Tunable Fr\"ohlich Polarons of slow-light polaritons in a two-dimensional Bose-Einstein condensate

When an impurity interacts with a bath of phonons it forms a polaron. For increasing interaction strengths the mass of the polaron increases and it can become self-trapped. For impurity atoms inside an atomic Bose-Einstein condensate (BEC) the nature of this transition is subject of debate. While Feynman's variational approach predicts a sharp transition for light impurities, renormalization group studies always predict an extended intermediate-coupling region characterized by large phonon correlations. To investigate this intricate regime we suggest a versatile experimental setup that allows to tune both the mass of the impurity and its interactions with the BEC. The impurity is realized as a dark-state polariton (DSP) inside a quasi two-dimensional BEC. We show that its interactions with the Bogoliubov phonons lead to photonic polarons, described by the Bogoliubov-Fr\"ohlich Hamiltonian, and make theoretical predictions using an extension of a recently introduced renormalization group approach to Fr\"ohlich polarons.