Theory of Three-Photon Transport Through a Weakly Coupled Atomic Ensemble

Understanding multi-photon interactions in non-equilibrium quantum systems is an outstanding challenge in quantum optics. In this work, we develop an analytical and diagrammatic framework to explore three-photon interactions in atomic ensembles weakly coupled to a one-dimensional waveguide. Taking advantage of the weak coupling, we use our diagrammatic framework to perform perturbation theory and calculate the leading-order contributions to the three-photon wavefunction, which would otherwise be intractable. We then compute the outgoing photon wavefunction of a resonantly driven atomic ensemble, with photon-photon interactions truncated up to three photons. Our formulation not only captures the individual transmission of photons but also isolates the connected S-matrix elements that embody genuine photon-photon correlations. Through detailed analysis, we obtain the analytic expressions of the connected third-order correlation function and the third-order electric-field-quadrature cumulant, which reveal non-Gaussian signatures emerging from the interplay of two- and three-photon processes. We also calculate the optical depth where non-Gaussian photon states can be observed. Numerical simulations based on a cascaded master equation validate our analytical predictions on a small-scale system. These results provide a formalism to further explore non-equilibrium quantum optics in atomic ensembles and extend this to the regime of non-Gaussian photon transport.