  We study the propagation of Rydberg slow light polaritons through an atomic
medium for intermediate interactions. Then, the dispersion relation for the
polaritons is well described by the slow light velocity alone, which allows for
an analytical solution for arbitrary shape of the atomic cloud. We demonstrate
the connection of Rydberg polaritons to the behavior of a conventional Kerr
nonlinearity for weak interactions, and determine the leading quantum
corrections for increasing interactions. We propose an experimental setup which
allows one to measure the effective two-body interaction potential between slow
light polaritons as well as higher body interactions.
