Consistency of optimizing finite-time Carnot engines with the low-dissipation model in the two-level
本站小编 Free考研考试/2022-01-02
Yu-Han Ma1, C P Sun1,2, Hui Dong,1,∗1Graduate School of China Academy of Engineering Physics, Beijing 100193, China 2Beijing Computational Science Research Center, Beijing 100193, China
First author contact:Author to whom any correspondence should be addressed. Received:2021-09-7Revised:2021-10-1Accepted:2021-10-5Online:2021-11-12
Abstract The efficiency at the maximum power (EMP) for finite-time Carnot engines established with the low-dissipation model, relies significantly on the assumption of the inverse proportion scaling of the irreversible entropy generation ΔS(ir) on the operation time τ, i.e. ΔS(ir) ∝ 1/τ. The optimal operation time of the finite-time isothermal process for EMP has to be within the valid regime of the inverse proportion scaling. Yet, such consistency was not tested due to the unknown coefficient of the 1/τ-scaling. In this paper, we reveal that the optimization of the finite-time two-level atomic Carnot engines with the low-dissipation model is consistent only in the regime of ηC ≪ 2(1 − δ)/(1 + δ), where ηC is the Carnot efficiency, and δ is the compression ratio in energy level difference of the heat engine cycle. In the large-ηC regime, the operation time for EMP obtained with the low-dissipation model is not within the valid regime of the 1/τ-scaling, and the exact EMP of the engine is found to surpass the well-known bound η+ = ηC/(2 − ηC). Keywords：finite-time thermodynamics;low-dissipation model;quantum heat engine;efficiency at maximum power;irreversible entropy generation
PDF (999KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Yu-Han Ma, C P Sun, Hui Dong. Consistency of optimizing finite-time Carnot engines with the low-dissipation model in the two-level atomic heat engine. Communications in Theoretical Physics, 2021, 73(12): 125101- doi:10.1088/1572-9494/ac2cb8
1. Introduction
Converting heat into useful work, a heat engine lies at the core of thermodynamics, both in classical and quantum regimes ([1–4]). Absorbing heat from a hot thermal bath with the temperature Th, the engine outputs work and release part of the heat to the cold bath with the temperature Tc. The upper limit of the heat engine working between two heat baths is given by the Carnot efficiency ηC = 1 − Tc/Th ([1]). Due to the limitation of the quasi-static cycle with an infinitely-long operation time, the heat engine with Carnot efficiency generally has vanishing output power and in turn is of no practical use. To design the heat engine cycles operating in finite-time, several practical heat engine models have been proposed ([5–7]), such as the endo-reversible model ([8–12]), the linear irreversible model ([13–15]), the stochastic model ([16, 17]), and the low-dissipation model ([18–25]). The efficiency at maximum power (EMP), is proposed as an important parameter to evaluate the performance of these heat engines in the finite-time cycles.
The utilization of the low-dissipation model ([18–21, 25, 26]) simplifies the optimization of the finite-time Carnot-like heat engines. As the model assumption, the heat transfer between the engine and the bath in the finite-time quasi-isothermal process is divided into two parts as follows$\begin{eqnarray}{Q}_{{\rm{h}},{\rm{c}}}({\tau }_{{\rm{h}}})={T}_{{\rm{h}},{\rm{c}}}({\rm{\Delta }}{S}_{{\rm{h}},{\rm{c}}}-{S}_{{\rm{h}},{\rm{c}}}^{(\mathrm{ir})}),\end{eqnarray}$where ΔSh = − ΔSc = ΔS is the reversible entropy change of the working substance and ${S}_{{\rm{h}},{\rm{c}}}^{(\mathrm{ir})}={{\rm{\Sigma }}}_{{\rm{h}},{\rm{c}}}/{\tau }_{{\rm{h}},{\rm{c}}}$ is the irreversible entropy generation which is inversely proportional to the process time τα. Optimizing the output power P(τh, τc) = [Qh(τh) + Qc(τc)]/(τh + τc) with respect to the operation time τh and τc, one gets the optimal operation times ([18]) as$\begin{eqnarray}{\tau }_{{\rm{h}}}^{* }=\displaystyle \frac{2{T}_{{\rm{h}}}{{\rm{\Sigma }}}_{{\rm{h}}}}{({T}_{{\rm{h}}}-{T}_{{\rm{c}}}){\rm{\Delta }}S}\left(1+\sqrt{\displaystyle \frac{{T}_{{\rm{c}}}{{\rm{\Sigma }}}_{{\rm{c}}}}{{T}_{{\rm{h}}}{{\rm{\Sigma }}}_{{\rm{h}}}}}\right),\end{eqnarray}$$\begin{eqnarray}{\tau }_{{\rm{c}}}^{* }=\displaystyle \frac{2{T}_{{\rm{c}}}{{\rm{\Sigma }}}_{{\rm{c}}}}{({T}_{{\rm{h}}}-{T}_{{\rm{c}}}){\rm{\Delta }}S}\left(1+\sqrt{\displaystyle \frac{{T}_{{\rm{h}}}{{\rm{\Sigma }}}_{{\rm{h}}}}{{T}_{{\rm{c}}}{{\rm{\Sigma }}}_{{\rm{c}}}}}\right),\end{eqnarray}$and the efficiency at the maximum power η* bounded by the following inequality as ([7, 18])$\begin{eqnarray}{\eta }_{-}\equiv \displaystyle \frac{{\eta }_{{\rm{C}}}}{2}\leqslant {\eta }^{* }\leqslant \displaystyle \frac{{\eta }_{{\rm{C}}}}{2-{\eta }_{{\rm{C}}}}\equiv {\eta }_{+}.\end{eqnarray}$Due to the simplicity of the model assumption and the universality of the obtained EMP, the low-dissipation model becomes one of the most studied finite-time heat engine models in recent years ([19–25]).
It is currently agreed that ([21, 25–29]) the low-dissipation assumption is valid in the long-time regime of τ/tr ≫ 1, where tr is the relaxation time characterizing the time scale for the work substance to reach its equilibrium with the heat bath. The quasi-static process is achieved with τ/tr → ∞ . The short-time deviation of irreversible entropy generation from the 1/τ-scaling has been clearly demonstrated theoretically ([21]) and experimentally ([29]). And the dissipation coefficient Σ of the 1/τ-scaling is determined by both the coupling strength γ ∼ 1/tr to the bath([21, 29]) and the control scheme ([26, 29]). Such a relation implies that the condition τ*/tr ≫ 1 is not fulfilled simply and should be justified to reveal the regime of validity. In this paper, we check the consistency of the obtained EMP with a minimal heat engine model consisting of a single two-level system. In section 2, we analytically obtain the regime, where the optimal operation times to achieve EMP are consistent with the low-dissipation assumption. And we further show the possibility of the exact EMP of the engine to surpass the upper bound of EMP, i.e. η+, obtained with the low-dissipation model in the large-ηC regime in section 3.
