Simple analytical model describing the collective nonlinear response of an ensemble of two-level emitters weakly coupled to a waveguide

We model and investigate the collective nonlinear optical response of an ensemble of two-level emitters that are weakly coupled to a single-mode waveguide. Our approach generalizes the insight that photon-photon correlations in the light scattered by a single two-level emitter result from two-photon interference to the case of many emitters. Using our model, we study different configurations for probing the nonlinear response of the ensemble, e.g., through the waveguide or via external illumination, and derive analytical expressions for the second-order quantum coherence function, $g^{(2)}(\tau)$, as well as for the squeezing spectrum of the output light in the waveguide, $S_\theta(\omega)$. For the transmission of resonant guided light, we recover the same predictions as previously made with far more involved theoretical models when analyzing experimental results regarding $g^{(2)}(\tau)$ (Prasad et al. [1]) and $S_\theta(\omega)$ (Hinney et al. [2]). We also study the transmission of light that is detuned from the transition of the two-level emitter, a situation that we recently studied experimentally (Cordier et al. [3]). Our model predictions show how the collectively enhanced nonlinear response of weakly coupled emitters can be harnessed to generate non-classical states of light using ensembles ranging from a few to many emitters.