Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: Random matrix approach for systems with broken time-reversal invariance

TLDR

The paper investigates how random‑matrix theory can describe statistical properties of the S‑matrix in open chaotic quantum systems, focusing on the stochastic (Heidelberg) approach to scattering. The authors employ random‑matrix theory and supersymmetry to analytically derive expressions for the density of S‑matrix poles, eigenphase correlation functions, and partial‑delay‑time distributions in chaotic quantum systems with broken time‑reversal symmetry coupled to M open channels. They use these results to obtain the distributions of partial delay times with respect to energy and arbitrary external parameters.

Abstract

Assuming the validity of random matrices for describing the statistics of a closed chaotic quantum system, we study analytically some statistical properties of the S-matrix characterizing scattering in its open counterpart. In the first part of the paper we attempt to expose systematically ideas underlying the so-called stochastic (Heidelberg) approach to chaotic quantum scattering. Then we concentrate on systems with broken time-reversal invariance coupled to continua via M open channels. By using the supersymmetry method we derive: (i) an explicit expression for the density of S-matrix poles (resonances) in the complex energy plane (ii) an explicit expression for the parametric correlation function of densities of eigenphases of the S-matrix. We use it to find the distribution of derivatives of these eigenphases with respect to the energy ("partial delay times" ) as well as with respect to an arbitrary external parameter.