Abstract

A method employing the second quantisation approach (analogous to [1]) is presented for obtaining the time-dependent Hartree-Fock-Bogoliubov equations for a trapped dilute weakly-interacting bosonic gas at temperatures well below the Bose-Einstein condensation temperature T. This method is based on the Heisenberg equation of motion governing quantum-mechanical boson field operators. For such low temperatures a mean field can be assumed, which allows for the decomposition of single-particle creation/annihilation operators into a complex part (expressing the mean field) and an operator part expressing the additional quantum- mechanical fluctuations [2]. Wick’s theorem for thermal averages can then be applied to decouple averages of products of the shifted operators. The resulting equations are shown to be consistent with previously derived results in the appropriate limits [3] and can be considered as a generalisation of the Gross-Pitaevskii equation governing condensate behaviour [4],[5] in the presence of single- and multiple-particle inchoerent excitation.

© 1996 Optical Society of America