Absence of "fractional ac Josephson effect" in superconducting junctions

Abstract

We develop a microscopic theory of ac Josephson effect in superconducting junctions described by an arbitrary scattering matrix that may include magnetic effects. In the limit of constant in time bias voltage $V$ applied to the junction we derive a formally exact current-phase relation (CPR) that is manifestly $2\pi$-periodic in the Josephson phase $\varphi$ in full accordance with general principles. This our result unambiguously argues against the idea of the so-called "fractional ac Josephson effect" admitting $4\pi$-periodic in $\varphi$ CPR. We also demonstrate that at any non-zero $V$ quantum dynamics of Andreev bound states becomes non-Hermitian which signals their instability, thus making any 'quasi-equilibrium' description of ac Josephson effect unreliable. We specifically address the limit of highly transparent junctions with magnetic scattering where -- along with super- and excess current terms -- at small $V$ we also recover a non-trivial $2\pi$-periodic dissipative current with the amplitude $\propto |V|^{1/3}$