  Second harmonic generation (SHG) is a non-linear optical process, where two
photons coherently combine into one photon of twice their energy. Efficient SHG
occurs for crystals with broken inversion symmetry, such as transition metal
dichalcogenide monolayers. Here we show tuning of non-linear optical processes
in an inversion symmetric crystal. This tunability is based on the unique
properties of bilayer MoS2, that shows strong optical oscillator strength for
the intra- but also inter-layer exciton resonances. As we tune the SHG signal
onto these resonances by varying the laser energy, the SHG amplitude is
enhanced by several orders of magnitude. In the resonant case the bilayer SHG
signal reaches amplitudes comparable to the off-resonant signal from a
monolayer. In applied electric fields the interlayer exciton energies can be
tuned due to their in-built electric dipole via the Stark effect. As a result
the interlayer exciton degeneracy is lifted and the bilayer SHG response is
further enhanced by an additional two orders of magnitude, well reproduced by
our model calculations.
