Energy Transfer at the Molecular Scale: Open Quantum ... · the onset of nonlinear thermal current...

Energy Transfer at the Molecular Scale:Open Quantum Systems Methodologies

by

Claire X. Yu

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Chemistry

University of Toronto

c© Copyright 2013 by Claire X. Yu

Abstract

Energy Transfer at the Molecular Scale:

Open Quantum Systems Methodologies

Claire X. Yu

Doctor of Philosophy

Graduate Department of Chemistry

University of Toronto

2013

Understanding energy transfer at the molecular scale is both essential for the design

of novel molecular level devices and vital for uncovering the fundamental properties of

non-equilibrium open quantum systems. In this thesis, we first establish the connection

between molecular scale devices – molecular electronics and phononics – and open quan-

tum system models. We then develop theoretical tools to study various properties of

these models. We extend the standard master equation method to calculate the steady

state thermal current and conductance coefficients. We then study the scaling laws of the

thermal current with molecular chain size and energy, and apply this tool to investigate

the onset of nonlinear thermal current - temperature characteristics, thermal rectifica-

tion and negative differential conductance. Our master equation technique is valid in the

“on-resonance” regime, referring to the situation in which bath modes in resonance with

the subsystem modes are thermally populated. In the opposite “off-resonance” limit, we

develop the Energy Transfer Born-Oppenheimer method to obtain the thermal current

scaling without the need to solve for the subsystem dynamics. Finally, we develop a

mapping scheme that allows the dynamics of a class of open quantum systems contain-

ing coupled subsystems to be treated by considering the separate dynamics in different

subsections of the Hilbert space. We combine this mapping scheme with path integral

numerical simulations to explore the rich phenomenon of entanglement dynamics within

ii

a dissipative two-qubit model. The formalisms developed in this thesis could be applied

for the study of energy transfer in different realizations, including molecular electronic

junctions, donor-acceptor molecules, artificial solid state qubits and cold-atom lattices.

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Dedication

To my parents

iv

Acknowledgements

First and foremost, I would like to thank my PhD advisor, Professor Dvira Segal, for

her encouragement, motivation and guidance. Her passion for research and her can-do

spirit are infectious. On top of all things technical, she taught me that the value of

derivations and calculations lies not in their complexity, but the messages they deliver

and the insights they provide.

I would like to thank my supervisory and examination committee members, Professor

Raymond Kapral, Professor Gregory Scholes, Professor Stuart Whittington, Professor

Jeremy Schofield and Professor Abhishek Dhar. Their stimulating questions and fresh

perspectives helped me appreciate both the depth and breadth of the subject.

I am grateful for the graduate opportunity fund from the Department of Chemistry.

It funded a research stay at the University of Basque Country, which led to the project

described in Chapter 3 of this thesis. I am thankful to the host, Professor Lian-Ao Wu,

for tirelessly answering my numerous questions and for connecting me with researchers

in other fields. I am grateful to Ben-Qiong Liu and Heng-Na Xiong for the friendship we

started during the stay.

I am grateful to my peers in the Chemical Physics Theory Group, present and past,

for their inspiration, companion and friendship. I would like to express my appreciation

to Lena Simine, who affected me with her curiosity and energy, and I am indebted

to her for making me a well-rounded person. I am truly grateful to Yaser Khan for

the numerous discussions on science, teaching and on life since my early undergraduate

years. Words can’t describe how fortunate I am to have him with me on this journey.

I thank Salil Bedkihal for the frequent discussions on recent scientific literature and for

sharing his pursuit for mathematical rigor. I am thankful to Cyrille Lavigne and Crystal

Chen for their infectious positive outlooks and for the stimulating discussions we had on

everything. I’d like to thank Zaheen Sadeq for his comradeship, and also Amro Dodin,

Ali Nassimi and Bryan Robertson for enriching my experience.

v

On my journey in science, several people have influenced me early on. I am forever

beholden to my long-time good friend Yi-Ran Wu for giving me the urge and the confi-

dence to embark on this journey, for sharing her passion and talent, and for showing that

a person can be unstoppable in the face of great adversity. I am grateful to my friend

and high school classmate Yu-Hong Yao for having encouraged me to pursue higher edu-

cation outside of my home country and for the comradery in our studies of physics. I was

fortunate to have had Joanne Yu as my demonstrator for the second year undergraduate

inorganic chemistry labs. She shaped my scientific thinking by driving me to read beyond

the written text and to never stop the inquisition into the whys and hows.

The graduate school experience became more colorful because of the on-campus or-

ganizations. I am grateful to have met wonderful people at the Hart House Singers and

the Hart House Chorus: Sam Huang, Ernest Ho, Henry Seeto, Constence Hsu, Jiayin

Sun, Amy Qu, Andrew Martinez, Usheer Kanjee, Jason Kereluk, Andree Monette, Allan

Olley, Al Thai, Ling Chen, Tania Lim, Gabrielle Woo, Quincy Poon, Vicki Peter, Rachel

Levy-Johnson, Angela Yoo, Jeremy Zung and many others. I am especially thankful

to Sam Huang and Ernest Ho for motivating me to attend physics colloquia, for the

countless discussions on physics, neuroscience and engineering, and for their continuing

friendship and comradery.

I would like to thank the amazing support staff at Department of Chemistry: Anna

Liza Villavelez, Nina Lee, Stefanie Steele, Violeta Gotcheva, Susan Arbuckle, Kelvin Ahn

and Denise Ing. Their timely help allowed me to focus on my study.

I am very grateful to Professor Gilbert Walker and Dr Shan Zou for their empathy

and support during a difficult time at the end of my undergraduate year.

The acknowledgement is incomplete without expressing my gratitude towards my

loving friends Janice Hilber, Jing Hao, Jeff Huang, Jason Chiu, Yuen-Lai Shek and Kai

Huang. These amazing individuals put up with my quirkiness and fickleness, and have

always been there with me, through many ups and downs.

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I thank Peter, who encourages me when I am in doubt, leads me when I am at lost,

and gives me all the love I’ve ever sought.

In closing, I am thankful to my parents Yuan-Fang and Feng for their love, constant

support and understanding. None of these would have been possible without them.

vii

Contents

1 Introduction 1

1.1 Moore’s laws and molecular-scale devices . . . . . . . . . . . . . . . . . . 1

1.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Theoretical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Weak Coupling Limit: Master Equation Method 15

2.1 Population dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Rate constants of bath-assisted transitions . . . . . . . . . . . . . . . . . 25

2.2.1 Collection of distinguishable non-interacting particles . . . . . . . 26

2.2.2 Harmonic oscillators . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.3 Fermionic particles . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.4 Three-level systems . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3 Thermal current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4.1 Nonlinear current-temperature characteristics . . . . . . . . . . . 38

2.4.2 Thermal rectification . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Spin chains in the on-resonance regime . . . . . . . . . . . . . . . . . . . 46

2.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

viii

2.5.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5.3 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Energy Transfer Born-Oppenheimer Method 56

3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1 Review of the Born-Oppenheimer approximation . . . . . . . . . . 57

3.1.2 The Principles of the Energy-Transfer Born-Oppenheimer method 58

3.1.3 Heat current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.1 Harmonic model . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.2 Open chain model in the off-resonance regime . . . . . . . . . . . 68

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.1 Transport mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.2 Heisenberg model . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 Exact Dynamics for Interacting Systems with QUAPI 81

4.1 Method: From interacting to non-interacting . . . . . . . . . . . . . . . . 82

4.1.1 Principles of the dynamics . . . . . . . . . . . . . . . . . . . . . . 83

4.1.2 Mapping procedure for a two-qubit model . . . . . . . . . . . . . 88

4.1.3 Numerical simulation with the influence functional formalism and

QUAPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2 Application: Entanglement dynamics . . . . . . . . . . . . . . . . . . . . 95

4.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5 Summary 110

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6 Appendices 114

6.0.1 Change of time variable . . . . . . . . . . . . . . . . . . . . . . . 114

6.0.2 Derivation of equation (3.17) . . . . . . . . . . . . . . . . . . . . . 115

6.0.3 Short proof of Tr[ĴL(t)ρB] = 0 . . . . . . . . . . . . . . . . . . . . 117

6.0.4 Derivation of subspace Pauli matrices . . . . . . . . . . . . . . . . 118

Bibliography 123

x

List of Figures

1.1 Polyphenylene-based conjugated oligomer with ethynyl groups. Adapted

from [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Bis-pyridinium derivative that functions as a photoswitch: light triggers

conversion between open and closed forms. Adapted from [2]. . . . . . . 4

1.3 The self-assembled monolayer of this molecule exhibits negative differential

resistance. Adapted from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 The schematic of a metal-molecule-metal junction. The molecule is repre-

sented by a two-level-system. The L- and R-terminals are maintained at

TL and TR, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 Nonlinear heat flow in various hybrid junctions. (a)-(c): ω = 1, Ta = 2.

