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.     

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.     

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.     

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.     

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.     

Properties of Some Chaotic Billiards with Time-Dependent Boundaries

A dispersing billiard (Lorentz gas) and focusing billiards (in the form of a stadium) with time-dependent boundaries are considered. The problem of a particle acceleration in such billiards is studied. For the Lorentz gas two cases of the time-dependence are investigated: stochastic perturbations of the boundary and its periodic oscillations. Two types of focusing billiards with periodically forced boundaries are explored: stadium with strong chaotic properties and a near-rectangle stadium. It is shown that in all cases billiard particles can reach unbounded velocities. Average velocities of the particle ensemble as functions of time and the number of collisions are obtained.     

二维Sinai台球系统的量子混沌研究

研究了二维Sinai台球系统的经典与量子的对应关系,运用定态展开法和Gutzwiller的周期轨道理论对Sinai台球系统的态密度经傅里叶变换得到的量子长度谱进行分析,并把量子长度谱中峰的位置与其所对应的经典体系的周期轨道长度做对比,发现两者之间存在很好的对应关系.观察到了一些量子态局域在短周期轨道附近形成量子scarred态或量子superscarred态.还研究了同心与非同心Sinai台球系统的能级最近邻间距分布,发现同心Sinai台球系统是近可积的,非同心Sinai台球系统在θ=3π/8下,随两中心间距离的增加,能级最近邻间距分布将由近可积向维格那分布过渡.     

On the signs of quantum chaos in a scattering billiard K system with kinks of the lateral boundary

Signs of quantum chaos in the spectra of linear Hamiltonian systems including scattering billiards of various configurations with kinks of the lateral surface have been experimentally studied. A billiard with kinks of the lateral surface at which the second derivative is indefinite constitutes a scattering K system. As a result, the spectrum of such a billiard and the corresponding model resonator becomes chaotic and the distribution of spectral intervals is close to a Wigner distribution. The spectral rigidity curves have been measured for a model microwave cavity whose shape is similar to the scattering billiard with kinks of the lateral surface. It has been found that the characteristics of the chaotic spectrum, the distribution of the spectral intervals, and the spectral rigidity curves for billiards with kinks of the lateral boundary exhibit signs of quantum chaos.     

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.     

E10, BE10 and arithmetical chaos in superstring cosmology

It is shown that the neverending oscillatory behavior of the generic solution, near a cosmological singularity, of the massless bosonic sector of superstring theory can be described as a billiard motion within a simplex in nine-dimensional hyperbolic space. The Coxeter group of reflections of this billiard is discrete and is the Weyl group of the hyperbolic Kac-Moody algebra E10 (for type II) or BE10 (for type I or heterotic), which are both arithmetic. These results lead to a proof of the chaotic ("Anosov") nature of the classical cosmological oscillations, and suggest a "chaotic quantum billiard" scenario of vacuum selection in string theory.     

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.     

Orbit Sum Rules for the Quantum Wave Functions of the Strongly Chaotic Hadamard Billiard in Arbitrary Dimensions

Sum rules are derived for the quantum wave functions of the Hadamard billiard in arbitrary dimensions. This billiard is a strongly chaotic (Anosov) system which consists of a point particle moving freely on a D-dimensional compact manifold (orbifold) of constant negative curvature. The sum rules express a general (two-point)correlation function of the quantum mechanical wave functions in terms of a sum over the orbits of the corresponding classical system. By taking the trace of the orbit sum rule or pre-trace formula, one obtains the Selberg trace formula. The sum rules are applied in two dimensions to a compact Riemann surface of genus two, and in three dimensions to the only non-arithmetic tetrahedron existing in hyperbolic 3-space. It is shown that the quantum wave functions can be computed from classical orbits. Conversely, we demonstrate that the structure of classical orbits can be extracted from the quantum mechanical energy levels and wave functions (inverse quantum chaology).     

