Triggered by a controversy surrounding a universal behaviour of the power spectrum in quantum systems exhibiting regular classical dynamics, we focus on a model of random diagonal matrices (RDM), often associated with the Poisson spectral universality class, and examine how the power spectrum and the form factor get affected by two-sided truncations of RDM spectra. Having developed a nonperturbative description of both statistics, we perform their detailed asymptotic analysis to demonstrate explicitly how a traditional assumption (lying at the heart of the controversy) -- that the power spectrum is merely determined by the spectral form factor -- breaks down for truncated spectra. This observation has important consequences as we further argue that bounded quantum systems with integrable classical dynamics are described by heavily truncated rather than complete RDM spectra. High-precision numerical simulations of semicircular and irrational rectangular billiards lend independent support to these conclusions.
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…
The pseudo-integrable barrier billiard invented by Hannay and McCraw [J. Phys. A 23, 887 (1990)] -- rectangular billiard with line-segment barrier placed on a symmetry axis -- is generalized. It is proven that the flow on invariant surfaces of genus two exhibits a singular continuous spectral component.
We investigate the semiclassical energy spectrum of quantum elliptic billiard. The nearest neighbor spacing distribution, level number variance and spectral rigidity support the notion that the elliptic billiard is a generic integrable system. However, second order statistics exhibit a novel property of long-range oscillations. Classical simulation shows that all the periodic orbits except two are not isolated. In Fourier analysis of the spectrum, all the peaks correspond to periodic orbits. The two isolated periodic orbits have small contribution to the fluctuation of level density, while non…
A crucial result in quantum chaos, which has been established for a long time, is that the spectral properties of classically integrable systems generically are described by Poisson statistics whereas those of time-reversal symmetric, classically chaotic systems coincide with those of random matrices from the Gaussian orthogonal ensemble (GOE). Does this result hold for two-dimensional Dirac material systems? To address this fundamen- tal question, we investigate the spectral properties in a representative class of graphene billiards with shapes of classically integrable circular-sector billia…
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…
The paper establishes the property of splittability of billiard boundary sequences in n dimensional cube into subsequences of fractional parts. This reveals a new property of integrable and weak perturbated Hamilton systems: under a simple assumption, the boundary motion of elliptic orbits on stable KAM tori, if considering in cartesian coordinates, can be splitted into a countable set of discrete rotations. The rate of the split process, expressed in terms of some exceptional sets density, in dependence of number-theoretical characteristics of the orbits frequencies, is also examined.
Semiclassical wave functions in billiards based on the Maslov-Fedoriuk approach are constructed. They are defined on classical constructions called skeletons which are the billiards generalization of the Arnold tori. Skeletons in the rational polygon billiards considered in the phase space can be closed with a definite genus or can be open being a cylinder-like or Moebius-like bands. The skeleton formulation is applied to calculate semiclassical wave functions and the corresponding energy spectra for the integrable and pseudointegrable billiards as well as in the limiting forms in some cases o…
Rounding border effects at the escape point of open integrable billiards are analyzed via the escape times statistics and emission angles. The model is the rectangular billiard and the shape of the escape point is assumed to have a semicircular form. Stickiness and self-similar structures for the escape times and emission angles are generated inside "backgammon" like stripes of initial conditions. These stripes are born at the boundary between two different emission angles but same escape times. As the rounding effects increase, backgammon stripes start to overlap and the escape times statisti…
Recently semiclassical approximations have been successfully applied to study the effect of a superconducting lead on the density of states and conductance in ballistic billiards. However, the summation over classical trajectories involved in such theories was carried out using the intuitive picture of Andreev reflection rather than the semiclassical reasoning. We propose a method to calculate the semiclassical sums which allows us to go beyond the diagonal approximation in these problems. In particular, we address the question of whether the off-diagonal corrections could explain the gap in t…
We study previously un-researched second order statistics - correlation function of spectral staircase and global level number variance - in generic integrable systems with no extra degeneracies. We show that the global level number variance oscillates persistently around the saturation spectral rigidity. Unlike other second order statistics - including correlation function of spectral staircase - which are calculated over energy scales much smaller than the running spectral energy, these oscillations cannot be explained within the diagonal approximation framework of the periodic orbit theory.…
We design a computational experiment in which a quantum particle tunnels into a billiard of variable shape and scatters out of it through a double-slit opening on the billiard's base. The interference patterns produced by the scattered probability currents for a range of energies are investigated in relation to the billiard's geometry which is connected to its classical integrability. Four billiards with hierarchical integrability levels are considered: integrable, pseudo-integrable, weak-mixing and strongly chaotic. In agreement with the earlier result by Casati and Prosen [1], we find the bi…
We study the effect on the density of states in mesoscopic ballistic billiards to which a superconducting lead is attached. The expression for the density of states is derived in the semiclassical S-matrix formalism shedding insight into the origin of the differences between the semiclassical theory and the corresponding result derived from random matrix models. Applications to a square billiard geometry and billiards with boundary roughness are discussed. The saturation of the quasiparticle excitation spectrum is related to the classical dynamics of the billiard. The influence of weak magneti…
Morphological image analysis is applied to the time evolution of the probability distribution of a quantum particle moving in two and three-dimensional billiards. It is shown that the time-averaged Euler characteristic of the probability density provides a well defined quantity to distinguish between classically integrable and non-integrable billiards. In three dimensions the time-averaged mean breadth of the probability density may also be used for this purpose.
