Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 343 / 3 / ii / Solution

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.