Statistics of time delay and scattering correlation functions in chaotic systems II. Semiclassical Approximation

TL;DR

This paper uses semiclassical methods to compute energy-averaged correlation functions of the S-matrix in chaotic systems, deriving results that align with random matrix theory predictions for moments of the time delay matrix.

Contribution

It introduces a semiclassical approximation approach to calculate S-matrix correlation functions and moments of the time delay matrix in chaotic cavities without time-reversal symmetry.

Findings

Derived an infinite series expansion in energy difference with rational coefficients in M.

Calculated moments of the time delay matrix Q.

Confirmed agreement with random matrix theory predictions for the first eight moments.

Abstract

We consider $S$-matrix correlation functions for a chaotic cavity having $M$ open channels, in the absence of time-reversal invariance. Relying on a semiclassical approximation, we compute the average over $E$ of the quantities ${\rm Tr}[S^\dag(E-\epsilon)S(E+\epsilon)]^n$, for general positive integer $n$. Our result is an infinite series in $\epsilon$, whose coefficients are rational functions of $M$. From this we extract moments of the time delay matrix $Q=-i\hbar S^\dag dS/dE$, and check that the first 8 of them agree with the random matrix theory prediction from our previous paper.