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Physical Review A

Domain formation and universally critical dynamics through phase separation in two-component Bose-Einstein condensates

Xunda Jiang, Yikai Ji, Xizhou Qin, Bin Liu, Bo Lü, Yongyao Li, Chaohong Lee

Published Dec 16, 2024• 0 citations
Publisher

Abstract

We investigate defect formation and the universal critical dynamics of two-dimensional two-component Bose-Einstein condensates (BECs) confined in both box and harmonic traps. Our focus is on the nonequilibrium universal dynamics of the miscible-immiscible phase transition, considering both linear and nonlinear quenching protocols. The general condition for a phase transition, ${g}_{11}{g}_{22}={g}_{12}^{2}$, depends exclusively on inter- and intraspecies interactions. In this system, simultaneous phase transitions occur in both the box and harmonic traps. Consequently, the nonequilibrium critical dynamics can still be described using the homogeneous Kibble-Zurek mechanism. Through an analysis of Bogoliubov excitations, we establish a power-law relationship linking the domain correlation length, phase transition delay, and quench time. Additionally, real-time simulations of phase transition dynamics demonstrate the formation of domain defects and the delay of phase transitions in nonequilibrium conditions. These simulations also reveal the universal scaling of the correlation length and phase transition delay across various quench times and nonlinear coefficients, showing well agreement with our analytical predictions. Our study confirms that the universality class of two-component BECs is independent of dimensionality. Additionally, larger nonlinear coefficients effectively suppress the growth of nonadiabatic excitations, offering a fresh perspective on achieving quasiadiabatic evolution.

DOI
10.1103/physreva.110.063316

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We investigate defect formation and the universal critical dynamics of two-dimensional two-component Bose-Einstein condensates (BECs) confined in both box and harmonic traps. Our focus is on the nonequilibrium universal dynamics of the miscible-immiscible phase transition, considering both linear and nonlinear quenching protocols. The general condition for a phase transition, g11g22=g122{g}_{11}{g}_{22}={g}_{12}^{2}g11​g22​=g122​, depends exclusively on inter- and intraspecies interactions. In this system, simultaneous phase transitions occur in both the box and harmonic traps. Consequently, the nonequilibrium critical dynamics can still be described using the homogeneous Kibble-Zurek mechanism. Through an analysis of Bogoliubov excitations, we establish a power-law relationship linking the domain correlation length, phase transition delay, and quench time. Additionally, real-time simulations of phase transition dynamics demonstrate the formation of domain defects and the delay of phase transitions in nonequilibrium conditions. These simulations also reveal the universal scaling of the correlation length and phase transition delay across various quench times and nonlinear coefficients, showing well agreement with our analytical predictions. Our study confirms that the universality class of two-component BECs is independent of dimensionality. Additionally, larger nonlinear coefficients effectively suppress the growth of nonadiabatic excitations, offering a fresh perspective on achieving quasiadiabatic evolution.
We explore the defect formation and universally critical dynamics in two-dimensional (2D) two-component Bose-Einstein condensates(BECs) subjected to two types of potential traps: a homogeneous trap and a harmonic trap.We focus on the non-equilibrium universal dynamics of the miscible-immiscible phase transition with both linear and nonlinear quenching types.Although there exists spatial independence of the critical point, we find that the inhomogeneity of trap doesn't affect the phase transition of system and the critical exponents can still be explained by the homogeneous Kibble-Zurek mechanism. By analyzing the Bogoliubov excitations, we establish a power-law relationship between the domain correlation length, the phase transition delay, and the quench time.Furthermore, through real-time simulations of phase transition dynamics, the formation of domain defects and the delay of phase transition in non-equilibrium dynamics are demonstrated, along with the corresponding universal scaling of correlation length and phase transition delay for various quench time and quench coefficients, which align well with our analytical predictions.Our study confirms that the universality class of two-component BECs remains unaffected by dimensionality, while the larger nonlinear coefficients effectively suppress non-adiabatic excitations, offering a novel perspective for addressing adiabatic evolution.
We investigate the non-equilibrium dynamics across the miscible–immiscible phase separation in a binary mixture of Bose–Einstein condensates. The excitation spectra reveal that the Landau critical velocity vanishes at the critical point, where the superfluidity spontaneously breaks down. We analytically extract the dynamical critical exponent z = 2 and static correlation length critical exponent v = 1/2 from the Landau critical velocity. Moreover, by simulating the real-time dynamics across the critical point, we find the average domain number and the average bifurcation delay show universal scaling laws with respect to the quench time. We then numerically extract the static correlation length critical exponent v = 1/2 and the dynamical critical exponent z = 2 according to Kibble–Zurek mechanism. The scaling exponents ( v = 1/2, z = 2) in the phase separation driven by quenching the atom–atom interaction are different from the ones ( v = 1/2, z = 1) in the phase separation driven by quenching the Rabi coupling strength (2009 Phys. Rev. Lett. 102 070401; 2011 Phys. Rev. Lett. 107 230402). Our study explores the connections between the spontaneous superfluidity breakdown and the spontaneous defect formation in the phase separation dynamics.
The miscibility-immiscibility phase transition in binary Bose-Einstein condensates (BECs) can be controlled by a coupling between the two components. Here we propose a new scheme that uses coupling-induced pattern formation to test the Kibble-Zurek mechanism (KZM) of topological-defect formation in a quantum phase transition. For a binary BEC in a ring trap we find that the number of domains forming the pattern scales as a function of the coupling quench rate with an exponent as predicted by the KZM. For a binary BEC in an elongated harmonic trap we find a different scaling law due to the transition being spatially inhomogeneous. We perform a "quantum simulation" of the harmonically trapped system in a ring trap to verify the scaling exponent.
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.
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.
Continuous phase transitions occur in a wide range of physical systems, and provide a context for the study of non-equilibrium dynamics and the formation of topological defects.The Kibble-Zurek (KZ) mechanism predicts the scaling of the resulting density of defects as a function of the quench rate through a critical point, and this can provide an estimate of the critical exponents of a phase transition.In this work we extend our previous study of the miscible-immiscible phase transition of a binary Bose-Einstein condensate (BEC) composed of two hyperfine states in which the spin dynamics are confined to one dimension [J.Sabbatini et al., Phys.Rev. Lett.107, 230402 (2011)].The transition is engineered by controlling a Hamiltonian quench of the coupling amplitude of the two hyperfine states, and results in the formation of a random pattern of spatial domains.Using the numerical truncated Wigner phase space method, we show that in a ring BEC the number of domains formed in the phase transitions scales as predicted by the KZ theory.We also consider the same experiment performed with a harmonically trapped BEC, and investigate how the density inhomogeneity modifies the dynamics of the phase transition and the KZ scaling law for the number of domains.We then make use of the symmetry between inhomogeneous phase transitions in anisotropic systems, and an inhomogeneous quench in a homogeneous system, to engineer coupling quenches that allow us to quantify several aspects of inhomogeneous phase transitions.In particular, we quantify the effect of causality in the propagation of the phase transition front on the resulting formation of domain walls, and find indications that the density of defects is determined during the impulse to adiabatic transition after the crossing of the critical point.