Relaxation Time of Quantized 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 ($\hslash$ -> 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 ${\hslash }^{-1}$. On the other hand, we show that in the ``quantum regime'' $\ep$ << $\hslash$ << 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.

LaTeX, 27 pages, former term dissipation time replaced by relaxation time, new introduction and references

References in corpus (11)

• Decoherence, Chaos, and the Second Law
• Ruelle-Perron-Frobenius spectrum for Anosov maps
• Scarred eigenstates for quantum cat maps of minimal periods
• Decoherence for classically chaotic quantum maps
• Classical decays in decoherent quantum maps
• Classical versus Quantum Time Evolution of Densities at Limited Phase-Space Resolution
• Spectral properties of noisy classical and quantum propagators
• On the multiplicativity of quantum cat maps
• Spectral properties and classical decays in quantum open systems
• Classical Limit of Quantum Dynamical Entropies
• Quantized chaotic dynamics and non-commutative KS entropy