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 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 improved resonance-assisted tunneling theory and derive a unified prediction which is valid from the quantum to the semiclassical regime. We obtain results which show a drastically improved accuracy of several orders of magnitude compared to previous studies. / 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 besitzen Gebiete regulärer und chaotischer Bewegung, die klassisch getrennt sind, aber quantenmechanisch durch den Prozess des dynamischen Tunnelns gekoppelt werden. In dieser Arbeit wird eine theoretische Vorhersage für dynamische Tunnelraten abgeleitet, die den Zerfall von Zuständen, die im regulären Gebiet lokalisiert sind, in die sogenannte chaotische See beschreibt. 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.

Generic Hamiltonian systems have a mixed phase space, where classically disjoint regions of regular and chaotic motion coexist. For many applications it is useful to approximate the regular dynamics of such a mixed system H by an integrable approximation Hreg. We present a new, iterative method to construct such integrable approximations. The method is based on the construction of an integrable approximation in action representation which is then improved in phase space by iterative applications of canonical transformations. In contrast to other known approaches, our method remains applicable to strongly non-integrable systems H. We present its application to 2D maps and 2D billiards. Based on the obtained integrable approximations we finally discuss the theoretical description of dynamical tunneling in mixed systems. / Typische Hamiltonsche Systeme haben einen gemischten Phasenraum, in dem disjunkte Bereiche klassisch regulärer und chaotischer Dynamik koexistieren. Für viele Anwendungen ist es zweckmäßig, die reguläre Dynamik eines solchen gemischten Systems H durch eine integrable Näherung Hreg zu beschreiben. Wir stellen eine neue, iterative Methode vor, um solche integrablen Näherungen zu konstruieren. Diese Methode basiert auf der Konstruktion einer integrablen Näherung in Winkel-Wirkungs-Variablen, die im Phasenraum durch iterative Anwendungen kanonischer Transformationen verbessert wird. Im Gegensatz zu bisher bekannten Verfahren bleibt unsere Methode auch auf stark nichtintegrable Systeme H anwendbar. Wir demonstrieren sie anhand von 2D-Abbildungen und 2D-Billards. Mit den gewonnenen integrablen Näherungen diskutieren wir schließlich die theoretische Beschreibung von dynamischem Tunneln in gemischten Systemen.

Für das Verständnis einer Vielzahl von Problemen von der Himmelsmechanik bis hin zur Beschreibung von Molekülen spielen Systeme mit mehr als zwei Freiheitsgraden eine entscheidende Rolle. Aufgrund der Dimensionalität gestaltet sich ein Verständnis dieser Systeme jedoch deutlich schwieriger als bei Systemen mit zwei oder weniger Freiheitsgraden. Die vorliegende Arbeit soll zum besseren Verständnis der klassischen und quantenmechanischen Eigenschaften getriebener Systeme mit zwei Freiheitsgraden beitragen. Hierzu werden dreidimensionale Schnitte durch den Phasenraum von 4D Abbildungen betrachtet. Anhand dreier Beispiele, deren Phasenräume zunehmend kompliziert sind, werden diese 3D Schnitte vorgestellt und untersucht. In einer sich anschließenden quantenmechanischen Untersuchung gehen wir auf zwei wichtige Aspekte ein. Zum einen untersuchen wir die quantenmechanischen Signaturen des klassischen "Arnold Webs". Es wird darauf eingegangen, wie die Quantenmechanik dieses Netz im semiklassischen Limes auflösen kann. Darüberhinaus widmen wir uns dem wichtigen Aspekt quantenmechanischer Kopplungen klassisch getrennter Phasenraumgebiete anhand der Untersuchung dynamischer Tunnelraten. Für diese wenden wir sowohl den in der Literatur bekannten "fictitious integrable system approach" als auch die Theorie des resonanz-unterstützen Tunnelns auf 4D Abbildungen an.:Contents ..... v 1 Introduction ..... 1 2 2D mappings ..... 5 2.1 Hamiltonian systems with 1.5 degrees of freedom ..... 5 2.2 The 2D standard map ..... 6 3 Classical dynamics of higher dimensional systems ..... 11 3.1 Coupled standard maps as paradigmatic example ..... 12 Stability of fixed points in 4D maps ..... 13 Center manifolds of elliptic degrees of freedom ..... 