Achieving the classical Carnot efficiency in a strongly coupled quantum heat engine

Generally, the efficiency of a heat engine strongly coupled with a heat bath is less than the classical Carnot efficiency. Through a model-independent method, we show that the classical Carnot efficiency is achieved in a strongly coupled quantum heat engine. First, we present the first law of quantum thermodynamics in strong coupling. Then, we show how to achieve the Carnot cycle and the classical Carnot efficiency at strong coupling. We find that this classical Carnot efficiency stems from the fact that the heat released in a nonequilibrium process is balanced by the absorbed heat. We also analyze the restrictions in the achievement of the Carnot cycle. The first restriction is that there must be two corresponding intervals of the controllable parameter in which the corresponding entropies of the work substance at the hot and cold temperatures are equal, and the second is that the entropy of the initial and final states in a nonequilibrium process must be equal. Through these restrictions, we obtain the positive work conditions, including the usual one in which the hot temperature should be higher than the cold, and a new one in which there must be an entropy interval at the hot temperature overlapping that at the cold. We demonstrate our result through a paradigmatic model—a two-level system in which a work substance strongly interacts with a heat bath. In this model, we find that the efficiency may abruptly decrease to zero due to the first restriction, and that the second restriction results in the control scheme becoming complex.

Figure 1

Temperature-entropy ($T\text{−}{S}_s$) diagram of a Carnot cycle. The thick line represents a quasistatic process, and the breakpoint at the ends of each stroke indicates the nonequilibrium processes $A$ and $B$. The operators $(H^i,H_s^i)$ are the Hamiltonian of the composite system and that of the work substance at the corresponding point, respectively.

Figure 2

Entropy of the work substance vs the transition energy $E$. The dotted, dashed, and thick lines show the entropy with inverse temperatures $\beta =10$, 5, and 0.01, respectively. The other parameters are $\mathrm{\Omega }=1.8$ and $\lambda =1$.

Figure 3

Contour plot of the entropy of the work substance with respect to the coupling strength $\lambda$ and the natural frequency of the reaction coordinate $\mathrm{\Omega }$. The parameters are $E=1$ and $\beta =0.8$.

Figure 4

Efficiency of the Carnot cycle with respect to the temperature of the hot bath $T_h$. The inset shows the overlap between the two entropy lines at the hot and cold temperatures. In this calculation, the range of the controlled parameter $E$ is chosen to be $[0,1]$. The other parameters are $\mathrm{\Omega }=1.8$, $T_c=0.2$, and $\lambda =0.9$.