Efficiency at maximum power of a laser quantum heat engine enhanced by noise-induced coherence

Quantum coherence has been demonstrated in various systems including organic solar cells and solid state devices. In this article, we report the lower and upper bounds for the performance of quantum heat engines determined by the efficiency at maximum power. Our prediction based on the canonical three-level Scovil and Schulz-Dubois maser model strongly depends on the ratio of system-bath couplings for the hot and cold baths and recovers the theoretical bounds established previously for the Carnot engine. Further, introducing a fourth level to the maser model can enhance the maximal power and its efficiency, thus demonstrating the importance of quantum coherence in the thermodynamics and operation of the heat engines beyond the classical limit.

Figure 1

(a) Three-level QHE model. The hot (cold) reservoir is coupled with the $|1\rangle -|g\rangle$ ($|0\rangle -|g\rangle$) transition with dissipative rate ${\mathrm{\Gamma }}_h$ (${\mathrm{\Gamma }}_c$). The single-mode laser field is driving between $|1\rangle$ and $|0\rangle$ with coupling strength $\lambda$. (b) Four-level QHE model. The difference from the three-level model is that there are two excited states $|1\rangle$ and $|2\rangle$ coupled with $|0\rangle$ via the hot reservoir, with dissipative rate ${\mathrm{\Gamma }}_{h1}$ and ${\mathrm{\Gamma }}_{h2}$, respectively. The energy gap between $|1\rangle$ and $|2\rangle$ is $2\mathrm{\Delta }$.

Figure 2

EMP ${\eta }_{\mathrm{SSD}}^{*}$ vs the Carnot efficiency ${\eta }_C$ for the three-level QHE model. When fixing ${\omega }_h$ and optimizing ${\omega }_c, {\eta }_{\mathrm{SSD}}^{*}$ lies between the bounds ${\eta }_C/2$ and ${\eta }_{\mathrm{CA}}$ (red area). When fixing ${\omega }_c$ and optimizing ${\omega }_h, {\eta }_{\mathrm{SSD}}^{*}$ lies between the bounds ${\eta }_{\mathrm{CA}}$ and ${\eta }_C/(2-{\eta }_C)$ (blue area). ${\eta }_{\mathrm{SSD}}^{*}$ approaches the lower (upper) bound when $\gamma \to 0\left(\infty \right)$. We present the ${\eta }_{\mathrm{SSD}}^{*}$ with $\gamma =1$ (green dash-dotted line) and $\gamma =0.05$ (blue dotted line) fixing ${\omega }_h$; $\gamma =1$ (yellow solid line) and $\gamma =0.05$ (purple dash line) fixing ${\omega }_c$. Here we set $k_{\mathrm{B}}{T}_h=100{\mathrm{\Gamma }}_c$ and $\lambda =1000{\mathrm{\Gamma }}_c$.

Figure 3

EMP vs the Carnot efficiency of the four-level QHE model in the high temperature for (a) fixing ${\omega }_h$, (b) fixing ${\omega }_c$, and in the low temperature for (c) fixing ${\omega }_h$ and (d) fixing ${\omega }_c$. In the high temperature we normalize the plots with respect to EMP at $p=0$ and set $k_{\mathrm{B}}{T}_h=100{\mathrm{\Gamma }}_c, \lambda =1000{\mathrm{\Gamma }}_c$; in the low temperature we normalize the plots with respect to ${\eta }_C/(2-{\eta }_C)$ and set $k_{\mathrm{B}}{T}_h=0.1{\mathrm{\Gamma }}_c, \lambda =10{\mathrm{\Gamma }}_c$. In the degenerate case $\mathrm{\Delta }=0$, the EMP is enhanced by increasing the coherence $p$ when $T_h$ is high, while the coherence does not affect the EMP when $T_h$ is low. Interchanging ${\mathrm{\Gamma }}_{h1}$ and ${\mathrm{\Gamma }}_{h2}$ ($\delta {\tilde{\mathrm{\Gamma }}}_h\equiv ({\mathrm{\Gamma }}_{h1}-{\mathrm{\Gamma }}_{h2})/{\mathrm{\Gamma }}_c=\pm 0.01$) gives the same result (blue circle line and red solid line) for $\mathrm{\Delta }=0$. While for $\mathrm{\Delta }=0.1$, the EMPs of $\delta {\tilde{\mathrm{\Gamma }}}_h=\pm 0.01$ (black dash-dotted line and green dash line) show obvious differences and are lower than their degenerate counterparts, especially for small ${\eta }_C$.