  We present an extension of our recent paper [Bienias et al., Phys. Rev. A 90,
053804 (2014)] in which we demonstrated the scattering properties and
bound-state structure of two Rydberg polaritons, as well as the derivation of
the effective low-energy many-body Hamiltonian. Here, we derive a microscopic
Hamiltonian describing the propagation of Rydberg slow light polaritons in one
dimension. We describe possible decoherence processes within a Master equation
approach, and derive equations of motion in a Schroedinger picture by using an
effective non-Hermitian Hamiltonian. We illustrate diagrammatic methods on two
examples: First, we show the solution for a single polariton in an external
potential by exact summation of Feynman diagrams. Secondly, we solve the two
body problem in a weakly interacting regime exactly.
