Distribution of the Wigner-Smith time-delay matrix for chaotic cavities with absorption and coupled Coulomb gases

TL;DR

This paper derives the distribution of the Wigner-Smith time delay matrix for chaotic cavities with absorption, using random matrix theory, and analyzes its large N behavior through Coulomb gas models, providing insights into quantum scattering properties.

Contribution

It presents a novel derivation of the full distribution of the Wigner-Smith matrix for chaotic cavities with absorption across different symmetry classes, including large N analysis.

Findings

Derived the distribution of the Wigner-Smith matrix for chaotic cavities with absorption.

Analyzed large N properties using Coulomb gas models.

Studied statistical properties of the Wigner time delay with absorption.

Abstract

Within the random matrix theory approach to quantum scattering, we derive the distribution of the Wigner-Smith time delay matrix $\mathcal{Q}$ for a chaotic cavity with uniform absorption, coupled via $N$ perfect channels. In the unitary class $\beta=2$ we obtain a compact expression for the distribution of the full matrix in terms of a matrix integral. In the other symmetry classes we derive the joint distribution of the eigenvalues. We show how the large $N$ properties of this distribution can be analysed in terms of two interacting Coulomb gases living on two different supports. As an application of our results, we study the statistical properties of the Wigner time delay $\tau_{\mathrm{W}} = \mathrm{tr}[\mathcal{Q}]/N$ in the presence of absorption.