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Merging $N$ independent condensates: Disentangling the Kibble-Zurek mechanism

Monika Aidelsburger, Jean-Loup Ville, R. Saint-Jalm, Sylvain Nascimbène, Jean Dalibard, J. Beugnon

Published May 7, 2017• 1 citation
OpenAlex

Abstract

The Kibble-Zurek theory predicts the scaling of the defect density with the quench rate across a second-order phase transition. The detailed dynamics is generally complex, which makes quantitative comparisons with experiments challenging. Here, we investigate the formation of supercurrents in an annulus by merging an adjustable number $N$ of equally-sized condensates with random phases. This circumvents uncertainties associated with random domain sizes and enables a direct investigation of the supercurrent formation. We study its scaling with $N$ and determine the underlying relaxation timescales.

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Merging NNN independent condensates: Disentangling the Kibble-Zurek mechanism
The Kibble-Zurek theory predicts the scaling of the defect density with the quench rate across a second-order phase transition. The detailed dynamics is generally complex, which makes quantitative comparisons with experiments challenging. Here, we investigate the formation of supercurrents in an annulus by merging an adjustable number NNN of equally-sized condensates with random phases. This circumvents uncertainties associated with random domain sizes and enables a direct investigation of the supercurrent formation. We study its scaling with NNN and determine the underlying relaxation timescales.
The Kibble-Zurek mechanism (KZM) describes the nonequilibrium dynamics and topological defect formation in systems undergoing second-order phase transitions. KZM has found applications in fields such as cosmology and condensed matter physics. However, it is generally not suitable for describing first-order phase transitions. It has been demonstrated that transitions in systems like superconductors or charged superfluids, typically classified as second order, can exhibit weakly first-order characteristics when the influence of fluctuations is taken into account. Moreover, the order of the phase transition (i.e., the extent to which it becomes first rather than second order) can be tuned. We explore quench-induced formation of topological defects in such tunable phase transitions and propose that their density can be predicted by combining KZM with nucleation theory.
Phase transitions are usually treated as equilibrium phenomena, which yields telltale universality classes with scaling behavior of relaxation time and healing length. However, in second-order phase transitions relaxation time diverges near the critical point ("critical slowing down"). Therefore, every such transition traversed at a finite rate is a non-equilibrium process. Kibble-Zurek mechanism (KZM) captures this basic physics, predicting sizes of domains - fragments of broken symmetry - and the density of topological defects, long-lived relics of symmetry breaking that can survive long after the transition. To test KZM we simulate Bose-Einstein condensation in a ring using stochastic Gross-Pitaevskii equation and show that BEC formation can spontaneously generate quantized circulation of the newborn condensate. The magnitude of the resulting winding numbers and the time-lag of BEC density growth - both experimentally measurable - follow scalings predicted by KZM. Our results may also facilitate measuring the dynamical critical exponent for the BEC transition.
We propose a theory to explain the experimental observed deviation from the Kibble-Zurek mechanism (KZM) scaling in rapidly quenched critical phase transition dynamics. There is a critical quench rate τQc1τ_{Q}^{c1}τQc1​ above it the KZM scaling begins to appear. Smaller than τQc1τ_Q^{c1}τQc1​, the defect density nnn is a constant independent of the quench rate but depends on the final temperature TfT_fTf​ as n∝LdεTfdνn \propto L^d ε_{T_f} ^{d ν}n∝LdεTf​dν​, the freeze out time t^\hat{t}t^ admits the scaling law t^∝εTf−νz\hat{t} \propto ε_{T_f}^{-νz}t^∝εTf​−νz​ where ddd is the spatial dimension, εTf=(1−Tf/Tc)ε_{T_f}= (1-T_f/T_c)εTf​​=(1−Tf​/Tc​) is the dimensionless reduced temperature, LLL is the sample size, ννν and zzz are spatial and dynamical critical exponents. Quench from TcT_cTc​, the critical rate is determined by the final temperature TfT_fTf​ as $τ_Q^{c1} \propto ε_{T_f}^{-(1+z ν)} $. All the scaling laws are verified in a rapidly quenched superconducting ring via the AdS/CFT correspondence.
The emergence of order from an initially uncorrelated state across a phase transition is a central problem in quantum many-body physics, particularly in multicomponent systems where interactions between components lead to rich nonequilibrium dynamics. While defect formation is known to follow universal scaling laws, prior studies have focused mainly on defect density, leaving their spatial organization largely unexplored. Here we show that gradually tuning the chemical potential in a two-dimensional binary Bose gas drives condensation into either a miscible or immiscible phase. In the immiscible regime, domains form whose number, size, and boundary length obey Kibble-Zurek (KZ) scaling and evolve self-similarly. In the miscible regime, vortices emerge with KZ scaling. In both cases, the spatial distribution of vortices and domains is well described by a Poisson point process with KZ-determined density. These results reveal universal features of far-from-equilibrium dynamics and provide a framework to characterize stochastic geometry in multicomponent quantum systems. Driving a binary quantum gas mixture through a continuous phase transition can create domains or vortices, depending on the interaction regime. Here, the authors show that their formation universally follow Kibble-Zurek scaling and Poisson spatial statistics with predictions testable in ultracold atom experiments.
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.