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A straight quantum vortex with integer winding number $\mathcal{N}$ has $\tilde{\psi }=R(r)e^{i\mathcal{N}\theta }$, independent of $z$. Here $r$ is measured in the unit of distance from the preceding part. The polar Laplacian supplies the centrifugal term, givingFor $\mathcal{N}\ne 0$, regularity requires $R\sim c{r}^{|\mathcal{N}|}$ near the axis; the bulk condition is $R\to 1$. The full wavefunction tends to $e^{i\mathcal{N}\theta }$ rather than one. This is the constant far-field phase excludes net vortex winding distinction: the equation remains applicable to a vortex, but the nonwinding boundary condition must be interpreted as an amplitude condition.

Let $F(R)=R-\alpha R^3-\beta R^5$. Since $\alpha +\beta =1$, one has $F(1)=0$ and $F^{\prime }(1)=1-3\alpha -5\beta =-2(1+\beta )$. Insert $R=1-p/r^2+\cdots$. Its derivative terms start at order $r^{-4}$, whileCancellation of the order-$r^{-2}$ terms proves the cubic-quintic quantum vortex tailFor a stable nondegenerate bulk state in this scaling, $1+\beta =n_0(V_0+2W_0{n}_0)/\mu >0$. At $\beta =-1$ the amplitude restoring derivative vanishes, and an inverse-square perturbation cannot cancel the centrifugal term, so the stated expansion fails. For zero winding the uniform solution has $p=0$.

For a spatially uniform pump and $\beta >0$, a nonzero homogeneous solution has constant number density determined by gain saturation. Its complex argument rotates uniformly:This uniform pumped polariton condensate requires $\alpha >0$, equivalentlyIn the cubic approximation its number density isIt is not the exact number density of the original two-field equations, which would be $(P-P_{\mathrm{th}})/{\gamma }_C$: the two coincide to first order in distance above threshold. At or below threshold there is no positive nonzero branch of this form; the vacuum remains a solution. This algebra establishes existence, not stability for every possible sign of $g$.

Choose ${\theta }_0=0$ and seek a time-independent normalized profile throughUsing $\beta n_{\infty }=\alpha$ and $\mu +s=g{n}_{\infty }$, its equation becomesDivision by $i$ therefore yieldsFor a localized zero-winding disturbance the bulk condition is $\psi \to 1$. Here $\xi$ is a complex coefficient with inverse-length-squared dimensions in the scaled equation, not the real wall-healing length denoted by the same letter in Question 1. The constant far-field phase excludes net vortex winding lemma explains why the next part requires interpreting the vortex boundary differently.