La lecture en ligne est gratuite
Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres
Télécharger Lire

Dynamical tunneling in systems with a mixed phase space [Elektronische Ressource] / vorgelegt von Steffen Löck

247 pages
Dynamical Tunnelingin Systems with a Mixed Phase SpaceDissertationzur Erlangung des akademischen GradesDoctor rerum naturaliumvorgelegt vonSteffen Löckgeboren am 22.03.1982 in RadebeulInstitut für Theoretische PhysikFachrichtung PhysikFakultät für Mathematik und NaturwissenschaftenTechnische Universität Dresden2009Eingereicht am 16.12.20091. Gutachter: Prof. Dr. Roland Ketzmerick2. Gutachter: Prof. Dr. Jan WiersigVerteidigt am 22.04.2010vAbstractTunneling is one of the most prominent features of quantum mechanics. While the tunnelingprocess in one-dimensional integrable systems is well understood, its quantitative prediction forsystems with a mixed phase space is a long-standing open challenge. In such systems regionsof regular and chaotic dynamics coexist in phase space, which are classically separated butquantum mechanically coupled by the process of dynamical tunneling. We derive a predictionof dynamical tunneling rates which describe the decay of states localized inside the regularregion towards the so-called chaotic sea. This approach uses a fictitious integrable systemwhich mimics the dynamics inside the regular domain and extends it into the chaotic region.Excellent agreement with numerical data is found for kicked systems, billiards, and opticalmicrocavities, if nonlinear resonances are negligible. Semiclassically, however, such nonlinearresonance chains dominate the tunneling process.
Voir plus Voir moins

Dynamical Tunneling
in Systems with a Mixed Phase Space
Dissertation
zur Erlangung des akademischen Grades
Doctor rerum naturalium
vorgelegt von
Steffen Löck
geboren am 22.03.1982 in Radebeul
Institut für Theoretische Physik
Fachrichtung Physik
Fakultät für Mathematik und Naturwissenschaften
Technische Universität Dresden
2009Eingereicht am 16.12.2009
1. Gutachter: Prof. Dr. Roland Ketzmerick
2. Gutachter: Prof. Dr. Jan Wiersig
Verteidigt am 22.04.2010v
Abstract
Tunneling is one of the most prominent features of quantum mechanics. While the tunneling
process in one-dimensional integrable systems is well understood, its quantitative prediction for
systems with a mixed phase space is a long-standing open challenge. In such systems regions
of regular and chaotic dynamics coexist in phase space, which are classically separated but
quantum mechanically coupled by the process of dynamical tunneling. We derive a prediction
of dynamical tunneling rates which describe the decay of states localized inside the regular
region towards the so-called chaotic sea. This approach uses a fictitious integrable system
which mimics the dynamics inside the regular domain and extends it into the chaotic region.
Excellent agreement with numerical data is found for kicked systems, billiards, and optical
microcavities, if nonlinear resonances are negligible. Semiclassically, however, such nonlinear
resonance chains dominate the tunneling process. Hence, we combine our approach with an
improvedresonance-assistedtunnelingtheoryandderiveaunifiedpredictionwhichisvalidfrom
the quantum to the semiclassical regime. We obtain results which show a drastically improved
accuracy of several orders of magnitude compared to previous studies.
Zusammenfassung
Der Tunnelprozess ist einer der bedeutensten Effekte in der Quantenmechanik. Während das
Tunneln in eindimensionalen integrablen Systemen gut verstanden ist, gestaltet sich dessen
Beschreibung für Systeme mit gemischtem Phasenraum weitaus schwieriger. Solche Systeme
besitzenGebieteregulärerundchaotischerBewegung, dieklassisch getrenntsind,
aberquantenmechanisch durch den Prozess des dynamischen Tunnelns gekoppelt werden. In dieser Arbeit
wird eine theoretische Vorhersage für dynamische Tunnelraten abgeleitet, die den Zerfall von
Zuständen, dieimregulärenGebiet lokalisiertsind, indiesogenannte chaotische Seebeschreibt.
Dazu wird ein fiktives integrables System konstruiert, das im regulären Bereich eine nahezu
gleiche Dynamik aufweist und diese Dynamik in das chaotische Gebiet fortsetzt. Die
Theorie zeigt eine ausgezeichnete Übereinstimmung mit numerischen Daten für gekickte Systeme,
Billards und optische Mikrokavitäten, falls nichtlineare Resonanzketten vernachlässigbar sind.
