Quantum Optics Model of Surface-Enhanced Raman Spectroscopy for Arbitrarily Shaped Plasmonic Resonators

We present a self-consistent quantum optics approach to calculating the surface-enhanced Raman spectrum of molecules coupled to arbitrarily shaped plasmonic systems. Our treatment is intuitive to use, provides fresh analytical insight into the physics of the Raman scattering near metallic surfaces, and can be applied to a wide range of geometries including resonators, waveguides, and hybrid photonic-plasmonic systems. Our general theory demonstrates that the detected Raman spectrum originates from an interplay between nonlinear light generation and propagation (which also includes the effects of optical quenching). Counterintuitively, at the nonlinear generation stage, we show that the Stokes (anti-Stokes) signal at the molecule location depends on the plasmonic enhancements, through the projected local density of photon states (LDOS), at the anti-Stokes (Stokes) frequency. However, when propagating from the vibrating molecule to the far field, the Stokes (anti-Stokes) emission experiences a plasmonic enhancement at the Stokes (anti-Stokes) frequency, as expected. We identify the limits of the commonly known E-4 electric-field rule for Raman signal enhancement near plasmonic surfaces at low pump powers, as well as a different E-8 rule at high pump powers, and we clarify the role of the LDOS. Our theory uses a generalized quantum master equation where the plasmonic system is treated as an environmental bath that is described through the photonic Green function of the medium. Therefore, any classical electrodynamics-related physics, such as quenching and propagation, are self-consistently included in the model. The presented formalism is also capable of describing the full spatial Raman response in a simple analytical way. This spatial analysis includes both the dependency of the Raman signals at a fixed detector location when the molecule is moved around the plasmonic platform and the dependency of the Raman signal on the detector location. We demonstrate the power of this approach by using a quasinormal mode expansion theory of localized plasmons to construct the photonic Green functions of the plasmonic resonators and explore several different nanoresonator systems.