Particle motion in a smoothly oscillating non-integrable billiard is known to result in unbounded energy growth. Though the asymptotic energy growth rate of an ensemble of particles in an oscillating chaotic billiard is known to be quadratic, there are no estimates available for smoothly oscillating pseudo-integrable billiards. The energy growth rate in such systems is so slow that it is very hard to predict the asymptotic rates from finite time computations. In this paper, a method is proposed to estimate the asymptotic energy growth rate in a system by using short-time data. The idea is appl…
This article presents a new method to calculate eigenvalues of right triangle billiards. Its efficiency is comparable to the boundary integral method and more recently developed variants. Its simplicity and explicitness however allow new insight into the statistical properties of the spectra. We analyse numerically the correlations in level sequences at high level numbers (>10^5) for several examples of right triangle billiards. We find that the strength of the correlations is closely related to the genus of the invariant surface of the classical billiard flow. Surprisingly, the genus plays an…
The barrier billiard is the simplest example of pseudo-integrable models with interesting and intricate classical and quantum properties. Using the Wiener-Hopf method it is demonstrated that quantum mechanics of a rectangular billiard with a barrier in the centre can be reduced to the investigation of a certain unitary matrix. Under heuristic assumptions this matrix is substituted by a special low-complexity random unitary matrix of independent interest. The main results of the paper are (i) spectral statistics of such billiards is insensitive to the barrier height and (ii) it is well describe…
Experimental study of coupled quantum billiards with integrable and chaotic classical dynamics and test of a special Rosenzweig-Porter model
Lanzhou University（兰州大学）；Institute for Basic Science（韩国基础科学研究院）；Qilu Normal University（齐鲁工业大学）；University of Electronic Science and Technology of China（电子科技大学）；Korea University of Science and Technology (UST)（韩国科学技术院）
The structure of the semiclassical trace formula can be used to construct a quasi-classical evolution operator whose spectrum has a one-to-one correspondence with the semiclassical quantum spectrum. We illustrate this for marginally unstable integrable and non-integrable billiards and demonstrate its utility by quantizing them using arbitrary non-periodic trajectories.
Some dynamical properties for a dissipative time-dependent oval-shaped billiard are studied. The system is described in terms of a four-dimensional nonlinear mapping. Dissipation is introduced via inelastic collisions of the particle with the boundary, thus implying that the particle has a fractional loss of energy upon collision. The dissipation causes profound modifications in the dynamics of the particle as well as in the phase space of the non dissipative system. In particular, inelastic collisions can be assumed as an efficient mechanism to suppress Fermi acceleration of the particle. The…
We have computed numerically the Casimir force between two identical pistons inside a very long cylinder, considering different shapes for the pistons. The pistons can be considered as quantum billiards, whose spectrum determines the vacuum force. The smooth part of the spectrum fixes the force at short distances, and depends only on geometric quantities like the area or perimeter of the piston. However, correcting terms to the force, coming from the oscillating part of the spectrum which is related to the classical dynamics of the billiard, are qualitatively different for classically integrab…
The Robnik billiard is investigated in detail both classically and quantally in the transition range from integrable to almost chaotic system. We find out that a remarkable correspondence between characteristic features of classical dynamics, especially topological structure of integrable regions in the Poincaré surface of section, and the statistics of energy level spacings appears with a system parameter $λ$ being varied. It is shown that the variance of the level spacing distribution changes its behavior at every particular values of $λ$ in such a way that classical dynamics changes its top…
Friedel oscillations of electron densities near step edges have an analog in microwave billiards. A random plane wave model, normally only appropriate for the eigenfunctions of a purely chaotic system, can be applied and is tested for non-purely-chaotic dynamical systems with measurements on pseudo-integrable and mixed dynamics geometries. It is found that the oscillations in the pseudo-integrable microwave cavity matches the random plane-wave modeling. Separating the chaotic from the regular states for the mixed system requires incorporating an appropriate phase space projection into the mode…
The eigenvalues of the Hyperspherical billiard are calculated in the semiclassical approximation. The eigenvalues where this approximation fails are identified and found to be related to caustics that approach the wall of the billiard. The fraction of energy levels for which the semiclassical error is larger than some given value is calculated analytically (and tested numerically) and found to be independent of energy. The implications for other systems, in particular integrable ones, are discussed.
