Effect of Magnetic Field and Rashba Spin-Orbit Interaction on the Josephson Tunneling between Superconducting Nanowires

We calculate the Josephson current between two one-dimensional (1D) nanowires oriented along \(x\) with proximity induced \(s\)-wave superconducting pairing and separated by a narrow dielectric barrier in the presence of both Rashba spin-orbit interaction (SOI) characterized by strength \(\alpha\) and Zeeman fields (\(h\) along \(\hat z\) and \({\bf B}\) in the \(x-y\) plane). We formulate a general method for computing the Andreev bound states energy which allows us to obtain analytical expressions for the energy of these states in several asymptotic cases. We find that in the absence of the magnetic fields the energy gap between the Andreev bound states decreases with increasing Rashba SOI constant leading eventually to touching of the levels. In the absence of Rashba SOI, the Andreev bound states depend on the magnetic fields and display oscillatory behavior with orientational angle of B leading to magneto-Josephson effect. We also present analytic expressions for the dc Josephson current charting out their dependence on \({\bf B}\), \(h\), and \(\alpha\). We demonstrate the existence of finite spin-Josephson current in these junctions in the presence of external magnetic fields and provide analytic expressions for its dependence on \(\alpha\), \(\bf B\) and \(h\). Finally, we study the AC Josephson effect in the presence of the SOI (for \(|{\bf B}|=h=0\)) and an external radiation and show that the width of the resulting Shapiro steps in such a system can be tuned by varying \(\alpha\). We discuss experiments which can test our theoretical results.