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arXiv:quant-ph/9805087 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Interfering resonances in a quantum billiard

Emil Persson, K. N. Pichugin, I. Rotter, Petr Šeba

Published Dec 1, 1998• 43 citations
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Abstract

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.

DOI
10.1103/physreve.58.8001

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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.
The poles of the S matrix and the wave functions of open two-dimensional quantum billiards with convex boundary of different shape are calculated by using the method of complex scaling. Two leads are attached to the cavities. The conductance of the cavities is calculated at energies with one, two, and three open channels in each lead. Bands of overlapping resonance states appear that are localized along the convex boundary of the cavities and contribute coherently to the conductance. These bands correspond to the whispering gallery modes known from classical calculations.
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.
The use of wave packet to analyze the dynamics of quantum mechanical systems is an increasingly important method to the study of the classical-quantum correspondence.Using the quantum Gaussian wave packet analysis method,we calculate the autocorrelation function of the rectangular billiard,the peak positions of the autocorrelation function match well with the periods of the classical periodic orbits,which show that the period of the classical orbits can be produced by the time-dependent quantum wave packet method.We also discuss wave packet revivals and fractional revivals in the rectangular billiard,the results show that there are exact revival for all wave packet at each revival time.We find additional cases of exact revivals with short revival times for zero-momentum wave packets initially located at special symmetry point inside the billiard.
We study kicked particle dynamics in a rectangular quantum billiard.The kicking potential is chosen as localized at the center of the billiard.The exact solution for the time-dependent Schrödinger equation for a single kicking period is derived.Using this solution, the time-dependence of the average kinetic energy and probability density as a function of spatial coordinates are computed.Different regimes for trapping of the particle in kicking area are analyzed.It is found that depending of the values of kicking parameters, the average kinetic energy can be a periodic or a monotonically growing function of time or can be suppressed.Such behavior is explained in terms of particle trapping regimes.Wave packet dynamics are also studied.
A simple expansion method for numerically calculating the energy levels and the corresponding wave functions of a quantum particle in a two-dimensional infinite potential well with arbitrary shape (quantum billiard) is presented. The method permits the study of quantum billiards in an introductory quantum mechanics course. According to the method, wave functions inside the billiard are expressed in terms of an expansion of a complete set of orthonormal functions defined in a surrounding rectangle for which the Dirichlet boundary conditions apply, while approximating the billiard boundary by a potential energy step of a sufficiently large size. Numerical implementations of the method are described and applied to determine the energies and wave functions for quarter-circle, circle, and triangle billiards. Finally, the expansion method is applied to investigate the quantum signatures of chaos in a classically chaotic generic-triangle billiard.