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 $Q=-i\hbar S^{-1} \partial S/\partial ε$ 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 $P[S(ε)]$ of energy dependent scattering matrices $S(ε)$. The distribution of the inverse of the eigenvalues $τ_1,...,τ_N$ of $Q$ is found to be the Laguerre ensemble from random-matrix theory. The eigenvalue density $ρ(τ)$ is computed using the method of orthogonal polynomials. This general theory has applications to the thermopower, magnetoconductance, and capacitance of a quantum dot.