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

TL;DR

This paper derives the joint probability distribution of the Wigner-Smith time-delay matrix and scattering matrix for a chaotic cavity, confirming a conjecture and connecting to random-matrix theory with applications in quantum dot physics.

Contribution

It proves Wigner's conjecture on the distribution's invariance and characterizes the eigenvalue distribution of the time-delay matrix using orthogonal polynomials.

Findings

Eigenvalues of the time-delay matrix follow the Laguerre ensemble.

Distribution functional of energy-dependent scattering matrices is unitary invariant.

Method applicable to quantum dot thermopower, magnetoconductance, and capacitance.

Abstract

We calculate the joint probability distribution of the Wigner-Smith time-delay matrix $Q=-i\hbar S^{-1} \partial S/\partial \epsilon$ 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.