Semiklassisch jedoch bestimmen diese nichtlinearen Resonanzketten den Tunnelprozess. Daher
kombinieren wir unseren Zugang mit einer verbesserten Theorie des Resonanz-unterstützten
Tunnelns und erhalten eine Vorhersage, die vom Quanten- bis in den semiklassischen Bereich
gültig ist. Ihre Resultate zeigen eine Genauigkeit, die verglichen mit früheren Theorien um
mehrere Größenordnungen verbessert wurde.Contents
1 Introduction 1
2 Barrier tunneling in one-dimensional systems 7
2.1 WKB method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Bohr-Sommerfeld quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Gamov theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Application to double-well potential . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Application to δ-potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Model systems with a mixed phase space 25
3.1 Kicked systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Classical maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.3 Classical perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.4 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.5 Numerical methods for the calculation of tunneling rates . . . . . . . . . 65
3.2 Billiard systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.1 Classical dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.2 Quantum billiards. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.3 Numerical methods for the calculation of tunneling rates . . . . . . . . . 87
3.3 Optical microcavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.3.1 Ray dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3.2 Wave mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4 Dynamical tunneling in quantum maps 97
4.1 Direct regular-to-chaotic tunneling . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.1.1 Theoretical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1.2 Fictitious integrable system and convergence . . . . . . . . . . . . . . . . 102
4.1.3 Semiclassical evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2 Unification with resonance-assisted tunneling . . . . . . . . . . . . . . . . . . . . 121
4.2.1 Theory of resonance-assisted tunneling . . . . . . . . . . . . . . . . . . . 122viii Contents
4.2.2 Improvements of resonance-assisted tunneling . . . . . . . . . . . . . . . 128
4.2.3 Multi-resonance effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5 Dynamical tunneling in billiard systems 143
5.1 Direct regular-to-chaotic tunneling . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.2.1 Mushroom billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.2.2 Annular billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.2.3 Dynamical tunneling of bouncing-ball modes . . . . . . . . . . . . . . . . 162
5.2.4 Two-dimensional nanowires with a magnetic field . . . . . . . . . . . . . 169
5.2.5 Cosine billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6 Quality factors of optical microcavities 179
6.1 The circular microcavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.2 Direct regular-to-chaotic tunneling in the annular microcavity . . . . . . . . . . 182
7 Summary and outlook 191
A Dimensionless variables 195
B Classical kicked systems 197
B.1 Smoothing of designed discontinuous mappings . . . . . . . . . . . . . . . . . . . 197
B.2 An example of the Lie transformation . . . . . . . . . . . . . . . . . . . . . . . . 200
B.3 An example of the normal-form analysis . . . . . . . . . . . . . . . . . . . . . . 204
B.4 Calculation of the pendulum parameters V , I , M . . . . . . . . . . . . . . 207r:s r:s r:s
C Quantum maps 211
C.1 Semiclassical energies of kicked systems . . . . . . . . . . . . . . . . . . . . . . . 211
C.2 Phase splittings and tunneling rates . . . . . . . . . . . . . . . . . . . . . . . . . 213
C.3 Tunneling rates and coupling matrix elements . . . . . . . . . . . . . . . . . . . 214
D Billiard systems 215
D.1 Coupling matrix elements of non-orthogonal states . . . . . . . . . . . . . . . . . 215
D.2 Derivation of A for the mushroom billiard . . . . . . . . . . . . . . . . . . . . 216ch
D.3 Eigenmodes of a wire in a magnetic field . . . . . . . . . . . . . . . . . . . . . . 217
D.4 Semiclassical description of localization lengths in wires with one-sided disorder 219
Bibliography 2211 Introduction
The exponential decay rates of radioactive substances were found experimentally by Elster
and Geitel [1] in 1899, three years after the discovery of natural radioactivity. One year later
Rutherford introduced the concept of half-life times [2]. The radioactive decay is a purely
quantum phenomenon. Due to the uncertainty principle for quantum particles the position
and the momentum cannot be specified at the same time with arbitrary precision as in
classical mechanics. Instead, it is described by a wave function which is given as a solution of the
Schrödinger equation. The modulus squared of this wave function is a probability amplitude
for the distribution of the particle. This probability can be positive even in regions where the
classical motion is forbidden. For example this occurs for an energy barrier of finite height
which classically cannot be passed if the energy of the particle is too low. Hence, a quantum
mechanical particle may pass through such a potential barrier which is called quantum
tunneling. This occurs for the α-decay which is described by the well-known Gamov formula [3–5].
Later, electron tunneling in solids was demonstrated by Esaki [6,7] in 1957 who discovered the
tunneling diode. In 1962 Josephson studied tunneling between two superconductors separated
by a thin layer of insulating oxide which serves as a barrier. He predicted the existence of a
supercurrent caused by the tunneling of electrons in pairs [8]. Today tunneling in individual
atoms and molecules can be observed [9–12] and Bose-Einstein condensates are used to study
the interplay of tunneling with effects caused by nonlinear interactions [13,14].
The theoretical description of tunneling in all its varieties is still a challenging open problem.
For one-dimensional time-independent systems, however, it is well understood. Therefore,
these simple systems provide a good starting point for the more complicated analysis of higher
dimensional problems. Classically, particles of low energy are trapped inbetween local maxima
ofapotentialV(q). One prominent example is the one-dimensional double-well potential which
consists of two symmetry related wells. The classical motion is confined to either side of the
wells. Quantum mechanically, however, a wave packet started in one ofthe wells performs Rabi
oscillations between the two sites with a frequency which depends on the width and the height
ofthe barrier and can be predicted using WKB theory [15]. The periodτ of these oscillations is
related to an energy splitting ΔE of quasi-degenerate eigenstates localized in the right and the
leftwell,τ =~/ΔE. Thisenergysplittingoccursduetothetunnelingprocesswhichcouplesthe
−S/~classically separated wells. It decreases exponentially, ΔE∼ e , with increasing height and
Rp
widthofthebarrier. ThisinformationiscontainedintheactionS = 2m(V(q)−E)dq. For2 Chapter 1. Introduction
small splittings ΔE the periodτ goes to infinity and the tunneling process is suppressed. This
happens when the imaginary part of the actionS divided by Planck’s constanth increases. To
study this behavior quantitatively we introduce an effective Planck constant h =h/S whicheff 0
is the ratio of Planck’s constant h and a typical action S of the system. When approaching0
the classical regime, the action S increases such that h will go to zero. This is called the0 eff
semiclassical limit. In this limit all tunneling processes have to vanish as they are forbidden in
classical mechanics. For the energy splittings ΔE in the double well potential this behavior is
predicted by its exponential decrease with h → 0.eff
Tunneling not only occurs for potential barriers but whenever a system consists of classically
disconnected regions. Such regions exist for higher dimensional integrable systems due to
additional integrals of motion [16,17]. Here, the tunneling process can be described by means
of the WKB or the instanton approach [18,19]. Typical Hamiltonian systems, however, are not
integrable. They have a mixed phase space where regions of regular motion, also called regular
islands, andregionsofchaoticmotion,calledthechaoticsea,coexist. Whiletheclassicalmotion
is confined to these regions, quantum mechanically the dynamically generated barrier can be
penetrated. According to the semiclassical eigenfunction hypothesis [20–22] the eigenstates
of systems with a mixed phase space are semiclassically concentrated either in the regular
islands or in the chaotic sea. For nonzero values of the effective Planck constant h thiseff
classificationstillapproximatelyholdssuchthatthecorrespondingeigenstatesarecalledregular
orchaotic. However, eacheigenstatehascontributionsinbothregionsofphasespace,duetothe
couplingbetweentheseregions. Thecorrespondingtunnelingprocesshasbeencalleddynamical
tunneling by Davis and Heller [23]. Finding a theoretical description of this tunneling process
is a challenging problem for systems with a mixed phase space and the main subject of this
thesis.
In order to study dynamical tunneling Tomsovic and Ullmo [24] considered a system whose
phase space consists of two symmetry related regular islands surrounded by a chaotic sea.
They observed that the energy splittings ΔE between two symmetry related regular states
are drastically changed compared to the integrable potential systems discussed before, due to
the appearing chaotic states. Under variation of external parameters or the effective Planck
constant the energy of such a chaotic state will come close to those of the regular doublet. If
this happens the chaotic state mediates the tunneling process between the two islands. The
two-step process coupling the regular state from one island to the chaotic state and then to the
other island dominates compared to the direct coupling of the two regular states. This leads
to an enhancement of the energy splittings and was called chaos-assisted tunneling [24]. The
spectrumoftheregularandthechaoticstatescanbemodeledbyrandommatrixtheory[25,26].
Hence, under variation of the effective Planck constant the distance between the energies of the
regular doublet and the closest chaotic state will vary in a random way. This leads to strong
fluctuations in the splittings ΔE [27–30] whose variance can be predicted by random matrix