(d)-(f): ω = 0.1, Ta = 15. (a) and (d) are harmonic junctions, the current

is given by equation (2.69) and (2.74a); (b) and (e) are spin junctions

obeying (2.71) and (2.74b); (c) and (f) are harmonic baths-spin junctions

obeying (2.72) and (2.74c). ΓL=10, ΓR = 1 in all figures. The x axis is

∆T . Reproduced from publication [4]. . . . . . . . . . . . . . . . . . . . 42

2.2 Schematic of the thermal conductance measurement set-up for a mass

gradient loaded nanotube (refer to reference [5]). One resistor serves as

the heater while the other as the sensor. . . . . . . . . . . . . . . . . . . 43

xi

2.3 Energy structures of the isotropic XX-spin chain model with N = 2, 4, 6

and 8 units. States in the same manifold have the same number of ex-

citations, ranging form 0 to N , from the bottom to the top manifold.

n = 1 . . . 2N labels the eigenenergy states. Other parameters are � = 2

and κ = 0.1. Reproduced from publication [6]. . . . . . . . . . . . . . . . 49

2.4 (Top) Schematic of a chain of TLS with energy spacing � and hopping

coefficient κ connecting two thermal baths at temperature βL and βR.

(Bottom) Two limiting regimes. The shaded area represents occupied bath

modes, with darker color reflecting larger occupation. At low temperature,

bath modes matching the chain modes are unpopulated, this is the off-

resonance limit. In the on-resonance chain modes overlap with populated

bath modes. Reproduced from publication [6]. . . . . . . . . . . . . . . . 50

2.5 Left panel: Energy current as a function of size for the XX chain in the

resonant limit, � = 0.5, 1, and 2, bottom to top. Right panel: Energy

current as a function of spin energy gap � for N = 6. Other parameters

are κ = 0.1, Γ = 0.01, TL = 4 and TR = 2. Reproduced from publication [6]. 53

3.1 Dependence of function D = BLBRfs(�, κ) on � for chain length N =

1, 2, 3, 4. Other parameters are BL = BR = 0.05 and κ = 0.1. Reproduced

from publication [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2 Dependence of function D = BLBRfs(�, κ) on κ for chain length N =

2, 3, 4. Other parameters are BL = BR = 0.05 and � = 4. Reproduced

from publication [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3 Excitation spectra of a Heisenberg spin chain with chain size N = 4 (a)

and N = 10 (b and c). Other parameters are � = 2, κ = 0.1, δ = 0 (◦),

δ = 1 (+), and δ = 2 (dotted). Panel (c) zooms in on a portion of panel

(b), highlighting the band gap closure with increasing δ for long chain.

Reproduced from publication [6]. . . . . . . . . . . . . . . . . . . . . . . 78

xii

4.1 Two particles initially prepared in state |ψ〉 exhibit non-classical corre-

lation in measurement results, even when physically displaced from each

other. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.2 Population dynamics (a) in the single excitation subspace P and (b) in the

zero and double excitation subspace Q. The initial state is the compos-

ite Bell state (equation (4.64)) with a = 1/2. Different lines correspond

to populations at different Kondo factors K = 0.05, 0.1, 0.2, 0.3, 0.4, 0.5.

Other parameters are �1 = �2 = 0.2, δ = 0.1, γ = 0.5, T = 0.2, J = 1.

The QUAPI time step is δt = 0.25 and the memory time is τc = 9δt.

Reproduced from publication [7]. . . . . . . . . . . . . . . . . . . . . . . 102

4.3 Real and imaginary parts of the coherences in the (a) and (c): P-subspace

(single excitation), and (b) and (d): Q-subspace (zero and double excita-

tion). Different lines correspond to coherences at different Kondo factors

K = 0.05, 0.1, 0.2, 0.3, 0.4, 0.5. Other parameters are the same as in fig-

ure 4.2. Reproduced from publication [7]. . . . . . . . . . . . . . . . . . . 103

4.4 Concurrence between the two qubits as a function of time, manifesting a

steady-state bath-induced entanglement generation. Different lines corre-

spond to concurrences at different Kondo factorsK = 0.05, 0.1, 0.2, 0.3, 0.4, 0.5.

The right panel shows the concurrence dynamics in the absence of a ther-

mal environment (K = 0) for the same set of qubit parameters. Repro-

duced from publication [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5 Concurrence between the two qubits as a function of time. Different lines

correspond to concurrences with different a values in the initial state, a = 0

(full), a = 0.2 (dashed), a = 0.5 (dash-dotted) and a = 0.8 (dotted). Left

panel: K = 0; middle panel: K = 0.05; right panel: K = 0.6. Other

parameters are the same as in figure 4.2. Reproduced from publication [7]. 106

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4.6 steady-state concurrence for different initial states and system-bath cou-

pling strength. The long-time limit was taken here as t = 100. Other

parameters are the same as in figure 4.2. Reproduced from publication [7]. 107

xiv

Chapter 1

Introduction

1.1 Moore’s laws and molecular-scale devices

In our information and computation based society, advancements in science and technol-

ogy require ever faster and cheaper computing devices. Applications such as miniature

diagnostic instruments, wearable devices and portable electronics are appearing on the

horizon [8, 9, 10]. Therefore, scientists and engineers are faced with the challenge of

packing more and more computing power into smaller and smaller devices. In an ordi-

nary electronic device with semiconductor processors, the processing speed is determined

by how fast a signal is transmitted between transistors. Two strategies exist in order to

increase computing speed: one is to increase the drift velocity of electrons in the circuit,

the other is to reduce the distance between adjacent transistors [11]. The first strategy

may be achieved by reducing the amount of defects in the semiconductor or by lowering

the operating temperature. While the former has become increasingly technologically

challenging and expensive, the latter is simply impractical. Therefore, only the second

strategy of reducing transistor-transistor distance is feasible. This results in the trend

of miniaturization in the semiconductor device industry, and the physical dimension of

the electronic circuit components has been steadily decreasing. This trend is captured

1

Chapter 1. Introduction 2

by the famous empirical rule called “Moore’s first law”, which states that the density of

transistors on integrated circuits doubles approximately every two years [12].

As successful as the semiconductor devices have been thus far, it may not be enough

for future computational needs. After four decades, the exponential growth rate ob-

served by Moore’s law has started to decrease in recent years [11]. The vast dissipated

power is restricting the increase of operating frequency of the central processing units

(CPUs) [13, 8]. Technology is approaching the physical limitation of balancing the scal-

ing of power supplied and dissipation in semiconductor devices. Even more detrimental

to a future of silicon semiconductors is the significant increase in cost of the extremely

sophisticated tools needed to manipulate and form the increasingly small features of the

devices. This phenomenon is noted in “Rock’s law”, or “Moore’s second law”, which

states that the cost of semiconductor chip fabrication plant doubles for every chip gen-

eration [11]. Consequently, the economics of chip production may ultimately become a

limiting factor to the growth of the semiconductor industry. As a result, a completely

different paradigm, independent of semiconductor processors, will be called for. The

emerging technology of molecular electronics [8, 9, 10, 11] and its counterpart utilizing

thermal components, the phononics [14], may be the solution.

While the efforts to miniaturize silicon semiconductor processors represent the top-

down approach, involving processes such as drilling, milling and molding, molecular

electronics and phononics are based on a drastically different idea, the “bottom-up”

approach [11, 9]. In 1959 Richard Feynman gave a visionary lecture titled “There is

plenty of space at the bottom” at an American Physical Society meeting [15]. At the

time, computers were still large enough to fill an entire room yet Feynman saw no rea-

son why they could not be made smaller. While his proposal to miniaturize computers

and machines was to manufacture smaller tools that manufacture even smaller tools by

means of cutting, molding, drilling, and soldering, that is, with the “top-down” method;

he also envisioned the use of electron microscopy to make patterns, arrange atoms at


10

throughout the molecules. Good examples of conjugated oligomers are polyphenylene

and polyphenylene-based molecules [7].

Polyphenylene and Polyphenylene-based Molecules

The delocalization in benzene can be extended to other adjacent atoms. For

example, one can bind benzene rings to each other (Figure 1.3a), forming a chain-like

structure called polyphenylene. Borrowing the idea of diblock copolymers1 from bulk

organic materials, one can also insert other types of molecular groups into a

polyphenylene chain, e.g., singly-bonded aliphatic groups (-CH2-CH2-), doubly-bonded

ethenyl groups (-HC=CH-), and triply-bonded ethynyl groups (-CΞC-) to obtain

1 Copolymer refers to the combination of two different monomers, A and B; diblock

indicates that they are combined in the ratio of one monomer each to make the new unit

cell (A+B). Diblock copolymer opens up the possibility of creating polymer

heterostructures with unique band structures.

Figure 1.3 (a) Polyphenylenes and (b) polyphenylene-based

conjugated oligomers.

(a)

(b)Figure 1.1: Polyphenylene-based conjugated oligomer with ethynyl groups. Adaptedfrom [1].

will, and synthesize molecules by putting atoms down where a chemist wishes them to

bond. Being able to see and manipulate matters on the atomic level utilizing molecular

self-assembly and recognition is the essence of the “bottom-up” approach. At present,

with the advancement of synthesis techniques, chemists have the ability to build complex

structures by assembling simple, elementary pieces [1, 8, 9]. In the context of molecular

electronics and phononics, the goal is to assemble sets of molecules to build structures

with properties and functionalities equal to and eventually surpassing those of present-

day microscopic and mesoscopic electronics. The introduction of instruments that are

capable of taking measurements at the molecular scale, such as the Scan Tunneling Micro-

scope (STM) and Atomic Force Microscope (AFM), as well as the availability of organic

synthesis and self-assembly methods, has opened up endless possibilities [16, 17]. Novel

organic synthesis and self-assembly methods lead to the realization of novel structures,

taking them from conceptual design to reality. STM can measure the electron transport

properties of single molecules; it is a tool to verify predictions and test designs generated

from the theoretical playground [9].

