Consistency of optimizing finite-time Carnot engines with the low-dissipation model in the two-level atomic heat engine

The efficiency at the maximum power (EMP) for finite-time Carnot engines established with the low-dissipation model, relies significantly on the assumption of the inverse proportion scaling of the irreversible entropy generation $\Delta S^{(\mathrm{ir})}$ on the operation time $\tau$, i.e., $\Delta S^{(\mathrm{ir})}\propto1/\tau$. The optimal operation time of the finite-time isothermal process for EMP has to be within the valid regime of the inverse proportion scaling. Yet, such consistency was not tested due to the unknown coefficient of the $1/\tau$-scaling. In this paper, we reveal that the optimization of the finite-time two-level atomic Carnot engines with the low-dissipation model is consistent only in the regime of $\eta_{\mathrm{C}}\ll2(1-\delta)/(1+\delta)$, where $\eta_{\mathrm{C}}$ is the Carnot efficiency, and $\delta$ is the compression ratio in energy level difference of the heat engine cycle. In the large-$\eta_{\mathrm{C}}$ regime, the operation time for EMP obtained with the low-dissipation model is not within the valid regime of the $1/\tau$-scaling, and the exact EMP of the engine is found to surpass the well-known bound $\eta_{+}=\eta_{\mathrm{C}}/(2 \eta_{\mathrm{C}})$.