Coefficient of performance under maximum $\chi$ criterion in a two-level atomic system as a refrigerator

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 $\varepsilon_{CA}=\sqrt{1+\varepsilon_C}-1$, 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 $\varepsilon^* = \varepsilon_+ \equiv (\sqrt{9+8\varepsilon_C}-3)/2$, which was derived previously as the upper bound of optimal COP for the low-dissipation or minimally nonlinear irreversible refrigerators. Within 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.