Completely Positive, Simple, and Possibly Highly Accurate Approximation of the Redfield Equation

Dragomir Davidovic

Published 2020-03-20Version 1

Here we present a Markovian master equation that approximates the Redfield equation, a well known non-Markovian master equation derived from first principles, without significantly compromising the range of applicability of the Redfield equation. Instead of full-scale coarse-graining, this approximation only truncates terms in the Redfield equation that average out over a time-scale typical of the heat bath. The key to this approximation is to properly renormalize the system Hamiltonian, followed by the Markovian arithmetic geometric mean approximation (MAGMA), that restores complete positivity of the master equation. Three applications of the Markovian master equation are presented and demonstrate its simplicity and accuracy. In an exactly solvable quantum dynamical problem of a three-level system, we find that the error of the approximate state is almost an order of magnitude lower than that obtained by solving the coarse-grained stochastic master equation. In the example of a four-level system, we find that the Markovian master equation is more accurate than the non-Markovian Redfield equation that it emulates. It is also shown that the Markovian master equation retains a near Born-Markov accuracy when applied to a complex many-body system, a ferromagnetic spin-1/2 chain with long-range, dipole-dipole interactions between 25 spins.