The correlation between coherent driving and non-Markovian dissipation plays a vital role in optical processes. To exhibit its effect on the simulation of optical spectroscopy, we explore the correlated driving-dissipation equation (CODDE) [R. X. Xu and Y. J. Yan, J. Chem. Phys. 116, 9196 (2002)], which modifies the conventional Redfield theory with the inclusion of correlated driving-dissipation effect at the second-order system-bath coupling level. With an exciton model mimicking the Fenna-Matthews-Olson pigment-protein complex, we compare between the Redfield theory, CODDE, and exact hierarchical dynamics, for their results on linear absorption and coherent two-dimensional spectroscopy. We clarify that the failure of Redfield approach originates mainly from the neglect of driving-dissipation correlation, rather than its second-order nature. We further propose a dynamical inhomogeneity parameter to quantify the applicable range of CODDE. Our results indicate that CODDE is an efficient and quantifiable theory for many light-harvesting complexes of interest. To facilitate the evaluation of multi-dimensional spectroscopy, we also develop the mixed Heisenberg-Schr?dinger picture scheme that is valid for any dynamics implementation on nonlinear response functions.
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