In the field of molecular electronics, some proposals for molecular structures with

functionalities analogous to elements in semiconductor devices have been realized by or-

ganic synthesis [1, 8]. These include molecular wires [1, 18, 19], switches [20, 2, 21],

rectifiers [22], transistors [23] and memory units [3, 24, 25]. One example of a molecular

conducting wire is the family of polyphenylene-based conjugated oligomers – polypheny-

lene chains with other molecular groups between neighboring benzene rings [1]. An

example with ethynyl groups is shown in figure 1.1. The delocalized π orbitals in the

aromatic rings and the triple bonds allow the electrons to be feasibly transported along


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Figure 1.2: Bis-pyridinium derivative that functions as a photoswitch: light triggersconversion between open and closed forms. Adapted from [2].

Figure 1.3: The self-assembled monolayer of this molecule exhibits negative differentialresistance. Adapted from [3].

the wire, leading to high currents. To operate, the molecular wire is connected to two

metallic (for example, Au) terminals, often through sulphur bonds, and a voltage bias is

applied [1]. This set-up is a “metal-molecule-metal” system (MMM). An example of a

molecular switch is shown in figure 1.2. Upon irradiation of UV light, it undergoes a con-

version between an open unconjugated form and a closed electron-conducting form, thus

functioning as a molecular electrical switch [2]. Negative differential resistance (NDR),

the characteristic of certain electronic components in which increasing voltage leads to

decreasing electrical current, has been shown to lead to an hysteresis behavior, which is


useful for information storage applications [26, 27]. A self-assembled monolayer (SAM)

of 2-amino-4-ethynylphenyl-4-ethynylphenyl-5-nitro-1-benzenethiol (shown in figure 1.3)

was reported to exhibit a high level of NDR [3].

Another direction in the miniaturization of computing devices is the development

of phononics, devices that utilize heat to carry and process information [8]. Proposals

have been put forward for controlling phonons, the quasi-particles of heat, similar to

controlling electrons and photons [14]. This turns out to be difficult experimentally:

because of the lack of a phonon charge, it cannot be manipulated with electrical fields.

However, theoretical works have already laid out designs for thermal diodes [28, 29, 30],

thermal transistors [31], thermal logic gates [32], and thermal memories [25].

1.2 Modeling

In our work, we concern ourselves with the problem of vibrational, electronic or radiative

heat flow in microscopic systems. We design theoretical models, develop analytical and

numerical tools to study the dynamics of quantum systems immersed in a macroscopic

environment in the context of thermal transport across molecular junctions. Such studies

are relevant to both molecular electronics and phononics [33]. They may also be applied

to energy transfer studies in proteins and other biomolecules [34], and to the study

of exciton transfer in semiconductors or biomolecules [35]. For molecular electronics,

electron energy dissipation during device operation may result in instability and even the

disintegration of the device [13, 36, 37, 38]. Thermal conductance studies may be used to

design cooling solutions to effectively remove excess heat from the system. For phononics,

the microscopic mechanism of desired phenomena can be uncovered by learning how heat

is transported through the system. One example is thermal rectification, which is the

phenomenon of the magnitude of thermal current being dependent on the sign of the

temperature gradient. Our study aims to answer some practical questions: What are the


possible mechanisms of heat conduction through a molecular scale system? What factors

affect the rate of heat conduction? What criteria does a system need to satisfy in order

to exhibit desired functionalities, such as thermal rectification and negative differential

thermal conductance? More fundamentally, this study translates into the exploration of

transport behaviour in open quantum systems: How do we evaluate the energy transport

characteristics of a system, quantum mechanically? What is the role of quantum effects

in molecular scale heat transport? Knowing the Hamiltonian describing the system, can

we predict the onset of certain phenomena, and the scaling behavior of the current with

system size and energetics? These questions are at the intersection of different topics, the

scope spanning quantum mechanics, quantum dynamics and quantum transport, as well

as statistical mechanics and thermodynamics. These topics also establish the groundwork

for quantum information processing, quantum computation and quantum device design.

We aim to uncover rich physical phenomena in elementary models with straightfor-

ward methods. The hope is that simple models will capture the essential features of a

realistic system while not overwhelming us with its intricate structural detail, and that

simple methods will produce transparent results, unobstructed by the cumbersome math-

ematical details of more involved methods. Motivated by the MMM set-up for molecular

electronics, the physical object studied in this work is a molecular junction, consisting

of a molecular wire connected to two meso- or macroscopic terminals. The terminals

are not necessarily metals – they may be made of solids or magnets, for example. We

are interested in the dynamics and transport properties of the wire, while the dynamical

details of terminals are not important for us. In the theory of open quantum systems,

the molecular wire forms an open system (“subsystem”), interacting with the external

environment formed by the two terminals. Instead of adopting a set-up consisting of a

subsystem connecting to only one terminal, for which the subsystem and environment

may eventually reach equilibrium, we are interested in the more involved non-equilibrium

set-up with two terminals prepared and maintained at different temperatures. To allow


theoretical treatment, we abstract the molecular wire as a single or a linear series of

central units, and each terminal is represented by a large collection of particles. While

the molecular chain consists a small number of atoms, the number of particles in the

terminals is huge. If one treats the entire MMM junction as a closed system, for which

each degree of freedom is considered, the study of dynamics soon becomes intractable.

Therefore in an open system, the subsystem and the environment are not treated with

the same level of detail. In particular, the influence of each environmental degree of

freedom is usually traced out. This is one feature of the open quantum system theory. A

short overview on established theoretical methods for describing open quantum systems

can be found in section 1.3. In this thesis, the words “system” and “subsystem” are used

interchangeably when the context is clear, referring to the small entity of interest.

For an open quantum system, the total Hamiltonian is typically divided into three

parts, describing the isolated system, the baths, and the system-bath interaction [39],

H = HS +HB +HSB. (1.1)

Here we call the environment a “bath” because in our applications it is usually maintained

at a fixed temperature, therefore representing a heat bath. The Hamiltonian HS contains

only subsystem operators, HB comprises bath operators, while HSB includes system-bath

coupling terms. Furthermore, in a two-terminal scenario, if the terminals do not interact

directly with each other but only through the mediation of the subsystem, then HB and

HSB may sometimes be divided into contributions at each terminal

HB = HL +HR, HSB = VL + VR. (1.2)

The exact form of each Hamiltonian is chosen according to the physical properties of

each component, as shown below. Because our aim in this work is to uncover thermal

properties of molecular junctions with simple models, details of the molecules, such as


their chemical composition and geometric arrangement are ignored. Particles that make

up the system and bath, which can be atoms, molecules or clusters, are simply represented

by a number of energy levels.

We now describe several standard models for the subsystem and the environment.

Beginning with the subsystem, one physically relevant model is the harmonic oscillator,

with the subsystem consisting of a collection of independent harmonic oscillators,

HHO =∑i

h̄ωib†ibi. (1.3)

Here i labels the oscillator mode, and b†i and bi are the bosonic creation and annihilation

operators for mode i, respectively. This model is used, for example, to represent the dis-

crete vibrational modes (phonons) of a solid lattice, or to provide a quantized description

of a radiation field. This simple model enjoys analytical solutions for many quantities,

such as its quantized energy eigenvalues and its partition function.

When the temperature is low relative to the modes frequency, higher excited states are

inaccessible and therefore only the lower levels contribute to the dynamics. In this case,

one may consider truncating the energy spectrum to include only the lowest states, the

ground and the first excited state, which forms the two level system (TLS) model. If we

consider a single oscillator mode and truncate it, then we obtain the TLS Hamiltonian,

given by

HTLS = �0 |0〉〈0|+ �1 |1〉〈1| , (1.4)

where �0 and �1 are the energies of the ground state and the excited state, respectively.

One can easily go beyond the single unit description for the subsystem and set up a

chain-like structure. The chain models Adapted in this work are based on the Heisenberg

model, a series of TLSs with nearest-neighbor couplings

HHeisenberg = −1

2

N∑j=1

(−hσzj + Jxσxj σxj+1 + Jyσyjσ

yj+1 + Jzσ

zjσ

zj+1). (1.5)


Here σij are the Pauli matrices of the j−th particle. The first term describes the on-site

energy of each particle, that is, the energy gap between its ground and first excited states.

The next three terms describe nearest-neighbor interactions. When Jz = 0 and Jx = Jy,

the model reduces to the isotropic XY model, or the XX model for short. The XX model

is studied in chapters 2 and 3, where the scaling of thermal current with the chain length

as well as with h and Jx, Jy, Jz is worked out under different environmental conditions.

