities for the construction of a two-qubit phase gate with microwave
photons (see also [57]).
Similarly, atomic systems like ions traps have been used to generate and ma-
nipulate entanglement [124] and to implement multi-qubit gates [125]. It is therefore
natural to attempt to construct hybrid systems comprising these two architectures.
107
Although the couplings of the dipoles generated by the motion of trapped ions to
the electric field of an LC circuit can be several hundred kHz, this coupling is far off
resonance. This is because the motional frequencies of ions are on the order of MHz
whereas the superconducting circuits are in the GHz (microwave) regime. Therefore,
implementation of a practical quantum device requires something additional.
Parametric processes are useful in this end, and they are common to many
physical systems. In the field of quantum optics, they are widely used in the fre-
quency conversion of photons using nonlinear media [21, 126]. In the realm of su-
perconducting quantum devices, one class of parametric amplifiers called Josephson
bifurcation amplifiers have been used to amplify signals, and also to perform very
sensitive quantum measurements, while adding very little noise [127]. Parametric
processes have also been used to generate controllable interactions between super-
conducting qubits and microwave resonators [128].
In [129] a successful attempt was made to generate a resonant coupling scheme
between ions and LC circuits. The ion, confined in a trap with frequency ωi, was
coupled to the driven sidebands of a high quality factor parametric LC circuit whose
capacitance was modulated at frequency ν = ωLC −ωi. This gave rise to a coupling
strength g/2pi = 60 kHz. Here I try a different approach. I drive a superconducting
loop comprising a Josephson junction and a capacitor confining a trapped ion, using
a time dependent external flux. This causes the system to act as a parametric
oscillator with a tunable inductance, and hence a tunable resonant frequency. In the
presence of a small nonlinearity in the junction, the parametric oscillator develops
sidebands and the ion can be resonantly coupled to these sidebands. However, one
108
EJ , CJ
x(t)
L
C

Flux Bias x(t)
C
L(t)
Figure 6.1: Depiction of an rf-SQUID with Josephson energy EJ and junction ca-
pacitance CJ driven by a time-dependent external flux Φx(t). The outer loop with
inductance L contributes an energy EL. The rf-SQUID is connected in parallel to a
capacitor C that confines an ion (green).
will see that the coupling strength cannot be made very large.
6.2 Model and Hamiltonian
I consider an ion in an ion-trap that generates a harmonic potential with fre-
quency ωz along the z-direction. However, the confinement of an ion in a trap also
leads to motion in the x and y directions. This is referred to as ion micromotion [130].
These motions correspond to a parametrically driven harmonic oscillator at para-
metric frequencies ωx(t) and ωy(t) respectively. I will ignore such x-y micromotion
here.
The ion motion in the z-direction interacts with the electric field of a capacitor
109
C of a circuit containing an rf-SQUID (Fig. 6.1) [37, 75]. The SQUID is driven
using a time-dependent magnetic flux Φx(t). I will show below that this SQUID
and capacitor system will act as a parametric LC circuit. For the purpose of the
following discussion, I define the dimensionlesss flux φx = 2pi(Φx/Φ0) and the phase
φ = 2pi(Φ/Φ0), where Φ0 = h/(2e) is the superconducting flux quantum.
I first construct the classical Lagrangian L = T − V of the system [50], so
that with the usual Legendre transformation prescription, one can determine the
Hamiltonian. The kinetic energy of the system is
T =
1
2
(C + CJ)Φ˙
2 +
1
2
mz˙2, (6.1)
where Φ denotes the node flux for the circuit and z describes the longitudinal ion
position. The potential energy is
V = −EJ cosφ+
1
2
EL(φ+ φx)
2 +
1
2
mω2zz
2 + VI . (6.2)
The parameters EJ and EL = Φ20(4pi
2L)−1 are the Josephson energy and the induc-
tive energy of the rf-SQUID respectively. The interaction potential VI between the
ion with charge Q and the capacitor is given in the dipole approximation by
VI = QEzz = Q
V
d
z = −Q
d
zΦ˙. (6.3)
Let CΣ = C + CJ . The canonical coordinates are
q =
∂L
∂Φ˙
= CΣΦ˙ +
Q
d
z, (6.4)
pz =
∂L
∂z˙
= mz˙. (6.5)
Let the effective ion harmonic frequency be
ωi =
(
ω2z +
Q2
d2CΣm
) 1
2
. (6.6)
110
The quantum Hamiltonian of the system can be written as
Hˆ(t) = Hˆion + Hˆq(t) + HˆI , (6.7)
corresponding to the ion, qubit, and interaction Hamiltonians respectively, where
Hˆion =
pˆ2z
2m
+
1
2
mω2i zˆ
2 →
(
bˆ†bˆ+
1
2
)
~ωi, (6.8)
Hˆq(t) =
qˆ2
2CΣ
− EJ cos φˆ+
1
2
EL(φˆ+ φx(t))
2, (6.9)
HˆI = −
Qqˆ
CΣ
zˆ
d
→ − Q
CΣ
(zp
d
)
qˆ(bˆ+ bˆ†). (6.10)
Here I introduced ionic operators bˆ and bˆ† satisfying [bˆ, bˆ†] = 1 and zero-point fluc-
tuations of the ion motion zp =
√
~/(2mωi). This ensures that [zˆ, pˆz] = i~ with
zˆ =
√
~
2mωi
(bˆ+ bˆ†) ≡ zp(bˆ+ bˆ†), (6.11)
pˆz = −i
√
mωi~
2
(bˆ− bˆ†). (6.12)
The canonical coordinates of the SQUID satisfy [φˆ, qˆ] = 2ei.
If EJ and Φx(t) where zero, one could quantize the SQUID just like the ion
motion, say with operators aˆ and aˆ† satisfying [aˆ, aˆ†] = 1. In that case, the inter-
action would be proportional to (aˆ− aˆ†)(bˆ+ bˆ†). However, in the frame rotating at
the frequencies of the SQUID (say ω0) and ion (ωi), one would get something like
HI = (aˆe
−iω0t − aˆ†eiω0t)(bˆe−iωit + bˆ†eiωit). (6.13)
One is interested in interactions of the form aˆbˆ† + aˆ†bˆ where a resonant exchange
of energy takes place between the two systems. However, since ω0  ωi, ω0 − ωi
is comparable to ω0 + ωi, and so a rotating-wave approximation is not possible in
(6.13). My goal is then the following: by tuning EJ and φx(t) adjust the spectrum
of qˆ(t) so that aˆbˆ† contains a time independent component, whereas aˆbˆ does not.
111
6.3 Linearization of the Parametric Oscillator
In the presence of EJ , Hˆq(t) is nonlinear and so aˆ and aˆ† are ill-defined.
Therefore, the first task is to linearize Hˆq(t) and to find the operators corresponding
to the the superconducting charge and flux variables. I let η = EJ/EL denote the
strength of nonlinearity. Linearization is possible in the regime where η/(1−η) 1
and when the quantum fluctuations of flux are much less than Φ0.
Let the time-dependent classical values of the reduced flux φˆ and the charge qˆ
be denoted by φc(t) and qc(t) respectively. Recall that they satisfy the commutation
relation [Φˆ, qˆ] = i~ or [φˆ, Nˆ ] = i with qˆ = 2eNˆ . Let the time-dependent generators
of charge and flux translations be denoted by
U1(t) = e
−iΦˆqc(t)/~, (6.14)
U2(t) = e
iqˆΦc(t)/~, (6.15)
and let
Vˆq(φˆ) = −EJ cos φˆ+
1
2
EL(φˆ+ φx)
2. (6.16)
Under U1(t) the Hamiltonian transforms to
Hˆ1 = U
†
1HˆU1 − i~U †1
∂U1
∂t
= Hˆion +
(qˆ − qc)2
2CΣ
+ Vˆq(φˆ)−
Q
dCΣ
zˆ(qˆ − qc)
−Φˆq˙c. (6.17)
112
Now under U2(t), H1 transforms to
Hˆ2 = U
†
2Hˆ1U2 − i~U †2
∂U2
∂t
= Hˆion +
(qˆ − qc)2
2CJ
+ Vˆq(φˆ− φc)
− Q
dCΣ
zˆ(qˆ − qc)− (Φˆ− Φc)q˙c + qˆΦ˙c. (6.18)
I approximate
Vˆq(φˆ− φc) = Vˆq(−φc) + Vˆ ′q (−φc)φˆ+
Vˆ ′′q (−φc)
2
φˆ2 +O(φˆ3). (6.19)
One can then set the terms linear in qˆ and φˆ in Hˆ2 to be zero. This gives
φ˙c −
2e
~
qc
CJ
= 0; Vˆ ′q (−φc)−
~
2e
q˙c = 0. (6.20)
These can be solved for φc(t) and qc(t). Let the effective charging energy be EC =
(2e)2/(2CΣ). The linearized Hamiltonian can then be written as
HˆL = Hˆion +
[
Nˆ2
2M
+
1
2
M~2ω(t)2φˆ2
]
− Qzˆ
dCΣ
(qˆ − qc),
(6.21)
the interaction term being
VˆL = −
Qzˆ
dCΣ
(qˆ − qc). (6.22)
The parametric frequency is denoted by ω(t)2 = ω20(1 + η cosφc(t)) with ω0 =
√
2ELEC/~. I have also defined an effective mass M ≡ 1/(2EC). One can modulate
φx(t) such that cosφc(t) = cosωdt for some frequency ωd. For instance, this can be
done by modulating φx(t) as a saw-tooth wave. With this approximation,
ω(t)2 = ω20(1 + η cosωdt), (6.23)
and my stated goal is now achieved.
113
EJ , CJ
x(t)
L
C

