An ultradilute quantum droplet is a self-bound liquid-like state recently observed in ultracold Bose-Einstein condensates. In most previous theoretical studies, it is described by a phenomenological low-energy effective theory, termed as the extended GrossPitaevskii equation. Here, we microscopically derive the GrossPitaevskii equation for the condensate and also for a pairing field in an inhomogeneous quantum droplet realized by Bose-Bose mixtures with attractive inter-species interaction. We show that the inclusion of the pairing field is essential, in order to have a consistent description of the droplet state. We clarify that, the extended GrossPitaevskii equation used earlier should be understood as the equation of motion for the pairing field, rather than the condensate. The fluctuations of the pairing field give rise to low-energy collective excitations of the droplet. We also present the Bogoliubov equations for gapless phonon modes and gapped modes due to pairing in real space, which characterizes single-particle-like excitations of the droplet. The equations of motion derived in this work for the condensate and the pairing field serve an ideal starting point to understand the structure and collective excitations of non-uniform ultradilute quantum droplets in on-going cold-atom experiments.

中文翻译：

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二元Bose混合物中量子液滴的扩展Gross-Pitaevskii方程的微观推导

最近在超冷的玻色-爱因斯坦凝聚物中观察到超稀量子滴是一种自结合的液体状状态。在大多数以前的理论研究中，它是由现象学的低能有效理论描述的，被称为扩展的GrossPitaevskii方程。在这里，我们从微观上导出了GrossPitaevskii方程，用于冷凝物以及在具有种间相互作用的Bose-Bose混合物实现的非均匀量子液滴中的配对场。我们展示了包含配对字段是必不可少的，以便对液滴状态具有一致的描述。我们阐明，更早使用的扩展的GrossPitaevskii方程应该理解为配对场的运动方程，而不是冷凝物。配对场的波动引起液滴的低能集体激发。我们还提出了由于真实空间中的配对而产生的无间隙声子模和带隙模的Bogoliubov方程，该方程表征了液滴的单粒子状激发。这项工作中针对冷凝物和配对场得出的运动方程式是了解正在进行的冷原子实验中非均匀超稀释量子液滴的结构和集体激发的理想起点。