arXiv subjects

Črt Lozej

Publications and source records attributed to Črt Lozej.

12 recordsLinked to original sources

arXiv2025

Quantum chaos and semiclassical behavior in mushroom billiards I: Spectral statistics

We study the aspects of quantum chaos in mushroom billiards introduced by Bunimovich. This family of billiards classically has the property of mixed phase space with precisely one entirely regular and one fully chaotic (ergodic) component, whose size depends on the width w of the stem, and has two limiting geometries, namely the circle (as the integrable system) and stadium (as the fully chaotic system). Therefore, this one-parameter system is ideal to study the semiclassical behavior of the quantum counterpart. Here, in paper I, we study the spectral statistics as a function of the geometry defined by w, and as a function of the semiclassical parameter k, which in this case is just the wavenumber k. We show that at sufficiently large k the level spacing distribution is excellently described by the Berry-Robnik (BR) distribution (without fitting). At lower k the small deviations from it can be well described by the Berry-Robnik-Brody (BRB) distribution, which captures the effects of weak dynamical localization of Poincaré-Husimi functions. We also employ the analytical theory of the level spacing ratios distribution P(r) for mixed-type systems, recently obtained by Yan (2025), which does not require a spectral unfolding procedure, and show excellent agreement with numerics in the semiclassical limit of large k. In paper II we shall analyze the eigenstates by means of Poincaré-Husimi functions.

quant-ph↗

arXiv2024

Intermediate spectral statistics of rational triangular quantum billiards

Triangular billiards whose angles are rational multiples of $π$ are one of the simplest examples of pseudo-integrable models with intriguing classical and quantum properties. We perform an extensive numerical study of spectral statistics of eight quantized rational triangles, six belonging to the family of right-angled Veech triangles and two obtuse rational triangles. Large spectral samples of up to one million energy levels were calculated for each triangle which permits to determine their spectral statistics with great accuracy. It is demonstrated that they are of the intermediate type, sharing some features with chaotic systems, like level repulsion and some with integrable systems, like exponential tails of the level spacing distributions. Another distinctive feature of intermediate spectral statistics is a finite value of the level compressibility. The short range statistics such as the level spacing distributions, and long-range statistics such as the number variance and spectral form factors were analyzed in detail. An excellent agreement between the numerical data and the model of gamma distributions is revealed.

nlin.CD↗

arXiv2023

Spectral form factors and dynamical localization

Quantum dynamical localization occurs when quantum interference stops the diffusion of wave packets in momentum space. The expectation is that dynamical localization will occur when the typical transport time of the momentum diffusion is greater than the Heisenberg time. The transport time is typically computed from the corresponding classical dynamics. In this paper, we present an alternative approach based purely on the study of spectral fluctuations of the quantum system. The information about the transport times is encoded in the spectral form factor, which is the Fourier transform of the two-point spectral autocorrelation function. We compute large samples of the energy spectra (of the order of $10^6$ levels) and spectral form factors of 22 stadium billiards with parameter values across the transition between the localized and extended eigenstate regimes. The transport time is obtained from the point when the spectral form factor transitions from the non-universal to the universal regime predicted by random matrix theory. We study the dependence of the transport time on the parameter value and show the level repulsion exponents, that are known to be a good measure of dynamical localization, depend linearly on the transport times obtained in this way.

nlin.CD↗

arXiv2022

Phenomenology of quantum eigenstates in mixed-type systems: lemon billiards with complex phase space structure

The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between their centers, as introduced by Heller and Tomsovic in Phys. Today 46 38 (1993). We study two classical and quantum lemon billiards, for the cases B = 0.1953, 0.083, which are mixed-type billiards with complex structure of phase space, without significant stickiness regions. A preliminary study of their spectra was published recently (Physics 1 1-14 (2021)). We calculate a great number $10^6$ of consecutive eigenstates and their Poincaré-Husimi (PH) functions, and analyze their localization properties by studying the entropy localization measure and the normalized inverse participation ratio. We also introduce an overlap index which measures the degree of the overlap of PH functions with classically regular and chaotic regions. We observe the existence of regular states associated with invariant tori and chaotic states associated with the classically chaotic regions, and also the mixed-type states. We show that in accordance with the Berry-Robnik picture and the principle of uniform semiclassical condensation of PH functions the relative fraction of mixed-type states decreases as a power law with increasing energy, thus in the strict semiclassical limit leaving only purely regular and chaotic states. Our approach offers a general very good phenomenological overview of the structural and localization properties of PH functions in quantum mixed-type Hamiltonian systems.

nlin.CD↗

arXiv2022

Quantum chaos in triangular billiards

We present an extensive numerical study of spectral statistics and eigenfunctions of quantized triangular billiards. We compute two million consecutive eigenvalues for six representative cases of triangular billiards, three with generic angles with irrational ratios with $π$, whose classical dynamics is presumably mixing, and three with exactly one angle rational with $π$, which are presumably only weakly mixing or even only non-ergodic in case of right-triangles. We find excellent agreement of short and long range spectral statistics with the Gaussian orthogonal ensemble of random matrix theory for the most irrational generic triangle, while the other cases show small but significant deviations which are attributed either to scarring or super-scarring mechanism. This result, which extends the quantum chaos conjecture to systems with dynamical mixing in the absence of hard (Lyapunov) chaos, has been corroborated by analysing distributions of phase-space localisation measures of eigenstates and inspecting the structure of characteristic typical and atypical eigenfunctions.