A thermal bath may be modeled with a collection of harmonic oscillators, whose pop-

ulation satisfies some statistical distribution, for example, the Boltzmann distribution,

indicative of an equilibrium temperature. In other applications, the baths may stand

for metals. A metal can be represented as a collection of fermionic particles, with the

energetics,

Hfermion =∑k

�kf†kfk, (1.6)

where f †k and fk are the spinless fermionic creation and annihilation operators to create

and destroy an electron with momentum k. Note that because we do not apply magnetic

fields in this work, the spin degrees of freedom do not contribute to the dynamics. There-

fore for simplicity, we do not include spins when considering electronic Hamiltonians.

The system-bath interaction Hamiltonians in this work all take on a simple bilinear

form

Vν = λνSBν , (1.7)

where ν = L,R denotes the L-and R-terminal, the c-number λν characterizes the strength

of the interaction, and S and Bν are the subsystem and bath operators, respectively.

For example, when the system and bath couple through their displacements from the

uncoupled equilibrium position, the interaction Hamiltonian can be written in the second

quantization notation as

Vν = (b†0 + b0)

∑j

λνj(b†νj

+ bνj). (1.8)


TL ; µL

J

TR ; µR

Figure 1.4: The schematic of a metal-molecule-metal junction. The molecule is repre-sented by a two-level-system. The L- and R-terminals are maintained at TL and TR,respectively.

Here λνj characterizes the system-bath coupling strength for each bath mode. In equa-

tion (1.7), the mode dependence is absorbed in the operator Bν . When the subsystem is

truncated to a two-level system, the corresponding interaction becomes

Vν = σx∑j

λνj(b†νj

+ bνj). (1.9)

In equations (1.8) and (1.9) we ascribed a coupling strength λνj to each mode. This

information can be summarized in the function of spectral density of ν bath, describing

the coupling strengths between the subsystem and every bath mode j

Jν(ω) =∑j

λ2νjδ(ω − ωνj), (1.10)

where ωνj is the frequency of the ν bath mode j.

A schematic of an MMM junction is shown in figure 1.4. It shows the microscopic

process of thermal energy transfer: one energy quantum is emitted from the L- terminal

and absorbed by the subsystem; it is subsequently emitted from the subsystem and

absorbed by the R-terminal. The net result is that one energy quantum is transferred

from the L- to the R-terminal, mediated by the subsystem.


1.3 Theoretical methods

In this section, we provide a concise review of several theoretical methods employed for

open quantum system thermal transport studies. Closed system dynamics is often ob-

tained by solving the Schrödinger equation for the system as a whole [40]. The dynamics

can be obtained, for example, by expressing the wavefunction at time t as a linear com-

bination of time-independent basis set functions with time-dependent coefficients and

then solving for these coefficients. However, the number of equations to solve scales

exponentially with the number of degrees of freedom. For an open system, the poten-

tially large number of environmental degrees of freedom renders the dynamical problem

intractable. Therefore, it cannot be settled simply by solving for the wavefunction from

the Schrödinger equation directly. The multilayer multiconfiguration time-dependent

Hartree (ML-MCTDH) theory [41, 42] overcomes the tractability issue by adopting time-

dependent basis functions (the “configurations”) and optimizing both the coefficients and

the basis functions using the variational principle. The time-dependent basis functions

are expanded by another set of basis functions and are also optimized via the variational

principle. This other set of basis functions can again be expanded and optimized and

so forth, forming multiple layers of basis functions. This scheme is capable of evolving

hundreds of wavefunctions in the transient regime. The advantages of this method are

that it is non-perturbative and numerically exact, provided that convergence is reached

with the application of sufficient number of layers. However, limited by the computing

power required to solve its equations of motion, applying the ML-MCTDH approach to

problems requiring long-time simulations is difficult. Note that the ML-MCTDH essen-

tially treats the entire system as one closed system, describing all degrees of freedom

in the same wavefunction, without making the distinction between the system and the

environment.

The master equation technique, on the other hand, evolves in time only the subsys-

tem reduced density matrix. This approach is typically based on the assumption that


the environment consists of an infinite number of degrees of freedom with a continuous

energy spectrum, and that the environment maintains an equilibrium state, described by

a certain thermal distribution [43]. The environment influences the subsystem through

the system-bath coupling, and due to its large size, the equilibrium state is not disturbed

by the back reaction from the subsystem. For this reason, the environment is regarded

as a thermal bath. A standard derivation of the Pauli master equation is presented

in Chapter 2. The starting point in this formalism is the Liouville equation of motion

for the full density matrix. It eventually provides closed equations of motion for the

subsystem populations, which are the diagonal elements of the reduced density matrix.

The equations are derived in a perturbative manner, often only up to the weak system-

bath coupling limit, assuming Markovian dynamics and using the secular approximation

(SA), ignoring the contribution from coherences to the population dynamics. The equa-

tions derived without SA are called the Redfield equations [43, 44, 45]. Non-Markovian

dynamics may be obtained by the master equation method by applying the projection

operator techniques [46, 47, 48]. Although it is possible to go beyond the weak coupling

limit, the master equation method is a perturbative method and therefore requires some

other parameter of the subsystem energetics to be small [49, 50]. While results obtained

from the master equations are only approximate, the method enjoys some advantages: It

is very versatile, capable of describing many different physical scenarios, as we show in

Chapter 2. Furthermore, analytical solutions can be obtained for simple models, clearly

revealing the functional dependence of observables on subsystem and bath parameters.

Therefore, the master equation method offers the transparency that many other methods

do not provide. Complementary to the master equation method, in Chapter 3 we develop

the Energy Transfer Born-Oppenheimer (ETBO) technique, which captures higher order

transport processes in a regime not suitable for the use of the master equation.

Tracing out the bath and extracting the dynamics of only a particular component is

practiced even in the context of classical mechanics. The most notable example is the


“Brownian motion” of large particles among much smaller solvent particles [51]. The

solute motion can be described by the “Langevin equation” [52], which is essentially an

equation of motion of the solute, affected by a dissipative friction force and a stochastic

force exerted by the solvent particles. In this treatment, the motion of each solvent

particle is not followed. Rather, one assumes that the solvent particles effectively drive

the solute in a stochastic manner. This formalism has been extended to the quantum

domain: (i) one can derive a generalized Langevin equation (GLE) for the quantum

mechanical coordinate operator of the subsystem [53, 54, 55], and (ii) one can arrive at

the quasiclassical Langevin equation (QLE) by accounting for the quantum statistical

fluctuations of the environment in the stochastic force, while retaining the form of the

classical Langevin equation [56, 57]. This QLE method may be used to obtain the thermal

current in the steady state limit [58]. Used in conjunction with molecular dynamics (MD)

simulations, it can produce the quantum thermal transport characteristics in both low and

high temperature limits, demonstrating crossover from ballistic to diffusive transport [59].

One should note that this formalism is applicable only when the environment is near

harmonic.

The Green’s function formalism has also been applied to thermal transport prob-

lems [60, 61, 62]. The corresponding method is the non-equilibrium Green’s function

(NEGF) technique, which evolves the Green’s functions under the full Hamiltonian along

the forward and backward time contour. Physical observables such as the thermal current

can be obtained by expressing them in terms of the Green’s functions. Both the transient

dynamics and the steady state limit are accessible through this method [62]. Aided by

diagrammatic expansions, the observable dynamics can be calculated to arbitrary order.

Therefore, the Green’s function is applicable to the strong system-bath coupling regime.

A drawback of the Green’s function method is that it is mathematically cumbersome,

and numerical studies are essential to reveal the parametric dependence of observables.

Furthermore, the extension to complex systems may be difficult.


Finally, we mention that different flavors of the path integral formalism have found

applications in the study of quantum dissipative systems [63, 64, 65, 66]. In Chapter 4

we apply a numerically exact method based on the discrete time path integral to treat

an open system with coupled subsystems.

1.4 Organization of the thesis

The main body of this thesis includes three separate theoretical methods and their ap-

plication to the open quantum system dynamics. In Chapter 2 we adopt the master

equation technique and use this standard approach to study heat current characteristics

in molecular junctions. In Chapter 3 we describe the Energy Transfer Born-Oppenheimer

method, suitable for the off-resonance regime. This method is complementary to the mas-

ter equation tool. In Chapter 4 we use the numerically exact quasi-adiabatic path integral

formalism to study the dynamics of entanglement within the subsystem. It is facilitated

by a mapping scheme that transforms a class of open system Hamiltonians with cou-

pled internal degrees of freedom into separate open systems. At last, in Chapter 5 we

summarize and comment on outlooks for future works.

Chapter 2

Weak Coupling Limit: Master

Equation Method

The quantum master equation is the equation of motion for the reduced density matrix

[43]. In this chapter we extend the standard master equation method to study energy

transport behavior in two-terminal molecular junctions. We first present the well-known,

regular derivation of population dynamics [43], then extend the method to calculate

steady state thermal current and conductance coefficients. We apply the method to

several molecular junction setups with the aim to identify the onset of nonlinear behavior,

including thermal rectification and negative differential resistance (NDR).

