  The existence of a paradoxical supersolid phase of matter, possessing the
apparently incompatible properties of crystalline order and superfluidity, was
predicted 50 years ago. Solid helium was the natural candidate, but there
supersolidity has not been observed yet, despite numerous attempts. Ultracold
quantum gases have recently shown the appearance of the periodic order typical
of a crystal, due to various types of controllable interactions. A crucial
feature of a D-dimensional supersolid is the occurrence of up to D+1 gapless
excitations reflecting the Goldstone modes associated with the spontaneous
breaking of two continuous symmetries: the breaking of phase invariance,
corresponding to the locking of the phase of the atomic wave functions at the
origin of superfluid phenomena, and the breaking of translational invariance
due to the lattice structure of the system. The occurrence of such modes has
been the object of intense theoretical investigations, but their experimental
observation is still missing. Here we demonstrate the supersolid symmetry
breaking through the appearance of two distinct compressional oscillation modes
in a harmonically trapped dipolar Bose-Einstein condensate, reflecting the
gapless Goldstone excitations of the homogeneous system. We observe that the
two modes have different natures, with the higher frequency mode associated
with an oscillation of the periodicity of the emergent lattice and the lower
one characterizing the superfluid oscillations. Our work paves the way to
explore the two quantum phase transitions between the superfluid, supersolid
and solid-like configurations that can be accessed by tuning a single
interaction parameter.
