T HE U NIVERSITY OF T OKYOP H D T HESIS

Many-body effect on quantum transport
in superconducting circuits

Tsuyoshi YAMAMOTO
February 7, 2022

Advisor: Takeo K ATO
Department of Physics, The University of Tokyo

iii

Abstract
A superconducting circuit is one of the prominent candidates for the realizationof quantum computation, which is the grand challenge in 21st-century science. The
superconducting circuit has several advantages in a high degree of freedom for designing, and its elementary ingredients, such as superconducting qubits, transmission lines, and hybrid structures called the circuit quantum electrodynamics (QED),
have been utilized for the application of quantum computers. The superconducting circuit also provides an ideal and useful platform for the study of fundamental
physics. In particular, coupling to environments and disorder, which are usually obstacles for the realization of quantum computers, provides fascinating problems on
many-body effects, quantum phase transition, localization, and heat transport.
In this thesis, we investigate three problems in superconducting circuits. First,
we consider a quantum phase transition in the sub-Ohmic spin-boson system, whichis realized by a charge qubit coupled to a transmission line, and study quantum
critical phenomena in the frequency dependence of microwave scattering off thiscircuit. By performing the quantum Monte Carlo simulation, we find signatures of
the quantum critical behavior as well as the Fermi-liquid-like feature indicated by
the generalized Shiba-Korringa relation. We also describe how to construct the subOhmic spin-boson system using superconducting circuits.
Second, we study the transmission properties through a Josephson-junction chain.
This system is not only useful for application as a superinductor but also suitable tostudy many-body effects induced by Coulomb interaction and disorder. When thedisorder is added to this system, the interplay of Coulomb interaction and disordermakes the Josephson-junction chain undergo a transition into an intriguing glassy
insulator: Bose glass. We focus on the deep insulating regime and show that thecollective pinning affects the dynamical properties of the Josephson-junction chain
in the whole frequency region.
Finally, we study heat transport through an assembly consisting of a superconducting qubit sandwiched between superconducting harmonic resonators, which
is described by the quantum Rabi model, a typical multi-level system. Using thenon-perturbative approximation, we find a two-peak structure of the linear thermal
conductance as a function temperature. This characteristic transport behavior comesfrom the fact that there are multiple levels in the assembly and is expected to be advantageous for the application of quantum heat devices.
These studies will provide a concrete foundation for further research on thephysics of superconducting circuits. We also expect that the results obtained in thisthesis will be observed in the near future.

v

Acknowledgements
I want to thank especially Professor Takeo Kato for supervising. He is very
friendly, and I could ask him any little things about physics. His broad knowledge
about mesoscopic physics was helpful for proceeding with my projects smoothly.
At the beginning of 2020, I stayed at CEA in Grenoble, France, for three months.
There, I worked on a disordered Josephson-junction chain with Professor ManuelHouzet. I want to thank him for welcoming me to his group and working with me.
His deep understanding of localization physics and great insight into the Josephsonjunction chain helped my project for proceeding. Although it was my first time
staying in a foreign country for over a week, I could enjoy Grenoble life thanks tomembers of IRIG/PHELIQS. I am also grateful to Professor Leonid I. Glazman. His
helpful comments made our work more sophisticated.
This thesis would not have been possible without the funding from JSPS Research Fellowship for Young Scientists. During my stay at Grenoble, I was financially supported by the Innovative Professional Development (IPD) program. Without this support, I could not stay at Grenoble and work with Professor ManuelHouzet. Some numerical results presented in this thesis have been calculated on
the supercomputer. I thank the Supercomputer Center, the Institute for Solid State
Physics, the University of Tokyo for the use of the facilities.
During my five years at the Institute for Solid State Physics, the members of the
Kato group have been continuously changing. I want to thank all members of the
Kato group. They have made my graduate student life enjoyable. Rui Sakano is anassistant professor in the Kato group. Chatting with him about any topic was fun
and relaxed me. In addition, he gave me useful comments for making presentations
for scientific conferences and documents for the fellowship. Tomohiro Tamaya is avery passionate person about physics. He gave me a lot of advice as an academic
researcher. Hiroko Eguchi is a secretary of the Kato group. She was always worried
about me and helped me with doing paperwork.
Finally, I want to thank my parents. While there is little understanding of the
Ph.D. in Japanese society, they accepted that I went on the Ph.D. course and encouraged me.

vii

List of Publications[1] “Microwave Scattering in the Subohmic Spin-Boson Systems of Superconducting Circuits”,
T. Yamamoto and T. Kato, J. Phys. Soc. Jpn. 88, 094601 (2019).
[2] “Transmission of waves through a pinned elastic medium”,
T. Yamamoto, L. I. Glazman, and M. Houzet, Phys. Rev. B 103, 224211 (2021).
[3] “Heat transport through a two-level system embedded between two harmonicresonators”,
T. Yamamoto and T. Kato, J. Phys.: Condens. Matter 33, 395303 (2021).

ix

Contents
Abstractiii

Acknowledgementsv

1

Introduction
1.1 Development of quantum computer . . . . . . . . . . . . . . . . . . . .
1.2 Superconducting circuits as a quantum simulator . . . . . . . . . . . .
1.2.1 Circuit-QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Josephson-junction chain . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Quantum heat transport . . . . . . . . . . . . . . . . . . . . . . .
1.3 Purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
2
2
2
3
3
4

2

Review of Superconducting Circuits
2.1 Linear circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 LC-circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Transmission line . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Josephson junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Charge qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Transmon qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Flux qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Circuit-QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Spin-boson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Kondo problem for the Ohmic case . . . . . . . . . . . . . . . . .
2.5.2 Recent experiment for the Ohmic case . . . . . . . . . . . . . . .
2.6 Josephson-junction chain . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Superinductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.3 Superconductor-insulator transition . . . . . . . . . . . . . . . .
2.7 Disordered Josephson-junction chain . . . . . . . . . . . . . . . . . . . .
2.7.1 Random offset charges . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2 Superfluid-Bose glass transition . . . . . . . . . . . . . . . . . .
2.7.3 Classical limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.4 Recent experiment . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Quantum heat transport . . . . . . . . . . . . . . . . . . . . . . . . . . .

5
5
5
6
7
9
9
11
11
12
14
15
16
17
17
18
19
20
20
21
22
23
24

3

Microwave Scattering through a Superconducting Qubit
3.1 Sub-Ohmic spin-boson model . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Formalism for microwave scattering . . . . . . . . . . . . . . . . . . . .
3.2.1 Circuit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Realization of the sub-Ohmic bath . . . . . . . . . . . . . . . . .
(i) case for s = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . .

27
27
29
29
32
33

x
(ii) case for 0 < s < 0.5 . . . . . . . . . . . . . . . . . . . . . . . .
(iii) case for 0 < s < 1 . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Microwave scattering . . . . . . . . . . . . . . . . . . . . . . . .
Continuous-time quantum Monte Carlo method . . . . . . . . . . . . .
3.3.1 Kink representation . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Calculation algorithm . . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Binder analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Dynamic spin susceptibility . . . . . . . . . . . . . . . . . . . . .
Quantum critical regime . . . . . . . . . . . . . . . . . . . . . . .
Delocalized regime . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Short summary of this chapter . . . . . . . . . . . . . . . . . . . . . . . .

33
34
35
36
36
37
39
40
40
40
42
42
44

4

Transmission through a Disordered Josephson-junction Chain
4.1 Disordered Josephson-junction chain . . . . . . . . . . . . . . . . . . . .
4.2 Formalism for scattering problem in the classical limit . . . . . . . . . .
4.2.1 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Effective disorder potential . . . . . . . . . . . . . . . . . . . . .
4.3 Weakly localized regime . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Mapping to the Dirac equation . . . . . . . . . . . . . . . . . . .
4.3.2 Fokker-Planck formalism . . . . . . . . . . . . . . . . . . . . . .
(i) perfect impedance matching (T0 = 1) . . . . . . . . . . . . . .
(ii) asymptote at d ≫ ℓ(ω ) and arbitrary T0 . . . . . . . . . . . .
(iii) asymptote at d ≪ ℓ(ω ) and T0 ≪ 1 . . . . . . . . . . . . . .
4.3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Strongly localized regime . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Lyapunov exponent . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Reflection coefficient . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Average transmission . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Short summary of this chapter . . . . . . . . . . . . . . . . . . . . . . . .

45
45
46
47
48
49
51
51
52
54
55
56
56
59
59
60
62
63

5

Heat Transport through the Quantum Rabi System
5.1 Quantum heat transport . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Formalism for heat transport . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Dissipative quantum Rabi model and spin-boson model . . . .
5.2.2 Linear thermal conductance . . . . . . . . . . . . . . . . . . . . .
5.2.3 Weak-dissipation regime . . . . . . . . . . . . . . . . . . . . . . .
5.3 Non-interacting blip approximation . . . . . . . . . . . . . . . . . . . .
5.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Symmetrized correlation function . . . . . . . . . . . . . . . . .
5.4.2 Linear thermal conductance . . . . . . . . . . . . . . . . . . . . .
5.4.3 Rotating-wave approximation . . . . . . . . . . . . . . . . . . .
5.5 Short summary of this chapter . . . . . . . . . . . . . . . . . . . . . . . .

65
65
67
68
70
71
72
73
74
74
77
79

6

Summary and Perspectives

81

3.3

3.4

3.5

A Lagrangian and Hamiltonian for a transmission line83

B Lagrangian for a Josephson-junction chain

87

xi
C Impedance of RLC Transmission Lines

91

D Fokker-Planck Equation

93

E Exact Solution of P2 ( x, θ2 )95

F Gaussian White-noise Disorder Potential

97

G Heat Current in the Keldysh Formalism

101

H Mapping to the Spin-boson Model

105

1

Chapter 1

Introduction
In this chapter, we briefly explain the research background of this thesis. After abrief overview of the many-body problems in superconducting circuits, we describe
the purpose of this thesis. Finally, we give the outline of this thesis.

1.1

Development of quantum computer

The realization of quantum computation is one of the grand challenges in 21stcentury science. The idea of the quantum computer was firstly proposed 40 years
ago by Richard P. Feynman for simulating quantum systems that are too hard to dousing classical computers [1]. While the information is expressed by a bit, whichtakes only two values, e.g., 0 or 1, in the classical computer, we can use their superposition α |0⟩ + β |1⟩ to perform computation in the quantum computer [2]. A set of
two quantum states is called a “qubit” or “quantum bit” in analogy to the classicalcomputer. Although this idea might sound like a fantasy at the time of the first proposal, quantum computers are now beginning to realize in various physical systems,e.g., superconducting circuits [3], semiconductors [4, 5, 6], or cold atoms [7, 8, 9].
A solid-state device using superconducting circuits is a leading candidate for therealization of quantum computers because of their high degree of freedom for designing, suppression of dissipation, and strong non-linearity. However, there still
remain several problems to be overcome. One of the major problems is the lossof the devices’ coherence. Since the solid devices inevitably interact with environments, the quantum coherence of qubits is lost in a finite amount of time. The coherence time of superconducting qubits was of the order of 1 ns at the beginning of
the development [10]. Thanks to technological development, it has been improved
for these twenty years to about 100 µs [11]. This remarkable improvement of coherence time has changed the situation of the development of quantum computerssignificantly. Recently, big companies, such as Google, IBM, Intel, and Microsoft,
also joined quantum computer development. This trend will be accelerated in the
future.
It is also worth noting that qubits can be coupled to an artificial environment fabricated on superconducting circuits in a controlled way. The circuit quantum electrodynamics (circuit-QED) has been studied intensively for the system composedof superconducting qubits and transmission lines [12]. The circuit-QED is now
an essential technology for control of inter-qubit coupling and quantum measurement [13, 14, 15].

2

1.2

Chapter 1. Introduction

Superconducting circuits as a quantum simulator

We have mentioned the superconducting qubit as a leading ingredient of the quantum computer. To operate correctly it as the quantum computer, it is needed to put
qubits on an ideal surrounding. However, in realistic systems, there are inevitable
troublesome sources, such as disorder of circuits and interaction with the environments. For the realization of the quantum computer, they have to be suppressed
as far as possible. However, in this thesis, we would like to promote the opposite
direction. Namely, we focus on many-body effects induced by the presence of badsituation, such as strong coupling to the environment and randomness of the system. The superconducting circuits indeed provide an attractive stage to investigate
fundamental physics related to many-body problems that emerge in such bad situations because of their high feasibility and controllability. In this sense, the superconducting circuit can be regarded as an ideal “quantum simulator” for solving thefundamental problems in quantum physics.
In the subsequent three subsections, we briefly overview many-body phenomena in superconducting circuits from three viewpoints. A detailed description, including experimental situations, will be given in Chapter 2.

1.2.1

Circuit-QED

Generally, it is challenging to realize a small quantum system strongly coupled toenvironments. For example, the interaction strength between an atom and the lightin a vacuum is characterized by the fine structure constant α = 1/137 ≪ 1. In thecavity-QED system, the atom-light interaction can be strengthened by introducing
a cavity for the light. Compared with the cavity-QED, the circuit-QED realized in
the superconducting circuit has several advantages for realizing strong interaction
between a superconducting qubit and electromagnetic field of the superconducting
resonator because the superconducting qubit has a much larger dipole moment anda higher quality factor [12]. Therefore, the circuit-QED system is an ideal platform
for realizing a strong coupling regime between the qubit and the resonator. Furthermore, the circuit-QED system allows us to realize a qubit system coupled to continuum environments that is described by a fundamental open quantum system called
the spin-boson model. It is notable that the spin-boson model has been studied theoretically for a long time as a useful prototype for the study of decoherence due tothe environment [16, 17, 18, 19, 20, 21, 22, 23]. It is also known that the spin-bosonmodel exhibits many-body effects such as the Kondo-like effect and the quantum
phase transition. Now, the recent technological development of superconducting
circuits has enabled us to directly access these many-body effects in a controllable
way.

1.2.2

Josephson-junction chain

Because of the flexibility in superconducting circuits, it is not difficult to fabricate
a Josephson-junction chain. Actually, the recent development of fabrication techniques has enabled us to realize the long Josephson-junction chain in which eachsuperconducting island is coupled to two neighboring islands via the Josephson coupling [24]. The Josephson-junction chain works as a high-impedance transmissionline, known as a superinductor, and is useful for various applications. Moreover,1.3. Purposes3

it exhibits the many-body phenomena, such as the superconductor-insulator transition, inherent to Coulomb interaction between Cooper pairs. Although it is possible to realize the Josephson junction chain with uniform junction parameters, thedisordered potential for superconducting islands is inevitable because of charged
impurities near the superconducting islands in the present technology. This disorder in the one-dimensional quantum system raises the localization, such as theAnderson localization for the non-interacting particles [25, 26, 27, 28]. Therefore,
the Josephson-junction chain is an important system for investigating the localization problem incorporated with the many-body effect. In particular, the disordered
Josephson-junction chain is an ideal system to study a compressible insulating state,
which is called “Bose glass” [29, 30, 31, 32].

1.2.3

Quantum heat transport

Recent development in the technology of heat measurement has enabled us to study
heat transport in superconducting devices [33, 34, 35]. The superconducting circuitprovides an ideal platform to study quantum heat transport via a small quantum
system. Heat transport in the superconducting circuit is mainly induced by energy
transfer between two or more thermal baths of photons at low temperatures where
phonon transport is negligible. In the analogy of the electronic devices, various
quantum heat devices such as a quantum heat rectifier, a quantum heat valve, aquantum heat transistor, and a quantum heat engine have been proposed, and someof them have been implemented on the superconducting circuits [35].
Heat transport through a single superconducting qubit is one of the most fundamental problems in the study of quantum heat transport. This kind of heat transport can be described by the spin-boson model, which exhibits many-body effects
as already described. The transport processes in heat transport through the single
superconducting qubit (the two-level system) have been studied theoretically in thewhole parameter region using the perturbative methods [36, 37] and the numericalsimulations [23, 38]. From the viewpoint of the function as quantum heat devices,
the system described by the spin-boson model is so simple that there is no room todemonstrate useful functions by controlling experimental parameters. Therefore, togain a useful function in quantum heat devices, we need to expand the experimental
setup beyond the single superconducting qubit.

1.3

PurposesIn this thesis, we theoretically study three types of many-body effects in superconducting circuits. First, we consider a single superconducting qubit (a two-level system) coupled to the environment based on the spin-boson model. In general, the environment of the spin-boson model is categorized into three types, i.e., sub-Ohimc,
Ohmic, and super-Ohmic baths (see Chapter 2 for a detail). In this thesis, we focus
on quantum phase transition in the sub-Ohmic spin-boson model and study how
quantum critical phenomena are observed in microwave spectroscopy, employing
the quantum Monte Carlo simulation and the renormalization-group analysis. In
addition, we propose superconducting circuits which realize the sub-Ohmic spinboson model.
Second, we study properties of microwave spectroscopy in a disordered Josephsonjunction chain with emphasis on the deep insulating phase where the disorder of thesystem and the elasticity due to the long-range Coulomb interaction compete with

4

Chapter 1. Introduction

each other. By studying the frequency dependence of the microwave transmission,
we find that the contribution of the forward and backward scatterings can be discussed separately and that the model for the Josephson-junction chain has a different
universality from the known disordered model in the Bose glass phase.
Finally, we study quantum heat transport through a superconducting qubit embedded between two harmonic resonators, which is described by the quantum Rabi
model. We show that this system has a sufficiently large degree of freedom for control of heat current compared with the standard spin-boson model. By employing
the non-perturbative approximation called the noninteracting-blip approximation,
we calculate the linear thermal conductance and discuss its temperature dependence. We seek out a parameter region where the transport property is peculiar
to multi-level systems and clarify its mechanism. We find that the linear thermal
conductance can be sensitive to a controllable parameter, which is advantageous forthe application of quantum heat devices.

1.4

OutlineThis thesis is organized as follows. In Chapter 2, we review how to treat the superconducting circuit theoretically. We explain theoretical descriptions for resonators,
transmission lines, Josephson junctions, superconducting qubits, hybrid systems,
and Josephson junction chains. We also review recent experiments for superconducting circuits and clarify the problems to be solved. In Chapter 3, we study the
microwave reflection in superconducting circuits described by the sub-Ohmic spinboson model [39]. We propose a circuit model to realize a sub-Ohmic spin-bosonmodel and discuss quantum critical behavior through frequency dependence of thereflection amplitude in microwave spectroscopy. In Chapter 4, we investigate themicrowave transmission through a disordered Josephson-junction chain [40]. By
combining analytical methods and numerical calculation, we clarify the frequency
dependence of the transmission in the localized regime and present several pieces of
information on the transmission, the Lyapunov exponent, and the reflection phase.
In Chapter 5, we study quantum heat transport through the quantum Rabi model [41].
Using the Keldysh formalism and the non-perturbative approach, we calculate thelinear thermal conductance and show its temperature dependence in detail and clarify useful properties as quantum heat devices. Finally, we summarize our works andgive future perspectives for them in Chapter 6.

5Chapter 2

Review of Superconducting
Circuits
In this chapter, we introduce various models for superconducting circuits, which
will be used in the theoretical studies described in later chapters. First, we explain basic linear superconducting elements, an LC-circuit and a transmission line,
in Section 2.1. We also introduce three types of superconducting qubits (charge
qubit, transmon qubit, and flux qubit) in Section 2.3. Finally, we introduce three
types of hybrid superconducting circuits, i.e., a dissipative superconducting qubit,a Josephson-junction chain, and quantum heat devices and briefly review recent experiments regarding each circuit.

2.1

Linear circuits

2.1.1

LC-circuit

First, we consider the LC-circuit, which is the simplest electrical circuit composed of
an inductor with the inductance L and a capacitor with the capacitance C in parallel(see Fig. 2.1 (a)) [42]. The equations of motion for the voltage V and the current I of
the LC-circuit are given by using the Kirchhoff’s circuit lawsdV (t),dt
dI (t)
V (t) = L.
dt
I (t) = −C

(2.1)
(2.2)

From these equations of motion, we can obtain an equation for the current
d2 I ( t )= −ω02 I (t)dt2

F IGURE 2.1: (a) LC-circuit and (b) lumped element model of the transmission line.

(2.3)

6

Chapter 2. Review of Superconducting Circuits√with ω0 = 1/ LC. This equation means that the current of the LC-circuit oscillateswith the angular frequency ω0 .
Energies stored at the capacitor and the inductor are given by (1/2)CV 2 and(1/2) LI 2 , respectively. Using relations between the charge and the voltage (Q = CV)and the charge and the current (I = − Q̇), the total energy is expressed in terms ofthe charge asH=

1 2 L 2Q + Q̇ .
2C
2

(2.4)

Now, introducing a flux variable ϕ defined as ϕ̇ = V, which is conjugate to thecharge,[ϕ, Q] = ih̄,

(2.5)

1 2
1 2
Q +
ϕ .
2C
2L

(2.6)

the total energy is rewritten as

H=This Hamiltonian is identical to one of a harmonic oscillator with the correspondence(p, x, m, k) ↔ (Q, ϕ, C, L−1 ). Using this identity, we can quantize the LC-circuit in
the same manner as the harmonic oscillator. Introducing bosonic annihilation andcreation operators a and a† , respectively, which satisfy the commutation relation[ a, a† ] = 1,rh̄Z0 
ϕ=a + a† ,
(2.7)
2
s

h̄ 
Q = −ia − a† ,
(2.8)
2Z0

√
with an impedance of the LC-circuit Z0 = L/C, the Hamiltonian reads1
H = h̄ω0 a† a +
,
2

(2.9)

which means that the LC-circuit is a resonator with eigen angular frequency ω0 .

2.1.2

Transmission line

A transmission line is a useful ingredient of superconducting circuits in which electromagnetic waves can propagate freely (see Fig. 2.1 (b)). The transmission line is anatural extension of the LC-circuit and its Hamiltonian is obtained from the continuum limit of a series of the LC-circuits as


Z d
1
1
2
2
(2.10)H=dx[ Q( x )] + [∂ x ϕ( x )] ,2c2l0
where Q( x ) and ϕ( x ) are the charge and flux variables which satisfy the commutation relation [ϕ( x ), Q(y)] = ih̄δ( x − y). In a similar procedure to the LC-circuit, we
can rewrite the Hamiltonian of the transmission line in terms of the bosonic annihilation and creation operators an and a†n , respectively, which satisfy the commutation

2.2. Josephson junction

7

F IGURE 2.2: Josephson junction. Superconductors are interrupted bya thin insulating barrier. Copper pairs induce the supercurrent without any voltage.

relation [ an , a†m ] = δn,m , as

H = ∑ h̄ωk

1a†k ak +

k

2


,

(2.11)√
where ωk = |k |/ lc is the natural frequency of the kth mode. This form of the
Hamiltonian means that transmission line is equivalent to a collection of harmonic
oscillators. For a detailed derivation from the Lagrangian, see Appendix A.

2.2

Josephson junction

Next, we introduce a Josephson junction, which is a non-linear circuit ingredient
to bring fruitful physics and complex functions of devices. A Josephson junction
consists of two superconductors interrupted by a thin insulating barrier [43, 44] asshown in Fig 2.2. Through the Josephson junction, a supercurrent can flow without
voltage drop across the barrier and its amplitude is given by the first Josephson
relation
Is = Ic sin φ,(2.12)

where φ is the phase difference between the two superconductors, which is called
the Josephson phase and Ic is the critical current. When the constant voltage difference V is applied to the Josephson junction, the time evolution of the Josephson
phase is governed by the second Josephson relation,
dφ(t)
2e
= V.
dt
h̄

(2.13)

From these relations, (2.12) and (2.13), we can see that the Josephson junction behaves as an inductor with the Josephson inductance
LJ ≡

V
h̄
h̄
p
=
=
.
dI/dt
2eIc cos φ
2eIc 1 − ( I/Ic )2(2.14)

This relation indicates the non-linearity of the Josephson junction, which plays acrucial role in superconducting circuits. We note that for Ic ≫ I, the Josephson
inductance becomes independent of the bias current as LJ = h̄/(2eIc ). This linear
inductor is useful for making lossless and high-impedance transmission lines.

8

Chapter 2. Review of Superconducting Circuits

F IGURE 2.3: Symbols for the Josephson junction. (a) The Josephson
junction with the capacitance CJ . (b) Symbol putting together theJosephson energy EJ and the self-capacitance for simplicity.

The energy stored in a Josephson junction can be calculated by integrating theelectrical power in the Josephson junction with respect to time,
U=

Z t
0

dt′ V Is =

h̄
Ic
2e

Z t

dφ(t)
sin φ(t) = EJ (1 − cos φ),
dt

dt′

0

(2.15)

where EJ = (h̄/2e) Ic is a Josephson energy. The Josephson inductance can be rewritten with the Josephson energy as

LJ =

h̄
2e

2

1
.
EJ

(2.16)

Besides the inductance, the Josephson junction has a capacitance, CJ , due to its
geometry of the two spatially-separated superconducting electrodes (see Fig 2.3).
This defines a charging energy at the Josephson junction
EC =

(2e)2
.
2CJ

(2.17)

Therefore, in the linear regime, the Josephson junction is equivalent to an LC-circuitwith a resonance frequency
ωp = p1.
LJ CJ(2.18)

This frequency is called the plasma frequency of the Josephson junction.
Here, we present a remark regarding operators used to quantize the electricalcircuit. For the linear circuits, we used the flux ϕ and the charge Q as conjugate variables (2.5). However, when treating the Josephson junction, the charge is quantizedin units of 2e because Cooper pairs pass across the insulating barrier and contributeto the supercurrent. Then, it is convenient to use the number of Cooper pairs,
n=

Q
,
2e

(2.19)

instead of the charge. Moreover, the phase variable φ is used as the conjugate variable to n. The phase is related to the flux through the second Josephson relation (2.13)
as
φ=

2e
ϕ.
h̄

(2.20)

2.3. Superconducting qubits

9

F IGURE 2.4: Circuits for three kinds of superconducting qubits; (a)charge qubit, (b) transmon qubit, (c) flux qubit. The red lines represent the Cooper-pair box.

Thus, the phase and the number of Cooper pairs satisfy the following commutation
relation:[n, φ] = i.

(2.21)

This non-linear element brings a quantum many-body effect under the dissipation. When a single Josephson junction is Ohmically shunted by a resistor, theJosephson junction shows a quantum phase transition due to the dissipation by theresistor, which is known as Schmid transition [45]. When the resistance R is larger
than a resistance quantum, RQ = h/(2e)2 , the Josephson junction becomes insulating at zero temperature while when R < RQ , it becomes superconducting. Thistransition is related to the charge localization and has been studied theoretically inmany literature [46, 47, 48, 49]. Recently, by replacing the normal resistor with thetransmission line with the tunable impedance on superconducting circuits, the dynamical property was addressed in the wide parameter region [50]. We note that the
Schmid transition has a different universality from a quantum phase transition in
the Ohmic spin-boson model, which will be discussed in Section 2.5.

2.3

Superconducting qubits

A qubit is realized by a small quantum system consisting of two states. Among
possible implementations of qubits in physical systems, a superconducting qubit is
one of the most hopeful candidates due to its high feasibility and readout accuracy.
Generally, there are many states more than two in closed superconducting circuits. To use them as qubits, they have to be reduced to the effective two-state systems, i.e., the lowest two energy states are separated well from other higher states
by utilizing the non-linearity of the Josephson junctions. In this section, we review
three types of superconducting qubits, i.e., a charge qubit, a transmon qubit, and aflux qubit.

