Non-equilibrium $8\pi$ Josephson Effect in Atomic Kitaev Wires

We theoretically study a Kitaev wire interrupted by an extra site which gives rise to super exchange coupling between two Majorana bound states. We show that this system hosts a tunable, non-equlibrium Josephson effect with a characteristic $8\pi$ periodicity of the Josephson current. We elucidate the physical mechanism deriving a minimal model for the junction and confirm its quantitative accuracy by comparison to the numerical solution of the full model. The visibility of the $8\pi$ periodicity of the Josephson current is then studied using time-dependent simulations including the effects of dephasing and particle losses. Our findings provide a novel signature of Majorana quasi-particles which is qualitatively different form the behavior of a conventional superconductor, and can be experimentally verified in cold atom systems using alkaline-earth-like atoms.