  We study a time-reversal-invariant topological superconductor island hosting
spatially separated Majorana Kramers pairs, with weak tunnel couplings to two
s-wave superconducting leads. When the topological superconductor island is in
the Coulomb blockade regime, we predict that a Josephson current flows between
the two leads due to a non-local transfer of Cooper pairs mediated by the
Majorana Kramers pairs. Interestingly, we find that the sign of the Josephson
current is controlled by the joint parity of all four Majorana bound states on
the island. Consequently, this parity-controlled Josephson effect can be used
for qubit read-out in Majorana-based quantum computing.

Mesoscale and Nanoscale Physics