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For a two-dimensional Thomas–Fermi condensate at fixed coupling, $N=\pi n_0{R}^2/2\propto {\mu }^2$, so $R\propto N^{1/4}$. Slow exponential atom loss shrinks the cloud while a fixed stirring radius becomes a larger fraction of its size. At $\Omega ={\omega }_r/\sqrt{2}$ and initial $D/R=1/2$, the local-sound-speed threshold is reached at $t_c=2\tau \ln 2$, before the shrinking cloud edge reaches the beam.

For a centered quantum vortex in a two-dimensional Thomas–Fermi condensate, use an intermediate radius ${\xi }_{\mathrm{core}}\ll L\ll R$. The approximately uniform core contributes $\pi n_c{\hslash }^2{\mathcal{N}}^2[\log (L/\xi )+L_{0|\mathcal{N}|}]/m$. The outer superfluid velocity is $\hslash \mathcal{N}/(mr)$, so its kinetic contribution is $(\pi {\hslash }^2{\mathcal{N}}^2/m){\int }_L^R{n}_c(1-r^2/R^2)dr/r$. Integration cancels $L$ and produces the constant $-1/2$. The displayed result is the matched small-core approximation at fixed particle number, not a formula for arbitrarily large winding at fixed cloud size. The angular momentum is exactly $L_z=N\hslash \mathcal{N}$ for a normalized axisymmetric single-winding state.

Write $g=V_0$ for the effective coupling and keep $N$ for the number obtained by the stated two-dimensional normalization. A repulsive interaction, $g>0$, permits the Thomas–Fermi approximation for a condensate: density-gradient kinetic energy is small compared with the trapping and interaction terms over most of the cloud. The stationary Gross–Pitaevskii equation then givesand $n_{\mathrm{TF}}=0$ outside the cloud, up to the narrow edge region. HereNormalize the parabolic number density directly:Consequently the two-dimensional Thomas–Fermi condensate parameters areWith $a_{\mathrm{ho}}=\sqrt{\hslash /(m\omega )}$ and central healing length $\xi =\hslash /\sqrt{2m\mu }$, the bulk approximation requiresIndeed $R/\xi =2\mu /(\hslash \omega )\gg 1$. In the coupling convention printed in the paper, formal substitution of $g=4\pi {\hslash }^{2}a/m$ givesThis is the requested relationship in that convention; $\omega$ sets the physical size and chemical potential, while the interaction-strength ratio to $\hslash \omega$ is independent of $\omega$.

There is a dimensional convention to make explicit. A physical three-dimensional scattering length $a$ makes $4\pi {\hslash }^{2}a/m$ a three-dimensional contact coupling, whereas a genuinely two-dimensional particle-number normalization requires coupling units of energy times area. For an axial factor $\chi (z)$ normalized to one, the effective two-dimensional contact coupling is $g_{2D}=(4\pi {\hslash }^{2}a/m)\int |\chi {|}^{4}dz$. For a Gaussian of width $a_z$ this is $4\pi {\hslash }^{2}a/(m\sqrt{2\pi }{a}_z)$, and the physical TF condition becomes $Na/a_z\gg 1$ up to numerical constants. Alternatively the printed coupling can be retained if $N$ is a line density. The paper supplies no axial scale, so its $Na$ expressions require that effective-unit or line-density convention; the formulas in terms of $g$ above are unambiguous. The approximation fails inside vortex cores and at the cloud edge, and it is not a repulsive TF ground-state construction for $g\le 0$.

In the large-$N$ Gross–Pitaevskii equation, dropping the kinetic energy gives $[\kappa r^2/2+gn(r)-\mu ]\Psi (r)=0$. The physical nonnegative Thomas–Fermi approximation for a condensate is thereforewhere $(z{)}_{+}=\max (z,0)$ and $R$ is the cloud radius. Its particle number isThus the two-dimensional Thomas–Fermi condensate obeysHere $N=\int n{d}^{2}r$ is a total particle number. Keeping the exact pair count in the fixed-$N$ Hartree product state would replace $g$ by $g(1-1/N)$ in these formulas expressed in terms of $n=N|\varphi {|}^2$.

The approximation is best at the centre, $r=0$, and worst near the edge, $r=R$. For example its omitted quantum pressure of a condensate, evaluated on the interior profile, isIt is smallest at the centre and diverges at the artificial sharp edge. The central healing length ${\xi }_0=\hslash /\sqrt{2m\mu }$ is much smaller than $R$ when $\mu \gg \hslash \omega$, since $R/{\xi }_0=2\mu /(\hslash \omega )$. This is the bulk validity condition; the full Gross–Pitaevskii equation smooths the edge and gives a decaying tail outside $R$.