Force measurement via dark-state cooling

We analyze the ‘‘polarization-gradient-assisted velocity-selective coherent population-trapping’’ laser cooling scheme in the presence of a homogeneous force. In the limit of a slow rate of cooling this system may be treated using a set of rate equations after applying the secular approximation. This uses a basis of states which form a band structure resulting from the periodic optical potentials. We show that in the presence of a force the secular approximation may be retained, and derive the required modification to the rate equations. This is a density operator method of treating the action of a force on a particle in a band structure, and clearly shows reflections at the band edge. We solve for the steady-state momentum distribution, which provides information on the amount to which the presence of a force limits the achievable temperature. We solve also for the steady-state fluorescence and suggest that the fluorescence from atoms cooled with this scheme could be used as a force meter. © 1996 The American Physical Society.