Plasmonic dimer cavities can induce extreme electric-field hot spots that allow one to access ultrastrong coupling regimes using Raman-type spectroscopy on single vibrating molecules. Using a generalized master equation, we study resonant Raman scattering in the strong coupling regime of cavity-QED, when also in the vibrational ultrastrong coupling regime, leading to phonon-dressed polaritons. The master equation rigorously includes spectral baths for the cavity and vibrational degrees of freedom, as well as a pure dephasing bath for the resonant two-level system, which play a significant role. Employing realistic parameters for gold dimer cavity modes, we investigate the emission spectra in several characteristic strong-coupling regimes, leading to extremely rich spectral resonances due to an interplay of phonon-modified polariton states and bath-induced resonances. We also show explicitly the failure of the standard master equation in these quantum nonlinear regimes.
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