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We investigate the excitation spectrum and compressibility of a dipolarBose-Einstein condensate in an infinite tube potential in the parameter regimewhere the transition between superfluid and supersolid phases occurs. Our studyfocuses on the density range in which crystalline order develops continuouslyacross the transition. Above the transition the superfluid shows a singlegapless excitation band, phononic at small momenta and with a roton at a finitemomentum. Below the transition, two gapless excitations branches (three at thetransition point) emerge in the supersolid. We examine the two gaplessexcitation bands and their associated speeds of sound in the supersolid phase.Our results show that the speeds of sound and the compressibility arediscontinuous at the transition, indicating a second-order phase transition.These results provide valuable insights into the identification of supersolidphenomena in dipolar quantum gases and the relationship to supersolidity inspin-orbit coupled gases.

Compressibility and speeds of sound across the superfluid to supersolid phase transition of an elongated dipolar gas