Texts and Readings in Physical Sciences

Volume 20

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H. S. Mani, Chennai Mathematical Institute, ChennaiRam Ramaswamy, Jawaharlal Nehru University, New Delhi

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Subhashish Banerjee

Open Quantum SystemsDynamics of Nonclassical Evolution

123

Subhashish BanerjeeDepartment of PhysicsIndian Institute of Technology JodhpurJodhpur, Rajasthan, India

ISSN 2366-8849 ISSN 2366-8857 (electronic)Texts and Readings in Physical SciencesISBN 978-981-13-3181-7 ISBN 978-981-13-3182-4 (eBook)https://doi.org/10.1007/978-981-13-3182-4

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Contents

List of Figures vii

Preface 1

1 Introduction 3

2 A Primer on Quantum Statistical Mechanics and Path Integrals 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . 7

2.2.1 States, Operators, Evolutions and Transformations . . . 82.2.2 Various Pictures . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Baker Campbell Hausdorff (BCH) Theorem . . . . . . . 122.2.4 Density Matrices . . . . . . . . . . . . . . . . . . . . . . 142.2.5 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . 172.2.6 Partition Function . . . . . . . . . . . . . . . . . . . . . 192.2.7 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.1 Introduction to the Path Integral . . . . . . . . . . . . . 242.3.2 Illustrative examples . . . . . . . . . . . . . . . . . . . . 272.3.3 Partition Function as a Path Integral . . . . . . . . . . . 302.3.4 Density Matrix Evolution as a Path Integral . . . . . . 32

2.4 Guide to advanced literature . . . . . . . . . . . . . . . . . . 33

3 Master Equations: A Prolegomenon to Open Quantum Systems 353.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Liouville Equation . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . 413.5 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . 443.6 Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . 453.7 Quantum Dynamical Semigroups and Markovian Master

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.7.1 Derivation of the Lindblad-Gorini-Kosakowoski-

Sudarshan Master Equation . . . . . . . . . . . . . . . . 49

v

3.7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 523.7.3 Connection to the Pauli Master Equation . . . . . . . . 54

3.8 Quantum Non-Demolition Master Equations . . . . . . . . . 553.9 Projection Operator Techniques . . . . . . . . . . . . . . . . 57

3.9.1 Nakajima-Zwanzig Technique . . . . . . . . . . . . . . . 583.9.2 Time-Convolutionless Technique . . . . . . . . . . . . . 59

3.10 Guide to advanced literature . . . . . . . . . . . . . . . . . . 64

4 Influence Functional Approach to Open Quantum Systems 654.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 A Primer to the Influence Functional (IF) formalism . . . . . 664.3 Influence Functionals: An Explicit Evaluation . . . . . . . . . 71

4.3.1 Conventional Derivation of IF . . . . . . . . . . . . . . . 714.3.2 Basis Independent Derivation of IF . . . . . . . . . . . . 764.3.3 Semiclassical Interpretation of the Influence Functional 85

4.4 Propagator for linear Quantum Brownian Motion . . . . . . 864.5 Master Equation for Quantum Brownian Motion . . . . . . . 894.6 Guide to advanced literature . . . . . . . . . . . . . . . . . . 92

5 Dissipative Harmonic Oscillator 935.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2 Lindbladian Approach to the Damped Oscillator . . . . . . . 935.3 Quantum Brownian Motion . . . . . . . . . . . . . . . . . . . 99

5.3.1 Weak coupling, high T regime . . . . . . . . . . . . . . . 995.3.2 Strong coupling regime . . . . . . . . . . . . . . . . . . 1055.3.3 Fluctuation-Dissipation Theorem . . . . . . . . . . . . . 108

5.4 Foundational Issues . . . . . . . . . . . . . . . . . . . . . . . 1095.4.1 Quantum Phase Distribution . . . . . . . . . . . . . . . 1095.4.2 Number-Phase Complementarity . . . . . . . . . . . . . 114

5.5 Guide to advanced literature . . . . . . . . . . . . . . . . . . 118

6 Dissipative Two-State System 1196.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.2 Spin-Boson Model . . . . . . . . . . . . . . . . . . . . . . . . 120

