Manipulation of Cooper pair entanglement in hybrid topological Josephson junctions

We investigated the supercurrent in a hybrid topological Josephson junction consisting of two planes of topological insulator (TI), which allows both local (LAR) and crossed (CAR) Andreev processes at the interfaces with two conventional s-wave superconductors. We describe the effects of local Rashba-like and Zeeman-like fields applied to the edge states. In particular, we demonstrate that the Rashba-like field, as induced by local gating, allows the manipulation of the entanglement symmetry of non-local Cooper pairs associated to the CAR process. We establish a connection between the Josephson current-phase relationship of the system and the action of the two local fields, finding that they selectively modify the LAR or the CAR contributions. Remarkably, we find that the critical current of the junction takes a very simple form which reflects the change in the symmetry occurred to the entangled state and allows to determine the microscopic parameters of the junction.