= 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> $M$, $\mathbf v_s=\hbar\nabla\theta/M$, hence the <quantized circulation>
$$
\boxed{\oint\mathbf v_s\cdot d\mathbf l=n\frac hM.}
$$
Here $M$ 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 $M\mathbf v_s=\hbar\nabla\theta-q\mathbf A$. Integrating gives
$$
M\oint\mathbf v_s\cdot d\mathbf l+q\Phi=nh,
\qquad \Phi+\frac Mq\oint\mathbf v_s\cdot d\mathbf l=\frac{nh}{q}.
$$
This is <fluxoid and flux quantization in a superconductor>. With Cooper-pair charge magnitude $2e$, the flux quantum's magnitude is
$$
\boxed{\Phi_0=\frac h{2e}.}
$$
The 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.
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