Signatures of time-reversal-invariant topological superconductivity in the Josephson effect

For Josephson junctions based on s-wave superconductors, time-reversal symmetry is known to allow for powerful relations between the normal-state junction properties, the excitation spectrum, and the Josephson current. Here we provide analogous relations for Josephson junctions involving one-dimensional time-reversal-invariant topological superconductors supporting Majorana-Kramers pairs, considering both topological-topological and s-wave-topological junctions. Working in the regime where the junction is much shorter than the superconducting coherence length, we obtain a number of analytical and numerical results that hold for arbitrary normal-state conductance and the most general forms of spin-orbit coupling. The signatures of topological superconductivity we find include the fractional ac Josephson effect, which arises in topological-topological junctions provided that the energy relaxation is sufficiently slow. We also show, for both junction types, that robust signatures of topological superconductivity arise in the dc Josephson effect in the form of switches in the Josephson current due to zero-energy crossings of Andreev levels. The junction spin-orbit coupling enters the Josephson current only in the topological-topological case and in a manner determined by the switch locations, thereby allowing quantitative predictions for experiments with the normal-state conductance, the induced gaps, and the switch locations as inputs.