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Physical Review Letters

Classical irregular scattering and its quantum-mechanical implications

R. Blümel, Uzy Smilansky

Published Feb 1, 1988• 317 citations
Publisher

Abstract

We analyze the effect of irregular classical scattering on the corresponding quantum-mechanical scattering matrix. Using semiclassical arguments, we show that the fluctuations in the S matrix and the cross sections are consistent with a random-matrix description (Ericson fluctuations). The results are illustrated by a numerical solution of a simple quantum problem, whose classical counterpart displays irregular scattering.

DOI
10.1103/physrevlett.60.477

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We analyze the effect of irregular classical scattering on the corresponding quantum-mechanical scattering matrix. Using semiclassical arguments, we show that the fluctuations in the S matrix and the cross sections are consistent with a random-matrix description (Ericson fluctuations). The results are illustrated by a numerical solution of a simple quantum problem, whose classical counterpart displays irregular scattering.
A two-dimensional scattering potential represents the quantum extension of a diffractive lattice: a Dirac delta function with a modulated permeability along the y -axis. This model does not have an explicit classical analogue and quantum effects such as tunneling and diffraction play an important role. An analytical solution for the one-harmonic case is found. For the general case of an arbitrary number of harmonics a simple criterion is derived for the range of parameters where quantum chaos is permitted (but does not necessarily occur). The statistical properties of the S -matrix for the given model have been investigated. The deviations from the usual predictions for irregular scattering in the random matrix theory (RMT) framework have been found and are discussed.
Quantized, compact graphs are excellent paradigms for quantum chaos in bounded systems. Connecting them with leads to infinity, we show that they display all the features which characterize quantum chaotic scattering. We derive exact expressions for the scattering matrix, and an exact trace formula for the density of resonances, in terms of classical orbits, analogous to the semiclassical theory of chaotic scattering. A statistical analysis of the cross sections and resonance parameters compares well with the predictions of random matrix theory. Hence, this system is proposed as a convenient tool to study the generic behavior of chaotic scattering systems and their semiclassical description.
In quantum open systems, transport coejcients such as crosssections or conductance fluctuate as functions of the relevant parameters. These fluctuations give rise to complicated patterns. An important question is their possible fractality. In this paper we present a historical review of fractal fluctuations in quantum systems and their relation to the properties of their classical counterparts. We discuss the original semiclassical predictions and various numerical and experimental results. Some numerical simulations claim explanations based on pure quantum grounds. We thus report an argument that traces fractality back to the statistical properties of the S -matrix poles. Although full understanding of the reasons for fractal fluctuations might seema secondary problem, it could lead to a deeper comprehension of quantum mechanics and its interplay with classical mechanics.
A quantum-mechanical scattering theory is developed for the scattering by a potential of which strength is randomly fluctuating between two values with a switching rate γ/2. Within this model the T matrix averaged over the fluctuation is derived exactly, from which the averaged scattering cross section (ACS) is obtained by using the optical theorem. Explicit calculations of the ACS are carried out for three cases; (i) within the Born approximation, (ii) for a separable potential, (iii) for a short-range potential in a lattice. It is found that the low-energy ACS is enhanced in comparison with the static case for E/ℏ≲γ≲ℏ/(2ma2)E/\hbar{\lesssim}\gamma{\lesssim}\hbar/(2ma^{2})E/ℏ≲γ≲ℏ/(2ma2) because of the large inelastic cross section, where E is the energy of the incident particle with a mass m and a is the range of the potential.