There are different flavors of master equations, Markovian or non-Markovian, ex-

panded to various orders of system-bath interaction [43, 49, 50, 46]. This work focuses

on the weak coupling regime, for which the “Born approximation” is appropriate: the

interaction with the system does not affect the state of the thermal baths appreciably,

therefore the total density matrix is separable into the system and bath parts at all times

ρ(t) = σ(t)⊗ ρB(t), where σ is the reduced density matrix of the system while ρB is the

density matrix of the bath; furthermore, the master equation is derived to the second

order in the system-bath interaction. In this presentation, we also take the Markovian

15

Chapter 2. Weak Coupling Limit: Master Equation Method 16

limit, at which the bath excitations caused by system-bath interaction decay on a much

shorter time scale than the system dynamics. These approximations allow us to ob-

tain simple analytical expressions for the current-temperature characteristics of various

important systems in thermal transport studies. Compared to other flavors of master

equations which demand more complicated calculations [46, 47], the results obtained from

our derivations are transparent, allowing us to gain insight into transport mechanisms.

They may also serve as guidelines to experiments and more detailed theoretical studies.

In section 2.5 we introduce the open quantum spin chain model and study the thermal

energy transport along the chain. This model is motivated by the physical system of a

molecular wire immersed in dissipative thermal environments, but it can also be used

to describe general excitation energy transfer problems, vibrational and electronic, in

bulk-molecule-bulk junctions and donor-bridge-acceptor systems [35, 67, 68]. Relevant

studies include works on thermal transport for spin chains subject to an external magnetic

field [69, 70, 71], quantum phase transitions for spin chains in a thermal environment [72,

73], and thermal transport along spin chains [74, 75]. We apply the master equation

technique to study the thermal energy transport mechanism along an open XX spin

chain, and our findings agree with known results [74].

The plan of this chapter is as follows. First, in section 2.1 we present the standard

derivation of the Pauli master equations and show the rate constant calculations for

various bath Hamiltonians. In section 2.3, expressions for thermal current are derived.

Then, in section 2.4, the master equation technique is applied to study the phenomena of

nonlinear thermal current and thermal rectification. The thermal transport mechanism

along an open spin chain is explored in section 2.5. At last in section 2.6 we summarize.

The work summarized here was presented in our works [4, 76] and [6].


2.1 Population dynamics

In this section we present the standard derivation of population dynamics using the

master equation method. First, we study the simple situation where a central subsystem

is connected to two heat baths maintained at temperatures Tν = β−1ν , (kB ≡ 1; ν =

L,R). The Hamiltonian can be written generally as

H = HS +HL +HR + VL + VR. (2.1)

In this derivation we do not specify the details of the subsystem, the structure of the

baths, or the functional form of the system-bath coupling, except to demand that the sub-

system Hamiltonian assumes a diagonal form and that system-bath interactions assume

a separable form,

HS =∑n

En |n〉〈n|

Vν = λνSBν , S =∑n,m

Sm,n |m〉〈n| . (2.2)

Here S is a subsystem operator and Bν is an operator of the ν-th bath. It is worth

pointing out that in this simple model both thermal baths are coupled to the same

subsystem operator. This is not necessarily true in practice. For example, each bath

may be coupled to a different vibrational mode of the subsystem, say, Sα and Sβ. This

situation is not captured by this simple model. A second example where this model

does not apply is when the thermal baths are connected by an extended central system,

for example, a molecular wire, and each bath is coupled to a different unit along the

subsystem. We generalize our derivation of the quantum master equation to this second

scenario in section 2.5.

We start with the equation of motion (Liouville equation) of the density matrix under


the interaction picture

˙̃ρ(t) = −i[Ṽ (t), ρ̃(t)], (2.3)

where Ṽ (t) = eiH0tV e−iH0t with H0 = HS+HB. Here V = VL+VR is the total interaction

induced by both thermal baths. Formally integrating the Liouville Equation gives

ρ̃(t) = ρ̃(0)− i∫ t

0

[Ṽ (τ), ρ̃(τ)]dτ. (2.4)

We insert this expression back into the Lioville equation and obtain the following integro-

differential equation for ρ̃(t)

˙̃ρ(t) = −i[Ṽ (t), ρ̃(0)]−∫ t

0

[Ṽ (t), [Ṽ (τ), ρ̃(τ)]]dτ. (2.5)

One can iterate this substitution process indefinitely, however, as we show in the dis-

cussion below equation (2.14), we truncate the iteration at the second term to obtain

equations of motion which are second order in the system-bath interaction. We will omit

the tildes from now on. Next, take trace over L-and R-bath degrees of freedom and as-

sume that the mean value of the interaction Hamiltonian averaged over the initial density

matrix is zero, such that Tr[V (t), ρ(0)] = 0. We now define the reduced density matrix

and write down its equation of motion

TrB[ρ̇(t)] ≡ σ̇(t) = −iTrB[V (t), ρ(0)]−∫ t

0

TrB[V (t), [V (τ), ρ(τ)]]dτ

= −∫ t

0

TrB[V (t), [V (τ), ρ(τ)]]dτ, (2.6)

where TrB stands for tracing over all bath degrees of freedom TrB = TrLTrR, and σ(t) is

the reduced density matrix of the subsystem. The time evolution of the mn−th element


of the reduced density matrix is given explicitly by the equation

σ̇m,n(t) = −∫ t

0

TrB[V (t), [V (τ), ρ(τ)]]m,ndτ (2.7)

= −∫ t

0

TrB[V (t)V (τ)ρ(τ)− V (t)ρ(τ)V (τ)

−V (τ)ρ(τ)V (t) + ρ(τ)V (τ)V (t)]m,ndτ.

To compute each term in the commutator, we first express the time-dependent system-

bath interaction in the interaction picture. We recall that

Vν = λνSBν = λν∑n,m

Sm,n |m〉〈n|Bν , (2.8)

and thus obtain

Vν(t) = eiH0tVνe

−iH0t

= eiH0tλν∑n,m

Sm,n |m〉〈n|BνeiH0t

= λν∑m,n

Sm,neiHSt |m〉〈n| e−iHSteiHBtBνe−iHBt

= λν∑m,n

Sm,neiEmt |m〉〈n| e−iEnteiHBtBνe−iHBt

= λν∑n,m

Sm,neiEm,nt |m〉〈n|Bν(t). (2.9)

Here Em is the eigenenergy of state |m〉 of the isolated subsystem, and Em,n is defined

as Em,n = Em − En. When the subsystem is weakly coupled to the baths, the influence

of the subsystem on the bath is small. The bath density matrix ρB is only marginally

affected and the total density matrix can be approximated with a factorizable form at

any given time

ρ(t) = σ(t)⊗ ρB. (2.10)

This factorization, a consequence of weak coupling, is called the Born approximation.


Initially and thus at all times both baths are at thermal equilibrium at temperature TL

and TR. The bath density matrix is a direct product of the L-and R-density matrices [43]

ρB = ρL ⊗ ρR, (2.11)

each given by the Boltzmann distribution

ρν =e−βνHν

Trνe−βνHν; ν = L,R. (2.12)

This factorization allows one to take the reduced density matrix of the subsystem out

of the bath trace operation. The four terms in the commutators in equation (2.7) then

become

TrB[V (t)V (τ)ρ(τ)]m,n =∑ν

λ2ν〈Bν(t)Bν(τ)〉Tν∑k,l

Sm,kSk,leiEm,kteiEk,lτσl,n(τ), (2.13a)

TrB[V (t)ρ(τ)V (τ)]m,n =∑ν

λ2ν〈Bν(τ)Bν(t)〉Tν∑k,l

Sm,kSl,neiEm,kteiEl,nτσk,l(τ), (2.13b)

TrB[V (τ)ρ(τ)V (t)]m,n =∑ν

λ2ν〈Bν(t)Bν(τ)〉Tν∑k,l

Sm,kSl,neiEm,kteiEl,nτσk,l(τ), (2.13c)

TrB[ρ(τ)V (τ)V (t)]m,n =∑ν

λ2ν〈Bν(τ)Bν(t)〉Tν∑k,l

Sk,lSl,neiEk,lteiEl,nτσm,k(τ), (2.13d)

where 〈Bν(t)Bν(τ)〉Tν = Trν [ρν(Tν)Bν(t)Bν(τ)] is the two-time correlation function of the

ν bath in the interaction picture. Putting all four terms together, we obtain a differential

equation of the reduced density matrix element σm,n at time t,

σ̇m,n(t) = −∫ t

0

∑ν λ

2ν〈Bν(t)Bν(τ)〉Tν

∑k,l Sm,kSk,le

iEm,kteiEk,lτσl,n(τ)dτ

+∫ t

0

∑ν λ

2ν〈Bν(τ)Bν(t)〉Tν

∑k,l Sm,kSl,ne

iEm,kteiEl,nτσk,l(τ)dτ

+∫ t

0

∑ν λ


∑k,l Sm,kSl,ne

iEm,kteiEl,nτσk,l(τ)dτ

−∫ t

0

∑ν λ


∑k,l Sk,lSl,ne

iEk,lteiEl,nτσm,k(τ)dτ. (2.14)


Next, we make the Markovian approximation, i.e. we assume that bath excitations decay

over a time scale much shorter than the subsystem relaxation time. The Markovian

approximation involves two steps: (i) we take the reduced density matrix out of the

integral on the right side, by replacing σ(τ) by σ(t); (ii) we extend the upper limit of the

time integral from t to ∞. This is justified because the bath correlation function falls

out to zero at short time, during which the subsystem state has not changed appreciably.

Therefore, the system state can be considered constant throughout the history of the

bath dynamics.