Flux Bias x(t)
C
L(t)
Figure 6.2: After linearization, the effective picture consists of an ion confined be-
tween the capacitance C of an LC circuit with parametric inductance L(t).
6.4 Time-Dependent Quantum Harmonic Oscillator
I now follow the approach in [131] to quantize the parametric Hamiltonian
HˆP =
Nˆ2
2M
+
1
2
M~2ω(t)2φˆ2, (6.24)
the details being provided in Appendix D. The parametric frequency ω(t) has period
τ = 2pi/ωd. The classical equation of motion for φ is
φ¨(t) + ω(t)2φ(t) = 0. (6.25)
Suppose some function f(t) is a solution of Equation (6.25). Then f ∗(t) is also a
linearly independent solution. Since f(t) is periodic, f(t + τ) = eiϕf(t) for some
real ϕ. It is helpful to write
f(t) = r(t)eiθ(t), (6.26)
114
with r(t) > 0 and θ(t) real-valued. The Wronskian corresponding to these solutions
is
W =
1
2i
∣
∣
∣
∣
∣
∣
∣
∣
f(t) f ∗(t)
f˙(t) f˙ ∗(t)
∣
∣
∣
∣
∣
∣
∣
∣
= −r(t)2θ˙(t). (6.27)
However, it can be shown that the time derivative of the Wronskian is zero and
so is a constant of the motion. Then apart from a time dependent factor |f(t)|−2,
the Hamiltonian (6.24) can be written as the Hamiltonian of a time-independent
harmonic oscillator with frequency W . One can write φˆ and Nˆ in terms of creation
and annihilation operators aˆ† and aˆ satisfying [aˆ, aˆ†] = 1 as
φˆ =
√
1
2M~W
(aˆ+ aˆ†), (6.28)
Nˆ = −i
√
M~W
2
(aˆ− aˆ†). (6.29)
and get a quantized Hamiltonian
HˆP =
1
|f(t)|2
(
aˆ†aˆ+
1
2
)
~W
= ~ωa(t)
(
aˆ†aˆ+
1
2
)
, (6.30)
where ωa(t) = W/|f(t)|2 = −θ˙(t) plays the role as a time-dependent oscillator
frequency.
6.5 Classical Solutions
To find the classical solutions, one can transform (6.25) with the substitution
2z = ωdt into
d2φ
dz2
+ (a− 2q cos 2z)φ = 0, (6.31)
115
where a = 4ω20/ω
2
d and q = −ηa/2 = −2ηω20/ω2d. This equation is known as the
Mathieu equation [132]. I am interested in solutions of Mathieu’s equation of the
form f(z+pi) = eiµpif(z), where µ is a real number. µ can in general be complex, but
that leads to solutions which decay in time, or to solutions which are unstable. If µ
is a rational or irrational, but positive number, then the Mathieu equation admits
solutions of fractional order [132]. These are solutions which reduce to sinµz or
cosµz when q = 0. They are
ceµ(z, q) = cosµz +
∞∑
i=1
qici(z), (6.32)
seµ(z, q) = sinµz +
∞∑
i=1
qisi(z), (6.33)
a = µ2 +
∞∑
i=1
αiq
i, (6.34)
where ci(z), si(z) are functions to be determined, and αi are constants to be deter-
mined. The parameter a has the same value for both solutions for any q. So (6.32)
and (6.33) coexist and are linearly independent. Thus, the general solution to (6.31)
with two arbitrary constants is
φ(z) = A ceµ(z, q) +B seµ(z, q). (6.35)
When µ > 0 and q2(µ2−1)/2 µ2, one approximately has µ2 = a. Addition-
ally, if |q|  1 one can ignore terms of O(q2). Then I get
ceµ(z, q) = cosµz −
q
4
[
cos(µ+ 2)z
µ+ 1
− cos(µ− 2)z
µ− 1
]
,
seµ(z, q) = sinµz −
q
4
[
sin(µ+ 2)z
µ+ 1
− sin(µ− 2)z
µ− 1
]
, (6.36)
Switching back to the original notation with 2z = ωdt, the linearly independent
116
solutions become
f1(t) = cosω0t+
ηω20
2ωd
[
cos(ω0 + ωd)t
ωd + 2ω0
+
cos(ω0 − ωd)t
ωd − 2ω0
]
,
f2(t) = sinω0t+
ηω20
2ωd
[
sin(ω0 + ωd)t
ωd + 2ω0
+
sin(ω0 − ωd)t
ωd − 2ω0
]
. (6.37)
These solutions are valid when
K1 ≡
η2ω20
2ω2d
(
4ω20
ω2d
− 1
)
 1. (6.38)
The time-independent Wronskian corresponding to f1(t) and f2(t) is
W = ω0 +O(η
2). (6.39)
Note that the Wronskian acts as an effective frequency. The general solution can be
written in terms of two real constants C and ϕ as
f(t) = Ce−i(ω0t+ϕ) +
Cηω20e
−iϕ
2ωd
[
e−i(ω0+ωd)t
ωd + 2ω0
+
e−i(ω0−ωd)t
ωd − 2ω0
]
. (6.40)
6.6 Derivation of the Interaction
As shown before, the time derivative of the Wronskian is zero and the only
time dependence is in f(t), the classical solution. Once again following the ap-
proach in [131], one can show that the interaction potential (6.22) of the linearized
Hamiltonian under the relevant unitary transformations becomes
HI = −
2Qe
dCΣ
z
(
N
|f(t)| − 2χ|f(t)|φ−Nc(t)
)
,
where χ is a dimensionless parameters that satisfies
χ = −~M
4
d
dt
ln |f |2. (6.41)
117
Define the energies
Ω1(t) = α
d|f(t)|
dt
= ~αr˙(t), (6.42)
Ω2(t) = β
1
|f(t)| =
β
r(t)
, (6.43)
Λ(t) =
2Qe
dCΣ
zpNc(t), (6.44)
with the parameters
α = −
√
2Qe
CΣ
(zp
d
)
√
M
~W
, (6.45)
β =
√
2Qe
CΣ
(zp
d
)√
M~W, (6.46)
where α is dimensionless and β has units of energy. The full quantized Hamiltonian
is Hˆ = Hˆ0 + Vˆ , where
Hˆ0 = ~ωa(t)aˆ†aˆ+ ~ωibˆ†bˆ, (6.47)
Vˆ =
(
Ω(t)aˆ+ Ω∗(t)aˆ†
)
(bˆ+ bˆ†) + Λ(t)(bˆ+ bˆ†), (6.48)
with coupling Ω(t) = Ω1(t) + iΩ2(t).
6.7 Sideband Coupling
I now examine the coupling to the ion motion in the z-direction. Coupling to
the ion micromotion can be done in a similar manner albeit much more involved.
Define the unitary operator
U = exp
[
−i
(
Ga(t)aˆ
†aˆ+ ωit bˆ
†bˆ
)]
(6.49)
where
Ga(t) =
∫ t
0
dt′ ωa(t
′) = −
∫ t
0
dt′ θ˙(t′) = −θ(t). (6.50)
118
Under this transformation, the original Hamiltonian
Hˆ →
[
Ω(t)aˆeiθ(t) + Ω∗(t)aˆ†e−iθ(t)
]
(bˆe−iωit + bˆ†eiωit)
+Λ(t)(bˆe−iωit + bˆ†eiωit). (6.51)
To find the coupling to the ion motion in the z-direction, one first needs to
evaluate the coefficient of aˆbˆ which is Ω(t)eiθ(t)e−iωit. Similarly, the coefficient of aˆbˆ†
is Ω(t)eiθ(t)eiωit. Note that
df(t)
dt
=
d
dt
(r(t)eiθ(t)) = r˙(t)eiθ(t) + iθ˙(t)r(t)eiθ(t). (6.52)
Then,
Ω1(t)e
iθ(t) = ~αr˙(t)eiθ(t)
= ~α
df(t)
dt
− i~αθ˙(t)r(t)eiθ(t)
= ~α
df(t)
dt
+ i~α
W
r(t)
eiθ(t), (6.53)
iΩ2(t)e
iθ(t) = i
β
r(t)
eiθ(t)
= −i~α W
r(t)
eiθ(t). (6.54)
Therefore,
Ωeiθ(t) = (Ω1(t) + iΩ2(t))e
iθ(t) = ~α
df(t)
dt
. (6.55)
Let C = 1 and ϕ = 0 for simplicity. The time derivative of f(t) is
df(t)
dt
= −iω0e−iω0t −
iηω20
2ωd
[
(ω0 + ωd)e−i(ω0+ωd)t
ωd + 2ω0
+
(ω0 − ωd)e−i(ω0−ωd)t
ωd − 2ω0
]
. (6.56)
Therefore, the time dependent coefficient of aˆbˆ is
−iω0e−i(ω0+ωi)t −
iηω20
2ωd
[
(ω0 + ωd)e−i(ω0+ωd+ωi)t
ωd + 2ω0
+
(ω0 − ωd)e−i(ω0−ωd+ωi)t
ωd − 2ω0
]
. (6.57)
119
Similarly the coefficient of aˆbˆ† is
−iω0e−i(ω0−ωi)t −
iηω20
2ωd
[
(ω0 + ωd)e−i(ω0+ωd−ωi)t
ωd + 2ω0
+
(ω0 − ωd)e−i(ω0−ωd−ωi)t
ωd − 2ω0
]
. (6.58)
If one assumes that ωd = ω0 − ωi, then only aˆbˆ† has a time independent term given
by
λ = ~α
(
−iηω
2
0
2ωd
)
ω0 − ωd
ωd − 2ω0
= ~α
(
iηω20
2
)
ωi
(ω0 + ωi)(ω0 − ωi)
≈ i~αη
2
ωi. (6.59)
where the approximation arises from the fact that ω0  ωi.
Ignoring all the coupling terms which oscillate rapidly compared to time scales
of the ion motion leads to a resonant coupling
VˆR = λ(aˆbˆ
† + aˆ†bˆ). (6.60)
Linearization requires η  1, and the quantum fluctuations in φˆ
φˆzp =
1√
2M~W
 1. (6.61)
Since W ≈ ω0,
α = −
√
2Qe
CΣ
(zp
d
)
√
M
~W
≈ −
√
2Qe
CΣ
(zp
d
) 1
(2EC)
3
4E
1
4
L
. (6.62)
Because the parameters η and α are small, λ is also small. With these constraints,
the coupling strength λ is much smaller than even a kHz.
6.8 Conclusions and Outlook
My analysis reveals that coupling a single trapped ion to a superconducting
circuit resonantly has limitations. It is the geometry which creates a severe limit-
ing factor, and not the dynamics. Therefore, a novel approach is required. Early
120
proposals for realizing a hybrid system comprising atoms or trapped ions coupled
to solid state systems include [133] and [134] respectively. However, these propos-
als have yet to be realized experimentally. Other approaches involve coupling solid
state systems to an ensemble of polar molecules with large dipole moments to en-
hance the coupling as proposed in [135], or using Rydberg atoms with large dipole
moments [136].
121
Chapter 7
Conclusions
Superconducting circuits may be used to implement many other interesting
Hamiltonians, and are an integral part of hybrid quantum sytems. In this thesis I
showed how to generate two-photon nonlinearities by coupling microwave photons
to a flux superconducting circuit. I also considered the implementation of a parent
Hamiltonian for the Pfaffian state using circuit-QED. Then I described a Hamilto-
nian that can emulate a chemical potential for light. I believe that these proposals
can be implemented experimentally, although the quantum control of these systems
will present challenges. Superconducting systems combined with other architectures
have already been used to demonstrate strong coupling to a single photon [30], and to
implement quantum error correcting codes [123]. They have also been used to gener-
ate Fock states of photons [64], and in the creation of low noise amplifiers [127]. The
lifetime and quality of superconducting quantum circuits have increased by many
orders of magnitude over the past decade and the trend continues [25]. Therefore,
it is reasonable to hope that they may eventually form scalable quantum simulation
and quantum information architectures in the future. As I discussed in the introduc-
122
tion, classical computation will encounter increasing challenges due to fundamental
laws of nature. By stepping into a domain where quantum processes occur, one can
cross these barriers [137].
123
Chapter 8
Appendix
8.1 Appendix A
In this appendix, I present a derivation of the effective Hamiltonian (3.85)
using a unitary transformation. I will ignore the state |0〉 ≡ |0¯0¯〉 and focus on the
nontrivial subspace for this part. The starting Hamiltonian is
H =








