Localization and splitting of a quantum droplet by immersing a heavy impurity

We investigate the existence, stability and nonequilibrium dynamics of one-dimensional harmonically trapped quantum droplets in the presence of a central (attractive or repulsive) potential well, which mimics the effect of a heavy impurity. It is demonstrated that an increasing repulsive potential barrier enforces droplet fragmentation into two distinct segments, while an attractive well facilitates substantial deformation of the droplet solution which ceases to exist above a critical potential strength. The stability properties and collective excitation modes of the ensuing droplet configurations are inferred through linearization analysis of the suitable extended Gross-Pitaevskii equation. A sudden ramp-up of the repulsive potential leads to droplet's dynamical splitting into two counter-propagating fragments. In contrast, following a quench to the attractive potential case favors spatial localization of the droplet accompanied by self-evaporation and eventual relaxation. Decomposing the droplet wave function into the underlying single-particle eigenstates allows to unveil and tune the participating dynamical modes. Our findings are expected to inspire engineered droplet dynamics with the aid of external potentials in forthcoming ultracold atom experiments.