Quantum Carnot thermal machines reexamined: Definition of efficiency and the effects of strong coupling

Whether the strong coupling to thermal baths can improve the performance of quantum thermal machines remains an open issue under active debate. Here we revisit quantum thermal machines operating with the quasistatic Carnot cycle and aim to unveil the role of strong coupling in maximum efficiency. Our analysis builds upon definitions of excess work and heat derived from an exact formulation of the first law of thermodynamics for the working substance, which captures the non-Gibbsian thermal equilibrium state that emerges at strong couplings during quasistatic isothermal processes. These excess definitions differ from conventional ones by an energetic cost for maintaining the non-Gibbsian characteristics. With this distinction, we point out that one can introduce two different yet thermodynamically allowed definitions for efficiency of both the heat engine and refrigerator modes. We dub them excess and hybrid definitions, which differ in the way of defining the gain for the thermal machines at strong couplings by either just analyzing the energetics of the working substance or instead evaluating the performance from an external system upon which the thermal machine acts, respectively. We analytically demonstrate that the excess definition predicts that the Carnot limit remains the upper bound for both operation modes at strong couplings, whereas the hybrid one reveals that strong coupling can suppress the maximum efficiency rendering the Carnot limit unattainable. These seemingly incompatible predictions thus indicate that it is imperative to first gauge the definition for efficiency before elucidating the exact role of strong coupling, thereby shedding light on the ongoing investigation on strong-coupling quantum thermal machines.

Figure 1

Sketch of the study. (a) Quantities and energetics during a quasistatic isothermal process. Upper panel: Weak coupling limit with conventional heat definition $Q$ due to bath energy loss and conventional work definition $W$ due to external drivings which are not shown in the figure. The first law of thermodynamics for the system reads $\mathrm{\Delta }{E}_S=Q+W$ [see Eq. (3)]. Lower panel: Strong coupling scenario with both conventional quantities $Q, W$ and excess quantities $Q_{\mathrm{ex}}=Q-Q_{\mathrm{ec}}$ [see Eq. (4)], $W_{\mathrm{ex}}=W+Q_{\mathrm{ec}}$ [see Eq. (5)] with $Q_{\mathrm{ec}}$ an energetic cost for maintaining the non-Gibbsian thermal state. In this scenario, the first law of thermodynamics for the system reads $\mathrm{\Delta }{E}_S=Q_{\mathrm{ex}}+W_{\mathrm{ex}}$ [Eq. (5)]. Directions of arrows reflect our positive sign convention of thermodynamic quantities. (b) Energy flow representation of a heat engine mode at strong couplings. Due to the difference between $W$ and $W_{\mathrm{ex}}$, one may come up with different definitions for thermodynamic efficiency $\eta$ which differ in gain as highlighted by the orange and green dashed circles (see details in Sec. 4). (c) Energy flow representation of a refrigerator mode at strong couplings. Due to the difference between $Q_c$ and $Q_{\mathrm{ex},c}$, one may carry out different definitions for the coefficient of performance ${\eta }_{\mathrm{COP}}$ which differ in gain as highlighted by the orange and green dashed circles (see details in Sec. 5).