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

TL;DR

This paper introduces a number-conserving Bogoliubov approximation for Bose-Einstein condensates using extended catalytic states, providing a framework that is compatible with Schrödinger picture calculations and adaptable to multi-component systems.

Contribution

The authors develop a novel approach to derive the number-conserving Bogoliubov approximation via extended catalytic states, enabling straightforward generalizations and Schrödinger picture analysis.

Findings

Derivation of the Gross-Pitaevskii equation from the effective Hamiltonian.

Formulation of number-conserving Bogoliubov equations at order N^0.

Method's applicability to multi-component BECs and higher-order corrections.

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 $\textbf{57}$, 3008 (1998)]. In contrast to other approaches, ours is well suited to calculating the state evolution in the Schr\"{o}dinger picture; moreover, it is straightforward to generalize our method to multi-component BECs and to higher-order corrections.