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 in-
distinguishability 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 experimen-
tally observable with Rydberg atoms [2].
[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. 93, 233001 (2004)