Ω1 λ1 0
λ1 2Ω1 λ2
0 λ2 Ω2








. (8.1)
Let P = |b〉 〈b|+|c〉 〈c| denote the projector onto the two-dimensional subspace
and Q = |a〉 〈a| be the projector onto the one dimensional subspace. I will find a
Hermitian matrix S such that the unitary transformation U = eiS generated by S,
renders H block diagonal in the respective subspaces P and Q [138]. That is
H˜ ≡ UHU † = P ⊕Q. (8.2)
124
Let S have the most general form
S =








d1 q r
q∗ d2 u
r∗ u∗ d3








. (8.3)
One can impose the constraints, PSP = QSQ = 0 [138]. This implies that d1 =
d2 = d3 = u = u∗ = 0. Thus, S reduces to
S =








0 q r
q∗ 0 0
r∗ 0 0








. (8.4)
S can be diagonalized with the unitary transformation
W =
1
√
|q|2 + |r|2















0 −
√
|q|2 + |r|2
√
2
√
|q|2 + |r|2
√
2
−r
q∗
√
2
q∗
√
2
q
r∗
√
2
r∗
√
2















, (8.5)
to a matrix Sd = W †SW = Diagonal(0,−
√
|q|2 + |r|2,
√
|q|2 + |r|2). W simply
consists of normalized eigenvectors of S as its columns.
After finding S one needs to evaluate U = eiS which can be expanded as
U = eiS =
∞∑
n=0
(iS)n
n!
. (8.6)
One can apply W to U to get
W †UW =
∞∑
n=0
(i)n
n!
W †SnW =
∞∑
n=0
(i)n
n!
(
W †SW
)n
=
∞∑
n=0
(i)n
n!
(Sd)
n, (8.7)
125
since W †W = WW † = 1. Define the matrices
M1 =








0 0 0
0 −1 0
0 0 1








; M2 =








0 0 0
0 1 0
0 0 1








, (8.8)
and let z =
√
|q|2 + |r|2. Therefore,
W †UW =
∞∑
n=0
(i)n
n!
(Sd)
n
= I +
∞∑
n=1
(i)n
n!
(zM1)
n
= I +M1
∑
n=1,3,...
(iz)n
n!
+M2
∑
n=2,4...
(iz)n
n!
= I + iM1
∞∑
n=0
(−1)n z
2n+1
(2n+ 1)!
+M2
∞∑
n=1
(−1)n z
2n
(2n)!
= I + iM1
∞∑
n=0
(−1)n z
2n+1
(2n+ 1)!
+M2
[
−1 +
∞∑
n=0
(−1)n z
2n
(2n)!
]
= I + iM1 sin z −M2(1− cos z). (8.9)
Hence,
U = I + iWM1W
† sin z −WM2W †(1− cos z). (8.10)
One can find the elements of the matrix S and that of U perturbatively in η′2 by
ensuring that H˜12 = H˜13 = 0. One then gets the effective Hamiltonian
H˜ = UHU † =












