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Spatial and momentum correlations are important in the analysis of thequantum states and different phases of trapped ultracold atom systems as afunction of the strength of interatomic interactions. Identification andunderstanding of spin-resolved patterns exhibited in two-point correlations,accessible directly by experiments, are key for uncovering the symmetry andstructure of the many-body wave functions of the trapped system. Using theconfiguration interaction method for exact diagonalization of the many-bodyHamiltonian of $N=2-4$ fermionic atoms trapped in single, double, triple, andquadruple wells, we analyze both two-point momentum and space correlations, aswell as associated noise distributions, for a broad range of interparticlecontact repulsion strengths and interwell separations, unveilingcharacteristics allowing insights into the transition, via an intermediatephase, from the non-interacting Bose-Einstein condensate to the weaklyinteracting quasi-Bose-Einstein regime, and from the latter to thestrong-repulsion Tonks-Girardeau (TG) one. The ab-initio numerical predictionsare shown to agree well with the results of a constructed analytical modelemploying localized displaced Gaussian functions to represent the $N$ fermions.The two-point momentum correlations are found to exhibit damped oscillatorydiffraction behavior. This diffraction behavior develops fully for atomstrapped in a single well with strong interatomic repulsion in the TG regime, orfor atoms in well-separated multi-well traps. Additionally, the two-bodymomentum correlation and noise distributions are found to exhibit"shortsightedness", with the main contribution coming from nearest-neighboringparticles.

Two-point momentum correlations of few ultracold quasi-one-dimensional trapped fermions: Diffraction patterns