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arXiv:0707.1568 · Journal of Mathematical Physics

Rapidly rotating Bose-Einstein condensates in homogeneous traps

Michele Correggi, Tanja Rindler-Daller, Jakob Yngvason

Published Oct 1, 2007• 30 citations
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Abstract

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.

DOI
10.1063/1.2789557

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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.
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 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 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.
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
I review free and trapped Bose‐Einstein condensates (BECs) [1, 2, 3, 4, 5]. In an isotropic harmonic potential V(r) = 12Mω2r2, the single‐particle ground state introduces a new intrinsic scale of length [the ground‐state size d = ℏ/(Mω)] and energy [the ground‐state energy E0 = 32ℏω]. When the trap rotates [6, 7, 8], there is a critical angular velocity for the appearance of one or a few quantized vortices. For more rapid rotation [9, 10, 11, 12, 13] the condensate contains a vortex array. The resulting centrifugal forces expand the condensate radially and shrink it axially; thus the condensate becomes effectively two‐dimensional. If the external rotation speed approaches the frequency of the radial harmonic confining potential, the condensate enters the “lowest‐Landau‐level” regime, and a simple description again becomes possible [14, 15]. Eventually, the system is predicted to make a quantum phase transition to a highly correlated state analogous to the fractional quantum Hall states of electrons in a strong magnetic field [16, 17, 18, 19, 20].
The state of a rotating Bose-Einstein condensate in a harmonic trap is modeled by a wave function that minimizes the Gross-Pitaevskii functional. The resulting minimization problem has two new features compared to other similar functionals arising in condensed matter physics, such as the Ginzburg-Landau functional. Namely, the wave function is defined in all the plane and is normalized relative to the L2L^2L2-norm. This paper deals with the situation when the coupling constant tends to 000 (Thomas-Fermi regime) and the rotation speed is large compared with the first critical speed. It is given the leading order estimate of the ground state energy together with the location of the vortices of the minimizing wave function in the bulk of the condensate. When the rotation speed is inversely proportional to the coupling constant, the condensate is confined in an elliptical region whose conjugate diameter shrinks and whose transverse diameter expands as the rotation speed increases.