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Physical Review A

Quantum corrections for chaotic scattering

J. H. Jensen

Published Jun 1, 1992• 9 citations
Publisher

Abstract

The leading quantum corrections for the scattering of a particle in one dimension from a pulsed potential are discussed. Depending on the model parameters, the scattering may be either regular or chaotic. The quantum corrections are obtained by applying a semiclassical formalism that gives the distribution of scattered momenta down to a resolution of order ${\mathrm{\ensuremath{\Elzxh}}}^{4/5}$. At this resolution, the main quantum effect is the softening of the classical rainbow singularities into peaks with widths of order ${\mathrm{\ensuremath{\Elzxh}}}^{2/3}$. Quantum corrections for examples of both regular and chaotic scattering are compared with exact quantum results, and a criterion is given for estimating the resolution at which the corrections first become significant.

DOI
10.1103/physreva.45.8530

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The leading quantum corrections for the scattering of a particle in one dimension from a pulsed potential are discussed. Depending on the model parameters, the scattering may be either regular or chaotic. The quantum corrections are obtained by applying a semiclassical formalism that gives the distribution of scattered momenta down to a resolution of order \ensuremath\Elzxh4/5{\mathrm{\ensuremath{\Elzxh}}}^{4/5}\ensuremath\Elzxh4/5. At this resolution, the main quantum effect is the softening of the classical rainbow singularities into peaks with widths of order \ensuremath\Elzxh2/3{\mathrm{\ensuremath{\Elzxh}}}^{2/3}\ensuremath\Elzxh2/3. Quantum corrections for examples of both regular and chaotic scattering are compared with exact quantum results, and a criterion is given for estimating the resolution at which the corrections first become significant.
A semiclassical formula that gives scattering cross sections down to an angular resolution of order ${\mathrm{\ensuremath{\Elzxh}}}^{4/5}$ is derived. The formula describes rainbow peaks, which have widths of order ${\mathrm{\ensuremath{\Elzxh}}}^{2/3}$, but not other interference effects, which give structure on a scale of order \ensuremath{\Elzxh}. For cross sections with many rainbow peaks, it can be easier to evaluate than conventional uniform approximations. As an example, the formula is applied to a simple two-dimensional model.
We elucidate the basic physical mechanisms responsible for the quantum-classical transition in one-dimensional, bounded chaotic systems subject to unconditioned environmental interactions. We show that such a transition occurs due to the dual role of noise in regularizing the semiclassical Wigner function and averaging over fine structures in classical phase space. The results are interpreted in the context of applying recent advances in the theory of measurement and open systems to the semiclassical quantum regime. We use these methods to show how a local semiclassical picture is stabilized and can then be approximated by a classical distribution at later times. The general results are demonstrated explicitly via high-resolution numerical simulations of the quantum master equation for a chaotic Duffing oscillator.
We consider scattering of a free quantum particle on a singular potential with rather arbitrary geometry of the boundaries of its support. In the classical limit ¯h = 0 this problem reduces to the well known problem of chaotic scattering. The universal estimates for the stability of the scattering amplitudes are derived. The application of the obtained results to the mesoscopic systems and quantum chaos are discussed. We also discuss a possibility of experimental verification of the obtained results.
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.