We examine three billiard systems -- the cardioid, diamond (Superman), and Sinai billiards -- all of which are known to be classically chaotic. We compute their classical Lyapunov exponents, and using out-of-time-order correlators (OTOCs) in the semi-classical regime, we also derive their quantum Lyapunov exponents. We observe that the classical and quantum Lyapunov exponents are in agreement, strengthening the role of OTOCs as a diagnostic for quantum chaos in billiard systems. At very low temperatures, the OTOC of the Sinai billiard shows sharp growth, a phenomenon absent in the other two billiards. We identify the source of this anomalous behaviour in the geometry of the ground state wave function of the Sinai billiard, which is more sensitive to the curvature of the billiard compared to the other billiards. We also remark on the late-time behaviour of the OTOCs and how the scrambling/Ehrenfest time is related to the temperature of quantum billiards.
Close to a spacelike singularity, pure gravity and supergravity in four to eleven spacetime dimensions admit a cosmological billiard description based on hyperbolic Kac-Moody groups. We investigate the quantum cosmological billiards of relativistic wavepackets towards the singularity, employing flat and hyperbolic space descriptions for the quantum billiards. We find that the strongly chaotic classical billiard motion of four-dimensional pure gravity corresponds to a spreading wavepacket subject to successive redshifts and tending to zero as the singularity is approached. We discuss the possib…
The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example…
We argue that the random-matrix like energy spectra found in pseudointegrable billiards with pointlike scatterers are related to the quantum violation of scale invariance of classical analogue system. It is shown that the behavior of the running coupling constant explains the key characteristics of the level statistics of pseudointegrable billiards.
We analyse the classical and quantum behaviour of a particle trapped in a diamond shaped billiard. We defined this billiard as a half stadium connected with a triangular billiard. A parameter $ξ$ which gradually change the shape of the billiard from a regular equilateral triangle ($ξ=1$) to a diamond ($ξ=0$) was used to control the transition between the regular and chaotic regimes. The classical behaviour is regular when the control parameter $ξ$ is one; in contrast, the system is chaotic when $ξ\neq 1$ even for values of $ξ$ close to one. The entropy grows fast as $ξ$ is decreased from 1 and…
We illustrate some of the techniques to identify chaos signatures at the quantum level using as a guiding examples some systems where a particle is constrained to move on a radial symmetric, but non planar, surface. In particular, two systems are studied: the case of a cone with an arbitrary contour or dunce hat billiard and the rectangular billiard with an inner Gaussian surface.
A billiard in the form of a stadium with periodically perturbed boundary is considered. Two types of such billiards are studied: stadium with strong chaotic properties and a near-rectangle billiard. Phase portraits of such billiards are investigated. In the phase plane areas corresponding to decrease and increase of the velocity of billiard particles are found. Average velocities of the particle ensemble as functions of the number of collisions are obtained.
For classical billiards we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distinguish chaotic from integrable systems. As examples of 2D chaotic billiards we considered the Bunimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral rigidity we found GOE behaviour consistent with predictions from random matrix theory. We studied transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, wh…
We study the low energy quantum spectra of two-dimensional rectangular billiards with a small but finite-size scatterer inside. We start by examining the spectral properties of billiards with a single pointlike scatterer. The problem is formulated in terms of self-adjoint extension theory of functional analysis. The condition for the appearance of so-called wave chaos is clarified. We then relate the pointlike scatterer to a finite-size scatterer through the appropriate truncation of basis. We show that the signature of wave chaos in low energy states is most prominent when the scatterer is we…
Using fractal analysis, we investigate how the size of openings affects the chaotic behavior of a classical closed billiard when two openings are made on the boundary of the billiard. This kind of open billiards retains chaotic properties of original closed billiards when openings are small compared to the size of the billiard. We calculate the fractal dimension using an one-dimensional subset of all possible initial conditions that will produce trajectories of a particle injected from an opening for the billiard, and then observe how the opening size can change the classical chaotic propertie…
Classical-quantum correspondence for conservative chaotic Hamiltonians is investigated in terms of the structure of the eigenfunctions and the local density of states, using as a model a 2D rippled billiard in the regime of global chaos. The influence of the observed localized and sparsed states in the quantum-classical correspondence is discussed.
