  Heterostructures made from van der Waals materials provide a template to
investigate proximity effects at atomically sharp heterointerfaces. In
particular, near-field charge and energy transfer in heterostructures made from
semiconducting transition metal dichalcogenides (TMD) have attracted interest
to design model 2D "donor-acceptor" systems and new optoelectronic components.
Here, using of Raman scattering and photoluminescence spectroscopies, we report
a comprehensive characterization of a molybedenum diselenide (MoSe$_2$)
monolayer deposited onto hexagonal boron nitride (hBN) and capped by mono- and
bilayer graphene. Along with the atomically flat hBN susbstrate, a single
graphene epilayer is sufficient to passivate the MoSe$_2$ layer and provides a
homogenous environment without the need for an extra capping layer. As a
result, we do not observe photo-induced doping in our heterostructure and the
MoSe$_2$ excitonic linewidth gets as narrow as 1.6~meV, hence approaching the
homogeneous limit. The semi-metallic graphene layer neutralizes the 2D
semiconductor and enables picosecond non-radiative energy transfer that
quenches radiative recombination from long-lived states. Hence, emission from
the neutral band edge exciton largely dominates the photoluminescence spectrum
of the MoSe$_2$/graphene heterostructure. Since this exciton has a picosecond
radiative lifetime at low temperature, comparable with the energy transfer
time, its low-temperature photoluminescence is only quenched by a factor of
$3.3 \pm 1$ and $4.4 \pm 1$ in the presence of mono- and bilayer graphene,
respectively. Finally, while our bare MoSe$_2$ on hBN exhibits negligible
valley polarization at low temperature and under near-resonant excitation, we
show that interfacing MoSe$_2$ with graphene yields a single-line emitter with
degrees of valley polarization and coherence up to $\sim 15\,\%$.