2.3.1

Charge qubit

The charge qubit is the simplest superconducting qubit which has been first implemented experimentally [10]. The charge qubit consists of a single Josephson junction
with the large charging energy, the gate capacitance Cg , and the gate voltage Vg as
shown Fig. 2.4 (a). The tiny superconducting island surrounded by the Josephson
junction and the gate capacitance is called a Cooper-pair box (indicated by the redline in Fig. 2.4 (a)). For the charge qubit, it is convenient to adopt the charge states as10Chapter 2. Review of Superconducting Circuits

F IGURE 2.5: Energy spectrum (n = 0; blue, 1; green, 2; red) of Hamiltonian (2.24) as a function of the number of Cooper pairs induced by
the gate voltage ng for different EJ /EC = (a) 0.25; charge qubit, (b) 10;
transmon qubit. The energy levels are normalized by the first transition energy E10 = E1 − E0 at ng = 1/2. The minimum energy is set tothe origin of energy.

basis states, where the states of the qubit are characterized by the number of Cooper
pairs across the Josephson junction.
The Hamiltonian of the charge qubit is given byH=( Q − Qg )2+ EJ (1 − cos φ) ,2(CJ + Cg )(2.22)

where Q is an excess charge in the Cooper-pair box, i.e., the charge measured fromits neutral state and φ is the phase difference across the Josephson junction. For
convenience, we introduce new variables, the number of excess Cooper pairs, n =Q/(2e) and ng = Qg /(2e). The latter can be tuned by the gate voltage as ng =Cg Vg /(2e). The operator e±iφ changes the number of Cooper pairs in the Cooperpair box by ∓1 because the phase and the charge are conjugate to each other (seeEq. (2.21)). The cos-term in the Josephson energy is then expressed in the charge
basis {|n⟩} as⟨n| cos φ |m⟩ = ⟨n|eiφ + e−iφδ+ δn,m−1|m⟩ = n,m+1.
2
2(2.23)

Therefore, the Hamiltonian of the charge qubit is written on the charge basis as
∞

H=

∞

EJ
(|n⟩ ⟨n + 1| + |n + 1⟩ ⟨n|) .
n=−∞ 2∑ EC (n − ng )2 |n⟩ ⟨n| − ∑n=−∞(2.24)

where EC = (2e2 )/(CJ + Cg ) and we have dropped the constant term.
The energy spectrum of the Hamiltonian (2.24) is shown as a function of ng in
Fig. 2.5. At large charging energies, EJ /EC < 1, it is clear that the charging energy
makes the energy spectrum with periodic parabolas and the Josephson energy opens
an energy gap EJ at crossing points. From Fig. 2.5 (a), one can observe two featuresof the energy spectrum for the charge qubit. First, in contrast with the LC-circuit (seeEq. (2.9)), the energy spectrum has an anharmonicity due to the non-linearity of theJosephson junction. This anharmonicity plays a crucial role in using this circuit as aqubit. Second, the energy spectrum is not flat. This indicates that the charge qubit is2.3. Superconducting qubits

11

sensitive to the charge noise, resulting in a reduction in the coherence time.
When we set ng to a half-integer, which is called a sweet spot, the lowest two energy levels are separated well from other energy levels. By neglecting higher energy
levels except for the lowest two levels, the Hamiltonian of the charge qubit becomes

H=

EJ
EC
(|0⟩ ⟨0| + |1⟩ ⟨1|) − (|0⟩ ⟨1| + |1⟩ ⟨0|) .
4
2(2.25)

From this form, it is clear that the circuit can be reduced to a quantum two-level system; two states being degenerate in the charge basis, |0⟩ or |1⟩, correspond that the
number of excess Cooper pairs in the Cooper-pair box is 0 or 1. Since the spectrum
curvature around sweet spots is small, the effect of the charge noise can be reduced
there.

2.3.2

Transmon qubit

In order to overcome the problem of the short coherence time, the transmon qubit,
which is an abbreviation of “transmission line shunted plasma oscillation qubit”,
was developed in 2007 [51, 52]. The idea to achieve a long coherence time is toincrease a ratio EJ /EC . At large ratios, EJ /EC ≫ 1, the energy spectrum becomes
flatpand the transition energy between the ground and first-excited states becomes∼ 2EJ EC for arbitrary values of ng [51], as shown in Fig. 2.5 (b). This means that thetransmon qubit is insensitive to the charge noise, and it allows the transmon qubit
to have a long coherent time, > 100 µs, as observed in a recent experiment [53].
To increase the ratio EJ /EC , the Cooper-pair box is shunted by a large capacitanceCs as shown in Fig.2.4 (b). This additional capacitance lowers the charging energy
EC = (2e2 )/(CJ + Cg + Cs ), resulting in the increase of the ratio EJ /EC . Note that thecircuit anharmonicity is reduced by increasing the ratio EJ /EC to realize the insensitivity to the charge noise. As a result, the transmon qubit gets a longer coherent
time at the expense of its anharmonicity. One might think that we need to consider
not only the two lowest levels but also the higher levels due to weak anharmonicity.
Actually, since the transmon qubit is coupled to the heat bath with a wide spectrum,
the thermal excitation to the higher levels is relevant in general. However, in a typical experimental situation, the splitting energy of the transmon qubit is of the orderof 0.5 K, and therefore the thermal population of the third level is sufficiently small
at low temperatures ≲ 0.2 K.

2.3.3

Flux qubit

Finally, we explain another famous superconducting qubit called the flux qubit [54,55]. While the charge basis is considered as a convenient basis in the charge qubit,
we use the flux (or phase) basis in the flux qubit. The flux qubit consists of a closed
superconducting circuit interrupted by three Josephson junctions with an external
flux ϕext through the loop as shown in Fig. 2.4 (c).
Let us start with the Lagrangian. The Lagrangian of the flux qubit is expressedby a sum of the Lagrangian for each Josephson junction as
CJ,i 2L= ∑ϕ̇ + EJ,i cos (ϕi /ϕ0 ) ,(2.26)2 i
i =1,2,3
where CJ,i , EJ,i , and ϕi are the Josephson self-capacitance, the Josephson energy, andthe flux difference of the Josephson junction i, respectively, and ϕ0 = h/(2e) is theChapter 2. Review of Superconducting Circuits

•••

12

F IGURE 2.6: (a) Schematic of the quantum Rabi model and (b) its
superconducting circuit. The transmon qubit is coupled capacitively
to the LC-circuit.

flux quantum. Here, we introduce an ununiformity to the circuit by setting that theJosephson junctions i = 1 and 3 are identical with EJ,1 = EJ,3 = EJ and CJ,1 = CJ,3 =CJ and the central Josephson junction i = 2 has reduced parameters EJ,2 = αEJ andCJ,2 = αCJ with α < 1. Since the flux through the superconducting loop is quantized
due to the single-valued order parameter, the fluxes in the Josephson junctions are
related to the external flux,
ϕ1 + ϕ2 − ϕ3 =

ϕext
.
2π

(2.27)

Eliminating the variable ϕ2 by this relation and introducing new variables ϕ± =(ϕ3 ± ϕ1 )/2, the Hamiltonian can be written as [56]

H=

Q2−
Q2
+ + − EJ [2 cos φ+ cos φ− + α cos(ϕext /ϕ0 + 2φ− )] ,8(1 + α)CJ 8CJ(2.28)

where φ± = ϕ± /ϕ0 is the phase variable and Q± are conjugate variables to the flux
ϕ± . When setting the external flux to be half of the flux quantum, ϕext = ϕ0 /2,
the sum of the Josephson energies takes two minima at φ+ = 0. The two lowest
energy states correspond to clockwise or anti-clockwise circular supercurrents in the
superconducting loop. The energy splitting between the two lowest energy states is
induced by the quantum tunneling, which is small enough to neglect other higher
energy states. Thus, the circuit for the flux qubit can be truncated into the two-levelsystem.

2.4

Circuit-QED

We have introduced several elements of the superconducting circuits (the LC-circuit,the transmission line, and the superconducting qubits) so far. By combining them
together, we can create an interacting quantum system without any resistive loss
experimentally. This endeavor is referred to as the circuit quantum electrodynamics (circuit-QED) [12]. In this section and the next section, we briefly introduce two
important models, i.e., the quantum Rabi model and the spin-boson model, both ofwhich will be studied in the later chapters; The former is investigated from the viewpoint of heat transport in Chapter 5 and the latter in the context of the microwave
scattering experiment in Chapter 3.

2.4. Circuit-QED

13

Quantum Rabi model
The quantum Rabi model describes a two-level system coupled to a single bosonicmode or one harmonic oscillator [57] (see Fig. 2.6 (a)). On the superconducting circuits, such a minimum hybrid system is realized by a transmon qubit capacitively
coupled to an LC-circuit as shown in Fig. 2.6 (b). The Hamiltonian is given by

H=−



h̄∆
σx + h̄Ωb† b + h̄gσz b + b† ,2(2.29)

with the transition frequency of the two-level system ∆, the Pauli matrix σi={ x,y,z} ,the resonant frequency of the bosonic mode Ω, the bosonic annihilation (creation)
operator b† (b), and the coupling strength g.
Although the quantum Rabi model looks very simple, the analytical solution
is non-trivial [58]. When the coupling between the two-level system and the single bosonic mode is weak (g ≪ Ω) and the resonant frequency is in the vicinityof the transition frequency (Ω ∼ ∆), we can employ the rotating wave approximation. Then, the quantum Rabi model is reduced to the Jaynes-Cummings model [59],whose Hamiltonian is given by

H=



h̄∆
τz + h̄Ωb† b + h̄g τ+ b + τ− b† .
2(2.30)

where the Pauli matrix τi is obtained by rotating σi along y-axis by π/2, i.e., σx =−τz , σy = τy , σz = τx . The Jaynes-Cummings model has been investigated for a longtime in the field of quantum optics because the rotating-wave approximation is valid
for usual experimental conditions. However, as the coupling increases, the counterrotating terms cannot be neglected and therefore the rotating wave approximation
breaks down. We should note that the ultra-strong coupling (g/Ω = 0.1 ∼ 1) or the
deep-strong coupling (g/Ω > 1) [60, 61] can be realized in the superconducting circuits. For example, the deep-strong coupling g/Ω = 1.3 has been achieved in the recent experiment on the superconducting circuit [62]. The ultra-stong (or deep strong)
coupling regime in the circuit-QED has been attracting the interest of researchers because rich physics associated with the higher-order and non-perturbative effect is
expected.
Now, it is instructive to derive the Hamiltonian (2.29) of the quantum Rabi modelfrom the superconducting circuit. Let us consider the transmon qubit with the Josephson energy EJ , the Josephson capacitance CJ , and the shunted capacitance Cs coupled
to the LC-circuit with the inductance L and the capacitance C through a capacitance
Cg , as shown in Fig. 2.6 (b). The Lagrangian of this circuit is written as

L=

2
Cg
C 2
1 2 Cq 2ϕ̇ −ϕ +ϕ̇q + EJ cos(ϕq /ϕ0 ) +ϕ̇ − ϕ̇q ,
2
2L
2
2

(2.31)

where ϕ and ϕq are the fluxes at the ends of the LC-circuit and the transmon qubit,
respectively, and Cq = Cs + CJ . By introducing the charge variables conjugate to thefluxes as
∂L= C ϕ̇ + Cg ϕ̇ − ϕ̇q ,
∂ϕ̇

∂L
Qq =
= Cq ϕ̇q + Cg ϕ̇ − ϕ̇q ,∂ϕ̇qQ=(2.32)(2.33)

14

Chapter 2. Review of Superconducting Circuits

we obtain the Hamiltonian as

H=

Cg
1
1 2
1
Q2 +
ϕ +
Q2q − EJ cos(ϕq /ϕ0 ) +
QQq ,
2C
2L
2C
CΣ
| eff {z} | J,eff{z} | {z }LC-circuit(2.34)

interaction

transmon qubit−1
where CΣ2 = CCg + CCq + Cg Cq , Ceff
= [C (Cg + Cq )2 + Cq Cg2 + Cg Cq2 ]/(2CΣ4 ), and−1
CJ,eff= [CCg2 + Cq (C + Cq )2 + Cg C2 ]/(2CΣ4 ).
By quantizing the charge and flux variables of the LC-circuit in the same manneras Section 2.1 and replacing Qq with 2en, we obtain
H = h̄Ωb† b + EC n2 − EJ cos(ϕq /ϕ0 ) + An b + b† ,(2.35)√√
where Ω = 1/ Ceff L, EC = e2 /(2CJ,eff ), and A = 2eCg /CΣ2 Ceff h̄Ω/2. Here, theconstant terms have been dropped. Finally, assuming EJ /EC > 1, the transmon
qubit is reduced to the two-level system as explained in Section 2.3, and then we
obtain the coupling strength in the model Hamiltonian (2.29) as
1/2 √Cg Ceffh̄g = |e|
h̄Ω
CΣ2

2.5



EJ
2EC

1/4
.

(2.36)

Spin-boson model

As described in Section 2.4, the quantum Rabi model is a two-level system coupled to a single electromagnetic mode. Here, we consider a two-level system coupled to many electromagnetic modes (harmonic oscillators), which is known as the
spin-boson model [16, 17]. The spin-boson model is one of the most fundamental models for open quantum systems. The description of the environment using
harmonic oscillators was originally considered in the context of the Brownian motion [63] and macroscopic quantum tunneling [64]. Subsequently, the spin-bosonmodel was intensively studied to understand the decoherence process in the twolevel system [16]. It has also been used in the fields of superconducting circuits todescribe the relaxation of a qubit due to coupling to the environment [65]. The spinboson model is described by the Hamiltonian


h̄∆
σz
H = − σx − ϵσz + ∑ h̄ωk bk† bk − ∑ h̄λk bk† + bk ,| 2 {z} |k {z } |2 k{z}
two-level systembath(2.37)

interaction

where σi={ x,y,z} is the Pauli matrix operator, ∆ and ϵ is a transition frequency and a
detunning energy of the two-level system, respectively, bk† (bk ) is an annihilation (acreation) bosonic operator in the bath, ωk is the frequency of the bosonic mode k,
and λk is the coupling strength between the two-level system and the bosonic modek. The properties of the bath are characterized by a spectral density
I (ω ) ≡ ∑ λ2k δ(ω − ωk ).
k(2.38)

2.5. Spin-boson model

15

F IGURE 2.7: Renormalization flow of the anisotropic Kondo model.

The spectral density is commonly parameterized asI (ω ) = 2αωωωc s −1e−ω/ωc ,(2.39)

where α is a dimensionless coupling strength and ωc is a cutoff frequency. The property of this model strongly depends on the exponent s. The cases of s < 1, s = 1, ands > 1 represent sub-Ohmic, Ohmic, and super-Ohmic dissipations, respectively.

2.5.1

Kondo problem for the Ohmic case

The quantum many-body effect becomes important in the spin-boson model if theinteraction between the two-level system and the environments is strong enough.
This is indicated by the fact that the Ohmic spin-boson model is equivalent to theanisotropic Kondo model [16, 17]. The Kondo problem has been a central topic
in condensed matter physics since the phenomenon of the electric resistivity minima in metals with magnetic impurities [66] was explained by Jun Kondo in the
1960s [67]. By using the perturbation theory, he showed that the resistivity minima
are caused by the spin-exchange process between a magnetic impurity and conduction electrons in the metal [68]. Nowadays, it is known that the ground state formsa many-body singlet state between the local spin of the magnetic impurities and the
conduction electrons, which is called the Kondo singlet. This has been clarified byseveral theoretical methods such as the poor man’s scaling [69], the Wilson’s numerical renormalization group [70]., the local Fermi liquid theory [71], and the exact
solutions [72, 73]. The many-body effect induced by an interaction between magnetic impurities and conduction electrons is called the Kondo effect. Recently, theKondo effect has been studied in various fields, e.g., metal surfaces [74, 75], quantum dots [76, 77], and cold atoms [78].
The Hamiltonian of the anisotropic Kondo model is given by

H=



J∥
J
σz ∑ sc†k,s ck′ ,s + ⊥ ∑ σ+ c†k,↓ ck′ ,↑ + σ− c†k,↑ ck′ ,↓ + ∑ ϵk,s c†k,s ck,s ,4L k,k′ ,s
2L k,k′k,s(2.40)

where σ± = (σx ± iσy )/2, L is a system size, and ck,s (c†k,s ) is an annihilation (acreation) operator of the conduction electron with the wavenumber k and the spins(=↑, ↓), and ϵk,s is an energy of the conduction electron measured from the Fermi
energy. The J∥ term describes the spin-conserved scattering and the J⊥ term describes the spin-flip one. The anisotropic Kondo model has a quantum phase transition between the ferromagnetic (J∥ < 0) and antiferromagnetic (J∥ > 0) phases atzero temperature as indicated by the renormalization group analysis (see the flow

16

Chapter 2. Review of Superconducting Circuits

F IGURE 2.8: Microwave spectroscopy of the Ohmic spin-bosonmodel. (a) Schematic of the spin-boson model in a superconducting
circuit consisting of the flux qubit and a transmission line. (b) The microwave transmission spectrum as a function of the probe frequency
f and the external flux Φε . These figures are taken from Ref. [81].

diagram in Fig. 2.7). This transition belongs to the Berezinskii-Kosterlitz-Thouless(BKT) universality class [79, 80]. In the antiferromagnetic phase, the divergence ofthe coupling strengths along the renormalization flow indicates the formation of the
Kondo singlet where the local spin is screened completely by the conduction electrons.
By using the bosonization technique [18], the anisotropic Kondo model can bemapped to the Ohmic spin-boson model with the following correspondence:



πρJ∥
∆
2
= ρJ⊥ cos arctan,(2.41)
ωc4πρJ∥ 22
α = 1 − arctan,(2.42)π4
where ρ is a density of states of the conduction electrons. From this equivalence,
the Ohmic spin-boson model also has a quantum phase transition at α = 1, whichcorresponds to J∥ = 0 in the anisotropic Kondo model.

2.5.2

Recent experiment for the Ohmic case

The superconducting circuit is an ideal platform to investigate the spin-boson modelusing microwave spectroscopy experimentally because a typical superconducting
qubit has an energy scale compatible with the microwave (∆/2π ∼ 10 GHz). Moreover, the superconducting qubit coupled to the transmission line can be controlled
precisely, and therefore we can obtain accurate experimental data.
In 2017, Pol Forn-Díaz et al. have realized a superconducting circuit describedby the Ohmic spin-boson model using a flux qubit and a transmission line. They
have observed clearly that the microwave transmission shows the parabolic trace(sharp dips), corresponding to the splitting tunneling frequency of the qubit, as afunction of the probe frequency and the external flux, which controls the splitting
tunneling frequency of the qubit (see Fig. 2.8) [81]. This sharp dip indicates that thequantum coherence of the two states of the flux qubit remains because the coupling
between the flux qubit and the transmission line is weak enough (α = 0.07). So far,the signatures of the Kondo effect have not been observed experimentally yet, but
it will be done in the near future. Recently, the experimental attempts to observemany-body effects using microwave spectroscopy are actively performed [24, 82].

2.6. Josephson-junction chain

17

•••

•••

F IGURE 2.9: Josephson-junction chain consisting of circuit elements,
Josephson junctions with the Josephson energy EJ and the Josephson
capacitance CJ , ground capacitances Cg . The elementary cell length isa. The blue dots represent the superconducting islands.

As described above, the Ohmic spin-boson model has been studied well boththeoretically and experimentally because it is related to the Kondo problem [21, 23]
and the strategy of its realization has been established in the superconducting circuit using standard superconducting elements [81, 83]. Compared with the Ohmic
case, there is little known about the microwave spectroscopic properties for the
non-Ohmic case (the sub-Ohmic and super-Ohmic cases) in the superconducting circuits. Particularly, the sub-Ohmic spin-boson model shows a quantum phase transition and then is an important model to study the dissipative quantum transitionin the well-controllable system. We study quantum critical phenomena through microwave spectroscopy in Chapter 3.

2.6

Josephson-junction chain

In the previous section, we considered the spin-boson model which is a dissipative zero-dimensional object. Here, we consider a one-dimensional object, i.e., theJosephson-junction chain, shown in Fig. 2.9. The one-dimensional Josephson-junction
chain is important not only for many practical applications in superconducting quantum circuits but also for fundamental quantum physics. Actually, the Josephsonjunction chain shows several many-body effects due to the Coulomb interaction between Cooper pairs.

2.6.1

Superinductor

In the linear regime, the Josephson junction behaves as the inductance, as discussed
in Section 2.2, and therefore the one-dimensional Josephson-junction array can be
regarded as a transmission line with the high impedance. This high-inductance lowloss element is known as a superinductor.
Using Kirchhoff’s current law to the lumped-element model, we can easily obtain
the dispersion relation of the Josephson-junction chain asω (k) = pv|k|
1 + (kv/Ω)2,(2.43)

pp
where v = a 2EJ Eg /h̄ is the plasmon velocity, Ω = 2EJ EC /h̄ is the plasma frequency, and a is the elementary unit length. Here, EJ is the Josephson junctionenergy, Eg = (2e)2 /(2Cg ) is the charging energy at the ground capacitance Cg ,and EC = (2e)2 /(2C ) is the charging energy at the Josephson capacitance C. As
shown in Fig. 2.10, for the low frequency, we obtain the linear dispersion relation,ω (k ) ≈ v|k |. These sound-like modes are called plasmons. For typical values of18

Chapter 2. Review of Superconducting Circuits

F IGURE 2.10: Dispersion relation of the Josephson-junction chain. v
is the plasmon velocity, Ω is the plasma frequency, a is the elementarycell length, and ℓsc is the screening length.

a ∼ 1 µm, EJ ∼ 1 GHz, and Cg ∼ 0.1 fF [24], the plasma velocity can be estimatesas v ∼ 106 ms−1 . Compared with the velocity of light in a vacuum, the photonicmodes in the Josephson-junction chain (plasmons) are slower by two orders. The
ultraslow plasmons lead to a high impedance Z ∼ 10 kΩ for the Josephson-junction
chain. As in the vacuum quantum electrodynamics, quantum fluctuations of fields
in the Josephson-junction chain are controlled by the effective fine structure constantα = Z/RQ , where RQ = h/(2e)2 ≈ 6.5 kΩ is the resistance quantum for Cooper
pairs [84]. While in the vacuum quantum electrodynamics, the fine structure constant is small (α = 1/137 corresponding to Z ≈ 377 Ω), the Josephson-junction chain
can have a large effective fine constant α is of the order of 1. Thus, the Josephsonjunction chain can achieve a one-dimensional superconducting element with strong
quantum fluctuations.
When focusing on the low-frequency modes with the linear dispersion relation,
we need to introduce the ultraviolet cutoff. It is the plasma frequency at which thedispersion relation becomes flat ω (k ) ≈ Ω as shown in Fig. 2.10. This means that theplasmon does not propagate through the Josephson-junction chain due to the charge
screening by the surrounding medium. Moreover, there is another cutoff scale thatcomes from the discreteness of the Josephson-junction chain where elementary circuits are located with the spacing of a. The shortest wavelength is λ = 2a whichmeans that the plasmons oscillate with π between the nearest-neighbor cells andcorresponds to k = 2π/λ = π/a.

2.6.2

Lagrangian

In 1984, Mark Bradley and Sebastian Doniach theoretically discussed the Josephsonjunction chain in their pioneering work and predicted that the Josephson-junction
chain exhibits the superconductor-insulator transition (SIT) at zero temperature [30].
This phase transition is driven by the competition between the charge fluctuationand the phase fluctuation and belongs to the Berezinski–Kosterlitz–Thouless (BKT)
universality class. For EJ ≪ Eg , the Josephson-junction chain becomes an insulator
because the charge fluctuation is strongly suppressed by the Coulomb blockade. On
the contrary, for EJ ≫ Eg , the Josephson-junction chain is in the superconducting
phase due to large charge fluctuation.

2.6. Josephson-junction chain

19

The Lagrangian for the one-dimensional Josephson-junction chain of the lengthd is given by


Z d
2
h̄ 
h̄v
Λ
2
L=
dx
θ̇ ( x, t) −(2.44)[∂ x θ ( x, t)] + cos [2θ ( x, t)] ,
2πKv
2πK
a
0
where θ ( x, t) is a local field associated with the accumulated charge,
q i.e., ρ( x, t) =−h̄/(2e)∂ x θ ( x, t) is the density of the Cooper pairs, and K = π EJ /(2Eg ) is the
Luttinger parameter (for a detailed derivation, see Appendix B). At the transmon
limit (EJ ≫ EC ), the quantum phase slip amplitude is written as [85, 86]
1/4 −√32EJ /E
8Ce.
Λ = √ 2EJ3 EC
π(2.45)

Note that, to derive the Lagrangian (2.44), we have used the continuous approximation which is valid for small ground capacitancep
Cg ≪ C (or Eg ≫ EC ), corresponding to the large charge screening length ℓsc = a C/Cg ≫ a [87]. The Lagrangian
for the Josephson-junction chain (2.44) is nothing but the sine-Gordon model [32].
The Luttinger parameter is related to the low-frequency impedance of the chain as
K = RQ /(2Z ). Namely, it is also related to the fine structure constant as α = 1/(2K ).
This relation indicates that the Luttinger parameter controls the quantum fluctuations of the Josephson-junction chain. Since the Josephson-junction chain can realizea large fine structure constant in the experiment [88], the Josephson-junction chain
provides an appropriate platform to investigate the property of modes in the insulating phase experimentally [89]. In this thesis, we focus on the classical limit K ≪ 1(EJ ≪ Eg ) in which the Josephson-junction chain is in the deep insulating phase.
We summarize the magnitude relation between three characteristic energy scales,i.e., the Josephson energy EJ , the charging energy at the ground capacitance Eg , andthe charging energy at the Josephson capacitance EC , in this thesis:• EJ ≪ Eg · · · Classical limit (K ≪ 1). The charge fluctuation is frozen and theJosephson-junction chain is in the deep insulating phase.
• EC ≪ EJ · · · Transmon limit. The energy spectrum is nearly harmonic andeach energy band is narrow. The amplitude of quantum phase slips is exponentially small (Λ ≪ Ω).
• EC ≪ Eg · · · Long-range Coulomb interaction (ℓsc ≫ a). The ground capacitance is small (Cg ≪ C).
In a typical experiment [88], we can confirm that the above conditions are satisfied;
EJ /EC ∼ 20 and EC /Eg ∼ 10−4 .

2.6.3

Superconductor-insulator transition

The sine-Gordon model (2.44) is one of the most studied models of one-dimensional
field theory [32]. It is well known that the sine-Gordon model processes a quantum phase transition at K = 2 between massless (the gapless spectrum; K > 2) andmassive (the gapped spectrum; K < 2) phases by changing the Luttinger parameter K, and it belongs to the BKT universality class. This BKT-type quantum phasetransition corresponds to the SIT in the Josephson-junction chain. In Fig. 2.11, weshow the renormalizationpgroup flow diagram of the sine-Gordon model on the mK plane, where m = 2π 2ΛEJ /h̄ is the mass. At K < 2, the cosine term in the

20

Chapter 2. Review of Superconducting Circuits

F IGURE 2.11: Flow diagram of the sine-Gordon model at zero temperature.

•••

•••

F IGURE 2.12: Josephson-junction chain with the disorder via random
offset charges induced by random gate voltages vn .

sine-Gordon model (2.44) is relevant and then the sine-Gordon model is always inthe massive phase (m → ∞). This phase corresponds to the insulating phase in theJosephson-junction chain. At K > 2, there are two fates. If the mass is larger thana critical mass mc , the sine-Gordon model is in the massive phase (the insulatingphase). If m < mc , the cosine term becomes irrelevant and the sine-Gordon model isin the massless phase (m → 0), which corresponds to the superconducting phase in
the Josephson-junction chain. Then, the Josephson-junction chain can be effectively
described by the Luttinger liquid, for which the charge density wave (CDW) can
propagate ballistically [32].

2.7

Disordered Josephson-junction chain

In realistic systems, the disorder is inevitable and plays an important role in theJosephson-junction chain. Among various types of disorder [90], the most commonand significant type of disorder is the random offset charges via the ground capacitance (see Fig. 2.12).

2.7.1

Random offset charges

In the presence of the random offset charges, the charging energy at the groundcapacitances (B.5) is modified as
ECoulomb =Z d
0dx aEg [∂ x q( x, t) − ρb ( x )] .

(2.46)

2.7. Disordered Josephson-junction chain

21

Here, ρb ( x ) is the time-independent density of coarse-grained offset charges withshort range correlations ⟨ρb ( x )ρb (y)⟩ ∝ δ( x − y), where ⟨· · ·⟩ indicates disorder averaging. Note that there are three characteristic timescales; the time scale of offset
charge fluctuation τc , that of the plasmon propagation τplasmon , and that of experimental measurement τexp . In the experiment, the timescale for the variation of theconfiguration of the offset charges is τc ∼ 1 min [91]. Then, the timescale for scrambling the random offset charges in the Josephson-junction chain can be estimated as
τc /N ∼ 0.05 sec, which is much larger than τplasmon . Therefore, we can assume thatthe random offset configuration is static in all subsequent calculations, i.e., the density of states of the random offset charges is independent of time. On contrary, thetime for gathering and averaging each experimental data, τexp , is sufficiently longer
than τc /N, which justified that observables are given by the disorder averaged values with respect to the random offset charge configuration.
Now, by introducing a phase field of the quantum phase slip amplitude (theAharonov-Casher phase) [92, 93]
χ( x ) = 2πZ xdy ρb (y),(2.47)

the total Lagrangian is expressed by the random phase sine-Gordon model:


Z d
2
h̄ 
h̄v
Λ
2
dx
L=
θ̇ ( x, t) −[∂ x θ ( x, t)] + cos [2θ ( x, t) + χ( x )] . (2.48)
2πKv
2πK
a
0
The random phase field is characterized by vanishing average,⟨cos χ( x )⟩ = ⟨sin χ( x )⟩ = 0,(2.49)and variance,
σ2 =

2.7.2

Z d
0

dx ⟨cos χ( x ) cos χ(0)⟩ =Z d
0dx ⟨sin χ( x ) sin χ(0)⟩ .

