We study the universal spatial statistics of point-like topological defects formed during the nonequilibrium condensation of a coherently coupled Bose gas using the stochastic projected Gross-Pitaevskii equation. The symmetry-breaking transition is driven by a linear quench of the chemical potential, leading to stochastic vortex nucleation in the individual condensate components. When the two components are considered together, these elementary defects may combine across components to emerge as composite topological defects known as full quantum vortices. Beyond the mean defect density predicted by the Kibble-Zurek mechanism (KZM), we investigate the spatial organization of both the elementary and composite defects and show that their positions are well described by a Poisson point process, revealing a universal stochastic geometry. This universality is further described through Voronoi tessellation, whose cell-area statistics follow Poisson-Voronoi predictions. We also introduce the spatial form factor for characterizing the vortex configurations and demonstrate the emergence of a characteristic dip-ramp-plateau structure. Our results establish universal stochastic geometry of topological defects beyond conventional Kibble-Zurek scaling and identify it as a fundamental feature of nonequilibrium condensation in coherently coupled Bose gases.
We study the universal spatial statistics of point-like topological defects formed during the nonequilibrium condensation of a coherently coupled Bose gas using the stochastic projected Gross-Pitaevskii equation. The symmetry-breaking transition is driven by a linear quench of the chemical potential, leading to stochastic vortex nucleation in the individual condensate components. When the two components are considered together, these elementary defects may combine across components to emerge as composite topological defects known as full quantum vortices. Beyond the mean defect density predicted by the Kibble-Zurek mechanism (KZM), we investigate the spatial organization of both the elementary and composite defects and show that their positions are well described by a Poisson point process, revealing a universal stochastic geometry. This universality is further described through Voronoi tessellation, whose cell-area statistics follow Poisson-Voronoi predictions. We also introduce the spatial form factor for characterizing the vortex configurations and demonstrate the emergence of a characteristic dip-ramp-plateau structure. Our results establish universal stochastic geometry of topological defects beyond conventional Kibble-Zurek scaling and identify it as a fundamental feature of nonequilibrium condensation in coherently coupled Bose gases.
The cooling of a Bose gas in finite time results in the formation of a Bose-Einstein condensate that is spontaneously proliferated with vortices. We propose that the vortex spatial statistics is described by a homogeneous Poisson point process (PPP) with a density dictated by the Kibble-Zurek mechanism (KZM). We validate this model using numerical simulations of the two-dimensional stochastic Gross-Pitaevskii equation (SGPE) for both a homogeneous and a hard-wall trapped condensate. The KZM scaling of the average vortex number with the cooling rate is established along with the universal character of the vortex number distribution. The spatial statistics between vortices is characterized by analyzing the two-point defect-defect correlation function, the corresponding spacing distributions, and the random tessellation of the vortex pattern using the Voronoi cell area statistics. Combining the PPP description with the KZM, we derive universal theoretical predictions for each of these quantities and find them in agreement with the SGPE simulations. Our results establish the universal character of the spatial statistics of point-like topological defects generated during a continuous phase transition and the associated stochastic geometry. Published by the American Physical Society 2024
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 Kibble-Zurek mechanism (KZM) predicts the spontaneous formation of topological defects in a continuous phase transition driven at a finite rate. We propose the generation of spontaneous quantum turbulence (SQT) via the KZM during Bose-Einstein condensation induced by a thermal quench. Using numerical simulations of the stochastic projected Gross-Pitaevskii equation in two spatial dimensions, we describe the formation of a newborn Bose-Einstein condensate proliferated by quantum vortices. We establish the nonequilibrium universality of SQT through the Kibble-Zurek and Kolmogorov scaling of the incompressible kinetic energy.
The crossing of a continuous phase transition results in the formation of topological defects with a density predicted by the Kibble-Zurek mechanism (KZM). We characterize the spatial distribution of pointlike topological defects in the resulting nonequilibrium state and model it using a Poisson point process in arbitrary spatial dimensions with KZM density. Numerical simulations in a one-dimensional ${\ensuremath{\phi}}^{4}$ theory unveil short-distance defect-defect corrections stemming from the kink excluded volume, while in two spatial dimensions, our model accurately describes the vortex spacing distribution in a strongly coupled superconductor indicating the suppression of defect-defect spatial correlations.
