Coherence and decoherence in quantum absorption refrigerators

Absorption refrigerators transfer thermal energy from a cold reservoir to a hot reservoir using input energy from a third, so-called work reservoir. We examine the operation of quantum absorption refrigerators when coherences between eigenstates survive in the steady state limit. In our model, the working medium comprises a discrete, four-level system. Several studies on related setups have demonstrated the performance-enhancing potential of steady-state eigenbasis quantum coherences. By contrast, in our model such coherences generally quench the cooling current in the refrigerator, while minimally affecting the coefficient of performance (cooling efficiency). We rationalize the behavior of the four-level refrigerator by studying three-level model systems for energy transport and refrigeration. Our calculations further illuminate the shortcomings of secular quantum master equations and the necessity of employing dynamical equations of motion that retain couplings between population and coherences.

Figure 1

Diagram of the VETS in the local site basis (left) and in the energy basis (right). Dashed arrows represent interactions with the hot ($h$) and cold ($c$) bath. The dotted arrow depicts the effect of the decoherence ($d$) bath. The baths' induced rate constants, denoted by ${\mathrm{\Gamma }}_h,{\mathrm{\Gamma }}_c$, and ${\mathrm{\Gamma }}_d$, are defined in Sec. 3.

Figure 2

Diagram of a three-level QAR. The dashed arrows represent interactions with the hot ($h$), cold ($c$), and work ($w$) baths with rate constants ${\mathrm{\Gamma }}_{h,c,w}$.

Figure 3

Diagram of the four-level QAR in the local-site basis (left) and energy basis (right). The dashed arrows represent interactions with the hot, cold, and work baths. The dotted arrow shows the effect of the decoherence bath.

Figure 4

Heat current in the VETS model. (a–d) Solution of the Redfield equation with an increasing decoherence strength between the excited states. (e) When decoherence is sufficiently strong, we retrieve the secular behavior. $\gamma \equiv {\gamma }_{h,c}=0.002$, ${\omega }_c=50$, $T_h=0.15$, $T_c=0.1$, $T_d=0.12$.

Figure 5

(a) Contour plot of current in the VETS model as a function of the intersite coupling $g$ and decoherence strength ${\gamma }_d$. (b) Demonstration of a nonmonotonic $J_q$ vs. ${\gamma }_d$ behavior. $\gamma \equiv {\gamma }_{h,c}=0.002$, ${\omega }_c=50$, $T_h=0.15$, $T_c=0.1$, $T_d=0.12$, $\theta =0.5$.

Figure 6

Heat current in the VETS model as a function of the intersite coupling $g$. Parameters are the same as in Fig. 4, with $\theta =1$, $\gamma \theta =0.002$.

Figure 7

(a) Current, (b) population, and (c) coherences in the VETS model at ${\gamma }_d=0$. Parameters are the same as in Fig. 6.

Figure 8

Cooling current in the 4lQAR. The white (empty) spaces are the no-cooling regions. (a–d) Solution of the Redfield equation with an increasing decoherence rate between the intermediate states. (e) When ${\gamma }_d$ is sufficiently strong, we retrieve the secular behavior. $\gamma \equiv {\gamma }_{c,h,w}=0.002$, ${\omega }_c=50$, $T_w=1$, $T_h=0.15$, $T_c=0.1$, $T_d=0.12$, ${\theta }_h=1$.

Figure 9

COP of the 4lQAR for ${\gamma }_d=0$ and ${\gamma }_d/\gamma =100$ (overlapping). Parameters are the same as in Fig. 8. The dashed line corresponds to the Carnot bound for a cooling machine, ${\eta }_C=\frac{{\beta }_h-{\beta }_w}{{\beta }_c-{\beta }_h}$, which equals 1.7 in our parameters.

Figure 10

(a) COP vs. cooling power for the 4lQAR at ${\gamma }_d=0$ while increasing $g$ from ${10}^{-3}$ to 0.4 as we move inward. The black dashed line corresponds to the secular limit at vanishing $g$, showing an endoreversible operation. To generate these plots, the frequency ${\theta }_c$ was varied within the cooling window. (b) Maximum cooling current and (c) COP at that value, optimized at every point with respect to ${\theta }_c$. Parameters are the same as in Fig. 8.

Figure 11

Schemes of nanoscale energy conversion devices that can be captured with our generic models. (a) The VETS model can be realized by a conducting junction with two degenerate electronic states. (b) The 3lQAR may correspond to a nondegenerate electronic junction, with the work bath responsible for internal transitions. (c) The 4lQAR can represent a photovoltaic cell. Heat absorbed from the work reservoir is used to transfer carriers—thus energy—from the cold metal to the hot metal. The full arrows represent energy exchange processes with the hot, cold and work heat baths. The effect of the decoherence bath is omitted for simplicity, but it can be further included.