Microwave experiments on chaotic billiards

We describe experiments using billiard-shaped microwave cavities, to test ideas in quantum chaos. The experimental method for observing cavity resonances to obtain the eigenvalues, and the advantages and limitations of the techniques, including the influence of absorption, are discussed. An experimental technique to obtain a 2D mapping of the wavefunction is described. Results are displayed for 36 of the low-lying wavefunctions of a Sinai billiard cavity consisting of a central disc in a rectangular enclosure. The wavefunctions demonstrate the influence of classical periodic orbits (PO), of which there are two types: non-isolated PO, which avoid the central disc, and isolated PO, which hit the central disc. Scarred states, including those associated with isolated PO, are directly observed.     

Effect of noise in open chaotic billiards

We investigate the effect of white-noise perturbations on chaotic trajectories in open billiards. We focus on the temporal decay of the survival probability for generic mixed-phase-space billiards. The survival probability has a total of five different decay regimes that prevail for different intermediate times. We combine new calculations and recent results on noise perturbed Hamiltonian systems to characterize the origin of these regimes and to compute how the parameters scale with noise intensity and billiard openness. Numerical simulations in the annular billiard support and illustrate our results.     

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.     

Numerical experiments on quantum chaotic billiards

A recently proposed numerical technique for generation of high-quality unstructured meshes is combined with a finite-element method to solve the Helmholtz equation that describes the quantum mechanics of a particle confined in two-dimensional cavities. Different shapes are treated on equal footing, including Sinai, stadium, annular, threefold symmetric, mushroom, cardioid, triangle, and coupled billiards. The results are shown to be in excellent agreement with available measurements in flat microwave resonator counterparts with nonintegrable geometries.     

Observation of chaotic and regular dynamics in atom-optics billiards

We report on experimental observations of chaotic and regular motion of ultracold atoms confined by a billiard-shaped optical dipole potential induced by a rapidly scanning laser beam. To investigate the dynamics of the atoms confined by such an "atom-optics" billiard we measure the decay of the number of trapped atoms through a hole on the boundary. A fast and purely exponential decay, the clear signature of chaotic motion, is found for a stadium billiard, but not for a circular or an elliptical billiard, in agreement with theory. We also investigated the effects of decoherence, velocity spread, and gravity on regular and chaotic motion.     

Stable regions and singular trajectories in chaotic soft-wall billiards

We present numerical and experimental results for the development of islands of stability in atom-optics billiards with soft walls. As the walls are soften, stable regions appear near singular periodic trajectories in converging (focusing) and dispersing billiards, and are surrounded by areas of “stickiness” in phase space. The size of these islands depends on the softness of the potential in a very sensitive way.     

Time-reversal symmetry breaking and decoherence in chaotic Dirac billiards

In this work, we perform a statistical study on Dirac Billiards in the extreme quantumlimit (a single open channel on the leads). Our numerical analysis uses a large ensembleof random matrices and demonstrates the preponderant role of dephasing mechanisms in suchchaotic billiards. Physical implementations of these billiards range from quantum dots ofgraphene to topological insulators structures. We show, in particular, that the role offinite crossover fields between the universal symmetries quickly leaves the conductance tothe asymptotic limit of unitary ensembles. Furthermore, we show that the dephasingmechanisms strikingly lead Dirac billiards from the extreme quantum regime to thesemiclassical Gaussian regime.     

Level spacing statistics for two-dimensional massless Dirac billiards

Classical-quantum correspondence has been an intriguing issue ever since quantum theory was proposed. The searching for signatures of classically nonintegrable dynamics in quantum systems comprises the interesting field of quantum chaos. In this short review, we shall go over recent efforts of extending the understanding of quantum chaos to relativistic cases. We shall focus on the level spacing statistics for two-dimensional massless Dirac billiards, i.e., particles confined in a closed region. We shall discuss the works for both the particle described by the massless Dirac equation(or Weyl equation)and the quasiparticle from graphene. Although the equations are the same, the boundary conditions are typically different,rendering distinct level spacing statistics.     

