Basis-dependent dynamics of trapped Bose-Einstein condensates and analogies with semi-classical laser theory

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We present a consistent second order perturbation theory for the lowest-lying condensed modes of very small, weakly-interacting Bose-Einstein condensates in terms of bare particle eigenstates in a harmonic trap. After presenting our general approach, we focus on explicit expressions for a simple three-level system, mainly in order to discuss the analogy of a single condensate occupying two modes of a trap with the semi-classical theory for two-mode photon lasers. A subsequent renormalization of the single-particle energies to include the dressing imposed by mean fields demonstrates clearly the consistency of our treatment with other kinetic approaches.