We report an analytical model to investigate an atom-cavity interaction phenomenon with the consideration of a two-level atom and a quantized mode of the cavity field in the form of harmonic oscillation. Accordingly, we consider the matrix form of a special type of two-level Hamiltonian consisting of three units, explaining a two-level atom, a quantum harmonic oscillator to be equivalent to a quantized cavity-mode field, and an interaction matrix between them, which are connected by some iteration parameters. The iteration parameters of the proposed Hamiltonian have judiciously been modulated to encounter an exceptional point (EP). A specific closed parameter space encircling the embedded EP has been considered to study the dynamics of the coupled states in the context of a flip-of-state phenomenon. We also incorporate a deliberate phase difference between the two unperturbed levels associated with the atomic system and study the phase induced interaction between them in the context of state-flipping around the identified EP. The proposed analytical model indeed opens up a fertile platform to explore a range of interesting applications associated with the atom-cavity interaction. © 2021 IOP Publishing Ltd.