Vortex Lattice Formation in a Rotating Condensate

Trap Rotation

Stirring frequency Ω / ω_⊥ 0.00

Mutual friction α 0.15

Core / healing length ξ 0.045

Lattice State

Vortices N_v

Feynman target N_eq

Six-fold order Ψ₆ (0 = disordered, 1 = crystal)

0.00

Thomas-Fermi radius R_TF

1.00 R₀

How it works

A trapped Bose-Einstein condensate is described by a single macroscopic wavefunction ψ(r,t) obeying the Gross-Pitaevskii equation. Stirred at angular frequency Ω, the lowest-energy state in the rotating frame is not smooth rotation but a lattice of quantized vortices, each carrying one quantum of circulation:

iħ ∂ψ/∂t = [-ħ²∇²/2m + V(r) + g|ψ|² - ΩL_z]ψ
∮ v_s · dl = h/m  (circulation is quantized, no partial vortices)

Feynman's rule fixes the equilibrium areal vortex density purely from the rotation rate, independent of interaction strength:

n_v = 2mΩ/h   →   N_eq = n_v · π R_TF²

Ω / ω_⊥ slider — the stirring rate. It sets both the equilibrium vortex number N_eq above and the centrifugally-expanded Thomas-Fermi radius R_TF(Ω) = R₀/√(1-(Ω/ω_⊥)²).
Nucleation — when N_v < N_eq, a new vortex is seeded at the condensate edge (real surface-mode instabilities inject vortices exactly this way) and migrates inward under the flow of its neighbors.
Point-vortex dynamics — each vortex is advected by the regularized Biot-Savart velocity of every other vortex plus an image vortex outside the boundary (method of images, enforcing zero flow through the condensate edge).
Mutual friction α — coupling to the thermal (non-condensed) cloud damps the vortex motion, rotating each vortex's velocity slightly inward-facing (v → (v − α ẑ×v)/(1+α²)). Without it vortices merely orbit forever; with it they dissipate energy and crystallize into the triangular Abrikosov lattice — exactly how real stirred-BEC experiments (JILA, ENS, MIT) form ordered lattices.
Ψ₆ order parameter — the magnitude of the averaged six-fold bond-orientational order ⟨e^(i6θ)⟩ over each vortex's nearest neighbors; it climbs toward 1 as the vortices settle into a hexagonal (triangular) crystal.

Condensate density (Thomas-Fermi) Quantized vortex core

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Vortex Lattice Formation in a Rotating Bose-Einstein Condensate