Long-range interaction of magnetic moments in a coupled system of superconductor-ferromagnet-superconductor Josephson junctions with anomalous ground-state phase shift

A mechanism of a superconductivity-mediated interaction of two magnets in a system of coupled superconductor-ferromagnet-superconductor (S-F-S) Josephson junctions (JJs) with spin-orbit interaction is proposed. The predicted indirect magnetic interaction favors the antiparallel orientation of the magnets. Its spatial scale is not restricted by the proximity length scales of the superconductor. Our estimates suggest that the interaction strength is not reduced considerably even at the macroscopic scales of the order of millimeters. At larger distances l between the magnets the coupling constant exhibits the long-range power-law 1/l behavior. The mechanism of the interaction is based on two key ingredients: (i) the anomalous ground-state phase shift in the S-F-S JJ provides a magnetoelectric coupling between the condensate phase and the magnetization, and (ii) the interaction is mediated by the condensate phase of the superconducting region connecting both JJs. In addition, we demonstrate high tunability of the total magnetic configuration of the system by the externally controlled superconducting phase between the leads.