Generic one-parameter billiards are studied both classically and quantally. The classical dynamics for the billiards makes a transition from regular to fully chaotic motion through intermediary soft chaotic system. The energy spectra of the billiards are computed using finite element method which has not been applied to the euclidean billiard. True generic quantum chaotic transitional behavior and its sensitive dependence on classical dynamics are uncovered for the first time. That is, this sensitive dependence of quantum spectral measures on classical dynamics is a genuine manifestation of qu…
We clarify from a general perspective, the condition for the appearance of chaotic energy spectrum in quantum pseudointegrable billiards with a point scatterer inside.
We consider a modification of isospectral cavities whereby the classical dynamics changes from pseudointegrable to chaotic. We construct an example where we can prove that isospectrality is retained. We then demonstrate this explicitly in microwave resonators.
A new type of classical billiard - the Andreev billiard - is investigated using the tangent map technique. Andreev billiards consist of a normal region surrounded by a superconducting region. In contrast with previously studied billiards, Andreev billiards are integrable in zero magnetic field, {\it regardless of their shape}. A magnetic field renders chaotic motion in a generically shaped billiard, which is demonstrated for the Bunimovich stadium by examination of both Poincaré sections and Lyapunov exponents. The issue of the feasibility of certain experimental realizations is addressed.
We construct an autonomous chaotic Hamiltonian ratchet as a channel billiard subdivided by equidistant walls attached perpendicularly to one side of the channel, leaving an opening on the opposite side. A static homogeneous magnetic field penetrating the billiard breaks time-reversal invariance and renders the classical motion partially chaotic. We show that the classical dynamics exhibits directed transport, owing to the asymmetric distribution of regular regions in phase space. The billiard is quantized by a numerical method based on a finite-element algorithm combined with the Landau gauge …
In generic Hamiltonian systems with a mixed phase space chaotic transport may be directed and ballistic rather than diffusive. We investigate one particular model showing this behaviour, namely a spatially periodic billiard chain in which electrons move under the influence of a perpendicular magnetic field. We analyze the phase-space structure and derive an explicit expression for the chaotic transport velocity. Unlike previous studies of directed chaos our model has a parameter regime in which the dispersion of an ensemble of chaotic trajectories around its moving center of mass is essentiall…
We perform echo spectroscopy on ultra cold atoms in atom optics billiards, to study their quantum dynamics. The detuning of the trapping laser is used to change the ``perturbation'', which causes a decay in the echo coherence. Two different regimes are observed: First, a perturbative regime in which the decay of echo coherence is non-monotonic and partial revivals of coherence are observed. These revivals are more pronounced in traps with mixed dynamics as compared to traps where the dynamics is fully chaotic. Next, for stronger perturbations, the decay becomes monotonic and independent of the…
The probability current statistics of two-dimensional open chaotic ballistic billiards is studied both analytically and numerically. Assuming that the real and imaginary parts of the scattering wave function are both random Gaussian fields, we find a universal distribution function for the probability current. In by-passing we recover previous analytic forms for wave function statistics. The expressions bridge the entire region from GOE to GUE type statistics. Our analytic expressions are verified numerically by explicit quantum-mechanical calculations of transport through a Bunimovich billiar…
We study the quantum behaviour of the stadium billiard. We discuss how the interplay between quantum localization and the rich structure of the classical phase space influences the quantum dynamics. The analysis of this model leads to new insight in the understanding of quantum properties of classically chaotic systems.
The Loschmidt echo (LE) measures the ability of a system to return to the initial state after a forward quantum evolution followed by a backward perturbed one. It has been conjectured that the echo of a classically chaotic system decays exponentially, with a decay rate given by the minimum between the width $Γ$ of the local density of states and the Lyapunov exponent. As the perturbation strength is increased one obtains a cross-over between both regimes. These predictions are based on situations where the Fermi Golden Rule (FGR) is valid. By considering a paradigmatic fully chaotic system, th…
We study classical and quantum dynamics of a particle in a circular billiard with a straight cut. This system can be integrable, nonintegrable with soft chaos, or nonintegrable with hard chaos, as we vary the size of the cut. We use a quantum web to show differences in the quantum manifestations of classical chaos for these three different regimes.
