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

Wave transport and statistical properties of an open non-Hermitian quantum dot with parity-time symmetry

A basic quantum-mechanical model for wave functions and current flow in open quantum dots or billiards is investigated. The model involves non-Hertmitian quantum mechanics, parity-time (PT) symmetry, and PT-symmetry breaking. Attached leads are represented by positive and negative imaginary potentials. Thus probability densities, currents flows, etc., for open quantum dots or billiards may be simulated in this way by solving the Schrödinger equation with a complex potential. Here we consider a nominally open ballistic quantum dot emulated by a planar microwave billiard. Results for probability distributions for densities, currents (Poynting vector), and stress tensor components are presented and compared with predictions based on Gaussian random wave theory. The results are also discussed in view of the corresponding measurements for the analogous microwave cavity. The model is of conceptual as well as of practical and educational interest.

Statistics of resonance states in a weakly open chaotic cavity with continuously distributed losses

In this Rapid Communication, we demonstrate that a non-Hermitian random matrix description can account for both spectral and spatial statistics of resonance states in a weakly open chaotic wave system with continuously distributed losses. More specifically, the statistics of resonance states in an open two-dimensional chaotic microwave cavity are investigated by solving the Maxwell equations with lossy boundaries subject to Ohmic dissipation. We successfully compare the statistics of its complex-valued resonance states and associated widths with analytical predictions based on a non-Hermitian effective Hamiltonian model defined by a finite number of fictitious open channels.

Superoscillation in speckle patterns

Waves are superoscillatory where their local phase gradient exceeds the maximum wavenumber in their Fourier spectrum. We consider the superoscillatory area fraction of random optical speckle patterns. This follows from the joint probability density function of intensity and phase gradient for isotropic Gaussian random wave superpositions. Strikingly, this fraction is 1/3 when all the waves in the two-dimensional superposition have the same wavenumber. The fraction is 1/5 for a disk spectrum. Although these superoscillations are weak compared with optical fields with designed superoscillations, they are more stable on paraxial propagation.

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.

Statistics of nodal points of in-plane random waves in elastic media

We consider the nodal points (NPs) u=0 and v=0 of the in-plane vectorial displacements u=(u,v) which obey the Navier-Cauchy equation. Similar to the Berry conjecture of quantum chaos, we present the in-plane eigenstates of chaotic billiards as the real part of the superposition of longitudinal and transverse plane waves with random phases. By an average over random phases we derive the mean density and correlation function of NPs. Consequently we consider the distribution of the nearest distances between NPs.

Spectral statistics in an open parametric billiard system

We present experimental results on the eigenfrequency statistics of a superconducting, chaotic microwave billiard containing a rotatable obstacle. Deviations of the spectral fluctuations from predictions based on Gaussian orthogonal ensembles of random matrices are found. They are explained by treating the billiard as an open scattering system in which microwave power is coupled in and out via antennas. To study the interaction of the quantum (or wave) system with its environment, a highly sensitive parametric correlator is used.

Emulation of quantum mechanical billiards by electrical resonance circuits

We propose that a two-dimensional electric network may be used for fundamental studies of wave function properties, transport, and related statistics. Using Kirchhoff's current law and the j omega method we find that the network is analogous to a discretized Schrödinger equation for quantum billiards and dots. Thus complex electric potentials play the role of quantum mechanical wave functions. Ways of realizing the electric network are discussed briefly. The role of symmetries is outlined, and a direct way of selecting states with a given symmetry is presented.

Electric circuit networks equivalent to chaotic quantum billiards

We consider two electric RLC resonance networks that are equivalent to quantum billiards. In a network of inductors grounded by capacitors, the eigenvalues of the quantum billiard correspond to the squared resonant frequencies. In a network of capacitors grounded by inductors, the eigenvalues of the billiard are given by the inverse of the squared resonant frequencies. In both cases, the local voltages play the role of the wave function of the quantum billiard. However, unlike for quantum billiards, there is a heat power because of the resistance of the inductors. In the equivalent chaotic billiards, we derive a distribution of the heat power which describes well the numerical statistics.

Statistics of wave functions and currents induced by spin-orbit interaction in chaotic billiards

We show that the wave function and current statistics in chaotic Robnik billiards crucially depend on the constant of the spin-orbit interaction (SOI). For small constant the current statistics is described by universal current distributions derived for slightly opened chaotic billiards [Saichev et al., J. Phys. A 35, L87 (2002)] although one of the components of the spinor eigenfunctions is not universal. For strong SOI both components of the spinor eigenstate are complex random Gaussian fields. This observation allows us to derive the distributions of spin-orbit persistent currents which well describe numerical statistics. For intermediate values of the statistics of the eigenstates and currents, both are deeply nonuniversal.

Current statistics for wave transmission through an open Sinai billiard: effects of net currents

Transport through quantum and microwave cavities is studied by analytic and numerical techniques. In particular, we consider the statistics for a finite net probability current (Poynting vector) <j> flowing through an open ballistic Sinai billiard to which two opposite leads/wave guides are attached. We show that if the net probability current is small, the scattering wave function inside the billiard is well approximated by a Gaussian random complex field. In this case, the current statistics are universal and obey simple analytic forms. For larger net currents, these forms still apply over several orders of magnitudes. However, small characteristic deviations appear in the tail regions. Although the focus is on electron and microwave billiards, the analysis is relevant also to other classical wave cavities as, for example, open planar acoustic reverberation rooms, elastic membranes, and water surface waves in irregularly shaped vessels.

Current statistics for transport through rectangular and circular billiards

We consider the statistics of currents for electron (microwave) transmission through rectangular and circular billiards. For the resonant transmission the current distribution is describing by the universal distribution [J. Phys. A 35, L87 (2002)]]. For the more typical case of nonresonant transmission the current statistics reveals features of the current channeling (corridor effect) interior of the billiard. The numerical statistics is compared with analytical distributions.

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.

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.

Current and vortex statistics in microwave billiards

Using the one-to-one correspondence between the Poynting vector in a microwave billiard and the probability current density in the corresponding quantum system, probability densities and currents were studied in a microwave billiard with a ferrite insert as well as in an open billiard. Distribution functions were obtained for probability densities, currents, and vorticities. In addition, the vortex pair correlation function could be extracted. For all studied quantities a complete agreement with the predictions from the approach using a random superposition of plane waves was obtained.