Experimental test of a trace formula for a chaotic three-dimensional microwave cavity

We have measured resonance spectra in a superconducting microwave cavity with the shape of a three-dimensional generalized Bunimovich stadium billiard and analyzed their spectral fluctuation properties. The experimental length spectrum exhibits contributions from periodic orbits of nongeneric modes and from unstable periodic orbits of the underlying classical system. It is well reproduced by our theoretical calculations based on the trace formula derived by Balian and Duplantier for chaotic electromagnetic cavities.     

Smoothed wave functions of chaotic quantum systems

It is shown that wave functions of quantum systems as ħ → 0 have an extra density near unstable periodic trajectories of the classical problem. The averaged wave function square is represented as the sum over a finite number of periodic trajectories. The contribution of each trajectory is expressed through the elements of the monodromy matrix of the trajectory. The results are compared with the numerical calculations of the wave functions for the stadium billiard.     

Cantori for the stadium billiard

Although almost all orbits in the stadium billiard are "chaotic," there are many regular orbits as well-the ordered periodic orbits and cantori. The symmetries of the stadium are exploited to find maximizing and saddle periodic orbits. Cantori are maximizing quasiperiodic orbits; they have caustics. Transport in the stadium should be impeded by cantori, just as in any twist map; this is particularly important for those orbits that are nearly glancing.     

Quantum-classical correspondence in the wave functions of andreev billiards

We present a classical and quantum mechanical study of an Andreev billiard with a chaotic normal dot. We demonstrate that the nonexact velocity reversal and the diffraction at the edges of the normal-superconductor contact render the classical dynamics of these systems mixed indicating the limitations of a widely used retracing approximation. We point out the close relation between the mixed classical phase space and the properties of the quantum states of Andreev billiards, including periodic orbit scarring and localization of the wave function onto other classical phase space objects such as intermittent regions and quantized tori.     

Quantifying intermittency in the open drivebelt billiard

A "drivebelt" stadium billiard with boundary consisting of circular arcs of differing radius connected by their common tangents shares many properties with the conventional "straight" stadium, including hyperbolicity and mixing, as well as intermittency due to marginally unstable periodic orbits (MUPOs). Interestingly, the roles of the straight and curved sides are reversed. Here, we discuss intermittent properties of the chaotic trajectories from the point of view of escape through a hole in the billiard, giving the exact leading order coefficient lim(t→∞)tP(t) of the survival probability P(t) which is algebraic for fixed hole size. However, in the natural scaling limit of small hole size inversely proportional to time, the decay remains exponential. The big distinction between the straight and drivebelt stadia is that in the drivebelt case, there are multiple families of MUPOs leading to qualitatively new effects. A further difference is that most marginal periodic orbits in this system are oblique to the boundary, thus permitting applications that utilise total internal reflection such as microlasers.     

Conductance fluctuations and quantum chaotic scattering in semiconductor microstructures

We review recent experiments on aperiodic conductance fluctuations in ballistic GaAs/AlGaAs microstructures in the shape of a stadium billiard and a circle with point-contact leads, measured at millikelvin temperatures. Much of the observed behavior can be analyzed within a semiclassical approach to quantum chaotic scattering. After a brief review of the Landauer-Buttiker formulation of coherent transport, a variety of novel experimental phenomena and comparisons to semiclassical theory are presented. In particular, we discuss quantum-enhanced backscattering, the power spectrum of conductance fluctuations, crossover to the high-magnetic-field and tunneling regimes, and an application allowing the rate of phase-randomizing scattering to be measured in chaotic ballistic microstructures.     

矩形弹子球中的量子波包分析(英文)

利用波包分析量子力学体系的动力学行为在研究经典和量子的对应关系方面越来越成为一个非常重要的方法.利用高斯波包分析方法,我们计算了矩形弹子球体系的自关联函数,自关联函数的峰和经典周期轨道的周期符合的很好,这表明经典周期轨道的周期可以通过含时的量子波包方法产生.我们还讨论了矩形弹子球的波包回归和波包的部分回归,计算结果表明在每一个回归时间,波包出现精确的回归.对于动量为零的波包,初始位置在弹子球内部的特殊对称点处,出现一些时间比较短的附加的回归.     

Characteristics of level-spacing statistics in chaotic graphene billiards

A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions.