  We show that the gravitational acceleration can be measured with the
matter-wave Ramsey interferometry, by using a nitrogen-vacancy center coupled
to a nano-mechanical resonator. We propose two experimental methods to realize
the Hamiltonian, by using either a cantilever resonator or a trapped
nanoparticle. The scheme is robust against the thermal noise, and could be
realized at the temperature much higher than the quantum regime one. The
effects of decoherence on the interferometry fringe visibility is caculated,
including the mechanical motional decay and dephasing of the nitrogen-vacancy
center. In addition, we demonstrate that under the various sources of random
and systematic noises, our gravimeter can be made on-chip and achieving a high
measurement precision. Under experimental feasible parameters, the proposed
gravimeter could achieve $10^{-10}$ relative precision.
