 Solution

Write the condensate order parameter as $\Psi =|\Psi |e^{i\theta }$. Away from vortex cores, single-valuedness around a closed loop gives $\oint \nabla \theta \cdot d\mathbf{l}=2\pi n$. For neutral condensed objects of mass $M$, ${\mathbf{v}}_s=\hslash \nabla \theta /M$, hence the quantized circulationHere $M$ is the atomic mass for a simple bosonic atomic condensate, or the pair mass for a paired-fermion condensate. A nonzero winding can surround a vortex core where the order-parameter amplitude falls to zero. The result concerns collective condensate phase winding, not separate orbit quantization of every constituent particle.

For charge $q$ per condensed object, the gauge-invariant relation is $M{\mathbf{v}}_s=\hslash \nabla \theta -q\mathbf{A}$. Integrating givesThis is fluxoid and flux quantization in a superconductor. With Cooper-pair charge magnitude $2e$, the flux quantum's magnitude isThe orientation fixes its sign. A loop taken deep in a thick superconductor has negligible superflow and therefore encloses an integer flux quantum. In a thin loop or one with appreciable screening current, it is the fluxoid, including the circulation term, that is quantized; the magnetic flux alone need not be exactly an integer multiple.