We study analytically and numerically the classical diffusive process which takes place in a chaotic billiard. This allows to estimate the conditions under which the statistical properties of eigenvalues and eigenfunctions can be described by Random Matrix Theory. In particular the phenomenon of quantum dynamical localization should be observable in real experiments.
The quantum dynamics of a chaotic billiard with moving boundary is considered in this work. We found a shape parameter Hamiltonian expansion which enables us to obtain the spectrum of the deformed billiard for deformations so large as the characteristic wave length. Then, for a specified time dependent shape variation, the quantum dynamics of a particle inside the billiard is integrated directly. In particular, the dispersion of the energy is studied in the Bunimovich stadium billiard with oscillating boundary. The results showed that the distribution of energy spreads diffusively for the firs…
We study the quantum mechanics of a billiard (Robnik 1983) in the regime of mixed-type classical phase space (the shape parameter λ=0.15) at very high-lying eigenstates, starting at about 1.000.000th eigenstate and including the consecutive 587654 eigenstates. By calculating the normalized Poincaré Husimi functions of the eigenstates and comparing them with the classical phase space structure, we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates, and the corresponding energies. The chaotic eigenstates appear all…
Dynamical billiards are paradigmatic examples of chaotic Hamiltonian dynamical systems with widespread applications in physics. We study how well their Lyapunov exponent, characterizing the chaotic dynamics, and its dependence on external parameters can be estimated from phase space volume arguments, with emphasis on billiards with mixed regular and chaotic phase spaces. We show that in the very diverse billiards considered here the leading contribution to the Lyapunov exponent is inversely proportional to the chaotic phase space volume, and subsequently discuss the generality of this relation…
We study, by numerical simulations and semi-rigorous arguments, a two-parameter family of convex, two-dimensional billiard tables, generalizing the one-parameter class of oval billiards of Benettin--Strelcyn [Phys. Rev. A 17, 773 (1978)]. We observe interesting dynamical phenomena when the billiard tables are continuously deformed from the integrable circular billiard to different versions of completely-chaotic stadia. In particular, we conjecture that a new class of ergodic billiard tables is obtained in certain regions of the two-dimensional parameter space, when the billiards are close to s…
We study the quantum behaviour of chaotic billiards which exhibit classically diffusive behaviour. In particular we consider the stadium billiard and discuss how the interplay between quantum localization and the rich structure of the classical phase space influences the quantum dynamics. The analysis of this model leads to new insight in the understanding of quantum properties of classically chaotic systems.
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 theo…
We present an experimental setup based on the normal modes of vibrating soap films which shows quantum features of integrable and chaotic billiards. In particular, we obtain the so-called scars -narrow linear regions with high probability along classical periodic orbits- for the classically chaotic billiards. We show that these scars are also visible at low frequencies. Finally, we suggest some applications of our experimental setup in other related two-dimensional wave phenomena.
We study theoretically and numerically the velocity dynamics of fully chaotic time-dependent shape-preserving billiards. The average velocity of an ensemble of initial conditions generally asymptotically follows the power law $v = n^β$ with respect to the number of collisions n. If the shape of the chaotic time-dependent billiard is not preserved it is well known that the acceleration exponent is $β= 1/2$. We show, on the other hand, that if the shape of the billiard is preserved then the only possible values of $β$ are $1/4$, $1/6$ and in a special case $0$. We show that the value of $β$ is d…
We investigate the possibility of quantum (or wave) chaos for the Bogoliubov excitations of a Bose-Einstein condensate in billiards. Because of the mean field interaction in the condensate, the Bogoliubov excitations are very different from the single particle excitations in a non-interacting system. Nevertheless, we predict that the statistical distribution of level spacings is unchanged by mapping the non-Hermitian Bogoliubov operator to a real symmetric matrix. We numerically test our prediction by using a phase shift method for calculating the excitation energies.
