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

Abstract

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) $\epsilon$. The COP at maximum $\chi (=\epsilon R)$ figure of merit is numerically determined, and it is proved to be in nice agreement with the so-called Curzon and Ahlborn COP ${\epsilon }_{CA}=\sqrt{1+{\epsilon }_C}-1$, where ${\epsilon }_C=T_c/(T_h-T_c)$ is the Carnot COP. In the high-temperature limit, the COP at maximum $\chi$ figure of merit, ${\epsilon }^{\ast }$, can be expressed analytically by ${\epsilon }^{\ast }={\epsilon }_{+}\equiv (\sqrt{9+8{\epsilon }_C}-3)/2$, 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 ${\epsilon }_{+}$ is also the upper bound of COP at maximum $\chi$ figure of merit when we regard our model as a linear irreversible refrigerator.

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