We rederive the dressed atom and the exciton polariton within the same framework to make clear that their major difference only comes from the number of electrons available for photoexcitations. This allows to analytically show how the decrease of the excited-state lifetime transforms the photon number time evolution from oscillations - at the stimulated Rabi frequency for an atom and the vacuum one for semiconductors - to an exponential decay identical for both. Although the matter ground state is in both cases coupled not to a continuum but to a discrete state, broadened by lifetime, this decay yet follows a Fermi golden rule, however having a different scale for the energy conservation. Our work actually provides a clean justification for using the golden rule for exciton absorption.