2. Self-consistency of the low-dissipation model in deriving efficiency at maximum power
The two-level atomic heat engine is the simplest quantum engine to demonstrate the relevant physical mechanisms ([21, 26, 30–32]). The energy spacing of the excited state $\left|e\right\rangle $ and ground state $\left|g\right\rangle $ is tuned by an outside agent to extract work with the Hamiltonian $H=\tfrac{1}{2}{\hslash }\omega \left(t\right){\sigma }_{z},$ where ${\sigma }_{z}=\left|e\right\rangle \left\langle e\right|-\left|g\right\rangle \left\langle g\right|$ is the Pauli matrix in the z-direction. The Planck's constant is taken as ℏ = 1 in the following discussion for convenience. For the finite-time quasi-isothermal process with the duration τ of the two-level system, the low-dissipation assumption of the 1/τ scaling is valid in the regime $\widetilde{\gamma }\tau \gg 1$ ([21]), where $\widetilde{\gamma }={t}_{{\rm{r}}}^{-1}=2\gamma T/{\omega }_{0}$ in the high temperature regime. Here γ is the coupling strength between the system and the bath with the temperature T and ω0 is the initial energy spacing of the system during the process.
The finite-time Carnot-like cycle for the two-level atomic heat engine of interest consists of four strokes, two isothermal and two adiabatic processes. The schematic diagram of the cycle is shown in figure 1. In the figure, ${\omega }_{{\rm{h}}}^{{\rm{i}}}$ and ${\omega }_{{\rm{h}}}^{{\rm{f}}}$ (${\omega }_{{\rm{c}}}^{{\rm{i}}}$ and ${\omega }_{{\rm{c}}}^{{\rm{f}}}$ ) are respectively the initial and final energy spacing of the working substance in the high (low) temperature finite-time quasi-isothermal process with duration τh (τc), which is shown with the red (right) [blue (left)] solid curve. The total operating time per cycle is t = τh + τc. Here, we have assumed that the interval of the adiabatic processes, plotted with the black (horizontal) solid lines, are ignored in comparison with τh and τc ([18, 21]). Such assumption can always be satisfied when the two-level system has no energy level crossing, namely, ω(t) = 0, during the whole cycle. (Since the eigen-states of such systems do not change with time, namely, $\dot{\left|e\right\rangle }=0$ ($\dot{\left|g\right\rangle }=0$), the quantum adiabatic conditions $\left|\,\left\langle e\right|\dot{\left.g\right\rangle }/\omega (t)\right|\ll 1$ and $\left|\,\left\langle g\right|\dot{\left.e\right\rangle }/\omega (t)\right|\ll 1$ are always satisfied with ω(t) ≠ 0 ([33, 34]). Therefore, we can tune the energy spacing of the system fast enough to make the corresponding duration of the adiabatic process negligible compared to the time scale ${\widetilde{\gamma }}^{-1}$ of the isothermal process). The quasi-isothermal process retains the normal isothermal process at the quasi-static limit of τh(c) → ∞ .
Figure 1.
New window|Download| PPT slide Figure 1.Schematic diagram of the finite-time Carnot-like cycle for a two-level atomic heat engine. The horizontal axis and the vertical axis represent respectively the energy spacing ω and excited state population pe of the two-level atom. The red (right) and blue (left) solid curves represent the high-temperature and low-temperature finite-time quasi-isothermal processes, respectively. The black (horizontal) solid lines represent the adiabatic processes.
For simplicity, we focus on the high-temperature regime, where the reversible entropy change ΔSα and the irreversible entropy generation coefficient Σα in equation (1) are analytically written as ([21])$\begin{eqnarray}{\rm{\Delta }}{S}_{\alpha }=\displaystyle \frac{\left[{\left({\omega }_{\alpha }^{{\rm{i}}}\right)}^{2}-{\left({\omega }_{\alpha }^{{\rm{f}}}\right)}^{2}\right]}{8{T}_{\alpha }^{2}},{{\rm{\Sigma }}}_{\alpha }=\displaystyle \frac{{\left({\omega }_{\alpha }^{{\rm{i}}}-{\omega }_{\alpha }^{{\rm{f}}}\right)}^{2}}{4{T}_{\alpha }^{2}{\widetilde{\gamma }}_{\alpha }},\end{eqnarray}$where α = h, c, ${\widetilde{\gamma }}_{\alpha }=1/{t}_{{\rm{r}}}^{(\alpha )}=2{\gamma }_{\alpha }{T}_{\alpha }/{\omega }_{\alpha }^{{\rm{i}}}$, and the natural unit kB = 1 and ℏ = 1 are used. To obtain the above equations, the relations ${\omega }_{{\rm{h}}}^{{\rm{i}}}/{T}_{{\rm{h}}}={\omega }_{{\rm{c}}}^{{\rm{f}}}/{T}_{{\rm{c}}}$, ${\omega }_{{\rm{h}}}^{{\rm{f}}}/{T}_{{\rm{h}}}={\omega }_{{\rm{c}}}^{{\rm{i}}}/{T}_{{\rm{c}}}$ have been used in the quantum adiabatic processes ([21, 31]). Substituting equation (5) into equations (2) and (3), we obtain the corresponding optimal operation time ${\tau }_{\alpha }^{* }$ for achieving the maximum power with the dimensionless time ${\widetilde{\tau }}_{\alpha }^{* }\equiv {\tau }_{\alpha }^{* }/{t}_{{\rm{r}}}^{(\alpha )}={\widetilde{\gamma }}_{\alpha }{\tau }_{\alpha }^{* }$ as$\begin{eqnarray}{\widetilde{\tau }}_{{\rm{h}}}^{* }=\displaystyle \frac{2}{{\eta }_{{\rm{C}}}}\displaystyle \frac{{\omega }_{{\rm{h}}}^{{\rm{i}}}-{\omega }_{{\rm{h}}}^{{\rm{f}}}}{{\omega }_{{\rm{h}}}^{{\rm{i}}}+{\omega }_{{\rm{h}}}^{{\rm{f}}}}\left[1+\sqrt{\left(1-{\eta }_{{\rm{C}}}\right)\displaystyle \frac{{\gamma }_{{\rm{h}}}}{{\gamma }_{{\rm{c}}}}}\right],\end{eqnarray}$$\begin{eqnarray}{\widetilde{\tau }}_{{\rm{c}}}^{* }=\displaystyle \frac{2}{{\eta }_{{\rm{C}}}}\displaystyle \frac{{\omega }_{{\rm{h}}}^{{\rm{i}}}-{\omega }_{{\rm{h}}}^{{\rm{f}}}}{{\omega }_{{\rm{h}}}^{{\rm{i}}}+{\omega }_{{\rm{h}}}^{{\rm{f}}}}\left[\sqrt{\left(1-{\eta }_{{\rm{C}}}\right)\displaystyle \frac{{\gamma }_{{\rm{c}}}}{{\gamma }_{{\rm{h}}}}}+1-{\eta }_{{\rm{C}}}\right].