Open Chaotic Billiards Research Articles

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.

Open chaotic Dirac billiards: Weak (anti)localization, conductance fluctuations, and decoherence

In this paper, we investigate the transport properties of open chaotic Dirac billiards and their intrinsic (chiral universal) symmetry classes. The prominent examples of these systems are some categories of topological insulators and graphene structures. We extend the diagrammatic method of integration over the unitary group and obtain analytical results for the semiclassical limit and for the high quantum limit in the universal regime. We show the emergence of quantum fingerprints characteristic of the chiral symmetries, which are amplified in the presence of a single open channel in each electronic terminals. We compare the chaotic Dirac billiards with the ``Schr\"odinger billiards'' in a myriad of regimes, exhibiting the differences between the chiral universal classes and the Wigner-Dyson classes. Two numerical methods were used to confirm our analytical findings, yielding also the distribution of conductances. We also investigate analytically the effect of dephasing using the characteristic time scales of the chaotic billiards and we show the appearance of peculiar numbers of chaos.

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.

Avoided-level-crossing statistics in open chaotic billiards

We investigate a two-level model with a large number of open decay channels in order to describe avoided level crossing statistics in open chaotic billiards. This model allows us to describe the fundamental changes in the probability distribution of the avoided level crossings compared with the closed case. Explicit expressions are derived for systems with preserved and broken time-reversal symmetry. We find that the decay process induces a modification at small spacings of the probability distribution of the avoided level crossings due to an attraction of the resonances. The theoretical predictions are in complete agreement with the recent experimental results of Dietz [Phys. Rev. E 73, 035201 (2006)].

Quantum stress in chaotic billiards

This paper reports on a joint theoretical and experimental study of the Pauli quantum-mechanical stress tensor T_{alphabeta}(x,y) for open two-dimensional chaotic billiards. In the case of a finite current flow through the system the interior wave function is expressed as psi=u+iv . With the assumption that u and v are Gaussian random fields we derive analytic expressions for the statistical distributions for the quantum stress tensor components T_{alphabeta} . The Gaussian random field model is tested for a Sinai billiard with two opposite leads by analyzing the scattering wave functions obtained numerically from the corresponding Schrödinger equation. Two-dimensional quantum billiards may be emulated from planar microwave analogs. Hence we report on microwave measurements for an open two-dimensional cavity and how the quantum stress tensor analog is extracted from the recorded electric field. The agreement with the theoretical predictions for the distributions for T_{alphabeta}(x,y) is quite satisfactory for small net currents. However, a distinct difference between experiments and theory is observed at higher net flow, which could be explained using a Gaussian random field, where the net current was taken into account by an additional plane wave with a preferential direction and amplitude.

The behaviour of resonances in Hecke triangular billiards under deformation

The right-hand boundary of Artin's billiard on the Poincaré half-plane is continuously deformed to generate a class of chaotic billiards which includes fundamental domains of the Hecke groups Γ(2, n) at certain values of the deformation parameter. The quantum scattering problem in these open chaotic billiards is described and the distributions of both real and imaginary parts of the resonant eigenvalues are investigated. The transitions to arithmetic chaos in the cases n ∊ {4, 6} are closely examined and the explicit analytic form for the scattering matrix is given together with the Fourier coefficients for the scattered wavefunction. The n = 4 and 6 cases have an additional set of regular equally spaced resonances compared to Artin's billiard (n = 3). For a general deformation, a numerical procedure is presented which generates the resonance eigenvalues and the evolution of the eigenvalues is followed as the boundary is varied continuously which leads to dramatic changes in their distribution. For deformations away from the non-generic arithmetic cases, including that of the tiling Hecke triangular billiard n = 5, the distributions of the positions and widths of the resonances are consistent with the predictions of a random matrix theory.

Evolution of wave-function statistics from closed quantum billiards up to the open quantum dot limit: Application to the measurement of dynamical properties through imaging experiments

We discuss the evolution of the electronic and scattering properties of a square billiard connected to the outside either by tunneling barriers or by progressively higher-conductance leads. The slightest connection already induces features of chaotic dynamics in the otherwise regular system. In the absence of large ensembles for energy level statistics or power spectrum analysis, we propose the distribution of the local densities to inspect the character of the underlying dynamics of a scattering state. We show that the wave-function statistics in wide open chaotic billiards strongly deviates from available predictions. The precursors of scarred wave functions are found in wave-function vortices that are tuned by the lead width.

Signatures of quantum chaos in complex wavefunctions describing open billiards

We discuss signatures of quantum chaos in open chaotic billiards. Solutions for such a system are given by complex scattering wavefunctions ψ = u + iv when a steady current flows through the billiard. For slightly opened chaotic billiards the current distributions are simple and universal. It is remarkable that the resonant transmission through integrable billiards also gives the universal current distribution. Currents induced by the Rashba spin–orbit interaction can flow even in closed billiards. Wavefunction and current distributions for a chaotic billiard with weak and strong spin–orbit interactions have been derived and compared with numerics. Similarities with classical waves are considered. In particular we propose that the networks of electric resonance RLC circuits may be used to study wave chaos. However, being different from quantum billiards, there is a resistance from the inductors which gives rise to heat power and decoherence.

