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

Effective pair-interaction of phase singularities in random waves

In two-dimensional random waves, phase singularities are point-like dislocations with a behavior reminiscent of interacting particles. This-qualitative-consideration stems from the spatial arrangement of these entities, which finds its hallmark in a pair correlation reminiscent of a liquid-like system. Starting from their pair correlation function, we derive an effective pair-interaction for phase singularities in random waves by using a reverse Monte Carlo method. This study initiates a new, to the best of our knowledge, approach for the treatment of singularities in random waves and can be generalized to topological defects in any system.

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.

Generation of two-dimensional chaotic vector fields from a surface-emitting semiconductor laser: analysis of vector singularities

We demonstrate experimental generation of two-dimensional (2D) chaotic vector fields from a surface-emitting microcavity laser with precisely control of the operating temperature and the operating current. The 2D optical vector field is found to be formed by frequency locking of two linearly polarized laser modes with different chaotic spatial structures. The eigenfunction expansion method is used to reconstruct the experimental wave function for analyzing the properties of the vector singularities. It is shown that the distribution of vector singularities obeys the sign rule of nearest-neighbor singularities in the phase space of the vector field.

Generation of polarization-entangled optical coherent waves and manifestation of vector singularity patterns

We use a high-level isotropic laser with off-axis focused and on-axis circular pumping to generate the high order polarization-entangled transverse modes. The main finding is that the complex transverse modes can be categorized into four types: square pattern, hyperbolic pattern, elliptic pattern, and circular pattern. Importantly, all types of the polarization-entangled modes can be well analyzed with the generalized coherent states. With the connection between theoretical analysis and experimental results, the formation of complex singularities can be clearly represented.

Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling

The propagation-dependent polarization vector fields are experimentally created from an isotropic microchip laser with a longitudinal-transverse coupling and entanglement of the polarization states. The experimental three-dimensional coherent vector fields are analytically reconstructed with a coherent superposition of orthogonal circularly polarized vortex modes. Each polarized component is found to comprise two Laguerre-Gaussian modes with different topological charges. With the analytical representation, the polarization singularities, on which the electric polarization ellipse is purely circular (C lines) or purely linear (L surfaces), are explored. The C line singularities are found to form an intriguing hyperboloidal structure.

Wave functions with localizations on classical periodic orbits in weakly perturbed quantum billiards

We analytically investigate quantum wave functions with localizations on classical periodic orbits (POs) in the circular billiards. We construct the coherent states which are the coherent superpositions of nearly degenerate eigenstates and found to be localized around single classical POs. With the constructed coherent states, we can analytically express the mesoscopic eigenstates in the weakly perturbed systems, which are found to be localized on the ensemble of classical POs. Furthermore, these coherent states can be utilized to study the quantum vortex structures in quantum billiards.

Observation of spatially coherent polarization vector fields and visualization of vector singularities

We present experimental observations of polarization vector fields from a highly isotropic microchip laser with a large Fresnel number. The experimental polarization vector field leads to complex entanglement of spatial structures and polarization states. We theoretically derive the representation of generalized coherent states to reconstruct the experimental polarization-resolved patterns. The analytical and precise reconstruction enables the vector singularities to be clearly visualized.

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.

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.

Vortex formation of coherent waves in nonseparable mesoscopic billiards

We analytically study the vortex formation associated with the classical periodic orbits in a nonseparable quantum billiard. The influence of vortex formation on the quantum flux in the mesoscopic region is clearly demonstrated. Remarkably, the high-order vortex structures in an equilateral triangle billiard display the triangular, swirl, and kagomè vortex lattices. The relationship between various vortex lattices is found to rely on the relative phase factor in the coherent state.