Absence of "fractional ac Josephson effect" in superconducting junctions

TL;DR

This paper develops a microscopic theory of the ac Josephson effect in superconducting junctions, demonstrating the absence of fractional (4π-periodic) effects and highlighting the instability of Andreev bound states at non-zero voltages.

Contribution

The authors provide a formal derivation showing the $2 ext{-} ext{pi}$-periodic current-phase relation and disprove the existence of fractional ac Josephson effect in arbitrary junctions.

Findings

The current-phase relation is exactly $2 ext{-} ext{pi}$-periodic.

At non-zero voltage, Andreev bound states become non-Hermitian and unstable.

A non-trivial $2 ext{-} ext{pi}$-periodic dissipative current appears at small voltages.

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 $\propto |V|^{1/3}$