A class of Hamiltonian impact systems exhibiting smooth near integrable behavior is presented. The underlying unperturbed model investigated is an integrable, separable, 2 degrees of freedom mechanical impact system with effectively bounded energy level sets and a single straight wall which preserves the separable structure. Singularities in the system appear either as trajectories with tangent impacts or as singularities in the underlying Hamiltonian structure (e.g. separatrices). It is shown that away from these singularities, a small perturbation from the integrable structure results in smo…
A two-dimensional circular quantum billiard with unusual boundary conditions introduced by Berry and Dennis (\emph{J Phys A} {\bf 41} (2008) 135203) is considered in detail. It is demonstrated that most of its eigenfunctions are strongly localized and the corresponding eigenvalues are close to eigenvalues of the circular billiard with Neumann boundary conditions. Deviations from strong localization are also discussed. These results agree well with numerical calculations.
We show that wave functions in planar rational polygonal billiards (all angles rationally related to Pi) can be expanded in a basis of quasi-stationary and spatially regular states. Unlike the energy eigenstates, these states are directly related to the classical invariant surfaces in the semiclassical limit. This is illustrated for the barrier billiard. We expect that these states are also present in integrable billiards with point scatterers or magnetic flux lines.
Diffractive systems are quantum-mechanical models with point-like singularities where usual semiclassical approximation breaks down. An overview of recent investigations of such systems is presented. The following examples are considered in details: (i) billiards (both integrable and chaotic) with small-size scatterers, (ii) pseudo-integrable polygonal plane billiards, and (iii) billiards with the Bohr-Aharonov flux lines. First, the diffractive trace formulas are discussed with particular emphasis on models where the diffractive coefficient diverges in certain directions. Second, it is demons…
An elementary application of Algorithmic Complexity Theory to the polygonal approximations of curved billiards-integrable and chaotic-unveils the equivalence of this problem to the procedure of quantization of classical systems: the scaling relations for the average complexity of symbolic trajectories are formally the same as those governing the semi-classical limit of quantum systems. Two cases-the circle, and the stadium-are examined in detail, and are presented as paradigms.
We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of o…
We propose the method for optical visualization of Bose-Hubbard model with two interacting bosons in the form of two-dimensional (2D) optical lattices consisting of optical waveguides, where the waveguides at the diagonal are characterized by different refractive index than others elsewhere, modeling the boson-boson interaction. We study the light intensity distribution function averaged over direction of propagation for both ordered and disordered cases, exploring sensitivity of the averaged picture with respect to the beam injection position. For our finite systems the resulting patterns rem…
An efficient finite element method (FEM) for calculating eigenvalues and eigenfunctions of quantum billiard systems is presented. We consider the FEM based on triangular $C_1$ continuity quartic interpolation. Various shapes of quantum billiards including an integrable unit circle are treated. The numerical results show that the applied method provides accurate set of eigenvalues exceeding a thousand levels for any shape of quantum billiards on a personal computer. Comparison with the results from the FEM based on well-known $C_0$ continuity quadratic interpolation proves the efficiency of the…
Nonperiodic tunable quantum echoes have been observed in experiments with an open microwave billiard whose geometry under certain conditions provides Fibonacci like sequences of classical delay times. These sequences combined with the reflection at the opening induced by the wave character of the experiment and the size of the opening allow to shape quantum pulses. The pulses are obtained by response of an integrable scattering system.
The properties of energy levels in a family of classically pseudointegrable systems, the barrier billiards, are investigated. An extensive numerical study of nearest-neighbor spacing distributions, next-to-nearest spacing distributions, number variances, spectral form factors, and the level dynamics is carried out. For a special member of the billiard family, the form factor is calculated analytically for small arguments in the diagonal approximation. All results together are consistent with the so-called semi-Poisson statistics.
Using heuristic arguments based on the trace formulas, we analytically calculate the semiclassical two-point correlation form factor for a family of rectangular billiards with a barrier of height irrational with respect to the side of the billiard and located at any rational position p/q from the side. To do this, we first obtain the asymptotic density of lengths for each family of periodic orbits by a Siegel-Veech formula. The result K(0)=1/2+1/q obtained for these pseudo-integrable, non-Veech billiards is different but not far from the value of 1/2 expected for semi-Poisson statistics and fr…
Integrability of a square billiard is spontaneously broken as it rotates about one of its corners. The system becomes quasi-integrable where the invariant tori are broken with respect to a certain parameter, $λ= 2E/ω^{2}$ where E is the energy of the particle inside the billiard and $ω$ is the angular frequency of rotation of billiard. We study the system classically and quantum mechanically in view of obtaining a correspondence in the two descriptions. Classical phase space in Poincaré surface of section shows transition from regular to chaotic motion as the parameter $λ$ is decreased. In the…
The Seba billiard, a rectangular torus with a point scatterer, is a popular model to study the transition between integrability and chaos in quantum systems. Whereas such billiards are classically essentially integrable, they may display features such as quantum ergodicity [KU] which are usually associated with quantum systems whose classical dynamics is chaotic. Seba proposed that the eigenfunctions of toral point scatterers should also satisfy Berry's random wave conjecture, which implies that the semiclassical moments of the eigenfunctions ought to be Gaussian. We prove a conjecture of Keat…
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.
