The perturbative master equation (Bloch–Redfield) is used extensively to study dissipative quantum mechanics—particularly for qubits—despite the 25-year-old criticism that it violates positivity (generating negative probabilities). We take an arbitrary system coupled to an environment containing many degrees-of-freedom and cast its perturbative master equation (derived from a perturbative treatment of Nakajima–Zwanzig or Schoeller–Schön equations) in the form of a Lindblad master equation. We find that the equation's parameters are time dependent . This time dependence is rarely accounted for and invalidates Lindblad's dynamical semigroup analysis. We analyse one such Bloch–Redfield master equation (for a two-level system coupled to an environment with a short but non-vanishing memory time), which apparently violates positivity. We analytically show that, once the time dependence of the parameters is accounted for, positivity is preserved.
Master equations play a pivotal role in investigating open quantum systems. In particular, the Bloch-Redfield equation stands out due to its relation to a concrete physical environment. However, without further approximations it does not lead to a Lindblad master equation that guarantees that the density matrix stays completely positive, which has raised some concerns regarding its use. This study builds on previous efforts to transform the Bloch-Redfield framework into a mathematically robust Lindblad equation while fully preserving the effects that are lost within the secular approximation that is commonly used to guarantee positivity. These previous approaches introduced two potential deficiencies: The environment-induced energy shift can be non-Hermitian and some decay rates can be negative, violating the assumptions of Lindblad's theorem. Here we propose and evaluate straightforward solutions to both problems. Our approach offers an effective and general procedure for obtaining a Lindblad equation, derived from a concrete physical environment, while mitigating the unphysical dynamics present in the Bloch-Redfield equation.
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.