Effects of arbitrary relaxation and strong-field dressing of energy levels on nonlinear optical susceptibilities

A nonlinear response theory of a quantum-mechanical system undergoing arbitrary relaxation and interacting with fields, some of which may be strong enough to saturate optical transitions, is developed. Explicit expressions for second-order and third-order susceptibilities are obtained. If the fields are weak, then these expressions show the existence of additional resonant contributions to ${\chi }^{\left(n\right)}$ which arise due to inelastic collisions. Various applications of these ${\chi }^{\left(n\right)}$’s to modulation spectroscopy, four-wave mixing, and pump-probe experiments are discussed. The general structure suggests how the additional resonances can be used to determine inelastic rates in a Doppler-broadened medium. For saturating fields, ${\chi }^{\left(n\right)}$’s become dependent on the intensity of such fields and can be formally obtained from weak-field ${\chi }^{\left(n\right)}$’s, provided proper identification of unperturbed eigenfunctions, eigenvalues, and relaxation times is made. Such intensity-dependent ${\chi }^{\left(n\right)}$’s have resonant denominators which lead to resonances at Rabi frequencies and submultiples of these frequencies, the widths of which are also dependent on the strength of the field. In the special cases intensity-dependent ${\chi }^{\mathrm{}\left(1\right)\mathrm{}}$ agrees with the work of Cohen-Tannoudji and co-workers.