Correlation functions and fidelity decay in chaotic systems

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 descri...

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Main Author: Schäfer, Rudi
Contributors: Stöckmann, Hans-Jürgen (Prof. Dr.) (advisor)
Format: Dissertation
Language: English
Published: Philipps-Universität Marburg 2004
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Summary: 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.