Clock shifts of optical transitions in ultracold atomic gases

We calculate the shift, due to interatomic interactions, of an optical transition in an atomic Fermi gas trapped in an optical lattice, as in recent experiments of Campbell {\it et al.}, Science {\bf 324}, 360 (2009). Using a pseudospin formalism to describe the density matrix of the internal two states of the optical transition, we derive a Bloch equation which incorporates both the spatial inhomogeneity of the probe laser field and the interatomic interactions. Expressions are given for the frequency shift as a function of the pulse duration, detuning of the probe laser, and the spatial dependence of the electric field of the probe beam. In the low temperature semiclassical regime, we find that the magnitude of the shift is proportional to the temperature.