Rapidly rotating Bose-Einstein condensates∗ Alexander Fetter ...

Large radial expansion means small central density n(0),
so that interaction energy gn(0) eventually becomes small
compared to gap 2ω ⊥
Hence focus on “lowest Landau level” (LLL), with n = 0
and general non-negative m ≥ 0
• ground-state wave function is Gaussian ψ 00 ∝ e −r2 /2d 2 ⊥
• general LLL eigenfunctions have a very simple form
ψ 0m (r) ∝ r m e imφ e −r2 /2d 2 ⊥
• here, d ⊥ = √ /Mω ⊥ is analogous to the “magnetic
length” in the Landau problem
• in terms of a complex variable ζ ≡ x + iy, these LLL
eigenfunctions become
ψ 0m ∝ ζ m e −r2 /2d 2 ⊥ ∝ ζ m ψ 00
with m ≥ 0 (note that ζ = r e iφ when expressed in
two-dimensional polar coordinates)
• apart from ground-state Gaussian ψ 00 , this is just ζ m
(a non-negative power of the complex variable)
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