(2.50)

Superfluid-Bose glass transition

Even when the random offset charges exist in the Josephson-junction chain, the SIT
still occurs. However, in this case, the insulating phase becomes a compressible state,
which is called Bose glass [31], instead of the incompressible Mott insulating staterealized in the absence of disorder. The disorder reduces the effect of the quantum
phase slips.
In 1988, Thierry Giamarchi and Heinz-Jürgen Schulz treated the one-dimensionalbosonic disordered model perturbatively and provided the renormalization groupequations (the Giamarchi-Schulz scaling) [31]:dK
K2= − D,dl2dD= (3 − 2K ) D,dl

(2.51)

where D describes the evolution of the quantum phase slip probability. The GiamarchiSchulz scaling (2.51) indicates that the transition between the superfluid and Bose
glass occurs at Kc = 3/2, which corresponds to Eg = (2π 2 /9) EJ and Z ∼ 2.2 kΩ.

22

Chapter 2. Review of Superconducting Circuits

F IGURE 2.13: Two frequency regimes in the disordered Josephsonjunction chain in the deep insulating phase (K ≪ 1); the strongly
localized (ω ≪ ω⋆ ) and weakly localized (ω ≳ ω⋆ ) regimes. In thestrongly (weakly) localized regimes, the collective pinning effect isstrong (weak).

2.7.3

Classical limit

In the classical limit (K → 0), the kinetic term in the total Lagrangian (2.48) can beneglected. Then, the disordered Josephson-junction chain is reduced to a problem
for the interplay of elasticity and disorder that is equivalent to that of the pinning
CDW [94].
Now, we introduce the Larkin length R⋆ [95, 96, 97] on which the static spatial
order is destroyed. For the field variable variation in the length scale of√ R, the elastic
energy and the pinning energy are estimated ∼ h̄v/(KR) and ∼ Λ R/a, respectively. We further incorporate quantum fluctuations due to the quantum phase slips
by replacing Λ with the renormalized phase slip amplitude Λ( R/ℓsc )−K . Then, thegeneralized Larkin length [98] is defined as the length scale at which the elastic energy and the pinning energy are compatible:

R⋆ =

h̄v
2πKΛσ

2/(3−2K).

(2.52)

Its corresponding frequency ω⋆ = v/R⋆ is called the Larkin frequency. The divergence of the Larkin length (2.52) at K = 3/2 indicates the superfluid-Bose glasstransition.
Elastic properties on a length scale shorter than the Larkin length (R ≲ R⋆ ) are
not much affected by the collective pinning. This regime corresponds to excitations
with high frequencies ω ≳ ω⋆ , leading to the almost-ballistic propagation of plasmons (see Fig. 2.13). On the contrary, at low frequencies ω ≪ ω⋆ , the collective
pinning effect due to the disorder drastically changes the excitation spectrum of thesystem. In this regime, the density of plasmon modes is determined by the statistics
of distinctive configurations of disorder, which leads to the strong suppression of
the density of modes at low frequencies as [99, 100, 101, 102]ν(ω ) ∝ ω 4 ,(2.53)

for a long chain (d ≫ R⋆ ).
We note that the collective pinning regime corresponds to R⋆ ≫ σ2 in the classical limit. Therefore, one may cross over from the collective pinning regime to the individual pinning regime by increasing the quantum phase slip amplitude Λ. When
we go beyond the classical limit (K ≪ 1), it is known that quantum fluctuations dueto the effect of a finite K increase the Larkin length [103].

2.7. Disordered Josephson-junction chain

23

F IGURE 2.14: (a) A Josephson-junction chain. Two chains consistingof more than 33,000 Josephson junctions are spaces parallel closely.
At the right end, the short-circuit is terminated and, at the left end, adipole antenna is connected. (b) The frequency of the reflection magnitude for different devices. The Luttinger parameters are K = 0.5(red), 0.171 (green), and 0.171 (purple). The ratios of the Josephsonenergy and the charging energy are EJ /EC = 582 (red), 16 (green),
and 7 (purple). (c) The frequency dependence of the quality factor forvarious devices. These figures are taken from Ref. [89].

2.7.4

Recent experiment

We introduce a recent experiment on ac transport through the disordered Josephsonjunction chain. Even in the deep insulating phase, ac current can flow like a superconductor for ω ≫ kB T/h̄. Nevertheless, we can observe traces of insulators in
the frequency dependence of the ac transport. By varying the probe frequency, theinteraction can be controlled effectively in the sense of the renormalization group.
Recently, Roman Kuzmin et al. have investigated the scattering reflection off
a Josephson junction using microwave spectroscopy to reveal the insulating statesfor K < 3/2 [89]. They have fabricated a long double-chain consisting of more than
33,000 junctions, as shown in Fig. 2.14 (a). When the ground capacitance of the superconducting islands is much smaller than the junction capacitance, the double-chain
can be mapped into a single-chain problem by considering only asymmetric modes
coupled to a dipole antenna [90]. By performing the microwave spectroscopic measurement for this Josephson-junction chain, they obtained the reflection magnitude
as a function of frequency for different devices, as shown in Fig. 2.14 (b). At K ≃ 0.5
and EJ /EC ≃ 500 (red color), we can observe the well-resolved plasmon spectrum
and no signature of the insulating states despite in a smaller K than the insulatortransition value, Kc = 3/2. We define a quality factor by the ratio of the mode frequency to its line width. As the frequency decreases, the quality factor grows andreaches a high value ∼ 105 , as shown in Fig. 2.14 (c). This large quality factor canbe explained by the dielectric loss in the Josephson-junction capacitance (solid line

24

Chapter 2. Review of Superconducting Circuits

in Fig. 2.14 (c)). On contrary, at K ≃ 0.171 and EJ /EC ≃ 7 (purple color), the wellresolved plasmons do not appear. Instead of them, we can observe the inhomogeneous broadening of the standing waves after disorder averaging in the reflection
magnitude. This inhomogeneous broadening, which is caused by the pinning effectof the CDW, is a signature of the insulating phase. It is remarkable that, opposite
to the well-resolved plasmon region, the quality factor decreases as the frequency islowered. This characteristic behavior of the quality factor is observed in the insulating states. Actually, in the limit of ω → 0, the quality factor is expected to approachzero, indicating strong suppression of the dc transport. Finally, we note that the device showing the signature of the insulating phase indeed has large dc resistance ofthe order of 1 MΩ, which is inconsistent with a superconductor.
Recently, Manuel Houzet and Leonid Glazman have theoretically clarified thecollective pinning effect of the CDW through the microwave reflection off the Josephson junction chain in the deep insulating phase [98]. However, since the reflection isrelated to the local density of modes at the edge of the chain, little is known about thestatistical properties of plasmon waves traveling along the entire chain. Moreover,from the reflection, we cannot distinguish the forward and backward scatterings.
In Chapter 4, we investigate the transmission of a long Josephson-junction chain
in the deep insulating phase and discuss the contributions of the forward and backward scatterings as well as the statistics of the transmission [40].

2.8Quantum heat transport

Finally, we introduce recent studies on quantum heat transport in superconducting circuits, which have been stimulated by the development of the technology ofcalorimeters in the last decade [33, 34, 35]. The superconducting circuit is also a
useful platform to study quantum heat transport because of flexible designing andfine-tuning by external fields. In the superconducting circuit, a circuit element (e.g.,a superconducting qubit, a superconducting resonator, or their hybridized circuit)mediates heat by microwave photons between two photon baths (e.g., transmission lines). Below, we introduce two recent experiments on quantum heat transport
through a resonator-qubit-resonator assembly.
In 2018, Jukka Pekola’s group has observed tunable heat transport through aresonator-qubit-resonator assembly as a quantum heat valve [104] (see Fig. 2.15).
Two resonators have identical resonant frequencies f r = 6.4 GHz and are coupled
to each other via a transmon qubit, whose splitting frequency f q can be controlled
by external magnetic flux Φ, with the coupling strength g. To study heat transport,
the two resonators are terminated by normal-metal resistors acting as heat baths
with the coupling strength γ, as shown in Fig. 2.15. When treating this circuit, there
are two regimes, i.e., the quasi-Hamiltonian (global) regime (Fig. 2.15 (a); γ ≪ g)and the non-Hamiltonian (local) regime (Fig. 2.15 (b); γ ≫ g). This separation isassociated with the location of the Heisenberg cut between the classical and quantum worlds [105]. In the quasi-Hamiltonian (QH) regime, the hybridized systemcomposed of the qubit and the resonators couples to the bare heat baths. Since thehybridized system shows the multilevel structure, the power is affected strongly
by this multilevel structure as indicated by the QH model in Fig. 2.15 (a). In the
non-Hamiltonian regime, the qubit is coupled to dissipative resonators whose spectral density is Lorentzian centered around the resonance frequency f r . Here, theheat current flows only for f q = f r , and thus the system acts as the quantum heatvalve. In the bottom panel of Fig. 2.15 (b), we show the experimental (left side) and

•••

•••

25

•••

•••

2.8. Quantum heat transport

F IGURE 2.15: Quantum heat valve. A transmon qubit (two-level system) is capacitively embedded between two superconducting transmission lines (harmonic resonators). Each transmission line is terminated by a normal-metal resistor (heat bath). Panels (a) and (b) show
the schematics of the model (top panels) and the external flux dependence of the heat current flowing into the right (drain) heat bathfor different left (source) heat bath temperatures (lower panels) in thequasi-Hamiltonian (global) and non-Hamiltonian, respectively. The
data plots are taken from Ref. [104].

theoretical (right side) results for the power as a function of the external flux for
different temperatures. It is observed that the power has several Lorentzian peaks
centered at fluxes corresponding to f q = f r . In both of the quasi-Hamiltonian andnon-Hamiltonian regimes, the theoretical calculation shows good agreement withthe experimental results.
Moreover, by breaking the symmetry of the circuit and using the non-linearityof the central element, the photonic heat rectification has been demonstrated [106]in a similar circuit to that in Ref. [104] (see Fig. 2.15). As shown in Fig. 2.16 (a), atransmon qubit is coupled to two unequal harmonic resonators, whose frequencies
are 2.8 GHz and 6.7 GHz, terminated by heat baths. In the same way as the quantum
heat valve, the splitting tunneling frequency of the transmon qubit can be tuned by
the external flux. Fig. 2.16 (b) shows the external flux dependence of the heat current
in forward (purple lines) and backward (green lines) directions for different threetemperatures. The forward direction corresponds to the flow from the left bath to
the right bath when TL = TR + |∆T |, while the backward direction corresponds tothe flow in the opposite direction when TL = TR − |∆T |. In Fig. 2.16 (c), we show
that, from the data in Fig. 2.16 (b), the external flux dependence of rectification ratio,
R = | Pf |/| Pb |, where Pf and Pb are the heat currents in the forward and backward
directions, respectively. From this figure, one can see that the flux-tunable photonic
heat rectification achieves 10% at maximum.
Quantum heat transport has also been studied in a number of theoretical works
for the last twenty years, and various quantum heat devices such as a quantum heatvalve [104], a quantum rectifiers [36, 106], a quantum heat transistor [107, 108], aquantum refrigerator [109], and a quantum heat engine [110, 111, 112] have been•••

Chapter 2. Review of Superconducting Circuits

•••

26

F IGURE 2.16: Quantum heat rectification. (a) Schematic of the photon diode. The circuit structure is similar to that in Fig. 2.15 but two
resonators are unequal, their resonant frequencies are 2.8 GHz (leftresonator) and 6.7 GHz (the right one), to break the symmetry of thecircuit. The forward and reverse directions are drawn in purple andgreen arrows, respectively. (b) The heat current as a function of theexternal flux in the forward (purple lines) and backward (green lines)
directions for different temperatures (380 mK, 400 mK, and 420 mKfrom bottom to top). (c) The external flux dependence of the rectification ratio from data in panel (b), where it is subtracted by minimalvalue Rmin . The data plots are taken from Ref. [106].

proposed. The most fundamental model for quantum heat transport is a single superconducting qubit (a two-level system) coupled to two baths described by the
spin-boson model [23, 38]. However, the single superconducting qubit is too simple to work functionally as a quantum heat device. We also note that the center partof superconducting circuits in the two experiments on a resonator-qubit-resonatorassembly, which are explained in the previous two paragraphs, looks like the quantum Rabi model introduced in Section 2.4. We point out that little is still known
about quantum heat transport through a resonator-qubit-resonator assembly in theultra-strong (or deep-strong) coupling regime.
In Chapter 5, we consider quantum heat transport through a superconducting
qubit embedded between two harmonic resonators, which is described by the quantum Rabi model. We show that the heat transport becomes sensitive to control parameters in the ultra-strong (or deep-strong) coupling regime. This property is advantageous for the application of quantum heat devices.

27

Chapter 3

Microwave Scattering through a
Superconducting Qubit
In this chapter, we study the microwave scattering in superconducting circuits described by the sub-Ohmic spin-boson model. First, we review quantum phase transition in the sub-Ohmic spin-boson model in Section 3.1. Next, we move to new
results obtained in Ref. [39]. We present a superconducting circuit to realize the subOhmic spin-boson model and formulate the microwave scattering in this circuit in
Section 3.2. Using the continuous-time quantum Monte Carlo simulation introducedin Section 3.3, we show that quantum critical phenomena emerge in the frequency
dependence of the microwave reflection in Section 3.4.

3.1

Sub-Ohmic spin-boson model

As described in Section 2.5, the spin-boson model is a fundamental model for describing a two-level system coupled to bosonic environments [16, 17]. We write the
Hamiltonian again, which consists of three terms, the two-level system HTLS , thebosonic bath HB , and their interaction HI ,H = HTLS + ∑ HB,ν + ∑ HI,ν ,
ν

(3.1)

ν

h̄∆
σx − ϵσz ,2†
HB,ν = ∑ h̄ων,k bν,kbν,k ,HTLS = −

(3.2)
(3.3)

k

HI,ν = −σz†h̄λν,k bν,k+ bν,k ,
∑
2 k

(3.4)

and the spectral density isIν (ω ) = 2αν ωωωc s ν −1e−ω/ωc ,(3.5)

where ν (= L, R) is the bath index and other parameters and operators are the sameas Section 2.5.
In this thesis, we focus on the sub-Ohmic case (s < 1), which has a second-orderphase transition at zero temperature [19, 20, 113, 114, 115, 116, 117, 118]. The quantum phase transition in the sub-Ohmic spin-boson model occurs when the couplingstrength between the two-state system and the environments is tuned to a criticalpoint αc . For α < αc , the ground state is described by a coherent superposition of
two wavefunctions localized at each well of the two-level system, which is called

28

Chapter 3. Microwave Scattering through a Superconducting Qubit

F IGURE 3.1: Schematic of the quantum phase transition in the subOhmic spin-boson model at zero temperature. For α < αc , the groundstate is a superposition of two states localized at each well (the delocalized phase). For α > αc , their coherence is broken and the state of
the system is localized at one of the two wells due to the disappearance of quantum tunneling (the localized phase).

F IGURE 3.2: The detunning energy ϵ dependence of the population⟨σz ⟩ϵ for α < αc (blue line), α = αc (green line), and α > αc (red line).
At the critical point, α = αc , while the population is continuous, its
derivative of ϵ diverges at ϵ = 0. As the coupling increasing, α > αc ,the population jumps discontinuously at ϵ = 0.

as a delocalized phase (see Fig. 3.1). For the stronger coupling, α > αc , the groundstate becomes two-fold degenerate because the coherent superposition is completely
broken due to the disappearance of quantum tunneling between the two wells. This
phase is called a localized phase.
The population of the two-level system ⟨σz ⟩ϵ as a function of the detunning energy ϵ for α < αc , α = αc , and α > αc is shown in Fig 3.2. We regard ⟨σz ⟩ϵ as the“magnetization” in the analogy of the order parameter in a magnetic phase transition. We also define the static “spin susceptibility” by the first derivative of themagnetization at ϵ = 0 as
χ0 ≡ limϵ →0⟨σz ⟩ϵ,ϵ(3.6)

where ⟨·⟩ indicates the thermal average. The static spin susceptibility is finite forα < αc , increases as α approaches αc from below, and finally diverges at the criticalpoint α = αc . In the localized phase (α > αc ), the magnetization ⟨σz ⟩ϵ jumps discontinuously from −mz to +mz at ϵ = 0, where the constant value mz ≡ limϵ→0+ ⟨σz ⟩ϵcorresponds to the spontaneous magnetization.
The thermal equilibrium properties at the quantum critical point (α = αc ) havebeen studied in several theoretical works [116, 19]. The critical exponent of the3.2. Formalism for microwave scattering

29

imaginary-time spin-spin correlation function, η, is defined as
C (τ ) ≡ ⟨σz (τ )σz (0)⟩ ∼ τ −η +1 .

(3.7)

For 0 < s ≤ 0.5, quantum critical behavior is described by the mean-field theory,
giving a critical exponent [119]η = 2 − s.

(3.8)

This critical exponent has been confirmed by the quantum Monte Carlo simulation [19] and the numerical renormalization group calculations [115, 116, 117, 20]. In
contrast, for 0.5 < s < 1, the critical exponents are non-trivial function of s [120]. So
far, experimental realization of the sub-Ohmic spin-boson model has been discussed
only for s = 0.5 using the transmission line with resistances [121, 122]. However, thesub-Ohmic bath with arbitrary s has not been discussed.
In the rest of this chapter, we propose a superconducting circuit to realize thesub-Ohmic spin-boson model with arbitrary s and investigate quantum critical phenomena from the viewpoint of the microwave scattering on that superconducting
circuit [39].

3.2

Formalism for microwave scattering

In this section, we introduce a superconducting circuit to realize the sub-Ohmic spinboson model using a charge qubit and an RLC transmission line and formulate thescattering problem on this superconducting circuit.

3.2.1

Circuit model

Let’s consider a superconducting circuit as shown in Fig. 3.3 for a realization of the
sub-Ohmic spin-boson model. In this circuit, to investigate the microwave scattering
off the sub-Ohmic spin-boson model, the Ohmic bath is also weakly coupled to the
two-level system as a probe system.
First, we consider a charge qubit as a two-level system. As discussed in Section 2.3, the charge qubit composed of two Josephson junctions with Josephsonenergies EJ,ν and self-capacitances CJ,ν (ν = L, R). The charge states of the qubit
is controlled by a gate voltage Vg via a gate capacitance Cg . Then, the Hamiltonian of the charge qubit is given by Eq. (2.24) with a modified charging energy
EC = (2e2 )/CΣ where CΣ = CJ,L + CJ,R + Cg . Assuming small Josephson junctionenergies, EJ,L , EJ,R ≪ EC , and setting the offset charge as ng = 0.5, we can reduce
the superconducting circuit to the two-level system as discussed in Section 2.3, andthen the Hamiltonian can be written as

HTLS = −h̄∆σx − ϵσz ,
2

(3.9)

where the first term describes the Josephson energy and the second term corresponds to detunning from the degenerate gate voltage n g = 0.5. Here, ∆ ≡ EJ /h̄ isthe tunneling frequency. The Pauli operators σi= x,y,z have a two-dimensional Hilbert
space and the eigenstates of σz correspond to the two charge states, |n = 0⟩ and|n = 1⟩, in the island. In this thesis, we consider the symmetric case (ϵ = 0) and use
only the detunning energy to define the static susceptibility (3.6).

30

Chapter 3. Microwave Scattering through a Superconducting Qubit

F IGURE 3.3: (a) A schematic of the model considered in this chapter. The spin-boson model, consisting of a two-level system and a
sub-Ohmic bath, is coupled to an Ohmic bath as a probe. (b) A superconducting circuit realizes the model in panel (a). The two-levelsystem and the Ohmic bath are realized by the charge qubit and thetransmission line, respectively. The realization of the sub-Ohmic bathwill be discussed later.

Second, we consider the sub-Ohmic bath coupled to the charge qubit, i.e., a system described by the spin-boson model with sL = s < 1. Explicit construction of
the sub-Ohmic bath using an RLC transmission line will be discussed in the next
subsection. The interaction between the charge qubit and the sub-Ohmic bath canbe expressed aseCJ,Lσz VL,0 ,
CΣ


CΣ
†
VL,0 = −h̄λL,k bL,k + bL,k,∑
2eCJ,L k

HI,L =(3.10)(3.11)

where VL,0 is the voltage at the end of the sub-Ohmic bath (see Fig. 3.3 (b)). This
form of the interaction corresponds to the system-environment coupling in the spinboson model. Here, we introduce the retarded voltage-voltage correlation function
defined byi
GVR (t) ≡ − θ (t) ⟨[VL,0 (t), VL,0 (0)]⟩B,L ,h̄(3.12)

where θ (t) is the Heaviside step function, VL,0 (t) = eiHB,L t/h̄ VL,0 e−iHB,L t/h̄ , and ⟨· · ·⟩B,L
indicates an ensemble average with respect to the sub-Ohmic bath, HB,L . Comparing the definition of the spectral density (2.38) and the imaginary part of the Fouriertransformation of GVR (t), the spectral density can be expressed as
IL (ω ) = −

1
πh̄



2eCJ,LCΣ2hi
Im GVR (ω ) .

(3.13)

3.2. Formalism for microwave scattering

31

Moreover, using the linear response theory [123], the voltage-voltage correlation
function, GVR (ω ), is related to the impedance of the sub-Ohmic bath, ZB,L (ω ), as
GVR (ω ) = −iωZsub (ω ). The relation allows the spectral function to be written interms of circuit parameters as
1
IL (ω ) =πh̄

2eCJ,LCΣ2
ωRe [ ZB,L (ω )] .

(3.14)

Note that this expression of the spectral density using the impedance of the bath isvalid for arbitrary baths.
Finally, let us consider a transmission line as an Ohmic bath. It is coupled tothe two-level system weakly as a probe system for the microwave scattering. The
transmission line is described by a continuum limit for a series of LC-circuits as
discussed in Section 2.1. Here, we briefly repeat the theoretical description. The
Hamiltonian of the transmission line is given as


Z d
1
1
2
2
HB,R =
dx(3.15)[ Q( x )] + [∂ x ϕ( x )] ,2c2l0
where c and l are the capacitance and the inductance per unit length, respectively,
and d is the length of the transmission line. The charge density operator Q( x ) and theflux operator ϕ( x ) at position x are conjugate to each other, [ϕ( x ), Q(y)] = ih̄δ( x −y). By introducing bosonic annihilation and creation operators as
"
#
r
Z d
2
1
Q
(
x
)
k
bR,k = pdx e−ikx √ − i
ϕ( x ) ,(3.16)
2lh̄ωR,k d 0
2c
#
"
r
Z d
2
1
k
Q
(
x
)
†
bR,k
=p
ϕ( x ) ,(3.17)
dx e+ikx √ + i
2lh̄ωR,k d 0
2crespectively, the Hamiltonian can be rewritten in the diagonal form:†
HR,B = ∑ h̄ωR,k bR,k
bR,k ,(3.18)k√† ] =where ωR,k = |k |/ lc. One can easily check that the commutation relation [bR,k , bR,k′δk,k′ holds.
Since this transmission line is capacitively coupled to the charge qubit, the coupling Hamiltonian can be written as
HI,R =eCJ,R
σz VR ( x = 0),CΣwith the voltage in the transmission linerh̄ωR,k †bR,k eikx + bR,ke−ikx .
VR ( x ) = ∑dckThus, we obtain the spin-boson-type interaction asr
2eCJ,R ωR,k.
λR,k = −CΣh̄dc(3.19)(3.20)(3.21)

Chapter 3. Microwave Scattering through a Superconducting Qubit=

32

F IGURE 3.4: An RLC transmission line to realize the sub-Ohmic bath.

Plugging λR,k into the definition of the spectral density (2.38), we obtain the Ohmic
spectral density of the transmission line asr 1l 2eCJ,R 2
ω = αR ω,(3.22)IR (ω ) =πh̄ cCΣin the limit d → ∞. This means that the transmission line plays the role of the
Ohmic bath, i.e., sR = 1. Note that the spectral function of the transmission line canbe also obtained from the generalform (3.14) by substituting the impedance of the√transmission line ZB,R (ω ) = l/c.

3.2.2

Realization of the sub-Ohmic bathHere, we provide a strategy to realize the sub-Ohmic bath using an RLC transmission line as shown in Fig. 3.4. As mentioned in the previous subsection, the spectral
density is related to the total impedance,
ZL−1 (ω ) = Z0−1 (ω ) + iωC0 ,(3.23)

where Z0 (ω ) is the impedance of the RLC transmission line and C0 is the capacitanceat the end of the RLC transmission line close to the charge qubit. The relation (3.14)
allows the spectral density of the RLC transmission line to be written as1
IL (ω ) =πh̄

2eCJ,LCΣ2ĨL (ω ),ĨL (ω ) = ωRe [ ZL (ω )] .

(3.24)

Therefore, to realize the sub-Ohmic bath, one needs to design the RLC transmission
line with appropriately frequency-dependent impedance. By solving the recurrence
relation for j ∈ [1, N ]:Z0 (ω ) = ZL,1 (ω ),ZL,j (ω ) = RL,j + iωLL,j +(3.25)1−1
ZL,j+1 ( ω ) + iωCL,j(3.26)−1with ZL,N+1 ( ω ) = 0, we obtain the impedance of the RLC transmission line, Z0 ( ω ),numerically.
We first consider a simple circuit to realize the sub-Ohmic bath with s = 0.5.
Then, we expand the circuit model for an arbitrary value of s smaller than 0.5. Finally, we mention an RLC transmission line for arbitrary s in the range 0 < s < 1.

3.2. Formalism for microwave scattering

33

F IGURE 3.5: Spectral density of the RLC transmission line as a function of frequency for different N. The exponent of the spectral density is 0.5. The circuit elements are set as R = 1 Ω, L = 1 pH, C =1 pF, C0 = 1 pF.

(i) case for s = 0.5
Let us first consider a uniform RLC transmission line:
R j = R,

L j = L,Cj = C.

(3.27)

Figure 3.5 shows the dimensionless spectral density of the RLC transmission line,ĨL (ω ), for N = 102 , 103 , 104 , and ∞ by solving the recurrence relation (3.26) numerically. Here, the circuit parameters are set as R = 1 Ω, L = 1 pH, C = 1 pF, andC0 = 1 pF. Note that the case of N = ∞ has been discussed in Ref. [121]. Figure 3.5
indicates that the circuit realizes the sub-Ohmic bath with s = 0.5 for the frequency
range, ω ∗ ≪ ω ≪ ωc , where ω ∗ and ωc are the low- and high-frequency cutoffs, respectively. From the recurrence relation in Eq. (3.26), these cutoffs can be estimatedas ω ∗ = π 2 /( N 2 RC ) and ωc = C/( RC02 ), respectively, which are shown for N = 103
in Fig. 3.5. Moreover, to observe the robustness in this circuit, we added 10% relative
randomness to the circuit parameters. As shown in Fig. 3.5, the fluctuation due tothe randomness is not visible. Thus, this uniform RLC transmission line is robust to
the randomness in the circuit elements.
(ii) case for 0 < s < 0.5
We next propose an RLC transmission line with spatial dependence circuit elements, njRj = R, L j = 0, Cj = C.
(3.28)
N
Here, n is a positive real number. Figure 3.6 shows the dimensionless spectral density, ĨL (ω ), for n = 2 and N = 102 , which is obtained by solving the recurrence
relation (3.26) numerically. The circuit parameters are set as R = 50 mΩ, L = 0 H,
C = 0.2 µF, and C0 = 0 F. According to this figure, the present circuit realizes the
sub-Ohmic bath for the frequency range of ω ∗ ≪ ω ≪ ωc , where ω ∗ and ωc are thelow- and high-frequency cutoffs, respectively. Randomness in the circuit leads tosmall fluctuations but does not change the overall feature of the spectral density forthe case of no randomness by comparing the circuit with 10% relative randomness

34

Chapter 3. Microwave Scattering through a Superconducting Qubit

F IGURE 3.6: The frequency dependence of the spectral density of the
RLC transmission line for n = 2 with the exponent 0.25. The circuit
parameters are R = 50 mΩ, L = 0 H, C = 0.2 µF, C0 = 0 F, N = 102 .

(blue line) and without randomness (red line). It is noteworthy that the sub-Ohmicbath can be realized for a wider frequency range extending to N = 102 , which ismuch less than the case for s = 0.5.
From Eq. (3.26), we can derive, for an arbitrary positive value of n, the spectral
density as (see Appendix C for the detailed derivation)ĨL (ω ) ∝ ω 1/(n+2) ,( ω ∗ ≪ ω ≪ ωc ) .

