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arXiv:cond-mat/0111443 · Physical Review Letters

Vortex States of Rapidly Rotating Dilute Bose-Einstein Condensates

Uwe R. Fischer, Gordon Baym

Published Apr 9, 2003• 181 citations
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Abstract

We show that, in the Thomas-Fermi regime, the cores of vortices in rotating dilute Bose-Einstein condensates adjust in radius as the rotation velocity, Omega, grows, thus precluding a phase transition associated with core overlap at high vortex density. In both a harmonic trap and a rotating hard-walled bucket, the core size approaches a limiting fraction of the intervortex spacing. At large rotation speeds, a system confined in a bucket develops, within Thomas-Fermi, a hole along the rotation axis, and eventually makes a transition to a giant vortex state with all the vorticity contained in the hole.

DOI
10.1103/physrevlett.90.140402

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We show that, in the Thomas-Fermi regime, the cores of vortices in rotating dilute Bose-Einstein condensates adjust in radius as the rotation velocity, Omega, grows, thus precluding a phase transition associated with core overlap at high vortex density. In both a harmonic trap and a rotating hard-walled bucket, the core size approaches a limiting fraction of the intervortex spacing. At large rotation speeds, a system confined in a bucket develops, within Thomas-Fermi, a hole along the rotation axis, and eventually makes a transition to a giant vortex state with all the vorticity contained in the hole.
We present an analytical solution for the vortex lattice in a rapidly rotating trapped Bose-Einstein condensate in the lowest Landau level and discuss deviations from the Thomas-Fermi density profile. This solution is exact in the limit of a large number of vortices and is obtained for the cases of circularly symmetric and narrow channel geometries. The latter is realized when the trapping frequencies in the plane perpendicular to the rotation axis are different from each other and the rotation frequency is equal to the smallest of them. This leads to the cancellation of the trapping potential in the direction of the weaker confinement and makes the system infinitely elongated in this direction. For this case we calculate the phase diagram as a function of the interaction strength and rotation frequency and identify the order of quantum phase transitions between the states with a different number of vortex rows.
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 $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Omega}}{\ensuremath{\Omega}}_{c2}.$
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.
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 develop an approach for calculating stationary states of rotating Bose-Einstein condensates in harmonic traps which is applicable for arbitrary ratios of the rotation frequency to the transverse frequency of the trap ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. Assuming the number of vortices to be large, we write the condensate wave function as the product of a function that describes the structure of individual vortices times an envelope function varying slowly on the scale of the vortex spacing. By minimizing the energy, we derive Gross-Pitaevskii equations that determine the properties of individual vortices and the global structure of the cloud. For low rotation rates, the structure of a vortex is that of an isolated vortex in a uniform medium, while for rotation rates approaching the frequency of the trap (the mean-field lowest-Landau-level regime), the structure is that of the lowest ppp-wave state of a particle in a harmonic trap with frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$. The global structure of the cloud is determined by minimizing the energy with respect to variations of the envelope function; for conditions appropriate to most experimental investigations to date, we predict that the transverse density profile of the cloud will be of the Thomas-Fermi form, rather than the Gaussian structure predicted on the assumption that the wave function consists only of components in the lowest Landau level for a regular array of vortices.