Origin of spectral holes in pump-probe studies of homogeneously broadened lines

The theory of pump-probe experiments is examined using the nonlinear optical susceptibility ${\chi }^{\mathrm{}\left(3\right)\mathrm{}}$. The results of this calculation are shown to differ from those based on a commonly employed heuristic model of pump-probe experiments which assumes that the pump beam "prepares" the system which the probe beam subsequently monitors. This model consequently accounts only for processes where the interactions with the pump field come first in time. The correct expression for the probe absorption line shape as predicted by $\mathrm{Im}{\chi }^{\mathrm{}\left(3\right)\mathrm{}}$ contains a sum over all the possible time orderings of the various radiative interactions. The interference terms missing from the simplified treatment are shown to account for the spectral hole in homogeneously broadened lines observed recently [L. W. Hillman et al., Opt. Commun. 45, 416 (1983)]. In addition we derive a generalized expression for ${\chi }^{\mathrm{}\left(3\right)\mathrm{}}$ that is valid away from the impact limit and contains a realistic population relaxation ($T_1$) matrix.