We explore the thermal properties of a system including a single-mode cavity interacting with a two-level atom at temperature T under the effect of the Stark shift. Thus, the partition function and then the thermal density matrix of the atom-field system will be needed. Using the generators of the \(\varvec{su(1,1)}\) algebra, we first compute the total excitation number which commutes with the Hamiltonian of the system. Considering the total excitation number, the Hamiltonian is represented by a block diagonal matrix. We solve exactly our model and calculate the energy eigenvalues and corresponding eigenstates. We analyze the fidelity as numerical criterion of the closeness of the thermal atom-cavity system at two temperatures. The effect of the Stark shift coefficients on the fidelity can be studied. Moreover, we will examine the thermodynamical properties of the thermal atom-field system using the Helmholtz free energy, entropy, internal energy and heat capacity. The dependence of the temperature and Stark shift parameters on the thermodynamical properties will be discussed.
