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Physical Review E

Manifesting the evolution of eigenstates from quantum billiards to singular billiards in the strongly coupled limit with a truncated basis by usingRLCnetworks

P. H. Tuan, H. C. Liang, J. C. Tung, P. Y. Chiang, K. F. Huang, Y. F. Chen

Published Dec 7, 2015• 7 citations

Abstract

The coupling interaction between the driving source and the RLC network is explored and characterized as the effective impedance. The mathematical form of the derived effective impedance is verified to be identical to the meromorphic function of the singular billiards with a truncated basis. By using the derived impedance function, the resonant modes of the RLC network can be divided into the open-circuit and short-circuit states to manifest the evolution of eigenvalues and eigenstates from closed quantum billiards to the singular billiards with a truncated basis in the strongly coupled limit. The substantial differences of the wave patterns between the uncoupled and strongly coupled eigenmodes in the two-dimensional wave systems can be clearly revealed with the RLC network. Finally, the short-circuit resonant states are exploited to confirm that the experimental Chladni nodal-line patterns in the vibrating plate are the resonant modes subject to the strong coupling between the oscillation system and the driving source.

DOI: 10.1103/physreve.92.062906

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The coupling interaction between the driving source and the RLC network is explored and characterized as the effective impedance. The mathematical form of the derived effective impedance is verified to be identical to the meromorphic function of the singular billiards with a truncated basis. By using the derived impedance function, the resonant modes of the RLC network can be divided into the open-circuit and short-circuit states to manifest the evolution of eigenvalues and eigenstates from closed quantum billiards to the singular billiards with a truncated basis in the strongly coupled limit. The substantial differences of the wave patterns between the uncoupled and strongly coupled eigenmodes in the two-dimensional wave systems can be clearly revealed with the RLC network. Finally, the short-circuit resonant states are exploited to confirm that the experimental Chladni nodal-line patterns in the vibrating plate are the resonant modes subject to the strong coupling between the oscillation system and the driving source.

A truncated basis is employed to analyze the influence of the point interaction on the eigenvalues and eigenfunctions in quantum billiards. The point interaction is numerically confirmed to cause the shift in the eigenvalue that leads the original eigenfunctions to be superposed to form the new eigenfunction. The amplitude and nodal-line patterns of the eigenfunctions are found to vary significantly with the coupling strength. It is further verified that the point-driven Chladni plates can be employed to analogously manifest the noticeable dependence of the nodal-line pattern on the coupling strength. More importantly, the dispersion relation between the frequency and the wave number for the flexural wave can be precisely determined in the process of utilizing the point interaction to model the modern Chladni plates.

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.

We derive an explicit expression for the coupling constants of individual eigenstates of a closed billiard that is opened by attaching a waveguide. The Wigner time delay and the resonance positions resulting from the coupling constants are compared to an exact numerical calculation. Deviations can be attributed to evanescent modes in the waveguide and to the finite number of eigenstates taken into account. The influence of the shape of the billiard and of the boundary conditions at the mouth of the waveguide are also discussed. Finally we show that the mean value of the dimensionless coupling constants tends to the critical value when the eigenstates of the billiard follow random-matrix theory.

We present a method for numerically obtaining the positions, widths, and wave functions of resonance states in a two-dimensional billiard connected to a waveguide. For a rectangular billiard, we study the dynamics of three resonance poles lying separated from the other ones. As a function of increasing coupling strength between the waveguide and the billiard two of the states become trapped while the width of the third one continues to increase for all coupling strengths. This behavior of the resonance poles is reflected in the time delay function, which can be studied experimentally.

Several key advances in the understanding of quantum chaos have resulted from the study of billiard systems, in which a particle is confined in a two‐dimensional potential well. There has been many numerical studies of wave function characteristics, nodal line and nodal point features and current distributions and correlations. One finds that chaotic states may often be represented by Gaussian random waves which consist of a superposition of monochromatic plane waves traveling in different directions.Direct studies of quantum billiards like 2D mesoscopic ballistic semiconductor quantum dots are, however, unfeasible because of their smallness and because they are normally imbedded deeply in a semiconductor structure. The 2D quantum billiard may, however, be mapped onto a flat microwave cavity, which has allowed for direct measurements of wave functions, current distributions etc. Here we discuss an alternative unconventional realization of a quantum billiard. Thus we outline how electrical networks may be used for fundamental studies of wave function properties and transport in general and, more specifically, their mapping onto open quantum dots. In addition to the scientific case the network model has obvious pedagogical merits.