Ground state and dynamics of one-dimensional quantum droplets

Quantum droplets-arising from the delicate balance between repulsive and attractive interactions-continue to be of significant interest in the study of ultracold atomic systems. In this work, we revisit the ground-state properties and collective dynamics of one-dimensional quantum droplets. We identify a critical effective particle number, [Formula: see text], at which the superfluid fraction [Formula: see text] exhibits a distinct inflection point, indicating a structural transition in the ground state. For [Formula: see text], the density profile is sharply peaked, whereas for [Formula: see text], it flattens into a plateau-like shape well-approximated by the Thomas-Fermi model. Additionally, we show that super-Gaussian functions provide excellent fits to the ground-state density profiles, offering a simple and accurate modeling approach. To study the system's dynamical behavior, we develop an analytical framework for quantum droplets subjected to a periodic lattice potential. In the weak-lattice limit ([Formula: see text]), the excitation spectrum reveals a Goldstone gapless phonon mode, characteristic of superfluidity. However, at low densities, the inclusion of Lee-Huang-Yang corrections leads to phonon instabilities, consistent with the transition from a peak- to a plateau-like ground state. In the strong-lattice regime (large [Formula: see text]), a gap opens in the lowest excitation modes, suggesting a crossover from a superfluid to a Mott-insulating phase. Our findings should shed light on key aspects of a low-dimensional quantum droplet.