(3.29)

This form means that the present RLC transmission line potentially realizes the subOhmic bath with s < 0.5 because n is a real positive number. It is consistent with
Fig. 3.6, in which the RLC transmission line with n = 2 realizes the sub-Ohmic bathwith s = 0.25. The low- and high-frequency cutoffs are obtained as
1  n 2
ω =
RC 2N
∗

√ ! n +2
2 2
1+
,
n

1
ωc =
RC



2N
n

n
,

(3.30)

respectively, and are shown in Fig. 3.6. Note that as n decreases, the frequency range
in which the spectral density behaves as the sub-Ohmic bath becomes narrower.
(iii) case for 0 < s < 1
Finally, we discuss how to realize the sub-Ohmic spin-boson model with 0 < s < 1.
We assume that the resistances and inductances depend on the position as follows:
 n
 p
j
j
Rj = R, Lj = L, Cj = C,(3.31)
N
N
where n and p are non-negative real numbers. From Eq. (3.26), we can derive ananalytic expression of the spectral density
ĨL (ω ) ∝ ω 2/( p+2) ,( ω ∗ ≪ ω ≪ ωc ) ,(3.32)

3.2. Formalism for microwave scattering

35

where the low-frequency cutoff, ω ∗ , is give by
∗

ω =



p  2( n − p ) R p +2
2N
L n +2 C n − p1/(2n− p+2),

(3.33)and the high-frequency cutoff, ωc , is a complex function of the circuit parameters.
Note that although this circuit can realize the sub-Ohmic bath with an arbitraryvalue of s in the range of 0 < s < 1, one needs a very large N ≳ 105 , to achievea sufficient frequency range where the spectral density behaves as the sub-Ohmic
one.

3.2.3

Microwave scattering

As shown in Fig. 3.3 (a), an incident microwave enters from the transmission line
into the charge qubit, and then the reflected microwave is allowed to propagate in
the same transmission line in the opposite direction. Here, we formulate the microwave scattering using the input-output theory [124, 21, 22] and derive a relation
between the reflection coefficient and the dynamic spin susceptibility of the subOhmic spin-boson model within the linear response theory.
Let us begin with the total Hamiltonian

H = Hsub + HB,R + HI,R ,(3.34)

where Hsub = HTLS + HB,L + HI,L is the Hamiltonian of the sub-Ohmic spin-bosonmodel. From the Heisenberg equation of motion, the time evolution of the annihilation operator of the transmission line isḃR,k (t) =λi[H, bR,k (t)] = −iωR,k bR,k (t) + i R,k σz (t),h̄2(3.35)

where O(t) = eiHt/h̄ O e−iHt/h̄ . By integrating the Heisenberg equation of motion (3.35),
its solution is given by [21]bR,k (t) = e−iωR,k (t−t0 )λR,kbR,k (t0 ) + i
2

Z t
t0

′

dt′ e−iωR,k (t−t ) σz (t′ ),(3.36)

where t0 is a past time before any excitation mode reaches the two-state system.
The first term represents the free time evolution in the transmission line and thesecond term describes the effect of the interaction between the charge qubit and thetransmission line. Similarly, we obtain an alternative solution to Eq. (3.35) as
bR,k (t) = e−iωR,k (t−t1 ) bR,k (t1 ) − i

λR,k
2

Z t1
t

′

dt′ e−iωR,k (t−t ) σz (t′ ),(3.37)

where t1 is a future time after the entire excitation mode leaves the two-state system.
The voltages of input and output modes are expressed asrih̄ωR,k h −iωR,k (t−t0 )Vin (t) = ∑ebR,k (t0 ) + h.c. ,(3.38)dckrih̄ωR,k h −iωR,k (t−t1 )Vout (t) = ∑ebR,k (t1 ) + h.c. ,(3.39)dck

36

Chapter 3. Microwave Scattering through a Superconducting Qubit

respectively. Using Eqs. (3.36)-(3.39), the input and output voltages in the Fourier
space relate to each other as follows:⟨Vout (ω )⟩ = ⟨Vin (ω )⟩ − iπh̄CΣIR (ω ) ⟨σz (ω )⟩ ,2eCJ,R(3.40)

where ⟨·⟩ indicates an average with respect to the total Hamiltonian and IR (ω ) is
linear to ω because the transmission line is Ohmic.
Assuming weak coupling between the charge qubit and the transmission line(αR ≪ 1), the population ⟨σz (ω )⟩ in Eq. (3.40) can be expressed within the linear
response theory as⟨σz (ω )⟩ = −eCJ,Rχ (ω ) ⟨Vin (ω )⟩ .
CΣ sub

(3.41)

Here, χsub (ω ) is the Fourier transformation of the dynamic spin susceptibility of the
sub-Ohmic spin-boson model defined asχsub (t) ≡iθ (t) ⟨[σz (t), σz (0)]⟩sub ,h̄(3.42)

where ⟨·⟩sub indicates an average to the Hamiltonian of the sub-Ohmic spin-bosonmodel Hsub . Hence, from Eqs. (3.40) and (3.41), we obtain a relation between thereflection coefficient and the dynamic spin susceptibility asr (ω ) ≡πh̄⟨Vout (ω )⟩= 1 + i IR (ω )χsub (ω ).
2⟨Vin (ω )⟩(3.43)

This form indicates that one can calculate the reflection coefficient once one getsthe dynamic spin susceptibility of the sub-Ohmic spin-boson model. In this thesis, we perform the continuous-time quantum Monte Carlo simulation to obtain the
dynamic spin susceptibility numerically.

3.3

Continuous-time quantum Monte Carlo method

In this section, we explain the continuous-time Monte Carlo (CTQMC) algorithm [125]and how to calculate the dynamic spin susceptibility of the sub-Ohmic spin-bosonmodel.

3.3.1

Kink representation

First, let us write the partition function of the spin-boson model in the kink representation. The partition function is given in the imaginary time path integral formas [17, 19]
h
i Z
− βH
Z = tr e= D σz (τ ) e−S[σz (τ )] ,(3.44)1
S [σz (τ )] = −
4

Z β
0

dτ

Z τ
0

dτ ′ σz (τ )K (τ − τ ′ )σz (τ ′ ),(3.45)

where σz (τ ) is a spin path along the imaginary-time axis, D σz (τ ) is an integral measure for all possible paths σz (τ ), S [σz (τ )] is an influence function, and K (τ ) is a

3.3. Continuous-time quantum Monte Carlo method

37

kernel defined as
K (τ ) =Z ∞0dω I (ω )cosh [ω ( β/2 − τ )],sinh( βω/2)(3.46)

where I (ω ) is the spectral density. Note that the kernel has the symmetric property,
K (τ ) = K ( β − τ ), and the asymptotic form K (τ ) ∼ τ −(1+s) for ωc−1 ≪ τ ≪ β/2.
Here, we move from the spin representation (3.44) to the kink representation.
The spin path σz (τ ) is characterized by an alternative configuration of kinks (jumps
from σz = −1 to σz = +1; qi = +1) and anti-kinks (opposite jumps; qi = −1) andtheir positions τi∈[1,2n] . Thus, the derivative of the spin path can be rewritten in termsof the kink variable qi as2ndσz (τ )= ∑ 2qi δ(τ − τi ),dτ
i =1(3.47)

where n is the number of the pairs of kinks and anti-kinks. By plugging this expression into Eq. (3.44), we obtain the partition function in the kink representation as
follows:
"
#
Z τ2
Z τ2n
2n
∞  2n Z β∆dτ1 exp ∑ qi q j W (τi − τj ) , (3.48)dτ2n−1 · · ·
Z= ∑
dτ2n
2
0
0
0
n =0
i,j
where W (τ ) is obtained from W ′′ (τ ) = −K (τ ) and reads
W (τ ) =

3.3.2

Z ∞
0

dω

I (ω ) cosh ( βω/2) − cosh [ω ( β/2 − τ )].
ω2sinh ( βω/2)(3.49)

Calculation algorithm

We apply the CTQMC method [125] to the partition function of the spin-bosonmodel in the kink representation. The CTQMC algorithm in this thesis employs acluster-flip update similar to the Swendsen–Wang cluster algorithm [126]. This algorithm allows us to overcome the critical slowing down, namely, the problem thatthe correlation time diverges near the critical point of second-order phase transition.
The cluster-flip update is constructed as follows [19] (see Fig. 3.7):
1. Insert new vertices with Poisson statics, P(∆τ ) = Γe−Γ∆τ with mean value
Γ−1 = 2/∆.
2. Connect two segments (the line segments between neighboring vertices), si
and s j , with the probabilityhip[si , s j ] = 1 − δσz (si ),σz (s j ) 1 − e−2A ,(3.50)

A = W (τi−1 − τj−1 ) − W (τi−1 − τj ) − W (τi − τj−1 ) + W (τi − τj ), (3.51)
where σz (si ) is the value of σz in the segment si , and positions of the vertices at
the two edges of the segment si are denoted by τi−1 and τi , respectively.
3. Flip each segment cluster with probability 1/2.
4. Remove the redundant vertices within segments.

38

Chapter 3. Microwave Scattering through a Superconducting Qubit

0

F IGURE 3.7: The configurations along the imaginary-time axis andthe CTQMC update process: (a) An initial configuration in the spin
representation, (b-i) the kink representation of panel (a), (b-ii) insertion of new cuts using the Poisson distribution, (b-iii) connection ofsegments with the probability (3.50), (b-iv) flipping each cluster withprobability 1/2, (b-v) removal of redundant cuts, and (c) the final configuration in the spin representation after the update.

The Monte Carlo data presented in this thesis represent averages over 105 -106 cluster
updates.
By using the CTQMC method, we can evaluate the Fourier transformation of thespin correlation function C (τ ) = ⟨σz (τ )σz (0)⟩sub defined on the imaginary-time axis
as follows:C (iωn ) =

Z h̄β
0

dτ eiωn τ C (τ ) =1⟨|ξ (iωn )|2 ⟩ ,h̄βωn2(3.52)

where ⟨·⟩ denotes the CTQMC sampling average, ωn = 2πn/(h̄β) is the Matsubara
frequency, and β = 1/(kB T ) is the inverse temperature. Here, ξ (iωn ) is the Fouriertransformation of the derivative of the spin path, ξ (τ ) = dσz (τ )/dτ, and can be
expressed in the kink representation, using Eq. (3.47), as2nξ (iωn ) = ∑ 2(−1) j eiωn τj .
j =1

Then, we evaluate the reflection coefficient numerically in the following way:
1. Get a set of vertex position {τj } at each configuration.

(3.53)

3.3. Continuous-time quantum Monte Carlo method

39

2. Calculate the Fourier transformation of the spin correlation function, C (iωn ),using Eqs. (3.52) and (3.53).
3. Perform the analytical continuation numerically using the Padé approximation [127] and then obtain the Fourier transformation of the dynamic spin susceptibility,χsub (ω ) = C (iωn → ω + iδ).

(3.54)

4. Evaluate the reflection coefficient using Eq. (3.43).
Moreover, we can calculate the static spin susceptibility using the CTQMC method
as follows:χ0 = β ⟨σ̄z2 ⟩ ,
1
h̄β

σ̄z =

3.3.3

Z h̄β
0

(3.55)dτ σz (τ ) =2 2n−1(−1) j τj + 1.
h̄β j∑=0(3.56)

Binder analysis

Finally, we explain how to determine the critical point of the quantum phase transition for the sub-Ohmic spin-boson model [19]. To study this quantum phase transition, we use the finite size (or temperature) scaling for observable A in the vicinityof the critical point,∗⟨A⟩ ( β, α) = β xA f A ( βyt δ),(3.57)

where δ = (α − αc )/αc denotes the dimensionless distance from the critical point,xA and f A are the scaling exponent and scaling function of the observable A, respectively. For the sub-Ohmic spin-boson model, the exponent y∗t is related to thecorrelation length exponent ν (ξ ∼ δ−ν at the critical point δ ≃ 0) as [19, 120, 119]

1

 ,
y∗t = ν
 1 + 1 − s,
ν 2(0 < s ≤ 0.5),(3.58)(s > 0.5).

Now, we introduce the Binder parameter defined as!
1⟨σ̄z4 ⟩
B=3−,2
⟨σ̄z2 ⟩2

(3.59)

R
where σ̄z = (h̄β)−1 dτ σz (τ ) is the magnetization for a spin configuration. The
Binder parameter takes values of 1 in the localized phase and 0 in the delocalized
phase and jumps between them when the coupling strength passes across the critical point. The scaling exponent of the Binder parameter is x B = 0 because it only
includes the ratio of the same order of parameters. Therefore, the critical point δ = 0,i.e., α = αc , is determined as a point at which the Binder parameter is independentof the temperature at sufficiently low temperatures.

40

Chapter 3. Microwave Scattering through a Superconducting Qubit

F IGURE 3.8: Frequency dependence of the dynamic spin susceptibility for s = 0.5 and ∆/ωc = 0.1. (a) and (b) describe the plot of numerical results for the real and imaginary parts of the dynamic spin susceptibility, respectively, calculated by the CTQMC simulations. The
three types of plots indicate the results at α = αc = 0.1074 for different temperatures, βh̄ωc = 212 (blue), 213 (green), and 214 (red), andthe other plot at α = 0.05 (< αc ) and βh̄ωc = 214 (purple). The blacksolid line in panel (b) is given by the generalized Shiba-Korringa relation (3.63).

3.4

Numerical results

In this section, we present the numerical results for the dynamic spin susceptibility
and the microwave reflection at the critical point, α = αc , and below it, α < αc , byperforming the CTQMC simulations, as explained in Section 3.3.

3.4.1

Dynamic spin susceptibility

Quantum critical regime
First, we discuss the dynamic spin susceptibility in the quantum critical regime, α =αc . In this regime, the dynamic spin susceptibility exhibits the power-law frequency
dependence reflecting the nature of the quantum phase transition,χsub (ω ) ∼ ω −y .

(3.60)

3.4. Numerical results

41

F IGURE 3.9: (a) The real and (b) imaginary parts of the dynamic spin
susceptibility for s = 0.3 as a function of frequency, which are obtained by the CTQMC calculation. For s = 0.3, the critical couplingstrength is αc = 0.034. Other parameters and plot colors are the sameas those in Fig. 3.8.

Here, y is a critical exponent of the dynamic spin susceptibility and related to thatof the imaginary-time spin-spin correlation function as y = 2 − η. Thus, for 0 < s ≤0.5, the critical exponent y can be obtained by using the mean-field result (3.8),
y = s.

(3.61)

Note that, as discussed in Section 2.5, the critical exponent is a complex function ofs for 0.5 < s < 1 because that case belongs to the non-trivial universality class.
To compare the analytical form (3.61) and numerical results, we first consider
the case of s = 0.5 and ∆/ωc = 0.1, for which the critical points are determined as
αc = 0.1074 by the Binder analysis, as discussed in Section 3.3. Figures 3.8 (a) and(b) show the frequency dependence of the real and imaginary parts of the dynamic
spin susceptibility obtained by the CTQMC simulations at α = αc for three different
temperatures, βh̄ωc = 212 , 213 , and 214 . As shown in Fig. 3.8, the dynamic spin
˜susceptibility is proportional to ω −0.5 for a frequency range, kB T/h̄ ≪ ω ≪ ∆,˜
where ∆ ≲ ∆ is the renormalized tunneling frequency dependent of s, ∆/ωc , and α.
This exponent for the low-frequency region is consistent with the critical exponent
in Eq. (3.61) for s = 0.5. As shown in Fig. 3.9, we can observe the critical powerlaw behavior (3.61) even for s = 0.3, for which the critical point is determined as

42

Chapter 3. Microwave Scattering through a Superconducting Qubit

αc = 0.034, although the frequency range showing the critical behavior is narrower
than the case of s = 0.5. Note that our results are in agreement with those of previous
studies [115, 128].
Delocalized regime
We next consider the delocalized regime, α < αc , in which the ground state is described by a coherent superposition of the two charge states. At low temperatures,
we can apply the generalized Shiba-Korringa relation [129, 38]h̄Im[χsub (ω )]lim= 2παωc1−s
ω →0
ωs



h̄χ0
2

2
.

(3.62)

This relation provides the low-frequency asymptotic behavior of the dynamic spin
susceptibility,
Im[χsub (ω )] ∼h̄παχ20 ωc1−s sω.
2(3.63)

This relation is extended from the Shiba-Korringa relation in the Kondo model [130],
which can be described by the local Fermi liquid theory [67]. Thus, what this relation
holds means the appearance of the many-body effect.
Fig. 3.8 shows the real and imaginary parts of the dynamic spin susceptibility fors = 0.5, ∆/ωc = 0.1, and βh̄ωc = 214 at α = 0.05, which is sufficiently smaller thanthe critical point αc = 0.1074. The imaginary part of the dynamic spin susceptibil˜ for the quantum critical regime
ity has a peak at a slightly higher frequency than ∆˜while the real one has a shoulder. This indicates that the renormalization effect in ∆due to the sub-Ohmic bath is stronger as the coupling strength increases. At suffi˜ the numerical results for the imaginary part of theciently small frequencies, ω ≪ ∆,dynamic spin susceptibility exhibits the power-law frequency dependence, and its
exponent is consistent with the generalized Shiba-Korringa relation (3.63), resulting
in Im[χsub (ω )] ∝ ω 0.5 for s = 0.5. Note that the static spin susceptibility in the prefactor of Eq. (3.63) is determined numerically by using the CTQMC, as discussed in
Section 3.3. One can observe this power-law behavior in the delocalized phase evenfor s = 0.3 in Fig. 3.9.

3.4.2

Reflection

Finally, we discuss the microwave reflection loss, 1 − |r (ω )|2 . From Eq. (3.43), thereflection loss can be written as
1 − |r (ω )|2 = πh̄IR (ω )Im [χsub (ω )] + O(α2R ).

(3.64)

Recalling that the bath R (the transmission line) is Ohmic, IR (ω ) ∝ ω, and the coupling between the two-level system and the Ohmic bath is sufficiently weak, αR ≪ 1,
the reflection loss is almost proportional to ωIm[χsub (ω )]. Furthermore, by using
Eqs. (3.61), (3.60), and (3.63), we obtain the the power-law behavior of the reflection
loss as follows:( 1− s˜ω , for α = αc , kB T/h̄ ≪ ω ≪ ∆,1 − |r (ω )|2 ∼(3.65)˜ω 1+s , for α < αc , ω ≪ ∆.

3.4. Numerical results

43

F IGURE 3.10: Frequency dependence of the reflection loss, 1 − |r (ω )|,calculated using the CTQMC simulations. The plots represent thenumerical results for (a) s = 0.5 and (b) s = 0.3. The insets in eachpanel represent the reflection as a function of frequency. The otherparameters are set as ∆/ωc = 0.1 and βh̄ωc = 214 . The red andpurple plots indicate the results at the critical point (α = 0.1074 for
s = 0.5 and α = 0.034 for s = 0.3) and below it (α = 0.05 for s = 0.5
and α = 0.01 for s = 0.3), respectively.

Figure 3.10 (a) and (b) show the frequency dependence of the reflection loss for
s = 0.5 and 0.3, respectively. The other parameters are set as ∆/ωc = 0.1, βh̄ωc =214 , and αR = 0.01. The insets in Fig. 3.10 represent the reflection as a functionof frequency. It has a small peak for both the quantum critical regime, α = αc ,˜and the delocalized regime, α < αc , near the renormalized tunneling amplitude ∆.
This indicates that the incident microwave is almost reflected but some part of the
˜ of the sub-Ohmic spin-boson model ismicrowave with characteristic frequency ∆˜ the reflection loss
absorbed in the two-level system. For the low frequency, ω ≪ ∆,shows the power-law frequency dependence and approaches zero more slowly forthe case of the quantum critical regime than the delocalized phase. Their exponents
are consistent with the analytical forms (3.65) both for s = 0.3 and 0.5. They are
signatures of the quantum phase transition and the quantum many-body effect.
At the end of this section, we add an interesting remark. The characteristic frequency behavior of the reflection loss coming from the quantum phase transitionor the quantum many-body effect is dependent on s. Combining this fact and the
proposition of the realization of the sub-Ohmic bath with arbitrary s using superconducting circuits in Section 3.2 leads to a possibility that one can control quantum

44

Chapter 3. Microwave Scattering through a Superconducting Qubit

critical phenomena and quantum many-body phenomena in well-controllable systems.

3.5

Short summary of this chapter

We studied quantum critical phenomena of the sub-Ohmic spin-boson model throughthe microwave scattering based on the superconducting circuit. By considering a
superconducting circuit composed of a charge qubit and an RLC transmission linewith space-dependent elements, we show that it is effectively described by the subOhmic spin-boson model. Moreover, we clarified the dynamic spin susceptibility
and the microwave reflection loss at and far from the quantum phase transition by
performing the continuous-time Monte Carlo method. As a result, the Fermi liquidlike feature (the Shiba-Korringa relation) and quantum critical behavior appear in
their frequency dependence, depending on the exponent of the spectral density s.
This indicates a possibility that quantum criticality can be studied in microwave
spectroscopy for superconducting circuits in a realistic setup. In this chapter, weconsidered only the case of s ≤ 0.5, which allows us to compare it with the analytical results by the mean-field universality class. It would be an important future
problem to check the critical exponents from the numerical approach for a wider
range of s.

45

Chapter 4

Transmission through a Disordered
Josephson-junction Chain
In this chapter, we investigate the scattering of elastic waves through a disordered
Josephson-junction chain. First, we briefly introduce the model of the disordered
Josephson-junction chain in Section 4.1. Next, we move to new results obtained inRef. [40]. We formulate the elastic scattering off the disordered Josephson-junction
chain in the classical limit in Section 4.2. Using both numerical and analytical approaches, we show the results for the transmission in the weakly localized regime
in Section 4.3. Finally, we present several pieces of information about the strongly
localized regime and compare them with the localization model with the Gaussian
white-noise disorder in Section 4.4.

4.1

Disordered Josephson-junction chain

As described in Sections 2.6 and 2.7, the disordered Josephson-junction chain of thelength d can be described by the random phase sine-Gordon model (2.48) (for thecorresponding circuit, see Fig. 4.1). We repeat the theoretical description of the disordered Josephson-junction chain. The Lagrangian is given as


Z d
2
h̄ 
Λ
h̄v
2
L=
dx
θ̇ ( x, t) −[∂ x θ ( x, t)] + cos [2θ ( x, t) + χ( x )] , (4.1)
2πKv
2πK
a
0
where
θ ( x, t) is the local field associated with the accumulated charge and v =p
a 2EJ Eg /h̄ is the plasmon velocity with the elementary unit length a and Eg =q(2e)2 /(2Cg ). The Luttinger parameter K = π EJ /(2Eg ) is related to the chain
impedance as Z = RQ /(2K ). The random phase field satisfies the following statistics; the average,⟨cos χ( x )⟩ = ⟨sin χ( x )⟩ = 0,

(4.2)

and variance,
σ2 =

Z d
0

dx ⟨cos χ( x ) cos χ(0)⟩ =Z d
0dx ⟨sin χ( x ) sin χ(0)⟩ .

(4.3)

At the transmon limit (EJ ≫ EC ), the quantum phase slips are rare events, and its
amplitude is written as
1/4 −√32EJ /E
8CeΛ = √ 2EJ3 EC
π

(4.4)

46

Chapter 4. Transmission through a Disordered Josephson-junction Chain

F IGURE 4.1: A disordered Josephson-junction chain of length d. The
Josephson energy and the Josephson capacitance of each junction are
EJ and C, respectively. The ground capacitance of each superconducting island (blue circle) is Cg . The random background charges are induced by the gate voltages Vg . An incident plasmon wave from one of
the waveguides contacted to the Josephson-junction chain is either reflected to the same waveguide or transmitted to the opposite waveguide with the amplitudes r (ω ) and t(ω ), respectively, at frequency
ω. The plasmon velocities in the waveguides and the Josephsonjunction chain are v0 and v. The impedances of the waveguides andthe Josephson-junction chain are RQ /2K0 and RQ /2K, respectively.

withEC = (2e)2 /(2C ).
In this thesis, we focus on the classical limit (K → 0). In this limit, as describedin Section 2.7, the kinetic term in the Lagrangian (4.1) can be neglected. Then, ourproblem is reduced to the interplay between the elasticity and the disorder. On thelength scale shorter than the Larkin length R⋆ defined in Eq. (2.52), the elastic properties still remain, which corresponds to excitations with higher frequencies (ω ≳ ω⋆ ).
However, at low frequencies (ω ≪ ω⋆ ), the collective pinning effect induced by thedisorder breaks the sound-like plasmon spectrum.

4.2

Formalism for scattering problem in the classical limit

In this section, we formulate the scattering problem for the disordered Josephsonjunction chain in the classical limit (K ≪ 1), i.e., in the deep Bose-glass phase. To
this end, we consider a setup where the Josephson-junction chain is coupled to two
waveguides (see Fig. 4.1). Classically, an incident plasmon wave from one of the
waveguides (the left one in Fig. 4.1) is either reflected to the same waveguide ortransmitted to the opposite waveguide, with the respective amplitudes r (ω ) andt(ω ). The two waveguides are the same and have the plasmon velocity v0 and the
Luttinger parameter K0 .
Using the continuity of θ ( x ) and (v/K )∂ x θ ( x ) at interfaces between the waveguides and the Josephson-junction chain, we find that the plasmon wave reflection
and transmission at the interfaces due to the impedance mismatch (Z ∝ 1/K) is
given byR0 =K − K0K + K0

2
,

T0 =

4KK0
.
( K + K0 )(4.5)

4.2. Formalism for scattering problem in the classical limit

4.2.1

47

Equation of motion

In the classical limit, the field θ ( x, t) can be divided into a static part θ̄ ( x ) and a small
oscillation around it with frequency ω,θ ( x, t) = θ̄ ( x ) + ψ( x )e−iωt .
The static field θ̄ ( x ) minimizes the classical energy functional


Z d
h̄v
2
E [θ ] =dx(∂ x θ ) − Λ cos(2θ + χ) ,
2πK
0

(4.6)

(4.7)

with boundary conditions ∂ x θ̄ (0) = ∂ x θ̄ (d) = 0 coming from the continuity of voltages at the both edges of the Josephson-junction chain at x = 0 and d:vv0∂ x θ (0+ , t ) =∂ x θ (0− , t ),
K
K0

v0
v
∂ x θ (d− , t) =∂ x θ ( d + , t ),
K
K0

(4.8)

respectively. The small oscillation ψ( x ) takes the asymptotic form of a scattering
state at frequency ω in the waveguides,( iωx/v
0e+ r (ω )e−iωx/v0 ,( x < 0),(4.9)
ψ( x ) =iω ( x −d)/v0t(ω )e,( x > d ),where r (ω ) and t(ω ) are the elastic reflection and transmission amplitudes, respectively. By linearizing the equation of motion, the small oscillation ψ( x ) solves the
Schrödinger-like wave equation
ω 2 ψ( x ) = −v2 ∂2x ψ( x ) + V ( x )ψ( x ),(4.10)

with boundary conditions
ψ (0) = 1 + r ( ω ),
ψ ( d ) = t ( ω ),Kv∂ x ψ(0) = iω [1 − r (ω )],K0
Kv∂ x ψ(d) = iω t(ω ),K0(4.11)(4.12)(4.13)(4.14)

which come from the continuity of current∂ t θ (0+ , t ) = ∂ t θ (0− , t ),∂ t θ ( d − , t ) = ∂ t θ ( d + , t ),and voltage (4.8). Here, the effective disorder potential V ( x ) takes 4πKvΛ cos 2θ̄ ( x ) + χ( x ) ,(0 < x < d ),h̄V (x) =
0,(otherwise),(4.15)(4.16)and its properties will be discussed later. Note that, in the waveguides, since theeffective disorder potential is vanishing, the linearized equation of motion (4.10)
becomes the usual wave equation, which is consistent with the asymptotic form of
the small oscillations (4.9).

