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In the study of open quantum systems, the polaron transformation has recentlyattracted a renewed interest as it offers the possibility to explore the strongsystem-bath coupling regime. Despite this interest, a clear and unambiguousanalysis of the regimes of validity of the polaron transformation is stilllacking. Here we provide such a benchmark, comparing second order perturbationtheory results in the original untransformed frame, the polaron frame and thevariational extension with numerically exact path integral calculations of theequilibrium reduced density matrix. Equilibrium quantities allow a directcomparison of the three methods without invoking any further approximations asis usually required in deriving master equations. It is found that the secondorder results in the original frame are accurate for weak system-bath coupling,the full polaron results are accurate in the opposite regime of strongcoupling, and the variational method is capable of interpolating between thesetwo extremes. As the bath becomes more non-Markovian (slow bath), all threeapproaches become less accurate.

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