Phase space Langevin equation for spin relaxation in a dc magnetic field

A Langevin equation for the quantum Brownian motion of a spin of arbitrary size in a uniform external dc magnetic field is derived from the phase space master equation in the weak coupling and narrowing limits, for the quasiprobability distribution (Wigner) function of spin orientations in the configuration space of polar and azimuthal angles following methods long familiar in quantum optics. The closed system of differential-recurrence equations for the statistical moments describing magnetic relaxation of the spin is obtained as an example of applications of this equation. Copyright c © EPLA, 2009 Phase space representations of quantum-mechanical evolution equations [1,2] (generally based on the coherent state representation of the density matrix introduced by Glauber and Sudarshan as widely used in quantum optics [3,4]) when applied to spins (e.g., [5–10]) enable one to study spin relaxation using a master equation. This equation governs the evolution of the quasiprobability distribution function W (θ, φ, t) of spin orientations in the relevant phase (here configuration) space (θ, φ), where θ and φ are the polar and azimuthal angles, constituting the canonical variables. Mapping of quantum spin dynamics onto c-number master equations for W (θ, φ, t) exemplifies how these equations reduce to the Fokker-Planck equation governing the rotational Brownian motion of an assembly of classical spins when the spin number S →∞ [6,7,10]. Phase space distribution functions for spins originally introduced by Stratonovich [11] for closed systems have been extensively developed for both closed and open spin systems (e.g., [12–15]) and are entirely analogous to the Wigner distribution W (x, p, t) for the translational motion of a particle in the phase space of positions and momenta (x, p) [1,2]. The Wigner function W (θ, φ, t) analogous (a)E-mail: [email protected] to W (x, p, t) enables the expected value 〈Â〉(t) of a spin operator  to be determined using the corresponding c-number (or Weyl symbol) A(θ, φ), viz, [12] 〈Â〉(t) = 2S+1 4π ∫ π