From spectral statistics to decay in quantum chaotic systems [Elektronische Ressource] : a semiclassical analysis beyond random matrix theory / vorgelegt von Martha Lucía Gutiérrez Márquez

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• Physik

Informations

• Publié par: universitat_regensburg
• Publié le: 01 janvier 2009
• Nombre de lectures: 20
• Poids de l'ouvrage: 5 Mo

Extrait

From spectral statistics to decay in
quantum chaotic systems: a
semiclassical analysis beyond Random
Matrix Theory
Dissertation
zur Erlangung des Doktorgrades
der Naturwissenschaften (Dr. rer. nat.)
der Naturwissenschaftlichen Fakult¨at II – Physik
der Universit¨at Regensburg
vorgelegt von
Martha Luc´ıa Guti´errez M´arquez
aus Bogota
Dezember 2008Die Arbeit wurde von Prof. Dr. Klaus Richter und von Prof. Dr. Matthias Brack
angeleitet.
Das Promotionsgesuch wurde am 07.11.2008 eingereicht.
Das Promotionskolloquium fand am 12.12.2008 statt.
Pru¨fungsausschuss:
Vorsitzender: Prof. Dr. Christian Back
1. Gutachter: Prof. Dr. Klaus Richter
2. Gutachter: Prof. Dr. John Schliemann
Weiterer Pru¨fer: Prof. Dr. Gunnar BaliContents
1 Introduction 1
1.1 The semiclassical approximation . . . . . . . . . . . . . . . . . . . 3
1.2 Spectral statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Semiclassical density of states . . . . . . . . . . . . . . . . . 5
1.2.2 The spectral form factor . . . . . . . . . . . . . . . . . . . . 6
1.2.3 The diagonal approximation . . . . . . . . . . . . . . . . . . 7
1.2.4 Quantum corrections in the semiclassical approximation . . 8
1.2.5 Deviations from universality . . . . . . . . . . . . . . . . . . 10
1.3 Semiclassical approximation near bifurcations . . . . . . . . . . . . 11
1.4 Open systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Spectral density for the quartic oscillator: from integrability to
hard chaos 19
2.1 Model system: the quartic oscillator . . . . . . . . . . . . . . . . . 19
2.2 Semiclassical density of states for discrete symmetries . . . . . . . 24
2.2.1 Integrable Systems . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 Isolated orbits . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Semiclassical approximation for bifurcating orbits . . . . . . . . . . 34
3 Effect of pitchfork bifurcations in the spectral statistics 39
3.1 Spectral Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Semiclassical theory for the spectral rigidity . . . . . . . . . . . . . 41
3.3 Rigidity for the Integrable QO . . . . . . . . . . . . . . . . . . . . 43
3.4 Bifurcation effects in the level statistics . . . . . . . . . . . . . . . 45
4 Semiclassical transport and open-orbits bifurcations 53
4.1 Semiclassical transport through mesoscopic systems . . . . . . . . 53
4.2 Open-orbits bifurcation theory . . . . . . . . . . . . . . . . . . . . 56
4.3 Uniform approximation for the transmission coefficient . . . . . . . 59
iiiiv CONTENTS
4.3.1 Uniform approximation for a tangent bifurcation . . . . . . 59
4.3.2 Uniform approximation for a pitchfork bifurcation . . . . . 62
4.4 Conductance and open-orbits bifurcations . . . . . . . . . . . . . . 63
5 Semiclassical approximation to the decay 67
5.1 Model system and important time scales . . . . . . . . . . . . . . . 67
5.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3 Survival probability . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Semiclassical approximation to the survival probability . . . . . . . 75
5.4.1 Diagonal approximation . . . . . . . . . . . . . . . . . . . . 76
5.4.2 Leading quantum corrections to the decay . . . . . . . . . . 79
5.5 Ehrenfest time effects . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.6 Variance of the survival probability . . . . . . . . . . . . . . . . . . 87
6 Semiclassical approximation to photo-dissociation statistics 91
6.1 Photo-dissociation statistics . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Semiclassical approximation to the cross-section form factor . . . . 93
6.2.1 Open trajectory contributions . . . . . . . . . . . . . . . . . 94
6.2.2 Periodic orbit contributions . . . . . . . . . . . . . . . . . . 96
6.3 Ehrenfest time effects . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3.1 Ehrenfest time dependence of the leading quantum correc-
tion to the open trajectories contribution . . . . . . . . . . 98
6.3.2 Ehrenfest time dependence of the leading quantum correc-
tion to the periodic orbits contribution . . . . . . . . . . . . 99
7 Conclusions and Outlook 101
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2 Open questions and outlook . . . . . . . . . . . . . . . . . . . . . . 103
A Reduced density of states of the separable quartic oscillator 107
