Distribution of the quantum mechanical time-delay matrix for a chaotic cavity

Abstract

We calculate the joint probability distribution of the Wigner-Smith time-delay matrix $Q=-i\hslash S^{-1}\partial S/\partial \epsilon$ and the scattering matrix $S$ for scattering from a chaotic cavity with ideal point contacts. Hereto we prove a conjecture by Wigner about the unitary invariance property of the distribution functional $P[S(\epsilon )]$ of energy dependent scattering matrices $S(\epsilon )$. The distribution of the inverse of the eigenvalues ${\tau }_1,...,{\tau }_N$ of $Q$ is found to be the Laguerre ensemble from random-matrix theory. The eigenvalue density $\rho (\tau )$ is computed using the method of orthogonal polynomials. This general theory has applications to the thermopower, magnetoconductance, and capacitance of a quantum dot.

17 pages, RevTeX; 3 figures included; To appear in Waves in Random Media (special issue on disordered electron systems)