Master equation for collective spontaneous emission with quantized atomic motion

We derive a markovian master equation for the internal dynamics of an ensemble of two-level atoms including the quantization of their motion. Our equation provides a unifying picture of the effects of recoil and indistinguishability of atoms beyond the Lamb-Dicke regime on both their dissipative and conservative dynamics. We give general expressions for the decay rates and the dipole-dipole shifts for any motional states, generalizing those in Ref. [1]. We find closed-form formulas for a number of relevant states (gaussian states, Fock states and thermal
states). In particular, we show that dipole-dipole interactions and cooperative photon emission [2] can be modulated through the external state of motion. As an application of our general formalism, we study the spatial Pauli blocking of two fermionic atoms beyond the Lamb-Dicke regime [3].
[1] G. S. Agarwal, Springer Tracts In Modern Physics 70, 1 (1974).
[2] R. H. Dicke, Phys. Rev. 93, 99 (1954).
[3] R. M. Sandner, M. Müller, A. J. Daley & P. Zoller, Phys. Rev. A 84, 043825 (2011).