  We investigate nonequilibrium phenomena in magnetic nano-junctions using a
numerical approach that combines classical spin dynamics with the hierarchical
equations of motion technique for quantum dynamics of conduction electrons. Our
focus lies on the spin dynamics, where we observe non-monotonic behavior in the
spin relaxation rates as a function of the coupling strength between the
localized spin and conduction electrons. Notably, we identify a distinct
maximum at intermediate coupling strength, which we attribute to a competition
that involves the increasing influence of the coupling between the classical
spin and electrons, as well as the influence of decreasing local density of
states at the Fermi level. Furthermore, we demonstrate that the spin dynamics
of a large open system can be accurately simulated by a short chain coupled to
semi-infinite metallic leads. In the case of a magnetic junction subjected to
an external DC voltage, we observe resonant features in the spin relaxation,
reflecting the electronic spectrum of the system. The precession of classical
spin gives rise to additional side energies in the electronic spectrum, which
in turn leads to a broadened range of enhanced damping in the voltage.

Mesoscale and Nanoscale Physics