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arXiv:nlin/0004006 · Physical Review Letters

Quantum Fingerprints of Classical Ruelle-Pollicott Resonances

Kristi Pance, Wentao Lu, S. Sridhar

Published Sep 25, 2000• 45 citations
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Abstract

Quantum and classical correlations are studied experimentally in model n-disk microwave billiards. The wave vector kappa autocorrelation C(kappa) of the quantum spectrum displays nonuniversal oscillations for large kappa, comparable to the universal random matrix theory behavior observed for small kappa. The nonuniversal behavior is shown to be completely determined by the classical Ruelle-Pollicott resonances, arising from the complex eigenvalues of the Perron-Frobenius operator, and calculated using periodic orbit theory. This work establishes a fundamental connection between the quantum and classical correlations of an open system.

DOI
10.1103/physrevlett.85.2737

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Quantum and classical correlations are studied experimentally in model n-disk microwave billiards. The wave vector kappa autocorrelation C(kappa) of the quantum spectrum displays nonuniversal oscillations for large kappa, comparable to the universal random matrix theory behavior observed for small kappa. The nonuniversal behavior is shown to be completely determined by the classical Ruelle-Pollicott resonances, arising from the complex eigenvalues of the Perron-Frobenius operator, and calculated using periodic orbit theory. This work establishes a fundamental connection between the quantum and classical correlations of an open system.
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.
We establish a duality between the quantum wave vector spectrum and the eigenmodes of the classical Liouvillian dynamics for integrable billiards. Signatures of the classical eigenmodes appear as peaks in the correlation function of the quantum wave vector spectrum. A semiclassical derivation and numerical calculations are presented in support of the results. These classical eigenmodes can be observed in physical experiments through the autocorrelation of the transmission coefficient of waves in quantum billiards. Exact classical trace formulas of the resolvent are derived for the rectangle, equilateral triangle, and circle billiards. We also establish a correspondence between the classical periodic orbit length spectrum and the quantum spectrum for integrable polygonal billiards.
Using the equivalence between the quantum-mechanical probability density in a quantum billiard and the Poynting vector in the corresponding microwave system, current distributions were studied in a quantum dot like cavity, as well as in a Robnik billiard with λ=0.4, and an introduced ferrite cylinder. Spatial auto correlation functions for currents and vorticity were studied and compared with predictions from the random-superposition-of-plane-waves hypothesis. In addition different types of vortex neighbour spacing distributions were determined and compared with theory.
In the preceding chapters we have learnt that random matrix theory is perfectly able to explain the universal properties of the spectra of chaotic systems, and this in spite of the oversimplifying assumptions applied. On the one hand it is very satisfactory that one single theory can cope with such a variety of systems as nuclei, mesoscopic structures, or microwave billiards, on the other hand this is a bit disappointing. If there is no possibility of discriminating between the spectra of a nucleus and a quantum dot, then there is little hope of learning anything of relevance about it. Fortunately, random matrix theory is only one side of the coin. We have already come across some examples demonstrating its limits of validity. Remember the spectral level dynamics where bouncing balls disturbed the otherwise universal Gaussian velocity distribution (see Section 5.2.2). Another example is the scarring phenomenon observed in many wave functions. Here obviously closed classical orbits have left their fingerprints in the amplitude patterns. We cannot expect that the universal random matrix theory can correctly account for individual features such as periodic orbits. We now come to an alternative approach to analysing the spectra. As we know from the correspondence principle, in the semiclassical limit quantum mechanics eventually turns into classical mechanics. That is why classical dynamics must be hidden somewhere in the spectra, at least in the limit of high quantum numbers. In the introduction we have already discussed this connection for a particle in a one-dimensional box.
The semi-classical method has become a necessary instrument to study the classical movement of the particle.Periodic orbit theory is rapidly becoming one of most useful semi-classical tools which can be used to make direct connections between the quantized energy eigenvalues of a bound state and the classical motions for the corresponding point particle.We use a quantum spectral function which contain rich information of classical orbits in well.We study the correspondence between quantum spectra and classical orbits in the circular.Two-dimensional billiard systems have provided easily visualization examples relevant for both types of analyses.As a simple example of the application to a billiard or infinite well system of Periodic orbit theory,we compute the Fourier transform(ρL) of the quantum mechanical energy level density of two-dimensional circular billiard systems.The resulting peaks in plots of ρL2 versus L are compared to the lengths of the classical trajectories in these geometries.The locations of peaks in ρL agree with the lengths of classical orbits perfectly,which testifies the correspondence of quantum mechanics and classical mechanics.This examples show evidently that semi-classical methods provides a bridge between quantum and classical mechanics.