Nonlinear gain and dispersion in the four-level Λ-type system

The nonlinear gain and dispersion spectra of a four-level Λ-type system, which driven by a weak probe field and coupling field, is investigated. Using the perturbation method, third-order perturbation iterative solutions of the steady density matrix equations are obtained. We find that effect of the SGC on the nonlinear gain and dispersion are negligible. At the resonant coupling, nonlinear dispersion spectra of the two transition paths are symmetrical about a line, but the nonlinear absorption spectra are symmetrical about a point. The degeneration of the two lower states greatly modifies the nonlinear gain and dispersion spectra.