Magnetic exchange interaction induced by a Josephson current

We show that a Josephson current flowing through a ferromagnet-normal-metal-ferromagnet trilayer connected to two superconducting electrodes induces an equilibrium exchange interaction between the magnetic moments of the ferromagnetic layers. The sign and magnitude of the interaction can be controlled by the phase difference between the order parameters of the two superconductors. We present a general framework to calculate the Josephson current induced magnetic exchange interaction in terms of the scattering matrices of the different layers. The effect should be observable as the periodic switching of the relative orientation of the magnetic moments of the ferromagnetic layers in the ac Josephson effect.