Collective excitations of self-bound droplets of a dipolar quantum fluid

We calculate the collective excitations of a dipolar Bose-Einstein condensate in the regime where it self-binds into droplets stabilized by quantum fluctuations. We show that the filament-shaped droplets act as a quasi-one-dimensional waveguide along which low angular momentum phonons propagate. The evaporation (unbinding) threshold occurring as the atom number $N$ is reduced to the critical value $N_c$ is associated with a monopole-like excitation going soft as ${ϵ}_0\sim (N-N_c{)}^{1\mathrm{/}4}$. Considering the system in the presence of a trapping potential, we quantify the crossover from a trap-bound condensate to a self-bound droplet.