  The chiral edge modes of a topological superconductor can transport fermionic
quasiparticles, with Abelian exchange statistics, but they can also transport
non-Abelian anyons: Edge-vortices bound to a $\pi$-phase domain wall that
propagates along the boundary. A pair of such edge-vortices is injected by the
application of an $h/2e$ flux bias over a Josephson junction. Existing
descriptions of the injection process rely on the instantaneous scattering
approximation of the adiabatic regime [Beenakker et al. Phys.Rev.Lett. 122,
(2019)], where the internal dynamics of the Josephson junction is ignored. Here
we go beyond that approximation in a time-dependent many-body simulation of the
injection process, followed by a braiding of mobile edge-vortices with a pair
of immobile Abrikosov vortices in the bulk of the superconductor. Our
simulation sheds light on the properties of the Josephson junction needed for a
successful implementation of a flying topological qubit.
