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We report resonant second harmonic generation (SHG) spectroscopy of anhBN-encapsulated monolayer of MoS$_2$. By tuning the energy of the excitationlaser, we identify a dark state transition (D) that is blue detuned by +25 meVfrom the neutral exciton X$^0$. We observe a splitting of the SHG spectrum intotwo distinct peaks and a clear anticrossing between them as the SHG resonanceis tuned through the energy of the dark exciton D. This observation isindicative of quantum interference arising from the strong two-photonlight-matter interaction. We further probe the incoherent relaxation from thedark state to the bright excitons, including X$^0$ and localized excitons LX,by the resonant enhancement of their intensities at the SHG-D resonance. Therelaxation of D to bright excitons is strongly suppressed on the bare substratewhilst enabled when the hBN/MoS$_2$/hBN heterostructure is integrated in ananobeam cavity. The relaxation enabled by the cavity is explained by thephonon scattering enhanced by the cavity phononic effects. Our work reveals thetwo-photon quantum interference with long-lived dark states and enables thecontrol through nanostructuring of the substrate. These results indicate thegreat potential of dark excitons in 2D-material based nonlinear quantumdevices.

Probing the Dark Exciton in Monolayer MoS$_2$ by Quantum Interference in Second Harmonic Generation Spectroscopy