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

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.Comment: Comments are wellcom

Scattering in the ultrastrong regime: nonlinear optics with one photon

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 rotating 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 statesComment: 5+4 page

Single photons by quenching the vacuum

Heisenberg's uncertainty principle implies that the quantum vacuum is not empty but fluctuates. These fluctuations can be converted into radiation through nonadiabatic changes in the Hamiltonian. Here, we discuss how to control this vacuum radiation, engineering a single-photon emitter out of a two-level system (2LS) ultrastrongly coupled to a finite-band waveguide in a vacuum state. More precisely, we show the 2LS nonlinearity shapes the vacuum radiation into a nonGaussian superposition of even and odd cat states. When the 2LS bare frequency lays within the band gaps, this emission can be well approximated by individual photons. This picture is confirmed by a characterization of the ground and bound states, and a study of the dynamics with matrix product states and polaron Hamiltonian methods.Comment: 9 pages, 10 figure

One- and two-photon scattering from generalized V-type atoms

The one- and two-photon scattering matrix S is obtained analytically for a one-dimensional waveguide and a point-like scatterer with N excited levels (generalized V -type atom). We argue that the two-photon scattering matrix contains sufficient information to distinguish between different level structures which are equivalent for single-photon scattering, such as a V -atom with N = 2 excited levels and two two-level systems. In particular, we show that the scattering with the V -type atom exhibits a destructive interference effect leading to two-photon Coupled-Resonator-Induced Transparency, where the nonlinear part of the two-photon scattering matrix vanishes when each incident photon fulfills a single-photon condition for transparency

Emergent Causality and the N-photon Scattering Matrix in Waveguide QED

In this work we discuss the emergence of approximate causality in a general setup from waveguide QED -i.e. a one-dimensional propagating field interacting with a scatterer. We prove that this emergent causality translates into a structure for the N-photon scattering matrix. Our work builds on the derivation of a Lieb-Robinson-type bound for continuous models and for all coupling strengths, as well as on several intermediate results, of which we highlight (i) the asymptotic independence of space-like separated wave packets, (ii) the proper definition of input and output scattering states, and (iii) the characterization of the ground state and correlations in the model. We illustrate our formal results by analyzing the two-photon scattering from a quantum impurity in the ultrastrong coupling regime, verifying the cluster decomposition and ground-state nature. Besides, we generalize the cluster decomposition if inelastic or Raman scattering occurs, finding the structure of the S-matrix in momentum space for linear dispersion relations. In this case, we compute the decay of the fluorescence (photon-photon correlations) caused by this S-matrix