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

Let $n=|\Psi {|}^2$. Two independent approximations are required for the reservoir reduction of a polariton condensate. First the exciton reservoir must follow the changing condensate number density rapidly: after its initial transient, its relaxation time $[{\gamma }_R+R_{R}n{]}^{-1}$ must be small compared with the local density-evolution time. A sufficient modewise condition is that the relevant number density frequencies and growth rates are much smaller than ${\gamma }_R+R_{R}n$; pump variations must be comparably slow. There is no reservoir diffusion in the given model. Adiabatic elimination then givesSecond, to obtain the particular cubic equation by expansion about the empty condensate, require $\epsilon =R_{R}n/{\gamma }_R\ll 1$. ThenKeeping the full denominator instead gives a saturable-gain equation, not exactly the requested cubic complex Ginzburg–Landau equation. A near-threshold condensate is one natural regime in which the second condition holds. Rapid reservoir relaxation alone does not justify this number density expansion.

Multiplying the condensate equation by $-i$ and inserting the expanded reservoir separates real gain from the conservative frequency shift:These are local functions if the pump is spatially varying. Their signs are important: the reservoir blueshift gives negative $s$ in the printed convention $+is\Psi$, while reservoir depletion subtracts from the effective interaction $g$. Positive rates and pump give $\beta >0$; the effective $g$ need not be positive merely because the original interactions are repulsive.

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.

For a straight singly quantized quantum vortex, use cylindrical distance $r$ from the line and writeThere is no dependence along the vortex line. Substitution into the normalized stationary equation gives, before separating real and imaginary parts,Consequently the wave amplitude and phase-gradient equations areThe second equation can also be written $(r{f}^{2}k{)}^{\prime }=-{\xi }_{I}r{f}^2(1-f^2)$, which remains convenient at the core.

To specify what is meant by radial velocity, use the condensate number current. The time-dependent equation has kinetic operator $-{\nabla }^2$ and therefore number current $\mathbf{j}=2|\Psi {|}^2\nabla S$. Its hydrodynamic superfluid velocity in these units is $\mathbf{u}=2\nabla S$, so $u_r=2k$ and $u_{\theta }=2/r$. In that convention the requested pair isIf instead velocity is defined as the complex argument gradient itself, the preceding $k$ pair is the corresponding convention; the distinction matters because the printed kinetic coefficient differs from that of the normalized conservative equation in Question 1.

Regularity of a unit-charge core gives $f(r)=ar+O(r^3)$ with $a>0$. Integrating the current equation from the origin, with no point source or singular radial flux, givesThe integral is $a^2{r}^4/4+O(r^6)$, so the radial flow near a driven vortex core isFor the phase-gradient convention, $k^{\prime }(0)=-{\xi }_I/4=\alpha /4$. Gain exceeds loss in the depleted core, hence this regular flow is outwards for $\alpha >0$. The real wave amplitude equation also gives $f=ar-{\xi }_{R}a{r}^3/8+O(r^5)$, independently consistent with the linear core behaviour. A nonzero integration constant in the radial-current identity would create a singular $u_r\sim r^{-3}$ and is excluded.

Finally, the literal boundary $\psi \to 1$ from the preceding part cannot apply to a multiplicity-one vortex: its complex argument changes by $2\pi$ around a large circle. Even $f(r)e^{i\theta }$ with $f\to 1$ has different limits along different rays. The usual intended vortex condition concerns the wave amplitude approaching its bulk value, with the winding complex argument retained; it is not a constant complex-field limit. The local equations and core slope derived here do not establish a global stationary vortex satisfying that literal boundary. In a driven system the far-field radial flow and oscillation frequency may also require selection, so no global zero-flow vortex is asserted.