Theory of molecule metal nano-particle interaction: Quantum description of plasmonic lasing

The recent quantum description of a few molecules interacting with plasmon excitations of a spherical metal nano-particle (MNP) as presented in the work of Zhang and May [Phys. Rev. B 89, 245441 (2014)] is extended to systems with up to 100 molecules. We demonstrate the possibility of multiple plasmon excitation and describe their conversion into far-field photons. The calculation of the steady-state photon emission spectrum results in an emission line-narrowing with an increasing number of molecules coupled to the MNP. This is considered as an essential criterion for the action of the molecule-MNP system as a nano-laser. To have exact results for systems with up to 20 molecules, we proceed as recently described by Richter et al. [Phys. Rev. B 91, 035306 (2015)] and study a highly symmetric system. It assumes an equatorial and regular position of identical molecules in such a way that their coupling is dominated by that to a single MNP dipole-plasmon excitation. Changing from the exact computation of the system's complete density matrix to an approximate theory based on the reduced plasmon density matrix, systems with more than 100 molecules can be described. Finally, nonlinear rate equations are proposed which reproduce the mean number of excited plasmons in their dependence of the number of molecules and of the used pump rate. The second order intensity correlation function of emitted photons is related to the respective plasmon correlation function which approaches unity when the system starts lasing.