We investigate the finite-time performance of a quantum endoreversible Carnot engine cycle and its inverse operation-Carnot refrigeration cycle, employing a spin-$1/2$ system as the working substance. The thermal machine is alternatively driven by a hot boson bath of inverse temperature $\beta_h$ and a cold boson bath at inverse temperature $\beta_c(>\beta_h)$. While for engine model the hot bath is constructed to be squeezed, in the refrigeration cycle the cold bath is established to be squeezed, with squeezing parameter $r$. We obtain the analytical expressions both for efficiency and power in heat engines and for coefficient of performance and cooling rate in refrigerators. We find that, in the high-temperature limit, the efficiency at maximum power is bounded by the analytical value, $\eta_+=1-\sqrt{{\rm{sech}}(2r)(1-\eta_C)}$, and the coefficient of performance at the maximum figure of merit is limited by $ \varepsilon_+=\frac{\sqrt{{\rm{sech}}(2r)(1+\varepsilon_C})}{\sqrt{{\rm{sech}}(2r)(1+\varepsilon_C)-\varepsilon_C}}-1$, where $\eta_C=1-\beta_h/\beta_c$ and $\varepsilon_C=\beta_h/(\beta_c-\beta_h)$ are the respective Carnot values of the engines and refrigerators, and $r$ is the squeezing parameter. These analytical results are identical to corresponding those obtained from the Carnot engines based on harmonic systems, indicating that the efficiency at maximum power and coefficient at maximum figure of merit are independent on the working substance.
We consider a quantum endoreversible Carnot engine cycle and its inverse operation–Carnot refrigeration cycle, working between a hot bath of inverse temperature [Formula: see text] and a cold bath at inverse temperature [Formula: see text]. For the engine model, the hot bath is constructed to be squeezed, whereas for the refrigeration cycle, the cold bath is set to be squeezed. In the high-temperature limit, we analyze efficiency at maximum power and coefficient of performance at maximum figure of merit, revealing the effects of the times allocated to two thermal-contact and two adiabatic processes on the machine performance. We show that, when the total time spent along the two adiabatic processes is negligible, the efficiency at maximum power reaches its upper bound, which can be analytically expressed in terms of squeezing parameter [Formula: see text]: [Formula: see text], with the Carnot efficiency [Formula: see text] and the coefficient of performance at maximum figure of merit is bounded from the upper side by the analytical function: [Formula: see text], where [Formula: see text].
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