48

4.2.2

Chapter 4. Transmission through a Disordered Josephson-junction Chain

Numerical method

Here, we describe our strategy to calculate the scattering amplitudes numerically.
First, we rewrite our problem with the dimensionless variables by rescaling the spatial dimension by the Larkin length R⋆ and the frequency dimension by the Larkinfrequency ω⋆ . Next, the dimensionless spatial variables are discretized with a small
spacing ϵ as d/R⋆ = ( M + 1)ϵ, where M is the division number. Then, the aboveprescription leads the classical energy functional (4.7), in units of (ϵh̄v)/(2πKR⋆ ), totake the discretized form:"#
M −1 
θ m +1 − θ m 2 V m
V0 + V M
Ẽ [θ ] = ∑−−.
(4.17)
ϵ
2
4
m =0
Here,


Vm = 2 Vm′ cos 2θm + Vm′′ sin 2θm(4.18)

is the discretized effective disorder potential in dimensionless units, and the correlators (2.50) are reproduced in continuum limit (ϵ → 0) if ⟨Vm′ ⟩ = ⟨Vm′′ ⟩ = ⟨Vm′ Vn′′ ⟩ = 0
and ⟨Vm′ Vn′ ⟩ = ⟨Vm′′ Vn′′ ⟩ = ϵ−1 δm,n . The last term in Eq. (4.17) is derived from considering an infinite periodic chain with the mirror symmetry Vm = V2M−m in unit chain(0 ≤ m ≤ 2M − 1). This ensures that the boundary conditions ∂ x θ̄ (0) = ∂ x θ̄ (d) = 0
are satisfied in continuum limit.
We compare two ways to generate the random fields Vm′ and Vm′′ . In the phasemodel reminiscent of the original model, we relate them with flatly distributed random phases χm ∈ [0, 2π ) at each site,
r
r
2
2
′
′′
Vm =
cos χm , Vm =sin χm .
(4.19)
ϵϵ′′′
Alternatively, in the box model, the √random√ fields Vm and Vm are independent andflatly distributed in the interval (− 3/ϵ, 3/ϵ). Our numerical results could not
distinguish between the two models (see Figs. 4.2).
The static fields {θ̄m } are determined by minimizing the classical energy functional (4.17). To perform the minimization, we adopted the minimization method
described in Ref. [102]. Note that a common π-shift of the static field, θm → θm +π, leaves the classical energy functional (4.17) invariant and the effective potential (4.18) only depends on θ̄m mod π. Thus, we can restrict for solutions θ̄m ∈(−π/2, π/2] by substituting (θm − θm+1 )2 with mink (θm − θm+1 − kπ )2 in Eq. (4.17).
Now, we calculate the scattering amplitudes using the static fields θ̄ obtained by
the above method. The linearized equation of motion (4.10) and boundary conditions (4.11)-(4.14) are discretized as



ω
ω⋆

2
ψm +

ψm+1 + ψm−1 − 2ψm− Vm ψm = 0,ϵ2(4.20)

4.2. Formalism for scattering problem in the classical limit49

for 0 < m ≤ M, andψ0 = 1 + r (ω ),(4.21)

ψ M = t ( ω ),
ψ1 − ψ0ω K=i[1 − r (ω )],ϵω ⋆ K0ω K
ψ M − ψ M −1=it ( ω ),ϵω ⋆ K0(4.22)(4.23)(4.24)

respectively. Using these boundary conditions, we can eliminate ψ0 and ψM andexpress r (ω ) and t(ω ) in terms of ψ1 and ψM−1 as
r (ω ) =t(ω ) =K ω
1 − iϵK0 ω ⋆

 −1 K ω
1 − iϵK0 ω ⋆

 −1K ω
ψ1 − 1 − iϵK0 ω ⋆

,ψ M −1 .

(4.25)(4.26)

Moreover, the equation of motion (4.20) reads eigenvalue equation with the vector {ψ1 , . . . , ψM−1 }. Thus, for a given disorder configuration yielding the potentials
Vm , by solving the eigenvalue problem, we obtain the scattering amplitudes from
Eqs. (4.25) and (4.26). The averaged scattering amplitudes are evaluated numericallyby iterating this procedure for a large number of disorder configurations, typically
104 disorder configurations in this thesis.

4.2.3

Effective disorder potential

The effective potential (4.16) depends on the solution θ̄ ( x ) of a nontrivial optimization problem defined by the functional (4.7), as discussed in the previous subsection.
Thus, it is expected to have a complex pattern of non-Gaussian correlations of which,to the best of our knowledge, little is known. From the reality condition ω 2 > 0 for
the eigen spectrum of Eq. (4.10), the local potential mean has a well defined positive
sign, ⟨V ( x )⟩ > 0. Figure 4.2 (a) shows the mean of the disorder potential as a function of space for d/R⋆ = 20. The mean reaches a constant value sufficiently far fromthe edges, on the scale of R⋆ . We also found numerically a fitting function v( x ) that
describes the exponential enhancement of ⟨V ( x )⟩ near the edges, see the legend of
Fig. 4.2 (a).
The potential V ( x ) inherits a finite correlation length R⋆ from θ̄, despite the underlying disorder χ( x ) being a short-ranged one. We characterize the spatial correlations in V ( x ) by considering its second cumulant and expressing it in the form⟨⟨V ( x )V (y)⟩⟩ ≡ ⟨(V ( x ) − ⟨V ⟩) (V (y) − ⟨V ⟩)⟩
 



x−y
x−y
4
= ω⋆ 4δ−w.
R⋆R⋆(4.27)

Here, the introduced function w( x ) is smooth and decays at large scales. We findw( x ) numerically as w( x ) = ce−|x|/a with c ≈ 3.24 and a ≈ 0.346 (see Fig. 4.2 (b)).
We note that the random potential V ( x ) is non-Gaussian. Its third-order cumulant
is shown in the inset of Fig. 4.2 (b). The Fourier component of ⟨⟨V ( x )V (0)⟩⟩ at small

50

Chapter 4. Transmission through a Disordered Josephson-junction Chain

F IGURE 4.2: (a) The mean effective disorder potential and (b) thesmooth part of the second-order correlation function at d/R⋆ = 20.
The green and red lines are the numerically obtained curves for the
random phase and box models, respectively; they cannot be distinguished from each other. The dashed line represents the fitting functions v( x ) = d[e− x/b + e−(d− x)/b ] + v0 with d ≈ 1.36, b ≈ 0.269, andv0 ≈ 2.44 in panel (a) and −w( x ) = −ce−| x|/a with c ≈ 3.24 anda ≈ 0.346 in panel (b). The inset of panel (b) shows the third-order
cumulant at noncoinciding points.

momentum on the scale of 1/R⋆ is reduced by a factor 1 − η withη=1
4

Z d/R⋆
0dx w( x ) ≈ 0.561,(4.28)

compared to its large-momentum value, which is determined by the delta-function
in Eq. (4.27). The property of η ̸= 0 is the consequence of the collective pinningmechanism. In the large-frequency range of interest in Section 4.3, η is the only parameter needed to quantitatively predict the statistics of the scattering amplitudes.
On the other hand, we illustrate in Section 4.4 that the signature properties of thereflection and transmission amplitudes at low frequencies cannot be accounted forby a white Gaussian disorder. Therefore, we attribute these properties to the nonGaussianity of the potential V ( x ) induced by the pinning effect.

4.3. Weakly localized regime

4.3

51

Weakly localized regime

In this section, we derive an analytical formula for the transmission ⟨ T (ω )⟩ ≡ ⟨|t(ω )|2 ⟩in the high-frequency regime (ω > ω⋆ ), where disorder can be treated perturbatively.

4.3.1

Mapping to the Dirac equation

First, we show that, at high frequency (ω > ω⋆ ), the equation of motion (4.10) is
equivalent to a Dirac equation in Gaussian random fields. Let us introduce twofunctions, R( x ) and L( x ), such that
ψ ( x ) = R ( x ) + L ( x ),ω
∂ x ψ( x ) = i [ R( x ) − L( x )] .
v

(4.29)(4.30)

Using ∂ x ψ( x ) = ∂ x R( x ) + ∂ x L( x ) and the equation of motion (4.10), the new functions are followed by

V (x)[ R( x ) + L( x )] ,2ωV (x)ωL( x ) = +iv∂ x L( x ) +[ R( x ) + L( x )] .
2ωωR( x ) = −iv∂ x R( x ) +(4.31)(4.32)

The functions R( x ) and L( x ) have the meaning of the right- and left-moving components of the wave function, respectively. The terms ∝ V ( x ) in Eqs. (4.31) and (4.32)
result in scattering between the components R( x ) and L( x ). In the weak-disorderregime, only the forward- and back-scattering amplitudes evaluated within the Born
approximation determine the scattering properties of the Josephson-junction chain.
At ω ≫ ω⋆ , since the forward- and backward-scattering amplitudes are associatedwith the harmonics of the effective disorder potential near momenta k = 0 and
k = 2ω/v ≫ 1/R⋆ , respectively, it allows us to expand the effective disorder potential as

V (x)≈ ∆0 ( x ) + ∆( x )e2iωx/v + ∆∗ ( x )e−2iωx/v ,2ω

(4.33)

where the real function ∆0 ( x ) and the complex function ∆( x ) are slowly varying random functions in the scale of v/ω. We find that the slow components of Eqs. (4.31)
and (4.32) yield a Dirac equation,[−iv∂ x τ3 + ∆0 ( x ) + ∆1 ( x )τ1 + ∆2 ( x )τ2 ] Φ( x ) = 0,(4.34)

where Φ( x ) = ( R( x )e−iω (x−d)/v , L( x )eiω (x−d)/v )t , ∆( x ) = ∆1 ( x ) + i∆2 ( x ), and Paulimatrices τ1,2,3 . Using Eq. (4.27), the Born approximation applied to scattering off apotential given by Eq. (4.33) provides the forward- and back-scattering lengths,ℓ0 ( ω ) =ℓπ (ω ),1−ηℓπ (ω ) = R⋆

ω
ω⋆

2
,

(4.35)

respectively. Thus, the correlation between the random offset charge (short-range
correlation) and the static charge density ∝ ∂ x θ̄ (long-range correlation) inducedby the scattering off the potential tends to weaken the forward-scattering rate andintroduces a quantitative difference between ℓ0 (ω ) and ℓπ (ω ) by a factor 1/(1 − η ).

52

Chapter 4. Transmission through a Disordered Josephson-junction Chain

As ℓ0 (ω ), ℓπ (ω ) ≫ R⋆ in the high-frequency regime, the random fields ∆0 ( x ) and∆( x ) in Eq. (4.34) are viewed as Gaussian ones each of which has a length scale ∼ R⋆ .
The Fourier components of the correlation function of V ( x ) are reproduced by the
Fourier components of correlation functions of the random fields:


ω⋆4
x−y
,
⟨∆0 ( x )∆0 (y)⟩ = (1 − η ) 2 δ
ω
R⋆


ω⋆4
x−y
∗
∆
(
x
)
∆
(
y
)⟩
=
δ
.
⟨
ω2
R⋆

(4.36)(4.37)

Both fields ∆0 ( x ) and ∆( x ) induce a random phase between right- and leftmovers. The contribution
from ∆0 ( x ) to the phase acquired over a distance x is
Rxestimated as (2/v) 0 dy ∆0 (y), and its variance grows as 4x/ℓ0 (ω ). As we will see
below, the additional contribution from ∆( x ) is 2x/ℓπ (ω ), and it only quantitatively
modifies the result about the random phase. Phase scrambling occurs in a longer
Josephson-junction chain with d ≫ ℓ0 (ω ), ℓπ (ω ). Using Eq. (4.35), this conditionis equivalent√ to a small frequency condition, ω ≪ ωcr with crossover frequency
ωcr = ω⋆ d/R⋆ . Thus, we can see loss of coherent phases for ω⋆ ≪ ω ≪ ωcr . Reference [98] addressed the role of ωcr in the visibility of oscillations in the reflection
amplitude. In this thesis, we find a similar effect in the frequency dependence of thetransmission.
In addition, the backward-scattering represented by ∆( x ) is responsible for theplasmons localization. The localization physics will be discussed in Section 4.4.

4.3.2

Fokker-Planck formalism

To proceed further, we use the Fokker-Planck formalism that was developed to
predict the statistics of scattering properties of waves subject to Gaussian whitenoise disorder. The previously known results [131, 132, 133, 134, 135] assume a perfect impedance matching. In this thesis, we generalize them to the case of a finite
impedance mismatch.
The wave functions in the waveguides and the Josephson-junction chain are related through








R (0)
1
t(ω )
R(d)
T
= S0
,
= S0
.
(4.38)r (ω )L (0)L(d)
0Here, a phase factor e−iωd/v was absorbed in the transmission coefficient t(ω ) andthe scattering matrix at each edge is


t0 r0
S0 =
(4.39)−r0 t0√√with r0 = R0 and t0 = T0 (see Eq. (4.5)). This allows us to relate the scattering
amplitudes with the Ricatti variable,z( x ) =L( x ) 2iω (x−d)/ve,
R( x )(4.40)

4.3. Weakly localized regime

53

at the edges of the chain (x = 0, d) as follows:r (ω ) =−r0 + z(0)e2iωd/v,
1 − r0 z(0)e2iωd/v

r0 = z ( d ).

(4.41)

From the Dirac equation (4.34), the Ricatti variable is a solution of a first-order nonlinear stochastic differential equation,−iv∂ x z( x ) = 2∆0 ( x )z( x ) + ∆∗ ( x ) + ∆( x )z2 ( x ).

(4.42)

Thus, the reflection coefficient r (ω ) and the transmission T (ω ) = 1 − |r (ω )|2 areexpressed in terms of a solution of Eq. (4.42) with a given boundary condition (4.41).
Since the Ricatti equation (4.42) can be seen as a Wiener process, by taking advantage of the Gaussian correlators (4.36) and (4.37), we derive the Fokker-Planck
equation for the conditional distribution probability P( x, θ1 , θ2 ), where θ1 and θ2 are
the amplitude and the phase of the Ricatti variable such that z = eiθ1 e−θ2 , (see Appendix D for the derivation)−

∂P
2 ∂2 P
1
∂2 
2
=
+
sinh
θ
P
,
2
∂x
ℓ(ω ) ∂θ12ℓπ (ω ) ∂θ22

(4.43)

for 0 < x < d, with the mean-free pathℓ(ω ) =

11+
ℓ0 ( ω ) 2ℓ π ( ω )

 −1

=

2
ℓπ (ω )
3 − 2η

(4.44)

Moreover, the solution of Eq. (4.43) is separable, P( x, θ1 , θ2 ) = P1 ( x, θ1 ) P2 ( x, θ2 ), andeach factor satisfies−∂P1
2 ∂2 P1=,∂xℓ(ω ) ∂θ12

(4.45)

−


∂P2
∂2 
1
2
,
sinh
θ
P
=
2
2
∂x
ℓπ (ω ) ∂θ22

(4.46)

with initial conditions P1 (d, θ1 ) = δ(θ1 ) and P2 (d, θ2 ) = δ(θ2 − θ0 ) with θ0 = − ln r0 .
Note that the statistics of θ2 (4.46) is only sensitive to backward-scattering, i.e., theamplitude of the Ricatti variable is reflected from the localization properties. By
contrast, the statistics of θ1 (4.45), which describes the random phase between rightand left-movers, depends both on forward- and backward-scattering.
Since Eq. (4.45) is a standard diffusion equation, its solution at x = 0 is easily
obtained by


θ12
1
P1 (θ1 ) ≡ P1 ( x = 0, θ1 ) = √
exp −
.
8d/ℓ
8πd/ℓ

(4.47)

Then, the variance of the θ1 , ⟨θ12 ⟩ = 4d/ℓ0 + 2d/ℓπ , includes the contributions from∆0 and ∆, which has the same order of magnitude, as discussed at the end of the54

Chapter 4. Transmission through a Disordered Josephson-junction Chain

F IGURE 4.3: Parameter space of transmission due to impedance mismatch T0 and frequency ω. The transmission ⟨ T ⟩ (4.49) can be expressed in simple forms for three parameter regions; (1) T0 = 1, (2)d ≫ ℓ(ω ), and (3) d ≪ ℓ(ω ) and T0 ≪ 1.

previous subsection. The solution of Eq. (4.46) can also be obtained as
P2 (θ2 ) ≡ P2 ( x = 0, θ2 )=e−d/4ℓπ24 sinh θ2

Z ∞0


dk k tanh

πk
2



2

P− 1 +i k (coth θ2 ) P− 1 +i k (coth θ0 )e−k d/4ℓπ ,
2

2

2

2

(4.48)
where Pν ( x ) is the Legendre function of the first kind (see Appendix E for the derivation). Finally, we obtain the ensemble-averaged transmission as⟨T ⟩ =ZZ

dθ2 P1 (θ1 ) P2 (θ2 ) T (θ1 , θ2 ),(4.49)

1 − e−2θ2
1 + r02 e−2θ2 − 2r0 sin(2ωd/v + θ1 )e−θ2(4.50)dθ1

where
T (θ1 , θ2 ) = T0

is obtained by plugging z(0) = eiθ1 −θ2 into Eq. (4.41) and using T = 1 − |r |2 .
In the rest of this subsection, using Eq. (4.50), we derive simple formulas for thetransmission in three cases (see Fig. ).
(i) perfect impedance matching (T0 = 1)
At T0 = 1, Eq. (4.50) simplifies T (θ1 , θ2 ) = 1 − e−2θ2 , which indicates that the statistics of θ2 fully determines the transmission coefficient. By using properties of the
Legendre function,Pν (1) = 1,

and

Z ∞
1

dx

1
π
P− 1 +i k ( x ) =
,
2
2
1+x
cosh(kπ/2)(4.51)(see Eq. 7.131.1 in Ref. [136]), we obtain the simple form for the transmission:⟨ T (ω )⟩ =

Z ∞
0

dk

πk tanh(πk/2) −(1+k2 )d/4ℓπ (ω )e.
2 cosh(πk/2)(4.52)

4.3. Weakly localized regime

55

F IGURE 4.4: f ( T0 ) = (4T0 /π 2 )K2 (1 − T0 ) (green line) and its asymptotic form (4.57) for T0 ≪ 1 (dashed line).

At d ≪ ℓπ (ω ), ⟨ T ⟩ ≈ 1 as expected for a ballistic junction. On contrary, at d ≫ℓπ (ω ), the ensemble-averaged transmission is
π 5/2⟨ T (ω )⟩ ≈2ℓπ (ω )d3/2

e−d/4ℓπ (ω ) ,(4.53)

which is agreement with Ref. [132]. Note that the frequency dependence of ⟨ T (ω )⟩is smooth (no oscillations) and fully captured by ℓπ (ω ).
(ii) asymptote at d ≫ ℓ(ω ) and arbitrary T0
The sine term regarding θ1 in Eq. (4.50) produces Fabry-Pérot oscillations of thetransmission with period πv/d when the impedance mismatch is finite K ̸= K0 .
At d ≫ ℓ(ω ), the dispersion of θ1 given by Eq. (4.47) washes out the oscillations on
average. Thus, in this regime, θ1 can be taken as uniformly distributed, and Eq. (4.49)
simplifies to⟨T ⟩ =Zdθ2 P2 (θ2 ) T (θ2 ),T ( θ2 ) =T0 (1 − e−2θ2 ).
1 − R0 e−2θ2(4.54)

Note that, at T0 = 1, it reproduces Eq. (4.52). Let us calculate the transmission for
an arbitrary value of T0 using Eqs. (4.54) and (4.48). For d ≫ ℓπ (ω ), the k-integral in
Eq. (4.48) is dominated by the region k ≪ 1, so that we can replace
√


2 2
x−1
P− 1 +i k ( x ) ≈ P− 1 ( x ) = √K,(4.55)
2
2
2
x+1
π x+1
where K(m) is the complete elliptic integral. After performing the k-integral, we
obtain
π 5/2 ℓπ (ω ) 3/2 −d/4ℓπ (ω )ef ( T0 ),(4.56)⟨ T (ω )⟩ =2dwith f ( T0 ) = (4T0 /π 2 )K2 (1 − T0 ) (see Fig. 4.4). In particular, at the perfect impedance
matching (T0 = 1), f ( T0 = 1) = 1, which reproduces Eq. (4.53), while, at large

56

Chapter 4. Transmission through a Disordered Josephson-junction Chain

impedance mismatches (T0 ≪ 1),
T0
f ( T0 ≪ 1) ≈ 2 ln2
π



16
T0


.

(4.57)

Note that, as in Eq. (4.53), the frequency dependence of the pre-exponential factor in
Eq. (4.56) comes only via ℓπ (ω ), i.e., the transmission is smooth (no oscillations) as
a function of frequency in this regime.
(iii) asymptote at d ≪ ℓ(ω ) and T0 ≪ 1
At finite impedance mismatch and d ≪ ℓ(ω ), Fabry-Pérot oscillations exist. For
large mismatch (T0 ≪ 1), we can use the initial value for the θ2 -distribution, P2 (θ2 ) ≈δ(θ2 − T0 /2), and then obtain⟨ T (ω )⟩ = ∑
n

Z

dθ1

T02
T02 + [2(ω − ωn )d/v + θ1 ]

2

P1 (θ1 )(4.58)

with P1 (θ1 ) of Eq. (4.47). Here, ωn = nπv/d are the frequencies at which thetransmission has local maxima. Equation (4.58) describes how, as the frequency
decreases, the oscillations’ Lorentzian lineshapes with half-width T0 v/2d, which isdeterminedp solely by the impedance mismatch, evolve into Gaussian ones with halfwidth v 2 ln 2/dℓ(ω ), which is determined by the randomness of the Josephsonjunction chain. The crossover frequency is given by ωcr /T0 . Remarkably, in thefrequency range ωcr ≪ ω ≪ ωcr /T0 the width of the oscillations is sensitive to η(see Eq. (4.44)), i.e., the collective pinning.

4.3.3

Numerical results

The brute-force numerical calculations for the transmission are compared with thepredictions from the Fokker-Planck formalism in Fig. 4.5. In that figure, Eq. (4.52) isused at perfect impedance matching (K/K0 = 1), while we useZ2( ω − ω n ) d2θ2
P2 (θ2 ),(4.59)dθ2⟨ T (ω )⟩ = 2πT0 ∑ P1v
2θ2 + T0nfor ω⋆ ≪ ω ≪ ωcr /T0 , at T0 ≪ 1 (see Figs. 4.5 (a) and (b), respectively). They are ingood agreement with each other for the high-frequency region.
The localization properties of a disordered Josephson-junction chain are frequently
characterized by its Lyapunov exponent, γ(ω ) = −(2d)−1 ln T (ω ) for a long chain.
In contrast with the transmission, the Lyapunov exponent has the self-averaging
property [131]. Here, for a random variable ⃗X = ( X1 , X2 , . . . , X N ), the self-averaging
property means
1 N∑ Xi ,
N →∞ Ni =1

X j = lim

(4.60)

for arbitrary j, where ⟨·⟩ denotes averaging over all disorder configurations. In
Fig. 4.6 (a), we show the frequency dependence of the averaged Lyapunov exponent, ⟨γ(ω )⟩ for different chain lengths at the impedance matching (K/K0 = 1).
Its inverse defines the localization length, Lloc (ω ) = 1/ ⟨γ(ω )⟩. We checked numerically that a celebrated Thouless relation [137], Lloc (ω ) = 2ℓπ (ω ), works well

4.3. Weakly localized regime

57

F IGURE 4.5: Frequency dependence of ⟨ T (ω )⟩ for (a) K/K0 = 1.0 and(b) K/K0 = 0.01 and d/R⋆ = 20. The green lines are the numericalresults, the dashed line is Eq. (4.52) in panel (a) and Eq. (4.59) in panel(b).

at large frequencies, ω ≫ ω⋆ . The self-averaging nature of γ(ω ) is associatedwith the log-normal character of the distribution of the transmission. To discuss
its statistics, the frequency dependence of the ratio between mean and variance,
Σ(ω ) = d ⟨⟨γ2 (ω )⟩⟩ / ⟨γ(ω )⟩, is plotted in Fig. 4.6 (b). According to the FokkerPlanck formalism [131], that ratio should be 1 at ω⋆ ≪ ω ≪ ωcr . Despite the tendency as d increases, the agreement is not perfect. (We attribute it to insufficient
length.) There was a renewed interest in Σ(ω ) [138, 139, 140, 141] to test the singleparameter scaling hypothesis of the theory of localization [26].
The presence of the factor 1/4 in the argument of the exponent in Eq. (4.53) or(4.56) reflects that the averaged transmission is dominated by rare, optimal disorder configurations – not simply captured by the log-normal distribution (tails), andwhich produce a resonant transmission, T ∼ 1 [142, 143, 144]. In other words,⟨ T ⟩ ≫ Ttyp where Ttyp = exp(⟨ln T ⟩) is the typical transmission. To check this effect,
in Fig. 4.7, we plot the frequency dependence of their ratio ln ⟨ T (ω )⟩ / ⟨ln T (ω )⟩ forvarious lengths at K = K0 . (We attribute to insufficient length the deviation of thatratio from 1/4 at ω ≫ ω⋆ .)Now, we compare the effect of disorder has on the waves’ transmission with thaton their reflection. By using the reflection coefficient (4.41) and assuming the flat
distribution of θ1 mentioned above Eq. (4.54), we find that, for ω⋆ ≪ ω ≪ ωcr , the58

Chapter 4. Transmission through a Disordered Josephson-junction Chain

F IGURE 4.6: (a) The average of the Lyapunov exponent and (b) its
variance as a function of frequency for different lengths d/R⋆ =20, 40, 80, and 160 at K/K0 = 1.0. Arrows indicate the crossover
frequency ωcr for each length. The inset is an enlarged view of the(a) low and (b) intermediate frequency regions. The dotted lines inpanels (a) and (b) show the result of the Gaussian white-noise potential at d → ∞ for comparisons. As d increases, the Lyapunov
exponent approaches a constant value. At ω ≫ ω⋆ , ⟨γ(ω )⟩ inthe two models coincide, in agreement with the Thouless relation,⟨γ(ω )⟩ = [2ℓπ (ω )]−1 . At ω ≪ ω⋆ , the average exponent and its variance saturate to different values in the two models.

real part of the reflection coefficient averages to a frequency-independent value,r ′ ( ω ) = −r0 .

(4.61)

This indicates that the averaged reflection coefficient is insensitive to disorder, incontrast to the averaged transmission, which is manifestly dependent on ℓπ (ω )(see Eqs. (4.53) and (4.56)). At higher frequency, ωcr ≪ ω ≪ ωcr /T0 , and large
impedance mismatch T0 ≪ 1, we use the same methods as the third case for thederivation of the simple transmission asymptote at d ≪ ℓ(ω ) and T0 ≪ 1 to find


T0
2ωn d
′
r ( ω ) = −1 +P1.
(4.62)
2 ∑vnThis form expresses the inhomogeneous broadening of plasmon standing waves

4.4. Strongly localized regime

59

F IGURE 4.7: The frequency dependence of the ratio of ln ⟨ T (ω )⟩and ⟨ln T (ω )⟩ for different lengths d/R⋆ = 20, 40, 80, and 160 at
K/K0 = 1.0. The ratio takes a frequency-independent value at low
frequency, ω ≪ ω⋆ . As the frequency increases, it decreases whileremaining above the value 1/4 (dashed line) predicted in the frequency range ω⋆ ≪ ω ≪ ωcr , then it increases to the value 1at larger frequencies. Arrows indicate the crossover frequency ωcr
for each length. A scaling analysis with the length gives the resultln ⟨ T (ω )⟩ / ⟨ln T (ω )⟩ ≈ 0.90 in the low-frequency limit ω/ω⋆ → 0
and d/R⋆ → ∞.

confined in the chain, in correspondence with Ref. [98]. Finally, the standard FabryPérot formula for the reflection is recovered at ω ≫ ωcr /T0 when the levels’ broadening is dominated by the radiation to waveguides, rather than inhomogeneous
broadening.

4.4

Strongly localized regime

At low frequencies ω ≪ ω⋆ , plasmons are localized over a typical length R⋆ , in thefluctuations of the effective disorder potential (4.16). Those fluctuations are nonGaussian, and we have not been able to make an analytical theory of the scattering
properties in that regime. However, we could gather several pieces of informationfrom our numerical calculations. We compare them with known analytical results
for the one-dimensional theory of localization in a Gaussian white-noise potential
for which the additional function w( x/R⋆ ) in Eq. (4.27) is replaced by zero. While
the scattering properties of the two models coincide in the weakly localized regime(ω ≫ ω⋆ ), important deviations appear in the strongly localized regime (ω ≪ ω⋆ ).
Below, we show the numerical results for the Lyapunov exponent, the reflection
phase, and the transmission.

4.4.1

Lyapunov exponent

First, we discuss the average of the Lyapunov exponent and its variance. The Lyapunov exponent remains a self-averaging quantity, which saturates to a frequencyindependent value, ⟨γ⟩ R⋆ ≈ 1.36 at low frequencies, as shown in Fig. 4.6 (a). Thatvalue
is larger than the one in the Gaussian white-noise model, ⟨γ⟩ R⋆ = 31/3 ×√
π/Γ(1/6) ≈ 0.46 [145].
In agreement with the central limit theorem, we checked that the variance of
the Lyapunov exponent scales inversely with d. The frequency dependence of the60

Chapter 4. Transmission through a Disordered Josephson-junction Chain

ratio between the variance and average Σ(ω ) = d ⟨⟨γ2 (ω )⟩⟩ / ⟨γ(ω )⟩ is plotted inFig. 4.6 (b). At low frequencies, numerical value marks Σ ≈ 0.218, which is significantly smaller than the one in the Gaussian white-noise model at vanishing frequency, Σ ≈ 1.1 [139]. Thus, a white-Gaussian disorder does not reproduce the rightvalue of Σ, i.e., it emphasizes the role of non-Gaussian correlations in the effective
potential created by the collective pinning effect.
The presented values of ⟨γ⟩ and Σ were obtained by brute-force numerical calculations of the dynamical plasmon propagation problem. Now, we show that thesevalues are reproduced by solving the plasmon transmission problem for the static
correlated random potential (4.16) upon averaging over its realizations. According
to problem §25.5 in Ref. [146], the low-frequency transmission at perfect impedance
matching (K = K0 ) for the wave equation (4.10) is given byT ( ω → 0) ≈4ω 2,
v2 [ψ0′ (d)]2(4.63)

where ψ0 ( x ) is the solution of Eq. (4.10) at ω = 0 satisfying the boundary conditions
ψ0 (0) = 1 and ψ0′ (0) = 0. Equation (4.63) works at the sufficiently small frequency
for typical disorder configurations such that ψ0′ (d) ̸= 0. Note that ψ0′ (d) = 0 would
signal a zero-energy bound state, and resonant transmission, T ≈ 1, is not captured
by Eq. (4.63). An explicit solution of Eq. (4.10) reads


Z x
−1
ψ0 ( x ) = exp ξdy Z (y) .
(4.64)
0Here, Z ( x ) = R⋆ ψ′ ( x )/ψ( x ) is another Ricatti variable from z defined in Eq. (4.40),
which solves an equation derived from Eq. (4.10) at ω = 0,R⋆ ∂ x Z ( x ) = − Z2 ( x ) +

V (x)
ω⋆2

with

Z (0) = 0.

