Consider a superconducting circuit with a "box-like 8" geometry like [|] (ie. two square loops which share one side of wire). Here we can have three different currents ($I_1=I_2+I_3$, see H. J. Fink and S. B. Haley, Phys. Rev. B 43, 10151 (1991)), one flowing along each wire segment, and we also have two different magnetic fluxes, one for each closed loop in the circuit.
In the case of a circuit with only a single closed wire loop, one can integrate the condensate phase around the loop to show the flux is quantized. In the above case, I want to be able to do this over three integration paths, two of which encircle one loop only and one which encircles both loops.
My question is the following. When we write the condensate wave function as $\Psi=\sqrt{n}\mathrm{e}^{i\theta}$, does the pair density $n$ need to be assumed constant in the superconductor in order to show the flux is quantized? If so, could the circuit I described above be expected to have uniform pair density in each wire segment, or would the different currents make this unlikely?