We now practice the first part of the Markovian approximation in equation (2.14),

σ̇m,n(t) = −∫ t

0

∑ν λ


∑k,l Sm,kSk,le

iEm,kteiEk,lτσl,n(t)dτ

+∫ t

0

∑ν λ


∑k,l Sm,kSl,ne

iEm,kteiEl,nτσk,l(t)dτ

+∫ t

0

∑ν λ


∑k,l Sm,kSl,ne

iEm,kteiEl,nτσk,l(t)dτ

−∫ t

0

∑ν λ


∑k,l Sk,lSl,ne

iEk,lteiEl,nτσm,k(t)dτ. (2.15)

Consequently, the evolution of the system state at time t only depends on the present state

σ(t) instead of the entire time evolution history. We say that this equation of motion is

local in time and it is called the Redfield equation [43]. We would like to point out that the

Born and Markov approximations go hand-in-hand in the above derivation. The Born

(weak coupling) approximation is justified when the disturbance from the subsystem

to the thermal baths is insignificant so that the density matrices of the baths can be

approximated as being time-independent. This does not imply that the interactions

with the subsystem do not cause excitations in the thermal baths. Instead, the Markov

approximation specifies that the baths excitations should relax on a timescale that is

unresolved compared to the subsystem dynamics. Therefore, the use of both assumptions

together is often termed the Born-Markov approximation [43].

We have given above the general form of the master equation of the reduced density


matrix elements as a system of coupled differential equations. While the Redfield equation

can be solved numerically both in the transient regime and directly in steady state, this

is not our goal here. For our applications, we are mostly interested in the population

dynamics of the subsystem, which are given by the diagonal elements of the reduced

density matrix. So we make the secular approximation (SA), also called the rotating wave

approximation (RWA), to remove the contribution of coherences from the population

dynamics and obtain a system of differential equations in terms of populations only.

SA is justified when the time scale of the intrinsic evolution of the subsystem is short

compared to the relaxation time of the subsystem [43]. Define the characteristic time

scale of the subsystem as the inverse of a typical subsystem transition energy τs ∼ E−1m,n

and denote the subsystem relaxation time as τR. When E−1m,n � τR, or equivalently when

Em,n is significant compared to the transition rate between subsystem states kmn, terms

in the form of eiEm,nt in equation (2.15), where m 6= n, will oscillate rapidly during the

time τR over which the subsystem varies appreciably. This means that the contribution

of the non-diagonal coherence terms will be averaged out in the integral. Therefore we

may include only the diagonal terms on the right hand side of the equation. The result

is still a system of coupled equations, but it is now manageable,

Ṗn(t) ' −∫ t

0

∑ν λ


∑m |Sm,n|2eiEn,m(t−τ)Pn(t)dτ

+∫ t

0

∑ν λ


∑m |Sm,n|2eiEn,m(t−τ)Pm(t)dτ

+∫ t

0

∑ν λ


∑m |Sm,n|2eiEn,m(τ−t)Pm(t)dτ

−∫ t

0

∑ν λ


∑m |Sm,n|2eiEn,m(τ−t)Pn(t)dτ. (2.16)

Pn stands for the population of the n-th state of the subsystem, and is the diagonal

element σn,n of the reduced density matrix. We assumed that Sm,n = S∗n,m. Next,

perform a change of variable, defining t − τ as x for the lines with bath correlation

〈Bν(t)Bν(τ)〉Tν and τ − t as x for the lines containing 〈Bν(τ)Bν(t)〉Tν . This allows one


to combine terms containing the same population. Furthermore, we practice here the

second part of the Markovian approximation. Since the bath correlation time is short

compared to the relaxation time of the system, the bath correlation function does not

contribute significantly at long times. Therefore we may extend the time integration

limit to infinity and obtain

Ṗn(t) = −Pn(t)∑ν

λ2ν∑m

|Sn,m|2∫ +∞−∞〈Bν(τ)Bν(0)〉TνeiEn,mτdτ

+Pm(t)∑ν

λ2ν∑m

|Sn,m|2∫ +∞−∞〈Bν(τ)Bν(0)〉TνeiEm,nτdτ, (2.17)

where ν = L,R. To re-emphasize, this expression was derived in the weak coupling limit,

with Markovian approximation and Rotating Wave Approximation. It is valid when

the bath-induced relaxation time is much longer than the subsystem characteristic time

τR � τS. We can now define the rate constants as

kνn→m = λ2ν

∫ ∞−∞〈Bν(τ)Bν(0)〉TνeiEn,mτdτ, (2.18)

where the rate for the reversed process can be calculated as

kνm→n = λ2ν

∫ ∞−∞〈Bν(τ)Bν(0)〉Tνe−iEn,mτdτ

= λ2ν

∫ ∞−∞〈Bν(0)Bν(τ)〉TνeiEn,mτdτ. (2.19)

Here we changed variable −τ → τ and used 〈Bν(−τ)Bν(0)〉Tν = 〈Bν(0)Bν(τ)〉Tν for a

stationary process. With the rate constants, the equation simplifies to

Ṗn(t) = −Pn(t)∑ν

∑m

|Sn,m|2kνn→m +∑ν

∑m

|Sn,m|2kνm→nPm(t). (2.20)

This is the master equation for the population of state n (also called the Pauli Master

Equation), coupled to the population of all other states m. This equation has a simple


physical interpretation: the rate of change of the population of state n depends on how

fast population transfers into state n from all other states m, minus how fast population

transfers out of state n into all other states m.

We can Fourier transform the rate constant expression and express it in terms of

frequencies. In its energy representation Hν =∑

j Eνj |j〉〈j|, the bath operator Bν ,

which couples to the system, is written in the form

Bν =∑j,j′

Bνj,j′ |j〉〈j′| . (2.21)

The bath time correlator can be expanded as

〈Bν(τ)Bν(0)〉 = TrB[eiHντBνe−iHντBνρν ]

= TrB[eiHντ

∑l,l′

Bνl,l′ |l〉〈l′| e−iHντBνρν ]

=∑j

〈j|∑l,l′

eiEl,l′τBνl,l′ |l〉〈l′|∑p,p′

Bνp,p′ |p〉〈p′|e−βνHν

Trν [e−βνHν ]|j〉

=∑j

∑l,l′

∑p,p′

eiEl,l′τBνl,l′Bνp,p′δj,lδl′,pδp′,j

e−βνEνj

Trν [e−βνHν ]

=∑j,j′

Bνj,j′Bνj′,je

iEj,j′τe−βνE

νj

Zν(βν), (2.22)

where Zν(βν) = Trν [e−βνHν ] is the partition function of the ν-bath. Plugging this ex-

pression into the time domain rate expression (2.18), we obtain its frequency domain

representation

kνn→m = λ2ν

∫ ∞−∞

dτ∑j,j′

ei(En,m+Eνj,j′ )τ |Bνj,j′|2

e−βνEνj

Zν(βν)

= 2πλ2ν∑j,j′

|Bνj,j′|2δ(En,m + Eνj − Eνj′)e−βνE

νj

Zν(βν). (2.23)

Note that n,m denote states of the subsystem, but the rate expression only includes the


information about the frequency of the transition En,m. The factor Sn,m in equation (2.20)

contains information on whether or not a transition actually takes place and its weight.

The rate constant of the reverse process is given by

kνm→n = 2πλ2ν

∑j,j′

|Bνj,j′ |2δ(Em,n + Eνj − Eνj′)e−βνE

νj

Zν(βν). (2.24)

Interchanging the bath state labels j ↔ j′, it is easy to see that the rate constants of

relaxation and excitation processes are related by

kνm→n = 2πλ2ν

∑j,j′

|Bνj,j′|2δ(En,m + Eνj − Eνj′)e−βνEνj′

Zν(βν)

= 2πλ2ν∑j,j′

|Bνj,j′|2δ(En,m + Eνj − Eνj′)e−βνEn,me−βνE

νj

Zν(βν)

= kνn→me−βνEn,m . (2.25)

This relation is called detailed balance. According to equation (2.25), if n → m denotes

a relaxation process from state n to state m, then En,m > 0 and the corresponding

excitation process from state m to n m → n is exponentially slower. The exponent

depends on both bath temperature and the energy gap between the two states.

2.2 Rate constants of bath-assisted transitions

Up to now the discussion has been based on a generic two-terminal model. We have

not yet specified the explicit form of the bath, system, or the system-bath interaction

Hamiltonians. In this section, we consider several physical thermal baths and derive the

corresponding rate constants using expression (2.23). Hamiltonian Hν and bath operator

Bν are those defined in equations (2.1) and (2.2). We consider a specific transition with

an energy difference ω = En,m, and provide next closed expressions for the relaxation

rate kνn→m with n > m, where ν = L,R is the index of the bath. From now on, we denote


the rate constants without explicitly specifying states of the transition, as

kν(Tν) = kνn→m. (2.26)

Here we specify the bath temperature Tν within the rate constant for the convenience in

discussing the thermal rectification effect in section 2.4.

