  We proposed utilizing a medium with a high optical depth (OD) and a Rydberg
state of low principal quantum number, $n$, to create a weakly-interacting
many-body system of Rydberg polaritons, based on the effect of
electromagnetically induced transparency (EIT). We experimentally verified the
mean field approach to weakly-interacting Rydberg polaritons, and observed the
phase shift and attenuation induced by the dipole-dipole interaction (DDI). The
DDI-induced phase shift or attenuation can be viewed as a consequence of the
elastic or inelastic collisions among the Rydberg polaritons. Using a
weakly-interacting system, we further observed that a larger DDI strength
caused a width of the momentum distribution of Rydberg polaritons at the exit
of the system to become notably smaller as compared with that at the entrance.
In this study, we took $n =32$ and the atomic (or polariton) density of
5$\times10^{10}$ (or 2$\times10^{9}$) cm$^{-3}$. The observations demonstrate
that the elastic collisions are sufficient to drive the thermalization process
in this weakly-interacting many-body system. The combination of the $\mu$s-long
interaction time due to the high-OD EIT medium and the $\mu$m$^2$-size
collision cross section due to the DDI suggests a new and feasible platform for
the Bose-Einstein condensation of the Rydberg polaritons.
