This study presents a comprehensive analysis of a quantum Stirling engine whose working substance is a two-level atom coupled to a single-mode optical cavity within the Jaynes–Cummings model. The non-extensive Tsallis entropy formalism, parameterized by \(q\) , is employed to investigate the influence of strong quantum correlations and non-Markovian effects. The performance metrics, including work output (W), thermodynamic efficiency (η), and the coefficient of performance (ϵ) for refrigeration, are rigorously evaluated as functions of the atom–field coupling ratio g2/g1, the bath temperature gradient Th/Tc, and the non-extensivity parameter q. It is demonstrated that a reduction in q, signifying stronger non-extensivity, systematically enhances all performance indicators. The engine efficiency and work output are found to increase with both the coupling strength and the temperature difference, revealing a synergistic quantum-thermodynamic effect. Notably, the coefficient of performance for refrigeration increases with the temperature gradient, a distinctive departure from classical behavior. Furthermore, the thermodynamic phase diagram of operational modes is shown to simplify under increased non-extensivity, with dissipative modes being suppressed in favor of stable engine and refrigerator operation.
