Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 84 / 3

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 $g\le 0$.

Use the central number density $n_c$, radius $R$ and central healing length $\xi$ from the preceding part. The vortex energy means the excess over the no-vortex ground state at fixed particle number. Choose an intermediate matching radius $\xi \ll L\ll R$. Inside $L$, the trapped number density differs little from $n_c$, so the resolved uniform vortex core givesOutside the core, the singly quantized superfluid velocity is $u_{\theta }=\hslash /(mr)$. Its leading flow kinetic energy in the parabolic background is thereforeAdding the regions cancels the arbitrary matching radius. Sending $L/R\to 0$ while $\xi /L\to 0$ gives the matched vortex energy in a parabolic condensateThe $-1/2$ comes from the decrease of background number density away from the centre. It would be missed by treating the whole trap as uniform. To this accuracy, using the unperturbed TF density in the outer region is appropriate at fixed $N$: the first variation of trapping plus interaction energy is $\mu \delta N$ and therefore vanishes after the small normalization adjustment. The resolved core energy supplies the finite core correction $L_{01}$, while remaining cloud-edge and density-relaxation corrections vanish in the small-core TF limit. This is an asymptotic energy estimate, not an exact formula at finite $R/\xi$.

For an axisymmetric centred unit quantum vortex, write $\psi =f(r)e^{i\theta }$, including its radial density depletion in $f$. The axial angular momentum operator is ${\hat{L}}_z=-i\hslash {\partial }_{\theta }$, so every particle in this single-winding state has eigenvalue $\hslash$. HenceThis is exact at the specified particle number and does not require neglecting the core or using the parabolic TF profile.

Let $\mathcal{N}\in \mathbb{Z}$ be the vortex winding number, distinct from the particle number $N$. Outside its core, the superfluid velocity is $u_{\theta }=\hslash \mathcal{N}/(mr)$. The outer flow energy is therefore multiplied by ${\mathcal{N}}^2$, while the resolved core contribution contains the appropriate constant $L_{0|\mathcal{N}|}$. The same matching integration givesThe core equation depends on ${\mathcal{N}}^2$, so reversing circulation leaves this energy unchanged. On the other hand, $-i\hslash {\partial }_{\theta }[f(r)e^{i\mathcal{N}\theta }]=\hslash \mathcal{N}f(r)e^{i\mathcal{N}\theta }$, giving the exact resultNegative winding reverses angular momentum; zero winding has zero excess vortex energy and zero angular momentum. The energy approximation applies to each fixed nonzero winding in the TF limit. More generally its core radius, of order $|\mathcal{N}|\xi$, must be much smaller than $R$ so that a matching interval exists. It is not valid uniformly for arbitrary growing $|\mathcal{N}|$ at fixed cloud size. The leading quadratic energy cost also explains why multiply quantized vortices can lower their energy by separating into singly quantized vortices; this energy comparison does not by itself establish a particular dynamical splitting rate.