OPEN QUANTUM SYSTEMS AND ITS APPLICATIONS

21 2.4 Time-Dependent Master Equation in Lindblad Form.. We will then turn our attention to driven systems, whereby we first investigate the e↵ects of dissipation and dephasing onpopulat

Applications TAN DA YANG B.Sc (Hons), NUS

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN SCIENCE

DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE

2016

I hereby declare that this thesis is my original work and it has been written by me in itsentirety I have duly acknowledged all the sources of information which have been used in thethesis This thesis has also not been submitted for any degree in any university previously

Tan Da Yang

19 August 2016

Summary i

1.1 What are Quantum Open Systems? 1

1.2 Overview of Main Fields of Research 3

1.2.1 Protection of Quantum Systems 4

1.2.2 Decoherence in Adiabatic Transport 7

1.2.3 Non-Markovianity in Open Quantum Systems 8

1.2.4 Aspects of Open Quantum Systems in Biological Systems 9

1.3 Outline of the Thesis 10

2 Master Equations 13 2.1 Overview 13

2.2 Derivation of Master Equation by Perturbation Theory 15

2.2.1 Master Equation in Integral Form 15

2.2.2 Master Equation in Integro-Di↵erential Form 18

2.2.3 Pure Dephasing Master Equation 20

2.2.4 Further Remarks 20

2.3 Driven Systems and its Challenges 21

2.4 Time-Dependent Master Equation in Lindblad Form 22

2.4.1 Dissipative Lindblad Equation 23

2.4.2 Dephasing Lindblad Equation 26

2.5 Concluding Remarks 27

2.A Alternative Derivation of the Master Equation in Eq (2.13) 29

2.B Derivation of the Interaction Unitary Operator UI(t) 32

3 Environment Induced Entanglement 34 3.1 The Spin-Boson Model 35

3.2 Bath Correlator and Spectral Density 37

3.3 Extension of The Spin-Boson Model 40

3.4 Concurrence as an Entanglement Measure 44

3.5 Entanglement Dynamics 46

3.5.1 Pure Dephasing Dynamics 46

3.5.2 More General Dynamics 49

3.5.3 Dependence with Temperature 54

3.6 Conclusion 55

3.A Entanglement dynamics with hard cuto↵ function 56

3.B Entanglement with respect to !c 57

3.C Numerical Check of the Master Equation 58

4 Environmental Induced Spin Squeezing 59

4.1 Basic Concept of Spin Squeezing 61

4.2 One-Axis Twisting Hamiltonian 63

4.3 Squeezing Dynamics in Bosonic Environment 64

4.3.1 The Model 64

4.3.2 Optimization of spin squeezing 66

4.4 Conclusion 69

4.A Derivation of squeezing parameter ⇠2 S 70

5 Population Transfer in Dephasing and Dissipation 71 5.1 Problem of Avoided Crossings 72

5.1.1 An Simple Illustration of the Problem 72

5.1.2 Significance of the Problem 75

5.1.3 Problem Statement 76

5.2 Population Transfer in Presence of Dissipation 77

5.2.1 The Derivation 77

5.2.2 Landau Zener Problem - An Example 81

5.3 Population Transfer under Dephasing 83

5.3.1 Example 85

5.4 Concluding Remarks 87

5.A Derivation of Density Matrix Elements for N Levels Systems Under Dissipation 89 5.B Proof of Generality of Eq (5.20) 91

5.C Alternative Derivation of the Dephasing Lindblad Equation 92

6.1 Introduction 94

6.2 Derivation of Pumping Formula 97

6.3 Chern Insulator - An Example 102

6.3.1 Transport Across Phase Transition Point 109

6.3.2 E↵ects of Initial State Preparation 111

6.4 Conclusion 118

6.A Comparison of the charge transport formula with numerics 119

6.B Relaxation of Even Function of k Assumption in Initial States 120

6.C Adiabatic Pumping Under Dephasing 121

7 Conclusion and Future Perspective 125 7.1 What Have We Achieved? 125

7.2 Outlook 127

In this thesis, we will investigate various aspects of open quantum systems, i.e systems thatare interacting with an external environment We will first study how the phenomenon ofentanglement between two qubits and spin squeezing of a large spin system can be optimised

by the environment, and find that contrary to conventional wisdom, the environment maysometimes assist with the formation of these quantum e↵ects We will then turn our attention

to driven systems, whereby we first investigate the e↵ects of dissipation and dephasing onpopulation transfer between energy levels as a result of adiabatic driving We will thenextend these results to investigate the e↵ects of dissipation on adiabatic quantum pumping

I would like to first thank my supervisor Prof Gong Jiangbin for his unwavering support ing my entire candidature Thank you for being such an inspiring teacher and understandingsupervisor who goes all the way out to help all your students.

dur-I would also like to thank my research group mates, past and present, Adam, Derek, YonShin, Hailong, Qi Fang, Longwen, Gaoyang, Neresh and Jia Wen, for all the meaningfuldiscussions in both office and over meal table In particular, I would like to especially thankAdam and Longwen for both of your guidance and pointers over the past few years, andhelping out with the technical difficulties that I encountered along the way

Special thanks also goes out to Junkai and Kendra for all the interesting and random cussions over meal table Thank you for being a big part in my life

dis-Last, but not least, this Ph.D journey could not have been possible without the support of

my family I will eternally be grateful for that

List of Publications

Da Yang Tan, Adam Zaman Chaudhry, and Jiangbin Gong Optimization of the environmentfor generating entanglement and spin squeezing, Journal of Physics B: Atomic, Molecular andOptical Physics 48, 11 (2015): 115505

Longwen Zhou, Da Yang Tan and Jiangbin Gong E↵ects of dephasing on quantum adiabaticpumping with nonequilibrium initial states, Physical Review B 92, 24 (2015): 245409

