The phenomenon of quantum localization in classically chaotic eigenstates is one of themain 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 endles S beta for small S, where 0_beta_1, and beta = 1corresponds to completely extended states. We show that there is a clear functional relation between the exponent betta 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.
COBISS.SI-ID: 76225025
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 k∈[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 βBR. (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 βloc in the interval [0,1]. The level repulsion parameters βBR and βloc are almost linearly related, close to the identity line. (3) We show the existence of a scaling law between βloc 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 [ Batistić and Robnik J. Phys. A: Math. Gen. 43 215101 (2010); arXiv:1302.7174 (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).
COBISS.SI-ID: 74771713
We study the quantum mechanics of a billiard (Robnik 1983 J. Phys. A: Math. Gen. 16 3971) in the regime of mixed-type classical phase space (the shape parameter lambda = 0.15) at very high-lying eigenstates, starting at about 1.000.000th eigenstate and including the consecutive 587654 eigenstates. By calculating the normalized Poincar´e-Husimi functions of the eigenstates and comparing them with the classical phase space structure, we introduce the overlap criterion which enables us to separatewith great accuracy and reliability the regular and chaotic eigenstates, and the corresponding energies. The chaotic eigenstates appear all to be dynamically localized, meaning that they do not uniformly occupy the entire available chaotic classical phase space component, but are localized on a proper subset.We find with unprecedented precision and statistical significance that the level spacing distributions of the regular levels obey the Poisson statistics, and the chaotic ones obey the Brody statistics, as anticipated in a recent paper by Batistić and Robnik (2010 J. Phys. A: Math. Theor. 43 215101), where the entire spectrum was found to obey the Berry-Robnik-Brody statistics. There are no effects of dynamical tunneling in this regime, due to the high energies, as they decay exponentially with the inverse effective Planck constant which is proportional to the square root of the energy.
COBISS.SI-ID: 75147009
We demonstrate that the energy or quasienergy level spacing distribution in dynamically localized chaotic eigenstates is excellently described by the Brody distribution, displaying the fractional power law level repulsion. This we show in two paradigmatic systems, namely for the fully chaotic eigenstates of the kicked rotator at K = 7, and for the chaotic eigenstates in the mixed-type billiard system (Robnik 1983), after separating the regular and chaotic eigenstates by means of the Poincar´e Husimi function, at very high energies with great statistical significance (587654 eigenstates, starting at about 1000000 above the ground state). This separation confirms the Berry-Robnik picture, and is performed for the first time at such high energies.
COBISS.SI-ID: 74806017
We perform an extensive and detailed analysis of the generalized diffusion processes in deterministic area preserving maps with noncompact phase space, exemplified by the standard map, with the special emphasis onunderstanding the anomalous diffusion arising due to the accelerator modes. The accelerator modes and theirimmediate neighborhood undergo ballistic transport in phase space, and also the greater vicinity of them is stillmuch affected (dragged) by them, giving rise to the non-Gaussian (accelerated) diffusion. The systematicapproach rests upon the following applications: the GALI method to detect the regular and chaotic regions andthus to describe in detail the structure of the phase space, the description of the momentum distribution in termsof the Levy stable distributions, and the numerical calculation of the diffusion exponent and of the correspondingdiffusion constant. We use this approach to analyze in detail and systematically the standard map at all valuesof the kick parameter K, up to K = 70. All complex features of the anomalous diffusion are well understoodin terms of the role of the accelerator modes, mainly of period 1 at large K)2 pi, but also of higher periods(2,3,4, . . .) at smaller values of K(2 pi.
COBISS.SI-ID: 77280257