Microscopic derivation of the extended Gross-Pitaevskii equation for quantum droplets in binary Bose mixtures

An ultradilute quantum droplet is a self-bound liquidlike 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 Gross-Pitaevskii equation. Here, we microscopically derive the Gross-Pitaevskii equation for the condensate and also for a pairing field in an inhomogeneous quantum droplet realized by Bose-Bose mixtures with attractive interspecies 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 as an ideal starting point to understand the structure and collective excitations of nonuniform ultradilute quantum droplets in ongoing cold-atom experiments.