The propagation of $N$ photons in one dimensional waveguides coupled to $M$ 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 $N$, $M$, 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 $N/M$) depends on the computed observable. While perfect reflection is obtained for $N/M cong 1$, photon-photon correlations are still resolved for ratios $N/M= 2/20$. 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.
The scattering of a flying photon by a two-level system ultrastrongly coupled to a one-dimensional photonic waveguide is studied numerically. The photonic medium is modeled as an array of coupled cavities and the whole system is analyzed beyond the r
otating wave approximation using Matrix Product States. It is found that the scattering is strongly influenced by the single- and multi-photon dressed bound states present in the system. In the ultrastrong coupling regime a new channel for inelastic scattering appears, where an incident photon deposits energy into the qubit, exciting a photon-bound state, and escaping with a lower frequency. This single-photon nonlinear frequency conversion process can reach up to 50% efficiency. Other remarkable features in the scattering induced by counter-rotating terms are a blueshift of the reflection resonance and a Fano resonance due to long-lived excited states