Ω1 −
r21η
′2
2
Ω1
+O(η′32 ) O(η
′3
2 ) O(η
′3
2 )
O(η′32 ) 2Ω1 +
r21η
′2
2
Ω1
+O(η′32 ) r2η
′
2 +O(η
′3
2 )
O(η′32 ) r2η
′
2 +O(η
′3
2 ) Ω2 +O(η
′3
2 )












.
(8.11)
126
Figure 8.1: A plot of log10[g
(3)(0)] as a function of U2 and U3 for ∆ = 0.
8.2 Appendix B
I briefly review the method to measure U2 and U3 using correlation functions.
I start with the system Hamiltonian Hˆ and input-ouput equations for a one-sided
driven system with a loss rate κ in a frame rotating at a frequency ωL of the input
field bˆin = Ee−iωLt.
Hˆ =
(
∆− iκ
2
)
aˆ†aˆ+
(
U2
2
aˆ†2aˆ2 +
U3
6
aˆ†3aˆ3
)
− i√κ(E aˆ† − E∗aˆ). (8.12)
˙ˆa = −i[aˆ, Hˆ]; bˆout = bˆin +
√
κaˆ. (8.13)
I will assume that the field is on resonance, that is ∆ = ω − ωL = 0 and β =
E/√κ 1. For numerical analysis, I let κ = 10 MHz and ∆ = 0.
I want to solve for |ψ〉 in
i
∂ |ψ〉
∂t
= Hˆ |ψ〉 . (8.14)
127
I start with the ansatz |ψ〉 = c0 |0〉+c1 |1〉+c2 |2〉+c3 |3〉+O(β4), where I assume that
c0 is O(1) and ci is of O(βi). I want to solve for the steady state of the system and
calculate the correlation functions. One can show that the second order correlation
function is
g(2)(0) =
〈aˆ†aˆ†aˆaˆ〉
〈aˆ†aˆ〉2 ≈
2|c2|2
|c1|4
=
4∆2 + κ2
(U2 + 2∆)2 + κ2
∆=0−−→ κ
2
U22 + κ2
, (8.15)
where the mean is taken in the steady state |ψ〉. When U2 = 0, one has g(2)(0) ≈ 1.
However, in the presence of U2, g2(0) < 1. Similarly, in the presence of both U2 and
U3, one can show that the third order correlation function is
g(3)(0) =
〈aˆ†aˆ†aˆ†aˆaˆaˆ〉
〈aˆ†aˆ〉3
≈ 6|c3|
2
|c1|6
=
9(4∆2 + κ2)2
[(U2 + 2∆)2 + κ2] [4(3U2 + U3 + 3∆)2 + 9κ2]
∆=0−−→ 9κ
4
(U22 + κ2) [4(3U2 + U3)2 + 9κ2]
. (8.16)
Note that when both U2, U3 → 0, g(3)(0)→ 1. The result is plotted in Figure 8.1.
8.3 Appendix C
In this appendix, I derive the Hamiltonian of a transmission line coupled to
external circuits following the approach in [100]. I assume that the left end of the
TL is connected to a left circuit via a capacitor CL, and the right end of the TL
is connected to a right circuit via an inductor L1. I also assume that the energies
associated with the capacitive coupling to the right Wheatstone circuit is small
compared to other energies in the system, and can be ignored.
128
.........
CL
L R
l dz
c dz
1 2 3 N-1 N
L1
Figure 8.2: A schematic of a transmission line. A left circuit is connected to its
left end via a capacitor CL, and a right circuit is connected to its right end via an
inductor L1.
In the discrete case where the TL is modeled by a series of inductors and
capacitors (Figure 8.2). The TL Lagrangian is
Ltl =
1
2
CLη˙
2
1 +
1
2
N∑
i=1
(cdz)η˙2i −
1
2(ldz)
N∑
i=2
(ηi − ηi−1)2 +
η2N
2L1
. (8.17)
I define a column vector η as η˜ = (η1, . . . , ηN) = ηT . The Lagrangian can then be
written as
Ltl = T − V =
1
2
˜˙ηTη˙ − 1
2
η˜Vη, (8.18)
where the kinetic energy matrix T has components
Tij = δij(cdz + CLδi1). (8.19)
129
The potential energy matrix is
V =
1
ldz




















1 −1 0 0 . . . 0
−1 2 −1 0 . . . 0
0 −1 2 −1 . . . 0
...
...
...
. . . . . .
...
0 0 0 −1 2 −1
0 0 0 0 −1 1− ldzL1




















