Relaxation Time of Quantized Toral Maps

Abstract

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 ( → 0) together with the limit of small noise strength (ǫ → 0).Focusing on quantized smooth Anosov maps, we exhibit a semiclassical régime < ǫ E ≪ 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 −1 . On the other hand, we show that in the "quantum régime" ǫ ≪ ≪ 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.

Dissipation enhancement by mixing

“…al. [FNW04] (see also [FW03,FNW06]) also obtain bounds on the dissipation time τ d assuming the time decay of the correlations of the diffusive operator e ν∆ U for sufficiently small ν. Explicitly they assume sufficient decay of (e ν∆ U ) n f, g as n → ∞, and then show that the dissipation time τ d is at most C/|ln ν|.…”

“…al. [FW03,FNW04,FNW06], we use Sobolev norms instead, as it is more convenient for our purposes. This is the content of the first part of Definition 2.3, and is repeated here for convenience.…”

“…If ν > 0 and ϕ is mixing, then Koopman operator U produces fine scales which are rapidly damped by the diffusion. We quantify this using the notion of dissipation time in [FW03] (see also [FNW04,FNW06]). Since U is unitary we clearly have θ n e −νλ1 θ n−1 , where λ 1 > 0 is the smallest non-zero eigenvalue of −∆ on M .…”

“…If ν > 0 and ϕ is mixing, then Koopman operator U produces fine scales which are rapidly damped by the diffusion. We quantify this using the notion of dissipation time in [FW03] (see also [FNW04,FNW06]). Definition 2.1 (Dissipation time).…”

Weyl Laws for Partially Open Quantum Maps

“…The behavior expressed by Eq. (3.5) is well known for flows [8], and we refer to [13], Lemma 1 for a detailed proof in the case of applications. The second part of the lemma follows easily from (3.5).…”