The low-energy Goldstone mode in a trapped dipolar supersolid.

A supersolid is a counter-intuitive state of matter that combines the frictionless flow of a superfluid with the crystal-like periodic density modulation of a solid1,2 . Since the first prediction in the 1950s3 , experimental efforts to realize this state have focused mainly on helium, where supersolidity remains elusive4 . Recently, supersolidity has also been studied intensively in ultracold quantum gases, and some of its defining properties have been induced in spin-orbit coupled Bose-Einstein condensates (BECs)5,6 and BECs coupled to two crossed optical cavities7,8 . However, no phonon modes have been observed in both systems. Recently, hallmark properties of a supersolid-the periodic density modulation and simultaneous global phase coherence-have been observed in arrays of dipolar quantum droplets9-11 , where the crystallization happens in a self-organized manner due to intrinsic interactions. In this letter, we prove the genuine supersolid nature of these droplet arrays by directly observing the low-energy Goldstone mode. The dynamics of this mode is reminiscent of the effect of second sound in other superfluid systems12,13 and features an out-of-phase oscillation of the crystal array and the superfluid density. This mode exists only due to the phase rigidity of the experimentally realized state, and therefore confirms the genuine superfluidity of the supersolid.