. (8.20)
In the continuum limit, the flux along the TL can be represented by a function
η(z) with 0 ≤ z ≤ L. In terms of the eigenmodes ϕν(z) = 0, 1, . . . of the TL,
η(z) =
∑
ν
ξνϕν(z), (8.21)
where the coefficients ξν satisfy
ξ¨ν + ω
2
νξν = 0. (8.22)
The functions ϕν(z) satisfy
∂2ϕν(z)
∂z2
= −(ων
√
lc)2ϕν(z). (8.23)
It has the solution
ϕν(z) = Aν cos(ων
√
lcz) +Bν sin(ων
√
lcz). (8.24)
Letting kν = ων
√
lc, the solution can also be written as
ϕν(z) = Cν cos(kνz + φν) = <[Cνei(kνz+φν)]. (8.25)
On the left, there is a boundary condition,
−
[
∂ϕν(z)
∂z
]
z=0
= lCLω
2
ν [ϕν(z)]z=0 , (8.26)
130
and on the right, there is a boundary condition
[
∂ϕν(z)
∂z
]
z=L
= − l
L1
[ϕν(z)]z=L . (8.27)
In the discrete picture, the orthonormalization condition is
(cdz + CL)ϕν(z1)ϕµ(z1) + cdzϕν(z2)ϕµ(z2) + · · ·+ cdzϕν(zN−1)ϕµ(zN−1)
+cdzϕν(zN)ϕµ(zN) = δµν . (8.28)
In the continuum limit, this becomes
CL [ϕν(z)ϕµ(z)]z=0 + c
∫ L
0
dz ϕν(z)ϕµ(z) = δµν . (8.29)
Note that the normal modes ϕν(z) have units of 1/
√
C where C has units of capac-
itance. With these results,
Ltl =
1
2
∑
µ,ν
[
ξ˙µξ˙νϕ
∗
µTϕν − ξµξνϕ∗µV ϕν
]
=
1
2
∑
µ,ν
[
ξ˙µξ˙νϕ
∗
µTϕν − ξµξνϕ∗µω2νTϕν
]
=
1
2
∑
µ,ν
[
ξ˙µξ˙ν − ω2νξµξν
]
ϕ∗µTϕν
︸ ︷︷ ︸
δµν
=
1
2
∑
ν
[
ξ˙2ν − ω2νξ2ν
]
. (8.30)
For convenience, one can let ξν =
√
CtlFν with Ctl = cL, and Fν has units of
flux. One can also define the dimensionless ψν =
√
Ctlϕν . The canonically conjugate
momentum is
qν =
∂L
∂F˙ν
. (8.31)
The Hamiltonian is then
Htl =
∑
ν
[
q2ν
2Ctl
+
1
2
Ctlω
2
νF
2
ν
]
. (8.32)
131
Introducing quantum operators bˆν that satisfy [bˆµ, bˆ†ν ] = δµν , and letting
Fν =
√
~
2Ctlων
(bˆν + bˆ
†
ν), (8.33)
qν = −i
√
Ctlων~
2
(bˆν − bˆ†ν), (8.34)
one gets a quantized Hamiltonian
H =
∑
ν
[
bˆ†ν bˆν +
1
2
]
~ων . (8.35)
The flux along the TL is
η(z) =
∑
ν
Fνψν(z) =
∑
ν
√
~
2Lcων
(bˆν + bˆ
†
ν)ψν(z). (8.36)
The time dependence of this flux comes from the normal mode operators bˆν and bˆ†ν .
Hence,
η(z, t) =
∑
ν
Fνψν(z) =
∑
ν
√
~
2Lcων
(bˆνe
−iωνt + bˆ†νe
iωνt)ψν(z). (8.37)
Hence, the voltage V (z, t) = −η˙(z, t) along the TL is
V (z, t) = i
√
1
2Lc
∑
ν
√
~ων(bˆνe−iωνt − bˆ†νeiωνt)ψν(z). (8.38)
Alternately,
V (z, t) = i
1√
2
∑
ν
√
~ων(bˆνe−iωνt − bˆ†νeiωνt)ϕν(z),
= i
1√
2
∑
ν
√
~ων(bˆνe−iωνt − bˆ†νeiωνt)<[Cνei(kνz+φν)]. (8.39)
One can decompose V (z, t) into left and right moving components
V ←(z, t) =
1√
2
∑
ν
√
~ων bˆ†νCνe
i(kνz+ωνt+φν−pi/2), (8.40)
V →(z, t) =
1√
2
∑
ν
√
~ων bˆνCνei(kνz−ωνt+φν+pi/2), (8.41)
so that V (z, t) = V →(z, t) + V ←(z, t).
132
8.4 Appendix D
In this appendix, I show in more detail how one can quantize a time-dependent
harmonic oscillator, following the approach of [131]. The Hamiltonian of a particle
in a Paul trap is given by
H(t) =
p2
2m
+
1
2
k(t)q2. (8.42)
The parameter k(t) has period τ . In experiments k(t) = a + b cos(2pit/τ). But in
general, k(t+ τ) = k(t). The classical equation of motion is
mq¨(t) + k(t)q(t) = 0. (8.43)
Suppose some function f(t) is a solution to the classical equation. Since f(t) is
periodic, f(t+ τ) = eiθf(t). Note that f ∗(t) is also a linearly independent solution.
The Wronskian is
W =
∣
∣
∣
∣
∣
∣
∣
∣
f(t) f ∗(t)
f˙(t) f˙ ∗(t)
∣
∣
∣
∣
∣
∣
∣
∣
. (8.44)
The time derivative of the Wronskian is zero. That is
d
dt
W =
d
dt
[
(ff˙ ∗ − f ∗f˙)
]
= (f˙ f˙ ∗ + ff¨ ∗)− (f˙ ∗f˙ + f ∗f¨)
= ff¨ ∗ − f ∗f¨
= − k
m
ff ∗ +
k
m
f ∗f
= 0. (8.45)
Henceforth, I will let (ff˙ ∗ − f ∗f˙) = 2iW with W > 0.
133
Let
χ(t) = −m
4
(
f˙
f
+
f˙ ∗
f ∗
)
= −m
4
d
dt
ln |f |2. (8.46)
In a new basis |Ψ′〉 = U1 |Ψ〉 with U1 = e−iχ(t)q2 , the coordinate and momentum
transform as
q → U †1qU1 = q, (8.47)
p → U †1pU1 = p+ iχ[q2, p] +
i2χ2
2!
[q2, [q2, p]] + ... = p− 2χq. (8.48)
I have used the commutation relation [q2, p] = i∂q
2
∂q = 2iq. Under U1 the Hamiltonian
transforms as H → H¯ where
H¯ = U †1HU1 − iU †1
dU
dt
,
=
(p− 2χq)2
2m
+
1
2
k(t)q2 − χ˙(t)q2,
=
p2
2m
− χ
m
(qp+ pq) +
2χ2q2
m
+
1
2
k(t)q2 − χ˙(t)q2. (8.49)
Now
χ˙ = −m
4
(
ff¨ − f˙ f˙
f 2
+
f ∗f¨ ∗ − f˙ ∗f˙ ∗
f ∗2
)
= −m
4
(
f¨
f
− f˙
2
f 2
+
f¨ ∗
f ∗
− f˙
∗2
f ∗2
)
= −m
4
(
−2k
m
)
+
m
4
(
f˙ 2
f 2
+
f˙ ∗2
f ∗2
)
=
k
2
+
m
4
(
f˙ 2
f 2
+
f˙ ∗2
f ∗2
)
. (8.50)
There is also the relation
(
f˙ 2
f 2
+
f˙ ∗2
f ∗2
)
= 2
|f˙ |2
|f |2 −
4W 2
|f |4 . (8.51)
134
So,
χ˙ =
k
2
+
m
4
(
2
|f˙ |2
|f |2 −
4W 2
|f |4
)
=
k
2
+
m
2
|f˙ |2
|f |2 −
mW 2
|f |4 . (8.52)
Thus,
1
2
kq2 − χ˙q2 = −m
2
|f˙ |2
|f |2 q
2 +
mW 2
|f |4 . (8.53)
There is one final term. That is
2χ2q2
m
=
2
m
m2
16
(
f˙
f
+
f˙ ∗
f ∗
)2
q2
=
m
8


