  We report on the investigation of the three-dimensional single-atom-resolved
distributions of bosonic Mott insulators in momentum-space. Firstly, we measure
the two-body and three-body correlations deep in the Mott regime, finding a
perfectly contrasted bunching whose periodicity reproduces the reciprocal
lattice. In addition, we show that the two-body correlation length is inversely
proportional to the in-trap size of the Mott state with a pre-factor in
agreement with the prediction for an incoherent state occupying a uniformly
filled lattice. Our findings indicate that the momentum-space correlations of a
Mott insulator at small tunnelling is that of a many-body ground-state with
Gaussian statistics. Secondly, in the Mott insulating regime with increasing
tunnelling, we extract the spectral weight of the quasi-particles from the
momentum density profiles. On approaching the transition towards a superfluid,
the momentum spread of the spectral weight is found to decrease as a result of
the increased mobility of the quasi-particles in the lattice. While the shapes
of the observed spectral weight agree with the ones predicted by perturbative
many-body calculations, the fitted mobilities are larger than the theoretical
ones. This discrepancy is similar to that previously reported on the
time-of-flight visibility.
