We simulate interaction quenches crossing from a superfluid to a supersolid state in a dipolar quantum gas of Dy 164 atoms, trapped in an elongated tube with periodic boundary conditions, via the extended Gross-Pitaevskii equation. A freeze-out time is observed through a delay in supersolid formation after crossing the critical point. We compute the density-density correlations at the freeze-out time and extract the frozen correlation length for the solid order. An analysis of the freeze-out time and correlation length versus the interaction quench rate allows us to extract universal exponents corresponding to the relaxation time and correlation length based on predictions of the Kibble-Zurek mechanism. Over several orders of magnitude, clear power-law scaling is observed for both the freeze-out time and the correlation length, and the corresponding exponents are compatible with predictions based on the excitation spectrum calculated via Bogoliubov theory. Defects due to independent local breaking of translational symmetry, contributing to globally incommensurate supersolid order, are identified, and their number at the freeze-out time is found to also scale as a power law. Our results support the hypothesis of a continuous transition whose universality class remains to be determined but appears to differ from that of the (1 + 1)-dimensional model.
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
164Dy 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±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
fs cross onset thresholds near
fs=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
0.037a0 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.
We characterize the role of roton instability in the formation of a supersolid state of an elongated dipolar condensate, following a quench of the contact interactions across the superfluid-supersolid transition, as observed in recent experiments. We perform dynamical simulations by means of the extended Gross-Pitaevskii equation including quantum corrections, for different final values of the $s\text{\ensuremath{-}}\mathrm{wave}$ scattering length. The corresponding excitation spectrum is computed using an effective one-dimensional description, which provides a reasonably accurate prediction for the growth rate of the most unstable mode observed in the simulations. To analyze the behavior of the system, we employ the inverse participation ratio, which conveniently characterizes the different degree of localization in the superfluid and supersolid phases. By means of a suitable effective ansatz for the density, we derive a simple yet effective expression for the formation time of the supersolid. This expression provides valuable insights regarding its scaling behavior with respect to the
s-wave scattering length.
The crossing of a continuous phase transition gives rise to the formation of topological defects described by the Kibble-Zurek mechanism (KZM) in the limit of slow quenches. The KZM predicts a universal power-law scaling of the defect density as a function of the quench time. We focus on the deviations from KZM experimentally observed in rapid quenches and establish their universality. While KZM scaling holds below a critical quench rate, for faster quenches the defect density and the freeze-out time become independent of the quench rate and exhibit a universal power-law scaling with the final value of the control parameter. These predictions are verified in several paradigmatic scenarios in both the classical and quantum domains.
Half a century ago, T. Kibble proposed a scenario for topological defect formation from symmetry breaking during the expansion of the early Universe. W. Zurek later crystallized the concept to superfluid helium, predicting a power-law relation between the number of quantum vortices and the rate at which the system passes through the lambda transition. Here, we report the observation of Kibble-Zurek scaling in a homogeneous, strongly interacting Fermi gas undergoing a superfluid phase transition. We investigate the superfluid transition using two distinct control parameters: temperature and interaction strength. The microscopic physics of condensate formation is markedly different for the two quench parameters, signaled by their two orders of magnitude difference in the condensate formation timescale. However, regardless of the thermodynamic direction in which the system passes through a phase transition, the Kibble-Zurek exponent is identically observed to be about 0.68 and shows good agreement with theoretical predictions that describe superfluid phase transitions. This work demonstrates the gedanken experiment Zurek proposed for liquid helium that shares the same universality class with strongly interacting Fermi gases.
We study the non-equilibrium quench dynamics crossing a continuous phase transition between the charge density wave (CDW) and supersolid (SS) phases of a bosonic lattice gas with cavity-mediated interactions. When changing the hopping amplitude in the Hamiltonian as a function of time, we investigate the scaling behavior of the correlation length and vortex density with respect to the quench time and find that there is a threshold of the quench rate separating two distinct scaling regimes. When slowly varying the system below that threshold, we find a power-law scaling as predicted by the Kibble-Zurek mechanism (KZM). While considering fast quench above that threshold, a deviation from the KZM prediction occurs, manifested by a saturation of the defect density. We further show that such distinct scaling behaviors during different dynamic procedures can be understood through comparing the relaxation time and the quench rate.
Kibble-Zurek theory models the dynamics of spontaneous symmetry breaking, which plays an important role in a wide variety of physical contexts, ranging from cosmology to superconductors. We explored these dynamics in a homogeneous system by thermally quenching an atomic gas with short-range interactions through the Bose-Einstein phase transition. Using homodyne matter-wave interferometry to measure first-order correlation functions, we verified the central quantitative prediction of the Kibble-Zurek theory, namely the homogeneous-system power-law scaling of the coherence length with the quench rate. Moreover, we directly confirmed its underlying hypothesis, the freezing of the correlation length near the transition. Our measurements agree with a beyond-mean-field theory and support the expectation that the dynamical critical exponent for this universality class is z = 3/2.