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

Physics of Fluids

Formation of vortex rings and hopfions in trapped Bose–Einstein condensates

Shan Feng Zou, Wen-Kai Bai, Tao Yang, Wu-Ming Liu

Published Feb 1, 2021• 13 citations
Publisher

Abstract

The topological transition of vortex lines to vortex rings and hopfions is numerically investigated by the Gross–Pitaevskii equation in three-dimensional trapped Bose–Einstein condensates. The shape of the vortex rings formed by the two vortex lines of the vortex dipole depends strongly on the initial separation of the lines. An approximately perfect vortex ring can be obtained by choosing some suitable values of the separation. The deformation of the formed rings depends on the shape of the rings in turn. Furthermore, we show a feasible approach to generate vortex hopfions by imprinting a vortex line in the center of the generated vortex rings. Specifically, the movement of the vortex rings can excite helical waves along the central vortex line of the hopfion structure if the vortex ring is not perfect.

DOI
10.1063/5.0035468

Related papers

Robust vortex lines, vortex rings, and hopfions in three-dimensional Bose-Einstein condensates

R. N. Bisset, Wenlong Wang, Christopher Ticknor +4 more

Performing a systematic Bogoliubov--de Gennes spectral analysis, we illustrate that stationary vortex lines, vortex rings, and more exotic states, such as hopfions, are robust in three-dimensional atomic Bose-Einstein condensates, for large parameter intervals. Importantly, we find that the hopfion can be stabilized in a simple parabolic trap, without the need for trap rotation or inhomogeneous interactions. We supplement our spectral analysis by studying the dynamics of such stationary states; we find them to be robust against significant perturbations of the initial state. In the unstable regimes, we not only identify the unstable mode, such as a quadrupolar or hexapolar mode, but we also observe the corresponding instability dynamics. Furthermore, deep in the Thomas-Fermi regime, we investigate the particlelike behavior of vortex rings and hopfions.

arXiv:1510.04344•Physical Review A•Dec 7, 2015•29 citations

Vortex rings in toroidal Bose-Einstein condensates

Marta Abad, M. Guilleumas, R. Mayol +1 more

We study vortex ring solutions of the Gross-Pitaevskii equation for Bose-Einstein condensates confined in toroidal traps. Analogously to the Feynman-Onsager ansatz for a single vortex line, we assume a condensate wavefunction that describes a quantized vortex ring. This leads to a modified Gross-Pitaevskii equation that incorporates vortex ring solutions. By numerically solving this equation, we have obtained the density profiles and the vortex core structure. We have also calculated the nucleation energy of a vortex ring, and have analyzed its dependence on the ring position. Finally, a qualitative description of the vortex ring dynamics is presented.

Laser Physics•May 1, 2008•7 citations

Generating vortex rings in Bose-Einstein condensates in the line-source approximation

M. Guilleumas, Dora M. Jezek, R. Mayol +2 more

We present a numerical method for generating vortex rings in Bose-Einstein condensates confined in axially symmetric traps. The vortex ring is generated using the line-source approximation for the vorticity, i.e., the curl of the superfluid velocity field is different from zero only on a circumference of a given radius located on a plane perpendicular to the symmetry axis and coaxial with it. The particle density is obtained by solving a modified Gross-Pitaevskii equation that incorporates the effect of the velocity field. We discuss the appearance of density profiles, the vortex core structure, and the vortex nucleation energy, i.e., the energy difference between vortical and ground-state configurations. This is used to present a qualitative description of the vortex dynamics.

