  We develop an approach to light-matter coupling in waveguide QED based upon
scattering amplitudes evaluated via Dyson series. For optical states containing
more than single photons, terms in this series become increasingly complex and
we provide a diagrammatic recipe for their evaluation, which is capable of
yielding analytic results. Our method fully specifies a combined
emitter-optical state that permits investigation of light-matter entanglement
generation protocols. We use our expressions to study two-photon scattering
from a $\Lambda$-system and find that the pole structure of the transition
amplitude is dramatically altered as the two ground states are tuned from
degeneracy.
