Universality of level spacing distributions in classical chaos

We suggest that random matrix theory applied to a matrix of lengths of classical trajectories can be used in classical billiards to distinguish chaotic from non-chaotic behavior. We consider in 2D the integrable circular and rectangular billiard, the chaotic cardioid, Sinai and stadium billiard as well as mixed billiards from the Limaçon/Robnik family. From the spectrum of the length matrix we compute the level spacing distribution, the spectral auto-correlation and spectral rigidity. We observe non-generic (Dirac comb) behavior in the integrable case and Wignerian behavior in the chaotic case. For the Robnik billiard close to the circle the distribution approaches a Poissonian distribution. The length matrix elements of chaotic billiards display approximate GOE behavior. Our findings provide evidence for universality of level fluctuations—known from quantum chaos—to hold also in classical physics.     

Test of Trace Formulas for Spectra of Superconducting Microwave Billiards

Experimental tests of various trace formulas, which in general relate the density of states for a given quantum mechanical system to the properties of the periodic orbits of its classical counterpart, for spectra of superconducting microwave billiards of varying chaoticity are reviewed by way of examples. For a two-dimensional Bunimovich stadium billiard the application of Gutzwiller's trace formula is shown to yield correctly locations and strengths of the peaks in the Fourier transformed quantum spectrum in terms of the shortest unstable classical periodic orbits. Furthermore, in two-dimensional billiards of the Limaçon family the transition from regular to chaotic dynamics is studied in terms of a recently derived general trace formula by Ullmo, Grinberg and Tomsovic. Finally, some salient features of wave dynamical chaos in a fully chaotic three-dimensional Sinai microwave billiard are discussed. Here the reconstruction of the spectrum is not as straightforward as in the two-dimensional cases and a modified trace formula as suggested by Balian and Duplantier will have eventually to be applied.     

The local geometry of chaotic billiards

For a billiard of a general shape a transformation is introduced which projects the boundary on the unit circle. This introduces a non-Euclidean metric on the plane which contains all relevant information of the shape of the boundary. Classically the straight lines of the free motion correspond to geodesics and quantum mechanically the energy spectrum is that of Laplace–Beltrami operator with Dirichlet boundary conditions on the unit circle. The geodesic equations are highly non-linear. Nevertheless for the interval between two consecutive scatterings we have two integrals of motion, the kinetic energy and the angular momentum. This fact helps to solve explicitly the geodesic equations. These solutions can be used to derive interesting properties for the classical scattering. Quantum mechanically the spectrum of the above billiards is obtained for certain parameter values both perturbatively for small values of the parameter and also using a diagonalization procedure. This method is applicable to any particular form of a billiard for which the transformation is invertible and can be used on one hand as a quick method of approximate spectral determination and as a theoretical tool to analyse specific properties of integrability and chaos through the associated connection form and the Laplace–Beltrami operator. Finally as a first indication of the potentiality of this method we present a graphical test where for very small deviations from the circular billiard an integrable and two non-integrable billiards can be distinguished by the distribution of the differences of the first order corrections while this distinction is not evident by the usual test for the nearest neighbor level spacings.     

Experiments on quantum chaos using microwave cavities: Results for the pseudo-integrable L-billiard

We describe microwave experiments used to study billiard geometries as model problems of non-integrability in quantum or wave mechanics. The experiments can study arbitrary 2-D geometries, including chaotic and even disordered billiards. Detailed results on an L-shaped pseudo-integrable billiard are discussed as an example. The eigenvalue statistics are well-described by empirical formulae incorporating the fraction of phase space that is non-integrable. The eigenfunctions are directly measured, and their statistical properties are shown to be influenced by non-isolated periodic orbits, similar to that for the chaotic Sinai billiard. These periodic orbits are directly observed in the Fourier transform of the eigenvalue spectrum.     

New mechanism of chaos in triangular billiards

A new mechanism of weak chaos in triangular billiards has been proposed owing to the effect of cutting of beams of rays. A similar mechanism is also implemented in other polygonal billiards. Cutting of beams results in the separation of initially close rays at a finite angle by jumps in the process of reflections of beams at the vertices of a billiard. The opposite effect of joining of beams of rays occurs in any triangular billiard along with cutting. It has been shown that the cutting of beams has an absolute character and is independent of the form of a triangular billiard or the parameters of a beam. On the contrary, joining has a relative character and depends on the commensurability of the angles of the triangle with π. Joining always suppresses cutting in triangular billiards whose angles are commensurable with π. For this reason, their dynamics cannot be chaotic. In triangular billiards whose angles are rationally incommensurable with π, cutting always dominates, leading to weak chaos. The revealed properties are confirmed by numerical experiments on the phase portraits of typical triangular billiards.     