6.2.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 1206.2.2 Shifted Oscillators . . . . . . . . . . . . . . . . . . . . . 1226.2.3 Polaron Transformation . . . . . . . . . . . . . . . . . . 124

6.3 Examples of two-state systems based on Josephson tunneljunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.3.1 Charge Qubit . . . . . . . . . . . . . . . . . . . . . . . . 1256.3.2 Flux Qubit . . . . . . . . . . . . . . . . . . . . . . . . . 1266.3.3 Phase Qubit . . . . . . . . . . . . . . . . . . . . . . . . 127

6.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 1286.4.1 Partition Function . . . . . . . . . . . . . . . . . . . . . 129

6.5 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Contentsvi

6.5.1 Exact Solution . . . . . . . . . . . . . . . . . . . . . . . 1306.5.2 Noninteracting-Blip Approximation . . . . . . . . . . . 135

6.6 Coupling to Reservoir via an Intermediate Harmonic Oscillator 1366.6.1 Effective Spectral Density . . . . . . . . . . . . . . . . . 1376.6.2 Application of the Effective Spectrum Method: Asymp-

totic behavior of the Spin-Boson Model . . . . . . . . . 1396.7 Guide to Advanced Literature . . . . . . . . . . . . . . . . . 142

7 Quantum Tunneling 145

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457.2 Semiclassical Approximation . . . . . . . . . . . . . . . . . . 1467.3 Double Well Potential . . . . . . . . . . . . . . . . . . . . . . 1527.4 Quantum Tunneling . . . . . . . . . . . . . . . . . . . . . . . 1597.5 Transition to Open Systems . . . . . . . . . . . . . . . . . . . 1657.6 Guide to further reading . . . . . . . . . . . . . . . . . . . . 169

8 Open Quantum System at Interface with Quantum Information 171

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.2 Role of Noise in Quantum Information: Introduction to Tools

and Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 1728.2.1 Quantum noise . . . . . . . . . . . . . . . . . . . . . . . 1758.2.2 Channel-State Duality . . . . . . . . . . . . . . . . . . . 179

8.3 Selected Applications of Open Quantum Systems to QuantumInformation Processing . . . . . . . . . . . . . . . . . . . . . 184

8.3.1 Environment-Mediated Quantum Deleter . . . . . . . . 1848.3.2 Geometric Phase (GP) in Open Quantum Systems . . . 1858.3.3 Classical Capacity of a Squeezed Generalized Amplitude

Damping Channel . . . . . . . . . . . . . . . . . . . . . 1898.3.4 Application to Quantum Cryptography . . . . . . . . . 1918.3.5 Dynamics of Quantum Correlations under Open Quan-

tum System evolutions . . . . . . . . . . . . . . . . . . . 1958.3.6 Quantum Walk . . . . . . . . . . . . . . . . . . . . . . . 2028.3.7 Quasiprobability distributions in open quantum systems 2048.3.8 Quantum error correction . . . . . . . . . . . . . . . . . 208

8.4 Guide to Further Literature . . . . . . . . . . . . . . . . . . . 219

9 Recent Trends 221

9.1 Application of Open Quantum System to Unruh Effect andSub-Atomic Physics . . . . . . . . . . . . . . . . . . . . . . . 221

9.1.1 Unruh Effect . . . . . . . . . . . . . . . . . . . . . . . . 2229.1.2 Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . 2269.1.3 Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

9.2 Non-Markovian Phenomena . . . . . . . . . . . . . . . . . . . 2349.2.1 Non-Markovian Master Equations . . . . . . . . . . . . 234

viiContents

9.2.2 Information backflow and breakdown of CP divisibilityof the intermediate map . . . . . . . . . . . . . . . . . . 236

9.2.3 Illustrative Examples . . . . . . . . . . . . . . . . . . . . 2399.3 Quantum Thermodynamics . . . . . . . . . . . . . . . . . . . 2459.4 What next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

