Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Phase ordering dynamics in a ferromagnetic spin-1 Bose-Einstein condensate

L. A. Williamson

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

Spinor Bose-Einstein condensates exhibit both superfluid and magnetic order, and accommodate phases with rich symmetry properties and topological defects. Transitions between these phases can be induced by tuning external fields. In this thesis we explore the dynamics of order formation in a quasi-2D spin-1 ferromagnetic condensate following a quench from an unmagnetised phase to one of three ferromagnetic phases. The ferromagnetic phases exhibit distinct symmetry properties (easy-plane, easy-axis or isotropic) and support distinct topological defects. In each phase we observe scale invariant ordering and identify the relevant topological defect affecting the order parameter growth. We find that each phase is characterised by a distinct dynamic critical exponent. In the easy-plane phase we identify a persistent turbulent cascade that affects spin ordering long after all topological defects have annihilated. In addition to our exploration of phase ordering dynamics, we study a microscopic model of spin vortex dynamics in the easy-plane phase. Our work provides a comprehensive theoretical study of phase ordering in a conservative system, provides a thorough foundation for studies of phase ordering in antiferromagnetic and higher spin condensates, and offers prospects for further research into fundamental questions regarding the ordering properties of spin systems. Our work is pertinent to current experiments, which have explored the initial stages of phase ordering in both ferromagnetic and antiferromagnetic spin-1 condensates.

Related papers

Nematic ordering dynamics of an antiferromagnetic spin-1 condensate

L. M. Symes, P. B. Blakie

We consider the formation of order in a quasi-two-dimensional antiferromagnetic spin-1 condensate quenched from an easy-axis to an easy-plane nematic phase. We define the relevant order parameter to quantify the spin-nematic degrees of freedom and study the evolution of the spin-nematic and superfluid order during the coarsening dynamics using numerical simulations. We observe dynamical scaling in the late-time dynamics, with both types of order extending across the system with a diffusive growth law. We identify half-quantum vortices as the relevant topological defects of the ordering dynamics and demonstrate that the growth of both types of order is determined by the mutual annihilation of these vortices.

arXiv:1704.00871•Physical Review A•Jul 5, 2017•26 citations

Superfluidity in a two-dimensional ferromagnetic Bose gas

Andrew Underwood

Spinor Bose gases may exhibit both phase and ferromagnetic order. Consequently, they may support superflow of both mass and spin currents. In this thesis we utilize a stochastic Gross-Pitaevskii model to investigate the superfluid properties of a two-dimensional spin-1 Bose gas with ferromagnetic interactions. Owing to the low dimensionality, the superfluidity of this system can not be attributed to the presence of a Bose-Einstein condensate, with thermal fluctuations precluding long-range order. Rather, mass and spin superfluidity emerge via respective Berezinskii-Kosterlitz-Thouless (BKT) transitions. Such transitions are characterized by the emergence of quasi-long-range order, driven by the binding of topological defect pairs. First, we consider an easy-plane ferromagnetic gas. We find the system superfluidity may be understood by direct application of existing BKT theory. In particular, mass and spin superfluidity are respectively associated with quasi-long-range order in the global phase and local magnetization. The former arises from the binding of mass vortex pairs, while the latter arises from the binding of polar-core spin vortex pairs. Following this, we introduce axial magnetization, and consider a gas in the broken-axisymmetric phase. We find axial magnetization acts to couple mass and spin superflows via a mechanism analogous to the superfluid drag exhibited by two-component superfluids. This coupling renders invalid the direct applicability of existing BKT theory: here mass and spin superfluidity may each be associated with both phase and magnetic order. We remedy this with a generalization making concrete the relation between superfluidity and quasi-long-range order in this system. In addition, through variations of the quadratic Zeeman energy and axial magnetization we identify three distinct superfluid phases, which serve as finite-temperature generalizations of the polar, broken-axisymmetric, and easy-axis ferromagnetic ground states.

