Nonequilibrium Fractional Josephson Effect

Josephson tunnel junctions exhibit a supercurrent typically proportional to the sine of the superconducting phase difference $\phi$. In general, a term proportional to $\cos(\phi)$ is also present, alongside microscopic electronic retardation effects. We show that voltage pulses sharply varying in time prompt a significant impact of the $\cos(\phi)$ term. Its interplay with the $\sin(\phi)$ term results in a nonequilibrium fractional Josephson effect (NFJE) $\sim\sin(\phi/2)$ in the presence of bound states close to zero frequency. Our microscopic analysis reveals that the interference of non-equilibrium virtual quasiparticle excitations is responsible for this phenomenon. We also analyse this phenomenon for topological Josephson junctions with Majorana bound states. Remarkably, the NFJE is independent of the ground state fermion parity unlike its equilibrium counterpart.