Ultracold Bosonic Atoms in Optical Lattices

This thesis covers most of my work in the field of ultracold atoms loaded in optical lattices. It makes a route through the physics of cold atoms in periodic potentials starting from the simple noninteracting system and going into the many-body physics that describes the strongly correlated Mott insulator regime. Even though this thesis is a theoretical work all the chapters are linked either with experiments already done or with ongoing experimental efforts. In the first part I investigate the validity of mean field approximations based on the Discrete Nonlinear Schr\"{o}dinger equation and quadratic approximations of the Hamiltonian to describe the approach of the system from the superfluid to the Mott insulator regime. In the second part I adopt the closed time path (CTP) and two particle irreducible (2PI) effective action formalism to study the non-equilibrium dynamics of a condensate loaded every third lattice site of an optical lattice. I show this formalism to be a powerful tool to describe far-from-equilibrium situations, particularly through its ability to incorporate the non-local and non-Markovian aspects characteristic of the quantum dynamics. In the last part I investigate the properties of the system deep in the Mott insulator regime. By using perturbation theory I study the Mott insulator ground state and its excitation spectrum, the response of the system to Bragg spectroscopy, and propose a mechanism to correct for the residual number fluctuations inherent to the Mott insulator ground state.