Photocount statistics of chaotic lasers

We derive the photocount statistics of the radiation emitted from a chaotic laser resonator in the regime of single-mode lasing. Random spatial variations of the resonator eigenfunctions lead to strong mode-to-mode fluctuations of the laser emission. The distribution of the mean photocount over an ensemble of modes changes qualitatively at the lasing transition, and displays up to three peaks above the lasing threshold.     

Correlation decay in quantum chaotic billiards with bulk or surface disorder

We study the decay properties of correlation functions in quantum billiards with surface or bulk disorder. The quantum system is modeled by means of a tight-binding Hamiltonian with diagonal disorder, solved on LxL clusters of the square lattice. The correlation function is calculated by launching the system at t=0 into a wave function of the regular (clean) system and following its time evolution. The results show that the correlation function decays exponentially with a characteristic correlation time (inverse of the Lyapunov exponent lambda). For small enough disorder the Lyapunov exponent is approximately given by the imaginary part of the self-energy induced by disorder. On the other hand, if the scaling of the Lyapunov exponent with L is investigated by keeping constant l/L, where l is the mean free path, the results show that lambda is proportional to 1/L.     

A new model of quantum chaotic billiards: application to granular metals

We discuss the properties of a recently proposed model of quantum chaotic billiards in two and three dimensions. The model is based on a tight-binding Hamiltonian in which the energies of the atomic levels at the boundary sites are chosen at random between -W/2 and W/2. The energy spectra show a complex behavior with regions that obey Wigner-Dyson statistics, and regions with localized and quasi-ideal states distributed according to Poisson statistics. Whereas at low energies long-range energy fluctuations follow Random Matrix Theory (RMT) for all W, at high energies fluctuations are below (above) RMT for small (large) W. For small W, the mean free path l is proportional to L/W ², L being the system size, and reaches a minimum for W of the order of the band width, at which l ≈ L/2. In 3D we found that the energy fluctuations of the highest occupied level are much larger than the average interlevel spacing. This provides an explanation for autoionization effects of the grains in granular metals.     

Decay of quantum correlations in atom optics billiards with chaotic and mixed dynamics

We perform echo spectroscopy on ultracold 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 nonmonotonic and partial revivals of coherence are observed in contrast with the predictions of random matrix theory. 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 strength of the perturbation. In this regime no clear distinction can be made between chaotic traps and traps with mixed dynamics.     

Statistics of eigenfunctions of chaotic billiards taking account of the Rashba spin-orbit interaction

It is demonstrated, both analytically and numerically, that eigenfunction statistics in chaotic billiards with spin-orbit interaction fundamentally depend on the ratio of the squared spin-orbit interaction constant. If this ratio is small, one of the eigenstate components is a random Gaussian field, whereas another is not universal and depends on the billiard type. In the opposite case, the statistics of both components is described by the independent random complex Gaussian fields with the same variances. In the intermediate case, both eigenfunction components do not satisfy Gaussian statistics.     

Electron energy level statistics in graphene quantum dots

Motivated by recent experimental observations of size quantization of electron energy levels in graphene quantum dots [7] we investigate the level statistics in the simplest tight-binding model for different dot shapes by computer simulation. The results are in a reasonable agreement with the experiment which confirms qualitatively interpretation of observed level statistics in terms of “Dirac billiards” without taking into account many-body effects. It is shown that edge effects are in general sufficient to produce the observed level distribution and that even strong bulk disorder does not change the results drastically. The article is published in the original.     

Electron energy level statistics in graphene quantum dots

Motivated by recent experimental observations of size quantization of electron energy levels in graphene quantum dots [7] we investigate the level statistics in the simplest tight-binding model for different dot shapes by computer simulation. The results are in a reasonable agreement with the experiment which confirms qualitatively interpretation of observed level statistics in terms of “Dirac billiards” without taking into account many-body effects. It is shown that edge effects are in general sufficient to produce the observed level distribution and that even strong bulk disorder does not change the results drastically.