# Ultrastrong coupling few-photon scattering theory

## TL;DR

This paper develops a theoretical framework for analyzing single-photon scattering in a system where a two-level quantum emitter is ultrastrongly coupled to a one-dimensional waveguide, relevant for quantum optics and superconducting circuits.

## Contribution

It introduces a combined polaron transformation and scattering theory approach to accurately compute photon scattering properties across a broad coupling range, including near the inelastic scattering regime.

## Key findings

- Accurately predicts resonance frequencies, lineshapes, and linewidths for ultrastrong coupling.
- Validates the method up to coupling strength α=0.3, close to the Toulouse point.
- Models experimental results with superconducting circuits.

## Abstract

We study the scattering of photons by a two-level system ultrastrongly coupled to a one-dimensional waveguide. Using a combination of the polaron transformation with scattering theory we can compute the one-photon scattering properties of the qubit for a broad range of coupling strengths, estimating resonance frequencies, lineshapes and linewidths. We validate numerically and analytically the accuracy of this technique up to $\alpha=0.3$, close to the Toulouse point $\alpha=1/2$, where inelastic scattering becomes relevant. These methods model recent experiments with superconducting circuits [P. Forn-D{\'\i}az et al., Nat. Phys. (2016)].