The asymptotic state of a quantum system, which is in contact with a heat bath, is strongly disturbed by a time-periodic driving in comparison to a time-independent system. In this thesis an extensive picture of the asymptotic state of time-periodic quantum systems is drawn by relating it to the structure of the corresponding classical phase space. To this end the occupation probabilities of the Floquet states are analyzed with respect to their semiclassical property of being either regular or chaotic. The regular Floquet states are occupied with exponential weights e^{-betaeff Ereg} similar to the canonical weights e^{-beta E} of time-independent systems. The regular energies Ereg are defined by the quantization of the time-periodic system, whose classical properties also determine the effective temperature 1/betaeff. In contrast, the chaotic Floquet states acquire almost equal probabilities, irrespective of their time-averaged energy. Beyond these semiclassical properties the existence of avoided crossings in the spectrum is an intrinsic quantum property of time-periodic systems. Avoided crossings can strongly influence the entire occupation distribution. As an impressive application a novel switching mechanism is proposed in a periodically driven double well potential coupled to a heat bath. By a weak variation of the driving amplitude its asymptotic state is switched from the ground state in one well to a state with higher average energy in the other well. / Der asymptotische Zustand eines Quantensystems, das in Kontakt mit einem Wärmebad steht, wird durch einen zeitlich periodischen Antrieb gegenüber einem zeitunabhängigen System nachhaltig verändert. In dieser Arbeit wird ein umfassendes Bild über den asymptotischen Zustand zeitlich periodischer Quantensysteme entworfen, indem es diesen zur Struktur des zugehörigen klassischen Phasenraums in Beziehung setzt. Dazu werden die Besetzungswahrscheinlichkeiten der Floquet-Zustände hinsichtlich ihrer semiklassischen Eigenschaft analysiert, nach welcher sie entweder regulär oder chaotisch sind. Die regulären Floquet-Zustände sind mit exponentiellen Gewichten e^{-betaeff Ereg} ähnlich der kanonischen Verteilung e^{-beta E} zeitunabhängiger Systeme besetzt. Dabei sind die reguläre Energien Ereg durch die Quantisierung des Systems vorgegeben, dessen klassische Eigenschaften auch die effektive Temperatur 1/betaeff bestimmen. Die chaotischen Zustände dagegen haben fast einheitliche Besetzungswahrscheinlichkeiten, welche unabhängig von ihrer mittleren Energie sind. Über diese semiklassischen Eigenschaften hinaus ist das Auftreten von vermiedenen Kreuzungen im Spektrum eine intrinsisch quantenmechanische Eigenschaft zeitlich periodischer Systeme. Diese können die gesamte Besetzungsverteilung nachhaltig beeinflussen und finden eine eindrucksvolle Anwendung in Form eines neuartigen Schaltmechanismus in einem harmonisch modulierten Doppelmuldenpotential in Kontakt mit einem Wärmebad. Der asymptotische Zustand kann unter geringer Variation der Antriebsamplitude vom Grundzustand der einen Mulde in einen Zustand höherer mittlerer Energie in der anderen Mulde geschaltet werden.

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

The main goal of our thesis is to understand qualitative properties of solutions to the Cauchy problem for the semi-linear wave model with time-dependent speed and dissipation. We greatly benefited from very precise estimates for the corresponding linear problem in order to obtain the global existence (in time) of small data solutions. This reason motivated us to introduce very carefully a complete description for classification of our models: scattering, non-effective, effective, over-damping. We have considered those separately.