Energetic cooling below the BEC transition: a quantum kinetic description within the Bogoliubov approximation

The dynamics of Bose-Einstein condensation in a three-dimensional harmonic trap is studied explicitly including the Bogoliubov approximation for temperatures below the critical one. To model the evolution towards equilibrium at each cooling step, we derive quantum kinetic equations that describe the dynamics of the gas for temperatures above and below the transition temperature. These equations, valid in the Born and Markov approximations, consider the essential role of the chemical potential as the main parameter that signals the transition. The kinetic equation that describes the growth of the condensate below the transition temperature is derived within the Bogoliubov approximation. To illustrate our results we propose an energetic cooling protocol and simulate the whole sequence of the formation of a condensate.