Analytical approach to higher-order correlation functions in U(1) symmetric systems

TL;DR

This paper presents a compact analytical method for calculating higher-order correlation functions in U(1) symmetric dissipative quantum systems, facilitating the analysis of photon statistics and correlations.

Contribution

It introduces a general $S$ matrix-based analytical solution for equal-time correlation functions applicable to a broad class of quantum systems with U(1) symmetry, including waveguide QED.

Findings

Provides a unified analytical framework for correlation functions

Develops a Python library for practical computations

Enables detailed study of photon correlations in complex systems

Abstract

We derive a compact analytical solution of the $n$th-order equal-time correlation functions by using scattering matrix ($S$ matrix) under a weak coherent state input. Our solution applies to any dissipative quantum system that respects the U(1) symmetry. We further extend our analytical solution into two categories depending on whether the input and output channels are identical. The first category provides a different path for studying cross-correlation and multiple-drive cases, while the second category is instrumental in studying waveguide quantum electrodynamics systems. Our analytical solution allows for easy investigation of the statistical properties of multiple photons even in complex systems. Furthermore, we have developed a user-friendly open-source library in Python known as the quantum correlation solver, and this tool provides a convenient means to study various dissipative quantum systems that satisfy the above-mentioned criteria. Our study enables using $S$ matrix to study the photonic correlation and advance the possibilities for exploring complex systems.