2.2.1 Collection of distinguishable non-interacting particles

Consider a collection of distinguishable non-interacting particles. Their energies and

interaction with the subsystem are described by

Hν |i〉ν,p = �νp(i) |i〉ν,p , Bν =

∑p

bν,p. (2.27)

Here p labels the p-th particle, i indicates the i-th state, and bν,p is an operator for

particle p in the ν-bath. Plugging the matrix element form of the bath operator |Bνi,j|2 =∑p | 〈i| bν,p |j〉 |2 into the frequency domain rate expression, we obtain the rate expression

for baths of distinguishable non-interacting particles. In terms of subsystem frequency,

the relaxation rate (2.26) is calculated as

kν(Tν) = 2πλ2ν

∑p

∑i,j

| 〈i| bν,p |j〉 |2δ(ω + �νp(i)− �νp(j))e−βν�

νp(i)∑

p e−βν�νp(i)

. (2.28)

For the specific case of the particles being spin-12

particles with states |0〉p and |1〉p, the

relaxation rate can be simplified into the form

kν(Tν) = kν1→0 = Γ

νS(ω)n

νS(−ω), (2.29)


where nνS(ω) is the spin occupation function nνS(ω) = [e

βνω+1]−1 and ΓS is a temperature-

independent coefficient

ΓνS(ω) = 2πλ2ν

∑p

| 〈0| bν,p |1〉 |2δ(ω + �p(0)− �p(1)). (2.30)

As suggested by the δ−function in equation (2.28), the only bath mode that contributes

to the rate is the one with frequency matching that of the subsystem frequency. There-

fore, the presence of other spin frequencies do not affect the calculation, and the rate

equation (2.28) is applicable whether the bath spectrum is continuous or discrete.

2.2.2 Harmonic oscillators

The harmonic oscillator is a versatile model for many physical systems. For exam-

ple, harmonic oscillators may model phonons in a solid and excitations in a radiation

field [77, 29]. They can also serve as a basis to describe a generic noisy environment [78].

Furthermore, since harmonic systems allow analytical solutions, they are widely used

in theoretical studies [39]. Consider a bath that consists of a collection of independent

harmonic oscillators, coupled to a subsystem through the oscillators’ displacements. The

bath Hamiltonian and coupling operator are written as

Hν =∑j

ωja†ν,jaν,j, Bν =

∑j

(aν,j + a†ν,j), (2.31)

where a†ν,j and aν,j are the bosonic creation and annihilation operators. The relaxation

rate (2.26) is given by

kν(Tν) = −ΓνB(ω)nνB(−ω), (2.32)


which contains the Bose-Einstein function nνB(ω) = [eβνω − 1]−1 and the temperature

independent coefficient

ΓνB(ω) = 2πλ2ν

∑j

δ(ωj − ω). (2.33)

Here we made use of the relations

nνB(−ω) =1

e−βνω − 1= − e

βνω

eβνω − 1, (2.34)

nνB(ω) + 1 = −nνB(−ω). (2.35)

2.2.3 Fermionic particles

Our next example is a collection of non-interacting spinless fermionic particles, represent-

ing electrons. For the interaction operator, we only consider transitions among states in

the same lead,

Hν =∑j

�jc†ν,jcν,j, Bν =

∑i,j

c†ν,icν,j, (2.36)

where c† and c are the fermionic creation and annihilation operators, respectively. The

relaxation rate induced by this bath is [79]

kν(Tν) = −2πλ2νnνB(−ω)∑i,j

δ(�i − �j + ω)[nνF (�i)− nνF (�i + ω)], (2.37)

with the Fermi-Dirac distribution function nνF (�) = [eβν(�−µν)+1]−1, where µν is the chem-

ical potential, or the Fermi level of the ν-th bath. In order to compact this expression,

we define

Λν(Tν , ω) = 2π

∫d�[nνF (�)− nνF (�+ ω)]Fν(�) (2.38)

where Fν(�) = λ2ν

∑δ(�−�j+ω)δ(�i−�) is a function that depends on the band structure.

This leads to

kν(Tν) = −Λν(Tν , ω)nνB(−ω). (2.39)


2.2.4 Three-level systems

In studying molecular junction transport, it may be of interest to consider the effect of

anharmonicity. So far we have provided the rate constants for a system coupled to a

fully harmonic bath, and for a system coupled to a spin bath. It is of interest to connect

these two limits and consider an intermediate structure. We now give the results for a

three-level system bath (3LS), with even or uneven energy spacings. For simplicity, we

assume the bath particles are of the same type and omit the particle label p from the

bath energies �(i), i = 0, 1, 2. Considering only transitions between neighboring states,

the relaxation rate between subsystem states n and m is given by

kνn→m(Tν) = 2πλ2ν ×

[∑p

|〈0|pbν,p|1〉p|2δ(En,m + �(0)− �(1))e−βν�(0)

Zp(βν)

+∑p

|〈1|pbν,p|2〉p|2δ(En,m + �(1)− �(2))e−βν�(1)

Zp(βν)

], (2.40)

with Zp(βν) =∑

i=0,1,2 e−βν�(i). When the 3LS particles have even spacing, �(2)− �(1) =

�(1)− �(0), the relaxation rate can be compactly written as

kνn→m(Tν) =

Γν3(En,m)

e−βν�(0)+e−βν�(1)

Zp(βν), En,m = �(2)− �(1);

0, else.

(2.41)

When bath particles have uneven spacing, assuming interaction matrix elements being

independent of level index | 〈0|p bν,p |1〉p |2 = | 〈1|p bν,p |2〉p |2, we obtain

kνn→m(Tν) =

Γν3(En,m)

e−βν�(0)

Zp(βν); En,m = �(1)− �(0)

Γν3(En,m)e−βν�(1)

Zp(βν); En,m = �(2)− �(1)

0 else.

(2.42)


2.3 Thermal current

While the quantum master equation is frequently applied to analytically solve for sub-

system dynamics and to compute kinetic rates; more recently it has been extended to

obtain thermal currents. Complementary to the rate calculations in the previous section,

we will derive the thermal current in this section. We first derive an expression for the

thermal current in a generic two-terminal junction, then apply it to junctions composed

of an harmonic oscillator or a TLS subsystem. Our starting point is the expression for the

thermal current operator derived in [80]. In this work, the thermal current is calculated

as the expectation value of the relevant operator. For a subsystem connected to two,

L- and R-heat baths, the steady-state heat current from the ν (ν = L,R) bath into the

subsystem is given by

jν = iTr([Vν , HS]ρ), (2.43)

where ρ is the total density matrix and Tr stands for the full trace over the entire open

quantum system space. Furthermore, because at steady-state L- and R-currents equal

in magnitude, jL = −jR, the average thermal current in the steady-state limit may be

defined with a symmetrized operator

j = Tr[Ĵρ], Ĵ =i

2[VL, HS] +

i

2[HS, VR], (2.44)

where the value is defined positive when the current flows from L- to R-bath. For the

Hamiltonian defined in equation (2.2), the steady-state thermal current is formally given

by

j =i

2

∑n,m

En,mSm,nTrB[λLρn,mBL]−i

2

∑n,m

En,mSm,nTrB[λRρn,mBR], (2.45)


where TrB represents tracing over the two baths TrLTrR. It is useful to rearrange (2.45)

as

j =i

2

∑n>m

En,mSm,nTrB[λL(ρn,m − ρm,n)BL

]− i

2

∑n>m

En,mSm,nTrB[λR(ρn,m − ρm,n)BR

]. (2.46)

We would like to express the current in terms of the long-time population Pn. To do

so, we first solve for ρn,m(t) and ρm,n(t) with the Liouville equation of motion under the

operator transformation Ô(t) = eiHBtÔe−iHBt,

ρ̇n,m = −iEn,mρn,m − i∑ν

∑p

λν[Bν(t)Sn,pρp,m(t)− Sp,mρn,p(t)Bν(t)

], (2.47)

Note that S does not depend on time in this representation. A closed expression for

ρn,m(t) is obtained by formally integrating equation (2.47)

ρn,m(t) = ρn,m(t = 0)−i∫ t

0

e−iEn,m(t−τ)[∑ν,p

λνBν(τ)Sn,pρp,m(τ)−∑ν,p

λνSp,mρn,p(τ)Bν(τ)]dτ.

