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Journal of Physics D Applied Physics

Phase space description of spin dynamics

Yuri P. Kalmykov, William T. Coffey, С. В. Титов

Published Jun 19, 2008• 4 citations
Publisher

Abstract

The spin system with Hamiltonian (ξ and σ are external and internal field parameters) is treated as an example of the phase space description of spin dynamics using a master equation for the quasiprobability distribution function of spin orientations in the representation (phase) space of the polar angles (analogous to the Wigner phase space distribution for translational motion). The master equation yields (via the Wigner–Stratonovich transformation of the density matrix) the solution as a Fourier series in the spherical harmonics with Fourier coefficients given by the statistical moments in a manner analogous to the classical distribution. The relaxation time of the longitudinal component of the spin can be estimated using a quantum generalization of the classical integral relaxation time via the stationary distribution and diffusion coefficient of the master equation.

DOI
10.1088/0022-3727/41/13/134005

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The spin system with Hamiltonian (ξ and σ are external and internal field parameters) is treated as an example of the phase space description of spin dynamics using a master equation for the quasiprobability distribution function of spin orientations in the representation (phase) space of the polar angles (analogous to the Wigner phase space distribution for translational motion). The master equation yields (via the Wigner–Stratonovich transformation of the density matrix) the solution as a Fourier series in the spherical harmonics with Fourier coefficients given by the statistical moments in a manner analogous to the classical distribution. The relaxation time of the longitudinal component of the spin can be estimated using a quantum generalization of the classical integral relaxation time via the stationary distribution and diffusion coefficient of the master equation.
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.
The Fourier series representation of the equilibrium quasiprobability density function ${W}_{S}(\ensuremath{\vartheta},\ensuremath{\varphi})$ or Wigner function of spin ``orientations'' for arbitrary spin Hamiltonians in a representation (phase) space of the polar angles $(\ensuremath{\vartheta},\ensuremath{\varphi})$ (analogous to the Wigner function for translational motion) arising from the generalized coherent state representation of the density operator is evaluated explicitly for some nonaxially symmetric problems including a uniaxial paramagnet in a transverse external field, a biaxial, and a cubic system. It is shown by generalizing transition state theory to spins [i.e., calculating the escape rate using the equilibrium density function ${W}_{S}(\ensuremath{\vartheta},\ensuremath{\varphi})$ only] that one may evaluate the reversal time of the magnetization. The quantum corrections to the transition state theory escape rate equation for classical magnetic dipoles appear both in the prefactor and in the exponential part of the escape rate and exhibit a marked dependence on the spin number. Furthermore, the phase-space representation allows us to estimate the switching field curves and/or surfaces for spin systems because quantum effects in these fields can be estimated via Thiaville's geometrical method [Phys. Rev. B 61, 12221 (2000)] for the study of the magnetization reversal of single-domain ferromagnetic particles. The calculation is accomplished (just as the determination of the equilibrium quasiprobability distributions in the phase space of the polar angles) by calculating switching field curves and/or surfaces using the Weyl symbol (ccc-number representation) of the Hamiltonian operator for given magnetocrystalline-Zeeman energy terms. Examples of such calculations for various spin systems are presented. Moreover, the reversal time of the magnetization allows us to estimate thermal effects on the switching fields for spin systems.
The quantum Brownian motion of a particle in a periodic potential V ( x )=− V 0 cos( x / x 0 ) is treated using the master equation for the Wigner distribution function W ( x , p , t ) in phase space ( x , p ). Explicit equations for the diffusion coefficients of the master equation for this dissipative quantum system are derived. The dynamic structure factor S ( k , t ) and escape rate Γ are evaluated using matrix continued fractions. The matrix continued fraction calculation for Γ agrees well with the analytical solution of the corresponding quantum Kramers turnover problem.
The Weyl-Wigner transformation enables us to construct a representation of quantum motion equations using functions in phase space. The states of the system are determined by the quasiprobability distribution in phase space and the motion is described by an orthogonal integral operator. This formalism is employed in the study of the classical meaning of the discrete symmetry groups, in the problem of defining the current probability and in the proving of the expression of cross section from the S-matrix.
A rigorous quantum theory is developed on the nonlinear spin relaxation process. With the use of the phase space method, the quantum mechanical equation for a density matrix is mapped onto a c -number space. A quasi-probability function is then expanded in terms of spherical harmonics. It is clarified that the nonlinear relaxation process is characterized by 2 J ( J being the magnitude of a spin) different relaxation times. For J =1, time-evolutions of the first moment, fluctuation and probability function are exactly solved. An enhancement of the fluctuation is observed in the course of time development: This is similar to the phenomenon called “anomalous fluctuation” found in certain nonequilibrium systems.
The concept of phase-space amplitudes for systems with continuous degrees of freedom is generalized to finite-dimensional spin systems. Complex amplitudes are obtained on both a sphere and a finite lattice, in each case enabling a more fundamental description of pure spin states than that previously given by Wigner functions. In each case the Wigner function can be expressed as the star product of the amplitude and its conjugate, so providing a generalized Born interpretation of amplitudes that emphasizes their more fundamental status. The ordinary product of the amplitude and its conjugate produces a (generalized) spin Husimi function. The case of spin-12\frac{1}{2}21​ is treated in detail, and it is shown that phase-space amplitudes on the sphere transform correctly as spinors under rotations, despite their expression in terms of spherical harmonics. Spin amplitudes on a lattice are also found to transform as spinors. Applications are given to the phase space description of state superposition, and to the evolution in phase space of the state of a spin-12\frac{1}{2}21​ magnetic dipole in a time-dependent magnetic field.