# Emergent Causality and the N-photon Scattering Matrix in Waveguide QED

## TL;DR

This paper explores how approximate causality emerges in waveguide QED systems and influences the structure of the N-photon scattering matrix, with implications for understanding photon interactions and correlations.

## Contribution

It proves a Lieb-Robinson-type bound for continuous models and characterizes the N-photon scattering matrix structure in waveguide QED.

## Key findings

- Verified cluster decomposition in two-photon scattering
- Analyzed ground state and correlations in ultrastrong coupling regime
- Derived S-matrix structure for inelastic and Raman scattering

## Abstract

In this work we discuss the emergence of approximate causality in a general setup from waveguide QED -i.e. a one-dimensional propagating field interacting with a scatterer. We prove that this emergent causality translates into a structure for the N-photon scattering matrix. Our work builds on the derivation of a Lieb-Robinson-type bound for continuous models and for all coupling strengths, as well as on several intermediate results, of which we highlight (i) the asymptotic independence of space-like separated wave packets, (ii) the proper definition of input and output scattering states, and (iii) the characterization of the ground state and correlations in the model. We illustrate our formal results by analyzing the two-photon scattering from a quantum impurity in the ultrastrong coupling regime, verifying the cluster decomposition and ground-state nature. Besides, we generalize the cluster decomposition if inelastic or Raman scattering occurs, finding the structure of the S-matrix in momentum space for linear dispersion relations. In this case, we compute the decay of the fluorescence (photon-photon correlations) caused by this S-matrix.