(
f˙
f
)2
+ 2
∣
∣
∣
∣
∣
f˙
f
∣
∣
∣
∣
∣
2
+
(
f˙ ∗
f ∗
)2

 q2
=
m
8
q2

4
∣
∣
∣
∣
∣
f˙
f
∣
∣
∣
∣
∣
2
− 4W
2
|f |4


=
mq2
2
∣
∣
∣
∣
∣
f˙
f
∣
∣
∣
∣
∣
2
− mq
2
2
W 2
|f |4 . (8.54)
Adding all three terms together, one gets
2χ2q2
m
+
1
2
kq2 − χ˙q2 = mW
2
2|f |4 q
2. (8.55)
Thus, the transformed Hamiltonian is
H¯ =
p2
2m
− χ
m
(pq + qp) +
mW 2
2|f |4 q
2. (8.56)
Now let λ = ln |f |2/4, and U2 = e−iλ(pq+qp). Using the commutation relations
[pq + qp, p] = 2ip, [pq + qp, q] = −2iq
U †2pU2 = p
∞∑
n=0
(2i)n(iλ)n
n!
= pe(2i)(iλ) = pe−2λ = pe− ln |f | =
p
|f | , (8.57)
U †2qU2 = q
∞∑
n=0
(−2i)n(iλ)n
n!
= qe(−2i)(iλ) = qe2λ = peln |f | = |f |q. (8.58)
135
Note that χ = −mλ˙. Then the Hamiltonian H¯ transforms again as
H¯ =
1
|f |2
p2
2m
− χ
m
(pq + qp) +
mW 2
2|f |2 q
2 − i(−iλ˙)(pq + qp)
=
1
|f |2
p2
2m
+ λ˙(pq + qp) +
mW 2
2|f |2 q
2 − λ˙(pq + qp)
=
1
|f |2
p2
2m
+
mW 2
2|f |2 q
2
=
1
|f |2
(
p2
2m
+
1
2
mWq2
)
. (8.59)
Apart from a time dependent factor |f(t)|−2 one now has a simple harmonic oscillator
Hamiltonian with frequency W . One can finally write q and p in terms of the usual
creation and annihilation operators aˆ† and aˆ to get
H =
1
|f(t)|2
(
aˆ†aˆ+
1
2
)
W. (8.60)
136
Bibliography
[1] M. A. Nielsen and I. Chuang, Quantum Computation and Quantum Informa-
tion. Cambridge University Press, 2000.
[2] A. M. Turing, “On computable numbers with an application to the entschei-
dungsproblem,” 1936.
[3] A. M. Turing, “Computability and λ-definability,” Journal of Symbolic Logic,
vol. 2, no. 04, pp. 153–163, 1937.
[4] J. von Neumann, “First draft of a report on the EDVAC,” tech. rep., Moore
School of Electrical Engineering, University of Pennsylvania, 1945.
[5] W. Shockley, “The theory of p-n junctions in semiconductors and p-n junction
transistors,” Bell System Technical Journal, vol. 28, pp. 435–489, 1949.
[6] M. Riordan, L. Hoddeson, and C. Herring, “The invention of the transistor,”
Rev. Mod. Phys., vol. 71, pp. S336–S345, Mar. 1999.
[7] W. H. Zurek, “Thermodynamic cost of computation, algorithmic complexity
and the information metric,” Nature, vol. 341, pp. 119–124, Sept. 1989.
[8] R. Landauer, “Irreversibility and heat generation in the computing process,”
IBM Journal of Research and Development, vol. 5, pp. 183–191, July 1961.
[9] E. L. Wolf, Nanophysics of Solar and Renewable Energy. Wiley-VCH, 2012.
[10] R. Feynman, “Simulating physics with computers,” International Journal of
Theoretical Physics, vol. 21, no. 6-7, pp. 467–488, 1982.
[11] S. P. Jordan, “Quantum algorithm zoo.”
[12] L. K. Grover, “A fast quantum mechanical algorithm for database search,”
in Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of
Computing, STOC ’96, (New York, NY, USA), pp. 212–219, ACM, 1996.
137
[13] P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete
logarithms on a quantum computer,” SIAM J. Comput., vol. 26, pp. 1484–
1509, Oct. 1997.
[14] A. Kitaev, S. A. H. Yu, and M. N. Vyalyi, Classical and Quantum Computa-
tion. American Mathematical Society, 2002.
[15] D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,”
Phys. Rev. A, vol. 51, pp. 1015–1022, Feb 1995.
[16] S. Lloyd, “Almost any quantum logic gate is universal,” Phys. Rev. Lett.,
vol. 75, pp. 346–349, Jul 1995.
[17] S. X. Wang, J. Labaziewicz, Y. Ge, R. Shewmon, and I. L. Chuang, “Demon-
stration of a quantum logic gate in a cryogenic surface-electrode ion trap,”
Phys. Rev. A, vol. 81, p. 062332, June 2010.
[18] G. S. Buller and R. J. Collins, “Single-photon generation and detection,”
Measurement Science and Technology, vol. 21, no. 1, p. 012002, 2010.
[19] G. J. Milburn, “Quantum optical Fredkin gates,” Phys. Rev. Lett., vol. 62,
pp. 2124–2127, May 1989.
[20] R. W. Boyd, Nonlinear Optics. Academic Press, Amsterdam, 2003.
[21] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics. Cambridge
University Press, September 29, 1995.
[22] J. I. Cirac and P. Zoller, “Quantum computations with cold trapped ions,”
Phys. Rev. Lett., vol. 74, pp. 4091–4094, May 1995.
[23] D. Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,”
Phys. Rev. A, vol. 57, pp. 120–126, Jan. 1998.
[24] J. Wrachtrup, S. Kilin, and A. Nizovtsev, “Quantum computation using the
13c nuclear spins near the single nv defect center in diamond,” Optics and
Spectroscopy, vol. 91, no. 3, pp. 429–437–, 2001.
[25] M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum
information: An outlook,” Science, vol. 339, pp. 1169–1174, Mar. 2013.
[26] D. P. DiVincenzo, “The physical implementation of quantum computation,”
Fortschr. Phys., vol. 48, pp. 771–783, Sept. 2000.
[27] S. Lloyd, “Universal quantum simulators,” Science, vol. 273, pp. 1073–1078,
Aug. 1996.
[28] I. M. Georgescu, S. Ashhab, and F. Nori, “Quantum simulation,” Rev. Mod.
Phys., vol. 86, pp. 153–185, Mar. 2014.
138
[29] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, “Hybrid quantum circuits:
Superconducting circuits interacting with other quantum systems,” Rev. Mod.
Phys., vol. 85, pp. 623–653, Apr. 2013.
[30] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, “Cavity
quantum electrodynamics for superconducting electrical circuits: An architec-
ture for quantum computation,” Phys. Rev. A, vol. 69, p. 062320, Jun 2004.
[31] P. Rabl, P. Cappellaro, M. V. G. Dutt, L. Jiang, J. R. Maze, and M. D. Lukin,
“Strong magnetic coupling between an electronic spin qubit and a mechanical
resonator,” Phys. Rev. B, vol. 79, p. 041302, Jan. 2009.
[32] K. H. Onnes Leiden Communications, vol. 120b, 1911.
[33] M. Tinkham, Introduction to Superconductivity. Dover, 2004.
[34] “30 - On the theory of superconductivity,” in Collected Papers of L.D. Landau
(D. T. HAAR, ed.), pp. 217–225, Pergamon, 1965.
[35] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,”
Phys. Rev., vol. 108, pp. 1175–1204, Dec. 1957.
[36] B. Josephson, “Possible new effects in superconductive tunnelling,” Physics
Letters, vol. 1, pp. 251–253, July 1962.
[37] K. K. Likharev, Dynamics of Josephson Junctions and Circuits. CRC Press,
1986.
[38] C. P. Slichter, Principles of Nuclear Magnetic Resonance. Springer, 3rd ed.,
1990.
[39] Y. Makhlin, G. Scho¨n, and A. Shnirman, “Quantum-state engineering with
Josephson-junction devices,” Rev. Mod. Phys., vol. 73, pp. 357–400, May 2001.
[40] J. Clarke and F. K. Wilhelm, “Superconducting quantum bits,” Nature,
vol. 453, pp. 1031–1042, June 2008.
[41] J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer,
A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Charge-
insensitive qubit design derived from the Cooper pair box,” Phys. Rev. A,
vol. 76, p. 042319, Oct 2007.
[42] R. C. Bialczak, R. McDermott, M. Ansmann, M. Hofheinz, N. Katz, E. Lucero,
M. Neeley, A. D. OConnell, H. Wang, A. N. Cleland, and J. M. Martinis, “Flux
noise in josephson phase qubits,” Phys. Rev. Lett., vol. 99, p. 187006, Nov.
2007.
[43] J. M. Martinis, S. Nam, J. Aumentado, K. M. Lang, and C. Urbina, “Deco-
herence of a superconducting qubit due to bias noise,” Phys. Rev. B, vol. 67,
p. 094510, Mar. 2003.
139
[44] J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G.
Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, “Noise spectroscopy through
dynamical decoupling with a superconducting flux qubit,” Nat Phys, vol. 7,
pp. 565–570, July 2011.
[45] L. Faoro and L. B. Ioffe, “Quantum two level systems and Kondo-like traps as
possible sources of decoherence in superconducting qubits,” Phys. Rev. Lett.,
vol. 96, p. 047001, Jan. 2006.
[46] M. Steffen, S. Kumar, D. P. DiVincenzo, J. R. Rozen, G. A. Keefe, M. B. Roth-
well, and M. B. Ketchen, “High-coherence hybrid superconducting qubit,”
Phys. Rev. Lett., vol. 105, p. 100502, Sept. 2010.
[47] R. J. Schoelkopf and S. M. Girvin, “Wiring up quantum systems,” Nature,
vol. 451, pp. 664–669, Feb. 2008.
[48] H. Walther, B. T. H. Varcoe, B.-G. Englert, and T. Becker, “Cavity quantum