Quantum transport and classical dynamics in open billiards

We report numerical results of an investigation of quantum transport for a weakly opened integrable circle and chaotic stadium billiards with a pair of conducting leads. While the statistics of spacings of resonance energies commonly follow the Wigner (GOE)-like distribution, the electric conductance as a function of the Fermi wavenumber shows characteristic noisy fluctuations associated with a typical set of classical orbits unique for both billiards. The wavenumber autocorrelation for the conductance is stronger in the stadium than the circle billiard, which we show is related to the length spectrum of classical short orbits. We propose an explanation of these contrasts in terms of the effect of phase decoherence due to the underlying chaotic dynamics.     

Optical billiards for atoms

One of the central paradigms for classical and quantum chaos in conservative systems is the two-dimensional billiard in which particles are confined to a closed region in the plane, undergoing elastic collisions with the walls and free motion in between. We report the first realization of billiards using ultracold atoms bouncing off beams of light. These beams create the desired spatial pattern, forming an "optical billiard." We find excellent agreement between theory and our experimental demonstration of chaotic and stable motion in optical billiards, establishing a new testing ground for classical and quantum chaos.     

Signatures of quantum chaos in the nodal points and streamlines in electron transport through billiards

Streamlines and the 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 the 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 the averaging is performed (over the positions of the leads, over an energy interval with a few conductance fluctuations, or both). The integrable rectangu-lar billiard, on the other hand, displays a nonuniversal distribution with a central peak related to partial order of nodal points for the case of symmetric attachment of the leads. However, cases with asymmetric leads tend to the universal form. Also, it is shown how nodal points in the rectangular billiard can lead to “channeling of quantum flows,” while disorder in the nodal points in the Sinai billiard gives rise to unstable irregular behavior of the flow. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 6, 398–404 (25 September 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Nodal domains statistics: a criterion for quantum chaos

We consider the distribution of the (properly normalized) numbers of nodal domains of wave functions in 2D quantum billiards. We show that these distributions distinguish clearly between systems with integrable (separable) or chaotic underlying classical dynamics, and for each case the limiting distribution is universal (system independent). Thus, a new criterion for quantum chaos is provided by the statistics of the wave functions, which complements the well-established criterion based on spectral statistics.     

Proximity-induced subgaps in andreev billiards

We examine the density of states of an Andreev billiard and show that any billiard with a finite upper cutoff in the path length distribution P(s) will possess an energy gap on the scale of the Thouless energy. An exact quantum mechanical calculation for different Andreev billiards gives good agreement with the semiclassical predictions when the energy dependent phase shift for Andreev reflections is properly taken into account. Based on this new semiclassical Bohr-Sommerfeld approximation of the density of states, we derive a simple formula for the energy gap. We show that the energy gap, in units of Thouless energy, may exceed the value predicted earlier from random matrix theory for chaotic billiards.     

Scars in nonintegrable and rational billiards

We numerically study quantum mechanical features of the Bunimovich stadium billiard and the rational billiards which approach the former as the number of their sides increases. The statistics of energy levels and eigenfunctions of the rational billiards becomes indistinguishable from that of the Bunimovich stadium billiard below a certain energy. This fact contradicts the classical picture in which the Bunimovich stadium billiard is chaotic, but the rational billiard is pseudointegrable. It is numerically confirmed that the wave functions do not detect the fine structure, which is much smaller than the wavelength.     

Quantum graphs: Applications to quantum chaos and universal spectral statistics

During the last few years quantum graphs have become a paradigm of quantum chaos with applications from spectral statistics to chaotic scattering and wavefunction statistics. In the first part of this review we give a detailed introduction to the spectral theory of quantum graphs and discuss exact trace formulae for the spectrum and the quantum-to-classical correspondence. The second part of this review is devoted to the spectral statistics of quantum graphs as an application to quantum chaos. In particular, we summarize recent developments on the spectral statistics of generic large quantum graphs based on two approaches: the periodic-orbit approach and the supersymmetry approach. The latter provides a condition and a proof for universal spectral statistics as predicted by random-matrix theory.     

Quantum spectra and classical periodic orbit in the cubic billiard

Quantum billiards have attracted much interest in many fields. People have made a lot of researches on the two-dimensional (2D) billiard systems. Contrary to the 2D billiard, due to the complication of its classical periodic orbits, no one has studied the correspondence between the quantum spectra and the classical orbits of the three-dimensional (3D) billiards. Taking the cubic billiard as an example, using the periodic orbit theory, we find the periodic orbit of the cubic billiard and study the correspondence between the quantum spectra and the length of the classical orbits in 3D system. The Fourier transformed spectrum of this system has allowed direct comparison between peaks in such plot and the length of the periodic orbits, which verifies the correctness of the periodic orbit theory. This is another example showing that semiclassical method provides a bridge between quantum and classical mechanics.     