(4.65)

Then, the transmission (4.63) can be expressed in terms of the Ricatti variable asT ( ω → 0) ≈ 4

ω
ω⋆

2



Z d
1
2
exp −
dy Z (y) .
Z2 (d)
R⋆ 0(4.66)

It was argued in Refs. [101, 102] that the solution of Eq. (4.65) with the correlated
random potential (4.16) has a positive mean value, ⟨ Z ( x )⟩ > 0. Thus, solving
Eq. (4.65) provides another way to find the Lyapunov exponent and its variancefrom Eq. (4.66), which only depends on V ( x ), as mentioned above. This approachyields ⟨γ⟩ R⋆ = 1.37 and Σ = 0.217, which are very close agreement with the resultsof the brute-force numerical evaluation of scattering amplitudes at a low, but finite
frequency, ⟨γ⟩ R⋆ = 1.36 and Σ = 0.218, respectively.

4.4.2

Reflection coefficient

When we enter the low-frequency regime, the transmission is exponentially suppressed and waves are almost perfectly reflected, r (ω ) ≈ 1. In the limit of perfect impedance matching (K = K0 ), we characterize the reflection coefficients withthe distribution of the reflection phase [147, 148] (see Fig. 4.8). While the distribution is uniform at ω ≫ ω⋆ , a single-peak structure near θ1 = ±π develops at alower frequency. At ω ≪ ω⋆ , the peak in the distribution obtained from the bruteforce numerics takes a universal frequency dependence both in our model and in4.4. Strongly localized regime

61

F IGURE 4.8: Probability distributions of the reflection phase θ1 at
K/K0 = 1.0, d/R⋆ = 20, ω/ω⋆ = 0.157 (green), 1.16 (red), and 8.58
(blue). The inset shows the universal frequency dependence of the
distribution found numerically in the vicinity of phase θ1 = −π either by the brute force numerical calculations at ω/ω⋆ = 0.1 (blue)and 0.25 (red), or from the solution of the Ricatti Eq. (4.65) (dashedline). The dotted line represents the analytical formula derived inRef. [147] for a Gaussian white-noise disorder.

the Gaussian white-noise model, as shown in the inset of Fig. 4.8, with importantdifferences between the two models. Furthermore, the universal dependence in ourmodel agrees with the one obtained by solving numerically Eq. (4.65) in the interval
0 < x < d with a given boundary condition at x = d for various disorder configurations. Then, we identify Z (0) = −2(ω/ω⋆ )/(θ1 + π ), which is the consequence ofthe boundary conditions (4.11) and (4.13) with K = K0 , ω/ω⋆ → 0, and r = eiθ1 . The
results do not depend on the boundary condition for Z (d) if d ≫ R⋆ , e.g., d/R⋆ = 20
in the inset of Fig. 4.8.
The distribution P1 (θ1 ) of the reflection phase θ1 can be seen as a measure of
the modes localized nearby an end of the pinned region and their frequencies lie
either above (θ1 > −π) or below (θ1 < −π) the one of the incoming waves. The
distribution P1 (θ1 ) appears to reach zero at θ1 = −π, unlike the white-noise Gaussian cases (see Appendix F), indicating the scarcity of low-frequency resonances.
This agrees with the strong suppression of the low-frequency modes’ density ofstates [100, 101, 102] in our pinned model, in contrast with the Gaussian models.
The distribution P1 (θ1 ), together with Eqs. (4.11) and (4.13), allows us to find thestatistics of the reflection phase at any impedance mismatch between the waveguideand half-infinite chain, i.e., d/R⋆ → ∞. In particular, the real part of the averaged
reflection coefficient is
r

′

+1 =

Z

dθ1 P1 (θ1 )

2.
1 + (K0 /K )2 tan2 (θ1 /2)(4.67)At ω ≪ ω⋆ , the frequency scaling of the peak in P1 (θ1 ) near θ1 = −π, which isdefined byp( φ = (θ1 + π )ω⋆ /ω ) = (ω/ω⋆ ) P1 ((θ1 + π )ω⋆ /ω ),(4.68)

62

Chapter 4. Transmission through a Disordered Josephson-junction Chain

F IGURE 4.9: The frequency dependence of the local density of states
at K/K0 = 0.01. The dashed line represents (ω/ω⋆ )4 . The pre-factor
is also agreement with Ref. [98]. The dotted line is the result of theGaussian white-noise model, (see Appendix F). Arrows indicate thecrossover frequency ωcr for each length, d/R⋆ = 20, 40, 80, and 160.

is independent of the frequency as demonstrated in the inset of Fig. 4.8. As a result,
we obtain 2  2
ωK′,(4.69)
r +1 = A
K0
ω⋆R
where the prefactor A can be evaluated by A = dφ p( φ) φ2 ≈ 2.4. At any K ≪ K0 ,another important contribution to Eq. (4.67), which scales linearly with K/K0 , comesfrom phase near θ1 = 0, yieldingr ′ + 1 = 4π

K
P1 (θ1 = 0).
K0(4.70)

Our numerical result for ⟨r ′ ⟩ at K ≪ K0 , shown in Fig. 4.9, confirms the result ofRef. [98] and Eq. (4.70) with P1 (θ1 = 0) ∝ ω 4 ; in this regime [⟨r ′ (ω )⟩ + 1]/(K/K0 )can be interpreted as the local density of plasmon modes. However, we were not able
to correlate this result with the result of a direct evaluation of P1 (θ1 = 0) illustrated
by Fig. 4.8, presumably due to an insufficient number of disorder configurations
because configuration yielding θ1 = 0 is rare. Furthermore, the deviation of thenumerical results from the ω 4 -scaling seen in Fig. 4.9 at the lowest frequencies issuggestive of a crossover to the ω 2 -scaling of Eq. (4.69); our numerical results did
not allow us to check this scaling quantitatively. The different (K/K0 )-dependence ofEq. (4.70), compared with Eq. (4.69), ensures that it dominates over a wide frequency
range. The scarcity of low-frequency modes reflects in a much stronger suppressionof the local density of states ∝ ω 4 at ω ≪ ω⋆ in our pinned model, in contrast
with the results of the white-noise Gaussian models where [⟨r ′ (ω )⟩ + 1]/(K/K0 ) ∝1.29ω/ω⋆ (see Appendix F).

4.4.3

Average transmission

At any frequency in the range, the averaged transmission is determined by the optimal configurations. Furthermore, the ratio ln ⟨ T ⟩ / ⟨ln T ⟩ remains smaller than 1, in
accordance with the inequality between the arithmetic and geometric means. In Section 4.3.3, we found ln ⟨ T ⟩ / ⟨ln T ⟩ = 1/4 in the diffusive regime (ω ≫ ω⋆ ). In the4.5. Short summary of this chapter

63

strongly localized regime, this ratio increases towards the value ln ⟨ T ⟩ / ⟨ln T ⟩ ≈ 0.9(see Fig. 4.7). Its closeness to 1 indicates a small difference between the typical andoptimal disorder configurations. This is further confirmed by the fact that ensembleaveraging of Eq. (4.63) yields a very close result for that ratio, ln ⟨ T ⟩ / ⟨ln T ⟩ ≈ 0.93.
As disorder configurations with nearly-perfect transmission [131] are not taken into
account in Eq. (4.63), they should have a negligible weight among the optimal disorder configurations.
Note that the frequency scaling ⟨ T (ω → 0)⟩ ∝ ω 2 in Eq. (4.66) confirms that, for
the waves at the bottom of the plasmon spectrum, i.e., in the infinite wavelength
limit, the disorder potential acts like a localized one for each realization of the disorder. Our numerical results do not have enough accuracy to check this asymptote
quantitatively and to establish its range of validity.
For comparison, we expect disorder configurations with nearly-perfect transmission to play a major role at arbitrarily low frequency in the Gaussian models. This
may result in a different frequency dependence of ⟨ T (ω )⟩ and ln ⟨ T (ω )⟩ / ⟨ln T (ω )⟩at ω ≪ ω⋆ . As far as we know, this question has not been studied systematically.
Our numerical results indicate ln ⟨ T (ω )⟩ / ⟨ln T (ω )⟩ ≈ 0.72 at ω ≪ ω⋆ in the Gaussian white-noise model. We attribute the difference of the result from our pinnedmodel result, ln ⟨ T (ω )⟩ / ⟨ln T (ω )⟩ ≈ 0.9, to the scarcity of low-frequency quasilocalized modes in the latter.

4.5

Short summary of this chapter

We investigated the transmission though the disordered Josephson-junction chain.
In this thesis, we focused on the classical limit (K ≪ 1), in which there is a competition between the elasticity and the pinning effect due to the disorder. By performing the numerical calculation in that regime, we obtained the frequency dependence of the transmission. For the high-frequency regime, the pinning effect is
expected to be neglected naively because the wavelength is shorter than the Larkinlength. Surprisingly, we found that the pinning-induced disorder still affects theforward-scattering length. On the other hand, for the low-frequency regime, we
could gather the numerical pieces of the important quantities, such as the Lyapunov
exponent, the reflection phase, the local density of states, and the ratio between the
Lyapunov exponent and the logarithmic transmission and clarified the universalityof the scattering property. Moreover, by comparing our model with the well-known
model with the Gaussian white-noise disorder, we revealed that the correlations of
the pinning-induced disorder play a crucial role in reproducing our results. Finally,
we would like to stress that recent experimental technology enables us to fabricate
the Josephson-junction chain with a small but finite K [88, 89]. Therefore, the theoretical prediction in this chapter can indeed be checked experimentally in the near
future.

65

Chapter 5

Heat Transport through the
Quantum Rabi System
In this chapter, we investigate heat transport through the quantum Rabi model [41].
First, we briefly review quantum heat transport in mesoscopic devices in Section 5.1.
Then, we introduce the quantum Rabi model and formulate heat transport throughit in Section 5.2. Using the noninteracting-blip approximation (NIBA), which is explained in Section 5.3, we present numerical results for the temperature dependenceof the thermal conductance in Section 5.4.

5.1

Quantum heat transport

In this section, we introduce recent theoretical and experimental advances on quantum heat transport through a small object for which quantum mechanical effects
become important at sufficiently low temperatures. In a pioneering theoretical work
by John Pendry in 1983, he predicted that heat conducted by a single transport channel has a universal maximum at sufficiently low temperatures, which is known as aquantized thermal conductance [149]. This quantized thermal conductance can beobserved for any type of carrier, e.g., phonons and photons. Based on the Landauer
theory [150, 151], which has been developed for electric transport, the heat current
for bosons between two heat baths, which are in thermal equilibrium, can be given
by
1
J=∑
n h

Z ∞
0

d(h̄ω ) h̄ω Tn (ω ) [nL (ω ) − nR (ω )] ,(5.1)

where ni (ω ) is the Bose-Einstein distribution of the heat bath i = L, R, and Tn (ω ) is
the transmission probability of the transport channel n. Here, assuming the perfecttransmission, Tn (ϵ) = 1, and a small temperature gradient, δT ≡ TL − TR ≪ T, we
obtain the linear thermal conductance asJ= N GQ ,δT →0 δTG ≡ lim(5.2)

where N is the number of transport channels and GQ is the thermal conductancequantum
GQ =π 2 k2B
T.
3h

(5.3)

In 2000, it has been observed experimentally for the first time that phonons in nanobridges
conduct heat with the linear thermal conductance GQ per a transport channel [152].

66

Chapter 5. Heat Transport through the Quantum Rabi System

F IGURE 5.1: View for the quantized thermal conductance in aphononic system. The inset shows the suspended device, in which a
4 × 4 µm2 phonon cavity (the bright area in the center) patterned fromthe membrane is connected to four thin-film Nb bridges, phonon
waveguides whose diameter at the narrowest point is less than
200 nm. In the dark regions, the membrane has been completely removed. The main panel shows the temperature dependence of thethermal conductance normalized by 16g0 (16GQ in the main text).
These figures are taken from Ref. [152].

F IGURE 5.2: Schematic of heat transport through a two-level system
between two bosonic heat baths. For TL > TR , the heat current flows
from the left heat bath to the right heat bath.

Fig. 5.1 presents the measured thermal conductance, renormalized by 16GQ , throughnanobridgeds as a function of temperature. At high temperatures, T ≳ 0.8 K, thethermal conductance shows the usual T 3 -dependence due to the phonon scatterings.
Below this temperature, T ≲ 0.8 K, the thermal conductance levels off a value near
the quantized thermal conductance, 16GQ . In Ref. [152], it was discussed that coefficient 16 consists of 4 from four independent bridges and 4 from four acoustic vibration modes in each bridge. Later, the quantized thermal conductance has also seen
in other systems; the superconducting circuit with microwave photons [153, 154].
As described in Section 2.8, the superconducting circuit is a convenient platform
for the study of quantum heat transport. One of the most fundamental systems is asuperconducting qubit embedded in two thermal photon baths fabricated in superconducting circuits, as shown in Fig 5.2. As described in Section 2.5, quantum heattransport for this system can be described by the spin-boson model. The theoretical
study by the author of this thesis for the spin-boson model [38] has systematicallyclarified the transport processes for all types of heat baths (the Ohmic bath s = 1,

5.2. Formalism for heat transport

67

F IGURE 5.3: Three transport processes of heat transport through a
two-level system. (a) At weak couplings and temperatures comparable to the renormalized tunneling splitting energy, one photon absorption (a-i) and emission (a-ii) processes are relevant; sequential
tunneling process. (b) At low temperatures, heat is carried by the virtual excitation processes; the co-tunneling process. (c) At high temperatures or strong couplings, heat transport is caused by the stochastic transition between two degenerate states localized in each well;incoherent tunneling process.

the sub-Ohmic bath s < 1, and the super-Ohmic bath s > 1 categorized by the exponent s of the spectral function; see Eq. (2.39)) by using the quantum Monte Carlosimulations. Depending on the exponent of the spectral function s, the temperature T = TL + 0 = TR − 0, and the coupling strength α = αL = αR , the relevanttransport process changes (see Fig. 5.3). At weak couplings, α ≪ 1, and the in˜ where ∆˜ is a renormalized tunneling splitting
termediate temperature, kB T ∼ h̄∆,frequency, one photon absorption and emission processes cause dominantly heattransport (Fig. 5.3 (a); sequential tunneling process). As the temperature decreases,
˜ one photon processes, i.e., the sequential tunneling process, are supkB T ≪ h̄∆,pressed and then a virtual excitation associated with higher-order processes is relevant for heat transport (Fig. 5.3 (b); co-tunneling process). This transport process is
associated with the Kondo effect, as discussed in Section 2.5. At high temperatures,
˜ or strong coupling, α > 1, the quantum coherence of the two-level system
kB T ≫ h̄∆,is broken and then heat transport is induced by a stochastic transition between two
degenerate states localized in each well (Fig. 5.3 (c); incoherent tunneling process).
Thus, heat transport through a two-level system provides a concrete theoretical basis for describing quantum heat transport. However, it is too simple for the
discussion of sophisticated control of quantum heat devices because the number of
independent model parameters is not large. We also note that recent experimentson heat transport through an assembly consisting of a superconducting qubit capacitively embedded between two superconducting resonators have indicated complex
functions as a quantum heat device [104, 106]. From the next section, we consider atwo-level system embedded between harmonic resonators, which is one of the more
ubiquitous devices in the current quantum computing toolbox, instead of a two-levelsystem for studying quantum heat transport.

5.2

Formalism for heat transport

In this section, we formulate heat transport through the quantum Rabi model inthe ultra-strong coupling regime. By using the exact mapping to the spin-boson

68

Chapter 5. Heat Transport through the Quantum Rabi Systemmodel with structured heat baths, we derive the linear thermal conductance and its
asymptotic form for the weak dissipation.

5.2.1

Dissipative quantum Rabi model and spin-boson model

To treat heat transport through a two-level system embedded between harmonicresonators, we consider the dissipative quantum Rabi model (see Fig. 5.4), whose
Hamiltonian consists of three parts:

H = HR + ∑(HB,r + HI,r ).

(5.4)

r

Here, HR represents the quantum Rabi model describing the two-level system embedded between harmonic resonators (indicated by the central part of Fig. 5.4 (a)enclosed by the dashed line), HB,r the heat bath r (= L, R), and HI,r the coupling
between the harmonic resonator r and the heat bath r. Although the Hamiltonianof the quantum Rabi model, HR , has already been given in Section 2.4, we briefly
repeat it here. The quantum Rabi model describes a two-level system coupled toharmonic resonators and its Hamiltonian is given by

HR = −h̄∆
σx + ∑ h̄Ωr ar† ar + ∑ h̄gr σz ar + ar† ,
2
r
r

(5.5)

where ar ( ar† ) is an annihilation (a creation) operator of the harmonic resonator r(= L, R), σi={ x,y,z } are the Pauli matrices, and ∆ is a tunneling splitting frequencyof the two-level system. Ωr is a natural frequency of the harmonic resonator r, andthe frequency gr represents the coupling strength between the two-level system andthe harmonic resonator r. Here, we have dropped a detunning term of the two-levelsystem for simplicity.
The two heat baths coupled to harmonic resonators are modeled as a collectionof harmonic oscillators. The Hamiltonian of the heat bath r (= L, R) is given as†
HB,r = ∑ h̄ωr,k br,k
br,k

(5.6)

k

† ) of the k-th bosonic mode with thewith an annihilation (a creation) operator br,k (br,kfrequency ωr,k in the heat bath r. The heat baths and the quantum Rabi model arecoupled to each other via the harmonic resonators:†HI,r = ∑ h̄λr,k ar + ar† br,k + br,k
,
(5.7)
k

where the frequency λr,k represents the coupling strength between the harmonicresonator r and the k-th bosonic mode in the heat bath r. As discussed in Section 2.5,
each of the heat baths is completely characterized by a spectral density
Ir (ω ) ≡ ∑ λ2r,k δ (ω − ωr,k ) .

(5.8)

k

In this thesis, we consider the Ohmic heat baths (see also Section 2.5):
Ir (ω ) = 2ηr ω.

(5.9)

69

•••

•••

•••

•••

5.2. Formalism for heat transport

F IGURE 5.4: Schematic illustrations of the model considered in thischapter. A two-level system with the tunneling splitting frequency∆ is embedded between two harmonic resonators with natural frequencies ΩL and ΩR which contacts to an Ohmic bath L and R, respectively. The finite temperature difference (TL > TR ) induces heattransport from the left (hotter) heat bath to the right (cooler) heat bath.
Two ways of viewing the model; (a) the quntum Rabi model coupled
to two Ohmic heat baths which is called the dissipative quantum
Rabi model and (b) the two-level system coupled to two structured
heat baths composed of the harmonic resonator and the Ohmic heatbath, which is equivalent to the spin-boson model. In this thesis, weconsider the symmetric case, ΩL = ΩR = Ω, gL = gR = g, andηL = ηR = η.

Here, ηr represents a dimensionless coupling strength between the heat bath r andthe harmonic resonator r.
Now, let us introduce the exact mapping of the model considered here by changing the way of viewing the model; from the quantum Rabi model coupled to Ohmicheat baths, as shown in Fig. 5.4 (a), to the two-level system coupled to structured heatbaths, as shown in Fig. 5.4 (b). Then, the dissipative quantum Rabi model (5.4) canbe mapped to the spin-boson model (see the detail derivation in Appendix H) [155,156, 83]:

Hsb = HTLS + HB + HI ,h̄∆
HTLS = − σx ,2†
HB = ∑ HB,r = ∑ h̄Ξr,k Br,k
Br,k ,
r(5.11)(5.12)

k,r

HI = ∑ HI,r = −r(5.10)



σz
†
h̄Λ
B
+
B
r,k
r,k
r,k ,
2 ∑
k,r

(5.13)† ) is an annihilation (a creation) operator for the k-th bosonic mode
where Br,k (Br,kwith the frequency Ξr,k in a new heat bath r, composed of the bare heat bath and70

Chapter 5. Heat Transport through the Quantum Rabi System

the harmonic resonator, and Λr,k represents the coupling strength between the twolevel system and the k-th bosonic mode in the new heat bath r. The spectral density
characterizing the effective heat baths is expressed byIeff,r (ω ) ≡ ∑ Λ2r,k δ (ω − Ξr,k ) = 2αr ωkΩ4r(Ω2r − ω 2 )2 + (2Γr ω )2,

(5.14)

which is called as a structured spectral density. The spectral density Ieff,r (ω ) has apeak at the natural frequency of the harmonic resonator, Ωr , with the width, Γr =πηr Ωr , and its height is characterized by the dimensionless coupling strength αr =ηr (4gr /Ωr )2 .
At the end of this subsection, we comment on the region of ∆ ≪ Ωr . In this region, since the lowest two states of the quantum Rabi model are separated well fromthe other, the dissipative quantum Rabi model can be regarded as the Ohmic spinboson model. This correspondence can also be seen from the spectral density (5.14),
i.e., the structured part of the spectral density can be neglected and then it becomes
Ohmic, Ieff,r (ω ) ∼ 2αr ω. Therefore, heat transport in this regime can be understoodby simply referring to the results on heat transport in the Ohmic spin-boson model,
as discussed in Section 5.1. This parameter regime is not preferable to control heattransport because this effective two-level system shows a rather simple behavior forheat transport with respect to the parameters. In fact, the degrees of freedom regarding the harmonic resonators, i.e., Ωr , gr , and η, are absorbed into a renormalized tunneling splitting frequency ∆∗ . In this thesis, we focus on the opposite region, ∆ ≳ Ω,
in which the present model is expected to exhibit non-trivial transport properties.

5.2.2

Linear thermal conductance

The heat current mediated via the quantum Rabi model from the heat bath r is defined as
Jr ≡ −

dHB,r.
dt(5.15)

Using the picture of the spin-boson model with the structured heat bath (5.10), thesteady state heat current is formulated by the Keldysh formalism as [157, 158, 107]

⟨ Jr ⟩ =

h̄2
2

Z ∞
0

 h

i
i
dω ωIeff,r (ω ) Im χR (ω ) nr (ω ) − χ< (ω ) ,2(5.16)

where nr (ω ) = 1/(e βr ω − 1) is the Bose-Einstein distribution for the structured heatbath r, β r = 1/kB Tr is the inverse temperature, kB is the Boltzmann constant, andTr is the temperature of the heat bath r (for a detailed derivation, see Appendix G).
The retarded and lesser components of the spin correlation function of the two-levelsystem are defined asiχR (t) ≡ − θ (t) ⟨[σz (t), σz (0)]⟩ ,h̄iχ< (t) ≡ − ⟨σz (t)σz (0)⟩ ,h̄(5.17)(5.18)

respectively. For simplicity, we consider the symmetric case, ΩL = ΩR ≡ Ω, gL =gR ≡ g, and ηL = ηR ≡ η in this thesis. Using the energy conservation law, ⟨ JL ⟩ +5.2. Formalism for heat transport

71⟨ JR ⟩ = 0, we finally obtain the heat current asαγh̄2⟨ JL ⟩ =
8

Z ∞
0

h
i
dω ω Ĩeff (ω )Im χR (ω ) [nL (ω ) − nR (ω )](5.19)

with α = αL + αR , γ = 4αL αR /α2 , and Ĩeff (ω ) = αr−1 Ieff,r (ω ).
Using the expression of the heat current, we obtain the linear thermal conductance asκ≡⟨ JL ⟩
TL ,TR → T TL − TR
lim

Z ∞

αγh̄kB=80(5.20)hRdω Ĩeff (ω )Im χ (ω )i2βh̄ω/2.
sinh( βh̄ω/2)(5.21)For the convenience of analysis, we introduce a symmetrized correlation function,
S(t) =1⟨σz (t)σz (0) + σz (0)σz (t)⟩ .
2(5.22)

From the fluctuation-dissipation theorem [17], the imaginary part of the retarded
component of the spin correlation function is related to the symmetrized correlation
function as


i
h
βh̄ω
S(ω ) = h̄ coth(5.23)Im χR (ω ) .
2Then, the linear thermal conductance is rewritten as
κ=

αγkB
16

Z ∞
0

dω Ĩeff (ω )S(ω )( βh̄ω )2.
sinh( βh̄ω )(5.24)

Due to the factor ( βh̄ω )2 / sinh( βh̄ω ) in the integral, the behavior of the linear thermal conductance is mainly determined by the symmetrized correlation function in
the region of 0 < ω ≲ kB T/h̄.

5.2.3

Weak-dissipation regime

Here, we consider the behavior of the linear thermal conductance for the weak dissipation, i.e., η ≪ 1. First, we look at the non-dissipative quantum Rabi model, η = 0,
for which the eigenstates and their energy are denoted as |n⟩ and h̄ωn , respectively:

HR |n⟩ = h̄ωn |n⟩ ,(n = 0, 1, 2, . . . ).

(5.25)

Then, the symmetrized correlation function S(ω ) is described asS(ω ) =

π
∑ e−βh̄ωm |⟨n|σz |m⟩|2 [δ(ω − ωnm ) + δ(ω + ωnm )] ,ZR n,m

(5.26)

where ZR is the partition function of the quantum Rabi model and ωnm = ωn − ωmis the transition frequency between the m-th and n-th states. When switching slowly
on the dissipation η, the peaks in S(ω ) have finite widths from the delta-function
ones. In general, their positions and widths vary as the dissipation increases. However, in the weak-dissipation regime (η ≪ 1) these effects are weak enough that the

72

Chapter 5. Heat Transport through the Quantum Rabi Systemexpression of S(ω ) in the non-dissipative quantum Rabi model (5.26) can be usedapproximately.
At low temperatures, the behavior of S(ω ) is determined mainly by transitions
from the ground state. Then, the symmetrized correlation function is approximatedasS(ω ) ≈ π ∑ |⟨n|σz |0⟩|2 [δ(ω − ωn0 ) + δ(ω + ωn0 )] .

(5.27)n

By substituting this expression into Eq. (5.24), the linear thermal conductance is obtained as
κ≈

παγkB
( βh̄ωn0 )2|⟨n|σz |0⟩|2 Ĩeff (ωn0 ).
∑16sinh( βh̄ωn0 )n(5.28)As a simple example, let us consider the case that the temperature is comparable
to the first transition energy, kB T ∼ h̄ω10 , and the first transition energy is muchsmaller than the other, ω10 ≪ ωn0 (n ≥ 2). For this case, the thermal conductance isapproximated as
κ≈

παγkB
( βh̄ω10 )2Ĩeff (ω10 ).
16sinh( βh̄ω10 )(5.29)

From this expression, the thermal conductance shows the Schottky-type temperature dependence [36, 37, 23, 38]; it has a peak near h̄ω10 /kB and shows exponential
suppression, κ ( T ) ∼ T −2 e−h̄ω10 /kB T , for kB T ≪ h̄ω10 . When the transition energy
to the second excited state, h̄ω20 , is also comparable to the temperature, the linear
thermal conductance is approximated as a sum of the two Schottky functions. Thisimplies that the linear thermal conductance is expected to have two peak structure
as a function of the temperature. We will discuss this characteristic behavior numerically in Section 5.4 in detail. The weak-dissipation approximation discussed here isuseful for qualitative discussion on the temperature dependence of the linear thermal conductance. However, since the dissipation indeed affects the peak positions
and peak broadening, we need a more sophisticated approximation for quantitative
discussion. In the next section, we will introduce an alternative approximation.

5.3

Non-interacting blip approximation

In this section, we introduce the noninteracting-blip approximation (NIBA), whichworks well in a wide parameter region including the strong system-bath coupling
regime [16, 17, 38], and apply it to the linear thermal conductance.
First, let us consider the time evolution of the population ⟨σz (t)⟩. It is describedby the generalized master equationd ⟨σz (t)⟩
=−
dt

Z t
0

dt′ Kz (t − t′ ) σz (t′ ) .

