The chiral edge modes of a topological superconductor can transport fermionic
quasiparticles, with Abelian exchange statistics, but they can also transport
non-Abelian anyons: Majorana zero-modes bound to a {\pi}-phase domain wall that
propagates along the boundary. Such an edge vortex 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, 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 the mobile edge vortex with an immobile Abrikosov vortex 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 Majorana
qubit.Comment: 13 pages 12 figure