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

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