Linking classical and molecular optomechanics descriptions of SERS

The surface-enhanced Raman scattering (SERS) of molecular species in plasmonic cavities can be described as an optomechanical process where plasmons constitute an optical cavity of reduced effective mode volume which effectively couples to the vibrations of the molecules. An optomechanical Hamiltonian can address the full quantum dynamics of the system, including the phonon population build-up, the vibrational pumping regime, and the Stokes-anti-Stokes correlations of the photons emitted. Here we describe in detail two different levels of approximation to the methodological solution of the optomechanical Hamiltonian of a generic SERS configuration, and compare the results of each model in light of recent experiments. Furthermore, a phenomenological semi-classical approach based on a rate equation of the phonon population is demonstrated to be formally equivalent to that obtained from the full quantum optomechanical approach. The evolution of the Raman signal with laser intensity (thermal, vibrational pumping and instability regimes) is accurately addressed when this phenomenological semi-classical approach is properly extended to account for the anti-Stokes process. The formal equivalence between semi-classical and molecular optomechanics descriptions allows us to describe the vibrational pumping regime of SERS through the classical cross sections which characterize a nanosystem, thus setting a roadmap to describing molecular optomechanical effects in a variety of experimental situations.