<p>We present a unified formulation of quantum spin relaxation in phase space based on the Wigner–Stratonovich transformation of the master equation for a uniaxial spin system in an external magnetic field. The reduced density matrix of a quantum spin of arbitrary magnitude S, interacting weakly with a thermal bath is mapped onto a quasiprobability distribution <i>W</i><sub><i>S</i></sub>(<i>ϑ</i>,<i>φ</i>,<i>t</i>) defined on the unit sphere. The resulting equation of motion is a quantum master equation that generalizes the classical Fokker–Planck equation for rotational diffusion in a potential. Analytical expressions are obtained for the stationary distribution and for the structure of the diffusion kernel, showing explicitly how quantum corrections appear as finite-series truncations in spherical harmonics. In the limit of large spin S→∞, the quantum master equation smoothly transforms into the classical rotational Fokker–Planck equation. The theory provides a transparent bridge between quantum and classical descriptions of spin relaxation and offers a practical route for evaluating quantum corrections to magnetization dynamics in nanoscale paramagnets.</p>

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Unified quantum - classical description of spin dynamics via the Wigner–Stratonovich transformation

  • Amina Benkaddour,
  • Bachir Ouari

摘要

We present a unified formulation of quantum spin relaxation in phase space based on the Wigner–Stratonovich transformation of the master equation for a uniaxial spin system in an external magnetic field. The reduced density matrix of a quantum spin of arbitrary magnitude S, interacting weakly with a thermal bath is mapped onto a quasiprobability distribution WS(ϑ,φ,t) defined on the unit sphere. The resulting equation of motion is a quantum master equation that generalizes the classical Fokker–Planck equation for rotational diffusion in a potential. Analytical expressions are obtained for the stationary distribution and for the structure of the diffusion kernel, showing explicitly how quantum corrections appear as finite-series truncations in spherical harmonics. In the limit of large spin S→∞, the quantum master equation smoothly transforms into the classical rotational Fokker–Planck equation. The theory provides a transparent bridge between quantum and classical descriptions of spin relaxation and offers a practical route for evaluating quantum corrections to magnetization dynamics in nanoscale paramagnets.