Absence of fractional ac Josephson effect in superconducting junctions

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 $2\ensuremath{\pi}$ periodic in the Josephson phase $\ensuremath{\varphi}$ in full accordance with general principles. Our result unambiguously argues against the idea of the so-called ``fractional ac Josephson effect'' admitting $4\ensuremath{\pi}$ periodic in $\ensuremath{\varphi}$ CPR. We also demonstrate that at any nonzero $V$ 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 nontrivial $2\ensuremath{\pi}$-periodic dissipative current with the amplitude ${\ensuremath{\propto}|V|}^{1/3}$.