Batistić Benjamin, Robnik Marko
CAMTP - Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia, European Union.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Nov;88(5):052913. doi: 10.1103/PhysRevE.88.052913. Epub 2013 Nov 20.
The phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos (or wave chaos), and thus plays an important role in general quantum mechanics or even in general wave mechanics. In this work we propose two different localization measures characterizing the degree of quantum localization, and study their relation to another fundamental aspect of quantum chaos, namely the (energy) spectral statistics. Our approach and method is quite general, and we apply it to billiard systems. One of the signatures of the localization of chaotic eigenstates is a fractional power-law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance S goes like [proportionality]S(β) for small S, where 0≤β≤1, and β=1 corresponds to completely extended states. We show that there is a clear functional relation between the exponent β and the two different localization measures. One is based on the information entropy and the other one on the correlation properties of the Husimi functions. We show that the two definitions are surprisingly linearly equivalent. The approach is applied in the case of a mixed-type billiard system [M. Robnik, J. Phys. A: Math. Gen. 16, 3971 (1983)], in which the separation of regular and chaotic eigenstates is performed.
经典混沌本征态中的量子局域化现象是量子混沌（或波混沌）的主要问题之一，因此在一般量子力学甚至一般波动力学中都起着重要作用。在这项工作中，我们提出了两种不同的局域化度量来表征量子局域化程度，并研究它们与量子混沌的另一个基本方面，即（能量）谱统计之间的关系。我们的方法和手段非常通用，并将其应用于台球系统。混沌本征态局域化的一个特征是相邻能级之间的分数幂律排斥，即对于小的距离S，在距离S处找到连续能级的概率密度与S(β)成正比，其中0≤β≤1，且β = 1对应于完全扩展态。我们表明，指数β与这两种不同的局域化度量之间存在明确的函数关系。一种基于信息熵，另一种基于胡西米函数的相关性质。我们表明这两个定义惊人地线性等价。该方法应用于一个混合型台球系统[M. 罗布尼克，《物理学杂志A：数学一般》16, 3971 (1983)]，其中对规则本征态和混沌本征态进行了分离。
