arXiv subjects

Marko Robnik

Publications and source records attributed to Marko Robnik.

At least 19 recordsLinked to original sources

arXiv2026

Mixed eigenstates in spin-boson systems with one-photon and two-photon interactions

Spin-boson systems have attracted increasing attention as accessible experimental platforms and for their potential applications in designing quantum technologies. One characteristic of these systems is the transition from regular to completely chaotic behavior when certain control parameters are varied. However, the characterization of their mixed phase space has not been thoroughly explored. In this work, we investigate the properties of mixed eigenstates in spin-boson systems, comparing one-photon interactions with two-photon interactions. We propose a generalized definition of the phase-space overlap index to identify genuine mixed eigenstates. Our study highlights the fundamental differences that arise when two-photon processes are considered compared to one-photon processes and provides complementary evidence supporting the validity of the principle of uniform semiclassical condensation (PUSC) of quasiprobability functions in spin-boson systems.

quant-ph↗

arXiv2026

Spectral statistics and localization properties of a $C_3$-symmetric billiard

We revisit the spectral statistics of the C$_3$--symmetric billiard introduced by Dembowski [Phys. Rev. E, R4516 (2000)], which exhibits both GOE and GUE statistics depending on the symmetry block. Using high--precision Beyn's contour--integral method for the nonlinear Fredholm eigenvalue problem with built-in separation of irreducible subspaces, we compute 2.8x10$^5$ eigenvalues in each symmetry subspace, enabling statistically meaningful comparisons with random matrix theory. The improved spectra reveal clear GOE--GUE correspondence and resolve previously observed deviations in long--range spectral correlations. Furthermore, we analyze phase--space eigenstate localization through the distribution of entropy localization measures, which, for chaotic states follow a Beta distribution whose standard deviation decays as a power--law with energy, consistent with the onset of quantum ergodicity as described by Schnirelman's theorem.

quant-ph↗

arXiv2025

Quantum chaos and semiclassical behavior in mushroom billiards II: Structure of quantum eigenstates and their phase space localization properties

We investigate eigenstate localization in the phase space of the Bunimovich mushroom billiard, a paradigmatic mixed-phase-space system whose piecewise-$C^{1}$ boundary yields a single clean separatrix between one regular and one chaotic region. By varying the stem half-width $w$, we continuously change the strength and extent of bouncing-ball stickiness in the stem, which for narrow stems gives rise to phase space localization of chaotic eigenstates. Using the Poincar\'e-Husimi (PH) representation of eigenstates we quantify localization via information entropies and inverse participation ratios of PH functions. For sufficiently wide stems the distribution of entropy localization measures converges to a two-parameter beta distribution, while entropy localization measures and inverse participation ratios across the chaotic ensemble exhibit an approximately linear relationship. Finally, the fraction of mixed (neither purely regular nor fully chaotic) eigenstates decays as a power-law in the effective semiclassical parameter, in precise agreement with the Principle of Uniform Semiclassical Condensation of Wigner functions (PUSC).

nlin.CD↗

arXiv2025

Correspondence principle, dissipation, and Ginibre ensemble

The correspondence between quantum and classical behavior has been essential since the advent of quantum mechanics. This principle serves as a cornerstone for understanding quantum chaos, which has garnered increased attention due to its strong impact in various theoretical and experimental fields. When dissipation is considered, quantum chaos takes concepts from isolated quantum chaos to link classical chaotic motion with spectral correlations of Ginibre ensembles. This correspondence was first identified in periodically kicked systems with damping, but it has been shown to break down in dissipative atom-photon systems [Phys. Rev. Lett. 133, 240404 (2024)]. In this contribution, we revisit the original kicked model and perform a systematic exploration across a broad parameter space, reaching a genuine semiclassical limit. Our results demonstrate that the correspondence principle, as defined through this spectral connection, fails even in this prototypical system. These findings provide conclusive evidence that Ginibre spectral correlations are neither a robust nor a universal diagnostic of dissipative quantum chaos.

quant-ph↗

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 Poincar\'e-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 Poincar\'e-Husimi functions.

quant-ph↗

arXiv2025

Characterizing the mixed eigenstates in kicked top model through the out-of-time-order correlator

Generic systems are associated with a mixed classical phase space.The question of the properties of the eigenstates for these systems remains less known, although it plays a key role for understanding several important quantum phenomena such as thermalization, scarring, tunneling, and (de-)localization. In this work, by employing the kicked top model, we perform a detailed investigation of the dynamical signatures of the mixed eigenstates via the out-of-time-order correlator (OTOC). We show how the types of the eigenstates get reflected in the short- and long-time behaviors of the OTOC and conjecture that the dynamics of the OTOC can be used as an indicator of the mixed eigenstates. Our findings further confirm the usefulness of the OTOC for studying quantum complex systems and also provide more insights into the characters the mixed eigenstates.

