Particle-Number-Conserving Bogoliubov Approximation for Bose-Einstein Condensates Using Extended Catalytic States

Abstract

We encode the many-body wavefunction 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 $\text{57}$, 3008 (1998)]. In contrast to other approaches, ours is well suited to calculating the state evolution in the Schrödinger picture; moreover, it is straightforward to generalize our method to multi-component BECs and to higher-order corrections.

29 pages, 1 figure

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• Nonlinear corrections in the quantization of a weakly nonideal Bose gas at zero temperature. II. The general case