Astute variations in the geometry of mathematical billiard tables have been and continue to be a source of understanding their wide range of dynamical behaviors, from regular to chaotic. Viewing standard specular billiards in the broader setting of no-slip (or rough) collisions, we show that an equally rich spectrum of dynamics can be called forth by varying the mass distribution of the colliding particle. We look at three two-parameter families of billiards varying both the geometry of the table and the particle, including as special cases examples of standard billiards demonstrating dynamics…
We use the semiclassical quantization scheme of Bogomolny to calculate eigenvalues of the Lima\c con quantum billiard corresponding to a conformal map of the circle billiard. We use the entire billiard boundary as the chosen surface of section (SOS) and use a finite approximation for the transfer operator in coordinate space. Computation of the eigenvalues of this matrix combined with a quantization condition, determines a set of semiclassical eigenvalues which are compared with those obtained by solving the Schrödinger equation. The classical dynamics of this billiard system undergoes a smoot…
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.
Dynamical properties of the elliptical stadium billiard, which is a generalization of the stadium billiard and a special case of the recently introduced mushroom billiards, are investigated analytically and numerically. In dependence on two shape parameters delta and gamma, this system reveals a rich interplay of integrable, mixed and fully chaotic behavior. Poincare sections, the box counting method and the stability analysis determine the structure of the parameter space and the borders between regions with different behavior. Results confirm the existence of a large fully chaotic region sur…
We study a two-particle circular billiard containing two finite-size circular particles that collide elastically with the billiard boundary and with each other. Such a two-particle circular billiard provides a clean example of an "intermittent" system. This billiard system behaves chaotically, but the time scale on which chaos manifests can become arbitrarily long as the sizes of the confined particles become smaller. The finite-time dynamics of this system depends on the relative frequencies of (chaotic) particle-particle collisions versus (integrable) particle-boundary collisions, and invest…
The persistent current of ballistic chaotic billiards is considered with the help of the Gutzwiller trace formula. We derive the semiclassical formula of a typical persistent current $I^{typ}$ for a single billiard and an average persistent current $<I>$ for an ensemble of billiards at finite temperature. These formulas are used to show that the persistent current for chaotic billiards is much smaller than that for integrable ones. The persistent currents in the ballistic regime therefore become an experimental tool to search for the quantum signature of classical chaotic and regular dynamics.
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…
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 …
Some of the subtleties of the integrability of the elliptic quantum billiard are discussed. A well known classical constant of the motion has in the quantum case an ill-defined commutator with the Hamiltonian. It is shown how this problem can be solved. A geometric picture is given revealing why levels of a separable system cross. It is shown that the repulsions found by Ayant and Arvieu are computational effects and that the method used by Traiber et al. is related to the present picture which explains the crossings they find. An asymptotic formula for the energy-levels is derived and it is f…
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…
High resolution eigenvalue spectra of several two- and three-dimensional superconducting microwave cavities have been measured in the frequency range below 20 GHz and analyzed using a statistical measure which is given by the distribution of the normalized mode fluctuations. For chaotic systems the limit distribution is conjectured to show a universal Gaussian, whereas integrable systems should exhibit a non-Gaussian limit distribution. For the investigated Bunimovich stadium and the 3D-Sinai billiard we find that the distribution is in good agreement with this prediction. We study members of …
In this work we study the nonlinear dynamics of the static and the driven ellipse. In the static case, we find numerically an asymptotical algebraic decay for the escape of an ensemble of non-interacting particles through a small hole due to the integrable structure of the phase space of the system. Furthermore, for a certain hole position a saturation value in the decay that can be tuned arbitrarily by varying the eccentricity of the ellipse is observed and explained. When applying harmonic boundary oscillations this saturation value caused by librator type orbits is gradually destroyed via t…
Determining the flow of rays or particles driven by a force or velocity field is fundamental to modelling many physical processes, including weather forecasting and the simulation of molecular dynamics. High frequency wave energy distributions can also be approximated using flow or transport equations. Applications arise in underwater and room acoustics, vibro-acoustics, seismology, electromagnetics, quantum mechanics and in producing computer generated imagery. In many practical applications, the driving field is not known exactly and the dynamics are determined only up to a degree of uncerta…
