Vortex structures in rotating Bose-Einstein condensates

We present an analytical solution for the vortex lattice in a rapidly rotating trapped Bose-Einstein condensate (BEC) 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 cancelation 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.

💡 Research Summary

The paper presents an analytical treatment of vortex lattices in rapidly rotating trapped Bose‑Einstein condensates (BECs) within the lowest Landau level (LLL) approximation. By assuming that the rotation frequency Ω approaches the transverse trapping frequency, the centrifugal force nearly cancels the harmonic confinement, and the system behaves as a two‑dimensional gas of particles confined to the LLL. The authors first consider a circularly symmetric trap. In this geometry the condensate wavefunction can be written as ψ(ζ)=f(ζ) exp(−|ζ|²/2ℓ²), where ζ=x+iy is a complex coordinate, ℓ=√(ħ/2mΩ) is the magnetic (or rotational) length, and f(ζ) is an analytic function whose zeros correspond to vortex cores. For a large number of vortices N_v≫1 the zeros arrange themselves into a regular triangular lattice. The authors derive the exact lattice spacing a≈√(2π)ℓ/√N_v and show that the coarse‑grained density reproduces the Thomas‑Fermi (TF) inverted‑parabola profile, while the fine‑grained density exhibits rapid oscillations due to the polynomial factor |f(ζ)|². This provides a quantitative description of the deviations from the TF approximation that have been observed experimentally in high‑rotation BEC images.

The second part of the work addresses a highly anisotropic (non‑cylindrically symmetric) trap where the transverse frequencies satisfy ω_x<ω_y and the rotation frequency is tuned to Ω=ω_x. In this “narrow‑channel” limit the confinement along the weak axis x is completely cancelled, rendering the system infinitely elongated along y. The vortex lattice then reorganizes into parallel rows (vortex “stripes”) that run along the y‑direction. The authors construct a trial wavefunction ψ_M(ζ)=e^{−x²/2ℓ²} ∏_{m=1}^{M}θ