Spin relaxation in phase space

TL;DR

This paper demonstrates how Wigner's phase-space formulation simplifies spin relaxation problems by using master equations similar to classical Fokker-Planck equations, enabling explicit solutions and computational techniques for quantum spins.

Contribution

It introduces a phase-space approach to spin relaxation that avoids operators, providing explicit solutions and extending classical Fokker-Planck methods to quantum spins.

Findings

Master equations resemble classical Fokker-Planck equations.

Explicit solutions via spherical harmonics expansion.

Potential for applying classical computational techniques to quantum spins.

Abstract

We have treated numerous illustrative examples of spin relaxation problems using Wigner's phase-space formulation of quantum mechanics of particles and spins. The merit of the phase space formalism as applied to spin relaxation problems is that only master equations for the phase-space distributions akin to Fokker-Planck equations for the evolution of classical phase-space distributions in configuration space are involved so that operators are unnecessary. The explicit solution of these equations can be expanded for an arbitrary spin Hamiltonian in a finite series of spherical harmonics like in the classical case. The expansion coefficients (statistical moments or averages of the spherical harmonics which are obviously by virtue of the Wigner-Stratonovich map the averages of the polarization operators) may be determined from differential-recurrence relations in a manner similar to the classical case. Furthermore, the phase space representation via the Weyl symbols of the relevant spin operators suggests how powerful computation techniques developed for Fokker-Planck equations (matrix continued fractions, mean first passage time, etc.) may be transparently extended to the quantum domain.