  Dark-state polaritons (DSPs) based on the effect of electromagnetically
induced transparency are bosonic quasiparticles, representing the
superpositions of photons and atomic ground-state coherences. It has been
proposed that stationary DSPs are governed by the equation of motion closely
similar to the Schr\"{o}dinger equation and can be employed to achieve
Bose-Einstein condensation (BEC) with transition temperature orders of
magnitude higher than that of the atomic BEC. The stationary-DSP BEC is a
three-dimensional system and has a far longer lifetime than the
exciton-polariton BEC. In this work, we experimentally demonstrated the
stationary DSP dressed by the Rydberg-state dipole-dipole interaction (DDI).
The DDI-induced phase shift of the stationary DSP was systematically studied.
Notably, the experimental data are consistent with the theoretical predictions.
The phase shift can be viewed as a consequence of elastic collisions. In terms
of thermalization to achieve BEC, the $\mu$m$^2$-size interaction cross-section
of the DDI can produce a sufficient elastic collision rate for the stationary
DSPs. This work makes a substantial advancement toward the realization of the
stationary-DSP BEC.
