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We provide a theoretical characterization of the dynamical crossing of thesuperfluid-supersolid phase transition for a dipolar condensate confined in anelongated trap, as observed in the recent experiment by G. Biagioni et al.[Phys. Rev. X 12, 021019 (2022)]. By means of the extended Gross-Pitaevskiitheory, which includes the Lee-Huang-Yang quantum fluctuation correction, wefirst analyze the ground state configurations of the system as a function ofthe interparticle scattering length, for both trap configurations employed inthe experiment. Then, we discuss the effects of the ramp velocity, by which thescattering length is tuned across the transition, on the collective excitationsof the system in both the superfluid and supersolid phases. We find that, whenthe transverse confinement is sufficiently strong and the transition has asmooth (continuous) character, the system essentially displays a (quasi) 1Dbehavior, its excitation dynamics being dominated by the axial breathing modes.Instead, for shallower transverse trapping, when the transition becomesdiscontinuous, the collective excitations of the supersolid display a couplingwith the transverse modes, signalling the onset of a dimensional crossover.

Crossing the superfluid-supersolid transition of an elongated dipolar condensate