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

Abstract

Here we present a Lindblad master equation that approximates the Redfield equation, a well known 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 quantum system. The first step in this approximation is to properly renormalize the system Hamiltonian, to symmetrize the gains and losses of the state due to the environmental coupling. In the second step, we swap out an arithmetic mean of the spectral density with a geometric one, in these gains and losses, thereby restoring complete positivity. This completely positive approximation, GAME (geometric-arithmetic master equation), is adaptable between its time-independent, time-dependent, and Floquet form. In the exactly solvable, three-level, Jaynes-Cummings model, 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. As a test-bed, we use a ferromagnetic Heisenberg spin-chain with long-range dipole-dipole coupling between up to 25-spins, and study the differences between various master equations. We find that GAME has the highest accuracy per computational resource.

Environmentally Induced Entanglement – Anomalous Behavior in the Adiabatic Regime

“…However, the suitability of these perturbative approaches has to be checked since an error estimation from within the perturbative formalism seems unfeasible. With the help of the exact dynamics obtained by the HOPS method we examine the applicability of the QOME (Born-Markov approximation and RWA) [33,35], its variation with only a partial rotating wave approximation (PRWA) [48][49][50], the Redfield equation (RFE) (no RWA at all) [51], the very recent geometric-arithmetic master equation (GAME) (GKSL kind equation based on the RFE) [52] and the coarse-graining approach [39,40,53], in the context of two independent qubits coupled to a common environment with sub-Ohmic SD.…”

“…In a recent publication the failure to quantify entanglement of a non-positive state in an approximative sense has been addressed as well [52]. The proposed GAME modifies the RFE such that it becomes a master equation of GKSL type.…”

A thermodynamically consistent Markovian master equation beyond the secular approximation

“…To arrive at the master equation in equation ( 66), we dropped terms that oscillate with frequencies much larger than 1/τ S . Unfortunately, these terms may be important when evaluating the standard expression for external power given in equation (33). Loosely speaking, the oscillations of the density matrix cancel oscillations of ∂ t ĤS (t), thereby contributing to the average power.…”

“…Note that the two thermodynamic Hamiltonians correspond to two possible choices for Ĥav . In this scenario, the external power can no longer be accessed by the standard expression given in equation (33). As a consequence of the secular approximation, this quantity evaluates to zero [13].…”

“…Recently, a novel approach termed PERLind (position and energy resolved Lindblad equation) was introduced [29] in an attempt to interpolate between the local and global approach. Since then, multiple microscopic derivations were given [30][31][32][33], showing that the PERLind approach does not require any additional assumptions going beyond the ones already present when performing Born-Markov approximations. Compared to the global and local approaches, the PERLind approach thus has an increased regime of validity, making it a very promising approach.…”