Master equation with quantized atomic motion including dipole-dipole interactions*

We derive a markovian master equation for the internal dynamics of an ensemble of two-level atoms including all effects related to the quantization of their motion [1]. Our equation provides a unifying picture of the consequences of recoil and indistinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics, and is relevant for experiments with ultracold trapped atoms. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, and we find analytical formulas for a number of relevant states (Gaussian states, Fock states and thermal states). In particular, we show that the dipole-dipole interactions and cooperative photon emission can be modulated through the external state of motion. The effects predicted should be experimentally observable with Rydberg atoms [2]. \\[5pt] {[1] F.\ Damanet, D.\ Braun, and J.\ Martin, arXiv:1512.06676v2.} \\ {[2] K.\ Afrousheh, P.\ Bohlouli-Zanjani, D.\ Vagale, A.\ Mugford, M.\ Fedorov, and J.\ D.\ D.\ Martin, Phys.\ Rev.\ Lett.\ \textbf{93}, 233001 (2004).}