Quantum Mechanical Treatment of Stimulated Raman Cross Sections

Stimulated Raman scattering (SRS) has played an increasingly pivotal role in chemistry and photonics. Recently, understanding of light-molecule interaction during SRS was brought to a new quantitative level through the introduction of stimulated Raman cross section, $\sigma_{SRS}$. Measurements of Raman-active molecules have revealed interesting insights, and theoretical consideration has suggested an Einstein-coefficient-like relation between $\sigma_{SRS}$ and the commonly used spontaneous Raman cross sections, $\sigma_{Raman}$. However, the theoretical underpinning of $\sigma_{SRS}$ is not known. Herein we provide a full quantum mechanical treatment for $\sigma_{SRS}$, via both a semi-classical method and a quantum electrodynamic (QED) method. The resulting formula provides a rigorous theory to predict experimental outcome from first principles, and unveils key physical factors rendering $\sigma_{SRS}$ inherently strong response. Through this formula, we also confirm the validity of the Einstein-coefficient-like equation connecting $\sigma_{Raman}$ and $\sigma_{SRS}$ reported earlier, and discuss the inherent symmetry between all spontaneous and stimulated optical processes. Hence the present treatment shall deepen the fundamental understanding of the molecular response during SRS, and facilitate quantitative applications in various experiments.