Real-time evolution of quenched quantum systems [Elektronische Ressource] / vorgelegt von Michael Möckel

Michael Möckel

Sujets

• Physik

Informations

• Publié par: ludwig-maximilians-universitat_munchen
• Publié le: 01 janvier 2009
• Nombre de lectures: 18
• Langue: English
• Poids de l'ouvrage: 8 Mo

Extrait

Real-time evolution of quenched
quantum systems
Michael M ockel
Munchen 2009c Michael M ockel 2009
Selbstverlag
Berger Stra e 19, 95119 Naila, Germany
All rights reserved
This publication is in copyright. Subject to statutory exceptions no reproduction
of any part may take place without the written permission of Michael M ockel.
In particular, this applies to any hardcopy, printout, storage, or distribution
of any electronic version of this publication.
ISBN 978-3-00-028464-9
First published 2009
Printed in GermanyReal-time evolution of quenched
quantum systems
Michael M ockel
Doktorarbeit
an der Fakult at fur Physik
der Ludwig{Maximilians{Universit at
Munc hen
vorgelegt von
Michael M ockel
aus Naila
Munc hen, den 30. April 2009Erstgutachter: Prof. Dr. Stefan Kehrein
Zweitgutachter: Prof. em. Dr. Herbert Wagner
Tag der mundlic hen Prufung: 24. Juni 2009Meinen Eltern
Siegfried und BrigittePercy Bysshe Shelley
(1792 { 1822)
Invocation
Rarely, rarely comest thou, I love all that thou lovest,
Spirit of Delight! Spirit of Delight!
Wherefore hast thou left me now The fresh Earth in new leaves drest
Many a day and night? And the starry night;
Many a weary night and day Autumn evening, and the morn
‘Tis since thou art ed away. When the golden mists are born.
How shall ever one like me I love snow and all the forms
Win thee back again? Of the radiant frost;
With the joyous and the free I love waves, and winds, and storms,
Thou wilt sco at pain. Everything almost
Spirit false! thou hast forgot Which is Nature‘s, and may be
All but those who need thee not. Untainted by man‘s misery.
As a lizard with the shade I love tranquil solitude,
Of a trembling leaf, And such society
Thou with sorrow art dismay‘d; As is quiet, wise, and good;
Even the sighs of grief Between thee and me
Reproach thee, that thou art not near, What di ‘rence? but thou dost possess
And reproach thou wilt not hear. The things I seek, nor love them less.
Let me set my mournful ditty I love Love - though he has wings,
To a merry measure; { And like light can ee,
Thou wilt never come for pity, But above all other things,
Thou wilt come for pleasure: { Spirit, I love thee -
Pity thou wilt cut away Thou art love and life! O come!
Those cruel wings, and thou wilt stay. Make once more my heart thy home!Contents
I Introduction 3
1 The world beyond equilibrium 5
1.1 Fundamental concepts of nonequilibrium physics . . . . . . . . . . . . . . . . 6
1.1.1 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.2 Metastable states of glasses and spin glasses: avoided relaxation . . . 7
1.1.3 Time dependent nonequilibrium phenomena . . . . . . . . . . . . . . . 8
1.2 De nition of a quantum quench . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Nonequilibrium initial conditions . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 initialization of a system by a quantum quench . . . . 9
1.2.3 Treatment of quench problems by Hamiltonian diagonalization . . . . 9
1.2.4 Energetic implications of a quench . . . . . . . . . . . . . . . . . . . . 10
1.3 Nonequilibrium phenomena in correlated many-body quantum systems . . . . 11
1.3.1 The X-ray edge problem . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Nonequilibrium dynamics of spin systems . . . . . . . . . . . . . . . . 11
1.3.3 Kibble-Zurek mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.4 Landau-Zener problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.5 Nonequilibrium dynamics of BCS systems . . . . . . . . . . . . . . . . 14
1.3.6 of lattice systems . . . . . . . . . . . . . . . 14
1.4 Questions and concepts in time-dependent nonequilibrium many-particle systems 15
1.4.1 Time scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.2 Time evolution and integrability . . . . . . . . . . . . . . . . . . . . . 16
