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We calculate the quantum state of the plasmon field excited by an ensemble ofmolecular emitters, which are driven by exchange of electrons with metallicnano-particle electrodes. Assuming identical emitters that are coupledcollectively to the plasmon mode but are otherwise subject to independentrelaxation channels, we show that symmetry constraints on the total systemdensity matrix imply a drastic reduction in the numerical complexity. For$N_{\text{m}}$ three-level molecules we may thus represent the density matrixby a number of terms scaling as $(N_{\rm m}+8)!/(8!N_{\rm m}!)$ instead of$9^{N_{\text{m}}}$, and this allows exact simulations of up to$N_{\text{m}}=10$ molecules. Our simulations demonstrate that many emitterscompensate strong plasmon damping and lead to the population of high plasmonnumber states and a narrowed linewidth of the plasmon field. For large$N_{\text{m}}$, our exact results are reproduced by an approximate approachbased on the plasmon reduced density matrix. With this approach, we haveextended the simulations to more than $50$ molecules and shown that the plasmonnumber state population follows a Poisson-like distribution. An alternativeapproach based on nonlinear rate equations for the molecular state populationsand the mean plasmon number also reproduce the main lasing characteristics ofthe system.

Theoretical Study of Plasmonic Lasing in Junctions with many Molecules