In this section, we discuss the PEN problem in two different physical conditions employing low-dimensional model Hamiltonians. The system comprises one nuclear mode and two singlet electronic states, S 0 and S 1 , and it is coupled to a single-mode cavity. Model A represents a typical nonadiabatic situation with the adiabatic PESs forming an avoided crossing at some nuclear geometries, and the cavity is in resonance with the S 0 -to-S 1 transition at the Franck-Condon point. In this case, we will study the effect of the cavity on the relaxation process of the initially photoexcited system. Model B is constructed to induce the spontaneous photon emission of the system from S 1 to S 0 , followed by photoexcitation due to the absorption of the emitted photon. In this case, the cavity is in resonance with the S 0 -to-S 1 energy gap at the minimum of the S 1 PES.