Measuring the full distribution of individual particles is of fundamental importance to characterize many-body quantum systems through correlation functions at any order. Here, we demonstrate the possibility to reconstruct the momentum-space distribution of three-dimensional interacting lattice gases atom by atom. This is achieved by detecting individual metastable
4He∗ atoms in the far-field regime of expansion, when released from an optical lattice. We benchmark our technique with quantum Monte Carlo calculations, demonstrating the ability to resolve momentum distributions of superfluids occupying
105 lattice sites. It permits a direct measure of the condensed fraction across phase transitions, as we illustrate on the superfluid-to-normal transition. Our single-atom-resolved approach opens a route to investigate interacting lattice gases through momentum correlations.
In this dissertation we make use of the many analogies between quantum optical and ultra-cold atomic and molecular systems in order to study the coherence properties of the latter with methods of non-linear optics. We adapt the XFROG method that has first been developed for the characterization of ultra-short laser pulses, to the problem of reconstructing both amplitude and phase of the condensate wavefunction of a Bose-Einstein-condensate (BEC). Using the example of a vortex state we study the dependence of the reconstruction quality on the number of measurements and different sources of noise and we find that the method is feasible with available experimental technology. Exploiting the similarity between the coherent formation of ultra-cold molecules and optical sum frequency generation we devise a scheme for measuring second-order correlations of atoms through density measurements of molecules. We use perturbation theory in the cases of weak and strong coupling between atoms and molecules to calculate the momentum distribution of the molecules for the cases where the molecules are formed from a BEC, a normal Fermi gas and a Fermi gas with superfluidity in a Bardeen-Cooper-Schrieffer (BCS) state. These calculations are supplemented by exact integrations of Schroedinger's equation in the single mode approximation for the molecules. Atoms in a BEC are collectively transformed into molecules with a narrow momentum distribution reflecting the long coherence length of atoms in the BEC. For the normal Fermi gas molecules are formed non-collectively and their momentum distribution is much wider. The momentum distribution of molecules from a BCS state looks similar to the BEC case: The superfluid component leads to collectively formed molecules with a very narrow momentum distribution and the unpaired fraction gives rise to non-collectively formed molecules with a much wider momentum distribution similar to the normal Fermi gas case. The counting statistics of the molecules from a BEC is that of a coherent state, from a normal Fermi gas it is that of a thermal state and the BCS case interpolates between the two.