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arXiv:1206.2198 · Physical Review A

Vortices in a rotating Bose-Einstein condensate under extreme elongation in a harmonic plus quartic trap

Chao Huang, C. H. Liu, W. C. Wu

Published Apr 8, 2010• 10 citations
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Abstract

The behaviors of a rapidly rotating Bose-Einstein condensate under extreme elongation in a two-dimensional anisotropic harmonic plus quartic trap are investigated. Due to the quartic trap, the system remains stable at high rotating velocity, $\ensuremath{\Omega}\ensuremath{\geqslant}{\ensuremath{\omega}}_{\ensuremath{\perp}}$ (${\ensuremath{\omega}}_{\ensuremath{\perp}}$ is the radial harmonic trap frequency), and vortex lattices form even in the absence of the repulsive $s$-wave interaction ($g$). When $g$ is present, the interplay between $g$ and the quartic trap potential can lead to rich vortex lattice transition states as a function of $\ensuremath{\Omega}$, to which vortex lattices vanish eventually at some higher $\ensuremath{\Omega}$.

DOI
10.1103/physreva.81.043605

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The behaviors of a rapidly rotating Bose-Einstein condensate under extreme elongation in a two-dimensional anisotropic harmonic plus quartic trap are investigated. Due to the quartic trap, the system remains stable at high rotating velocity, \ensuremathΩ\ensuremath⩾\ensuremathω\ensuremath⊥\ensuremath{\Omega}\ensuremath{\geqslant}{\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremathΩ\ensuremath⩾\ensuremathω\ensuremath⊥​ (\ensuremathω\ensuremath⊥{\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremathω\ensuremath⊥​ is the radial harmonic trap frequency), and vortex lattices form even in the absence of the repulsive sss-wave interaction (ggg). When ggg is present, the interplay between ggg and the quartic trap potential can lead to rich vortex lattice transition states as a function of \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ, to which vortex lattices vanish eventually at some higher \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ.
A two-dimensional rapidly rotating Bose-Einstein condensate in a harmonic plus quartic trap is expected to have unusual vortex states that do not occur in a pure harmonic trap. At a critical rotation speed ${\ensuremath{\Omega}}_{h}$, a central hole appears in the condensate, and at some faster rotation speed ${\ensuremath{\Omega}}_{g}$, the system undergoes a transition to a giant vortex state with pure irrotational flow. Using a time-dependent variational analysis, we study the behavior of an annular condensate with a single concentric ring of vortices. The transition to a giant vortex state is investigated by comparing the energy of the two equilibrium states (the ring of vortices and the giant vortex) and also by studying the dynamical stability of small excitation modes of the ring of vortices.
Abstract We consider the ground-state properties of a rotating spin-orbit-coupled Bose—Einstein condensate under extreme elongation in a harmonic plus quartic potential. The effects of spin-orbit coupling and rotation on the ground-state vortex structures are investigated. In the absence of spin-orbit coupling, new nucleated vortices gradually form vortex lines and annular vortex structures with the increase of the rotation frequency. In the presence of spin-orbit coupling, part of the vortices arrange in a line and form a stable vortex chain, and the remanent vortices coexist in pairs aside such vortex chain. More specially, the remanent vortices of each component repel each other and form vortex pair for isotropic spin-orbit coupling, while attract each other and locate in the same positions for anisotropic spin-orbit coupling.
We consider a rotating Bose-Einstein condensate confined in combined harmonic and quartic traps, following recent experiments [V. Bretin, S. Stock, Y. Seurin, and J. Dalibard, Phys. Rev. Lett. 92, 050403 (2004)]. We investigate numerically the behavior of the wave function which solves the three-dimensional Gross Pitaevskii equation and analyze in detail the structure of vortices. For a quartic-plus-harmonic potential, as the angular velocity increases, the vortex lattice evolves into a vortex array with hole. The merging of vortices into the hole is highly three dimensional, starting from the top and bottom of the condensate to reach the center. We also investigate the case of a quartic-minus-harmonic potential, not covered by experiments or previous numerical works. For intermediate repulsive potentials, we show that the transition to a vortex array with hole takes place for lower angular velocities, when the lattice is made up of a small number of vortices. For the strong repulsive case, a transition from a giant vortex to a hole with a circle of vortices around is observed.
A two-dimensional rapidly rotating Bose-Einstein condensate in an anharmonic trap with quadratic and quartic radial confinement is studied analytically with the Thomas-Fermi approximation and numerically with the full time-independent Gross-Pitaevskii equation. The quartic trap potential allows the rotation speed $\ensuremath{\Omega}$ to exceed the radial harmonic frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. In the regime $\ensuremath{\Omega}\ensuremath{\gtrsim}{\ensuremath{\omega}}_{\ensuremath{\perp}}$, the condensate contains a dense vortex array (approximated as solid-body rotation for the analytical studies). At a critical angular velocity ${\ensuremath{\Omega}}_{h}$, a central hole appears in the condensate. Numerical studies confirm the predicted value of ${\ensuremath{\Omega}}_{h}$, even for interaction parameters that are not in the Thomas-Fermi limit. The behavior is also investigated at larger angular velocities, where the system is expected to undergo a transition to a giant vortex (with pure irrotational flow).
We study rotational motion of an interacting atomic Bose-Einstein condensate confined in a quadratic-plus-quartic potential. We calculate the lowest energy surface mode frequency and show that a symmetric trapped (harmonic and quartic) Bose-Einstein condensate breaks the rotational symmetry of the Hamiltonian when rotational frequency is greater than one-half of the lowest energy surface mode frequency. We argue that the formation of a vortex is not possible in a non-interacting as well as in an attractive Bose-Einstein condensate confined in a harmonic trap due to the absence of the spontaneous shape deformation, but it can occur which leads to the vortex formation if we add an additional quartic potential. Moreover, the spontaneous shape deformation and consequently the formation of a vortex in an attractive system depends on the strengths of the two-body interaction and the quartic potential.
A dense vortex lattice in a rotating dilute Bose-Einstein condensate is studied with the Thomas-Fermi approximation. The upper critical angular velocity ${\ensuremath{\Omega}}_{c2}$ occurs when the intervortex separation b becomes comparable with the vortex-core radius $\ensuremath{\xi}.$ For a radial harmonic trap, the loss of confinement as $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Omega}}{\ensuremath{\omega}}_{\ensuremath{\perp}}$ implies a singular behavior. In contrast, an additional radial quartic potential provides a simple model for which ${\ensuremath{\Omega}}_{c2}$ is readily determined. Unlike the case of a type-II superconductor at fixed temperature, the onset of ${\ensuremath{\Omega}}_{c2}$ arises not only from decreasing b but also from increasing $\ensuremath{\xi}$ caused by the vanishing of the chemical potential as $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Omega}}{\ensuremath{\Omega}}_{c2}.$