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arXiv:math-ph/0606058 · Journal of Mathematical Physics

Rapidly rotating Bose-Einstein condensates in strongly anharmonic traps

Michele Correggi, Tanja Rindler-Daller, Jakob Yngvason

Published Apr 1, 2007• 37 citations
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Abstract

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.

DOI
10.1063/1.2712421

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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.
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.
We study a rapidly rotating Bose–Einstein condensate confined to a finite trap in the framework of two-dimensional Gross–Pitaevskii theory in the strong coupling (Thomas–Fermi) limit. Denoting the coupling parameter by 1/ε 2 and the rotational velocity by Ω, we evaluate exactly the next to the leading-order contribution to the ground-state energy in the parameter regime |log ε| ≪ Ω ≪ 1/(ε 2 |log ε|) with ε → 0. While the TF energy includes only the contribution of the centrifugal forces the next order corresponds to a lattice of vortices whose density is proportional to the rotational velocity.
We extend the results of a previous paper on the Gross-Pitaevskii description of rotating Bose-Einstein condensates in two-dimensional traps to confining potentials of the form V(r)=rs, 2<s<∞. Writing the coupling constant as 1∕ε2, we study the limit ε→0. We derive rigorously the leading asymptotics of the ground state energy and the density profile when the rotation velocity Ω tends to infinity as a power of 1∕ε. The case of asymptotically homogeneous potentials is also discussed.
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).
Starting from the full many body Hamiltonian we derive the leading order energy and density asymptotics for the ground state of a dilute, rotating Bose gas in an anharmonic trap in the ‘Thomas Fermi ’ (TF) limit when the Gross-Pitaevskii coupling parameter and/or the rotation velocity tend to infinity. Although the many-body wave function is expected to have a complicated phase, the leading order contribution to the energy can be computed by minimizing a simple functional of the density alone. 1
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.