Voltage controllable superconducting state in the multi-terminal superconductor-normal metal bridge

We study voltage controllable superconducting state in multi-terminal bridge composed of the dirty superconductor/pure normal metal (SN) bilayer and pure normal metal. In the proposed system small control current $I_{ctrl}$ flows via normal bridge, creates voltage drop $V$ and modifies distribution function of electrons in connected SN bilayer. In case of long normal bridge the voltage induced nonequilibrium effects could be interpreted in terms of increased local electron temperature. In this limit we experimentally find large sensitivity of critical curent $I_c$ of Cu/MoN/Pt-Cu bridge to $I_{ctrl}$ and relatively large current gain which originate from steep dependence of $I_c$ on temperature and large $I_c$ (comparable with theoretical depairing current of superconducting bridge). In the short normal bridge deviation from equilibrium cannot be described by simple increase of local temperature but we also theoretically find large sensitivity of $I_c$ to control current/voltage. In this limit we predict existence at finite $V$ of so called in-plane Fulde-Ferrell state with spontaneous currents in SN bilayer. We argue that its appearance is connected with voltage induced paramagnetic response in N layer.