Sum and difference frequency generation in a two-level system with permanent dipole moments

We calculate the sum- and difference-frequency-generation (SFG and DFG) susceptibilities, (2)(-(1+2);1,2) and (2)(-(1-2);1,-2), of a homogeneously broadened two-level system with permanent dipole moments in the presence of pump fields at frequencies 1 and 2. Both SFG and DFG susceptibilities have two maxima at frequencies corresponding to the one- and two-photon resonances of the two-level system. Knowledge of the magnitude of one of the maxima of either SFG or DFG is sufficient for determining the magnitude of the second peak of that process and also the two peaks of the other process. In both processes the peak at the two-photon resonance has dominant contributions arising from off-diagonal elements of the density matrix. The interference between these contributions is constructive in SFG and destructive in DFG. Therefore, SFG is more efficient at the two-photon resonance than DFG. An expression for the ratio between these peaks is obtained. The peak of the spectrum of each process corresponding to the one-photon resonance has two main contributions, one of which is diagonal and the other off diagonal. In DFG the interference between the two contributions is constructive and in SFG the interference is destructive. Hence, DFG is more efficient in the vicinity of the one-photon resonance. However, since the off-diagonal contributions to SFG and DFG on the one-photon resonance are equal in magnitude (and differ by a phase difference of) the efficiencies of the two processes are equal when the duration of the pulses is comparable to or shorter than T1.