quant-ph↗

arXiv2024

Finite time quantum-classical correspondence in quantum chaotic systems

Although the importance of the quantum-classical correspondence has been recognized in numerous studies of quantum chaos, whether it still holds for finite time dynamics remains less known. We address this question in this work by performing a detailed analysis of how the quantum chaotic measure relates to the chaoticity of the finite time classical trajectories. A good correspondence between them has been revealed in both time dependent and many-body systems. In particular, we show that the dependence of the quantum chaotic measure on the chaoticity of finite time trajectories can be well captured by a function that is independent of the system. This strongly implies the universal validity of the finite time quantum-classical correspondence. Our findings provide a deeper understanding of the quantum-classical correspondence and highlight the role of time for studying quantum ergodicity.

quant-ph↗

arXiv2024

Further results on the power-law decay of the fraction of the mixed eigenstates in kicked-top model with mixed-type classical phase space

By using the Krylov subspace technique to generate the spin coherent states in kicked top model, a prototype model for studying quantum chaos, the accessible system size for studying the Husimi functions of eigenstates can be much larger than that reported in the literature and our previous study Phys. Rev. E 108, 054217 (2023) [arXiv:2308.04824]. In the fully chaotic kicked top, we find that the mean Wehrl entropy localization measure approaches the prediction given by the Circular Unitary Ensemble. In the mixed-type case, we identify mixed eigenstates by the overlap of the Husimi function with regular and chaotic regions in classical compact phase space. Numerically, we show that the fraction of mixed eigenstates scales as $j^{-\zeta}$, a power-law decay as the system size $j$ increases, across nearly two orders of magnitude. This provides supporting evidence for the principle of uniform semiclassical condensation of Husimi functions and the Berry-Robnik picture in the semiclassical limit.

quant-ph↗

arXiv2024

Chaos and quantization of the three-particle generic Fermi-Pasta-Ulam-Tsingou model II: phenomenology of quantum eigenstates

We undertake a thorough investigation into the phenomenology of quantum eigenstates, in the three-particle FPUT model. Employing different Husimi functions, our study focuses on both the $\alpha$-type, which is canonically equivalent to the celebrated H\'enon-Heiles Hamiltonian, a nonintegrable and mixed-type system, and the general case at the saddle energy where the system is fully chaotic. Based on Husimi quantum surface of sections (QSOS), we find that in the mixed-type system, the fraction of mixed eigenstates in an energy shell $[E-\delta E/2, E+\delta E/2]$ with $\delta E\ll E$ shows a power-law decay with respect to the decreasing Planck constant $\hbar$. Defining the localization measures in terms of the R\'enyi-Wehrl entropy, in both the mixed-type and fully chaotic systems, we find a better fit with the beta distribution and a lesser degree of localization, in the distribution of localization measures of chaotic eigenstates, as the controlling ratio $\alpha_\mathcal{L} = t_H /t_T$ between the Heisenberg time $t_H$ and the classical transport time $t_T$ increases. This transition with respect to $\alpha_\mathcal{L}$ and the power-law decay of the mixed states, together provide supporting evidence for the principle of uniform semiclassical condensation (PUSC) in the semiclassical limit. Moreover, we find that in the general case which is fully chaotic, the maximally localized state, is influenced by the stable and unstable manifold of the saddles (hyperbolic fixed points), while the maximally extended state notably avoids these points, extending across the remaining space, complementing each other.

quant-ph↗

arXiv2024

Chaos and quantization of the three-particle generic Fermi-Pasta-Ulam-Tsingou model I: Density of states and spectral statistics

We study the mixed-type classical dynamics of the three-particle Fermi-Pasta-Ulam-Tsingou (FPUT) model in relationship with its quantum counterpart, and present new results on aspects of quantum chaos in this system. First we derive for the general N-particle FPUT system the transformation to the normal mode representation. Then we specialize to the three-particle FPUT case, and derive analytically the semiclassical energy density of states, and its derivatives in which different singularies are determined, using the Thomas-Fermi rule. The result perfectly agrees with the numerical energy density from the Krylov subspace method, as well as with the energy density obtained by the method of quantum typicality. Here, in paper I, we concentrate on the energy level statistics (level spacing and spacing ratios), in all classical dynamical regimes of interest: the almost entirely regular, the entirely chaotic, and the mixed-type regimes. We clearly confirm, correspondingly, the Poissonian statistics, the GOE statistics, and the Berry-Robnik-Brody (BRB) statistics in the mixed-type regime. It is found that the BRB level spacing distribution perfectly fits the numerical data. The extracted quantum Berry-Robnik parameter is found to agree with the classical value within better than one percent. We discuss the role of localization of chaotic eigenstates, and its appearances, in relation to the classical phase space structure (Poincar\'e and SALI plots), whose details will be presented in paper II, where the structure and the statistical properties of the Husimi functions in the quantum phase space will be studied.

cond-mat.stat-mech↗

arXiv2023

Mixed eigenstates in the Dicke model: Statistics and power-law decay of the relative proportion in the semiclassical limit

