Path integral formulation of out-of-equilibrium interacting bosons.

This thesis explores the dynamics of weakly interacting out-of-equilibrium Bose gases. The theoretical framework to describe this system is known as Keldysh formalism, and it can be thought of as a generalization of the techniques used in equilibrium Quantum Field Theory. The student will derive, using a path integral formulation, two coupled equations that fully describe the behavior of Bose gases near the critical temperature. The first is an extension to the well-known Gross-Pitaevskii equation that describes the time evolution of the condensate, with an additional term to account for particle exchanges with the excited states. The second one is a Boltzmann-like equation for the distribution function of the thermal cloud, which is treated in a semi-classical approximation. The student will write his own code to solve these two equations, which will require the usage of a variety of techniques, ranging from Monte Carlo sampling to the Split Step Fourier method and imaginary time evolution. Finally, he will use his implementation to study systems of experimental interest, and compare his results with the literature.