Phonon-mediated interactions and polaron formation of slow-light polaritons in a BEC

We study the motion of dark-state polaritons (DSP) in a Bose-Einstein condensate. DSPs are formed in an atomic ensemble interacting in a $\Lambda$-type configuration with two light fields under conditions of electromagnetically induced transparency. In particular, we consider the ground-state atoms to form a BEC which can be well described by a macroscopic Gross-Pitaevskii wavefunction. Taking into account the interaction of pairs of ground-state atoms and between ground and spin-state atoms leads to the formation of polaronic quasi-particles consisting of DSPs and Bogoliubov phonons. In additon, the coupling to phonons results into a coupling between dark and bright-state polaritons as well as into phonon-mediated interactions between DSPs.