Quantum dynamical semigroups and applications

Reinvigorated by advances and insights, in particular from the active fields of quantum information and computing, the quantum theory of irreversible processes has recently attracted growing attention.

This volume introduces the very basic concepts of semigroup dynamics of open quantum systems and reviews a variety of modern applications. The present book, originally published as Vol. 286 (1987) in Lecture in Physics, has been newly typeset, revised and corrected and also expanded to include a review on recent developments.

Robert AlickiKarl LendiRobert Alicki and Karl Lendi

General Theory and Applications to Unstable Particles	p. 1
1 General Theory	p. 1
1.1 Introduction	p. 1
1.2 Completely Positive Dynamical Semigroups	p. 2
1.2.1 Reduced Dynamics	p. 2
1.2.2 Completely Positive Maps	p. 3
1.2.3 Generalized H-theorem	p. 5
1.2.4 Generators of Quantum Dynamical Semigroups	p. 7
1.2.5 How to Construct Generators?	p. 9
1.3 Hamiltonian Models and Markovian Approximation	p. 10
1.3.1 Generalized Master Equation	p. 10
1.3.2 Weak Coupling Limit	p. 11
1.3.3 Low Density Limit	p. 14
1.3.4 Heat Bath, Detailed Balance and Return to Equilibrium	p. 16
1.3.5 Singular Coupling Limit	p. 18
1.4 Extensions of the Formalism	p. 19
1.4.1 Nonconservative Dynamical Semigroups	p. 19
1.4.2 Time-dependent Generators	p. 20
1.4.3 Nonlinear Quantum Dynamical Semigroups	p. 21
1.4.4 Discrete Quantum Boltzmann Equation	p. 22
1.4.5 Nonlinear Schrodinger Equation	p. 23
1.5 A System of N Two-level Atoms	p. 24
1.5.1 The Hamiltonian of the System	p. 24
1.5.2 The Markovian Master Equation	p. 25
1.5.3 Return to Equilibrium and Superradiance	p. 27
2 Quantum Dynamical Semigroups for Unstable Particles	p. 29
2.1 Introduction	p. 29
2.2 Damped and Pumped Quantum Harmonic Oscillator	p. 30
2.2.1 Derivation of the Markovian Master Equation	p. 30
2.2.2 Birth and Death Process, Kinetic Equation	p. 31
2.2.3 Explicit Solutions	p. 31
2.3 Models of Unstable Particles	p. 32
2.3.1 Fock Spaces and Quantum Fields	p. 32
2.3.2 Construction of Markovian Master Equation	p. 34
2.3.3 Single-particle Description	p. 35
2.3.4 Explicit Solutions	p. 36
2.3.5 Hamiltonian Models of Unstable Particles	p. 38
2.3.6 Relativistic Unstable Particles	p. 40
Appendix
A.1 Banach Spaces B(H) and T(H)	p. 41
A.2 One-parameter Semigroups	p. 42
A.3 Quantum Correlation Functions	p. 44
References	p. 45
N-Level Systems and Applications to Spectroscopy	p. 47
1 Introduction	p. 47
2 General Structure of Quantum Markovian Master Equations for N-level Systems	p. 48
2.1 The Kossakowski-Generator of Infinitesimal Time-evolution	p. 48
2.2 Positive-semidefiniteness of the Relaxation Matrix	p. 49
2.3 Complete Orthonormal Matrix Sets	p. 50
2.4 Coherence-vector Formulation	p. 53
2.5 Relaxing Semigroups	p. 57
3 Two-level Systems: Generalized Magnetic or Optical Bloch-equations	p. 61
3.1 Details of the Full Relaxation Equations for Static External Fields	p. 61
3.2 Alternating External Fields and Constant Relaxation	p. 64
3.3 Modified Lineshapes and Free Induction Decay	p. 66
4 Three-level Systems	p. 69
4.1 General Equations	p. 69
4.2 Bloch-equations for Decaying Systems	p. 70
5 Comparison with Common Versions of Master Equations	p. 73
5.1 General Considerations	p. 73
5.2 Equations for Spontaneous Emission	p. 74
5.3 Equations of Lamb-type	p. 76
6 Open Quantum Systems with Non-constant Relaxation in Time-dependent External Fields	p. 77
6.1 Modifications of the Original Semigroup Generator	p. 77
6.2 A Model with Field-Strength-dependent Relaxation	p. 79
7 Determination of Relaxation Parameters from First Principles	p. 81
7.1 Relationship between Kossakowski- and Davies-generators	p. 81
7.2 A Model for Spin-relaxation by Spin-waves	p. 85
8 Entropy and Irreversibility	p. 89
8.1 Entropy Production	p. 89
8.2 Measure of Irreversibility	p. 94
9 Conclusion	p. 98
Appendix
A.1 Generators and Structure Constants for SU(N), N = 2,3,4	p. 99
A.2 Eigenvalues of the General Two-level Evolution Matrix	p. 102
A.3 Elements of the Time-dependent Two-level Evolution Matrix	p. 104
References	p. 104
Recent Developments	p. 109
1 Complete Positivity, Entanglement and Decoherence	p. 109
2 Unbounded Generators and Stochastic Equations	p. 110
3 Nonlinear QDS	p. 111
4 Geometry of States and Symmetries of Generators	p. 111
5 QDS and Thermodynamics	p. 112
6 Applications in Atomic and Molecular Physics	p. 113
7 Beyond a Markovian Approximation	p. 114
References	p. 117
Index	p. 123

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