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Electromotive interference in a mechanically oscillating superconductor: generalized Josephson relations and self-sustained oscillations of a torsional SQUID

We consider the superconducting phase in a moving superconductor and show that it depends on the displacement flux. Generalized constitutive relations between the phase of a superconducting interference device (SQUID) and the position of the oscillating loop are then established. In particular, we show that the Josephson current and voltage depend on both the SQUID position and velocity. The two proposed relativistic corrections to the Josephson relations come from the macroscopic displacement of a quantum condensate according to the (non-inertial) Galilean covariance of the Schrödinger equation, and the kinematic displacement of the quasi-classical interfering path. In particular, we propose an alternative demonstration for the London rotating superconductor effect (also known as the London momentum) using the covariance properties of the Schrödinger equation. As an illustration, we show how these electromotive effects can induce self-sustained oscillations of a torsional SQUID, when the entire loop oscillates due to an applied dc-current.