How the mixed eigenstates vary with approaching the semiclassical limit in mixed-type many-body quantum systems is an interesting but still less known question. Here, we address this question in the Dicke model, a celebrated many-body model that has a well defined semiclassical limit and undergoes a transition to chaos in both quantum and classical case. Using the Husimi function, we show that the eigenstates of the Dicke model with mixed-type classical phase space can be classified into different types. To quantitatively characterize the types of eigenstates, we study the phase space overlap index, which is defined in terms of Husimi function. We look at the probability distribution of the phase space overlap index and investigate how it changes with increasing system size, that is, when approaching the semiclassical limit. We show that increasing the system size gives rise to a power-law decay in the behavior of the relative proportion of mixed eigenstates. Our findings shed more light on the properties of eigenstates in mixed-type many-body systems and suggest that the principle of uniform semiclassical condensation of Husimi functions should also be valid for many-body quantum systems.

quant-ph↗

arXiv2023

Power-law decay of the fraction of the mixed eigenstates in kicked top model with mixed-type classical phase space

The properties of mixed eigenstates in a generic quantum system with classical counterpart that has mixed-type phase space, although important to understand several fundamental questions that arise in both theoretical and experimental studies, are still not clear. Here, following a recent work [\v{C}.~Lozej {\it et al}. Phys. Rev. E {\bf 106}, 054203 (2022)], we perform an analysis of the features of mixed eigenstates in a time-dependent Hamiltonian system, the celebrated kicked top model. As a paradigmatic model for studying quantum chaos, kicked top model is known to exhibit both classical and quantum chaos. The types of eigenstates are identified by means of the phase space overlap index, which is defined as the overlap of the Husimi function with regular and chaotic regions in classical phase space. We show that the mixed eigenstates appear due to various tunneling precesses between different phase space structures, while the regular and chaotic eigenstates are, respectively, associated with invariant tori and chaotic component in phase space. We examine how the probability distribution of the phase space overlap index evolves with increasing system size for different kicking strengths. In particular, we find that the relative fraction of mixed states exhibits a power-law decay as the system size increases, indicating that only purely regular and chaotic eigenstates are left in the strict semiclassical limit. We thus provide further verification of the principle of uniform semiclassical condensation of Husimi functions and confirm the correctness of the Berry-Robnik picture.

quant-ph↗

arXiv2023

Statistics of phase space localization measures and quantum chaos in the kicked top model

Quantum chaos plays a significant role in understanding several important questions of recent theoretical and experimental studies. Here, by focusing on the localization properties of eigenstates in phase space (by means of Husimi functions),we explore the characterizations of quantum chaos using the statistics of the localization measures. We consider the paradigmatic kicked top model, which shows a transition to chaos with increasing the kicking strength. We demonstrate that the distributions of the localization measures exhibit a drastic change as the system undergoes the crossover from integrability to chaos. We also show how to identify the signatures of quantum chaos from the central moments of the distributions of localization measures. Moreover, we find that the localization measures in the fully chaotic regime apparently exhibit universally the beta distribution, in agreement with previous studies in the billiard systems and the Dicke model. Our results contribute to a further understanding of quantum chaos and shed light on the usefulness of the statistics of phase space localization measures in diagnosing the presence of quantum chaos, as well as the localization properties of eigenstates in quantum chaotic systems.

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 Poincar\'e-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↗

arXiv2021

Multifractality in quasienergy space of coherent states as a signature of quantum chaos

We present the multifractal analysis of coherent states in kicked top model by expanding them in the basis of Floquet operator eigenstates. We demonstrate the manifestation of phase space structures in the multifractal properties of coherent states. In the classical limit, the classical dynamical map can be constructed, allowing us to explore the corresponding phase space portraits and to calculate Lyapunov exponent. By tuning the kicking strength, the system undergoes a transition from regularity to chaos. We show that the variation of multifractal dimensions of coherent states with kicking strength is able to capture the structural changes of the phase space. The onset of chaos is clearly identified by the phase space averaged multifractal dimensions, which are well described by random matrix theory in strongly chaotic regime. We further investigate the probability distribution of expansion coefficients, and show that the deviation between the numerical results and the prediction of random matrix theory behaves as a reliable detector of quantum chaos.

quant-ph↗

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 $t_H =2\pi \hbar/\Delta E$, where $\Delta E$ 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 $\alpha=t_H/t_T$) 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 $\propto S^\beta$ for small $S$, where $0\leq\beta\leq1$, and $\beta=1$ corresponds to completely extended states. We show that the level repulsion exponent $\beta$ is empirically a rational function of $\alpha$, and the mean $\langle A \rangle$ as a function of $\alpha$ 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 $\alpha\rightarrow \infty$ well described by the beta distribution. Like in other systems, $\beta$ goes from $0$ to $1$ when $\alpha$ goes from $0$ to $\infty$. $\beta$ 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 $t_H =2\pi \hbar/\Delta E$, where $\Delta E$ 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 $\alpha=t_H/t_T$) 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 $\propto S^\beta$ for small $S$, where $0\leq\beta\leq1$, and $\beta=1$ corresponds to completely extended states. We show that the level repulsion exponent $\beta$ is a unique rational function of $\alpha$, and $A$ is a unique rational function of $\alpha$. $\beta$ goes from $0$ to $1$ when $\alpha$ goes from $0$ to $\infty$. Also, $\beta$ 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 Poincar\'e-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 $\beta$, 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↗