The frequency-domain pump-probe signal of a material system interacting with two quantum modes of the radiation field is recast in terms of products of scattering amplitudes ($T$ matrix elements) rather than the third-order susceptibility $\text{Im}{\chi }^{\left(3\right)}$. The resulting expression offers a more intuitive physical picture for the optical process compared with the semiclassical approach which treats the radiation field as classical. It can be derived and interpreted using closed-time-path-loop diagrams which represent the joint state of the matter and the field for each contribution to the signal. The signal has two components representing stimulated Raman scattering ${\omega }_1-{\omega }_2$ and two-photon absorption ${\omega }_1+{\omega }_2$ two-photon resonances. Both are expressed as nonequilibrium steady-state photon and matter fluxes, as is common in the description of dissipative processes in open quantum systems.

• Received 27 January 2009

DOI:https://doi.org/10.1103/PhysRevA.79.063827