  An ultradilute quantum droplet is a self-bound liquid-like state recently
observed in ultracold Bose-Einstein condensates. In all previous theoretical
studies, it is described by a phenomenological low-energy effective theory,
termed as the extended Gross\textendash Pitaevskii equation. Here, we
microscopically derive the Gross\textendash Pitaevskii 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
Gross\textendash Pitaevskii 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.
