  The time-dependent state of a classical spin locally exchange coupled to an
edge site of a Kane-Mele model in the topologically non-trivial phase is
studied numerically by solving the full set of coupled microscopic equations of
motion for the spin and the electron system. Dynamics in the long-time limit is
accessible thanks to dissipative boundary conditions, applied to all but the
zigzag edge of interest. We study means to control the state of the spin via
transport of a spin-polarization cloud through the helical edge states. The
cloud is formed at a distant edge site using a local magnetic field to inject
an electron spin density and released by suddenly switching off the injection
field. This basic process, consisting of spin injection, propagation of the
spin-polarization cloud, and scattering of the cloud from the classical spin,
can be used to steer the spin state in a controlled way. We find that the
effect of a single basic process can be reverted to a high degree with a
subsequent process. Furthermore, we show that by concatenating several basic
injection-propagation-scattering processes, the spin state can be switched
completely and that a full reversal can be achieved.
