A two-level atomic system as a working substance is used to set up a refrigerator consisting of two quantum adiabatic and two isochoric processes (two constant-frequency processes ${\omega }_a$ and ${\omega }_b$ with ${\omega }_a<{\omega }_b)$, during which the two-level system is in contact with two heat reservoirs at temperatures $T_h$ and $T_c(<T_h)$. Considering finite-time operation of two isochoric processes, we derive analytical expressions for cooling rate $R$ and coefficient of performance (COP) $\varepsilon$. The COP at maximum $\chi (=\varepsilon R)$ figure of merit is numerically determined, and it is proved to be in nice agreement with the so-called Curzon and Ahlborn COP ${\varepsilon }_{\text{CA}}=\sqrt{1+{\varepsilon }_C}-1$, where ${\varepsilon }_C=T_c/(T_h-T_c)$ is the Carnot COP. In the high-temperature limit, the COP at maximum $\chi$ figure of merit, ${\varepsilon }^{*}$, can be expressed analytically by ${\varepsilon }^{*}={\varepsilon }_{+}\equiv \left(\sqrt{9+8{\varepsilon }_C}-3\right)/2$, which was derived previously as the upper bound of optimal COP for the low-dissipation or minimally nonlinear irreversible refrigerators. Within the context of irreversible thermodynamics, we prove that the value of ${\varepsilon }_{+}$ is also the upper bound of COP at maximum $\chi$ figure of merit when we regard our model as a linear irreversible refrigerator.

• Received 20 May 2014
• Revised 21 August 2014

DOI:https://doi.org/10.1103/PhysRevE.90.052151