3.1 (Colour online) Behavior of the concurrence as a function of time with s = 0.5(solid, black line) and s = 1 (dashed, red line) Inset shows the concurrence

at s = 4 (dash-dotted, orange line) and s = 6 (short-dashed, blue line) spectively Here we set !0 = 0.1, !c = 20, = 1 and g = 0.01 47

re-3.2 (Colour online) Maximum concurrence with varying coupling strength g andOhmicity parameter s Here !0 = 0.1, !c = 2 and = 1 48

3.3 Variation of (t) with respect to Ohmicity parameter s at finite long time

t = 500 from s = 3 to s = 3.2 The inset shows the variation between s = 1.2

to s = 6 Here !0 = 0.1, !c = 2, = 1 and g = 0.005 50

3.4 Variation of maximum concurrence Cmax with respect to Ohmicity parameter

s from s = 1 to s = 4 for varying values of " Other parameters used are

!c = 50, !0 = 0.1, = 1 and g = 0.01 Here the lines showing the puredephasing case and " = 0.01 are almost indistinguishable, while the di↵erencebetween the pure dephasing case and " = 0.08 is also not very appreciable 51

LIST OF FIGURES

3.5 (Colour online) Evolution of concurrence for s = 1 (solid, black line) and

s = 3 (dashed, red line) We set !0 = 0.1 and use !c = 20 for s = 1 and

!c = 2 for s = 3 Also, we have g = 0.005 and = 1, and p1 = p3 = 0.9,

p2 = p4 = 0.1 We note that for the sub-Ohmic case, s = 0.5, the concurrenceremains zero throughout, hence is not plotted here The inset shows the longtime evolution for s = 3 Here we note that there is a finite time intervalbetween each cycle of revival of entanglement 52

3.6 (Colour online) Evolution of purity for the mixed state given by Eq (3.45).Parameters used are same as Fig 3.5 53

3.7 (Colour online) Variation of maximum concurrence with respect to for s =3.1 The inset shows the variation of maximum concurrence with respect toand s Here g = 0.03, !0 = 0.1 and !c = 2 If !0 is in the GHz regime (forinstance, trapped ions), then = 1 corresponds to a temperature in the µKregime 54

3.8 (Colour online) Concurrence using F (!, !c) = exp (!/!c)2 at s = 0.5 (solid,black line), s = 1 (dashed, red line), s = 4 (dash-dotted, orange line) and

s = 6 (short-dashed, blue line) respectively Parameters used are !c = 50,

pro-4.1 (Colour Online) Coherent spin state (left) and spin squeezed state (right) inthe Bloch sphere representation Here, the radius of the sphere gives D

~

JE

We observe that one main e↵ect of squeezing is that it reduces the tainty in one axis while increasing the uncertainty in the other (see the rightdiagram) Figure is adapted from [157] 624.2 (Colour online) Variation of the optimised squeezing parameter with the Ohmic-ity parameter s in the presence of the OAT Hamiltonian (circle, blue dottedlines) and without the OAT Hamiltonian (square, black solid lines) Here

uncer-N = 10, g = 0.05, !0 = 0.1, !c = 10 and = 1 For the OAT case, = 1 674.3 Variation of minimum ⇠2

S(t) with coupling strength g Here N = 10, !0 = 0.1,

s = 2.5 and = 1 and we consider the spin squeezing generated at time T = 1 68

5.1 Schematics of the adiabatic eigenstates (represented by solid lines) and batic eigenstates |"i and |#i (represented by dashed lines) The regime withthe minimum gap correspond to the avoided crossing Figure is adapted fromRef [85] 735.2 (Colour online) Population P (s) of ⇢++(s) with respect to rescaled time sfor (red, solid line) = 1 and (blue, dashed line) = 0.1 respectively Thecircle and star symbols represent the numerical results obtained by evolvingthe master equation directly Here we set v = 0.001, = 1, |A +|2 = 1 and

dia-|A+ |2 = 0 The initial state of the system is given by (0) =p

0.8|+(0)i +p

0.2| (0)i A strong agreement between the numerical results and theory isobserved 825.3 Transition probability of ⇢11 as a function of The coherent state is initiallyprepared in | (s0)i = 0.8 |E1i + 0.1 |E2i + 0.1 |E3i, and the mixed state isinitially in ⇢(s0) = 0.8|E1i hE1| + 0.1 |E2i hE2| + 0.1 |E3i hE3| Here we set

g0 = 1 and v = 10 3 Inset shows the transition probability when the state isprepared in a pure state | i = |E1i 86

LIST OF FIGURES

5.4 Percentage di↵erence of the transition probability of ⇢11between the numericalresults and our theory as a function of The coherent state is initiallyprepared in | (s0)i = 0.4 |E1i + 0.3 |E2i + 0.3 |E3i, and the mixed state isinitially in ⇢(s0) = 0.4|E1i hE1| + 0.3 |E2i hE2| + 0.3 |E3i hE3| Here we set

g0 = 1 and v = 10 3 Inset shows the transition probability when the state isprepared in a pure state | i = |E1i 87

6.1 (Colour online) Population P (s) of state ⇢++(s) as the system is driven withrespect to s Blue solid lines corresponds to the analytical results and blackcrosses corresponds to the results obtained by numerically evolving the Lind-blad master equation directly Here k = 0.25, v = 10 4, = 1, = 0.1,

|A +|2 = 1 and |A+ |2 = 0 The initial state of the system is given by (0) =p

0.8|+(0)i +p0.2| (0)i Here we note that P (s) approaches zero ataround s = 0.006 Both results obtained by numerics and analytical formulasare in strong agreement with each other 103

6.2 (Colour online) Same as Fig (6.1), except that = 1 Here we note that

P (s) approaches zero at around s = 0.001 Both results obtained by numericsand analytical formulas are in strong agreement with each other 104

