Table of contents for Coherent dynamics of complex quantum systems / Vladimir M. Akulin.


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1   Complex Systems and Their Statistical Description .......  1
2   Examples of Complex Systems..........................     17
2.1 Molecules and Atoms in Laser Fields... ................... 18
2.1.1 Laser Breaking of a Weakly Bonded Complex ........ 18
2.1.2 Laser-Induced Electronic Transitions in Molecules .... 21
2.1.3 Vibrational Excitation of Polyatomic Molecules ...... 22
2.1.4 Transitions Among Levels with Fine Structure ....... 24
2.1.5 Excitation of Rydberg States in Atoms.............. 26
2.1.6 Competition of Multiphoton Processes of Different
Orders .......................................... 27
2.2  Collisions and Reactions of Molecules .....................  28
2.2.1 Collisional Redistribution of Energy ................ 28
2.2.2  Chemical Reactions..............................  32
2.2.3 Intermolecular Conversion and Photochemistry ....... 35
2.3 Rydberg Molecules ................................... 38
2.3.1  Subthreshold Photoionization  ......................  38
2.3.2  Collisional Ionization  ............................  40
2.4  Atomic and  Molecular Clusters ...........................  41
2.4.1 Ground Electronic State of Hot Metallic Clusters..... 43
2.4.2  Optical Properties of Clusters .....................  46
2.5  Some  Other  Examples  ................................ .  48
2.5.1 Ion Traps .....................................  48
2.5.2 Disordered Solids and Surfaces ..................... 51
2.5.3  Nonlinear  Optics  .............................. .  55
2.5.4 Cooperative Effect ............................... 56
2.5.5 Many-Body Effects in Cold Rydberg Gas ............ 58
3   Two-Level and Level-Band Systems ...................... 61
3.1 Two-Level System ..................................... 62
3.2 Level-band System .................................... 68
3.2.1  General Consideration  ............................  68
3.2.2  Continuous Band  Model ...........................  70
3.2.3 Measurements and Relaxation as Processes in
Level-Continuum  Systems .........................  83
3.3 Long-Time Behavior ................................... 88
3.3.1 General Consideration of the Long-Time Limit ....... 88
3.3.2  Quantum  Recurrences  ............................  91
3.3.3  Quantum  Revivals ............................. .  95
3.3.4  Fractional Revivals  ............................. .  99
3.3.5 Revivals and the Classical Limit .................. . 104
S3.4  P opulation  of  Inhom ogeneousıB ands  .-; : -  .- ;.. . . ...  .  1-05
3.4.1  Statistically Independent Levels ..................  . 106
3.4.2 Factorization of the Level Population and the
Ensemble  Average  ................................ 108
3.4.3  The Long-Time Asymptotic  ....................... 111
4   Two-Band System .....................................    . 123
4.1  General Consideration  .................................. 124
4.1.1 Series and Diagrams for the Level-Band Problem..... 124
4.1.2 Series and Diagrams for the Two-Band Problem...... 128
4.1.3  The  Renormalization  ........................... . 133
4.2  Non-Degenerate Bands .................................. 139
4.2.1 General Remarks and the Main Questions ........... 139
4.2.2 Renormalized Energies and the Population Distribution140
4.2.3 Dynamics of the Total Populations of Bands ......... 143
4.2.4  Different  Bands  .................................. 144
4.3  Two  Degenerate  Levels ............... ................. . 145
4.3.1 Degenerate Levels as a Complex System............. 145
4.3.2 The Bands as an Ensemble of Two-Level Systems .... 148
4.4 A Band Coupled to a Degenerate Level.................... 151
4.4.1  Total Population  of the Bands  ..................... 151
4.4.2 Population Distribution over the Band ............... 154
4.4.3  Role of the Interaction  Rank.........................  159
4.5 The Role of Correlations ............................. 161
4.5.1  Two  levels and  a  band  ...........................  161
4.5.2 Two Bands With a Correlated Coupling ............. 169
4.5.3 Regime of Stabilization for the Correlated Coupling... 178
4.5.4 Correlation Between the Mean Squared Coupling and
the  Energy  Position  .............................. 179
5   Soluble Time-Dependent Systems ....................... 187
5.1 Algebraic Structure of Time Dependent Systems ........... 188
5.2 Time-Dependent Two-Level Systems ...................... 194
5.2.1  Landau-Zener Problem  ........................... 194
5.2.2 Landau-Zener Transition to a Decaying State ........ 199
5.2.3 Landau-Zener Transition in the Presence of
Transversal Relaxation  ............................ 202
5.2.4  Excitation  by  a  Pulse  ............................. 208
5.2.5  Exponentially Rising Coupling  ..................... 210
5.3 Semiclassical Analysis of Time-Dependent Systems ......... 213
