  We benchmark a selection of semiclassical and perturbative dynamics
techniques by investigating the correlated evolution of a cavity-bound atomic
system to assess their applicability to study problems involving strong
light-matter interactions in quantum cavities. The model system of interest
features spontaneous emission, interference, and strong coupling behaviour, and
necessitates the consideration of vacuum fluctuations and correlated
light-matter dynamics. We compare a selection of approximate dynamics
approaches including fewest switches surface hopping, multi-trajectory
Ehrenfest dynamics, linearized semiclasical dynamics, and partially linearized
semiclassical dynamics. Furthermore, investigating self-consistent perturbative
methods, we apply the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy in the
second Born approximation. With the exception of fewest switches surface
hopping, all methods provide a reasonable level of accuracy for the correlated
light-matter dynamics, with most methods lacking the capacity to fully capture
interference effects.
