Exciting the Goldstone Modes of a Supersolid Spin-Orbit-Coupled Bose Gas

Supersolidity is deeply connected with the emergence of Goldstone modes, reflecting the spontaneous breaking of both phase and translational symmetry. Here, we propose accessible signatures of these modes in harmonically trapped spin-orbit-coupled Bose-Einstein condensates, where supersolidity appears in the form of stripes. By suddenly changing the trapping frequency, an axial breathing oscillation is generated, whose behavior changes drastically at the critical Raman coupling. Above the transition, a single mode of hybridized density and spin nature is excited, while below it, we predict a beating effect signaling the excitation of a Goldstone spin-dipole mode. We further provide evidence for the Goldstone mode associated with the translational motion of stripes. Our results open up new perspectives for probing supersolid properties in experimentally relevant configurations with both symmetric as well as highly asymmetric intraspecies interactions.