Many-photon scattering and entangling in a waveguide with a Λ-type atom

We develop the analytical theory that describes simultaneous transmission of several photons through a waveguide coupled to a $\mathrm{\Lambda }$-type atom. We show that after transmission of a short few-photon pulse, the final state of the atom and all the photons is a genuine multipartite entangled state belonging to the W class. The parameters of the input pulse are optimized to maximize the efficiency of three- and four-partite W-state production.