6.3 (Colour online) Same as Fig (6.1), except that = 1, |A +|2 = 1 and

|A+ |2 = 1 Here we note that P (s) approaches 0.5 asymptotically instead ofzero, indicating the e↵ect of the Lindblad operators on the adiabatic probabil-ity Both results obtained by numerics and analytical formulas are in strongagreement with each other 105

6.4 (Colour online) Number of pumped particles Q vs the dissipation rate for(blue, solid line) = 1 and (red,dashed line) = 1.6 using the equationdescribed in (6.14) Here |A +|2 = 1 and|A+ |2 = 0 106

6.5 (Colour online) Number of pumped particles Q vs the dissipation rate for

= 2.5 using the equation described in (6.14) Here|A +|2 = 1 and|A+ |2 =

⌘

| (0)i h (0)| respectively Here,

= 1 1136.11 (Colour online) Number of pumped particles Q vs energy bias using theequation described in (6.14) Here |A +|2 = 1 and |A+ |2 = 0 Here, weprepare the initial state of the system to be in a superposition state of (bluesolid line)q

3

4 + k 4⇡|+(0)i +q1

4

k 4⇡| (0)i and mixed state of (red dashedline) 34 + 4⇡k |+(0)i h+(0)| + 1

4

k 4⇡ | (0)i h (0)| respectively Here, =1.6 Here we note that maximum di↵erence of Q for a particular choice ofbetween superposition and mixed state is about 0.025 116

LIST OF FIGURES

6.12 (Colour online) Number of pumped particles Q vs energy bias using theequation described in (6.14) Here |A +|2 = 1 and |A+ |2 = 0 Here, weprepare the initial state of the system to be in a superposition state of (redsolid line)p

0.6|+(0)i +p0.4eik| (0)i and its mixed state counterpart (bluedashed line) respectively The dark yellow dashed dot line corresponds tothe contribution due to Eq.(6.10) We further note that Eq.(6.9) has nocontribution for this particular choice of initial states Here, = 1.6 1176.13 (Colour online)fk vs the k for the QWZ model Here we set v = 10 4, = 2,

= 1, |A +|2 = 1 and |A+ |2 = 0 The initial state of the system is given by (0) =p

0.8|+(0)i+p0.2| (0)i A strong agreement between the numericalresults and theory is observed 1196.14 (Colour online) Number of pumped particles Q vs dephasing rate in theQWZ model Here the initial state is chosen to be 12h

| (1)k (0)i + | (2)k (0)ii⌦h

h (1)k (0)| + h (2)k (0)|i Other parameters used are v = 10 3 and = 0.5.Figure is adapted from Ref [185] 1236.15 (Colour online) Number of pumped particles Q vs energy bias through thephase transition point = 0 for various dephasing rate in the QWZ model.Figure is adapted from Ref [185] 124

Chapter 1

Introduction

In a real world, no system is completely isolated and every system interacts with its ronment For instance, a cup of co↵ee, interacts with its surrounding and loses energy to it,and eventually cools down and comes to an equilibrium state that is described by classicalstatistical mechanics Such kind of interaction is known as dissipation or relaxation and can

envi-be found in both classical and quantum systems

If we were to study the system-environment interaction in the quantum regime, there isanother phenomenon, known as decoherence1, that will appear due to its interaction withthe environment It refers to the destruction of superposition between two quantum statedue to its interaction with the environment When a quantum system becomes completelydecoherent, the quantum state will then become a mixture and any information about itssuperposition will be lost

Decoherence has been widely studied for mainly two reasons: Firstly, many quantumtechnologies depend on the preservation of quantum superposition and hence protecting

1 In literature, sometimes decoherence refers to the interaction with the environment whereby there is both loss of superposition and relaxation To avoid confusion, in subsequent chapters, we will refer such loss

of superposition as dephasing.

1.1 WHAT ARE QUANTUM OPEN SYSTEMS?

such superposition from decoherence becomes an important issue at hand For instance,many quantum applications such as quantum computing and quantum crytography relyheavily on the coherence of quantum states, and such destruction of the states will result

in a lowered or non-efficiency of these applications As a result, e↵orts have been devoted

to eliminate or reduce the e↵ects of decoherence on these quantum mechanical devices, forinstance, protocols such as dynamical decoupling [1–4], use of decoherence free subspace[5, 6] or combination of protocols [7] All these methodologies aim to preserve the quantumstate of the system, and prevent the superposition states from decaying into a mixture, due

to the influence from their environment

More fundamentally, given the e↵ectiveness of the quantum mechanical formalism inexplaining the behaviours in the microscopic world, it becomes a question as to why macro-scopic objects do not behave like a quantum object in our everyday life In other words,the laws governing the macroscopic and microscopic world may be di↵erent and the divisionbetween the two worlds is known as the Heisenberg cut [8, 9] However, it has also been pro-posed that such cut does not exist and that quantum mechanical properties should manifestitself at all scales [10] While the interpretation of quantum mechanics remains open [11, 12],the decoherence framework nonetheless is able to provide a partial answer to the questionposed at the start of the paragraph In a quantum mechanical setting, the environment acts

as a probe and continuously monitor the system of interest, leading to a correlation betweenthe system and the environment Any information about the coherence of the system is nowquantum mechanically entangled with the infinite degrees of freedom of the environment,hence in practice we will no longer be able to measure any information about the coherencethat is embedded in the environment In a more technical sense, in practice we are not able

to have a complete description of the environmental degree of freedom, even if we do, theamount of information from the environment is too much for us to make any reasonablecalculations by treating the environment and system as a closed system [13]

As a simple illustration, consider a double slit experiment using a photon Let | 1i and

| 2i be the paths of the photon through slit 1 and slit 2 respectively; and |Di be the initial

state of the detector at the screen When the photon reaches the detector, the state will now

be entangled with the detector and this is represented as follows:

