Chaotic eigenfunctions in phase space

TL;DR

This paper investigates the phase space structure of eigenstates in quantum chaotic maps on the torus, using semiclassical and statistical methods to understand their universal properties and the influence of classical dynamics.

Contribution

It introduces a rigorous link between Husimi density distributions and constellation patterns of zeros, and models chaotic eigenstates with random ensembles to predict their statistical properties.

Findings

Semiclassical uniform Husimi distributions imply equidistributed zeros in phase space.

Chaotic eigenfunctions exhibit universal phase patterns similar to WKB states.

Random state ensembles accurately predict the statistical behavior of eigenconstellations.

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.