Work extremum principle: Structure and function of quantum heat engines

We consider a class of quantum heat engines consisting of two subsystems interacting with a work-source and coupled to two separate baths at different temperatures $T_h>T_c$. The purpose of the engine is to extract work due to the temperature difference. Its dynamics is not restricted to the near equilibrium regime. The engine structure is determined by maximizing the extracted work under various constraints. When this maximization is carried out at finite power, the engine dynamics is described by well-defined temperatures and satisfies the local version of the second law. In addition, its efficiency is bounded from below by the Curzon-Ahlborn value $1-\sqrt{T_c/T_h}$ and from above by the Carnot value $1-\left(T_c/T_h\right)$. The latter is reached—at finite power—for a macroscopic engine, while the former is achieved in the equilibrium limit $T_h\to T_c$. The efficiency that maximizes the power is strictly larger than the Curzon-Ahloborn value. When the work is maximized at a zero power, even a small (few-level) engine extracts work right at the Carnot efficiency.

Figure 1

Schematic representation of the considered quantum engine. Quantum systems $\mathit{R}$ and $\mathit{S}$ interact with thermal baths at temperatures $T_h$ and $T_c$, respectively. The mutual interaction between them is governed by the potential $V\left(t\right)$ that also serves to deliver work to an external source. The mutual interaction is switched on for a short time only. Once it is switched off, the systems $\mathit{S}$ and $\mathit{R}$ do relax to equilibrium under influence of the respective thermal baths. Similar constructions of quantum heat engines appeared in [9, 12, 24].

Figure 2

Normal curves: efficiency ${\eta }_{nn}$ at the maximal work versus the temperature ratio $\theta =\frac{T_c}{T_h}$ for various values of the energy level number $n$. From bottom to top: $n=2,501,1001,10001,\text{and}100001$. Upper dashed line: Carnot efficiency ${\eta }_{\text{Carnot}}=1-\theta$. Lower dashed line: Curzon-Ahlbronefficiency ${\eta }_{\text{CA}}=1-\sqrt{\theta }$. It is seen that ${\eta }_{\text{Carnot}}>{\eta }_{nn}>{\eta }_{\text{CA}}$.

Figure 3

Dimensionless maximal work $\frac{W_{nn}}{T_h}$ versus the temperature ratio $\theta =\frac{T_c}{T_h}$ for various values of the energy level number $n$. From bottom to top: $n=2,51,251,501,\text{and}1001$.

Figure 4

The dimensionless work $\frac{W_{nn}\left(u,v_{\text{m}}\right)}{T_h}$ versus the parameter $u$. The parameter $v$ is set at its optimal value $v_{\text{m}}$ that provides the unconditional maximum of $\frac{W_{nn}}{T_h}$. Here $T_c/T_h=0.5$. Heavy line: $n=100$, normal line: $n=50$, and dashed line: $n=10$. It is seen that the maximum of $\frac{W_{nn}}{T_h}$ gets larger and sharper for larger values of $n$. As explained after Eq. (50), for some $u\ne u_{\text{m}}$ no work can be extracted at all.

Figure 5

The dimensionless work $\frac{W_{nn}\left(\eta \right)}{T_h}$ defined in Eq. (81) versus the efficiency $\eta$. Recall from Eq. (81) that $W_{nn}\left(\eta \right)$ is obtained by maximizing the work over all parameters for a fixed efficiency $\eta$. For this figure $\theta =\frac{T_c}{T_h}=0.5$, and from bottom to top: $n=2,51,\text{and}101$. Note that $\frac{W_{nn}\left(\eta \right)}{T_h}$ turns to zero two times: for $\eta =0$ and for $\eta ={\eta }_{\text{Carnot}}=1-\theta$. The maximum of $\frac{W_{nn}\left(\eta \right)}{T_h}$ corresponds to the overall maximization of work; see Fig. 3.