The momentum distribution and dynamical structure factor in a weakly interacting Bose gas with a time-dependent periodic modulation in terms of the Bogoliubov treatment are investigated. The evolution equation related with Bogoliubov weights happens to be a solvable Mathieu equation when the coupling strength is periodically modulated. An exact relation between the time derivatives of momentum distribution and dynamical structure factor is derived, which indicates that the single-particle property strongly related to the two-body property in the evolutions of Bose-Einstein condensates. It is found that the momentum distribution and dynamical structure factor cannot display periodical behavior. For stable dynamics, some particular peaks in the curves of momentum distribution and dynamical structure factor appear synchronously, which is consistent with the derivative relation. However there is no evident corresponding peaks in the unstable dynamics.

中文翻译：

---

具有周期性调制的弱相互作用 Bose 气体中动量分布和结构因子的动力学

研究了在 Bogoliubov 处理方面具有时间相关周期性调制的弱相互作用 Bose 气体中的动量分布和动力学结构因子。当耦合强度被周期性调制时，与 Bogoliubov 权重相关的演化方程恰好是一个可解的 Mathieu 方程。推导出动量分布的时间导数与动力学结构因子之间的精确关系，表明玻色-爱因斯坦凝聚演化过程中单粒子性质与二体性质密切相关。发现动量分布和动力结构因子不能表现出周期性行为。对于稳定动力学，动量分布曲线和动力学结构因子曲线中的某些特定峰值同步出现，这与导数关系一致。然而，在不稳定动力学中没有明显的对应峰值。