Density-matrix equations and photon recoil for multistate atoms

We derive a Fokker-Planck equation (FPE) for the center-of-mass Wigner function of an atom with an arbitrary level structure, moving in a given three-dimensional light field. This FPE is obtained by eliminating adiabatically the internal degrees of freedom from the expansion of the equation of motion of the atomic density operator into a power series of photon momentum. The force and the diffusion tensor in the FPE may be computed by solving a set of differential equations closely related to the time-evolution equations of the density matrix for the internal state of the atom, as encountered in the conventional recoilless spectroscopy. By studying spontaneous, induced, and free-evolution terms in the equations of motion of the density operator for various level configurations, we devise algorithms that enable a computer to build all the required equations automatically for an arbitrary atomic level scheme.