Langevin methods for spin systems

Joachim A. Messer

Scott D. Linz, Jochen Gemmer

A system of spins coupled to a bath is a traditional setup in open quantum systems. Through Heisenberg's equation, the spin dynamics can be modeled by a set of first-order differential equations. Interpreting the terms as colored noise and non-Markovian damping, one can write them as quantummechanical Heisenberg-Langevin (HL) equations. These are notoriously difficult to solve because of the high dimensionality of the Hilbert space. Classical generalized Langevin equations, involving non-Markovian damping and colored noise, are well understood and can be treated numerically with relative ease. Thus, a classical ansatz can be made by substituting quantum expectation values with classical functions. This allows the application of standard methods developed for classical stochastic dynamical systems to tackle spin dynamics. However, this approach is uncontrolled and should be benchmarked against known quantum dynamics. In this investigation, a Hamiltonian for spin dynamics is modified to obtain a setup analogous to the Weisskopf-Wigner (WW) theory of spontaneous emission, enabling a comparison of the results. This will be compared for T = 0 and with a slight adaptation in the high-temperature limit.

Fumiaki Shibata, Natsuki Hashitsume

In this article is proposed a method of constructing the quantal Boltzmann-Langevin equation, i.e. the stochastic kinetic equation for a suitably chosen statistical operator with a fluctuating force term. This method is based on the use of the generalized phase-space method, which makes calculations very simple. As an example, the method of spin coherent states is applied to a model of a localized single spin placed under actions of its magnetic environment, and the Bloch-Langevin equation, i.e. the well-known Bloch equation for a spin magnetic moment with a fluctuating force term, is derived. The drift and the relaxation terms contained in the Boltzmann-Langevin and the Bloch-Langevin equations are completely in agreement with those previously obtained. An expression of the fluctuation-dissipation theorem generalized so as to include non-linear dissipation terms is given.

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

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.