Simplified solution for the energy eigenvalues of a two-state atom in a sinusoidal field

In a famous 1965 paper [Phys. Rev. 138, B979], J. H. Shirley calculates the time-averaged transition probability for an atom in a periodic field in terms of the eigenvectors and eigenvalues of the time-independent Floquet Hamiltonian. It is shown in the present paper that, for the specific case considered by Shirley, a two-state atom in a sinusoidal field, Shirley's solution can be simplified considerably. Also, Shirley's results are generalized to include, for an arbitrary atom-field system, the time-varying components of the transition probability as well.