(5.30)

Here, the kernel Kz (t) can describe non-Markivian dynamics of the population ⟨σz (t)⟩.
Although, in general, the non-local kernel is a complicated structure, it takes a simple form in the NIBA [16, 17],′Kz (t) = ∆2 e−Q (t) cos Q′′ (t),(5.31)

5.4. Numerical results

73

where Q(t) = Q′ (t) + iQ′′ (t) is the complex bath correlation function, the real andimaginary parts are


Z ∞
βh̄ω
Ieff (ω )′Q (t) =dωcoth
(1 − cos ωt),(5.32)
ω2
2
0
Z ∞I (ω )(5.33)
Q′′ (t) =dω eff 2 sin ωt,ω0respectively, with Ieff (ω ) = Ieff,L (ω ) + Ieff,R (ω ). When the spectral density is the
structured one as Eq. (5.14), we can calculate the complex bath correlation function
explicitly as follows [159, 83]:




2
′
−Γt
−Γt
′
Q (t) = παt−L ecos Ω̄t − 1 − Zesin Ω̄t + QMats (t) ,(5.34)h̄βhiQ′′ (t) = πα 1 − e−Γt (cos Ω̄t − N sin Ω̄t) ,(5.35)whereΩ2 − 2Γ2,
2ΓΩ̄N sinh( βh̄Ω̄) − sin( βh̄Γ)L=,cosh( βh̄Ω̄) − cos( βh̄Γ)sinh( βh̄Ω̄) + N sin( βh̄Γ)Z=,
cosh( βh̄Ω̄) − cos( βh̄Γ)

Ω̄ =

p

Ω2 − Γ2 ,N=(5.36)(5.37)(5.38)′and QMats(t) is a series with respect to the Matsubara frequency, νn = 2πn/(h̄β),
′
QMats
(t) =

4Ω4 ∞ 11 − e−νn t.
∑h̄β n=1 νn (Ω2 + νn2 )2 − (2Γνn )2(5.39)The symmetrized correlation function S(ω ) can be related to the non-local kernel
as [17]
2
S(ω ) = Re,
(5.40)−iω + K̂z (−iω )where K̂z (λ) is the Laplace transformation of Kz (t). By using Eqs. (5.31)-(5.40) and(5.24), we can calculate the linear thermal conductance numerically within the NIBA.
In general, the NIBA is justified at arbitrary temperatures for the weak system-bathcoupling or for incoherent transport realized at the high temperature or the strong
system-bath coupling [17, 16]. Moreover, even when the system-bath coupling is
not weak, the NIBA fails at sufficiently low temperatures where heat transport isinduced by virtual excitations, i.e., the co-tunneling process is dominant (see Section 5.1) [38]. However, the deviation from the exact results remains small as far as
the system-bath coupling is sufficiently weak.

5.4

Numerical results

In this section, we present the numerical results for the symmetrized correlation
function and the linear thermal conductance using the NIBA for Ω ≲ ∆, where heattransport is expected to be non-trivial, as discussed in Section 5.2.

74

Chapter 5. Heat Transport through the Quantum Rabi System

F IGURE 5.5: Symmetrized correlation function S(ω ) as a function of
the frequency for different Ω/∆ at fixed other parameters as g/∆ =0.5, η = 0.01, kB T/(h̄∆) = 0.01. The symmetrized correlation function has several peaks and the peaks at the lower frequency are more∗ /∆ = 0.096
dominant. The black arrows indicate the first peak at ω10∗ /∆ = 0.54 at Ω/∆ = 0.5. The solid and dotand the second one at ω20
ted arrows represent the lowest two transition frequencies of the nondissipative quantum Rabi model, ω10 and ω20 , respectively, which are
obtained by the exact diagonalization of the non-dissipative quantum
Rabi model. The inset represents a ratio between the frequencies of∗ /ω ∗ , as a function of Ω/∆.
the first and second peaks, ω10
20

5.4.1

Symmetrized correlation function

First, we show the symmetrized correlation function, S(ω ), in Fig. 5.5 for g/∆ = 0.5,
η = 0.01, and kB T/(h̄∆) = 0.01. From this figure, the symmetrized correlation function has several peaks corresponding to the transition frequencies of the dissipative
quantum Rabi model, as indicated by the approximation for the weak dissipation(see Section 5.2). This multi-peak structure derives from the multilevel property ofthe quantum Rabi model. However, the peak positions are slightly shifted from thetransition frequencies of the non-dissipative quantum Rabi model due to the dissipation even for small dissipation, η = 0.01. For later discussion, we introduce the∗ corresponding to the n-th peak in the symmetrized
renormalized frequencies ωn0
correlation function.

5.4.2

Linear thermal conductance

Next, we consider the linear thermal conductance. Fig. 5.6 (a) shows the temperaturedependence of the linear thermal conductance for g/∆ = 0.5, η = 0.01, and differentΩ/∆. For Ω/∆ = 0.4, it is clear that there are two peaks in the temperature dependence. We also see that the two-peak structure becomes less significant as Ω/∆
increases from 0.4 to 0.8. This feature originates from the first and second peaks of∗ and ω ∗ , respectively (see Fig. 5.5). Then,the symmetrized correlation function, ω1020
its temperature dependence can be qualitatively described by the sum of the two∗∗
Schottky-type functions, T −2 e−h̄ω10 /kB T and T −2 e−h̄ω20 /kB T , which leads to the twopeak structure. The two-peak structure is a signature of the multilevel property ofthe quantum Rabi model and can be clearly observed only when the symmetrized
correlation function has sharp well-separated peaks. We show the temperature dependence of the linear thermal conductance for Ω/∆ = 0.5, g/∆ = 0.5, and different

5.4. Numerical results

F IGURE 5.6: Temperature dependence of the linear thermal conductance for Ω/∆ = 0.5, g/∆ = 0.5, and η = 0.01. When varying (a)Ω/∆ = 0.4 − 0.8, (b) η = 0.01 − 0.1, and (c) g/∆ = 0.3 − 0.7, the linear thermal conductance considerably changes between the one- andtwo-peak structures.

75

76

Chapter 5. Heat Transport through the Quantum Rabi System

F IGURE 5.7: Linear thermal conductance as a function of Ω/∆ forg/∆ = 0.5, η = 0.01, and different temperatures kB T/(h̄∆) = 0.01
and 0.02. The inset is an enlarged view of the high-frequency side of
the peak frequency. The dashed line represents a guide for the powerlaw (Ω/∆)−6 .

η in Fig. 5.6 (b). As η increases from 0.01 to 0.1, the peak at the low-temperatureside becomes less significant. This is because the peaks in the symmetrized correlation function are broadened due to the dissipation and are smoothed out them after
performing the integral in Eq. (5.24).
Now, let us discuss the detailed condition for the appearance of the two-peakstructure in the linear thermal conductance (5.24). If the peaks in the symmetrized
correlation function are sufficiently sharp, the linear thermal conductance is approximately evaluated as∗∗e−(ω20 /ω10 )/T̃
e−1/T̃+a,κ ( T ) ∼ a1
2
( k B T )2
( k B T )2

(5.41)∗ ) and the prefactors arewhere T̃ = kB T/(h̄ω10

an =

παγkB
∗ 2
∗
| ⟨n| σz |0⟩ |2 (h̄ωn0) Ĩeff (ωn0).
8

(5.42)∗ /ω ∗ ≫ 1 is required to
From this expression, one can see that the condition ω20
10have the two-peak structure in the temperature dependence of the linear thermal
conductance. Note that although the ratio between a1 and a2 contributes to the appearance of the two peaks, it is almost irrelevant. This condition can be confirmedby the comparison between Fig. 5.5 and Fig. 5.6 (a). As Ω/∆ increases from 0.4 to∗ /ω ∗ is reduced from 13.9 to 2.87, as shown in the inset of Fig. 5.5.
0.8, the ratio ω2010∗ /ω ∗ , the two-peak structure becomes less significant (seeWith the decreasing of ω20
10
main panel in Fig. 5.6 (a)). This feature can be also observed from the dependenceof the hybridization constant g. Fig. 5.6 (c) shows the temperature dependence ofthe linear thermal conductance for Ω/∆ = 0.5, η = 0.01, and different g/∆. It isobserved that as g/∆ increases from 0.3 to 0.7, the two-peak structure becomes more∗ /ω ∗ is enlarged due to the level repulsion
significant. This is because the ratio ω20
10
as the hybridization g increases.
When we choose the parameters for which the two-peak structure is clearly observed, the linear thermal conductance is sensitive to the parameters, particularly
to Ω/∆. In Fig. 5.7, we show the linear thermal conductance as a function of Ω/∆

5.4. Numerical results

77

for two different temperatures. The linear thermal conductance has a sharp peak atΩ = Ω0 , where Ω0 /∆ ≈ 0.40 and 0.44 for kB T/(h̄∆) = 0.01 and 0.02, respectively.
While it decays exponentially in the smaller Ω/∆ side of the peak, it shows a powerlaw decay in the opposite side, whose exponent is −6 (see the inset in Fig. 5.7). The
strong dependence of Ω/∆ has an advantage for the application of heat transistor orheat valve in mesoscopic heat devices.
Now, we comment on the power-law decay for large Ω/∆ in the linear thermal
conductance as a function of Ω/∆. It is a signature of the co-tunneling process inwhich heat transport is induced by the tunneling process via virtual excitations, as∗ , it is supposed
discussed in Section 5.1. At low temperatures, kB T ≪ h̄Γ, h̄Ω̄, h̄ω20
the system behaves effectively like the Ohmic spin-boson model with a single pa∗ because the high-energy states are almost irrelevant. Furthermore, forrameter ω10∗ , thermal excitation from the ground state to the first excited state is
kB T ≲ 0.1h̄ω10strongly suppressed, and then heat transport is governed by the co-tunneling process for which the linear thermal conductance is written as [37, 23, 38]κ≈

π 3 γk4B 2 2 3
α χ0 T ,30(5.43)

where the static spin susceptibility χ0 of the effective Ohmic spin-boson model is∗ ) for the weak coupling, α ≪ 1. Using α =roughly approximated as 2/(h̄ω102∗
η (4g/Ω) and ω10 ∝ Ω for Ω ≲ ∆, we obtain κ (Ω) ∝ Ω−6 , which is consistent
with the numerical result shown in the inset of Fig. 5.7.

5.4.3

Rotating-wave approximation

Finally, we comment on the contribution of the counter-rotating terms. By neglecting the counter-rotating terms, i.e., the rotating-wave approximation to the quantum
Rabi model, we obtain the Jaynes-Cummings model, as discussed Section 2.4. Whilethese two models give almost the same results in weak coupling regime, g/∆ ≪ 1,
they are different in the ultrastrong coupling regime, g/∆ ≳ 1. We stress that thequbit-resonator system in the ultrastrong coupling regime can be described not by
the Jaynes-Cumming model but by the quantum Rabi model because the rotatingwave approximation fails in that regime. However, it would be instructive and useful to consider the Jaynes-Cummings model as a mathematical model or a modelthat may be realized in systems other than superconducting circuits. Since it is cumbersome to construct the NIBA for the Jaynes-Cummings model, we employ theapproximation for the weak dissipation (η ≪ 1) with the same procedure in Section 5.2.
For the convenience of later discussion, we introduced new Pauli operators, τx,y,z ,defined asτx = σz ,τy = σy ,τz = −σx .

(5.44)

This corresponds to a rotation of the coordinate around the y-axis by π/2. We also
introduce the state vectors of the two-level system, |0⟩ and |1⟩, as eigenstates of τz as
τz |0⟩ = − |0⟩ ,τz |1⟩ = + |1⟩ .

(5.45)For a decoupled system (g = 0), |0⟩ and |1⟩ correspond to the ground and excitedstates of the two-level system, respectively. Moreover, we define τ+ = |1⟩ ⟨0| and78

Chapter 5. Heat Transport through the Quantum Rabi System

τ− = |0⟩ ⟨1|. Using these new operators and the identity τx = τ+ + τ− , the Hamiltonian of the quantum Rabi model can be rewritten as

HR =



h̄∆
τz + ∑ h̄Ωr ar† ar + ∑ h̄gr (τ+ + τ− ) ar + ar† .
2rr(5.46)

After dropping the counter-rotating terms including τ+ a† and τ− a by applying therotating-wave approximation, we finally obtain the Jaynes-Cummings model as

HJC =



h̄∆
τz + ∑ h̄Ωr ar† ar + ∑ h̄gr τ+ ar + τ− ar† .
2rr(5.47)For simplicity, we consider the symmetric case, ΩL = ΩR ≡ Ω and ηL = ηR ≡ η.
Introducing new operators
A=

aL + aR√ ,2B=

aL − aR√ ,2(5.48)

corresponding to symmetric and antisymmetric modes, respectively, the Hamiltonian is expressed as

HJC = H A + H B ,


√
h̄∆
HA =
τz + h̄ΩA† A + 2h̄g τ+ A + τ− A† ,2
H B = h̄ΩB† B.

(5.49)(5.50)(5.51)

From this expression, since the antisymmetric mode is completely decoupled withthe two-level system, it does not contribute heat transport through a two-level system. Thus, it is sufficient to consider only the Hamiltonian H A .
The Jaynes-Cummings model conserves the total number of particles, whose operator is N̂ = τ+ τ− + A† A while the quantum Rabi model does not. The basis of the
Jaynes-Cummings model, |τ ⟩ ⊗ |n⟩, has an eigenvalue N = τ + n, where τ (= 0, 1)
and n (= 0, 1, 2, . . . ) assign the states of the two-level system and the symmetricmode, respectively. It is easy to show the ground state is given as |0⟩ ⊗ |0⟩. For theexcited states, there are two eigenstates belonging to the same N (≥ 1):|i; N ⟩ = a Ni |0⟩ ⊗ | N ⟩ + b Ni |1⟩ ⊗ | N − 1⟩ ,(i = 1, 2),(5.52)

where a Ni and b Ni are quantum amplitudes.
At low temperatures, the symmetrized correlation function is determined by thetransitions from the ground state dominantly. Then, the symmetrized correlation
function is written as
SJC (ω ) ≈ π ∑ | ⟨i; 1| τx |0; 0⟩ |2 δ(ω − ωi,1 + ω0,0 ),(5.53)

i =1,2where h̄ωi,N is an energy eigenvalue of |i; N ⟩. Here, we have used the fact that theoperator τx changes the eigenvalue N only by one. Thus, the existence of the conserved quantity, N, strongly restricts the possible transitions from the ground state
to the excited states. However, the expression of the symmetrized correlation function in the Jaynes-Cummings model (5.53) indicates that it has indeed two peaks atω = ωi,1 − ω0,0 (i = 1, 2). Therefore, we can conclude that the two-peak structure appears in the temperature dependence of the linear thermal conductance even for the5.5. Short summary of this chapter

79

Jaynes-Cummings model when ω20 /ω10 ≫ 1. Note that since the energy spectrum
differs between the Jaynes-Cummings and the quantum Rabi models, particularly inthe ultra-strong coupling regime [60], they give qualitatively different temperaturedependences for the linear thermal conductance there.

5.5

Short summary of this chapter

We studied heat transport through an assembly, a two-level system embedded between harmonic resonators, described by the quantum Rabi model. By mapping thequantum Rabi model to the spin-boson model with the structured bath and using thenoninteracting-blip approximation (NIBA), we calculated the linear thermal conductance. As a result, we discovered that the linear thermal conductance shows the two
peaks as a function of the temperature in the ultrastrong coupling regime. We foundthat this characteristic behavior comes from the multiple levels in the quantum Rabi
model. In addition, we found that the linear thermal conductance changes sensitively to the tunable parameter, particularly the natural frequency of the harmonic
oscillators.

81

Chapter 6

Summary and Perspectives
In this thesis, we have studied three problems on transport properties of the superconducting circuits from several viewpoints. We have theoretically discussed various transport properties, in particular, with emphasis on many-body effects inducedby system-environment coupling, Coulomb interaction, and disorder.
In Chapter 3, we investigated the quantum critical phenomena in the microwave
scattering off a superconducting charge qubit coupled to an RLC transmission line,
which is effectively described by the sub-Ohmic spin-boson model. This model
exhibits a zero-temperature quantum phase transition at zero temperature due tothe coupling between the two-level system and the heat bath. By performing thecontinuous-time quantum Monte Carlo simulation, we observed quantum criticalbehavior in the frequency dependence of the reflection loss consistent with analytic
results using the renormalization group analysis. Moreover, by using the generalrelation between the circuit impedance and the spectral density of the heat bath, we
showed that the sub-Ohmic heat bath could be constructed by the arrays of the RLC
circuits whose circuit constants depend on the spacial positions. This study paves
a path toward the observation of quantum critical phenomena in superconducting
circuits in a controllable way. Because various types of quantum measurement on
qubits are possible in superconducting circuits, it would be a challenging task to
study the effect of its back-action on quantum phase transition.
In Chapter 4, we considered a one-dimensional Josephson-junction chain withthe disorder and studied the transmission through it. The disordered Josephsonjunction chain exhibit a superfluid-Bose glass transition at a critical Luttinger parameter Kc = 3/2, where the Luttinger parameter K is related to the circuit impedanceas Z = RQ /(2K ). We focused on the classical limit (K ≪ 1), corresponding to
the deep insulating regime, where the collective pinning affects an elastic scattering
property in a wide frequency range. In the high-frequency regime, we investigated
the transmission using the Fokker-Planck approach. The analytical formulas for thetransmission agree with numerical results obtained by solving the Schödinger equation. Surprisingly, we found that the correlations of the pinning-induced disorderstill affect the forward-scattering length despite the wavelength being shorter thanthe Larkin length. In the low-frequency regime, the localization becomes strong
and the analytical approach does not work. To obtain information on this stronglylocalized regime, we performed a brute-force numerical calculation. We found that
the Lyapunov exponent is significantly increased, and its variance is suppressed incomparison with the well-known model with the Gaussian white-noise disorder. We
show that the ratio ln ⟨ T (ω )⟩ / ⟨ln T (ω )⟩ takes a value near 1, indicating the strong
correlation between the typical disorder configurations and rare disorder configurations optimal for the transmission determines the low-frequency transport properties. From the fact that the transmission is exponentially suppressed, we also foundsignatures of the scarcity of the localized low-frequency plasmon modes. It also

82

Chapter 6. Summary and Perspectives

affects the distribution of the reflection phase. These observables make out uniquesignatures of the non-Gaussianity of the pinned chain. We expect that our study provides benchmarks for further developments of the dynamics in quantum Bose glass.
It is an important future problem to investigate the effect of large quantum fluctuation (K ∼ 1) beyond the classical regime (K ≪ 1). It is also a fascinating problem toextend the present work to the experiment of inelastic microwave scattering.
Finally, we studied heat transport through the quantum Rabi model consistingof a superconducting qubit embedded between superconducting LC resonators. By
using the exact mapping to the spin-boson model and the non-perturbative approximation, we evaluated the linear thermal conductance. In the ultra-strong coupling
regime, the temperature dependence of the linear thermal conductance has a twopeak structure. This characteristic behavior comes from the fact that there are multiple levels in the quantum Rabi model and can not be observed in heat transport
through a simple two-level system. Moreover, we found that the linear thermal conductance is highly sensitive to the controllable parameters, particularly to the natural
frequency of the harmonic resonators. This property is advantageous to the application of well-controllable nanoscale heat devices such as a heat transistor. In thisthesis, we restricted ourselves to the linear response theory to study quantum heattransport. For studying the rectification effect in heat transport through nanoscale
objects, we need to expand our theory beyond the linear response.
In the last part of this chapter, we describe future perspectives of the present
theoretical work. There are several future problems that may bring fruitful physics.
First of all, this thesis treated only thermal equilibrium properties regarding to manybody effects in the superconducting circuits. While equilibrium properties are fundamental and important, the systems being in equilibrium is just a part of realistic
systems. Therefore, it is important to expand our results toward non-equilibrium
states under large temperature/voltage biases or strong excitations by microwaves.
Next, it will be a fascinating problem to combine the many-body effects with quantum measurement. Thanks to the technological development in the superconducting
circuit, the environment (e.g., a transmission line and a Josephson-junction chain)can be monitored mode by mode experimentally. It allows us to consider the effect of the quantum measurement of a small quantum system on the environments,
i.e., “backaction”. For example, by performing the quantum measurement to a superconducting qubit coupled to environments, we can trace how the quantum measurement affects modes in the environments. It is not only important in the aspect offundamental physics but also expected to be helpful for quantum processing. Third,in the superconducting circuit, quantum dynamics away from the equilibrium states
can be simulated. For example, in the superconducting circuit, we can switch on andoff the interaction between an impurity and the Josephson transmission line by tuning a flux through. This setup enables us to study quantum quench dynamics ofthe many-body states in a controllable way. The theoretical investigation of such
dynamics in superconducting circuits is also an important future problem. We expect that these general future problems will attract the interest of researchers in thenear future. We hope that the present theoretical work will contribute to taking animportant step towards future development in this direction.

83

Appendix A

Lagrangian and Hamiltonian for atransmission line
In this appendix, we derive the Lagrangian and Hamiltonian of the transmission
line.
The transmission line is a long circuit in which electromagnetic waves can propagate freely. In the transmission line, the voltage and the current change over its
length. Therefore, the capacitance and the inductance are represented in the density
at a position, c and l, respectively. The lumped element model of the transmission
line is a series of the LC-circuit with the inductance ldx and the capacitance cdx,
where dx is a spacing, per unit length as shown in Fig. A.1.
This formalism is valid not only for the transmission line but also for any linear
circuit system. The Lagrangian of the lumped element model can be written downas1
1
L = ⃗ϕ˙ t C⃗ϕ˙ − ⃗ϕt L−1⃗ϕ,22where ⃗ϕ is a vector whose components are the flux at node n = 1, 2, . . . , N,
 
 
ϕ1
ϕ̇1
 ϕ2 
 ϕ̇2  
⃗ϕ = 
 ..  , ⃗ϕ˙ =  ..  .
 . 
 . 
ϕN

(A.1)

(A.2)

ϕ̇N

The capacitance matrix C and the inductance matrix L are real and symmetric. Notethat what the Lagrangian can be written in the quadratic form means that the system
is linear. Using the Euler-Lagrange equation, the evolution of the fluxes is describedby⃗ϕ¨ = −C−1 L−1⃗ϕ.

F IGURE A.1: Lumped element model of the transmission line.

(A.3)

84

Appendix A. Lagrangian and Hamiltonian for a transmission line

From this equation, the system has resonance frequencies, and they are given bypositive square root of the eigenvalues of the matrix C−1 L−1 .
By performing the Legendre transformation, the Lagrangian provides the Hamiltonian1 ⃗ t −1 ⃗1
H= Q
C Q + ⃗ϕt L−1⃗ϕ,
2
2

(A.4)

⃗ is a vector of charges at each node and has the relation to the flux,
where Q
⃗ = C⃗ϕ˙ .
Q

(A.5)

The charge component at node n is conjugate to the flux at the same node,[ Qn , ϕm ] = ih̄δn,m .

(A.6)

Note that the Hamiltonian (A.4) and the commutation relation (A.6) recover ones of
the LC-circuit (2.6) and (2.5) when one restricts a single node.
Now, we treat the transmission line as a quantum mechanical problem by re⃗ and ⃗ϕ with operators [160]. First, let us introduce bosonic annihilation
placing Q
and creation operators an and a†n , respectively, which satisfy the commutation relation [ an , a†m ] = δn,m ,
r


h̄ N
ϕn =
An,m am + a†m ,(A.7)∑
2 m =1
r


h̄ N
†
(A.8)
B
a
−
a
Qn = −i
n,m
m
m .
2 m∑
=1
To hold the commutation relation, the coefficient matrices A and B has a relation,
ABt = 1.

(A.9)

Using new variables, the Hamiltonian becomes

i 
i 
h
 
h

h̄ N
†
t −1
†
†
t −1
†
H= ∑
aj + aj B C B
ak + ak − a j − a j A L A
ak − ak.
4 j,k=1j,kj,k(A.10)
The coefficient matrices A and B are chosen as matrices that diagonalize L and C,
respectively, such that
ω1 0 · · · 0 0 ω2 · · · 0 At L−1 A = Bt C−1 B =  .
(A.11)..
..  .
.
.
.
 .
.
. 
.
0
0 · · · ωN
In this choice, one can easily see that the columns of the coefficient matrix A are theeigenvectors of C−1 L−1 with eigenvalues ωn2 .

Appendix A. Lagrangian and Hamiltonian for a transmission line

85

Finally, we can obtain the Hamiltonian on the diagonal basis,N

H = ∑ h̄ωn
n =1



1
a†n an +

2,(A.12)

which means that a linear circuit with the inductances and capacitances is equivalent
to a collection of harmonic oscillators.
Now, we consider transmission line with the constant element density. In thetransmission line, the capacitance and inductance matrices are
1 −1 00 −1 2 −1 0
 0 −1 2 −1C−1
= L ldx =  ..
..
..
..
 .
cdx
.
.
.

 0
0 · · · −100 ··· 0


···
···
···
..
.


0
0 

0 

,
.. 
. 

2 −1
−1 1

(A.13)√and eigenvalues of L−1 C−1 are constant ωn2 = [1/( lcdx )]2 .
Finally, we consider the continuum limit, dx → 0, of the transmission line. In
this limit, the flux (charge) variable becomes a functions of position x, ϕn → ϕ( x )(Qn → Q( x )dx). Thus, the Lagrangian (A.1) reads


Z d
c
1
2
2
L=
dx
(A.14)[∂t ϕ( x, t)] − [∂ x ϕ( x, t)] ,22l0
where d is the length of the transmission line and Euler-Lagrange equation for the
Lagrangian provides the wave equation for the fluxv2 ∂2x ϕ( x, t) − ∂2t ϕ( x, t) = 0,

(A.15)√
with the velocity v = 1/ lc. Note that when imposing the open boundary condition, ∂ x ϕ(0, t) = ∂ x ϕ(d, t) = 0, the position dependence of the flux is separable,ϕ( x, t) = ∑n ϕ( x ) cos(πnx/d), and thus one can recover the equation (A.3). Finally, by performing the Legendre transformation, we obtain the Hamiltonian for
the transmission line in the continuum limit, Eq. (2.10) in the main text. It is clearthat the Hamiltonian and the commutation relation of the transmission line in thecontinuum limit are natural extensions of the LC-circuit.

87

Appendix B

Lagrangian for a
Josephson-junction chain
In this appendix, we derive the Lagrangian for the one-dimensional Josephsonjunction chain shown in Fig. B.1. As discussed in Section 2.2, the Hamiltonian ofa single Josephson junction is given by

H = EJ (1 − cos φ) + EC q2 ,(B.1)

where EJ is the Josephson energy, EC = (2e)2 /(2C ) is the charging energy at theJosephson junction, q is the quasicharge on the Josephson junction, and φ is thephase difference across the Josephson junction. In this thesis, we consider the limit
EJ ≫ EC , which corresponds to the transmon limit (see Section 2.3), to decrease
quantum fluctuations of the phase across a single Josephson junction due to thequantum phase slips. In the transmon limit, each energy band is almost flat andthe transition
frequency between each band is estimated as the plasma frequencypΩ = 2EJ EC /h̄ (see Fig. 2.5). When considering only the lowest energy band, theplasma frequency plays the role of the ultraviolet cutoff in the low-energy theory.
Then, eigenenergies in the lowest band are given by
EQPS = −Λ cos(2πq),(B.2)

where the half-band width Λ is determined by quantum phase slip, i.e., a nonperturbative process in which the phase difference across the Josephson junction
φn jumps by ±2π due to quantum tunneling between the minima of the Josephson

•••

•••

F IGURE B.1: Josephson-junction chain consisting of Josephson junctions with the Josephson energy EJ and the Josephson capacitance CJ ,ground capacitances Cg , and phenomenological inductances L. The
elementary cell length is a. The quasicharge on the n-th Josephson
junction is qn . The blue dots represent the superconducting islands.

88

Appendix B. Lagrangian for a Josephson-junction chain

energy. The amplitude of the quantum phase slip is given as
1/4 −√32EJ /E
8C.
Λ = √ 2EJ3 EC
e
π

(B.3)

This is associated with the conjuration relation between the phase difference φ andquasi charge q as [ φ, q] = i , i.e., operators e±2πiq translate the phase difference by±2π, φ → φ ± 2π. In the transmon limit (EJ ≫ EC ), the amplitude is exponentiallysmall, Λ ≪ 1, and the quantum phase slips become rare events. When the numberof the Josephson junctions in the chain is small enough, the contribution of the quantum phase slip is negligible and the Josephson-junction chain faithfully behaves as
the superinductor consisting of Josephson inductances LL . However, the recent technological development enables the realization of longer chains with N ∼ 104 [89] forwhich a finite density of quantum phase slips can appear.
Since the total charge in the elementary cell, inside of the red-dashed box in
Fig. B.1, is zero, the charge on the n-th ground capacitance is 2e(qn+1 − qn ), andthe Coulomb energy at the ground capacitance along the Josephson-junction chain
is

(2e)2
( q n +1 − q n )2 .
2Cgn

ECoulomb = ∑(B.4)

We assume that all characteristic frequencies are much smaller than the plasma
frequency and employ the adiabatic approximation in which the
R quasicharge is asmooth function of the position. In the continuum limit (∑n a → dx), the Coulomb
energy is expressed as
ECoulomb =Z d
0dx aEg [∂ x q( x, t)]2 .

