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arXiv:2412.18850 · Physical Review A

Dynamics of topological defects in a Rashba spin-orbit-coupled Bose-Einstein condensate

Sheng Liu, Yong-Sheng Zhang

Published Aug 15, 2025• 2 citations
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Abstract

We investigate the quench dynamics of a two-dimensional Rashba spin-orbit-coupled Bose-Einstein condensate. Our study focuses on quenching the system from a zero-momentum phase to a stripe phase. During this quench, topological defects emerge in the form of vortices. These vortices and antivortices exhibit a random spatial distribution with equal numbers. In a uniform system, we observe an exponential scaling of both the vortex production time and the vortex number with the quench rate, consistent with the conventional Kibble-Zurek mechanism, the decay of which adheres to a logarithmic law, aligning with experimental observations.

DOI
10.1103/qlx5-dqr3

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We investigate the quench dynamics of a two-dimensional Rashba spin-orbit-coupled Bose-Einstein condensate. Our study focuses on quenching the system from a zero-momentum phase to a stripe phase. During this quench, topological defects emerge in the form of vortices. These vortices and antivortices exhibit a random spatial distribution with equal numbers. In a uniform system, we observe an exponential scaling of both the vortex production time and the vortex number with the quench rate, consistent with the conventional Kibble-Zurek mechanism, the decay of which adheres to a logarithmic law, aligning with experimental observations.
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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.
We investigate the saturation of defect density in an atomic Bose gas rapidly cooled into a superfluid phase. The number of quantum vortices, which are spontaneously created in the quenched gas, exhibits a Poissonian distribution not only for a slow quench in the Kibble-Zurek (KZ) scaling regime but also for a fast quench, in which case the mean vortex number is saturated. This shows that the saturation is not caused by destructive vortex collisions, but by the early-time coarsening in an emerging condensate, which is further supported by the observation that the condensate growth lags the quenching in the saturation regime. Our results demonstrate that the defect saturation is an effect beyond the KZ mechanism, opening a path for studying critical phase transition dynamics using the defect number distribution.
Quench dynamics of first-order phase transitions in a homogeneous Bose-Einstein condensate with Raman-type spin-orbit coupling is investigated. By adjusting the coupling strength, the system can be driven across the critical point between the plane-wave phase and stripe phase in two opposite directions, and we find the scaling-law behaviors for both processes of phase transitions. This is different from previous studies, which only focus on the quench dynamics in the unidirectional transitions from a higher symmetric phase to a lower symmetric phase. Subsequently, we obtain the power law of the domain numbers (correlation length) and formation time (correlation time) of these phase transitions and extract the critical exponents. The results reveal the effect of the directionality of the phase transition on the dynamic scaling law.
In this work, we study the non-equilibrium quench dynamics from the superfluid stripe phase to the supersolid phase of a two dimensional spin-orbital coupled interacting Bose-Einstein condensate in the presence of a one dimensional optical lattice. The quench protocol here is constructed through varying the lattice depth linearly with the evolution time. By using the time-dependent Gutzwiller method, various physical quantities, such as the vortex number and the overlap of wave-function, have been investigated with respect to the quench time. Through analyzing the dynamical behavior of the above physical quantities, we find out the transition time of the quench procedure, which captures the freeze out time indicating the moment that the system catches the quench speed beginning to evolve quickly. Before the transition time, the dynamics is frozen and the state of the system cannot follow the changes in the Hamiltonian. While passing the transition time, we find that there are significant alterations to both the vortex number and the wave-function. At the transition time, on one hand the vortex number abruptly increases from zero; on the other hand the overlap of wave-function departures from 1 shortly. These signatures indicate that the system evolves rapidly when passing the transition time. Furthermore, we also find that due to the presence of spin-orbital coupling, the spin texture represents a periodic magnetic structure accompanying with the emergence of the supersolid dynamically. It is shown that during the quench procedure, the density distribution of the system are always accompanied with the spatial structure of spin texture, i.e., the central position of topological spin skyrmion (antiskyrmion) corresponding to the minimum position of the density distribution. The topological charge of the above spin structures also shows interesting dynamical properties. We find that the quantized topological charge appears with the emergence of the supersolid dynamically.