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This again is an argument by F. London, although we formulate it in G-L terms, by expliciting the phase in the G-L wave function, {$\psi=|\psi|e^{i\phi}$}. Then we can calculate the average velocity by means of

{$\qquad\qquad m\mathbf{v}_s|\psi| e^{i\phi}=\left(-i\hbar\mathbf{\nabla} -e\mathbf{ A}\right)|\psi| e^{i\phi}$}

from which we derive

{$ (1)\qquad\qquad m\mathbf{v}_s = -i \frac {\hbar} {|\psi|} \mathbf{\nabla}|\psi|+(\hbar\mathbf{\nabla}\phi - e\mathbf{ A}) $}

where the imaginary part arises from boundaries, whereas the real part is due to the supercurrents. One could associate therefore a momentum {$\mathbf{p}_s=\hbar\mathbf{\nabla}\phi$} to the supercurrent.

Since travelling along any closed curve {$\Gamma$} fully immersed in the superconductor we must come back to the initial phase, modulo {$2\pi$},

{$(2) \qquad\qquad\oint_\Gamma\mathbf{\nabla}\phi\cdot d\mathbf{l}= 2n\pi, $}

the following important consequence can be drawn: {$\mathbf{p}_s$} must be similarly quantized.

So, let us consider a quantity equivalent to the magnetic flux through a surface {$\Sigma$}, which is equal by Stokes theorem to the circulation of {$\mathbf{ A}$} along the closed path {$\Gamma$}, the contour of {$\Sigma$}

{$ (3) \qquad\qquad\Phi^\prime=\frac 1 e \oint_\Gamma \mathbf{p}\cdot d\mathbf{l} = \oint_\Gamma \left( \frac m e \mathbf{v}_s+\mathbf{ A}\right)\cdot d\mathbf{l} = \oint_\Gamma \left( \frac {\lambda^2} {\mu_0} \mathbf{J}_s+\mathbf{ A}\right)\cdot d\mathbf{l}$}

This quantity defines the fluxoid and the last equality links it to the supercurrent density by means of the penetration depth {$\lambda$}. In its definition we can use indifferently {$\mathbf{p}_s$} or {$\mathbf{p}$}: their difference is connected by Eq. 1 to the gradient of {$|\psi|$}, which has zero circulation. Let's start considering a bulk type I superconductor where London Eq. 4 holds: then {$\Phi^\prime=0$}. This simply reflects the fact that inside the superconductor there is zero field (Meissner effect), hence constant vector potential, zero magnetic flux and zero supercurrents along the closed path.

Let's next consider a cylindrical hole in the bulk. In this case Eq. 2 can take its non vanishing {$n$} values. Whatever the case, we can conclude that a cylinder of flux within a superconductor (e.g. one with a hole) and the supercurrents shielding it may be both signalled by a nonvanishing circulation of the phase gradient, Eq. (2).

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