Chaotic eigenfunctions in phase space

Abstract

We study individual eigenstates of quantized area-preserving maps on the 2-torus which are classically chaotic. In order to analyze their semiclassical behavior, we use the Bargmann-Husimi representations for quantum states, as well as their stellar parametrization, which encodes states through a minimal set of points in phase space (the constellation of zeros of the Husimi density). We rigorously prove that a semiclassical uniform distribution of Husimi densities on the torus entails a similar equidistribution for the corresponding constellations. We deduce from this property a universal behavior for the phase patterns of chaotic Bargmann eigenfunctions, which reminds of the WKB approximation for eigenstates of integrable systems (though in a weaker sense). In order to obtain more precise information on ``chaotic eigenconstellations", we then model their properties by ensembles of random states, generalizing former results on the 2-sphere to the torus geometry. This approach yields statistical predictions for the constellations, which fit quite well the chaotic data. We finally observe that specific dynamical information, e.g. the presence of high peaks (like scars) in Husimi densities, can be recovered from the knowledge of a few long-wavelength Fourier coefficients, which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions.

61 pages, LaTeX, plus 20 eps figures, two of them in color. The updated version contains the variances of L^s norms (see Table I), and updates some references

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• Distribution of zeros of random and quantum chaotic sections of positive line bundles
• Quantum Chaotic Dynamics and Random Polynomials
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Cited by in corpus (9)

• Distribution of zeros of random and quantum chaotic sections of positive line bundles
• Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems
• Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles
• Entropy of semiclassical measures of the Walsh-quantized baker's map
• Dimension reduction techniques for the minimization of theta functions on lattices
• Wigner function statistics in classically chaotic systems
• Classical route to ergodicity and scarring in collective quantum systems
• Phase-space representations of SIC-POVM fiducial states
• Log-scale equidistribution of zeros of quantum ergodic eigensections