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Hybrid structures of quantum spin-Hall insulators (QSHIs) and superconductors(Ss) present a unique opportunity to access dissipationless topological statesof matter, which, however, is frequently hindered by the lack of control overthe spin polarization in QSHIs. We propose a very efficient spin-polarizationmechanism based on the magnetoelectric (Edelstein) effect in superconductingQSHI structures. It acts akin to the Zeeman splitting in an external magneticfield, but with an effective $g$-factor of order of 1000, resulting in anunprecedented spin-splitting effect. It allows a magnetic control of the QSHI/Shybrids without destroying superconductivity. As an example, we demonstrate arecurrent crossover from $\Phi_0$ - to $\Phi_0/2$ - periodic oscillations ofthe Josephson current in an rf superconducting quantum interference device($\Phi_0=h/2e$ is the magnetic flux quantum). The predicted period halving is astriking manifestation of $0-\pi$ Josephson transitions with a superharmonic$\pi$-periodic current-phase relationship at the transition. Such controllable$0-\pi$ transitions may offer new perspectives for dissipationless spintronicsand engineering flux qubits.

Giant spin splitting and $0 - \pi$ Josephson transitions from the Edelstein effect in quantum spin-Hall insulators