electrodynamics,” Reports on Progress in Physics, vol. 69, no. 5, p. 1325, 2006.
[49] D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett., vol. 47,
pp. 233–236, July 1981.
[50] M. Devoret, “Quantum fluctuations in electrical circuits,” Les Houches Session
LXIII, Elsevier Science B. V., 1995.
[51] G. Burkard, R. H. Koch, and D. P. DiVincenzo, “Multilevel quantum de-
scription of decoherence in superconducting qubits,” Phys. Rev. B, vol. 69,
p. 064503, Feb 2004.
[52] E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum
computation with linear optics,” Nature, vol. 409, pp. 46–52, Jan. 2001.
[53] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Mil-
burn, “Linear optical quantum computing with photonic qubits,” Rev. Mod.
Phys., vol. 79, pp. 135–174, Jan. 2007.
[54] M. A. Nielsen, “Optical quantum computation using cluster states,” Phys.
Rev. Lett., vol. 93, p. 040503, July 2004.
[55] D. E. Browne and T. Rudolph, “Resource-efficient linear optical quantum
computation,” Phys. Rev. Lett., vol. 95, p. 010501, June 2005.
[56] D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta,
A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin,
and R. J. Schoelkopf, “Resolving photon number states in a superconducting
circuit,” Nature, vol. 445, pp. 515–518, Feb. 2007.
[57] P. Adhikari, M. Hafezi, and J. M. Taylor, “Nonlinear optics quantum comput-
ing with circuit QED,” Phys. Rev. Lett., vol. 110, p. 060503, Feb 2013.
140
[58] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Ku-
mar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon
to a superconducting qubit using circuit quantum electrodynamics,” Nature,
vol. 431, pp. 162–167, Sept. 2004.
[59] E. Zakka-Bajjani, F. Nguyen, M. Lee, L. R. Vale, R. W. Simmonds, and
J. Aumentado, “Quantum superposition of a single microwave photon in two
different “colour” states,” Nat Phys, vol. 7, pp. 599–603, Aug. 2011.
[60] J. M. Martinis and O. Kevin, “Superconducting qubits and the physics of
Josephson junctions,” 2004.
[61] M. Boissonneault, J. M. Gambetta, and A. Blais, “Improved qubit bifurca-
tion readout in the straddling regime of circuit QED,” Phys. Rev. A, vol. 86,
p. 022326, Aug. 2012.
[62] C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, “Exper-
imental state tomography of itinerant single microwave photons,” Phys. Rev.
Lett., vol. 106, p. 220503, June 2011.
[63] A. A. Houck, D. I. Schuster, J. M. Gambetta, J. A. Schreier, B. R. John-
son, J. M. Chow, L. Frunzio, J. Majer, M. H. Devoret, S. M. Girvin, and
R. J. Schoelkopf, “Generating single microwave photons in a circuit,” Nature,
vol. 449, pp. 328–331, Sept. 2007.
[64] M. Hofheinz, E. M. Weig, M. Ansmann, R. C. Bialczak, E. Lucero, M. Neeley,
A. D. O’Connell, H. Wang, J. M. Martinis, and A. N. Cleland, “Generation of
Fock states in a superconducting quantum circuits,” Nature, vol. 454, pp. 310–
314, July 2008.
[65] Y. Xiao, M. Klein, M. Hohensee, L. Jiang, D. F. Phillips, M. D. Lukin,
and R. L. Walsworth, “Slow light beam splitter,” Phys. Rev. Lett., vol. 101,
p. 043601, July 2008.
[66] B. R. Johnson, M. D. Reed, A. A. Houck, D. I. Schuster, L. S. Bishop, E. Gi-
nossar, J. M. Gambetta, L. DiCarlo, L. Frunzio, S. M. Girvin, and R. J.
Schoelkopf, “Quantum non-demolition detection of single microwave photons
in a circuit,” Nat Phys, vol. 6, pp. 663–667, Sept. 2010.
[67] G. Romero, J. J. Garca-Ripoll, and E. Solano, “Microwave photon detector in
circuit QED,” Phys. Rev. Lett., vol. 102, p. 173602, Apr. 2009.
[68] Y. Yin, H. Wang, M. Mariantoni, R. C. Bialczak, R. Barends, Y. Chen,
M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides,
J. Wenner, T. Yamamoto, J. Zhao, A. N. Cleland, and J. M. Martinis, “Dy-
namic quantum Kerr effect in circuit quantum electrodynamics,” Phys. Rev.
A, vol. 85, p. 023826, Feb 2012.
141
[69] A. A. Abdumalikov, O. Astafiev, Y. Nakamura, Y. A. Pashkin, and J. Tsai,
“Vacuum rabi splitting due to strong coupling of a flux qubit and a coplanar-
waveguide resonator,” Phys. Rev. B, vol. 78, p. 180502, Nov. 2008.
[70] H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears,
B. R. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, S. M. Girvin, M. H.
Devoret, and R. J. Schoelkopf, “Observation of high coherence in Josephson
junction qubits measured in a three-dimensional circuit qed architecture,”
Phys. Rev. Lett., vol. 107, p. 240501, Dec. 2011.
[71] K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-
not gate,” Phys. Rev. Lett., vol. 93, p. 250502, Dec. 2004.
[72] I.-C. Hoi, A. F. Kockum, T. Palomaki, T. M. Stace, B. Fan, L. Tornberg,
S. R. Sathyamoorthy, G. Johansson, P. Delsing, and C. M. Wilson, “Giant
cross-Kerr effect for propagating microwaves induced by an artificial atoms,”
Phys. Rev. Lett., vol. 111, p. 053601, Aug. 2013.
[73] S. Rebi, J. Twamley, and G. J. Milburn, “Giant Kerr nonlinearities in circuit
quantum electrodynamics,” Phys. Rev. Lett., vol. 103, p. 150503, Oct. 2009.
[74] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically in-
duced transparency: Optics in coherent media,” Rev. Mod. Phys., vol. 77,
pp. 633–673, July 2005.
[75] J. Clarke and B. A. I., The Squid Handbook: Applications of Squids and Squid
Systems, vol. 2. Wiley-VCH, 2007.
[76] J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker,
and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling
regime,” Nature, vol. 471, pp. 204–208, Mar. 2011.
[77] K. Kakuyanagi, T. Meno, S. Saito, H. Nakano, K. Semba, H. Takayanagi,
F. Deppe, and A. Shnirman, “Dephasing of a superconducting flux qubit,”
Phys. Rev. Lett., vol. 98, p. 047004, Jan. 2007.
[78] S. M. Anton, C. Mller, J. S. Birenbaum, S. R. OKelley, A. D. Fefferman, D. S.
Golubev, G. C. Hilton, H.-M. Cho, K. D. Irwin, F. C. Wellstood, G. Schn,
A. Shnirman, and J. Clarke, “Pure dephasing in flux qubits due to flux noise
with spectral density scaling as 1/falpha.,” Phys. Rev. B, vol. 85, p. 224505,
June 2012.
[79] F. Wilczek, “Quantum mechanics of fractional-spin particles,” Phys. Rev.
Lett., vol. 49, pp. 957–959, Oct. 1982.
[80] A. Kitaev, “Anyons in an exactly solved model and beyond,” Annals of
Physics, vol. 321, pp. 2–111, Jan. 2006.
142
[81] R. B. Laughlin, “Anomalous quantum Hall effect: An incompressible quantum
fluid with fractionally charged excitations,” Phys. Rev. Lett., vol. 50, pp. 1395–
1398, May 1983.
[82] G. Moore and N. Read, “Nonabelions in the fractional quantum Hall effects,”
Nuclear Physics B, vol. 360, pp. 362–396, Aug. 1991.
[83] M. H. Freedman, M. Larsen, and Z. Wang, “A modular functor which is uni-
versal for quantum computation,” Communications in Mathematical Physics,
vol. 227, no. 3, pp. 605–622, 2002.
[84] A. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of
Physics, vol. 303, pp. 2–30, Jan. 2003.
[85] M. Greiter, X.-G. Wen, and F. Wilczek, “Paired Hall state at half filling,”
Phys. Rev. Lett., vol. 66, pp. 3205–3208, June 1991.
[86] A. S. S(ø)rensen, E. Demler, and M. D. Lukin, “Fractional quantum hall states
of atoms in optical lattices,” Phys. Rev. Lett., vol. 94, p. 086803, Mar. 2005.
[87] Y.-F. Wang, Z.-C. Gu, C.-D. Gong, and D. N. Sheng, “Fractional quantum
hall effect of hard-core bosons in topological flat bands,” Phys. Rev. Lett.,
vol. 107, p. 146803, Sept. 2011.
[88] T. Senthil and M. Levin, “Integer quantum hall effect for bosons,” Phys. Rev.
Lett., vol. 110, p. 046801, Jan. 2013.
[89] I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics with ultracold
gases,” Rev. Mod. Phys., vol. 80, pp. 885–964, July 2008.
[90] N. Cooper, “Rapidly rotating atomic gases,” Advances in Physics, vol. 57,
pp. 539–616, Nov. 2008.
[91] H. P. Buchler, A. Micheli, and P. Zoller, “Three-body interactions with cold
polar molecules,” Nat Phys, vol. 3, pp. 726–731, Oct. 2007.
