  We rederive the dressed atom and the exciton polariton within the {\it same}
framework to make clear that their difference only comes from the number of
electrons available for photoexcitations. Using it, we analytically show how
the time dependence of the photon number transforms from an oscillating
behavior (at the stimulated or vacuum Rabi frequency) to an exponential decay
(identical for atom and semiconductor) when the excited state lifetime
decreases. Although the matter ground state is in both cases coupled by
monochromatic photons {\it not to a continuum but to a discrete state}, this
decay yet follows a kind of Fermi Golden rule. The energy conservation it
contains, is however conceptually different.
