${\epsilon}_0+\Delta$, see Fig \[Fig:tophat-width\]. When we take $\Delta\big/({k_{\rm B}}T_{L,R}) \to 0$, one has Carnot efficiency, however we will see (leading order term in Eq. (\[Eq:Pgen-eng-lowpower\])) that $$\begin{aligned}
P_{\rm gen} \propto {1 \over \hbar} \Delta^2,
\label{Eq:power-for-Delta-to-zero}\end{aligned}$$ which vanishes as $\Delta\big/({k_{\rm B}}T_{L,R}) \to 0$.

Heat-engine efficiency at finite power output and Curzon-Ahlborn efficiency {#Sect:eff-CA}
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To increase the power output beyond that of a reversible system, one has to consider irreversible machines which generate a finite amount of entropy per unit of work generated. Curzon and Ahlborn[@Curzon-Ahlborn1975] popularized the idea of studying the efficiency of a heat-engine running at its maximum power output. For classical pumps, this efficiency is $\eta_{\rm eng}^{\rm CA} = 1- \sqrt{T_L/T_R}$, which is now called the Curzon-Ahlborn efficiency, although already given in Refs. \[\]. As refrigerators, these pumps have an efficiency at maximum cooling power of zero, although Refs. \[\] discuss ways around this.

The response of a given heat-engine is typically a “loop” of efficiency versus power (see Fig. \[Fig:summary\]) as one varies the load on the system[@Casati-review]. For a peaked transmission function with width $\Delta$ (see e.g. Fig. \[Fig:tophat-width\]), the loop moves to the left as one reduces $\Delta$. In the limit $\Delta \to 0$, the 