Quantum correlations of ultracold atoms in optical lattices

An optical lattice is the periodic potential that atoms experience via the ac-Stark shift when\ they are illuminated by counter-propagating laser beams that form a standing-wave pattern. Optical\ lattices have been widely used as a versatile platform in cooling, trapping, controlling atoms,\ and for the study of a variety of problems in physics. The many-body states of ultracold\ atoms in optical lattices can be characterized by the quantum correlations encoded in time-of-flight\ images. In this thesis, we mainly discuss the use of the correlations function as a natural framework\ for characterizing quantum states in optical lattices.