In the present work, the problem of an all-coupling analytic description for the optical conductivity of the Froehlich polaron is treated, with the goal being to bridge the gap in validity range that exists between two complementary methods: on the one hand the memory function formalism and on the other hand the strong-coupling expansion based on the Franck-Condon picture for the polaron response. At intermediate coupling, both methods were found to fail as they do not reproduce Diagrammatic Quantum Monte Carlo results. To resolve this, we modify the memory function formalism with respect to the Feynman-Hellwarth-Iddings-Platzman (FHIP) approach, in order to take into account a non-quadratic interaction in a model system for the polaron. The strong-coupling expansion is extended beyond the adiabatic approximation, by including into the treatment non-adiabatic transitions between excited polaron states. The polaron optical conductivity that we obtain by combining the two extended methods agree well, both qualitatively and quantitatively, with the Diagrammatic Quantum Monte Carlo results in the whole available range of the electron-phonon coupling strength.
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