  Phonon interactions are inevitable in cavity quantum electrodynamical systems
based on solid-state emitters or fluorescent molecules, where vibrations of the
lattice or chemical bonds couple to the electronic degrees of freedom. Due to
the non-Markovian response of the vibrational environment, it remains a
significant theoretical challenge to describe such effects in a computationally
efficient manner. This is particularly pronounced when the emitter-cavity
coupling is comparable to or larger than the typical phonon energy range, and
polariton formation coincides with vibrational dressing of the optical
transitions. In this Article, we consider four non-Markovian perturbative
master equation approaches to describe such dynamics over a broad range of
light-matter coupling strengths and compare them to numerically exact reference
calculations using a tensor network. The master equations are derived using
different basis transformations and a perturbative expansion in the new basis
is subsequently introduced and analyzed. We find that two approaches are
particularly successful and robust. The first of these is suggested and
developed in this Article and is based on a vibrational dressing of the
exciton-cavity polaritons. This enables the description of distinct
phonon-polariton sidebands that appear when the polariton splitting exceeds the
typical phonon frequency scale in the environment. The second approach is based
on a variationally optimized polaronic vibrational dressing of the electronic
state. Both of these approaches demonstrate good qualitative and quantitative
agreement with reference calculations of the emission spectrum and are
numerically robust, even at elevated temperatures, where the thermal phonon
population is significant.
