  We study the effect of degenerate parametric down-conversion (DPDC) in an
ensemble of two-level quantum emitters (QEs) coupled via near-field
interactions to a single surface plasmon (SP) mode of a nonlinear plasmonic
cavity. For this purpose, we develop a quantum driven-dissipative model
capturing non-equilibrium dynamics of the system in which incoherently pumped
QEs have transition frequency tuned near the second-harmonic response of the
SPs. Considering the strong coupling regime, i.e., the SP-QE interaction rate
exceeds system dissipation rates, we find a critical SP-QE coupling attributed
to the phase transition between normal and lasing steady states. Examining
fluctuations above the system's steady states, we predict new elementary
excitations, namely, the exciton-plasmon polaritons formed by the two-SP quanta
and single-exciton states of QEs. The contribution of two-SP quanta results in
the linear scaling of the SP-QE interaction rate with the number of QEs, ${\cal
N}_o$, as opposed to the $\sqrt{{\cal N}_o}$-scaling known for the Dicke and
Tavis-Cummings models. We further examine how SP-QE interaction scaling affects
the polariton dispersions and power spectra in the vicinity of the critical
coupling. For this purpose, we compare the calculation results assuming a
finite ensemble of QEs and the model thermodynamic limit. The calculated power
spectra predict an interplay of coherent photon emission by QEs near the
second-harmonic frequency and correlated photon-pair emission at the
fundamental frequency by the SPs (i.e., the photonic DPDC effect).
