In this work several aspects of quantum chaos are studied in the time domain. The wave equation for flat microwave cavities is equivalent to the Schrödinger equation in quantum mechanics. Therefore microwave measurements provide an experimental approach to quantum chaos. The experiments are described in terms of scattering theory, to take the coupling of the antennas to the system as well as the dissipation in the cavity walls into account. In the first part the scattering matrix of several microwave cavities is analyzed in the regime of overlapping resonances. The difference between regular and chaotic systems can be observed both in the autocorrelation functions of the same S-matrix elements, and in the cross-correlation functions of different S-matrix elements. In accordance with literature, this difference is more pronounced in the cross-correlation functions. To describe the experimental correlation functions, the absorption in the cavity walls is modeled by an infinite number of weak decay channels. In the second part the focus is shifted to the stability of quantum time-evolution. The fidelity amplitude is a standard benchmark for the stability of a quantum system against a change of the Hamiltonian. It is defined as the overlap of the perturbed and unperturbed time-evolution of the same initial state. Exact theoretical results for a random matrix model are compared with numerical simulations and the linear-response results. For strong perturbations a partial recovery of the fidelity amplitude is found, and an intuitive explanation for this behavior is given in terms of a spectral Debye-Waller factor. Further, in the third part, experimental results for the fidelity amplitude are presented for two microwave cavities with classically chaotic dynamics. The perturbation of the systems is realized by applying small changes to their geometry. The results are well described by the linear-response expression, and the perturbation strength can be related to the change of the geometry of the cavities. In the fourth part microwave measurements on dielectric quadrupole billiards with mixed phase space are discussed. The internal dynamics is analyzed by means of Husimi distributions, while for the outer region the Poynting vector is determined to obtain the emission pattern. The emission pattern of the quadrupole billiard is strongly influenced by the structures of its mixed phase space.
In this work several aspects of quantum chaos are studied in the time domain. The wave equation for flat microwave cavities is equivalent to the Schrödinger equation in quantum mechanics. Therefore microwave measurements provide an experimental approach to quantum chaos. The experiments are described in terms of scattering theory, to take the coupling of the antennas to the system as well as the dissipation in the cavity walls into account. In the first part the scattering matrix of several microwave cavities is analyzed in the regime of overlapping resonances. The difference between regular and chaotic systems can be observed both in the autocorrelation functions of the same S-matrix elements, and in the cross-correlation functions of different S-matrix elements. In accordance with literature, this difference is more pronounced in the cross-correlation functions. To describe the experimental correlation functions, the absorption in the cavity walls is modeled by an infinite number of weak decay channels. In the second part the focus is shifted to the stability of quantum time-evolution. The fidelity amplitude is a standard benchmark for the stability of a quantum system against a change of the Hamiltonian. It is defined as the overlap of the perturbed and unperturbed time-evolution of the same initial state. Exact theoretical results for a random matrix model are compared with numerical simulations and the linear-response results. For strong perturbations a partial recovery of the fidelity amplitude is found, and an intuitive explanation for this behavior is given in terms of a spectral Debye-Waller factor. Further, in the third part, experimental results for the fidelity amplitude are presented for two microwave cavities with classically chaotic dynamics. The perturbation of the systems is realized by applying small changes to their geometry. The results are well described by the linear-response expression, and the perturbation strength can be related to the change of the geometry of the cavities. In the fourth part microwave measurements on dielectric quadrupole billiards with mixed phase space are discussed. The internal dynamics is analyzed by means of Husimi distributions, while for the outer region the Poynting vector is determined to obtain the emission pattern. The emission pattern of the quadrupole billiard is strongly influenced by the structures of its mixed phase space.
The present doctoral thesis is concerned, on the one hand, with the investigation of the decay probability in a quantum billiard, and, on the other, the realization of the double-slit experiment in microwave billiards. For both purposes transmission elements of the scattering matrix are experimentally measured in microwave cavities with regular and chaotic dynamics. One or more pieces of the billiards' boundary are removed and the leaked outward eld is investigated. The general aim is to distinguish the eect of the billiard dynamics on the temporal decay behaviour and on the interference patterns. The decay probability in an open quantum billiard can be explored by means of the scattering matrix formalismus derived in the framework of nuclear physics. In a circular billiard, archetype of regular systems, the opening breaks the spatial symmetry causing a splitting of the degenerate states observable at low frequencies. One state of the doublet remains unaected contributing to the long-time behavior inside the cavity (survival probability), while the short-lived one is strongly influenced by the opening and decays much faster. The case of chaotic dynamics is studied by means of a tilted stadium billiard. For sufficiently long times the decay probability shows an essentially nonexponential behaviour. Agreement with theoretical results derived by Harney, Dittes and Muller is obtained for a few of the open channels and intermediate times. For both, the circle and the stadium,the short time decay resembles the classical behaviour. Furthermore, a double-slit experiment is performed with a wave packet starting inside a microwave billiard domain, both for regular and chaotic dynamics. Stationary and temporal patterns formed by the leaking of the wave packet are experimentally determined. The evolution of a directed initial state is constructed by means of the electromagnetic propagator. Also single-slit experiments are carried out and compared with the corresponding double-slit experiments. The present work sheds light on the fundamental question of the decay of open quantum systems and provides plentiful information about the structure of the diracted eld from microwave billards with a double-slit on the boundary connecting classical properties with the corresponding quantum features in both cases.