References 249

Index 277

Contentsviii

List of Figures

5.1 Quantum phase distributions P(θ), for a harmonic oscillator sys-tem starting in a squeezed coherent state, for dissipative system-bath interaction (Eq. (5.147)). Temperature (in units where� ≡ kB ≡ 1) T = 0, the squeezing parameters r = r1 = 1,bath exposure time t = 0.1, γ0 = 0.025, ω = 1 and Φ = 0. . . . 114

5.2 Plot of number entropy H[m] (dot-dashed line), phase knowledgeR[θ] (dashed line) and entropy excess X[m, θ] (bold line) for aharmonic oscillator, initially in a coherent state superposition(5.157), as a function of the state parameter φ (5.157). Figure (a)pertains to the pure state case. Figure (b) represents the systemsubjected to a dissipative interaction with the environment for anevolution time t = 0.1 and temperature T = 2. The parametersused are ω = 1.0, γ0 = 0.026, |α|2 = 2.2, φ0 (5.157) = 0. . . . . 117

6.1 Double Well representing the TSS; (a). Symmetric, (b). Asym-metric Well. Figure adapted from [1]. . . . . . . . . . . . . . . . 121

6.2 Circuit representations of the CPB. . . . . . . . . . . . . . . . . 125

6.3 Circuit representations of the rf-SQuID. . . . . . . . . . . . . . 127

6.4 Circuit representations of the current-biased Josephson junction. 128

7.1 Double well potential with local minima at ∓a. Figure adaptedfrom [42]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.2 The two instanton solutions depicted in Eqs. (7.35) and (7.36). 155

7.3 Quadratic plus cubic potential V (x) = 12mω2

0x2(1− 2x

3xb

)with a

stable local minima at ω0 at x = 0, unstable local maxima at ωb

at x = xb and a barrier height of Vb. . . . . . . . . . . . . . . . 160

8.1 Fidelity (f(t)) falls as a function of temperature (T , in unitswhere � ≡ kB ≡ 1) until it reaches the value 1/

√2 corresponding

to a maximally mixed state. The case shown here correspondsto θ0 = 0, γ0 = 0.6, ω = 1.0 and time t = 12. Here we set thesqueezing parameters s and Φ to zero. Figure adapted from [205]. 186

ix

8.2 GP (ΦGP in radians) as function of temperature (T ) for dissipa-tive interaction with a bath of harmonic oscillators. Here ω = 1,θ0 = π/2, the large-dashed, dot-dashed, small-dashed and solidcurves, represent, γ0 = 0.005, 0.01, 0.03 and 0.05, respectively.Fig. (A) zero squeezing, Fig. (B): squeezing non–vanishing, withs = 0.4 and Φ = 0. GP falls with T . Effect of squeezing: GPvaries more slowly with T , by broadening peaks and flatteningtails. Counteractive action of squeezing on influence of T on GPuseful for practical implementation of GP phase gates. Figureadapted from [80]. . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.3 Holevo bound χ for a squeezed generalized amplitude dampingchannel with Φ = 0, over the set {θ0, φ0}, which parametrizes theensemble of input states {(θ0, φ0), (θ0+π, φ0)}, corresponding tothe symbols 0 and 1, respectively, with probability of the inputsymbol 0 being f = 0.5. Here temperature T = 5, γ0 = 0.1,time t = 5.0 and bath squeezing parameter s = 1. The figure isadapted from [80]. . . . . . . . . . . . . . . . . . . . . . . . . . 190

8.4 Optimal source coding for the squeezed amplitude dampingchannel, with χ plotted against θ0 corresponding to the “0” sym-bol. Here Φ = 0, γ0 = 0.05 and f = 0.5. The solid and small-dashed curves represent temperature T = 0 and bath squeezingparameter s = 0, but t = 1 and 4, respectively. The large-dashedand dot-dashed curves represent T = 2.5 and t = 2, but withs = 0 and 2, respectively. The figure is adapted from [80]. . . . 191

8.5 Interplay of squeezing and temperature on the classical capacityC of the squeezed amplitude damping channel (with input states1√2(|0〉±|1〉), and f = 1/2, corresponding to the optimal coding).

