# Propagating wave correlations in complex systems

## TL;DR

This paper introduces a semiclassical method for computing wave correlation functions in complex systems, linking classical dynamics with wave fluctuations, applicable in quantum and classical wave contexts.

## Contribution

It provides explicit algorithms for calculating wave correlation functions using classical dynamics and characterizes fluctuations through ensemble averages, applicable to complex quantum and wave systems.

## Key findings

- Derived explicit expressions relating wave correlation fluctuations to classical correlations.
- Applied methods to quantum maps modeling chaotic systems, demonstrating practical utility.
- Showed the approach's relevance to both quantum mechanics and classical wave problems.

## Abstract

We describe a novel approach for computing wave correlation functions inside finite spatial domains driven by complex and statistical sources. By exploiting semiclassical approximations, we provide explicit algorithms to calculate the local mean of these correlation functions in terms of the underlying classical dynamics. By defining appropriate ensemble averages, we show that fluctuations about the mean can be characterised in terms of classical correlations. We give in particular an explicit expression relating fluctuations of diagonal contributions to those of the full wave correlation function. The methods have a wide range of applications both in quantum mechanics and for classical wave problems such as in vibro-acoustics and electromagnetism. We apply the methods here to simple quantum systems, so-called quantum maps, which model the behaviour of generic problems on Poincar\'e sections. Although low-dimensional, these models exhibit a chaotic classical limit and share common characteristics with wave propagation in complex structures.