Giant enhancement of the optical second-harmonic emission of WSe(2) monolayers by laser excitation at exciton resonances.

We show that the light-matter interaction in monolayer WSe_{2} is strongly enhanced when the incoming electromagnetic wave is in resonance with the energy of the exciton states of strongly Coulomb bound electron-hole pairs below the electronic band gap. We perform second harmonic generation (SHG) spectroscopy as a function of laser energy and polarization at T=4  K. At the exciton resonance energies we record an enhancement by up to 3 orders of magnitude of the SHG efficiency, due to the unusual combination of electric dipole and magnetic dipole transitions. The energy and parity of the exciton states showing the strong resonance effects are identified in 1- and 2-photon photoluminescence excitation experiments, corroborated by first principles calculations. Targeting the identified exciton states in resonant 2-photon excitation allows us to maximize k-valley coherence and polarization.