Crossover-time in quantum boson and spin systems

Cover ......Page 1
Lecture Notes in Physics: Monographs 21 ......Page 2
Crossover-Time in Quantum Boson and Spin Systems ......Page 4
Contents ......Page 8
1 Introduction ......Page 14
2.1 Introduction to the Coherent States ......Page 17
2.2 Some Properties of Boson Coherent States ......Page 21
2.3 Some Properties of Spin Coherent States ......Page 26
3 The Exact C-Number Equation for Time-Dependent Expectation Values in Coherent States ......Page 29
3.1 Exact C-Number Equation for Quantum Expectation Values: Time-Independent Boson Hamiltonians in Coherent States ......Page 30
3.2 Generalization for the Case of Any Number of Degrees of Freedom ......Page 33
3.3 Example: Dynamics of Expectation Values for a Nonlinear Quantum Oscillator ......Page 34
3.4 Nonlinear Representation of Expectation-Value Dynamics ......Page 37
3.5 Exact C-Number Equation for Quantum Expectation Values: Time-Independent Spin Hamiltonians in Coherent States ......Page 39
3.6 Exact C-Number Equation for Quantum Time-Dependent Expectation Values for Boson-Spin Hamiltonians with Arbitrary Number of Degrees of Freedom ......Page 41
3.8 Generalization of C-Number Equations in the Case of Time-Dependent Hamiltonians ......Page 43
4.1 Quasiclassical Dynamical Perturbation Theory for Linear C-Number Equations ......Page 45
4.2 Nonlinear Dynamical Perturbation Theory for Time-Dependent Expectation Values ......Page 48
5 The Characteristic Times of Violation of Quasiclassical Approximation for Integrable Boson and Spin Systems ......Page 53
5.1 The Characteristic Times of Violation of Quasiclassical Approximation for Integrable Boson Systems ......Page 54
5.2 Violation of Quasiclassical Approximation upon Taking into Account an Additional Averaging over the Initial Density Matrix ......Page 59
5.3 Time-Behavior of Quantum Correlation Functions ......Page 64
5.4 The Characteristic Times of Violation of Quasiclassical Approximation for Integrable Quantum Nonlinear Spin Systems ......Page 65
5.5 Application of the Dynamical Perturbation Theory in Deriving the Time-Scale Tn ......Page 71
5.6 The Time-Scale r^ for the One-Dimensional Einstein Gas ......Page 74
5.7 Conclusion ......Page 75
6 Time-Scale tau_hbar for a Kicked Quantum Nonlinear Oscillator ......Page 77
6.1 Quantum Map for C-Number Expectation Values ......Page 79
6.2 Classical Limit ......Page 82
6.3 The Time-Scale tau_hbar of Applicability of Classical Chaotic Motion ......Page 85
6.4 Quantum Boundary of Stochasticity ......Page 88
6.5 Additional Averaging over the Initial Density Matrix ......Page 91
7 The Time of Applicability of the Quasiclassical Approach in a Boson System of Two Interacting Quantum Nonlinear Resonances ......Page 92
7.1 The Main Equations ......Page 93
7.2 Classical Limit ......Page 96
7.3 The Results of Numerical Simulations of Quantum Dynamics ......Page 97
7.4 Conclusion ......Page 101
8.1 Introduction ......Page 102
8.2 Time-Independent Hamiltonian and Closed C-Number Equations ......Page 103
8.3 The Approximation of Constant Amplitude of the External Field ......Page 105
8.4 Time-Dependent Quantum Correlation Functions ......Page 107
8.5 Quantum Perturbation Theory ......Page 108
9.1 Introduction ......Page 116
9.2 Quantum Map ......Page 117
9.3 Classical Limit ......Page 118
9.4 Equation for the Generating Function ......Page 120
9.5 The Region of Parameters for Quasiclassical Consideration ......Page 122
9.6 Time-Scale of Quasiclassical Description for Strong Chaos ......Page 125
9.7.1 Time-Scale rift for Quantum Map ......Page 127
9.7.2 The Influence of the Initial Conditions ......Page 130
9.8 Conclusion ......Page 140
10.1 Introduction ......Page 142
10.2 The S-Ordered Quantum Density Matrix ......Page 143
10.3 Wigner Representation in \"Action-Angle\" Variables ......Page 148
10.4 Evolution of the Quantum Kicked Rotator in the Wigner Representation ......Page 151
10.5 Quantum Equations of Motion ......Page 155
10.6 Classical Model of Quantum Stochasticity ......Page 159
10.7 Discussion ......Page 169
11.1 Introduction ......Page 170
11.2 Quantum Equations of Motion ......Page 173
11.3 Closed Equations of Motion for the Quantum-Mechanical Expectation Values ......Page 176
11.4 Quantum Correlation Functions ......Page 181
11.5 Breaking the Semiclassical Approximation ......Page 182
11.6 Results of Numerical Analysis ......Page 184
11.7 Conclusion ......Page 190
12.1 Introduction ......Page 191
12.2 The Formulation of the Model ......Page 193
12.3 Time-Independent Hamiltonian for Slow Dynamics ......Page 197
12.4 Closed C-Number Equations for Quantum Expectation Values in Coherent States ......Page 199
12.5 Semiclassical Approximation and Chaotic Dynamics ......Page 203
12.6 The Peculiarities of Quantum Dynamics in the Regime of Strong Semiclassical Chaos ......Page 208
12.7 Additional Averaging over the Initial Density Matrix ......Page 213
12.8 Conclusion ......Page 215
13.1 Introduction ......Page 217
13.2 Description of the Paramagnetic Spin Model ......Page 218
13.3 Slow Dynamics and Its Time-Independent Hamiltonian ......Page 224
13.4 Conclusion ......Page 225
14 Weak Quantum Chaos in a System of N Atoms in a Resonant Cavity Interacting with an External Resonant Field ......Page 227
14.1 The Nonsingular Representation in the Vicinity of Separatrix ......Page 229
14.2 Equations for Quantum Expectation Values ......Page 231
14.4 Dynamics of Expectation Values and Quantum Correlation Functions in the Integrable Limit ......Page 232
14.5 Numerical Calculations of Weak Quantum Chaos ......Page 245
14.6 Conclusion ......Page 248
15.1 Introduction ......Page 249
15.2 Stationary Coherent States. Basic Equations ......Page 250
15.3 Dynamical Chaos in SCS ......Page 253
15.5 Diffusion in SCS ......Page 256
15.6 Strange Attractor in SCS ......Page 257
16 Discussion ......Page 260
Appendix A ......Page 264
Appendix B ......Page 268
Appendix C ......Page 269
References ......Page 273