Spinodal-controlled freeze-out and nucleation at the first-order superfluid-supersolid transition of a dipolar gas

Abstract

Dipolar quantum gases in elongated traps become supersolid through a transition that is continuous at intermediate density but first order at low and high density; at the continuous point, ramps of the scattering length follow Kibble-Zurek scaling. We study ramps across the first-order superfluid-supersolid transition of $^{164}$Dy in an infinite tube using the 3D extended Gross-Pitaevskii equation with truncated-Wigner noise. With quantum noise only, the uniform superfluid survives the metastable window and transforms after its spinodal, where the roton softens. For onsets occurring during the linear ramp, the delay measured from this spinodal has a fitted exponent $ζ= 0.36 \pm 0.02$, close to the continuous-transition result and the mean-field value $1/3$. Reverse ramps show a comparable delay beyond the fold of the crystal branch, producing dynamical hysteresis. Initial thermal noise with nominal temperatures of 2.5-10 nK produces crystalline regions inside the metastable window during holds, with faster nucleation closer to the spinodal and at higher initial temperature. In ramps at 5 nK, thermal fluctuations make the superfluid fraction $f_s$ cross onset thresholds near $f_s = 1$ early, whereas the bulk of the crystal still forms after the zero-temperature spinodal with a similar scaling. Without a heat bath these effective rates depend on preparation: ramps crossing the window at the same rate nucleate early in 2 of 32 or 11 of 16 realizations depending on their starting point. At low density, slower ramps select larger spacings in a crystal of separated droplets, and droplets survive far above the transition on reverse ramps. A Maxwell construction gives a coexistence window $0.037 a_0$ wide around the first-order transition point, in which localized states exist; in the simulated tube they lie above the homogeneous states. At the continuous point there is no coexistence.