•Mar 26, 2025•0 citations

Exact nonlinear dynamics of Spinor BECs applied to nematic quenches

L. M. Symes

In this thesis we study the nonlinear dynamics of spin-1 and spin-2 Bose-Einstein condensates, with particular application to antiferromagnetic systems exhibiting nematic (beyond magnetic) order. Firstly, we give a derivation of the spinor energy functionals with a focus on the connections between the nonlinear terms. We derive a hierarchy of nonlinear irreducible multipole observables sensitive to different levels of nematic order, and explore the various nematic states in terms of their multipolar order, representations of their symmetries, and topological defects.\n\nWe then develop an exact solution to the nonlinear dynamics of spinor Bose-Einstein condensates. We use this solution to construct efficient and accurate numerical algorithms to evolve the spinor Gross-Pitaevskii equation in time. We demonstrate the advantages of our algorithms with several 1D numerical test problems, comparing with existing methods in the literature.\n\nWe apply our numerical methods to simulating quenches of the condensate between various antiferromagnetic phases for spin-1 and spin-2. For spin-1, we carry out quenches for a theoretical uniform system in 2D, and then specialize to the parameters used in a recent harmonically trapped experiment in 3D. We connect the long-time coarsening growth law of the relevant order parameter to the decay of half-quantum vortices, which are the relevant topological defects of the ground state.\n\nFor the spin-2 system, we investigate a novel quench from two different quadrupolar-nematic phases to an octupolar-nematic “cyclic” phase which supports 1/3 fractional vortices. We develop appropriate order parameter observables which couple to the spin and superfluid currents generated by these defects, and show that a new growth law appears with exponent 1/3.

•Jan 1, 2019•0 citations

Coarsening Dynamics of an Isotropic Ferromagnetic Superfluid

L. A. Williamson, P. B. Blakie

In zero magnetic field the ground-state manifold of a ferromagnetic spin-1 condensate is SO(3) and exhibits Z_{2} vortices as topological defects. We investigate the phase-ordering dynamics of this system after being quenched into this ferromagnetic phase from a zero-temperature unmagnetized phase. Following the quench, we observe the ordering of both magnetic and gauge domains. We find that these domains grow diffusively, i.e., with domain size L(t)∼t^{1/2}, and exhibit dynamic scale invariance. The coarsening dynamics progresses as Z_{2} vortices annihilate; however, we find that at finite energy a number of these vortices persist in small clumps without influencing magnetic or gauge order. We consider the influence of a small nonzero magnetic field, which reduces the ground-state symmetry, and show that this sets a critical length scale such that when the domains reach this size the system dynamically transitions in order parameter and scaling behavior from an isotropic to an anisotropic ferromagnetic superfluid.

arXiv:1703.09360•Physical Review Letters•Dec 22, 2017•28 citations

Coarsening and thermalization properties of a quenched ferromagnetic spin-1 condensate

L. A. Williamson, P. B. Blakie

We examine the dynamics of a quasi-two-dimensional spin-1 condensate in which the quadratic Zeeman energy $q$ is suddenly quenched to a value where the system has a ferromagnetic ground state. There are two distinct types of ferromagnetic phases, i.e., a range of $q$ values where the magnetization prefers to be in the direction of the external field (easy axis) and a range of $q$ values where it prefers to be transverse to the field (easy plane). We study the quench dynamics for a variety of $q$ values and show that there is a single dynamic critical exponent to characterize the scale-invariant domain growth for each ferromagnetic phase. For both quenches we give simple analytic models that capture the essential scale-invariant dynamics and correctly predict the exponents. Because the order parameter for each phase is different, the natures of the domains and the relevant topological defects in each type of coarsening are also different. To explore these differences we characterize the fractal dimension of the domain walls and the relationship of polar-core spin vortices to the domains in the easy-plane phase. Finally, we consider how the energy liberated from the quench thermalizes in the easy-axis quench. We show that local equilibrium is established in the spin waves on moderate time scales, but continues to evolve as the domains anneal.

arXiv:1605.04016•Physical Review A•Aug 8, 2016•42 citations

Emergence and scaling of spin turbulence in quenched antiferromagnetic spinor Bose-Einstein condensates

