A new approach to number-conserving Bogoliubov approximation for Bose-Einstein condensates

We consider a BEC of $N$ ultra-cold atoms in a trapping potential. The many-body wave function of the BEC is ``encoded" in the $N$-particle sector of an extended catalytic state, coherent state for the condensate mode and a state for the orthogonal modes. Using a time-dependent interaction picture, we move the coherent state to the vacuum, where all the field operators are small compared to ${N}^{1/2}$. The resulting Hamiltonian can then be organized by powers of ${N}^{-1/2}$. Requiring the terms of order ${N}^{1/2}$ to vanish, we get the GP equation for the condensate wave function. Going to the next order, $N^0$, we are able to derive equations equivalent to those found by Castin and Dum [Phys. Rev. A \textbf{57}, 3008 (1998)] for a number-conserving Bogoliubov approximation. In contrast to other approaches, ours allows one to calculate the state evolution in the Schr\"{o}dinger picture, and it also has advantages in discussing higher-order corrections and multi-component cases.