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

TL;DR

This paper analyzes the efficiency of a quantum two-level atomic refrigerator operating under finite-time conditions, deriving analytical expressions for the COP at maximum figure of merit and comparing it with established theoretical bounds.

Contribution

It provides an analytical derivation of the COP at maximum  for a quantum refrigerator model, confirming its relation to the Curzon-Ahlborn COP and established thermodynamic bounds.

Findings

COP at maximum  matches Curzon-Ahlborn COP .

Analytical expression for  in high-temperature limit .

Upper bound of  confirmed within linear irreversible thermodynamics.

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) $\varepsilon$. 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.