Details and Observables of Three-Dimensional Scattering between Two Rydberg Polaritons

The properties and applications of Rydberg polaritons, whereby novel quantum optical phenomena can arise from combining the interactions between Rydberg atoms with control associated with electromagnetically-induced transparency (EIT), have received copious attention recently. Previous works on scattering between two polaritons assumed a 1-D geometry\footnote{P. Bienias, et al., Scattering resonances and bound states for strongly interacting Rydberg polaritons, Phys. Rev. A \textbf{90}, 053804 (2014)}, and then a 3-D treatment was undertaken with the explicit goal of predicting the Efimov effect between three equal-mass polaritons\footnote{M. J. Gullans, et al., Efimov States of Strongly Interacting Photons, Phys. Rev. Lett. \textbf{119}, 233601 (2017)}. We present a formalism for understanding the details of basic two-polariton scattering in three dimensions (including such observables as the cross section and scattering lengths), with the goal of supplementing the current theories of the bound states of Rydberg polaritons.