Theory of the nonreciprocal Josephson effect

The heterojunction between different materials often exhibits a rectifying effect; e.g., pn junction is used for diode. On the other hand, Josephson two superconductors assumed to show symmetric response directions of current, i.e., voltage drop $V$ antisymmetric with respect sign change current $I$. However, there should be an asymmetry states charge accumulation on right and left sides junction, which can lead nonreciprocal responses. Here we demonstrate theoretically that $I\text{\ensuremath{-}}V$ characteristic appears due this charging energy difference both in classical quantum regimes.