Four-wave parametric interactions in a strongly driven two-level system

We present an analysis of four-wave parametric amplification resulting from the nonlinear response of a two-level atomic system. The atomic dipole moment induced by weak optical fields at frequencies ${\omega }_3$ and ${\omega }_4$ in the presence of an optical field of arbitrary intensity at frequency ${\omega }_1$, where ${\omega }_3+{\omega }_4=2\mathrm{}{\omega }_1$, is obtained by solving the density-matrix equations of motion with phenomenological damping constants. The resulting nonlinear polarization can induce loss or three-photon gain in either weak wave and can act to parametrically couple these waves. The spatial evolution of the weak-field amplitudes is obtained by finding approximate solutions to the Helmholtz wave equations using the nonlinear polarization for the source terms. These solutions predict large gain for either or both of the weak waves under experimentally attainable conditions. Depending on the circumstance, this gain can be attributed solely to the three-photon gain, solely to the parametric coupling, or to an interplay between these effects. In addition, the solutions show an enhancement in the gain when $|{\omega }_3-{\omega }_1\mathrm{}|=|\mathrm{}{\omega }_4-{\omega }_1|={\Omega }^{\prime }$, where ${\Omega }^{\prime }$ is the generalized Rabi frequency associated with the driving of the atoms by the wave at frequency ${\omega }_1$.