We report on the investigation of the three-dimensional single-atom-resolved distributions of bosonic Mott insulators in momentum space. First, 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 prefactor 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 tunneling are those of a many-body ground state with Gaussian statistics. Second, in the Mott insulating regime with increasing tunneling, we extract the spectral weight of the quasiparticles 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 quasiparticles. While the shapes of the observed spectral weight agree with those predicted by perturbative many-body calculations for homogeneous systems, the fitted mobilities are larger than the theoretical ones, mostly because of the coexistence of various phases in the trap.
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