Non-Markovian Second-Order Quantum Master Equation and Its Markovian Limit: Electronic Energy Transfer in Model Photosynthetic Systems

TL;DR

This paper introduces a numerical algorithm for solving non-Markovian quantum master equations in photosynthetic models, analyzing the validity of Markovian approximations and their impact on energy transfer dynamics.

Contribution

The paper presents an efficient auxiliary function method for solving non-Markovian equations and assesses the validity of Markovian approximations in photosynthetic energy transfer models.

Findings

Redfield theory is inaccurate for λ > 10 cm^{-1} in FMO systems.

Analytic inequalities define the validity regimes of Markov approximation.

Initial coherence influences the evolution of the dimer system.

Abstract

A direct numerical algorithm for solving the time-nonlocal non-Markovian master equation in the second Born approximation is introduced and the range of utility of this approximation, and of the Markov approximation, is analyzed for the traditional dimer system that models excitation energy transfer in photosynthesis. Specifically, the coupled integro-differential equations for the reduced density matrix are solved by an efficient auxiliary function method in both the energy and site representations. In addition to giving exact results to this order, the approach allows us to computationally assess the range of the reorganization energy and decay rates of the phonon auto-correlation function for which the Markovian Redfield theory and the second order approximation is valid. For example, the use of Redfield theory for $\lambda> 10 \textrm{cm}^{-1}$ in systems like Fenna-Mathews-Olson (FMO) type systems is shown to be in error. In addition, analytic inequalities are obtained for the regime of validity of the Markov approximation in cases of weak and strong resonance coupling, allowing for a quick determination of the utility of the Markovian dynamics in parameter regions. Finally, results for the evolution of states in a dimer system, with and without initial coherence, are compared in order to assess the role of initial coherences.