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arXiv:chao-dyn/9902002 · Physical Review Letters

Quantum Resonances and Decay of a Chaotic Fractal Repeller Observed Using Microwaves

Wentao Lu, M. Rose, Kristi Pance, S. Sridhar

Published Jun 28, 1999• 31 citations
arXivHTMLPDF

Abstract

The quantum resonances of classically chaotic $n$-disk geometries were studied experimentally utilizing thin 2D microwave geometries. The experiments yield the frequencies and widths of low-lying resonances, which are compared with semiclassical calculations. The long time or small energy behavior of the wave-vector autocorrelation gives information about the quantum decay rate, which is in good agreement with that obtained from classical scattering theory. The intermediate energy behavior shows nonuniversal oscillations determined by periodic orbits.

DOI
10.1103/physrevlett.82.5233

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The quantum resonances of classically chaotic nnn-disk geometries were studied experimentally utilizing thin 2D microwave geometries. The experiments yield the frequencies and widths of low-lying resonances, which are compared with semiclassical calculations. The long time or small energy behavior of the wave-vector autocorrelation gives information about the quantum decay rate, which is in good agreement with that obtained from classical scattering theory. The intermediate energy behavior shows nonuniversal oscillations determined by periodic orbits.
We describe a wave-mechanical implementation of classically chaotic n-disk scattering based on thin two-dimensional microwave cavities. Two-, three-, and four-disk scatterings are investigated in detail. The experiments, which are able to probe the stationary Green's function of the system, yield both frequencies and widths of the low-lying quantum resonances. The observed spectra are found to be in good agreement with calculations based on semiclassical periodic orbit theory. Wave-vector autocorrelation functions are analyzed for various scattering geometries, the small wave-vector behavior allowing one to extract the escape rate from the quantum repeller. Quantitative agreement is found with the value predicted from classical scattering theory. For intermediate energies, nonuniversal oscillations are detected in the autocorrelation function, reflecting the presence of periodic orbits.
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.
We study chaotic quantum billiards using both microwave cavities and numerical simulations. For the same geometry, viz., a Sinai billiard, agreement to remarkable precision is found for both the eigenvalue magnitudes and the spatial detail of the eigenfunctions. The association of the eigenfunctions with classical periodic orbits is demonstrated, and scarred states are identified. Desymmetrizing the Sinai billiard by slightly moving the central disk is shown to lead to strong localization of the eigenfunction. The calculated eigenstates of the symmetric billiard show an even- and odd-parity pair whose linear combination gives the localized state.
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We study the effect on quantum spectra of the existence of small circular disks in a billiard system. In the limit where the disk radii vanish there is no effect, however this limit is approached very slowly so that even very small radii have comparatively large effects. We include diffractive orbits which scatter off the small disks in the periodic orbit expansion. This situation is formally similar to edge diffraction except that the disk radii introduce a length scale in the problem such that for wave lengths smaller than the disk radius we recover the usual semi-classical approximation; however, for wave lengths larger than the disk radius there is a qualitatively different behaviour. We test the theory by successfully estimating the positions of scattering resonances in geometries consisting of three and four small disks. PACS numbers: 0.320.+i, 03.65.Sq 1 Typeset using REVTEXThe presence of discontinuities in classical Hamiltonian systems implies the necessity of