5.3.1 Two-Level Systems and the Dykhne Formula ........ 213
5.3.2  M ultilevel Systems  ............................... 218
5.4 Time-Dependent Level-Band System................. .  . 221
5.4.1 The Demkov-Osherov Problem.................. 222
5.4.2 The Landau-Zener Transition at the Continuum Edge 227
6   Time-Dependent Complex Systems ...................... 237
6.1 Degenerate Level Crosses an Infinite Band ................. 238
6.2 Perturbation Proportional to a Random Matrix ........... 244
6.2.1 Population Distribution ........................ 246
6.2.2 Response to Perturbation Proportional to a Random
M atrix  .................... . . ............... . 250
6.3 Harmonic Perturbation of Complex Systems ............... 261
6.3.1 Population Distribution over a Uniform Spectrum .... 261
6.3.2 Response of the Uniform Spectrum to a Harmonic
Perturbation .................................. . 265
6.4 Two-Frequency Excitation of Complex Systems ............ 270
6.4.1 Population Dynamics for Bi-Harmonic Excitation .... 270
6.4.2 Response to Bi-Harmonic Excitation ............... 273
6.5 Two-Band System in a Periodic Field .................... 277
6.5.1 Dynamics of Total Band Populations ......... ..... 277
6.5.2 Population Distribution over the Bands ....... ..... 282
6.6 Control of Complex Quantum Systems ................... 284
6.6.1 Control of Two-Level Systems.................. 286
6.6.2 Holonom and Non-Holonom Systems ................ 294
6.6.3  Control of Coherence Loss..........  ............. 302
7 The Dynamics of One-Dimensional Relay-Type Systems .. 309
7.1 Exactly Soluble Relays of Isolated Levels ... ............... 309
7.1.1 Uniform Coupling and Linear Detuning ............. 310
7.1.2 The Harmonic Oscillator in an Arbitrary
Time-Dependent Field  .......................... . 311
7.1.3 Raman Pumping of a Harmonic Oscillator ........... 313
7.1.4 The Harmonic Oscillator in the Simultaneous
Presence of Dipole and Raman Pumping ............ 316
7.2 General Case of an Exactly Soluble Relay ................. 317
7.2.1 Conditions for the Existence of a Polynomial Solution. 318
7.2.2 The Increasing Coupling IV,I =  a (n - b) (n - c) ... 322
7.2.3 Decreasing Coupling IV,n = 1/van + b .............. 324
7.3 Smooth Variation of the Parameters ...................... 325
7.3.1 WKB approximation ............................ 326
7.3.2 Position and Width of the Erenfest Wavepacket ...... 327
7.3.3 The Tunneling Probability ....................... .329
7.3.4 Applicability of the WKB Approximation .......... 330
7.4  Relay with  disordered parameters  ........................ 331
7.4.1 Ensemble Averaged Amplitudes and Corresponding
Populations ................ .................... 333
7.4.2  Ensemble Averaged Spectrum  ...................... 335
7.4.3 Distribution of the Amplitude Ratios ............... 335
7.4.4 Distribution of the Populations for Long Times ...... 340
7.4.5 Dynamics-of the-Asymptotic Populations . .......... 342
7.5 Field Theory Method for Disordered Systems .............. 344
7.5.1 Tunneling Transparency and Classical Bosonic Fields . 344
7.5.2 An Analogy With the Liouville Equation ............ 352
7.5.3 Classical Fermionic Fields for the Population Dynamics 354
7.6 Population Dynamics in a Disordered Chain ............... 360
7.6.1 Population Dynamics and Propagating Fictitious
Particles  ..................................... . 360
7.6.2 Mapping over a Period and the Ensemble Average .... 364
7.6.3 Time Dependence of the Population Distribution ..... 369
8   Composite Complex Quantum Systems ................... 373
8.1  Relay  of M ultilevel Bands  ............................. . 373
8.1.1 Degenerate Bands With Random Coupling .......... 376
8.1.2 Non-Degenerate Bands With Random Coupling ...... 380
8.1.3  Correlated  Coupling  .............................. 387
8.2 Random Walks and Coherent Behavior .................... 392
8.2.1 Level Decay to a Band of Random Walks ............ 392
8.2.2 Interference of Random Returns at Long Times.
General Consideration  ............................ 397
8.2.3 Three Types of Random Walk and the Asymptotic
Decay .................  ................... .... . 406
8.3 Manifestation of Quantum Complexity in the State Density. . 413
8.3.1 Spectrum Transformation Induced by Random
Perturbation ...................................  414
8.3.2 The Effect of Quantum Recurrences on the State
D ensity  Profiles  .................................. 422
8.3.3 The Density of Quantum States of Fractals .......... 431
9   Bibliography  and  Problems ............................... 439
9.1 Bookshelf ............................................. 440
9.2 Problems .......................... .................... 449



Library of Congress subject headings for this publication: Quantum statistics, Differentiable dynamical systems, Quantum theory, Multilevel models (Statistics)