Ray-wave correspondence in bent waveguides

The present paper is concerned with wave propagation at high frequency in bent waveguides. The multimodal formalism proposed in earlier papers by the authors is shown to be suitable for investigating very high frequency problems. Then, following works made by Luna-Acosta et al. (G. Luna-Acosta, J.A. Méndez-Bermúdez, P. S˘eba, K.N. Pichugin, Classical versus quantum structure of the scattering probability matrix: chaotic waveguides, Phys. Rev. E 65 (2002) 046605; J.A. Méndez-Bermúdez, G. Luna-Acosta, P. P˘eba, K.N. Pichugin, Understanding quantum scattering properties in terms of purely classical dynamics: two-dimensional open chaotic billiards, Phys. Rev. E 66 (2002) 046207) for open quantum billiards, the ray-wave correspondence of the scattering matrix ( S matrix) is studied by first constructing a ray- S matrix and then comparing its structure with the structure of the wave- S matrix that is obtained with the exact multimodal formalism, for different geometries of curved waveguides. A great similarity between these two matrices is observed and it is shown that the scattering matrix constructed only by counting rays allows us to predict and understand numbers of the wave scattering properties in waveguides.

Wave function statistics in open chaotic billiards.

We study the statistical properties of wave functions in a chaotic billiard that is opened up to the outside world. Upon increasing the openings, the billiard wave functions cross over from real to complex. Each wave function is characterized by a phase rigidity, which is itself a fluctuating quantity. We calculate the probability distribution of the phase rigidity and discuss how phase rigidity fluctuations cause long-range correlations of intensity and current density. We also find that phase rigidities for wave functions with different incoming wave boundary conditions are statistically correlated.

Understanding quantum scattering properties in terms of purely classical dynamics: two-dimensional open chaotic billiards.

We study classical and quantum scattering properties of particles in the ballistic regime in two-dimensional chaotic billiards that are models of electron- or micro-waveguides. To this end we construct the purely classical counterparts of the scattering probability (SP) matrix |S(n,m)|(2) and Husimi distributions specializing to the case of mixed chaotic motion (incomplete horseshoe). Comparison between classical and quantum quantities allows us to discover the purely classical dynamical origin of certain general as well as particular features that appear in the quantum description of the system. On the other hand, at certain values of energy the tunneling of the wave function into classically forbidden regions produces striking differences between the classical and quantum quantities. A potential application of this phenomenon in the field of microlasers is discussed briefly. We also see the manifestation of whispering gallery orbits as a self-similar structure in the transmission part of the classical SP matrix.

Crossover from regular to irregular behavior in current flow through open billiards.

We discuss signatures of quantum chaos in terms of distributions of nodal points, saddle points, and streamlines for coherent electron transport through two-dimensional billiards, which are either nominally integrable or chaotic. As typical examples of the two cases we select rectangular and Sinai billiards. We have numerically evaluated distribution functions for nearest distances between nodal points and found that there is a generic form for open chaotic billiards through which a net current is passed. We have also evaluated the distribution functions for nodal points with specific vorticity (winding number) as well as for saddle points. The distributions may be used as signatures of quantum chaos in open systems. All distributions are well reproduced using random complex linear combinations of nearly monochromatic states in nominally closed billiards. In the case of rectangular billiards with simple sharp-cornered leads the distributions have characteristic features related to order among the nodal points. A flaring or rounding of the contact regions may, however, induce a crossover to nodal point distributions and current flow typical for quantum chaos. For an irregular arrangement of nodal points, as for example in the Sinai billiard, the quantum flow lines become very complex and volatile, recalling chaos among classical trajectories. Similarities with percolation are pointed out.

Statistics of interior current distributions in two-dimensional open chaotic billiards

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 interior probability current. As a by-product 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 billiard.

Effects of Coulomb Interaction on the Density of States in Open Chaotic Billiards

The density of states (DOS) in open chaotic billiards is studied semiclassically in the presence of Coulomb interaction. The exchange and Hartree corrections to the DOS as a function of energy ε relative to the Fermi energy are calculated at zero temperature including higher-order interaction effects. It is shown that the DOS decreases with increasing | ε | in the low-energy regime and begins to increase around | ε |≈ t B -1 , where t B is the classical escape time. For t B -1 ≪| ε |, a logarithmic increase of the DOS is shown to appear. This anomalous ε -dependence is due to competition between the exchange and Hartree corrections.

Phase Relaxation Time in Open Chaotic Billiards

Phase relaxation time τ φ due to Coulomb interactions is studied semiclassically for electrons in open chaotic billiards with two leads of width W . When W is much larger than the Fermi wavelength λ F , the phase relaxation time at temperature T is shown to be τ φ -1 = c (λ F / W ) T , where c is a numerical constant of the order of unity. The behavior τ φ ∝ T -1 , similar to that in two-dimensional diffusive systems, is in agreement with the experimental results.

Quantum conductance fluctuations and classical short-path dynamics.

We present numerical results for ballistic-electron quantum transport through weakly open integrable circle and chaotic stadium billiards. The geometry of the pair of conducting leads is chosen in accordance with recent experiments for semiconductor microstructures [Marcus [ital et] [ital al]., Phys. Rev. Lett. [bold 69], 506 (1992)]. The conductance as a function of the Fermi wave number displays characteristic noisy fluctuations for both the integrable and the chaotic systems. We show that structures in the conductance autocorrelation function as a function of the Fermi wave number are related to short-length classical orbits. This correspondence permits incorporation of effects of phase decoherence due to incoherent scattering into the quantum calculation.