13 3.2 Near-integrable systems ..... 15 3.2.1 Analytical description of multidimensional, near-integrable systems ..... 15 Resonance structures in 4D maps ..... 16 3.2.2 Pendulum approximation ..... 18 3.2.3 Normal forms ..... 24 3.2.4 Arnold diffusion and Arnold web ..... 24 3.3 Numerical tools for the analysis of regular and chaotic motion ..... 26 3.3.1 Frequency analysis ..... 26 Aim of the frequency analysis ..... 26 Realizations of the frequency analysis ..... 27 Wavelet transforms ..... 30 3.3.2 Fast Lyapunov indicator ..... 31 3.3.3 Phase-space sections ..... 33 Skew phase-space sections containing invariant eigenspaces ..... 34 3.4 Systems with regular dynamics and a large chaotic sea ..... 35 3.4.1 Designed maps: Map with linear regular region, P_llu ..... 36 Phase space of the designed map with linear regular region ..... 38 FLI values ..... 41 Estimating the size of the regular region ..... 43 3.4.2 Designed maps: Islands with resonances, P_nnc ..... 46 Frequency analysis ..... 46 FLI values and volume of the regular and stochastic region ..... 50 Frequency analysis for rank-2 resonance ..... 52 Phase-space sections at different positions p_1 and p_2 ..... 53 Using color to provide the 4-th coordinate ..... 53 Skew phase-space sections containing invariant eigenspaces ..... 57 Arnold diffusion ..... 58 3.4.3 Generic maps: Coupled standard maps, P_csm ..... 63 FLI values and volume of the regular and stochastic region ..... 63 Analysis of fundamental frequencies ..... 66 Skew phase-space sections containing invariant eigenspaces ..... 69 4 Quantum Mechanics ..... 75 4.1 Quantization of Classical Maps ..... 77 4.2 Eigenstates of the time evolution operator U ..... 79 4.2.1 Eigenstates of P_llu ..... 80 4.2.2 Eigenstates of P_nnc ..... 84 4.2.3 Eigenstates of P_csm ..... 87 4.3 Quantum signatures of the stochastic layer ..... 89 4.3.1 Eigenstates resolving the stochastic layer ..... 90 4.3.2 Wave-packet dynamics into the stochastic layer ..... 94 4.4 Dynamical tunneling rates ..... 98 4.4.1 Numerical calculation of dynamical tunneling rates ..... 99 4.4.2 Direct regular-to-chaotic tunneling rates gamma^d of P_llu ..... 101 4.4.3 Prediction of gamma^d using the fictitious integrable system approach ..... 103 4.4.4 Dynamical tunneling rates of P_nnc ..... 105 4.4.5 Interlude: Theory of resonance assisted tunneling (RAT) ..... 106 4.4.6 Prediction of tunneling rates for P_nnc, RAT ..... 111 Selection rules from nonlinear resonances ..... 111 Energy denominators ..... 114 Estimating the parameters of the pendulum approximation from phase-space properties ..... 116 Prediction ..... 118 4.4.7 Dynamical tunneling rates of P_csm ..... 120 5 Summary and outlook ..... 123 Appendix ..... 125 A Potential of the designed map ..... 125 B Quantum-number assignment-algorithm ..... 128 C Alternate paths due to alternate resonances in the description of RAT ..... 131 D Alternate resonances in the description of RAT leading to different tunneling rates ..... 133 E Tunneling rates of map with nonlinear resonances but uncoupled regular region ..... 133 F Interpolation of quasienergies ..... 135 G 2D Poincar'e map for the pendulum approximation ..... 137 H RAT prediction broken down to single paths ..... 139 I Linearization of the pendulum approximation ..... 140 J Iterative diagonalization schemes for the semiclassical limit ..... 143 Inverse iteration ..... 143 Arnoldi method ..... 144 Lanczos algorithm ..... 144 List of figures ..... 148 Bibliography ..... 163 / Systems with more than two degrees of freedom are of fundamental importance for the understanding of problems ranging from celestial mechanics to molecules. Due to the dimensionality the classical phase-space structure of such systems is more difficult to understand than for systems with two or fewer degrees of freedom. This thesis aims for a better insight into the classical as well as the quantum mechanics of 4D mappings representing driven systems with two degrees of freedom. In order to analyze such systems, we introduce 3D sections through the 4D phase space which reveal the regular and chaotic structures. We introduce these concepts by means of three example mappings of increasing complexity. After a classical analysis the systems are investigated quantum mechanically. We focus especially on two important aspects: First, we address quantum mechanical consequences of the classical Arnold web and demonstrate how quantum mechanics can resolve this web in the semiclassical limit. Second, we investigate the quantum mechanical tunneling couplings between regular and chaotic regions in phase space. We determine regular-to-chaotic tunneling rates numerically and extend the fictitious integrable system approach to higher dimensions for their prediction. Finally, we study resonance-assisted tunneling in 4D maps.:Contents ..... v 1 Introduction ..... 1 2 2D mappings ..... 5 2.1 Hamiltonian systems with 1.5 degrees of freedom ..... 5 2.2 The 2D standard map ..... 6 3 Classical dynamics of higher dimensional systems ..... 11 3.1 Coupled standard maps as paradigmatic example ..... 12 Stability of fixed points in 4D maps ..... 13 Center manifolds of elliptic degrees of freedom ..... 13 3.2 Near-integrable systems ..... 15 3.2.1 Analytical description of multidimensional, near-integrable systems ..... 15 Resonance structures in 4D maps ..... 16 3.2.2 Pendulum approximation ..... 18 3.2.3 Normal forms ..... 24 3.2.4 Arnold diffusion and Arnold web ..... 24 3.3 Numerical tools for the analysis of regular and chaotic motion ..... 26 3.3.1 Frequency analysis ..... 26 Aim of the frequency analysis ..... 26 Realizations of the frequency analysis ..... 27 Wavelet transforms ..... 30 3.3.2 Fast Lyapunov indicator ..... 31 3.3.3 Phase-space sections ..... 33 Skew phase-space sections containing invariant eigenspaces ..... 34 3.4 Systems with regular dynamics and a large chaotic sea ..... 35 3.4.1 Designed maps: Map with linear regular region, P_llu ..... 36 Phase space of the designed map with linear regular region ..... 38 FLI values ..... 41 Estimating the size of the regular region ..... 43 3.4.2 Designed maps: Islands with resonances, P_nnc ..... 46 Frequency analysis ..... 46 FLI values and volume of the regular and stochastic region ..... 50 Frequency analysis for rank-2 resonance ..... 52 Phase-space sections at different positions p_1 and p_2 ..... 53 Using color to provide the 4-th coordinate ..... 53 Skew phase-space sections containing invariant eigenspaces ..... 57 Arnold diffusion ..... 58 3.4.3 Generic maps: Coupled standard maps, P_csm ..... 63 FLI values and volume of the regular and stochastic region ..... 63 Analysis of fundamental frequencies ..... 66 Skew phase-space sections containing invariant eigenspaces ..... 69 4 Quantum Mechanics ..... 75 4.1 Quantization of Classical Maps ..... 77 4.2 Eigenstates of the time evolution operator U ..... 79 4.2.1 Eigenstates of P_llu ..... 80 4.2.2 Eigenstates of P_nnc ..... 84 4.2.3 Eigenstates of P_csm ..... 87 4.3 Quantum signatures of the stochastic layer ..... 89 4.3.1 Eigenstates resolving the stochastic layer ..... 90 4.3.2 Wave-packet dynamics into the stochastic layer ..... 94 4.4 Dynamical tunneling rates ..... 98 4.4.1 Numerical calculation of dynamical tunneling rates ..... 99 4.4.2 Direct regular-to-chaotic tunneling rates gamma^d of P_llu ..... 101 4.4.3 Prediction of gamma^d using the fictitious integrable system approach ..... 103 4.4.4 Dynamical tunneling rates of P_nnc ..... 105 4.4.5 Interlude: Theory of resonance assisted tunneling (RAT) ..... 106 4.4.6 Prediction of tunneling rates for P_nnc, RAT ..... 111 Selection rules from nonlinear resonances ..... 111 Energy denominators ..... 114 Estimating the parameters of the pendulum approximation from phase-space properties ..... 116 Prediction ..... 118 4.4.7 Dynamical tunneling rates of P_csm ..... 120 5 Summary and outlook ..... 123 Appendix ..... 125 A Potential of the designed map ..... 125 B Quantum-number assignment-algorithm ..... 128 C Alternate paths due to alternate resonances in the description of RAT ..... 131 D Alternate resonances in the description of RAT leading to different tunneling rates ..... 133 E Tunneling rates of map with nonlinear resonances but uncoupled regular region ..... 133 F Interpolation of quasienergies ..... 135 G 2D Poincar'e map for the pendulum approximation ..... 137 H RAT prediction broken down to single paths ..... 139 I Linearization of the pendulum approximation ..... 140 J Iterative diagonalization schemes for the semiclassical limit ..... 143 Inverse iteration ..... 143 Arnoldi method ..... 144 Lanczos algorithm ..... 144 List of figures ..... 148 Bibliography ..... 163