  We present a method to systematically study multi-photon transmission in one
dimensional systems comprised of correlated quantum emitters coupled to input
and output waveguides. Within the Green's function approach of the scattering
matrix (S-matrix), we develop a diagrammatic technique to analytically obtain
the system's scattering amplitudes while at the same time visualise all the
possible absorption and emission processes. Our method helps to reduce the
significant effort in finding the general response of a many-body bosonic
system, particularly the nonlinear response embedded in the Green's functions.
We demonstrate our proposal through physically relevant examples involving
scattering of multi-photon states from two-level emitters as well as from
arrays of correlated Kerr nonlinear resonators in the Bose-Hubbard model.
