Statistics of Impedance, Local Density of States, and Reflection in Quantum Chaotic Systems with Absorption

TL;DR

This paper derives the joint distribution of the real and imaginary parts of the local Green function in quantum chaotic systems with absorption, linking it to impedance and reflection statistics using random matrix theory.

Contribution

It provides exact distributions for beta=2 and beta=4 symmetry classes and an interpolation formula for beta=1, connecting theory with experimental and numerical data.

Findings

Exact distributions for beta=2 and beta=4 symmetry classes.

An interpolation formula for beta=1 that fits experimental data.

Established links between Green function statistics and impedance in chaotic systems.

Abstract

We are interested in finding the joint distribution function of the real and imaginary parts of the local Green function for a system with chaotic internal wave scattering and a uniform energy loss (absorption). For a microwave cavity attached to a single-mode antenna the same quantity has a meaning of the complex cavity impedance. Using the random matrix approach, we relate its statistics to that of the reflection coefficient and scattering phase and provide exact distributions for systems with beta=2 and beta=4 symmetry class. In the case of beta=1 we provide an interpolation formula which incorporates all known limiting cases and fits excellently available experimental data as well as diverse numeric tests.