Abstract
The propagation of photons in one dimensional waveguides coupled to qubits is discussed, both in the strong and ultrastrong qubit-waveguide coupling. Special emphasis is placed on the characterisation of the nonlinear response and its linear limit for the scattered photons as a function of , , qubit inter distance and light-matter coupling. The quantum evolution is numerically solved via the Matrix Product States technique. Both the time evolution for the field and qubits is computed. The nonlinear character (as a function of ) depends on the computed observable. While perfect reflection is obtained for , photon-photon correlations are still resolved for ratios . Inter-qubit distance enhances the nonlinear response. Moving to the ultrastrong coupling regime, we observe that inelastic processes are \emph{robust} against the number of qubits and that the qubit-qubit interaction mediated by the photons is qualitatively modified. The theory developed in this work modelises experiments in circuit QED, photonic crystals and dielectric waveguides.