  Unique structural and optical properties of atomically thin two-dimensional
semiconducting transition metal dichalcogenides enable in principle their
efficient coupling to photonic cavities with optical modes volumes close to or
below the diffraction limit. Recently, it has become possible to make
all-dielectric nano-cavities with reduced mode volumes and negligible
non-radiative losses. Here, we realise low-loss high-refractive-index
dielectric gallium phosphide (GaP) nano-antennas with small mode volumes
coupled to atomic mono- and bilayers of WSe$_2$. We observe a photoluminescence
enhancement exceeding 10$^4$ compared with WSe$_2$ placed on planar GaP, and
trace its origin to a combination of enhancement of the spontaneous emission
rate, favourable modification of the photoluminescence directionality and
enhanced optical excitation efficiency. A further effect of the coupling is
observed in the photoluminescence polarisation dependence and in the Raman
scattering signal enhancement exceeding 10$^3$. Our findings reveal dielectric
nano-antennas as a promising platform for engineering light-matter coupling in
two-dimensional semiconductors.

Applied Physics;Mesoscale and Nanoscale Physics