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We study Andreev reflection and Josephson currents in topological bilayerexciton condensates (TEC). These systems can create 100% spin entanglednonlocal currents with high amplitudes due to perfect nonlocal Andreevreflection. This Andreev reflection process can be gate tuned from a regime ofpurely retro reflection to purely specular reflection. We have studied thebound states in TEC-TI-TEC Josephson junctions and find a gapless dispersionfor perpendicular incidence. The presence of a sharp transition in thesupercurrent-phase relationship when the system is in equilibrium is asignature of fractional charge, which can be further revealed in acmeasurements faster than relaxation processes via Landau-Zener processes.

Nonlocal Andreev reflection, fractional charge and current-phase relation in topological bilayer exciton condensate junctions