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The propagation of $N$ photons in one dimensional waveguides coupled to $M$qubits is discussed, both in the strong and ultrastrong qubit-waveguidecoupling. Special emphasis is placed on the characterisation of the nonlinearresponse and its linear limit for the scattered photons as a function of $N$,$M$, qubit inter distance and light-matter coupling. The quantum evolution isnumerically solved via the Matrix Product States technique. Both the timeevolution for the field and qubits is computed. The nonlinear character (as afunction of $N/M$) depends on the computed observable. While perfect reflectionis obtained for $N/M \cong 1$, photon-photon correlations are still resolvedfor ratios $N/M= 2/20$. Inter-qubit distance enhances the nonlinear response.Moving to the ultrastrong coupling regime, we observe that inelastic processesare \emph{robust} against the number of qubits and that the qubit-qubitinteraction mediated by the photons is qualitatively modified. The theorydeveloped in this work modelises experiments in circuit QED, photonic crystalsand dielectric waveguides.

Nonlinear quantum optics in the (ultra)strong light-matter coupling