Theoretical Study of Plasmonic Lasing in Junctions with many Molecules

We calculate the quantum state of the plasmon field excited by an ensemble of molecular emitters, which are driven by exchange of electrons with metallic nano-particle electrodes. Assuming identical emitters that are coupled collectively to the plasmon mode but are otherwise subject to independent relaxation channels, we show that symmetry constraints on the total system density matrix imply a drastic reduction in the numerical complexity. For $N_{\text{m}}$ three-level molecules we may thus represent the density matrix by 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 emitters compensate strong plasmon damping and lead to the population of high plasmon number states and a narrowed linewidth of the plasmon field. For large $N_{\text{m}}$, our exact results are reproduced by an approximate approach based on the plasmon reduced density matrix. With this approach, we have extended the simulations to more than $50$ molecules and shown that the plasmon number state population follows a Poisson-like distribution. An alternative approach based on nonlinear rate equations for the molecular state populations and the mean plasmon number also reproduce the main lasing characteristics of the system.