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arXiv:cond-mat/0503541 · Open Physics

Rapidly rotating Bose-Einstein condensates in anharmonic confinement

Li-Hua Lü, You‐Quan Li

Published Jun 1, 2006• 0 citations
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Abstract

Abstract We study a rapidly rotating Bose-Einstein condensate in anharmonic confinement and find that many properties, such as the critical rotating frequency and phase diagram, are different from those in a harmonic trap. We investigate the phase transitions between various vortex lattices and find that a hole emerges in the center of the cloud when the rotating frequency Θ reaches Θh but it becomes invisible when Θ > 1.0842ω ⊥.

DOI
10.2478/s11534-006-0013-2

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Abstract We study a rapidly rotating Bose-Einstein condensate in anharmonic confinement and find that many properties, such as the critical rotating frequency and phase diagram, are different from those in a harmonic trap. We investigate the phase transitions between various vortex lattices and find that a hole emerges in the center of the cloud when the rotating frequency Θ reaches Θh but it becomes invisible when Θ > 1.0842ω ⊥.
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.
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 examine an effectively repulsive Bose-Einstein condensate of atoms that rotates in a quadratic-plus-quartic trapping potential. We investigate the phase diagram of the system as a function of the angular frequency of rotation and of the coupling constant, demonstrating that there are phase transitions between multiply and singly quantized vortex states. We also show that states of different circulation can be metastable and, as a result, the gas can exhibit hysteresis as the angular frequency of rotation of the trap is varied. The simplicity of the picture that emerges for small coupling strengths suggests that this system may be attractive for studies of phase transitions.
We address the challenging proposition of using real experimental parameters in a three-dimensional (3D) numerical simulation of fast rotating Bose-Einstein condensates. We simulate recent experiments [V. Bretin, S. Stock, Y. Seurin, and J. Dalibard, Phys. Rev. Lett. 92, 050403 (2004); S. Stock, V. Bretin, S. Stock, F. Chevy, and J. Dalibard, Europhys. Lett. 65, 594 (2004)] using an anharmonic (quadratic-plus-quartic) confining potential to reach rotation frequencies $(\ensuremath{\Omega})$ above the trap frequency $({\ensuremath{\omega}}_{\ensuremath{\perp}})$. Our numerical results are obtained by propagating the 3D Gross-Pitaevskii equation in imaginary time. For $\ensuremath{\Omega}\ensuremath{\leqslant}{\ensuremath{\omega}}_{\ensuremath{\perp}}$, we obtain an equilibrium vortex lattice similar (as the size and number of vortices) to experimental observations. For $\ensuremath{\Omega}>{\ensuremath{\omega}}_{\ensuremath{\perp}}$ we observe the evolution of the vortex lattice into an array of vortices with a central hole. Since this evolution was not visible in experiments, we investigate the 3D structure of vortex configurations and 3D effects on vortex contrast. Numerical data are also compared to recent theory [D. E. Sheehy and L. Radzihovsky, Phys. Rev. A 70, 063620 (2004)] describing vortex lattice inhomogeneities and a remarkably good agreement is found.
We study a rotating Bose-Einstein condensate in a strongly anharmonic trap (flat trap with a finite radius) in the framework of two-dimensional Gross-Pitaevskii theory. We write the coupling constant for the interactions between the gas atoms as 1∕ε2 and we are interested in the limit ε→0 (Thomas-Fermi limit) with the angular velocity Ω depending on ε. We derive rigorously the leading asymptotics of the ground state energy and the density profile when Ω tends to infinity as a power of 1∕ε. If Ω(ε)=Ω0∕ε a “hole” (i.e., a region where the density becomes exponentially small as 1∕ε→∞) develops for Ω0 above a certain critical value. If Ω(ε)⪢1∕ε the hole essentially exhausts the container and a “giant vortex” develops with the density concentrated in a thin layer at the boundary. While we do not analyze the detailed vortex structure we prove that rotational symmetry is broken in the ground state for const∣logε∣<Ω(ε)≲const∕ε.