(2.48)

Recall the assumption that the mean value of the interaction Hamiltonian, averaged

over the initial density matrix is zero, namely TrB[Vν(t), ρ(0)] = 0. This means that

the initial condition ρn,m(t = 0) in equation (2.48) can be neglected. Now plug ρn,m(t)

of equation (2.48) back into the thermal current expression (2.46) and compute the

trace. Similarly to the population calculation, we assume the full density matrix to be

factorizable at any point in time, ρ(t) = σ(t)ρL(TL)ρR(TR), and take the Markovian limit

by making the reduced density at the right-hand side of the equation time-local. This


results in

TrB[ρn,m(t)Bν(t)] = −iλν∑p

∫ t0

e−iEn,m(t−τ)[〈Bν(t)Bν(τ)〉Tνσp,m(t)Sn,p

−〈Bν(τ)Bν(t)〉Tνσn,p(t)Sp,m]dτ. (2.49)

Note that cross terms such as 〈BL(t)BR(τ)〉 have been omitted since we assume that the

L- and R-bath correlator is zero for all models we consider. Next, we make the secular

approximation by assuming that coherences σn 6=m are negligible under the weak-coupling

approximation for an initial preparation such that σn 6=m(0) ∼ 0. Therefore, the only

terms that remain in the sum on the right hand side of equation (2.49) are the diagonal

terms. After applying the Markovian limit, we reach

TrB[ρn,m(t)Bν(t)] = −iλν∫ 0−∞

eiEn,mx[〈Bν(0)Bν(x)〉Tνσm,m(t)Sn,m

−〈Bν(x)Bν(0)〉Tνσn,n(t)Sn,m]dx, (2.50)

TrB[ρm,n(t)Bν(t)] = −iλν∫ ∞

0

eiEn,mx

[〈Bν(x)Bν(0)〉Tνσn,n(t)Sm,n

−〈Bν(0)Bν(x)〉Tνσm,m(t)Sm,n

]dx. (2.51)

Assuming Sn,m = Sm,n and identifying the population of the subsystem state n as Pn =

σn,n, these two expressions combine to give

TrB[(ρn,m(t)− ρm,n(t))Bν(t)] = iλνSn,m

[Pn(t)

∫ ∞−∞

eiEn,mx〈Bν(x)Bν(0)〉Tνdx

−Pm(t)∫ ∞−∞

eiEn,mx〈Bν(0)Bν(x)〉Tνdx

]. (2.52)


We plug the above equation back into equation (2.46) and find that the heat current

expression reduces to

j = −12

∑n>mEn,m|Sm,n|2

[Pn(t)λ

2L

∫ ∞−∞

eiEn,mτ 〈BL(τ)BL(0)〉TL

−Pm(t)λ2L∫ ∞−∞

eiEn,mτ 〈BL(0)BL(τ)〉TLdτ

]

+12

∑n>mEn,m|Sm,n|2

[Pn(t)λ

2R

∫ ∞−∞

eiEn,mτ 〈BR(τ)BR(0)〉TR

−Pm(t)λ2R∫ ∞−∞

eiEn,mτ 〈BR(0)BR(τ)〉TRdτ

]. (2.53)

This expression is second order in λν instead of being exact because we truncated the

iteration of the reduced density matrix in its Liouville equation of motion (2.5) at the

second order term. We again identify the relaxation rate constant as equation (2.18) and

finally obtain the heat current flowing from L to R as

j = −12

∑n>m

En,m|Sm,n|2[Pnk

Ln→m − PmkLm→n

]+

1

2

∑n>m

En,m|Sm,n|2[Pnk

Rn→m − PmkRm→n

],

(2.54)

or more compactly

j =1

2

∑n,m

Em,n|Sn,m|2Pn[kLn→m − kRn→m

]. (2.55)

Given that the definition of j is a meaningful quantity only in the steady state limit,

the populations Pn and Pm in the current expressions should be the steady-state popu-

lations. Similarly, below in equation (2.56) all reduced density matrix elements should

be evaluated at the steady-state limit. The steady-state limit for thermal current exists,

as long as a solution exists for the steady-state population Ṗn = 0, where Ṗn is given by

expression (2.20). Expression (2.55) is often used as a phenomenological equation, yet

here we have given a full microscopic derivation at the level of second order perturbation

theory in the system-bath interaction. It can be interpreted as follows: the magnitude


of the thermal current is proportional to the per-quanta energy transferred (Em,n), how

many quanta are available for transferring (Pn), and how fast energy is transferred kνn→m;

we consider the net current as the difference between the left-going excitations and the

right-going excitations.

Our derivation is for a coupling form that is linear in bath displacement operator

Vν = λνSBν . However, the coupling does not have to be in this form for the current

expression (2.55) to hold. As long as the system and bath operators are separable in Vν ,

the thermal current expression will assume the same structure. Variations in the bath

operator are reflected in the explicit expression of the rate constants.

A natural extension of this derivation is to retain the coherences in the current ex-

pression [81]. This can be easily done by circumventing step (2.50) and keeping the

summation over all elements of the reduced density matrix in all subsequent equations.

The steady-state current expression becomes

j =1

2

∑n,m

∑p

Em,nSn,m{

[λ2L

∫ ∞0

eiEn,mx〈BL(0)BL(x)〉dx− λ2R∫ ∞

0

eiEn,mx〈BR(0)BR(x)〉dx]Sp,mσn,p

−[λ2L

∫ ∞0

eiEn,mx〈BL(x)BL(0)〉dx− λ2R∫ ∞

0

eiEn,mx〈BR(x)BR(0)〉dx]Sn,pσp,m

]}.

(2.56)

Due to the absence of integration in the negative half of time axis, the integrals in this

equation do not reduce to the rate constant derived before. Nonetheless, they can be

evaluated explicitly once the bath coupling operators Bν are specified.

We would like to emphasize that because we began our derivation with definitions (2.43)

and (2.44) for the thermal current, our results are valid only at steady-state thus the pop-

ulations and coherences in equation (2.55) and (2.56), should be calculated at the long-

time limit. The time-dependent current can be derived from the more general thermal


current operator Ĵν =i2([H0ν−HS, Vν ]) following a similar procedure and maintaining the

time-dependency of the reduced density matrix elements [81]. It is worth mentioning that

in other methods the population dynamics and thermal current often need to be derived

separately, in different calculations. For example, instead of using the density matrix,

the thermal current is often calculated as the expectation value of the time derivative

of the bath Hamiltonian 〈 ddtHν〉 [41, 39]. The advantage of our derivation is that once

the subsystem dynamics is obtained, either analytically or numerically, the steady-state

thermal current is attained immediately.

2.3.1 Examples

Next we demonstrate the calculation of the steady-state thermal current for two rep-

resentative models of the subsystem, the harmonic oscillator (HO) model and the two-

level-system (TLS) model. Consider a harmonic oscillator subsystem with frequency

ω,

HS =∑n

nω |n〉〈n| . (2.57)

This model can describe either a local radiation mode or an active vibrational mode of

a molecule trapped between solids. At this stage, we do not specify the bath Hamilto-

nian or the bath operator in the system-bath interaction term, except to demand that

the coupling is bilinear through the subsystem displacement. Therefore, the interaction

Hamiltonian should take the form Vν ∝ xBν , where x is a subsystem coordinate. Recall

that in the second quantization language, the position operator is given by x ∼ (a† + a),

where a† and a are the bosonic creation and annihilation operators. Using the energy

representation of the creation and annihilation operators a†+a, we see that the subsystem


interacts with the bath through an operator of the form

S =∑n

√n |n〉〈n− 1|+ h.c.. (2.58)

This implies that within second order perturbation theory, i.e. when the rate constants

contain bath operators up to the second power, only transitions between nearest levels

are allowed, namely, only |Sn,n±1| 6= 0. These neighboring-state transition rates are given

by

kν(Tν) ≡ kνn→n−1(Tν) = λ2ν∫ ∞−∞

dτeiωτ 〈Bν(τ)Bν(0)〉Tν , (2.59)

with the excitation and relaxation rates satisfying the detailed balance relation

kνn−1→n(Tν) = e−βνωkνn→n−1(Tν). (2.60)

Using the structure (2.58) for the subsystem interaction operator S, the master equation

for the population dynamics (2.20) reduces to

Ṗn = −[nkn→n−1 + (n+ 1)kn→n+1]Pn + (n+ 1)kn+1→nPn+1 + nkn−1→nPn−1. (2.61)

At steady-state Ṗn = 0, the normalized state population is given by

Pn = yn(1− y), y = k

Le−βLω + kRe−βRω

kL + kR. (2.62)

Substituting these populations into equation (2.55), the steady-state thermal current for

the harmonic oscillator model becomes

j(HO) = − ω[nLB(ω)− nRB(ω)]

nLB(−ω)/kL(TL) + nRB(−ω)/kR(TR), (2.63)


with Bose-Einstein distribution function nνB(ω) = [eβνω−1]−1 at temperature Tν = 1/βν .

So far the bath operators have not been specified yet. A well-known limit is when the

baths also contain only harmonic oscillators. Then the current should reduce to the Lan-

dauer expression for heat transfer. This situation is discussed in section 2.4.

The second subsystem under consideration is a two-level system (spin). It is inter-

acting with each bath through the transition operator σx

HS =ω

2σz, S = σx. (2.64)

Here the σs are the Pauli matrices. The two-level system can be viewed as a truncated

harmonic oscillator retaining only the lowest two levels, n = 0 and n = 1. Therefore,

the population master equation for the harmonic oscillator is also applicable here. The

populations are given by

P1 =k0→1

k0→1 + k1→0, P0 =

k1→0k0→1 + k1→0

, (2.65)

where k0→1 = kL0→1 + k

R0→1. With this at hand, the steady-state thermal current for the

two-level system reduces to

j(TLS) =ω[nLS(ω)− nRS (ω)]

nLS(−ω)/kL(TL) + nRS (−ω)/kR(TR), (2.66)

with the spin occupation factor nνS(ω) = [eβνω + 1]−1.

We emphasize that in deriving the thermal current expressions (2.63) and (2.66), we

have only specified the structure of the subsystem. These expressions do not depend on

the specific form of the bath Hamiltonian HB or the bath operator B in the interaction

Hamiltonian. Information on baths and their coupling operators are however required to

calculate the rate constants kν .


2.4 Applications

2.4.1 Nonlinear current-temperature characteristics

In the previous two sections we have presented expressions for the thermal current and the

relevant rate constants calculations. We are now able to construct various two-ter

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