Theory of nonreciprocal Josephson effect

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