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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 trappedspin-orbit-coupled Bose-Einstein condensates, where supersolidity appears inthe form of stripes. By suddenly changing the trapping frequency, an axialbreathing oscillation is generated, whose behavior changes drastically at thecritical Raman coupling. Above the transition, a single mode of hybridizeddensity and spin nature is excited, while below it, we predict a beating effectsignaling the excitation of a Goldstone spin-dipole mode. We further provideevidence for the Goldstone mode associated with the translational motion ofstripes. Our results open up new perspectives for probing supersolid propertiesin experimentally relevant configurations with both symmetric as well as highlyasymmetric intraspecies interactions.

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