B Ehrenfest time dependence of the decay probability 111
C Variance of the decay for a Gaussian initial state 113
D Ehrenfest time dependence of the spectral form factor for open
systems 117Chapter 1
Introduction
Inclassicalmechanicsthestateofaparticleatatimetisfullydescribedbyapoint
x(t) =(r(t),p(t)) in phasespace. Themotion isgiven bythe solution oftheequa-
tions of motion (Newton, Lagrange or Hamilton). This solution is unique given
theinitialconditions,andthereforethemotioniscompletelydeterministic,though
the dynamics can be extremely complicated. Let us consider an autonomous con-
servative system with f degrees of freedom, described by a Hamiltonian function
H(x). Due to conservation of energy the motion in phase-space is restricted to a
2f−1-dimensionalhyper-surface. Classically, thetypeofmotion can beseparated
into integrable and non - integrable. The first situation appears if there are, apart
from the energy, f −1 other independent constants of motion [1]. In this case,
it is possible to perform a canonical transformation to a new set of phase-space
coordinates, called the action-angle variables, such that the Hamiltonian depends
onlyontheaction variables, thereforethemotion inphasespaceisrestricted toan
f-dimensional hyper-surface [2]. In the new coordinates, the dynamics is trivial,
the angles vary linearly with time, while the actions remain constant, so, all the
solutions are periodicor quasi-periodic, dependingon thefrequencyratio between
the different angular degrees of freedom. Opposite to this situation, where there
are apart from energy no other constants of motion, the system can display hard
chaos: there is an extreme sensitivity to initial conditions, i.e. perturbing slightly
theinitial conditions leadsto exponential separation intime ofthesolutions. This
makes impossible to predict the long time behaviour of the solutions if the initial
conditions are not known exactly.
On the other hand, a more general theory is the quantum theory, in which a
physical state is depicted by a vector |ψ(t)i in Hilbert space, whose evolution is
given through the Schr¨odinger equation:
d ˆi~ |ψ(t)i =H|ψi, (1.1)
dt
12 CHAPTER 1. INTRODUCTION
which has also a unique solution once the initial condition (state) is given. A
first connection between the quantum and the classical dynamics, can be found
by making the ansatz ψ(r,t) = A(r,t)exp(iS(r,t)/~) in position representation
2for a Hamiltonian of the form H(r,p) =p /2m+V(r), as first done in Ref. [3].
After substituting in Eq. (1.1), and neglecting all ~-dependent terms, one arrives
to the Hamilton-Jacobi equation of motion
∂S(r,t)− =H(r,∇S(r,t)). (1.2)
∂t
This equation is satisfied by the action principal functionZ
t
′ ′ ′′ ′′ ′′ ′′S(r,t) =S(r,r,t−t) = dt L(r(t ),r˙(t ),t ), (1.3)
′t
′ ′with the condition r(t) = r, where L(r(t),r˙(t),t) is the Lagrangian. Moreover,
thenextorderin~correspondstotheclassical continuityequation. Soapparently,
for small values of ~ the quantum solutions are closely related to the classical
ones. This ansatz that we have just mentioned above, is the main ingredient to
obtain a semiclassical quantization of integrable systems. For integrable systems
one can relate the classical conserved quantities to “good” quantum numbers,
in the sense that each invariant torus, characterized by its frequencies, can be
linked to a quantum wave function through the quantization condition where
each classical action coordinate is a multiple integer of Planck’s constant ~ (plus
phases). ThisisthewellknownEBKquantizationintroducedinRef. [4](basedon
WKBquantizationforone-dimensionalsystems[5])relatingclassicalandquantum
solutions in a direct way.
AspointedoutbyEinstein,thetoriquantization isnotapplicableifthesystem
is not longer integrable, and the way of quantizing this type of systems is still an
issue in the semiclassical community. However, the complexity in the dynamics
of chaotic systems is compensated by the simplicity in the statistical properties:
all chaotic systems satisfy universal properties on the classical side, which are (i)
ergodicity, i.e. almost any trajectory (apart from a set of zero measure), will ho-
mogeneously fill the energy shell after long times; (ii) mixing, i.e. correlations of
functions in phase space decay exponentially fast; and (iii) hyperbolicity, i.e. a
small initial separation between almost any two trajectories will grow exponen-
tially fast [6]. On the quantumsideit hasbeen widelyshown that chaotic systems
display universal properties as well [7], e.g. the energy eigenvalues of a confined
system display universal statistics, conductance and shot noise in chaotic open
systems are also universal.
The quest of “quantum chaos” [8] is the study of quantum systems whose