  We report on theoretical study of second-harmonic generation (SHG) in
plasmonic nanostructures interacting with two-level quantum emitters (QE) under
incoherent energy pump. We generalize driven-dissipative Tavis-Cummings model
by introducing anharmonic surface plasmon-polariton (SPP) mode coupled to QEs
and examine physical properties of corresponding SPP-QE polariton states. Our
calculations of the SHG efficiency for strong QE-SPP coupling demonstrate
orders of magnitude enhancement facilitated by the polariton gain. We further
discuss time-domain numerical simulations of SHG in a square lattice comprised
of Ag nanopillars coupled to QEs utilizing fully vectorial nonperturbative
nonlinear hydrodynamic model for conduction electrons coupled to Maxwell-Bloch
equations for QEs. The simulations support the idea of gain enhanced SHG and
show orders of magnitude increase in the SHG efficiency as the QEs are tuned in
resonance with the lattice plasmon mode and brought above the population
inversion threshold by incoherent pump. By varying pump frequency and tuning
QEs to a localized plasmon mode, we demonstrate further enhancement of the SHG
efficiency facilitated by strong local electric fields. The incident light
polarization dependence of the SHG is examined and related to the symmetries of
participating plasmon modes.