Seji Kang, Sang Won Seo, Joon Hyun Kim +1 more

We investigate the phase-transition dynamics of a quasi-two-dimensional antiferromagnetic spin-1 Bose-Einstein condensate from the easy-axis polar phase to the easy-plane polar phase, which is initiated by suddenly changing the sign of the quadratic Zeeman energy $q$. We observe the emergence and decay of spin turbulence and the formation of half-quantum vortices (HQVs) in the quenched condensate. The characteristic time and length scales of the turbulence generation dynamics are proportional to ${|q|}^{\ensuremath{-}1/2}$ as inherited from the dynamic instability of the initial state. In the evolution of the spin turbulence, spin-wave excitations develop from large to small length scales, suggesting a direct energy cascade, and the spin population for the axial polar domains exhibit a nonexponential decay. The final equilibrated condensate contains HQVs, and the number is found to increase and saturate with increasing $|q|$. Our results demonstrate the time-space scaling properties of the phase-transition dynamics near the critical point and the peculiarities of the spin-turbulence state of the antiferromagnetic spinor condensate.

arXiv:1701.01559•Physical Review A•May 30, 2017•35 citations
Spinor Bose-Einstein condensates exhibit both superfluid and magnetic order, and accommodate phases with rich symmetry properties and topological defects. Transitions between these phases can be induced by tuning external fields. In this thesis we explore the dynamics of order formation in a quasi-2D spin-1 ferromagnetic condensate following a quench from an unmagnetised phase to one of three ferromagnetic phases. The ferromagnetic phases exhibit distinct symmetry properties (easy-plane, easy-axis or isotropic) and support distinct topological defects. In each phase we observe scale invariant ordering and identify the relevant topological defect affecting the order parameter growth. We find that each phase is characterised by a distinct dynamic critical exponent. In the easy-plane phase we identify a persistent turbulent cascade that affects spin ordering long after all topological defects have annihilated. In addition to our exploration of phase ordering dynamics, we study a microscopic model of spin vortex dynamics in the easy-plane phase. Our work provides a comprehensive theoretical study of phase ordering in a conservative system, provides a thorough foundation for studies of phase ordering in antiferromagnetic and higher spin condensates, and offers prospects for further research into fundamental questions regarding the ordering properties of spin systems. Our work is pertinent to current experiments, which have explored the initial stages of phase ordering in both ferromagnetic and antiferromagnetic spin-1 condensates.
We consider the formation of order in a quasi-two-dimensional antiferromagnetic spin-1 condensate quenched from an easy-axis to an easy-plane nematic phase. We define the relevant order parameter to quantify the spin-nematic degrees of freedom and study the evolution of the spin-nematic and superfluid order during the coarsening dynamics using numerical simulations. We observe dynamical scaling in the late-time dynamics, with both types of order extending across the system with a diffusive growth law. We identify half-quantum vortices as the relevant topological defects of the ordering dynamics and demonstrate that the growth of both types of order is determined by the mutual annihilation of these vortices.
Spinor Bose gases may exhibit both phase and ferromagnetic order. Consequently, they may support superflow of both mass and spin currents. In this thesis we utilize a stochastic Gross-Pitaevskii model to investigate the superfluid properties of a two-dimensional spin-1 Bose gas with ferromagnetic interactions. Owing to the low dimensionality, the superfluidity of this system can not be attributed to the presence of a Bose-Einstein condensate, with thermal fluctuations precluding long-range order. Rather, mass and spin superfluidity emerge via respective Berezinskii-Kosterlitz-Thouless (BKT) transitions. Such transitions are characterized by the emergence of quasi-long-range order, driven by the binding of topological defect pairs. First, we consider an easy-plane ferromagnetic gas. We find the system superfluidity may be understood by direct application of existing BKT theory. In particular, mass and spin superfluidity are respectively associated with quasi-long-range order in the global phase and local magnetization. The former arises from the binding of mass vortex pairs, while the latter arises from the binding of polar-core spin vortex pairs. Following this, we introduce axial magnetization, and consider a gas in the broken-axisymmetric phase. We find axial magnetization acts to couple mass and spin superflows via a mechanism analogous to the superfluid drag exhibited by two-component superfluids. This coupling renders invalid the direct applicability of existing BKT theory: here mass and spin superfluidity may each be associated with both phase and magnetic order. We remedy this with a generalization making concrete the relation between superfluidity and quasi-long-range order in this system. In addition, through variations of the quadratic Zeeman energy and axial magnetization we identify three distinct superfluid phases, which serve as finite-temperature generalizations of the polar, broken-axisymmetric, and easy-axis ferromagnetic ground states.