arXiv:cond-mat/0202312•Physical Review A•May 2, 2002•16 citations

Vortices in trapped Bose-Einstein condensates

Brian Jackson

In this thesis we solve the Gross-Pitaevskii equation numerically in order to model the response of trapped Bose-Einstein condensed gases to perturbations by electromagnetic fields. First, we simulate output coupling of pulses from the condensate and compare our results to experiments. The excitation and separation of eigenmodes on flow through a constriction is also studied. We then move on to the main theme of this thesis: the important subject of quantised vortices in Bose condensates, and the relation between Bose-Einstein condensation and superfluidity. We propose methods of producing vortex pairs and rings by controlled motion of objects. Full three-dimensional simulations under realistic experimental conditions are performed in order to test the validity of these ideas. We link vortex formation to drag forces on the object, which in turn is connected with energy transfer to the condensate. We therefore argue that vortex formation by moving objects is intimately related to the onset of dissipation in superfluids. We discuss this idea in the context of a recent experiment, using simulations to provide evidence of vortex formation in the experimental scenario. Superfluidity is also manifest in the property of persistent currents, which is linked to vortex stability and dynamics. We simulate vortex line and ring motion, and find in both cases precessional motion and thermodynamic instability to dissipation. Strictly speaking, the Gross-Pitaevskii equation is valid only for temperatures far below the BEC transition. We end the thesis by describing a simple finite-temperature model to describe mean-field coupling between condensed and non-condensed components of the gas. We show that our hybrid Monte-Carlo/FFT technique can describe damping of the lowest energy excitations of the system. Extensions to this model and future research directions are discussed in the conclusion. (author)

•Jan 1, 2000•1 citation

Vortex-ring solutions of the Gross–Pitaevskii equation for an axisymmtrically trapped Bose–Einstein condensate

C-H Hsueh, S-C. Gou, T-L Horng +1 more

We propose a numerical scheme for obtaining the stationary vortex-ring solutions of the Gross–Pitaevskii (GP) equation for an axisymmetrically trapped Bose–Einstein condensate (BEC). The effective energy functional and the associated GP equation are derived by assuming a trial phase profile for the wavefunction that is subject to the condition of circulation quantization on the rz plane. The wavefunction of the vortex ring is determined by solving the ground state of the effective GP equation numerically. Application of our method to the formation of a three-dimensional Skyrmion in a trapped two-component BEC is demonstrated.

Journal of Physics B Atomic Molecular and Optical Physics•Nov 19, 2007•17 citations

Vortex ring dynamics in trapped Bose-Einstein condensates

Matthew D. Reichl, Erich J. Mueller

We use the time-dependent Gross-Pitaevskii equation to study the motion of a vortex ring produced by phase imprinting on an elongated cloud of cold atoms. Our approach models the experiments of Yefsah et al. [Nature (London) 499, 426 (2013)] on ${}^{6}$Li in the Bose-Einstein-condensate regime where the fermions are tightly bound into bosonic dimers. We find ring oscillation periods which are much larger than the period of the axial harmonic trap. Our results lend further strength to Bulgac et al.'s arguments (arXiv:1306.4266) that the ``heavy solitons'' seen in those experiments are actually vortex rings. We numerically calculate the periods of oscillation for the vortex rings as a function of interaction strength, trap aspect ratio, and minimum vortex ring radius. In the presence of axial anisotropies the rings undergo complicated internal dynamics where they break into sets of vortex lines, then later combine into rings. These structures oscillate with a similar frequency to simple axially symmetric rings.

