Relaxation Time of Quantized Toral Maps

Abstract

We introduce the notion of relaxation time for noisy quantum maps on the 2d-dimensional torus – generalization of previously studied dissipation time. We show that the relaxation time is sensitive to the chaotic behavior of the corresponding classical system if one simultaneously considers the semiclassical limit $$({\hbar } \to 0)$$ together with the limit of small noise strength (ε → 0). Focusing on quantized smooth Anosov maps, we exhibit a semiclassical régime $${\hbar } < \epsilon ^{E} \ll 1$$ (where E > 1) in which classical and quantum relaxation times share the same asymptotics: in this régime, 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 régime” $$\epsilon \ll {\hbar } \ll 1,$$ 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 régime of correspondence between quantum and classical relaxations. Communicated by Jens Marklof