We propose a method for deriving Lindblad-like master equations when the environment/reservoir is consigned to a classical description. As a proof of concept, we apply the method to continuous wave (CW) magnetic resonance. We make use of a perturbation scheme we have termed "affine commutation perturbation" (ACP). Unlike traditional perturbation methods, ACP has the advantage of incorporating some effects of the perturbation even at the zeroth-order approximation. Indeed, we concentrate here on the zeroth-order, and show how -- even at this lowest order -- the ACP scheme can still yield non-trivial and equally important results. In contradistinction to the purely quantum Markovian master equations in the literature, we explicitly keep the term linear in the system-environment interaction -- at all orders of the perturbation. At the zeroth-order, we show that this results in a dynamics whose map is non-CP (Completely Positive) but approaches asymptotically a CP map as
t→+∞. We also argue that this linear term accounts for the linear response of the system to the presence of the environment -- thus the harbinger for a linear response theory (LRT) within the confines of such (semiclassical) Lindblad-like master equations. The adiabatic process limit of the dynamics is also defined, and considerably explored in the context of CW magnetic resonance. Here, the same linear term emerges as the preeminent link between standard (adiabatic process) LRT (as formulated by Kubo and co.) and Lindblad-like master equations. And with it, we show how simple stick-plot CW magnetic resonance spectra of multispin systems can be easily generated under certain conditions.
We consider a system perturbed by an external field and subject to dissipative processes. From the von Neumann equation for such a system in the weak coupling limit we derive an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig’s projection operator technique in Liouville space. From this equation the response function, as well as expressions for the generalized conductivity and susceptibility, is obtained. It is shown that for large times only the diagonal part of the density operator is required. The various expressions are found to be in complete harmony with previous results (Part I) obtained via the van Hove limit of the Kubo–Green linear response formulas. In order to account for the properties at quantum frequencies, the evolution of the nondiagonal part in the weak coupling limit is also established. The complete time dependent behavior of the dynamic variables in the van Hove limit is expressed by B (t) =exp[−(Λd−iℒ0) t] B, where Λd is the master operator and L0 the Liouville operator in the interaction picture. The cause of irreversibility is discussed. Finally, the inhomogeneous master equation is employed to obtain as first moment equation a Boltzmann equation with streaming terms, applicable to quantum systems.