The coupling of orbital and spin degrees of freedom is the source of many interesting phenomena. Here, we study the electron dynamics in a quantum billiard --a mesoscopic rectangular quantum dot-- with spin-orbit coupling driven by a periodic electric field. We find that both the spatial and temporal profiles of the observables demonstrate the transition to chaotic dynamics with qualitative modifications of the power spectra and patterns of probability and spin density. The time dependence of the wavefunctions and spin density distributions indicates spin-charge separation {seen in the decay o…
We investigate the effective field theory of a quantum chaotic billiard from a new perspective of quantum anomalies, which result from the absence of continuous spectral symmetry in quantized systems. It is shown that commutators of composite operators on the energy shell acquire anomalous part. The presence of the anomaly allows one to introduce effective dual fields as phase variables without any additional coarse-graining nor ensemble averaging in a ballistic system. The spectral Husimi function plays a role as the corresponding amplitude.
We introduce the spherical wedge billiard, a dynamical system consisting of a particle moving along a geodesic on a closed non-Euclidean surface of a spherical wedge. We derive the analytic form of the corresponding Poincaré map and find very complex dynamics, ranging from completely chaotic to very regular, exhibiting fractal features. Further, we show that upon changing the billiard parameter, the fixed points of the Poincaré map merge in complex ways, which has origin in the spherical aberration of the billiard mapping. We also analyze in detail the regular regime when phase space diagram i…
We present the expanded boundary integral method for solving the planar Helmholtz problem, which combines the ideas of the boundary integral method and the scaling method and is applicable to arbitrary shapes. We apply the method to a chaotic billiard with unidirectional transport, where we demonstrate existence of doublets of chaotic eigenstates, which are quasi-degenerate due to time-reversal symmetry, and a very particular level spacing distribution that attains a chaotic Shnirelman peak at short energy ranges and exhibits GUE-like statistics for large energy ranges. We show that, as a cons…
The connection of a superconductor to a chaotic ballistic quantum dot leads to interesting phenomena, most notably the appearance of a hard gap in its excitation spectrum. Here we treat such an Andreev billiard semiclassically where the density of states is expressed in terms of the classical trajectories of electrons (and holes) that leave and return to the superconductor. We show how classical orbit correlations lead to the formation of the hard gap, as predicted by random matrix theory in the limit of negligible Ehrenfest time $\tE$, and how the influence of a finite $\tE$ causes the gap to…
Chaos, namely exponential sensitivity to initial conditions, is generally considered a nuisance, inasmuch as it prevents long-term predictions in physical systems. Here, we present an easily accessible approach to undo deterministic chaos and tailor ray trajectories in arbitrary two-dimensional optical billiards, by introducing spatially varying refractive index therein. A new refractive index landscape is obtained by a conformal mapping, which makes the trajectories of the chaotic billiard fully predictable and the billiard fully integrable. Moreover, trajectory rectification can be pushed a …
Streamlines and distributions of nodal points are used as signatures of chaos in coherent electron transport through three types of billiards, Sinai, Bunimovich and rectangular. Numerical averaged distribution functions of nearest distances between nodal points are presented. We find the same form for the Sinai and Bunimovich billiards and suggest that there is a universal form that can be used as a signature of quantum chaos for electron transport in open billiards. The universal distribution function is found to be insensitive to the way avaraging is performed (over positions of leads, over …
Chaotic orbits of mushroom billiards display intermittent behaviors. We investigate statistical properties of this system by constructing an infinite partition on the chaotic part of a Poincaré surface which illustrates details of chaotic dynamics. Each piece of the infinite partition has an unique escape time from the half disk region, and from this result it is shown that, for fixed values of the system parameters, the escape time distribution obeys power law $1/t_{\rm esc}^3$.