Here Φ = 0 and γ0 = 0.1. The dotted and dot-dashed curvescorrespond to zero squeezing s, and temperature T = 0 and 7,respectively. The solid curve corresponds to T = 7 and s = 3.The figure is adapted from [80]. . . . . . . . . . . . . . . . . . . 192

8.6 Information recovered by Bob, quantified by the Holevo quan-tity χ, as a function of the key information κ communicated byCharlie, in the noiseless case. The figure is adapted from [252]. 194

8.7 Information recovered by Bob, quantified by the Holevo quantityχ, as a function of the squeezing parameter s, coming from theSGAD Channel, and key information κ communicated by Char-lie. The time of evolution t = 0.8, while temperature T = 0.2 (inunits where � ≡ kB ≡ 1). The figure is adapted from [252]. . . . 194

List of Figuresx

8.8 Information recovered by Bob, quantified by the Holevo quantityχ, as a function of the SGAD channel parameters s (squeezing)and t (time of evolution), assuming Charlie communicates onebit of information. We note that, for sufficiently early times,squeezing fights thermal effects (here T = 0.2) to cause an in-crease in the recovered information. The figure is adapted from[252]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

8.9 Concurrence C as a function of time of evolution t. The left fig-ure deals with the case of vacuum bath (T = s = 0), while thefigure in the right panel considers concurrence in the two-qubitsystem interacting with a squeezed thermal bath, for a temper-ature T = 1 and bath squeezing parameter s equal to 0.1. Inthe left figure the bold curve depicts the collective decoherencemodel (k0r12 = 0.01), while the dashed curve represents the in-dependent decoherence model (k0r12 = 1.1). In the right figure,for the given settings, concurrence for the collective model isdepicted by the dashed curve. Here, for the independent deco-herence model, concurrence is negligible and is thus not seen.The figure is adapted from [264]. . . . . . . . . . . . . . . . . . 200

8.10 Concurrence C with respect to inter-qubit distance r12. The fig-ure on the left deals with the case of vacuum bath (T = s = 0),while the figure on the right considers concurrence in the two-qubit system interacting with a squeezed thermal bath, forT = 1, evolution time t = 1 and bath squeezing parameter sequal to 0.2. In the left figure the oscillatory behavior of concur-rence is stronger in the collective decoherence regime, in compar-ison with the independent decoherence regime (k0r12 ≥ 1); herek0 is set equal to one. In the right figure, the effect of finite bathsqueezing and T has the effect of diminishing the concurrence toa great extent in comparison to the vacuum bath case. Here theconcurrence for the localized decoherence regime is negligible, inagreement with the previous figure. The figure is adapted from[264]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

8.11 Dynamics of concurrence and maximum teleportation fidelityFmax as a function of the inter-qubit separation r12. The figureis adapted from [266]. . . . . . . . . . . . . . . . . . . . . . . . 201

8.12 Dynamics of Bell’s inequality factor M(ρ) and discord as a func-tion of the inter-qubit separation r12. The figure is adapted from[266]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

8.13 Variance of classical and quantum walk as a function of the timesteps. The figure is adapted from [287]. . . . . . . . . . . . . . . 204

xiList of Figures

8.14 The effect of increasing the environmental squeezing parameters of the SGAD channel, on the position probability distributionof a particle with the initial state (1/

√2)(|0〉 + i|1〉) ⊗ |ψ0〉 and

coin toss instruction given by B(0◦, 45◦, 0◦). Other channel pa-rameters are fixed at T = 3, γ0 = 0.03 and Δ = 0.2. (a) Theprobability distribution on a line for different values of s, at timet = 100 steps. With increasing noise (parametrized by s), thedistribution transforms from the characteristic QW twin-peakeddistribution to the classical Gaussian. The figure is adapted from[291]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

8.15 The variation of all the QDs with time (t) are depicted for asingle spin- 1

2 atomic coherent state in the presence of the SGADnoise for, zero bath squeezing angle, in units of � = kB = 1,with ω = 1.0, γ0 = 0.05, and α = π

