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 Neq above and the centrifugally-expanded Thomas-Fermi radius RTF(Ω) = R₀/√(1-(Ω/ω⊥)²).
- Nucleation — when Nv < Neq, 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). This 2D engine solves the same dzi/dt = (iΓ/2π)Σj 1/(zi-zj)* equations directly on the complex plane of the canvas.
- 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.