One-dimensional quantum droplets under a linear gravitational-like trap

We investigate the influence of a constant and time-dependent linear gravitational-like potential on one-dimensional quantum droplets, governed by an extended Gross-Pitaevskii equation incorporating a repulsive cubic effective mean-field (EMF) term and an attractive quadratic beyond-mean-field (BMF) correction. By constructing a tailored external confinement, we derive an exact analytical family of droplet wave functions and explicitly characterize the effective interaction contributions. A key result is that the droplet's c.m. follows a strictly Newtonian equation of the form ${\stackrel{\ifmmode \ddot{}\else \"{}\fi{}}{X}}_{\mathrm{c}.\mathrm{m}.}(t)=\ensuremath{-}a(t)$, demonstrating that its falling trajectory depends solely on the applied linear potential. Notably, this motion is completely independent of atom number and of the EMF-BMF balance, providing a clean classical baseline for dynamics in linear-potential environments. Temporal modulation of the potential induces clear deviations from static trajectories and leaves measurable imprints on internal coherence, captured through the Shannon entropy and the Wigner phase-space distribution, with the modulation strength controlling phase-space localization, deformation, and coherence degradation. Numerical simulations substantiate the stability of the analytical solutions, demonstrating their robustness. These findings suggest promising implications for quantum sensing and metrological applications using ultradilute quantum fluids.