  The combination of high optical nonlinearity in the electromagnetically
induced transparency (EIT) effect and strong electric dipole-dipole interaction
(DDI) among the Rydberg-state atoms can lead to important applications in
quantum information processing and many-body physics. One can utilize the
Rydberg-EIT system in the strongly-interacting regime to mediate photon-photon
interaction or qubit-qubit operation. One can also employ the Rydberg-EIT
system in the weaklyinteracting regime to study the Bose-Einstein condensation
of Rydberg polaritons. Most of the present theoretical models dealt with the
strongly-interacting cases. Here, we consider the weaklyinteracting regime and
develop a mean field model based on the nearest-neighbor distribution. Using
the mean field model, we further derive the analytical formulas for the
attenuation coefficient and phase shift of the output probe field. The
predictions from the formulas are consistent with the experimental data in the
weakly-interacting regime, verifying the validity of our model. As the
DDI-induced phase shift and attenuation can be seen as the consequences of
elastic and inelastic collisions among particles, this work provides a very
useful tool for conceiving ideas relevant to the EIT system of
weakly-interacting Rydberg polaritons, and for evaluating experimental
feasibility.
