量子化Toral映射的弛豫时间
Relaxation Time of Quantized Toral Maps
摘要：
我们介绍的概念的噪声量子映射的二维环面上的弛豫时间-以前研究的耗散时间的推广。我们表明，弛豫时间是敏感的混沌行为的相应的经典系统，如果一个同时考虑半经典极限（$\hbar$ -> 0）与小噪声强度的限制（$\ep$ -> 0）。聚焦于量子化光滑Anosov映射，我们展示了一个半经典区域$\hbar<\ep^{E}$ << 1（其中E>1），其中经典和量子弛豫时间共享相同的渐近性：在这个区域中，量子化Anosov映射像经典映射一样快速弛豫到平衡态。作为一个中间结果，我们得到严格的估计的量子经典对应的噪声地图上的环面，倍对数$\hbar^{-1}$。另一方面，我们表明，在“量子制度”$\ep$ $\hbar$ << 1，量子和经典弛豫时间表现得非常不同。在特殊的情况下，遍历toral辛同胚（广义的“阿诺德的猫”映射），我们得到的量子弛豫时间的精确渐近和精确的制度之间的对应量子和经典的弛豫。
英文摘要：
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.
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