\end{eqnarray}$The low-dissipation assumption is valid in the regime ${\widetilde{\tau }}_{{\rm{h}}}^{* }\gg 1$ and ${\widetilde{\tau }}_{{\rm{c}}}^{* }\gg 1$, namely,$\begin{eqnarray}1+\sqrt{\left(1-{\eta }_{{\rm{C}}}\right)\displaystyle \frac{{\gamma }_{{\rm{h}}}}{{\gamma }_{{\rm{c}}}}}\gg \displaystyle \frac{{\eta }_{{\rm{C}}}}{2}\displaystyle \frac{{\omega }_{{\rm{h}}}^{{\rm{i}}}+{\omega }_{{\rm{h}}}^{{\rm{f}}}}{{\omega }_{{\rm{h}}}^{{\rm{i}}}-{\omega }_{{\rm{h}}}^{{\rm{f}}}},\end{eqnarray}$$\begin{eqnarray}\sqrt{\left(1-{\eta }_{{\rm{C}}}\right)\displaystyle \frac{{\gamma }_{{\rm{c}}}}{{\gamma }_{{\rm{h}}}}}+1-{\eta }_{{\rm{C}}}\gg \displaystyle \frac{{\eta }_{{\rm{C}}}}{2}\displaystyle \frac{{\omega }_{{\rm{h}}}^{{\rm{i}}}+{\omega }_{{\rm{h}}}^{{\rm{f}}}}{{\omega }_{{\rm{h}}}^{{\rm{i}}}-{\omega }_{{\rm{h}}}^{{\rm{f}}}}.\end{eqnarray}$The above two inequalities are fulfilled when$\begin{eqnarray}{\eta }_{{\rm{C}}}\ll 2\displaystyle \frac{1-\delta }{1+\delta },\end{eqnarray}$where $\delta \equiv {\omega }_{{\rm{h}}}^{{\rm{f}}}/{\omega }_{{\rm{h}}}^{{\rm{i}}}$ is the compression ratio of the heat engine cycle in the quasi-isothermal process, and the restriction δ > 0 is required to avoid energy level crossing of the system as we mentioned before. Similar discussion can be applied to the low-temperature regime, where ΔSα and Σα have different expressions ([21]) and equation (10) will change accordingly, please see appendix B for details. The above relation is one of the main results of the current work and reveals the range of ηC in which the low-dissipation model is applicable for finding EMP. The bound for EMP obtained in the low-dissipation regime, as given by equation (4), thus may be not unconditionally applicable to such two-level atomic engine. Indeed, we will show that, out of the low-dissipation regime, the EMP of the two-level atomic heat engine is larger than the upper bound η+ predicted by the low-dissipation model in the next section.
3. Efficiency at maximum power: beyond the low dissipation model
With the analytical discussion above, we find the EMP obtained with the low-dissipation model is only consistent with the assumption of the low-dissipation model in the low-ηC regime for the two-level system. The question is whether the bound provided by the low-dissipation model, i.e. η+, is still the upper bound for the achievable efficiency of the system out of the low-ηC regime. Unfortunately, the answer is no. In this section, we will focus the efficiency at the maximum power in the regime with large ηC.
By numerically simulating the dynamics of the two-level system engine with a different cycle time, we obtain the exact power and efficiency to find the EMP. The results in the large-ηC regime show that: (i) the optimal operation time corresponding to the maximum power of the heat engine does not meet the low-dissipation assumption; (ii) the EMP surpass the upper bound obtained with the low-dissipation model, namely, ηMP > η+.
The dynamics of the two-level atom in the finite-time quasi-isothermal process is given by the master equation as follows ([21])$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{p}_{{\rm{e}}}(t)}{{\rm{d}}t}=-\kappa (t){p}_{{\rm{e}}}(t)+C(t),\end{eqnarray}$where pe(t) is the excited state population and C(t) = γn[ω(t)]. $\kappa (t)=\gamma \left(2n[\omega (t)]+1\right)$ is the effective dissipation rate with the mean occupation number $n[\omega (t)]=1/\left(\exp [\beta \omega (t)]-1\right)$ for the bath mode ω(t). The dissipation rate γ equals to γh (γc) in the high (low) temperature quasi-isothermal process with the inverse temperature βh = 1/(kBTh) (βc = 1/(kBTc)). The energy spacing of the two-level atom is tuned linearly as $\omega (t)={\omega }_{{\rm{h}}}^{{\rm{i}}}+({\omega }_{{\rm{h}}}^{{\rm{f}}}-{\omega }_{{\rm{h}}}^{{\rm{i}}})t/{\tau }_{{\rm{h}}},t\in [0,{\tau }_{{\rm{h}}}]$ in the high-temperature finite-time quasi-isothermal process and as $\omega (t)={\omega }_{{\rm{c}}}^{{\rm{i}}}+({\omega }_{{\rm{c}}}^{{\rm{f}}}-{\omega }_{{\rm{c}}}^{{\rm{i}}})t/{\tau }_{{\rm{c}}},t\in [{\tau }_{{\rm{c}}},{\tau }_{{\rm{c}}}+{\tau }_{{\rm{h}}}]$ in the low-temperature finite-time quasi-isothermal process. The population of the two-level system keeps unchanged during the adiabatic processes whose operation time is ignored in comparison with τh and τc.
