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arXiv:cond-mat/0407119 · Physical Review A

Rapid rotation of a Bose-Einstein condensate in a harmonic plus quartic trap

Alexander L. Fetter, B. Jackson, S. Stringari

Published Jan 28, 2005• 127 citations
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Abstract

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).

DOI
10.1103/physreva.71.013605

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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 \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ to exceed the radial harmonic frequency \ensuremathω\ensuremath⊥{\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremathω\ensuremath⊥​. In the regime \ensuremathΩ\ensuremath≳\ensuremathω\ensuremath⊥\ensuremath{\Omega}\ensuremath{\gtrsim}{\ensuremath{\omega}}_{\ensuremath{\perp}}\ensuremathΩ\ensuremath≳\ensuremathω\ensuremath⊥​, the condensate contains a dense vortex array (approximated as solid-body rotation for the analytical studies). At a critical angular velocity \ensuremathΩh{\ensuremath{\Omega}}_{h}\ensuremathΩh​, a central hole appears in the condensate. Numerical studies confirm the predicted value of \ensuremathΩh{\ensuremath{\Omega}}_{h}\ensuremathΩh​, 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).
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.
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 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 $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Omega}}{\ensuremath{\Omega}}_{c2}.$
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 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 $\ensuremath{\Omega}$, to which vortex lattices vanish eventually at some higher $\ensuremath{\Omega}$.
Rapidly rotating Bose-Einstein condensates confined in anharmonic traps can exhibit a rich variety of vortex phases, including a vortex lattice, a vortex lattice with a hole, and a giant vortex. Using an augmented Thomas-Fermi variational approach to determine the ground state of the condensate in the rotating frame -- valid for sufficiently strongly interacting condensates -- we determine the transitions between these three phases for a quadratic-plus-quartic confining potential. Combining the present results with previous numerical simulations of small rotating condensates in such anharmonic potentials, we delineate the general structure of the zero temperature phase diagram.
We study a two-dimensional rotating Bose-Einstein condensate confined by an anharmonic trap in the framework of the Gross-Pitaevskii theory. We consider a rapid rotation regime close to the transition to a giant vortex state. It was proven in Correggi et al. [J. Math. Phys. 53, 095203 (2012)] that such a transition occurs when the angular velocity is of order ε−4, with ε−2 denoting the coefficient of the nonlinear term in the Gross-Pitaevskii functional and ε ≪ 1 (Thomas-Fermi regime). In this paper, we identify a finite value Ωc such that if Ω = Ω0/ε4 with Ω0 > Ωc, the condensate is in the giant vortex phase. Under the same condition, we prove a refined energy asymptotics and an estimate of the winding number of any Gross-Pitaevskii minimizer.