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A prominent signature of Majorana bound states is the exotic Josephsoneffects they produce, the classic example being a fractional Josephson currentwith 4\pi periodicity in the phase difference across the junction. Recent workestablished that topological insulator edges support a novel `magneto-Josephsoneffect', whereby a dissipationless current exhibits 4\pi-periodic dependencealso on the relative orientation of the Zeeman fields in the two banks of thejunction. Here, we explore the magneto-Josephson effect in junctions based onspin-orbit coupled quantum wires. In contrast to the topological insulatorcase, the periodicities of the magneto-Josephson effect no longer follow froman exact superconductor-magnetism duality of the Hamiltonian. We employnumerical calculations as well as analytical arguments to identify the domainconfigurations that display exotic Josephson physics for quantum-wirejunctions, and elucidate the characteristic differences with the correspondingsetups for topological insulators edges. To provide guidance to experiments, wealso estimate the magnitude of the magneto-Josephson effects in realisticparameter regimes, and compare the Majorana-related contribution to thecoexisting 2\pi-periodic effects emerging from non-Majorana states.

Magneto-Josephson effects and Majorana bound states in quantum wires