1p

2(| 1i + | 2i) ⌦ |Di 7! p1

2(| 1i ⌦ |1i + | 2i ⌦ |2i) = | i (1.1)where |1i and |2i are the states of the detector detecting the photon from slit 1 and 2respectively

Here, if we want to obtain information about the probability distribution of the photon

on the screen, we will need to find the reduced density matrix of the photon by performing

a trace over the detector’s degree of freedom

The study and application of open quantum system spans over a wide range of subjects,including quantum computation [14], condensed matter physics [15–19] and biological sys-

1.2 OVERVIEW OF MAIN FIELDS OF RESEARCH

tems [20–23], to name a few Given the wide spectrum of research areas in this topic, it is

a daunting task to list down all the topics associated with open quantum systems Instead,here we will note down a list of areas of research that are currently active, bearing in mindthat the list is non-exhaustive

1.2.1 Protection of Quantum Systems

One of the major topics in the subject of open quantum system is on the control of unwantedinteractions between the system and environment As illustrated in the previous section, thecorrelation established between the environment and system will wash away the coherence ofthe system, therefore protecting the system from any loss of coherence becomes an importanttask at hand In the following we will give a review of two of the main methods of protectingthe quantum system from interaction with environment, namely reservoir engineering anddynamical decoupling

Reservoir Engineering

The idea of reservoir engineering was first proposed by Potayos et al [24], whereby thecoupling between a single ion and the environment was controlled by the absorption andspontaneous emission of the laser photon The idea was then explicitly discussed and ex-tended to combat decoherence in the paper by Carvalho et al [25] The essential idea

of reservoir engineering is that the system is made to couple to a reservoir at which itspointer state includes the target state of the system The system is then made e↵ectivelydecoupled from the environment by making the engineered reservoir’s coupling significantlystronger One of the key advantages of such form of scheme is that since the reservoir isprepared beforehand, there is no longer an external intervention within a small time scale.Furthermore, compared to measurement based feedback schemes, one no longer needs toknow the measurement outcome in order to control the system However, one of the keychallenges of such engineering is to be able to find a suitable coupling such that the system

will be driven to its target state in the steady state limit The authors in Ref [26] tackledthe problem by finding the necessary and sufficient conditions for a unique steady state toexist in the Lindblad formalism, whereby it can be, in principle, used to stabilise a quan-tum state through reservoir engineering A scheme that was made up of a stream of twolevel systems undergoing dispersive, resonant and then dispersive atom-cavity interactionwas proposed as a possible candidate for an engineered reservoir [27] The authors then usedthis reservoir and illustrated that in a cavity with finite damping time, a stabilised squeezedstate and superposition of multiple coherent component state could be created as the result

of the system-engineered reservoir coupling The same group of authors then showed thatthe engineered reservoir was fairly robust against experimental imperfections and could beimplemented in the context of microwave cavity and circuit quantum electrodynamics [28]

In Ref [29], the authors considered a local interaction of N particles with its individualenvironment by organising the system in a particular geometry, and the local interactionwill drive the system to the desired steady state In the aspect of bosonic reservoir engi-neering, one can in fact tune the so called Ohmicity2 by modifying the scattering length

of the Bose-Einstein condensate (BEC) [30] Such form of reservoir engineering has foundapplications in the control of entanglement of two impurity qubits, where the authors inRef [31] showed that by controlling the reservoir parameters, one could in fact produce therich entanglement dynamics, such as sudden death and revival, entanglement trapping andBEC-mediated entanglement generation As a brief remark, we note that it is the richness

in the studies of reservoir engineering described above that motivates us in our own studies

of the environmental e↵ect of entanglement and spin squeezing to be presented in Chapter

1.2 OVERVIEW OF MAIN FIELDS OF RESEARCH

interaction term between the system and environment changes sign rapidly This will thenresult in the interaction term being averaged out to zero, thus cancelling the e↵ect of envi-ronment on the system Such scheme was first proposed in Ref [32], whereby the authorsproposed using pulsed DD applied at equal interval for a single qubit coupled to a quantumenvironment

A significant advancement was made when Uhrig showed that the coherence of a singlequbit can be protected up to N th order by using N aperiodic instantaneous pulses for apure dephasing model (dubbed as the Uhrig DD, or UDD), thus significantly reducing thedifficulty in performing DD [33] The idea has since been extended and it was furtherillustrated that such DD is independent of the choice of system-environmental coupling [34],and it was further shown that a nested UDD sequence can in fact protect a single qubit fromboth dephasing and relaxation [35] UDD has also been implemented experimentally andstudied in Ref [36–38]

Another direction in this problem would be to investigate the means of protecting multiplequbits from the undesirable e↵ects of decoherence, especially since multi qubits can haveinteractions of other kinds that do not exist in single qubit systems, such as sudden deathphenomenon [39, 40] In this aspect, it has been found that even with the lack of informationabout the system-environment coupling, it is still possible to construct a N pulse sequence

to protect two qubits system up to the N th order [41] It was then further found that byusing four layers of nested pulsed UDD, one can protect a completely unknown two-qubitstate up to a high fidelity [42]

The third direction in the topic of dynamical decoupling is the use of continuous field,rather than a pulse sequence in the process The main advantages of a continuous pulsesequence are that: (i) it can be more easily implemented in practice; (ii) one no longer has

to be concerned about the type of pulse sequence anymore; (iii) the problem of imperfections

in the pulse sequence due to the finite time pulse is no longer relevant The authors in Ref.[43] have found that one can also achieve universal protection with respect to all types

of decoherence e↵ects by using a relatively simple form of continuous DD, i.e via local

continuous and periodic fields It has also been found that such form of continuous DD can

in fact be used to protect and enhance spin squeezing in multi qubit systems [44]