In the following simulation, we set γh = 1 and focus on the regime of γc/γh → ∞ , i.e. Σc/Σh → 0, where the upper bound η+ = ηC/(2 − ηC) of EMP of the engine is achieved according to the prediction with the low-dissipation model ([18]). In this regime, the low-temperature quasi-isothermal process approaches the isothermal process quickly enough that the operation time τc is further ignored for the optimization of the cycle's output power, and the relaxation time corresponding to the high-temperature quasi-isothermal process is ${t}_{{\rm{r}}}^{({\rm{h}})}={\widetilde{\gamma }}_{{\rm{h}}}^{-1}={\omega }_{{\rm{h}}}^{{\rm{i}}}/\left(2{\gamma }_{{\rm{h}}}{T}_{{\rm{h}}}\right)$. In this case, there is only one relaxation time in the cycle. For brevity, in the following discussion, the superscript of ${t}_{{\rm{r}}}^{({\rm{h}})}$ is removed, namely, ${t}_{{\rm{r}}}^{({\rm{h}})}$ is denoted as tr. The optimization of the heat engine cycle is thus simplified as a single parameter optimization problem: find the maximum value ${P}_{\max }$ of the cycle's output power with respect to τh, and obtain the EMP of the engine, ${\eta }_{\mathrm{MP}}\equiv \eta (P={P}_{\max })$.
The cycles with different τh are illustrated in figure 2, where ${\omega }_{{\rm{h}}}^{{\rm{i}}}=1$ and ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.9$ are fixed. The temperatures for the hot and cold bath are chosen as Th = 10 and Tc = 9 as an example. The relaxation time is ${t}_{{\rm{r}}}={\omega }_{{\rm{h}}}^{{\rm{i}}}/\left(2{\gamma }_{{\rm{h}}}{T}_{{\rm{h}}}\right)=0.05$. The quasi-static cycles with τh = 200tr, 10tr and 2tr are represented by the dash-dotted line, dashed line, and solid line, respectively. The figure shows that the output work represented by the cycle area decreases with τh. Here, we emphasize that tr is not an independent parameter, and determined by ${\omega }_{{\rm{h}}}^{{\rm{i}}}$, γh, and Th as ${t}_{{\rm{r}}}={\omega }_{{\rm{h}}}^{{\rm{i}}}/\left(2{\gamma }_{{\rm{h}}}{T}_{{\rm{h}}}\right)$. The dynamic behavior of the system in finite-time isothermal process is typically characterized by the dimensionless time τ/tr ([21, 29]). In order to unify the discussion in this section, we did not choose different values of tr when plotting the figures below, but fix the value of tr = 0.05. (fix the value of ${\omega }_{{\rm{h}}}^{{\rm{i}}}$, γh, and Th).
Figure 2.
New window|Download| PPT slide Figure 2.The finite-time Carnot-like cycles for a two-level atomic heat engine with different operation time τh. The red (right) curves represent the high-temperature finite-time quasi-isothermal processes with the duration τh, while the blue (left) curves represent the low-temperature isothermal processes. The adiabatic processes are plotted with the black (horizontal) lines. The outermost dash-dotted curves relate to the quasi-static cycle with τh = 200tr, while the middle dashed cycle and inner solid cycle are obtained with τh = 10tr and τh = 2tr, respectively. In this example, we choose ${\omega }_{{\rm{h}}}^{{\rm{i}}}=1$, ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.9$, γh = 1, Th = 10, and Tc = 9. ${t}_{{\rm{r}}}={\omega }_{{\rm{h}}}^{{\rm{i}}}/\left(2{\gamma }_{{\rm{h}}}{T}_{{\rm{h}}}\right)=0.05$ is the relaxation time related to the high-temperature finite-time quasi-isothermal process.
In figure 3, we show the normalized power of the engine $\widetilde{P}\equiv P/{P}_{\max }$ as the function of τh/tr with ηC = 0.1 (blue solid line), ηC = 0.12 (orange dash-dotted line), and ηC = 0.15 (purple dashed line). In the simulation, the parameters are set as ${\omega }_{{\rm{h}}}^{{\rm{i}}}=1$, ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.9$, and Th = 10 with changing Tc = 9, 8.8 and 8.5. The relaxation time is tr = 0.05. The maximum output power ${P}_{\max }$ is obtained numerically for different ηC. It is observed from the figure that the dependence of $\widetilde{P}$ on operation time τh changes with ηC. In the figure, the optimal ${\tau }_{{\rm{h}}}^{* }$ decreases with ηC and is away from the low-dissipation regime of τh/tr ≫ 1, illustrated with the orange dash-dotted line (ηC = 0.12, ${\tau }_{{\rm{h}}}^{* }/{t}_{{\rm{r}}}\approx 0.5$) and the blue solid line (ηC = 0.1, ${\tau }_{{\rm{h}}}^{* }/{t}_{{\rm{r}}}\approx 1$). As shown clearly by the purple dashed line with ηC = 0.15, the maximum power $\widetilde{P}=1$ is achieved in the short-time regime of τh/tr ≪ 1, where the 1/τ-scaling of irreversible entropy generation is invalid ([21, 29]).
Figure 3.
New window|Download| PPT slide Figure 3.The normalized power of the engine $\widetilde{P}=P/{P}_{\max }$ as a function of τh/tr. The blue solid line, the orange dash-dotted line, and the purple dashed line are respectively obtained with ηC = 0.1, ηC = 0.12, and ηC = 0.15. In this example, we choose ${\omega }_{{\rm{h}}}^{{\rm{i}}}=1$, ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.9$, γh = 1, and Th = 10 with changing Tc = 9, 8.8 and 8.5. The relaxation time is ${t}_{{\rm{r}}}={\omega }_{{\rm{h}}}^{{\rm{i}}}/\left(2{\gamma }_{{\rm{h}}}{T}_{{\rm{h}}}\right)=0.05$.