Chaos-assisted tunneling and 1/falpha spectral fluctuations in the order-chaos transition

It has been shown that the spectral fluctuations of different quantum systems are characterized by 1/falpha noise, with 1< or =alpha< or =2, in the transition from integrability to chaos. This result is not well understood. We show that chaos-assisted tunneling gives rise to this power-law behavior. We develop a random matrix model for intermediate quantum systems, based on chaos-assisted tunneling, and we discuss under which conditions it displays 1/falpha noise in the transition from integrability to chaos. We conclude that the variance of the elements that connect regular with chaotic states must decay with the difference of energy between them. We compare the characteristics of the transition modeled in this way with what is obtained for the Robnik billiard.     

Semiclassical Analysis of Quarter Stadium Billiards

An expansion method for stationary states is applied to obtain the eigenfunctions and the eigenenergies of the quarter stadium billiard, and its nearest energy-level spacing distribution is obtained. The histogram is consistent with the standard Wigner distribution, which indicates that the stadium billiard system is chaotic. Particular attention is paid to pursuing the quantum manifestations of such classical chaos. The correspondences between the Fourier transformation of quantum spectra and classical orbits are investigated by using the closed-orbit theory. The analytical and numerical results are in agreement with the required resolution, which corroborates that the semiclassical method provides a physically meaningful image to understand such chaotic systems.     

The Spectrum of the Billiard Laplacian of a Family of Random Billiards

Random billiards are billiard dynamical systems for which the reflection law giving the post-collision direction of a billiard particle as a function of the pre-collision direction is specified by a Markov (scattering) operator P. Billiards with microstructure are random billiards whose Markov operator is derived from a “microscopic surface structure” on the boundary of the billiard table. The microstructure in turn is defined in terms of what we call a billiard cellQ, the shape of which completely determines the operator P. This operator, defined on an appropriate Hilbert space, is bounded self-adjoint and, for the examples considered here, a Hilbert-Schmidt operator. A central problem in the statistical theory of such random billiards is to relate the geometric characteristics of Q and the spectrum of P. We show, for a particular family of billiard cell shapes parametrized by a scale invariant curvature K (Fig. 2), that the billiard Laplacian P−I is closely related to the ordinary spherical Laplacian, and indicate, by partly analytical and partly numerical means, how this provides asymptotic information about the spectrum of P for small values of K. It is shown, in particular, that the second moment of scattering about the incidence angle closely approximates the spectral gap of P.     

Spectral Gap for a Class of Random Billiards

A random billiard is a random dynamical system similar to an ordinary billiard system except that the standard specular reflection law is replaced with a more general stochastic operator specifying the post-collision distribution of velocities for any given pre-collision velocity. We consider such collision operators for certain random billiards that we call billiards with microstructure. Collisions modeled by these operators can still be thought of as elastic and time reversible. The operators are canonically determined by a second (deterministic) billiard system that models “microscopic roughness” on the billiard table boundary. Our main purpose here is to develop some general tools for the analysis of the collision operator of such random billiards. Among the main results, we give geometric conditions for these operators to be Hilbert-Schmidt and relate their spectrum and speed of convergence to stationary Markov chains with geometric features of the microscopic billiard structure. The relationship between spectral gap and the shape of the microstructure is illustrated with several simple examples.     

Entanglement distribution statistic in Andreev billiards

We investigate statistical aspects of the entanglement production for open chaotic mesoscopic billiards in contact with superconducting parts, known as Andreev billiards. The complete distributions of concurrence and entanglement of formation are obtained by using the Altland–Zirnbauer symmetry classes of circular ensembles of scattering matrices, which complements previous studies in chaotic universal billiards belonging to other classes of random matrix theory. Our results show a unique and very peculiar behavior: the realization of entanglement in a Andreev billiard always results in non-separable state, regardless of the time reversal symmetry. The analytical calculations are supported by a numerical Monte Carlo simulation.     

Influence of phase-space localization on the energy diffusion in a quantum chaotic billiard

The quantum dynamics of a chaotic billiard with moving boundary is considered in this paper. 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 wavelength. 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 first oscillations of the boundary (=2Dt). We studied the diffusion constant D as a function of the boundary velocity and found differences with theoretical predictions based on random matrix theory. By extracting highly phase-space localized structures from the spectrum, previous differences were reduced significantly. This fact provides numerical evidence of the influence of phase-space localization on the quantum diffusion of a chaotic system.