This comparison leads to some deviations between the pQED-TD-DFT and scQED-TD-DFT schemes. Here, however, the pQED Hamiltonian was treated with a multi-state generalization of Jaynes-Cummings model Hamiltonian (Eq. 83) while the sc-QED was treated with Pauli-Fierz QED Hamiltonian (Eq. 106). As such, the deviation might due to the use of different QED Hamiltonians.