Inhomogeneous broadening of a ground hyperfine transition in an optical trap

We report a theoretical study of the resonance frequency shift and the linewidth broadening of a ground hyperfine transition for an optically trapped atom. We use an adiabatic approximation to disentangle the internal and the external degrees of freedom of the atom. A magnetic dipole transition driven by a resonant microwave is considered in the Lamb-Dicke approximation. We obtain the lineshape of the transition as a sum of Rabi transitions between bound states of upper and lower hyperfine levels and make a transformation into an integral form. We consider the cesium clock transition in a far-detuned optical trap as a model case and find the frequency shift and line broadening by numerical integration. We find the general dependence of the shift and the broadening on the temperature, well depth, and the wavelength of the trap beam.