  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.
