Quantum Scattering and Transport in Classically Chaotic Cavities: An Overview of Past and New Results

TL;DR

This paper develops a statistical framework for quantum scattering in chaotic cavities, applying maximum-entropy methods to predict conductance properties in mesoscopic systems, with strong agreement to numerical simulations.

Contribution

It introduces the Poisson's kernel ensemble for scattering matrices, incorporating system-specific averages and providing accurate predictions for quantum conductance statistics.

Findings

Poisson's kernel effectively models quantum conductance fluctuations.

The theory matches numerical solutions of the Schrödinger equation.

It describes systems even when stationarity and ergodicity are not fully met.

Abstract

We develop a statistical theory that describes quantum-mechanical scattering of a particle by a cavity when the geometry is such that the classical dynamics is chaotic. This picture is relevant to a variety of physical systems, ranging from atomic nuclei to mesoscopic systems and microwave cavities; the main application to be discussed in this contribution is to electronic transport through mesoscopic ballistic structures or quantum dots. The theory describes the regime in which there are two distinct time scales, associated with a prompt and an equilibrated response, and is cast in terms of the matrix of scattering amplitudes S. We construct the ensemble of S matrices using a maximum-entropy approach which incorporates the requirements of flux conservation, causality and ergodicity, and the system-specific average of S which quantifies the effect of prompt processes. The resulting ensemble, known as Poisson's kernel, is meant to describe those situations in which any other information is irrelevant. The results of this formulation have been compared with the numerical solution of the Schroedinger equation for cavities in which the assumptions of the theory hold. The model has a remarkable predictive power: it describes statistical properties of the quantum conductance of quantum dots, like its average, its fluctuations, and its full distribution in several cases. We also discuss situations that have been found recently, in which the notion of stationarity and ergodicity is not fulfilled, and yet Poisson's kernel gives a good description of the data. At the present moment we are unable to give an explanation of this fact.