1.4.3 Ergodicity and thermalization in classical systems . . . . . . . . . . . 17
1.4.4 Thermalization debate in quantum systems . . . . . . . . . . . . . . . 20
1.4.5 (Quantum) Boltzmann equation . . . . . . . . . . . . . . . . . . . . . 25
2 Landau’s theory of a Fermi liquid 29
2.1 Concept and prerequisite: Adiabatic connection between interacting and non-
interacting degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Macroscopic approach to Fermi liquid theory . . . . . . . . . . . . . . . . . . 30
2.2.1 The momentum distribution for interacting and noninteracting cases . 30
2.2.2 Expansion of the free energy functional . . . . . . . . . . . . . . . . . 31
2.2.3 Quasiparticle picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.4 Boltzmann dynamics of the momentum distribution . . . . . . . . . . 32
2.2.5 Range of validity of the quasiparticle picture . . . . . . . . . . . . . . 33
2.2.6 Thermodynamic properties of a Fermi liquid . . . . . . . . . . . . . . 34
2.3 Microscopic foundation of Fermi liquid theory . . . . . . . . . . . . . . . . . . 34
ixCONTENTS CONTENTS
3 Hubbard model 39
3.1 De nition of the Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Properties of the model . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Experimental motivation 45
4.1 Ultrafast spectroscopy of condensed matter systems . . . . . . . . . . . . . . 45
4.1.1 Ultrafast spectroscopy for semiconductors . . . . . . . . . . . . . . . . 45
4.1.2 spectroscopy for metals . . . . . . . . . . . . . . . . . . . . . 47
4.1.3 Discussion of the observed time scales . . . . . . . . . . . . . . . . . . 47
4.2 Ultracold atoms on optical lattices . . . . . . . . . . . . . . . . . . . . . . . . 47
5 Time evolution in quantum mechanics 53
5.1 Schr odinger dynamics and Heisenberg picture . . . . . . . . . . . . . . . . . . 53
5.2 Greens function formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3 Time dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 56
5.3.1 Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3.2 Coherent perturbation theory with respect to the ground state . . . . 57
5.3.3 Keldysh nonequilibrium perturbation theory and secular terms . . . . 58
5.3.4 Canonical perturbation theory . . . . . . . . . . . . . . . . . . . . . . 60
5.3.5 Unitary p theory . . . . . . . . . . . . . . . . . . . . . . . . 62
6 Unitary perturbation theory for the squeezed oscillator 65
6.1 Squeezed one-particle oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2 Perturbative study of squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.3 Exact (Bogoliubov) treatment of squeezing . . . . . . . . . . . . . . . . . . . 69
7 Generic mismatch of equilibrium and nonequilibrium expectation values 73
7.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7.2 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.3 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.3.1 First proof of the theorem by analyzing overlap matrix elements . . . 75
7.3.2 Second proof of the by applying unitary perturbation theory 76
7.4 Corollary to the Theorem regarding the kinetic energy of quenched systems . 78
8 Flow equations 79
8.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.1.1 Renormalization group ideas. . . . . . . . . . . . . . . . . . . . . . . . 80
8.1.2 Philosophy of the ow equations. . . . . . . . . . . . . . . . . . . . . . 80
8.1.3 Limited diagonalization of the ow equation method. . . . . . . . . . . 81
8.2 De nition of the in nitesimal transformations . . . . . . . . . . . . . . . . . . 81
8.2.1 Setup of a di erential ow equation for observables . . . . . . . . . . . 81
8.2.2 Canonical generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.2.3 Intrinsic exibility and ne-tuning of the method. . . . . . . . . . . . 82
8.3 Representations of observables and normal ordering. . . . . . . . . . . . . . . 82
8.3.1 Representations of owing observables . . . . . . . . . . . . . . . . . . 83
8.3.2 Normal ordering and truncations . . . . . . . . . . . . . . . . . . . . . 83
8.4 Continuous sequence of in nitesimal transformations . . . . . . . . . . . . . . 85
x