We study the quantum localization in the chaotic eigenstates of a billiard with mixed-type phase space (J. Phys. A: Math. Gen. 16, 3971 (1983); 17, 1049 (1984)), after separating the regular and chaotic eigenstates, in the regime of slightly distorted circle billiard where the classical transport time in the momentum space is still large enough, although the diffusion is not normal. This is a continuation of our recent papers (Phys. Rev. E 88, 052913 (2013); 98, 022220 (2018)). In quantum systems with discrete energy spectrum the Heisenberg time tH = 2π /∆E, where ∆E is the mean level spacing (inverse energy level density), is an important time scale. The classical transport time scale tT (transport time) in relation to the Heisenberg time scale tH (their ratio is the parameter α = tH /tT ) determines the degree of localization of the chaotic eigenstates, whose measure A is based on the information entropy. We show that A is linearly related to normalized inverse participation ratio. The localization of chaotic eigenstates is reflected also in the fractional power-law repulsion between the nearest energy levels in the sense that the probability density (level spacing distribution) to find successive levels on a distance S goes like ∝ S β for small S, where 0 ≤ β ≤ 1, and β = 1 corresponds to completely extended states. We show that the level repulsion exponent β is empirically a rational function of α, and the mean A (averaged over more than 1000 eigenstates) as a function of α is also well approximated by a rational function. In both cases there is some scattering of the empirical data around the mean curve, which is due to the fact that A actually has a distribution, typically with quite complex structure, but in the limit α → ∞ well described by the beta distribution. The scattering is significantly stronger than (but similar as) in the stadium billiard (Nonl.Phen.Compl.Sys. 21, No3, 225 (2018)) and the kicked rotator (Phys.Rev. E 91, 042904 ( 2015)). Like in other systems, β goes from 0 to 1 when α goes from 0 to ∞. β is a function of A , similar to the quantum kicked rotator and the stadium billiard.

We perform a detailed numerical study of diffusion in the ɛ stadium of Bunimovich, and propose an empirical model of the local and global diffusion for various values of ɛ with the following conclusions: (i) the diffusion is normal for all values of ɛ (≤0.3) and all initial conditions, (ii) the diffusion constant is a parabolic function of the momentum (i.e., we have inhomogeneous diffusion), (iii) the model describes the diffusion very well including the boundary effects, (iv) the approach to the asymptotic equilibrium steady state is exponential, (v) the so-called random model (Robnik et al., 1997) is confirmed to apply very well, (vi) the diffusion constant extracted from the distribution function in momentum space and the one derived from the second moment agree very well. The classical transport time, an important parameter in quantum chaos, is thus determined.

The localization measures A (based on the information entropy) of localized chaotic eigenstates in the Poincaré-Husimi representation have a distribution on a compact interval [0;A0], which is well approximated by the beta distribution, based on our extensive numerical calculations. The system under study is the Bunimovich' stadium billiard, which is a classically ergodic system, also fully chaotic (positive Lyapunov exponent), but in the regime of a slightly distorted circle billiard (small shape parameter ") the diffusion in the momentum space is very slow. The parameter α = tH/tT , where tH and tT are the Heisenberg time and the classical transport time (diffusion time), respectively, is the important control parameter of the system, as in all quantum systems with the discrete energy spectrum. The measures A and their distributions have been calculated for a large number of ε and eigenenergies. The dependence of the standard deviation σ on α is analyzed, as well as on the spectral parameter β (level repulsion exponent of the relevant Brody level spacing distribution). The paper is a continuation of our recent paper (B. Batistić, Č. Lozej and M. Robnik, Nonlinear Phenomena in Complex Systems 21, 225 (2018)), where the spectral statistics and validity of the Brody level spacing distribution has been studied for the same system, namely the dependence of β and of the mean value < A > on α.

We study the classical and quantum ergodic lemon billiard introduced by Heller and Tomsovic in Phys. Today 46 38 (1993), for the case B=1/2, which is a classically ergodic system (without a rigorous proof) exhibiting strong stickiness regions around a zero-measure bouncing ball modes. The structure of the classical stickiness regions is uncovered in the S-plots introduced by Lozej in Phys.Rev. E 101 052204 (2020). A unique classical transport or diffusion time cannot be defined. As a consequence the quantum states are characterized by the following nonuniversal properties: (i) All eigenstates are chaotic but localized as exhibited in the Poincaré-Husimi (PH) functions. (ii) The entropy localization measure A (also the normalized inverse participation ratio) has a nonuniversal distribution, typically bimodal, thus deviating from the beta distribution, the latter one being characteristic of uniformly chaotic systems with no stickiness regions. (iii) The energy level spacing distribution is Berry-Robnik-Brody (BRB), capturing two effects: the quantally divided phase space (because most of the PH functions are either the inner-ones or the outer-ones, dictated by the classical stickiness, with an effective parameter µ1 measuring the size of the inner region bordered by the sticky invariant object, namely a cantorus), and the localization of PH functions characterized by the level repulsion (Brody) parameter β. (iv) In the energy range considered (between 20.000 states to 400.000 states above the ground state) the picture (the structure of the eigenstates and the statistics of the energy spectra) is not changing qualitatively, as β fluctuates around 0.8, while µ1 decreases almost monotonically, with increasing energy.

We analyze the structure and stickiness in the chaotic components of generic Hamiltonian systems with divided phase space. Following the method proposed recently in (Lozej, Robnik, Phys. Rev. E 98, 022220 (2018)), the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid. We perform extensive numerical studies of three example systems: the Chirikov standard map, the family of Robnik billiards and the family of lemon billiards. The filling of the cells is compared to the random model of chaotic diffusion, introduced in (Robnik et al. J. Phys. A: Math. Gen. 30, L803 (1997)) for the description of transport in the phase spaces of ergodic systems. The model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential. In generic systems the stickiness induces correlations in the cell visits. The distribution of recurrence times exhibits a separation of time scales because of the dynamical trapping. We model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times. We introduce the variable S, which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness. We use S to globally assess the distributions of recurrence times. We find that in the bulk of the chaotic sea S = 1, while S > 1 in areas of stickiness. We present the results in the form of animated grayscale plots of the variable S in the largest chaotic component for the three example systems, included as supplemental material to this paper.