The formation of topological defects during continuous phase transitions exhibits nonequilibrium universality. While the Kibble-Zurek mechanism (KZM) predicts universal scaling of point-like defect numbers under slow driving, the statistical properties of extended defects remain largely unexplored across both slow and fast protocols. We investigate vortex string formation in a three-dimensional holographic superfluid. For slow quenches, the vortex string number follows KZM scaling, while for rapid quenches, it exhibits complementary universal scaling governed by the final temperature. Beyond the vortex string number, the loop-length distribution reveals a richer structure: individual loops follow the first-return statistics of three-dimensional random walks, P(ℓ)∼ℓ−5/2. While the total vortex length distribution remains Gaussian, its cumulants obey universal scaling laws with varying power-law exponents, and thus differ markedly from those observed in point-defect systems, indicating distinct statistical features of extended topological defects.
We investigate the dynamics of formation of topological defects in ultra-cold quantum gases. Using classical field methods, such as the truncated Wigner approximation and the stochastic Gross-Pitaevskii equation, we explore the connection between the rate of a phase transition and the number of topological defects formed as a consequence of it. We show that in various systems such defects form in number comparable with the predictions of the Kibble-Zurek theory, known for thirty years but as yet unverified experimentally. We also consider the spontaneous production of entangled vortices following a quantum phase transition in a multi-component spin-1 Bose-Einstein condensate (BEC). To begin, we provide a literature review and describe some of the background theory of multi-component condensates, stochastic methods and Kibble-Zurek theory. Part of this thesis focus on a two component Bose-Einstein condensates of the same atomic species. In particular on the quantum critical point separating the miscible from the immiscible phase of such systems. We show that such phase transition cannot only be achieved with the tuning of the interaction constants between the components, but also through changes in the intensity of the coupling between atoms in different hyperfine states. We first determine the value of the critical coupling, and also the critical exponents defining the universality class of the coupling—induced phase transition. We then proceed to simulate the dynamics of the transition in effective 1D ring and harmonically trapped condensates, and test the results against predictions of the Kibble-Zurek (KZ) mechanism. We find good agreement between the KZ theory and the numerical results in the ring geometry, and we motivate the origin of the discrepancies between the two in harmonically trapped BECs. We use the ”equivalence” of inhomogeneous phase transitions in anisotropic systems and spatially inhomogeneous quenches to engineer coupling quenches and systematically study key aspects of the inhomogeneous KZ mechanism. The coupling induced miscible—immiscible transition results in the formation of a stable pattern of domains that easily be detected and counted in current ultra-cold gases experiment, and the scheme is a viable candidate for the first experimental testing of the KZ predictions. We then examine the spontaneous formation of a superflow as result of the thermal Bose-Einstein phase transition in ring traps. According to the Kibble-Zurek theory, the fast production of a condensate in a ring trap should result in the formation of solitons, with the number depending on the rate of the temperature quench. Solitons are short-living topological defects and their observation in ultra-cold gases experiments remains challenging. We show here that even after they decay, solitons leave a signature in the variance of the final winding number of the condensate. Solitons fragment a 1D condensate into regions of random phase, hence the larger the number of solitons, the higher the probability that the resulting phase wraps around the ring by a multiple of 2π. We numerically simulate the BEC transition in this system with the stochastic Gross-Pitaevskii equation, and observe that the winding number scales with the quenching time as predicted by the KZ mechanism. We examine the effect of inhomogeneity on the expectation value of the winding number, and find that the tilting of the ring greatly suppress the spontaneous formation of vortices. This has implications for the necessary horizontal alignment of any experiment on this system. Finally, we consider the polar to anti-ferromagnetic phase transition in magnetically quenched spin-1 BEC. In this quantum phase transition a pure condensate of atoms in the mF = 0 Zeeman sublevel is converted into a three component BEC with atoms in mF = 0, ±1 through spin changing collisions. An analogous mechanism, four—wave mixing, in quantum optics produces entangled photons, and it has been proposed in ultra-cold gases as a method to obtain Einstein-Podolski-Rosen entanglement of macroscopic atomic ensembles. We begin with a rotating 1D spinor condensate of atoms in the mF = 0 component, and perform a quench of the magnetic field through the critical point. We show how the transition produces a pair of entangled components with different winding numbers, and quantify the entanglement as a function of the modes considered and of the type of magnetic quench, sudden or instantaneous. Lastly, we propose a method to use the original spin zero component as coherent source to perform homodyne detection and consider the two rotating entangled components as a resource for the sensing of rotations.