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 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. Our findings show that the quantum statistics of ideal bosons uniformly distributed in the lattice sites quantitatively captures momentum-space correlations of a Mott insulator at vanishing tunnelling. Secondly, we extract the spectral weight of the Mott quasi-particles from the momentum distributions. On approaching the transition towards a superfluid, the momentum spread of the spectral weight decreases as a result of the increased mobility of the quasi-particles.
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