Towards all electrical control of topological Josephson junctions and Majorana zero modes via spin-orbit interactions

We study the current-phase relation of topological Josephson junctions with spin-orbit interactions, and show that the coupling between Majorana zero modes (MZMs) can be controlled via gate tunable spin-orbit couplings (SOCs). The spin-triplet pairings in the presence of MZMs at the two ends of a one-dimensional topological superconductor, are shown to have a $\pi$ phase difference, from which a Josephson $\pi$-junction can be created. This $\pi$ phase is unambiguously manifested to be a spin-dependent superconducting phase, dubbed spin-phase. We demonstrate that SOC can induce such spin-phase in spin-triplet superconducting condensates which can tune the MZM coupling energy and allow a finite topological Josephson current without a magnetic flux in superconducting circuits. We further establish the linkage between this Josephson current and the fermion parity in a topological Josephson junction and propose an all-electronically controlled superconductor-semiconductor hybrid circuit to detect the non-Ableian nature of MZMs.