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.

The outline of the thesis is as follows. Chapter 1 gives a brief introduction to optical lattice\ potentials and ways for describing particles moving in a periodic potential. Chapter 2 explains the\ importance of correlations and discusses common methods to detect them in cold atom experiments.

Chapter 3 presents work done to study the many-body Schrodinger equation in a quasiperiodic\ potential and discusses its connection with the Kolmogorov-Arnold-Moser (KAM) problem\ of classical mechanics. We posed a possible visualization of such a connection in experimentally\ accessible many-body observables. These observables are useful probes for the three characteristic\ phases of the problem: the metallic, Anderson, and band-insulator phases. In addition, they exhibit\ fingerprints of nonlinear phenomena such as bifurcations and devil{\textquoteright}s staircases. Our numerical\ treatment is complemented with a perturbative analysis that provides insight on the underlying\ physics. The perturbative-theory approach is particularly useful in illuminating the distinction\ between the Anderson-insulator and the band-insulator phases in terms of paired sets of dimerized\ states.

Chapter 4 discusses several theoretical procedures developed to understand a recent experiment\ on macroscopic quantum self-trapping (ST) performed in a 2D optical lattice. Mean field\ and truncated-Wigner-approximation (TWA) calculations are performed trying to reproduce the\ experimental observations. The discrepancy between the theory and the experiment lead to the\ hypothesis of a new type of ST caused by strong correlations. We analyze toy models to support\ it.

}, author = {Shuming Li} }