Quantized Vortices and Sound Velocities in Dipolar Supersolids

Supersolids are an intriguing phase of matter that simultaneously exhibit crystalline order and superfluidity. They have recently been realized experimentally in dipolar Bose-Einstein condensates. This thesis investigates two fundamental features of dipolar supersolids: quantized vortices and the propagation of sound modes. Although quantized vortices constitute a key probe of superfluidity, their observability in dipolar supersolids is largely inhibited by the strong density depletion caused by droplet formation. We propose a protocol for vortex nucleation and detection, based on a ramp of the s-wave scattering length across the superfluid-supersolid phase transition. Starting from a slowly rotating, vortex-free configuration in the superfluid phase, a vortex is nucleated as the system enters the supersolid phase, due to the strong reduction of the critical angular velocity. Once created, the vortex is robustly preserved as the condensate is brought back into the superfluid phase, where it can be readily observed. This protocol was recently implemented in experiments, confirming its feasibility. The second part explores sound propagation in a dipolar supersolid confined in a tubular potential with periodic boundary conditions, corresponding to a ring geometry. We propose a method for exciting Goldstone modes, arising from the spontaneous breaking of phase and translational symmetries. By abruptly removing a weak periodic perturbation, the resulting oscillations are studied by numerically solving the extended Gross-Pitaevskii equation. The values of the two longitudinal sound velocities exhibited in the supersolid phase are then analyzed using the hydrodynamic theory of supersolids at zero temperature. This approach allows for the determination of key hydrodynamic parameters, such as the layer compressibility modulus and the superfluid fraction, the latter found to be consistent with the Leggett upper bound.