 Gauge-invariant Josephson phase

Write the superconducting order parameters as ${\psi }_j=|{\psi }_j|e^{i{\theta }_j}$, using a field of pair electric charge $q=-2e$. For a specified path across the superconducting weak link, defineHere $\mathbf{A}$ is the magnetic vector potential. To check electromagnetic gauge invariance, use a scalar function $\chi$ and transformThe change of ${\theta }_2-{\theta }_1$ is $q({\chi }_2-{\chi }_1)/\hslash$, exactly cancelled by the change of the line integral. Thus $\varphi$ is unchanged although each local quantum phase changes. This is the same charge/sign bookkeeping discussed in gauge-coupling sign in Ginzburg-Landau superconductivity. Different physical paths can enclose magnetic flux, so fixing the path matters.

For stationary reservoirs with equal internal pair chemical potentials and a time-independent vector potential, the local Schrödinger equation gives ${\dot{\theta }}_j=-q{U}_j/\hslash$ after dropping the common internal term. With electric potentials $U_j$ and $V=U_2-U_1$, subtraction gives $\dot{\varphi }=-qV/\hslash =2eV/\hslash$. For a general circuit, $V$ denotes the corresponding measured electrochemical voltage and the electromagnetic contribution must be treated consistently. A bare difference of phases without the vector potential term is insufficient in an arbitrary gauge.