Magneto-electric coupling in a two-dimensional ballistic Josephson junction with in-plane magnetic texture

We study a Josephson junction made with a spin-textured bridge, when both Rashba and Zeeman interactions combine to generate a magneto-electric coupling between the superconducting current and the in-plane magnetic texture in the normal region. In particular, we unambiguously obtain the so-called anomalous current-phase relation $j=j_{c}\sin\varphi+j_{a}\cos\varphi$ in a two-dimensional ballistic Josephson junction close to the critical temperature of the heterostructure, when an anomalous current $j_{a}\neq0$ subsists even at zero phase-difference between the superconductors, and is responsible for the coupling between the magnetic and electric degrees of freedom of the junction. The anomalous magneto-electric current is due to the combination of the chirality of the propagating modes and the anisotropy of the in-plane magnetic texture.