1.2.2 Decoherence in Adiabatic Transport

As we will see in Chapter 6, adiabatic transport is a phenomenon whereby the chargedparticles are being pumped through the system when it is subjected to a cycle of slowperiodic driving In particular, Thouless [45] showed that when a one-dimensional lattice

is being driven by a slow periodic external field, the amount of charge passing through thecross section perpendicular to the lattice will always be given by a quantised value andcan be expressed in terms of an integral of the Berry curvature when the initial state isgiven by an uniformly filled Bloch band Such form of quantised adiabatic pumping isknown to be resilient against disorder in the substrate, as well as multi-body interactions[46] In fact, Thouless pump has been experimentally proposed in many di↵erent setups,such as cold atoms and photonic systems [47–56] Furthermore, it has also been shownrecently that the non-adiabatic correction to such transport is in fact dependent on the statepreparation, whereby the correction factor scales with respect to driving speed, v, if thebands are coherently filled, and v2 is the band is singly filled [57]

In terms of environmental e↵ects on adiabatic transport, most of the studies thus farhave involved systems that are coupled to leads, rather than a truly open system [58, 59].One main e↵ect of open systems in the adiabatic pumping is that the so-called time reversalsymmetry will be broken, and as a result the charge transport will become a direct current[60] Nonetheless, the quantum master equation approach had been utilised to study thee↵ects of the quantum leads in the transport process For instance, in Ref [61], it wasdemonstrated for interacting electrons in quantum dot systems, one could control the pump-ing by modulating the chemical potential In the same paper, they also derived expressionsfor the cumulant generating function for the pumping and found that it was related to thegeometrical Berry-phase-like quantities in parameter space The use of quantum masterequation was further extended to an anharmonic junction model recently where the system

1.2 OVERVIEW OF MAIN FIELDS OF RESEARCH

interacts with two di↵erent bosonic environments [62] In the work, the author found thatunder such setup, the pumping current displays a non-trivial relationship with respect toboth the intial states, as well as the environmental parameters, and one can then optimisethe current by controlling these parameters Similar approach was taken in Ref [63] in thecontext of single quantum dots and it was found that in the non-adiabatic regime the chargetransport is given by a trinomial distribution However, most of the work described aboveinvolved quantum systems coupled to two leads, and there has been a limited number ofstudies involving the coupling with a general reservoir, which is the focus of our work in thisthesis

The remaining two sections, though having no relevance to the rest of the chapters, arediscussed nonetheless as these topics have been surveyed during the formation of this thesis,and may be still interesting for some readers

1.2.3 Non-Markovianity in Open Quantum Systems

Historically, the modelling of open quantum systems relied on the Markovian approximation,i.e the memoryless e↵ect between system and environment, in order to gain analytical insight

on the system Such neglect of the back action of the environment, while widely studied,

is proven to be inadequate in situations where the system-environment coupling is strong,when the temperature is low and when the environment is of a finite size or structured.However, unlike its classical counterpart, the concept of Markovianity at this stage has noclear and uniquely model-independent definitions It is also unclear if one should regardthe non-Markovianity as a mathematical property of the dynamical map, or as a relevantphysical quantity that evolves with time [64] In terms of its definition, various quantitativemeasures have been proposed to define non-Markovianity, with the main criteria being that

it has to be independent of the choice of the model For example, Rivas et al proposed ameasure, known as the RHP measure, that was dependent on the divisibility of the quantumdynamical map [65] On the other hand, Breuer et al proposed another more physicallyintuitive measure [66], the BLP measure, that was based on the fact that since the system’s

interaction with the environment will typically reduce the distinguishability of the quantumstates, any moment in time when the distinguishability increases between the quantum stateswill be due to the backflow of information from the environment, and by quantifying suchamount, one can in fact measure the degree of non-Markovianity However, at this stage there

is still no consensus on the appropriate measure to use and the suitability of the di↵erentmeasures is still an open question [67–69]

Another aspect of the problem is the possibility of using the non-Markovian property as aresource and exploiting it in quantum processes It has been demonstrated that by exploitingthe memory time of the environment, one can in fact generate entanglement that are long-lived even with the presence of environment [70] It has also been shown that in the Jaynes-Cumming model, the non-Markovianity of the environment can be used to speed up quantumevolutions resulting in a shorter quantum speed limit time [71] It has further been shownrecently [72] that such non-Markovianity is related to the so called coherence trapping of thequantum states, where the coherence of the steady state is found to be maximised wheneverthe qubit undergoes a non-Markovian dynamics These results have further been extendedrecently to include systems with initial system-environment correlation [73] Furthermore,

it has also been proposed recently that non-Markovianity can be harnessed as a resource forquantum technologies [74–77]

1.2.4 Aspects of Open Quantum Systems in Biological Systems

The union between quantum mechanics and biological systems is indeed one that is ing For a long time, the warm and wet environment that biological systems are subjected tohas been thought to prevent any sort of quantum mechanical e↵ects from persisting, and it

intrigu-is such ideas that partially explain the stability of certain molecules However, the ments in the recent decade has shown that not only that such quantum e↵ects are important

develop-to biological systems, but also it is paradoxically the interaction with its environment thatallows numerous biological processes to take place One distinctive example is the energytransfer in photosynethesis processes [21–23, 78–80] The seminal works in Refs [21, 79]

1.3 OUTLINE OF THE THESIS

showed that energy transport in photosynthesis can be maximised by the interplay betweenthe coherence and decoherence of system as a result of its interaction with the dephasingenvironment Furthermore, the authors in Ref [79] showed that such transport is the mostefficient if decoherence rate and the energy scale of the system matches This opened up theexploration of such light harvesting mechanism and is currently an active field of research[81, 82]

The thesis is generally categorised into two main theme: Firstly we will study the influence

of environment on certain static systems, in particular the entanglement and spin squeezingdynamics of the system Secondly, we will extend our results to driven systems and studyhow dephasing and dissipation will a↵ect the population transfer and charge transport oflattice systems

