Coherent dynamics of complex quantum systems

476 Pages·2006·6.843 MB·English

by  Vladimir M. Akulin, V.M. Akulin

Texts and Monographs in Physics
SeriesEditors:
R.Balian,Gif-sur-Yvette,France
W.Beiglböck,Heidelberg,Germany
H.Grosse,Wien,Austria
W.Thirring,Wien,Austria
Vladimir M. Akulin
Coherent Dynamics
of Complex Quantum
Systems
With140Figures
123
ProfessorVladimirM.Akulin
LaboratoireAiméCotton
Bat505
Campusd’Orsay
91405OrsayCedex,France
LibraryofCongressControlNumber:2005929409
ISSN0172-5998
ISBN-10 3-540-21052-0 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-21052-8 SpringerBerlinHeidelbergNewYork
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Preface
This book is an attempt to put together a large number of similar problems
that one encounters in different fields of modern quantum physics and that
have common features considering multilevel quantum systems. The main
motivation was to present from the same standpoints various models and
approachesthathavebeendevelopedinatomic,molecular,condensedmatter,
chemical, laser and nuclear physics in various contexts.
The book is based on my lectures in the Moscow Institute of Physics and
Technology,intheAimeCottonLaboratoryofCNRS,andsomeothercourses
that I have delivered during last two decades. It includes the original results
obtained in collaboration with my collegues V. Aquilanti, I. Averbukh, A.
Belousov, M. Blaauboer, E. Borsela, E. Brion, B. Brunetti, C. Brechignac,
P. Cahuzac, F. Carlier, I. Dumer, V. Gershkovich, G. Esadze, G. Garsevan-
ishvili, G. Harel, E. Khokhlov, G. Kurizki, R. Larciprete, I. Murachko, A.
Nesterenko, A. Orlov, S. Pelegrin, P. Pillet, F. Rebentrost, A. Sarfati, E.
Schlag, W. Schleich, F. Vecchiocativi from different scientific centres in the
world and with whom I had the pleasure to work on the dynamical aspects
of the behaviour of complex quantum systems. I express my deep gratitude
to them for their collaboration. The book also contains numerous results of
otherauthorsthathave,however,beenexpoundedindifferentnotationscon-
sistent with the present text, and that sometimes even rely on an alternative
derivation as compared to the original version.
In preparing the text I decided to add several results both scientific, yet
unpublished, and pedagogical that I feel are necessary for giving the entire
picture of the processes in complex quantum systems. Some of these results,
presentedinChap.6,havebeenobtainedincollaborationwithV.Kravtsovto
whomIexpressmysincereacknowledgments.Ialsoverymuchappreciatethe
discussionswithV.Agranovich,E.Bogomolny,B.Chirikov,T.Gallagher,P.
Golovinskii, M. Fedorov, Y. Fyodorov, C. Jungen, J. Jortner, L. Maksimov,
V. Man’ko, I. Mazets, V. Kac, A. Kazantsev, D. Khmelnitski, A. Kofman,
I. Lerner, E. Nikitin, M. Shapiro, D. Shepelyanski, V. Pokrovskii and A.
Prokhorov on the different aspects of the results presented in the book.
I want to worship the memory of G. Askaryan who has influenced my
choice of profession, showing me the beauty of physics presented in our life.
VI Preface
Finally,IexpressmyprofoundgratitudetomyteachersAlexandrDyknne
and Nikolay Karlov.
Paris, May 2004 Vladimir Akulin
Contents
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
VIII Contents
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
3.4 Population of Inhomogeneous Bands ...................... 105
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
Contents IX
5.3 Semiclassical Analysis of Time-Dependent Systems ......... 213
5.3.1 Two-Level Systems and the Dykhne Formula ........ 213
5.3.2 Multilevel 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
Matrix .......................................... 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(cid:1)Polynomial Solution. 318
7.2.2 The Increasing Coupling |Vn|=√ a(n−b)(n−c). ... 322
7.2.3 Decreasing Coupling |V |=1/ an+b .............. 324
n
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
X Contents
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 Dynamics354
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 Multilevel 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
Density 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
References.................................................... 459
Index......................................................... 467
1 Complex Systems
and Their Statistical Description
We often meet in physics various quantum systems comprising many quan-
tum levels. One may say that the majority of problems of contemporary
interestareofthistype.Statisticsisanaturalwaytodescribelargesystems.
Initially,thestatisticalapproachtoindividualcomplexquantumobjectshad
beenformulatedinthecontextoftheinterpretationofnuclearspectrawhere
theaveragepositionsoftheenergylevelsandtheiraveragerelativedistances
were the principal concern. Later, the effects of complexity related to dis-
order were studied in solid state physics and this gave rise to the theory
of dynamical localization, focused on the structure of wavefunctions. The
rapid development of laser physics, the molecular beam technique, experi-
ments with cold atoms, femtosecond physics, quantum optics and quantum
informatics has formulated a series of theoretical problems centered on the
time evolution of multilevel quantum objects. Analytical description of the
dynamics of multilevel quantum systems, including the results which can be
obtainedwiththehelpofthestatisticalapproach,isthemainsubjectofthis
book.
The dynamics of multilevel systems is a complicated process, which in-
volvespopulationexchangeamongmanyquantumstates.Thoughafewtypes
of multilevel problems can be solved analytically, the Schro¨dinger equation
foramultilevelquantumsystemtypicallydoesnotpossesanexactanalytical
solution in terms of elementary or special functions nor in terms of integrals
over a reasonably small number of variables. Therefore the exact solution
implies the numerical integration of the Schro¨dinger equation. However, the
numerical solution of such problems usually encounters difficulties of three
different kinds. The first one is rather evident: as the number of states in a
systemincreases,thenumericalapproachbecomesawkward,sinceitbecomes
more and more difficult to handle the increasing amount of information. In
somecases,thisdifficultymayariseevenearlier,wheninordertosimplyfind
the matrix elements of the Hamiltonian and to write down the Schro¨dinger
equationinafinitebasisrepresentation,onehastoperformcomplicatedmul-
tidimensional integrations. The second difficulty arises due to the influence
of higher-order resonances which may manifest themselves in a hierarchy of
interactions that are different by many orders of magnitude. This makes the
dynamics of a multilevel quantum system considerably different at different