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arXiv:cond-mat/0109344 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Density of proper delay times in chaotic and integrable quantum billiards

M. G. A. Crawford, Piet W. Brouwer

Published Jan 25, 2002• 7 citations
arXivHTMLPDF

Abstract

We calculate the density P(tau) of the eigenvalues of the Wigner-Smith time delay matrix for two-dimensional rectangular and circular billiards with one opening. For long times, the density of these so-called "proper delay times" decays algebraically, in contradistinction to chaotic quantum billiards for which P(tau) exhibits a long-time cutoff.

DOI
10.1103/physreve.65.026221

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We calculate the density P(tau) of the eigenvalues of the Wigner-Smith time delay matrix for two-dimensional rectangular and circular billiards with one opening. For long times, the density of these so-called "proper delay times" decays algebraically, in contradistinction to chaotic quantum billiards for which P(tau) exhibits a long-time cutoff.
The study of classical Liouville density arises naturally in chaotic systems where a probabilistic treatment is more appropriate. In this review, we show that the evolution of a density projected onto the configuration space has a quantum connection in billiard systems. The eigenvalues and eigenfunctions of the concerned evolution operator have an approximate one-to-one correspondence with the quantum Neumann eigenvalues and eigenfunctions. For exceptional billiard shapes such as the rectangle, this correspondence is even exact. Despite the approximate nature of the correspondence, we demonstrate that the exact quantum Neumann eigenstates can be used to expand and evolve an arbitrary classical projected-density. For the rectangular and stadium billiards, results are compared with the actual evolution of the density using classical trajectories and are found to be satisfactory.
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.
We perform numerical studies of wave packet propagation through open quantum billiards whose classical counterparts exhibit regular and chaotic dynamics. We show that for $t\ensuremath{\lesssim}{\ensuremath{\tau}}_{H}({\ensuremath{\tau}}_{H}$ being the Heisenberg time), the features in the transmitted and reflected currents are directly related to specific classical trajectories connecting the billiard leads. When $t\ensuremath{\gtrsim}{\ensuremath{\tau}}_{H},$ the calculated quantum-mechanical current starts to deviate from its classical counterpart, with the decay rate obeying a power law that depends on the number of decay channels. In a striking contrast to the classical escape from chaotic and regular systems (exponentially fast ${e}^{\ensuremath{-}\ensuremath{\gamma}t}$ for the former versus power-law ${t}^{\ensuremath{-}\ensuremath{\xi}}$ for the latter), the asymptotic decay of the corresponding quantum systems does not show a qualitative difference.
Two-dimensional quantum billiards are one of the most important paradigms for exploring the connection between quantum and classical worlds. Researchers are mainly focused on nonintegrable and irregular shapes to understand the quantum characteristics of chaotic billiards. The emergence of the scarred modes relevant to unstable periodic orbits (POs) is one intriguing finding in nonintegrable quantum billiards. On the other hand, stable POs are abundant in integrable billiards. The quantum wavefunctions associated with stable POs have been shown to play a key role in ballistic transport. A variety of physical systems, such as microwave cavities, optical fibers, optical resonators, vibrating plates, acoustic waves, and liquid surface waves, are used to analogously simulate the wave properties of quantum billiards. This article gives a comprehensive review for the subtle connection between the quantum level clustering and the classical POs for three integrable billiards including square, equilateral triangle, and circular billiards.
We study the low-energy quantum spectra of two-dimensional rectangular billiards with a small but finite-size scatterer inside. We start by examining the spectral properties of billiards with a single pointlike scatterer. The problem is formulated in terms of the self-adjoint extension theory of functional analysis. The condition for the appearance of so-called wave chaos is clarified. We then relate the pointlike scatterer to a finite-size scatterer through an appropriate truncation of the basis. We show that the signature of wave chaos in low-energy states is most prominent when the scatterer is weakly attractive. As an illustration, numerical results of a rectangular billiard with a small rectangular scatterer inside are exhibited. \textcopyright{} 1996 The American Physical Society.