2 , β = π3 , θ = π

2 , φ = π3 . In

(a) the variation with time is shown for temperature T = 5.0 inthe absence of squeezing parameter, i.e., s = 0. In (b) the effectof the change in squeezing parameter for same temperature, i.e.,T = 5.0 is shown by using the squeezing parameter s = 1.5,keeping all the other values as same as that used in (a). Further,in (c) keeping s = 1.5 as in (b), the temperature is increased toT = 15 to show the effect of variation in T. In (c) time is variedonly up to t = 5 to emphasize the effect of temperature. In allthe three plots, dashed, solid and dotted lines correspond to theW , P and Q functions, respectively. The figure is adapted from[322]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

9.1 Degradation of QMID QM (dark bold line), maximum tele-portation fidelity Fmax (large dashed curve), Bell quantity M/2(small dashed curve)) and concurrence (light bold line) C as afunction of Unruh acceleration a, for ω = 0.1 (in units where� ≡ c ≡ 1). Figure adapted from [371]. . . . . . . . . . . . . . . 225

List of Figuresxii

9.2 Average correlation measures c, i.e., the various measures modu-lated by the exponential factor e−2Γt, as a function of time t. Theupper row corresponds to the correlations of a KK̄ and BdB̄d

pair, respectively, while the correlations of BsB̄s pair is depictedin the bottom row figure; here these pairs are created at t = 0.The four correlation measures are (top to bottom): M(ρ) (Bell’sinequality; blue band), Fmax (teleportation fidelity; red band),EF (entanglement of formation; grey band) and DG (geometricdiscord; green band). For KK̄ pairs, left panel, time is in unitsof 10−10 seconds whereas for the BdB̄d and BsB̄s pairs, timeis in units of 10−12 seconds (in all cases, the approximate life-time of the particles). In the figures in the upper row, the bandsrepresent the effect of decoherence corresponding to a 3σ upperbound on the decoherence parameter λ. The bottom row has nosuch bands because there is currently no experimental evidencefor decoherence in the case of Bs mesons. Figure adapted from[372]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

9.3 The eigenvalues of the Choi matrix obtained from intermedi-ate dynamical map in both the Markovian and non-Markovianregime for RTN. We can observe that the eigenvalue (λ3) of RTNbecome negative only in the non-Markovian (solid line) regimeindicative of NCP character, whereas in the Markovian regime(dashed line) the eigenvalue is always positive. Note that boththe intermediate time t1 = 1 and the noise amplitude a = 0.6 iscommon for both the curves. Figure adapted from [397]. . . . . 244

9.4 Plot of TD (Trace Distance) evolution under the influence ofRTN, between the initial states |ψ(±π

4 , 0)〉c. (a) Noiseless quan-tum walk: Oscillations in TD are observed due to the interactionwith position environment. Effect of RTN on the QW: (b). inthe Markovian regime (middle curve) damping of the positioninduced oscillations is observed, while (c). in the non-Markovianregime (right) an additional oscillatory frequency component ispresent due to RTN-induced decoherence. Figure adapted from[397]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

xiiiList of Figures

xv

About the Author

Subhashish Banerjee is teaching at the Department of Physics, Indian Institute of Technology Jodhpur, India. He obtained his PhD degree from the School of Physical Sciences, Jawaharlal Nehru University, New Delhi, India, in 2003. He has been involved in the study of the theory of open quantum systems since his Ph.D. A major theme of his work has been to see how the theory of open quantum sys­tems provides a common umbrella to understand various facets of quantum optics, quantum information processing, quantum computing, quantum cryptography, the foundations of quantum mechanics, relativistic quantum mechanics and field theory. He has published two books and more than 70 articles in international journals of repute.

Preface

Open quantum systems is the study of quantum dynamics of the system ofinterest, taking into account the effects of the ambient environment. It is ubiq-uitous in the sense that any system could be envisaged to be surrounded by itsenvironment which could naturally exert its influence on it. It traces its rootsfrom Quantum Optics and has found applications in diverse areas, ranging fromcondensed matter to quantum cosmology. Open Quantum Systems allows fora systematic understanding of irreversible processes such as decoherence anddissipation, of essence in order to have a correct understanding of realistic quan-tum dynamics and also for possible implementations. This would be essentialfor a possible development of quantum technologies. Interest has been revivedin recent times due to the upsurge of theoretical and experimental progress.

