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Accuracy of Second Order Perturbation theory in the Polaron and Variational Polaron Frames

In the study of open quantum systems, the polaron transformation has recently attracted a renewed interest as it offers the possibility to explore the strong system-bath coupling regime. Despite this interest, a clear and unambiguous analysis of the regimes of validity of the polaron transformation is still lacking. Here we provide such a benchmark, comparing second order perturbation theory results in the original untransformed frame, the polaron frame and the variational extension with numerically exact path integral calculations of the equilibrium reduced density matrix. Equilibrium quantities allow a direct comparison of the three methods without invoking any further approximations as is usually required in deriving master equations. It is found that the second order results in the original frame are accurate for weak system-bath coupling, the full polaron results are accurate in the opposite regime of strong coupling, and the variational method is capable of interpolating between these two extremes. As the bath becomes more non-Markovian (slow bath), all three approaches become less accurate.