Open Quantum Systems (eBook)

SUBHASHISH BANERJEE is an assistant professor and head of 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 quantum theory of open systems since his PhD. 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 67 articles in national and international journals of repute. 

Contents 6
List of Figures 10
About the Author 15
Preface 16
1 Introduction 18
2 A Primer on Quantum Statistical Mechanics and Path Integrals 21
2.1 Introduction 21
2.2 Quantum Statistical Mechanics 21
2.2.1 States, Operators, Evolutions and Transformations 22
2.2.2 Various Pictures 25
2.2.3 Baker Campbell Hausdorff (BCH) Theorem 26
2.2.4 Density Matrices 28
2.2.5 Harmonic Oscillator 31
2.2.6 Partition Function 33
2.2.7 Entropy 35
2.3 Path Integrals 37
2.3.1 Introduction to the Path Integral 38
2.3.2 Illustrative examples 41
2.3.3 Partition Function as a Path Integral 44
2.3.4 Density Matrix Evolution as a Path Integral 46
2.4 Guide to advanced literature 47
3 Master Equations: A Prolegomenon to Open Quantum Systems 48
3.1 Introduction 48
3.2 Liouville Equation 49
3.3 Langevin Equation 50
3.4 Fokker-Planck Equation 54
3.5 Boltzmann Equation 57
3.6 Master Equation 58
3.7 Quantum Dynamical Semigroups and Markovian Master Equation 61
3.7.1 Derivation of the Lindblad-Gorini-Kosakowoski-Sudarshan Master Equation 62
3.7.2 Examples 65
3.7.3 Connection to the Pauli Master Equation 67
3.8 Quantum Non-Demolition Master Equations 68
3.9 Projection Operator Techniques 70
3.9.1 Nakajima-Zwanzig Technique 71
3.9.2 Time-Convolutionless Technique 72
3.10 Guide to advanced literature 77
4 Influence Functional Approach to Open Quantum Systems 78
4.1 Introduction 78
4.2 A Primer to the Influence Functional (IF) formalism 79
4.3 Influence Functionals: An Explicit Evaluation 84
4.3.1 Conventional Derivation of IF 84
4.3.2 Basis Independent Derivation of IF 89
4.3.3 Semiclassical Interpretation of the Influence Functional 98
4.4 Propagator for linear Quantum Brownian Motion 99
4.5 Master Equation for Quantum Brownian Motion 102
4.6 Guide to advanced literature 105
5 Dissipative Harmonic Oscillator 106
5.1 Introduction 106
5.2 Lindbladian Approach to the Damped Oscillator 106
5.3 Quantum Brownian Motion 112
5.3.1 Weak coupling, high T regime 112
5.3.2 Strong coupling regime 118
5.3.3 Fluctuation-Dissipation Theorem 121
5.4 Foundational Issues 122
5.4.1 Quantum Phase Distribution 122
5.4.2 Number-Phase Complementarity 127
5.5 Guide to advanced literature 131
6 Dissipative Two-State System 132
6.1 Introduction 132
6.2 Spin-Boson Model 133
6.2.1 Hamiltonian 133
6.2.2 Shifted Oscillators 135
6.2.3 Polaron Transformation 137
6.3 Examples of two-state systems based on Josephson tunnel junctions 137
6.3.1 Charge Qubit 138
6.3.2 Flux Qubit 139
6.3.3 Phase Qubit 140
6.4 Thermodynamics 141
6.4.1 Partition Function 142
6.5 Dynamics 143
6.5.1 Exact Solution 143
6.5.2 Noninteracting-Blip Approximation 148
6.6 Coupling to Reservoir via an Intermediate Harmonic Oscillator 149
6.6.1 Effective Spectral Density 150
6.6.2 Application of the Effective Spectrum Method: Asymptotic behavior of the Spin-Boson Model 152
6.7 Guide to Advanced Literature 155
7 Quantum Tunneling 157
7.1 Introduction 157
7.2 Semiclassical Approximation 158
7.3 Double Well Potential 164
7.4 Quantum Tunneling 171
7.5 Transition to Open Systems 177
7.6 Guide to further reading 181
8 Open Quantum System at Interface with Quantum Information 183
8.1 Introduction 183
8.2 Role of Noise in Quantum Information: Introduction to Tools and Techniques 184
8.2.1 Quantum noise 187
8.2.2 Channel-State Duality 191
8.3 Selected Applications of Open Quantum Systems to Quantum Information Processing 196
8.3.1 Environment-Mediated Quantum Deleter 196
8.3.2 Geometric Phase (GP) in Open Quantum Systems 197
8.3.3 Classical Capacity of a Squeezed Generalized Amplitude Damping Channel 201
8.3.4 Application to Quantum Cryptography 203
8.3.5 Dynamics of Quantum Correlations under Open Quantum System evolutions 207
8.3.6 Quantum Walk 214
8.3.7 Quasiprobability distributions in open quantum systems 216
8.3.8 Quantum error correction 220
8.4 Guide to Further Literature 231
9 Recent Trends 232
9.1 Application of Open Quantum System to Unruh Effect and Sub-Atomic Physics 232
9.1.1 Unruh Effect 233
9.1.2 Neutrinos 237
9.1.3 Mesons 241
9.2 Non-Markovian Phenomena 245
9.2.1 Non-Markovian Master Equations 245
9.2.2 Information backflow and breakdown of CP divisibility of the intermediate map 247
9.2.3 Illustrative Examples 250
9.3 Quantum Thermodynamics 256
9.4 What next? 258
References 259
Index 286