We try to put down in this book, in a comprehensive manner, the basicideas of open quantum systems and the tools needed for the same. Emphasis isgiven to both the traditional master equation as well as the functional (path)integral approaches. In fact, this book can be used as a beginning guide forunderstanding and use of path integrals. The basic paradigm of open systems,the harmonic oscillator and the two-level system are discussed in detail. Thetraditional topics of dissipation and tunneling as well as the modern field ofquantum information find a prominent place in the text.

Despite its importance, the subject of Open Quantum Systems is notpresent in the curriculum of Indian Universities and Institutes; it is treated, atbest, as an abstruse subject. One of the main goals, and hopes, of this bookwould be to bring a change in this scheme of things. Assuming a basic back-ground of quantum and statistical mechanics, this book will help to familiarizethe reader with the basic tools of open quantum systems.

This book is aimed at taking a reader with a basic background of quan-tum and statistical mechanics to the level where he/she can start appreciatingresearch problems of current interest. A good background of undergraduatephysics should suffice to begin with the present book. In any case, an introduc-tory chapter on quantum statistical mechanics and path integrals are included,with references to more advanced literature. The book aims to highlight theubiquity of Open Quantum Systems based on simple models and calculations.

As I reflect back, I find that there are a number of people to whom I owethe development of this book. The first person who comes to mind is Prof. R.

1

Ramaswamy who has been a teacher and friend to me for a long time. It was hissuggestion that started this project. His constant help and advice, along withthat of Prof. Debashis Ghoshal and Mr. D. K. Jain of Hindustan Book Agencymade this book possible. If I were to trace the roots of my involvement with thissubject, then I would say that it started with the works of Profs. A. O. Caldeiraand A. J. Leggett. I also immensely benefited from the works of Prof. Vinay Am-begaokar, with whom I was fortunate to have a brief interaction, and Profs. H.Grabert, G. Ingold, P. Hanggi, G. S. Agarwal and B. L. Hu. My sincere thanksto all of them. I have also benefitted very much from the classic book on thesubject by Profs. H-P. Breuer and F. Petruccione, with both of whom I have hadthe opportunity of some interaction. Over the years there have been a number ofpeople in the scientific community to whom I have looked up to for inspirationand this would be an appropriate juncture to thank them. They are Profs. R.Ghosh, R. Rajaraman, (late) D. Kumar, R. Simon, J. Kupsch, G. Rajasekaran,H. S. Mani and C. S. Seshadri. My journey in this field would not have beenpossible without discussions and collaborations with a number of colleagues:Richard Mackenzie, Andreas Buchleitner, Christophe Couteau, Sibasish Ghosh,C. M. Chandrashekar, V. V. Sreedhar, R. Jagannathan, R. Parthasarathy, Ab-hishek Dhar, R. Srikanth, (late) N. Kumar, Hema Ramachandran, A. R. UshaDevi, A. K. Rajagopal, Subhash Chaturvedi, Subhasish Dutta Gupta, ArunJayanavar, Pankaj Agrawal, Arun Pati, Debasis Sarkar, Archan S. Majumdar,Guruprasad Kar, Somshubhro Bandyopadhyay, Prasanta K. Panigrahi, Anir-ban Pathak, V. Ravishankar, Sankalpa Ghosh, Sandeep Goyal, George Thomasand S. Uma Sankar. In particular, I need to thank S. Omkar, Pradeep Ku-mar, N. Siddharth, Javid A. Naikoo and Supriyo Dutta for their extensive helpwith the numerical and formatting related issues associated with the book. Ialso want to express my gratitude to my family and friends for all their helpand support. I conclude by thanking my wife Pallavi and son Shubhonkar formaking me realize there is more to life than open quantum systems.

2 Preface