We show the obtained efficiency ηMP at the maximum power of the engine as a function of ηC in figures 4(a) and (b), and plot the corresponding optimal operation time ${\tau }_{{\rm{h}}}^{* }$ in figure 4(c). We choose the final energy spacing of the two level system as ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.6$ and ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.9$ respectively for (a) and (b), and other parameters are set as ${\omega }_{{\rm{h}}}^{{\rm{i}}}=1$, γh = 1, Th = 10. The parameters used in this figure are in the high temperature regime, and the results in the low-temperature regime are illustrated in appendix B. As shown in figures 4(a) and (b), the EMP of the engine ηMP(orange solid line) in the large-ηC regime surpasses the upper bound of EMP, η+ = ηC/(2 − ηC) (black dashed line) obtained with the low-dissipation model. The lower bound of EMP, η− = ηC/2, obtained with the low-dissipation model is plotted with the black dash-dotted line. The gray area represents the consistent regime as demonstrated by equation (10). The figure shows that ηMP is bounded by η+ and η− of equation (4) in the gray area with relatively small ηC. Additionally, by comparing (b) and (a) of figure 4, with the larger the compression rate $\delta ={\omega }_{{\rm{h}}}^{{\rm{f}}}/{\omega }_{{\rm{h}}}^{{\rm{i}}}$ (δ = 0.9 for (a) and δ = 0.6 for (b)), we illustrate the narrower the range of ηC in which ηMP is bounded by η+. With the increase of the compression ratio δ, the valid regime of optimization of the engine with the low-dissipation model becomes smaller. And it is consistent with the theoretical analysis of equation (10). Here we emphasize that the gray area only represents the approximate regime where the low-dissipation model is self-consistent, not the exact criterion for ηMP to exceed η+. Moreover, in the large-ηC regime, the EMP of the heat engine ηMP can be analytically obtained as$\begin{eqnarray}{\eta }_{\mathrm{MP}}=\displaystyle \frac{2{\eta }_{{\rm{C}}}-1+\delta }{1+\delta },\end{eqnarray}$which increase linearly with ηC as shown by the orange solid lines in figures 4(a) and (b). The detailed derivation and discussion of the above result are presented in appendix A. We note that the achievable regime of the EMP, i.e. ηMP ∈ (ηC/2, ηC), obtained in the current work is consistent with that of the sub-dissipative engines introduced in ([35]).
Figure 4.
New window|Download| PPT slide Figure 4.Efficiency at the maximum power ηMP (orange solid line) of the heat engine as the function of the Carnot efficiency ηC for different final energy spacing of the two level system(a) ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.6$ and (b) ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.9$. The black dashed line (black dash-dotted line) represents the upper bound η+ (lower bound η−) of EMP obtained with the low-dissipation model [equation (4)], and the Carnot efficiency ηC is plotted with the black dotted line. The gray area represents the low-dissipation regime predicted by equation (10). (c) Optimal operation time ${\tau }_{{\rm{h}}}^{* }$ at the maximum power as the function of ηC. The blue solid curve is obtained with ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.6$ (δ = 0.6) while the red dash-dotted curve is obtained with ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.9$ (δ = 0.9). The other parameters in this figure are chosen as ${\omega }_{{\rm{h}}}^{{\rm{i}}}=1$, γh = 1, and Th = 10. The relaxation time is ${t}_{{\rm{r}}}={\omega }_{{\rm{h}}}^{{\rm{i}}}/\left(2{\gamma }_{{\rm{h}}}{T}_{{\rm{h}}}\right)=0.05$.
In figure 4(c), the optimal operation time ${\tau }_{{\rm{h}}}^{* }$ at the maximum power (blue solid curve for ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.6$ and red dash-dotted curve for ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.9$) decreases monotonically with increasing ηC. For the relatively large ηC, the operation time at maximum power ${\tau }_{{\rm{h}}}^{* }$ of the engine is not satisfy the low-dissipation assumption holds in the long-time regime of τh/tr ≫ 1. Beyond such regime, the irreversible entropy generation will deviate from the 1/τ-scaling ([21, 29]). This explains why ηMP is no longer satisfies the bound provided by the low-dissipation model in large-ηC regime, and verifies our analytical analysis in section 2. In addition, one can find in figure 4(c) that the red dash-dotted curve is lower than the blue solid curve. This leads to a narrower parameter range of ηC, in which the optimal operation time ${\tau }_{{\rm{h}}}^{* }$ satisfies the low-dissipation assumption, for the heat engine with ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.9$ than that with ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.6$. Therefore, the phenomenon that the gray area in figure 4(a) is wider than that in figure 4(b) is explained from the perspective of the operation time.
Here we stress the connection of the current result on ηMP in the large-ηC regime with the linear irreversible thermodynamics (LIT) theory: Since the bounds obtained with the low-dissipation model have been tested in the framework of LIT ([36]), it seems that our results contradict linear thermodynamics. However, it should be noted that the equivalence of the low-dissipation model and linear irreversible heat engine model is not unconditional, but depends on specific conditions such as long-time approximation([14]), tight-coupling condition ([36]), and it only holds in the low Carnot efficiency regime with a small temperature difference([14, 36]). Therefore, our discussion in the large-ηC regime goes beyond the applicable regime of LIT, and it is not surprising that the obtained results are inconsistent with the predictions of LIT works.
4. Conclusions and discussions
In summary, we checked whether the optimal operation time for achieving the maximum power is consistent with the requirement of the low-dissipation model for the finite-time two-level atomic Carnot-like heat engines in this paper. The low-dissipation model, widely used in the finite-time thermodynamics to study EMP, relies on the assumption that the irreversible entropy generation in the finite-time quasi-isothermal process of duration τ follows the 1/τ scaling in the long-time regime. The operation time for the maximum power obtained from the model should fulfill the requirement of the low-dissipation model assumption. Due to the unknown coefficient of the 1/τ scaling, the consistency of the model in optimizing finite-time Carnot engines had not been tested before.
In this paper, we proved that, in the high-temperature regime, the optimal operation times for a finite-time two-level atomic Carnot engine achieving EMP satisfy the low-dissipation assumption only in the low Carnot efficiency regime of ηC ≪ 2(1 − δ)/(1 + δ), such bound is determined by compression ratio in energy level difference δ of the heat engine cycle. This observation motivated us to check the EMP in the regime with large ηC. We calculated the EMP of the two-level atomic heat engine in the full parameter space of ηC. It is found that, in the large-ηC regime, the true EMP of the heat engine can surpass the upper bound for EMP, i.e. η+ = ηC/(2 − ηC) obtained with the low-dissipation model. Moreover, in this regime, we found that the true EMP, which is achieved in the short-time limit of τh/tr ≪ 1, depends linearly on ηC.
