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The time-dependent state of a classical spin locally exchange coupled to anedge site of a Kane-Mele model in the topologically non-trivial phase isstudied numerically by solving the full set of coupled microscopic equations ofmotion for the spin and the electron system. Dynamics in the long-time limit isaccessible thanks to dissipative boundary conditions, applied to all but thezigzag edge of interest. We study means to control the state of the spin viatransport of a spin-polarization cloud through the helical edge states. Thecloud is formed at a distant edge site using a local magnetic field to injectan electron spin density and released by suddenly switching off the injectionfield. This basic process, consisting of spin injection, propagation of thespin-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 theeffect of a single basic process can be reverted to a high degree with asubsequent process. Furthermore, we show that by concatenating several basicinjection-propagation-scattering processes, the spin state can be switchedcompletely and that a full reversal can be achieved.

Controlling the real-time dynamics of a spin coupled to the helical edge states of the Kane-Mele model