Localization and splitting of a quantum droplet with a potential defect

We unravel the existence and nonequilibrium response of one-dimensional harmonically trapped droplet configurations in the presence of a central potential barrier or well. For fixed negative chemical potentials, it is shown that droplets fragment into two for increasing potential barrier heights, a process that occurs faster for larger widths. However, atoms from the droplet accumulate at the potential well, especially for wider ones, leading to a deformed droplet and eventually to the termination of the solution. Linearization analysis yields the underlying excitation spectrum which dictates stability and the behavior of the ensuing collective modes. Quenches in the potential height are used to demonstrate dynamical fragmentation of the droplet for potential barriers as well as self-evaporation along with droplet localization and eventual relaxation for longer evolution times in the case of potential wells. The presence of selective excitation processes emanating from quantum superposition in the induced droplet dynamics is explicated by evaluating the contribution of the participating single-particle eigenstates. Our results should be detectable by current ultracold atom experiments and may inspire engineered droplet dynamics with the aid of external potentials.