Our study on EMP in the large-ηC regime shall provide a new insight for designing heat engines with better performance working between two heat baths with a large temperature difference. Similar investigation to this work on the two-level atomic heat engine can be extended to some relevant scenarios, such as the optimal cycle of the refrigerator mode beyond the low-dissipation regime. Besides, in addition to affecting the EMP of the heat engine, the short-time effects caused by fast driving may also influence the trade-off between power and efficiency ([20, 21, 27, 28]), which needs further exploration. The predictions of this paper can be tested on some experimental platforms ([29, 37–41]) in the short-time regime.
Appendix A. Efficiency at maximum power in the large-ηC regime
The heat absorbed from the high temperature reservoir reads ([21])$\begin{eqnarray}{Q}_{{\rm{h}}}=\int {\omega }_{{\rm{h}}}(t){\rm{d}}{p}_{{\rm{e}}}={\int }_{0}^{{\tau }_{{\rm{h}}}}{\omega }_{{\rm{h}}}(t)\displaystyle \frac{{\rm{d}}{p}_{{\rm{e}}}}{{\rm{d}}t}{\rm{d}}t.\end{eqnarray}$As shown in figure 3, the maximum power $\widetilde{P}=1$ is achieved in the short-time regime of τh/tr ≪ 1 for large ηC. In this case, the excited state population can be approximated as$\begin{eqnarray}{p}_{{\rm{e}}}\left(t/{t}_{{\rm{r}}}\ll 1\right)\approx {p}_{{\rm{e}}}(0)+t{\left(\displaystyle \frac{{\rm{d}}{p}_{{\rm{e}}}}{{\rm{d}}t}\right)}_{t=0},\end{eqnarray}$where$\begin{eqnarray}{\left(\displaystyle \frac{{\rm{d}}{p}_{{\rm{e}}}}{{\rm{d}}t}\right)}_{t=0}=-\kappa (0){p}_{{\rm{e}}}(0)+C(0)\equiv \zeta .\end{eqnarray}$With this approximation, the heat absorbed is written as$\begin{eqnarray}\begin{array}{rcl}{Q}_{{\rm{h}}}&=&{\displaystyle \int }_{0}^{{\tau }_{{\rm{h}}}}{\omega }_{{\rm{h}}}(t)\displaystyle \frac{{\rm{d}}{p}_{{\rm{e}}}}{{\rm{d}}t}{\rm{d}}t={\displaystyle \int }_{0}^{{\tau }_{{\rm{h}}}}{\omega }_{{\rm{h}}}(t)\zeta {\rm{d}}t\\ &=&\zeta \left[{\displaystyle \int }_{0}^{{t}_{{\rm{h}}}}{\omega }_{{\rm{h}}}^{{\rm{i}}}+({\omega }_{{\rm{h}}}^{{\rm{f}}}-{\omega }_{{\rm{h}}}^{{\rm{i}}})\displaystyle \frac{t}{{\tau }_{{\rm{h}}}}\right]{\rm{d}}t\\ &=&\zeta {t}_{{\rm{h}}}\left({\omega }_{{\rm{h}}}^{{\rm{i}}}+\displaystyle \frac{{\omega }_{{\rm{h}}}^{{\rm{f}}}-{\omega }_{{\rm{h}}}^{{\rm{i}}}}{2}\right)=\displaystyle \frac{\zeta {t}_{{\rm{h}}}\left({\omega }_{{\rm{h}}}^{{\rm{i}}}+{\omega }_{{\rm{h}}}^{{\rm{f}}}\right)}{2}.\end{array}\end{eqnarray}$On the other hand, the heat released to the low temperature reservoir is$\begin{eqnarray}{Q}_{{\rm{c}}}={\int }_{{p}_{{\rm{e}}}({\tau }_{{\rm{h}}})}^{{p}_{{\rm{e}}}(0)}{\omega }_{{\rm{c}}}{\rm{d}}{p}_{{\rm{e}}}.\end{eqnarray}$Since we focus on the regime of γc/γh → ∞ , the low-temperature quasi-isothermal process can be considered as the isothermal process with$\begin{eqnarray}{p}_{{\rm{e}}}\left({\omega }_{{\rm{c}}}\right)=\displaystyle \frac{\exp \left(-{\beta }_{{\rm{c}}}{\omega }_{{\rm{c}}}\right)}{1+\exp \left(-{\beta }_{{\rm{c}}}{\omega }_{{\rm{c}}}\right)},\end{eqnarray}$which gives$\begin{eqnarray}{\omega }_{{\rm{c}}}={\omega }_{{\rm{c}}}\left({p}_{{\rm{e}}}\right)=-\displaystyle \frac{1}{{\beta }_{{\rm{c}}}}\mathrm{ln}\left(\displaystyle \frac{{p}_{{\rm{e}}}}{1-{p}_{{\rm{e}}}}\right).\end{eqnarray}$In the two adiabatic processes, we have the following relations$\begin{eqnarray}{p}_{{\rm{e}}}\left({\omega }_{{\rm{c}}}^{{\rm{f}}}\right)={p}_{{\rm{e}}}(0),{p}_{{\rm{e}}}\left({\omega }_{{\rm{c}}}^{{\rm{i}}}\right)={p}_{{\rm{e}}}({\tau }_{{\rm{h}}}).\end{eqnarray}$Substituting equation (A7) into equation (A5), we have$\begin{eqnarray}\begin{array}{rcl}{Q}_{{\rm{c}}}&=&-\displaystyle \frac{1}{{\beta }_{{\rm{c}}}}{\displaystyle \int }_{{p}_{{\rm{e}}}({\tau }_{{\rm{h}}})}^{{p}_{{\rm{e}}}(0)}\mathrm{ln}\left(\displaystyle \frac{{p}_{{\rm{e}}}}{1-{p}_{{\rm{e}}}}\right){\rm{d}}{p}_{{\rm{e}}}\\ &=&-\displaystyle \frac{1}{{\beta }_{{\rm{c}}}}{\left[\mathrm{ln}\left(1-{p}_{{\rm{e}}}\right)+{p}_{{\rm{e}}}\mathrm{ln}\left(\displaystyle \frac{{p}_{{\rm{e}}}}{1-{p}_{{\rm{e}}}}\right)\right]}_{{p}_{{\rm{e}}}({\tau }_{{\rm{h}}})}^{{p}_{{\rm{e}}}(0)}\\ &=&\displaystyle \frac{1}{{\beta }_{{\rm{c}}}}\displaystyle \frac{d}{{{dp}}_{{\rm{e}}}}{\left[\mathrm{ln}\left(1-{p}_{{\rm{e}}}\right)+{p}_{{\rm{e}}}\mathrm{ln}\left(\displaystyle \frac{{p}_{{\rm{e}}}}{1-{p}_{{\rm{e}}}}\right)\right]}_{{p}_{{\rm{e}}}(0)}{\rm{\Delta }}{p}_{{\rm{e}}}\\ & & +\ O\left({\rm{\Delta }}{p}_{{\rm{e}}}^{2}\right)\\ & \approx & \displaystyle \frac{1}{{\beta }_{{\rm{c}}}}\mathrm{ln}\left[\displaystyle \frac{{p}_{{\rm{e}}}(0)}{1-{p}_{{\rm{e}}}(0)}\right]{\rm{\Delta }}{p}_{{\rm{e}}}\\ &=&-{\omega }_{{\rm{c}}}\left[{p}_{{\rm{e}}}(0)\right]\left[{p}_{{\rm{e}}}({\tau }_{{\rm{h}}})-{p}_{{\rm{e}}}(0)\right]=-\zeta {t}_{{\rm{h}}}{\omega }_{{\rm{c}}}^{{\rm{f}}}.