Relaxation Time of Quantized Toral Maps

TL;DR

This paper investigates the relaxation times of noisy quantum maps on the torus, revealing their dependence on classical chaos, and identifies regimes where quantum and classical relaxation behaviors align or diverge.

Contribution

It introduces the concept of relaxation time for noisy quantum maps, establishes quantum-classical correspondence regimes, and provides rigorous estimates for quantum relaxation times on the torus.

Findings

Quantum and classical relaxation times coincide in a specific semiclassical regime.

Quantum relaxation times differ significantly from classical ones when noise strength is much smaller than Planck's constant.

Exact asymptotics for quantum relaxation times are derived for ergodic toral maps.

Abstract

We introduce the notion of the relaxation time for noisy quantum maps on the 2d-dimensional torus - a generalization of previously studied dissipation time. We show that relaxation time is sensitive to the chaotic behavior of the corresponding classical system if one simultaneously considers the semiclassical limit ($\hbar$ -> 0) together with the limit of small noise strength ($\ep$ -> 0). Focusing on quantized smooth Anosov maps, we exhibit a semiclassical regime $\hbar<\ep^{E}$ << 1 (where E>1) in which classical and quantum relaxation times share the same asymptotics: in this regime, a quantized Anosov map relaxes to equilibrium fast, as the classical map does. As an intermediate result, we obtain rigorous estimates of the quantum-classical correspondence for noisy maps on the torus, up to times logarithmic in $\hbar^{-1}$. On the other hand, we show that in the ``quantum regime'' $\ep$ << $\hbar$ << 1, quantum and classical relaxation times behave very differently. In the special case of ergodic toral symplectomorphisms (generalized ``Arnold's cat'' maps), we obtain the exact asymptotics of the quantum relaxation time and precise the regime of correspondence between quantum and classical relaxations.