Two experiments with super- and normal conducting microwave cavities which are used as an analog to two-dimensional, infinitely deep quantum potentials are presented. The eigenvalues of such a potential can be measured directly through the resonance frequencies of the cavity, while the eigenvectors can be determined by measuring the field distributions inside the cavity. In the case of open systems another information – the imaginary part of the eigenvalues – is observable by measuring the widths of the resonances. As examples for this experimental approach to quantum chaos the semiclassical reconstruction of length spectra of billiards with varying chaoticity based of various trace formulas proposed by Gutzwiller, by Ullmo, Grinberg and Tomsovic and by Berry and Tabor for systems with chaotic, mixed and regular dynamics, respectively, and the first observation of square root singularities – so called exceptional points recently proposed by Heiss – in the energy spectrum of coupled cavities are presented.
We discuss the stability of quantum motion under system's perturbations in the light of the corresponding classical behavior. In particular we focus our attention on the so called "fidelity" or Loschmidt echo, its relation with the decay of correlations, and discuss the quantum-classical correspondence. We then report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.
This book introduces the quantum mechanics of classically chaotic systems, or quantum chaos for short. The author's philosophy has been to keep the discussion simple and to illustrate theory, wherever possible, with experimental or numerical examples. The microwave billiard experiments, initiated by the author and his group, play a major role in this respect. Topics covered include the various types of billiard experiment, random matrix theory, systems with periodic time dependences, the analogy between the dynamics of a one-dimensional gas with a repulsive interaction and spectral level dynamics, where an external parameter takes the role of time, scattering theory distributions and fluctuation, properties of scattering matrix elements, semiclassical quantum mechanics, periodic orbit theory, and the Gutzwiller trace formula. This book will be of great value to anyone working in quantum chaos.
Quantum chaos concerns the emergence of universal statistical and dynamical prop- erties in quantum systems whose classical counterparts exhibit chaotic behavior. Unlike classical chaos, which can be characterized by sensitivity to initial conditions and exponential divergence of trajectories, quantum chaos manifests itself through a variety of indirect diagnostics, including spectral correlations, operator growth, and information scrambling. Understanding how these signatures arise from microscopic dynamics, and how they can be accessed in realistic many-body systems, is a central challenge in modern physics. This thesis investigates quantum chaos from complementary theoretical and practical perspectives. On the theoretical side, it develops effective descriptions of chaotic dy- namics in many-body systems with approximate symmetries, focusing on how weak symmetry breaking controls the exploration of Hilbert space and the crossover to random-matrix universality. Using perturbative arguments, numerical simulations, and a nonlinear sigma-model formulation, we identify a regime of Hilbert-space dif- fusion in which chaotic dynamics unfolds through a slow, collective spreading across nearly decoupled symmetry sectors. On the practical side, the thesis addresses the question of how paradigmatic chaotic models can be realized and probed in controlled experimental platforms. In partic- ular, it explores quantum simulation schemes for Sachdev–Ye–Kitaev (SYK) models using cavity quantum electrodynamics. By employing time-dependent disorder and Trotterized evolution in single-mode optical cavities, we demonstrate how low-rank interactions can be systematically promoted to effective all-to-all randomness, re- producing the spectral and dynamical signatures of SYK physics. Extensions to open-system dynamics are discussed, including Lindbladian realizations and the role of dissipation. The thesis thus highlights the connections between effective descriptions of quantum chaos and their implementation in controlled quantum simulation settings.
This thesis investigates quantum transport in the energy space of two paradigm systems of quantum chaos theory. These are highly excited hydrogen atoms subject to a microwave field, and kicked atoms which mimic the delta-kicked rotor model. Both of these systems show a complex dynamical evolution arising from the interaction with an external time-periodic driving force. In particular two quantum phenomena, which have no counterpart on the classical level, are studied: the suppression of classical diffusion, known as dynamical localisation, and quantum resonances as a regime of enhanced transport for the delta-kicked rotor. The first part of the thesis provides new support for the quantitative analogy between energy transport in strongly driven highly excited atoms and particle transport in Anderson-localised solids. A comprehensive numerical analysis of the atomic ionisation rates shows that they obey a universal power-law distribution, in agreement with Anderson localisation theory. This is demonstrated for a one-dimensional model as well as for the real three-dimensional atom. We also discuss the implications of the universal decay-rate distributions for the asymptotic time-decay of the survival probability of the atoms. The second part of the thesis clarifies the effect of decoherence, induced by spontaneous emission, on the quantum resonances which have been observed in a recent experiment with delta-kicked atoms. Scaling laws are derived, based on a quasi-classical approximation of the quantum evolution. These laws describe the shape of the resonance peaks in the mean energy of an experimental ensemble of kicked atoms. Our analytical results match perfectly numerical computations and explain the initially surprising experimental observations. Furthermore, they open the door to the study of the competing effects of decoherence and chaos on the stability of the time evolution of kicked atoms. This stability may be characterised by the overlap of two identical initial states which are subject to different time evolutions. This overlap, called fidelity, is investigated in an experimentally accessible situation.