We will first begin our studies by deriving the master equation to encompass the e↵ects

of environment in Chapter 2 Starting from the standard Liouville von-Neumann equation

of motion in the density matrix representation, we utilise the standard perturbation theory

to arrive at the master equation in both the integral form and the integro-di↵erential form

In the second part of Chapter 2, we will derive an expression for the density matrix elementsusing the master equation in the Lindblad form, in order to handle driven system problems inChapter 5 and 6 The expressions are purposefully split into dissipative and dephasing parts

so as to allow us to consider each phenomenon individually Indeed, as we will illustrate, thetwo phenomenons demonstrate very di↵erent and interesting dynamics

As our first application, we first investigate the influence of the bosonic environment onthe entanglement of the two-level spin system in Chapter 3 As we will discuss later, such

a model (known as the spin-boson model) has found applications in many fields of researchand thus making such form of studies relevant We then consider the system subjected topure dephasing from the environment We find that the degree of entanglement can, in

fact, be maximised by finding an optimal value of the so-called Ohmicity parameter (we willdiscuss more of this in Chapter 3), thus providing a proof of principle of such non-trivialinterplay between the system and the environment We then utilise the master equationapproach derived in Chapter 2 to include an additional tunnelling term and we find that

we are able to arrive at similar conclusion As a final remark, we then briefly discuss theinterplay between temperature and Ohmicity parameter on the degree of entanglement.With the insight from Chapter 3, we extend our model to a large spin system and probeinto the dynamics of spin squeezing of large spin system in the bosonic environment inChapter 4 We will first provide a brief description on the concept of spin squeezing, aswell as the commonly used one-axis twisting (OAT) Hamiltonian that is used to generatespin-squeezing in a closed system With these basics in place, we find that the degree ofsqueezing can once again be optimised by the Ohmicity parameter We further argue thatwhile a reduction in the coupling strength with the environment will allow the system toreach the maximum squeezing as dictated by the OAT Hamiltonian, the duration to reachthat particular value too increases Thus, given a fixed duration, we also find that there is

an optimisation of such squeezing with respect to the coupling strength

We will then turn our attention to the driven systems As we will explain, driven systemstypically form a computational challenge and hence further approximations may be neces-sary to make the problem solvable In this regard, we consider a finite-time Landau-Zenerproblem and incorporate the e↵ects of dissipation and dephasing through the master equa-tion of Lindblad type in Chapter 5 We first derive expressions for the population in theexcited instantaneous eigenstate under dissipation by recasting the Lindblad master equa-tion into its instantaneous eigenbasis We illustrate that the final population of the system

is always dictated by the Lindblad operators, and demonstrate this using the Landau-ZenerHamiltonian as an example We will then review the population transfer under the e↵ect ofdephasing As the expression for dephasing has a significantly simpler structure, we are able

to consider a more general j-th level system and construct the expression for the populationprobability to be in the j-th state We then illustrate the e↵ects of dephasing by considering

1.3 OUTLINE OF THE THESIS

a three-level Landau-Zener-like system

With all these results in place, the penultimate chapter (Chapter 6) seeks to extend thefindings in Chapter 5 and discuss the e↵ects of dissipation on adiabatic pumping in a onedimensional lattice We then derive an expression for the amount of pumped particle over

an adiabatic cycle up to first order with respect to the driving speed and find that unlikethe closed and dephasing case, the expression does not depend on the choice of the initialstate Instead, the amount of charge transport now depends on the Berry curvature of thesystem and because of this, the amount of charge transport can now present interestingnon-monotonic behaviour with respect to the coupling strength with the environment Wethen finally demonstrate this using the well known Qi-Wu-Zhang model

In the last chapter, we will then summarise our findings and provide some outlook forfuture research directions

1 The density matrix formalism is actually a more useful in describing open quantum systems, since now

it can represent states that loses its coherence and becomes mixed

2 As we will observe later, the choice of representation can be freely made depending on the context of

A general recipe of solving the master equation starts from the Liouville von-Neumannequation, where the Hamltonian is broken into the system, environment and interactioncomponents The interaction Hamiltonian correlates the two components and approxima-tions have to be made to this term in order to render the equation solvable, and this is thepart where various approaches in the literature start to di↵er Perturbative treatment ofthe interaction Hamiltonian is one of the more typical approaches in dealing with such aproblem, although there have been also numerous non-perturbative approaches developed.

At this stage, we note that despite many e↵orts in the development of the master equationapproach, there is still no general solution to the problem Depending on the degree ofapproximations made and the types of system of interest, di↵erent equations have beendeveloped For instance, one of the most common type of master equation found in literature

is the Redfield equation [83], where the Markovian approximation has been made in thederivation By further making the rotating wave approximation, we would arrive at theLindblad master equation3 [84] To study periodic driven dissipative systems, formalismsuch as the Floquet-Markov master equation [85] has also been derived Non-markovian

the system of interest More subtly, this means that a density matrix state in a particular representation may be completely mixed, but retains some coherence in another representation.

3 We will be using this form of master equation in this thesis.

equations for quantum walk [86] and quantum jumps [87] have also been developed However,the generality of master equations is nonetheless an open question.