We report on the experimental investigation of the properties of the eigenvalues and wavefunctions and the fluctuation properties of the scattering matrix of closed and open billiards, respectively, of which the classical dynamics undergoes a transition from integrable via almost integrable to fully chaotic. To realize such a system we chose a billiard with a 60 degree sector shape of which the classical dynamics is integrable, and introduced circular scatterers of varying number, size and position. The spectral properties of generic quantum systems of which the classical counterpart is either…
In this paper we have tested several general numerical methods in solving the quantum billiards, such as the boundary integral method (BIM) and the plane wave decomposition method (PWDM). We performed extensive numerical investigations of these two methods in a variety of quantum billiards: integrable systens (circles, rectangles, and segments of circular annulus), Kolmogorov-Armold-Moser (KAM) systems (Robnik billiards), and fully chaotic systems (ergodic, such as Bunimovich stadium, Sinai billiard and cardiod billiard). We have analyzed the scaling of the average absolute value of the system…
We investigate the properties of eigenstates and local density of states (LDOS) for a periodic 2D rippled billiard, focusing on their quantum-classical correspondence in energy representation. To construct the classical counterparts of LDOS and the structure of eigenstates (SES), the effects of the boundary are first incorporated (via a canonical transformation) into an effective potential, rendering the one-particle motion in the 2D rippled billiard equivalent to that of two-interacting particles in 1D geometry. We show that classical counterparts of SES and LDOS in the case of strong chaotic…
Based on very accurate measurements performed on a superconducting microwave resonator shaped like a desymmetrized three-dimensional (3D) Sinai billiard, we investigate for the first time spectral properties of the vectorial Helmholtz, i.e. non-quantum wave equation for a classically totally chaotic and theoretically precisely studied system. We are thereby able to generalize some aspects of quantum chaos and present some results which are consequences of the polarization features of the electromagnetic waves.
It has recently been shown that the computing abilities of Boltzmann machines, or Ising spin-glass models, can be implemented by chaotic billiard dynamics without any use of random numbers. In this paper, we further numerically investigate the capabilities of the chaotic billiard dynamics as a deterministic alternative to random Monte Carlo methods by applying it to classical spin models in statistical physics. First, we verify that the billiard dynamics can yield samples that converge to the true distribution of the Ising model on a small lattice, and we show that it appears to have the same …
We report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two correspondi…
Experiments with superconducting microwave cavities have been performed in our laboratory for more than two decades. The purpose of the present article is to recapitulate some of the highlights achieved. We briefly review (i) results obtained with flat, cylindrical microwave resonators, so-called microwave billiards, concerning the universal fluctuation properties of the eigenvalues of classically chaotic systems with no, a threefold and a broken symmetry; (ii) summarize our findings concerning the wave-dynamical chaos in three-dimensional microwave cavities; (iii) present a new approach for t…
We study the quantum interference effect for the single ballistic Aharonov-Bohm billiard in the presence of a weak magnetic field B. The diagonal part of the wave-number averaged reflection coefficient $δ{\cal R}_D$ is calculated by use of semi-classical scattering theory. In addition to the appearance of "h/2e oscillation" that are caused by interference between time-reversed coherent backscattering classical trajectories, B in the conducting region leads to negative magnetoresistance and dampening of the h/2e oscillation amplitude. The B dependence of the results reflects the underlying clas…
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 $…
The boundary integral method (BIM) is a formulation of Helmholtz equation in the form of an integral equation suitable for numerical discretization to solve the quantum billiard. This paper is an extensive numerical survey of BIM in a variety of quantum billiards, integrable (circle, rectangle), KAM systems (Robnik billiard) and fully chaotic (ergodic, such as stadium, Sinai billiard and cardioid billiard). On the theoretical side we point out some serious flaws in the derivation of BIM in the literature and show how the final formula (which nevertheless was correct) should be derived in a sou…
We describe conditions under which higher-dimensional billiard models in bounded, convex regions are fully chaotic, generalizing the Bunimovich stadium to dimensions above two. An example is a three-dimensional stadium bounded by a cylinder and several planes; the combination of these elements may give rise to defocusing, allowing large chaotic regions in phase space. By studying families of marginally-stable periodic orbits that populate the residual part of phase space, we identify conditions under which a nonlinear instability mechanism arises in their vicinity. For particular geometries, t…
