Breathing mode of a quantum droplet in a quasi-one-dimensional dipolar Bose gas

We investigate the breathing mode and the stability of a quantum droplet in a tightly trapped one-dimensional dipolar gas of bosonic atoms. When the droplet with a flat-top density profile is formed, the breathing-mode frequency scales as the inverse of the number of atoms in the cloud. This is straightforwardly derived within a phenomenological hydrodynamical approach and confirmed using both a variational method based on a generalized Gross-Pitaevskii action functional and the sum-rule approach. We extend our analysis also to the presence of axial confinement showing the effect of the trap on the density profile and therefore on the breathing-mode frequency scaling. Our analysis confirms the stability of the quantum droplet against the particle emission when the flat-top density profile is observed. Our results can be used as a guide to the experimental investigations of collective modes to detect the formation of quantum droplets in quasi-one-dimensional dipolar gases.