\end{array}\end{eqnarray}$Here, only the first order of Δpe = pe(τh) − pe(0) is kept and ${\omega }_{{\rm{c}}}\left[{p}_{{\rm{e}}}(0)\right]={\omega }_{{\rm{c}}}^{{\rm{f}}}$ is used by noticing equation (A8). Combining equation (A4) and equation (A9), the EMP of the heat engine in the large-ηC regime (τh/tr ≪ 1 regime) is obtained as$\begin{eqnarray}\begin{array}{rcl}{\eta }_{\mathrm{MP}}\left({\tau }_{{\rm{h}}}/{t}_{{\rm{r}}}\ll 1\right)&=&\displaystyle \frac{{Q}_{{\rm{h}}}+{Q}_{{\rm{c}}}}{{Q}_{{\rm{h}}}}=\displaystyle \frac{\zeta {t}_{{\rm{h}}}\left(\tfrac{{\omega }_{{\rm{h}}}^{{\rm{f}}}+{\omega }_{{\rm{h}}}^{{\rm{i}}}}{2}-{\omega }_{{\rm{c}}}^{{\rm{f}}}\right)}{\tfrac{\zeta {t}_{{\rm{h}}}\left({\omega }_{{\rm{h}}}^{{\rm{i}}}+{\omega }_{{\rm{h}}}^{{\rm{f}}}\right)}{2}}\\ &=&\displaystyle \frac{{\omega }_{{\rm{h}}}^{{\rm{f}}}+{\omega }_{{\rm{h}}}^{{\rm{i}}}-2{\omega }_{{\rm{c}}}^{{\rm{f}}}}{{\omega }_{{\rm{h}}}^{{\rm{i}}}+{\omega }_{{\rm{h}}}^{{\rm{f}}}}\\ &=&\displaystyle \frac{2{\eta }_{{\rm{C}}}-1+\delta }{1+\delta },\end{array}\end{eqnarray}$where $\delta ={\omega }_{{\rm{h}}}^{{\rm{f}}}/{\omega }_{{\rm{h}}}^{{\rm{i}}}$ is the compression ratio, and the relation ${\omega }_{{\rm{h}}}^{{\rm{i}}}/{T}_{{\rm{h}}}={\omega }_{{\rm{c}}}^{{\rm{f}}}/{T}_{{\rm{c}}}$ has been used ([21, 31]). As shown in figure 5, in the large-ηC regime, equation (A10) (orange dashed line) matches the numerically obtained EMP (blue solid line) well.
Figure 5.
New window|Download| PPT slide Figure 5.Efficiency at the maximum power ηMP (blue solid line) of the heat engine as the function of the Carnot efficiency ηC for different final energy spacing of the two level system (a) ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.9$ and (b) ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.8$. The black dotted line represents the upper bound η+ and the Carnot efficiency ηC is plotted with the black dash-dotted line. The numerically obtained ηMP is plotted with the blue solid line, while the orange dashed line represents the approximated analytical result of equation (A10).
Appendix B. Low-temperature regime
In the low-temperature regime with ω/T ≫ 1 (the natural unit kB = 1 and ℏ = 1 are used here), the reversible entropy change ΔSα and the irreversible entropy generation coefficient Σα in the long-time limit are analytically written as ([21])$\begin{eqnarray}{\rm{\Delta }}{S}_{\alpha }={\beta }_{\alpha }{\omega }_{\alpha }^{{\rm{f}}}{{\rm{e}}}^{-{\beta }_{\alpha }{\omega }_{\alpha }^{{\rm{f}}}}-{\beta }_{\alpha }{\omega }_{\alpha }^{{\rm{i}}}{{\rm{e}}}^{-{\beta }_{\alpha }{\omega }_{\alpha }^{{\rm{i}}}},\end{eqnarray}$and$\begin{eqnarray}{{\rm{\Sigma }}}_{\alpha }=\displaystyle \frac{{{\rm{e}}}^{-{\beta }_{\alpha }{\omega }_{\alpha }^{{\rm{f}}}}-{{\rm{e}}}^{-{\beta }_{\alpha }{\omega }_{\alpha }^{{\rm{i}}}}}{{\widetilde{\gamma }}_{\alpha }},\end{eqnarray}$respectively, with α = h, c, ${\beta }_{\alpha }=1/\left({k}_{{\rm{B}}}{T}_{\alpha }\right)$, and ${\widetilde{\gamma }}_{\alpha }=1/{t}_{{\rm{r}}}^{(\alpha )}={\gamma }_{\alpha }{T}_{\alpha }/\left({\omega }_{\alpha }^{{\rm{i}}}-{\omega }_{\alpha }^{{\rm{f}}}\right)$. Substituting equations (B1) and (B2) into equations (2) and (3), we obtain the corresponding optimal operation time ${\tau }_{\alpha }^{* }$ for achieving the maximum power with the dimensionless time ${\widetilde{\tau }}_{\alpha }^{* }\equiv {\tau }_{\alpha }^{* }/{t}_{{\rm{r}}}^{(\alpha )}={\widetilde{\gamma }}_{\alpha }{\tau }_{\alpha }^{* }$ as$\begin{eqnarray}{\widetilde{\tau }}_{{\rm{h}}}^{* }=\displaystyle \frac{2}{{\eta }_{{\rm{C}}}}\displaystyle \frac{{T}_{{\rm{h}}}\left({{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{f}}}}-{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{i}}}}\right)}{{\omega }_{{\rm{h}}}^{{\rm{f}}}{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{f}}}}-{\omega }_{{\rm{h}}}^{{\rm{i}}}{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{i}}}}}\left(1+\sqrt{\left(1-{\eta }_{{\rm{C}}}\right)\displaystyle \frac{{\gamma }_{{\rm{h}}}}{{\gamma }_{{\rm{c}}}}}\right),\end{eqnarray}$$\begin{eqnarray}{\widetilde{\tau }}_{{\rm{c}}}^{* }=\displaystyle \frac{2}{{\eta }_{{\rm{C}}}}\displaystyle \frac{{T}_{{\rm{h}}}\left({{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{f}}}}-{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{i}}}}\right)}{{\omega }_{{\rm{h}}}^{{\rm{f}}}{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{f}}}}-{\omega }_{{\rm{h}}}^{{\rm{i}}}{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{i}}}}}\left(1-{\eta }_{{\rm{C}}}+\sqrt{\left(1-{\eta }_{{\rm{C}}}\right)\displaystyle \frac{{\gamma }_{{\rm{c}}}}{{\gamma }_{{\rm{h}}}}}\right).