nlin.CD↗

arXiv2021

Statistical properties of the localization measure of chaotic eigenstates and the spectral statistics in a mixed-type billiard

We study the quantum localization in the chaotic eigenstates of a billiard with mixed-type phase space, 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. In quantum systems with discrete energy spectrum the Heisenberg time $t_H =2π\hbar/ΔE$, where $ΔE$ is the mean level spacing, is an important time scale.The classical transport time scale $t_T$ in relation to the Heisenberg time scale $t_H$ (their ratio is the parameter $α=t_H/t_T$) 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 to find successive levels on a distance $S$ goes like $\propto S^β$ for small $S$, where $0\leqβ\leq1$, 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 $\langle A \rangle$ 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 $α\rightarrow \infty$ well described by the beta distribution. Like in other systems, $β$ goes from $0$ to $1$ when $α$ goes from $0$ to $\infty$. $β$ is a function of $\langle A \rangle$, similar to the quantum kicked rotator and the stadium billiard.

quant-ph↗

arXiv2021

The level repulsion exponent of localized chaotic eigenstates as a function of the classical transport time scales in the stadium billiard

We study the aspects of quantum localization in the stadium billiard, which is a classically chaotic ergodic system, but in the regime of slightly distorted circle billiard the diffusion in the momentum space is very slow. In quantum systems with discrete energy spectrum the Heisenberg time $t_H =2π\hbar/ΔE$, where $ΔE$ is the mean level spacing (inverse energy level density), is an important time scale. The classical transport time scale $t_T$ (diffusion time) in relation to the Heisenberg time scale $t_H$ (their ratio is the parameter $α=t_H/t_T$) determines the degree of localization of the chaotic eigenstates, whose measure $A$ is based on the information entropy. 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 $\propto S^β$ for small $S$, where $0\leqβ\leq1$, and $β=1$ corresponds to completely extended states. We show that the level repulsion exponent $β$ is a unique rational function of $α$, and $A$ is a unique rational function of $α$. $β$ goes from $0$ to $1$ when $α$ goes from $0$ to $\infty$. Also, $β$ is a linear function of $A$, which is similar as in the quantum kicked rotator, but different from a mixed type billiard.

nlin.CD↗

arXiv2021

Classical and quantum mixed-type lemon billiards without stickiness

The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance $2B$ between their centers, as introduced by Heller and Tomsovic in Phys. Today {\bf 46} 38 (1993). This paper is a continuation of our recent paper on classical and quantum ergodic lemon billiard ($B=0.5$) with strong stickiness effects published in Phys. Rev. E {\bf 103} 012204 (2021). Here we study the classical and quantum lemon billiards, for the cases $B=0.42,\;0.55,\; 0.6$, which are mixed-type billiards without stickiness regions and thus serve as ideal examples of systems with simple divided phase space. The classical phase portraits show the structure of one large chaotic sea with uniform chaoticity (no stickiness regions) surrounding a large regular island with almost no further substructure, being entirely covered by invariant tori. The boundary between the chaotic sea and the regular island is smooth, except for a few points. The classical transport time is estimated to be very short (just a few collisions), therefore the localization of the chaotic eigenstates is rather weak. The quantum states are characterized by the following {\em universal} properties of mixed-type systems without stickiness in the chaotic regions:(i) Using the Poincaré-Husimi (PH) functions the eigenstates are separated to the regular ones and chaotic ones. The regular eigenenergies obey the Poissonian statistics, while the chaotic ones exhibit the Brody distribution with various values of the level repulsion exponent $β$, its value depending on the strength of the localization of the chaotic eigenstates. Consequently, the total spectrum is well described by the Berry-Robnik-Brody (BRB) distribution.

nlin.CD↗

arXiv2021

Effects of stickiness in the classical and quantum ergodic lemon billiard

We study the classical and quantum ergodic lemon billiard introduced by Heller and Tomsovic in Phys. Today 46(7), 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 [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 $μ_1$ 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 $μ_1$ decreases almost monotonically, with increasing energy.

nlin.CD↗

arXiv2021

Aspects of diffusion in the stadium billiard

We perform a detailed numerical study of diffusion in the $\varepsilon$ stadium of Bunimovich, and propose an empirical model of the local and global diffusion for various values of $\varepsilon$ with the following conclusions: (i) the diffusion is normal for all values of $\varepsilon \leq(0.3)$ 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.

nlin.CD↗

arXiv2021

Structure, size, and statistical properties of chaotic components in a mixed-type Hamiltonian system

We perform a detailed study of the chaotic component in mixed-type Hamiltonian systems on the example of a family of billiards [introduced by Robnik in J. Phys. A: Math. Gen. 16, 3971 (1983)]. The phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times. The structure of the chaotic component is discerned from the cells visited by the chaotic orbit. The fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box-counting method. The cell-filling dynamics is compared to a model of uncorrelated motion, the so-called random model [Robnik et al. J. Phys. A: Math. Gen. 30, L803 (1997)], and deviations attributed to sticky objects in the phase space are found. The statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run. The stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times. The recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed. The standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale. Three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter. Lastly, the decay of correlations and the diffusion of momenta is analyzed.

nlin.CD↗

arXiv2021

Stickiness in generic low-dimensional Hamiltonian systems: A recurrence-time statistics approach

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 and 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 timescales 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.

nlin.CD↗