Analytic solution to N vs. M photon phase control in an open two-level system

Decoherence effects on the traditional N vs. M photon coherent control of a two-level system are investigated, with 1 vs. 3 used as a specific example. The problem reduces to that of a two-level system interacting with a single mode field, but with an effective Rabi frequency that depends upon the fundamental and third harmonic fields. The resultant analytic control solution is explored for a variety of parameters, with emphasis on the dependence of control on the relative phase of the lasers. The generalization to off-resonant cases is noted.