\end{eqnarray}$Here, the relations ${\omega }_{{\rm{h}}}^{{\rm{i}}}/{T}_{{\rm{h}}}={\omega }_{{\rm{c}}}^{{\rm{f}}}/{T}_{{\rm{c}}}$, ${\omega }_{{\rm{h}}}^{{\rm{f}}}/{T}_{{\rm{h}}}={\omega }_{{\rm{c}}}^{{\rm{i}}}/{T}_{{\rm{c}}}$ have been used in the quantum adiabatic processes. The low-dissipation assumption is valid in the regime ${\widetilde{\tau }}_{{\rm{h}}}^{* }\gg 1$ and ${\widetilde{\tau }}_{{\rm{c}}}^{* }\gg 1$, namely,$\begin{eqnarray}1+\sqrt{\left(1-{\eta }_{{\rm{C}}}\right)\displaystyle \frac{{\gamma }_{{\rm{h}}}}{{\gamma }_{{\rm{c}}}}}\gg \displaystyle \frac{{\eta }_{{\rm{C}}}}{2{T}_{{\rm{h}}}}\displaystyle \frac{\left[{\omega }_{{\rm{h}}}^{{\rm{f}}}-{\omega }_{{\rm{h}}}^{{\rm{i}}}{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{i}}}\left(1-\delta \right)}\right]}{1-{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{i}}}\left(1-\delta \right)}},\end{eqnarray}$$\begin{eqnarray}\sqrt{\left(1-{\eta }_{{\rm{C}}}\right)\displaystyle \frac{{\gamma }_{{\rm{c}}}}{{\gamma }_{{\rm{h}}}}}+1-{\eta }_{{\rm{C}}}\gg \displaystyle \frac{{\eta }_{{\rm{C}}}}{2{T}_{{\rm{h}}}}\displaystyle \frac{\left[{\omega }_{{\rm{h}}}^{{\rm{f}}}-{\omega }_{{\rm{h}}}^{{\rm{i}}}{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{i}}}\left(1-\delta \right)}\right]}{1-{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{i}}}\left(1-\delta \right)}},\end{eqnarray}$where $\delta ={\omega }_{{\rm{h}}}^{{\rm{f}}}/{\omega }_{{\rm{h}}}^{{\rm{i}}}$ is the compression ratio of the heat engine cycle. The above two inequalities are fulfilled when$\begin{eqnarray}{\eta }_{{\rm{C}}}\ll 2\displaystyle \frac{{T}_{{\rm{h}}}}{{\omega }_{{\rm{h}}}^{{\rm{f}}}}\displaystyle \frac{1-{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{i}}}\left(1-\delta \right)}}{1-{\delta }^{-1}{{\rm{e}}}^{-{\beta }_{{\rm{h}}}{\omega }_{{\rm{h}}}^{{\rm{i}}}\left(1-\delta \right)}}.\end{eqnarray}$In this inequality, the right hand term is in the order of ${T}_{{\rm{h}}}/{\omega }_{{\rm{h}}}^{{\rm{f}}}$, which is much small than 1 in the low-temperature regime. While in equation (10) derived in the high-temperature regime, $2\left(1-\delta \right)/\left(1+\delta \right)$ is in the order of 1. This implies that the range of ηC in which the low-dissipation model is applicable in the low-temperature regime is much narrower that that in the high-temperature regime.
In comparison with the results illustrated in figures 4(a) and (b), we plot ηMP (orange solid line) of the heat engine as the function of the Carnot efficiency ηC in the low-temperature regime in figure 6, where we also focus on the regime of γc/γh → ∞ as in the main text. In this figure, except Th is changed from 10 to 0.05, the values of other parameters are the same as those in figure 4, and the gray area represents the region in equation (B7). As we mentioned in the main text that the gray area only shows the approximate regime where the low-dissipation model is self-consistent, not the exact criterion for ηMP to exceed η+. It can be observed in figure 6 that the applicable regime of the low-dissipation model for the two-level atomic heat engine in the low-temperature case is narrower than that in the high-temperature case, as demonstrated in figure 4.
Figure 6.
New window|Download| PPT slide Figure 6.Efficiency at the maximum power ηMP (orange solid line) of the heat engine as the function of the Carnot efficiency ηC in the low-temperature regime with Th = 0.05. (a) ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.6$ and (b) ${\omega }_{{\rm{h}}}^{{\rm{f}}}=0.9$. The black dashed line (black dash-dotted line) represents the upper bound η+ (lower bound η−) of EMP obtained with the low-dissipation model [equation (4)], and the Carnot efficiency ηC is plotted with the black dotted line. The gray area represents the low-dissipation regime predicted by equation (B7). The other parameters in this figure are chosen as ${\omega }_{{\rm{h}}}^{{\rm{i}}}=1$ and γh = 1.
Acknowledgments
We gratefully acknowledge discussions with D. Xu at Beijing Institute of Technology. We would like to thank the anonymous referees for helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (NSFC) (Grants No. 11 534 002, No. 11 875 049, No. U1730449, No. U1530401, and No. U1930403), the National Basic Research Program of China (Grant No. 2016YFA0301201), and the China Postdoctoral Science Foundation (Grant No. BX2021030).
,1,∗1Graduate School of China Academy of Engineering Physics, Beijing 100193, 
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