Bose-Hubbard model of cold atoms to higher orders

The aim of this work is to study the dynamics of cold atoms trapped in a one-dimensional periodic lattice generated by an optical potential. Such a system is described by a standard model in condensed matter, namely the so-called Bose-Hubbard model. Arrays of cold neutral atoms or ions are the basic elements of some proposals of quantum computers, which motivates the interest of understanding their dynamics from a mathematical point of view. The main targets of this Master Thesis are the following. First, starting from a classical Hamiltonian of the Frenkel-Kontorova type and thanks to Perturbation Theory, to derive the discrete Bougolyubov-Gross-Pitaevskii (dBGP) equation. Second, starting from first principle formulation, to provide a systematic, quantum mechanical, perturbative derivation of the Bose-Hubbard (BH) model to orders higher than the first one in the small parameter.