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arXiv:1903.02042 · Physical Review A

Quasistable quantum vortex knots and links in anisotropic harmonically trapped Bose-Einstein condensates

Christopher Ticknor, V. P. Ruban, P. G. Kevrekidis

Published Jun 10, 2019• 13 citations
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Abstract

The long-time existence of topologically nontrivial configurations of quantum vortices in the form of torus knots and links in trapped Bose-Einstein condensates is demonstrated numerically within the three-dimensional Gross-Pitaevskii equation with an external anisotropic parabolic potential. We identify parametric domains of trap anisotropy, characterized by the axial over planar frequency ratio $\ensuremath{\lambda}\ensuremath{\approx}1.5--1.6$, where the lifetime of such quasistationary rotating vortex structures is many hundreds of typical rotation times. This suggests the potential experimental observability of the structures. We quantify the relevant lifetimes as a function of the model parameters (e.g., $\ensuremath{\lambda}$) and initial condition parameters of the knot profile.

DOI
10.1103/physreva.99.063604

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The long-time existence of topologically nontrivial configurations of quantum vortices in the form of torus knots and links in trapped Bose-Einstein condensates is demonstrated numerically within the three-dimensional Gross-Pitaevskii equation with an external anisotropic parabolic potential. We identify parametric domains of trap anisotropy, characterized by the axial over planar frequency ratio \ensuremathλ\ensuremath≈1.5−−1.6\ensuremath{\lambda}\ensuremath{\approx}1.5--1.6\ensuremathλ\ensuremath≈1.5−−1.6, where the lifetime of such quasistationary rotating vortex structures is many hundreds of typical rotation times. This suggests the potential experimental observability of the structures. We quantify the relevant lifetimes as a function of the model parameters (e.g., \ensuremathλ\ensuremath{\lambda}\ensuremathλ) and initial condition parameters of the knot profile.
The dynamics of the simplest vortex knots, “unknots,” and torus links in an atomic Bose condensate at zero temperature in an anisotropic harmonic trap has been simulated numerically within the three-dimensional Gross–Pitaevskii equation. It has been found that such quasistationary rotating vortex structures exist for a very long time in wide ranges of the parameters of the system. This new result is qualitatively consistent with a previous prediction based on a simplified one-dimensional model approximately describing the motion of knotted vortex filaments.
Vortex states of weakly interacting Bose-Einstein condensates confined in three-dimensional rotating harmonic traps are investigated numerically at zero temperature. The ground state in the rotating frame is obtained by propagating the Gross-Pitaevskii equation for the condensate in imaginary time. The total energies between states with and without a vortex are compared, yielding critical rotation frequencies that depend on the anisotropy of the trap and the number of atoms. Vortices displaced from the center of nonrotating traps are found to have long lifetimes for sufficiently large numbers of atoms. The relationship between vortex stability and bound core states is explored.
Abstract We study three-dimensional vortex lattice structures in purely dipolar Bose–Einstein condensate (BEC). By using the mean-field approximation, we obtain a stability diagram for the vortex states in purely dipolar BECs as a function of harmonic trap aspect ratio ( λ ) and dipole–dipole interaction strength ( D ) under rotation. Rotating the condensate within the unstable region leads to collapse while in the stable region furnishes stable vortex lattices of dipolar BECs. We analyse stable vortex lattice structures by solving the three-dimensional time-dependent Gross–Pitaevskii equation in imaginary time. Further, the stability of vortex states is examined by evolution in real-time. We also investigate the distribution of vortices in a fully anisotropic trap by increasing eccentricity of the external trapping potential. We observe the breaking up of the condensate in two parts with an equal number of vortices on each when the trap is sufficiently weak, and the rotation frequency is high.
We study the vortex lattices in a Bose-Einstein condensate in a rotating anisotropic harmonic trap. We first investigate the single-particle wave functions obtained by the exact solution of the problem and give simple expressions for these wave functions in the small anisotropy limit. Depending on the strength of the interactions, a few or a large number of vortices can be formed. In the limit of many vortices, we calculate the density profile of the cloud and show that the vortex lattice stays triangular. We also find that the vortex lattice planes align themselves with the weak axis of the external potential. For a small number of vortices, we numerically solve the Gross-Pitaevskii equation and find vortex configurations that are very different from the vortex configurations in an axisymmetric rotating trap.
The nucleation of vortices and the resulting structures of vortex arrays in dilute, trapped, zero-temperature Bose-Einstein condensates are investigated numerically. Vortices are generated by rotating a three-dimensional, anisotropic harmonic atom trap. The condensate ground state is obtained by propagating the Gross-Pitaevskii equation in imaginary time. Vortices first appear at a rotation frequency significantly larger than the critical frequency for vortex stabilization, consistent with a critical velocity mechanism for vortex nucleation. At higher frequencies, the structures of the vortex arrays are strongly influenced by trap geometry.
In the present work, motivated by numerous recent experimental developments we revisit the dynamics of a single vortex ring in anisotropic harmonic traps. At the theoretical level, we start from a general Lagrangian dynamically capturing the evolution of a vortex ring and not only consider its spectrum of linearized excitations, but also explore the full nonlinear dynamical evolution of the ring as a vortical filament. The theory predicts that the ring is stable for $1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}2$, where $\ensuremath{\lambda}={\ensuremath{\omega}}_{z}/{\ensuremath{\omega}}_{r}$ is the ratio of the trapping frequencies along the zzz and rrr axes, i.e., for spherical to slightly oblate condensates. We compare this prediction with direct numerical simulations of the full three-dimensional Gross-Pitaevskii equation (GPE) capturing the linearization spectrum of the ring for different values of the chemical potential and as a function of the anisotropy parameter $\ensuremath{\lambda}$. We identify this result as being only asymptotically valid as the chemical potential $\ensuremath{\mu}\ensuremath{\rightarrow}\ensuremath{\infty}$, revealing how the stability interval narrows and, in particular, its upper bound decreases for finite $\ensuremath{\mu}$. Finally, we compare at the dynamical level the results of the GPE with the ones effectively capturing the ring dynamics, revealing the unstable evolution for different values of $\ensuremath{\lambda}$, as well as the good agreement between the two.