Signatures of the topological spin of Josephson vortices in topological superconductors

We consider a modified setup for measuring the Aharonov-Casher phase which consists of a Josephson vortex trapped in an annular topological superconducting junction. The junction encloses both electric charge and magnetic flux. We discover a deviation from the Aharonov-Casher prediction whose origin we identify in an additive universal topological phase that remarkably depends only on the parity of the number of vortices enclosed by the junction. We show that this phase is $\pm 2\pi$ times the topological spin of the Josephson vortex and is proportional to the Chern number. The presence of this phase can be measured through its effect on the junction's voltage characteristics, thus revealing the topological properties of the Josephson vortex and the superconducting state.