扩展催化态下玻色-爱因斯坦凝聚体的粒子数守恒Bogoliubov近似
Particle-Number-Conserving Bogoliubov Approximation for Bose-Einstein Condensates Using Extended Catalytic States
摘要：
我们在扩展催化态的$N$ -粒子扇区中对玻色-爱因斯坦凝聚体（BEC）的多体波函数进行编码。该催化态对于凝聚态模是相干态，对于与凝聚态模正交的模是任意态。对于一个含时相互作用图，其中凝聚态模的状态被转移到真空中，我们可以用${N}^{-1/2}$的幂来组织有效哈密顿量。要求${N}^{1/2}$阶项为零，得到Gross-Pitaevskii方程。进入下一阶，$N^0$，我们推导出数量守恒的Bogoliubov近似的方程，该方程首先由Castin和Dum [Phys. Rev. A $\textbf{57}$，3008（1998）]给出。与其他方法相比，我们的方法非常适合于计算薛定谔图像中的状态演化;此外，将我们的方法推广到多分量BEC和高阶校正是简单的。
英文摘要：
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 Schrödinger picture; moreover, it is straightforward to generalize our method to multi-component BECs and to higher-order corrections.
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