In this thesis we study the nonlinear dynamics of spin-1 and spin-2 Bose-Einstein condensates, with particular application to antiferromagnetic systems exhibiting nematic (beyond magnetic) order. Firstly, we give a derivation of the spinor energy functionals with a focus on the connections between the nonlinear terms. We derive a hierarchy of nonlinear irreducible multipole observables sensitive to different levels of nematic order, and explore the various nematic states in terms of their multipolar order, representations of their symmetries, and topological defects.\n\nWe then develop an exact solution to the nonlinear dynamics of spinor Bose-Einstein condensates. We use this solution to construct efficient and accurate numerical algorithms to evolve the spinor Gross-Pitaevskii equation in time. We demonstrate the advantages of our algorithms with several 1D numerical test problems, comparing with existing methods in the literature.\n\nWe apply our numerical methods to simulating quenches of the condensate between various antiferromagnetic phases for spin-1 and spin-2. For spin-1, we carry out quenches for a theoretical uniform system in 2D, and then specialize to the parameters used in a recent harmonically trapped experiment in 3D. We connect the long-time coarsening growth law of the relevant order parameter to the decay of half-quantum vortices, which are the relevant topological defects of the ground state.\n\nFor the spin-2 system, we investigate a novel quench from two different quadrupolar-nematic phases to an octupolar-nematic “cyclic” phase which supports 1/3 fractional vortices. We develop appropriate order parameter observables which couple to the spin and superfluid currents generated by these defects, and show that a new growth law appears with exponent 1/3.
In zero magnetic field the ground-state manifold of a ferromagnetic spin-1 condensate is SO(3) and exhibits Z_{2} vortices as topological defects. We investigate the phase-ordering dynamics of this system after being quenched into this ferromagnetic phase from a zero-temperature unmagnetized phase. Following the quench, we observe the ordering of both magnetic and gauge domains. We find that these domains grow diffusively, i.e., with domain size L(t)∼t^{1/2}, and exhibit dynamic scale invariance. The coarsening dynamics progresses as Z_{2} vortices annihilate; however, we find that at finite energy a number of these vortices persist in small clumps without influencing magnetic or gauge order. We consider the influence of a small nonzero magnetic field, which reduces the ground-state symmetry, and show that this sets a critical length scale such that when the domains reach this size the system dynamically transitions in order parameter and scaling behavior from an isotropic to an anisotropic ferromagnetic superfluid.
We examine the dynamics of a quasi-two-dimensional spin-1 condensate in which the quadratic Zeeman energy qqq is suddenly quenched to a value where the system has a ferromagnetic ground state. There are two distinct types of ferromagnetic phases, i.e., a range of qqq values where the magnetization prefers to be in the direction of the external field (easy axis) and a range of qqq values where it prefers to be transverse to the field (easy plane). We study the quench dynamics for a variety of qqq values and show that there is a single dynamic critical exponent to characterize the scale-invariant domain growth for each ferromagnetic phase. For both quenches we give simple analytic models that capture the essential scale-invariant dynamics and correctly predict the exponents. Because the order parameter for each phase is different, the natures of the domains and the relevant topological defects in each type of coarsening are also different. To explore these differences we characterize the fractal dimension of the domain walls and the relationship of polar-core spin vortices to the domains in the easy-plane phase. Finally, we consider how the energy liberated from the quench thermalizes in the easy-axis quench. We show that local equilibrium is established in the spin waves on moderate time scales, but continues to evolve as the domains anneal.
We investigate the phase-transition dynamics of a quasi-two-dimensional antiferromagnetic spin-1 Bose-Einstein condensate from the easy-axis polar phase to the easy-plane polar phase, which is initiated by suddenly changing the sign of the quadratic Zeeman energy qqq. We observe the emergence and decay of spin turbulence and the formation of half-quantum vortices (HQVs) in the quenched condensate. The characteristic time and length scales of the turbulence generation dynamics are proportional to ${|q|}^{\ensuremath{-}1/2}$ as inherited from the dynamic instability of the initial state. In the evolution of the spin turbulence, spin-wave excitations develop from large to small length scales, suggesting a direct energy cascade, and the spin population for the axial polar domains exhibit a nonexponential decay. The final equilibrated condensate contains HQVs, and the number is found to increase and saturate with increasing ∣q∣|q|∣q∣. Our results demonstrate the time-space scaling properties of the phase-transition dynamics near the critical point and the peculiarities of the spin-turbulence state of the antiferromagnetic spinor condensate.