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 quantum kicked rotator in the classically fully chaotic regime, in the domain of the semiclassical behaviour. We use Izrailev's N-dimensional model for various N<=4000, which in the limit N-> infinity tends to the quantized kicked rotator, not only for K=5 as studied previously, but for many different values of the classical kick parameter 5<=K<=35, and also of the quantum parameter k. We describe the dynamical localization of chaotic eigenstates as a paradigm for other both time-periodic and time-independent fully chaotic or/and mixed type Hamilton systems. We generalize the localization length L and the scaling variable (L/N) to the case of anomalous classical diffusion. We study the generalized classical diffusion also in the regime where the simple minded theory of the normal diffusion fails. We greatly improve the accuracy of the numerical calculations with the following conclusions: The level spacings distribution of the eigenphases is very well described by the Brody distribution, systematically better than by other proposals, for various Brody exponents. When N->infinity and L is fixed we have always Poisson, even in fully chaotic regime. We study the eigenfunctions of the Floquet operator and characterize their localization properties using the information entropy measure describing the degree of dynamical localization of the eigenfunction. The resulting localization parameter is found to be almost equal to the Brody parameter. We show the existence of a scaling law between the localization parameter and the scaling variable L/N, now including the regimes of anomalous diffusion. The above findings are important also in time-independent Hamilton systems, like in mixed type billiards, where the Brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates.