= Gauge-invariant Josephson phase
{c}
{title2=$\phi=\theta_2-\theta_1-(q/\hbar)\int_1^2\mathbf A\cdot d\mathbf l$}
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>, define
$$
\phi=\theta_2-\theta_1-\frac{q}{\hbar}\int_1^2\mathbf A\cdot d\mathbf l
=\theta_2-\theta_1+\frac{2e}{\hbar}\int_1^2\mathbf A\cdot d\mathbf l.
$$
Here $\mathbf A$ is the magnetic <vector potential>. To check electromagnetic <gauge invariance>, use a scalar function $\chi$ and transform
$$
\mathbf A'=\mathbf A+\nabla\chi,\qquad
\psi_j'=e^{iq\chi_j/\hbar}\psi_j.
$$
The change of $\theta_2-\theta_1$ is $q(\chi_2-\chi_1)/\hbar$, exactly cancelled by the change of the <line integral>. Thus $\phi$ 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 $\dot\theta_j=-qU_j/\hbar$ after dropping the common internal term. With <electric potentials> $U_j$ and $V=U_2-U_1$, subtraction gives $\dot\phi=-qV/\hbar=2eV/\hbar$. 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.
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