Spin-orbit coupling (SOC), which is inherent to a Dirac particle that moves under the influence of electromagnetic fields, manifests itself in a variety of physical systems including non-relativistic ones. For instance, it plays an essential role in spintronics developed in the past few decades, particularly by controlling spin current generation and relaxation. In the present work, by using an extended Caldeira-Leggett model, we elucidate how the interplay between spin relaxation and momentum dissipation of an open system of a single spin-$1/2$ particle with a Rashba type SOC is induced by the interactions with a spinless, three-dimensional environment. Staring from the path integral formulation for the reduced density matrix of the system, we have derived a set of coupled nonlinear equations that consists of a quasi-classical Langevin equation for the momentum with a frictional term and a spin precession equation. The spin precesses around the effective magnetic field generated by both the SOC and the frictional term. It is found from analytical and numerical solutions to these equations that a spin torque effect included in the effective magnetic field causes a spin relaxation and that the spin and momentum orientations after a long time evolution are largely controlled by the Rashba coupling strength. Such a spin relaxation mechanism is qualitatively different from, e.g., the one encountered in semiconductors where essentially no momentum dissipation occurs due to the Pauli blocking.
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