# Analytic Few Photon Scattering in Waveguide QED

## TL;DR

This paper introduces a diagrammatic Dyson series approach to analyze multi-photon scattering in waveguide QED, enabling analytic solutions and insights into light-matter entanglement and transition amplitude structures.

## Contribution

It develops a novel diagrammatic method for evaluating complex scattering amplitudes in waveguide QED with multiple photons, advancing analytic understanding.

## Key findings

- Two-photon scattering from a $ ext{Lambda}$-system shows altered pole structures when ground states are tuned.
- The method allows full specification of emitter-optical states for entanglement studies.
- Analytic results are obtained for multi-photon scattering processes.

## Abstract

We develop an approach to light-matter coupling in waveguide QED based upon scattering amplitudes evaluated via Dyson series. For optical states containing more than single photons, terms in this series become increasingly complex and we provide a diagrammatic recipe for their evaluation, which is capable of yielding analytic results. Our method fully specifies a combined emitter-optical state that permits investigation of light-matter entanglement generation protocols. We use our expressions to study two-photon scattering from a $\Lambda$-system and find that the pole structure of the transition amplitude is dramatically altered as the two ground states are tuned from degeneracy.