The demonstration of the non-Abelian properties of Majorana bound states (MBS) is a crucial step toward topological quantum computing. We theoretically investigate how Majorana fusion rules manifest themselves in the current-voltage characteristics of a topological Josephson junction. The junction is built on U-shaped quantum spin Hall edges and hosts a Majorana qubit formed by four MBS. Owing to Majorana fusion rules, inter- and intra-edge couplings among adjacent MBS provide two orthogonal components in the rotation axis of the Majorana qubit. We show that the interplay of the dynamics of the superconductor phase difference and the Majorana qubit governs the Josephson effect. Strikingly, we identify sequential jumps of the voltage across the junction with increasing DC current bias without external AC driving. Its role is replaced by the intrinsic Rabi oscillations of the Majorana qubit. This phenomenon, DC Shapiro steps, is a manifestation of the non-trivial fusion rules of MBS.
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