In this chapter, we will first review and derive the master equation using time-dependentperturbation theory We will then cast the master equation in the integral form, as well as

in terms of an integro-di↵erential equation [100] In Section 2.4, we will further consider atime-dependent master equation in the Lindblad form, where we cast the master equation

in terms of a time-dependent basis that evolves together with time These derivations willset the foundation of our discussion in the next few chapters

The-ory

2.2.1 Master Equation in Integral Form

In this section, we will derive the master equation using time-dependent perturbation theory

To retain the generality and applicability of the master equation, the equation of motion will

be derived with minimum amount of approximations, in which they will be stated explicitly

in the section

First, we consider the system and the bath to be given by the Hamiltonian

where HS(t) is the Hamiltonian of the system of interest, while HB(t) is the Hamiltonian

of the environment, and V (t) is the system-environment interaction Hamiltonian is aconstant that is used to track the order of the perturbation that will be set to unity at theend of the derivation

From standard quantum mechanics, the evolution of the density matrix at later time will

be given by

2.2 DERIVATION OF MASTER EQUATION BY PERTURBATION THEORY

⇢tot(t) = Utot(t)⇢tot(t0)Utot† (t)

= US(t)UB(t)UI(t)⇢tot(t0)UI†(t)UB†(t)US†(t)

(2.3)

where US(t), UB(t) and UI(t) are the unitary operators associated with the Hamiltonian ofthe system, environment and the interaction term respectively In particular, we note thatthe interaction term in the interaction picture is given by ˜V (t) = U0†(t)V (t)U0(t), with ˜V (t)the interaction term in the interaction picture and U0(t) = US(t)UB(t), therefore it alwayscontains explicit time dependence As a result, UI(t) can be formally be written as follows4:

is common used as in many systems, the system and environment is sufficiently weaklycoupled such that the system would have little influence on the environment statistics initially[88] Nonetheless, systems with initial system-environment correlation has also been studiedrecently [89, 90] We also assume that the interaction Hamiltonian can be written in afactorised form V (t) = S(t)⌦ B(t), where S(t) and B(t) are the system and environmentoperators respectively With such approximations, the system reduced density matrix cannow be written as follows:

⇢S(t) = TrB[US(t)UB(t)UI(t)(⇢S(t0)⌦ ⇢B)UI†(t)UB†(t)US†(t)] (2.5)

Furthermore, because of the weak coupling between system and environment, we can alsoperform a Dyson expansion on Eq (2.4), such that

4 See the appendix for the details of derivation.

By substituting Eq (2.6) into the Eq (2.5), we obtain the following with respect to :For the term independent of :

TrB[U0(t)(⇢S(t0)⌦ ⇢B)U0†(t)] = US(t)⇢S(t0)US†(t) = ¯⇢S(t) (2.7)For the term proportional to :

B yields zero, and in an event that it does not, we can alwaysredefine the environment operator B(t)! B(t) DB(t)˜ E

B such that the expectation valuewill once again be zero In this case, the other term that is proportional to will also yieldzero, hence the first order terms have null contribution to the ⇢S(t)

For the term proportional to 2, there will be two similar terms that will arise from thederivation, for conciseness, we will demonstrate the derivation for one of the terms below:

B

US(t) ˜S(t2)⇢S(t0) ˜S(t1)US†(t)D

˜B(t2) ˜B(t1)E

B

(2.9)

2.2 DERIVATION OF MASTER EQUATION BY PERTURBATION THEORY

As a result, by setting to unity, the system density matrix ⇢S(t) is given by:

B+ h.c (2.10)Here h.c denotes the hermitian conjugate

At this stage, such form of master equation has already encapsulated all the essentialproperties of the e↵ects of the environment onto the system through the bath correlation

In principle, so long one specifies the type of system operator and the details of the bath,this equation can be solved directly to reveal the dynamics of the system By casting themaster equation into an integral form, one realises that the system unitary operator is now

an expression in terms of t, and hence this term needs to be computed once for every timestep during calculations Such an expression may be useful for driven systems, where theevaluation of the next time step depends on the values of the current time step (such aproperty is known in literature as the back-propagation [91, 92]) Hence, any reduction inthe amount of integrals that are required to be computed at each time step will potentiallysignificantly reduce the computational time Nonetheless, in practice, this form of masterequation still involves a double integral and may be computationally challenging to evaluatefor large Hilbert spaces

In chapter 3, the system that we are dealing with are time-independent of nature, hencethe integral form in Eq (2.10) does not serve any computational advantage during theevaluation due to the double integrals involved Hence, in this case we will attempt toremove the double integrals by recasting the master equation in a di↵erential form

2.2.2 Master Equation in Integro-Di↵erential Form

From Eq (2.10), we first multiply both sides by US†(t) on the left and US(t) on the right, weyield the following:

Here we use the fact that dUS (t)

dt = iHS(t)US(t) in the above derivation

On the right hand, the derivative with respect to t will remove the integral with respect

to t1 according to calculus, and by exchanging the index of t2 with t1, we get the followingresults:

B+ h.c.(2.13)Here, US(t, t1) = US(t)US†(t1), and we have utilised the completeness relation US†(t)US(t) =I

to obtain the above equation We have further make the replacement of ˜⇢S(t) ! ⇢S(t)5 inthe above In this case, we have successfully recasted the master equation into a di↵erentialform Unlike the integral form, this form of master equation only deals with a single integraland can be numerically computed quickly for time-independent systems via Runge-Kuttamethods

Here, the first term on the right hand side gives the coherent evolution due to the closedsystem that is of the Liouville-von Neumann form, while the second term provides thecorrection due to its interaction with the environment We also further reiterate that in thederivation of the master equation, we only assumed that the system and environment areweakly coupled (also known as Born’s approximation) and hence we were allowed to perform

5 This is justified as the corrections are of higher order with respect to

2.2 DERIVATION OF MASTER EQUATION BY PERTURBATION THEORY

perturbation and expand our interaction term in the form of Eq (2.6) Due to the weakcoupling, we hence can also further assume a factorised initial state at t0, ⇢S(t0)⌦ ⇢B, both

of which allows us to simplify our expression into a solvable form

2.2.3 Pure Dephasing Master Equation

Lastly, in relevance to the subsequent chapters, we can make further simplification to ourmaster equation by considering a pure dephasing case As mentioned in the earlier chapter,

a pure dephasing mechanism allows the system to reach a completely mixed state withoutany change in the population probability or energy loss In this case, the system operatorS(t) and the unitary operator US(t) of the system then commutes with each other suchthat [US(t), S(t)] = 0 Hence, the above master equation can be further simplified and thesystem’s unitary operator can now be eliminated due to completeness relation as shownbelow:

˜B(t) ˜B(t1)E

B decaysvery quickly and we can push the upper limit of our integration to infinity Furthermore,the second term of Eq (2.13) has to be computed only once, further saving computationaltime Second assumption that is commonly made is the rotating wave approximation In this

case, the fast oscillating terms, which average to zero, in the master equation are droppedfrom the beginning With these two further assumptions, the master equation is furthersimplified into the Lindblad form [84], which ensures the preservation of trace, hermiticityand complete-positivity of the reduced density matrix In Section 2.4, we will be derivinganother expression of master equation based on the Lindblad’s approach, and apply it totime-dependent driven systems In the Appendix 2.A, we will also present another directway to derive the master equation in Eq (2.13) for completeness of the discussion.

In many real systems, the system Hamiltonian carries explicit time dependence Famousexamples of such time-dependent systems include the Landau-Zener transitions [93], the Rabimodel [94] and the Jaynes-Cummings model [95] Such driven systems have been motivated

by the development of the laser and maser systems which allows for strong electromagneticinteractions between the laser fields, atoms and molecules, where the laser and maser systemsact like a driving field on the system This in turn leads to an growing interest, for instance,towards periodic driven systems such as the kicked rotor model [96, 97], which is known as

a paradigm to study quantum chaos and quantum-classical correspondence

Given the rich dynamics in the driven systems, it becomes extremely important to be able

to solve the equation of motion involving such driven systems Unfortunately, such systemsusually cannot be solved in a straightforward manner and have to be approached numerically.Various methods, such as the Generalised Bloch vector approach [98, 99], Floquet theory forperiodic driven systems [85, 100] and split operator method [101, 102] have been developed

in order to tackle such time-dependent problems

In the open quantum systems context, many techniques have been developed to solve suchproblems, such as hierarchy equation approach [103, 104], Green’s function renormalisationapproach [105], Floquet-Markov master equation approach [85], just to name a few In fact,one can also, in principle, utilise the master equations approach described in Chapter 2 to

2.4 TIME-DEPENDENT MASTER EQUATION IN LINDBLAD FORM

solve the the equations of motion for the reduced density matrix of the system However,regardless the choice of approach, one key challenge is the computational e↵ort required

in computing such systems We will briefly illustrate this below for the master equationapproach derived in previous section

As we observe in Eq (2.13), the integral term consists of a system unitary operator term

US(t, t1) For a time-independent system, we can simplify this term into exp [ iHS(t t1)],which can be typically recasted in terms of the eigenvalues of HS after projecting the entireequation into its energybasis Therefore, the overall term involves only a single integralwhich can be easily solved numerically For time-dependent systems, however, the unitaryoperator is instead given by exp [ iT Rtt1HS(s)ds], where T is the time ordering operator

By inspection, we can easily observe that one has to repeat the integrals for each densitymatrix element for the unitary operator for each time step, and as such this will eventually

be a daunting task for systems with long driving time (which is of our interest in subsequentchapters) or of large Hilbert space We therefore will have to resort to further approximations

to our master equations, as mentioned in the concluding remarks of the previous section

We have reviewed and derived the master equation in Eq (2.13) using nothing more thanthe standard time-dependent perturbation theory in quantum mechanics However, as men-tioned in the previous section, in many time-dependent driven systems, such general form

of master equation may make the problem intractable, or at least require an extremely longcomputation time to be solvable practically Hence, to simplify the problem at hand, asmentioned earlier, we perform Markovian approximation and rotating wave approximation

to reduce the master equation into a more solvable form, otherwise known as the Lindbladmaster equation

In this section we first consider a quantum system described by a time-dependent tonian H(s), where s = vt is the rescaled time, v is the driving speed of the parameter s and

Hamil-t is Hamil-the real Hamil-time To include Hamil-the e↵ecHamil-ts of dissipaHamil-tion in Hamil-the adiabaHamil-tic driving, we considerthe master equation in Lindblad form:

v d

ds⇢(s) = i[H(s), ⇢(s)] +

⇢A(s)⇢(s)A†(s) 1

2

⇥

A†(s)A(s)⇢(s) + ⇢(s)A†(s)A(s)⇤

(2.15)where A(s) = P

mnAmn(s)|m(s)i hn(s)| and also that A†(s) = P

mnA⇤

nm(s)|n(s)i hm(s)|.Here, |m(s)i and |n(s)i are instantaneous eigenstates of H(s) at given s The subject ofinterest here is the probability that the system is in the jth energy level, ⇢jj(s), where ⇢jj(s) =hj(s)| ⇢(s) |j(s)i Here, is the coupling strength between the system and the environment

To proceed further, we cast the entire master equation in the time-dependent basis of thesystem Hamltionian and consider the e↵ect of dissipation and dephasing individually

2.4.1 Dissipative Lindblad Equation

In a dissipative system, the energy levels of the system will undergo transition due to itsinteraction with the environment, and this can be represented by the choice of Lindbladoperator whereby the indices m 6= n Such choice of Lindblad operator will always induce

a transition between the m and n levels and mimic the e↵ect of dissipation to the ment Furthermore, such form of Lindblad operator will not be able to commute with theHamiltonian To proceed, we project the instantaneous eigenstate of H(s) into the masterequation and we obtain the following:

environ-For the diagonal terms, we have

Cjm(s) = X

m6=j

(⇢jm(s)h ˙m(s)|j(s)i + ⇢mj(s)hj(s)| ˙m(s)i) (2.17)

2.4 TIME-DEPENDENT MASTER EQUATION IN LINDBLAD FORM

At this stage, to solve for the diagonal terms, we will now multiply both sides by a factor