Ultracold atoms trapped in optical lattice

In this Thesis we study a paradigmatic experimental set up established for the study of quantum many-body system, namely ultracold atoms trapped in optical lattice. In particular, we consider interacting bosons placed into a periodic optical lattice generated by a set of counter-propagating laser beams that form a stationary electric potential. The work will be organized as follows: first, we consider the system Hamiltonian in the formalism of first quantization. We introduce the second quantization formalism in order to derive the Bose-Hubbard Hamiltonian and study the proprieties of the system by doing a numerical analysis. Also, in this case we numerically study this system in order to gather informations about its proprieties.