Recent scanning tunnelling microscopy (STM) experiments reported single-molecule fluorescence induced by tunneling currents in the nanoplasmonic cavity formed by the STM tip and the substrate.The electric field of the cavity mode couples with the current-induced charge fluctuations of the molecule, allowing the excitation of the mode. We investigate theoretically this system for the experimentally relevant limit of large damping rate $kappa$ for the cavity mode and arbitrary coupling strength to a single-electronic level. We find that for bias voltages close to the first inelastic threshold of photon emission, the emitted light displays anti-bunching behavior with vanishing second-order photon correlation function. At the same time, the current and the intensity of emitted light display Franck--Condon steps at multiples of the cavity frequency $omega_c$ with a width controlled by $kappa$ rather than the temperature $T$. For large bias voltages, we predict strong photon bunching of the order of the $kappa/Gamma$ where $Gamma$ is the electronic tunneling rate. Our theory thus predicts that strong coupling to a single level allows current-driven non-classical light emission.
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