A nearly closed ballistic billiard with random boundary transmission

TL;DR

This paper develops a supersymmetric nonlinear sigma-model to analyze the statistical properties of a circular billiard with random boundary coupling, relevant for mesoscopic systems like quantum dots and optical cavities.

Contribution

It introduces a novel theoretical framework for studying mesoscopic billiards with random boundary transmission, enabling analysis of measurable quantities like the local density of states.

Findings

Derived a supersymmetric nonlinear sigma-model for random boundary coupling

Provided analytical results for the local density of states in the system

Applicable to various mesoscopic systems with boundary randomness

Abstract

A variety of mesoscopic systems can be represented as a billiard with a random coupling to the exterior at the boundary. Examples include quantum dots with multiple leads, quantum corrals with different kinds of atoms forming the boundary, and optical cavities with random surface refractive index. The specific example we study is a circular (integrable) billiard with no internal impurities weakly coupled to the exterior by a large number of leads with one channel open in each lead. We construct a supersymmetric nonlinear $\sigma$-model by averaging over the random coupling strengths between bound states and channels. The resulting theory can be used to evaluate the statistical properties of any physically measurable quantity in a billiard. As an illustration, we present results for the local density of states.