  Stabilized by quantum fluctuations, dipolar Bose-Einstein condensates can
form self-bound liquidlike droplets in the mean-field unstable regime. However
in the Bogoliubov theory, some phonon energies are imaginary in the
long-wavelength limit, implying dynamical instability of this system. A similar
instability appears in the Bogoliubov theory of a binary quantum droplet, and
is removed due to higher-order quantum fluctuations as shown recently [1]. In
this work, we study the phonon energy of a dipolar quantum droplet in the
Beliaev formalism, and find that quantum fluctuations can enhance the phonon
stability. We obtain the anisotropic sound velocity which can be tested in
experiment.
