Particle-number-conserving Bogoliubov approximation for Bose-Einstein condensates using extended catalytic states

We encode the many-body wave function of a Bose-Einstein condensate (BEC) in the N-particle sector of an extended catalytic state. This catalytic state is a coherent state for the condensate mode and an arbitrary state for the modes orthogonal to the condensate mode. Going to a time-dependent interaction picture where the state of the condensate mode is displaced to the vacuum, we can organize the effective Hamiltonian by powers of N-1/2. Requiring the terms of order N-1/2 to vanish gives the Gross-Pitaevskii equation. Going to the next order, N-0, we derive equations for the number-conserving Bogoliubov approximation, first given by Castin and Dum [Phys. Rev. A 57, 3008 (1998)]. In contrast to other approaches, ours is well suited to calculating the state evolution in the Schrodinger picture; moreover, it is straightforward to generalize our method to multicomponent BECs and to higher-order corrections.