Methods of quantum field theory for trapped Bose-Einstein condensates

In this tutorial we present an introduction to some theoretical methods of quantum field theory applied to the description of a trapped Bose-Einstein condensate. First of all, we give a brief account of the main characteristics of the phenomenon of condensation and present the many-body Hamiltonian of the system. We outline some of the most important approaches used in the characterization of a condensed Bose gas, including the mean-field theory and the Hartree-Fock-Bogoliubov method. Finally we illustrate the use of these techniques addressing some important issues in quantum atom optics. We characterize the quantum state of a Bose-Einstein condensate (BEC) at zero temperature. We also describe a process of Beliaev coupling between quasiparticles using a method that includes terms beyond the usual Bogoliubov approach.