(B.5)

where d is the chain length and Eg = (2e)2 /(2Cg ) is the charging energy at the
ground capacitance. Note that the continuous approximation is valid when the characteristic wave vectors are much smaller than 1/a and the plasma frequency is muchsmaller than v/a [87]. The latter condition is equivalent to small ground capacitanceCg ≪ Cp(or Eg ≫ EC ), which corresponds to the large charge screening lengthℓsc = a C/Cg ≫ a, i.e., long-range Coulomb interaction. The inductive energy
along the chain isL(2eq̇n )2 ,n 2

Einductive = ∑(B.6)

and it can be expressed in the continuum limit as
Einductive =

Z d
0

dx

1 (2e)2 L[q̇( x, t)]2 .
a 2

(B.7)

Here, we have introduced the phenomenological inductance L between the Josephson junction for simplicity [87]. We note that the mass term, ∝ q̇2 , can be derived
from the Hubbard-Stratonovich transformation [161, 162] without introducing thephenomenological inductance. In this derivation, the phenomenological inductancecoincides with the Josephson inductance, Eq. (2.16), in the transmon limit (EJ ≫ EC ).

Appendix B. Lagrangian for a Josephson-junction chain

89

Therefore, the inductive energy reads
1 (2e)2 LJ
Einductive =dx[q̇( x, t)]2 =
a
2
0
Z d

Z d
0

dx

1 h̄2
[q̇( x, t)]2 .
a 2EJCombining Eqs. (B.2), (B.5), and (B.8), we obtain the full Lagrangian as
(
)
Z d
1 h̄2
1
L=
dx
[q̇( x, t)]2 − aEg [∂ x q( x, t)]2 + Λ cos [2πq( x, t)] .
a 2EJ
a
0

(B.8)

(B.9)

Finally, by introducing a new field θ ( x, t) = πq( x, t), we obtain the Lagrangian,Eq. (2.44) in the main text.

91

Appendix C

Impedance of RLC Transmission
Lines
In this appendix, we derive the analytical expression of the impedance of the RLC
transmission lines with spatially-dependent resistances (3.28).
Assuming |ωCL,j ZL,j+1 (ω )| ≪ 1, the recurrence relation (3.26) becomes a differential equation in the continuous limit (N → ∞):∂Z (ω, x )= −r ( x ) + iωl ( x ) + iωc( x ) Z2 (ω, x ),
∂x

(C.1)

where Z (ω, x = j/N ) ≡ ZL,j (ω ), and r ( x ), l ( x ), and c( x ) are the resistance, capacitance, and inductance per unit length at x = j/N, respectively. The total impedanceof the RLC transmission line is obtained from the relation ZL (ω ) = Z (ω, x → 0).
The spatial dependence of the circuit elements given in Eq. (3.28) can be rewritten in
the continuous limit asr ( x ) = rx n ,l ( x ) = 0,c( x ) = c.

(C.2)

At x ≫ x ∗ ≡ (n2 /4ωrc)1/(n+2) , we can neglect ∂ x Z (ω, x ) and then obtain the
impedance asrr n/2
Z (ω, x ) ≈ Z A (ω, x ) =
x .
(C.3)
iωc
At x ≪ x ∗ , where the first term, rx n , on the right-hand side of Eq. (C.1), can beneglected, the impedance is given by
Z (ω, x ) ≈ ZB (ω, x ) =1,−iωcx + f (ω )(C.4)

where f (ω ) is the constant of integration determined from the condition Z A (ω, x ∗ ) =ZB (ω, x ∗ ). Finally, we obtain the impedance of the RLC transmission line as
ZL (ω ) ∼ ZB (ω, x → 0) = √

n
2ωc



4ωrc
n2

1/(n+2)√ ! −1
n+ 2
1+i √
.
2

(C.5)

for ω ∗ ≪ ω ≪ ωc . Therefore, the spectral density is obtained as
IL (ω ) ∝ ĨL (ω ) = ωRe[ ZL (ω )] ∝ ω 1/(n+2) .

(C.6)

This expression is consistent with Eq. (3.29). Note that the high-frequency cutoff ωc

92

Appendix C. Impedance of RLC Transmission Lines

is obtained through the condition ωC | Z A (ω, x ∗ )| ≪ 1, where the recurrence relation (3.26) can be reduced to the differential equation (C.1) for x < x ∗ . The imaginary part of the impedance shows a sharp peak for x ∼ 1. Then, for x > x ∗ , it can beneglected in the above analysis, given a condition ωC (1 − x ∗ )Im [ Z A (ω, x ∗ )] ≫ 1,
which results in the low-frequency cutoff ω ∗ (3.30).

93

Appendix D

Fokker-Planck Equation
In this appendix, we provide the general recipe to derive the Fokker-Planck equation (4.43) from the Ricatti equation (4.42) and Gaussian correlators (4.36) and (4.37).
First, we consider the general dynamical equation for the multidimensional variable ⃗X = ( X1 , X2 , . . . , X N ) ,∂ t Xi ( t ) = a i ( ⃗X (t)) + ∑ bij ( ⃗X (t))ξ j (t),

(D.1)

j

where ξ j (t) is the random force characterized by the Gaussian correlator,
ξ i (t)ξ j (t′ ) = 2Dij δ(t − t′ ).

(D.2)

Then, the joint probability P(t, ⃗X ) of the variables X1 , X2 , . . . , X N satisfies the FokkerPlanck equation [163]i∂P(t, ⃗
X)
∂ h ⃗
∂
= −∑
ai ( X ) P(t, ⃗X ) + ∑ Dlm
∂t
∂Xi
∂Xi
i
ijlm



i
∂ h
⃗
⃗
⃗
bil ( X )b jm ( X ) P(t, X ) .
∂X j
(D.3)

This general formula has the correspondence with our model as follows: t = x,
⃗X (t) = z( x ), a1 = 0, (b11 , b12 , b13 ) = (2iz, i, iz2 ), (ξ 1 , ξ 2 , ξ 3 ) = (∆0 , ∆∗ , ∆), and ω⋆ /2ℓ0 (ω )00
D11 D12 D13
 D21 D22 D23  = 00ω⋆ /2ℓπ (ω )  .
(D.4)
D31 D32 D33
0ω⋆ /2ℓπ (ω )0
By using the above correspondence and introducing z = eiθ1 e−θ2 , we obtain Eq. (4.43).

95

Appendix E

Exact Solution of P2 ( x, θ2 )In this appendix, according to Ref. [135], we find the solution of Eq. (4.46) for theconditional probability P2 ( x, θ2 ) with the initial condition P2 ( x = d, θ2 ) = δ(θ2 +ln r0 ).
To this end, we use various changes of variables to recover equivalent formulations of that equation. First, introducing τ = (d − x )/ℓπ , λ (> 0) such thate−2θ2 = λ/(1 + λ), and P̃(τ, λ) = P2 (τ, θ2 )|dθ2 /dλ|, Eq. (4.46) reads


∂ P̃
∂
∂ P̃
=
λ (1 + λ )
.
∂τ
∂λ
∂λ

(E.1)

Then, introducing X (> 0) such that λ = sinh2 X and P̄(τ, X ) = P̃(τ, λ)|dλ/dX |, we
find



∂
∂ P̄
1 ∂2 P̄
2 cosh 2X
−
=
P̄ .
(E.2)
∂τ
4 ∂X 2∂X
sinh 2X√
Furthermore, using a new probability variable Q( X ) = P̄( X )/ sinh 2X, it satisfies


∂Q
1
= H+
Q
(E.3)
−
∂τ
4
with an effective Hamiltonian

H=−

1 ∂2
1
−
.
4 ∂X 2 4 sinh2 2X(E.4)

Together with Eq. (E.2), the probability conservation,
Z ∞0

dX P̄(τ, X ) = 1,

(E.5)

imposes

lim

X →0

∂ P̄
P̄
−
∂X
X



∂Q
Q
−
∂X 2X



=0

(E.6)

= 0.

(E.7)

and, subsequently,

lim

X →0

Appendix E. Exact Solution of P2 ( x, θ2 )96

The knowledge about a normalized eigenspectrum of Eq. (E.4), H ψk = ε k ψk , withthe boundary condition (E.7) allows us to express the conditional probability asrsinh 2X
ψk ( X )ψk∗ ( X ′ )e−(ε k +1/4)τ ,(E.8)
P̄(τ, X ) =sinh 2X ′ ∑ksuch that P(0, X ) = δ( X − X ′ ) with cosh 2X ′ = (2 − T0 )/T0 .
Now, we consider the solution of the eigenvalue problem, H ψk = ε k ψk , with theboundary condition(E.7). By changing variable z = cosh 2X (> 1) and f k (z) such√that ψk ( X ) = sinh 2X f k (cosh 2X ), the new variable f k (z) satisfies the Legendre
differential equationd2 d fk(1 − z )+ ν(ν + 1) f k = 0,
(E.9)
dz
dz
with (ν + 1/2)2 = −ε k . The boundary condition (E.7) translates into

3/4 d f k ( z )= 0.
( z − 1)dz
z →1

(E.10)The solution of Eq. (E.9) is expressed by the Legendre functions of the first and second kind, Pν (z) and Qν (z), respectively. The boundary condition (E.10) excludesQν (z) from the set of solutions because it diverges logarithmically at z → 1. AsPν (z) ∼ z−ν−1 for |z| ≫ 1, the normalizability condition of the wavefunctions also
requires Re[ν] > 1/2, which results in that only solutions with ε k > 0 are allowedfrom the relation (ν + 1/2)2 = −ε k . Thus, the set of solutions is given by the conicalfunctions, f k (z) = Ck P−1/2+ik/2 (z) with ε k = k2 /4 (k > 0) and the pre-factor Ck [asP−1/2+ik/2 (z) = P−1/2−ik/2 (z)]. The normalization condition
Z ∞0dxψk ( x )ψk′ ( x ) = δ(k − k′ )is then obtained by choosing Ck =(E.11)p

πk tanh(πk/2). Finally, Eq. (E.8) reads Zsinh 2X ∞πkP̄(τ, X ) =dk k tanhP− 1 +i k (cosh 2X )
2
2
2
2
0
2

× P− 1 +i k (cosh 2X ′ )e−(1+k )τ/4 .
2

2

Reverting to variable θ2 and setting x = 0, i.e., τ = d/ℓπ , we obtain Eq. (4.48).

(E.12)

97

Appendix F

Gaussian White-noise Disorder
Potential
In this appendix, we show analytical results from the literature on a model with
Gaussian white-noise disorder potential, which was used in plotting some of thelines in Figs. 4.6, 4.9 and 4.8 [148, 145, 139]. We assume that the correlation of thedisorder potential corresponds to the first term of Eq. (4.27) including the delta function as a white-noise disorder potential. Note that the literature mostly addresses
the stationary Schrödinger equation at energy E. In the context of Eq. (4.10), we set
E = ω 2 (≥ 0) and then use positive-energy results in literature.
The Lyapunov exponent for the white-noise disorder potential is given by [145]R∞ √
1 0 dx x exp − x3 /24 − x/2(ω/ω⋆ )2.
(F.1)⟨γ(ω )⟩ R⋆ = R ∞√2 0 dx 1/ x exp [− x3 /24 − x/2(ω/ω⋆ )2 ]This formula is plotted by the dotted line in Fig. 4.6 (a). Then, it yields the zerofrequency result√31/3 π≈ 0.46,⟨γ(0)⟩ R⋆ =
Γ(1/6)(F.2)

while, at ω ≫ ω⋆ , the Thouless relation holds ⟨γ(ω )⟩ R⋆ = (ω⋆ /ω )2 /2 as well as
our pinned model.
The variance of the Lyapunov exponent is given by [141]



Z ∞
1
d
2
2
γ (ω ) R⋆ d =ds Re 2 ⟨γ(ω )⟩ R⋆ − if (s) ,
(F.3)
s
ds
0
with f (s) = ϕ(s)/ϕ(0) and(ω/ω⋆ )2 + 2is
ϕ(s) = Ai −
22/3




(ω/ω⋆ )2 + 2is− i Bi −
22/3



,

(F.4)

where Ai and Bi are Airy functions. This formula is plotted by the dotted line in
Fig. 4.6 (b). It yields the zero-frequency result [139]



5π
7 3 3 32√
≈ 1.1 ⟨γ(0)⟩ ,(F.5)
γ (0) d = ⟨γ(0)⟩− 3 F2 1, 1, ; , ;6 2 2 4
3 3
where 3 F2 is an hypergeometric function, while, at ω ≫ ω⋆ , ⟨⟨γ2 (ω )⟩⟩ d = ⟨γ(ω )⟩.

98

Appendix F. Gaussian White-noise Disorder Potential

The distribution of the reflection phase at the perfect impedance matching (K =K0 ) between a half-infinite chain and a waveguide [147],
P1 (θ1 ) =(ω/ω⋆ )2 + Z2
P( Z ),
2|ω |/ω⋆Z =−(ω/ω⋆ ) tan(θ1 /2)

is related with the distribution of the Ricatti variable Z,R∞dx exp − x3 /24 − x/2(ω/ω⋆ )2 − x/2( Z − x/2)20
R∞ √P( Z ) =.
dx 2π/x exp [− x3 /24 − x/2(ω/ω⋆ )2 ]
0

(F.6)

(F.7)

From Eqs. (F.1) and (F.7), we can check that the relation⟨γ(ω )⟩ R⋆ =Z

dZ ZP( Z ),(F.8)

holds at any ω. While, at ω ≫ ω⋆ , the distribution is uniform, P1 (θ1 ) ≈ 1/(2π ), at
0 < ω ≪ ω⋆ , the distribution is concentrated around phase θ1 = −π with
ω⋆ 
ω⋆ 
P1 (θ1 ) ≈p ( θ1 + π )
,
(F.9)
ω
ω


Z ∞
35/6
1
x3
x2
2x
p( φ) = √dx 2 exp − +− 2 ,(F.10)φ6φφπΓ(1/6) 0
if |θ1 + π | ≪ 1. The function p( φ) is plotted by the dotted line in the inset of Fig. 4.8.
The tails of the distribution P1 (θ1 ) at |θ1 + π | ≫ ω/ω⋆ are given by
P1 (θ1 ) ≈αω/ω⋆,cos2 (θ1 /2)

α=31/6 Γ(2/3)≈ 0.10.
27/3 π

(F.11)

From Eq. (F.6), we can find the real part of the disorder-averaged reflection coefficient at any impedance mismatch (K < K0 ),r′ + 1 =2.
1 + (K0 /K )2 tan2 (θ1 /2)(F.12)

While, at ω ≫ ω⋆ , Eq. (4.61) is reproduced with the uniform distribution, at ω ≪ ω⋆ ,the average is contributed by the tails of the distribution P1 (θ1 ), Eq. (F.11), whichyieldsr ′ + 1 = 4παK ω.
K0 ω ⋆(F.13)At large impedance mismatch (K ≪ K0 ), Eq. (F.12) is approximated by r ′ + 1 ≈4π (K/K0 )δ(θ1 ), which results inr ′ + 1 = 4π

K
P1 (θ1 = 0)
K0

(F.14)

with P1 (θ1 ) of Eq. (F.6). This formula is plotted by dotted line in Fig. 4.9. We cancheck that Eq. (F.14) reproduces Eq. (F.13) using Eq. (F.11) for θ1 ≈ 0.
The linear frequency dependence in Eq. (F.13) at low frequency, as well as the saturation at ω ≫ ω⋆ , reflects the frequency dependence of the bulk plasmon density

Appendix F. Gaussian White-noise Disorder Potential

99

of modes. Indeed by using the particle density of states of the Schrödinger problem [164, 165, 145] at energy E = ω 2 (> 0), we findR∞ √dx x exp − x3 /24 − x/2(ω/ω⋆ )2
ν(ω )
ω√
0
=
2π R ∞,
(F.15)√2ν0ω⋆dx 1/ x exp [− x3 /24 − x/2(ω/ω⋆ )2 ]0

where ν0 = 1/(πv). In particular,(1,
ν(ω )/ν0 =6π 31/6 /Γ2 (1/6) (ω/ω⋆ ) ≈ 0.73 ω/ω⋆ ,( ω ≫ ω ⋆ ),( ω ≪ ω ⋆ ).

(F.16)The linear frequency dependence at low frequencies reflects the finite particle density of states ν̃(0) of the Schrödinger problem at E = 0 due to the relation ν(ω ) =2ω ν̃(ω 2 ), which connects the waves’ and particles’ densities of states. The finitenessof ν̃(0) is a precursor of the Lifshits tail of states at E < 0 , which is an inaccessible
region for waves. Note that the overall frequency dependences of ⟨r ′ (ω )⟩ + 1 andν(ω ) are different.
Note that Eqs. (F.1) and (F.15) alternatively read
(F.17)⟨γ(ω )⟩ R⋆ = −Im f ′ (s = 0) ,d ′ν(ω )/ν0 = −ω⋆ Re
f ( s = 0) ,(F.18)dω
respectively. Actually, the relation of the Lyapunov exponent ⟨γ(ω )⟩ and the densityof modes ν(ω ) to a common analytic function in the upper complex plane is valid
beyond the Gaussian white-noise model. It is a consequence of the Herbert-JonesThouless relation [166] for any potential with translational invariance,γ( E) − γ0 ( E) =Z
dE′ ln | E − E′ | ν̃( E′ ) − ν̃0 ( E′ ) .

(F.19)√√
Here, γ0 ( E) = Θ(− E) − E/v and ν̃0 ( E) = Θ( E)/(2πv E) are for a free particle.
In the presence of a random potential, γ( E) and ν̃( E) are the respective functions
averaged over the disorder realizations, and the translational invariance property
refers to the correlation function of the potential. Equation (F.8), which relates the
Lyapunov exponent with the average the Ricatti variable, also holds for a translationally invariant disorder potential, under an additional assumption of inversion
symmetric disorder [131].

101

Appendix G

Heat Current in the Keldysh
Formalism
In this appendix, we derive the heat current mediated through the quantum Rabi
model using the picture of the spin-boson model (5.10). The heat current from theheat bath r into the central system is defined as
Jr ≡ −

dHB,r
.
dt

(G.1)

Using the Heisenberg equation, the heat current operator readsσz†
Jr = −ih̄ ∑ Λr,k Ξr,k − Br,k + Br,k
.
2 k

(G.2)

Then, the expectation value of the heat current is⟨ Jr (t)⟩ = −

D

E
ih̄
†
Λ
Ξ
σ
(
t
)
−
B
(
t
)
+
B
(
t
)
,
z
r,k r,k
r,k
r,k
2 ∑
k

(G.3)

where O(t) = e+iHsb t/h̄ O e−iHsb t/h̄ and ⟨O⟩ = tr[ρ0 O] with the initial density matrix
ρ0 . Here, we assume that the initial density matrix can be described by productstates:ρ0 = ρTLS ⊗ ∏r

e− βr HB,r
,
ZB,r

(G.4)

where ρTLS is the density matrix of the two-level system, β r = 1/(kB Tr ) is the inverse
temperature of the structured heat bath r, and ZB,r is the partition function of thestructured heat bath r. The heat current can be expressed in terms of Green functionsas"#⟨ Jr (t)⟩ = Re

∑ Ξr,k Λr,k Gσ<,B (t1 , t2 )
k

z

,

†
r,k

(G.5)

t1 = t2 = t

where Gσ< ,B† (t1 , t2 ) is the lesser component of the Green function defined by
z

r,k

i
G<
A,B ( t1 , t2 ) ≡ − ⟨ B ( t2 ) A ( t1 )⟩ .
h̄

(G.6)

102

Appendix G. Heat Current in the Keldysh Formalism

F IGURE G.1: Keldysh contour C = C+ ∪ C− . C+ and C− are forward
and backward branches, respectively.

For later, we introduce other components of the Green function, the retarded andadvanced components,i
GRA,B (t1 , t2 ) ≡ − θ (t1 − t2 ) ⟨[ A(t1 ), B(t2 )]⟩ ,h̄iGAA,B ( t1 , t2 ) ≡ + θ ( t2 − t1 ) ⟨[ A ( t1 ), B ( t2 )]⟩ ,
h̄

(G.7)
(G.8)

respectively, where θ (t) is the Heaviside step function.
Now, we move to the Keldysh contour and calculate the lesser component using
the Keldysh formalism. As shown in Fig G.1, the Keldysh contour is composed offorward and backward branches, where the forward branch, C+ , is running from t =−∞ to t = +∞ and the backward one, C− , is going back in the opposite direction.
First, we introduce the Green function defined on the Keldysh contour asi
G A,B (τ1 , τ2 ) ≡ − ⟨ TC A(τ1 ) B(τ2 )⟩ ,
h̄

(G.9)

where τ is the Keldysh time defined on the Keldysh contour and TC is the Keldysh
time-ordered product operator. The formal expansion with respect to the coupling
providesGσz ,B† (τ1 , τ2 ) = −
r,k

E
i D
†
TC SC σ̃z (τ1 ) B̃r,k(τ2 ) ,
h̄(G.10)

where Õ(τ ) = e+i(HTLS +HB )τ/h̄ O e−i(HTLS +HB )τ/h̄ is the time evolution in the interaction picture. The S-matrix on the Keldysh contour SC is∞(−i )nn=0 n!

SC = ∑

ZC

du1 · · ·ZC

dun H̃I (u1 ) · · · H̃I (un ),(G.11)

R
where the Keldysh time u and C du indicates a line integral along the Keldysh contour. By using the Wick’s theorem [167], the Green function can be reduced toΛr,kGσz ,B† (τ1 , τ2 ) =
r,k
2

Z
C

du Gσz ,σz (τ1 , u) gBr,k B† (u, τ2 ),r,k(G.12)

where gBr,k ,B† (τ, τ ′ ) is the Green function of the isolated structured heat bath r.
r,kNow, we back to the real-time axis from the Keldysh contour by using the Langreth
rule [168]. The lesser component can be expressed asZh̄Λr,k ∞ ′
<
R
′ <
′
<
′ A
Gσ ,B† (t1 , t2 ) =dt Gσz ,σz (t1 , t ) gB ,B† (t , t2 ) + Gσz ,σz (t1 , t ) gB ,B† (t′ ,t ) .
2z r,kr,k r,kr,k r,k2−∞(G.13)

Appendix G. Heat Current in the Keldysh Formalism

103

Therefore, by plugging this expression into Eq. (G.5), the heat current can be written
as
Z ∞
i
∂ h R
′
dt′ (−i )Gσz ,σz (t1 , t′ )Σr< (t′ , t2 ) + Gσ<z ,σz (t1 , t′ )ΣA,⟨ Jr (t)⟩ = Rer ( t , t2 )∂t2
t1 = t2 = t−∞(G.14)′
where Σr< (t, t′ ) and ΣAr ( t, t ) are the lesser and advanced components of the selfenergy,Σr< (t, t′ ) ≡

∑
k

′
ΣA
r ( t, t ) ≡∑kh̄Λ2r,k2h̄Λ2r,k2ih̄′g<† ( t, t ) = −Br,k ,Br,k
2

Z ∞
0

′

dω e−iω (t−t ) Ieff,r (ω )nr (ω ),

ih̄
′
gA
θ (t − t′ )† ( t, t ) =Br,k ,Br,k
2

Z ∞
0

′

dω e+iω (t−t ) Ieff,r (ω ),(G.15)(G.16)

with the Bose-Einstein distribution, nr (ω ) = 1/(e βr ω − 1), in the structured heatbath r. Here, we have used the definition of the spectral density (2.38). Finally, by
performing the Fourier transformation, we obtain the formula for the heat current,
Eq. (5.16) in the main text.

105

Appendix H

Mapping to the Spin-boson ModelIn this appendix, we provide the detail of the mapping from the dissipative quantum Rabi model (5.4) to the spin-boson model (5.10) with the structured spectral
density (5.14).
Let us start with the dissipative quantum Rabi model given as

H = HS + ∑ (Hres,r + Hbath,r ) + ∑ (HS-res,r + Hres-bath,r ) ,
r

h̄∆
σx ,
2
1
P2
Hres,r = r + Mr Ω2r Xr2 ,2mr,0 2HS = −

Hbath,r = ∑
k

(H.1)

r

(H.2)
(H.3)

p2r,k

1
2 2
+ mr,k ωr,k
xr,k
2mr,k
2

!
,

(H.4)

where HS , Hres,r , and Hbath,r describe the two-level system, the harmonic oscillatorr (= L, R), and the heat bath r, respectively. For the harmonic oscillator r, the Hamiltonian is rewritten using a natural frequency Ωr , a mass Mr , a momentum operator
Pr , and a position operator Xr . The heat bath r is described by a collection of theharmonic oscillators, where its Hamiltonian is similarly rewritten by using a natural
frequency ωr,k , a mass mr,k , a momentum operator pr,k , and a position operator xr,k .
The coupling between the two-level system and the harmonic oscillator is describedby1
HS-res,r = − Cr σz Xr ,
2

(H.5)

where Cr denotes a coupling strength. Similarly, the coupling between the harmonic
oscillator and the heat bath is by

Hres-bath,r = − ∑ cr,k Xr xr,k ,

(H.6)

k

where cr,k denotes a coupling strength. It is easy to check that the Hamiltonian (H.1)
reproduces the original form given in the main text (5.4) by introducing annihilation
operators defined as
r


Mr Ωr
i
ar =
Xr +
Pr ,
(H.7)
2h̄
Mr Ωr
r


mr,k ωr,ki
br,k =
xr,k +
p
.
(H.8)
2h̄
mr,k ωr,k r,k106

Appendix H. Mapping to the Spin-boson Model

We note that the coupling strengths, Cr and cr,k , are related to the parameters usedin the main text as
gr = − √Cr,
8h̄Mr Ωr

λr,k = − pcr,k,4Mr Ωr mr,k ωr,k(H.9)

respectively.
Now, we consider a composite system composed of the harmonic oscillator r andthe heat bath r, whose Hamiltonian is given as

Hres+bath,r = Hres,r + Hbath,r + Hres-bath,r .

(H.10)The key idea of the present mapping is that the Hamiltonian Hres+bath,r is exactly
solvable because it only includes the harmonic oscillators. Indeed, the Hamiltonian
Hres+bath,r can be diagonalized by the canonical transformation [169, 170] as!
p̃2r,k1
2 2
Hres+bath,r = ∑+ m̃r,k ω̃r,k
x̃r,k ,
(H.11)2m̃
2r,kkwhere p̃r,k and x̃r,k are new momentum and position operators, respectively. This
expression indicates that the composite system can be regarded as an effective heatbath for the spin-boson model. Then, the coupling between the effective heat bathand the two-level system is obtained by rewriting HS-res,r with the new operator x̃r,kas
1
H̃S-bath,r = − σz ∑ c̃r,k x̃r,k ,2
k(H.12)

where c̃r,k is a coupling strength between the two-level system and the effective heatbath. This coupling form between the two-level system and the heat bath takes thebilinear form after the canonical transformation. Thus, the resultant model becomes
the spin-boson model. By using the relation∑ c̃r,k x̃r,k = Cr Xr ,(H.13)kthe spectral density of the effective heat bath can be written byIeff,r (ω ) =Cr2Im [ Gr (ω )] .
πh̄

(H.14)

Here, Gr (ω ) is the Fourier transformation of a correlation function of the harmonic
oscillator
Gr (t) = −1
θ (t) ⟨[ Xr (t), Xr (0)]⟩bath,r ,ih̄(H.15)

where Xr (t) and ⟨ Xr ⟩bath,r indicate the time evolution and the thermal average withrespect to Hres+bath,r . The linear response theory allows the Fourier transformationof the correlation function to find [17]Gr (ω ) =

1
1
,
2
2
Mr Ωr − ω − iωγr (ω )(H.16)

Appendix H. Mapping to the Spin-boson Model107with the friction kernel
γr (t) = θ (t)c2r,k1
cos(ωr,k t).
2Mr ∑k mr,k ωr,k

(H.17)

Assuming the heat baths of the original model as the Ohmic baths (5.9), the Fouriertransformation of the friction kernel reads constant, γr (ω ) = 2πηr Ωr = 2Γr . Therefore, the effective spectral density (H.14) is calculated as
Ieff,r (ω ) =

Cr2
2Γr ω
.
2
2
πh̄Mr (Ωr − ω )2 + (2Γr ω )2

Using Eq. (H.9), we obtain Eq. (5.14) in the main text.

(H.18)

109

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