Moments of the Wigner delay times

TL;DR

This paper provides a semiclassical derivation confirming that the moments of Wigner delay times in chaotic systems align with predictions from random matrix theory, using classical trajectories and correlation functions.

Contribution

It introduces a semiclassical approach to derive moments of delay times, validating random matrix theory results through trajectory correlations and polynomial equations.

Findings

Semiclassical derivation matches random matrix predictions.

Correlation functions of scattering matrices are key to the analysis.

Scattering matrix unitarity holds to all orders in semiclassical approximation.

Abstract

The Wigner time delay is a measure of the time spent by a particle inside the scattering region of an open system. For chaotic systems, the statistics of the individual delay times (whose average is the Wigner time delay) are thought to be well described by random matrix theory. Here we present a semiclassical derivation showing the validity of random matrix results. In order to simplify the semiclassical treatment, we express the moments of the delay times in terms of correlation functions of scattering matrices at different energies. In the semiclassical approximation, the elements of the scattering matrix are given in terms of the classical scattering trajectories, requiring one to study correlations between sets of such trajectories. We describe the structure of correlated sets of trajectories and formulate the rules for their evaluation to the leading order in inverse channel number. This allows us to derive a polynomial equation satisfied by the generating function of the moments. Along with showing the agreement of our semiclassical results with the moments predicted by random matrix theory, we infer that the scattering matrix is unitary to all orders in the semiclassical approximation.