  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.
