Distribution of resonances for open quantum maps

TL;DR

This paper studies open quantum maps on the torus, confirming the fractal Weyl law for quantum resonances and analyzing conductance fluctuations, linking classical chaos to quantum spectral properties.

Contribution

It provides numerical and rigorous evidence for the fractal Weyl law in open quantum maps and connects classical trapped set dimensions to quantum resonance distributions.

Findings

Numerical confirmation of the fractal Weyl law.

Full resonance spectrum satisfying the law in a simplified model.

Shot noise power close to random matrix theory predictions.

Abstract

We analyze simple models of classical chaotic open systems and of their quantizations (open quantum maps on the torus). Our models are similar to models recently studied in atomic and mesoscopic physics. They provide a numerical confirmation of the fractal Weyl law for the density of quantum resonances of such systems. The exponent in that law is related to the dimension of the classical repeller (or trapped set) of the system. In a simplified model, a rigorous argument gives the full resonance spectrum, which satisfies the fractal Weyl law. For this model, we can also compute a quantity characterizing the fluctuations of conductance through the system, namely the shot noise power: the value we obtain is close to the prediction of random matrix theory.