Dipole-dipole interaction in three-dimensional optical lattices

We compute the near-resonant dipole-dipole interaction between atoms in a three-dimensional optical lattice. After a general derivation, we concentrate specifically on the case of a J = 1/2 --> J = 3/2 off-resonant transition, in which case the excited electronic states can be adiabatically eliminated. We present in detail the spatial dependence of the resulting effective dipole-dipole potential, and discuss the physical origin of its anisotropy. In addition to impacting the center-of-mass motion of the atoms, the effective dipole-dipole potential can induce electronic transitions within one or both atoms. The symmetries of the matrix elements corresponding to these various mechanisms are also commented upon.