[92] B. Paredes, T. Keilmann, and J. I. Cirac, “Pfaffian-like ground state for three-
body hard-core bosons in one-dimensional lattices,” Phys. Rev. A, vol. 75,
p. 053611, May 2007.
[93] A. J. Daley, J. M. Taylor, S. Diehl, M. Baranov, and P. Zoller, “Atomic three-
body loss as a dynamical three-body interaction,” Phys. Rev. Lett., vol. 102,
p. 040402, Jan. 2009.
[94] L. Mazza, M. Rizzi, M. Lewenstein, and J. I. Cirac, “Emerging bosons with
three-body interactions from spin-1 atoms in optical lattices,” Phys. Rev. A,
vol. 82, p. 043629, Oct. 2010.
143
[95] K. W. Mahmud and E. Tiesinga, “Dynamics of spin-1 bosons in an optical
lattice: Spin mixing, quantum-phase-revival spectroscopy, and effective three-
body interactions,” Phys. Rev. A, vol. 88, p. 023602, Aug. 2013.
[96] S. P. Jordan and E. Farhi, “Perturbative gadgets at arbitrary orders,” Phys.
Rev. A, vol. 77, p. 062329, June 2008.
[97] D. R. Hofstadter, “Energy levels and wave functions of Bloch electrons in
rational and irrational magnetic fields,” Phys. Rev. B, vol. 14, pp. 2239–2249,
Sept. 1976.
[98] M. Hafezi, A. S. S(ø)rensen, E. Demler, and M. D. Lukin, “Fractional quantum
Hall effect in optical lattices,” Phys. Rev. A, vol. 76, p. 023613, Aug. 2007.
[99] M. Hafezi, P. Adhikari, and J. M. Taylor, “Engineering three-body interaction
and Pfaffian states in circuit QED systems,” Phys. Rev. B, vol. 90, p. 060503,
Aug. 2014.
[100] J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, “Time-reversal-
symmetry breaking in circuit-QED-based photon lattices,” Phys. Rev. A,
vol. 82, p. 043811, Oct 2010.
[101] E. Kapit, “Quantum simulation architecture for lattice bosons in arbitrary,
tunable, external gauge fields,” Phys. Rev. A, vol. 87, p. 062336, June 2013.
[102] M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical
delay lines with topological protection,” Nat Phys, vol. 7, pp. 907–912, Nov.
2011.
[103] R. O. Umucallar and I. Carusotto, “Artificial gauge field for photons in coupled
cavity arrays,” Phys. Rev. A, vol. 84, p. 043804, Oct. 2011.
[104] V. E. Manucharyan, J. Koch, L. I. Glazman, and M. H. Devoret, “Fluxonium:
Single Cooper-pair circuit free of charge offsets,” Science, vol. 326, no. 5949,
pp. 113–116, 2009.
[105] D. Bozyigit, C. Lang, L. Steffen, J. M. Fink, C. Eichler, M. Baur,
R. Bianchetti, P. J. Leek, S. Filipp, M. P. da Silva, A. Blais, and A. Wall-
raff, “Antibunching of microwave-frequency photons observed in correlation
measurements using linear detectors,” Nat Phys, vol. 7, pp. 154–158, Feb.
2011.
[106] J. M. Fink, M. Goppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and
A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlin-
earity in a cavity Qed systems,” Nature, vol. 454, pp. 315–318, July 2008.
[107] K. Huang, Statistical Mechanics. New York: Wiley, 1987.
144
[108] P. Wu¨rfel, “The chemical potential of radiation,” Journal of Physics C: Solid
State Physics, vol. 15, no. 18, p. 3967, 1982.
[109] H. Ries and A. McEvoy, “Chemical potential and temperature of light,” Jour-
nal of Photochemistry and Photobiology A: Chemistry, vol. 59, pp. 11–18, June
1991.
[110] F. Herrmann and P. Wu¨rfel, “Light with nonzero chemical potential,” Amer-
ican Journal of Physics, vol. 73, pp. 717–721, 2005.
[111] G. Job and F. Herrmann, “Chemical potential–a quantity in search of recog-
nition,” European Journal of Physics, vol. 27, no. 2, p. 353, 2006.
[112] J. Klaers, J. Schmitt, F. Vewinger, and M. Weitz, “Bose-Einstein condensation
of photons in an optical microcavity,” Nature, vol. 468, pp. 545–548, Nov. 2010.
[113] J. Klaers, J. Schmitt, T. Damm, F. Vewinger, and M. Weitz, “Statistical
physics of Bose-Einstein-condensed light in a dye microcavity,” Phys. Rev.
Lett., vol. 108, p. 160403, Apr. 2012.
[114] C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. W. Fleischer, “Observation
of the kinetic condensation of classical waves,” Nat Phys, vol. 8, pp. 470–474,
June 2012.
[115] D. Snoke and S. Girvin, “Dynamics of phase coherence onset in Bose conden-
sates of photons by incoherent phonon emission,” Journal of Low Temperature
Physics, vol. 171, no. 1-2, pp. 1–12, 2013.
[116] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of bose-
einstein condensation in trapped gases,” Rev. Mod. Phys., vol. 71, pp. 463–512,
Apr. 1999.
[117] P. Kirton and J. Keeling, “Nonequilibrium model of photon condensation,”
Phys. Rev. Lett., vol. 111, p. 100404, Sept. 2013.
[118] A.-W. de Leeuw, H. T. C. Stoof, and R. A. Duine, “Schwinger-Keldysh theory
for Bose-Einstein condensation of photons in a dye-filled optical microcavity,”
Phys. Rev. A, vol. 88, p. 033829, Sept. 2013.
[119] D. N. Sob’yanin, “Bose-Einstein condensation of light: General theory,” Phys.
Rev. E, vol. 88, p. 022132, Aug. 2013.
[120] Y. Subasi, C. H. Fleming, J. M. Taylor, and B. L. Hu, “Equilibrium states of
open quantum systems in the strong coupling regime,” Phys. Rev. E, vol. 86,
p. 061132, Dec. 2012.
[121] N. Bergeal, R. Vijay, V. E. Manucharyan, I. Siddiqi, R. J. Schoelkopf, S. M.
Girvin, and M. H. Devoret, “Analog information processing at the quantum
limit with a Josephson ring modulator,” Nat Phys, vol. 6, pp. 296–302, Apr.
2010.
145
[122] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechan-
ics,” arxiv:1303.0733, 2013.
[123] M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and
R. J. Schoelkopf, “Realization of three-qubit quantum error correction with
superconducting circuits,” Nature, vol. 482, pp. 382–385, Feb. 2012.
[124] R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature,
vol. 453, pp. 1008–1015, June 2008.
[125] T. Monz, K. Kim, W. Hnsel, M. Riebe, A. S. Villar, P. Schindler, M. Chwalla,
M. Hennrich, and R. Blatt, “Realization of the quantum Toffoli gate with
trapped ions,” Phys. Rev. Lett., vol. 102, p. 040501, Jan. 2009.
[126] D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in paramet-
ric production of optical photon pairs,” Phys. Rev. Lett., vol. 25, pp. 84–87,
July 1970.
[127] R. Vijay, M. Devoret, and S. I., “Invited review article: The Josephson bifur-
cation amplifiers,” Rev. Sci. Instrum., vol. 80, 2009.
[128] F. Beaudoin, M. P. da Silva, Z. Dutton, and A. Blais, “First-order sidebands
in circuit QED using qubit frequency modulation,” Phys. Rev. A, vol. 86,
p. 022305, Aug. 2012.
[129] D. Kielpinski, D. Kafri, M. J. Woolley, G. J. Milburn, and J. M. Taylor,
“Quantum interface between an electrical circuit and a single atom,” Phys.
Rev. Lett., vol. 108, p. 130504, Mar. 2012.
[130] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of
single trapped ions,” Rev. Mod. Phys., vol. 75, pp. 281–324, Mar. 2003.
[131] L. S. Brown, “Quantum motion in a Paul trap,” Phys. Rev. Lett., vol. 66,
pp. 527–529, Feb. 1991.
[132] N. W. McLachlan, Theory and Application of Mathieu Functions. Oxford
University Press, 1947.
[133] A. S. S(ø)rensen, C. H. van der Wal, L. I. Childress, and M. D. Lukin, “Capac-
itive coupling of atomic systems to mesoscopic conductors,” Phys. Rev. Lett.,
vol. 92, p. 063601, Feb. 2004.
[134] L. Tian, P. Rabl, R. Blatt, and P. Zoller, “Interfacing quantum-optical and
solid-state qubits,” Phys. Rev. Lett., vol. 92, p. 247902, June 2004.
[135] P. Rabl, D. DeMille, J. M. Doyle, M. D. Lukin, R. J. Schoelkopf, and P. Zoller,
“Hybrid quantum processors: Molecular ensembles as quantum memory for
solid state circuits,” Phys. Rev. Lett., vol. 97, p. 033003, July 2006.
146
[136] D. Petrosyan, G. Bensky, G. Kurizki, I. Mazets, J. Majer, and J. Schmied-
mayer, “Reversible state transfer between superconducting qubits and atomic
ensembles,” Phys. Rev. A, vol. 79, p. 040304, Apr. 2009.
[137] J. Preskill, “Quantum computing and the entanglement frontier,” Arxiv-1203-
5813-v3, 2012.
[138] C.-C. Tannoudji and J. Dupont-Roc, Atom Photon Interactions: Basis Pro-
cesses and Applications. Wiley-VCH, 1992.
147