arXiv:1309.7012•Physical Review A•Nov 22, 2013•26 citations
The topological transition of vortex lines to vortex rings and hopfions is numerically investigated by the Gross–Pitaevskii equation in three-dimensional trapped Bose–Einstein condensates. The shape of the vortex rings formed by the two vortex lines of the vortex dipole depends strongly on the initial separation of the lines. An approximately perfect vortex ring can be obtained by choosing some suitable values of the separation. The deformation of the formed rings depends on the shape of the rings in turn. Furthermore, we show a feasible approach to generate vortex hopfions by imprinting a vortex line in the center of the generated vortex rings. Specifically, the movement of the vortex rings can excite helical waves along the central vortex line of the hopfion structure if the vortex ring is not perfect.
Performing a systematic Bogoliubov--de Gennes spectral analysis, we illustrate that stationary vortex lines, vortex rings, and more exotic states, such as hopfions, are robust in three-dimensional atomic Bose-Einstein condensates, for large parameter intervals. Importantly, we find that the hopfion can be stabilized in a simple parabolic trap, without the need for trap rotation or inhomogeneous interactions. We supplement our spectral analysis by studying the dynamics of such stationary states; we find them to be robust against significant perturbations of the initial state. In the unstable regimes, we not only identify the unstable mode, such as a quadrupolar or hexapolar mode, but we also observe the corresponding instability dynamics. Furthermore, deep in the Thomas-Fermi regime, we investigate the particlelike behavior of vortex rings and hopfions.
We study vortex ring solutions of the Gross-Pitaevskii equation for Bose-Einstein condensates confined in toroidal traps. Analogously to the Feynman-Onsager ansatz for a single vortex line, we assume a condensate wavefunction that describes a quantized vortex ring. This leads to a modified Gross-Pitaevskii equation that incorporates vortex ring solutions. By numerically solving this equation, we have obtained the density profiles and the vortex core structure. We have also calculated the nucleation energy of a vortex ring, and have analyzed its dependence on the ring position. Finally, a qualitative description of the vortex ring dynamics is presented.
We present a numerical method for generating vortex rings in Bose-Einstein condensates confined in axially symmetric traps. The vortex ring is generated using the line-source approximation for the vorticity, i.e., the curl of the superfluid velocity field is different from zero only on a circumference of a given radius located on a plane perpendicular to the symmetry axis and coaxial with it. The particle density is obtained by solving a modified Gross-Pitaevskii equation that incorporates the effect of the velocity field. We discuss the appearance of density profiles, the vortex core structure, and the vortex nucleation energy, i.e., the energy difference between vortical and ground-state configurations. This is used to present a qualitative description of the vortex dynamics.
In this thesis we solve the Gross-Pitaevskii equation numerically in order to model the response of trapped Bose-Einstein condensed gases to perturbations by electromagnetic fields. First, we simulate output coupling of pulses from the condensate and compare our results to experiments. The excitation and separation of eigenmodes on flow through a constriction is also studied. We then move on to the main theme of this thesis: the important subject of quantised vortices in Bose condensates, and the relation between Bose-Einstein condensation and superfluidity. We propose methods of producing vortex pairs and rings by controlled motion of objects. Full three-dimensional simulations under realistic experimental conditions are performed in order to test the validity of these ideas. We link vortex formation to drag forces on the object, which in turn is connected with energy transfer to the condensate. We therefore argue that vortex formation by moving objects is intimately related to the onset of dissipation in superfluids. We discuss this idea in the context of a recent experiment, using simulations to provide evidence of vortex formation in the experimental scenario. Superfluidity is also manifest in the property of persistent currents, which is linked to vortex stability and dynamics. We simulate vortex line and ring motion, and find in both cases precessional motion and thermodynamic instability to dissipation. Strictly speaking, the Gross-Pitaevskii equation is valid only for temperatures far below the BEC transition. We end the thesis by describing a simple finite-temperature model to describe mean-field coupling between condensed and non-condensed components of the gas. We show that our hybrid Monte-Carlo/FFT technique can describe damping of the lowest energy excitations of the system. Extensions to this model and future research directions are discussed in the conclusion. (author)
We propose a numerical scheme for obtaining the stationary vortex-ring solutions of the Gross–Pitaevskii (GP) equation for an axisymmetrically trapped Bose–Einstein condensate (BEC). The effective energy functional and the associated GP equation are derived by assuming a trial phase profile for the wavefunction that is subject to the condition of circulation quantization on the rz plane. The wavefunction of the vortex ring is determined by solving the ground state of the effective GP equation numerically. Application of our method to the formation of a three-dimensional Skyrmion in a trapped two-component BEC is demonstrated.
We use the time-dependent Gross-Pitaevskii equation to study the motion of a vortex ring produced by phase imprinting on an elongated cloud of cold atoms. Our approach models the experiments of Yefsah et al. [Nature (London) 499, 426 (2013)] on 6{}^{6}6Li in the Bose-Einstein-condensate regime where the fermions are tightly bound into bosonic dimers. We find ring oscillation periods which are much larger than the period of the axial harmonic trap. Our results lend further strength to Bulgac et al.'s arguments (arXiv:1306.4266) that the ``heavy solitons'' seen in those experiments are actually vortex rings. We numerically calculate the periods of oscillation for the vortex rings as a function of interaction strength, trap aspect ratio, and minimum vortex ring radius. In the presence of axial anisotropies the rings undergo complicated internal dynamics where they break into sets of vortex lines, then later combine into rings. These structures oscillate with a similar frequency to simple axially symmetric rings.