A gauge invariant derivation of the AC Josephson frequency and a reconsideration of the origin of the phase of the order-parameter
We derive the AC Josephson frequency using the gauge invariant equations of motion with including the battery-contact effect. The frequency is given as a sum of a contribution from an Aharonov-Bohm phase that arises when charged-partices pass through an electric field in the insulator and that from a chemical potential difference arising form the battery contact; each of them contributes $|q|V/h$ to the frequency where $q$ is the charge, thus, the sum is $2|q|V/h$. The observed Josepshon frequency, $2|e|V/h$, hence, means that the charge on the tunneling particles is $e$. A variety of derivations so far miss one of the above two contributions; the original derivation misses the first contribution due to the lack of inclusion of the electric field in the insulator. The present result indicates that the phase of the order-parameter does not arise from the number fluctuations of Cooper pairs. We present an alternative origin for it; we argue that a Berry phase that arises from spin-vortices is the origin.