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

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