Diagrammatic Approach to Scattering of Multi-Photon States in Waveguide QED

We give an exposure to diagrammatic techniques in waveguide QED systems. A particular emphasis is placed on the systems with delayed coherent quantum feedback. Specifically, we show that the $N$-photon scattering matrices in single-qubit waveguide QED systems, within the rotating wave approximation, admit for a parametrization in terms of $N-1$-photon effective vertex functions and provide a detailed derivation of a closed hierarchy of generalized Bethe-Salpeter equations satisfied by these vertex functions. The advantage of this method is that the above mentioned integral equations hold independently of the number of radiation channels, their bandwidth, the dispersion of the modes they are supporting, and the structure of the radiation-qubit coupling interaction, thus enabling one to study multi-photon scattering problems beyond the Born-Markov approximation. Further, we generalize the diagrammatic techniques to the systems containing more than a single emitter by presenting an exact set of equations governing the generic two and three-photon scattering operators. The above described theoretical machinery is then showcased on the example of a three-photon scattering on a giant acoustic atom, recently studied experimentally [Nat. Phys. 15, 1123 (2019)].