Dynamical localization in chaotic systems: spectral statistics and localization measure in the kicked rotator as a paradigm for time-dependent and time-independent systems.

ABSTRACT

We study the kicked rotator in the classically fully chaotic regime using Izrailev's N-dimensional model for various N≤4000, which in the limit N→∞ tends to the quantized kicked rotator. We do treat not only the case K=5, as studied previously, but also many different values of the classical kick parameter 5≤K≤35 and many different values of the quantum parameter kε[5,60]. We describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time-periodic and time-independent (autonomous) fully chaotic or/and mixed-type Hamilton systems. We generalize the scaling variable Λ=l(∞)/N to the case of anomalous diffusion in the classical phase space by deriving the localization length l(∞) for the case of generalized classical diffusion. We greatly improve the accuracy and statistical significance of the numerical calculations, giving rise to the following conclusions: (1) The level-spacing distribution of the eigenphases (or quasienergies) is very well described by the Brody distribution, systematically better than by other proposed models, for various Brody exponents ß(BR). (2) We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure, which after normalization is given by ß(loc) in the interval [0,1]. The level repulsion parameters ß(BR) and ß(loc) are almost linearly related, close to the identity line. (3) We show the existence of a scaling law between ß(loc) and the relative localization length Λ, now including the regimes of anomalous diffusion. The above findings are important also for chaotic eigenstates in time-independent systems [Batistic and Robnik, J. Phys. A Math. Gen. 43, 215101 (2010); arXiv1302.7174 (2013)], where the Brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates, even in the mixed-type systems (after separation of regular and chaotic eigenstates).